AN IMPROVED ABSORPTION CORRECTION FOR QUANTITATIVE ANALYSIS

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AN IMPROVED ABSORPTION CORRECTION FOR QUANTITATIVE ANALYSIS

G. Love, D. Sewell, V. Scott

To cite this version:

G. Love, D. Sewell, V. Scott. AN IMPROVED ABSORPTION CORRECTION FOR QUANTITATIVE ANALYSIS. Journal de Physique Colloques, 1984, 45 (C2), pp.C2-21-C2-24.

�10.1051/jphyscol:1984205�. �jpa-00223760�

JOURNAL DE PHYSIQUE

Colloque C2, suppl6ment au n02, Tome 45, f6vrier 1984 page C2-21

AN IMPROVED ABSORPTION CORRECTION FOR QUANTITATIVE ANALYSIS

G. Love, D.A. Sewell and V.D. S c o t t

SchooZ of Materials Science, University of Bath, U.K.

Rdsumd - On propose une c o r r e c t i o n n o u v e l l e e t p o l y v a l e n t e pour l ' a b s o r p - t i o n . E l l e e s t fondee s u r une r e p r d s e n t a t i o n approchde de l a d i s t r i b u t i o n en profondeur du rayonnement X c a r a c t e r i s t i q u e , sous l a forme d'un p r o f i l q u a d r i l a t e r a l .

A b s t r a c t - A new and v e r s a t i l e c o r r e c t i o n f o r a b s o r p t i o n i s proposed which i s based upon s i m p l i f y i n g t h e x-ray d e p t h d i s t r i b u t i o n i n a t a r g e t t o a q u a d r i l a t e r a l p r o f i l e .

1. INTRODUCTION

When c a r r y i n g o u t a n a l y s i s of u l t r a l i g h t elements (boron, carbon, n i t r o g e n , oxygen and f l u o r i n e ) very l a r g e mass a b s o r p t i o n c o e f f i c i e n t s a r e involved. (For example a ,mass a b s o r p t i o n c o e f f i c i e n t of 37000 i s given f o r

r a d i a t i o n i n s i l i c o n ( 1 ) ) . Consequently t h e a b s o r p t i o n c o r r e c t i o n f a c t o r t e n d s t o b e l a r g e and its a c c u r a c y s t r o n g l y i n f l u e n c e s the o v e r a l l r e l i a b i l i t y of c o r r e c t e d m i c r o a n a l y s ~ s d a t a . The u s u a l way of reducing t h e magnitude of t h e a b s o r p t i o n c o r r e c t i o n i s t o lower t h e probe v o l t a g e , t h e r e b y d e c r e a s i n g t h e d e p t h t o which e l e c t r o n s p e n e t r a t e i n t o t h e specimen. However, even i f probe v o l t a g e s of lOkV a r e used f o r l i g h t element work, it i s found t h a t t h e a b s o r p t i o n c o r r e c t i o n f a c t o r , f ( X ) , may s t i l l be

0.2 o r l e s s ( i . e . 80% of t h e x-rays a r e a b s o r b e d ) . Reducing t h e probe v o l t a g e

s u b s t a n t i a l l y below 10kV i s g e n e r a l l y n o t f e a s i b l e because high beam c u r r e n t s a r e r e q u i r e d t o g e n e r a t e s u f f i c i e n t x-rays f o r a n a l y t i c a l purposes and image q u a l i t y i s degraded. Moreover it i s v e r y inconvenient t o o p e r a t e a t low k i l o v o l t a g e s

because many o t h e r x-ray l i n e s of i n t e r e s t may not b e e x c i t e d . Thus any a b s o r p t i o n c o r r e c t i o n which can b e c o n s i d e r e d t o b e u n i v e r s a l l y a p p l i c a b l e must perform a c c u r a t e l y over t h e range 1 > f ( X ) > 0.2.

2. THE ABSORPTION CORRECTION

The method most commonly used f o r c o r r e c t i n g f o r a b s o r p t i o n i n t h e specimen i s t h a t of P h i l i b e r t ( 2 ) . Although based upon t h e p h y s i c s of x-ray g e n e r a t i o n it i n c l u d e s s e v e r a l d r a s t i c approximations and i s g e n e r a l l y c o n s i d e r e d t o b e i n a p p r o p r i a t e f o r l i g h t element work ( 3 ) .

An a l t e r n a t i v e approach t o o b t a i n i n g f ( X ) h a s been by d e r i v i n g , wholly e m p i r i c a l l y , a n a l y t i c a l e x p r e s s i o n s t o r e p r e s e n t t h e g e n e r a t e d x-ray d i s t r i b u t i o n with d e p t h i n t h e t a r g e t ( h e r e a f t e r r e f e r r e d t o a s t h e $ ( p z ) c u r v e ) . The a b s o r p t i o n c o r r e c t i o n f a c t o r i s t h e n given by

~ ; ( D z ) e x p ( - X p z ) d p z

where X = p/p cosec $, p/p b e i n g t h e mass a b s o r p t i o n c o e f f i c i e n t and

t h e x-ray take-off a n g l e . This $ ( p z ) c u r v e f i t t i n g method i s becoming i n c r e a s i n g l y popular because t h e r e a r e now a l a r g e number of d a t a on x-ray depth d i s t r i b u t i o n s

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1984205

C2-22 JOURNAL DE PHYSIQUE

which have been o b t a i n e d e x p e r i m e n t a l l y using t r a c e r t e c h n i q u e s o r Monte Carlo methods.

Before a t t e m p t i n g t o model a @ ( p z ) curve it i s necessary t o e s t a b l i s h t h e most important parameters t o i n c l u d e i n a n e q u a t i o n f o r @ ( p z ) and how a c c u r a t e l y t h e s e parameters have t o b e d e f i n e d .

i s obviously unneeessary t o d e s c r i b e (P(pz) e x a c t l y i f appraximations t o t h e shape i n t r o d u c e i n s i g n i f i c a n t e r r o r s i n f ( x ) . Bishop ( 4 ) showed t h a t when f ( X ) > 0.5 t h e s i n g l e most important parameter

c o n t r o l l i n g t h e - magnitude of t h e a b s o r p t i o n c o r r e c t i o n was t h e mean depth of x-ray g e n e r a t i o n , pz. H i s p r o p o s a l was t o use a simple r e c t a n g u l a r p r o f i l e t o r e p r e s e n t U p z ) whereby t h e x-ray production remained c o n s t a n t u n t i l t h e mass d e p t h was t w i c e pz and t h e n f e l l a b r u p t l y t o z e r o . Love and S c o t t ( 5 ) developed t h e i d e a f u r t h e r , u s i n g an e x p r e s s i o n f o r d e r i v e d from Monte Carlo c a l c u l a t i o n s , and were a b l e t o show t h a t t h e i r a b s o r p t i o n c o r r e c t i o n model was s i g n i f i c a n t l y s u p e r i o r t o t h a t of P h i l i b e r t , e s p e c i a l l y when X i s l a r g e . Such r e s u l t s i n d i c a t e d t h a t it i s probably n o t n e c e s s a r y t o f i t @ ( p z ) e x a c t l y although a s l i g h t l y more r e a l i s t i c p r o f i l e t h a n t h e r e c t a n g l e should l e a d t o a markedly improved a b s o r p t i o n c o r r e c t i o n .

3 . THE QUADRILATERAL MODEL

F i g u r e 1 shows a t y p i c a l @ ( p z ) v e r s u s depth c u r v e superimposed upon which i s t h e p r o f i l e we s h a l l use i n our new a b s o r p t i o n c o r r e c t i o n . P o i n t

of t h e quadri- l a t e r a l i s d e f i n e d by t h e c o o r d i n a t e s ( 0 , @ ( O ) ) . P o i n t B i s given by t h e maximum v a l u e of ( P ( p z ) , i . e . @ ( p z m ) and by t h e corresponding mass d e p t h , pzm. P o i n t C

(pzn,O) l i e s c l o s e t o t h e value f o r t h e x-ray range, pzr. Given a s e r i e s of @ ( p z ) c u r v e s , p o i n t s

and B a r e e a s i l y d e f i n e d b u t l o c a t i n g

p r e s e n t s some d i f f i c u l t y because of t h e long tail which o c c u r s , p a r t i c u l a r > y f o r h i g h atomic number elements.

Also, i n t h i s form, t h e ~ u a d r i l a t e r a l model does n o t c o n t a i n a term r e p r e s e n t i n g t h e mean depth of x-ray g e n e r a t i o n a n d , a s i n d i c a t e d e a r l i e r , it i s e s s e n t i a l t h a t t h i s be included. Both t h e s e problems a r e overcome by r e p r e s e n t i n g pzn i n terms of - pz and t h e o t h ~ ? r t h r e e parameters @ ( O ) , @ ( p z m ) and pzm. Geometrical c o n s i d e r a t i o n s r e v e a l t h e following r e l a t i o n s h i p .

With t h i s simple q u a d r i l a t e r a l shape it i s p o s s i b l e t o d e r i v e d i r e c t l y an a n a l y t i c a l expression f o r £ ( X I , v i z

f ( X )

2 [m (n-m) (m+kn)x2 I-'

x

(n-kn-m)exp(-xm) + km exp (-xn) + ~(rnn-m2) + kn- n - km + ml ( 2 ) where k

@ ( p z m ) / @ ( 0 ) , m

pz and n

pz

m n '

Although t h i s formula may appear complicated it i s r e a d i l y solved with t h e a i d of a mini-computer a n d , u n l i k e most o t h e r @ ( p z ) c u r v e f i t t i n g t e c h n i q u e s , lengthy numerical i n t e g r a t i o n i s n o t r e q u i r e d .

The n e x t s t a g e i s t o t e s t t h e performance .of t h e model o v e r a range of X v a l u e s . I n i t i a l l y we w i l l make t h e assumption t h a t a l l f o u r parameters ( ( P ( O ) , (P(pzm), pZm and p ) c a n b e d e f i n e d e x a c t l v . This h a s been done usinq t h e Monte Carlo (P(0z) c u r v e f o r aluminium K r a d i a t i o n a t 30keV shown i n f i q u r e 1 , althouqh any

r e u r e s e n t a t i v e (P(pz) p r o f i l e i s s u i t a b l e f o r e v a l u a t i o n puruoses. Values f o r f ( X )

o b t a i n e d u s i n g t h e Q u a d r i l a t e r a l model a r e compared with t h o s e o b t a i n e d from t h e

Monte C a r l o (P(pz) c u r v e f o r a range of X v a l u e s between 100 and 100,000. In f i g u r e

2 t h e r e s u l t s a r e p l o t t e d a s f(X)QUAD/f(X)MC v e r s u s l o g X. When t h e r a t i o of t h e

f ( X ) v a l u e s i s u n i t y it may b e a d ~ u d g e d t h a t t h e Q u a d r i l a t e r a l model i s performing

p e r f e c t l y . Also shown, f o r t h e purposes of comparison, a r e t h e e q u i v a l e n t r e s u l t s

o b t a i n e d from t h e P h i l i b e r t and Bishop a b s o r p t i o n models, b o t h of which have been

a d j u s t e d t o g i v e t h e c o r r e c t mean depth of x-ray g e n e r a t i o n .

i s e v i d e n t t h a t

t h e Q u a d r i l a t e r a l model g i v e s by f a r t h e b e s t f ( X ) d a t a and e r r o r s i n excess of 5%

r e l a t i v e do n o t occur u n t i l x exceeds 20,000. In c o n t r a s t a 5% e r r o r c r i t e r i o n i s exceeded when X > 3,000 f o r t h e Bishop model and X > 1000 f o r t h e P h i l i b e r t model.

T h i s ranking o r d e r of performance w i l l remain t h e same f o r o t h e r a n a l y t i c a l c o n d i t i o n s although much h i g h e r X v a l u e s can b e t o l e r a t e d a s t h e probe v o l t a g e i s reduced. This i s i l l u s t r a t e d i n f i g u r e 3 where 5% e r r o r s a r e n o t exceeded u n t i l

X > 7500 ( s i m p l i f i e d P h i l i b e r t ) and X > 30,000 ( B i s h o p ) ; f o r t h e Q u a d r i l a t e r a l model e r r o r s a r e l e s s t h a n 1 % up t o X

100,000. Q u i t e c l e a r l y t h e n t h e simple q u a d r i l a t e r a l shape i s p e r f e c t l y adequate t o d e s c r i b e t h e $ ( p z ) curve with t h e necessary degree of accuracy and it merely remains t o d e r i v e a p p r o p r i a t e e q u a t i o n s f o r @, @ ( O ) , $(pzm) and pzm.

A s with a l l a b s o r p t i o n c o r r e c t i o n s , t h e Q u a d r i l a t e r a l model depends s e n s i t i v e l y on t h e v a l u e of pz and f i g u r e 4 i l l u s t r a t e s t h e e f f e c t of a 10% e r r o r i n i t s v a l u e on t h e aluminium d a t a a t 30keV. A 5% e r r o r i n t h e a b s o r p t i o n c o r r e c t i o n now o c c u r s a t

X

780 compared with X

20,000, t h e v a l u e o b t a i n e d when @2 was c o r r e c t . Thus any a n a l y t i c a l expression f o r p must r e p r e s e n t t h e mean depth with t h e utmost accuracy p o s s i b l e . An e x p r e s s i o n h a s a l r e a d y been developed s p e c i f i c a l l y t o d e s c r i b e t h e mean depth o f x-ray g e n e r a t i o n ( 6 ) . However, i n view of t h e importance of g e t t i n g t h e value e x a c t l y r i g h t it i s being r e - e v a l u a t e d u s i n g new $ ( p z ) c u r v e s from t r a c e r experiments and Monte Carlo c a l c u l a t i o n s .

F o r t u n a t e l y t h e Q u a d r i l a t e r a l model i s much l e s s s e n s i t i v e t o e r r o r s i n t h e o t h e r t h r e e parameters and performance i s n o t s e r i o u s l y degraded provided t h a t t h e s e can b e s p e c i f i e d t o w i t h i n 10% o f t h e ' t r u e ' v a l u e s . Various formulae f o r @ ( 0 ) a l r e a d y e x i s t ( 7 , 8 , 9 ) and one of t h e s e may b e s u i t a b l e f o r i n c o r p o r a t i o n i n t h e

Q u a d r i l a t e r a l model. Furthermore, c u r r e n t work s u g g e s t s t h a t $ ( p z

can b e r e p r e s e n t e d a s m

and pz a s

m

where U i s t h e o v e r v o l t a g e r a t i o ,

t h e mean atomic number of t h e t a r g e t and r, t h e b a c k s c a g t e r c o e f f i c i e n t .

When t h e f o u r parameters on which t h e Q u a d r i l a t e r a l model i s based have been p r e c i s e l y d e f i n e d in terms of a p p r o p r i a t e e q u a t i o n s we s h a l l b e r e p o r t i n g upon t h e performance of o u r a b s o r p t i o n correction,when a p p l i e d t o a wide range of

experimental m i c r o a n a l y s i s d a t a . Acknowledgements

To AERE Harwell and t o SERC f o r t h e i r s u p p o r t . References

1. HENKE, B.L. and EBISU, E.S., Adv. i n X-ray Anal. - 17 (1974) 150.

2 . PHILIBERT, J . , X-ray O p t i c s and X-ray ~ i c r o a n a l y s i s ( N e w York: Academic P r e s s ) (1963) 379.

3. LOVE,

and SCOTT,

, Scanning 4 (198 1

11 1 . 4 . BISHOP,

J.Phys.D: Appl.Phys. 2 (1974) 2009.

5. LOVE,

and SCOTT,

J.Phys.D: Appl. Phys. 11, (1978) 1369.

6 . LOVE,

and SCOTT,V.D. J . P h y s . D : ~ p p l . P h y ' s . s (1977) 7.

7 . REUTER, w., X-ray O p t i c s and Microanalysis (Tokyo: ~ o k y o UP) (1972) 121.

8. LOVE, G . , COX, M.G.C. and SCOTT, V.D., J.Phys.D:Appl.Phys. 11, (1978) 23.

9. PACKWOOD, R.H. and BROWN, J . D . X-ray Spectrom. 10, (1981) 138.

JOURNAL DE PHYSIQUE

F i g . 1 . Q u a d r i l a t e r a l p r o f i l e super- imposed upon Monte C a r l o + ( p z ) c u r v e f o r aluminium Ka r a d i a t i o n a t 30 keV.

F i g . 3 . As Fig.2. b u t t h e c a l c u l a t i o n s c a r r i e d o u t f o r Fig.2. f ( x ) v a l u e s c a l c u l a t e d f o r alummiurn a t 10 keV.

aluminium a t 30 keV u s i n g t h e d i f f e r e n t c o r r e c t i o n models (f(x)mod) compared w i t h Monte C a r l o c a l c u l a t i o n s ( f ( x )

EI-C)

Fig.4. f (X) v a l u e s c a l c u l a t e d f o r aluminium a t 30 keV u s i n g g e Q u a d r i l a t e r a l model; - ^pz v a l u e c o r r e c t ; --- 10% e r r o r i n E .

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