H

RFNC

NBS TECHNICAL NOTE 769

1 75107 -fe^ octi 41975

A Rigorous Correction Procedure

for Quantitative Electron

Probe Microanalysis (COR 2)
NATIONAL BUREAU OF STANDARDS

The National Bureau of Standards 1 was established by an act of Congress March 3, 1901. The Bureau's overall goal is to strengthen and advance the Nation's science and technology

and facilitate their effective application for public benefit. To this end, the Bureau conducts

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THE INSTITUTE FOR BASIC STANDARDS provides the central basis within the United

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* Part of the Center for Building Technology.
Bureau of Standards

8 1974

53

A Rigorous Correction Procedure for

Quantitative Electron Probe Microanalysis (COR 2)

u.*<

c

Jean He'noc

Centre National d Etudea de Telecommunication

196 Rue de Paris

92-Bagneux, France

and

Kurt F. J. Heinrich and Robert L. Myklebust

Analytical Chemistry Division

Institute for Materials Research

National Bureau of Standards

Washington, D.C. 20234

v WW \

*fli

U.S. DEPARTMENT OF COMMERCE, Frederick B. Dent, Secretary

NATIONAL BUREAU OF STANDARDS, Richard W. Roberts, Director

Issued August 1973
National Bureau of Standards Technical Note 769

Nat. Bur. Stand. (U.S.), Tech. Note 769, 132 pages (Aug. 1973) CODEN: NBTNAE

For sale by the Superintendent of Documents, U.S. Government Printing Office, Washington, D.C. 20402 (Order by SD Catalog No. 03.46:769). Price $2.10
TABLE OF CONTENTS

PAGE

Abstract . 1

1. Foreword 2

2. Specifications and Theoretical Foundations for

COR 2 4

3. Structure of the Program 6

4. Explanation of the Program u

4.1 MAIN Program 11

4.2 SUBROUTINE C0CA (J0,JM,JJA) 17

4.3 FUNCTION C0 ( J0, JM, JJA, FC , FQC) 22

4.4 FUNCTIONS F (X, J0 , JM, JJA) and FT(X) 23

4.5 FUNCTION AC(CL,J) 23

4.6 FUNCTION PRA( J0 , JM, LX , KX, JX,WAA) 23

4.7 FUNCTION RB(U0,Z) 23

4.8 FUNCTION FQI (J0 , JM,LX , KX, JX,WAA) 23

4.9 FUNCTION FCA ( J0, JM, II , JJA, FCA1 , FCA0) ... 23

4.10 SUBROUTINE CALI (J0,JM, JJA, AIT , SWITCH

,

FQCF,FQCA) 24

4.11 FUNCTION EI(X) 24

4.12 SUBROUTINE AINP (C , I J , I IMAX, SD) 24

5. Statements of the Program 25

6. Table of Formats for the Data Input Cards .... 53

7. Outputs ...... 65

8. References 70

APPENDIX 1: NBS Technical Note 521 71

APPENDIX 2: Index of Terms and Symbols 88

iii
PAGE

APPENDIX 3: Permanent Data File 112

LIST OF FIGURES

Figure No. 1: Structure of COR 2 7 Figure No. 2: Flowsheet of Subroutine C0CA 19

IV
A Rigorous Correction Procedure for Quantitative

Electron Probe Microanalysis (COR 2)

Jean Henoc

Centre National des Etudes de Telecommunication

Issy-les -Moulineaux, France

and

Kurt F. J. Heinrich and Robert L. Myklebust

Institute for Materials Research

National Bureau of Standards

Washington, D. C. 20234

This publication describes a rigorous data reduction

procedure for quantitative electron probe microanalysis,

which avoids simplifications present in conventional schemes.

In addition, the program contains a correction for fluorescence

due to the continuum, and separate computation of the fluor-

)escent effects of each exciting primary x-ray line. Constants

characteristic of elements are stored in a permanent data

file, and the constants and parameters needed for the calculation of mass absorption coefficients are contained in

a sub-program. These provisions and decision-making sections

in the development of the fluorescence corrections reduce the

required input considerably.

I Key words : Continuum fluorescence; data reduction; electron

probe microanalysis; fluorescence correction; quantitative

' analysis

.
1 . Foreword

Quantitative electron probe analysis requires elaborate

computer procedures for data reduction, unless standards of

composition close to that of the specimen are available. For

several years in the electron probe laboratory at NBS, a program called MULTI was used. The most recent version of this

program - MULTI8 - was described in a recent Technical Note [1]

The program MULTI8 contains "corrections" for the primary

generation ("atomic number correction") and absorption, and

for fluorescence caused by characteristic lines (according to

Reed [2]). There are, however, no provisions in MULTI8 for

the treatment of the effects of fluorescence due to continuous

radiation. These effects are usually not very severe; tests

on known materials have shown that MULTI8 performs to our

entire satisfaction in analytical practice. Nevertheless,

there is need for a more rigorous and complete program which

permits an estimation of the errors due to the neglect of the

fluorescence caused by the continuum, and to other approximations used in MULTI8 and other simplified programs which are

used in routine work.

The project to obtain such a rigorous procedure, incorporating all known refinements of the theory, was initiated when

J. Henoc, of the C.N.E.T., Issy- les -Moulineaux, France, spent

one year in the electron probe laboratory of NBS as a guest

worker in 1969. Dr. Henoc has previously developed a method

for the calculation of the fluorescence due to the continuum

[3]. He agreed to develop, in consultation with us, a program

which would satisfy our needs.

The scheme of formal integration previously used in

reference [3] was replaced with a numerical integration with

respect to the wavelength of the exciting radiation. This

change was necessary for the use of the expressions for

x-ray mass absorption coefficients previously developed by

one of us [4] . It also became evident that it would be easier

to write an entirely new program for data reduction than to
incorporate the continuum- fluorescence calculation into

existing procedures. The need for locating all x-ray absorption edges in the specimen for the treatment of fluorescence

from the continuum suggested that the program could also

determine which fluorescence excitations due to characteristic

lines should apply. The procedure of Reed [2] was abandoned

in favor of a separate computation of the effects of each

exciting characteristic line ; the intensity of every exciting

line was computed by a full generation calculation. The

resulting program was called COR.

Although we did not originally conceive of COR as a tool

for routine analysis, we find that it can be used advantageously for this purpose wherever batch processing with

a reasonably large computer is acceptable. The fact that

the program calculates all necessary mass absorption coefficients and makes appropriate decisions concerning the applicability of fluorescence calculations is a distinct advantage

over MULTI8 and similar programs, since the data input is considerably simplified. The difference in the results obtained

by COR and by MULT 1 8 is frequently insignificant, except

where fluorescence by the continuum is unusually large.

After a period of testing and use, we have introduced

some minor changes into the program, which are mainly additions

that enable us to use the M-lines for analysis, and an alternative in the input and readout procedures. In order to

distinguish the amended program from the original, we call

the new version COR 2.

The new version was cast into its present form on the

occasion of a second visit by Dr. Henoc in November 1972.

We will describe this program in detail. It is useful to

recall, however, that the main purpose of this publication

As not only the description of a computer program, but also

the presentation of a data evaluation procedure which is

not used elsewhere, to our knowledge, in any form except that

of COR and COR 2.

K. Heinrich

3
2 . S pecifications and Theoretical Foundations for COR 2

COR 2 is written for batch processing in FORTRAN IV,

and occupies about 50K words. At NBS, it is run on the UNIVAC

1108 computer*, and elsewhere, with minor adaptation, on computers such as the IBM 360 and GE 635.

The program permits the loading, in a single operation,

of repeated measurements at one point on the sample, of measurements on several points (up to 100), and of several types

of analyses or specimens ( cases )

.

In its present form, COR 2 permits the presence of up

to 15 chemical elements per point. The limits indicated here

can be changed simply by altering the corresponding dimension

statements

.

The program can calculate either theoretical relative

intensities for specimens of known composition - without

need for iteration - or the composition of analyzed specimens. For the analysis of specimens of unknown composition,

either relative x-ray intensities or uncorrected x-ray intensity measurements are acceptable.

The composition of analyzed specimens is expressed both

in weight fractions and in atomic fractions (in the conventional output) . Each element can be measured at a different

operating voltage. The iteration proceeds until two successive

sets of mass fractions differ, for all elements, by less than

10 (0.001%), or, if convergence is not reached, until completion of 20 iterations. In such case, a warning message

would be printed. In practice, convergence is usually obtained

in three to six iterations; lack of convergence has never been

observed

.

*The computers are identified in order to adequately specify

the use of the program. In no case does such identification

imply recommendation or endorsement by the National Bureau

of Standards, nor does it imply that the equipment identified is necessarily the best available for the purpose.

4
The following options of COR 2 can be used by means

of switches provided in the program.

1. One element can be determined by difference.

2. One element can be determined by stoichiometric

relations (and another by difference, if desired). In such

cases, the results are also printed in terms of binaries

(e.g., as oxides), and the stoichiometric formula is printed.

(See the section concerning output, p. 65)

3. The concentration of one or more elements can be

assumed known, and be maintained constant throughout the

analysis. (e.g., for the analysis of impurities in a stoichiometric compound or in an element.)

4. In any of these variants, the correction for fluorescence due to the continuum can be suppressed.

5. The standards for measured elements may either be

elements or multi-element materials of known composition.

6. A special output particularly useful for diagnostic

purposes is available. This output is the one used in the

original version of COR (output 2 or extended output).

As in other schemes of data reduction for electron probe

microanalysis, the basic process imbedded in COR 2 is the

calculation of a theoretical relative x-ray intensity emitted

from a target of known or postulated composition. We will

call this part of the procedure the "theoretical calculation", although we recognize the presence of empirical adjustment in almost all steps. The designation as "theoretical"

jmerely distinguishes a procedure which contains separate corrections for primary generation, absorption, and fluorescence,

from other approaches which require, at the time of the

analysis, several compound standards to generate an empirical

calibration model. For the analysis of specimens of unknown

(Icompos ition, the "theoretical calculation" is coupled with an

iteration procedure which, starting with a rough estimate of

specimen composition, proceeds to establish the best estimate

by a method of successive approximations.
The theoretical or semi -empirical foundations of the

method which we have followed were described in the NBS

Technical Note 521, reproduced at the end of the present

publication (see Appendix 1). The comments referring to

COR in TN 521 are equally valid for the present version,

COR 2. By modifying the corresponding statements in the

correction functions, variants in the correction models can

be introduced. The pertinent literature references are found

in the Appendix.

3 . Structure of the Program

COR 2 consists of a main program (MAIN), and a series

of subprograms linked to the main program as shown in

Figure 1. The input consists of a series of option specifications, a permanent data file, and the experimental x-ray

intensities or known concentrations. The purpose of the

specifications is to select among the options available for

the execution of the program, such as use of stoichiometric

relations, calculation of an element by difference, use of

known concentration of some components (e.g., in the analysis

of impurities of an element or of a stoichiometric compound)

,

corrections for background and dead-time, and calculation of

theoretical x-ray intensities for a target of known composition

There are two versions of the output which can be

selected by another code in the input. The conventional

output (output 1) has been developed for COR 2. The extended

output (output 2), which was used in the original version of

COR, is particularly useful for the diagnosis of programming

errors or similar problems. References to output, unless

specified otherwise, refer to output 1.

The data file which forms part of the input contains

constants specific for particular elements present in the

specimens. We have added elements to this data file, as required, and keep it stored on a magnetic drum. Other
INPUT $ OUTPUT

AINP

DEADTIME,

BACKGROUND

COCA SELECTS CONSTANTS

CALI ESTABLISHES

SEQUENCE OF PROCEDURES

CORRECTION FUNCTIONS

PARAMETERS

AND INTEGRALS

PRA PRIMARY EMISSION

BACKSCATTER

FACTOR \

FQI PRIMARY ABSORPTION

EI EXPONENTIAL

INTEGRAL

FCA CHARACTER. FLUORESCENCE

XCO CONTINUUM

FLUORESCENCE

X-RAY ABSORP- TION COEFFIC,

NUMERICAL

INTEGRATION

NO ABSORPTION

NUMERICAL

INTEGRATION

Figure 1. Structure of COR 2
systems may require different procedures. On page 114 the

reader will find instructions for adding the constants for

elements not present in our file to the data file.

The input data can either be mass fractions of a known

target, or relative x-ray intensities, corrected for background and dead-time ("k-values") , or uncorrected x-ray

intensities (counts), for line, background and standard.

In the latter case, an input specification directs the main

program to transfer the x-ray intensities to the subprogram

AINP, which performs the corrections for deadtime and background. If repeated measurements were performed at the same

specimen location, AINP also provides statistics of the uncorrected intensities. The corrected relative intensities are

returned to the main program.

The principal functions of MAIN are to perform the

theoretical calculations of the x-ray intensities corresponding

to an estimated target composition, and to refine the estimated

mass fractions by means of the previously mentioned iteration

procedure. The theoretical calculations are based on the

equation:

I ' *f * + ZI ' *f * + I ' *f * k = -£-J2 f—I c c (i)

I ' S

f

S + ZI ' JfJ + I'

S

f

S

P P f f C C

in which k is the background- and deadtime-corrected relative x-ray intensity, I

1 signifies x-ray intensities

generated within the specimen, f denotes the absorption

factors, the subscripts p, f, and c indicate, respectively, primary emission, fluorescent emission excited by

characteristic lines, and fluorescent emission excited by

the continuum. The superscripts * and s denote, respectively, the specimen and the standard. The summation sign

in the characteristic fluorescence terms indicates the possibility of excitation of a line by more than one exciting
line. If there is more than one exciting line, the fluorescence due to each line is calculated separately. The use of

equation (1) replaces that of multiplicative correction factors

according to the ZAF procedure, which is impractical when the

fluorescence due to the continuum is considered (see p. 80,

Appendix 1)

.

In the computation of k, MAIN is assisted by several

subprograms. COCA starts by selecting the constants relevant

to the elements and lines from the data provided by the data

file. With the aid of these constants, MAIN directs CALI

to establish a sequence of procedures for calculating the

parameters necessary for equation (1). In this process, CALI

utilizes the following subprograms:

PRA, to determine the intensities of primary emission

I

1 , (which are also used in the characteristic fluorescence

P'

^correction) , FQI, to determine the primary absorption factors, f , FCA, to determine the characteristic fluorescence

contributions, 1',-f.p, and

CO, to determine the fluorescent intensity due to the

continuum, I ' f . ' c c These subprograms in turn use the following subprograms

for the calculation of needed parameters and integrals:

RB , to determine the backscatter correction function,

EI, to calculate exponential integrals,

AC, to calculate x-ray mass attenuation coefficients,

F, to perform the numerical integrations for the continuum fluorescence correction, and

FT, to perform the same integrations, with omission of

[the x-ray absorption (this is required only to determine

the absorption term f , and could be omitted if this term

is of no interest).

CALI, assisted by these subprograms, assembles the

numerator and the denominator of the fraction on the right
side of equation (1). These are sent to MAIN, which calculates the relative intensity k for all elements which

have been measured, and initiates a step of iteration by the

method outlined in the Appendix. If the purpose of the

calculation was to establish theoretical intensities for a known composition, or if the results of the theoretical

calculations for all elements match the results of the previous iteration (with a tolerance of 10 ) , then the calculation is terminated, and the results are printed in the

OUTPUT procedure.

Besides making the pertinent decisions concerning the

application of fluorescence correction, the program incorporates

warning messages which are transmitted to the output if needed

parameters (e.g., mass absorption coefficients) are not

available or cannot be reliably calculated, or if the

operating conditions were chosen improperly. These two

features of the COR program render it less susceptible to

mistakes by the operator.

10
4 . Explanation of the Program

4.1 MAIN Program

4.11 Input of the Contents of the Permanent Data File

At this point, the complete permanent data file is

loaded

.

READ (5,30) NELM Read first card which contains

the number of elements of which the constants are

tabulated

loop 350

READ (5,40) Read a card which contains the constants that depend on the atomic number only:

J atomic number of element

KM (J) number of x-ray absorption edges

A(J) atomic weight

FCK12(J) Coster Kronig coefficients, fl2, f23,

FCK23(J) £13

FCK13(J)

CI (J) multiplicative constant for the mass

absorption coefficients [4]:

C K if J<50, C KL if J>50

ANK(J) exponent nK , for the mass absorption

coefficients

ANKL(J) exponent nKL » f° r the mass absorption

coefficients (u) according to y = CA

loop 350

READ (5,50) Read a card containing the constants

which depend on the atomic level:

WAV(K,J)

R(K,J)

0ME(K,J)

LM(K,J)

implied loop

wavelength of absorption jump

jump ratio

fluorescence yield

number of tabulated lines for this level

READ (5,70) Read cards containing line wavelengths

and relative intensities

CL(L,K,J) wavelength of line

ZL(L,K,J) relative intensity of line

r

All the variables used in this program are defined in

Appendix 2

.

11
for K,Lj , Lj-j- , Lj j -ji empty card must 1 Note: The levels are indexed 1,2,3,4,.

If there is no information for a level, an empty card J"must

x be

inserted. Do not insert a blank card at the end of the table

of constants for an element; otherwise, difficulties will

appear in calculating the absorption coefficients

.

4.1.2 Input Specific to the Particular Analysis, to the

Specimen and to the Standards

READ (5,30) NECM Read a card containing the number

of cases. Two cases are considered different if

the elements contained in them differ, or if the

conditions of analysis (voltage, line, etc.) differ.

loop 1010, concerning the handling of one case.

The tally of the numbers of iterations, NIT, is set to

zero, since intermediate parameters which are independent

of concentration are only printed after the first iteration

The switch KST0P* is set equal to 1, to assure the

start of the calculation. If an absorption coefficient

cannot be calculated in subprogram AC(CL,J), this switch

goes to . READ (5,6 0)

the case:

Read first card with information on

JJM

JJAM

ND

NPM

LOUT

ICOF

number of elements contained in the

specimen

number of elements analyzed

switch indicating the mode of ratio

input

if it is or if the field is empty:

direct input of relative x-ray

intensities (k-values)

if it is 1, k-values are obtained by

the subroutine AINP

number of points in the case,

switch for choosing the version of

output

switch for suppression of continuum

fluorescence correction.

FIRST is a logical variable to detect the first

card of a case: punch a T. If the execution

of the program is stopped because of any difficulty, the program proceeds to the next case.

•Switch, m the context of this paper, is a variable which determines the option to be followed from a group of options

12
ignoring the rest of the cards which belong to

the discarded case.

WRITE (6,160) NEC Print case number (NEC).

Loop 430, for an analyzed element, calculates the intensity

of the standard.

READ (5,80) Read a card identifying the element:

NZ(JJA)

E0

KA(JJA)

LA(JJA)

COSEC(JJA)

NV(JJA)

NEST

CON(JJA)

atomic number

operating voltage

index for excited level

index for analyzed line

cosecant of the x-ray emergence angle

valence, punch only if s toichiometry

is to be calculated

if a composite standard is used,

NEST is the number of elements

present, besides that which is

measured

switch for entering known concentrations

The value of each concentration is now put to 1, for

all real or hypothetical standards. Then, the intensities

emerging from the standards are calculated as follows.

I A call is made to C0CA ( JJA, JJA, JJA) to obtain the input

necessary to calculate fluorescence due to the continuum,

and to introduce it into the C0MM0N memory.

If execution of C0CA (JJA, JJA, JJA) has not caused an

error message to be printed, the execution continues.

Otherwise, control is transferred to the next case.

Note that the switch NC0 (JJA) is tested immediately

after each execution of the program C0CA.

A call is made to CALI (JJA, JJA, JJA, AIT ... ) to calculate

the total emerging intensity of the standard AI . Then

the switch KST0P is tested. If it has turned to

(i.e. we have a meaningless mass absorption coefficient),

a message is printed and control is transferred to the

point of reading data for the next case.

READ (5,100) When the standard for an element is

composite (NEST^O) then a card gives the mass fraction

of this element in the standard. As many cards as

there are additional elements in the standard, give

for each element:

13
READ (5,90)

NZ(JJA) its atomic number

C(JJ) its mass fraction

The subprograms C0CA ( JJA, JST , JJA) and CALI (JJA,JST,

JJA,AIT...) produce the ratio AIR (JJA) = I A

/I(A); this

value is 1 for a simple standard.

The specimen is now characterized as follows:

A series of IF statements switches NSW into one of five

possible positions:

if all elements are analyzed, NSW->1

if there are two unanalyzed elements,

NSW->4 (one element by stoichiometry

,

another by difference)

if there is one unanalyzed element,

and valences were listed, NSW->2

(one element by stoichiometry)

if a concentration is listed for the

unanalyzed element, NSW+5 (calculate k-value for known composition)

if none of the preceding is true,

NSW+3 (calculate one element by

difference)

If NSW was not switched to 1, the following READ statement

is used to set NSW to the proper value:

READ (5,110)

NZ(JJ) atomic number of unanalyzed element

NV(JJ) its valence (empty unless stoichiometry

is calculated)

CD(1,JJ) its concentration (empty unless

k-values are calculated)

The decision to calculate k-values from known concentrations is signaled through the input of a "known" concentration value for one element (JJA1) in CD(1,JJ) . This

tells the program that the C(JJA) values entered for the

elements other than JJA1 are concentrations rather than

k-values, and the k-values of these elements are calculated. But, JJA1 is an "unanalyzed" element, and its

k-value is not calculated. In order to obtain the

14
k-values for all elements of a specimen of known composition, one must run it in two cases, changing the order

of input of the element between two cases since the last

element being entered is always the "unanalyzed" element

JJA1.

The subprogram C0CA (1,JJM,JJM) is called to determine

the constants for fluorescence, in iteration no. only

for the measured lines of elements to be analyzed,

(loop 500)

.

The value of ND (p. 12 ) indicates if the experimental

k-values are entered by cards or through subprogram AINP

(CD,NPM,JJAM,SD)

.

READ (5,100) If the input is by cards, this card,

and the following, give the mass fractions or k-values

of all measured elements for all points to be calculated. If relative intensities are input for more

than one point of the same specimen, the k-values

are read on the next card or cards for all points

after the first point.

AINP (CD,NPM,JJAM,SD) is called if ND = 1 (output from

the electron probe; output on punched cards or magnetic

tape)

4.1.3 Calculation of Concentrations for Each Point

oop 1000; the subscript denotes the point on the specimen.

loop 550

As a first approximation, C = k. The k-value for each

measured element is stored in a permanent memory location.

The sum of concentrations is calculated.

The position of switch NSW now leads to the following

instructions

NSW=1: All elements present must be analyzed. The

number of each point and the concentration

of the elements are printed: instruction

630

NSW=2: One element is determined by stoichiometry

.

Instructions 560 and 630

NSW=3: One element is determined by difference;

instructions 580 and 630

NSW=4: One element by difference, and another by

stoichiometry; instructions 590 and 630

NSW=5: Calculation of k-values; instructions 610

and 630.

15
loop 620

Input of concentration of elements which are not calculated.

loop 650: Normalization of concentrations for calculation

of k-values

.

after 650: Switch NSW2 , which will cause the advance to

next iteration, is positioned to 2.

660: If NSW2 is equal to 1, print results, and pass

on to the next point.

670: If NSW2 = 2, initiate another iteration (max.

no. of iterations: 20), and, in the extended

output, print the number of the iteration.

loop 730

Calculate k-values for each analyzed element.

Print the main steps of calculation and k-values, in

iteration only, if extended output (output 2) is used.

Obtain next approximation by the hyperbolic equation.

Sum concentrations.

According to the position of switch NSW, command is now

transferred to one of the following instructions:

790: NSW2 switches to 1. Test for convergence follows.

750: Calculate one element by stoichiometry

.

740: Calculate one element by difference.

770: Calculate one element by difference,

another by s toichiometry

.

loop 810: normalization of concentrations, test for difference of concentrations found in two successive iterations (convergence test).

Normalization of concentrations for elements which were

not measured.

Printing of results

.

End of loop 1000

End of loop 1010

END

16
4.2 SUBROUTINE C0CA (J0,JM,JJA)

Arguments

J0

JM

JJA

label of first element considered in the case

label of last element considered in the case

label of analyzed element in the case

4.2.1 This Subroutine Prepares the Fluorescence

Calculations and Determines the Value of the

Constants Needed

Limitations and warnings:

a) If operating voltage is high enough to excite higher

levels corresponding to a different series, a warning

is given.

b) If the operating voltage is above 30 kV, a message

is given.

c) If the wavelength chosen passes beyond the N3 edge,

or if the operating voltage is too low to excite

the selected line, a message will be printed and

the case will be aborted.

1) Fluorescence excited by the continuum

If the characteristic

produced by the excit

edge of wavelength X

tinuous radiation of

limit, X Qt and A . ment present, the c8n

coefficients change,

elements between X these limits constitu

to be performed; the

and all parameters re

edge are entered in m

line of the measured element is

ation level corresponding to the

, it can be excited by cona wavelength between the Douane-Hunt

As one crosses an edge of any elestants for calculating mass absorption

One must consider all edges of all

and Xq. Any of the edges between

tes a sublimit for the integration

corresponding element is registered,

ferring to wavelengths above this

emory

.

If the element under consideration is the one we analyze,

we also enter in memory a new efficiency factor, (r--l)/r.

and a new effective fluorescence yield, 00 ,-,-. ^

J 'eft

2) F_luor_es_ce_n£e_exci^te_d_by characteristic lines

For each element, except the analyzed element, the

characteristic lines at each level are compared with

A ; if the shell emitting the analyzed line can be

excited with a sufficient yield, the exciting line is

17
counted and the label of the corresponding element and

the index of the series are stored.

All wavelengths previously stored in memory, with the

associated parameters, are rearranged in increasing

order, for the calculation of the fluorescence produced

by the continuous spectrum, and to determine (r--l)/r.

and we ff of the analyzed element for fluorescence -

1 by characteristic lines.

The output of the program consists of: number of sublimits and associated parameters, number of lines which

excite fluorescence, and associated parameters; error

messages, in case the user has caused the switch NC0(JJA)

to be changed.

4.2.2 Commentary on C0CA (J0,JM,JJA) (see Figure 2)

The following preliminary steps are performed in the

first block of instructions:

The switch NC0(JJA) is set to 1. This switch will go to

if the calculation becomes impossible.

The tally of starting points for integrations is set to 1.

The first point will be A Q

. The tally 13 which counts the number of lines exciting

fluorescent radiation, is set to zero.

The following constants related to the analyzed element,

JJA, are stored:

NZA its atomic number

WAA the wavelength associated with the operating

voltage (X Q ) KAA the subscript of the shell emitting the characteristic lines (A ) F,

? the Coster-Kronig coefficients

F

12

F

13

Loop 450

The subscript of the loop (JJ) is the tally in the

sequence of the elements contained in the specimen.

Beginning at the starting point of interval (WAA,

WAV (KAA, NZA) ) , (A ,A ) the program searches for cases

of characteristic fluorescent excitation.

The collection of information for fluorescence from the

continuum concerning the integration interval ends before

the signal instruction 260.
COCA(J0,JM,JJA)

PARAM.ELEM.JJA

I

, >_.

do 450

JJ-*-JO,JM

FOR ALL ELEMENTS

EXPO(JJ),CC(JJ)

r—>• do 440

K^-1,KMJ

(WAVtK^JKWAAV1

If

(WAV(K,J)> WAV(KAA,NZA))

1 F

FOR ALL EDGES

-260

450

WAVE (I) ,JJJ(I)

i

(l2*l)S[RR(I),EX(I)|—^260

RR(I),EX(I)

,260,

(jj g jjaJF/ND L/NES EXCITING

CHAR- FLUORESCENCE

^:

RRR(I),OMEG(I) do 430

1,LMKJ ~l

XcL(L,K,J)>WAV(KAA,NZA))

430 CHARAC; FLUOR |

._. I

Figure 2. Flowsheet of Subroutine C0CA

19
Tally 12: The tally for the number of discontinuities

within the interval (WAA, WAV(KAA,NZA) ) is set to zero.

The atomic number of element NZ(JJ) is renamed J, for

the purpose of entering the permanent data file.

The number of absorption edges KM (J) stored in the permanent data file is renamed as a non-subscripted integer,

KMJ

.

The following instructions, until and including instruction 60, have the purpose to define the coefficients Ci

and n^ for computing the absorption coefficient of the

element JJ at the wavelength WAA (A Q

) . These coefficients are called CC(JJ,JJA) and EXPO (JJ,JJA) for the

element JJ of a specimen in which the element JJA is

being determined.

Loop 440:

The subscript (K) is that of the absorption edge.

The positions of switches KR1, KR2 , Kl, are set, as a function of K.

The wavelength WAV (K,J) of the edge having the subscript K of the element of atomic number J, is compared

with diverse wavelengths:

1. If the permanent data files have indicated that

WAV (K,J) <. 10~6 A, the program is instructed that the

corresponding level does not exist and proceeds to the

next edge, K+l.

2. If WAV (K,J) <. WAA (i.e. the edge is at a wavelength

shorter than \q) , transfer is made, depending on the

position of switch KR1 , to one of the sets of instructions subscripted 80 to 570, to readjust CC(JJ,JJA) and

EXPO (JJ,JJA).

A message is printed if one passes the edge Mjy. Switch

NC0 (JJA) is thrown if R(KMJ) is missing in the input or

if K=12 (i.e., the analyzed line falls beyond the N TTT edge of any element)

.

3. If WAA < WAV(KMJ) < wAv(KAA,NZA) , the tally 12, which

counts the absorption edges of element J, is advanced one

unit. The tally of the starting points for integration,

I, is also advanced one unit. The edge, and the corresponding element, are stored in WAVE (I, JJA) and JJJ(I,JJA);

the remaining memory locations are reset to 0.

20
If the edge WAV(K,J) is the first of the respective element to fall within the interval, control goes to one of

the sets of instructions 190 to 570, depending on switch

KR2 . If not so, switch Kl leads to 150-570. In either

case, the absorption edge jump and the exponent n,

valid at the right of the edge, are calculated. The

values are stored as RR(I,JJ) and EX(I,JJA).

Switch NC0 goes to if one passes the edge Nyjj. If

the MTV edge is crossed, a warning message is printed.

Following instruction 260, if the element is the one

which is being measured, the switches KAS and KK determine a set of instructions for computing the efficiency

ratio (r-j_-l)/r-, and the effective fluorescence yield

a) £f . These ate stored in RRR(I,JJA) and 0MEG(I,JJA).

If the element being considered differs from that of the

analyzed element, its strongest lines are compared to the

edge WAV(KAA,NZA)

.

If excitation can occur, the tallies I and 13 increase

by one; the wavelength and the subscripts for the element

shell, and line are stored in WAVE (I , JJA) , JJJ(I,JJA),

KB (I, JJA) , LB (I, JJA)

.

4. If WAV(K,J) > WAV (KAA,NZA), the program passes to

the end of loop 450 and goes on to next element.

When all elements have been scanned, WAA is stored in

WAVE (1,JJA), and JJJ(1,JJA) = 0.

At this point, for each value of I<JM, one has assembled

the following table:

numb er

[M--I3M line exciting

starting points the fluorescence

WAVE (I, J JA)

(continuum fluorescence)

X

of JJA

X

RR(I,JJA) X (X)

JJJ(I,JJA-) X X

RRR(I,JJA) (X)

0MEG(I,JJA) (X)

EX (I, JJA) X (X)

KB (I, JJA) X

f LB(I,JJA) X

X Storage in memory.

,i(X) Storage not needed or empty

21
The nested loops 470 and 480 arrange the set of wavelengths, together with the associated quantities, in

increasing order. Then, a permutation of the indices I ensues

.

The memory locations in the preceding table, for 0MEG(I,JJA)

and RRR(I,JJA) associated with characteristic lines are

empty. In loop 500 one transmits stepwise these quantities

so as to define the efficiency ratio and effective fluorescence yield for every characteristic line.

Output

After the preceding process, the computer performs a rearrangement such that the first wavelengths, in number

IMA(JJA) , are the sublimits for integration (for the

continuum) and that those following, in number MF(JJA)

,

are those of the characteristic lines capable of exciting

the fluorescence of the analyzed element JJA.

The subprogram returns the following quantities:

WAVE (I ,JJA)

JJJ(I,JJA)

RR(I,JJA) if 1 < i < IMA(JJA)

EX(I,JJA) (for continuum fluorescence)

RRR(I,JJA)

0MEG(I,JJA)

WAVE (I, JJA)

JJJ(I ,JJA)

RRR(I,JJA) if tma(JJA) + 1 < I ^MF(JJA) + IMA(JJA) + 1 0MEG(I,JJA) (for characteristic fluorescence)

KB (I, JJA)

LB(I,JJA)

4.3 FUNCTION C0 ( J0, JM, JJA, FC , FQC)

This program calculates the intensity of fluorescence

due to the continuum, by means of a numerical integration.

C0 is separate from C0CA( J0, JM, JJA) because it requires

concentrations; therefore it is used in each iteration,

while C0CA(J0,JM,JJA) is used only once.

22
4.4 FUNCTIONS F (X, J0, JM, JJA) and FT(X)

Function F calculates the expression for continuum fluorescence to be numerically integrated in C0(J0, JM, JJA)

.

Function FT(X) does the same calculation, with omission

of the absorption terms for continuum fluorescence.

The purpose of this calculation is the computation

of the absorption correction factor for fluorescence from

the continuum (f

c

, FQC) . If this information is not of

interest, the calculation of FT(X) can be omitted.

4.5 FUNCTION AC(CL,J)

Calculates the x-ray absorption coefficient of atomic

number J and wavelength CL; and acts upon the switch

KST0P if an error is detected (i.e., if the coefficient

cannot be calculated) . The parameters given by Heinrich

[7] are used.

4.6 FUNCTION PRA( J0, JM,LX, KX, JX ,WAA)

Calculates the intensity of primary emission of the line

LX, shell KX, of the element of atomic number JX, for a case containing the elements subscripted J0 to JM, where

the Duane-Hunt limit associated with the energy of

the electrons, X , is WAA. ' o'

4.7 FUNCTION RB(U0,Z)

Calculates the backscatter correction R of the element

of atomic number Z for the overvoltage U0

.

4.8 FUNCTION FQI ( J0, JM, LX, KX, JX,WAA)

Calculates the absorption correction f commonly called

f (x) , corresponding to the primary intensity PRA; the

arguments have the same meaning as in PRA.

4.9 SUBROUTINE FCA ( J0, JM, I I

, JJA, FCA1 ,FCA0)

Calculates the fluorescent intensity FCA1 for a case containing the elements J0 to JM, where the radiation is

excited by the line II, and emitted as characteristic

radiation of the element JJA.

23
4.10 SUBROUTINE CALI (J0, JM.JJA ,AIT , SWITCH, FQCF , FQCA)

Calculates the total emergent intensity, AIT, of the

line of the element JJA, for a case containing the

elements J0 to JM . CALI establishes the sequence of

procedures involving the correction function (see

Figure 1)

.

4.11 FUNCTION EI(X)

\ f-dt, u« J o: Calculates EI (x) = / J-dt , used for the integration

if stopping power.

4.12 SUBROUTINE AINP (C , I J , I IMAX , SD)

This subroutine is needed for the input of intensity

data in any form except of k-values. If intensity data

are on cards, this subroutine will be applicable as is.

Other subroutines, such as PTLOAD subroutine which enters

data from a magnetic tape into the UNIVAC 1108, can be

used to prepare data for AINP. The subroutine PTLOAD

is reproduced in the statements of the program, but not

discussed elsewhere, since it is specifically designed

for use with the UNIVAC 1108 computer.

Function DEADT , which performs the dead-time corrections

for counts, is attached to subroutine AINP.

Use of the Program

In the present form, the program should be acceptable

to all computers having a FORTRAN IV compiler. The

executive command cards are, of course, specific for

a system and must be adapted to it.

24
5. Statements of the Program

25
10

20

30

40

50

60

70

80

90

100

110

120

130

140

150

160

170

180

C0R2

A CORRECTION PROGRAM FOR ELECTRON PROBE MICROANALYSIS

BY HENOC, HEINRICH, AND MYKLEBUST

MAIN PROGRAM

THIS SECTION HANDLES ALL INPUT AND OUTPUT. THE ANALYSIS METHOD

IS DETERMINED HERE AND THE PROPER ORDER IS SET UP. THE

ITERATION PROCEDURE IS ALSO HERE.

LOGICAL FIRST

INTEGER SWITCH

DIMENSION NC0(15) ,MF(15) ,A(100) ,WAV(12,100) , CL(3 , 12, 100) ,ZL(3,12,1

100) ,NZ(15) ,AK(15) ,LA(15) ,C0SEC(15) ,C(15)

DIMENSION KM(IOO) ,LM(12,100) ,EXP0(15,15) ,CC(15,15) ,CI(100) ,ANK(100

1) ,ANKL(100) ,R(12,100)

DIMENSION FCK12(100) ,FCK23(100) ,FCK13(100) ,0ME(12,100) ,IMA(15) ,WAV

1E(120,15) ,RR(120,15) ,EX(120,15) , MESS (15, 15)

DIMENSION KB (120 ,15) ,LB(120,15) , WA ( 15 ) ,KA( 15 ) ,JJJ(120,15) ,RRR(120,

115) ,0MEG(120,15)

DIMENSION FQF ( 15 ), FCF ( 15 ), FCASF ( 15 ), RPF ( 15 ), NV ( 15 ), AI ( 15 ), AKEX ( 15

)

1,CT(15) , AIR (15) , CD (100, 15) ,SD(100,15) ,ELNA(100) ,LI(9,3) , CON (15) , FQ

2C(15) ,FQCA(15)

COMMON KSTOP , NCO , NIT , MF , A , WAV , CL , ZL , NZ , AK , LA , COSEC , C/B12/KM , LM/B12

13/EXPO , CC/B125/CI , ANK , ANKL , R/B126/FCK12 , FCK23 , FCK13 , 0ME/B1239/IMA

,

2WAVE , RR , EX , MESS/B1289/KB , LB/B5P/WA , KA , JJJ , RRR , OMEG/BICF/ICOF

COMMON /BINPT/ III , NTAPE/BLT/LOUT , FQF , FCF , FCASF , RPF

DATA ( ELNA ( KK )

, KK=3 , 92 ) /2HLI , 2HBE , 1HB , 1HC , 1HN , 1H0 , 1HF , 2HNE , 2HNA , 2H

IMG , 2HAL , 2HSI , 1HP , 1HS , 2HCL , 1HA , 1HK , 2HCA , 2HSC , 2HTI , 1HV , 2HCR , 2HMN , 2HF

2E , 2HC0 , 2HNI , 2HCU , 2HZN , 2HGA , 2HGE , 2HAS , 2HSE , 2HBR , 2HKR , 2HRB , 2HSR , 1HY

,

32HZR , 2HNB , 2HM0 , 2HTC , 2HRU , 2HRH , 2HPD , 2HAG , 2HCD , 2HIN , 2HSN , 2HSB , 2HTE ,

1

4HI , 2HXE , 2HCS , 2HBA , 2HLA , 2HCE , 2HPR , 2HND , 2HPM , 2HSM , 2HEU , 2HGD , 2HTB , 2HD

5Y , 2HH0 , 2HER , 2HTM , 2HYB , 2HLU , 2HHF , 2HTA , 1HW , 2HRE , 2H0S , 2HIR , 2HPT , 2HAU

,

62HHG , 2HTL , 2HPB , 2HBI , 2HP0 , 2HAT , 2HRN , 2HFR , 2HRA , 2HAC , 2HTH , 2HPA , 1HU/

DATA LI ( 1 , 1 ) /3HKB1/LI (1,2) /3HKA1/LI (1,3) /3HKA2/LI (2,1) /3HLB3/LI ( 2

,

12 ) /3HLB4/LI (3,1) /3HLG1/LI (3,2) /3HLB1/LI (4,1) /3HLB2/LI (4,2) /3HLA1/L

21(4,3) /3HLA2/LI (9,1) /3HMA1/LI (7,1) /3HMB1/

FORMAT

FORMAT

FORMAT

FORMAT

FORMAT

FORMAT

FORMAT

FORMAT

FORMAT

FORMAT

FORMAT

FORMAT

1K)=,F8

FORMAT

FORMAT

FORMAT

FORMAT

FORMAT

FORMAT

,5X,2HK( ,I1,2H)=,F5.4)

,19X,A2,I1,A1,I1,5X,F6.4)

1H

1H

13)

212 , F6 . 3 , 3F6 . 4 , F6 . 2 , 2F4 . 2

)

F7.5,2F6.4,I1)

212, II, 312, 68X, LI)

F7 . 5 , F6 . 4

)

12 , 10X , F7 . 3 , 211 , F6 . 4 , 211 , F6 . 4

)

I2,F5.4)

15F5.4)

I2,10X,I1,F5.4)

/7HELEMENT,I2,4X,9HAT0MIC NO , 12 , /30X.9HABS COEFF ,/30X, 5HC

(

,3X,5HN(K)=,F4.2)

18H COMPOUND STANDARD/5H JJA=, 12 ,4X,4HNZA=, 12/ ( 9X, 3HNZ( , II

1 , 2H ) = , 12 , 3X , 2HC= , F5 . 3 , 3X , 3HCC= , F8 . 2 , 3X , 5HEXP0= , F4 . 2 )

)

11X,4HAIR( ,I1,2H)=,F6.4)

/12X,3HAI( ,I1,2H)=,E10.5)

1H0.10H SAMPLE NO , 13 , /5X , 9HSTANDARDS

//5X.8HP0INT NO, 12)

/7X.13HITERATI0N NO ,12)

26

A 1 A 2 A 3

A 4

A 5

A 6

A 7

A 8

A 9

A 10

A 11

A 12

A 13

A 14

A 15

A 16

A 17

A 18

A 19

A 20

A 21

A 22

A 23

A 24

A 25

A 26

A 27

A 28

A 29

A 30

A 31

A 32

A 33

A 34

A 35

A 36

A 37

A 38

A 39

A 40

A 41

A 42

A 43

A 44

A 45 ' A 46

A 47

A 48

A 49

A 50

A 51

A 52 ! A 53

A 54 j A 55

A 56

A 57

A 58
FORMAT (/5H JJA=, 12 ,4X,4HNZA=, 12/ ( 9X , 3HNZ( , II , 2H)=, 12 , 3X, 2HC=, F5 .

3

1 , 3X , 3HCC= , F8 . 2 , 3X , 5HEXP0= , F4 . 2 )

)

FORMAT (12X,3HCT( , II , 2H)=, F5 . 3

)

FORMAT (/10(1X,3HCT( , II , 2H)=, F6.4)

)

FORMAT ( 1H , 5X , A2 , 2X , 12 , 3 ( 5X , F6 . 4 )

)

FORMAT ( 1H+ , 46X , F6 . 4 , 4X , ElO . 5 , 6X , ElO . 5 , 4X . F6 . 4

)

FORMAT (1H ,10X,29HC0MP0UND STANDARD FOR ELEMENT, 12/)

FORMAT ( 1HO , 4X , 7HELEMENT , 4X , 7HAT CONC,4X,7HWT CONC ,4X,7HSTD DEV.4X

1 , 4HF ( X )

, 4X , 13HI ( P ) / ( F ( X

) *C ) , 4X , 9HI ( C ) /I ( P ) , 4X , 7HF ( X ) CON , 4X , 10HCHAR

2 FLUOR , 4X , 7HF ( X ) CHA

)

FORMAT (1H0.24X.12HITERATI0N NO, 13)

FORMAT ( 1H+ , 46X , F6 . 4 , 4X , ElO . 5 , 6X , ElO . 5 , 4X , F6 . 4 , 4X , ElO . 5 , 5X , F6 . 4

)

FORMAT (1H ,10X,A2,A3)

FORMAT (80H

1 ) FORMAT (1H1)

FORMAT (1H0,19X,7HT0TAL =,2X,F6.4)

FORMAT ( 1HO , 9X , 13HST0ICHI0METRY//20X , 5H0XIDE , 5X , 5HC0NC . /

)

FORMAT ( 1HO , 4X , 7HELEMENT , 15X , 7HK-VALUE , 14X4HF ( X ), 4X , 13HI ( P ) / ( F ( X

)

1C) ,4X,9HI(C)/I(P) ,4X,7HF(X)CON,4X,10HCHAR FLUOR , 4X , 7HF ( X ) CHA

)

FORMAT (1H ,5X,A2,2X,I2,16X,F6.4)

CALL DEFTAB

READ DATA INTO THE ELEMENT TABLE

READ (5,30) NELM

DO 350 N0E=1,NELM

READ (5,40) J,KM(J) ,A(J) ,FCK12(J) ,FCK23(J) ,FCK13(J) ,CI(J) , ANK(J) ,A

1NKL(J)

KMJ=KM(J)

DO 350 K=1,KMJ

READ (5,50) WAV(K,J) ,R(K,J) ,OME(K,J) ,LM(K,J)

LMKJ=LM(K,J)

IF (LMKJ.EQ.O) GO TO 350

READ (5,70) (CL(L.K.J) ,ZL(L,K,J) , L=l , LMKJ

)

CONTINUE

READ IN INITIAL DATA FOR EACH SAMPLE

READ (5,30) NECM

DO 1010 NEC=1,NECM

NIT=0

KST0P=1

READ (5,60) JJM, J JAM, ND,NPM, LOUT, ICOF, FIRST

NP1=NPM-1

IF (.NOT. FIRST) GO TO 360

READ (5,290)

SWITCH=0

WRITE (6,300)

WRITE (6,290)

WRITE (6,160) NEC

DO 430 JJA=1,JJAM

CON(JJA)=0.

READ ELEMENT CARDS FOR EACH SAMPLE

READ (5,80) NZ(JJA) ,EO,KA(JJA) ,LA(JJA) ,COSEC(JJA) ,NV(JJA) , NEST, CON

l(JJA)

NC0(JJA)=1

WA(JJA)=12.398/E0

C(JJA)=1.

27

A 59

A 60

A 61

A 62

A 63

A 64

A 65

A 66

A 67

A 68

A 69

A 70

A 71

A 72

A 73

A 74

A 75

A 76

A 77

A 78

A 79

A 80

A 81

A 82

A 83

A 84

A 85

A 86

A 87

A 88

A 89

A 90

A 91

A 92

A 93

A 94

A 95

A 96

A 97

A 98

A 99

A 100

A 101

A 102

A 103

A 104

A 105

A 106

A 107

A 108

A 109

A 110

A 111

A 112

A 113

A 114

A 115

A 116

A 117

A 118

A 119
C A 120

C CALCULATE FLUORESCENCE CONSTANTS FOR PURE ELEMENTS A 121

C A 122

CALL COCA (JJA, JJA, JJA) A 123

IF (LOUT.EQ.O) GO TO 370 A 124

WRITE (6,120) JJA,NZ(JJA) ,CC(JJA,JJA) , EXPO ( JJA , JJA) A 125

370 CONTINUE A 126

IF (NCO(JJA) .EQ.O) GO TO 1010 A 127

NZA=NZ(JJA) A 128

KAA=KA(JJA) A 129

LAA=LA(JJA) A 130

IF (NZA.LT.10) LAA=2 A 131

IF (NZA.LT.40.AND.KAA.EQ.4) LAA=2 A 132

WRITE (6,280) ELNA(NZA )

, LI ( KAA , LAA) A 133

C A 134

C CALCULATION OF ABSORPTION, ATOMIC NUMBER, AND CONTINUUM A 135

C FLUORESCENCE CORRECTIONS FOR THE STANDARDS. A 136

C AI = THE TOTAL CORRECTION FACTOR A 137

C A 138

CALL CALI (JJA,JJA,JJA,AIT,SWITCH,FQC,FQCA) A 139

IF (KSTOP.EQ.O) GO TO 1010 A 140

AI(JJA)=AIT A 141

AIR(JJA)=1. A 142

IF (LOUT.EQ.O) GO TO 380 A 143

WRITE (6,150) JJA, AI (JJA) A 144

380 CONTINUE A 145

IF (NEST. EQ.O) GO TO 410 A 146

C A 147

C READ DATA INPUT FOR COMPOUND STANDARDS AND CALCULATE A 148

C FLUORESCENCE CONSTANTS FOR THE COMPOUND STANDARDS. A 149

C A 150

READ (5,100) C(JJA) A 151

JST=JJA+NEST A 152

JJAS=JJA+1 A 153

READ (5,90) ( NZ ( JJ )

, C ( J J ) , J J=JJ AS , JST

)

A 154

CALL COCA (JJA, JST, JJA) A 155

IF (LOUT.EQ.O) GO TO 390 A 156

WRITE (6,130) JJA.NZ(JJA) , (JJ.NZ(JJ) ,C(JJ) ,CC(JJ,JJA) , EXPO (JJ , JJA) A 157

1,JJ=JJA,JST) A 158

GO TO 400 A 159

390 WRITE (6,240) JJA A 160

400 CONTINUE A 161

IF (NCO(JJA) .EQ.O) GO TO 1010 A 162

C A 163

C CALCULATE CORRECTIONS FOR COMPOUND STANDARDS. A 164

C AIR = TOTAL CORRECTION A 165

C A 166

CALL CALI (JJA, JST, JJA, AIT, SWITCH, FQCFQCA) A 167

IF (KSTOP.EQ.O) GO TO 1010 A 168

AIR(JJA)=AIT/AI(JJA) A 169

410 IF (LOUT.EQ.O) GO TO 420 A 170

WRITE (6,140) JJA, AIR (JJA) A 171

420 CONTINUE A 172

430 CONTINUE A 173

C A 174

C SORTING ROUTINE FOR ANALYSIS METHOD - NSW DETERMINES METHOD A 175

C 1 - ALL ELEMENTS ANALYZED A 176

C 2 - ONE ELEMENT BY STOICHIOMETRY A 177

C 3 - ONE ELEMENT BY DIFFERENCE A 178

C 4 - TWO OR MORE ELEMENTS NOT ANALYZED A 179

C 5 - CALCULATE K-VALUES FROM KNOWN CONCENTRATIONS A 180

28
: a i8i

JJA1=JJAM+1 A 182

IF (JJM-JJA1) 480,450,440 A 183

140 NSW=4 A 184

150 READ (5,110) (NZ( JJ

)

,NV( JJ ), CD( 1 , JJ ) , JJ=JJA1 , JJM) A 185

IF (CD(l.JJAl) .GT.l.E-7) GO TO 460 A 186

IF (NV(JJAl) .EQ.O) GO TO 470 A 187

IF ( (JJM-JJA1) .NE.O) GO TO 490 A 188

NSW=2 A 189

GO TO 490 A 190

160 NSW=5 A 191

GO TO 490 A 192

170 NSW=3 A 193

GO TO 490 A 194

180 NSW=1 A 195

190 DO 500 JJA=1,JJAM A 196

) A 197

) CALCULATE FLUORESCENCE CONSTANTS FOR UNKNOWNS A 198

) A 199

CALL COCA (l.JJM.JJA) A 200

IF (NCO(JJA) .EQ.O) GO TO 1010 A 201

>00 CONTINUE A 202

SWITCH=1 A 203

IF (ND.EQ.l) GO TO 530 A 204

) A 205

: READ IN K-VALUES FROM CARDS A 206

) A 207

DO 510 NP=1,NPM A 208

>10 READ (5,100) (CD(NP,JJA) , JJA=1 , JJAM

)

A 209

IF (NSW.NE.5) GO TO 540 A 210

IF (NPM.EQ.l) GO TO 540 A 211

DO 520 NP=2,NPM A 212

>20 READ (5,100) ( CD(NP, JJ ) , JJ=J JA1 , JJM) A 213

GO TO 540 A 214

) A 215

J DATA ARE READ BY THE INPUT SUBROUTINE (AINP) A 216

) A 217

>30 CALL AINP (CD , NPM, JJAM,SD) A 218

NP1=NPM-1 A 219

3 A 220

: BEGIN CALCULATION OF CONCENTRATIONS FOR EACH UNKNOWN A 221

) A 222

>40 DO 1000 NP=1,NPM A 223

NIT=0 A 224

SC=0.0 A 225

DO 550 JJA=1,JJAM A 226

C(JJA)=CD(NP,JJA)*AIR(JJA) A 227

IF (CON(JJA) .GT.l.OE-7) C ( JJA)=CD( NP , JJA) A 228

AKEX(JJA)=C(JJA) A 229

SC=SC+C(JJA) A 230

S50 CONTINUE A 231

IF (NSW.EQ.5) SC=SC+CD(1,JJA1) A 232

GO TO (630,560,580,590,610), NSW A 233

A 234

STOICHIOMETRY CALCULATION A 235

A 236

} C(JJA1)=0.0 A 237

NZ0=NZ(JJA1) A 238

DO 570 JJA=1,JJAM A 239

NZA=NZ(JJA) A 240

C0=C(JJA)/A(NZA)*A(NZ0)*NV(JJA)/NV(JJA1) A 241

29
C(JJA1)=C(JJA1)+C0

570 CONTINUE

SC=SC+C(JJA1)

GO TO 630

c

c

c580

C

C

C590

CALCULATE ONE ELEMENT BY DIFFERENCE

C(JJA1)=1.-SC

SC=1.

GO TO 630

TWO OR MORE ELEMENTS NOT ANALYZED

C(JJM)=0.0

NZO=NZ(JJM)

NZEX=NZ(JJA1)

DO 600 JJA=1,JJAM

NZA=NZ(JJA)

CO=C ( JJA ) /A ( NZA ) *A ( NZO ) *NV ( JJA ) /NV ( J JM

600

C

C

c610

620

630

640

650

C

Cc660

670

680

690

CCC

700

C(JJM)=C(JJM)+CO

CONTINUE

SC=SC+C(JJM)

C ( J J Al ) = ( 1

. -SC ) *A ( NZEX ) *NV ( JJM ) / ( A ( NZEX ) *NV ( JJM ) +A ( NZO ) *NV ( JJA1 )

)

C ( J JM ) =C ( JJM ) + ( 1

. -SC ) *A ( NZO ) *NV ( JJ Al ) / ( A ( NZEX ) *NV ( J JM ) +A ( NZO ) *NV (

J

1JA1)

)

SC-1.

GO TO 630

CALCULATE K-VALUES FROM KNOWN CONCENTRATIONS

DO 620 JJ=JJA1,JJM

C(JJ)=CD(NP,JJ)

CONTINUE

WRITE (6,170) NP

DO 640 JJ=1,JJAM

WRITE (6,10) JJ,C(JJ)

CONTINUE

DO 650 JJ=1,JJM

C(JJ)=C(JJ)/SC

CONTINUE

NSW2=2

START OF ITERATION LOOP

GO TO (840,670) , NSW2

IF (NIT.GT.20) GO TO 840

IF (LOUT.EQ.O) GO TO 680

WRITE (6,180) NIT

CONTINUE

IF (NSW.NE.5) SC=0.0

DO 730 JJA=1,JJAM

IF (CD(NP,JJA) .LT.1.0E-7

IF (LOUT.EQ.O) GO TO 700

IF (NIT.EQ.O) GO TO 690

GO TO 700

WRITE (6,190) JJA,NZ(JJA

1, JJ=1,JJM)

GO TO 720

( JJ , NZ ( J J ) , C ( J J )

, CC ( J J , J J A )

, EXPO ( JJ , J JA

)

calculate corrections for unknowns

call cali (i,jjm,jja,ait,switch,fqc,fqca;

30

A

A

A

A

A

A

A

A

A

A

A

A

\

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

AA

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

242

243

244

245

246

247

248

249

250

251

252

253

254

255

256

257

258

259

260

261

262

263

264

265

266

267

268

269

270

271

272

273

274

275

276

277

278

279

280

281

282

283

284

285

286

287

288

289

290

291

292

293

294

295

296

297

298

299

300

301

302
IF (KSTOP.EQ.O) GO TO 1010 A 303

CT(JJA)=AIT/AI(JJA) A 304

IF (CON(JJA) .GT.l.OE-7) CT( JJA)=CD(NP, JJA) A 305

IF (LOUT.EQ.O) GO TO 710 A 306

IF (NIT.EQ.O) WRITE (6,200) JJA.CT(JJA) A 307

710 CONTINUE A 308

IF (NSW.EQ.5) GO TO 730 A 309

C A 310

C HYPERBOLIC APPROXIMATION FOR CONCENTRATIONS A 311

C A 312

CT(JJA)=AKEX(JJA)*C(JJA)*(1.-CT(JJA) )/( AKEX( JJA) * ( C ( JJA)-CT( JJA) )+ A 313

1CT(JJA)*(1.-C(JJA) )

)

A 314

IF (CON(JJA) .GT.l.OE-7) CT( JJA)=CD(NP, JJA) A 315

SC=SC+CT(JJA) A 316

GO TO 730 A 317

CT(JJA)=0. A 318

CONTINUE A 319

GO TO (790,750,740,770,820), NSW A 320

:740 CT(JJA1)=1.-SC A 321

SC=1. A 322

GO TO 790 A 323

r,750 CT(JJA1)=0.0 A 324

DO 760 JJA=1,JJAM A 325

NZA=NZ(JJA) A 326

C0=CT(JJA)/A(NZA)*A(NZ0)*NV(JJA)/NV(JJA1) A 327

CT(JJA1)=CT(JJA1)+C0 A 328

i760 CONTINUE A 329

SC=SC+CT(JJA1) A 330

GO TO 790 A 331

:770 CT(JJM)=0.0 A 332

DO 780 JJA=1,JJAM A 333

NZA=NZ(JJA) A 334

CO=CT ( JJA ) /A ( NZA ) * A ( NZO ) *NV ( JJA ) /NV ( JJM

)

A 335

CT(JJM)=CT(JJM)+CO A 336

CONTINUE A 337

SC=SC+CT(JJM) A 338

CT ( JJA1 ) = ( 1

. -SC ) *A ( NZEX ) *NV ( JJM ) / ( A ( NZEX ) *NV ( J JM ) +A ( NZO ) *NV ( JJA1 )

) A 339

CT ( JJM ) =CT ( JJM ) + ( 1

. -SC ) *A ( NZO ) *NV ( JJA1 ) / ( A ( NZEX ) *NV ( J JM ) +A ( NZO ) *NV A 340

1(JJA1)

)

A 341

SC=1. A 342

^90 NSW2=1 A 343

IF (LOUT.EQ.O) GO TO 800 A 344

WRITE (6,210) (JJ,CT(JJ) ,JJ=1,JJM) A 345

300 CONTINUE A 346

DO 810 JJA=1,JJAM A 347

CT(JJA)=CT(JJA)/SC A 348

IF (CON(JJA). GT.l.OE-7) CT( JJA)=CD(NP, JJA) A 349

IF (ABS(CT(JJA)-C(JJA) ) .GT.l.E-5) NSW2=2 A 350

C(JJA)=CT(JJA) A 351

310 CONTINUE A 352

C(JJA1)=CT(JJA1)/SC A 353

JJA2=JJA1+1 A 354

C(JJA2)=CT(JJA2)/SC A 355

NIT=NIT+1 A 356

: A 357

£ END OF ITERATION LOOP - IF CONVERGENT , BEGIN WRITING OUTPUT A 358

C A 359

GO TO 660 A 360

320 WRITE (6,330) A 361

DO 830 JJA=1,JJAM A 362

•: A 363

31
C HYPERBOLIC APPROXIMATION FOR K-VALUES A 364

c A 365

CT(JJA)=1./(1-+(1-/AKEX(JJA)-1. )*(1./CT(JJA)-1. )/(l./C(JJA)-l. )) A 366

830 CONTINUE A 367

JJMX=JJAM A 368

GO TO 890 A 369

840 WRITE (6,250) A 370

SACO=0. A 371

DO 870 JJ=1,JJM A 372

IF (NSW.EQ.5) GO TO 860 A 373

IF (CON(JJ) .GT.1.0E-7) GO TO 860 A 374

IF (NSW.NE.2) GO TO 850 A 375

IF (JJ.EQ.JJM) GO TO 860 A 376

850 CT(JJ)=CT(JJ)*SC A 377

860 CONTINUE A 378

870 CONTINUE A 379

DO 880 JJ=1,JJM A 380

NZA=NZ(JJ) A 381

SACO=SACO+CT(JJ)/A(NZA) A 382

880 CONTINUE A 383

JJMX=JJM A 384

890 DO 970 JJ=1,JJMX A 385

NZA=NZ(JJ) A 386

IF (NSW.EQ.5) GO TO 900 A 387

SA1=SAC0-CT(JJ)/A(NZA) A 388

ATC0=CT(JJ)/(CT(JJ)+A(NZA)*SA1) A 389

WRITE ( 6 , 220 ) ELNA ( NZA )

, NZA , ATCO , CT ( JJ )

, SD ( NP , JJ

)

A 390

GO TO 910 A 391

900 WRITE (6,340) ELNA ( NZA ), NZA, CT(JJ

)

A 392

CT(JJ)=AKEX(JJ) A 393

910 IF (JJAM-JJM) 920,930,930 A 394

920 IF (JJ.EQ.JJM) GO TO 940 A 395

930 IF (CT(JJ) .GT.l.OE-7) GO TO 950 A 396

940 RPFX=0. A 397

FCFX=0. A 398

GO TO 970 A 399

950 RPFX=RPF ( J J ) / ( FQF ( JJ

) *CT ( JJ )

)

A 400

FCFX=FCF(JJ)/RPF(JJ) A 401

IF (FCASF(JJ) .GT.1.0E-12) GO TO 960 A 402

WRITE (6,230) FQF( J J ) ,RPFX, FCFX, FQC( JJ

)

A 403

GO TO 970 A 404

960 FCASFX=FCASF(JJ)/RPF(JJ) A 405

WRITE (6,270) FQF ( J J ) ,RPFX, FCFX, FQC (JJ ), FCASFX,FQCA( JJ

)

A 406

970 CONTINUE A 407

WRITE (6,310) SC A 408

WRITE (6,260) NIT A 409

IF (NSW.NE.2) GO TO 1000 A 410

C A 411

C WRITE STOICHIOMETRY OUTPUT IF THIS OPTION WAS USED A 412

C A 413

WRITE (6,320) A 414

NZA1=NZ(JJM) A 415

DO 990 JJ=1,JJAM A 416

NZA=NZ(JJ) A 417

ST0IC=(A(NZA)*NV(JJM)+A(NZA1)*NV(JJ))/(A(NZA)*NV(JJM))*CT(JJ) A 418

NY=NV(JJM) A 419

NX=NV(JJ) A 420

IF (NX.LT.NY) GO TO 980 A 421

IF (MOD (NX, NY) . EQ . 1 ) GO TO 980 A 422

NX=NX/NY A 423

NY=1 A 424

32
>0

,CI(100) ,ANK(100

, IMA (15) ,WAV

RRR(120

12, 1H, ,I2,1H, ,12,1

(//30H BAD CHOICE FOR EO, ARGUMENTS (, 12 , 1H, , 12 , 1H ,, 12 , 1H)

)

(3H R( , I2,1H, ,I2,28H) MISSING IN DATA .ARGUMENTS (, 12 , 1H, , 12,

380 WRITE ( 6 , 20 ) ELNA ( NZA ), NY , ELNA ( NZA1 ), NX , STOIC

)90 CONTINUE

.000 CONTINUE

.010 CONTINUE

STOP

END

SUBROUTINE COCA (JO,JM,JJA)

1 DETERMINE ALL THE CONSTANTS NECESSARY FOR A CALCULATION

DIMENSION NC0(15) ,MF ( 15 ) ,D3( 100 ), WAV (12, 100) , CL(3 , 12 , 100 ) ,D4(3 , 12

1100) ,NZ(15) ,D5(15) ,N6(15) ,D7(15) ,D8(15)

DIMENSION KM(IOO) ,LM(12,100) ,EXP0(15,15) ,CC(15,15

1) ,ANKL(100) ,R(12,100)

DIMENSION FCK12 ( 100 ), FCK23 ( 100 ), FCK13 ( 100 ), OME (12,100

IE (120, 15) ,RR(120,15) ,EX(120,15) , MESS (15, 15)

DIMENSION KB(120,15) , LB( 120, 15 )

, WA( 15 ) ,KA(15) ,JJJ(120,15

115) ,0MEG(120,15)

COMMON Nl , NCO , N2 , MF , D3 , WAV , CL , D4 , NZ , D5 , N6 , D7 , D8/B12/KM , LM/B1239/IM

1A , WAVE , RR , EX , MESS/B123/EXP0 , CC/B5P/WA , KA , JJ J , RRR , 0MEG/B1289/KB , LB/

2B126/FCK12 , FCK23 , FCK13 , 0ME/B125/CI , ANK , ANKL ,

R

.0 FORMAT (//34H NUMBER OF EDGES GE 12 ARGUMENTS I 11H) ELEMENT. ,12, 16H DISCONTINUITY ., 12

)

FORMAT (//34H EO GREATER THAN 30 KV , ARGUMENTS (, 12 , 1H, , 12 , 1H ,, 12 ,

1

1H))

FORMAT

FORMAT

11H, ,I2,1H)

)

NC0(JJA)=1

NSW3=1

1=1

13=0

NZA=NZ ( JJA

)

WAA=WA ( JJA

)

KAA=KA ( JJA

)

F12=FCK12(NZA)

F23=FCK23(NZA)

F13=FCK13(NZA)

IF (WAA.GT.WAV(KAA.NZA) ) WRITE (6,30) JO,JM,JJA

DO 450 JJ=JO,JM

12=0

MESS(JJ,JJA)=0

J=NZ(JJ)

KMJ=KM ( J

)

IF (J.GT.50) GO TO 50

INITIALIZATION OF FACTORS FOR THE CALCULATION OF

MASS ABSORPTION COEFFICIENTS.

EXPO(JJ,JJA)=ANK(J)

GO TO 60

'30 EXPO(JJ,JJA)=ANKL(J)

50 CC(JJ,JJA)=CI(J)

DO 440 K=1,KMJ

KRl=l+K/2+K/8- ( K/3 ) * ( K/5 ) + (

( K-8 ) * ( K/8 )

) *2+K/10- ( K/ll ) *2

KR2=l+K/2+K/5- ( K/5 ) * ( K/6 ) +K/9+ ( K/12 ) *2

K1=KR1-1- ( K/10 ) *4+ ( K/12 ) *3

IF (WAV(K.J) .LE.l.E-6) GO TO 440

IF (WAV(K.J) .LE.WAA) GO TO 70

IF (WAV(K.J) .GT.WAV(KAA.NZA) ) GO TO 450

GO TO 140

A 425

A 426

A 427

A 428

A 429

A 430-

B 1 B 2

BB

BB

B

B

B

B

B

B

BB

BB

B

B

B

B

B

BB

B

B

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

B 18

B 19

20

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

B 43

B 44

45

46

47

48

B 49

B 50

51

52

53

54

55

33
70 NNR=R ( K , J

)

IF (KR1.NE.7.AND.NNR.EQ.0) GO TO 560

GOTO (80,90,100,110,120,130,570), KR1

80 IF (J. GT. 50) GO TO 440

CC ( J J , J JA ) =CI ( J ) /R ( K , J

) *WAV ( K , J) **ANK ( J ) /WAV ( K , J

) **ANKL ( J

)

EXPO(JJ,JJA)=ANKL(J)

GO TO 440

90 CC(JJ,JJA)=CC(JJ,JJA)/R(K,J)

GO TO 440

100 CC ( J J , J JA ) =CC ( J J , JJA ) /R ( K , J

) *WAV ( K , J

) **EXPO ( J J , J JA ) /WAV ( K , J

) **2 .

6

EXPO ( J J, J JA) =2.

6

GO TO 440

110 CC(JJ,JJA)=CC(JJ,JJA)/R(K,J)

MESS ( JJ , J JA ) =MESS ( JJ , J JA ) +1

GO TO 440

120 CC ( J J , J JA ) =CC ( JJ , JJA ) /R ( K , J

) *WAV ( K , J

) **2 . 6/WAV ( K , J

) **2 . 22

EXPO ( JJ, J JA) =2. 22

MESS(JJ,JJA)=0

GO TO 440

130 CC(JJ,JJA)=CC(JJ,JJA)/R(K,J)

GO TO 440

CC DETERMINE JUMP RATIOS (R) AND EXPONENTS (N) NEEDED BY THE

C CONTINUUM FLUORESCENCE CORRECTION

C140 12=12+1

1=1+1

WAVE(I,JJA)=WAV(K,J)

JJJ(I,JJA)=JJ

RRR ( I , JJA ) =0 . OMEG(I,JJA)=0.0

KB(I,JJA)=0

LB(I,JJA)=0

IF (I2.EQ.1) GO TO (190,200,210,220,230,240,250,570), KR2

GO TO (150,160,170,180,570), Kl

150 RR(I,JJA)=R(K,J)

EX(I,JJA)=EX(I-1,JJA)

GO TO 260

160 RR ( I , JJA ) =R ( K , J

) *WAV ( K , J

) **2 . 6/WAV ( K , J

) * *EX ( 1-1 , JJA

)

EX (I, JJA) =2.

6

GO TO 260

170 RR(I,JJA)=R(K,J)

EX(I,JJA)=EX(I-1,JJA)

MESS ( J J , JJA ) =MESS ( J J , JJA ) +1

GO TO 260

180 RR ( I , J JA ) =R ( K , J

) *WAV ( K , J

) **2 . 22/WAV ( K , J

) **EX ( 1-1 , JJA

)

EX(I,JJA)=2.22

MESS(JJ,JJA)=0

GO TO 260

190 IF (J.GT.50) GO TO 600

RR(I,JJA)=R(K,J)*WAV(K,J)**ANKL(J)/WAV(K,J)**ANK(J)

EX(I,JJA)=ANKL(J)

GO TO 260

200 RR(I,JJA)=R(K,J)

EX ( I , JJA ) =EXPO ( JJ , JJA

)

GO TO 260

210 RR ( I , J JA )=R ( K , J

) *WAV (K, J

) **2 . 6/WAV ( K , J

) **EXPO ( JJ , JJA

)

220

RR(I JJA):=R(K J)

EX (I JJA):=2.6

GO TO 260

RR(I JJA) = =R(K, J)

EX (I JJA):=2.6

B 56

B 57

B 58

B 59

B 60

B 61

B 62

B 63

B 64

B 65

B 66

B 67

B 68

B 69

B 70

B 71

B 72

B 73

B 74

B 75

B 76

B 77

B 78

B 79

B 80

B 81

B 82

B 83

B 84

B 85

B 86

B 87

B 88

B 89

B 90

B 91

B 92

B 93

B 94

B 95

B 96

B 97

B 98

B 99

34

B 95

B 96

B 97

B 98

B 99 j B 100

B 101

B 102

B 103

B 104

B 105

B 106

B 107

B 108

B 109

B 110

B 111

B 112

B 113

B 114

B 115

B 116 i
GO TO 260

RR(I,JJA)=R(K,J)

EX(I,JJA)=2.6

MESS ( J J , JJ A

) =MESS ( JJ , JJA ) +1

GO TO 260

RR ( I , JJA )=R ( K , J

) *WAV ( K , J) **2 . 22/WAV ( K , J

) **2 .

6

EX(I,JJA)=2.22

MESS(JJ,JJA)=0

GO TO 260

RR(I,JJA)=R(K,J)

EX ( I , JJA ) =EXPO ( J J , JJA

)

GO TO 260

IF (JJ.NE.JJA) GO TO 420

KAS=KAA-KAA/6-KAA/7-KAA/9

GO TO (270,280,290,300,580,380), KAS

EFFICIENCY RATIOS —

(R-l)/R FOR THE CONTINUUM FLUORESCENCE FOR K-LINES

RRR(1,JJA)=(R(K,J)-1. )/R(K,J)

0MEG(1,JJA)=0ME(K,J)

GO TO 440

KK=K

GOTO (310,320,330,340,350,360,370), KK

KK=K+K/2

GO TO (310,320,330,340,350,360,370), KK

KK=(K+4)/6*3+K

GOTO (310,320,330,340,350,360,370), KK

WRITE (6,30) JO,JM,JJA

NSW3=2

RKR=1 . /R ( K , J

)

GO TO 440

(R-l)/R FACTORS FOR THE CONTINUUM FLUORESCENCE CORRECTION FOR

L- AND M-LINES

Ll-EDGE

RRR ( 1 , J J A

) = ( R ( K , J

) -1 . ) /R ( K , J

)

0MEG(1,JJA)=0ME(K,J)

GO TO (440,410) , NSW3

L2-EDGE

RRR ( 1 , JJA ) = ( R ( K+l , J

) -1 . ) /R ( K , J ) /R ( K+l , J

)

OMEG ( 1 , J JA ) =OME ( K+l , J

) * ( 1 . +F12* ( R ( K , J

) -1 .

) *R ( K+l . J ) / ( R ( K+l . J

) -1 . )

)

GO TO (440,410) , NSW3

RRR(I-1,JJA)=(R(K,J)-1. )/R(K,J)

OMEG ( 1-1 , J J A

) =OME ( K , J

)

GO TO 440

L3-EDGE

RRR ( 1 , J JA ) = ( R ( K+2 , J

) -1 . ) /R ( K , J ) /R ( K+l , J ) /R ( K+2 , J

)

OMEG ( 1 , J JA ) =OME ( K+2 , J

) * ( 1 . +F13*R ( K+l , J

) *R ( K+2 , J) * ( R ( K , J

) -1 . ) / ( R ( K+

12 , J

) -1 . ) +F23* ( R ( K+2 , J

) * ( R ( K+l , J

) -1 . ) / ( R ( K+2 , J

) -1 . ) +F12*R ( K+l , J

) *R

(

2K+2 , J

) * ( R ( K , J

) -1 . ) / ( R ( K+2 , J

) -1 . ) )

)

GO TO (440,410) , NSW3

RRR ( 1-1 , J J A

) = ( R ( K+l , J

) -1 . ) /R ( K , J ) /R ( K+l , J

)

OMEG ( 1-1 , J JA ) =OME ( K+l , J

) * ( 1 . +F23* ( R ( K , J

) -1 .

) *R ( K+l , J ) / ( R ( K+l , J

) -1

.

D)

35

B 117

B 118

B 119

B 120

B 121

B 122

B 123

B 124

B 125

B 126

B 127

B 128

B 129

B 130

B 131

B 132

B 133

B 134

B 135

B 136

B 137

B 138

B 139

B 140

B 141

B 142

B 143

B 144

B 145

B 146

B 147

B 148

B 149

B 150

B 151

B 152

B 153

B 154

B 155

B 156

B 157

B 158

B 159

B 160

B 161

B 162

B 163

B 164

B 165

B 166

B 167

B 168

B 169

B 170

B 171

B 172

B 173

B 174

B 175

B 176

B 177
GO TO 440 B 178

370 RRR ( 1-1 , J J A

) = ( R ( K , J

) -1 . ) /R ( K , J

)

B 179

OMEG ( 1-1 , J JA ) =OME ( K , J

)

B 180

GO TO 440 B 181

C B 182 '

C M4 - M5 EDGES B 183

C B 184

380 IF (K.EQ.l) WRITE (6,30) JO,JM,JJA B 185

RRRR=R ( KAA , J

) -1

.

B 186

DO 390 LLL=K,KAA B 187

RRRR=RRRR/R ( LLL , J

)

B 188 ! 390 CONTINUE B 189

IF (I2.EQ.1) GO TO 400 B 190

RRR(I-1,JJA)=RRRR B 191

OMEG ( 1-1 , J JA ) =OMEG ( 1 , J J A

)

B 192

GO TO 440 B 193

400 OMEG ( 1 , J JA ) =OME ( KAA , J

)

B 194

RRR(1,JJA)=RRRR B 195

GO TO 440 B 196

410 RRR ( 1-1 , J J A

) =RRR ( 1 , JJA

)

B 197

OMEG ( 1-1 , J J A

) =OMEG ( 1 , J JA

)

B 198

RRR ( 1 , J J A

) =RRR ( 1 , J J A

) *RKR B 199

NSW3=1 B 200

GO TO 440 B 201

420 LMKJ=LM ( K , J ) B 202

IF (LMKJ.EQ.O) GO TO 440 B 203

DO 430 L=1,LMKJ B 204

C B 205

c TEST FOR ANY CHARACTERISTIC FLUORESCENCE B 206

c B 207

c IF THERE IS A CHARACTERISTIC FLUORESCENCE CORRECTION, THEN B 208

c DETERMINE THE (R-l)/R, FLUORESCENT YIELD, LINE NUMBER, EDGE B 209

c NUMBER, ELEMENT NUMBER, AND WAVELENGTH FOR THE PARTICULAR B 210

c CORRECTION. B 211

c B 212

IF ( CL ( L , K , J ) . GT . WAV ( KAA , NZA ) . OR . CL ( L , K , J ) /WAV ( KAA , NZA ) . LT . . 5 ) GO B 213

1 TO 430 B 214

1=1+1 B 215

13=13+1 B 216

RRR(I,JJA)=0.0 B 217

OMEG(I,JJA)=0.0 B 218

WAVE(I,JJA)=CL(L,K,J) B 219

JJJ(I,JJA)=JJ B 220

EX(I,JJA)=EX(I-1,JJA) B 221

KB(I,JJA)=K B 222

LB(I,JJA)=L B 223

430 CONTINUE B 224

440 CONTINUE B 225

450 CONTINUE B 226

WAVE(1,JJA)=WAA B 227

JJJ(1,JJA)=0 B 228

RR(1,JJA)=0. B 229

EX(1,JJA)=0. B 230

LB(1,JJA)=0 B 231

KB(1,JJA)=0 B 232

IM=I B 233

I3M=I3 B 234

IM1=IM-1 B 235

IF (IM1.LT.2) GO TO 550 B 236

DO 480 1=2, IM1 B 237

IL=IM-I+1 B 238

36
DO 470 11=2, IL

B 2 39

IF (WAVE(II.JJA) .GT.WAVE(II+1,JJA)) GO TO 460 - B 240

GO TO 470

B 241

T1=WAVE(II,JJA)

B 242

T2=RR(II,JJA)

B 243

T3=JJJ(II,JJA)

B 244

T4=RRR(II,JJA)

B 245

T5=0MEG(II,JJA)

B 246

T6=EX(II,JJA)

B 247

T7=KB(II,JJA)

B 248

T8=LB(II,JJA)

B 249

WAVE(II,JJA)=WAVE(II+1,JJA)

B 250

RR(II,JJA)=RR(II+1,JJA)

B 251

JJJ(II,JJA)=JJJ(II+1,JJA)

B 252

RRR(II,JJA)=RRR(II+1,JJA)

B 253

0MEG(II ) JJA)=0MEG(II+1 ) JJA)

B 254

EX(II,JJA)=EX(II+1,JJA)

B 255

KB(II,JJA)=KB(II+1,JJA)

B 256

LB(II,JJA)=LB(II+1,JJA)

B 257

WAVE(II+1 ( JJA)=T1

B 258

RR(II+1,JJA)=T2

B 259

JJJ(II+1,JJA)=T3

B 260

RRR(II+1,JJA)=T4

B 261

0MEG(II+1,JJA)=T5

B 262

EX(II+1,JJA)=T6

B 263

KB(II+1,JJA)=T7

B 264

LB(II+1,JJA)=T8

B 265

CONTINUE

B 266

CONTINUE

B 267

DO 500 1=2, IM1

B 268

IF (RRR(I, JJA) .LE.l.E—7) GO TO 490

B 269

GO TO 500

B 270

RRR ( I , JJA ) =RRR ( I—1,JJA)

B 271

OMEG(I,JJA)=OMEG(I—1,JJA)

B 272

CONTINUE

B 273

114=1111

B 274

1=2

B 275

)10 IF (LB (I, JJA) .EQ.O) GO TO 530

B 276

T1=WAVE(I,JJA)

B 277

T2=JJJ(I,JJA)

B 278

T3=KB(I,JJA)

B 279

T4=LB(I,JJA)

B 280

T5=RRR ( I , J JA ) B 281

T6=0MEG(I,JJA)

B 282

DO 520 II=I,M4

B 283

WAVE(II,JJA)=WAVE(II+1,JJA)

B 284

RR(II,JJA)=RR(II+1,JJA)

B 285

JJJ(II,JJA)=JJJ(II+1,JJA)

B 286

RRR(II,JJA)=RRR(II+1,JJA)

B 287

0MEG(II,JJA)=0MEG(II+1,JJA)

B 288

EX(II,JJA)=EX(II+1,JJA)

B 289

KB(II,JJA)=KB(II+1,JJA)

B 290

LB(II,JJA)=LB(II+1,JJA)

B 291

)20 CONTINUE

B 292

WAVE(M4+1,JJA)=T1

B 293

JJJ(M4+1,JJA)=T2

B 294

KB(M4+1,JJA)=T3

B 295

LB(M4+1,JJA)=T4

B 296

RRR(M4+1,JJA)=T5

B 297

0MEG(M4+1,JJA)=T6

B 298

M4=M4-1

B 299

37
530

540

550

C

CC

560

570

580

590

600

610

620

10

GO TO 540

1=1+1

IF (I.LE.M4) GO TO 510

IMA(JJA)=IM-I3M

SET THE SWITCH FOR THE CHARACTERISTIC FLUORESCENCE CORRECTION

MF(JJA)=I3M

GO TO 620

WRITE (6,40) K,J,JO,JM,JJA

GO TO 610

WRITE (6,10) JO,JM,JJA,J,K

GO TO 610

WRITE (6,590)

1H ,'NO PROVISION IS MADE FOR LINES ORIGINATING'/'

OR M3 EDGES. '/)

FORMAT (1H FRO

1M THE Ml,

GO TO 560

WRITE (6,20) JO,JM,JJA

NCO(JJA)=0

IMA(JJA)=I

MF(JJA)=0

RETURN

END

SUBROUTINE CO ( JO , JM, JJA, FC , FQC)

CALCULATION OF CONTINUUM FLUORESCENCE CORRECTION

DIMENSION N2(15) ,N4(15) ,D5(100) ,D6(12,100) , CL(3 , 12 , 100 ) ,ZL(3,12,10

10) ,NZ(15) ,AK(15) ,LA(15) ,C0SEC(15) ,C(15)

DIMENSION EXPO (15, 15) ,CC(15,15)

DIMENSION IMA (15) , WAVE ( 120 , 15 ) ,RR(120,15) , EX (120, 15) , MESS (15, 15)

DIMENSION WA(15) ,KA(15) ,JJJ(120,15) ,RRR(120,15) , OMEG( 120 , 15

)

DIMENSION EX0(15) ,CC0(15)

COMMON Nl , N2 , N3 , N4 , D5 , D6 , CL , ZL , NZ , AK , LA , COSEC , C/B1239/IMA , WAVE , RR

,

1EX , MESS/B123/EXP0 , CC/B5P/WA , KA , JJ J , RRR , 0MEG/B34/EX0 , CCO , WAA , ACS

IM1=IMA(JJA)-1

FCO=0 . FC=0 . NZA=NZ(JJA)

KAA=KA(JJA)

LAA=LA(JJA)

WAA=WAVE ( 1 , JJA

)

ZMK=0 . ACS=0 . RRRI=RRR(1,JJA)

0MEGI=0MEG(1,JJA)

DO 10 J=JO,JM

EXO(J)=EXPO(J,JJA)

CCO(J)=CC(J,JJA)

I=NZ(J)

CALCULATE CONSTANTS FOR CONTINUUM FLUORESCENCE

ACS=ACS+C ( J

) * AC ( CL ( LAA , KAA , NZA ) , I

) *COSEC ( JJA

)

ANZ=NZ(J)

AK(J)=3.0E-5

ZMK=ZMK+C ( J

) *ANZ*AK ( J

)

CONTINUE

DO 70 1=1, IM1

DO 40 J=JO,JM

IF (J.EQ.JJJ(I.JJA) ) GO TO 20

B 300

B 301

B 302

B 303

B 304

B 305

B 306

B 307

B 308

B 309

B 310

B 311

B 312

B 313

B 314

B 315

B 316

B 317

B 318

B 319

B 320

B 321

B 322-

C 1 C 2 C 3

C 4

c 5

C 6

c 7

C 8

c 9

c 10

c 11 c 12

c 13

c 14

c 15

c 16

c 17

c 18

c 19

c 20

c 21

c 22

c 23

c 24

c 25

c 26

c 27

c 28

c 29

c 30

c 31

c 32

c 33

c 34

c 35

c 36

c 37

c 38

38
GO TO 40

[ 20 EXO(J)=EX(I,JJA)

CCO ( J

) =CCO ( J ) /RR ( I , JJA

)

IF (J.EQ.JJA) GO TO 30

GO TO 40

-30 OMEGI=OMEG(I,JJA)

RRRI=RRR ( I , J JA

)

40 CONTINUE

C

: C SET UP INTERVALS FOR NUMERICAL INTEGRATION

C

NIN2= ( WAVE ( 1+1 , J J A

) -WAVE ( I , JJA ) ) /O . 02

NIN=2*NIN2

H=0.01

IF (NIN.GE.10) GO TO 50

H= ( WAVE ( 1+1 , JJ A

) -WAVE ( I , J JA ) ) /10

.

NIN=10

50 ANIN=NIN

RL= ( WAVE ( 1+1 , J J A

) -WAVE ( I , JJA )) -ANIN*H

BO=WAVE(I,JJA)+RL

BN=WAVE(I+1,JJA)

BN1=BN-H

LKM=NIN-3

NUMERICAL INTEGRATION BETWEEN ABSORPTION EDGES

TRAPEZOIDAL RULE

S=0 . 5* ( F ( WAVE ( I , J JA ) , JO , JM , J JA ) +F ( BO , JO , JM , JJA )

) *RL

S0=0 . 5* ( FT ( WAVE ( I , JJA ) ) +FT ( BO )

) *RL

C

C SIMPSONS RULE

C S=S+H* ( F ( BO , JO , JM , JJA ) +F ( BN , JO , JM , JJA ) +4 . *F ( BN1 , JO , JM , JJA ) ) /3

.

SO=SO+H* ( FT ( BO ) +FT ( BN ) +4 . *FT ( BN1 ) ) /3

.

DO 60 LK=1,LKM,2

ALK=LK

BK1=B0+ALK*H

BK2=BK1+H

S=S+2 . * ( 2

. *F ( BK1 , JO , JM , JJA ) +F ( BK2 , JO , JM , JJA ) ) *H/3

.

S0=S0+2 . * ( 2

. *FT ( BK1 ) +FT ( BK2 )

) *H/3

.

: 60 CONTINUE

EABI=0 . 5*ZL ( LAA , KAA , NZA ) *ZMK*C ( JJA ) *OMEGI*RRRI

FI=EABI*S

FIO=EABI*SO

C

JC FINAL VALUES OF EMITTED AND GENERATED CONTINUUM FLUORESCENCE

C

FC=FC+FI

FCO=FCO+FIO

70 CONTINUE

C

C F(X) OF THE CONTINUUM

C

FQC=FC/FCO

RETURN

END

FUNCTION F (X,JO,JM,JJA)

CC CALCULATION OF THE INTEGRAL OF THE CONTINUUM CORRECTION FOR

C THE EMITTED X-RAYS

C

39

c 39

c 40

c 41

c 42

c 43

c 44

c 45

c 46

c 47

c 48

c 49

c 50

c 51

c 52

c 53

c 54

c 55

c 56

c 57

c 58

c 59

c 60

c 61

c 62

c 63

c 64

c 65

c 66

c 67

c 68

c 69

c 70

c 71

c 72

c 73

c 74

c 75

c 76

c 77

c 78

c 79

c 80

c 81

c 82

c 83

c 84

c 85

c 86

c 87

c 88

c 89

c 90

c 91

c 92

c 93

c 94

D 1 D 2

D 3

D 4

D 5
10

10

20

30

Cc

c40

C

C

C50

DIMENSION N2(15) ,N4(15) ,D5(100) ,D6(12,100) ,D7( 3 , 12 , 100 ) ,D8(3,12,10

10) ,N9(15) ,D10(15) , Nil (15) ,D12(15) ,C(15)

DIMENSION EX0(15),CC0(15)

COMMON Nl , N2 , N3 , N4 , D5 , D6 , D7 , D8 , N9 , D10 , Nil . D12 , C/B34/EX0 , CCO , WAA , AC

1S/B410/ACAX.ACX

ACAX=CCO ( JJA ) *X**EXO ( JJA

)

ACX=0 . DO 10 J=JO,JM

ACX=ACX+C ( J

) *CCO ( J

) *X**EXO ( J

)

CONTINUE

F=ACAX/ACS* ( X-WAA ) /WAA/X/X*ALOG ( 1 . +ACS/ACX

)

RETURN

END

FUNCTION AC (CL,J)

CALCULATION OF THE MASS ABSORPTION COEFFICIENTS

DIMENSION Nl ( 15 ), N3 ( 15 ), D4 ( 100 ), WAV ( 12 , 100 )

, D5 ( 3 , 12 , 100 ), D6 ( 3 , 12 ,

1

100) ,N7(15) ,D8(15) ,N9(15) ,D10(15) ,D11(15)

DIMENSION CI(IOO) ,ANK(100) .ANKL(IOO) ,R(12,100)

COMMON KSTOP , Nl , N2 , N3 , D4 , WAV , D5 , D6 , N7 , D8 , N9 , D10 , D11/B125/CI , ANK , AN

1KL.R

FORMAT (//4H MU( , F7 . 5 , 1H, , 12 ,42H) UNKNOWN .MORE THAN 12 EDGES ARE

1 CROSSED)

FORMAT (//4H MU( ,F7.5,1H, ,I2,14H) UNDEFINED ,R( , 12 , 1H ,

, 12 , 17H) MISS

1ING IN DATA)

K=l

CONSTANTS FROM OPERATING POTENTIAL TO K-EDGE

CC=CI(J)

AN=ANK(J)

IF (CL.LT.WAV(K.J) ) GO TO 100

CONSTANTS FROM K-EDGE TO Ll-EDGE FOR ELEMENTS ATOMIC NUMBER 50

AND BELOW

IF ( J . LE . 50 ) CC=CC/R ( K , J

) *WAV ( K , J

) ** ( ANK ( J

) -ANKL ( J )

)

AN=ANKL ( J

)

K=K+1

IF (WAV(K.J) .LE.l.E-7) GO TO 30

IF (CL.LT.WAV(K,J) ) GO TO 100

L=l+K/4-(K/4)*((K-l)/4)+(K/9)*4-(K/10)*2+(K/12)*4

NNR=R ( K , J

)

IF (L.NE.3.AND.NNR.EQ.0) GO TO 80

GO TO (40,50,60,70) , L

ALL CONSTANTS ABOVE Ll-EDGE

CC=CC/R(K,J)

GO TO 30

ALL CONSTANTS ABOVE L3-EDGE

CC=CC/R(K,J)

CC=CC*WAV ( K , J

) ** ( ANKL ( J

) -2 . 6

)

AN=2 .

6

GO TO 30

ALL CONSTANTS ABOVE M5-EDGE

D 6

D 7

D 8

D 9

D 10

D 11

D 12

D 13

D 14

D 15

D 16

D 17

D 18- E 1 E 2

E 3

E 4

E 5

E 6

E 7

E 8

E 9

E 10

E 11

E 12

E 13

E 14

E 15

E 16

E 17

E 18

E 19

E 20

E 21

E 22

E 23

E 24

E 25

E 26

E 27

E 28

E 29

E 30

E 31

E 32

E 33

E 34

E 35

E 36

E 37

E 38

E 39

E 40

E 41

E 42

E 43

E 44

E 45

E 46

E 47

E 48

40
60 CC=CC/R(K,J) E 49

CC=CC*WAV(K,J)**(2.6-2.22) - E 50

AN=2 .22 E 51

GO TO 30 E 52

70 WRITE (6,10) CL,

J

E 53

GO TO 90 E 54

80 WRITE (6,20) CL,J,K,J E 55

90 AC=l.E-3 E 56

KSTOP=0 E 57

RETURN E 58

AC=CC*CL**AN E 59

RETURN E 60

END E 61-

FUNCTION PRA ( JO , JM, LX, KX, JX, WAA) F 1 C F 2

C CALCULATION OF PRIMARY INTENSITIES F 3

F 4

DIMENSION N2(15) ,N4(15) ,A(100) ,WAV(12,100) ,D5 ( 3 , 12 , 100 ) ,ZL(3,12,10 F 5

10) ,NZ(15) ,D6(15) ,N7(15) ,D8(15) ,C(15) F 6

DIMENSION FCK12 ( 100 ) , FCK23 ( 100 )

, FCK13 ( 100 ), OME ( 12 , 100

)

F 7

COMMON N1,N2,N3,N4,A,WAV,D5,ZL,NZ,D6,N7,D8,C/B126/FCK12,FCK23,FCK1 F 8

13, OME F 9

PRA=0 .0 F 10

UO=WAV ( KX , JX ) /WAA F 11

EM=12.398/WAV(KX,JX) F 12

AM=0.0 F 13

AMLW=0 .0 F 14

DO 10 J=JO,JM F 15

I=NZ(J) F 16

IF (I.EQ.JX) CX=C(J) F 17

RZ=NZ(J) F 18

AM=AM+C ( J

) *RZ/A ( I

)

F 1

9

F 20

CALCULATE MEAN IONIZATION POTENTIAL - J F 21

F 22

ZJ=9.76*RZ+58.5*(RZ**(-.19)

)

F 23

AMLW=AMLW+C(J)*RZ/A(I)*AL0G(1166.*EM/ZJ) F 24

CONTINUE F 25

F 26

ALW = LN(V) AW = V ALUOW = LN(UOV) = LN(UO) + LN(V) F 27

F 28

ALW=AMLW/AM F 29

AW=EXP ( ALW

)

F 30

ALUOW=ALOG(UO)+ALW F 31

Tl=( EI (ALUOW) -EI (ALW) ) *ALW/AW F 32

|C F 33

b T2 = THE INTEGRATED VOLTAGE DEPENDENT TERM F 34

]C F 35

T2=(U0-1.-T1)*0.76*0.5/AM F 36

IF (KX.GT.4) GO TO 90 F 37

GO TO (20,30,40,60) , KX F 38

F 39

SELECT THE NUMBER OF ELECTRONS (ZNL) AND THE FLUORESCENT YIELD F 40

(OMED) FOR THE ABSORPTION EDGE PRODUCING THE MEASURED LINE F 41

C F 42

K-EDGE F 43

F 44

ZNL=2

.

F 45

OMED=OME ( KX , JX

)

F 46

GO TO 130 F 47

F 48

41
c Ll-EDGE

c30 ZNL=2

.

0MED=0ME(KX,JX)

GO TO 130

C

C L2-EDGE

C40 ZNL=2

.

IF ( WAV (KX-l.JX) .GT.WAA) GO TO 50

OMED=OME(KX,JX)

GO TO 130

50 OMED=OME ( KX , JX ) * ( 1 . +FCK12 ( JX )

)

GO TO 130

C

C L3-EDGE

C60 ZNL=4

.

IF ( WAV (KX-2.JX) .GT.WAA) GO TO 80

IF ( WAV (KX-l.JX) .GT.WAA) GO TO 70

OMED=OME ( KX , JX

)

GO TO 130

70 OMED=OME ( KX , JX ) * ( 1 . +0 . 5*FCK23 ( JX )

)

GO TO 130

80 OMED=OME ( KX , JX ) * ( 1 . +0 . 5*FCK13 ( JX ) +0 . 5*FCK23 ( JX ) * ( 1 . +FCK12 ( JX ) )

)

GO TO 130

C

C M-EDGES NONE OF THE LISTED LINES ARE PRODUCED BY THE Ml AND

C M2-EDGES

C90 KXX=l+(KX-4)/3+(KX-4)/5

OMED=OME(KX,JX)

GO TO (100,110,120) , KXX

C

C M3-EDGE

C100 ZNL=2.

GO TO 130

C

C M4-EDGE

C110 ZNL=4.

GO TO 130

C

C M5-EDGE

C

120 ZNL=6.

130 T0=0MED*ZL ( LX , KX , JX ) *CX*ZNL/A ( JX

)

PRA=T0*T2

RBA=0.0

DO 140 J=J0,JM

Z=NZ(J)

RBA=RBA+RB ( UO , Z

) *C ( J

)

140 CONTINUE

C

C PRA = THE PRIMARY INTENSITY

C

PRA=PRA*RBA

RETURN

END

FUNCTION RB (UO,Z)

42

F 49

F 50

F 51 [ F 52

F 53

F 54

F 55

F 56

F 57

F 58

F 59

F 60

F 61

F 62

F 63

F 64

F 65

F 66

F 67

F 68

F 69

F 70

F 71

F 72

F 73

F 74

F 75

F 76

F 77

F 78

F 79

F 80

F 81

F 82

F 83

F 84 I F 85 I F 86

F 87

F 88 fl 89

90

F

F

F 91 I F 92

F 93

F 94

F 95

F 96

F 97 : F 98 i F 99 : F 100

F 101

F 102

F 103

F 104

F 105

F 106

F 107 \ F 108-

G 1
G 2

CALCULATION OF BACKSCATTER -R- FROM DUNCUMBS FIT . G 3

G 4

W=l . /UO G 5

W2=W**2 G 6

W3=W**3 G 7

W4=W**4 G 8

W5=W**5 G 9

Z1=1.0E-02*Z G 10

Z2=1.0E-04*Z**2 G 11

Z3=1.0E-06*Z**3 G 12

Z4=l . 0E-08*Z**4 G 13

Z5=1.0E-10*Z**5 G 14

Rl=-. 581+2. 162*W-5.137*W2+9.213*W3-8.619*W4+2.962*W5 G 15

R2=-l. 609-8. 298*W+28.791*W2-47.744*W3+46.540*W4-17.676*W5 G 16

R3=5.4+19.184*W-75.733*W2+120.05*W3-110.7*W4+41.792*W5 G 17

R4=-5. 725-21. 645*W+88.128*W2-136.06*W3+1 17. 75*W4-42.445*W5 G 18

R5=2. 095+8. 947*W-36.51*W2+55.694*W3-46.079*W4+15.851*W5 G 19

RB=1.+R1*Z1+R2*Z2+R3*Z3+R4*Z4+R5*Z5 G 20

RETURN G 21

END G 22-

FUNCTION FQI ( JO . JM , LX, KX , JX, WAA) H 1 H 2

CALCULATION OF F(X) H 3

H 4

DIMENSION N2 ( 15 ), N4 ( 15 ), A ( 100 ), WAV ( 12 , 100 ) , CL ( 3 , 12 , 100 ), D5 ( 3 , 12 , 10 H 5

10) ,NZ(15) ,D6(15) ,N7(15) ,C0SEC(15) ,C(15) H 6

COMMON N1,N2,N3,N4,A,WAV,CL,D5,NZ,D6,N7,C0SEC,C . H 7

H=0 .0 H 8

ACS=0 .0 H 9

DO 10 JJ=J0,JM H 10

J=NZ(JJ) H 11

IF (J.EQ.JX) JJA=JJ H 12

RZ=NZ(JJ) H 13

H 14

CALCULATE THE H OF PHILIBERT AND THE VALUE OF CHI H 15

H 16

H=H+1.2*A(J)*C(JJ)/RZ**2 H 17

ACS=ACS+C(JJ)*AC(CL(LX,KX,JX) , J

) *COSEC ( JJA ) H 18

CONTINUE H 19

H 20

CALCULATE SIGMA AND FQI = F(X) H 21

H 22

SIG=4.5E05/( (12.398/WAA)**1.65-(12.398/WAV(KX, JX) )**1.65) H 23

FQI=(1.+H)/(1.+ACS/SIG)/(1.+H*(1.+ACS/SIG) ) H 24

RETURN H 25

END H 26-

SUBROUTINE FCA ( JO . JM , II , JJA , FCA1 . FCAO ) I 1 I 2

CALCULATION OF CHARACTERISTIC FLUORESCENCE CORRECTION I 3

I 4

DIMENSION N2(15) ,N3(15) ,A(100) ,WAV(12,100) , CL( 3 , 12 , 100 ) ,ZL(3,12,10 I 5

10) ,NZ(15) .D4(15) ,LA(15) ,C0SEC(15) ,C(15) ,CA(2) , CB ( 2 ) I 6

DIMENSION KB (120, 15) ,LB(120,15) ,WA(15) ,KA(15) , JJJ(120,15) ,RRR(120. I 7

115) ,0MEG(120,15) I 8

DIMENSION FQF ( 15 ) , FCF ( 15 ) . FCASF ( 15 )

, RPF ( 15

)

19

COMMON N1,N2,NIT,N3,A,WAV,CL,ZL,NZ,D4,LA,C0SEC,C/B5P/WA,KA,JJJ,RRR I 10

1.0MEG/B1289/KB.LB/BLT/L0UT, FQF, FCF, FCASF, RPF I 11

FORMAT (/13X,5HWAVE=,F7.5,4X,5HACBA=,F6.1,4X,4HACA=,F6.1,4X,4HACB= I 12

1,F6.1/(30X,4HCAI=.E10.5,14X,4HCBI=,E10.5) ) I 13

NZA=NZ(JJA) I 14

43
KAA=KA(JJA)

LAA=LA ( JJA

)

JJB=JJJ(II,JJA)

NZB=NZ(JJB)

KBB=KB(II,JJA)

LBB=LB(II,JJA)

CC ABSORPTION COEFFICIENT OF EXCITING RADIATION (B) IN THE

C EXCITED ELEMENT (A)

C ACBA=AC ( CL ( LBB , KBB , NZB )

, NZA

)

H=0.0

ACA=0 . ACB=0 . DO 20 I=JO,JM

J=NZ ( I

)

RJ=J

C

C ABSORPTION COEFFICIENT OF RADIATION A IN THE SAMPLE X COSEC

C THETA

C ACA=ACA+AC ( CL ( LAA , KAA , NZA ) , J

) *COSEC ( J JA ) *C ( I

)

C

C ABSORPTION COEFFICIENT OF RADIATION B IN THE SAMPLE

C ACB=ACB+AC ( CL ( LBB , KBB , NZB )

, J

) *C ( I

)

H=H+1 . 2*A ( J

) *C ( I ) /RJ**2

20 CONTINUE

SIG=4.5E05/( (12.398/WA(JJA) ) **1 . 65- ( 12 . 398/WAV (KBB, NZB ) )**1.65)

FQIB= ( 1 . +H ) / ( 1 . +ACA/SIG ) / ( 1 . +H* ( 1 . +ACA/SIG )

)

CA(1)=(1.+H)*SIG

CA(2)=-CA(1)

CB(1)=SIG

CB(2)=SIG*(1.+1./H)

C

C CALCULATE CONSTANTS IN EQUATION AND THE PRIMARY INTENSITY

C FCA1=0MEG ( II , J JA ) *RRR ( II , J JA ) *ZL ( LAA , KAA , NZA ) *ACBA*C ( JJA ) *0 . 5*PRA

(

1 JO , JM , LBB , KBB , NZB , WA ( J JA )

)

FCA0=FCA1/ACB

C

C SI AND SO ARE THE INTEGRATIONS OF THE EMITTED AND GENERATED

C FACTORS RESPECTIVELY IN THE CRISS EQUATION

C S2=FQIB*ALOG ( 1 . +ACA/ACB ) /ACA

IF (LOUT.EQ.O) GO TO 30

IF (NIT.EQ.O) WRITE (6,10) CL(LBB, KBB , NZB) , ACBA, ACA, ACB , CA ( 1 )

, CB (

1

1) ,CA(2) ,CB(2)

30 CONTINUE

S1=0

S0=0

DO 40 1=1,2

S1=S1+CA ( I ) / ( CB ( I ) +ACA ) * ALOG ( 1 . +CB ( I ) /ACB ) /CB ( I

)

SO=SO+CA ( I ) /CB ( I

) *ALOG ( 1 . +CB ( I ) /ACB ) /CB ( I

) *ACB

40 CONTINUE

C

C FCA1 IS THE CHARACTERISTIC FLUORESCENCE FACTOR EMITTED

C FCA1=FCA1*(S1+S2)

FCAO=FCAO*(SO+FQIB)

RETURN

44

I 15

I 16

I 17

I 18

I 19

I 20

I 21

I 22

I 23

I 24

I 25

I 26

I 27

I 28

I 29

I 30

I 31

I 32

I 33

I 34

I 35

I 36

I 37

I 38

I 39

I 40

I 41

I 42

I 43

I 44

I 45

I 46

I 47

I 48

I 49

I 50

I 51

I 52

I 53

I 54

I 55

I 56

I 57

I 58

I 59

I 60

I 61

I 62

I 63

I 64

I 65

I 66

I 67

I 68

I 69

I 70

I 71

I 72

I 73

I 74

I 75
c

c

'Cc

10

1 20

60

100

END

SUBROUTINE CALI ( JO , JM, JJA , AIT , SWITCH , FQCF , FQCA)

DETERMINE SUBROUTINES TO BE CALLED TO CALCULATE THE TOTAL

EMITTED RADIATION FOR EITHER STANDARDS OR SAMPLES

,D4(12,100) ,D5(3,12,100) ,D6(3,12,10

C(15) ,FQCF(15) , FQCA (15)

,WA(15) ,KA(15) ,JJJ(120,15) , RRR (120,

INTEGER SWITCH

DIMENSION N2 ( 15 )

, MF ( 15 )

, D3 ( 100

10) ,NZ(15) ,D7(15) ,LA(15) ,D8(15)

DIMENSION KB(120,15) ,LB(120,15

115) ,0MEG(120,15)

DIMENSION IMA ( 15 ), WAVE ( 120 , 15 )

, RR ( 120 , 15 )

, EX ( 120 , 15 ), MESS ( 15 , 15

)

DIMENSION FQF ( 15 ), FCF ( 15 )

, FCASF ( 15 ), RPF ( 15

)

COMMON Nl , N2 , NIT , MF , D3 , D4 , D5 , D6 , NZ , D7 , LA , D8 , C/B1239/IMA , WAVE , RR , EX

1 , MESS/B5P/WA , KA , JJ J , RRR , 0MEG/B1289/KB , LB/BLT/LOUT , FQF , FCF , FCASF , RP

2F

COMMON /BICF/ ICOF

FORMAT ( 6X , 2HI= , 12 , 3X , 5HWAVE= , F7 . 5 , 3X , 3HRR= , F5 . 3 , 3X , 4HJJJ= , 12 , 3X ,

4

1HRRR= , F5 . 3 , 3X , 5H0MEG= , F5 . 3 , 3X , 3HEX= , F5 . 3 , 3X , 3HLB= , II , 3X, 3HKB= , II

)

FORMAT ( /5H J JA= , II , 8X , 3HRP= , E10 . 5 , 3X , 3HFQ= , F6 . 4 , 3X , 3HFC= , E10 . 5 , 3X

1,4HFQC=,F6.4/)

FORMAT ( 71X , 5HFCA1= , E10 . 5 , 2X , 6HFQCA1= , F6 . 4

)

FORMAT (//6X,5HMESS( ,I2,1H, , 12 , 2H)=, 12 , 67H . M4 EDGE IS CROSSED, ABS

10RPTI0N COEFFICIENTS MAY NOT BE MEANINGFUL ) FORMAT ( 1H+ , 19X , 5HAT NO , 13 , 4X , 5HF ( X

) = , F6 . 4 , 4X , 14HI ( P ) / ( F ( X

) *C ) = , El

10 . 5 , 4X , 10HI ( C ) /I ( P

) = , E10 . 5 , 4X , 10HF ( X )

, CONT= , F6 . 4

)

IMAM=IMA ( JJA ) +MF ( JJA

)

IF (LOUT.EQ.O) GO TO 60

IF ( NIT . EQ . ) WRITE (6,10) ( I , WAVE ( I , JJA ), RR ( I , JJA )

, JJ J ( I , JJA )

, RRR

1(1, JJA )

, OMEG ( I , JJA )

, EX ( I , JJA )

, LB ( I , JJA )

, KB ( I , JJA )

, 1=1 , IMAM

)

GO TO 60

CONTINUE

DO 70 JJ=JO,JM

IF (MESS (J J, JJA) .EQ.O) GO TO 70

WRITE (6,40) J J, JJA, MESS (J J, JJA)

CONTINUE

IF (ICOF.EQ.l) GO TO 80

FC = EMITTED RADIATION DUE TO FLUORESCENCE BY THE CONTINUUM

CALL CO (JO, JM,JJA,FC,FQC)

GO TO 90

FC=0.

FQC=0

.

RPO=PRA ( JO , JM , LA ( JJA ), KA ( JJA

;

FQ=FQI ( JO , JM , LA ( JJA ), KA ( JJA )

,

,NZ(JJA) ,WA(JJA

NZ(JJA) ,WA(JJA)

RP = THE EMITTED PRIMARY INTENSITY

RP=RPO*FQ

FCAS=0.0

FCASO=0 . IF (LOUT.EQ.O) GO TO 100

WRITE (6,20) JJA,RP,FQ,FC,FQC

CONTINUE

FCFX=FC/RP

RPFX = THE NORMALIZED GENERATED PRIMARY INTENSITY

RPFX=RP/(FQ*C(JJA)

)

I 76

J 1 J 2

J 3

J 4

J 5

J 6

J 7

J 8

J 9

J 10

J 11

J 12

J 13

J 14

J 15

J 16

J 17

J 18

J 19

J 20

J 21

J 22

J 23

J 24

J 25

J 26

J 27

J 28

J 29

J 30

J 31

J 32

J 33

J 34

J 35

J 36

J 37

J 38

J 39

J 40

J 41

J 42

J 43

J 44

J 45

J 46

J 47

J 48

J 49

J 50

J 51

J 52

J 53

J 54

J 55

J 56

J 57

J 58

J 59

J 60

45
C WRITE DATA FOR STANDARDS

C IF (SWITCH. EQ.O) WRITE (6,50) NZ( JJA) , FQ ,RPFX, FCFX.FQC

FQCF(JJA)=FQC

FQF(JJA)=FQ

FCF(JJA)=FC

RPF(JJA)=RP

IF (MF( JJA) .EQ.O) GO TO 130

CC CHARACTERISTIC FLUORESCENCE CORRECTION - THE VALUE OF MF IS SET

C IN SUBROUTINE COCA

C I0=IMA(JJA)+1

DO 120 1=10, IMAM

CALL FCA (J0,JM,I,JJA,FCA1,FCA0)

IF (FCA0.LT.1.0E-12) GO TO 110

FQCA1=FCA1/FCA0

IF (LOUT. EQ.O) GO TO 110

WRITE (6,30) FCA1.FQCA1

110 CONTINUE

C

C FCAS = TOTAL EMITTED RADIATION DUE TO CHARACTERISTIC

C FLUORESCENCE

C FCAS=FCAS+FCA1

FCASO=FCASO+FCAO

120 CONTINUE

IF (FCAS0.LT.1.0E-12) GO TO 130

FQCA ( JJA ) =FCAS/FCASO

CC AIT = TOTAL EMITTED RADIATION

C

130 AIT=RP+FC+FCAS

FCASF(JJA)=FCAS

RETURN

END

FUNCTION EI (X)

C

C EXPONENTIAL INTEGRAL

C

EI=ALOG(X)+X

TN=X

N=l

10 AN=N

IF (AN.GT.2.*X.AND.TN.LT.l.E-06) GO TO 20

TN=TN*X*AN/ ( AN+1 .

) **2

EI=EI+TN

N=N+1

GO TO 10

20 RETURN

END

FUNCTION FT (X)

C

C CALCULATION OF THE INTEGRAL OF THE CONTINUUM FLUORESCENCE

C CORRECTION FOR GENERATED X-RAYS

C DIMENSION Dl(15) ,D2(15)

COMMON /B34/ Dl ,D2 , WAA ,DACS/B410/ACAX, ACX

FT=ACAX/ACX* ( X-WAA ) /WAA/X/X

RETURN

END

46

J 61

J 62

J 63

J 64

J 65

J 66

J 67 i J 68

J 69

J 70

J 71

J 72

J 73

J 74

J 75

J 76

J 77

J 78

J 79

J 80

J 81

J 82

J 83

J 84

J 85

J 86

J 87

J 88

J 89

J 90

J 91

J 92

J 93

J 94

J 95

J 96-

K 1 K 2

K 3

K 4

K 5

K 6

K 7

K 8

K 9

K 10

K 11

K 12

K 13

K 14

K 15-

L 1 L 2

L 3

L 4

L 5

L 6

L 7

L 8

L 9

L 10-'
1 90

100

110

C

C

C

SUBROUTINE AINP ( C , IJ , UMAX, SD)

DOUBLE PRECISION SUM , SSUM , SUMS , FSD

DIMENSION NUMBER ( 1000 )

, NTABL ( 100 , 8 ), MCO ( 8 ), NCODE ( 100 ,7

)

, T ( 100 ), ICO

IN ( 7 , 2 )

, NSTD (12,2,7,2), SNSTD (7,2), CSTD (7,2), SPH (5,7), SPHC ( 5 , 7 )

, PHC

(

2100,7) ,C( 100,6) ,PEAK(7) ,KK(5) ,TIM(5) ,SD( 100,7) , SUM (6, 7) , SSUM (6, 7)

,

3SUMS(6,7) , FSD (6, 7)

COMMON /BINPT/ III.NTAPE

FORMAT (412,15,12)

FORMAT (1018)

FORMAT ( 1H0 , 8HELEMENT

1.11H POINTS IS ,F6.4)

FORMAT ( 1H0 , 8HELEMENT

10R , I2,11H POINTS IS

111=510

NIND=0

.I1.34H - STANDARD DEVIATION OF PEAK FOR ,12

,I1,40H - STANDARD DEVIATION OF BACKGROUND F

F6.4)

READ DATA FROM MAG TAPE AND SET UP TABLE

READ (5,10) NCNM.NTC NAV,ND,NTRA

NC1=NC-1

NS=6

M=NC*ND

IF (NTC.EQ.l) GO TO 60

INDEX=0

CALL PTLOAD (NUMBER, M, NS ,NC

)

DO 50 NR0W=1,ND

DO 50 NC0L=1,NC

INDEX=INDEX+1

NTABL ( NROW , NCOL ) =NUMBER ( INDEX

)

CONTINUE

GO TO 80

READ DATA FROM CARDS AND SET UP TABLE

DO 70 NR0W=1,ND

READ (5,20) (NTABL (NROW, NCOL

CONTINUE

CONTINUE

NC0L=1,NC

SEPARATE MANUAL DATA ENTRY INTO 6 VARIABLES

DO 110 NR0W=1,ND

MCO ( 1

) =NTABL ( NROW , NM

)

DO 100 1=1,6

IK=6-I

IF (MCO(I) .EQ.O) GO TO 90

ITEN=10**IK

IF (MCO(I) .LT.ITEN) GO TO 90

NCODE ( NROW , I

) =MCO ( I ) /ITEN

IF (IK. EQ.O) GO TO 100

I) -NCODE (NROW,

I

MC0(I+1)=MC0 ITEN

GO TO 100

NCODE (NROW, I )=0

MC0(I+1)=MC0(I)

CONTINUE

CONTINUE

SET UP TABLES OF STANDARDS AND BACKGROUNDS FOR ALL ELEMENTS

DO 130 1=1, NCI

DO 120 J=l,2

M 1 M 2

M 3

M 4

M 5

M 6

M 7

M 8

M 9

M 10

M 11

M 12

M 13

M 14

M 15

M 16

M 17

M 18

M 19

M 20

M 21

M 22

M 23

M 24

M 25

M 26

M 27

M 28

M 29

M 30

M 31

M 32

M 33

M 34

M 35

M 36

M 37

M 38

M 39

M 40

M 41

M 42

M 43

M 44

M 45

M 46

M 47

M 48

M 49

M 50

M 51

M 52

M 53

M 54

M 55

M 56

M 57

M 58

M 59

M 60

M 61

47
SUM(I,J)=Q

SSUM(I,J)=0

ICON(I,J)=0

SNSTD(I,J)=0

120 CONTINUE

130 CONTINUE

DO 160 NR0W=1,ND

DO 150 1=1, NCI

PHC(NROW,I)=0

IF (NCODE(NROW.I) . EQ . ) GO TO 150

T(NR0W)=1000

IF (NCODE(NROW.I) .NE.6) GO TO 140

IC0N(I,1)=IC0N(I,1)+1

IIC=IC0N(I,1)

NSTD(IIC,1,I,1)=NR0W

NSTD ( I IC , 2 , I , 1

) =NTABL ( NROW I)

GO TO 150

140 CONTINUE

IF ( NCODE ( NROW , I ) . NE . 7 ) GO TO 150

IC0N(I,2)=IC0N(I,2)+1

IIC=IC0N(I,2)

NSTD(IIC,1,I,2)=NR0W

NSTD (110,2,1,2) =NTABL ( NROW I)

150 CONTINUE

160

C

C

c

CONTINUE

AVERAGE STANDARDS AND B^ICKC;rou

170

180

190

200

CC

C

DO 200 1=1, NCI

DO 190 J=l,2

IF (ICON (I, J) .EQ.O) GO TO 190

IC=IC0N(I,J)

DO 170 IIC=1,IC

NR=NSTD(IIC,1,I,J)

SNSTD ( I , J

) =SNSTD ( I , J ) +NSTD ( IIC , 2 . I , J ) / ( ICON ( I , J

) *T ( NR

'

SUM ( I , J

) =SUM ( I , J ) +NSTD ( I IC , 2 , I , J

)

SSUM ( I , J

) =SSUM ( I , J ) +NSTD ( IIC , 2 , I , J

) **2

CSTD ( I , J

) =DEADT ( SNSTD ( I , J

)

IF (NTRA.EQ.l) GO TO 170

CSTD ( I , J

) =CSTD ( I , J

) *T ( NR

)

CONTINUE

SUMS ( I , J )=SUM ( I , J

) **2/IC0N

ASUM=SUM ( I , J ) /ICON ( I , J

)

STSD=DSQRT ( ( SSUM ( I , J

) -SUMS

SSTSD=SNGL(STSD)

IF (J.EQ.2) GO TO 180

I,ICON(I,

I.NIND

(i,j:

I , J ) ) / ( ICON ( I , J

) -1 ) ) /ASUM

WRITE (6,30)

GO TO 190

WRITE (6,40)

CONTINUE

CONTINUE

IF (NAV.EQ.l GO TO 260

SSTSD

I, ICON (I, J) , SSTSD

AVERAGE DATA ON INDIVIDUAL PHASES AND CORRECT FOR DEAD TIME

DO 250 I DO 210 J KK ( J

) =0

SUM(J,I)=0

SSUM(J,I)=0

TIM(J)=0

NCI

5

M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

M 100

M 101

M 102

M 103

M 104

M 105

M 106

M 107

M 108

M 109

M 110

M 111

M 112

M 113

M 114

M 115

M 116

M 117

M 118

M 119

M 120

M 121

M 122

62

63

64

65

66

67

68

69

70

71

72

73

74

75

76

77

78

79

80

81

82

83

84

85

86

87

88

89

90

91

92

93

94

95

96

97

98

99

48
SPH(J,I)=0

CONTINUE

DO 220 NR0W=1,ND

IF (NCODE (NROW, I) . EQ . . OR. NCODE( NROW , I ) .GE.6) GO TO 220

J=NCODE(NROW,I)

KK(J)=KK(J)+1

SPH ( J , I

) =SPH ( J , I ) +NTABL ( NROW , I ) /T ( NROW

)

TIM(J)=TIM(J)+T(NROW)

SUM ( J , I

) =SUM ( J , I ) +NTABL ( NROW , I

)

SSUM ( J , I

) =SSUM ( J , I ) +NTABL ( NROW , I

) **2

CONTINUE

DO 240 J=l,5

SPHC ( J , I

) =DEADT ( SPH ( J , I ) /KK ( J )

, I , NIND

)

IF (NTRA.EQ.l) GO TO 230

SPHC ( J , I

) =SPHC ( J , I

) *TIM ( J ) /KK ( J

)

CONTINUE

SUMS ( J , I

) =SUM ( J , I

) * *2/KK ( J

)

ASUM=SUM ( J , I ) /KK ( J

)

FSD(J,I)=DSQRT( ( SSUM( J , I )-SUMS ( J , I ) )/(KK(J)-l) )/ASUM

CONTINUE

CONTINUE

GO TO 290

CORRECT FOR DEAD TIME ON INDIVIDUAL DETERMINATIONS

DO 280 NR0W=1,ND

DO 270 1=1, NCI

IF ( NCODE ( NROW , I ) . EQ . . OR . NCODE ( NROW , I ) . GE . 6 ) GO TO 270

PHC ( NROW , I

) =DEADT ( NTABL ( NROW , I ) /T ( NROW ) , I , NIND

)

IF (NTRA.EQ.l) GO TO 270

PHC ( NROW , I

) =PHC ( NROW , I

) *T ( NROW

)

I? 270 CONTINUE

| 280 CONTINUE

290 CONTINUE

C PEAK MINUS BACKGROUND ON STANDARDS

C DO 300 1=1, NCI

IF (ICON(I.l) .EQ.O) GO TO 300

PEAK ( I

) =CSTD (1,1) -CSTD (1,2)

300 CONTINUE

IF (NAV.EQ.l) GO TO 340

C

C DETERMINE K-VALUES ON AVERAGED PHASES

C

IJ=0

DO 330 J=l,5

11=0

DO 320 1=1, NCI

IF (SPHC(J.I)) 320,320,310

310 11=11+1

IF (II.EQ.l) IJ=IJ+1

C ( IJ , II ) = ( SPHC ( J , I

) -CSTD (1,2)) /PEAK ( I

)

SD(IJ,II)=SNGL(FSD(J,I)

)

IIMAX=II

320 CONTINUE

330 CONTINUE

RETURN

C

C DETERMINE K-VALUES ON INDIVIDUAL POINTS

C

49

M 123

M 124

M 125

M 126

M 127

M 128

M 129

M 130

M 131

M 132

M 133

M 134

M 135

M 136

M 137

M 138

M 139

M 140

M 141

M 142

M 143

M 144

M 145

M 146

M 147

M 148

M 149

M 150

M 151

M 152

M 153

M 154

M 155

M 156

M 157

M 158

M 159

M 160

M 161

M 162

M 163

M 164

M 165

M 166

M 167

M 168

M 169

M 170

M 171

M 172

M 173

M 174

M 175

M 176

M 177

M 178

M 179

M 180

M 181

M 182

M 183
340 IJ=0

DO 370 NR0W=1,ND

11=0

DO 360 1=1, NCI

IF (PHC(NROW.I)) 360,360,350

350 11=11+1

IF (II.EQ.l) IJ=IJ+1

C(IJ,II)=(PHC(NR0W,I)-CSTD(I,2) )/PEAK(I)

SD ( I J , II ) =1 . /SQRT ( NTABL ( NROW , I )

)

IIMAX=II

360 CONTINUE

370 CONTINUE

RETURN

END

FUNCTION DEADT ( COUNT, IND, NIND)

C

C CORRECTION OF RAW INTENSITIES FOR DEAD TIME

C THIS MAY BE EXPANDED TO INCLUDE MORE THAN THE TWO DETECTORS

C USED HERE

C

C IF (NIND.EQ.O) GO TO 10

IF ( IND. GT. NIND) GO TO 20

CC DEAD TIME OF FIRST DETECTOR

C

10 DEADT=C0UNT/(l-C0UNT*3.2E-6)

NIND=IND

GO TO 30

CC DEAD TIME OF SECOND DETECTOR

C20 DEADT=C0UNT/(l-C0UNT*2.8E-6)

30 CONTINUE

RETURN

END

SUBROUTINE PTLOAD (NUMBER ,M,NS , NC)

C

C READ DATA FROM MAGNETIC TAPE - USE ONLY ON UNIVAC 1108 F0RTRAN5

C COMMON /BINPT/ III.NTAPE

DIMENSION NUMBER ( M )

, NEOF ( 1 ) /056/ , ICHK ( 2 ) /040 , 015/ , IOCT ( 10 ) /060 , 061

1 , 062 , 063 , 064 , 065 , 066 , 067 , 070 , 071/

DIMENSION FMT(2)

DATA (FMT(I) ,1=1,2)/' ( 1H , ','9110) '/

10 FORMAT (1H1)

20 FORMAT ( 1H0

)

30 FORMAT ( 1H0 , 20X , 'DIRECT LISTING OF DATA INPUT VIA PTLOAD')

IF (NC.GT.5) GO TO 40

FLD(0,6,FMT(2) )=FLD( 6*NC , 6

, "012345'

)

GO TO 50

40 FLD(0,6,FMT(2) )=FLD( 6* (NC-6) ,6, '678900'

)

50 CONTINUE

DO 60 1=1,

M

60 NUMBER(I)=0

NTAPE=7

WRITE (6,10)

DO 130 1=1,

M

70 IC0UNT=1

DO 120 J=1,NS

NCHK=0

50

M 184

M 185

M 186

M 187

M 188

M 189

M 190

M 191

M 192

M 193

M 194

M 195

M 196

M 197-

N 1 N 2

N 3

N 4

N 5 N 6

N 7

N 8

N 9

N 10

N 11

N 12

N 13

N 14

N 15

N 16

N 17

N 18

N 19

N 20

N 21

N 22-

12

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

21

22

23

24

25
80

90

100

110

140

150

160

170

C

C

180

190

200

210

10

20

CALL CHAR (N)

IF (N.EQ.NEOF(l) ) GO TO 140

IF ( . NOT . ( N . EQ . ICHK ( 1 ) . OR . N . EQ . ICHK (

2

IF (ICOUNT.EQ.l) GO TO 80

IF (IC0UNT.EQ.2) GO TO 70

NUMBER ( I )=0

GO TO 130

DO 100 K=l.ll

IF (K.EQ.ll) NCHK=NCHK+1

IF (NCHK.EQ.5000) GO TO 160

IF (K.EQ.ll) GO TO 80

IF (N.EQ.IOCT(K) ) GO TO 110

CONTINUE

NUMBER ( I

) =NUMBER ( I

) *10+IABS ( K-l

)

IC0UNT=IC0UNT+1

CONTINUE

CONTINUE

WRITE (6,30)

WRITE (6,20)

WRITE (6,FMT;

GO TO 170

FORMAT (1H , WRITE (6,150

STOP

RETURN

SUBROUTINE CHAR (N)

COMMON /BINPT/ III.NTAPE

DIMENSION INCOME (510) .INPUT (170)

IF (III.NE.510) GO TO 200

CALL NTRAN (NTAPE , 2 , 170 , INPUT, L)

IF (L+l) ,180

IF (L) 210

1=1

DO 190 J=l,510,3

INC0ME(J)=(AND(FLD(6,6,INPUT(I)

)

,7]

) GO TO 90

; NUMBER ( I ) , 1=1 , M

)

INPUT ERROR TERMINATION

*16)+(AND(FLD(0,6,INPUT(I) ) ,15)

,7)*16)+(AND(FLD(12,6,INPUT(I)

)

INCOME ( J+l ) = ( AND ( FLD (18,6, INPUT (

I

1,15))

INCOME ( J+2 ) = ( AND ( FLD (30,6, INPUT ( I ) )

, 7

) *16 ) + ( AND ( FLD ( 24 , 6 , INPUT ( I )

)

1.15))

1=1+1

111=0

111=111+1

N=INCOME(III)

RETURN

N=-l

RETURN

END

SUBROUTINE DEFTAB

FORMAT (1H1,9X, 'DEFINITIONS OF HEADINGS USED IN OUTPUT'//)

FORMAT (1H ,4X, 'F(X) ' ,16X, 'ABSORPTION CORRECTION FACTORS FOR EITHE

1R STANDARDS OR SAMPLES .

'

)

FORMAT (1H ,4X, ' I ( P)/(F(X) *C )

* ,7X, 'PRIMARY X-RAY GENERATION FROM E

1ITHER STANDARD OR SAMPLE IN PHOTONS PER ELECTRON. ')

FORMAT (1H ,4X, 'I(C)/I(P) ' ,11X, 'RATIO OF EMITTED X-RAYS EXCITED BY

1 THE CONTINUUM TO THE EMITTED PRIMARY X-RAYS .

'

)

FORMAT (1H ,4X, 'F(X) ,CONT' ,11X, 'ABSORPTION CORRECTION FACTOR FOR T

1HE CONTINUUM IN THE STANDARD .

'

)

FORMAT (1H ,4X, 'F(X) CON' ,13X, 'SAME AS F(X),CONT BUT FOR THE SAMPLE

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65

66

67

68

69

70

71

72

73

74

75

p 1 p 2

p 3

p 4

p 5

p 6

p 7

p 8

p 9

p 10

p 11

51
CHAR FLUOR MOX, 'CHARACTERISTIC FLUORESCENCE CORRE

1.')

70 FORMAT ( 1H ,4X,

1CTI0N FACTOR. '

)

80 FORMAT ( 1H ,4X, ' F(X) CHA \13X, ' ABSORPTION CORRECTION FACTOR FOR THE

1 CHARACTERISTIC FLUORESCENCE CORRECTION.')

90 FORMAT ( 1H 4X, 'AT CONC M3X, ' ATOM FRACTION .' /5X, ' WT CONC ' , 13X, 'MAS

IS FRACTION. '

)

100 FORMAT (1H ,4X,'STD DEV ', 13X, 'STANDARD DEVIATION OF THE DATA - NOT

1 USED WHEN INPUT IS INTENSITY RATIOS.')

110 FORMAT (1H ,4X, 'K( 1 )... K(N) ', 9X, ' INTENSITY RATIOS .'

)

120 FORMAT ( 1H ,4X, 'TOTAL' , 15X, ' TOTAL OF MASS FRACTIONS OBTAINED.*)

130 FORMAT ( 1H ,4X, ' ITERATION N0',8X,'THE NUMBER OF ITERATIONS REQUIRE

ID FOR CONVERGENCE. '

)

140 FORMAT ( 1H0 , 9X ,

' WHEN THE STOICHIOMETRY OPTION IS USED,'/9X,'A TABL

1E0F THE FORMULAS OF THE POSTULATED '/9X, ' COMPOUNDS , AND THEIR MASS

2FRACTI0NS AS ' /9X, ' FOUND IN THE ANALYSIS IS PRINTED OUT.'//////////

3)

WRITE (6,10)

WRITE (6,20)

WRITE (6,30)

WRITE (6,40)

WRITE (6,50)

WRITE (6,60)

WRITE (6,70)

WRITE (6,80)

WRITE (6,90)

WRITE (6,100)

WRITE (6,110)

WRITE (6,120)

WRITE (6,130)

WRITE (6,140)

RETURN

END

p 12

p 13

p 14

p 15

p 16

p 17

p 18

p 19

p 20

p 21

p 22

p 23

p 24

p 25

p 26

p 27

p 28

p 29

p 30

p 31

p 32

p 33

p 34

p 35

p 36

p 37

p 38

p 39

p 40

p 41

p 42

p 43

p 44-

52
6 . Table of Formats for the Data Input Cards

Card

no

.

When used

1 always

2 always

always

always

Cols . 1-3*

6-7*

9

11

Format

(13)

1-2* (12)

3-4* (12)

5 (11)

(12)

(ID

(ID

80 (Al)

1-80 (80H)

1-2* (12)

13-19 (F7.3)

20 (ID

Description

Number of sets of

data

Number of elements

in case

Number of elements

analyzed

Blank-input is either

k -values or known

concentrations

.

1 - input is raw data

via subroutine AINP.

Number of data points

Blank-normal output.

1 - extended output.

Blank-normal calculation

1 - omits continuum

correction

Must contain a "T"

Title - anything may

be input

Atomic number of

first element

Operating kilovoltage

Edge no. of measured

line

1 - K-edge

2 - L

T -edge

L-,

T -edge

T

, TTT -edge

1

L: M^edge

21

Right justified in field

(ID

9 - My-edge

Line no. associated

with the edge.

(Depends on element

table)

53
Card

no

.

When used Cols . Format Description

If K-edge:

1 " K

RP 1 , 3 K

ai

If L

III

1 - L

-edge

3 2

'ai

22-27

28

29

30-35 (F6.4)

5a(l) Element

was from a compound

standard

etc

.

(F6.4) Cosecant of the x-rayemergence angle.

(II) Valence of element

if one element is

calculated by stoichiometry

.

(II) If a compound standard was used, enter

the number of additional elements in

the standard.

Any number greater

than will fix the

concentration of this

element at whatever

value is entered on

cards 6a.

(F5.4) If present, must

follow card 4.

Weight fraction of

analyzed element in

compound standard.

54
Card

no

.

5a(2)

5b

6a

!6b(l)

When used

Element

was from a compound

standard

Element

not analyzed

Cols .

l-2 ;

3-7

1-2*

13

14-1

Format

(12)

(F5.4)

(12)

(ID

(F5.4)

Input is

either

k-values

or weight

fractions

Input is

raw data

through

subroutine

AINP

Fields (15F5.4)

of 5

1-2* (12)

Description

If present, must

follow card 5a(l).

Atomic number of

the other element in

compound standard.

Weight fraction of

this element in

compound standard.

If more than one

other element is

present, this card

is repeated for each.

Atomic number of

element

.

Valence if calculation by stoichiometry

.

Blank - calculate by

difference or stoichiometry . If the weight fraction

of the element is input here, the program

will calculate the

k-value of all other

elements

.

The k-values or weight

fractions for each

element (element 1 is

first, etc.) measured

are on one card for

each point

.

Total number of

6-digit numbers

read per data point.

(Max. of 6 plus code)

55
Card

no. When used Cols . Format Description

3-4* (12) Position of manual

data entry for

coding the line.

(Must be in the

last position.)

5-6* (12) Blank - read data

from magnetic tape

via PTLOAD (or tape

reading routine supplied by user)

.

1 - read data from

cards

.

7-8* (12) Blank - average data

on each phase for

each element.

1 - do each point

individually

.

9-13* (12) Total number of lines

of input including

all samples and

standards

14-15* (12) Blank - calculate

k-values from total

counts

.

1 - calculate k-values

from counts/sec.

If the data are read from magnetic tape, no further

cards are required.

6b (2) Raw data 10 (1018) Raw data for each

element, time in

seconds, and the

data code. (Data

code must be in

last column used.)

Raw data 10

read from fields

cards . One of 8*

card per

line

.

56
6b Data POINT 2 6b Data POINT 1

5c Element 6 (Done

by difference or

s toichiometry)

4 Element 5

5b2 "not analyzed"

element in compound

standard

5bl Element 4 in com- pound standard

4 Element 4, Compound

standard with 1 other

element

5b3 Second "not analyzed'

element in compound

standard

5b2 First "not analyzed" ele

ment in compound standard

5b 1 Element 3 in compound

standard

4 Element 3, compound standard with

2 other elements

4 Element 2 4 Element 1

3 Title

2 No. of elements, etc

1 No. of cases

Example of the order of the data input cards

57
Options for Data Code

for Raw Data

(Subroutine AINP)

The program is set up to handle six entries besides the data

code. The data code is a 6 -digit number and must be the last

entry on any line. Each digit in the code identifies the

contents of its respective field. (If the data code is in

the seventh field, the first digit refers to the first field,

the second to the second field, etc.) The codes have the

following meanings

Code Contents of field contains

Field empty or deleted in this

calculation.

1-5 Data from specimen. (May code

up to 5 separate points. See

below .

)

6 Standard for an element.

7 Background for an element.

8 Not used at present.

9 Time in tenths of seconds.

Data may be sent from AINP to the main program in two different ways. First, it may send k-values for each element

at each data point in the usual manner. Or, if each phase

analyzed was coded with a different number (phases 1-5

above) , the program will calculate an average k-value for

each element in each point and send only these values to the

main program.

The data need not be in any specific order. All measurements

on any particular standard will be averaged (peak value or

background) . Specimens and standards may be mixed in any

way provided the data code is correct.

58
(A

<U

<AnJUmou

,0s

59
T3

5-1

CJ

(I) rH

4-> •HH

a:

aJ T3

Piod

+-> CO

4-> Pi

CD m

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P! •HM

aJ5h

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t+H LHoI CD

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T— CS

*— CM

«— CNJ

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60
Pi

CDS

•Huo

4-> rt

CD6

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t3

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rHn

CO

+-> P!CDS

<d

pi m

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CO s

rt 3 U f

-4

— — CNI

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aJ 'd

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aJ

+-> inno

p<eo

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U)

fi

0)

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p<

rt 1 d

p!

n3

4->

C/)

P)H

T3

f-. 4-> OJ Pi T3 CD

Pi 6

CTJ CD P t-H W CD a V) •H • HX +-> P

PI

0) <-H S O

CD i—1 a

CD o• H ^ -P

<DJ ^ rt •M M O 4-1 m f) o /)

tj fH 3 o

X3S^

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4-) aj

r--

<u

^?^

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|J".

'•.I..' 0..I

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V^X.

63
urr P

CD6

CD

r-HCDOmo

•HP

OJ P

•H

to

Pi

CD PC

•H

64
7 . Outputs

The output originally developed, now called output 2,

or extended output, contains many intermediate results and

parameters that are very useful for troubleshooting. For

routine analysis, a more compact output called output 1 was

developed. This output is also printed when output 2 is

requested.

Notes on output 1:

1. It is preceded by a title, which can contain whatever

identification is desired.

2. "Sample No" X gives the number of the "case".

3. Then follows information on the standards:

F(X) is the attenuation factor for p

I(P)/(FX(X)*C) gives the emitted pri

(photons/electron) divided by the

the emitting element. The ratio o standard over the term for the spe

"atomic number correction factor".

I(C)/I(P) gives the ratio of the emi

excited by the continuum, over the

intensity

.

F(X) ,CONT is

escence by

rimary emission,

mary intensity

mass fraction of

f this term for the

cimen gives the

tted fluorescence

emitted primary

the absorption correction factor for fluor

the continuum

After indicating the number of the p

prints the k-values for the elements

is printed here for elements determi

stoichiome try

.

oint, the program

measured. Nothing

ned by difference or

Next, the results are printed, one line per element.

The headings have the same meaning as for the standards

with the following additions:

AT CONC

WT CONC

STD DEV

Atomic fraction of element in the specimen

Weight fraction

If there are several measurements made per

point, the true calculated standard deviation is given. If the number of counts for

a single measurement is input, the standard

deviation is estimated, assuming a Poissonian

distribution. If k-values are input, the

program prints .0000.

65
CHAR FLUOR: Emitted fluorescent intensity due to

characteristic excitation, divided by

emitted primary intensity.

F(X) CHA: Absorption correction factor for fluorescence due to characteristic lines.

If there is no characteristic fluorescence of an element, nothing is printed under CHAR FLUOR or F(X) CHA.

5. After the list by elements, the total of calculated mass

fractions is printed, as well as the number of iterations

In the case of stoichiometry , a table of mass fractions

of oxides is appended.

Notes on output 2:

1. Output 2 is requested by punching 1 in column 9 of

card 2 as described under Table of Options for Sample

Input

.

2. The variable names printed out by output 2 are the same

as those in the Index of Terms and Symbols.

66
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c> \D

n C3- IT. z • • »- cr a ^ • * »- cr cr

ii ii z <v r- il II ZWN i UI t- a. «^ UI Z»* CJ s z «— M y

»-,ww UJ Z> 13 l-H

w ui 3 3 o y y _J «J « e> v: y _l o «r a. Ui 0. UI o

CO z o Ita — z T CJ 3 o o o c. c c •H (J <v cc • 1- • » w <

1- or i

II_1<

UI

1-

u NO 3" »- z cc ^i c c F) »D »- €J 3- IT • • 1-<

«\

m 3 vD CNJ a- in O c\j r- Z • • 1- o- oII 11 z cj r» : . -, UJ Z -i CJ 5

h-

—— UI 3 3 O X- V _l l_> < a UJ

68
v ir e 9-1 dr^ r-- n iC IT

IT r^ o c fIT f ir n r» • • • • •

II ii ii li ii H i- K 1- K

*- ;^r 2 z 2 C c e c C o <j «_ o o m » 4k * »

** ««. «-h *•* *^ X X X X >

<k- *»*-

w >fc- *—

L. li. li.

u

u

T

—

cj er — a x ir

oro CM 3 c

-J I U C\j cv a m. < vC a *-

<_< •

l«- f CV (V

—> c C C o c 1 1 1 1 1 vC ir a d- r~- C O r- c rd- o o ~ ec

cr- f- h- or. *

oc o .- •-< cv • • • • •

II it li it li •— *-* **** c— -^. D a o a d

** ^* *-* <**

w

»-* «-< »-i t-l

n

V ^vVVV

—

^^ ** *—. <-^ <- C; <_ o <-

z. o cr- it tv oh tf c — f^ -t K X Kj IO IC a c o o

*-

C

—M

<-

C lf;»<

CK

O CV

—

H <V ^

c\ Cv CV cv. cv c c c c c 1 1 1 1 1 3- c\ r- f IO c c\ C vC

co c\. r- CVj vO c .- c <? o a vc vf —

II II II II

<J «_! CJO «_) * * * * *

*». ^» ^"» *— **-. X > > > X w »* *- *— s—'

Li

uuLU

^* ^- ^~ •w

W

VVV

V \ — *-» *-. ** ** Q_ a o. cl a w ^ ^» w w - - r-< i— *~ cv. cv. ^ x c o c- — I I I u r- .- c — ^ cr cr sir ir s-

-- c ^ r>- a. c \C -* x i^ ir r- >- r-- vd i^- u. a- ir c >C

CVJ

K5

c\cr »- • T.

ii

n > — u x _i ~ Ll' L ac

li.

cv c

<r

C

C

<r

f- IvD 1^

IT- »h CVJ ^ r^

vc cr I- • * ^*

Li'

ll II 5 X X _l w — Ll

Ll U. crO

li.

CV. ^ c

-h «-i tr< c c c 2 ? Z

<r

K I- k- c <: u*

cc vD

r- IT n ii X X U Ll ^ CVJ c o

z: z < c

>ccc

C

U' c c cr c C c c c o c o c c c • • • • 1- IT L0

IT.

C

U IT

C IO co to 2 ~ •— «-t ^ \C c o c vr a r<- «-. c »-* u o — — i" • 1- 9 • • • <r

1-3 Ll o Q o ^o ^ r- vo J if «- cv c c r s ir s c c — cr ir

O

3.

1/1

CV' \S

V > r o c p > t_

O <3 3 Q

Ll! <

_l »-

CL l/l 2 < r.

>- c

LT

O

<\ ^ r

—i -i vO o

IT'

— cv

C - r* ~4 • •• - c. if vr cr

ii ii ii r~

—• —i -r

.' I- — -^ -^ lLl J" ^- cv; to 5" •— — U. (."' +-• b.J Ovyv J J l/l L.

C

n. Ld oc

69
8 . References

1. Heinrich, K. F. J., Myklebust, R. L., Yakowitz, H. , and

Rasberry, S. D., "A Simple Correction Procedure £orN Quantitative Electron Probe Microanalysis," U.S.

National Bureau of Standards Technical Note 719, U.S.

Government Printing Office, Washington, D. C, (May 1972).

2. Reed, S. J. B., Brit. J. Appl. Phys . 16, 913 (1965).

3. Henoc, J. in "Quantitative Electron Probe Microanalysis,"

U.S. National Bureau of Standards Special Publication

298, Heinrich, K. F. J., Editor, p. 197, U.S. Government

Printing Office, Washington, D. C. (October 1968).

4. Heinrich, K. F. J. in "The Electron Microprobe," McKinley,

T. D., Heinrich, K. F. J., and Wittry, D. B., Editors,

p. 296, John Wiley $ Sons, Inc., New York (1966).

5. Criss, J. in "Quantitative Electron Probe Microanalysis,"

U.S. National Bureau of Standards Special Publication

298, Heinrich, K. F. J., Editor, p. 61, U.S. Government

Printing Office, Washington, D. C. (October 1968).

70
APPENDIX 1

NBS Technical Note 521

71
PRESENT STATE OF THE CLASSICAL THEORY OF

QUANTITATIVE ELECTRON PROBE MICROANALYSIS*

Kurt F. J. Heinrich

Analytical Chemistry Division

Institute for Materials Research

National Bureau of Standards

Washington, D a C. 20224

Although the foundations for a procedure of data reductior

in quantitative electron probe analysis have not been changed

for several years, there has been progress in the choice of

expressions, parameters, and constants. A brief account of

recommended expressions and procedures is given. Reference is

made to the Standard Reference Materials of Au-Ag and Au-Cu

alloys issued for electron probe microanalysis. These are

especially useful for investigating the application of correction procedures.

Key words: corrections; electron probe microanalysis; quantitative analysis; x-ray spectroscopy.

INTRODUCTION

In his doctoral thesis, published in 1951 [1]* Castaing

laid the groundwork for a theory of quantitative electron

probe microanalysis. He pointed out that the intensity of

characteristic primary radiation generated in multi-element

specimens is approximately proportional to the mass fraction

of the emitting element. He therefore proposed that the

corrections which need to be applied to measured x-ray intensity ratios to obtain mass fractions ("concentrations") were

as follows:

#

This paper is to appear in Mikrochim. Acta, in German, as part

of the proceedings of a meeting on electron probe microanalysis and metallography, held in Vienna, (Technische Hochschule,

Wien), October 22-25, 1969.

72
:::

irt

1. An absorption correction, which takes into account

the absorption of x-rays emerging from the specimen, and

2. A fluorescence correction which cancels the effects

of indirect x-ray generation,

Castaing also suggested that for accurate analysis the

proportionality between concentration and intensity of emission may have to be considered an insufficient approximation.

Hence, he proposed a form of atomic number correction to

correlate more accurately the mass fraction with the measured

intensity of x-ray emission.

This "classical" scheme of data reduction has essentially

remained the commonly used approach to quantitative analysis.

It has been described in many publications, including the

National Bureau of Standards Special Publication 298 (1968)

[2], which contains several references used in this paper. In

the recent past there has been considerable discussion of the

choice of parameters and constants to be employed within the

framework of this procedure. The problem remains if, instead

of the classical procedure, a Monte-Carlo method or a transport

equation procedure is used to calculate the mass-fractions of

the elements being measured.

It is the purpose of the present publication to give an

jaccount of the specific choices of parameters and constants

'which are available at present for the computation of corrections for quantitative electron probe microanalysis. For a general description of the correction procedures, as well as

for a detailed discussion of the origin of these expressions,

the reader is referred to the cited literature.

73
ATOMIC NUMBER CORRECTION

Thanks to the work of many investigators [.e.g. 2-6] the

processes resulting in primary x-ray emission are now quite

well understood. The number of ionizations produced by a

single electron coming to rest within the target can be

written as follows ;

n A = N A „.p.^y* ° -^W- dE (1)

o For the ionization cross-section, Q, and the stopping power,

-dE/d(px), we can substitute in the above equation the expressions proposed by Bethe et at. [7,8]:

Q = b • n e

4 • z£ . MI . E- 2 (2)

s;.;-l.£-W.N4.:f.fc*.£) <3)

The values of some parameters of these equations are, however,

poorly known D Henoc has concluded from the study of experimental results on the ionization cross section that the value

b = 0.76 can be considered valid for all levels of ionization

[9]. The value of the coefficient a in equation (3) was frequently considered to be equal to two. However, according to

Nelms [10], for electrons the value a = 1.166 should be used.

There is some controversy concerning the best expression

for the mean ionization potential, J. The value (J = 11»5Z).

used frequently in the past, is certainly inaccurate. We use

at present an expression obtained by Berger and Seltzer [11],

who averaged a large number of experimental observations by

several authors:

J = 9.76 Z + 58.5 Z-°- 19 (4)

1

A list of symbols used is appended

74
Another expression for J has been proposed by Duncumb et al.

[12]:

J/Z = 14.0[l-exp(-0.1Z)] + 75.5/Z z/75 - Z/dOO + Z) (5)

This equation has been derived empirically to optimize the

results of a large number of electron probe analyses of specimens of known composition. We withold judgment on the best

choice of expression for J, pending further investigations

„

In order to obtain an algebraic expression for ionization

within a multi-element target, we introduce the variables M

and V, defined by the following equations:

M = SC.-i ; V = exp

i

1 A. — y c

M i 1 A; - (f)]

(6)

As shown by Springer [13] 5 if one introduces equations (2),

(3) j and (6), the integration of equation (1) yields the

expression

n A = 1 — z 2AME bliuo -l-!nY[EI(£n(UV))-EI(£nV)]l

1 V

' (7)

in which EI is the exponential integral:

EI (x ) = const. + In x + 2

s = 1 s. s ! (8)

Expression (7) is used in an extensive program for data processing in quantitative electron probe microanalysis, called

COR, recently developed by Henoc and Heinrich at the National

Bureau of Standards [14].

A further correction is necessary to take account of the

effects of electron backscattering. A fraction of the impinging electrons is reemitted with energies sufficient to excite

x-rays. This reduces the generated x-ray intensity by a fac75
tor R (backscatter effect), which decreases rapidly with increasing atomic number. The average number of ionizations per

impinging electron is thus n.R. We believe that the most

reliable expression for the factor R is that proposed by

Duncumb [15]

(9)

R = 1.0000 +

+ (-.58U2.162W-5.137W 2+9.213W 3 -8.619W4 +2.962W5

) x 10~ 2 Z +

+ (-1.609-8. 298W+ 28.791W 2 -47.744W3 +46.540W4 -17.676W 5

) x 10" 4 Z

2 +

+ (5.400+19. 184W-75.733W 2+ 120.050W 3-110.700W4 +41.792W 5

) x 10- 6 Z

3 +

+ (-5.725-21.645W+88.128W 2-136.060W 3+117.750W 4 -42.445W5

) x 10- 8 Z

4 +

+ (2.095+8.947W-36.510W 2+55.694W 3 -46.079W4+15.851W 5

) x 10- 10 Z

5

where W = 1/U = E /E, . This expression agrees well with the

o o k

^ &

experimental measurements of the factor R by Derian [16],

The number of photons of the x-ray line n, generated by ionization of the electron shell m, is obtained by multiplying

the number of ionizations, n n R j with the fluorescence yield,

oo , and with the ratio of the intensity of the line of interest

m

5 J to the intensity of all lines originating from this shell

(weight of the line, p ):

II - nA . R .a> • p (10)

Ap A m Fmn

THE ATTENUATION OP PRIMARY RADIATION

The attenuation of primary radiation on emergence from

the specimen can be calculated using an equation proposed by

Philibert [17]

l/f( x) = (1 + *)(1 + _h_ . 21 )

; h = 1.2 A. (11)

1+h ° Z

2

The factor f(x) is the ratio between attenuated and unattenuated primary radiation: F(x) = I /I'. Expressions for the

coefficient a, which depends on the electron energy, were pro76
posed by Duncumb and Shields [18] and by Heinrich [19]. We

presently use the following equation:

a = 4.5 x 10 5 E

l - 65 - E, l - 65

o k

(12)

We have not evaluated an alternative equation for the absorption correction, by Wittry and Andersen [20].

The attenuation of primary radiation is large when x-rays

of low photon energy are measured. It is important to note,

however, that the magnitude of this effect, and hence, the uncertainty of its computation, depends upon experimental conditions under our control. Primary absorption can be minimized

by working at low electron energies and using high x-ray

emergence angles; and these conditions are desirable when soft

x-rays are measured. It should be added that further experiments on the depth distribution of primary x-rays should be

^encouraged. The uncertainties in the absorption correction

have been analyzed by Yakowitz and Heinrich [21] who suggested

Ithat the experimental conditions should be adjusted, wherever

possible, to maintain the value of f(x) above 0.8.

SECONDARY RADIATION EXCITED BY CHARACTERISTIC LINES

The effects of secondary radiation excited by characteristic lines are usually treated according to the equation

proposed by Castaing [1]:

r

f , fA c, PA rA - 1 w B fi (A,BK°=)

'pA

P A

2 ^(AB,BK«) B

£n(l + u) fnd + v) +

U V

(13)

In this equation, r

f is the ratio of the attenuated fluorescent

/ radiation to the attenuated primary radiation of the element

I which is being determined. I' /I\ is the ratio of the gener- ° pB pA

77
ated (not attenuated) primary emissions of the exciting and

the excited lines as obtained from pure elements . Castaing

proposed to approximate this ratio by the ratio of the absorption edge wavelengths of these elements, while Reed [22] replaced it by the expression [(U B - 1)/(U A - D] L67 in which U

A

and UD are the corresponding overvoltage ratios.

B

This procedure is amenable to several improvements which

were incorporated in the program COR. In this program, the

ratio I D/I» is fully calculated, using equation (10) for

B A

primary intensities, just as in the atomic number correction.

This simplifies the calculation of fluorescence of K lines by

L lines and viae versa, as well as the calculation of fluoresence of Ka lines by KS lines, which is performed separately.

As pointed out by Criss [23], the exponential approximation to

primary distribution in depth of x-ray emission, used by

Castaing to calculate the term v in equation (13) s can be replaced by a more accurate model. We use in COR the model contained in Philibert's calculation for the function f(x) [17].

With the computer facilities presently available to most

analysts, this more rigorous approach is possible without an

excessive increase in the cost of computation.

SECONDARY RADIATION EXCITED BY THE CONTINUUM

Henoc has described a method for calculating the effects

of secondary radiation excited by the continuum [24], In

order to perform the formal integration of the expressions

contained in his procedure, he had to make the simplifying

assumption that between absorption edges the mass attenuation

coefficients vary with the cube of wavelength. However, the

wavelength dependence of mass attenuation coefficients is always less than cubic [25], This has been taken into account

in the modification of Henoc's original method incorporated

78
In the program COR. The formal integration is replaced by a umerical integration, using the parameters for attenuation

coefficients published in reference [25].

The main practical difficulty in performing the correction for fluorescence excited by the continuum is that the

position of the absorption edges of all elements contained in

specimen and standard must be taken into account. Each wavelength range between adjacent absorption edges must be calculated separately. Therefore, the procedure is lengthy, and

in machine computation it requires extensive data input.

Errors in the input occur frequently and may escape detection,

since it is difficult to acquire an intuitive feeling for the

magnitude of this correction.

In the program COR the relations between lines and absorption edges are inspected by the computer which selects the

wavelength ranges for the integration. These decisions are

made and the computer completes the calculation without active

intervention of the operator. In the same process, the computer also determines which operations must be performed for

calculating secondary excitation by characteristic lines. This

enormously simplifies the execution of the fluorescence

corrections, and renders unnecessary further attempts to

simplify them, as advocated by Springer [26],

THE ITERATIVE PROCESS

We will now discuss how the individual corrections are

combined in calculating the composition of the specimen.

The usual form employs multiplicative correction factors:

C* = k. k| • k*A

• k* (14)

79
This procedure is based on the concept that ideally the method

should yield a linear correction curve. The multiplicative

factors correct for deviations from the ideal model. However,

the values of the correction factors depend on the composition

of the specimen, which is unknown. The problem is traditionally resolved by using an estimate of composition to calculate

the correction factors, and using the resulting mass fractions

as a new estimate of composition and iterating until convergence of results is obtainedo

As stated by Criss [23], the formulation of the correction by multiplicative factors unnecessarily complicates the

structure of the correction equations, and obscures the underlying physical events. Moreover, the iteration frequently

fails to converge, as observed by Reed [27]. For these reasons,

the program COR uses a different iterative concept, first proposed by Criss and Birks [28], and somewhat modified by Henoc

and Heinrich.

Each iteration contains two steps. In the first step, we

establish an estimate of the specimen composition and calculate

the intensity ratios to be expected for each element, according

the theory. The intensity ratios are obtained by the simple

relation

I* I* + I* + XI*

K =- = (15)

I Ip + I c + SI£ The correction procedures previously discussed are used to calculate the absolute intensities (in photons per electron) due

to direct and indirect excitation. As a first estimate of the

specimen composition we assume that the weight fractions of the

elements are proportional to the experimentally obtained x-ray

intensity ratios. This estimate, as well as the subsequent

estimates is normalized so as to render the sum of mass fractions equal to unity. Thus, in each iteration, we calculate

80
the signal intensities to be expected from a theoretically

possible composition.

Since equation (15) does not yield an explicit term for

mass fractions, we must use another relation to obtain, in the

second step of each iteration, a new estimate of composition.

We use for this purpose a hyperbolic approximation to the

analytical calibration curve for each element:

1-k 1-C (l rs

= a (16)

Ziebold and Ogilvie observed that the analytical curves in

electron probe microanalysis are indeed approximately hyperbolic [29]. On the plot of intensity ratio versus mass frac-

. tion for each element, the hyperbola can be defined by three

points (0,0), (1,1), and (C,k), where C and k are the estimate

of mass fraction used in the iteration, and of the intensity

I ratio calculated for it. Our next estimate for the mass

fraction, C , corresponding to the observed intensity ratio,

k , can thus be calculated by the equation

[ 'C m

=

k "

- C(1 - k) (17)

km (C-k) + k(l-C)

Contrary to the procedure suggested by Criss, we do not normalize the observed intensity ratios k in equation (16), or the

final results after convergence. Thus, if the presence of an

element has been overlooked, or if there are experimental

errors or failures of the correction procedures, this will

become apparent since the calculated mass fractions fail to

sum to unity

This iteration procedure has been extensively tested. We

have never observed failure to converge. In fact, the convergence is always rapid, as is to be expected in view of the good

fit of the hyperbolic model observed by Ziebold and Ogilvie.
The program COR can also compute results of analysis with

use of multi-element standards; furthermore it can determine

one element by difference or several unmeasured elements on

the basis of stoichiometric relations.

STANDARD REFERENCE MATERIALS

In spite of the progress made in recent years, there are

still uncertainties in several aspects of the correction procedures o I believe that the availability of reliable standard

reference materials will be instrumental in pointing out the

areas in which further Investigation is most needed. For this

reason, the National Bureau of Standards has devoted considerable effort to preparing and characterizing materials specially

selected for electron probe microanalysis [30], Recently, NBS

certified and issued an alloy of nominal composition 0.8 W,

0.2 Mo, a series of binary alloys in the Au-Ag system, in nominal steps of 0.2 (20 weight percent), and a similar series of

alloys in the system Au-Cu. All these alloys were carefully

analyzed chemically by several laboratories, and painstakingly

Investigated for homogeneity. They should be particularly useful in testing the atomic number correction,, Extension of the

studies of oxydic systems is presently in progress.

82
REFERENCES

[1] Castaing, R., Doctoral Thesis, Univ. Paris (1951).

[2] Green, M., Doctoral Thesis, Univ. Cambridge, England

(1962).

[3] Philibert, J. and Tixier, R., Nat. Bur. Stand. (U.S.),

Spec. Publ. 298, 13-33 (1968).

[4] Brown, D. B., Nat. Bur. Stand. (U.S.), Spec. Publ. 298,

63-79 (1968).

[5] Poole, D. M., Nat. Bur. Stand. (U.S.), Spec'. Publ. 298,

93-131 (1968).

[6] Duncumb, P. and Reed, S. J. B., Nat. Bur. Stand. (U.S.),

Spec. Publ. 298, 133-154 (1968).

[71 Bethe, H. A., Ann. Phys . Lpz . 5, 325-400 (1930).

[8] Bethe, H. A., Rose M. E. and Smith, L. P., Proc. Am.

Phil. Soc. 78, 573-585 (1938).

[9] Henoc, J., private communication.

[10] Nelms, A. T., Nat. Bur. Stand. (U.S.), Circ 577 (1956).

[11] Berger, M. J. and Seltzer, S. M.,. Nat. Acad. Sci., Nat.

Res. Council Publ. 1133, 205, Washington, D. C,

(1964).

[12] Duncumb, P., Shields-Mason, P. K. and da Casa, C, Proc.

Fifth International Congress on X-ray Optics and

Microanalysis, Springer-Verlag, Berlin, 146-150 (1969)

[13] Springer, G., N. Jb . Miner. Abh. 1061, 304-317 (1967).

[14] Henoc, J. and Heinrich, K. F. J., to be published.

[15] Duncumb, P., private communication.

[16] Derian and Castaing, R., Optique des rayons X et microanalyse, Castaing, Deschamps, Philibert, editors,

193-199-, Paris, Herman (1966).

[17] Philibert, Proc. Third International Congress on X-ray

ODtics and X-ray Microanalysis, Academic Press, New

York, 379-392 (1963).

83
[18] Duncumb, P. and Shields, P. K., The Electron Microprobe, McKinley, Heinrlch, Wittry, editors, Plennum

Press, New York, 40-55 (1968).

[19] Heinrlch, K. F. J., Adv. in X-ray Anal. 11, pp. 40-55

(1968).

[20] Andersen, C. A. and Wittry, D. B., Brit. J. Appl. Phys.

Ser. 2, 1, 529-540 (1968).

[21] Yakowitz, H. and Heinrlch, K. P. J., Mikrochim. Acta

1968 , 182-200.

[22] Reed, S. J. B. Brit. J. Appl. Phys. 16_, 913-926 (1965).

[23] Criss, J. W., Nat. Bur. Stand. (U.S.), Spec. Publ. 298,

52-62 (1968) .

[24] Henoc, J., Nat. Bur. Stand. (U.S.), Spec. Publ. 298,

197-214 (1968)

.

[25] Heinrlch, K. F. J., The Electron Microprobe , McKinley,

Heinrlch, Wittry, editors, John Wiley, New York,

296-377 (1966).

[26] Springer, G., N. Jb . Miner. Abh. 106, 241-256 (1967).

[27] Reed, S. J. B. and Mason, P. K., Paper presented at the

Second National Conference on Electron Microprobe

Analysis, June 14-16, 1967, Boston, Massachusetts.

[28] Criss, J. W. and Birks, L. S., The Electron Microprobe,

McKinley, Heinrich, Wittry, editors, John Wiley, New

York, 217-236 (1966).

[29] Ziebold, T. 0. and Ogilvie, R. E., The Electron Microprobe, McKinley, Heinrich, Wittry, editors, John

Wiley, New York, 378-389 (1966).

[30] NBS Certificates of Analysis, Standard Reference

Materials 480 (1968), 48l, 482 (1969).

84
List of Symbols

A,B atomic weights of elements A, B

C mass fraction ("concentration")

E kinetic energy of the electron

E initial kinetic energy of the electron, operating

voltage

E, excitation potential of line k

k

I attenuated (emergent) intensity of radiation

I intensity of generated radiation, before attenuation

J mean ionization potential

M variable used for stopping power calculation (eq.

6-7)

N n Avogadro ' s number

Av &

Q ionization cross-section

R correction factor for electron backscattering

S stopping power

U = E/Ek

U = E /E,

o ok

V variable used for stopping power calculation (eq.

6-7)

W = 1/U

o

Z atomic number

a constant in the hyperbolic approximation

b constant in the calculation of the ionization

cross-section

c constant in the calculation of a

I e charge of electron
f(x) correction factor for absorption of primary radiation

p h = 1.2(A/Z ) in Philibert's equation. For multielement targets, , _ ~ ~ . ,„ 2

i i I

1

EI exponential integral

In natural logarithm

n. number of ionizations produced by one electron

which comes to rest within the specimen

p relative frequency of the line n, generated on

mn ionization of the shell m, with respect to

generation of all lines originating from shell m

r absorption edge jump of the absorption coefficient

r„ intensity of emerging fluorescent radiation,

relative to emerging primary radiation of the

same line

s tally in the exponential integral

k relative intensity of emergent radiation

k„,k ,k correction factors for atomic number, absorption,

and fluorescence effects

u,v auxiliary variables for attenuation of fluorescent

radiation

a constant In Bethe's equation for energy loss of

electrons

G emergence angle of measured x-rays

]s x-ray mass attenuation coefficient

y(AB,AKa) mass attenuation coefficient of specimen containing

elements A and B, for Ka radiation of element A

p density of specimen

a coefficient for electron energy effect in Philibert's

equation

X

= u cosec 6

oj = fluorescence yield

86
Subscripts and superscripts:

A,B elements A, B

i tally in integration

1 electron shell 1

* composition dependent
APPENDIX 2 Index of Terms and Symbols

88
Index of Terms and Symbols

This index is a list of variable names in the main program and in the subprograms, with an explanation of the use

and meaning of the variable, and, where pertinent, with the

equivalent symbol most commonly used in literature. Reference

is made several times to the PTLOAD routine, which serves to

enter data from a magnetic tape. This routine is specifically

designed for the type of computer we use, and might have to

be redesigned to enter data from magnetic tape in other

installations

.

In the manuscript, underlining has been used to distinguish the number 1_ from the letter 1. The following definitions, used in the explanatory notes, should be given here.

A run is a complete load of input cards. It usually

consists of several cases . A case is a set of measurements

of the same lines and elements, under identical conditions

of excitation (operating voltage); in some of the measurements, one or more of the elements included in the case may

be absent. Each case consists of a set of points . A point

is a set of data corresponding to a single measurement, or to

repeated measurements at the same location or at equivalent

locations. For instance, in a run we may have one case

consisting of measurements on stainless steel , and containing

many points which may represent different specimens or points

on specimens, and another case of brass analyses, again containing many points.

The summation signs in the explanations denote summation

jof the corresponding terms over all elements present in the

specimen. Dummy variables have no effective functions, except

to occupy spaces in memory arrays, so that other variables

fall into the right places.

89
Main Program

Program Variable

Name

A(J)

AI(JJA)

AIR(JJA)

AIT

AK(JJA)

AKEX(JJA)

ANK(J)

ANKL(J)

AT CO

C(JJA) ,

Common Symbols

A, B, etc.

1(A)

yi(A)

"A

KA

kA=I*/I(A)

n

K

n KL

kA,k

fi

. .

.

cU c A

,L

B

. .

.

Remarks

Atomic weight of element,

the atomic number of which

is J

Total emerging x-ray intensity from pure element A

(See definition for JJA)

Ratio of emerging x-ray

intensities from a compound

standard, s, and from pure A

Total emerging x-ray intensity from the specimen for

element A

Continuum efficiency constant

(Kramer's constant)

Ratio of experimental x-ray

intensities emerging from

specimen and pure A

Exponent for calculating

absorption coefficients for

element J; in the equation:

for A<A

y(J,A) = CA

K edge

n

same, for A„<A<A T Jv L j

Atomic fraction of each

analyzed element

1. At start of the first

iteration, and in the calculation of k-values: x-ray

intensity ratio for each

element

.

2. In all other instances:

successive estimates of

weight fraction of each

element

.

90
CC(JJ,JJA)

CD(NP,JJA)

C

1

CI (J)

CL(L,K,J)

CO

CON(JJA)

COSEC(JJA)

CT(JJA)

ELNA(NZA)

IEO

C.

n

C . (from u = C . A ) 3

v. 3

J

CSC ty

CA

alphabetic

variable

Absorption coefficient constant in y(JJ,Xjj^) (definitions for JJ and JJA

below)

.

1. Experimental x-ray intensity ratio for point NP, and

line of JJA.

2. (In the calculation of

k-values) given concentration of element JJ

.

Absorption coefficient constant for element of atomic

number J

Wavelength of line:

L = line number

K = series number (edge)

J = atomic number

Weight fraction of an element determined by stoichiometry

If the value of this variable

is greater than zero for any

element, the weight fraction

of that element will be

fixed at whatever is input

on data cards

.

Cosecant of the x-ray emergence angle

Approximation obtained as the result of each iteration

(approximation number 1+1

at the end of the Ith iteration)

Chemical symbol for an element of atomic number NZA

(alphabetic)

Operating potential, in kV

91
EX(II,JJA) n Value of exponent for the

absorption coefficient in

y=CAn , selected and stored

by C0CA. II = subinterval

number, JJA = element label

EXPO(JJ,JJA)

FCASF(JJ)

n

EI

Value of the exponent for

the absorption coefficient

at the wavelength A , corresponding to E [kv]

r

, 12.398,

(A

o E

3 o Sum of emitted fluorescence

intensities excited by char

acteristic lines, for element JJ

FCASFX 11 J

I

f P

FCF(JJ)

FCFX

FCK12(J)

FCK13(J)

FCK2 3(J)

FIRST

FQC(JJ)

FQCA(JJ)

I /I

c p

.12

-1_3

23

f

c

, f c

(x)

f

f

. f

f

Cx)

Sum of emitted characteristic fluorescence intensities, divided by the emitted

primary intensity.

Emitted fluorescence intensity excited by the continuum, for element JJ

Emitted continuum-fluorescence

intensity divided by the

emitted primary intensity

Coster-Kronig coefficients

stored in the permanent

data file for each element

Logical variable used to

label the first input card

of a sample

Absorption factor, f (x) > for fluorescence excited

by the continuum

Absorption factor, f (x)

,

for the fluorescence x-ray

intensity excited by characteristic lines

92
FQF(JJ)

ICOF

III

f

p

. f

p

(x),f(x)

IMA(JJA)

J

JJ

JJA

]jJJAl,JJA2

J JAM

JJAS

JJJ(I,JJA)

|jJM

iJJMX

JST

Absorption factor for primary

radiation, element JJ

Switch to eliminate correction for continuum fluorescence

Constant needed for reading

magnetic tape input via

PTLOAD (not operative unless

PTLOAD is used)

Number of sublimits need in

the calculation of fluorescence excited by the continuum

Subscript for atomic number

Subscript for order in which

the element appears in the

input data (arbitrary) = "element label"

Label JJ of the element

being measured

Element labels of unanalyzed

elements

Number of analyzed elements

JJA + 1

Label of the element which

emits the line of wavelength

WAVE (I, JJA)

Number of elements in a case

Either JJAM or JJM, depending

on the variant of the program

Number of elements contained

in a compound standard

Subscript: level number

(edge)

93
KA(JJA)

KAA

KB(I,JJA)

III'

K , L j

, L

j

j ,

K , L j

, L

j

j

, . . .

Number of the level of the

analyzed element which emits

the measured line

Number of the level emitting the line WAVE (I, JJA)

KK

KM(J)

KMJ

KSTOP

LA(JJA)

LB(I,JJA)

LI (KAA,LAA)

LM(K,J)

LMKJ

LOUT

MESS(JJ,JJA)

MF(JJA)

alphabetic

Diverse switches

Number of absorption edges

listed in the permanent data

file, for element J

Switch for termination of

program

Subscript for CL(L,K,J).

Position in the sequence

in which the lines are

filed in the permanent

data file. This sequence

restarts after each edge.

Subscript: Position of

measured line in permanent

data file.

Subscript: Position of

exciting line in permanent

data file.

Name of analyzed line (KA1_,

MB1_, . . . , etc.)

Number of lines listed in

the series K, of the element

of atomic number J (K=level

number of the edge)

.

Switch for output 2.

Switch for writing error

message

Number of characteristic

lines which excite fluorescent emission

NCO(JJA)

94

Switch for error termination

due to insufficient data
ND

NEC

NECM

NELM

NEST

NIT

NOE

NP

NP1 e NPM-1

NPM

NSW

i NSW2

3 NTAPE

|NV(JJ)

NX

NY

NZ(JJ)

= JST-1

z z

Switch for input method

Case number

Number of cases in a run

Number of elements listed

in table

Number of additional

elements in a compound

standard (not including

the analyzed element)

Number of iteration

Running number of element

for reading permanent data

file.

Index: data point number

in any case

Total number of data points

minus 1_ Total number of data points

Switch determining the

variant of the program to

be used

Switch set for terminating

iterations

Magnetic tape assignment

(for PTLOAD only)

Valence of element number JJ

Subscripts for s toichiometry

printout

Atomic number of element JJ

95
NZA

NZA1

NZEX

NZO

OME(K,J)

OMEG(I,JJA)

R(K,J)

RPF(JJ)

RPFX

RR(I ,JJA)

RRR(I,JJA)

SA1

SACO

SC

SD(NP,JJ)

U> (J)

CO eff

P

r

.

1

(r.-l)/r.

I(C/A.)

EC

l

\

Atomic number for a particular element

Fluorescence yield for

level number K of element

of atomic number J

Effective yield for ionization produced by photons

(corrected for CosterKronig effects)

Jump ratio at edge number

K of element of atomic

number J

Emitted primary intensity

of element number JJ

Generated primary intensity

of an element

Absorption jump ratio associated with line of wavelength WAVE (I, JJA)

Jump-efficiency ratio for

element JJA

Auxiliary variable in calculation of atomic concentration

Sum of the weight fractions

divided by their respective

atomic weights

Summation of the weight

fractions

Relative standard deviation

of the k-value of element JJ

for point number NP (used

only when raw data are input)

96
STOIC

SWITCH

WA(JJA)

WAV(K,J)

WAVE (I, J JA)

ZL(L,K,J)

q,m

qm

12.398

Weight fraction of oxide

in stoichiometric compound

Switch used in composing

the output

Minimum wavelength of

continuum emission, related

to the incident electron

energyto/ E for element JJA

Edge wavelength of the

absorption edge of level

number K, of the element

of atomic number J

Wavelength of a characteristic line of element JJA

Weight of line with respect

to the sum of line intensities in its series: L = line number, K = series or

level number, J = atomic

number

All variables named:

Dl_, D2 , ... or Nl_, N2 , ... are dummy variables used only

to keep variables in COMMON in order.

97
Additional Variables in AINP

ASUM

C(IJ,II)

COUNT

CSTD(I,J)

DEADT(---)

FSD(J,I)

I

IC

ICON(I,J)

II, IIC

UMAX

IJ

IK

IND

INDEX

ITEN

Index

Average value of a set of

peak- or background-intensity

measurements

C(IJ,II) = CD(NP,JJA)

Count-rate, in cts/sec, for

dead-time correction

Intensity of standard,

corrected for dead-time

Function used to make deadtime correction

Double-precision standard

relative deviation of counts

collected for each element

in each point

Index

IC = ICON(I,J)

Number of times standards

or backgrounds are read

UMAX = JJAM (number of

elements analyzed)

IJ = NPM (total number of

data points)

IK = 6-1 (tally for separating of codes for scalers)

Variable for dead-time

determination

An index for setting up

a table of all the input

data

I

K

ITEN =.-10 auxiliary variable for scaler code separation

98
KK(J)

M

MCO(I)

NAV

NC

NCI

NCODE(NROW,I)

NCOL

ND

NIND

NM

NSTD(. . .)

NTABL(NROW,I)

Counter for number of determinations in each point

M + NC*ND

Variable used in separating

data code

Output code: if 0, print:

average of measurement for

each point

if 1_, : handle

all data individually as separate points

Number of scalers used in

a measurement

NCI = NC-1

NC0DE(NR0W,I) = MC0(I)/ITEN

Index (number of columns)

Total number of lines or

cards in input (including

standards and background

measurement)

Indicator which selects

the proper dead-time for

each spectrometer

Position of data coder

(NM = NC-- (coding must be

in the last position))

NR = NSTD(IIC,1, I, J)

Index (number of lines)

Number of digits per scaler

(NS=6)

Various sub-tables of

NTABL (NR0W,I)

Complete table of all input

data

99
NTC

NTRA

NUMBER (INDEX)

PEAK (I)

PHC(NROW,I)

SNSTD(I,J)

SPH(J,I)

SPHC(J,I)

SSTSD

SSUM(I,J)

STSD

SUM(I,J)

SUMS (I, J)

T(NR)

TIM(J)

100

If NTC=0 , data are on magnetic tape

If NTC=1_, data are on cards

If NTRA=0, calculate using

total counts

If NTRA=1_, calculate using

counts/sec

List of numbers returned by

magnetic -tape reading subroutine PTLOAD

Peak minus background for

standards

Peak intensity of specimen

Sum of the standard intensities (peak or background)

Sum of peak intensities on

each point of the specimen

Average peak intensity on

each point of the specimen

Conversion of STSD from

double precision to single

Sum of the squares of all

peak and background measure

ments for each standard

(double precision)

Relative double precision

standard deviation for

signals from each standard

and background

Sum of all peak and background measurements for

each standard (double

precis ion)

SUMS (I, J) = [SUM(I,J)] 2 /

IC0N(I,J) (double precision

Time in seconds

Summation of times in each

point
Additional Variables in COCA

F13 t\\

F2 3 fi

F^ £-, j

Coster -Kronig coefficients

13

23

I Subscript: index of sublimits of the wavelength

ranges

12 Number of absorption edges

of one element between A and A m

13 Number of the exciting line

(char, fluor.)

I3M Total number of exciting

lines (char, fluor.)

II Subscript: index of sublimits

IL IM-I+1

;IM Variable value for I in loop

IM1 IM1 = IM-1_

JM For pure element standard:

JM = JJA

For compound standard:

JM = JST

For specimen: JM = JJM

JO For all standards: JO = JJA

For all specimens: JO = 1

101
Kl

KR1_

KR2

M4

NNR

Tl to Ti

WAA

Auxiliary variables related

to shells and subshells:

K KR1 KR2 Kl

1 1 1

2 2 2 1

3 2 2 1

4 3 3 2 5 2 4 1

6 2 4 1

7 2 4 1 8 4 5 3

9 5 6 4

10 6 7 1

11 6 7 1

12 7 8 5

Varia.ble M4 =- I Ml

Jump ratio: NNR = R(K,J)

Temporary storage

\ wavelength: WAA = WA(JJA)

102
Additional Variables in CALI

II*

11

'J

fp.fCx)

f

f ,f(x) ch

f

c f(x)c

ar

I»

P

Total fluorescence intensity

excited by characteristic

lines (emitted)

Total fluorescence intensity

excited by characteristic

lines (generated)

Absorption factor for

primary emission

F0CA1 = FQCA(JJ)

FQCF(JJA) = FQC(JJ)

IMAM = IMA(JJA)

10 = IMA(JJA)+1

Emerging primary intensity = RPF(JJ)

Generated primary intensity

103
Additional Variables for FCA

ACA X^ X% = A^B... C

B y(B ' A qm) X

X cosec ip

ACB Xg Xj = A^ B> # C A y(A,B qm) X

X cosec i>

ACBA y£ y£ = C^ (A

fi ) n

i

CA(I),CB(I) a.,b. Coefficients of expansion

of the depth-distribution

function <|>(pz) (according

to Criss [5])

cf>(pz) = 2 a

i exp [-b

i pz]

i

FCA1_ L Fluorescence intensity

excited by a characteristic

line (emitted)

FCAO Ir } Fluorescence intensity

excited by a characteristic

line (generated)

FQIB f_p Absorption factor in fluorescence intensity excited

by a characteristic line

H h h from Philibert's absorption correction equation

JJB Number of the exciting

element

KBB KBB = KB(I,JJA)

LBB LBB = LB(I,JJA)

NZB Atomic number of exciting

element

RJ Conversion of J from integer

to real mode

104
Sl_ Step in calculation of

fluorescence intensity

excited by characteristic

lines (emitted)

S2 Step in calculation of

fluorescence intensity

excited by characteristic

lines (emitted)

SO Step in calculation of

fluorescence intensity

excited by characteristic

lines (generated)

105
Additional Variables in CO

AC(CL,J)

ACS A

Mass absorption coefficient

of element J, for the wavelength: X = CL (function)

* = A^B... C

B

^B ' A <& X

X cosec ip

ALK

AN IN

ANZ

bki

BK2

BN

BN1

BO

CCO(J)

EABI

EXO(J)

F(X,JO,JM,JJA)

FC

FCO

FI

n

r f

Conversion of LK from

integer to real mode

Conversion of NIN from

integer to real mode

Conversion of NZ from

integer to real mode

Sublimits used in the

computation of the definite

integral over wavelength

Value of C (constant for

calculation of y) for element J, as a function of

wavelength

Auxiliary variable in the

calculation of fluorescence

excited by the continuum

Value of exponent n- for

element J

Auxiliary function for

numerical integration

(emitted)

Fluorescence intensity

excited by the continuum

(emitted)

Fluorescence intensity

excited by the continuum

(generated)

Step in calculation of FC

106
FIO

FT(X,JO,JM,JJA)

AA

0) eff

(r.-l)/r.

KA,K

Step in calculation of FCO

Auxiliary function for

numerical integration, case

without absorption

Wavelength increment for

integration

IM_1 = IMA(JJA)

Index

Limit of LK

Number of intervals used

in the integration

NIN = 2*NIN2

Effective yield in a given

interval of wavelength

Remainder after wavelength

interval is divided

RRRI = RRR(I,JJA)

Step in calculation of

fluorescence excited by

the continuum (emitted)

Step in calculation of

fluorescence excited by

the continuum (generated)

Efficiency constant for

continuum production

107
Additional Variables in RB

Rl to R5 Auxiliary variables in —

calculation of backscatter

W 1/U Reciprocal of overvoltage

W2 to W5 (jj-)

2 to (^)

5 Powers of W

o o

Zl to Z5 Zl = lxlO~ 2

Z; . . . Z5 =

Ixl0~-°Z 5

Additional Variables in FQI

H h h from Philibert's equation

A c in^/rE 1*65 -n 1*65.

SIG a o= 4.5x10 /(E

Q- -E - J

Additional Variables in EI

AN Conversion of N from

integer to real mode

N Index

TN Step in calculation of

exponential integral

X Function to be integrated

(ALUOW or ALW)

108
Additional Variables in FT

DACS Dummy variable in COMMON

Additional Variables in AC

AN Conversion of N from

integer to real mode

Additional Variables in PRA

ALUOW ln(U V) ln(U V) = ln(U ) + ln(V)

Z.

ALW ln(V) ln(V) =

A Z R

C. ^-ln X A , r> . . . 1 A .

1

Y r l »166E. , X ( ~ v pi

T. 3/ A^B... C

i A7

i ' l

m M M =A^B... C i^

z. AMLW Mln(V) Mln(V) =

A Z R C —In X

A,B. . . i A

A

X (i^6E)

i

iAW V V = exp [ALW]

CX C A Concentration of analyzed

element v V

f EI ei(x) Function: ei(x) =

/ y—dV cV

-00 109
EM E-. 1 energy level

JX Z. Atomic number

KX Subscript: level number

(edge) = K

KXX KX must be greater than 4:

KX KXX

5 1

6 1

7 2 8 2 9 3

LX Subscript: line number = L

OMED co rr Effective fluorescence yielc

for ionization produced by

electrons

RB R Backscatter factor

RBA RBA = A^B... C

ARA

j RZ Z. Conversion of NZ from

integer to real

E

UO ^— = U Overvoltage

E-, o

&

Z Z. Atomic number

ZJ J A Mean excitation potential

Numbe:

shell

A

ZNL Z n Number of electrons in a nl

110
Additional Variables in F

A

u. Mass absorption coefficient,

pure element:

u.(A,A ) = C (A 7 ) ni

M X Mass absorption coefficient,

specimen

^*>V " A^B... CA

t C i(^

n

i]

Whichever sublimit is used

in the calling routine

(BK1, BK2, BN, BN1, or BO)

111
APPENDIX 3

Permanent Data File

112
Permanent Data File

Types of cards for data input to the permanent data table

When Used

once at

beginning

of table

Cols

l-3 ;

Format Description

(13) Number of elements

in permanent data

table

first card

for each

element

1-2*

3-4*

5-10

3 fields of 6

(11-28)

29-34

(12) atomic number

(12) number of absorption

edges listed in table

(F6.3) atomic weight

(3F6.4) Cos ter -Kronig coefficients

f f f 12' 23' 13

(F6.2) absorption constant

C

2 fields of 4

(35-42) (2F4.2) absorption exponents

n

K n KL

card for

each edge

card for

each line

1-7

8-13

14-19

20

1-7

8-13

(F7.5) wavelength at

absorption edge

(F6.4) jump for the edge

(F6.5) fluorescent yield oo (II) number of lines

listed for the edge

(F7.5) wavelength of line

(F6.4) weight of the line

with respect to the

sum of line intensities in its series

*Number must be right justified in the field.

113
67

1 1 1007

200.

3 1 693 9

120.

4 1 90 12

111.

5 1 1081

676

6 2 120 1 1 4368 2622 00091

447 1 99

7 2 14007

3099 2576 00151

316 1 99

8 2 16

2 332 24 00221

2362 1 99

9 2 19

17913 1933 1 1832 1 99

1 1 2 2299

1 1569 1 84 025 2

1 1575 12

119101 88

99

12 2 2431

95122 15 028 2

95207 12

9 89 88

99

13 2 2698

79481 14 038 2

79605 12

83393 88

99

1 290

14 2 88

35 286

74 285

135 284 273

221 283 273

38 282 273

49 281 273

905 279 273

1175 279 273

1487 278 273

14 2 2809 185 2 77 273

6738 1296 038 2

67530 12

712542 88

99

15 3 30974 225 277 273

5784 1232 08 2

114
5796 12

61568 88

00

94

16 2 32064

5019 1 167

50316 12

53722 88

757

17 2 35453

43971 1 1 12

44034 12

47278 88

609

19 3 39102

34365 101

34539 12

37414 88

00

421

20 3 4008

3070 3 96

30897 12

33584 88

00

3513

21 3 4496

2762 929

27796 12

30309 88

00

3014

22 3 479

24973 887

25139 12

27485 88

00

2729

23 3 5094

22694 8703

22844 12

250356 88

00

238882

24 2 52

20702 8497

208487 12

22897 88

167

25 2 5494

18964 824

27 276 273

83 2

317 2 76 2 73

425 275 273

155 2

484 274 273

165 2

551 274 273

17

621 273 273

213 2

698 273 273

242 2

78 273 273

272 2

867 272 273

291 2

115
19102 126

2101 8 874

1624

26 3 5585

174346 8108 32 2 175661 102

193604 895

00

17202 1 1726 857

27 3 5894

160815 7962 35 2 162079 10

178897 88

00

15618

28 3 5869

148807 7769 38 2

150014 107

165791 893

00

14242 1 14271 857

29 5 6354

138059 7659 415 2 139222 12

1 540 56 88

1 12709 1 17

13014 139

13288 573 00561

13336 1 110.6

30 4 6538

12834 7505 45 2

129525 12

143516 88

1 106 1 17

1 1862 139

12131

3 1 5 6972

1 1958 7376 469 2

120789 12

134008 88

9517 1 17 1 10365 1 10828 139 1

1 1023 1

1 1 1 4246 006 1

1 1292 1 99

32 5 7259

1 1 1658 7299 51 2

1 12894 12

9580 272 273

1055 271 273

11590 271 273

12680 271 273

138 270 273

1498 270 273

1622 270 273

116
125405 88

8773 1 17

9581 59

9640 41

9924 139

10 175 1 10187 4605

104361 1 99

33 5 7492

1045 722

10573 12

1 17588 88

8107 1 17

8929 1 9125 139

94141 1 9367 4661

96709 1 99

34 5 7896

97974 718

992 18 12

1 10477 88

7503 1 17

8321 1 8407 139

87358 1 8646 4583

899

99

35 5 799C9

9204 7142

93279 12

103974 88

6959 1 17

7767 1 7753 139

81251 1 7984 4524

83746 1 99

37 5 8547

8157 7018

82868 12

92555 88

5998 1 17

67876 59

68207 41

6644 139

70759 1 6865 439

73183 1 38265

006 1

548 2

006 1

585 2

006 1

628 2

1754 269 273

1894 269 273

205 269 273

2355 268 273

006 1

68 2

008 2

024 1 024 1

38 5 8762 2513 268 273

117
76973 6976 7 2 78292 16

87526 84

5592 1 17 008 2

63672 59

64026 41

6173 139 024 1 66239 1 6387 4314 24 1 68628 1 345

39 5 8891

72766 695 719 2 74072 16

82884 84

5217 1 17 008 2 59832 59

60186 41

5756 139 024 1 6212 1 5962 424 024 1 64488 1 3147

40 5 9122

68883 696 737 2

70173 16

78593 84

4879 1 17 008 2 5633 59

56681 41

5378 139 024 2

53843 10

5 836 90

5579 4 166 024 2

55863 7

60705 93

288

41 5 9291

65298 7035 754 2 66576 16

7462 84

4575 1 17 008 2

53102 59

53455 41

5031 139 024 2

50361 10

54923 90

523 4 123 24 2

52379 7

57243 93

2627

42 5 9595

61978 694 77 2 63229 16

7093 84

4304 1 17 001 2

57 268 1 267 273

57 2851 267 273

56 3039 267 273

55 3241 266 272

118
50133 59

50488 41

4719 139 ( 303 2

47258 10

517708 90

4913 3979 03 2

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127
FORM NBS-114A (1-71) U.S. DEPT. OF COMM. LIBLIOGRAPHIC DATA

SHEET

1. PUBLICATION OR REPORT NO.

NBS TN-769

2. Gov't Accession No.

3. Recipient's Accession No.

4. TITLE AND SUBTITLE

A Rigorous Correction Procedure for Quantitative

Electron Probe Microanalysis (COR 2)

5. Publication Date August 1973

6. Performing Organization Code

7. author(s) Jean Henoc, Kurt F. J. Heinrich, and

Robert L. Myklebust

8. Performing Organization

10. Project/Task/Work Unit No. 9. PERFORMING ORGANIZATION NAME AND ADDRESS

NATIONAL BUREAU OF STANDARDS

DEPARTMENT OF COMMERCE

WASHINGTON, D.C. 20234

11. Contract/Grant No.

12. Sponsoring Organization Name and Address

Same as No

13. Type of Report & Period Covered

Final

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15. SUPPLEMENTARY NOTES

16. ABSTRACT (A 200-word or less factual summary of most significant information. If document includes a significant

bibliography or literature survey, mention it here.) This publication descr

quantitative electron probe

present in conventional sch

correction for fluorescence

of the fluorescent effects

characteristic of elements

constants and parameters ne

coefficients are contained

decision-making sections in

rections reduce the require

ibes a rigorous data reduction procedure for

microanalysis, which avoids simplifications

ernes. In addition, the program contains a due to the continuu, and separate computation

of each exciting primary x-ray line. Constants

are stored in a permanent data file, and the

eded for the calculation of mass absorption

in a sub-program. These provisions and

the development of the fluorescence cord input considerably.

17. KEY WORDS (Alphabetical order, separated by semicolons) Continuum f lUOT eS Cence ; data

reduction; electron probe microanalysis; fluorescence correction; quanti

tative analysis.

18. AVAILABILITY STATEMENT

Q UNLIMITED.

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132

22. Price$2.10

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