National Bureau of Standards Uim .

HITED STATES kRTMENT OF 1MERCE

3UCATION

Lt»T0'cQ

|>«ris o* *

AUG 18 1970

NBS TECHNICAL NOTE 521

U.S.

ARTMENT

OF

OMMERCE

National

Bureau

of

Standards

Present State of the

Classical Theory of

Quantitative Electron

Probe Microanalysis
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NBS TECHNICAL NOTE 521

ISSUED AUGUST 1970

Nat. Bur. Stand. (U.S.), Tech. Note 521, 17 pages (Aug. 1970)

CODEN: NBTNA

Present State of the Classical Theory of

Quantitative Electron Probe Microanalysis

Kurt F. J. Heinrich

Spectrochemical Analysis Section

Analytical Chemistry Division

Institute for Basic Standards

National Bureau of Standards

Washington, D.C. 20234

NBS Technical Notes are designed to supplement the Bureau's regular publications program. They provide a means for making available scientific data that are of transient or limited interest. Technical Notes may be

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(Order by SD Catalog No. C 13.46: 521). Price 30 cents

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, Do 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.
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.
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 ; "-"i/'i^ <»

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£ . Ml . E~ 2 (2)

S.-i.i|S.W.NAT .|.It. feV <3)

P dx Av A E \J /

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

poorly known „ 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)

* A list of symbols used is appended.
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/75 -Z/(100 + 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:

= 2c3 i

l A. V exp

M i CiA.

£n(!

aE^

J

(6)

As shown by Springer [13] > if one introduces equations (2),

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

expression

n A = I JL 2AMH bIiu -l-inY[EI(£n(U V))-EI(£nV)]l

1 V

'

in which EI is the exponential integral:

(7)

EI (x ) = const. + £n x + 2

8 = 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 fac4
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]

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 +41.792W 5

) x 10- 6 Z

3 + (9)

+ (-5.725-21.645W+88.128W 2-136.060W 3+117.750W4 -42.445W 5

) x 10~ 8 Z

4 +

+ (2.095+8.947W-36.510W 2+55.694W 3 -46.079W 4+15.85lW 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.R, with the fluorescence yield,

a) , 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 &> of the line,i p ): ^mn

I' = nA

. R.o) . 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]

i/f( x) = (i + £>(i + rr-r- -£ >; h = 1-2 A.; (11)

The factor f(x) is the ratio between attenuated and unattenuated primary radiation: F(x) = In /I f • Expressions for the

coefficient a, which depends on the electron energy, were pro5
posed by Duncumb and Shields [18] and by Heinrich [19] • We

presently use the following equation:

4.5 x 10 5 E

!- 65 - E, 1 - 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

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]:

r

f . h± = C

'pa

! p a rA - * w b /* <A,BK«) A

B

"

XpA

2 ^(AB,BK«) B

Ml + u) (1 + 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
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 [(UB - 1)/(U A - D] L67 in which U"A

and uV. .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,-,/ In 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 viae versa, as well as the calculation of fluoresence of Ka lines by K3 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) 9 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

7
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* . k^ • k* (14)
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 D

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* + SI*

K =- = (15)

I Ip + I c + Sif 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

9
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 (1 ,.

= 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 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

c

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.

10
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.

11
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.

[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. 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, New

York, 379-392 (1963).

12
[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. 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).

13
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 n Avogadro's number

Av

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

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

e charge of electron

14
f(x) correction factor for absorption of primary radiation

2

h = 1.2(A/Z ) in Philibert's equation. For multielement targets, , _ y r

. .„ 2

i i i

1

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„ jk. 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

6 emergence angle of measured x-rays

y 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

oo = fluorescence yield

15
Subscripts and superscripts:

A,B elements A, B

i tally in integration

1 electron shell 1

* composition dependent

*U. S. GOVERNMENT PRINTING OFFICE : 1970— 431-988/6

16
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