'EFERENi

UNITED STATES PARTMENT OF 3MMERCE

JBLICATION

^O'Co.

NBS TECHNICAL NOTE 719

'^rts o«

A Simple Correction Procedure

For Quantitative Electron Probe

Microanalysis

U.S.

>ARTMENT

OF

OMMERCE

100

U5753
NATIONAL BUREAU OF STANDARDS

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nal Bureau of Standards JUL 2 4 1972

-Aa^ act- &/

&G(OG A Simple Correction Procedure

For Quantitative Electron Probe

Microanalysis

Kurt F. J. Heinrich and Robert L. Myklebust

Analytical Chemistry Division

Harvey Yakowitz

Metallurgy Division, and

Stanley D. Rasberry

Analytical Chemistry Division

Institute for Materials Research

National Bureau of Standards

Washington, D.C. 20234

(Supersedes and extends NBS Technical Note 521)

.e*»T OF _

<f WW \

Q, 4^

U.S. DEPARTMENT OF COMMERCE, Peter G. Peterson, Secretary

NATIONAL BUREAU OF STANDARDS, Lawrence AA. Kushner, Acting Director

Issued May 1972
National Bureau of Standards Technical Note 719

Nat. Bur. Stand. (U.S.), Tech. Note 719, 49 pages (May 1972) CODEN: NBTNAE

Issued May 1972

For sale by the Superintendent of Documents, U.S. Government Printing Office Washington, D.C. 20402 (Order by SD Catalog No. C 13.46 : 719). Price 55 cents. Stock Number 0303-0984
A Simple Correction Procedure for Quantitative Electron

Probe Microanalysis

Kurt P. J. Heinrich, Robert L. Myklebust

,

Harvey Yakowitz,* and Stanley D. Rasberry

Analytical Chemistry Division

Institute for Materials Research

National Bureau of Standards

Washington, D. C. 2023^

A calculation technique for data reduction

in quantitative electron probe microanalysis is

described. This technique is embodied in a computer program, called MULTI8, written in FORTRAN IV

for batch processing. Six chemical elements may be

determined simultaneously without redimensioning

the program. The relative x-ray intensity data are

corrected for atomic number, absorption and secondary

fluorescence due to characteristic lines. Options

for calculating the mass fraction of one chemical

element by difference or by stoichiometry are available in the program.

Key words: Absorption; atomic number; chemical

elements; computation procedure; data reduction;

electron probe microanalysis; fluorescence;

time-sharing; x-ray analysis.

Introduction

This Technical Note details the practical aspects of

applying the concepts and models for data reduction which were

presented in NBS Technical Note 521 [1] which is reproduced in

Appendix II for the convenience of the reader. For the computation of mass fractions of chemical elements in quantitative

electron probe microanalysis, we use two calculation techniques,

which are contained, respectively, in programs entitled MULTI8

and COR. The MULTI8 program, described herein, was originally

* Metallurgy Division, Institute for Materials Research

I
written in BASIC language for use with a time-shared computer.

MULTI8 has had several predecessors dating back to 1965 when we

first had access to a time-shared computer. Until recently, all

of the quantitative electron probe microanalysis results generated in this laboratory were calculated with these programs.

The reduced computation capacity available with the early timeshared computer that we used forced certain simplifications, the

most important being the omission of the fluorescence correction

for excitation by the continuum. The present version of MULTI8

is a FORTRAN IV program suitable for batch processing.

A more complete treatment of the theoretical models,

contained in the program COR, will be the subject of another

Technical Note currently in preparation. The COR program includes the correction for fluorescence due to the continuum,

and the integration of the stopping power term according to

Springer [2] and to Philibert and Tixier [3]. COR requires a larger computer and has therefore not been adapted to timeshared computing. In practice, we usually find little difference between the results from MULTI8 and from the more extensive program COR. However, the input into COR is much simpler

and requires less operator time, since the mass attenuation

coefficients are internally generated, and all decisions concerning applicability of fluorescence corrections are also

made internally.

1The original BASIC version is available on request.

2
MULTI8 Computation Methods

The number of chemical elements for which mass fractions

may be simultaneously calculated by the program MULTT8 is

limited only by the size of the computer. In its present form,

MULTI8 occupies approximately 3.5 K words of storage and is

dimensioned for a maximum of six chemical elements. The background and dead-time corrections as well as the specimen/standard intensity ratios must be precalculated.

MULTI8 performs the following corrections

1. Absorption correction according to Philibert [43 and

Duncumb [53. The expression for sigma developed by Duncumb

has been modified by Heinrich [63.

2. Fluorescence by characteristic lines according to

Reed [73.

3. Atomic number correction by the method of Thomas [83

as modified by Heinrich. The backscatter factor, R, is that

.of Duncumb and Reed. The stopping power is calculated according

jto Bethe [93, using the mean excitation potentials of Berger and

Seltzer [103. An equivalent electron energy is calculated using

an empirical fit obtained by H. Yakowitz. This yields the same

'.result as full integration of the stopping power term according

to Springer [2] and Philibert and Tixier [33.

There is no provision for correcting the effects of fluorescence produced by continuous radiation. For this reason, the

customary form of correction by multiplicative factors (ZAF

procedure) was used.

The iteration procedure followed is that of Criss [113.

However, we have not adopted Criss' suggestion of normalizing

the intensity ratios. Only the concentration estimates are

normalized before applying the corrections in the iteration.

The iteration stops when variations for all elements between

3
two consecutive loops are less than 0.0001 (0.001$), which

almost always is achieved in 3 to 6 loops. Should convergence

within the specified limit not occur after 19 loops, the iteration is stopped, the results are listed, and the comment "NO

CONVERGENCE" is printed. Failure to converge has not been

observed in practice.

The program will accept any number of sets of data and

calculate the results in sequence. The following special cases

are acceptable:

1. Secondary (fluorescence) excitation of several chemical

elements by one element, or of one or more elements by more than

one element

.

2. One or more elements included in the program may be

absent

.

3. One element may be determined by difference.

4. One or more elements, although not measured, may be

calculated on the basis of stoichiometric relationships with

the elements measured and/or with the element determimed by

difference

.

5. The program allows for using different operating

voltages in the determination of each element.

With slight modification, this program can also be used

for the following cases:

1. The standard used for one or more elements is not an

element but a compound;

2. The background and dead-time corrected intensity

ratios of a specimen of known or postulated composition can

be calculated.
Comments on the Program

The statement numbers below refer to the statement

numbers appearing on the program listing.

Main Program

Statements Comments

START-110

Between 110 and

120

Dimension^ formats and related

statements

Read initial data and set up

headings

Call input subroutine.

Call subroutine to calculate

backscatter coefficients.

Loop for calculation of the

electron absorption coefficient,

o, the h factor' of Philibert,

and the overvoltage term of

Reed's fluorescence correction.

120-150

150-300

150-180

180-200

200-270

220-250

Write headings and read intensity

ratios. Make first estimate of

concentration or terminate if

end of data has been reached.

Iteration loop.

Loop for determination by difference

or stoichiometry . Obtain sum of con- centration . Adjust sum of concentrations to 1

for the purpose of calculating

corrections

Loop for calculating atomic number,

absorption and fluorescence corrections . Calculate stopping power terms.

Calculate intermediate terms for

atomic number, absorption and

fluorescence corrections.

Calculate characteristic fluorescence correction.
Statements Comments

250-260 Calculate atomic number correction.

Calculate f(x) values.

Calculate the emergent intensity

ratios for the assumed concentrations . By hyperbolic fit, obtain new

estimates for concentrations,

comparing the experimental intensity ratio with the one obtained

by calculation.

270-310 Stopping criterion for interation

loop. If not convergent after 19

loops, stop iteration and print

"NO CONVERGENCE".

320-370 Print results.

Go to next set of data (line 130

Stop.

Input Subroutine

10-30 Dimension, formats and related

statements

.

Between 30 and Read all data required for

40 each element (but not intensity

ratios )

.

40-END Print out the characteristic

fluorescence correction that

has been selected.

Return to main program.

Backscatter Subroutine

All Calculate the backscatter

coefficient, R, from the equation of Duncumb (as referenced

by Heinrich [1] )

.
Input Requirements

The numbers and letters below each card number are the

FORTRAN IV formats for the card. (Card #2 is listed in full. )

Card #1

II col 2 number of element - Nl (max. of 6)

F4 . 1 col 6-9 x-ray emergence angle in degrees - N2

col 15-26 any title

Card #2 a namelist card written as follows beginning in

col 1:

X$NAM1 P=0, 2*1., .24, 4.2, .88, 2*. 75, .88$

Card #3

10F8.4 Read the data for each element:

(each element requires 5 fields)

atomic no. A (1,1), atomic weight A (2,1), operating

kV

A(3,I), min. excitation potential A(4,l), and

fluorescence yield A(5,I). (In the input

subroutine these variables are called AM

instead of A .

)

Card #4

10F8.4 Read mass absorption coefficients needed:

y(A,A), y(A,B), y(B,A), y(B,B)

(first letter is the line - second is the absorber.)

- M(I,J)

Card #5

10F8.4 Matrix for characteristic fluorescence correction - Q(I,J). (In the input subroutine this matrix is

called QM.

)

7

,

461-929 O - 72 - 2
Card #6

10P8.4 Matrix for s toichiometry calculation. - B(I,J).

The values in this matrix are valences. Let us

assume that we have listed the elements of the

compound TiO *BaO in the order: Ti, Ba , 0.

If oxygen is determined stoichiometrically

,

and the other elements directly, the matrix

B is as follows:

Ti Ba

TiKa

BaLa

(stoich. ) 4 2 2

Card #7 & above

8P10.5 Intensity ratios * one card per analysis with

intensities of all elements on each card - K(I).

To calculate intensity ratios from known compositions,

these compositions should be substituted for the k-values on

the appropriate cards and the following changes made to the

main program:

Insert the three statements:

DABS = 1/D(J)

WRITE (10, 70) Z(J), K(J)

WRITE (10, 90) DABS

immediately before the statement which reads GO TO 270 (which

precedes statement 260).

After statement number 270, insert the statement:

GO TO 130

Then the remainder of the main program cannot be reached

and can be deleted except for the END statement.
Dictionary of Variables in MULTI8

The following notations are used:

1. If an * precedes a variable name, that variable

must be input.

2. In the list of common symbols, i refers to the

iteration number, j to the element number, and q to the

measured line.

3. In the list of common symbols, an * in the symbol

signifies the analyzed specimen.
Main Program

Variable Common

Name Symbols

* A(1,J) Z . J

* A(2,J) A.

J

* A(3,J) E

o

* A(4,J) E„ (q=K

Description

Atomic number of the Jth element . Atomic weight of the Jth element . Operating kilovoltage in the

determination of the Jth element .

,...) Minimum excitation potential,

M in keV, for the line used in

determining element J.

* A(5,J) w Fluorescent yields of elements

q which excite secondary emission of other elements.

* B(IjJ) Valence of element J used to

determine element (I). B(I,I)

the valence of element determined by stoichiometry , must

be included.

Bl Stoichiometric contribution

to concentration.

C(I,J) C. Estimate of concentration of

J element J in the Ith iteration

CON C. Final concentration of C(I,J).

J D(J) 1/f* l/f(x)* Philibert's l/f(x) for specimen.

E(I) EC. Sum of concentrations for Ith

iteration.

F(I) o Sigma in Philibert's equation.

P

G(J) 1/f^ l/f(x) Philibert's l/f(x) for the

standard

10
Variable Common

Name Symbols Description

Gl Factor for determining the

equivalent electron energy

for calculation of stopping

power

H(I) h Philibert T s h=1.2^-

I J N N4 Running index numbers in loops

IA Z. Integer conversion of A(1,J)

( at omic numb er )

II Logical unit number for input

(11=5 for card reader).

10 Logical unit number for output

(10=6 for line printer).

jl Mean ionization potential

from Berger-Seltzer equation.

* K(I) k Background and dead-time corrected intensity ratios, in

groups ordered according to

the sequence of atomic numbers

A(ljl). Special cases:

Case Value of K(I)

Element not (Zero)

present

Element determined -1

by difference

Element determined 1

by stoichiometry

End of data 2

Kl Reed's characteristic fluorescence correction factor.

K9 1 + Z(K1).

KK Integer of the absolute value

of K(I).

11
Variable Common

Name Symbols Description

* M(I,J) u(I J) — Mass absorption coefficient

P for the line of element I and

absorber J.

* Ni Number of elements in specimen.

* N2 ^ 9 X-ray emergence angle in

degrees

jjo Subscript for selecting the

proper characteristic fluorescence (N9=Q(J,N4)+1).

0(j) p(A B)* ^n line A79: absorption coefficient for the specimen.

In line A96: \/ c for the specimen.

* p cu 2) P(k,j) Constants of Reed's equation,

for L-*K and K->L fluorescence

These are input on the NAMELIST card NAM1.

* q(j j) Switches for fluorescence of

element (I) by element (J).

Case Value of Q(I,J)

No fluorescence

K excites K 1

L " L 2

K " L 3

L " K 4

r/j j) r Backscatter factors according

to Duncumb equation for radiation

I and element J.

jq r* Backscatter factor from specimen.

S(IMK) S Stopping power factors calculated from Bethe's equation

I.67 f°r radiation I and element J.

U B~ 1

A T(I,J) u /\ _1 Overvoltage term in Reed's

equation.

12
Variable Common

Name Symbols Description

U(J) h* Philibert's h-value for the

specimen

Ul E /E Overvoltage.

o q

&

V(J) Equivalent electron energy in

the calculation of stopping

power. (Pit by H. Yakowitz).

W(J) (R/S)* Atomic number correction for

the specimen.

XO Factor for determining the

equivalent electron energy for

stopping power calculation.

st XI (x/°) (In Philibert's equation for

the standard.

)

X5,Y5 x, y Factors x and y in Reed's

equation [ 7]

Y8,Y9 u Mass absorption coefficients

in the characteristic fluorescence correction.

Z(J) k T Intensity ratio for element J,

calculated in iteration I.

L J

Zl ZC.Z. Effective atomic number of the

specimen.

Z9 Intermediate step in hyperbolic approximation of Criss.

13
Input Subroutine "iNP"

Additional Variables

Variable

Name Description

* AM (I, J) Same as A (I, J) in Main

Program.

CH(I) Variable "K" or "L" as defined

by DATA statement B6

IA Part of switch for selecting

proper printout of characteristic fluorescence correction.

IAM Integer value of AM(1 3 J).

IBM Integer value of AM(1,I).

IC1,IC2 Selected value of CH(I)

(either "K" or "L" )

IQ Value of Q(I,J)+1.

IX IX=5 (from Main Program).

jj JJ=N1 (from Main Program).

* QM(I,J) Same as Q(I,J) in Main Program,

Backscatter Subroutine "BACK"

Additional Variables

R1,R2,R3 Temporary factors used in the

R4,R5,T1 calculation of the backscatter

T2,T3,T4,T5 factors R(I,J).

W Reciprocal of the overvoltage

(E /E ).

q o

J W2,W3 3 Vj4,W5 W

2

, W

3

, l/ , and W

5 respect ively

14
References

1. Heinrich, K. P. J., U.S. National Bureau of Standards

Technical Note 521, U.S. Government Printing Office,

Washington, D. C. 20402 (1970).

2. Springer, G. , N. Jb . Miner. Mh . 1967 , 304-317-

3. Philibert, J. and Tixier, R. , U.S. National Bureau of

Standards Special Publication 298, K. F. J. Heinrich,

Editor, 13-33 (1968).

4. Philibert, J., Proceedings Third International Congress

on X-ray Optics and X-ray Microanalysis, Pattee, H. H.

,

Cosslett, V. E., and Engstrttm, A., Editors, 379-392,

Academic Press, New York (1963).

5. Duncumb, P. and Shields, P. K., The Electron Microprobe,

McKinley, T. D., Heinrich, K. F. J., and Wittry, D. B.

,

Editors, 40-55, J. Wiley and Sons,- New York (1964).

6. Heinrich, K. F. J., Advances in X-Ray Analysis 11 , 40-55, Plenum Press, New York (1967).

7. Reed, S. J. B., Brit. J. Appl. Phys. 16_, 913-926 (1965).

8. Thomas, P. M. , AERE Rept . 4593, Atomic Energy Research

Establishment, U.K.A.E.A. (1964).

9. Bethe, H. A., Ann. Phys. Lpz . 5_, 325-400 (1930).

10. Berger, M. J. and Seltzer, S. M., Nat. Acad. Sci.,

Nat. Res. Council Publ. 1133 , p. 205, Washington, D. C.

(1964).

11. Criss, J. W. and Birks, L. S., The Electron Microprobe,

McKinley, T. D., Heinrich, K. F. J., and Wittry, D. B.,

Editors, 40-55, J. Wiley and Sons, New York (1964).

15

461-929 O - 72 - 3

APPENDIX I

Program and an Example for MULTI8

s

CD H

-P•HP

CD H

hO

cti

<m

CmOI CD P

-P

CDS

CD H

CD CmOM

,Q6C

ui y1

inin

ru

CDcHH

CmOP

.C +•> hO CO

•H CO

CD

I'm 5 in

rn3 '" a C *

cti OJ

V.

I'll CO CO p • C ' ^.

cti

'*"*

p o

CO ' r'mt. C

iru

O *f o v.

-<j- CD OJ o

'""• c >

CDO *

CO OJ

CD '"•% M

1(1 "« o n 3 c» rH x

<M »HE

<r

a1

CMpC

CD6

CD HH

H

CT1 *-i

N CT. a1 a1 W

toX

pC

CDS

CD HH

W

O

kO

N

19
oH

-PO

CDm

?Hoo

CO -pC

CD •Ho

•HOh

<h

CDOocO

•H

-Po

CO ,Q

cd

co

COcd

in

r--

cxi

coa

CXI

in

in

•aCDocO

to

CD

IhOH

OnO

•HP

CO •HM

CD PO

cdM

cd &oMo

CmX

•HM

-P

cd o

•T O

CO

in cO

•HP

cd HoH

cdo

>jMP

CDBo

•H

.C

cj •HOP

COMO

<HX

•HMP

cdB oo

20
wo

•H

-P

cd

-p•H

CO£

(D -Pc

•H

CO

in

'It o

03

OJ

CO

CiJ CO KO ID

in ^

co |oj

(XI

•— CNI " • !'*'•! S"'

CO in OJ CO i""J cu CO M3

'£< U3 -T

'•£> vf CU i r .

111 ft vvi^

cd -p

cdo

CD OJ

21
2 52.5 AULA - CUKA

THE L -LINE OF 79 EXCITES THE K -LINE OF 2 9

Zl= 29 £0 * 2Q.Q

Z2« 79 EO = 20»0

K C F(X )

SPECIMEN NUMBER 1 .8M01 .8060 .97H6

.1570 .1973 .9H87

TOTAL =1.0033 ITERATION H

SPECIMEN NUMBER 2

•6622 .6059 .9658

.3323 .3976 .95M3

TOTAL =1.0035 ITERATION H

SPECIMEN NUMBER 3

.M63H .M015 .9573

•5286 »5991 .9600

TOTAL =1,0006 ITERATION M

SPECIMEN NUMBER H •2M62 .2008 .9H91

•7525 .8030 .9655

TOTAL =1.0038 ITERATION *4

Figure 1. Results of MULTI8 calculation of mass

fractions (C) from relative x-ray intensities (K). P(X) is the absorption correction

factor. Total mass fraction and iteration

number are also printed out.

22
2 52,5 AULA - CUKA

THE L -LINE Op 79 EXCITES THE K ^LINE OF 29

Zl« 29 EO « 2Q#0

Z2« 79 EO a 20#0

K C F(X)

SPECIMEN NUMBER I .8378 .8060 ,9746

.1565 .1973 .9487

SPECIMEN NUMBER 2

.6603 .6059 .9656

.3311 .3976 .95*13

SPECIMEN NUMBER 3

.4631 .4015 .9573

.5282 .5991 .9600

SPECIMEN NUMBER 4

.2453 .2008 .9491

.7489 .8030 .9655

Figure 2. Results of MULTI8 calculation of relative

x-ray intensities (K) from mass fractions

(C). F(X) is the absorption correction

factor

23

461-929 0-72-4

Complete Listing of Program MULTI8

120

C

C

130

140

R(6,6),S(6,6),C(20,6),B(6,6),Q(6,6),T(6,6;

,H(6),F(6),W(6),U(6),0(6),G(6),D(6),Z(6)

10 FORMAT ( 20 FORMAT ( 30 FORMAT ( 40 FORMAT ( 50 FORMAT ( 60 FORMAT ( 70 FORMAT ( 80 FORMAT ( 90 FORMAT ( 100 FORMAT ( 110 FORMAT (

MULTI8

MAIN PROGRAM - THIS PROGRAM DETERMINES CHEMICAL CONCENTRATIONS

FROM RELATIVE X-RAY INTENSITY DATA MEASURED ON AN ELECTRON PROBE

MICROANALYZER

REAL M,N2,H,K,J1,K9,K1

DIMENSION A(5,6),M(6,6)

1,P(5,2) ,V(6) ,E(20) ,K(6)

COMMON 11,10

NAMELIST /NAM1/ P

1H1)

IX , II , 3X , F4 . 1 , 5X , ' TITLE OF RUN '

)

1H0 ,

' Z Ml ,' = • , 13 , 5X ,' EO = ' , F5 . 1

)

1H0.9X, 'K' , 15X, 'C ,13X, 'F(X) '/)

1H0, 'NO CONVERGENCE'

)

1H ,7X,F6.4,9X,F6.4)

1H+ , T40 ,

' N . D

.

)

1H+,T38,F6.4)

1H0, 'TOTAL =' ,F6.4, ' ITERATION ',12/

)

8F10.5)

1H0, 'SPECIMEN NUMBER M3

)

11=5

10=6

READ INITIAL DATA AND SET UP HEADINGS

WRITE (10,10)

READ (11,20) N1,N2

WRITE (10,20) N1,N2

READ (II.NAM1)

CALL SUBROUTINE TO READ INPUT DATA

CALL INP (A,M,Q,B,5,N1)

CALL SUBROUTINE TO CALCULATE BACKSCATTER COEFFICIENTS

CALL BACK (A,R,N1)

CALCULATE THE ELECTRON ABSORPTION COEFFICIENT , SIGMA , THE H-FACTOR

OF PHILIBERT, AND THE OVERVOLTAGE TERM OF REEDS FLUORESCENCE

CORRECTION

DO 120 1=1, Nl

IA=A(1,I)

WRITE (10,30) I, I A, A (3, I)

H(I)=1.2*A(2,I)/(A(1,I)**2)

F(I)=4.5E5/(A(3,I)**1.65-A(4,I)**1.65)

DO 120 J=1,N1

T(I,J)=(((A(3,J)-A(4,J))*A(4,I))/((A(3,I)-A(4,I))*A(4,J)))**1.67

WRITE HEADINGS AND READ INTENSITY RATIOS. MAKE THE FIRST ESTIMATE

OF CONCENTRATION OR TERMINATE IF END OF DATA HAS BEEN REACHED.

WRITE (10,40)

NS=0

READ (11,100) (K(I) ,I=1,N1)

NS=NS+1

DO 150 1=1, Nl

IF (K(I)-2.0) 140,370,140

C(1,I)=K(I)

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

27
KK=INT(ABS(K(I) )

)

A

IF (KK.EQ.l) C(1,I)=.5 A

150 CONTINUE A

WRITE (10,110) NS A

C A

C BEGIN ITERATION LOOP A

C A

DO 300 1=1,20 A

Z1=0 A

E(I)=0 A

C LOOP FOR THE DETERMINATION AN ELEMENT BY DIFFERENCE OR A

C BY STOICHIOMETRY. OBTAIN THE SUM OF CONCENTRATIONS. A

DO 180 J=1,N1 A

IF (K(J) .LT.O. ) GO TO 180 A

IF (K(J)-l.O.LT.O. ) GO TO 170 A

C(I,J)=0 A

DO 160 N=1,N1 A

IF (J-N.EQ.O) GO TO 160 A

B1=C ( I , N ) /A ( 2 , N

) * A ( 2 , J) *B ( J , N ) /B ( J , J

)

A

C(I,J)=C(I,J)+B1 A

160 CONTINUE A

170 E(I)=E(I)+C(I,J) A

180 CONTINUE A

C ADJUST THE SUM OF CONCENTRATIONS TO 1 FOR THE PURPOSE OF A

C CALCULATING CORRECTIONS. A

DO 190 J=1,N1 A

IF (K(J) .GE.O. ) GO TO 190 A

C(I,J)=1-E(I) A

E(I)=1 A

190 CONTINUE A

DO 200 J=1,N1 A

C(I,J)=C(I,J)/E(I) A

200 Z1=Z1+C(I,J)*A(1,J) A

DO 270 J=1,N1 A

IF (ABS(K(J) )-1.0) 210,260,210 A

210 W(J)=0 A

U(J)=0 A

S1=0 A

0(J)=0 A

R1=0 A

G1=0 A

K9=l A

U1=A(3,J)/A(4,J) A

IF (Z1/A(1.J) .GT.l) GO TO 220 A

G1=.225-.722*Z1/A(1,J)+.789*(Z1/A(1,J) ) **2-.291*(Zl/A(l , J) ) **3 A

220 X0=(1+G1)*( .0604*AL0G(50*U1-16)+.1262) A

V(J)=X0*A(3,J)+(1-X0)*A(4,J) A

IF (C (I, J) ) 230,260,230 A

230 DO 250 N4=1,N1 A 100

C CALCULATE MEAN EXCITATION POTENTIAL J. A 101

Jl=9. 76*A ( 1, N4 ) +58. 5*A(1,N4 )**(-. 19) A 102

C CALCULATE STOPPING POWER. A 103

S(J,N4)=1/V(J)*A(1,N4)/A(2,N4)*AL0G((1166*V(J))/J1) A 104

0(J)=0(J)+C(I,N4)*M(J,N4) A 105

U(J)=U(J)+C(I,N4)*H(N4) A 106

N9=Q(J,N4)+1 A 107

28
530

540

Y8=0 A 108

Y9=0 A 109

DO 240 N=1,N1 A 110

Y8=Y8+C(I,N)*M(J,N) A 111

Y9=Y9+C(I,N)*M(N4,N) A 112

S1=S1+C(I,N4)*S(J,N4) A 113

R1=R1+C(I,N4)*R(J,N4) A 114

X5= (Y8/Y9) /SIN (N2/57. 2958) A 115

Y5=F(J)/Y9 A 116

CALCULATE SECONDARY FLUORESCENCE CORRECTION. A 117

K1=.5*A(5,N4)*P(N9,1)*C(I,N4)*P(N9,2)*A(2,J)*T(J,N4) A 118

K1=K1/A ( 2 , N4 ) * ( ALOG ( 1+X5 ) /X5+AL0G ( 1+Y5 ) /Y5 ) *M ( N4 , J ) /Y9 A 1 19

K9=K9+K1 A 120

DETERMINE ATOMIC NUMBER CORRECTION. A 121

W(J)=(R(J,J)*S1)/(S(J,J)*R1) A 122

CALCULATE ABSORPTION FACTOR. A 123

Xl=M(J,J)/F(J)/SIN(N2/57.2958) A 124

0(J)=0(J)/F(J)/SIN(N2/57.2958) A 125

G(J)=(1+X1)*(1+H(J)*(1+X1))/(H(J)+1) A 126

D(J)=(1+0(J))*(1+U(J)*(1+0(J)))/(U(J)+1) A 127

CALCULATE THE EMERGENT INTENSITY RATIOS FOR THE ASSUMED A 128

CONCENTRATIONS. A 129

Z(J)=C(I,J)/W(J)*K9*G(J)/D(J) A 130

Z9=K(J)*C(I,J)*(1-Z(J)) A 131

BY HYPERBOLIC FIT, OBTAIN NEW ESTIMATES FOR CONCENTRATIONS, A 132

COMPARING THE EXPERIMENTAL INTENSITY RATIO WITH THE ONE OBTAINED A 133

BY CALCULATION. A 134

C(I+1,J)=Z9/(K(J)*(C(I,J)-Z(J))+Z(J)*(1-C(I,J))) A 135

GO TO 270 A 136

C(I+1,J)=C(I,J) A 137

CONTINUE A 138

IF (I.EQ.l) GO TO 300 A 139

STOPPING CRITERION FOR ITERATION LOOP. IF NOT CONVERGENT AFTER A 140

19 LOOPS, STOP ITERATION AND PRINT 'NO CONVERGENCE'. A 141

PRINT RESULTS. A 142

DO 280 J=1,N1 A 143

IF (ABS(C(I,J)-C(I-1,J) ) .GT. .00001) GO TO 290 A 144

CONTINUE A 145

GO TO 320 A 146

IF (I.EQ.19) GO TO 310 A 147

CONTINUE A 148

WRITE (10,50) A 149

DO 360 N=1,N1 A 150

CON=C(I,N)*E(I) A 151

WRITE (10,60) K(N), CON A 152

IF (ABS(K(N) )-1.0) 330,340,330 A 153

IF (C( I.N)) 350,340,350 A 154

WRITE (10,70) A 155

GO TO 360 A 156

DABS=1/D(N) A 157

WRITE (10,80) DABS A 158

CONTINUE A 159

WRITE (10,90) E(I),I A 160

GO TO NEXT SET OF DATA (LINE 130). A 161

GO TO 130 A 162

STOP A 163

END A 164-
SUBROUTINE INP ( AM,M, QM,B , IX, JJ

)

B

C THIS SUBROUTINE READS ALL THE REQUIRED DATA FOR A CALCULATION. B

INTEGER CH B

REAL M B

DIMENSION AM(IX,JJ) ,M(6,6) ,QM(6,6) ,B(6,6) ,CH(2) B

COMMON 11,10 B

DATA CH(1) ,CH(2)/1HK,1HL/ B

10 FORMAT (1H ,

' THE ',A2,'-LINE OF ' , 13 ,' EXCITES THE ' ,A2,'-LINE OF', B

113) B

20 FORMAT ( 1H ,'NO FLUORESCENCE CORRECTION')

30 FORMAT (10F8.4)

C READ DATA FOR EACH ELEMENT.

READ (11,30) AM

READ (11,30) ( (M(I,J) ,1=1, JJ) ,J=1,JJ)

READ (11,30) ((QM(I,J) ,I=1,JJ) , J=l , JJ

)

READ (11,30) ( (B(I,J) ,I=1,JJ) ,J=1,JJ)

IA=0

DO 80 1=1, J

J

B

DO 80 J=1,JJ B

IQ=QM(I,J)+1 B

IC1=CH(1)

IC2=CH(1)

GO TO (80,70,60,50,40) , IQ

40 IC1=CH(2)

GO TO 70 B

50 IC2=CH(2) B

GO TO 70 B

60 IC1=CH(2) B

IC2=CH(2) B

70 IAM=AM(1,J) B

IBM=AM(1,I)

C PRINT OUT THE CHARACTERISTIC FLUORESCENCE CORRECTION THAT HAS

C BEEN SELECTED.

IF (IQ.GT.l) WRITE (10,10) IC1 , IAM, IC2 , IBM

80 IA=IA+QM ( I , J

)

IF (IA.EQ.O) WRITE (10,20)

C RETURN TO MAIN PROGRAM.

RETURN

END

30
SUBROUTINE BACK (A.R.N1)

THIS SUBROUTINE CALCULATES THE BACKSCATTER COEFFICIENT, R, FROM

THE EQUATION OF DUNCUMB

DIMENSION A(5,N1),R(6,6)

DO 10 .1=1, Nl

W=A ( 4 , I ) /A ( 3 , I

)

W2=W**2

W3=W**3

W4=W**4

W5=W**5

DO 10 J=1,N1

Rl= (- . 581+2 . 162*W-5 . 137*W2+9 . 213*W3-8 . 619*W4+2 . 962*W5 ) *1E-2*A ( 1 , J

)

R2= ( -1 . 609-8 . 298*W+28 . 791 *W2-47 . 744*W3+46 . 54*W4-17 . 676*W5 ) * 1E-4*A

(

11,J)**2

R3= ( 5 . 4+19 . 184*W-75 . 733*W2+120 . 05*W3-110 . 7*W4+41 . 792*W5 ) *1E-6*A ( 1

,

1J)**3

R4=(-5 . 725-21 . 645*W+88 . 128*W2-136 . 06*W3+117 . 75*W4-42 . 445*W5 ) *lE-8*

1A(1,J)**4

R5=( 2 . 095+8 . 947*W-36 . 51*W2+55 . 694*W3-46 . 079*W4+15 . 851 *W5 ) *1E-10*A

(

11,J)**5

R ( I , J

) =1+R1+R2+R3+R4+R5

RETURN

END

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-

31

APPENDIX II

NBS TECHNICAL NOTE 521

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. 20234

Although the foundations for a procedure of data reduction

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.

35
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

account 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.

36
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 ;

nA

= NAv .,.^/°-5|5-dE (1)

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 al. [7,8]:

Q = b- 77 e

4

. z£ . MI . E- 2 (2)

S. -i.iE = 2ffe 4. N ZK /oE\ ( 3 ) P dx Av A E \J /

The values of some parameters of these equations are, however,

poorly known c 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)

3

A list of symbols used is appended.

57
Another expression for J has been proposed by Duncumb et at.

[12]:

J/Z = 14.0 [1 - exp (-0.1 Z)] + 75.5/Z z/7 - 5 - ZA100 + 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

i l A. exp '4**5 e" (f)

(6)

As shown by Springer [13l s if one introduces equations (2),

(3), and (6), the integration of equation (1) yields the

expression

n A = 1 _A z 2AM£ bIiu -l-M[EI(£n(U V))-EI(£nV)]i

1 V

'

in which EI is the exponential integral:

(7)

EI (x ) = const. + In x + 2

8 = 1 s-s ! (8)

Expression (7) is used in an extensive program for data processing in quantitative electron probe microanalysis 3 called

COR, recently developed by Henoc and Heinrich at the National

Bureau of Standards [1-4],

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 fac58
tor R (backscatter effect), which decreases rapidly with increasing atomic number. The average number of ionizations per

impinging electron is thus n

fl R. We believe that the most

reliable expression for the factor R is that proposed by

Duncumb [15] :

R = 1.0000 +

+ (-.581+2. 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 +41.792W5

) x 10~ 6 Z

3 + (9)

+ (_5.725-21.645W+88.128W 2-136.060W 3+117.750W4 -42.445W5

) x 10~ 8 Z

4 +

+ (2.095+8.947W-36.510W 2+55.694W 3 -46.079W4+15.851W5

) x 10- 10 Z

5

where W = 1/U = E /E, . This expression agrees well with the O OK!

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,

(jo , and with the ratio of the intensity of the line of interest m' J to the intensity of all lines originating from this shell

(weight a of the line, p ): j ^mn

II = nA . R..fl> • p (10)

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 + j!_ . 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 1 . Expressions for the

coefficient a, which depends on the electron energy, were pro59
posed by Duncumb and Shields [18] and by Heinrich [19]. We

presently use the following equation:

4.5 x 10 5 (12)

E

i- 65 - E, 1 - 65

o k 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

that 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]:

l£i C . *PA /A" 1 "B V- <A,BK«) A

IpA L

' a rA

2 ,z(AB,BK«) B

r tA

. C

J pA

'£n(l + U> En(l + V) (13)

In this equation, r_ is the ratio of the attenuated fluorescent

radiation to the attenuated primary radiation of the element

which is being& determined. I' /I' is the ratio of the generpB pA

40
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 U 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 R/I

A is fully calculated, using equation (10) for

primary intensities, just as in the atomic number correction.

This simplifies the calculation of fluorescence of K lines by

L lines and vioe versa, as well as the calculation of fluoresence of Ka lines by K3 lines, which is performed separately.

As pointed out by Criss [23] 5 the exponential approximation to

primary distribution in depth of x-ray emission, used by

Castaing to calculate the term v in equation (13) > 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

41
in the program COR. The formal integration is replaced by a numerical 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*z .k*A

.k* (14)

42
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 obtained,,

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* + 21*

(15)

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

*3

K =

I*

I

I* + I* + 21*

Ip + h

+ 2lf
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:

i^.ai^ (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 fraction 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

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

Cm

=

k m .C(l-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 c

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.

44
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.

45
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).

[7] 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.

110] 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. and Miner, N. J. B., Mh., 1967, 304-317

[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

Optics and X-ray Microanalysis, Academic Press, NewYork, 379-392 (1963).

46
[18] Duncumb, P. and Shields, P. K. Shields, The Electron

Microprobe, McKinley, Heinrich, Wittry, editors,

Plennum Press, New York, 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 Heinrich, K. F. 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] Heinrich, K. F. J., The Electron Microprobe, McKinley,

Heinrich, Wittry, editors, John Wiley, New York,

296-377 (1966).

[26] Springer, G., N. Jb . Miner. Abh. 10_6, 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).

47
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

t I intensity of generated radiation, before attenuation

J mean ionization potential

M variable used for stopping power calculation (eq.

6-7)

N . Avogadro ' s number Av to

Q ionization cross-section

R correction factor for electron backscattering

S stopping power

U = E/E,

k

U = E /E,

o ok

V variable used for stopping power calculation (eq.

6-7)

W = 1/U

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

e charge of electron

48
f(x) correction factor for absorption of primary radiation

2

h = 1.2(A/Z ) in Philibert's equation. For multielement targets, , „ _ . to /r7 2 h = E C . A. /Z

1

111

EI exponential integral

In natural logarithm

n fl number of ionizations produced by one electron

which comes to rest within the specimen

p relative frequency of the line n, generated on

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.jk-p 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

emergence angle of measured x-rays

u 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 = y cosec 6

go = fluorescence yield

49
Subscripts and superscripts:

A 3 B elements A, B

i tally in integration

1 electron shell 1

* composition dependent

50
FORM NBS-114A (1-71) U.S. DEPT. OF COMM. BIBLIOGRAPHIC DATA

SHEET

1. PUBLICATION OR REPORT NO. NBS TN-7L9

2. Gov't Accession

No.

3. Recipient's Accession No.

4. TITLE AND SUBTITLE

A Simple Correction Procedure for Quantitative

Electron Probe Microanalysis

5. Publication Date May 1972

6. Performing Organization Code

7. author(s) Kurt F. J. Heinrich, Robert L. Myklebust,

Harvey Yakowitz, and Stanley D. Rasberry

8. Performing Organization

9. PERFORMING ORGANIZATION NAME AND ADDRESS 10. Project/Task/Work Unit No. NATIONAL BUREAU OF STANDARDS

DEPARTMENT OF COMMERCE

WASHINGTON, D.C. 20234

11. Contract/Grant No.

12. Sponsoring Organization Name and Address

Same as No . 9

.

13. Type of Report & Period Coveted

Final

14. Sponsoring Agency Code

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.)

A calculation technique for data reduction in quantitative

electron probe microanalysis is described. This technique is

embodied in a computer program, called MULTI8, written in FORTRAN IV

for batch processing. Six chemical elements may be determined

simultaneously without redimensioning the program. The relative

x-ray intensity data are corrected for atomic number, absorption

and secondary fluorescence due to characteristic lines. Options

for calculating the mass fraction of one chemical element by

difference or by stoichiometry are available in the program.

17. key words (Alphabetical order, separated by semicolons) Absorption; atomic number; chemical

elements; computation procedure; data reduction; electron probe microanalysis; fluorescence ; time-sharing; x-ray analysis . 18. AVAILABILITY STATEMENT

(23 UNLIMITED.

I I FOR OFFICIAL DISTRIBUTION. DO NOT RELEASE

TO NTIS.

19. SECURITY CLASS

(THIS REPORT)

UNCLASSIFIED

20. SECURITY CLASS

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21. NO. OF PAGES

49

22. Price 55 cents

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