AlllQM bSflim PUBUCATIONS

NIST Monograph 180

The Gage Block Handbook

Ted Doiron and John S. Beers

NETUnited States Department of Commerce

Technology Administration

National Institute of Standards and Technology
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NIST Monograph 180

The Gage Block Handbook

Ted Doiron and John S. Beers

Precision Engineering Division

Manufacturing Engineering Laboratory

National Institute of Standards and Technology

Gaithersburg, MD 20899-0001

U.S. Department of Commerce

Ronald H. Brown, Secretary

Technology Administration

Mary L. Good, Under Secretary for Technology

National Institute of Standards and Technology

Arati Prabhakar, Director

June 1995
National Institute of Standards

and Technology

Monograph 180

Natl. Inst. Stand. Technol. Mono. 180

154 pages (June 1995)

CODEN: NIMOEZ

U.S. Government Printing Office

Washington: 1995

For sale by the Superintendent

of Documents

U.S. Government Printing Office

Washington, DC 20402
The Gage Block Handbook

by

Ted Doiron and John Beers

Dimensional Metrology Group

Precision Engineering Division

National Institute of Standards and Technology

Preface

The Dimensional Metrology Group, and its predecessors at the National Institute of

Standards and Technology (formerly the National Bureau of Standards) have been

involved in documenting the science of gage block calibration almost continuously since

the seminal work of Peters and Boyd in 1926 [^]\ Unfortunately, most of this

documentation has been in the form of reports and other internal documents which are

difficult for the interested metrologist outside the Institute to obtain.

On the occasion of the latest major revision of our calibration procedures we decided to

assemble and extend the existing documentation of the NIST gage block calibration

program into one document. We use the word assemble rather than write because

most of the techniques described have been documented by various members of the

Dimensional Metrology Group over the last 20 years. Unfortunately, much of the work

is spread over multiple documents, many of the details of the measurement process

have changed since the publications were written, and many large gaps in coverage

exist. It is our hope that this handbook has assembled the best of the previous

documentation and extended the coverage to completely describe the current gage

block calibration process.

Many of the sections are based on previous documents since very little could be added

in coverage. In particular, the entire discussion of single wavelength interferometry is due to John Beers [2]; the section on preparation of gage blocks is due to Clyde Tucker

[3]; the section on the mechanical comparator techniques is predominantly from Beers

and Tucker [4]; and the appendix on drift eliminating designs is an adaptation for

dimensional calibrations of the work of Joseph Cameron [5] on weighing designs.

^ The numbers in brackets refer to the reference section, Chapter 7.

iii
They have, however, been rewritten to make the handbook consistent in style and

coverage. The measurement assurance program has been extensively modified over

the last 10 years by one of the authors (TD), and chapter 4 reflects these changes.

We would like to thank Mr. Ralph Veale, Mr. John Stoup, Mrs. Trish Snoots, Mr. Eric

Stanfield, Mr. Dennis Everett, Mr. Jay Zimmerman, Ms. Kelly Warfield and Dr. Jack

Stone, the members of the Dimensional Metrology Group who have assisted in both the

development and testing of the current gage block calibration system and the

production of this document.

TDandJSB

iv
CONTENTS

Page

Preface iii

Introduction 1

1. Length

1.1 The meter 2

1.2 The inch 3

2. Gage blocks

2.1 A short history of gage blocks 4

2.2 Gage block standards (U.S.) 4

2.2.1 Scope 5

2.2.2 Nomenclature and definitions 5

2.2.3 Tolerance grades 8

2.2.4 Recalibration requirements 10

2.3 International standards 10

3. Physical and thermal properties of gage blocks

3.1 Materials 13

3.2 Flatness and parallelism

3.2.1 Flatness measurement 14

3.2.2 Parallelism measurement 16

3.3 Thermal expansion 20

3.3.1 Thermal expansion of gage block materials 20

3.3.2 Thermal expansion uncertainty 24

3.4 Elastic properties 25

3.4.1 Contact deformation in mechanical comparisons 27

3.4.2 Measurement of probe force and tip radius 28

3.5 Stability 30

V
4. Measurement assurance programs

4.1 Introduction 33

4.2 A comparison: traditional metrology vs measurement

assurance programs 33

4.2.1 Tradition 33

4.2.2 Process control: a paradigm shift 34

4.2.3 Measurement assurance: building a measurement

process model 36

4.3 Determining Uncertainty

4.3.1 Stability 37

4.3.2 Uncertainty 38

4.3.3 Random error 40

4.3.4 Systematic error and type B uncertainty 41

4.3.5 Error budgets 44

4.3.6 Combining type A and type B uncertainties 46

4.3.7 Combining random and systematic errors 47

4.4 The NIST gage block measurement assurance program

4.4.1 Establishing interferometric master values 49

4.4.2 The comparison process 53

4.4.2.1 Measurement schemes - drift eliminating designs 53

4.4.2.2 Control parameter for repeatability 56

4.4.2.3 Control test for variance 59

4.4.2.4 Control parameter (S-C) . 60

4.4.2.5 Control test for (S-C), the check standard ... 63

4.4.2.6 Control test for drift 64

4.4.3 Calculating total uncertainty 64

4.5 Summary of the NIST measurement assurance program 66

vi
5. The NIST mechanical comparison procedure

5.1 Introduction 68

5.2 Preparation and inspection 68

5.2.1 Cleaning procedures 68

5.2.2 Cleaning interval 69

5.2.3 Storage 69

5.2.4 Deburring gage blocks 69

5.3 The comparative principle 70

5.3.1 Examples 71

5.4 Gage block comparators 73

5.4.1 Scale and contact force control 75

5.4.2 Stylus force and penetration corrections 76

5.4.3 Environmental factors 77

5.4.3.1 Temperature effects 78

5.4.3.2 Control of temperature effects 80

5.5 Intercomparison procedures 81

5.5.1 Handling techniques 82

5.6 Comparison designs 83

5.6.1 Drift eliminating designs 83

5.6.1.1 The 12/4 design 84

5.6.1.2 The 6/3 design 84

5.6.1.3 The 8/4 design 85

5.6.1.4 The ABBA design 86

5.6.2 Example of calibration output using the 12/4 design 86

5.7 Current NIST system performance 88

5.7.1 Summary 91

6. Gage block interferometry

6.1 Introduction 92

vii
6.2 Interferometers 93

6.2.1 The Kosters type interferometer 93

6.2 2 The NPL interferometer 95

6.2.3 Testing optical quality of interferometers 97

6.2.4 Interferometer corrections . 98

6.2.5 Laser light sources 101

6.3 Environmental conditions and their measurement 102

6.3.1 Temperature 102

6.3.2 Atmospheric pressure 104

6.3.3 Water vapor . 104

6.4 Gage block measurement procedure 105

6.5 Computation of gage block length 107

6.5.1 Calculation of the wavelength 107

6.5.2 Calculation of the whole number of fringes 109

6.5.3 Calculation of the block length from observed data 109

6.6 Type A and B errors 110

6.7 Process evaluation 112

6.8 Multiple wavelength interferometry 115

6.9 Use of the line scale interferometer for end standard calibration . . 117

7. References 120

Appendix A. Drift eliminating designs for non-simultaneous

comparison calibrations 124

Appendix B. Wringing films 138

Appendix C. Phase shifts in gage block interferometry 141

Appendix D. Deformation corrections 145

viii
Gage Block Handbook

Introduction

Gage block calibration is onexif the oldest high precision calibrations made in dimensional metrology. Since their invention at the turn of the century gage blocks

have been the major source of length standardization for industry. In most

measurements of such enduring importance it is to be expected that the measurement

would become much more accurate and sophisticated over 80 years of development.

Because of the extreme simpleness of gage blocks this has only been partly true. The

most accurate measurements of gage blocks have not changed appreciably in accuracy in the last 70 years. What has changed is the much more widespread

necessity of such accuracy. Measurements which previously could only be made with

the equipment and expertise of a national metrology laboratory are routinely expected

in private industrial laboratories.

To meet this widespread need for higher accuracy, the calibration methods used for

gage blocks have been continuously upgraded. This handbook is a both a description

of the current practice at the National Institute of Standards and Technology, and a compilation of the theory and lore of gage block calibration. Most of the chapters are

nearly self-contained so that the interested reader can, for example, get information on

the cleaning and handling of gage blocks without having to read the chapters on

measurement schemes or process control, etc. This partitioning of the material has led

to some unavoidable repetition of material between chapters.

The basic structure of the handbook is from the theoretical to the practical. Chapter 1 concerns the basic concepts and definitions of length and units. Chapter 2 contains a

short history of gage blocks, appropriate definitions and a discussion of pertinent

national and international standards. Chapter 3 discusses the physical characteristics

of gage blocks, including thermal, mechanical and optical properties. Chapter 4 is a

description of statistical process control (SPC) and measurement assurance (MA)

concepts. The general concepts are followed by details of the SPC and MA used at

NIST on gage blocks.

Chapters 5 and 6 cover the details of the mechanical comparisons and interferometric

techniques used for gage block calibrations. Full discussions of the related

uncertainties and corrections are included. Finally, the appendices cover in more detail

some important topics in metrology and gage block calibration.

1
1.0 Length

1.1 The Meter

At the turn of 19th century there were two distinct major length systems. The metric

length unit was the meter which was originally defined as 1/10,000,000 of the great arc from the pole to the equator, through Paris. Data from a very precise measurement of

part of that great arc was used to define an artifact meter bar, which became the

practical and later legal definition of the meter. The English system of units was based

on a yard bar, another artifact standard [6].

These artifact standards were used for over 150 years. The problem with an artifact

standard for length is that nearly all materials are slightly unstable and change length

with time. For example, by repeated measurements it was found that the British yard

standard was slightly unstable. The consequence of this instability was that the British

inch ( 1/35 yard) shrank [7], as shown in table 1.1.

Table 1.1

1895 - 25.399978 mm

1922 - 25.399956 mm

1932 - 25.399950 mm

1947 - 25.399931 mm

The first step toward replacing the artifact meter was taken by Albert Michelson, at the

request of the International Committee of Weights and Measures (CIPM). In 1 892

Michelson measured the meter in terms of the wavelength of red light emitted by

cadmium. This wavelength was chosen because it has high coherence, that is, it will form fringes over a reasonable distance. Despite the work of Michelson, the artifact

standard was kept until 1 960 when the meter was finally redefined in terms of the

wavelength of light, specifically the red-orange light emitted by excited krypton-86 gas.

Even as this definition was accepted, the newly invented helium-neon laser was

beginning to be used for interferometry. By the 1970's a number of wavelengths of

stabilized lasers were considered much better sources of light than krypton red-orange

for the definition of the meter. Since there were a number of equally qualified

candidates the International Committee on Weights and Measures (CIPM) decided not

to use any particular wavelength, but to make a change in the measurement hierarchy.

2
The solution was to define the speed of light in vacuum as exactly 299,792,458 m/s,

and make length a derived unit. In theory, a meter can be produced by anyone vi/ith an

accurate clock[8].

In practice, the time-of-flight method is impractical for most measurements, and the

meter is measured using known wavelengths of light. The CIPM lists a number of laser

and atomic sources and recommended frequencies for the light. Given the defined

speed of light, the wavelength of the light can be calculated and a meter can be

generated by counting wavelengths of the light. Methods for this measurement are discussed in the chapter on interferometry.

1.2 The Inch

In 1866, the United Stated Surveyor General decided to base all geodetic

measurements on an inch defined from the international meter. This inch was defined

such that there were exactly 39.37 inches in the meter. England continued to use the

yard bar to define the inch. These different inches continued to coexist for nearly 100

years until quality control problems during World War II showed that the various inches

in use were too different for completely interchangeable parts from the English

speaking nations. Meetings were held in the 1950's and in 1959 the directors of the

national metrology laboratories of the United States, Canada, England, Australia and

South Africa agreed to define the inch as 25.4 millimeters, exactly[9]. This definition

was a compromise; the English inch being somewhat longer, and the U.S. inch smaller.

The old U.S. inch is still in use for commercial surveying of land in the form of the

"surveyor's foot," which is 12 old U.S. inches.

3
2.0 Gage Blocks

2.1 A Short History of Gage Blocks

By the end of the nineteenth century the idea of interchangeable parts begun by Eli Whitney had been accepted by industrial nations as the model for industrial

manufacturing. One of the drawbacks to this new system was that in order to control

the size of parts numerous gages were needed to check the parts and set the

calibrations of measuring instruments. The number of gages needed for complex

products, and the effort needed to make and maintain the gages was a significant

expense. The major step toward simplifying this situation was made by C.E.

Johannson, a Swedish machinist.

Johannson's idea, first formulated in 1 896 [1 0], was that a small set of gages which

could be combined to form composite gages could reduce the number of gages needed

in the shop. For example, if four gages of sizes 1 mm, 2 mm, 4 mm, and 8 mm could

be combined in any combination, all of the millimeter sizes from 1 mm to 1 5 mm could

be made from only these four gages. Johannson found that if two opposite faces of a piece of steel were lapped very flat and parallel, two blocks would stick together when

they were slid together with a very small amount of grease between them. The width of

this "wringing" layer is about 25 nm, and was so small for the tolerances needed at the

time, that the block lengths could be added together with no correction for interface

thickness. Eventually the wringing layer was defined as part of the length of the block,

allowing the use of an unlimited number of wrings without correction for the size of the

wringing layer.

In the United States, the idea was enthusiastically adopted by Henry Ford, and from his

example the use of gage blocks was eventually adopted as the primary transfer

standard for length in industry. By the beginning of World War I, the gage block was

already so important to industry that the Federal Government had to take steps to

insure the availability of blocks. At the outbreak of the war, the only supply of gage

blocks was from Europe, and this supply was interrupted.

In 1917 inventor William Hoke came to NBS proposing a method to manufacture gage

blocks equivalent to those of Johannson [11]. Funds were obtained from the Ordnance

Department for the project and 50 sets of 81 blocks each were made at NBS. These

blocks were cylindrical and had a hole in the center, the hole being the most prominent

feature of the design. The current generation of square cross-section blocks have this

hole and are referred to as "Hoke blocks."

2.2 Gage Block Standards (U.S.)

There are two main American standards for gage blocks, the Federal Specification
GGG-G-15C [12] and the American National Standard ANSI/ASME B89.1.9M [13].

There are very few differences between these standards, the major ones being the

organization of the material and the listing of standard sets of blocks given in the GGGG-15C specification. The material in the ASME specification which is pertinent to a discussion of calibration is summarized below.

2.2.1 Scope

The ASME standard defines all of the relevant physical properties of gage blocks up to

20 inches and 500 mm long. The properties include the block geometry (length,

parallelism, flatness and surface finish), standard nominal lengths, and a tolerance

grade system for classifying the accuracy level of blocks and sets of blocks.

The tolerancing system was invented as a way to simplify the use of blocks. For

example, suppose gage blocks are used to calibrate a certain size fixed gage, and the

required accuracy of the gage is 0.5 pm. If the size of the gage requires a stack of five

blocks to make up the nominal size of the gage the accuracy of each block must be

known to 0.5/5 or 0.1 pm. This is near the average accuracy of an industrial gage block

calibration, and the tolerance could be made with any length gage blocks if the

calibrated lengths were used to calculate the length of the stack. But having the

calibration report for the gage blocks on hand and calculating the length of the block

stack are a nuisance. Suppose we have a set of blocks which are guaranteed to have

the property that each block is within 0.05 pm of its nominal length. With this

knowledge we can use the blocks, assume the nominal lengths and still be accurate

enough for the measurement.

The tolerance grades are defined in detail in section 2.2.3, but it is important to recognize the difference between gage block calibration and certification. At NIST,

gage blocks are calibrated, that is, the measured length of each block is reported in the

calibration report. The report does not state which tolerance grade the blocks satisfy.

In many industrial calibrations only the certified tolerance grade is reported since the

corrections will not be used.

2.2.2 Nomenclature and Definitions

A gage block is a length standard having flat and parallel opposing surfaces. The

cross-sectional shape is not very important, although the standard does give suggested

dimensions for rectangular, square and circular cross-sections. Gage blocks have

nominal lengths defined in either the metric system (millimeters) or in the English

system (1 inch = 25.4 mm).

5
The length of the gage block is defined at standard reference conditions:

temperature = 20 °C (68°F)

barometric pressure = 101,325 Pa (1 atmosphere)

water vapor pressure = 1 ,333 Pa (10 mm of mercury)

CO2 content of air = 0.03%.

Of these conditions only the temperature has a measurable effect on the physical

length of the block. The other conditions are needed because the primary

measurement of gage block length is a comparison with the wavelength of light. For

standard light sources the frequency of the light is constant, but the wavelength is dependent on the temperature, pressure, humidity, and CO2 content of the air. These

effects are described in detail later.

The length of a gage block is defined as the perpendicular distance from a gaging point

on one end of the block to an auxiliary true plane wrung to the other end of the block, as shown in figure 2.1 (from B89.1.9).

Figure 2.1 . The length of a gage block is the distance from the gaging point on the top

surface to the plane of the platten adjacent to the wrung gage block.

This length is measured interferometrically, as described later, and corrected to standard

Reference point on

free gaging surface

Interferonnetric

gage length

Auxiliary surface

6
conditions. It is worth noting that gage blocks are NEVER measured at standard conditions

because the standard vapor pressure of water of 10 mrr of mercury is nearly 60% relative

humidity which would allow steel to rust. The standard conditions are actually

spectroscopic standard conditions, i.e., the conditions at which spectroscopists define the

wavelengths of light.

This definition of gage block length which uses a wringing plane seems odd at first, but is very important for two reasons. First, light appears to penetrate slightly into the gage block

surface, a result of the surface finish of the block and the electromagnetic properties of

metals. If the wringing plane and the gage block are made of the same material and have

the same surface finish, then the light will penetrate equally into the block top surface and

the reference plane, and the errors cancel. If the block length was defined as the distance

between the gage block surfaces the penetration errors would add, not cancel, and the

penetration would have to be measured so a correction could be made. These extra

measurements would, of course, reduce the accuracy of the calibration.

The second reason is that in actual use gage blocks are wrung together. Suppose the

length of gage blocks was defined as the actual distance between the two ends of the gage

block, not wrung to a plane. For example, if a length of 6.523 mm is needed gage blocks

of length 2.003 mm, 2.4 mm, and 2.12 mm are wrung together. The length of this stack

is 6.523 plus the length of two wringing layers. It could also be made using the set (1 mm,

1 mm, 1 mm, 1.003 mm, 1.4 mm, 1.12 mm) which would have the length of 6.523 mm plus

the length of 5 wringing layers. In order to use the blocks these wringing layer lengths

must be known. If, however, the length of each block contains one wringing layer length

then both stacks will be of the same defined length.

NIST master gage blocks are calibrated by interferometry in accordance with the definition

of gage block length. Each master block carries a wringing layer with it, and this wringing

layer is transferred to every block calibrated at NIST by mechanical comparison

techniques.

The mechanical length of a gage block is the length determined by mechanical

comparison of a block to a another block of known interferometrically determined length.

The mechanical comparison must be a measurement using two designated points, one on

each end of the block. Since most gage block comparators use mechanical contact for the

comparison, if the blocks are not of the same material corrections must be made for the

deformation of the blocks due to the force of the comparator contact.

The reference points for gage block mechanical comparison are shown in figure 2.2.

7
REFERENCE POINT, RECTANGULAR

REFERENCE POINT, SQUARE

Figure 2,2 a and b. Definition of the gaging point on gage blocks.

For rectangular and round blocks the reference point is the center of gaging face. For

round or square blocks which have a center hole, the point is midway between the hole

edge and the edge of the block nearest to the size marking.

2.2.3 Tolerance Grades

There are 4 tolerance grades; 0.5, 1 , 2, and 3. Grades 0.5 and 1 gage blocks have lengths

very close to their nominal values. These blocks are generally used as calibration masters.

Grades 2 and 3 are of lower quality and are used for measurement and gaging purposes.

Table 2.1 shows the length, flatness and parallelism requirements for each grade. The

table shows that grade 0.5 blocks are within 1 millionth of an inch (1 pin) of their nominal

length, with grades 1, 2, and 3 each roughly doubling the size of the maximum allowed

deviation.

8
Table 2.1a Tolerance Grades for Inch Blocks (in Microinches)

iNominai oraae .o oraae i oraae z (jrade o

<1 inch -I- 'I +4, -2 +0, -4 z ±Z 4-/1 +0, -4 +1b, -o Qo ±D + 1 U, -O +zU,-1U

4

4. /I -l-K ±D +1 z, -o_i_0 >1 "1 o +24, -1z

5

1 "7 ±7 1 >1 yf "7 +14, -7 +28,-14

6 ±8 +16, -8 +32,-16

7 ±9 +18, -9 +36,-18

8 ±10 +20,-10 +40,-20

10 ±12 +24,-12 +48,-24

12 ±14 +28,-14 +56,-28

16 ±18 +36,-18 +72,-36

20 ±20 +40,-20 +80,-40

Table 2.1b Tolerance Grades for Metric Blocks (in Micrometers)

Nominal Grade .5 Grade 1 Grade 2 Grade 3

< 10 mm ±0.03 ±0.05 +0.10, 0.05 +0.20, -0.10 < 25 mm ±0.03 ±0.05 +0.10, -0.05 +0.30, -0.15 < 50 mm ±0.05 ±0.10 +0.20, -0.10 +0.40, -0.20 < 75 mm ±0.08 ±0.13 +0.25, -0.13 +0.45, -0.23 < 100 mm ±0.10 ±0.15 +0.30, -0.15 +0.60, -0.30

125 mm ±0.18 +0.36, -0.18 +0.70, -0.35

150 mm ±0.20 +0.41, -0.20 +0.80, -0.40

175 mm ±0.23 +0.46, -0.23 +0.90, -0.45

200 mm ±0,25 +0.51, -0.25 +1.00, -0.50

250 mm ±0.30 +0.60, -0.30 +1.20, -0.60

300 mm ±0.35 +0.70, -0.35 +1.40, -0.70

400 mm ±0.45 +0.90, -0.45 +1.80, -0.90

500 mm ±0.50 +1.00, -0.50 +2.00, -1.90

Since there is uncertainty in any measurement, the standard allows for an additional

tolerance for length, flatness, and parallelism. These additional tolerances are given in table 2.2.

9
Table 2.2 Additional Deviations for Measurement Uncertainty

Nominal

inches (millimeters)

Grade .5 Grade 1 Grade 2

microinches (micrometers)

Grade 3

<4(100)

< 8 (200) < 12 (300) < 20 (500)

1 (0.03) 2 (0.05)

3 (0.08)

4(0.10)

5 (0.13)

3 (0.08)

6 (0.15)

8 (0.20)

10 (0.25)

4(0.10)

8 (0.20)

10(0.25)

12 (0.30)

For example, for a grade 1 gage block of nominally 1 inch length the length tolerance is ±2 microinches. With the additional tolerance for measurement uncertainty from table 2

of ±2 microinches, a 1 inch grade 1 block must have a measured length within ±4

microinches of nominal.

2.2.4 Recalibration Requirements

There is no required schedule for recalibration of gage blocks, but both the ASME and

Federal standards recommend recalibration periods for each tolerance grade, as shown

below:

Most NIST customers send master blocks for recalibration every 2 years. Since most

master blocks are not used extensively, and are used in a clean, dry environment this

schedule is probably adequate. However, despite popular misconceptions, NIST has no

regulatory power in these matters. The rules for tolerance grades, recalibration, and

replacement rest entirely with the appropriate government agency inspectors.

Gage blocks are defined internationally by ISO Standard 3650 [14], the current edition

being the first edition 1 978-07-1 5. This standard is much like the ANSI standard in spirit,

but differs in most details, and of course does not define English size blocks.

The length of the gage block is defined as the distance between a flat surface wrung to

Grade Recalibration Period

Annually

Annually

Monthly to semi-annually

Monthly to semi-annually

0.5

12

3

2.3 International Standards

10
one end of the block, and a gaging point on the opposite end. The ISO specification only

defines rectangular cross-sectioned blocks and the gaging point is the center of the gaging

face. The non-gaging dimensions of the blocks are somewhat smaller than the

corresponding ANSI dimensions.

There are four defined tolerance grades in ISO 3650; 00, 0, 1 and 2. The algorithm for the

length tolerances are shown in table 2.3, and there are rules for rounding stated to derive

the tables included in the standard.

Table 2.3

Grade Deviation from Nominal

Length (pm)

00 (0.05 + 0.0001 L)

0 (0.10 + 0.0002L)

1 (0.20 + 0.0004L)

2 (0.40 + 0.0008L)

Where L is the block nominal length in millimeters.

The ISO standard does not have an added tolerance for measurement uncertainty;

however, the ISO tolerances are comparable to those of the ANSI specification when the

additional ANSI tolerance for measurement uncertainty is added to the tolerances of Table

2.1.

A graph of the length tolerance versus nominal length is shown in figure 2.3. The different

class tolerance for ISO and ANSI do not match up directly. The ANSI grade 1 is slightly

tighter than the ISO class 00, but if the additional ANSI tolerance for measurement

uncertainty is used the ISO Grade 00 is slightly tighter. The practical difference between

these specifications are negligible.

In many countries the method for testing the variation in length is also standardized. For

example, in Germany [15] the test block is measured in 5 places: the center and near each

corner (2 mm from each edge). The center gives the length of the block and the four corner measurements are used to calculate the shortest and longest lengths of the block.

Some of the newer gage block comparators have a very small lower contact point to

facilitate these measurements very near the edge of the block.

11
E

c

0)o

c

03

i_

0)o

H

100 200 300 400 500

Length (mm)

Figure 2.3. Comparison of the ISO and ANSI gage block

tolerance grades. The bottom solid line is the ANSI grade

1 tolerance, the top black line the grade 1 tolerance with

the allowable added uncertainty, and the dashed line the

ISO grade 00 tolerance

12
3. Physical and Thermal Properties of Gage Blocks

3.1 Materials

From the very beginning gage blocks were made of steel. The lapping process used to

finish the ends, and the common uses of blocks demand a hard surface. A second virtue

of steel is that most industrial products are made of steel. If the steel gage block has the

same thermal expansion coefficient as the part to be gaged, a thermometer is not needed

to obtain accurate measurements. This last point will be discussed in detail later.

The major problem with gage blocks was always the stability of the material. Because of

the hardening process and the crystal structure of the steel used, most blocks changed

length in time. For long blocks, over a few inches, the stability was a major limitation.

During the 1950s and 1960s a program to study the stability problem was sponsored by

the National Bureau of Standards and the ASTM [16,17]. A large number of types of steel

and hardening processes were tested to discover manufacturing methods which would

produce stable blocks. The current general use of 52100 hardened steel is the product of

this research. Length changes of less than 1 part in 10"^/decade are now common.

Over the years, a number of other materials were tried as gage blocks. Of these, tungsten

carbide, chrome carbide, and Cervit are the most interesting cases.

The carbide blocks are very hard and therefore do not scratch easily. The finish of the

gaging surfaces is as good as steel, and the lengths appear to be at least as stable as

steel, perhaps even more stable. Tungsten carbide has a very low expansion coefficient

(1/3 of steel) and because of the high density the blocks are deceptively heavy. Chrome

carbide has an intermediate thermal expansion coefficient (2/3 of steel) and is roughly the

same density as steel. Carbide blocks have become very popular as master blocks

because of their durability and because in a controlled laboratory environment the thermal

expansion difference between carbide and steel is easily manageable.

Cervit is a glassy ceramic which was designed to have nearly zero thermal expansion

coefficient. This properfy, plus a zero phase shift on quartz platens (phase shift will be

discussed later), made the material attractive for use as master blocks. The drawbacks

are that the material is softer than steel, making scratches a danger, and by nature the

ceramic is brittle. While a steel block might be damaged by dropping, and may even need

stoning or recalibration, Cervit blocks tended to crack or chip. Because the zero

coefficient was not always useful and because of the combination of softness and

brittleness they never became popular and are no longer manufactured.

A number of companies are experimenting with alumina based ceramics, and one type is being marketed. These blocks are very hard and have thermal expansion coefficient of

approximately 9 x lO'^/X, about 20% lower than steel.

13
3.2 Flatness and Parallelism

We will describe a few methods which are useful to characterize the geometry of gage

blocks. It is important to remember, however, that these methods provide only a limited

amount of data about what can be, in some cases, a complex geometric shape. When

more precise measurements or a permanent record is needed, the interference fringe

patterns can be photographed. The usefulness of each of the methods must be judged

in the light of the user's measurement problem.

3.2.1 Flatness Measurements.

Various forms of interferometers are applicable to measuring gage block flatness. All produce interference fringe patterns formed with monochromatic light by the gage block

face and a reference optical flat of known flatness. Since modest accuracies (25 nm or 1 pin) are generally needed, the demands on the light source are also modest. Generally

a fluorescent light with a green filter will suffice as an illumination source. For more

demanding accuracies, a laser or atomic spectral lamp must be used.

The reference surface must satisfy two requirements. First, it must be large enough to cover the entire surface of the gage block. Usually a 70 mm diameter or larger is sufficient. Secondly, the reference surface of the flat should be sufficiently planar that any

fringe curvature can be attributed solely to the gage block. Typical commercially available

reference flats, flat to 25 nm over a 70 mm diameter, are usually adequate.

Gage blocks 2 mm (0.1 in.) and greater can be measured in a free state, that is, not wrung

to a platen. Gage blocks less than 2 mm are generally flexible and have warped surfaces.

There is no completely meaningful way to define flatness. One method commonly used

to evaluate the "flatness" is by "wringing" the block to another more mechanically stable

surface. When the block is wrung to the surface the wrung side will assume the shape of

the surface, thus this surface will be as planar as the reference flat. We wring these thin blocks to a fused silica optical flat so that the wrung surface can be

viewed through the back surface of the flat. The interface between the block and flat, if wrung properly, should be a uniform gray color. Any light or colored areas indicate poor

wringing contact which will cause erroneous flatness measurements. After satisfactory

wringing is achieved the upper (non-wrung) surface is measured for flatness. This process

is repeated for the remaining surface of the block.

Figures 3.1a and 3.1b illustrate typical fringe patterns. The angle between the reference

flat and gage block is adjusted so that 4 or 5 fringes lie across the width of the face of the

block, as in figure 3.1a, or 2 or 3 fringes lie along the length of the face as in figure 3.1b.

Four fringes in each direction are adequate for square blocks.

14
Figure 3.1 a, b, and c. Typical fringe patterns used to measure gage block

flatness. Curvature can be measured as shown in the figures.

The fringe patterns can be interpreted as contour maps. Points along a fringe are points

of equal elevation and the amount of fringe curvature is thus a measure of planarity.

Curvature = a/b in fringes.

For example, a/b is about 0.2 fringe in figure 3.1a and 0.6 fringe in figure 3.1b. Conversion

to length units is accomplished using the known wavelength of the light. Each fringe

represents a one-half wavelength difference in the distance between the reference flat and

the gage block. Green light is often used for flatness measurements. Light in the green

range is approximately 250 nm (10 microinches) per fringe, therefore the two illustrations

indicate flatness deviations of 50 nm and 150 nm (2 microinches and 6 microinches)

respectively.

Another common fringe configuration is shown in figure 3.1c. This indicates a twisted

gaging face. It can be evaluated by orienting the uppermost fringe parallel to the upper

gage block edge and then measuring "a" and "b" in the two bottom fringes, the magnitude

of the twist is a/b which in this case is 75 nm (3 microinches) in green.

In manufacturing gage blocks, the gaging face edges are slightly bevelled or rounded to eliminate damaging burrs and sharpness. Allowance should be made for this in flatness

measurements by excluding the fringe tips where they drop off at the edge. Allowances

vary, but 0.5 mm ( 0.02 in) is a reasonable bevel width to allow.

15
3.2.2 Parallelism measurement

Parallelism between the faces of a gage block can be measured in two ways; with

interferometry or with an electro-mechanical gage block comparator.

Interferometer Technique

The gage blocks are first wrung to what the standards call an auxiliary surface. We will call

these surfaces platens. The platen can be made of any hard material, but are usually steel or glass. An optical flat is positioned above the gage block, as in the flatness

measurement, and the fringe patterns are observed. Figures 3.2a and 3.2b illustrate

typical fringe patterns. The angle between the reference flat and gage block is adjusted

to orient the fringes across the width of the face as in Figure 3.2a or along the length of the

face, as in figure 3.4b. The reference flat is also adjusted to control the number of fringes,

preferably 4 or 5 across, and 2 or 3 along. Four fringes in each direction are satisfactory

for square blocks.

Figure 3.2 a and b. Typical fringe patterns for measuring gage block

parallelism using the interferometer method.

A parallelism error between the two faces is indicated by the slope of the gage block

fringes relative to the platen fringes. Parallelism across the block width is illustrated in figure 3.2a where

Slope = (a/b) - (aVb) = 0.8 - 0.3 = 0.5 fringe (3.1)

16
Parallelism along the block length in figure 3.2b is

Slope = (a/b) + (aVb) = 0.4 + 0.3 = 0.7 fringe (3.2)

Note that the fringe fractions are subtracted for figure 3.2a and added for figure 3.2b. The

reason for this is clear from looking at the patterns - the block fringe stays within the same

two platen fringes in the first case and it extends into the next pair in the latter case. Conversion to length units is made with the value of A/2 appropriate to the illumination.

Since a fringe represents points of equal elevation it is easy to visualize the blocks in figure

3.2 as being slightly wedge shaped.

This method depends on the wringing characteristics of the block. If the wringing is such

that the platen represents an extension of the lower surface of the block then the

procedure is reliable. There are a number of problems which can cause this method to fail.

If there is a burr on the block or platen, if there is particle of dust between the block and

platen, or if the block is seriously warped, the entire face of the block may not wring down

to the platen properly and a false measurement will result. For this reason usually a fused

silica platen is used so that the wring can be examined by looking through the back of the

platen, as discussed in the section on flatness measurements. If the wring is good, the

block-platen interface will be a fairly homogeneous gray color.

Gage Block Comparator Technique

Electro-mechanical gage block comparators with opposing measuring styli can be used to measure parallelism. A gage block is inserted in the comparator, as shown in figure 3.3,

after sufficient temperature stabilization has occurred to insure that the block is not

distorfed by internal temperature gradients. Variations in the block thickness from edge

to edge in both directions are measured, that is, across the width and along the length of

the gaging face through the gaging point. Insulated tongs are recommended for handling

the blocks to minimize temperature effects during the measuring procedure.

17
X=DEFINED GAGE POINT X=DEFINED GAGE POINT

o

c

b

O ® X o

d

O

o X o

d

o

UPPER GAGING FACE UPPER GAGING FACE

Figure 3.3. Basic geometry of measurements using a mechanical comparator.

Figures 3.4a and 3.4b stiow locations of points to be measured with the comparator on the

two principle styles of gage blocks. The points designated a, b, c, and d are midway along

the edges and in from the edge about 0.5 mm ( 0.02 in) to allow for the normal rounding

of the edges.

Figure 3.4 a and b. Location of gaging points on gage blocks

for both length and parallelism measurements.

18
A consistent procedure is recommended for making the measurements:

(1 ) Face the side of the blocl< associated with point "a" toward the

comparator measuring tips, push the block in until the upper tip contacts

point "a", record meter reading and withdraw the block.

(2) Rotate the block 180° so the side associated with point "b" faces the

measuring tips, push the block in until tip contacts point "b", record meter

reading and withdraw block.

(3) Rotate block 90° so side associated with point "c" faces the tips and proceed

as in previous steps.

(4) Finally rotate block 180° and follow this procedure to measure at point "d".

The estimates of parallelism are then computed from the readings as follows:

Parallelism across width of block = a-b

Parallelism along length of block = c-d

The parallelism tolerances, as given in the GGG and ANSI standards, are shown in table

3.1.

Table 3.1 ANSI tolerances for parallelism in microinches

Size Grade .5 Grade 1 Grade 2 Grade 3

(in)

<1 ±1 ±2 ±4 ±5

2 ±2 ±2 ±4 ±5

3 ±3 ±3 ±4 ±5

4 ±4 ±3 ±4 ±5

5-8 • • ±3 ±4 ±5

10-20 • • ±4 ±5 ±6

Referring back to the length tolerance table, you will see that the allowed parallelism and

flatness errors are very substantial for blocks under 25 mm (or 1 in). For both

interferometry and mechanical comparisons, if measurements are made with little attention

to the true gage point significant errors can result when large parallelism errors exist.

19
3.3 Thermal Expansion

In most materials, a change in temperature causes a change in dimensions. This change

depends both on the size of the temperature change and the temperature at which the

change occurs. The equation describing this effect is

AL/L = Ql at (3.3)

where L is the length, AL is the change in length of the object, AT is the temperature

change and a,, is the coefficient of thermal expansion(CTE).

3.3.1 Thermal Expansion of Gage Block Materials

In the simplest case, where AT is small, can be considered a constant. In truth,

depends on the absolute temperature of the material. Figure 3.5 [18] shows the

measured expansion coefficient of gage block steel. This diagram is typical of most

metals, the thermal expansion rises with temperature.

12.0

•9° 1 1 .5 X

o

o0 10.5 c

o

't/i D 10.0 aX

UJ1 9.5

0) I—

-40 -20 0 20 40 60 80 100

Temperature ( ° C)

Figure 3.5. Variation of the thermal expansion coefficient of gage block steel

with temperature.

As a numerical example, gage block steel has an expansion coefficient of 11 .5 x 1 0"^/°C.

This means that a 100 mm gage block will grow 11.5 x 10"^ times 100 mm , or 1.15

micrometer, when its temperature is raised 1 °C. This is a significant change in length,

since even class 3 blocks are expected to be within 0.2 pm of nominal. For long standards

20
the temperature effects can be dramatic. Working backwards, to produce a 0.25 um

change in a 500 mm gage block, a temperature change of only 43 millidegrees (0.043 °C)

is needed.

Despite the large thermal expansion coefficient, steel has always been the material of

choice for gage blocks. The reason for this is that most measuring and manufacturing

machines are made of steel, and the thermal effects tend to cancel.

To see how this is true, suppose we wish to have a cube made in the shop, with a side

length of 100 mm. The first question to be answered is at what temperature should the

length be 100 mm. As we have seen, the dimension of most objects depends on its temperature, and therefore a dimension without a defined temperature is meaningless.

For dimensional measurements the standard temperature is 20 °C (68 °F). If we call for a 100 mm cube, what we want is a cube which at 20 °C will measure 100 mm on a side.

Suppose the shop floor is at 25 °C and we have a perfect gage block with zero thermal

expansion coefficient. If we make the cube so that each side is exactly the same length

as the gage block, what length is it? When the cube is taken into the metrology lab at 20°C, it will shrink 11.5x10"^ /°C, which for our block is 5.75 pm, i.e., it will be 5.75 |jm

undersized.

Now suppose we had used a steel gage block. When we brought the gage block out onto

the shop floor it would have grown 5.75 [jm. The cube, being made to the dimension of

the gage block would have been oversized by 5.75 pm. And finally, when the block and

cube were brought into the gage lab they would both shrink the same amount, 5.75 |jm,

and be exactly the length called for in the specification.

What this points out is that the difference in thermal expansion between the workpiece and

the gage is the important parameter. Ideally, when making brass or aluminum parts, brass

or aluminum gages would be used. This is impractical for a number of reasons, not the

least of which is that it is nearly impossible to make gage blocks out of soft materials, and

once made the surface would be so easily damaged that its working life would be on the

order of days. Another reason is that most machined parts are made from steel. This was

particularly true in the first half of the century when gage blocks were invented because

aluminum and plastic were still undeveloped technologies.

Finally, the steel gage block can be used to gage any material if corrections are made for

the differential thermal expansion of the two materials involved. If a steel gage block is used to gage a 100 mm aluminum part at 25 °C, a correction factor must be used. Since

the expansion coefficient of aluminum is about twice that of steel, when the part is brought

to 20 °C it will shrink twice as much as the steel. Thus the aluminum block must be made

oversized by the amount

21
AL = {a^_^^^— - Ql"*""') X L X at (3.4)

= (24 - 11 .5) X 1 0-^ X 1 00 mm X 5°C

= 6.25 |jm

So if we make the cube 6.25 pm larger than the steel gage block it will be exactly 1 00 mm

when brought to standard conditions (20 °C).

There are a few mixtures of materials, alloys such as invar and crystalline/glass mixtures

such as Zerodur, which have small thermal expansion coefficients. These materials are a combination of two components, one which expands with increasing temperature, and

one which shrinks with increasing temperature. A mixture is made for which the expansion

of one component matches the shrinkage of the other. Since the two materials never have

truly opposite temperature dependencies, the matching of expansion and shrinkage can

be made at only one temperature. It is important to remember that these materials are designed for one temperature, usually 20 °C, and the thermal expansion coefficient can

be much different even a few degrees away. Examples of such materials, super-invar [19],

Zerodur and Cervii [20], are shown in figure 3.6.

Figure 3.6. Variation of the thermal expansion coefficient for selected low

expansion materials with temperature.

The thermal expansion coefficients, at 20 °C, of commonly used materials in dimensional

metrology are shown in table 3.2.

22
Table 3.2

Material Thermal Expansion Coefficient

(10-^/°C )

Aluminum 24

Free Cutting Brass 20.5

Steel Gage Block ( <25mm ) 11.5

Steel Gage Block ( 500 mm ) 10.6

Ceramic Gage Block (zirconia) 9.2

Chrome Carbide 8.4

Granite 6.3

Oak (across grain) 5.4

Oak (along grain) 4.9

Tungsten Carbide 4.5

Invar 1.2

Fused Silica 0.55

Zerodur 0.05

To give a more intuitive feel for these numbers, figure 3.7 shows a bar graph of the relative

changes in length of 100 mm samples of various materials when taken from 20 °C to 25 °C (68 °F to 77 °F).

5 °C Expansion of 100 mm Samples

Aluminum

Brass

Steel Chrome Carbide

Granite Tungsten Carbide

Invar Fused Silica Zerodur

0 0.4 0.6 0.8 1.0 1.2 Micrometers

1.4

Figure 3.7 Thermal Expansion of 100 mm blocks of various materials from

20 °Cto 25 °C

23
3.3.2 Thermal Expansion Uncertainty

There are two sources of uncertainty in measurements due to the thermal expansion of

gages. These are apparent in thermal expansion equation, 3.2, where we can see that the

length of the block depends on our knowledge of both the temperature and the thermal

expansion coefficient of the gage. For most measurement systems the uncertainty in the

thermometer calibrations is known, either from the manufacturers specifications or the

known variations from previous calibrations. For simple industrial thermocouple or thermistor based systems this uncertainty is a few tenths of a degree. For gaging at the

sub-micrometer level this is generally insufficient and more sophisticated thermometry is needed.

The uncertainty in the expansion coefficient of the gage or workpiece is more difficult to

estimate. Most steel gage blocks under 1 00 mm are within a five tenths of 1 1 .5 x 1 0'^/°C,

although there is some variation from manufacturer to manufacturer, and even from batch

to batch from the same manufacturer. For long blocks, over 100 mm, the situation is more

complicated. Steel gage blocks have the gaging surfaces hardened during manufacturing

so that the surfaces can be properly lapped. This hardening process affects only the 30

to 60 mm of the block near the surfaces. For blocks under 1 00 mm this is the entire block,

and there is no problem. For longer blocks, there is a variable amount of the block in the

center which is partially hardened or unhardened. Hardened steel has a higher thermal

expansion coefficient than unhardened steel, which means that the longer the block the

greater is the unhardened portion and the lower is the coefficient. The measured

expansion coefficients of the NIST long gage blocks are shown in table 3.3.

Table 3.3. Thermal Expansion Coefficients of NIST Master Steel Gage Blocks (10"^/°C)

Size Set 1 Set 2

(in.)

5 11.41 11.27

6 11.33 11.25

7 11.06 11.35

8 11.22 10.92

10 10.84 10.64

12 10.71 10.64

16 10.80' 10.58

20 10.64 10.77

Table 3.3 shows that as the blocks get longer, the thermal expansion coefficient becomes

systematically smaller. It also shows that the differences between blocks of the same size can be as large as a few percent. Because of these variations, it is important to use long

24
length standards as near to 20 °C as possible to eliminate uncertainties due to the

variation in the expansion coefficient.

As an example, suppose we have a 500 mm gage block, a thermometer with an

uncertainty of 0.1 °C, and the thermal expansion coefficient is known to +/-0.3 x10"^. The

uncertainties when the thermometer reads 20 and 25 degrees are

AL = QlL X 6(AT) + 5(aL)L x AT (3.4)

where AT is the temperature difference (T-20), and 6() denotes the uncertainty of the

quantity within the parentheses.

At 25 °C:

AL =(11.5x10-')x500x0.1 + 0.3x10-^x 500x 5 (3.5) = 0.58 pm + .75 pm

= 1.33 pm

At 20°C: when the thermometer reads 20 °C, the worst case error is 0.1 °C

AL = (1 1.5x10-^) X 500x0.1 + 0.3x1 0"" x 500 x .1 (3.6) = 0.58 pm + 0.02 pm

= 0.60 pm

This points out the general need to keep dimensional metrology labs at, or very near

20 °C.

3.4 Elastic Properties

When a force is exerted on any material, the material deforms. For steel and other gage

block materials this effect is small, but not completely negligible. There are two

dimensional effects due to the elastic properties of gage blocks. The first, and least

important, is the compression of blocks under their own weight. When a block is supported

horizontally, the force on each point is the weight of the steel above it, and the steel is slightly compressed. The compression is, however, not in the direction of the gaging

dimension of the block and the effect is negligible. If the block is set upright, the force is now in the direction of the gaging surfaces, and for very long blocks the weight of the block

can become significant. Solved analytically, the change in length of a blocks is found to

be

AL = pgL2/2E (3.7)

25
Where

AL = length of shortening

p = density of material

g = acceleration of gravity

L = total length of block

E = Young's modulus for material

For steel gage blocks, the shrinkage is

AL = (7.8x1 0'(kg/m') x 9.8 m/s^ x L^)/{2 x 21 0x1 0^ N/m^) (3.8)

= 0.18x10"^ X in meters.

For a 500 mm gage block the correction is 45 nm (1.8 pin). The corrections from this

formula are made at NIST on demand, but are negligible for blocks less than 300 mm

(12 in).

When using long gage blocks supported horizontally, some care is needed to assure that

the block bends properly. Since the sides of the gage block are not precision surfaces, no

matter how flat the surface where it is placed it will touch only at a few points, therefore

bending, and in general producing some small angle between the two gaging faces. The

proper way to supporf the block so that the two end faces are parallel, and thereby

produce an unambiguous length, is shown in figure 3.8. This assumes, however, that the

gaging faces are parallel when the block is vertical.

Figure 3.8. Long gage block supported at i's Airy points.

26
When a block of length L is supported at two positions, 0.577L apart, the end faces will be

parallel. These positions are called the Airy points.

3.4.1 Contact Deformation in Mechanical Comparisons

Nearly all gage block length comparisons or length measurements of objects with gage

blocks are made with contact type comparators where a probe tip contacts a surface under

an applied force. Contact between a spherical tip and a plane surface results in local

deformation of small but significant magnitude. If the gage blocks or objects being

compared are made of the same material, the measured length difference between them

will be correct, since the deformation in each case will be the same. If the materials are

different, the length difference will be incorrect by the difference in the amount of

deformation for the materials. In such cases, a deformation correction may be applied if

its magnitude is significant to the measurement.

Total deformation (probe plus object) is a function of the geometry and elastic properties

of the two contacting surfaces, and contact force. Hertz [21 ] developed formulas for total

uniaxial deformation based on the theory of elasticity and by assuming that the bodies are

isotropic, that there is no tangential force at contact, and that the elastic limit is not

exceeded in the contact area. Many experimenters [22, 23] have verified the reliability of

the Hertzian formulas. The formulas given below are from a CSIRO (Australian metrology

laboratory) publication which contains formulas for a number of combinations of geometric

contact between planes, cylinders and spheres [24]. The gage block deformations have

been tested against other calculations and agree to a few nanometers.

For a spherical probe tip and a flat object surface the uniaxial deformation of the probe and

surface together is given by:

a = 3n2'^/(2p2'^(V,+V2)Di-^'^) (3.9)

Where

(1 -a,^)/nE,

^1 = Poisson ratio of sphere

Ei = elastic modulus of sphere

(1 -02')/nE2

02 = Poisson ratio of block

elastic modulus of block

p = force

D = diameter of sphere

27
Some example deformations for a 6 mm diameter diamond stylus, and common gage

block materials, at various pressures are given in table 3.4.

Table 3.4 Deformations at interface in micrometers

(pin in parenthesis)

Material Force in Newtons

0.25 0.50 0.75

Fused Silica

Steel

Chrome Carbide

Tungsten Carbide

0.13(5.2)

0.07 (2.7)

0.06 (2.2)

0.04 (1.6)

0.21 (8.3)

0.11 (4.4)

0.12 (3.4)

0.06 (2.5)

0.28(11.2)

0.14(5.7)

0.12 (4.6)

0.08 (3.2)

The gage block comparators at NIST use 6 mm diameter tips , with forces of 0.25 N on the

bottom probe and 0.75 N on the top probe. The trade-offs involved in tip radius selection

are:

1. The larger the probe radius the smaller the penetration, thus the correction will be

smaller and less dependent on the exact geometry of the tip radius.

2. The smaller the tip radius the greater its ability to push foreign matter, such as traces

of oil, water vapor, or dust out of the way when the block is dragged between the tips.

There are comparator tips used by some laboratories which have radii as large as 10 to

20 mm. These large radius tips have small penetrations, although generally more than

20 nm, and thus some correction still must be made.

3.4.2 Measurement of Probe Force and Tip Radius

Reliability of computed deformation values depends on careful measurement of probe

force and, especially, of probe tip radius. Probe force is easily measured with a force gage

(or a double pan balance and a set of weights) reading the force when the probe indicator

meter is at mid-scale on the highest magnification range.

Probe tip inspection and radius measurement are critical. If the tip geometry is flawed in any way it will not follow the Hertz predictions. Tips having cracks, flat spots, chips, or

ellipticity should be replaced and regular tip inspection must be made to insure reliability.

An interference microscope employing multiple beam interferometry is the inspection and

measurement method used at NIST [25]. In this instrument an optical flat is brought close

28
to the tip, normal to the probe axis, and monochromatic light produces a Newton Ring

fringe pattern which is magnified though the microscope lens system. The multiple beam

aspect of this instrument is produced by special optical components and results in very

sharp interference fringes which reveal fine details in the topography of the tip.

Figure 3.9 are multiple beam interference micrographs of diamond probe tips. The pictures

have been skeleltonized so that the important features are clear. The micrograph is a

"contour map" of the tip so that all points on a given ring are equidistant from the reference

optical flat. This can be expressed mathematically as

NA = 2t (3.10)

where N is the number (order) of the fringe, counting from the center, A is the wavelength

of the light and t is the distance of the ring from the optical flat. The zero order fringe at the center is where the probe tip is in light contact with the optical flat and t is nearly zero. This relationship is used to calculate the tip radius.

Discontinuities

from tip crack

Figure 3.9 a, b, and c. Examples of microinterferograms of diamond stylus tips.

Tip condition is readily observed from the micrograph. In figure 3.9a a crack is seen in the

tip as well as some ellipticity. The crack produces sharp breaks and lateral displacement

of the fringes along the crack. Figure 3.9b shows a sharp edge at the tip center. An

acceptable tip is shown in figure 3.9c. It is very difficult to produce a perfectly spherical

surface on diamond because the diamond hardness is not isotropic, i.e., there are "hard"

directions and "soft" directions, and material tends to be lapped preferentially from the

"soft" directions.

29
Radius measurement of a good tip is relatively simple. Diameters of the first five rings are measured from the photograph along axes A and B, as in figure 3.9c. If there is a small

amount of ellipticity, A and B are selected as the major and minor axes. Then

fd = (3.11)

2M

where r^ is the actual radius of the nth Nev\/ton ring, d^ is the ring average diameter (of A

or B) measured on the micrograph, and M is the microscope magnification. Substituting

the hng diameter measurements in the equation will result in 5 radii, r^ through and from

these a radius of curvature between consecutive rings is calculated:

R. = (3.12)

A

for i = 1 to 4. The average of these four values is used as the tip radius.

The preceding measured and calculated values will also serve to evaluate tip sphericity.

If the average difference between the five A and B ring diameters exceeds 10 percent of

the average ring diameter, there is significant lack of sphericity in the tip. Also, if the total

spread among the four tip radius values exceeds 10 percent of the average R there is significant lack of sphericity. These tests check sphericity around two axes so it is important that a tip meet both requirements or it will not follow the Hertz prediction.

Our current laboratory practice uses only like materials as master blocks for comparisons.

By having one set of steel masters and one set of chrome carbide masters, the only blocks

which have deformation corrections are tungsten carbide. We have too small a customer

base in this material to justify the expense of a third master set. This practice makes the

shape of the comparator tips unimportant, except for cracks or other abnormalities which

would scratch the blocks.

3.5 Stability

No material is completely stable. Due to processes at the atomic level all materials tend

to shrink or grow with time. The size and direction of dimensional change are dependent

on the fabrication processes, both the bulk material processing as well as the finishing

processing. During the 1950's the gage block manufacturers and an interdisciplinary group

of metallurgists, metrologists and statisticians from NIST (NBS at the time) did extensive

studies of the properties of steel gage blocks to optimize their dimensional stability. Blocks

made since that era are remarkably stable compared to their predecessors, but not perfect.

Nearly all NIST master gage blocks are very stable. Two typical examples are shown in figure 3.10. Because most blocks are so stable, we demand a measurement history of at

30
least 5 years before accepting a non-zero slope as real.

100

50

0

-50

-100

-150

-200

ID: 2110 Size: 2 Inch 72 Pts. Avg. = -1.821

+ \ + 4.

+

_l I L_ 1970 1980 1990

>

(D Q

D

C

E

o

250

200

150

100

50

0

-50

-100

-150

-200

ID:33e.O Size: 1 Inch 66 Pts. Slope = -€.10^ Pred. = -0.336

+ +

+

I + -t

r 1 J 1

+ i —

+

1 1 1 1

tT

1970 1980 1990

Figure 3.10. Examples of the dimensional stability of NIST master gage blocks.

A histogram of the growth rates of our master blocks, which have over 15 years of

measurement history, is shown in figure 3.11. Note that the rate of change/unit of length

is the pertinent parameter, because most materials exhibit a constant growth rate per unit

of length.

31
stability of Gage Blocks

4

0

Rate of Change (ppm/year)

Figure 3.1 1 . Histogram of the growth rates of NIST master blocks.

There have been studies of dimensional stability for other materials, and a table of typical

values [26-31] is given in table 3.5, In general, alloys and glassy materials are less stable

than composite materials. For the average user of gage blocks the dimensional changes

in the time between calibration is negligible.

Tab e 3.5

Material

Zerodur

Corning 7971 ULE

Corning 7940(fused Silica)

Cer-Vit C-1010

LR-35 Invar

Super Invar

52100 Gage Block Steel

Brass

Stability (1 part in 10^/yr)

-0.7 to -0.03

-0.14, 0.07, 0.06

-0.18, -0.18

0.54, -0.18

2.05

0.0

0.01

-1.2

Steel, chrome carbide and tungsten carbide have all proven to be very stable materials for

gage blocks. The newer ceramic materials may also be stable, but no studies have been

made of either the material or gage blocks made from it. Until studies are available, a

shorter calibration cycle for the first few years should be considered to assure that the

blocks are suitably stable.

32
4. Measurement Assurance Program

4.1 Introduction

One of the primary problems in metrology is to estimate the uncertainty of a measurement. Traditional methods, beginning with simple estimates by the metrologist

based on experience gradually developed into the more formal error budget method.

The error budget method, while adequate for many purposes, is slowly being replaced

by more objective methods based on statistical process control ideas. In this chapter

we will trace the development of these methods and derive both an error budget for

gage block calibrations and present the measurement assurance (MAP) method

currently used at NIST.

4.2 A comparison: Traditional Metrology versus Measurement Assurance

Programs

4.2.1 Tradition

Each measurement, in traditional metrology, was in essence a "work of art." The result

was accepted mainly on the basis of the method used and the reputation of the person

making the measurement. This view is still prevalent in calibrations, although the trend

to statistical process control in industry and the evolution of standards towards

demanding supportable uncertainty statements will eventually end the practice.

There are circumstances for which the "work of art" paradigm is appropriate or even

inevitable as, for example, in the sciences where most experiments are seldom

repeated because of the time or cost involved, or for one of a kind calibrations . Since

there is no repetition with which to estimate the repeatability of the measurement the

scientist or metrologist is reduced to making a list of possible error sources and

estimating the magnitude of these errors. This list, called the error budget, is often the

only way to derive an uncertainty for an experiment.

Making an accurate error budget is a difficult and time consuming task. As a simple

example, one facet of the error budget is the repeatability of a gage block comparison.

A simple estimation method is to measure the same block repeatedly 20 or more times.

The standard deviation from the mean then might be taken as the short term

repeatability of the measurement. To get a more realistic estimate which takes into

consideration the effects of the operator and equipment, the metrologist makes a GR&R (gage repeatability and reproducibility) study. In this test several blocks of

different length are calibrated using the normal calibration procedure. The calibrations are repeated a number of times by different operators using different comparators.

This data is then analyzed to produce a variability for the measurement process. Even

33
a study as large as this will not detect the effects of long term variability from sources

such as drift in the comparator or thermometer calibrations. Unfortunately, these

auxiliary experiments are seldom done in practice and estimates based on the

experimenter's experience are substituted for experimentally verified values. These

estimates often reflect the experimenter's optimism more than reality.

The dividing line between the era of traditional metrology and the era of measurement

assurance programs is clearly associated with the development of computers. Even

with simple comparison procedures, the traditionalist had to make long, detailed hand

computations. A large percentage of calibration time was spent on mathematical

procedures, checking and double checking hand computations. The calibration of a meter bar could generate over a hundred pages of data and calculations. Theoretical

work revolved around finding more efficient methods, efficient in the sense of fewer

measurements and simpler calculations. ^

4.2.2 Process Control: A Paradigm Shift

Measurement assurance has now become an accepted part of the national standards

system. According to the ANSI/ASQC Standard M-1 , "American National Standard for

Calibration Systems" the definition of measurement assurance is as follows:

2.12 Measurement Assurance Method

A method to determine calibration or measurement uncertainty based on

systematic observations of achieved results. Calibration uncertainty limits

resulting from the application of Measurement Assurance Methods are considered estimates based upon objective verification of uncertainty

components.

Admittedly, this definition does not convey much information to readers who do not

already understand measurement. To explain what the definition means, we will examine the more basic concept of process control and show how it relates to

measurement and calibration. Measurement assurance methods can be thought of as an extension of process control.

The following is an example of control of a process using a model from the chemical

industry. In a chemical plant the raw materials are put into what is essentially a black

box. At the molecular level, where reactions occur, the action is on a scale so small

and so fast that no human intervention is possible. What can be measured are various

bulk properties of the chemicals; the temperature, pressure, fluid flow rate, etc. The

basic model of process control is that once the factory output is suitable, if the

measurable quantities (process control variables) are kept constant the output will

34
remain constant. Thus the reactions which are not observable are monitored and

characterized by these observable control variables.

Choosing control variables is the most important part of the control scheme. A

thermometer placed on a pipeline or vat may or may not be a useful measure of the

process inside. Some chemical reactions occur over a very large range of

temperatures and the measurement and control of the temperature might have virtually

no effect on the process. In another part of the plant the temperature may be vitally

important. The art and science of choosing the process control measurements is a

large part of the chemical engineering profession.

Defining the measurement process as a production process evolved in the early 1960's

[33], a time coincident with the introduction of general purpose computers on a commercial basis. Both the philosophy and scope of measurement assurance

programs are a direct result of being able to store and recall large amounts of data, and

to analyze and format the results in many different ways. In this model, the calibration

procedure is a process with the numbers as the process output. In our case, the length

of a gage block is the output.

The idea of a measurement assurance program is to control and statistically

characterize a measurement process so that a rational judgement about the accuracy

the process can be made. Once we accept the model of measurement as a production

process we must examine the process for possible control variables, i.e., measurable

quantities other than the gage block length which characterize the process.

For example, suppose we compare one gage block with one "master" block. If one

comparison is made, the process will establish a number for the length of the block. An

obvious extension of this process is to compare the blocks twice, the second

measurement providing a measure of the repeatability of the process. Averaging the

two measurements also will give a more accurate answer for two reasons: first,

statistics assures us that the mean of several measurements has a higher probability of

being the accurate than any single measurement and secondly, if the two

measurements differ greatly we can repeat the measurement and discard the outlying

measurement as containing a blunder. Thus the repeatability can be used as a process control variable.

The next extension of the method is to make more measurements. It takes at least four

measurements for statistical measures of variability to make much sense. At this point

the standard deviation becomes a possible statistical process control (SPC) parameter.

The standard deviation of each measurement is a measure of the short term

repeatability of the process which can be recorded and compared to the repeatability of

previous measurements. When this short term repeatability is much higher than its historical value we can suspect that there is a problem with the measurement, i.e., the

35
process is out of control. Since the test is statistical in nature we can assign a confidence level to the decision that the process is in or out of control. An analysis of

the economic consequences of accepting bad or rejecting good calibrations can be

used to mal<e rational decisions about the control limits and confidence levels

appropriate to the decisions.

Another extension is to have two master blocks and compare the unknown to both of

them. Such a procedure can also give information about the process, namely the

observed difference between the two masters. As in the previous example, this

difference can be recorded and compared to the differences from previous calibrations.

Unlike the previous case this process control variable measures long term changes in the process. If one of the blocks is unstable and grows, over a year the difference

between the masters will change and an examination of this control parameter will detect it. Short term process repeatability is not affected by the length changes of the

blocks because the blocks do not change by detectable amounts during the few

minutes of a comparison.

Thus, comparing a gage block with two masters according to a measurement design

provides, not only a value for the block, but also an estimate of short term process

repeatability, and in time, an estimate of long term process variability and a check on

the constancy of the masters. All of this is inexpensive only if the storage and

manipulation of data is easy to perform and fast, tasks for which computers are ideal.

The complexity of a measurement assurance program depends upon the purpose a

particular measurement is to serve. NIST calibration services generally aim at providing

the highest practical level of accuracy. At lower accuracy levels the developer of a measurement system must make choices, balancing the level of accuracy needed with

costs. A number of different approaches to measurement assurance system design

are discussed by Croarkin [34].

4.2.3 Measurement Assurance: Building a Measurement Process Model

Assigning a length value to a gage block and determining the uncertainty of that value

is not a simple task. As discussed above, the short and long term variability of the

measurement process are easily measured by redundant measurements and a control

measurement (the difference between two masters) in each calibration. This is not

enough to assure measurement accuracy. SPC only attempts to guarantee that the

process today is the same as the process in the past. If the process begins flawed, i.e., gives the wrong answer, process control will only guarantee that the process continues

to give wrong answers. Obviously we need to ascertain the sources of error other than

the variability measured by the SPC scheme.

36
Many auxiliary parameters are used as corrections to the output of the measurement

procedure. These corrections must be measured separately to provide a complete

physical model for the measurement procedure. Examples of these parameters are the

phase change of light on reflection from the gage block during interferometric

measurements, the penetration depth of the gage block stylus into the block during

mechanical comparisons, and thermal expansion coefficients of the blocks during all measurements. These auxiliary parameters are sources of systematic error.

To assess systematic errors, studies must be made of the effects of factors not subject

to control, such as the uncertainty of thermometer and barometer calibrations, or

variations in the deformability of gage blocks. Combining these systematic errors with

the known random error, one arrives at a realistic statement of the accuracy of the

calibration.

In summary, the MAP approach enables us to clearly establish limitations for a

particular measurement method. It combines process control with detailed process

modeling. The process control component provides a means to monitor various

measurement parameters throughout the system and provides estimates of the random

error. The calibration model allows us to obtain reasonable estimates of the systematic

uncertainties which are not sampled by the statistics derived from an analysis of the

process control parameters. This leads to a detailed understanding of the

measurement system and an objective estimate of the measurement uncertainty. It also creates a reliable road map for making process changes that will increase

calibration accuracy.

4.3 Determining Uncertainty

4.3.1 Stability

All length measurement processes are, directly or indirectly, comparative operations.

Even the simplest concept of such a process assumes that the basic measurement unit

(in our case the meter) is constant and the object, procedures, equipment, etc., are

stable.

As an example, before 1959 the English inch was defined in terms of an official yard

bar and the American inch was defined in terms of the meter. By 1959 the English inch

had shrunk compared to the American inch because the yard bar was not stable and

was shrinking with respect to the meter. This problem was solved by adopting the

meter as the primary unit of length and defining the international inch as exactly 25.4

mm.

37
Similarly, the property to be measured must be predictable. If a measurement process

detects a difference between two things, it is expected that repeated measures of that

difference should agree reasonably well. In the absence of severe external influence,

one does not expect things to change rapidly.

There is a difference between stability and predictability as used above. Repeated

measurements over time can exhibit a random-like variability about a constant value, or about a time dependent value. In either case, if the results are not erratic (with no

unexpected large changes), the process is considered to be predictable. Gage blocks

which are changing length at a constant rate can be used because they have a

predictable length at any given time. Stability means that the coefficients of time

dependent terms are essentially zero. Stability is desirable for certain usages, but it is not a necessary restriction on the ability to make good measurements.

4.3.2 Uncertainty

A measurement process is continually affected by perturbations from a variety of sources. The random-like variability of repeated measurements is a result of these

perturbations. Random variability implies a probability distribution with a range of

variability that is not likely to exceed test limits. Generally a normal distribution is assumed.

Traditionally the second type of error is called systematic error, and includes

uncertainties which come from constants which are in error and discrepancies in operational techniques. The systematic error, expressed as a single number, is an

estimate of the offset of the measurement result from the true value. For example, for

secondary laboratories using NIST values for their master blocks the uncertainty

reported by NIST is a systematic error. No matter how often the master blocks are used the offset between the NIST values and the true lengths of the blocks remains the

same. The random error and systematic error are combined to determine the

uncertainty of a calibration.

A new internationally accepted method for classifying and combining errors has been

developed by the International Bureau of Weights and Measures (BIPM) and the

International Organization for Standardization(ISO). This method seems more

complicated than the traditional method, and is presented in detail in the ISO "Guide to the Expression of Uncertainty in Measurement" [35]. A short summary of the method

and how it will be applied at NIST is given in reference [36]. Quoting ISO, the core concepts are:

1 . The uncertainty in the result of a measurement generally consists of several

components which may be grouped into two categories according to the way in which their numerical value is estimated:

38
A - those which are evaluated by statistical methods,

B - those which are evaluated by other means.

There is not always a simple correspondence between the classification into

categories A or B and the previously used classification into "random" and

"systematic" uncertainties. The term "systematic uncertainty" can be misleading

and should be avoided.

Any detailed report of the uncertainty should consist of a complete list of the

components, specifying for each the method used to obtain its numerical value.

2. The components in category A are characterized by estimated variances, Sj^, (or the estimated "standard deviations" S|) and the number of degrees of

freedom, v, . Where appropriate, the estimated covariances should be given.

3. The components in category B should be characterized by quantities uf-, which may be considered as approximations to the corresponding variances, the

existence of which is assumed. The quantities may be treated like variances

and the quantities Uj like standard deviations. Where appropriate, the

covariances should be treated in a similar way.

4. The combined uncertainty should be characterized by the numerical value

obtained by applying the usual method for the combination of variances. The

combined uncertainty and its components should be expressed in the form of

"standard deviations."

5. If, for particular applications, it is necessary to multiply the combined

uncertainty by a factor to obtain an overall uncertainty, the multiplying factor

used must always be stated.

The random errors, being determined statistically, are type A errors. The errors which

made up the systematic errors become the sources of type B uncertainty. The main

difference is the method for estimating the type B uncertainty. The traditional

systematic error was estimated by the metrologist with little or no guidance. The ISO

guide gives specific criteria for the estimation of type B uncertainty. NIST has adopted

the new international method for reporting calibration uncertainties. In addition to adopting the new method, NIST has adopted k=2 as the coverage factor, rather than

the more traditional 3a, to align its reported uncertainty with the current European

practice.

39
4.3.3 Random Error

There are very few sources of truly random error. Some examples are 1/f and shot

noise in electronic components. These contributions to the uncertainty of most

dimensional measurements are very small. Most of the contributions to the random

errors are systematic effects from sources which are not under experimental control.

Examples are floor and air vibrations, thermal drift, and operator variability. Many of

these error sources are too costly to control.

These effects can be grouped into two categories: long term effects in which the effect

does not significantly change in the time required for a given sequence of

measurements and short term effects which vary significantly in the course of a single

measurement or measurements made over a short time interval. Typical long term

effects are changes in length of the master blocks or slow changes in the comparator

electronic magnification.

An example of smooth continuous short term effects are thermal drift of the

environment during the measurement. Another category of short term effects are those

which are instantaneous, or step-like, in nature. A simple example is a dust particle left on a gage block. Depending on the exact point of contact the gage length may be

shifted by the size of the particle. In many cases, "shocks" or vibrations can cause

temporary shifts in the instrument output.

These short term variations make up the within-group variability usually expressed as a

standard deviation, o^. The within-group variability of the measurement process is the

most familiar process parameter, and is generally called repeatability. It is easily

demonstrated in a repeated sequence of measurements of the same thing in a short

time interval. Practically all measurements should be repeated several times. Withingroup variability is generally limited by the degree that certain types of perturbations are

controlled and by factors such as operator skills, instrument quality, and attention to

detailed procedure. Sources contributing to within variability are not easily identified

and process improvement in terms of reducing is obtained more frequently by trial and error than by design.

Process accuracy is often judged on the basis of within-group variability. Such a judgment, however, is usually erroneous. A more reliable assessment is the total

variability which is the variability of a long sequence of measurements. Repeating a

given measurement over a time interval Sufficiently long to reflect the influence of all possible perturbations establishes a total process standard deviation, Oj.

The primary way to improve total variability is to modify the process by reducing the

variability associated with all identifiable perturbations. This can be very tedious and

expensive, but is necessary to produce the highest quality measurements.

40
Another straightforward way to reduce variability is to place severe restrictions on the

characteristics of objects being measured and on the measurement environment.

Reducing the complexity of the object reduces the possible error sources. Thus

traditional metrology focused on simple geometric forms, spheres, cylinders and blocks as the mainstays of length standards. By making all measurements at or near standard

conditions, 20 °C for example, many other errors are reduced. Restricting the number

and severity of possible errors brings the uncertainty estimate closer to the truth.

4.3.4 Systematic Error and Type B Uncertainty

The sources of systematic error and type B uncertainty are the same. The only major

difference is in how the uncertainties are estimated. The estimates for systematic error which have been part of the NIST error budget for gage blocks have been traditionally

obtained by methods which are identical or closely related to the methods

recommended for type B uncertainties.

There are several classes of type B uncertainty. Perhaps the most familiar is instrument scale zero error (offset). Offset systematic errors are not present in comparative measurements provided that the instrument response is reasonably linear over the range of difference which must be measured. A second class of type B

uncertainty comes from supplemental measurements. Barometric pressure,

temperature, and humidity are examples of interferometric supplemental

measurements used to calculate the refractive index of air, and subsequently the

wavelength of light at ambient conditions. Other examples are the phase shift of light

on reflection from the block and platen, and the stylus penetration in mechanical

comparisons. Each supplemental measurement is a separate process with both

random variability and systematic effects. The random variability of the supplemental

measurements is reflected in the total process variability. Type B Uncertainty

associated with supplemental data must be carefully documented and evaluated.

One practical action is to "randomize" the type B uncertainty by using different

instruments, operators, environmental and other factors. Thus, variation from these

sources becomes part of the type A uncertainty. This is done when we derive the

length of master blocks from their interferometric history. At NIST, we measure blocks

using both the Hilger^ and Zeiss^ interferometers and recalibrate the barometer,

thermometers, and humidity meters before starting a round of calibrations. As a measurement history is produced, the effects of these factors are randomized. When a

block is replaced this "randomization" of various effects is lost and can be regained only

after a number of calibrations.

^ Certain commercial instruments are identified in this paper in order to adequately specify our measurement process. Sucfi identification does not imply recommendation or endorsement by NIST.

41
Generally, replacement blocks are not put into service until they have been calibrated

interferometrically a number of times over three or more years. Since one calibration

consists of 3 or more wrings, the minimum history consists of nine or more

measurements using three different calibrations of the environmental sensors. While

this is merely a rule of thumb, examination of the first 1 0 calibrations of blocks with long

histories appear to assume values and variances which are consistent with their long

term values.

The traditional procedure for blocks which do not have long calibration histories is to evaluate the type B uncertainty by direct experiment and make appropriate corrections

to the data. In many cases the corrections are small and the uncertainty of each

correction is much smaller than the process standard deviation. For example, in correcting for thermal expansion, the error introduced by a temperature error of 0.1 °C

for a steel gage block is about 1 part in 10^. This is a significant error for most

calibration labs and a correction must be applied. The uncertainty in this correction is due to the thermometer calibration and the uncertainty in the thermal expansion

coefficient of the steel. If the measurements are made near 20 °C the uncertainty in the thermal coefficient (about 10%) leads to an uncertainty in the correction of 1 part in 10'' or less; a negligible uncertainty for many labs.

Unfortunately, in many cases the auxiliary experiments are difficult and time

consuming. As an example, the phase correction applied to interferometric

measurements is generally about 25 nm (1 pin). The 2a random error associated with

measuring the phase of a block is about 50 nm (2pin). The standard deviation of the

mean of n measurements is n"^'^ times the standard deviation of the original

measurements. A large number of measurements is needed to determine an effect

which is very small compared to the standard deviation of the measurement. For the

uncertainty to be 10% of the correction, as in the thermal expansion coefficient

example, n must satisfy the relation:

5 0

10% of 2b < {in nanometers) (4.1)

Since this leads to n=400, it is obvious this measurement cannot be made on every

customer block. Phase measurements made for a random sample of blocks from each

manufacturer have shown that the phase is nearly the same for blocks made: (1 ) the

same material, (2) by the same lapping procedure and (3) by the same manufacturer.

At NIST, we have made supplementary measurements under varying conditions and

calculated compensation factors for each manufacturer (see appendix C). A corollary

is that if you measure your master blocks by interferometry, it helps to use blocks from

only one or two manufacturers.

42
Our next concern is whether the corrections derived from these supplementary

measurements are adequate. A collection of values from repeated measurements

should be tested for correlation with each of the supplementary measurements, and

their various combinations. If correlation is indicated then the corrections which are being applied are not adequate. For example, if a correlation exists between the length

of gage blocks previously corrected for temperature and the observed temperature of

the block, then either the temperature is being measured at an inappropriate location or the correction scheme does not describe the effect that is actually occurring.

Corrective action is necessary.

Low correlation does not necessarily indicate that no residual type B uncertainty are

present, but only that the combined type B uncertainty are not large relative to the total

standard deviation of the process. There may be long term systematic error effects

from sources not associated with the current supplemental measurements. It is relatively easy to demonstrate the presence or absence of such effects, but it may be

difficult to reduce them. If both the short and long term standard deviations of the

process are available, they are expected to be nearly equal if there are no unresolved

sources of variation. Frequently such is not the case. In the cases where the long term

SD is distinctly larger than the short term SD there are uncorrected error sources at work.

The use of comparison designs, described in chapter 5, facilitates this type of analysis.

The within group variability, o^, is computed for each measurement sequence. Each

measurement sequence includes a check standard which is measured in each

calibration. The total standard deviation, a, is computed for the collection of values for

the check standard. The inequality o{»o^ is taken as evidence of a long term

systematic effect, perhaps as yet unidentified. For the NIST system a very large

number of measurements have been made over the years and we have found, as shown in table 4.1 , that except for long blocks there appear to be no uncorrected long

term error sources. The differences for long blocks are probably due to inadequate

thermal preparation or operator effects caused by the difficult nature of the

measurement.

43

11
Table 4.1

Comparison of Short Term and Long Term Standard Deviations

for Mechanical Gage Block Comparisons (in nanometers)

Group ^shortterm ^ olong term

14 1.00 mm to 1.09 mm 5 6

15 1.10 mm to 1.29 mm 5 5

16 1.30 mm to 1 .49 mm 5 5

17 1.50 mm to 2.09 mm 5 5

- 18 2.10 mm to 2.29 mm 5 5

19 2.30 mm to 2.49 mm 5 5

20 2.50 mm to 10 mm 5 5

21 10.5 mm to 20 mm 5 7

22 20.5 mm to 50 mm 6 8

23 60 mm to 100 mm 8 18

24 125 mm 9 19

150 mm 11 38

175 mm 10 22

200 mm 14 37

250 mm 11 48

300 mm 7 64

400 mm 11 58

500 mm 9 56

4.3.5 Error Budgets

An error budget is a systematic method for estimating the limit of error or uncertainty for a measurement. In this method, one lists all known sources of error which might affect

the measurement result. In the traditional method, a theoretical analysis of error sources is made to provide estimates of the expected variability term by term. These

estimates are then combined to obtain an estimate of the total expected error bounds.

The items in an error budget can be classified according to the way they are most likely

to affect the uncertainty. One category can contain items related to type B uncertainty;

another can relate to a^; and a third category can relate to Oy. Since all sources of short

term variability also affect the long term variability, short term effects are sometimes

subtracted to give an estimate of the effects of only the sources of long term variability.

This long term or "between group variability", Oq, is simply calculated as (Oi-o^). The

example in table 4.2 shows a tentative disposition for the mechanical comparison of

44
gage blocks.

Table 4.2

Error Source Category

Uncertainty of Master Block

Stylus diameter, pressure and condition

Assigned Temperature Coefficients

Temperature difference between blocks

Stability of Blocks

Environment: vibration, thermal changes,

operator effects

Block flatness/ anvil design

Block geometry: effect of flatness

and parallelism on measurement

with gaging point variability

Comparator calibration

Comparator repeatability

systematic

long term

systematic

long term

long term

short term

systematic

short term

long term

short term

Most of the error sources in table 4.2 can be monitored or evaluated by a judicious

choice of a comparison design, to be used over a long enough time so that the

perturbations can exert their full effect on the measurement process. Two exceptions

are the uncertainties associated with the assigned starting values, or restraint values,

and the instrument calibration.

Table 4.3 shows error budget items associated with gage block measurement

processes. Listed (1 o level) uncertainties are typical of the NIST measurement process

for blocks under 25 mm long. The block nominal length, L, is in meters.

45
Table 4.3

Error Sources

Resulting Length

Uncertainty in |jm Note

Repeatability 0.004 + 0.12 L (1)

Contact Deformation 0.002 (2)

Value of Master Block 0.012 + 0.08 L

Temperature Difference

Between Blocks 0.17L (3)

Uncertainty of Expansion

Coefficient 0.20 L (4)

Note 1. Repeatability is determined by analyzing the long term control data. The

number reported is the 1o, value for blocks under 25 mm.

Note 2. For steel and chrome carbide, blocks of the same material are used in the

comparator process and the only error is from the variation in the surface hardness

from block to block. The number quoted is an estimate. For other materials the

estimate will be higher.

Note 3. A maximum temperature difference of 0.030 °C is assumed. Using this as the

limit of a rectangular distribution yields a standard uncertainty of 0.1 7L.

Note 4. A maximum uncertainty of 1x10"^/°C is assumed in the thermal expansion

coefficient, and a temperature offset of 0.3 °C is assumed. For steel and chrome

carbide blocks we have no major differential expansion error since blocks of the same

material are compared. For other materials the estimate will be higher.

These errors can be combined in various ways, as discussed in the next section.

r 4.3.6 Combining Type A and Type B uncertainties

Type A and type B uncertainties are treated as equal, statistically determined and if uncorrelated are combined in quadrature.

46
U = Expanded Uncertainty (4.3)

In keeping with current European practice, the coverage factor k is set equal to two

(k=2) on NIST gage block uncertainty statements.

Combining the uncertainty components in Table 4.3 according to these rules gives a standard uncertainty (coverage factor k=2)

U = Expanded Uncertainty = 0.026 + 0.32L

Generally, secondary laboratories can treat the NIST stated uncertainty as an

uncorrelated type B uncertainty. There are certain practices where the NIST

uncertainty should be used with caution. For example, if a commercial laboratory has a gage block measured at NIST every year for nine years, it is good practice for the lab to use the mean of the nine measurements as the length of the block rather than the last

reported measurement. In estimating the uncertainty of the mean, however, some

caution is needed. The nine NIST measurements contain type B uncertainties which

are highly correlated, and therefore cannot be handled in a routine manner. Since the

phase assigned to the NIST master blocks, the penetration, wringing film thickness and

thermal expansion differences between the NIST master and the customer block do not

change over the nine calibrations, the uncertainties due to these factors are correlated.

In addition, the length of the NIST master blocks are taken from a fit to all of it's measurement history. Since most blocks have over 20 years of measurement history,

the new master value obtained when a few new measurements are made every 3 years

is highly correlated to the old value since the fit is heavily weighted by the older data.

To take the NIST uncertainty and divide it by 3 (square root of 9) assumes that all of the

uncertainty sources are uncorrelated, which is not true.

Even if the standard deviation from the mean was calculated from the nine NIST

calibrations, the resulting number would not be a complete estimate of the uncertainty

since it would not include the effects of the type B uncertainties mentioned above. The

NIST staff should be consulted before attempting to estimate the uncertainty in averaged NIST data.

4.3.7 Combining Random and Systematic Errors

There is no universally accepted method of combining these two error estimates. The

traditional way to combine random errors is to sum the variances characteristic of each

error source. Since the standard deviation is the square root of the variance, the

practice is to sum the standard deviations in quadrature. For example, if the

47
measurement has two sources of random error characterized by and O2, the total

random error is

cjt„, = [o,' + 02']"^. (4.4)

This formula reflects the assumption that the various random error sources are independent. If there are a number of sources of variation and their effects are independent and random, the probability of all of the effects being large and the same

sign is very small. Thus the combined error will be less than the sum of the maximum

error possible from each source.

Traditionally, systematic errors are not random (they are constant for all calibrations)

and may be highly correlated, so nothing can be assumed about their relative sizes or

signs. It is logical then to simply add the systematic errors to the total random error:

Total Uncertainty (traditions) = 3a,ot + Systematic Error. (4.5)

The use of 3o as the total random error is fairly standard in the United States. This

corresponds to the 99.7% confidence level, i.e., we claim that we are 99.7% confident

that the answer is within the ±3a bounds. Many people, particularly in Europe, use the

95% confidence level (2a) or the 99% confidence level (2.5a) rather than the 99.7%

level (3a) used above. This is a small matter since any time an uncertainty is stated the

confidence level should also be stated. The uncertainty components in Table 4.3 are

all systematic except the repeatability, and the uncertainty calculated according to the

traditional method is

Total Uncertainty (traditional) = 3(0.004 + 0.1 2L) + 0.002 + 0.012 + 0.08L + 0.17L +

0.2L

Total Uncertainty (traditional) = 0.026 + 0.81 L

4.4 The NIST Gage Block Measurement Assurance Program

The NIST gage block measurement assurance program (MAP) has three parts. The

first is a continual interferometric measurement of the master gage blocks under

process control supplied by control charts on each individual gage block. As a measurement history is built up for each block, the variability can be checked against

both the previous history of the block and the typical behavior of other similar sized

blocks. A least squares fit to the historical data for each block provides estimates of the

current length of the block as well as the uncertainty in the assigned length. When the

master blocks are used in the mechanical comparison process for customer blocks, the

48
interferometric uncertainty is combined in quadrature with the Intercomparison random

error. This combined uncertainty is reported as the uncertainty on the calibration

certificate.

The second part of the MAP is the process control of the mechanical comparisons with

customer blocks. Two master blocks are used in each calibration and the measured

difference in length between the master blocks is recorded for every calibration. Also,

the standard deviation derived from the comparison data is recorded. These two sets

of data provide process control parameters for the comparison process as well as

estimates of short and long term process variability.

The last test of the system is a comparison of each customer block length to its history,

if one exists. This comparison had been restricted to the last calibration of each

customer set, but since early 1988 every customer calibration is recorded in a database. Thus, in the future we can compare the new calibration with all of the

previous calibrations. Blocks with large changes are re-measured, and if the

discrepancy repeats the customer is contacted to explore the problem in more detail.

When repeated measurements do not match the history, usually the block has been

damaged, or even replaced by a customer. The history of such a block is then purged

from the database.

4.4.1 Establishing Interferometric Master Values

The general practice in the past was to use the last interferometric calibration of a gage

block as its "best" length. When master blocks are calibrated repeatedly over a period

of years the practice of using a few data points is clearly not the optimum use of the

information available. If the blocks are stable, a suitable average of the history data will produce a mean length with a much lower uncertainty than the last measurement value.

Reviewing the calibration history since the late fifties indicates a reasonably stable

length for most gage blocks. More information on stability is given in section 3.4. The

general patterns for NIST master gage blocks are a very small constant rate of change

in length over the time covered. In order to establish a "predicted value," a line of the

form:

U = U + K,(t-tJ (4.6)

is fit to the data. In this relation the correction to nominal length (Lt) at any time (t) is a

function of the correction (LJ at an arbitrary time (tj, the rate of change (Ki), and the

time interval (t-tj.

49
When the calibration history is short (occasionally blocks are replaced because of wear)

there may not be enough data to make a reliable estimate of the rate of change K^. We

use a few simple rules of thumb to handle such cases:

If the calibration history spans less than 3 years the rate of change is assumed to be zero. Few blocks under a few inches have rates of

change over 5 nm/year, and thus no appreciable error will accumulate.

A linear least squares fit is made to the history (over 3 years) data. If the

calculated slope of the line does not exceed three times the standard

deviation of the slope, we do not consider the slope to be statistically

significant. The data is then refitted to a line with zero slope.

ID: 5-83 SIZE: 0.1 inch 56 Pts. Avg. = 0.296

0

o

c2

o

E

c

co

>

0Q

D

c

e

o

LlL*

100

o3

50 Q

O

I

Q

O

0 g

Q

O

H -50 I

©

1960 1970 1980 1990

YEAR

Figure 4.1a. Interferometric history of the 0. 1 inch steel master block

showing typical stability of steel gage blocks.

50
ID; 3161 Size: 0.1 12 Inch 21 Pts. Slope = -0.154 Pred.=0,613

Figure 4. 1 a shows the history of the 0. 1 inch block, and it is typical of most NIST master

blocks. The block has over 20 years of history and the material is very stable. Figure,

4.1b, shows a block which has enough history to be considered unstable; it appears to be

shrinking. New blocks which show a non-negligible slope are remeasured on a faster cycle

and if the slope continues for over 20 measurements and/or 5 years the block is replaced.

The uncertainty of predicted values, shown in figure 4.1 by dashed lines above and below

the fitted line, is a function of the number of points in each collection, the degree of

extrapolation beyond the time span encompassed by these points, and the standard

deviation of the fit.

The uncertainty of the predicted value is computed by the relation

3o = 3a 1 (t-tj2 - + (4.7)

^ it -t )^

where n = the number of points in the collection;

t = time/date of the prediction;

ta= average time/date ( the centroid of time span covered by the measurement

51

II I
history, that is ta^It/n);

tj= time/date associated with each of the n values;

a = process standard deviation (s.d. about the fitted line);

at= s.d. of predicted value at time t.

In figure 4.2 the standard deviations of our steel inch master blocks are shown as a

function of nominal lengths. The graph shows that there are two types of contributions to

the uncertainty, one constant and the other length dependent.

CDo

c

oo

1.5

1.2

0.9

0.6

0.0

+

+

I I I I I I 1

1

J I I

I I I I I

30

20 §

o

3

%

©

J3 10

0,1 1.0 10

Nominal Length (Inches)

0.0 30

Figure 4.2. Plot of standard deviations of interferometric length history of

NIST master steel gage blocks versus nominal lengths.

The contributions which are not length dependent are primarily variations in the wringing

film thickness and random errors in reading the fringe fractions. The major length

dependent component are the refractive index and thermal expansion compensation

errors.

The interferometric values assigned to each block from this analysis forms the basis of the

NIST master gage block calibration procedure. For each size gage block commonly in use

up to 100 mm (4 in.), we have two master blocks, one steel and one chrome carbide.

Blocks in the long series, 125 mm (5 in.) to 500 mm (20 in.) are all made of steel, so there

are two steel masters for each size. Lengths of customer blocks are then determined by

52
mechanical comparison with these masters in specially designed sequences.

4.4.2 The Comparison Process

There are currently three comparison schemes used for gage block measurements. The

main calibration scheme uses four gage blocks, two NIST blocks and two customer blocks,

measured six times each in an order designed to eliminate the effects of linear drift in the

measurement system. For gage blocks longer than 100mm (4 in.) there are two different

schemes used. For most of these long blocks a procedure using two NIST blocks and two

customer blocks measured four times each in a drift eliminating design is used.

Rectangular blocks over 250 mm are very difficult to manipulate in the comparators, and

a reduced scheme of two NIST masters and one customer block is used.

Comparators and associated measurement techniques are described in chapter 5. In this

chapter, only the measurement assurance aspects of processes are discussed. The MAP

for mechanical comparisons consists of a measurement scheme, extraction of relevant

process control parameters, process controls, and continuing analysis of process

performance to determine process uncertainty.

4.4.2.1 Measurement Schemes - Drift Eliminating Designs

Gage block comparison schemes are designed with a number of characteristics in mind.

Unless there are extenuating circumstances, each scheme should be drift eliminating,

provide one or more control measurements, and include enough repeated comparisons

to provide valid statistics. Each of these characteristics is discussed briefly in this section.

The only scheme used at NIST which does not meet these specifications is for long

rectangular blocks, where the difficulty in handling the blocks makes multiple

measurements impractical.

In any length measurement both the gage being measured and the measuring instrument

may change size during the measurement. Usually this is due to thermal interactions with

the room or operator, although with some laser based devices changes in the atmospheric

pressure will cause apparent length changes. The instrumental drift can be obtained from

any comparison scheme with more measurements than the number of unknown lengths

plus one for drift, since the drift rate can be included as a parameter in the model fit. For

example, the NIST comparisons include six measurements each of four blocks for a total

of 24 measurements. Since there are only three unknown block lengths and the drift, least

squares techniques can be used to obtain the best estimate of the lengths and drift. It has

been customary at NIST to analyze the measurements in pairs, recording the difference

between the first two measurements, the difference between the next two measurements,

and so on. Since some information is not used, the comparison scheme must meet

specific criteria to avoid increasing the uncertainty of the calibration. These designs are

called drift eliminating designs.

53
Besides eliminating the effects of linear instrumental drift during the measurements, drift eliminating designs allow the linear dnft itself to be measured. This measured drift can be

used as a process control parameter. For small drift rates an assumption of linear drift will certainly be adequate. For high drift rates and/or long measurement times the assumption

of linear drift may not be true. The length of time for a measurement could be used as a

control parameter, but we have found it easier to restrict the measurements in a calibration

to a small number which reduces the interval and makes recording the time unnecessary.

As an example of drift eliminating designs, suppose we wish to compare two gage blocks,

A and B. At least two measurements for each block are needed to verify repeatability.

There are two possible measurement schemes: [ A B A B ] and [ABBA]. Suppose the

measurements are made equally spaced in time and the instrument is drifting by some A

during each interval. The measurements made are:

Measurement Scheme ABAB Scheme ABBA

m2

m4

A

B + A

A + 2A

B + 3A

A

B + A

B + 2A

A + 3A

(4.8)

If we group the measurements into difference measurements:

Scheme ABAB Scheme ABBA

yi= mi-m2 = A-(B + A) A - (B + A) (4.9a)

Y2= m3-m4 = (A+2A)-(B+3A) (B + 2A)-(A+3A) (4.9b)

Solving for B in terms of A:

B= A-(y,+y2)/2-A A - (y,-y2)/2 (4.10)

In the first case there is no way to find B in terms of A without knowing A, and in the

second case the drift terms cancel out. In reality, the drift terms will cancel only if the four

measurements are done quickly in a uniformly timed sequence so that the drift is most

likely to be linear.

A general method for making drift eliminating designs and using the least squares method

to analyze the data is given in appendix A.

54
The current scheme for most gage block comparisons uses two NIST blocks, one as a master block (S) and one as a control (C), in the drift eliminating design shown below. Two

blocks, one from each of two customers, are used as the unknowns X and Y. The 12

measured differences, y,, are given below:

QO - u

y2

VY - co

VA - VY

y4 c - S

c -X

Ve Y -c

V7 S -X

Ys C - Y

Ys S - Y

Yio X -C

Yii X -S

Y12 Y -X

The calibration software allows the data to be analyzed using either the S block or the C

block as the known length (restraint) for the comparison. This permits customer blocks of

different materials to be calibrated at the same time. If one customer block is steel and the

other carbide, the steel master can be used as the restraint for the steel unknown, and the

carbide master can be used for the carbide unknown. The length difference (S-C) is used

as a control. It is independent of which master block is the restraint.

There is a complete discussion of our calibration schemes and their analysis in appendix

A. Here we will only examine the solution equations:

S = L (reference length) (4.12a)

C = (1/8) (-2yi + y2 + 2y, + Ys - Ye - Vy + Vs - Ys - Yio + Yii) + L (4.12b)

X = (1/8) (-yi + Ys + Y3 + Y4 - Ys - 2y7 - Y9 + Y10 + 2y,, - y,^) + L (4.12c)

Y = (1/8) i-y, + 2y2 - Y3 + Y4 + Ye - Y7 - Ys - 2y9 + Yn + Y12) + L (4.1 2d)

A = (-1/12)(yi+y2+y3+Y4+Y5+Y6+Y7+Y8+Y9+Yio+Yii+Yi2) (4.12e)

55
The standard deviation is calculated from the ohginal comparison data. The value of each

comparison, <yi>, is calculated from the best fit parameters and subtracted from the actual

data, Yj. The square root of the sum of these differences squared divided by the square

root of the degrees of freedom is taken as the (short term or within) standard deviation of

the calibration:

For example.

Ay,2 =(C-B)-y,2 (4.13a)

= (1/8) iVi -2y3 + Vs +y6 + yy - ys - yg - yio - yn) - yi2 (4.i3b)

These deviations are used to calculate the variance.

12

o ^ = Y i^Y ) /8 (4.14)

i = 0 .

'

The standard deviation (square root of the variance), a measure of the short term variation

of the comparison process, is recorded for each calibration in the MAP file, along with other

pertinent data from the calibration. These data are later used to determine the value of

one control parameter, the short term variation, as described in the next section.

4.4.2.2 Control Parameter for Repeatability

There are a number of ways to monitor the comparator process. The simplest is to

compare the variance of each calibration to an accepted level based on the history. This

accepted level (average variance from calibration history) is characteristic of the short term

repeatability of the comparator process, and is called the "within" variance.

In our system the within variance is used for two purposes. One is as a control point for

the calibration, as discussed below. The second is that being a measure of short term

variation of the entire calibration process, it is used to calculate the process uncertainty, a part of the total uncertainty reported to the customer.

In our system the variance for each calibration is recorded in the MAP file. Each year the

last three years of data are analyzed to update the process control parameters used for

the next year. Figure 4.3 shows a typical MAP file data plot.

56
O = 8.5 nm a = 7.5 nm C = 5.0 nm

30

©

25

E 20

O

c

o

c

15

c

10

I—CO 5

0

1992 YEAR

Figure 4.3. Variation of the short term standard deviation with time for the

0.1 inch size.

The most prominent characteristic of the graph is the continuing reduction of the standard

deviation since 1981. The installation of new comparators during 1987 and the

replacement of one of the old master sets with a new chrome carbide set in 1989 have

resulted in significant reductions in the process uncertainty. When the current process

parameter, o^, is calculated, only the data since the last process change (new master set

in 1989) will be used.

If the calibration variance was independent of the block length only one value of the control

parameter would be needed, and the average of the last few years of data from all of the

calibrations could be used as the control point. The variance is, however, made up of

many effects, some length dependent and some not.

Some of the non-length dependent part of the variance is due to electronic noise in the

comparator. Another source is the contact point of the comparator sampling slightly

different points on the surface of the block each time. Due to variations in flatness,

parallelism and probe penetration into the surface, different lengths are measured.

The length dependent part of the variance is due to factors such as thermal variations in the blocks. The variation caused by this factor is not so strong that every block size must

be considered separately. For example, the effects of thermal variations for 1 mm and 1.1

mm blocks should be nearly identical, although the same thermal variation on the 5 mm

and 50 mm blocks will be measurably different. By separating the blocks into small groups

of nearly the same size the number of measurements is increased, making the

57
determination of tlie variance more accurate for the group than it would be if done on an

individual basis.

Each group is made of about 20 sizes, except for blocks over 25 mm (1 in) where each

size is analyzed separately. The groupings in current use are shown in table 4.4.

For each block size the average variance for the last three years is calculated. A weighted

average of the variances from individual blocks is calculated, as shown in eq (4.4).

771 E in.-2)S^

oj = (4.15)

£ in. -2)

1=1

In the above equation:

Op is the pooled standard deviation of the group,

m is the number of sizes in the group,

nj is the number of measurements of size j,

Sj is the average standard deviation calculated from the MAP file for size j.

The within, is calculated for each group.

Table 4.4. Current Groupings for Calculating o,

Group Number Sizes Units

0.010 to 0.09375 inch (thins)

2 0.100 to 0.107 inch

3 0.108 to 0.126 inch

4 0.027 to 0.146 inch

5 0.147 to 0.500 inch

6* ; 0.550 to 4.000 inch

7* 5.000 to 20.00 inch

8 0.0101 to 0.019 inch (.010 thins)

9 0.0201 to 0.029 inch (.020 thins)

10 0.0501 to 0.059 inch (0.050 thins;

58
1

1

1 1 nu.zuu onn TO U.zUU 1 inch (step

1

1 Z9 1 1 1^ O O 1 ^*\ unassignea

1 O n -1 n U. 1 u to n oc u.yo millimeter

1 /I

1 ^ 1 .uu to 'I nn 1 .uy millimeter

1 O 1 . 1 u to d on 1 .zy millimeter

1 D 1 .ou TO 'I /I Q 1 .4y millimeter

1 / 1 .ou TO o no z.uy millimeter

1 O Z. 1 u TO o on z.zy millimeter

1 Q z. ou +1-1 TO O /I Q z.4y millimeter

on z.oU to 10 millimeter

21 10.5 to 20 millimeter

22 20.5 to 50 millimeter

23* 60 to 100 millimeter

24' 125 to 500 millimeter

NOTE: for sizes over 1 inch and 50 mm each block history is analyzed individually, the

groupings here are merely to simplify the data file structure.

4.4.2.3 Control test for the variance

To test if the variance of an individual calibration is consistent v\/ith the short term variance

derived from the calibration history we use the "F-test." The F-Test compares the ratio of

these two variances to the F distribution, or equivalently the ratio of the standard deviations

squared (variance) to the F distribution. Since the variance is always positive it is a one

sided test:

o'cai/o\< Fi.„(k-1,v) (4.16)

where is the historical (pooled) short term group standard deviation, a^ai is the standard

deviation from the calibration, a is the upper percentage point of the F distribution with k-1

degrees of freedom in the numerator and v degrees of freedom in the denominator. Using

the 12 comparison calibration scheme, the number of degrees of freedom is 8, i.e. 12

measurements minus the number of unknowns (S,C,X,Y,A) plus one for the restraint. The

number of degrees of freedom in finding from the history is generally a few hundred,

large enough to use infinity as an approximation. The a used is 0.01. Using these

parameters the control test becomes

o\Jo\, < 2.5 (4.17)

If this condition is violated the calibration fails, and is repeated. If the calibration fails more

than once the test blocks are re-inspected and the instrument checked and recalibrated.

59
All calibrations, pass or fail, are entered into the history file. A flag is appended to the

record to show that an F test failed.

4.4.2.4 Control parameter (S-C)

Control parameter will monitor certain aspects of the calibration, but not necessarily the

accuracy. That is, measurements can have good repeatability and still be wrong. It is possible, for example, that the comparator scale calibration could be completely wrong and

the data could still be repeatable enough that the variance (as measured by o^) would be

in control. Our second control parameter samples the accuracy of the system by

monitoring, for each size, the length difference between two NIST master blocks (S-C).

No matter which master block is taken as the restraint this difference is the same.

Consider the measurement sequence shown in the previous sections. The customer block

lengths X and Y, as well as the length of one of our master blocks are determined by a model fit to the comparison data using the known length of the other NIST master block

(the restraint). The idea of this test is that while the customer block length is not known,

the length of our control block is known and if the calibration process produces the correct

length for the control block it probably has also produced the correct length for the

customer block.

Figure 4.4. (S-C) data from the MAP files for the 0. 1 in. blocks from 1 982 to

1992. Note the process change in 1988-89 when the "C" block was

replaced.

A plot, for the 0.1" size, of all of the S-C measurements from 1982 to the end of 1992 is shown in figure 4.4. There are three distinct control regions shown in the graph by

60
vertical lines. The first, which extends from 1982 to the middle of 1987 shows the

performance of the system when we used two steel master blocks and our old (circa 1 974)

comparators. During the next period, until the beginning of 1990, we used new

comparators with modern electronics. The higher repeatability of the new system resulted

in a slightly smaller spread in the (S-C) values, in 1989 we replaced the "C" set of steel

blocks with chrome carbide, resulting in a shift in the (S-C) values. When we analyze the

MAP data to get the accepted value of S-C to use in the statistical test we obviously ignore

all data before 1989.

CT = 10 nm 10 nm o = 8nnn

i2

0)£

oc

D

C

cU

CO

1982 1984 1986 1988 YEAR

1990 1992

Figure 4.5. MAP data for 1 .0 in. calibrations from 1 982 to 1 992. Note that

the spread is somewhat larger than that of the 0.1 in size data in figure 4.4

because of temperature related uncertainties. The C block was replaced in 1989.

Figures 4.5 and 4.6 show the corresponding S-C graphs for the 1.0 in. and 10 in. sizes.

The most prominent feature is that the data spread gets larger with length, a reflection of

thermal variations during the calibrations. The 10 in. blocks show no process jump

because the long block comparator and master blocks have not been changed during the

time period shown.

61
1760

J2®

E

o

c

oc

co

1710

1660

1610

1560

++

+++

++

1510

1982 1984 1986 1988 1990 1992

YEAR

Figure 4.6. MAP data for 1 0 in. calibrations from 1 982 to 1 992. There is no

process change for this size because the long block comparator and the two

master blocks have not changed during the time period shown.

These charts provide a wealth of information about the calibration. Since the control block

is treated exactly as the customer blocks, the variation in the control block length is identical to long term variation in the customer block calibrations. The within uncertainty,

calculated from deviations from the model fit for each calibration only samples the random

errors within characteristic times (less than 1 or 2 minutes) it takes for a single calibration.

The (S-C) history samples both the short term variations and the long term randomness

caused by comparator scale variations, thermometer calibration changes, or operator

differences.

The long term value of (S-C) is not necessarily constant for all sizes. Some gage blocks, as discussed earlier, are not stable. If one master block of a size is not stable, the (S-C)

recorded for each calibration will gradually change. This change will be detected when the

history for each size is analyzed each year to update the control parameters. Usually

blocks with large instabilities are replaced.

Because of possible instability, the long term variation in (S-C) is calculated by the

standard deviation of a linear fit to the (S-C) history data collected from past calibrations.

Data for each size is fit to the form

Li = mj + \, (4.19)

where L| is the length of the i'th block, rrji is the best fit slope, T is the time of the

62
measurement, and I, is the best fit intercept. The standard deviation of the slope, o^i^pe, is determined by the fit procedure. If the block has a "significant" drift as determined by the

test:

mi > 3a3,„pe (4.20)

then the value of the control parameter is taken as the fit value from equation 4. 1 9 when

T is equal to half way till the next update. If the slope is not significant the slope is assumed to be zero, and the data is simply averaged. The standard deviation of the data

from the average is then determined.

These deviations are pooled for every block in a group and the pooled standard deviation,

Ot, is determined. This ,a is recorded in the calibration master file to be used in the

uncertainty calculation reported to the customer.

Finally, as discussed earlier short term variability is characterized by the within standard

deviation, recorded at each calibration. Long term variability is characterized bytO

calculated from the (S-C) history data. These two numbers can be compared and any

non-negligible difference can be attributed to uncorrected long term variations or errors in the system. Based on the size of these uncorrected errors it may or may not be

economically useful to search for their sources and reduce their size.

4.4.2.5 Control test for (S-C), the Check Standard

The length difference between the S and C blocks of each size are called check standards^

and they monitor process accuracy. The control test for accuracy is based on a comparison of the (S-C) value of the current calibration with the accepted value of (S-C)

derived from the (S-C) mechanical comparison data that has been accumulated for several

years. This comparison is the second control test for the calibration, and the historically

accepted mechanical value of (S-C) is the control parameter.

To be in statistical control, the current calibration must be consistent with the previous

population of measurements. The t-test compares the current measurement of (S-C) to the accepted (S-C) value using the standard deviation derived from previous calibrations,

a^, as a scale. If the difference between the current and accepted value is large compared

to this scale the measurement system is judged to be out-of-control.

The actual test statistic is:

63
{Measured-Accepted) — < t^,,iv) (4.21

ow

Where v is the number of degrees of freedom in the accepted total standard deviation, and

a is the significance level. At NIST we use the 3a, or 0.3 % level in calculating the total

error, and therefore a=.003 is used in the t-test. The total number of measurements used

to calculate the accepted total standard deviation is usually over 200, and we can use

infinity as an approximation of v.

With these assignments a t-test critical value of 2.62 is used. As a comparison, if v were

chosen lower, due to having only a short calibration history, the changes would be small.

For v=120 and 60 the critical values are only increased to 2.77 and 2.83 respectively.

(Measured-Accepted) ^ ^ < z . bz (4.22)

aw

If this condition is violated the calibration fails, and is repeated. If the calibration fails more

than once the test blocks are re-inspected and the instrument checked and recalibrated.

All calibrations, pass or fail, are entered into the history file. A flag is appended to the

record to show that the t-test failed.

4.4.2.6 Control test for drift

As a secondary result of analyzing comparison data using a drift eliminating design, linear

drift is measured and can be used as another control parameter. If the drift shows

nonlinear behavior in time, the design will not completely eliminate the drift, and the

answers will be suspect. However, nonlinear drift will accentuate the deviations from the

best line fit of the data and the calibration will fail the F-test. Thus a control based on the

drift rate would be redundant. In either case, because of the environmental control used

in our lab ( ±0.1 °C) drift has never been significant enough to consider such a test.

4.4.3 Calculating the total uncertainty

In the general method for combining uncertainties discussed in section 4.3.6 total

uncertainty is the root-sum-of-squares of the uncertainty components times a coverage

factor k=2.

64
For NIST calibrations the total error is made up of random error from the master block

interferometric history and long term random error of the comparator process derived from

control data. Other sources of uncertainty is the phase correction assigned to each block

in the interferometric measurements and the temperature difference between the customer

block and the NIST master block. For blocks over 50 mm the temperature gradients and

non-negligible sources of uncertainty..

The long term random error of the comparison process is calculated from the (S-C) data

collected in the MAP file. Since each block, master and unknown, is handled and

measured in exactly the same manner, the variance of the difference (S-C) is a reasonable

estimator of the long term variance of the customer calibration (S-X) or (C-X). The major

difference is that since the S and C blocks are of different materials, the variation will be

slightly larger than for customer calibrations, for which (Standard-X) is always derived from

the master block of the same materia! as the customer block. Although we expect this

difference is small, we have not collected enough customer data to analyze the difference,

and our total uncertainty must be considered slightly inflated.

The total (long term) standard deviation (Otot) is calculated by groups, in much the same

manner as a^. One additional step is included because of the possible instability in the

length of one of the master blocks. The (S-C) history of each size is fit to a straight line

and the deviations from the line are pooled with the deviations from the other sizes in the

group, as shown in eq (4.15).

a

\| {m-2)

(4.23)

In the above equation:

Qtot is the pooled long term standard deviation of the group,

m is the number of measurements of all sizes in the group,

Dj is the deviation from the best fit to the (S-C) history data for each size.

As discussed by Cameron [5], the random error for calibration schemes using incomplete

block designs is a combination of the short and long term repeatability. In these cases

short term repeatability is the average of the standard deviations of each calibration

recorded in the calibration history, a^, and the long term repeatability is Oto,. For complete

65

11
block designs only Otot is needed.

The interferometric uncertainty is essentially a random error. If each block were measured

only a few times on a single date the pressure, temperature and humidity calibrations used

to make corrections for the index of refraction and thermal expansion would be sources

of systematic error. However, each time our master blocks are remeasured ( 3 year

intervals) the pressure, temperature and humidity sensors are recalibrated. After the

blocks have been measured using three or more sensor calibrations, errors of the

environmental sensors have been sampled and the variations due to them have been

absorbed into the gage block calibration data.

The interferometric phase correction has a fairly high uncertainty because of the difficulty

of the measurements. In addition, phase measurement is much too difficult to make on

each block and so only a few blocks from each manufacturer are tested and the average

of these measurements are assigned to all blocks from the same manufacturer. Estimated

uncertainty in the phase correction is around 8 nm ( 0.3 |jin).

We have measured temperature gradients on all of our comparators. On short block

comparators (<100 mm) we have found gradients up to 0.030 °C between blocks. Since

we do not have extensive measurements we consider this as a rectangular distribution of

total width ±0.030 °C, Using the rules of NIST Technical Note 1297 we divide the width

by the square root of 3, giving an uncertainty of ±0.020 °C. The long block comparators

are partially enclosed which reduces the gradients below ±0.010 °C. The standard

uncertainty from these gradients is only ±0.005 °C.

4.5 Summary of the NIST Measurement Assurance Program

Two major goals of the gage block MAP program are to provide a well controlled

calibration process to ensure consistently accurate results, and to quantitatively

characterize the process so that a rational estimate of process accuracy can be made.

Process control is maintained with three tests and data collected from the calibrations is used for both the tests and the estimation of process uncertainty.

In the first test interferometric history of each master block is continually updated and

checked for consistency. Suspicious values are re-measured. As the number of

measurements in each history increase, the dimensional stability of each block can be

determined. If the block is changing length, the projected length at the time of the

comparison measurement can be used, or the block can be replaced. The standard

deviation of the history data for each block is used as an estimate of the long term random

uncertainty of the master block value.

In the second test the standard deviation of each mechanical comparison is checked

against the average value of the standard deviations from previous calibrations. This

66
average standard deviation for an individual calibration is called the "within" or short term

standard deviation (aj. Since the error sources for similar size blocks are similar, v\/e pool

the historical data for groups of similar sized blocks to increase statistical precision. The

standard deviation from each calibration is tested against the accepted by the formula:

0\j0\ < 2.5 (4.24)

In the final test each mechanical comparison includes a check standard as well as the

master block. The difference between these two NIST blocks, (S-C), is tested against an

accepted value obtained by a linear fit to the recorded (S-C) values for the last few years.

The difference between the value obtained in the calibration and the accepted value is measured against the expected standard deviation of the measurement, a^, using the

formula:

(Measured- Accepted) , , ^ <

The uncertainty reported to the customer is a combination of the type A and type B

uncertainties. The major components of uncertainty are the standard deviation of the

interferometric history of the master block used in the calibration, the long term standard

deviation of the mechanical comparison process, o^x, and the estimated type B uncertainty.

The primary components of type B uncertainty are the phase correction applied during

interferometric calibrations of our master gage blocks and the temperature gradients

between blocks during the comparison. All of these errors are combined in quadrature and

multiplied by a coverage factor (k) of two, thus arriving at the reported uncertainty.

Total Uncertainty = k ^/^^ (4.26)

67
5. The NIST Mechanical Comparison Procedure

5.1 Introduction

Techniques described here for intercomparing gage blocks with electro-mechanical gage

block comparators are those used at NIST and are the results of considerable

experimental work, statistical design and process analysis. Some laboratories will find

these techniques ideally suited to their needs while others may wish to modify them, but

in either event the process control methods described here will promote the analysis of

measurement requirements and the application of statistical principles.

Length values are assigned to unknown gage blocks by a transfer process from blocks of

known length. Briefly, the transfer process is a systematized intercomparison of 3 or 4

blocks of the same nominal size using an electro-mechanical comparator. One or two

blocks are unknowns and two are standards. Such a process is ideally suited to

comparing objects with nearly identical characteristics, such as gage blocks, and the

process provides the redundancy needed for statistical analysis.

We will also describe the corrections needed for the case where the blocks are not made

of identical materials. The correction methods apply to all sizes but special precautions

will be described for long gage blocks where temperature effects predominate. When the

master and unknown blocks are of the same material, which for our calibrations include

both steel and chrome carbide blocks, no corrections are needed.

5.2 Preparation and Inspection

It is essential that the gage blocks being calibrated, as well as the standards being used,

be clean and free from burrs, nicks and corrosion if the calibration is to be valid.

Therefore, it is necessary to institute an inspection, cleaning and deburring procedure as

part of every calibration. Such a procedure will also protect the comparator working

surface and the diamond styli.

5.2.1 Cleaning Procedures

Gage blocks should be cleaned in an organic solvent such as mineral spirits. Solvents

such as trichloroethylene or benzene should be avoided because of health risks involved

with their use. Freons should not be used because of environmental hazards they pose.

The bottom of the bath should be lined with a waffled mat of neoprene rubber to minimize

scratches and burrs to the gage block surfaces caused by grit or metal to metal contact.

Rectangular blocks should be immersed in the bath and a soft bristle brush used to

remove the layer of protecting oil or grease from all surfaces. Lint-free towels should be

used to dry the blocks._ A second cleaning using ethyl alcohol is recommended to remove

68
any remaining residue. Again, wipe with lint-free towels.

Square type blocks are cleaned in the same manner as the rectangular blocks except that

the center hole needs special attention. The hole must be thoroughly cleaned and dned

to prevent "bleeding out" of an oil film onto the contact surfaces and supporfing platen

when the block is positioned vertically. A .22 caliber gun cleaning rod makes an ideal

instrument for cleaning the bore of blocks in the size range of 2 mm to 500 mm. A solvent

moistened piece of lint-free paper towel can be pushed through the hole repeatedly until

all visible residue is removed.

5.2.2 Cleaning Interval

It is recommended that the solvent and alcohol cleaning process be completed several

hours prior to use because the cleaning process will disturb the gage block thermal state.

If blocks are to be joined together or to platens by wringing, recleaning is necessary prior

to introducing a wringing film to ensure that no foreign substances are on the wringing

surfaces that could corrode or mar the contact surfaces.

5.2.3 Storage

Cleaned blocks should be stored in trays covered with lint-free paper or on temperature

equalization plates that are clean. A cover of lint-free towels should be used to secure the

blocks from dust and other airborne solids and liquids. When blocks are not in daily use,

they should be coated with any one of the many block preservatives recommended by the

gage block manufacturers. It is recommended that steel gage blocks be coated to prevent

rust when the work area relative humidity exceeds 50%.

Foodstuffs should be banned from all calibration areas, as acids and salts will corrode steel

gage surfaces. Hands should be washed to reduce corrosion on blocks when gloves or tongs cannot be used conveniently.

5.2.4 Deburring Gage Blocks

Gage blocks can often be restored to serviceable condition by stoning if the damage is not

severe. Three different optically flat gage block stones are available: black granite, natural

Arkansas, or sintered aluminum oxide. All stones are used in the same manner, but are used for different purposes or on different materials.

The black granite deburring stone is used on steel blocks to push a burr of metal back into

the area from which it was gouged. The natural Arkansas deburring stone is a white finegrained stone which is used on steel blocks to remove burrs and nicks caused by everyday

mishaps. It is especially effective on damaged edges of gaging faces. The sintered

aluminum oxide stone is used to treat either steel or carbide gage blocks. It is most

69
effective on hard materials such as tungsten and chromium carbide.

The stone and block should both be cleaned with alcohol before stoning. Sweep the

surfaces with a camel's hair brush; then slide the damaged surface of the block onto the

stone, taking precautions that no foreign matter is trapped between the two surfaces, and

with firm downward pressure the block is moved over the stone in either a circular or a back and forth motion until the metal is restored to its original placement. Generally, there

is a lessening of drag between the two surfaces as the burr is reduced.

After stoning and cleaning a block, one can check the surface by wringing the restored

surface to a fused silica flat. Looking through the back side of the flat, the interface

between the block and flat should appear uniformly grey when it reflects white light. Any

light areas or color in the interface indicates poor wringing caused by a burr that has not

been fully reduced to the normal gage surface.

Periodically, depending on use, the stone should be cleaned with a stiff natural fiber brush,

soap and water to remove embedded pieces of metal and other foreign matter that

contaminates the surface with use. Allow moisture absorbed by the stone surface to evaporate before using the stone.

5.3 The Comparative Principle

A comparator is a device for measuring differences. The unknown length of a block is determined by measuring the difference between it and a reference block of the same

nominal size and then calculating the unknown length. Let L^ equal the comparator length

when its indicator reads zero (the zero may be at the left end or the center of the scale).

Then, by inserting the two blocks of length L^ (unknown) and L^ (reference) and reading

numerical values x and r from the comparator scale, we get

U = L, + X (5.1)

Lr=U + r (5.2)

Solving these equations for L^:

L, = L,+ (x-r). (5.3)

Note that the quantity (x-r) is the difference between the two blocks, if the blocks are made of different materials we add a quantity, 5, to each comparator reading to

compensate for elastic deformation of the contact surfaces. We must also add a correction

for the thermal expansion if the calibration temperature T is not 20 °C. The equation

becomes

70
U = L, + (X - r) + (5, - 6,) + L{a, - aJ(t-20). (5.4)

The quantity (6^ - in this equation is the deformation correction, Cp. and are the

thermal expansion coefficients, and L the blocks' nominal lengths. Using C, for this thermal

correction the equation becomes

L, = U + (x-r) + Cp + C, (5.5)

If two probes are used, one lower and one upper, there are two penetration corrections

and Cp must contain a lower and upper component.

A key result of the comparative principle is that if the two blocks being compared are the

same material, Cp and C, are zero.

5.3.1 Examples

1. A single probe comparator is used to measure a 50 mm tungsten carbide gage

block with a 50 mm steel reference standard gage block. The measurement data

are:

Stylus force (top): 0.75 N

Stylus force (bottom): 0.25 N

Stylus tip material: diamond

Stylus tip radius: 6 mm

L, of reference at 20 °C 50.00060 mm

Temperature of blocks: 20.4 °C

Expansion coefficient:

Steel 11.5x10-^/ C°

Tungsten carbide 6 x1 0"^/ C

°

Comparator reading on tungsten block 1 .25 pm

Comparator reading on steel block 1 .06 pm

The unknown length (LJ at 20 °C can be found using the last equation and the penetration

corrections from table 3.4:

Penetration, pm (pin)

Steel 0.14(5.7)

Tungsten Carbide 0.08 (3.2)

71
Then

L, = 50.00060 + (1.25-1.06)x10-' + 50x(11 .5-6)x10-^x(20.4 - 20) + (08 - .14)x10-^ (5.6a)

L, = 50.00060 + .00019 + .00011 - .00006 (5.6b)

L, = 50.00084 mm (5.6c)

A dual probe comparator is used to measure a 10 mm steel block with a 10 mm

chrome carbide reference block. From the following data find the length of the steel

block at 20°C.

Stylus force: 0.75 N upper, 0.25 N lower

Stylus tip material: diamond

Stylus tip radius: 6 mm

I, of reference at 20 °C 9.99996 mm

Temperature of blocks: 23.6 °C

Expansion coefficient:

Steel 1 1.5x10-'/ C°

Chrome carbide 8.6x1 0'^/ C

°

Comparator reading on chrome block 0.00093 mm

Comparator reading on steel block 0.001 10 mm

penetration (0.25 N) penetration (0.75 N) TOTAL

Steel .07 Mm 0.14 pm 0.21 Mm

Chrome Carbide .06 Mm 0.12 Mm 0.18 Mm

Solution:

measurements thermal correction penetration correction

U = 9.99996+(.00110-.00093) +10x(8.6-1 1 .5)x10-'x(23.6 - 20) + (.21-.18)x10-3 (5.7a)

L, = 9.99996 + .00017 - .00010 + 0.00003 (5.7b)

L, = 10.00006 mm (5.7c)

In summary, deformation corrections can be easily and accurately applied if the operative

parameters of probe tip radius and probe force are known and if the probe tip is spherical

and remains undamaged. Extremely low forces should be avoided, however, because

72
poor contact is a risk. It is also important to know that thermal conditions, if ignored, can

cause far greater errors than deformation, especially for long lengths.

Currently the NIST comparator program uses steel master blocks for steel unknowns and

chrome carbide masters for chrome or tungsten carbide unknowns. This allows the

penetration correction to be ignored for all of our calibrations except for a very few sets of

tungsten carbide and other materials that we receive. Because of the small number of

gage blocks sent for calibration it is not economically feasible to have master sets for every

possible gage block material. For these blocks the correction is minimized by the use of

the master blocks with the most closely matched elastic properties.

5.4 Gage Block Comparators

There are a number of suitable comparator designs and a typical one is shown

schematically in figure 5.1 . An upper stylus and a lower stylus contact the gaging faces

of a block supported on an anvil. Each stylus is attached to a linear variable differential

transformer (LVDT) core. The amplifier displays the difference between the two LVDT

signals.

Figure 5.1.

Schematic

drawing of a gage block comparator showing the component parts.

73
The NIST comparators have a more complex mechanical arrangement, shown in figure

5.2. The use of pivots allows measurement of the relative displacement of two measuring

styli using only one LVDT. Since one of the main causes of variations in measurements

is the LVDT repeatability, reducing the number of LVDTs in the system enhances

repeatability.

Figure 5.2. NIST gage block comparator.

An important comparator feature is that the block is held up by the anvil (also called

platen), removing the weight of the block from the lower stylus. The force of the gaging

stylus on the bottom of the block is therefore the same for every block measured. Were

this not the case the corrections needed for deformation at the point of contact would

depend on the weight of the block as well as it's material. There are, however, systems

which support the block on the lower stylus. In these cases a penetration correction which

depends on the material, size and geometry of the block must be made for each block.

Since the LVDT range is small, the height of one of the sensors must be mechanically

changed for each size gage block. The highest feature on figure 5.2 is the crank used to

rotate a fine pitch screw attached to the tdp stylus. For each size the top stylus is set and

locked so that the LVDT output is on the proper scale. The lifting lever raises the top stylus

from the block and locks the pivot system so that the stylus cannot crash into the gaging

surface by mistake during the gross adjustment to the next size.

For the most precise measurements, it is important that the defined gage point of a block

PIVOTS

COUNTER BALANCE

PIVOTS

74
be contacted by the comparator stylus. If the two gaging surfaces are not exactly parallel

the measured block length will vary with the gaging point position. This is particularly

important for long blocks, which tend to have less parallel gaging surfaces than short

blocks. To ensure that contact will be made at the proper point a metal or plastic bar about

1/4 inch thick is fastened to the anvil behind the stylus and is positioned to stop the gage

blocks so the stylus will contact the gage point. The stop used on our long block

comparator is shown in figure 5.3.

Figure 5.3. The mechanical stop is used to force the gage

block to a nearly identical position above the stylus for each

measurement.

The LVDT signal (or difference between two LVDTs) is displayed on a meter graduated

in length units. The analog output is also read by a digital voltmeter which, when triggered

by a foot switch, sends the measured voltage to a microcomputer. The difference in length

between two gage blocks is obtained by inserting the blocks, one at a time, between the

stylus tips and taking the difference between the two readings.

5.4.1 Scale and Contact Force Control

The comparator transducer must be calibrated to relate output voltage to probe

displacement from some arbitrary zero. Our comparators are calibrated regularly using a set of 11 gage blocks which vary in length from 0.200000 inches to 0.2001 00 inches in 10 microinch increments. Each calibration is recorded in a probe calibration file and the

last 20 calibrations are available to the operator as a control chart with appropriate control

limits. If a probe calibration is inconsistent with the previous calibrations the instrument is examined to find the cause of the discrepancy.

75
The NIST comparators for blocks less than 100 mm long use contact forces of 0.75 N (3

ounces) for the upper contact and 0.25 N (1 ounce) for the lower contact. The

corresponding forces for the long gage block comparator (over 100 mm) are 1.2 N (5 oz)

and 0.75 N (3 oz). The force on both upper and lower contacts are checked daily using

a calibrated, hand held, force gage. If the force is found to be in error by more than 0.1 N,

adjustments are made.

5.4.2 Stylus Force and Penetration Corrections

The penetration of the spherical stylus into the planar gage block has already been

presented in table 3.4. Values are correct to a few nanometers. Calculation of the value

assumes a perfectly spherical surface with a nominal radius of curvature of 3 mm. This

assumption is not always true. Effects of deviations from the assumed probe geometry

can be significant and have plagued comparison procedures for many years.

Usually the probe is not exactly spherical, even when new, because of the characteristics

of diamond. The probe is manufactured by lapping, which if diamond were an isotropic

media would produce a spherical form. Diamond, however, is harder in some directions

than in others. The lapping process removes more material in the soft directions producing

an ellipsoidal surface. Even worse, the diamond surface slowly wears and becomes

flattened at the contact point. Thus, even though the calculation for deformation of a

spherical-planar contact is nearly exact, the real correction differs because the actual

contact is not a perfect sphere against a plane.

!n the past these effects were a problem because all NIST master blocks were steel. As

long as the customer blocks were also steel no correction was needed. If, however, other

materials were measured a penetration correction was needed and the accuracy of this

correction was a source of systematic error (bias).

A gage block set bias is determined from the history of each block. First, the yearly

deviation of each block is calculated from its historical average length. Then the deviations

of all the blocks are combined by years to arrive at the average change in the length of the

set by year. This average should be close to zero, in fact if there are 81 blocks in the set

the average will, if in control, fall within the limits ±3a/y81 or ±a/3. The method of

examining set averages is useful for many types of systematic errors and is more fully

described in Tan and Miller [36].

Figure 5.4 shows the set bias for three customer gage block sets having lengthy histories.

Two sets, represented by squares are steel and the third, represented by asterisks, is chrome carbide.

The two steel set calibration histories are considerably better than those for the chrome

76
carbide. Also, before the early 1970's all calibrations were done by interferometry. The

chrome carbide calibrations have degenerated since 1970 when mechanical compansons

were introduced as the calibration method.

COQ

0

E

o

c

o

c

g

en0

CO

30

20

10

0

-10

-20

-30

1960

+

++

+

^ n r-, Dgo- ^ ° +++ + + + ++

+ +

+

1970 1980 1990

Figure 5.4. Bias of three gage block sets since 1960. The squares are steel

blocks and the crosses are chrome carbide blocks.

In the past it was believed that carbide calibrations were not as accurate as steel and one

extra microinch of uncertainty was added to carbide calibrations. The graph shows that

this belief gained from experience was accurate. The NIST calibration procedures were

changed in 1989 to eliminate this systematic error by having one set of master blocks

made from chrome carbide. When steel blocks are measured the steel master is used as

the reference block and when carbide blocks are measured the chrome carbide block is used as the reference. In each case the difference in length between the steel and carbide

masters is used as the check standard.

5.4.3 Environmental Factors

A temperature controlled laboratory is essential for high precision intercomparisons. The

degree of temperature control needed depends on the length of the blocks being

compared, differences in thermal expansion coefficients among the blocks, and the limiting

uncertainty required for the unknown blocks. At NIST, the gage block calibration areas are

specially designed laboratories with temperature control at 20±0.1 °C. Special

precautions, described in this chapter, are taken for blocks over 100 mm.

77
5.4.3.1 Temperature Effects

The correction equation for thermal expansion is

6L/L = a(20-T) (5.8)

Where L is the lenght of the block, 6L the change in length due to thermal expansion, a

is the thermal expansion coefficient, and (20-T) the difference between the measurement

temperature and 20 °C.

When two blocks are compared, the length difference (m=S-X) between the blocks may

need correction due to the difference in thermal expansion coefficients and temperatures

of the blocks. If corrected the calibration result becomes

Kai = Sref " m + 6L(therma!) (5.9)

5L/L = (ax-as)(20-Ts)+ax(Ts-Tx) (5.10)

Uncertainty of the calculated 5L arises from uncertainty in a, AT, and the generally

unmeasured temperature difference between S and X. An estimate of the upper limit of

this uncertainty is:

Uncertainty = L(ax-as)(6Tx) + L(6as+5ax)(20-Ts) + Lax(Ts-Tx) (5.11)

where 5a is the uncertainty in the thermal expansion coefficient and 5T is the uncertainty

in the temperature measurement.

We illustrate the effects with a simple example. Three 250 mm gage blocks are being

intercompared, a steel block is the reference master designated S, and one steel block and

one chrome carbide block are designated X and Y. The thermal expansion coefficients

and related uncertainties are:

S: (11.5±0.1)x 10-^/°C

X: (11,.5±0.5)x 10"^/°C

Y: ( 8,4 ± 0.5) X 10"^/°C

The blocks are measured at 20.5 °C with an uncertainty of ±0.2 °C, and from previous

practice it is known that there is the possibility of a temperature difference between blocks

of up to 0.1 °C.

78
Length errors in three categories can be illustrated with these data: those caused by (1)

temperature measurement uncertainty, a5T; (2) gage block expansion coefficient

uncertainty, L5a(t-20); and (3) temperature differences between the reference and

unknown blocks.

(1) The temperature measurement uncertainty of 0.2 °C together with the

expansion coefficients of the unknown blocks yields:

Uncertainty = L(ax-as)5T (5.12)

U{X) = 250x(11.5x10^ - 11.5x10 ^) X 0.2 = 0 (5.13)

U{Y) = 250x(8.4x10"^-11.5xl0 ^)x0.2 = 0.00015 mm (6 \iin) (5.14)

(2) The uncertainty in the thermal expansion coefficients together with a measurement temperature different than 20 °C yields:

Uncertainty = L(6as+5ax) |20-Ts| (5.15)

U{X or y)=250x(0. 1x10^+0.5x10 ^)x(0.5)=0. 00007 mm (3 \iin) (5.16

(3) If the master and unknown blocks are not at the same temperature, there will be an

error depending on the size of the temperature gradient:

Uncertainty = Lax(Ts-Tx) (5.17)

'J{X) = 250 X 11.5 X 10 ^x 0.1 = 0.0003 mm (12 \iin) (5.18;

J{Y) = 250 X 8.4 X 10 ^ X 0.1 = 0.0002 mm (8 \iin) (5.19

The example shows that the uncertainty in the thermal expansion coefficient is not

important if the temperature is kept close to 20 °C. The uncertainty in the temperature

measurement is very important for blocks longer than 25 mm, and is critical for blocks

79
longer than 100 mm. Finally, temperature differences between blocks are a serious matter

and measures to prevent these gradients must be taken. Ways of controlling temperature

effects are discussed in the next section.

5.4.3.2 Control of Temperature Effects

In the NIST comparison process it is assumed that the gage blocks are all at the same

temperature, i.e. the temperature of the stabilization plate or comparator anvil. A large error in the comparison process can be introduced by temperature gradients between

blocks, or even within a single block. For example, a temperature difference of 0.5 °C

between two 100 mm steel blocks will cause an error of nearly 0.6 pm (24 pin) in the

comparison. Two causes of temperature differences between blocks are noteworthy:

1. Room temperature gradients from nearby heat sources such as electronic

equipment can cause significant temperature differences between blocks even

when they are stored relatively close to each other before comparison.

2. Blocks with different surface finishes on their non-gaging faces can absorb

radiant heat at different rates and reach different equilibrium temperatures.

The magnitude of these effects is proportional to gage block length.

A number of remedies have been developed. For short blocks (up to 50 mm) the remedies

are quite simple. For example, storing the blocks, both standards and unknowns, on a thermal equalization plate of smooth surface and good heat conductivity close to the

comparator but away from heat sources. Also, the use of insulated tweezers or tongs to

handle the blocks and a systematic, rhythmic block handling technique in the comparison

procedure to ensure identical thermal environment for each block. The use of a special

measurement sequence called a drift eliminating design is discussed in section 5.6.

Precautions for long blocks (over 50 mm) are more elaborate. At NIST the comparator is enclosed in an insulated housing partially open on the front to allow handling the blocks

with tongs. Blocks are stored on the comparator anvil and each block is wrapped in three

layers of aluminized mylar to assure that each block has the same reflectivity to incident

radiation. The insulation, although very thin, also limits the largest heat transfer

mechanism, convection, during the measurement. This reduces thermal changes due to

operator presence during calibration.

Laboratory lights remain off except for one lamp well removed from the comparator, but

giving enough illumination to work by. The visible light is a secondary radiation problem,

the infrared radiation from the lights is the primary problem. Heat sources are kept as far

away from the comparator as possible and the comparator is located away from other

80
activities where laboratory personnel might congregate. As a further precaution during

intercomparisons the operator wears a cape of Mylar reflective film and heavy cotton

gloves while handling the blocks with tongs. Finally, as with short blocks, the handling

procedure is systematic, rhythmic and quick but not rushed. Rectangular blocks longer

than 100 mm are measured using a reduced number of intercomparisons because of

difficulty in handling this type of long block with its tendency to tip on its narrow base.

Temperature problems can be detected in long block intercomparisons by reversing the

storage positions of the 4 blocks and repeating the intercomparison after suitable

equalization time. Temperature gradients in the storage area will be revealed by a

significant change in the relative lengths of the blocks. Still another method is to measure

the temperature of each block with a thermocouple. A simple two junction thermocouple

used with a nanovoltmeter can measure temperature differences of 0.001 °C or less.

Tests should be made for gradients between blocks and internal gradients (top to bottom)

within each block. When a gradient free environment is established a thermometer

mounted on the comparator will be adequate for detecting unusual or gross changes.

Relative humidity is held below 50% to prevent corrosion of blocks and instruments. A full discussion of environmental requirements for calibration laboratories can be found in the

ANSI standard B89.6.2.

5.5 Intercomparison Procedures

Once the blocks have been cleaned, as described in section 5.2 they are laid out on an

aluminum soaking plate to allow all of the blocks to come to the same temperature. The

blocks are laid out in 4 parallel rows in each size sequence, each row being a different set. The two NIST master blocks, S and C, are the top and second row respectively because

each comparison scheme begins S-C. The customer sets, X and Y are set out as the third

and fourth rows. It is not necessary to use this specific setup, but as in all facets of

calibration work the setup should be exactly the same for each calibration to reduce errors and allow the operators to evolve a consistent rhythm for the comparison procedure.

The length of time needed for blocks to thermally equilibrate varies depending on block

size and the contact area with the soaking plate. For blocks less than 10 mm long the

same time can be used for all sizes. For longer sizes the contact area is important. If long

blocks are laid down so their entire side is resting on the soaking plate then they will equilibrate much like the short blocks. If the blocks are wrapped in mylar this is no longer

the case because the mylar and rubber bands used to hold the mylar on insulate the

blocks from the plate. Wrapped blocks thus must stand upright on the plate. Since the

thermal mass of the block increases but the contact area does not, the longer the block,

the longer time it will take to come to thermal equilibrium. The minimum times used in our

lab are shown in table 9.

81
Table 5.1

Minimum Thermal Soaking Times in Minutes

Length (metric) Length(English) Time (min)

0 - 10 mm

10 - 25 mm

25- 100 mm

125- 500 mm

0

0.3

2.0

5.0

0.25

1.0"

•4.0"

20"

30

60

100

8 h

Experiments establishing optimum equalization times should be conducted in your own

laboratory because of the many variables involved and differing measurement uncertainty

requirements.

5.5.1 Handling Technique

The success of any intercomparison scheme is largely dependent on block handling

techniques. These techniques include the insertion of all blocks betv\/een the styli in a like

manner. The operator should develop a rhythm, after practice, that will ensure that each

block is handled for approximately the same length of time. The time to make 24

observations should be approximately 2 to 4 minutes for an experienced operator with

automatic data recording or an assistant doing the recording.

A camel's hair brush or an air bulb is useful for sweeping or blowing dust particles from the

blocks and the anvil just before insertion.

Short blocks are moved about by grasping them with rubber tipped tweezers or tongs. For

handling square style blocks, the tips of the tweezers may be bent to accommodate this

configuration.

For added insulation, light cotton gloves may be worn in addition to using tweezers when

handling blocks from 10 mm to 100 mm (0.4 inch through 4 inches). Blocks above the

100 mm size are compared on a different instrument situated in a more precisely controlled

temperature environment and the operator may wear a Mylar reflective cape as well as heavy gloves. A special pair of tongs is used to facilitate moving long blocks about the

anvil and between the measuring probes. The design used is shown in figure 5.5.

Operators have found that the least awkward place to grasp long blocks is about 10 mm

above the bottom.

82
Figure 5.5. Tongs

for moving long square cross section gage blocks.

For our long block comparator the anvil stop (see figure 5.3) plays an important part in seating the probe on the block surface as well as positioning the gage point between the

probes. The block is moved tangentially toward the stop allowing one corner to touch the

stop and then with an angular motion (which seats the probe on the block surface) proceed

to abut the entire edge of the block to the stop.

5.6 Comparison Designs

5.6.1 Drift Eliminating Designs

The generation and analysis of drift eliminating designs is described in appendix A. In this

section we will discuss the four designs used for gage block calibrations at NIST. The

normal calibration scheme is the 12/4 design which uses two master blocks and two

customer blocks. The design consists of all 12 possible comparisons between the four

blocks. When only one set of customer blocks is available we use the 6/3 design which

uses two master blocks and one customer block. The design consists of all six possible

comparisons between the three blocks.

There are also two special designs used only for long gage blocks. For long square cross

section blocks (Hoke blocks) we use the 8/4 design which uses 8 comparisons among four

blocks; two master blocks and two customer blocks. Long rectangular blocks are difficult

to manipulate, particularly when over 200 mm. For these blocks we use a simplified

measurement scheme using a master, a check standard, and one customer block in 4

measurements. For historical reasons this scheme is called "ABBA."

83
5.6.1.1 The 12/4 Design

Gage blocks were calibrated using the 8/4 design for nearly 15 years. It was replaced in 1989 with a new design, the 12/4 design. The new scheme using 12 intercomparisons

replaced the earlier scheme which used 8 intercomparisons. Increasing from 8 to 12

intercomparisons evolved from a determination to eliminate the systematic error caused

by the varying contact deformation noted earlier. This design was described in detail in section 4.4.2.1 . Here we will only discuss the reason for its development.

Previously, the restraint was the sum of the known lengths of block S and C, (S+C). The

benefit of this restraint was that the percentage uncertainty is proportional to the

uncertainties of the two blocks added together in quadrature. Since the uncertainty in S

and C were generally about the same, the uncertainty in the restraint was reduced by a

factor of 72. The drawback was that if the S, C, and customer blocks were not all the same

material, corrections for the difference in stylus penetration was required. As discussed

in the previous section this problem was found to result in measurable systematic errors. A study of our calibration history showed that contact pressure variations and the changing

stylus geometry due to wear could not be controlled well enough to prevent nearly 25 nm

of additional calibration uncertainty.

To reduce this problem one set of steel masters was replaced with chrome carbide

masters. A new scheme was then designed using only the steel masters as the restraint

for steel customer blocks, and the carbide masters as the restraint for carbide customer

blocks. This eliminates the penetration correction when steel or chrome carbide blocks are

calibrated, but it also increases the uncertainty in the restraint. To lower the total

uncertainty the number of measurements was increased, offsetting the increased

uncertainty in the restraint by reducing the comparison process random error

.

5.6.1.2 The 6/3 Design

On some occasions two sets with the same sizes are not available for calibration. In the

past the one set was used as both X and Y, giving two answers for each block, the

average of the two being used for the report. To eliminate excess measurements a new

design for one customer block was developed. The six comparisons are given below:

yi

y2

S

X

C

C

C

S

X

S

(5.20)

84
= x-c

= s-x

When the length L is the master block, the lengths of the other blocks are given by the

following equations. The parameter A is the length drift/(time of one comparison).

S = L (5.21a)

C = (1/6)(-2y, - y2 + Vs + 2y, + ys

- ye) + L (5.21 b)

X = (1/6)(-yi - 2y2 - ys + y4 + 2ys + ye) + L (5.21c)

<A> = (1/6)(-yi - y2 - Vs - y4 - Ys - Ye) (5.21 d)

Formulas for the standard deviation for these quantities are in appendix A.

5.6.1.3 The 8/4 Design

The original 8/4 comparison design is currently only used for long rectangular blocks (100

mm and longer). This original design v\/as drift eliminating but one comparison was used

twice and there were several consecutive measurements of the same block. In the current

(and new) 8/4 scheme, like the new 12/4 design, the length of only one block is used as

the restraint, resulting in a slightly larger uncertainty. For long blocks the 12/4 design is not appropriate because the added time to make the extra comparisons allows drift to

become less linear, an effect which increases the apparent random component of

uncertainty.

The current 8/4 design is as follows:

Yi S-C

Y2 X

Ys Y-S

Va C-X

Ys C-S

Ye Y-X

Y7 S-Y

Ys X-C

85
For the case where S is the master block (known length) the values of the other blocks are:

s = L (5.23a)

c = (1/12) (-4y,- 2y3 + 2y4 + 4y5 - 2si, + 2y8) + L (5.23b)

x = (1/12) (-2y, 4 2y2 + 3y3 + 3y4 + 2y5 - Sy^ - ys) + L (5.23c)

Y = (1/12) (-2y,

2y2 + y3 + 5y4 + 2y6 - Syy - Syg) + L (5.23d)

A =

(-1/8)(yi +y2 + y3 + y4 + ys + ye + yy + ys) (5.23e)

5.6.1.4 The ABBA Design

The actual measurement scheme for this design is X-S-C-X, where X is the customer

block, S and C are the NIST standard and control blocks. The name ABBA is somewhat

of a misnomer, but has been attached to this design for long enough to have historical

precedence.

The design is used only for long (over 200 mm or 8 in) rectangular blocks. These blocks

have a very severe aspect ratio (height/width) and are very difficult to slide across the

platen without risk of tipping. The X block is set on the anvil and under the measurement

stylus when the blocks are set up the evening before they are measured. The S and C

master blocks are also arranged on the anvil. The next morning the X block need only be

move twice during the measurement, reducing the possibility of tipping. The measurement

is repeated later and the two answers averaged. Obviously the F-test cannot be used, but

the value of S-C is used as the control in a manner similar to the other measurement

schemes.

5.6.2 Example of Calibration Output Using the 12/4 Design

12GAGE PROGRAM Tuesday September 10, 1991 11:01:28

3 CaUbratioii: 4.000000 Incli Gage Blocks Observer: TS

4

5 Block Id's: S: 4361 C: lUNG X: XXXXXXX Y: YYYYY Federal: F4

6 Observations Observations

7

8

Block.s at Ainbieiil Conditions at Standard Conditions Differences Residuals

9 S ** C 52.88 62.51 66.28 72.92 -6.64 10 Y ** S 62.50 53.32 72.91 66.71 6.20 .22 11 X ** Y 63.82 62.28 77.21 72.68 4.52 1.07 12 C S 62.18 54.15 72.59 67.54 5.05

86
1314

1516

17

181920

2122

23

24

25

26

27

28

29

30

31

32

33

34

35

C ** X

Y

**

S ** X

g ** Y

X ** C

X ** s Y

**

62.19

63.15

53.87

62.06

53.57

62.76

62.86

62.49

63.36

62.26

62.69

62.19

62.87

62.21 53.77

62.77

72.60

73.55 67.26

72.46

66.96

76.15

76.25 72.90

Obs Withiii S.D. Acc Within S.D. Acc Group S.D.=.68 .26 .36

76.75

72.66

76.09

72.60

73.27

72.62

67.16

76.17

Obs Control=

Acc Control=

Temperature=

-4.15

.89 -8,80

-.14 -6.32 3.53 9.09

-3.27

-5.64

-8.90

19.91

T-test

Serial Number

Deviation

Nominal Size from Nominal

(inches) (microinch)

Total Uncertainty Coef.

(microinch) ppm/C Material

-.35

.55 .62

.22

-.33

-.27 -.34

.19

F-test

= 9.07

6.89

Block

4361

1UN6

MARTMA

TELEDY

4.000000

4.000000

4.000000 -1.86 4.000000 .65

-8.32

-2.68

1.89 2.53

1.86 11.50 steel 1.86 8.40 clirome carbide

1 1 .50 steel 8.40 clu-oine carbide

Process not in statistical control ...

Comments referenced by line numbers:

1 . The header shows which computer program was used (GAGE), the date and time of

the calibration. The date is recorded in the MAP file.

3. Block size and operator are identified and later recorded in the MAP file.

4. Block IDs are given here. The S and C blocks are NIST master blocks. The X and Y

blocks are shortened forms of customer company names, represented here by strings of

Xs and Ys. The ID of the comparator used in the calibration, F4, is also given and

recorded in the MAP file.

9-20. These lines provide the comparison data at ambient temperature and corrected to

20 °C using the thermal expansion coefficients in lines 30-34, and the temperature

recorded in line 24. The differences are those between the corrected data, in the final

column are the residuals, i.e., the difference between the best fit to the calibration data and

the actual measured values.

22-25 These lines present statistical control data and test results.

The first column of numbers shows observed and accepted short term (within) standard

deviation. The ratio of (observed/accepted) squared is the F-test value, and is shown in

87
column 3. The last number, group S.D. is the long term standard deviation derived from

(S-C) data in the MAP file.

The top line in the second column shows the observed difference (S-C), between the two

NIST masters. The second line shows the accepted value derived from our history data

in the MAP file. The difference between these two numbers is compared to the long term

accepted standard deviation (group S.D. in column 1 ) by means of the t-test. The ratio of

the difference and the group S.D. is the value of the t-test shown in column 3.

30-33 These lines present the calibration result, gage block material and thermal

expansion coefficient used to correct the raw data. Note that in this test the two customer

blocks are of different materials. The same data is used to obtain the results, but the

length of the steel customer block is derived from the NIST steel master and the length of

the chrome carbide customer block is derived from the NIST chrome carbide master.

35 Since the F-test and t-test are far beyond the SPC limits of 2.62 and 2.5 respectively,

the calibration is failed. The results are not recorded in the customer file, but the

calibration is recorded in the MAP file.

The software system allows two options following a failed test: repeating the test or passing on to the next size. For short blocks the normal procedure is to reclean the blocks

and repeat the test because a failure is usually due to a dirty block or operator error. For

blocks over 25 mm a failure is usually related to temperature problems. In this case the

blocks are usually placed back on the thermal soaking plate and the next size is measured.

The test is repeated after a suitable time for the blocks to become thermally stable.

5.7 Current System Performance

For each calibration the data needed for our process control is sent to the MAP file. The

information recorded is:

1. Block size

2. Date

3. Operator

4. Comparator

5. Flag for passing or failing the F-test, t-test, or both

6. Value of (S-C) from the calibration

7. Value of from the calibration

Process parameters for short term random error are derived from this information, as discussed in chapter 4 (see sections 4.4.2.1 and 4.4.2.2). The (S-C) data are fit to a

88
straight line and deviations from this fit are used to find the standard deviation O^ot

4.4.3). This is taken as an estimate of long term process variability. Recorded values of

are averaged and taken as the estimate of short term process variability. Except for

long blocks, these estimates are then pooled into groups of about 20 similar sizes to give

test parameters for the F--test and t-test, and to calculate the uncertainty reported to the

customer.

Current values for these process parameters are shown in figure 5.6 {0^Q^) and table 5.2.

In general the short term standard deviation has a weak dependence on block length, but

long blocks show a more interesting behavior.

>

0) Q

b

T3c

D

80

60

40

20

Total SD= +

Within SD =

± +

0.1

++

+ +

J 1 I M ill I I I I I I 10 30

Figure 5.6. Dependence of short term standard deviation, O^^,

and long term standard deviation, O^q^, on gage block length.

89
Table 5.2

Table of o^jthin and o^tai by groups, in nanometers

Group O^j^hin Ototal

2 0\J . 100 \ \J\J innil. to 0\J . 107 I \J 1 in111. 5 4

3 0 108 in to 0 196 in 6

4 0 127 in to 0\J . 146 I inIII. vJ 5 0.147 in. to 0.500 in. 5 5

6 0.550 in. to 2.00 in. 6 8

7 3 in. 1

181 \J 4 in. 15 41

5 in. 10 18

6 in. 12 48

7 in. 11 26

8 in. 10 76

10 in. 10 43

- 12 in. 10 25

16 in. 17 66

20 in. 13 29

Group ^Within ^t<

14 1 00 mm to 1.09 mm R\j

1 10 mm to 1.29 mm >j Ifi 1 '^0 mm to 1.49 mm c; c;\j 17 1 '^O mm to 2.09 mm c.\J

1

9Z.. 10mm 1 W 1 1 II 1 1 to 2.29 mm c. 1Q Z9 . '^n OU mm1 1 1 1 1

to 2.49 mm c; on z.oU mm to 10 mm ro

cO

21 10.5 mm to 20 mm 5 7

22 20.5 mm to 50 mm 6 8

23 60 mm to 100 mm 8 18

24 125 mm 9 19

150 mm 11 38

175 mm 10 22

200 mm 14 37

250 mm 11 48

300 mm 7 64

400 mm 11 58

500 mm 9 56

90
The apparent lack of a strong length dependence in short term variability, as measured by

o^, is the result of extra precautions taken to protect the thermal integrity of longer blocks.

Since the major cause of variation for longer blocks is thermal instability the precautions

effectively reduce the length dependence.

One notable feature of the above table is that short term variability measured as o^ and

long term variability, measured as a^t, are identical within the measurement uncertainties

until the size gets larger than 50 mm (2 in). For short sizes, this implies that long term

variations due to comparators, environmental control and operator variability are very

small.

A large difference between long and short term variability, as exhibited by the long sizes,

can be taken as a signal that there are unresolved systematic differences in the state of

the equipment or the skill levels of the operators. Our software system records the identity

of both the operator and the comparator for each calibration for use in analyzing such

problems. We find that the differences between operators and instruments are negligible.

We are left with long term variations of the thermal state of the blocks as the cause of the

larger long term variability. Since our current level of accuracy appears adequate at this

time and our thermal preparations, as described earlier, are already numerous and time

consuming we have decided not to pursue these effects.

5.7.1 Summary

Transferring length from master blocks to customer blocks always involves an uncertainty

which depends pnmanly on comparator repeatability and the number of comparisons, and

the accuracy of the correction factors used.

The random component of uncertainty (Oto,) ranges from 5 nm (0.2 pin) for blocks under

25 mm to about 75 nm (3 pin) for 500 mm blocks. This uncertainty could be reduced by

adding more comparisons, but we have decided that the gain would be economically

unjustified at this time.

Under our current practices no correction factors are needed for steel and chrome carbide

blocks. For other materials a small added uncertainty based on our experience with

correction factors is used. At this time the only materials other than steel and chrome

carbide which we calibrate are tungsten carbide, chrome plated steel and ceramic, and

these occur in very small numbers.

91
6. Gage Block Interferometry

6.1 Introduction

Gage Block calibration at NIST depends on interferometric measurements where the

unit of length is transferred from its definition in terms of the speed of light to a set of

master gage blocks which are then used for the intercomparison process.

Gage blocks have been measured by interferometry for nearly 100 years and the only

major change in gage block interferometry since the 1950's has been the development

of the stabilized laser as a light source. Our measurement process employs standard

techniques of gage block interferometry coupled with an analysis program designed to

reveal random and systematic errors. The analysis program fosters refinements aimed

at reducing these errors. A practical limit is eventually reached in making refinements,

but the analysis program is continued as a measurement assurance program to monitor

measurement process reliability.

Briefly, static interferometry is employed to compare NIST master gage blocks with a

calibrated, stabilized laser wavelength. The blocks are wrung to a platen and mounted

in an interferometer maintained in a temperature controlled environment. The fringe

pattern is photographed and at the same moment those ambient conditions are measured which influence block length and wavelength. A block length computed from

these data together with the date of measurement is a single record in the history file for the block. Analysis of this history file provides an estimate of process precision

(long term repeatability), a rate of change of length with time, and an accepted value for

the block length at any given time.

Gage block length in this measurement process is defined as the perpendicular

distance from a gaging point on the top face of the block to a plane surface (platen) of

identical material and finish wrung to the bottom face. This definition has two

advantages. First, specifying a platen identical to the block in material and surface

finish minimizes phase shift effects which may occur in interferometry. Second, it duplicates practical use where blocks of identical material and finish (from the same

set) are wrung together to produce a desired length. The defined length of each block

includes a wringing layer, eliminating the need for a wringing layer correction when

blocks are wrung together.

The NIST master gage blocks are commercially produced gage blocks and possess no

unusual qualities except that they have a history of calibration from frequent and

systematic comparisons with wavelengths of light.

92
6.2 Interferometers

Two types of interferometers are used at NIST for gage block calibration. The oldest is a Kosters type interferometer, and the newest, an NPL Gage Block Interferometer.

They date from the late 1930's and 1950's respectively and are no longer made. Both

were designed for multiple wavelength interferometry and are much more complicated

than is necessary for single wavelength laser interferometry. The differences are

pointed out as we discuss the geometry and operation of both interferometers.

6.2.1 The Kosters Type Interferometer

Figure 6.1 Kosters type gage block interferometer.

The light beam from laser L1 passes through polarization isolator B and spatial filter B', and is diverged by lens C. This divergence lens is needed because the interferometer was

designed to use an extended light source (a single element discharge tube). The laser

produces a well collimated beam which is very narrow, about 1 mm. The spatial filter and

lens diverges the beam to the proper diameter so that lens D can focus the beam on the

entrance slit in the same manner as the light from a discharge tube. The entrance slit (Si)

is located at the principle focus of lenses D and F. Lens F collimates the expanded beam

at a diameter of about 35 mm, large enough to encompass a gage block and an area of

the platen around it.

The extended collimated beam then enters a rotatable dispersion prism, W. This prism

allows one wavelength to be selected from the beam, refracting the other colors at angles

which will not produce interference fringes. An atomic source, such as cadmium (l) and

a removable mirror M, can be used in place of the laser as a source of 4 to 6 different

93
calibrated wavelengths. If only a laser is used the prism obviously has no function, and it could be replaced with a mirror.

The rest of the optics is a standard Michelson interferometer. The compensator plate, I,

is necessary when using atomic sources because the coherence length is generally only a few centimeters and the effective light path in the reference and measurement arms

must be nearly the same to see fringes. Any helium-neon laser will have a coherence

length of many meters, so when a laser is used the compensator plate is not necessary.

The light beam is divided at the beam splitter top surface, BS, into two beams of nearly the

same intensity. One beam (the measuring beam) continues through to the gage block

surface G and the platen surface P. The other beam (reference beam) is directed through

the compensating plate i to the plane mirror R. The beams are reflected by the surfaces

of the mirror, platen, and gage block, recombined at the beam splitter, and focused at the

exit aperture, S2. A camera or human eye can view the fringes through the exit aperture.

Interference in this instrument is most readily explained by assuming that an image of

reference mirror R is formed at R' by the beam splitter. A small angle, controlled by

adjustment screws on table T, between the image and the gage block platen combination

creates a Fizeau fringe pattern. When this wedge angle is properly adjusted for reading

a gage block length, the interference pattern will appear as in figure 6.2.

Fig. 6.2 Observed fringe pattern for a gage block wrung to a platen.

The table T has its center offset from the optical axis and is rotatable by remote control.

Several blocks can be wrung to individual small platens or a single large platen and placed

on the table to be moved, one at a time, into the interferometer axis for measurement.

94
Two criteria must be met to minimize systematic errors originating in the interferometer:

(1) the optical components must be precisely aligned and rigidly mounted and (2) the

optical components must be of high quality, i.e., the reference mirror, beam splitter and

compensator must be plane and the lenses free of aberrations.

Alignment is attained when the entrance aperture, exit aperture and laser beam are on a common axis normal to, and centered on, the reference mirror. This is accomplished with a Gaussian eyepiece, having vertical and horizontal center-intersecting cross-hairs,

temporarily mounted in place of the exit aperture, and vertical and horizontal centerintersecting cross-hairs permanently mounted on the reference mirror. Through a process

of autocollimation with illumination at both entrance aperture and Gaussian eyepiece, the

reference mirror is set perpendicular to an axis through the intersections of the two sets

of cross-hairs and the entrance, and exit apertures are set coincident with this axis.

In addition, the laser beam is aligned coincident with the axis and the prism adjusted so

the laser light coming through the entrance aperture is aligned precisely with the exit

aperture. Having an exact 90° angle between the measuring leg and the reference leg is not essential as can be seen from instrument geometry. This angle is governed by the

beam splitter mounting. All adjustments are checked regularly.

The gage block and its platen are easily aligned by autocollimation at the time of

measurement and no fringes are seen until this is done. Final adjustment is made while

observing the fringe pattern, producing the configuration of figure 6.2.

Temperature stability of the interferometer and especially of the gage block being

measured is important to precise measurement. For this reason an insulated housing

encloses the entire Kosters interferometer to reduce the effects of normal cyclic laboratory

air temperature changes, radiated heat from associated equipment, operator, and other

sources. A box of 2 cm plywood with a hinged access door forms a rigid structure which

is lined with 2.5 cm thick foam plastic. Reflective aluminum foil covers the housing both

inside and out.

6.2.2 The NPL Interferometer

The NPL interferometer is shown schematically in figure 6.3. The system is similar in principle to the Kosters type interferometer, although the geometry is rather different.

95
The NIST version differs slightly from the original interferometer in the addition of a

stabilized laser, L^. The laser beam is sent through a spatial filter B (consisting of a

rotating ground glass plate) to destroy the temporal coherence of the beam and hence

reduce the laser speckle. It is then sent through a diverging lens C to emulate the

divergent nature of the atomic source L2. The atomic source, usually cadmium, is used for

multicolor interferometry.

The beam is focused onto slit, S^, which is at the focal plane of the D. Note that the

entrance slit and exit slit are separated, and neither is at the focal point of the lens. Thus,

the light path is not perpendicular to the platen. This is the major practical difference

between the Kosters and NPL interferometers. The Kosters interferometer slit is on the

optical axis which eliminates the need for an obliquity correction discussed in the next

section. The NPL interferometer, with the slit off the optical axis does have an obliquity

correction. By clever design of the slits and reflecting prism the obliquity is small, only a few parts in 1 0^.

The beam diverges after passing through slit Si and is collimated by lens F. It then passes

through the wavelength selector (prism) W and down onto the Fizeau type interferometer

formed by flat R and platen P. The height of the flat can be adjusted to accommodate

blocks up to 100 mm. The adjustable mount which holds the flat can be tilted to obtain

proper fringe spacing and orientation shown in figure 6.2.

The light passes back through the prism to a second slit S2, behind which is a small

reflecting prism (RP) to redirect the light to an eyepiece for viewing.

96
The optical path beyond the slit is inside a metal enclosure with a large door to allow a gage block platen with gage blocks wrung down to be inserted. There are no heat sources

inside the enclosure and all adjustments can be made with knobs outside the enclosure.

This allows a fairly homogeneous thermal environment for the platen and blocks.

Multicolor interferometry is seldom needed, even for customer blocks. A single wavelength

is adequate if the gage block length is known better than 1/2 fringe, about 0.15 pm (6 pin).

For customer calibrations, blocks are first measured by mechanical comparison. The

length of the block is then known to better than 0.05 pm (2 pin), much better than is necessary for single wavelength interferometry.

Since the NPL interferometer is limited to relatively short blocks the need for thermal

isolation is reduced. The temperature is monitored by a thermistor attached to the gage

block platen. One thermometer has proven adequate and no additional insulation has

been needed.

6.2.3 Testing Optical Quality of Interferometers

Distortion caused by the system optics, is tested by evaluating the fringe pattern produced

on a master optical flat mounted on the gage block platen support plate. Photographs

showing fringes oriented vertically and horizontally are shown in figure 6.4. Measurements

of the fringe pattern indicate a total distortion of 0.1 fringe in the field. A correction could

be applied to each block depending on the section of the field used, but it would be

relatively small because the field section used in fringe fraction measurement is small.

Generally the corrections for optical distortion in both instruments are too small to matter.

Figure 6.4 The quality of the optics of the interferometer can be measured

by observations of the fringes formed by a calibrated reference surface such

as an optical flat or gage block platen.
6.2.4 Interferometer Corrections

In some interferometers, the entrance and exit apertures are off the optical axis so light

falls obliquely on the gage block. The NPL interferometer is one such design.

In the Kosters type interferometer these apertures are aligned on the optical axis and

consequently the illumination is perpendicular to the gage block and platen surfaces. In figure 6.5 the case for normal incidence is given. Note that the light reflected from the

surfaces follows the same path before and after reflection. At each point the intensity of

the light depends on the difference in phase between the light reflected from the top

surface and that reflected from the lower surface. Whenever this phase difference is 180°

there will be destructive interference between the light from the two surfaces and a dark

area will be seen. The total phase difference has two components. The first component

results from the electromagnetic phase difference of reflections between the glass to air

interface (top flat) and the air to glass interface (bottom flat). This phase shift is (for nonconducting platens) 180°. The second phase difference is caused by the extra distance

traveled by the light reflected at the bottom surface. The linear distance is twice the

distance between the surfaces, and the phase difference is this difference divided by the

wavelength, A.

Figure 6.5 In the ideal case of normal incidence , a dark fringe will appear wherever

the total path difference between the gage block and platen surfaces is a multiple

of the wavelength.

98
At every position of a dark fringe this total phase difference due to the path difference is a multiple of A:

A(path) = nA = 2D (6.1)

where n is an integer.

If the wedge angle is a, then the position on the nth fringe is

X = (nA)/(2tan(a)) (6.2)

For the second case, where the light is not normal to the block and platen, we must

examine the extra distance traveled by the light reflecting from the lower flat and apply a compensating factor called an obliquity correction. Figure 6.6 shows this case. The

difference in path between the light reflected from the upper reference surface and the

lower reference surface must be a multiple of A to provide a dark fringe. For the nth dark

fringe the distance 2L must equal nA.

Figure 6.6 When the light is not normal to the surface the

fringes are slightly displaced by an amount proportional to the

distance between the surfaces.

99
The equation for a dark fringe then becomes

A(path) = nA = 2L

From the diagram we see that L is:

L = D cos(e) = nA

Thus the new perpendicular distance, D', between the two flats must be larger than D.

This implies that the nth fringe moves away from the apex of the wedge, and the distance

moved is proportional to cos(6) and distance D.

When a gage block is being measured, the distance from the top flat to the bottom flat is larger than the distance from the top flat to the gage block surface. Thus the fringes on

the block do not shift as far as the fringes on the flat. This larger shift for the reference

fringes causes the fringe fraction to be reduced by a factor proportional the height of the

block, H, and cos(e). Note that this effect causes the block to appear shorter than it really

is. Since the angle 6 is small the cosine can be approximated by the first terms in its expansion

cos(e)= 1 - — + + - (6.5)

2 24

and the correction becomes:

Obliquity Correction ~ {U/2)Q^ (6.6)

This correction is important for the NPL Hilger interferometer where the entrance slit and

observation slit are separated. In the Kosters type interferometer the apertures are

precisely on the optical axis, the obliquity angle is zero, and thus no correction is needed.

In most interferometers, the entrance aperture is of finite size, therefore ideal collimation

does not occur. Light not exactly on the optical axis causes small obliquity effects

proportional to the size of the aperture area. A laser, as used with these interferometers,

is almost a point source because the aperture is at the common focal point of lens D, and

collimating lens E. At this point the beam diameter is the effective aperture. When using

diffuse sources, such as discharge lamps, the effective aperture is the physical size of the

aperture.

(6.3)

(6.4)

100
The theory of corrections for common entrance aperture geometries has been worked out

in detail [37,38,39]. The correction factor for a gage block of length L, measured with

wavelength A, using a circular aperture is approximately [40]

Slit Correction (Circle) (6.7)

where L is the block length, D is the aperture diameter, and f is the focal length of the

collimating lens.

For rectangular apertures the approximate correction is

where L is the block length, h and I the height and length of the slit, and f the focal length

of the collimating lens.

As aperture diameter approaches zero, the correction approaches zero. The correction

is very small when using a laser because the beam diameter is small at the aperture.

6.2.5 Laser Light Sources

The advantage of a laser light source lies in its unequalled coherence. Conventional

spectral light sources have such low coherence that blocks longer than 300 mm have, in the past, been measured by stepping up from shorter blocks. Laser fringe patterns in these interferometers compared with patterns from other spectral lamps are of superior

contrast and definition through the length range 0 to 500 mm.

The disadvantage stems from the somewhat unstable wavelength of lasers. This problem

has been overcome by a number of different stabilization schemes, and by using the .6328

pm line of HeNe lasers. These lasers can be related to the definition of the meter by

calibration against an iodine stabilized HeNe laser [41 ,42]. The wavelength of the iodine

stabilized laser is known to a few parfs in 1 0^ and is one of the recommended sources for

realizing the meter [8]. The lasers currently used for gage block calibration are polarization

stabilized [43].

Slit Correction{Rectangle) L^jh^+I^)

24f^

(6.8)

101
Single wavelength static interferometry, in contrast with multiple wavelength interferometry,

requires that the gage block length be known to within ±.25 wavelength (±.5 fringe) either

from its history or from another measurement process. This is no problem for NiST

reference blocks, or for blocks calibrated by the NIST mechanical comparison technique.

Laser light must not be allowed to reflect back into its own cavity from the interferometer

optical components because this will disturb the lasing action and may cause a wavelength

shift, wavelength jittering or it may stop the lasing altogether. The reflected light is prevented from re-entering the laser by a polarization isolator consisting of a Glan

Thompson prism and a quarter wave plate in the beam as it emerges from the cavity. This

assembly is tilted just enough to deflect reflections from its faces to the side of the laser

exit aperture.

6.3 Environmental Conditions and their Measurement

Environmental control of the laboratory holds temperature at 20 ±.05 °C and water vapor

content below 50% relative humidity. Temperature variations within the insulated

interferometer housing are attenuated by a factor of about 10 from those in the room, thus

insuring stability of both interferometer and blocks.

Temperature, atmospheric pressure, and water vapor content of the ambient air in the

interferometer light path must be measured at the time the gage block is measured. From

these properties the refractive index of the air is calculated, and in turn, the laser

wavelength. Since interferometry is a comparison of the light wavelength to the block

length the accuracy of the wavelength is of primary importance. The gage block

temperature is measured so that the block length at 20 °C, the standard temperature for

dimensional metrology, can be computed.

6.3.1 Temperature

The measuring system accuracy for air and block temperature depends on the length of

the gage block to be measured. Since both the wavelength and block temperature

corrections are length dependent, a 500 mm measurement must have correction factors

10 times as good as those for a 50 mm measurement to achieve the same calibration

accuracy,

For blocks measured with the NPL interferometer (less than 100 mm) a calibrated

thermistor probe is used to measure temperature to an accuracy of 0.01 °C and a

calibrated aneroid barometer is used to measure atmospheric pressure to 0.1 mm of

mercury.

102
For long blocks the system is much more complicated. The measunng system for air and

block temperature consists of copper-constantan ther ru couples referenced to a platinum

resistance thermometer. Figure 6.7 is a schematic diagram of the system.

A solid copper cube provides a stable reference temperature measured by a Standard

Platinum Resistance Thermometer (SPRT) and a Mueller resistance bridge. A well in the

cube holds the thermometer stem, the reference thermocouple junction, and a liquid of

high heat conductivity but low electrical conductivity. There is also a second well whose

function will be described later. An enclosed thermocouple selector switch connects the

measuring junctions, one at a time, to the reference junction and the thermal emfs

generated by the temperature differences between measuring junction and reference

junction are indicated on a nanovoltmeter. The nanovoltmeter is shared with the bridge

where it is used as a nullmeter. To keep the copper block in a stable state and to minimize

thermal emfs in the selector switch, both block and switch are enclosed in an insulated

box. The switch is operated through a fiber shaft extending to the exterior and the

protruding part of the thermometer is protected by an insulated tube.

DPDT SWITCH

THERMOCOUPLE

SELECTOR

SWITCH

MEASURING THERMOCOUPLE

JUNCTIONS

Figure 6.7 The temperature measurement system used for gage block

calibration consists of multiple thermocouples referenced to a calibrated

platinum resistance thermometer.

Thermocouples generate emfs proportional to the temperature gradient along the wire:

E = f^' (\) dT (6.9)

where T^ and J are the temperatures of the reference and measuring junctions

respectively, and (j) is the thermocouple constant. Minimizing this gradient reduces

10.3
uncertainties.

Relating the system to the International Temperature Scale of 1990 (ITS '90) is a three

step procedure. First, the SPRT is calibrated using methods described in NIST Technical

Note 1265 [44]. Second, the bridge is calibrated as described the same monograph. The

third step is the calibration of the thermocouples, which is accomplished with a second

SPRT and insulated copper cube. The second cube has an extra well for the

thermocouples being calibrated and is heated one or two degrees to equilibrium.

Measured thermal emfs generated by the temperature difference between the two cubes

together with the temperatures of the cubes allow computation of a calibration factor for

each junction.

Integrity is maintained by periodic check of the SPRT with a triple point cell, checking the

bridge against a calibrated resistor and checking the thermocouples for equality when they

are all at the same temperature in a well of an insulated copper cube.

6,3.2 Atmospheric Pressure

The interferometer housing is not airtight and pressure inside it is the same as laboratory

pressure. An aneroid barometer is located adjacent to and at the same elevation as the

interferometer. The aneroid used for gage block measurements has good stability, and

frequent comparisons with NIST reference barometers insure its reliability.

For long blocks more precise measurement of the pressure is useful. There are a number

of commercially available atmospheric pressure sensors available which are of adequate

accuracy for gage block calibration. Remembehng that measurements to one part in 10^

requires pressure measurement accuracy of .3mm of Hg, a calibrated aneroid barometer

can be adequate. For systems requiring automatic reading of pressure by a computer

there are a number of barometers which use capacitance or vibrating cylinders to measure

pressure and have RS-232 or IEEE 488 interfaces.

6.3.3 Water Vapor

The humidity is derived from a chilled mirror dew point hygrometer. The system is calibrated by the NIST humidity calibration service and is accurate to about 1 % R.H. Other

systems have adequate accuracy but need more frequent checking for drift. The

disadvantage of the chilled mirror is that it requires a specified air flow rate not usually

available inside an interferometer. Humidity inside an interferometer enclosure may differ

from the laboratory value by as much as 6% R.H., especially during periods of changing

laboratory humidity. If this presents a serious problem (see table 6.1 ) then a hygrometer

type less dependent on air flow rate must be mounted inside the interferometer.

104
6.4 Gage Block Measurement Procedure

In preparation for measurement, the gaging faces of the NIST standards are cleaned with

ethyl alcohol. Wiping the alcohol-rinsed gaging faces is done with paper tissue or a clean

cotton towel. Lint or dust is removed with a camel's hair brush.

The gaging faces are examined for burrs and, if present, these are removed with a

deburring stone. The bottom end is tested for wringability with a quartz optical flat. The

transparency of the flat makes possible a judgment of the wring. A uniform grey color at the interface indicates a good wring. Any colored or light areas indicate a gap of more

than 25 nm between the faces in that area, and such a gap may result in an erroneous

length measurement. Deburring and cleaning is continued until a good wring is achieved.

The quartz flat is left wrung to the gaging face to keep it clean until preparations are completed for wringing the block to the steel optical flat (platen).

Preparation of the steel platen is similar. Ethyl alcohol is used to rinse the face, then

deburring if necessary, followed by more rinsing and drying. After sweeping with a camel's

hair brush, a thin uniform film of light grease is applied to the wringing surface with a bit of tissue. This film is rubbed and finally polished with tissue or towel until the film appears

to be gone. After carefully sweeping the platen with a brush, it is ready for wringing.

The block is removed from the quartz flat and the exposed gaging face is immediately

wrung by carefully sliding it onto the edge of the platen, maintaining perpendicularity by

feel. Sliding is continued until wringing resistance is felt, and then with slight downward

pressure the block is slowly worked into its final position. Square style blocks, such as the

NIST long blocks, are positioned so that the gage point is at the right in the viewing field

of the interferometer. One to four blocks can be wrung to the platens used in the Kosters

interferometer. Ten square blocks or 20 rectangular blocks can be wrung onto the platens

at one time for use in the NPL interferometer.

For long blocks measured in the Kosters interferometer, the platen with its wrung blocks

is placed in the interferometer and two thermocouples are fed down the center hole of

each block, one about three quarters down and one about one quarter down the length of

the block. A small wad of polyurethane is pushed into the hole to seal and hold the wires.

For small blocks, where there can be no significant vertical thermal gradient, the

temperature sensor is attached to the platen.

A preliminary alignment by autocollimation with the Gaussian eyepiece as the support table

is adjusted, will produce fringes. Rapid changes in the fringe pattern occur as the blocks

and platen cool from handling.

It is convenient to wring the blocks in late afternoon and allow overnight temperature

105
stabilization. Two length observations are made several hours apart, but the first measurement is not taken until the laser is in equilibrium. Leaving the laser on

continuously when observations are being made over several days eliminates delays.

Final fringe pattern adjustment is made so that the platen fringes are parallel to the top and

bottom edges of the block and one fringe on the block goes through its defined gage point.

For the Kosters interferometer the direction of increasing fringe order is verified by

pressing down on the eyepiece and observing the fringe movement. In the NPL, the

reference mirror is pushed and the fringe movement is observed. The fringe fraction can

be estimated by eye or a photograph can be taken of the pattern. Block temperature, air

temperature, barometric pressure, and humidity are measured immediately before and

after the observation and then averaged.

The photograph can be either a positive or a negative. A compromise is made between

image size and exposure time, but short exposure is desirable because the fringe pattern

may shift during a long exposure. Further image magnification takes place in the fringe

measuring instrument. A compromise is also made between magnification in the

photograph and magnification in the instrument used to measure the photograph. Too

much total magnification degrades image sharpness, making fringe measurements more

difficult.

Fringe fractions are obtained from the photograph with a coordinate comparator having a

fixed stage in the focal plane of a microscope movable on X-Y slides by precision screws

with drum readouts. Four measurements of distance "a" and"b" are recorded and

averaged to obtain fringe fraction f, as shown in figure 6.8.

FIELD OF VIEW

Figure 6.8. The fringe fraction is the fractional offset of the fringes on the

block from those of the platen.

106
Settings on the block fringe to obtain "a" are made at the point where the fringe intersects

the gage point. Settings on the platen fringes to obtain "a" and "b" are made as close as

practical to the edges of the block because it is here that the platen best represents an

extension of the bottom face of the block. Parallelism between gage block and platen is also read from the photo because this information is useful in detecting a defective wring.

Poor wringing contact is indicated if the parallelism is different from its usual value (or

mechanically determined value using the comparator) and this is sufficient cause to discard a particular length measurement.

Photographing fringe patterns has several advantages. The photograph is a permanent

record which can be reinterpreted at any time with the same or other techniques. Changes

in block geometry are readily seen by comparing photographs in time sequence. The

coordinate comparator is superior to the common practice of fringe fraction estimation by

eye because it is more objective, lends itself to multiple measurements and averaging, and

is more precise. Finally, photography is fast and thus permits readings of the ambient

conditions to closely bracket the fringe recording. This is especially important because

atmospheric pressure is generally not constant.

The advent of computer vision systems provides another method of recording and

analyzing fringes. There are commercially available software packages for measuring

fringe fractions which can be incorporated into an automated gage block measuring

system.

6.5 Computation of Gage Block Length

Calculating gage block length from the data is done in 3 steps:

1 . Calculation of wavelength, A,pf, at observed ambient conditions.

2. Calculation of the whole number of fringes in the gage block length from its predicted length and the laser wavelength in ambient air.

3. Calculation of the gage block length from the whole number of fringes, the

observed fraction, wavelength in ambient air, gage block temperature, and

interferometric correction factors.

6.5.1 Calculation of the Wavelength

Edien's 1966 formula [45] for the refractive index of air is used to calculate the wavelength.

Comparison of absolute refractometers showed that the Edien equation agreed with

refractometers as well as the refractometers agreed with each other, to about 1 part in

107
10^ [46]. More recent work by Birch and Downs [47] has shown that the original data used

for the water vapor correction was slightly in error. The new correction (the parameter C

below) is thought to increase the accuracy of the Edien formula to about 3x1 0'®

. This latter

figure can be taken as an estimate of the accuracy of the formula.

The latest revision of the EdIen formula is [48]:

K - "tpfV (6.10)

and thus

V = = Ao[1+A*B-Cr (6.11)

n,p,

where

A = p[8342.74 + 2406205(1 + 15998 (38.9-a^V^] 10'^ (6.12)

96095.45

B = 1 + p (0.601 -Q.QQ972t) 10'^ (6. 1 3)

1+0. 0036611

C = f(3. 7345-0. 0401 o^) 10'^ (6.14)

The symbols used are:

Aq = Vacuum wavelength in pm

Aptf = Wavelength at observed conditions t,p,f

rip^f = Index of refraction of air at t,p,f

p = Atmospheric pressure in Pa

t = Temperature in degrees Celsius

f = water vapor pressure in Pa

O = 1/Ao in pm.

108
6.5.2 Calculation of the Whole Number of Fringes

This method for determining the whole number of fringes is valid if the block length is known to better than ±0.5 fringe ( 1 fringe is about 0.32 |jm for helium neon laser light)

from either its history or from an independent measurement process. The calculation is as follows:

2Lp[1+C(f-20)]

F= (6.15)

where F is the number of interference fringes in the length of the block at temperature

t', and in air of temperature t, pressure p, and vapor pressure f.

Lp is the predicted block length at 20 °C taken from its history or an

independent measurement.

C is the linear thermal expansion coefficient per degree Celsius of the block.

t' is the block temperature at the time of measurement.

Xtpf is the wavelength in the ambient air at the time of measurement.

The fractional part of F is retained temporarily for the reasons explained below.

6.5.3 Calculation of Block Length from the Observed Data

Generally, the observed fringe fraction 0 is simply substituted for the fractional part of F,

but there are cases where the last digit in whole number F must be raised or lowered by

one. For example if the fractional part of F is 0.98 and the observed fraction is 0.03

obviously the last whole number in F must be raised by one before substituting the

observed fraction.

The new measured block value, in fringes, is

F' = F + 4). (6.16)

Finally, the interferometer aperture correction, 62, is added, the block is normalized to

109
20 °C, and a conversion to length units is made to arrive at the final value at 20 °C as

follows:

L2o= —-- F'[1+C(20-r)] + 52. (6.17)

2

6.6 Type A and B Errors

Potential sources of errors are as follows:

1. Wavelength

a. Vacuum wavelength of laser

b. Refractive index of air equation

c. Refractive index determination

Air temperature measurement

Atmospheric pressure measurement

Humidity measurement

2. Interferometer

a. Alignment

b. Aperture correction

c. Obliquity correction

3. Gage Block

a. Gage block temperature measurement

b. Thermal expansion coefficient

c. Phase shift difference between block and platen

Length proportional errors are potentially the most serious in long block measurement.

Their relative influence is illustrated in table 10. One example will help explain the table:

if air temperature measurement is in error by 0.1 °C a systematic error of 1 part in 10'' results. For a 250 mm gage block the error is 25 nm, a significant error. On a 10 mm

gage block the same temperature measurement leads to only a 1 nm error, which is negligible, f

In our measurement process every practical means, as previously discussed, was used

to reduce these errors to a minimum and, at worst, errors in measuring the parameters

listed above do not exceed those listed in table 10. Further evaluation of these errors is described in section 4.3 on process control and our master block history.

110
A few sources of error in the list of systematic errors but not in the table have not been

completely discussed. The first of these is phase shift difference between block and

platen. When light is reflected from a surface the phase of the reflected light is shifted from

the phase of the incident by some amount which is characteristic of the surface's material

composition and surface finish. The explanation and measurement of phase is given in appendix C. Most gage blocks have phase shifts which differ from quartz by an equivalent

of 25 to 75 nm. This is a large source of error, although if both the block and platen are made of the same metal the error is reduced below 20 nm. Since the phase shift occurs

only at the metal surface it is not length dependent.

Table 10

Effects of Environmental Factor Errors on Length

Parameter Change Which Leads to a Length

Error of 1 part in 10''

Wavelength

Vacuum wavelength 1 part in 10^

Refractive index formula 1 part in 10^

Air temperature 0.1 °C

Atmospheric Pressure 40 pa

Relative Humidity 12% R.H.

Interferometer Alignment 0.45 mm/m

Gage Block

Steel gage block temperature 0.009 °C

Chrome Carbide block temp. 0.012 °C

Thermal expansion coefficients of gage blocks have an uncertainty that is, to a large

extent, nullified by measuring at temperatures very close to the 20 °C reporting

temperature. Gage block manufacturers give 11.5x1 0'^rC for steel blocks. This average

value may be uncertain by up to 0.6 x 10'^/°C for individual blocks. The expansion

coefficient for steel blocks is also length dependent for long blocks, as discussed earlier

in section 3.3. The expansion coefficient for longer sizes can differ from the nominal 1 1 .5 x ^0'^/°C by nearly 1 x 10'^/°C. These uncertainties can cause significant errors if blocks are used at other than 20 °C.

There are, of course, random errors which arise from variability in any parameter in the

table, and in addition, they can arise in the fringe fraction measurement and wringing

variability. Wringing is discussed in more detail in appendix B. Gage blocks are generally

111
measured for 3 to 5 separate wrings to sample variability due to these sources.

6.7 Process Evaluation

Figure 6.9 shov\/s some examples of the interferometric history of NIST master gage

blocks. As expected, the scatter in the data is larger for longer blocks because of the

uncertainties in the length dependent thermal expansion and index of refraction

corrections.

ID: F15 Size: 25 mm 15Pts. Avg. = 0.018

1960 1970 1980 1990

ID: 5 83 Size: 0.1 inch 56 Pts. Avg. = 0.296

gQ

O

c

eo

100

o

3

Q

O

1960 1970 1980 1990

Figure 6.9. Examples of the interferometric history of NIST master gage blocks.

112
The length dependence of the variability is shown graphically in figure 6,10. This

variability has two components: (1 ) non-length related variabilities due to variations in the

wringing film between the block and the platen, and random errors in reading the fringes;

and (2) a length dependent part primarily due to environmental effects.

Nominal Length (inches)

Figure 6. 1 0. The variability (a) of NIST master gage blocks as a function of length.

The major components of the interferometric measurement uncertainty are shown in table

6.2. The errors can be divided into two classes: those which are correlated ( the same for

all measurements) and those which are uncorrelated (vary for each measurement). The

classification depends on the measurement history. NIST gage blocks which are measured over a long period of time, during which the thermometers and barometers are

recalibrated. In this case the errors due to thermal expansion and refractive index

corrections vary. For a customer block, measured 4 times over a 4 day period, they do not

vary. Table 6.2 classifies the errors for both NIST masters and customer calibrations.

113
Table 6.2

Source of Uncertainty

Reading error

Obliquity/Slit Correction

Wringing Film Thickness

Phase Correction

Thermal Expansion Correction

Temperature Measurement

Expansion Coefficient Value

Index of Refraction

Edien Equation

Environmental Measurements

Laser Frequency

Uncertainty

Type for NIST Masters

uncorrelated

correlated

r uncorrelated

correlated

uncorrelated

correlated

correlated

uncorrelated

uncorrelated

Uncertainty

Type for Customers

uncorrelated

correlated

uncorrelated

correlated

correlated

correlated

correlated

uncorrelated

correlated

The interferometric history provides a statistical sample of all of the uncorrelated errors. Using the information from figure 6.10 we obtain an estimate of the uncorrelated errors

to be:

U, (pm)- 0.008 + 0.03x1 (L in meters) (6.20)

Multiple measurements of a gage block can reduce this figure; if n measurements are made the uncertainty of the average length becomes UJ/n. Unfortunately, the correlated

errors are not sampled by repeated measurements and are not reduced by multiple

measurements. An estimate of the correlated uncertainties for NIST master gage block

calibrations is:

• U,(Mm) « 0.007 + 0.07x 1 (L in meters) (6.21)

Because customer blocks have more correlated errors and more importantly do not have

measured thermal expansion coefficients, the use of multiple measurements is less

effective in reducing measurement uncertainty. For most customers, interferometric

measurements are not cost-effective since the uncertainty for a three wring interferometric

measurement is not significantly lower than the mechanical comparison measurement, but

114
costs twice as much. If, however, a customer sends in the same blocks for interferometric

measurement over a number of years and uses the accumulated history then the

uncertainty of the blocks will approach that of NIST master blocks.

6.8 Multiple wavelength Interferometry

If we think of gage block interferometry as the measurement of height with a ruler marked

in units of A/2, we quickly realize that our ruler has units unlike a normal ruler. The lines

which show a distance of A/2 are there but there are no numbers to tell us which mark we

are looking at. This ruler can be used if we know the length of the block to better than A/4

since we can use this distance and make the small correction (less than A/4) found by

interpolating between the lines on our scale. Suppose, however, we do not know the

length this well ahead of time.

The correct method in this case is to use a number of rulers with different and

incommensurate scales. In interferometry we can use several light sources ranging from

blue to red to provide these different scales. A simple example is shown in figure 6.1 1

.

f = 0 f=.71 f=.87 f=.62 —i j

u

_oca

03O)

ni O

'2 10 ^2 17 ^2 23 ^2 29

Figure 6.11. Four color interferometry is equivalent to

measuring a length with four scales of known pitch which have

indeterminate zeros.

For each scale (color) there will be a fringe fraction f|. With only one color the block might

have any length satisfying the formula:

115
L = (rii + fi)*Ai/2 where is any integer. (6.22)

If we look at a second color, there will be another fringe fraction fs. It also will be consistent

with any block length satisfying the formula:

L = + f2)*A2/2 where n2 is any integer. (6.23)

However, we have gained something, since the number of lengths which satisfy both

relations is very much reduced, and in fact are considerably further apart than A/2. With

more colors the number of possible matches are further reduced until a knowledge of the

length of the block to the nearest centimeter or more is sufficient to determine the exact

length.

In theory, of course, knowing the exact fringe fractions for two colors is sufficient to know

any length since the two wavelengths are not commensurate. In practice, our knowledge

of the fringe fractions is limited by the sensitivity and reproducibility of our equipment. In practice, 1/20 of a fringe is a conservative estimate for the fringe fraction uncertainty. A

complete analysis of the effects of the uncertainty and choice of wavelengths on multicolor

interferometry is given by Tilford [49].

Before the general availability of computers, the analysis of multicolor interferometry was

a time consuming task [50]. There was a large effort made to produce calculational aids

in the form of books of fringe fractions for each popular source wavelength, correction

tables for the index of refraction, and even fringe fraction coincidence rules built somewhat

like a slide rule. Since the advent of computers it is much easier to take the brute force

approach since the calculations are quite simple for the computer.

Figure 6.1 1 shows a graphical output of the NIST multicolor interferometry program using a cadmium light source. The program calculates the actual wavelengths for each color

using the environmental factors (air temperature, pressure and humidity). Then, using the

observed fringe fraction, shows the possible lengths of the gage block which are near the

nominal length for each color. Note that the possible lengths are shown as small bars, with

their width corresponding to the uncertainty in the fringe fraction.

116
Laser

Cd Red

Cd Green

Cd Blue

Cd Violet

I

I

I

I I

I

I I I

I

I

I I

I

I

I I

I

I I I

I

I I I

I

I I I

I

I j I I I

-.4 -.3 -.2 -.1 0 .1 .2 .3 .4

Micrometers from Nominal

Figure 6.12. Graphical output of the NIST multicolor interferometry

program. The length at which all the colors match within 0.030 jjm is assumed to be the true gage block length. The match is at -0.09 pm in this

example.

The length where all of the fringes overlap is the actual length of the block. If all the fringes

do not overlap in the set with the best fit, the inconsistency is taken as evidence of an

operator error and the block is remeasured. The computer is also programmed to examine

the data and decide if there is a length reasonably close to the nominal length for which

the different wavelengths agree to a given tolerance. As a rule of thumb, all of the

wavelengths should agree to better than 0.030 pm to be acceptable.

Analytic methods for analyzing multicolor interferometry have also been developed [49].

Our implementations of these types of methods have not performed well. The problem is probably that the span of wavelengths available, being restricted to the visible, is not wide

enough and the fringe fraction measurement not precise enough for the algorithms to work

unambiguously.

6.9 Use of the Linescale Interferometer for End Standard Calibration

There are a number of methods to calibrate a gage block of completely unknown length.

The multiple wavelength interferometry of the previous section is used extensively, but has

the limitation that most atomic sources have very limited coherence lengths, usually under

25 mm. The method can be used by measuring a set of blocks against each other in a sequence to generate the longest length. For example, for a 10 inch block, a 1 inch block

can be measured absolutely followed by differential measurements of a 2 inch block with

the 1 inch block, a 3 inch block with the 2, a 4 inch block with the 3, a 5 inch block with the

4, and the 10 inch block with the 2, 3 and 5 inch blocks wrung together. Needless to say

this method is tedious and involves the uncertainties of a large number of measurements.

117
Another, simpler, method is to convert a long end standard into a line standard and

measure it with an instrument designed to measure or compare line scales ( for example,

meter bars) [51]. The NIST linescale interferometer, shown schematically below, is generally used to check our master blocks over 250 mm long to assure that the length is known within the 1/2 fringe needed for single wavelength interferometry.

The linescale interferometer consists of a 2 m long waybed which moves a scale up to a meter in length, under a microscope. An automated photoelectric microscope, sends a

servo signal to the machine controller which moves the scale so that the graduation is at a null position on the microscope field of view. A laser interferometer measures the

distances between the marks on the scale via a corner cube attached to one end of the

scale supporf. This system is described in detail elsewhere [52,53].

To measure an end standard, two small gage blocks which have linescale graduations on

one side, are wrung to the ends of the end standard, as shown in figure 6. 1 3. This "scale"

is then measured on the linescale interferometer. The gage blocks are then removed from

the end standard and wrung together, forming a short scale. This "scale" is also measured

on the interferometer. The difference in length between the two measurements is the

physical distance between the end faces of the end standard plus one wringing film. This

distance is the defined length of the end standard.

M

a

m = a + b

Bar Length = M - m

Figure 6.13. Two small gage blocks with linescale graduations

on one side are wrung to the ends of the end standard,

allowing the end standard to be measured as a linescale.

118
The only significant problem with this method is that it is not a platen to gage point

measurement like a normal interferometric measurement. If the end standard faces are

not flat and parallel the measurement will not give the exact same length, although a knowledge of the parallelism and flatness will allow corrections to be made. Since the

method is only used to determine the length within 1/2 fringe of the true length this

correction is seldom needed.
7. References

[I] C.G. Peters and H.S. Boyd, "Interference methods for standardizing and testing

precision gage blocks," Scientific Papers of the Bureau of Standards, Vol. 17, p.691

(1922).

[2] Beers, J.S. "A Gage Block Measurement Process Using Single Wavelength

Interferometry," NBS Monograph 152, 1975.

[3] Tucker, CD. "Preparations for Gage Block Comparison Measurements," NBSIR

74-523.

[4] Beers, J.S. and CD. Tucker. "Intercomparison Procedures for Gage Blocks Using

Electromechanical Comparators," NBSIR 76-979.

[5] Cameron, J.M. and G.E Hailes. "Designs for the Calibration of Small Groups of

Standards in the Presence of Drift," NBS Technical Note 844, 1974.

[6] Klein, Herbert A., The Science of Measurement. Dover Publications, 1988.

[7] Galyer, J.F.W. and CR. Shotbolt, Metrology for Enoineers . Cassel & Company,

Ltd., London, 1964.

[8] "Documents Concerning the New Definition of the Meter," Metrologia, Vol. 19, 1984.

[9] "Use of the International Inch for Reporting Lengths of Gage Blocks," National

Bureau of Standards (U.S.) Letter Circular LC-1033, May, 1959.

[10] T.K.W. Althin, CE. Johansson. 1864-1943. Stockolm, 1948.

[II] Cochrane, Rexmond C, "Measures for Progress," National Bureau of Standards

(U.S.), 1966.

[12] Federal Specification: Gage Blocks and Accessories (Inch and Metric) . Federal

Specification GGG-G-15C, March 20, 1975.

[13] Precision Gage Blocks for Length Measurement (Through 20 in. and 500 mm). ANSI/ASME B89.1.9M-1984, The American Society of Mechanical Engineers, 1984.

[14] International Standard 3650, Gauge Blocks . First Edition, 1978-07-15, 1978..

[15] DIN 861, part 1. Gauge Blocks: Concepts, requirements, testing . January 1983.

120
M.R. Meyerson, T.R. Young and W.R. Ney, "Gage Blocks of Superior Stability:

Initial Developments in Materials and Measurement," J. of Research of the National

Bureau of Standards, Vol. 64C, No. 3, 1960.

Meyerson, M.R., P.M. Giles and P.P. Newfeld, "Dimensional Stability of Gage Block

Materials," J. of Materials, Vol. 3, No. 4, 1968.

Birch, K.P., "An automatic absolute interferometric dilatometer," J. Phys. E: Sci.

Instrum, Vol. 20, 1987.

J.W. Berthold, S.F. Jacobs and M.A. Norton, "Dimensional Stability of Fused Silica,

Invar, and Several Ultra-low Thermal Expansion Materials," Metrologia, Vol. 13, pp.

9-16 (1977).

C.W. Marshall and R E. Maringer, Dimensional Instability. An Introduction . Pergamon Press, New York (1977).

Hertz, H., "On the contact of elastic solids," English translation in Miscellaneous

Papers. Macmillan, N.Y., 1896.

Poole, S.P., "New method of measuring the diameters of balls to a high precision,"

Machinery, Vol 101, 1961.

Norden, Nelson B., "On the Compression of a Cylinder in Contact with a Plane

Surface," NBSIR 73-243, 1973.

Puttock, M.J. and E.G. Thwaite, "Elastic Compression of Spheres and Cylinders at Point and Line Contact," National Standards Laboratory Technical Paper No. 25,

CSIRO, 1969.

Beers, John, and James E. Taylor, "Contact Deformation in Gage Block

Comparisons," NBS Technical Note 962, 1978

Beyer-Helms, P., H. Darnedde, and G. Exner. "Langenstabilitat be!

Raumtemperatur von Proben er Glaskeramik 'Zerodur'," Metrologia Vol. 21 , p49-57

(1985).

Berthold, J.W. Ill, S.F. Jacobs, and M.A. Norton. "Dimensional Stability of Fused

Silica, invar, and Several Ultra-low Thermal Expansion Materials," Metrologia, Vol.

13, p9-16 (1977).

Justice, B., "Precision Measurements of the Dimensional Stability of Four Mirror

Materials," Journal of Research of the National Bureau of Standards - A: Physics

121
and Chemistry, Vol. 79A, No. 4, 1975.

[29] Bruce, C.F., Duffy, R.M., Applied Optics Vol.9, p743-747 (1970).

[30] Average of 1 ,2,3 and 4 inch steel master gage blocks at N.I.S.T.

[31] Doiron, T., Stoup, J., Chaconas, G. and Snoots, P. "stability paper, SPIE".

[32] Eisenhart, Churchill, "Realistic Evaluation of the Precision and Accuracy of

Instrument Calibration Systems," Journal of Research of the National Bureau of

Standards, Vol. 67C, No. 2, pp. 161-187, 1963.

[33] Croarkin, Carroll, "Measurement Assurance Programs, Part II: Development and

Implementation," NBS Special Publication 676-11, 1984.

[34] ISO, "Guide to the Expression of Uncertainty in Measurement," October 1993.

[35] Taylor, Barry N. and Chris E. Kuyatt, "Guidelines for Evaluating and Expressing the

Uncertainty of NIST Measurement Results," NIST Technical Note 1297, 1994

Edition, September 1994.

[36] Tan, A. and J.R. Miller, "Trend studies of standard and regular gauge block sets,"

Review of Scientific Instruments, V.62, No. 1, pp. 233-237, 1991.

[37] C.F. Bruce, "The Effects of CoHimation and Oblique Incidence in Length

Interferometers. I," Australian Journal of Physics, Vol. 8, pp. 224-240 (1955).

[38] C.F. Bruce, "Obliquity Correction Curves for Use in Length Interferometry," J. of the

Optical Society of America, Vol. 45, No. 12, pp. 1084-1085 (1955).

[39] B.S. Thornton, "The Effects of Collimation and Oblique Incidence in Length

Interferometry," Australian Journal of Physics, Vol. 8, pp. 241-247 (1955).

[40] L. Miller, Engineenna Dimensional Metrology. Edward Arnold, Ltd., London (1962).

[41] Schweitzer, W.G., et.al., "Description, Performance and Wavelengths of Iodine

Stabilized Lasers," Applied Optics, Vol 12, 1973.

[42] Chartier, J.M., et. al., "Intercomparison of Northern European ^^^{2 - Stabilized HeNe Lasers at A = 633 nm," Metrologia, Vol 29, 1992.

[43] Balhorn, R., H. Kunzmann and F. Lebowsky, "Frequency Stabilization of Internal122
Mirror Helium-Neon Lasers," Applied Optics, Vol 11/4, April 1972.

[44] Mangum, B.W. and G.T. Furukawa, "Guidelines for Realizing the International

Temperature Scale of 1990 (ITS-90)," NIST Technical Note 1265, National Institute

of Standards and Tecnology, 1990.

[45] Edien, B., 'The Refractive Index of Air," Metrologia, Vol. 2, No. 2, 1966.

[46] Schellekens, P., G. Wllkening, F. Reinboth, M.J. Downs, K.P. Birch, and J. Spronck,

"Measurements of the Refractive Index of Air Using Interference Refractometers,"

Metrologia, Vol. 22, 1986.

[47] Birch, K.P. and Downs, M.J., 'The results of a comparison between calculated and

measured values of the refractive index of air," J. Phys. E: Sci. Instrum., Vol. 21, pp.

694-695, 1988.

[48] Birch, K.P. and M.J. Downs, "An Updated EdIen Equation for the Refractive Index

of Air," Metrologia, Vol. 30, 1993.

[49] C.R. Tilford, "Analytical Procedure for determining lengths from fractional fringes,"

Applied Optics, Vol. 16, No. 7, pp. 1857-1860 (1977).

[50] F.H. Rolt, Gauges and Fine Measurements . Macmillan and Co., Limited, 1929.

[51] Beers, J.S. and Kang B. Lee, "Interferometric measurement of length scales at the

National Bureau of Standards," Precision Engineering, Vol. 4, No. 4, 1982.

[52] Beers, J.S., "Length Scale Measurement Procedures at the National Bureau of

Standards," NBSIR 87-3625, 1987.

[53] Beers, John S. and William B. Penzes, "NIST Length Scale Interferometer

Measurement Assurance," NISTIR 4998, 1992.

123
APPENDIX A. Drift Eliminating Designs for Non-simultaneous Comparison

Calibrations

Introduction

The sources of variation in measurements are numerous. Some of the sources are truly

random noise, 1/f noise in electronic circuits for example. Usually the "noise" of a measurement is actually due to uncontrolled systematic effects such as instability of the

mechanical setup or variations in the conditions or procedures of the test. Many of these

variations are random in the sense that they are describable by a normal distribution. Like

true noise in the measurement system, the effects can be reduced by making additional

measurements.

Another source of serious problems, which is not random in nature is drift in the instrument

readings. This effect cannot be minimized by additional measurement because it is not

generally pseudo-random, but a nearly monotonic shift in the readings. In dimensional

metrology the most import cause of drift is thermal changes in the equipment during the

test. In this paper we will demonstrate techniques to address this problem of instrument

drift.

A simple example of the techniques for eliminating the effects of drift by looking at two

different ways of comparing 2 gage blocks, one standard (A) and one unknown (B).

Now let us suppose we make the measurements regularly spaced in time, 1 time unit

apart, and there is an instrumental drift of A. The actual readings (yi) from scheme 1 are:

Scheme 1: A B A B

Scheme 2: ABBA

y2

y4

B + A

A + 2A

B + 3A

A (A.

(A.

(A,

(A

1a)

1b)

1c)

Id)

Solving for B in terms of A we get:

124
B =-A - - {Y1 + Y3 - Y2 - Y4) - A

2

0

which depends on the drift rate A.

Now look at scheme 2. Under the identical conditions the readings are:

V4

= A (A.3a)

(A.3b)

(A. 3c)

(A.3d)

= B + A

= B + 2A

= A + 3A

Here we see that if we add the second and third readings and subtract the first and fourth

readings we find that the A drops out:

Thus if the drift rate is constant - a fair approximation for most measurements if the time

is properly restricted - the analysis both eliminates the drift and supplies a numerical

approximation of the drift rate.

The calibration of a small number of "unknown" objects relative to one or two reference

standards involves determining differences among the group of objects. Instrumental drift,

due most often to temperature effects, can bias both the values assigned to the objects

and the estimate of the effect of random errors. This appendix presents schedules for

sequential measurements of differences that eliminate the bias from these sources and at the same time gives estimates of the magnitude of these extraneous components.

Previous works have [A1,A2] discussed schemes which eliminate the effects of drift for

simultaneous comparisons of objects. For these types of measurements the difference

between two objects is determined at one instant of time. Examples of these types of

measurements are comparisons of masses with a double pan balance, comparison of

standard voltage cells, and thermometers which are all placed in the same thermalizing

environment. Many comparisons, especially those in dimensional metrology, cannot be

done simultaneously. For example, using a gage block comparator, the standard, control

(check standard) and test blocks are moved one at a time under the measurement stylus.

For these comparisons each measurement is made at a different time. Schemes which

assume simultaneous measurements will, in fact, eliminate the drift from the analysis of

B = A - - i Y1 + Y4 - Y2 ~ Y3)

2

{A.

125
the test objects but will produce a measurement variance which is drift dependent and an

erroneous value for the drift, A.

In these calibration designs only differences between items are measured so that unless

one or more of them are standards for which values are known, one cannot assign values

for the remaining "unknown" items. Algebraically, one has a system of equations that is not of full rank and needs the value for one item or the sum of several items as the

restraint to lead to a unique solution. The least squares method used in solving these

equations has been presented [A3] and refined [A4] in the literature and will not be

repeated in detail here. The analyses presented of particular measurement designs

presented later in this paper conform to the method and notation presented in detail by

Hughes [A3].

The schemes used as examples in this paper are those currently used at NIST for gage

block comparisons. In our calibrations a control (check standard) is always used to generate data for our measurement assurance plan [A5]. It is not necessary, however, to use a control in every measurement and the schemes presented can be used with any of

the objects as the standard and the rest as unknowns. A number of schemes of various

numbers of unknowns and measurements is presented in the appendix.

Calibration Designs

The term calibration design has been applied to experiments where only differences

between nominally equal objects or groups of objects can be measured. Perhaps the

simplest such experiment consists in measuring the differences between the two objects

of the n(n-1) distinct pairings that can be formed from n objects. If the order is unimportant, X compared to Y is the negative of Y compared to X, there are only n(n-1 )/2

distinct pairings. Of course only 1 measurement per unknown is needed to determine the

unknown, but many more measurements are generally taken for statistical reasons.

Ordinarily the order in which these measurements are made is of no consequence.

However, when the response of the comparator is time dependent, attention to the order

is important if one wished to minimize the effect of these changes.

When this effect can be adequately represented by a linear drift, it is possible to balance

out the effect by proper ordering of the observations. The drift can be represented by the

series, ...-3,-2,-1,0, 1 , 2 , 3 , ... if there are an odd number of comparisons and by

... -5/2 , -3/2 ,-1/2,1/2,3/2,5/2,... if there are an even number of comparisons.

As and example let us take n=3. If we make all possible n(n-1 )=6 comparisons we get a scheme like that below, denoting the three objects by A, B, C.

126
Observation Measurement

ITIi m2

^3

m4

ms

nrig

my

nrig

mg

m^o

m11 m12

A- 11/2A

B - 9/2A

C - 7/2A

A - 5/2A

B - 3/ 2A

C- 1/2 A

A + 1/2 A

C + 3/2A

B + 5/2A

A + 7/2A

C + 9/2A

B + 11/2A

(A.5)

If we analyze these measurements by pairs, in analogy to the weighing designs of

Cameron we see that:

A B C A

Vi = mi - m2 = A-B-A +1 -1

y2 = mg - m4 = C-A-A -1 +1

y3 = nis - nie = B-C-A +1 -1

y4 = ^7 - mg = A-C-A +1 -1

y5 = mg - mio = B-A-A -1 +1

y6 = mii -mi2 = C-B-A -1 +1

(A.6)

The notation used here, the plus and minus signs, indicate the items entering into the

difference measurement. Thus, yj is a measurement of the difference between object C

and object A.

Note the difference between the above table and that of simultaneous comparisons in reference 2 is that the drift column is constant. It is simple to see by inspection that the

drift is balanced out since each object has two (+) and two (-) measurements and the drift column is constant. By extension, the effects of linear drift is eliminated in all complete

block measurement schemes (those for which all objects are measured in all possible n(n127
1) combinations).

Although all schemes in which each object has equal numbers of (+) and (-) measurements is drift eliminating, there are practical criteria which must be met for the

scheme to work. First, the actual drift must be linear. For dimensional measurements the

instrument drift is usually due to changes in temperature. The usefulness of drift eliminating designs depends on the stability of the thermal environment and the accuracy

required in the calibration. In the NIST gage block lab the environment is stable enough

that the drift is linear at the 0.1 microinch level over periods of 5 to 10 minutes. Our

comparison plans are chosen so that the measurements can be made in this period.

Secondly, each measurement must be made in about the same amount of time so that the

measurements are made at fairly regular intervals. In a completely automated system this

is simple, but with human operators there is a natural tendency to make measurements

simpler and quicker is the opportunity presents itself. For example, if the scheme has a

single block measured two or more times in succession it is tempting to measure the object

without removing it from the apparatus, placing it in its normal resting position, and

returning it to the apparatus for the next measurement.

Finally, the measurements of each block are spread as evenly as possible across the

design. Suppose in the scheme above where each block is measured 4 times block A is measured as the first measurement of y^, yj, ys, and y^. There is a tendency to leave block

A near the measuring point rather than its normal resting position because it is used so

often in the first part of the scheme. This allows block A to have a different thermal

handling than the other blocks which can result in a thermal drift which is not the same as

the other blocks.

Restraints

Since all of the measurements made in a calibration are relative comparisons, at least one

value must be known to solve the system of equations. In the design of the last section,

for example, if one has a single standard and two unknowns, the standard can be assigned

to any one of the letters. (The same would be true of three standards and one unknown.)

If there are two standards and one unknown, the choice of which pair of letters to assign

for the standards is important in terms of minimizing the uncertainty in the unknown.

For full block designs (all possible comparisons are made) there is no difference which

label is used for the standards or unknowns. For incomplete block designs the uncertainty

of the results can depend on which letter the standard and unknowns are assigned. In these cases the customer blocks are assigned to minimize their variance and allow the

larger variance for the measurement of the extra master (control).

128
This asymmetry occurs because every possible comparison between the four items has

not been measured. For 4 objects there are 12 possible intercomparisons. If an 8

measurement scheme is used all three unknowns cannot be compared directly to the

standard the same number of times. For example, two unknowns can be compared

directly with the standard twice, but the other unknown will have no direct comparisons.

This indirect comparison to the standard results in a slightly larger variance for the block

compared indirectly. Complete block plans, which compare each block to every other

block equal number of times, has no such asymmetry, and thus removes any restriction

on the measurement position of the control.

Example: 4 block, 12 comparison, Single Restraint Design for NIST Gage Block

Calibration

The gage block comparison scheme put into operation in 1989 consists of two standards

blocks, denoted S and C, and two customer blocks to be calibrated, denoted X and Y. In order to decrease the random error of the comparison process a new scheme was devised

consisting of all 12 possible comparisons between the four blocks. Because of continuing

problems making penetration corrections, the scheme was designed to use either the S

or C block as the restraint and the difference (S-C) as the control parameter. The S blocks

are all steel, and are used as the restraint for all steel customer blocks. The C blocks are chrome carbide, and are used as the restraint for chrome and tungsten carbide blocks.

The difference (S-C) is independent of the choice of restraint.

We chose a complete block scheme which assures that the design is drift eliminating and

the blocks can be assigned to the letters of the design arbitrarily. We chose (S-C) as the

first comparison. Since there are a large number of ways to arrange the 1 2 measurements

for a complete block design, we added two restrictions as a guide to choose a "best"

design.

1. Since the blocks are measured one at a time, it was decided to avoid schemes

which measured the same block two or more times consecutively. In the scheme

presented earlier blocks D, A and C are all measured twice consecutively. There is a

great temptation to not remove and replace the blocks under these conditions, and the

scheme assumes that each measurement is made with the same motion and are evenly spaced in time. This repetition threatens both these assumptions.

2. We decided that schemes in which the six measurements of each block were

spread out as evenly as possible in time would be less likely to be affected by small

non-linearities in the drift. For example, some schemes had one block finished its 6

measurements by the 8th comparison, leaving the final 1/3 of the comparisons with no

sampling of that block.

129
The new scheme is as follows:

S C X Y

yi = s-c 1 -1 0 0

y2 = Y-S -1 0 0 1 V2 = X-Y 0 0 1 -1

Va = C-S -1 1 0 0

Vs = C -X 0 1 -1 0

ye = Y-C X = 0 -1 0 1 y? = S-X 1 0 -1 0

Vs = C-Y 0 1 0 1 ys = S-Y 1 0 0 -1

yio = X-C 0 -1 1 0

yii = X-S -1 0 1 0

yi2 = Y-X 0 0 -1 1

When the S block is the restraint the matrix equation to solve is

-1 x*x a X'Y

a' 0 L

[a*] = [ 1 0 0 0]

6 -2 -2 -2 0 1 -2 6 -2 -2 0 0

2 -2 6 -2 0 0

-2 -2 -2 6 0 0

0 0 0 0 12 0

1 0 0 0 0 0

[Bn] (1/24)

0 0 •0 0 0 24

0 6 3 3 0 24

0 3 6 3 0 24

' 0 3 3 6 0 24

0 0 0 0 2 0

24 24 24 24 0 0

130
Using S as the restraint, the solution to the equations is:

S = L (A.12a)

C = (1/8) (-2yi + y2 + 2y4 + Vs - Ye - Yr + - Vs - Yio + Yii) + L (A.12b)

X = (1/8) i-y, + Y2 + Ya + Y4 - Ys - 2Y7 - Ys + Yio + 2y,, - y,^) + L (A.12c)

Y = (1/8) i-y, + 2y2 - Y3 + Y4 + Ye - Y/ - Ys - Syg + Yn + Yis) + L (A.12d)

A = (-2)(Yi + Y2 + Y3 + Y4 + Ys + Ye + Y7 + Ys + Ys + Yio + Y11 + Y12) (A.12e)

The deviations, d^, dj, d^j can be determined from the equations above, or can be

calculated directly using matrix methods. For example,

These deviations provide the information needed to obtain a value, s, which is the

experiment's value for the short term process standard deviation, or within standard

deviation o^.

n m

= E (i^i - E (A^x X,^ ))^} / (n - m + 1) (^.13

/ = 0 /- = 0

The number of degrees of freedom results from taking the number of observations (n=12)

less the number of unknowns (m=5; S, C, X, Y, A ), and then adding one for the restraint.

Because of the complete block structure (all 12 possible combinations measured) all of the

standard deviations are the same:

{AAA

Process Control: F - Test

Continued monitoring of the measurement process is required to assure that predictions

131
based on the accepted values for process parameters are still valid. For gage block

calibration at NIST, the process is monitored for precision by comparison of the observed

standard deviation, a^, to the average of previous values. For this purpose the value of

is recorded for every calibration done, and is periodically analyzed to provide an

updated value of the accepted process for each gage block size.

The comparison is made using the F distribution, which governs the comparison of

variances. The ratio of the variances s^ (derived from the model fit to each calibration) and

o^'^ derived from the history is compared to the critical value F(8,oo,a), which is the a

probability point of the F distribution for degrees of freedom 8 and oo. For calibrations at

NIST, a is chosen as 0.01 to give F(8,oo,.01) = 2.5.

If this condition is violated the calibration fails, and is repeated. If the calibration fails more

than once the test blocks are re-inspected and the instrument checked and recalibrated.

All calibrations, pass or fail, are entered into the history file.

Process Control: T - Test

At NIST a control measurement is made with each calibration by using two known master

blocks in each calibration. One of the master blocks is steel and the other chrome carbide.

When a customer block is steel the steel master is used as the restraint, and when a customer block is carbide, the carbide master is used as the restraint. The use of a control

measurement for calibrations is necessary in order to provide assurance of the continuing

accuracy of the measurements. The F-test, while providing some process control, only

attempts to control the repeatability of the process, not the accuracy. The use of a control

is also the easiest method to find the long term variability of the measurement process.

While the use of a control in each calibration is not absolutely necessary, the practice is highly recommended. There are systems which use intermittent tests, for example

measurements of a control set once a week. This is a good strategy for automated

systems because the chances of block to block operator errors is small. For manual

measurements the process variability, and of course the occurrence of operator error is much higher.

The check for systematic error is given by comparison of the observed value of the

difference between the standard and control blocks. If S is the standard it becomes the

restraint, and if A is used as the control (S-A) is the control parameter for the calibration.

132
This observed control is recorded for every calibration, and is used to periodically used to update the accepted, or average value, of the control. The process control step involves

the comparison of the observed value of the control to the accepted (historical) value. The

comparison is made using the Student t-distribution.

The control test demands that the observed difference between the control and its accepted value be less than 3 times the accepted long term standard deviation, a,, of the

calibration process. This value of the t-distribution implies that a good calibration will not

be rejected with a confidence level of 99.7%.

T = — < 3 (>A.16)

The value of a, is obtained directly from the sequence of values of (S-A) arising in regular

calibrations. The recorded (S-C) values are fit to a straight line, and the square root of the

variance of the deviations from this line is used as the total standard deviation, (aj).

If both the precision (F-test) and accuracy (t-test) criteria are satisfied, the process is regarded as being "in control" and values for the unknown, X, and its associated

uncertainty are regarded as valid. Failure on either criterion is an "out-of-control" signal

and the measurements are repeated.

The value for drift serves as an indicator of possible trouble if it changes markedly from its usual range of values. However, because any linear drift is balanced out, a change in the

value does not of itself invalidate the result.

Conclusion

The choice of the order of comparisons is an important facet of calibrations, in particular

if chosen properly the comparison scheme can be made immune to linear drifts in the

measurement equipment. The idea of making a measurement scheme robust is a

powerful one. What is needed to implement the idea is an understanding of the sources

of variability in the measurement system. While such a study is sometimes difficult and

time consuming because of the lack of reference material about many fields of metrology,

the NIST experience has been that such efforts are rewarded with measurement

procedures which, for about the same amount of effort, produce higher accuracy.

133
References

[A1] J.M. Cameron, M.C. Croarkin, and R.C. Raybold, "Designs for the Calibration of

Standards of Mass," NBS Technical Note 952, 1977.

[A2] Cameron, J.M. and G.E Hailes. "Designs for the Calibration of Small Groups of

Standards in the Presence of Drift," NBS Technical Note 844, 1974.

[A3] CO. Hughes, III and H.A. Musk. "A Least Squares Method for Analysis of Pair

Comparison Measurements," Metrologia, Volume 8, pp. 109-1 13 (1972).

[A4] C. Croarkin, "An Extended Error Model for Comparison Calibration," Metrologia,

Volume 26, pp. 1 07-1 1 3, 1 989.

[A5] C. Croarkin, "Measurement Assurance Programs, Part II: Development and

Implementation," NBS Special Publication 676-11, 1984.

134
A Selection of Other Drift Eliminating Designs

The following designs can be used with or without a control block. The standard block is denoted S, and the unknown blocks A, B, C, etc. If a check standard block is used it can

be assigned to any of the unknown block positions. The name of the design is simply the

number of blocks in the design and the total number of comparisons made.

3-6 Design 3-9 Design 4-8 Design

(One master block,

2 unknowns

4 measurements each)

(One master block,

2 unknowns,

6 measurements each)

(One master block,

3 unknowns,

4 measurements each)

yi = S-A Yi S-A Yi = S-A

y2 = B-S y2 B-A Y2 = B-C

= A-B S-B Y3 = C-S

y4 = A-S y4 A-S Y4 = A- B

= B-A B-S Ys = A-S

Ve = S-B Ye A-B Ye = C-B

Y7 A-S Y7 = S-C

Ys B-A Ys = B-A

Yg S-B

4-12 Design

(One master block,

3 unknowns

6 measurements each)

Yi S -A

Y2 C -S

Y3 B -0

Y4 A -S

Ys A - B

Ye C -A

Y7 S -B

Ys A -C

Ys S -0

Yio B -A

Yii B -S

Yl2 C -B

5-10 Design

(One master block,

4 unknowns,

4 measurements each)

Yi S -A

Y2 D -C

Y3 S -B

Y4 D -A

Ys C -B

Ye A -C

Yt B -S

Ys B -D

Y9 C -S

Yio A -D

6-12 Design

(One master block,

5 unknowns,

4 measurements each)

Yi S -A

Y2 D -0

Ys E - B

Y4 E - D

Ys C -A

Ye B -C

Yi S -E

Ys A -D

Yg A -B

Yio D -S

Yii B -E

Yl2 C -S

135
7-14 Design

(One master block,

6 unknowns

4 measurements each)

8-16 Design

(One master block,

7 unknowns,

4 measurements each)

9-18 Design

(One master block,

7 unknowns,

4 measurements each)

yi = S - A

y2 = E-C

y3 = B-D

y4 = A - F

Vs = S - E

Ve = D-B

V/ = A - C

Vs = B-f

yg = D-E

Yio = F - S

yii = E-A

y,2= C - B

yi3 = c - s

yu= F - D

— s -A

y2 — E - G

— F - C

y4 — D - S

Vs B - E

Ve G -F

Vy C -B

Vs E -A

F -D

Vio C -S

Vii A -G

yi2 D -B

yi3 C -S

yi4 G -C

yi5 B -D

yi6 A -F

yi = s -A

Vi = H - F

y3 = A - B

y4 = D - C

ys = E - G

ye = C -A

yy = B - F

ys = G -H

y9 = D -S

yio = C -E

yu= H -S

Vn = G -D

V^3 = C -S

Vu = A -C

yi5 = F -D

yi6 = S -H

yi7 = E -B

yi8 = F -G

136
10-20 Design

(One master block,

9 unknowns

4 measurements each)

11-22 Design

(One master block,

10 unknowns,

4 measurements each)

yi = s - A

F - G

= 1 - C

y4 = D

1

- E

Vs = A - H

Ve = B - C

y/ = G - H

Va = 1 - S

yg = E - F

y^o = H - 1

yii = D - F

Vm = A - B

y^3 = C - 1

y^4 = H - E

V^5 = B -G

yi6 = S -D

y^7 = F - B

yi8 = C - D

yi9 = G -S

Vio = E -A

= S-A

y2 = D-E

y3 = G-l

y4 = C - H

Vs = A-B

Ye = l-J

yy = H - F

ya = D-s

yg = B-C

yio= s-E

yii = A-G

yi2 = F - B

yi3= E-F

yi4= J -A

y,,= C - D

yi6 = H - J yi7= F-G

yi8 = I - s

yi9= B-H

Vso = G - D

y2i = J - c

y22= E -

1

137
Appendix B: Wringing Films

In recent years it has been found possible to polish plane surfaces of

hardened steel to a degree of accuracy which had previously been

approached only in the finest optical work, and to produce steel blocks in the

form of end gauges which can be made to adhere or "wring" together in combinations. Considerable interest has been aroused by the fact that these

blocks will often cling together with such tenacity that a far greater force must

be employed to separate them than would be required if the adhesion were

solely due to atmospheric pressure. It is proposed in this paper to examine

the various causes which produce this adhesion: firstly, showing that by far

the greater portion of the effect is due to the presence of a liquid film

between the faces of the steel; and , secondly, endeavoring to account for

the force which can be resisted by such a film.

Thus began the article 'The Adherence of Flat Surfaces" by H.M. Budgett in 1912 [B1], the

first scientific attack on the problem of gage block wringing films. Unfortunately for those

wishing tidy solutions, the field has not progressed much since 1 91 2. The work since then

has, of course, added much to our qualitative understanding of various phenomena

associated with wringing, but there is still no clear quantitative or predictive model of

wringing film thickness or its stability in time. In this appendix we will only describe some

properties of wringing films, and make recommendations about strategies to minimize

problems due to film variations.

Physics of Wringing Films

What causes wrung gage blocks to stick together? The earliest conjectures were that

sliding blocks together squeezed the air out, creating a vacuum. This view was shown to

be wrong as early as 1912 by Budgett [B1] but still manages to creep into even modern

text books [B2]. It is probable that wringing is due to a number of forces, the relative

strengths of which depend on the exact nature of the block surface and the liquid adhering

to the surface. The known facts about wringing are summarized below.

1 . The force of adhesion between blocks can be up to 300 N (75 lb). The force of the

atmosphere, 101 KPa (14 psi), is much weaker than an average wring, and studies

have shown that there is no significant vacuum between the blocks.

2. There is some metal-metal contact between the blocks, although too small for a

significant metallic bond to form. Wrung gage blocks show an electrical resistance of

about 0.003O [B3] which corresponds to an area of contact of 1 0'^ cm^.

138
3. The average wringing film thickness depends on the fluid and surface finish as well as the amount of time blocks are left wrung together. Generally the thickness is about

10 nm (0.4 pin), but some wrings will be over 25 nm (1 pin) and some less than 0. (Yes,

less than zero.) [B3,B4.B5,B6]

4. The fluid between blocks seems to provide much of the cohesive force No matter

how a block is cleaned, there will be some small amount of adsorbed water vapor. The

normal wringing procedure, of course, adds minute amounts of grease which allows a more consistent wringing force. The force exerted by the fluid is of two types. Fluid,

trapped in the very small space between blocks, has internal bonds which resist being

pulled apart. The fluid also has a surface tension which tends to pull blocks together.

Both of these forces are large enough to provide the observed adhesion of gage

blocks.

5. The thickness of the wringing film is not stable, but evolves over time. First changes

are due to thermal relaxation, since some heat is transferred from the technician's

hands during wringing. Later, after the blocks have come to thermal equilibrium, the

wring will still change slowly. Over a period of days a wring can grow, shrink or even

complicated combinations of growth and shrinkage [B5,B6].

6. As a new block is wrung repeatedly the film thickness tends to shrink. This is due

to mechanical wear of the high points of the gage block surface [B5,B6].

7. As blocks become worn and scratched the wringing process becomes more erratic,

until they will not wring well. At this point the blocks should be retired.

There may never be a definitive physical description for gage block wringing. Besides the

papers mentioned above, which span 60 years, there was a large project at the National

Bureau of Standards during the 1960's. This program studied wringing films by a number

of means, including ellipsometry [B8]. The results were very much in line with the 7 points

given above, i.e., on a practical level we can describe the length properties of wringing

films but lack a deeper understanding of the physics involved in the process.

Fortunately, standards writers have understood this problem and included the length of

one wringing film in the defined block length. This relieves us of determining the film

thickness separately since it is automatically included whenever the block is measured

interferometrically. There is some uncertainty left for blocks which are measured by

mechanical comparison, since the length of the master block wringing film is assumed to be the same as the unknown block. This uncertainty is probably less than 5 nm ( .2 pin)

for blocks in good condition.

139
REFERENCES

[B1 ] "H.M. Budgett, 'The Adherence of Flat Surfaces," Proceedings of the Royal Society,

Vol. 86A, pp. 25-36 (1912).

[82] D.M. Anthony, Engineering Metrology. Peragamon Press, New York, N.Y. (1986).

[83] C.F. Bruce and B.S. Thornton, "Adhesion and Contact Error in Length Metrology,"

Journal of Applied Physics, Vol. 17, No.8 pp. 853-858 (1956).

[84] C.G. Peters and H.S. Boyd, "Interference methods for standardizing and testing

precision gage blocks," Scientific Papers of the Bureau of Standards, Vol. 17, p. 691

(1922).

[85] F.H. Rolt and H. Barrell, "Contact of Flat Surfaces," Proc. of the Royal Society

(London), Vol. 106A, pp. 401-425 (1927).

[86] G.J. Siddall and P.C. Willey, "Flat-surface wringing and contact error variability," J. of Physics D: Applied Physics, Vol. 3, pp. 8-28 (1970).

[87] J.M. Path, "Determination of the Total Phase, Basic Plus Surface Finish Effect, for

Gage Blocks," National Bureau of Standards Report 9819 (1967).

140
Appendix C. Phase Shifts in Gage block Interferometry

When light reflects from a surface there is a shift in phase; the reflected light appears to have reflected from a plane below the mechanical surface, as shown in figure C. Many

years ago, when surface finishes were rougher, the main component of phase differences

between metal surfaces was due to surface finish. Current manufacturing practice has

reduced this component to a small part of the phase shift. There are still significant phase

shift differences between blocks of different manufacturers and different materials.

Nonconducting materials, such as quartz and glass have phase shifts of 180°. Electrical

conductors will have phase shifts less than 180° by 10° to 30°. The theoretical foundation

for this shift is given in Ditchburn [CI]. From an understanding of electromagnetic behavior

of electrons near the metal surface the phase shift could be calculated. There have been

such calculations using the Drude model for electrons. Unfortunately this classical model

of electrons in metals does not describe electron behavior at very high frequencies, and

is only useful for calculations of phase shifts for wavelengths in the infrared and beyond.

There have been no successful calculations for phase shifts at higher (visible light)

frequencies, and there is no currently available theory to predict phase shifts from any

other measurable attributes of a metal.

Given this fact, phase shifts must be measured indirectly. There is one traditional method

to measure phase shift, the "slave block" method [C2]. In this method an auxiliary block,

called the slave block, is used to help find the phase shift difference between a block and

a platen. The method consists of two steps, shown schematically in figure C2 and C3.

„ block

L

ttest L

I yech

J

1L . wrin

platen

Figure C. The interferometric length, Ljest, includes the mechanical length, the

wringing film thickness and the phase change at each surface.

141
step 1 . The test and slave blocks are wrung down to the same platen and measured

independently. The two lengths measured consist of the mechanical length of the block,

the wringing film and the phase changes at the top of the block and platen, as in C.

Slave

Slave

Test Test

Step 1 Step 2

Figure C2. Measurements for determining the phase shift difference

between a block and platen by the slave block method.

The general formula for the measured length of a wrung block is:

^est ^mechanical ^wring ^platen phase ~ ^block phase

For the test and slave blocks the formulas are:

Ltest = U + Lt,w + (4>platen-4>test)

-slave w + (4^p|aten"4^slave)

(C.1)

(C.2)

(C.3)

Step 2. Either the slave block or both blocks are taken off the platen, cleaned, and

rewrung as a stack on the platen. The new length measured is: -

Uest + slave ~ U Ls U w Lg.w (4^platen~0slave) (^-4)

If this result is subtracted from the sum of the two previous measurements we find that:

142
Uest + slave " ^test " ^s\a\/e (4^test"^platen) (C.5)

The weakness of this method is the uncertainty of the measurements. The uncertainty of

one measurement of a wrung gage block is about 30 nm (1.2 pin). Since the phase

measurement depends on the difference between 3 measurements, the phase

measurement is uncertain to about /3 x uncertainty of one measurement, or about 50 nm.

Since the phase difference between block and platen is generally about 20 nm, the

uncertainty is larger than the effect. To reduce the uncertainty a large number of

measurements must be made, generally between 50 and 100. This is, of course, very time

consuming.

Table C shows a typical result of the slave block method for blocks wrung to quartz. The

uncertainty is, unfortunately large; we estimate the standard uncertainty (95% confidence

level) to be 8 nm.

Table C

Corrections to Nominal Lengths and Phase in Nanometers

Manufacturer Material Measured Phase

(nanometers)

DoAII steel 25

C.E. Johansson steel 50

Matrix steel 50

Pratt & Whitney steel 50

Webber steel 55

Webber chrome carbide 30

Mitutoyo steel 40

We have made attempts to measure the phase for the ceramic (zirconia) gage blocks but

have had problems with residual stresses in the block stack. We have measured the

phase to be 35 nm, but the geometry leads us to suspect that the measured phase might

be a distortion of the block rather than the interferometric phase.

There have been a number of attempts to measure phase by other means [C3], but none

have been particularly successful. One method, using Newton's rings formed by a

spherical quartz surface in contact with a metal surface is under investigation [C4]. With

computer vision systems to analyze fringe patterns, this method may become useful in the

future.

As a rule, blocks from a single manufacturer have the same phase since the material and

lapping method (and thus surface finishes) are the same for all blocks. Blocks from the

143
same manufacturer but with different materials, or tine same material from different

manufacturers generally have different phases. One way to reduce phase problems is to

restrict your master blocks to one manufacturer and one material.

Another way to reduce effects of phase change is to use blocks and platens of the same

material and manufacturer. While this seems simple in principle, it is not as easy in practice. Most manufacturers of gage blocks do not make gage block platens, and even

when they do, lapping procedures for large platens are not always the same as procedures

for blocks. At NIST we currently do not have any block/platen combinations which have

matched phase.

REFERENCES

[C1] R.W. Ditchburn, Light. Interscience Publications, 2nd. edition, 1963.

[C2] Miller, L., Engineering Dimensional Metroloav. Edward Arnold, Ltd., London, 1962.

[C3] J.M. Path, "Determination of the Total Phase, Basic Plus Surface Finish Effect, for

Gage Blocks," National Bureau of Standards Report 9819 (1967).

[C4] K.G. Birch and R. Li, "A Practical Determination of the Phase Change at Reflection,"

IMEKO Symposium on Laser Applications in Precision Measurement, pp. 5-25,

Budapest, 1986.

144
Appendix D. Deformation Corrections

Whenever two materials are forced into contact there is an elastic deformation. For gage

block comparators the contact is between a spherical probe and the flat gage block

surface. The following formula, from a CSIRO Technical Note [D1] has been found to be

identical to the earlier NBS nomographs [D2]. The nomographs were developed to avoid

the cumbersome calculations needed for deformation corrections. However,

microcomputers have made the formula easier to use than nomographs, and we therefore

give only the deformation formula.

2 1^1

a ^ 2.231 {V^ + V^)^ D ^ (D.1)

a is the deformation in millimeters

P is the force in NEWTONS,

D is the diameter of the probe in MILLIMETERS,

Vi is a number characteristic of the probe or block material from the table below:

Material V(10-^)

Steel 139

Chrome Carbide 86

Tungsten Carbide 40

Ceramic 139

Diamond 43

Example 1: A steel gage block is measured with a two point probe system each with a

spherical diamond contact of 6 mm diameter. The top force is 1 N and the bottom force

is 1/3 N.

Top Contact: D = 6 mm

F = 1 N

Vi= 139x10'^ steel block

V2= 43 X 10"^ diamond sphere

145
a = 2.231 * (1f3 * (139 X 10-' + 43 X lO'^f^ * (6)-^'^ (D.2)

= 0.000181 mm = 0.181 |jm

Bottom Contact: D = 6 mm

F = 1/3 N

Vi= 139x10-' steel block

43x10"' diamond sphere

a = 2.231 * {M3f'^ * (139 x 10"' + 43 x lO'Y^ * (gyi/3

3^

= 0.000087 mm = 0.087 |jm

Thus the total deformation is 0.00027 mm or 0.27 pm.

Example 2: Same apparatus but measuring a chrome carbide gage block:

Top Contact: D = 6 mm - F = 1 N

Vi= 86x10 ' chrome carbide block

\/2= 43 X 10 ' diamond sphere

a = 2.231 *

(1 f'^

* (86 x 10"' + 43 x 10"'^' * (6)-^'^ (D.4)

= 0.000145 mm = 0.145 pm

Bottom Contact: D = 6 mm

F = 1/3 N

\/i= 86x10-' chrome carbide block

\/2= 43 X 10"' diamond sphere

a = 2.231 * (1/3f^ * (86 x 10'' + 43 x 10"'^^ * {lo)-"^ (D.5)

= 0.000070 mm = 0.070 Mm

Thus the total deformation is 0.00022 mm or 0.22 |jm.

If we were to use a steel master block to calibrate a chrome carbide block under the above

146
conditions we see that the penetration correction for steel and chrome carbide blocks differ

substantially. The total length correction of the calibr -ted block is given by;

Lu = M, - M3 + L3 + {a, - aj (D.6)

where is the calibrated length of the unknown block, is the measured value of the

unknown block, Mg is the measured value of the reference standard, and and are penetration corrections of the standard and unknown respectively, and 05-0^=0. 05 pm

(2.0 pin).

REFERENCES

[D1] Puttock, M.J. and E.G. Thwaite, "Elastic Compression of Spheres and Cylinders at Point and Line Contact," Natoinal Standards Laboratory Technical Paper No. 25,

CSIRO, 1969.

[D2] Beers, J.S. and J.E. Taylor. "Contact deformation in Gage Block Comparisons,"

NBS Technical Note 962, 1978.

Il;

147
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