ON 'tHE EQUIVALENCE OF METHODS OF DARWIN

AND EWALD--LAU E

GENERALIZATION OF DARWIN'S METHOD TO AN ARBITRARY

NUMBER OF STRONG WAVES

A. V. Kuznetsov and A. D. Fofanov UDC 539.1.01:548.0.53

The fundamental sys tem of equations of the dynamical theory of scat ter ing by an ideal c ry s t a l is given by Darwin ' s method for any number of s t rong waves . The equivalence of this sys tem and the fundan'mntal sys tem of equations of Ewald--Laue is demonst ra ted .

i. I n t r o d u c t i o n

Various methods are used for the computation of the diffraction pat tern scat tered by an ideal c rys ta l . The f i rs t method was proposed by Darwin in 1914 [1]. However, he considered only the symmet r i c Bragg case . In 1917 Ewald [2] developed a dynamic theory based on the concept of oscil lat ing dipoles. In 1931 Laue [3] gave his t rea tment of the dynamical theory . In subsequent y e a r s the main mass of theoret ical and experimental works was based on the theor ies of Ewald and Laue, and in 1968 Wagenfeld [4] demon- strated their equivalence.

Darwin ' s method has come into use again for the computation of the pat tern of sca t ter ing by an ideal c rys ta l in a number of works of the Iast decade [5-8]o However, here par t i cu la r problems have been solved and in each of these it is shown that the same resul t s are obtained by Darwin ' s method as by Ewald--Laue method. But even now a number of inves t igators s t r e s s the lack of r igor in the method of Darwin and the l imitat ions of i ts application.

In the presen t work an attempt is made to show that the r ecu r ren t equations in the method of Darwin, written in vec tor fo rm for any number of s t rong waves, can be reduced to the fundamental sys tem of equations obtained by Ewald--Laue method, so that any conclusions derived f rom Ewald--Lane theory can be obtained also by Darwin' s method.

2 . S c a t t e r i n g b y A t o m i c P l a n e

A crys ta l can be represented as a set of any group of atomic planes, in par t icu lar , paral lel to the surface of the c rys t a l . We s t r e s s that this is not a Bragg plane in the usual sense. One of these planes is shown in Fig. 1: x, y, z is a rec tangular sys tem of coordinates chosen in such a way that the unit vector of the incident wave s o l ies in the yz plane; a and b are the axial scales of an e lementary cell in the plane; s H is the unit vec tor of one of the scat tered waves; ~H is the angle at which the plane wave is sca t te red .

We note that fo r x - r a y s the unit vec tors of the incident and scat tered waves may not even lie in one plane containing the normal to the atomic p l a n e .

The amplitude of the scat tered wave is

D. ~ D ~ e x p -- i 2-~A.. = ( I )

Under the condition that the size of the atomic plane is considerably l a r g e r than the f i r s t Fresnel zone and the incident wave is plane, the path difference AH is of the fo rm

0 . V. Kuusinen Pet rozavodski i State Univers i ty . Trans la ted f rom Izvest iya Vysshikh Uchebnykh Zavedenii, Fizika, No. 4, pp. 108-113, April , 1972. Original ar t ic le submitted J anua ry5 , 1971.

0 1974 Consultants Bureau a division of Plenum Publishing Corporation, 227 West 17th Street New York, N. Y. 10011. [ No part of this publication may be reproduced stored in a retrieval system, or transmitted, in any form or by any means, ] electronic, mechanical, photocopying microfilming, recording or otherwise withoat written permission of the publisher. A ] copy of this article is available from the publisher for $15.00. l

559

4t

r

. . . . J----- 7

/ / ! \ ../' I '

I l I I I I

Fig. 1, Pa t t e rn of sca t t e r ing of x - r a y s by an a tomic p lane .

rD2,r 117~r\\ " ' r+1

\\

I] D',~§ \&r, t ~/.r+/ (h~k~l~)

Fig . 2. Wave field inside of c rys taUine media in the case of th ree s t rong waves .

r2 - - ( s . , r) ~=-($.-$o)r+ (2) 2R

The summat ion in the genera l f o r m is not poss ib le and it mus t be rep laced by in tegra t ion .

Fo r an a tomic plane of l a rge d imensions the sca t t e red waves will have apprec iab le in tens i t ies only for those d i rec t ions fo r which Laue conditions a re s t r i c t ly sa t i s f ied . In this case the f i r s t t e r m in fo rmula (2) may be neglected because of the cyclic na ture of the exponent, s ince

2~ ,8

Then

Using the usual a s sumpt ions (possibil i ty of taMng out Dat, the ampli tude of sca t t e r ing of a toms , outside the in tegra l sign, extending: the l imi t s of in tegra t ion to infinity), we obtain

~R DH = iD~t,

The quantity ab[l--(s H, a/a) 2-. . . ]i/2 is the volume of the parallelopiped constructed on the vectors a, b,

~H and is equal to (~ x b], SH).

Then

~R (3) D. = iDat , ([a X b] s.)

The quantity [a x b] i s a vec to r no rm a l to the a tomic plane and equal to z0ab sin (ab) (z 0 is the unit no rma l to the a tomic plane) . [ab sin (ab)] J = n is the n u m b e r of sca t t e r ing c e n t e r s p e r unit a r e a of the a tomic p lane .

The ampli tude of the wave sca t t e red by an a tom [9] i s

e21 Dat = ~c2 ~j%_L Is. ){ [$. X DL]], (4)

where f H - L is the a tomic fac tor ; the index "H" denotes the di rect ion of sca t te r ing , and the index L denotes the di rect ion of incidence.

560

We substitute (4) into (3) and for the amplitude of the wave scat tered by an "infinite" atomic plane we obtain the following express ion:

3 .

D . = i e" n ) . f . - L Is" X Is. X DLI] mc" " (Zo, s . ) (5)

S c a t t e r i n g b y " T h i n " C r y s t a l l i n e P l a t e

We form a crys ta l l ine plate f rom such planes . Each plane of the plate gives severa l scat tered waves whose amplitudes are given by formula (5), but only those waves which satisfy the third equation of Lane C(SH--So) = lX will be amplified. Here c is the distance between the planes of the crys ta l l ine plate. The value of this quantity its not required for the final computat ions.

If the thickness of this plate is small , it can be shown that for direct ions close to the direct ions s H the amplitude of the wave scat tered by such a p l a t e i s A r t i m e s l a r g e r than the amplitude of the wave scat tered by a single plane. (The direct ion s H sat is f ies the in te r fe rence equation of Lane for a t h r e e - dimensional grat ing SH--S 0 = XI-I, where H is the vector of the inverse grating; Ar is the number of atomic planes in the plate).

Fu r the rmore , the atomic factor can be replaced by the s t ruc ture fac to r . Then the amplitude of the wave scat tered by a thin crys ta l l ine plate in direct ions close to the direct ions s H is equal to

e ~ [s, • [s, • DLI] D . = i - - Nc ) ,F ._r h r

m c ~ (Zo, s . )

= r'r'c-- %-L Ar [s. X [ s . • DrlI = i G ~ _ r A r [s. 'X [s.X DLll (6) z (Zo, s.) (Zo, s~)

The quantities GH_ L and CH-L are introduced to cut short the subsequent computat ions.

4 . S c a t t e r i n g b y " T h i c k " C r y s t a l l i n e P l a t e ( C r y s t a l )

We fo rm a c rys ta l f rom "thin" crysta l l ine plates , l aminas . The concept of "thin" and "thick" is highly re la t ive . The crys ta l can be a tonce regarded as a set of planes paral le l to i ts surface. However, the introduction of the in termediate unit, i . e . a "thin" crys ta l l ine plate o r "lamina" in the investigation of which the interact ion of the scat tered waves is not considered, offers the possibil i ty of introducing the s t ruc ture fac tor and the representa t ion in the inverse space~

A crys ta l composed of such " laminas" a distance Ar apar t is shown in Fig. 2. Each such "lamina" is depicted by a plane perpendicular to the plane of the f igure .

Let n s trong waves appear when a plane wave is incident on such a plane, " l amina , "a t an angle (the case of th ree s t rong waves is shown in Fig. 2). In this case three scat tered waves at angles a, ill, fi2 appear with the incidence of a plane wave on the "lamina" at an angle fil. Next when the plane wave is incident on the "lamina" at an angle fil (point 5 in Fig. 2), then in this case also three scat tered waves appear: two obvious waves at angles tr and fll and the third at an angle fi2. This is easily shown f rom the analysis of the in te r fe rence equation of Laue SH--S 0 = Xtt. The indices of the plane, which gives reflection at an angle f12, a re (hi--h2, kl--k2, ll--/2). Here (hi, kl, ll) and (h2, k2, lz) a re the indices of Bragg planes shown in Fig. 2. The same conclusions can be extended also to the case of any number of s t rong waves, i . e . each scat tered wave when incident on a "lamina" gives the same number of scat tered waves as the p r i m a r y wave.

The sys tem of r ecu r r en t equat ions for any number of s t rong waves can be easi ly written f rom the construct ion of Fig. 2.

Let us c o n s i d e r the principle of compar ison of r e c u r r e n t equations for the wave Do, r + 1 which is scat tered by the lamina with number (r + 1) at an angle ~. The wave Do, r+ i is made up of the wave Do, r , then of the wave scat tered by the lamina with number (r + 1) during the passage of the wave Do, r through it; according to formula (6) the amplitude of this wave is equal to

i Go Ar [so X [So X Do, r] ]* (Z0, SO)

561

Next the wave s ca t t e r ed by the l amina with n u m b e r (r + 1) during the p a s s a g e of the wave D1, r through it g ives contr ibution to the wave Do, r + l . The ampli tude of this wave i s i (G-1Ar/(z 0, so)) [so • [So x D l , r ] ] and so for th .

I t i s n e c e s s a r y to cons ider succes s ive ly all the waves and take account of the path d i f ference between the wave Do, r + l and all o t h e r s . In the same way we can wri te the r e c u r r e n t equations for any o ther wave DH, r + l . The change of phase during the p a s s a g e of the wave DH between the p lanes is wri t ten in the f o r m

~ (c, s.). 9, (7)

Then taking into cons idera t ion all the d iscuss ion above, the s y s t e m of r e c u r r e n t equations will have the f o r m

Do,~+~ = Do,~exp(i%Ar) -4-i GeAr [So X [SeX Do,~]]exp(i%Ar) ( Zo, So)

G-1 Ar, [so X ['So X D1, r]] exp (iglAr) + i (zo, So)

~- i G-~ Ar [So X [So X D2,,] ] exp (ig~Ar) + �9 . . . (Zo, So)

D~, ,+~ = Dr,, exp (i~,Ar) + i Go Ar [s~ X [s~ X D~, r]] exp ( ig~r) (Zo, s~)

+ i G ~ r , [si X [s~ X Do, r]] exp (i%Ar) (z0, s , )

. G,-2 ~r . . . (8) t: [sl X [sl X D2, r]] exp (i92Ar) -4- (z0, si)

. . . . �9 �9 �9 . �9 �9 �9 �9 �9 �9 �9 �9 �9 * o' . �9

The s y s t e m of equations (8) for any n u m b e r of s t rong waves can be rewr i t t en in a m o r e compact fo rm:

Ar ~ 6. 'L [s. X [s. X DLll exp (i~,Ar). (9) D.,r+l = D . , r exp ( ig , Ar) + i (z---~o, s.---( z

The s y s t e m of r e c u r r e n t equations (9) can be solved by usual methods , i . e . by making the substi tution Do, r + l = x D0, r , where x i s a complex quantity which is independent of r . However , sy s t em (9) can be reduced to a s y s t e m of homogeneous di f ferent ia l equat ions . For th is pu rpose the phase d i f ference can be wri t ten in the following fo rm:

2= ~" = T c ( s . ~ + As,) = ~ + a~ , . ( io )

H e r e the vec to r s ~ c o r r e s p o n d s to the d i rec t ion for which the path d i f ference i s exact ly equal to an in tegra l mult iple of the wavelength ( i . e . (2~/k) c s ~ = 2~n, n is an integer) and ASH is the depa r tu re f r o m this

quanti ty. Then

exp (i9. h r ) = exp (ihg. hr) ~ 1 + lAg.hr. (11)

In an analogous way exp( igLAr) can also be e x p r e s s e d . We subst i tute the phase f ac to r s in f o r m (11) into fo rmu la (9) and d i sca rd all t e r m s of second o r d e r to sma l lne s s , i . e . t e r m s An which product of two smal l quant i t ies A9 .G o c c u r . In th is case fo rmula (9) becomes

i D.. r+l - - D . ;r = iA~. D . r -~ - ~ O . - , [ s . X [s . M DL. ,11. ( i2) ~r ' (Zo, s . ) L

By tl~e method of f inite d i f fe rences the lef t side of f o rmu la (12) can be rep laced by the f i r s t der iva t ive ;

then

dDH' =~A~HOH" ~ ~ ~L~ G H - L [ S H X [SI'I X D F ] ] " ( 1 3 ) dr ' ( Zo~-S.)

The solution of th is type of s y s t e m is sought in the f o r m

D. = D ~ exp ( - - izr). (14)

562

Whence dDH/dr = --icoI)H and sys tem (13) can be writ ten in the form

[Go -- (zo, s.) (~ + h,?.)] D. = ~ G.-L [s. X [s. X DLI] L ~ K

(15)

or in the fo rm (see formula (6))

In o r de r to de te rmine the wave field it is n e c e s s a r y to find ~o f rom system (16). However , instead of w one can introduce the new var iable v = co + Ag01 then formula (16) can be rewri t ten in the following form:

[ *~ - ~ (z~ s") (~ + ~ c h%--h~'~ o'= L§ r163 [s. X [ s ,X DL]]. (17)

The fundamental sys tem of dynamical theory of Ewald--Laue in the same notations can be writ ten in the following form [10]:

(~o -- 2~.) D . = ~ *._L [S. X [s. X D d ] . H ~ L

In o rde r to prove the equivalence of sys tems (!7) and (18) it is n e c e s s a r y to show that

2~. = --~ (z0, s . ) (~ + A~. - @0). ~ C

For the case of two strong waves this was done by us in [8]. (19) can be proved also for the most general case .

(18)

(19)

By some elementary manipulations identity

CONCLUSIONS

Thus the system of equations (17) obtained by Darwin's method is equivalent to the fundamental system of the dynamical theory of Ewald--Laue.

These conclusions can be obtained even after the transformation of the system of recurrent equations (9) omitting the system of homogeneous differential equations; however, in our opinion the system of homogeneous differential equations (13) is more convenient for practical computations.

In general case t]he method of Darwin does not have, in our opinion, any advantages over the method of Ewald--Laue for the computation of the diffraction pattern scattered by an ideal crystal, but in some special cases for crystals with imperfections (one or several dislocations) Darwin's method can give a simpler way of solving the posed problem.

LITERATURE CITED

1. C. G, Darwin, Phi l , 1Kag., 27, 315, 675 (1914). 2, P . P . Ewald, Ann. P h y s . , 54, 519 (1917). 3, M. Lane, Ergebn. Exakt . Na tu rwiss . , 10, 133 (1931). 4. H. Wagenfeld, A e t a C r y s t a l l o g r . , A 2 4 , 1, 170 (1968). 5. A. Howie and M. J , Whelan, P r o c . Roy. Soc . , A263, 217 (1961). 6. B. Bor ic , Acta C rys t a l l og r . , 2_33, 2, 210 (1967). 7. H. J , Holloway, J . Appl. P h y s . , 4___0, 5, 2187 (1969). 8. A . V . Kuznetsov and A. D. Fofanov, Izv . Vuzov SSSR, Fizika, No. 10, 12 (1970). 9. A.I . Kitaigorodskii, X-Ray Structure Analysis [in Russian], GITTL (1950), p . 211,

10. R. J a me s , Optical P r inc ip les of the Diffract ion of X-Rays , Cornel l Univ. P r e s s (1948),

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