- Email: [email protected]
A general theory of lattice-distortions
P h y s i c a I X , t~o I
A GENERAL
J a n u a r i 1942
THEORY
OF
LATTICE-DISTORTIONS
b y J. B O U M A N Gymnasium Delft
Summary A n e x p r e s s i o n for t h e i n t e n s i t y 9f a d i f f r a c t e d b e a m of X - r a y s is d e r i v e d f o r a d i s t o r t e d c r y s t a l . T h i s f o r m u l a i n c l u d e s t h e r e s u l t s of D e h 1 i n g e r a n d B o a s, a n d t h e i n t e n s i t y c a u s e d b y t h e d i s t o r t i o n , w h i c h is k n o w n as t h e f r o z e n h e a t m o t i o n . A t h e o r e m , v a l i d for all i n h o m o g e n e o u s d i s t o r t i o n s , is p r o v e d w i t h t h e h e l p of t h i s e x p r e s s i o n . T h i s t h e o r e m c a n b e u s e d for t h e d i s c u s s i o n of t h e b r o a d e n i n g or d e c r e a s e of i n t e n s i t y of t h e reflect i o n s for d i s t o r t e d c r y s t a l s .
§ I. Introduction. The broadening of D e b y e-S c h e r r e r lines, caused b y cold-working on metals, has been explained in several ways. One of t h e m represents the t h e o r y of the inhomogeneous distortion, which has been considered b y D e h I i n g e r *), B o a s 5), B r i n d 1 e y and R i d 1 e y 6) a.o. In these theories it is assumed t h a t the a t o m s are displaced from their ideal positions over small distances. This distortion must be distinguished from the homogeneous distortion, e.g. a single stretch. Here the displacements cannot be small for all the atoms, as t h e y are linear functions of the ideal coordinates of the atoms, the crystal retaining after distortion an ideal crystal s t r u c t u r e with new parameters. Now the papers referred to deal with special inhomogeneous distortions. These distortions m a y cause a broadening of the reflection, or a decrease of intensity, or both. I t is h o w e v e r not quite clear how this broadening, or the decrease of i n t e n s i t y can be derived from the chosen t y p e of distortion. A general t h e o r y does not exist at present. In this paper an a t t e m p t will be made 1° to formulate a general theory, from which special theories m a y be deduced, and 2 ° to p u t the several m e t h o d s of evaluating the broadening of reflections on a c o m m o n basis. T h o u g h it seems likely t h a t not much can be said --
29
j. BOUMAN
30
a b o u t a general distortion, at least one t h e o r e m can be p r o v e d (though a p p r o x i m a t e l y ) , a n d this in t u r n m a y be used to elucidate t h e prob l e m of t h e b r o a d e n i n g of reflections•
§ 2. Preliminary remarks. L e t us consider first a perfect crystal, m e a s u r i n g N I p r i m i t i v e t r a n s l a t i o n s along t h e a-axis, N2 along the b-axis, Na along the c-axis. F o r the sake of s i m p l i c i t y we shall a s s u m e t h a t e a c h cell c o n t a i n s one a t o m only. A parallel, m o n o c h r o m a t i c b e a m of X - r a y s falls on t h e crystal. I t s direction-cosines are %, ~p, 3% F o r the a m p l i t u d e of the d i f f r a c t e d b e a m h a v i n g the direction ct,, [~,, y~ we find Nt--I Nt--I
F = Z,,, 0
Ns--I
Z,,, Z., e x p 2~ri [A. ix'i' + A2x ~ + A3x~]. 0
(I)
0
F is p r o p o r t i o n a l to the a m p l i t u d e , we can h ~ w e v e r leave out the several factors, as a t o m f a c t o r etc. x"i, x"2, x~ are the c o o r d i n a t e s of the a t o m nln2n3, r e l a t i v e l y to the e l e m e n t a r y axes a, b, c. In the case of an u n d i s t o r t e d c r y s t a l x7 =
nl,
x~ =
n 2 , x~ =
ns.
At, A2, Aa a r e . t h e c o m p o n e n t s in t h e reciprocal lattice of the vect o r w i t h C a r t e s i a n c o m p o n e n t s 7) ~,
'
X
'
X
W r i t i n g instead of Na--I N t - - I N , - - I
X-,
)2-, ~ . ,
0
0
------E-
0
the f o r m u l a ( 1) m a y be t r a n s f o r m e d for an u n d i s t o r t e d c r y s t a l into F = Z - e x p 27ti [Alnl + A 2 ~ + A3n3] e 21riAtNt - -
=
I
e 2~iAtNt --
e2*rla' - - 1
]
e 2~'iAtN* - -
e2=*~, - - 1
l
e2=~a' - - 1
(2)
T h e i n t e n s i t y I will t h e n be I -----F •
F * ---- sin2 =A lNl sin 2 ~A 1
sin 2 x A 2 N 2 sin 2 r~A2
sin 2 rcA3N s sin 2 rcA3
(2a)
N o w this i n t e n s i t y will h a v e a v e r y s t e e p m a x i m u m for
A i : h, A 2 = k, As = l (h, k, I being integral n u m b e r s )
A GENERAL THEORY
OF LATTICE-DISTORTIONS
31
where We can visualize this function b y ascribing to each point A~, A2, As in the reciprocal lattice the value of I. T h e m a x i m u m values are to be found at the lattice points h, k, l, b u t there will be a d o m a i n r o u n d each point h, k, l, where I has a finite intensity. F o r the b o u n d a r i e s of this d o m a i n we shall take the surfaces, where I a t t a i n s for the first time the value zero, 1
1
A t=h4-~,
A 2=k
4- N 2 '
1
A3=14-
N3
As is well known, we can find the t o t a l e n e r g y of a reflection b y integrating I over the domain r o u n d h, k, l, and m u l t i p l y i n g the integral with the L o r e n t z factor. F o r our purpose we m a y disregard the L o r e n t z factor too, the total energy then being
E = fffI
dA1 dA2 dA3.
We have to integrate over all parts of the reciprocal space, where I has a finite value. Of course we have to take care not to include those parts of the energy, t h a t belong to a n o t h e r line, e.g. h + 1, k, I. We have therefore to take as boundaries h+
1
"o,
"+"T T I
E =f
k +. - .L .
.
.
l+½.
dA, dA2dA3 --=-N ~ N 2 N 3 * ).
(3)
h---4t k--~ ~ - 4
Let us call this d o m a i n the domain, connected to hkl, and the domain h + 1/Nl, k -~: 1IN 2, l + l/N3 the domain o/intensity. The extension of this d o m a i n of i n t e n s i t y is also one of the determining factors for the b r e a d t h of the reflection. In order to deduce a formula for this b r e a d t h of the reflection, we have to refer to a formula for a cubical crystal derived b y v o n L a u e. He finds for the angular breadth ofa Debye-Scherrer line~) b=
v'=-z N . a cos 8 "
*) T h i s f o r m u l a differs m s o m e r e s p e c t s f r o m the o r d i n a r y one. H e r e the i n t e g r a l s e x t e n d f r o m - - oo to 0% b u t t h e d e n o m i n a t o r sin t ~ A 1 is r e p l a c e d b y sin t ~r (A~ - - h) a n d this in t u r n b y ~rI (AI - - h) t. O u r f o r m u l a is r i g o r o u s l y t r u e , a n d can be p r o v e d b y p u t t i n g I = Y.n Y.n' exp 2~i [Altt t + .42n 2 + Astt J -- Atn't a n d b y i n t e g r a t i n g before s u m m i n g .
--
.4~n'~_ - -
.-Isn't]
32
1. B O U M A N
N being a kind of average of NtN2N3, a the primitive translation, 2~ the angle between the primary and the secondary beam, X the wave-length. (The angular breadth is b y definition the total energy of the li.ne, divided b y the radius of the film and b y the energy in the m .aximum, which is supposed~"to be i n t h e centre.) If we introduce the theoretical breadth bt, i.e. an average dimension of the domain of intensity, 2 b, = - ~ we can write,
b=
½v/n.X a ~-o-~- b,
(4)
expressing the breadth of the line with the aid .of the domain of intensity. The problem is now reduced to the question : How are the function I, and the domain of intensity affected b y a distortion of the lattice ?
§ 3. The general distortion. We will first state some results. For a distorted crystal we have to subdivide the reciprocal space dividing a* into Nx parts, b* into N2 parts and c* into N3 parts. The domain, connected with hkl, has now the following lattice-points h+
, k+-~-~2,
l+
N--~'
where q lies in the range s
in
the range
in the range
--
N l -2
l
N2 - -
1
N32
1
+
N 1- - -
1
N2 --
I
N3-
1
-
2
~
(taking for N l, N 2, N3 odd numbers. For even numbers we shall find the same results, but the description of the results is less simple). Other values of q, s, t will lead into another domain. Each point has its own domain of intensity 1
s
1
t
I
A GENERAL THEORY OF LATTICE-DISTORTIONS
33
It will be seen that the domains of intensity overlap in such a manner that each new lattice point lies on the boundary of a neighbouring domain. For a perfect crystal the intensity in a point hkl will have the value N~t. N~. N~ and the total energy of the domain will be NI . N2. N3. Now the intensities of the new points will be
A ¢st A *a N~ N~2 N] and the intensity of the original line
A ooo A *oo N~t N~2 N~
Aqa a r e
complex numbers with the property [ Aqa I < 1. This holds also for Aooo. We can now integrate ~he energy of a single domain of intensity, the result being A~,, A*a Nt Nz N3. In general this value will have no special significance, because we cannot neglect the energies of the other domains in the neighbourhood of this domain. It will be shown, however, that the sum o] the energies A qs~A *n N tN2N3 will be the total energy o[ the domain belonging to hkl, and that this energy will have the value NINaNa, i.e. the energy o[ the reflection o] the undistorted crystal. The first part will be ipso facto true, if each point is surrounded by domains, for which Aqst = 0. The second part of the proposition is approximately true, the approximation being all the better, if only the domains in the neighbourhood of hkl have finite values of Aq,t. We shall now derive the expression for the amplitude of the diffracted beam for a distorted crystal, in order to prove the results, mentioned above. The displacement of the atom ntn2n ~ may have the components u . , v., w., relatively to the axes a, b, c. Then X~ =
n l + Un,
X~=
n 2 + v,,,
x] =
n a + wn,
and F = Z- exp 2~i [At(nt + u.) + A2(n~ + v.) + Aa(n3 + w.)]. Physica IX
3
34
j. BOUMAN
I n this f o r m u l a nln2n3 assume v e r y great values, u, v, w are b o u n d ed, a n d are small for all values of nln2n3. We can separate the two factors of the s u m m a n d b y the following device. NI--I
Nt '--I
2
2
F = Y." Z-' Y.q
N.--I 2
Z,
Nx~l 2
~t .
Nl--I 2
Nt--] 2
l
×
NI N2 N3
xexp2~i[Alnt+A2*h+A3n3+ x
2hi [A lu: +
exp
,
.
,
nl + ~-~2n2 + ~-~3n3] × . q
,
s
,
l
A 2 v . + A 3 w n - - ~ ] n, - - N-2 ~ - - -~3 n~].
This t r a n s f o r m a t i o n can be ;¢erified with the aid of the following relation N--I 2
Z, exp 2hi ~s- ( n -
n') = 0 . . . . n . ~ n',
N--I 2
~-N
....
~=~'
(sa)
(n < N, n' < N) We shall also m a k e f r e q u e n t use of the relation NZ, -I
s (n - - n ' ) = 0 . . . . n : # n ' exp 2~i -~-
0
(sb) I n t r o d u c i n g the symbol N~.~I 2 S -Y.~ Nt--] 2
Nt--I 2 Z, Mr--| 2
Na--I 2 Zt Nt--I 2
we can write
1 3 X,, exp A,a=.Ntl~2N
2~i
[Alu,,+A2v,,+A3w,,--~ nl---~2s ha--N-3 t n3]
+
+ (A2+
+ (A3+
(6)
,7o,
A GENERAL
THEORY
OF LATTICE-DISTORTIONS
35
The summation Y.- can be effected ; we find in this way
F = SAq~
'exp[2~i(A, + ~)Nl]--I
exp[2rd(A2 + N2)N2]--I
oxp[~!S,+ k)]--~
exp[~(S~+ ~)]--,
§ 4. Discussion o~the results. Evaluating the intensity for a given direction At. A2, A3, we find sinn(A~+~-~)N~, sinTr(A2+~2) N2 F ----- S A a,t
sinTr(Ai+Nfi~),
(
sinn(A2+~)
')
t' sin =(A3+ ~3 )
× expTri Al+
(Nt--l) +
s
+ (A2+ ~-=)(N2--1) + (A3+ N--a)(N3--1) ] and
I ~ FF* -=
sin~(A,+~,) ~, ~in~(A. ~) ~ ~in~(~+~)~ ~A,.,A.*~ s)" ( ~) + sin2~(A,+N~ ) sin2~(A2+~-~2
sin[=(A'+ ~)N']sin[=(A'+~)N'I
sin2~ A3+~-~3
.(q___q')
~-SS'Av,,A**vv sin[~r(Al+~)]sin[Tr( A'------fr~+~Nll/j ]| expa~----XY--× ~ (N'--I) ' 1 x terms with the index 2 x terms with the index 3.. (8) *) T h e n u m e r a t o r s in (7b) a n d in t h e f o l l o w i n g f o r m u l a s m a y be s i m p l i f i e d , as e x p
[2~i ( A t N L + q)] = e x p 2rri A vNl. We h a v e r e f r a i n e d f r o m t h i s s i m p l i f i c a t i o n , in o r d e r t o m a k e t h e s t r u c t u r e of the f o r m u l a s m o r e clear.
36
J . BOUMAN
Each term of the first summation can be Compared with formula (2a). If we could neglect the second summation SS', the intensity would be the sum o f a series of functions, each of ~:hem having a steep m a x i m u m in a point
A~+
=h, A 2 + ~ = k ,
A3+~=l
with the value * 2 Aq. Aq,, NI/~2 N]3.
The total energy in the domain of intensity belonging to At = h
q NI'
A 2 ---- k
s --N--~'
Az = l
t --N--~
A3 = I
t --N-3"
would be , Aqa Aqa NI N2 N3,
if we take for Aqn the value for At = h
q Nl'
A2 = k
s --~'
The total energy of the domain, connected with hkl, would be
S Aqa A*a NI N2 N3. As we have already remarked, this reasoning is not quite sound, for we cannot, in general, neglect the second term of I. Now we turn to Aqn. It is evident from formula (6) that 1
IAq,t I < NtN2N3 Z" I = 1.
Aq,t is a function of AtAzA3. The variation with respect to these variables is but small: aA qn I exp 2rd [At u . + A2 v. + aAt = N IN 2N 3 2r~i Y.- u. s t + A~w.--~n~ -- ~n~ -- ~n~]
~A qst I 2~ I -~'AI < NtN2N3 E,, I u. I
'2= I u. I
We shall now evaluate the total energy in the domain, connected with hkl, making use of the formula (7a).
A G E N E R A L T H E O R Y OF L A T T I C E - D I S T O R T I O N S h+½ k+½ t+½
E = S Z - S ' Z,; f
f
t
f dA, dA2dA~AqnA~,vv
h--~ k--t ~--t
x exp2rd
[(,) A, +-~l
37
n, +
(s) A2+-~2
'
n2+
×
(A 3 + ~ ') n3l x
~).'
s', '--(A~
The i n t e g r a t i o n can be effected, if we t r e a t the t e r m s Aqn as constants. Now we h a v e seen t h a t the values Aqn v a r y slowly w i t h A 1A2A 3. W e can also infer f r o m f o r m u l a (8) t h a t the terms, in which Aqn occurs, have a m a x i m u m for
q A2=k A, = h - - N--~l,
s t - - N----2' A 3 = l - - N--33" /
The error m a d e is, therefore, not a serious one, if we s u b s t i t u t e for the function Aoa its value for the values of A 1, A2, .43, m e n t i o n e d above. Aqa --
N,N2N~I
Z exp 27ti [(h _ _ ~ ) l
u,, + (k ___~2) v. +
t
__ s
T h e integrals being zero, if n v~ n', and equal to 1 if n = n', we can write
E : SS' ~,,, Aqa Aq,* s,t, × x e,,p 2,~i 7 ,
(q-
q') +
-~
-~
•
The s u m m a t i o n Z,, will also lead to the value zero, if q :/: q', s :/: s', t -7s t', and to NIN2N3 if q = q', s -----s', t = t'. E = S Aqs t
A*:iN1 N2 Na
(9)
This is the first p a r t of the proposition, which has been s t a t e d in § 3. To prove the second part, we h a v e to show t h a t S Aq, t Aq*t =
i,
or
1
q
s
t
S Y., Z,,'.N~tN~N~ exp 2rd [A t u . + A2v.+ A3wn'-'--XT-nt--~-~--n'z---~n3]'~.t "'2 3
exp2~[--A,u,',.---A2v,',--A3w:,+
+ x /r n 3 3,_ = N~I n,' + xv-n6 s 1v2
.LV3
Z.
38
J. BOUMAN Now Al, A2, As have in Aq,t the values h
q
N, '
k- s N2'
1
t N3"
We shall neglect the second terms, and put
At=h,
Az=k,
A3=l.
This means that we take the values of Aqs, for the point hkl. This "approximation will be justified, if only the lattice-points in the neighbourhood of hk/ have. finite Aqs,, i.e. if for great values of qst Ava = 0. We m a y remark that in the speci~ theories this approxi'mation always occurs. In this case the truth of the proposition can be seen immediately by summing over qst. We shall retain this last approximation through out the following part of this paper.
§ 5. Some examples. 'Frozen heat motion'. The c o n s t a n t s Aqa , and not the displacements u,, v,, w,, define the broadenings of reflections. Let us consider a-distorted crystal, for which A qa = 0, except for--2
39
A G E N E R A L T H E O R Y OF L A T T I C E - D I S T O R T I O N S
Now an analogous effect c a n n o t be e x p e c t e d tbr a single distorted crystal. T h e crystal will have one distinct configuration. If we take into account, t h a t the D e b y e-S c h e r r e r lines are due to a cong l o m e r a t e of m a n y small crystals, we can e x p e c t the frozen heat m o t i o n to be represented b y a r a n d o m distribution of several types of d i s t o r t i o n o v e r the various crystals. To test this hypothesis, we will assume t h a t the displacements u,,, v., w. are v e r y small. The formula for Aqn can be developed into a series. Before giving this expression, it will be well to consider a n o t h e r m e t h o d of dealing with u . . v., w,,, which is always valid. These n u m b e r s can be developed into finite F o u r i e r series.
u. -----S(U)o.,exp2"m;
q n, + N-~n2 + ~-~3n3
~. = s(v),., exp. 2=i [q ~ , . , + ~s. ~ + ~ 1 .~
s
(,o~)
,]
As the sum S is t a k e n over the range - - ( N t 1)/2 -+ ( N t - 1)/2 etc., there are NIN2N3 n u m b e r s (U)qa etc. These F o u r i e r components m a y be used to define the displacements. With the aid of (5a) we find 1 £,, u,, exp - - 2~i [NI q nl (U)qst = .NIN2Na and
+
~S n2 + h na [(10b) *'a J
1 (U)ooo -- NIN=Na Z,, u. =- u.
In the same m a n n e r we can express u2., u . v . , etc.
If
'
u~ =-- S(U2),,,exp 2rd -~ln, + ~ n 2
lq
u,,v,, = S ( U V ) o a e x p 2rd - ~ l " '
,4.1 I
+ *,a
+ ~s; n 2 + . ,, , , n a l *'3
J
( U2)ooo = u'~. " (UV)ooo-- u.v. Now, from (6), a n d replacing A1, A=, Aa b y hkl, A qa ~
1 £,, exp 2rff [ - - q--s t NIN2N3 Ni nt - - N2 n2 - - -N-an3] x × I1 + 2 n i ( h u . + kv. + l w . ) -
2re2 (hu. + kv. + lw.) 2 ...].
40
j. BOUMAN Les us consider Aooo first.
Aooo = 1 + ~
[ h ( U ) ~ + k(V)ooo + Z(W)ooo] - --27,2 W(O)ooo + ~k(UV)ooo + . . . .
+ t2(w2)ooo].
Now (00ooo, (V)ooo, (W)ooo, being u,,, v., w,,, can always be put e q u a l to zero. And, as (U2)ooo = u--~.etc. Aooo ---- ! - - 2 ~ : 2 [(hu, + kv,, + lw,,)2]. Now hkl are the c o m p o n e n t s in the reciprocal lattice of the v e c t o r p ] d ~ , if dh,~ is the distance between two lattice planes (hkl) and p the c o m m o n d e n o m i n a t o r of h, k, and l. Its direction is the normal to the plane (hkl). I f u~ denotes the c o m p o n e n t of the displacement of the n ~ a t o m along the n o r m a l to (hkl), it follows t h a t :
hu,, + kv, + lw,,
Pu~" -
Aooo = 1
8~2 X2 sin2 ~ ~
d~
2 sin = ~
~
~-~ exp [ - - 8 ~
u~
sin2 ~ 2 -~ (-~7~] •
We will write ~ = u~, the m e a n square of the d i s p l a c e m e n t along the normal to the plane (hkl), t h e n : sin2 $ 2Aooo ---- exp [ - - 8~ ~ ~ uv]. N o w we t u r n to Aqa.
Aq,, = 27ri [h(U)qn + k(V)on + l(W)o,,j + . . . . The form between brackets will be the F o u r i e r c o m p o n e n t qst of 2 sin -
Unv '
-
The distribution of e n e r g y for a single crystal will be defined by 16n 2 sin 2 I Aooo 12 -----exp [ - X2 . u~ (l la) [A,,, 12 = 4re2 I h(O),,,
+ k(V)q,, + Z(W)o,, 12
(l lb)
If we take At = h, A2 = k, A 3 = l, as we have done before, then •
S [aqal2+ IAoool 2--- 1 qst~O
or, denoting S b y S ' q,s,t~O
S' ) Aqa 12
--
167ta s i n 2 ~
x~
u~.
(12)
A GENERAL THEORY
OF LATTICE-DISTORTIONS
41
We m a y add the following remark. The energy of the original line, given b y (11 a), is decreased i n t h e same m a n n e r as in the case of the heat effect. B u t as the neighbouring domains of i n t e n s i t y m a y have high values of Aqa, and so are to be reckoned to the original line, the energy m a y be greater t h a n in the case of the heat-effect. The expression is only valid" for small values t)f u, v, w, or r a t h e r of hu,, + kv, + lw,,. If we turn now to an aggregate with a r a n d o m distribution of deformations over the individual crystals, we m u s t take the average values of [ Aoo0 ]2 and [ Aqa ]2. We m a y assume in this case t h a t [ (V)qn [~,
] (V)q,, (V)*a ],~ etc. *)
will not depend on qst. (In general the second form will not be zero). Therefore: J A,,, 1,2~= 492 [h.2O 2, + 2hk (UV),, . . . .
+ 12W2v]
The formula for hA00o [2 will not bc altered, but u 2 will now stand for the mean square displacement over the atoms of one crystal, taking then the average over all crystals. F r o m formula (12) 1
1671:2 s i n 2
',A [~,. = [.Aq,, J2,v -- N , N 2 N 3 __ 1 l 16re2 sin 2 -- N I N 2 N 3 . X2
X2
(uv2)"o
(U~)av.
(12a)
Though this relation is not strictly true, as I Aq,t [2 is a function of A t, A2, As, and we have replaced these variable numbers b y hkl, w e m a y infer t h a t [Aqs t I~v will be very small. The original line will s t a n d out against a weak, continuous background. It is not difficult to calculate the intensity of this background. For this calculation it will not be necessary to assume t h a t the displacements are very small. The average of [Aot [2 will still be independent of qst, but the formula (12a) must now be written I
[Aqst 12v - - I A I~ -- N~N2N3 [1 - - ] A0oo 1,2~]
(12b)
*) !We d e n o t e b y a h o r i z o n t a l b a r an a v e r a g e over the a t o m s of one c r y s t a l , and b y av. the average o v e r the crystals of a n aggregate.
42
J. BOUMAN
We may assume too that the average of Aq,t Acvv * will be zero. Making use of (8), we find I,° = IAooo12,o sin2 ~ A tNi sin 2 n A2N2 sin 2 ~ AaN3 • sin2r~At sin2~A2 sin2nAz + sin2n(A2+ ~ ) N 2
sinZ~(Aa+ ~--~a)Na
sin2~(A2+ ~ )
sin2n (A3+ ~-~3)
+ I A t,vS 2 , The sum can be calculated easily• For sinZx(A,+ k ) N l = Y,- exp 2rri (A (cf. (2), (2a,)), and sinZx(A2 + ~ ) N 2
(
+ A2+
s
s!n2.n(A3 + ~ ) N 3
(
(n2--n~) + A a +
'F
(na--n3) .
The summation over qst will make each term zero, if not nt = n;, n2 ---- n2, na ---- n3. In this case the value of the exponential function 2 ~dll be 1 and the total sum will be N~tN2N 32 . e
t
t N3 sin2~ (A l + --~t)Nl sinZn (A2 + N~)N2 sin27r (Aa +. N~) S, S
sin2n (AI + ~ ) __
--
We find I,v
=
sin 2 rc A iNt sin2x At
AT2~T2AT2
,t v l , t , 2 ~ . 3
o sin2~ AtN1
I A o o o I~v
s i n 2 rc A i
+ IA 12,v[N~tN~2N]
sin 27:A2N2 sin 2n AaN 3 sin2~ A2 sin27r As
sin2~r N2A2 sin2~ AsN3 + sin 27r A2 sin 27r As sin2~AtNl sin27rA2N2 sin2nAaNa] sin2~A| sin2rcA2 " s-~r~A-~ ]"
A GENERAL
THEORY
OF LATTICE-DISTORTIONS
43
As I A [~ can be neglected, c o m p a r e d with [ Aooo I~, we find the final expression I~=
tAoool.2v sin2 ~ A t N ' • sin 2 ~ A t
sin 2 ~ A 2 N 2 sin 2 h A 2
sin 27:AaN3 sin 2 n A a +
+ NtN2N 3 [I - - 1Aoool2].
(13~
T h e first p a r t of I,v is the expression for the i n t e n s i t y of the original line, the second p a r t t h a t for the continuous scattering• R e t u r n i n g to our assumption of small values of u, v, w and introducing M --
8n 2 sin 2 X2
(u2~).v
we can write I . . = e -2M sin2~ A iN, sin2r~ A2N2 sin2~ AaNa • sin2~A I sin2~A 2 s i n 2 ~ A a +2MNIN2Na.(13a) This r a t h e r l e n g t h y c a l c u l a t i o n , leading to the well-known result
(13a), m a y serve as a check on our theory, and m a y m a k e clear, how the c o n t i n u o u s scattering m a y be conceived and which assumptions had to be made.
§6. Comparison with the work o/ D e h l i n g e r and B o a s . F o r the sake of simplicity, the previous papers on this subject assume t h a t the displacement has b u t one c o m p o n e n t u . . and f u r t h e r t h a t the displacement is a function of nl only. We find 1 Z . exp 2hi [A I u., --A¢a = NtN2N3
n, - - Nt n2 - - ~ n3]
This is in general = 0, but Aqoo = ~
Y'", exp 2m' [A, u. - -
n,].
We will suppress the index 1, and replace A, b y h. The formula
(7b) will then be t r a n s f o r m e d into exp [2rci (Al + ~ t ) N I ] - - I V
::
S Ago0
exp (2~i A2N2) - - 1 exp (2hi A2) - - l
exp (2hi A3Na) - - 1 exp (2~:i Aa) - - 1
44
J. BOUMAN
The t w o l a t t e r factors of the p r o d u c t are the same as for an und i s t o r t e d crystal ; we m a y leave t h e m out too. T h e t h i r d c o m m o n feature of the papers, m e n t i o n e d above, is t h a t the displacement is p_eriodicaJ with a periotl of r i d t e r a t o m i c distances (Nit beiiag a integer). This mekns t h a t Aq0o has o n l y then a finite value," when q is a multiple of N/r. To prove this proposition, we r e m a r k t h a t the F o u r i e r series (10a) m a y now be written as
u,, --= S(u)~ exp q
V
2~i -- n, r
N
~
V m
T h e t r u t h of the proposition is evident, from the expansion of Ag into a series (pag. 012). T h e formula's becor0e therefore A~oo = --1 Z,, , - i exp 2hi 0
(hu,, --7 n)
(l 4)
exp [2ni (A, + r) N , ] - - I F = SA,,oo This m e a n s t h a t the domains of i n t e n s i t y are s e p a r a t e d b y e m p t y domain.s, and a crystal of this kind will show separate lines in the n e i g h b o u r h o o d of the original line. These are the so-called ghosts. W e can now calculate A for the several t y p e s of distortion, and we shall find the intensities of the ghosts, as c o m p u t e d b y D e h 1 i ng e r a n d B o a s. We will, however, t r e a t one distortion in particular, as t h e r e is a discrepancy b e t w e e n our results and the results of D e h 1 i n g e r 2) 5). This d e p l a c e m e n t is for a p a r t (l - - p) of the period r S ~t~ =
a COS 2 ~ n - -
The o t h e r p a r t is not distorted. We will assume at once t h a t r and s are large numbers; e.g. ~ ~ I0', s = 102, and t h a t (1 - - p ) r is a multiple of the period of the cosine-distortion r/s. It follows t h a t (1 -- p)s a n d therefore ps are integers.
A GENERAL
THEORY
OF
LATTICE-DISTORTIONS
45
N o w . f r o m (14) A ~oo = .2
Z e x p 2~:i h a
r
r
r
r
+
2
+ -=
e x p - - 27ri
r
o
Z~ Z* - -
o
o
.
o
(l--k)!
2; e x p ~ 2 r c i
----
(l~p)r
Es(/--2k)--v)]
k!exp
+
+ --1 e x p [ - - 2~i(1 - - p)v] e x p [ - - 2rri pv] - - 1
I n t h e first s u m we r e p l a c e v b y ~s +
A,oo = rl ~1o Eko - -
" (l--k)!kt. ×
x
e x p 2~i (1 - - p ) [ s ( l - - 2 k - - ~ ) - - -r] - -
1
+
e x p 2hi s ( l - - 2 k - - tL) - - "r _ _ l ¥
+ - - I -exp [ - - 27riv] - - e x p [ - - 2hi (1 - - p)v] r
exp [--2~irl--1
N o w t a k i n g , i n s t e a d of l a n d k k andre = l--2k--~, a n d r e a r r a n g i n g t h e d o u b l e s u m m a t i o n , we c a n w r i t e A~oo = 1r +Z~ i~'+" Jo~+..) (27rah) e x p [2ni (1 - - p ) ( s r n ---¢o S~ -- T exp 27ri--
z)] - - 1
1
1 e x p [ - - 2~:i (1 - - p) (s~ + ~)] - - e x p [ - - 2~i (s~ + z)] r
exp[ -2nis~+r
*]--I (v = s~ + .).
W e will t r e a t s e p a r a t e l y t h e cases v = 0, v = Fs, v = ~, v = Fs + . 1°. t L = T = 0 , ~=0 A v0o =
-
1 ~
-
r
--0o
i " J . , (2nah) e x p 2 h i (1 - - p ) ( s i n ) - e x p 27ri
$~ 1"
--
1
1 + p
.
46
J. BOUMAN
As (! - - p ) s is an integer, the t e r m s of the sum will be zero, e x c e p t when the d e n o m i n a t o r is zero too. This will occur for m -----0, m = ± r (or 7"Is if this n u m b e r is a integer). B u t the B e s s e 1 functions with a large i n d e x h a v e v e r y small values for small a r g u m e n t s . We retain m = 0 only, and find Ao0o = (I - - p) J o (2nab) + p. In the o t h e r cases (v :# 0) we m a y write • A~o
o o i v + " j(~,+,,) (27rah) e x p [ - - 2rci(1 - - p) x] - - 1 _ _ = --1 +Zm /'
"
--oo
exp 2~i
S?H, - -
T
1
7,
1 exp [ - - 2~i(1 - - p) x] - - 1 exp[--2~isV'+r ~]--1 2 ° .
T ~--- O ,
v ~
~l.S.
The secofid p a r t will be zero, in the first p a r t we retain m = 0 only Am,o,o = (1 - - p) iv I v (2nah). These results h a v e also been o b t a i n e d b y D e h I i n g e r. It is clear, however, t h a t also for o t h e r values of v, finite values for A~oo m a y be found. W e replace again the series b y the t e r m m = 0. 3 °. t L = 0 , v = x . As x m a y be a small number, we will not omit the second p a r t of A~oo; we find I A~oo 12 ---- 7,21 sin 2 n px [Jo (2~ah) - - l ]2. sin2 ~rx 7,
4 °. v = ~s + x. N o w we can omit the second p a r t J A~00 12 --
I sin27rp, -2 (2nab).
r ~ s~n2~ Jv r
If we now take X I A~0o j2 we m u s t find 1, in accordance with our general result: B o a s 6) calculates Y. lAv.,.o.o js and finds a value for this sum, which is less t h a n I. As this sum is used for calculating the b r e a d t h of the line, the i m p o r t a n c e of our general proposition is clear (vid § 7).
A G E N E R A L T H E O R Y OF L A T T I C E - D I S T O R T I O N S
47
§ 7. Conclusion. In the a b o v e the problem has been solved as to how the distribution of i n t e n s i t y in the reciprocal lattice is affected b y distortion. The following step should be to evaluate the distribution of e n e r g y on the film b u t this problem presents considerable difficulties. W i t h the aid of simplifying assumptions, however, it can be solved for an u n d i s t o r t e d crystal. This leads to the f o r m u l a for the b r e a d t h of a line (4) as a function of the size of the particles. Now the size of the d o m a i n of i n t e n s i t y is inversely p r o p o r t i o n a l to the size of the particle. W e have seen t h a t in general the d o m a i n of i n t e n s i t y will increase for a d i s t o r t e d crystal, so t h a t the effect of distortion will in some respects be the same as t h a t of a breaking u p of the particles. B u t the distribution of energy in the reciprocal lattice will bear in general no resemblance to the distribution for a small, und i s t o r t e d particle. This will occur only if the Aqa, considered as a function of qsl, have a m a x i m u m for q = 0, s = 0, t = 0, and decrease with increasing values of q, s, t, to zero. In this case B o a s 5) uses the following m e t h o d . We consider a one-dimensional reciprocal space. F o r an u n d i s t o r t e d crystal we find for the distribution of energy sin 2 ~ A N sin 2 ~ A
I-As h+½
f
I.dA--N,
I(A = h ) = N 2 = I m ~
h--i
so we can write for the theoretical b r e a d t h b,
2 bt-
N
f I . dA -- 2
Imp,
In our case the domain of the distorted crystal will have a distrib u t i o n of energy, which resembles the distribution for an u n d i s t o r t e d crystal. We can take the integral over the range h - - ½ ~ h + ½, as all Aq,t values are zero, e x c e p t in the neighbourhood of 0, 0, 0. As this integral is N, and Ima~ = N 2 [ Aooo 12
2N bi --
N2 I Aoool 2
and b--
1
acos~
2
" IAooo[2"X"
48
J. BOUMAN
tn general it will be uncertain, which d o m a ~ s of intensity' must be reckoned to the original line. This means thai the range of the' ivitegr~d is uncertain. But, if we integrate over the domain, connected with hkl, the breadth must be inversely proportional to [ A0oo [2. We have mentioned this calculation to elucidate our remark in § 6, concerning f I . dA, but do not intend to discuss the breadth of the line for. all possible cases. The final problem must be the determination of the distortion u., 'v., w. from the intensity and the breadth of the D e b y e-S c h e rr e r fines, or from the distribution of energy over these lines. The intermediate step will be the determination of the values of Aq,,. If ~these were known, then u . , v., q. could be determined. This is easy for small displacements. Then the formula (I I b) can be used, and (U)q;,, (V)qst, (Wqa) can be found from the reflections, and from these the distortion. If only [Aqst [ is k~own, the phases of the F o u r i e r components of the distortion still remain unknown, and we know only whether the components with short or with long wavelength are preponderant. Without determining the connection between the Aq,t values and the properties of the reflection, we can formulate some rules (cf. § 5). A reflection, with the original breadth, b u t diminished intensity, appearing on a continuous background, will indicate a frozen heat motion. All terms of the F o u r i e r series will occur, with the same average amplitude. A broadened llne, with the original intensity, can only be caused b y a distortion of the type, already mentioned. In the F o u r i e r series~only the components with long wavelength appear, and in the reciprocal space we shall find energy only in the neighbourhood of hkl. The total energy will be found in the line, and there will be no scattered energy at all. An example is the cosine-distortion of D e hlinger. If the broadened line has a diminished intensity, then energy must b e found between the lines, and the F o u r i e r series will have at teast a few terms with short wavelength, as there will be factors Aq. with large values of q, s, t. We have shown some consequences of the proposition, that the total energy in a domain, connected to hkl, remains approximately constant. The proposition will be a guiding principle for determining the deformation of a crystal. Received November 3rd, 1941.
A G E N E R A L T H E O R Y OF L A T T I C E - D I S T O R T I O N S
49
REFF_.RENCES 1) 2) 3) 4) 5) 61
M. y o n L a u e , Z. Kristallogr. 64, 115, 1926. U. D e h l i n g e r , Z. Kristallogr. 65,615, 1927. J. H e n g s t e n b e r g a n d H . M a r k , Z. Phys. 61, 435,1930. W. B o a s, Z. Kristallogr. 1t6, 214~ 1937. W. B o a s, Z. Kristallogr. 97,345, 1937. G.W. Brindley and P. R i d l e y , Proc. phys. Soe. (London) 50, 501, 1938; 51,432, 1939: P) J. B o u m a n a n d W . F. d e J o n g , P h y s i c a S , 817, 1938.
Physica IX
A GENERAL
J a n u a r i 1942
THEORY
OF
LATTICE-DISTORTIONS
b y J. B O U M A N Gymnasium Delft
Summary A n e x p r e s s i o n for t h e i n t e n s i t y 9f a d i f f r a c t e d b e a m of X - r a y s is d e r i v e d f o r a d i s t o r t e d c r y s t a l . T h i s f o r m u l a i n c l u d e s t h e r e s u l t s of D e h 1 i n g e r a n d B o a s, a n d t h e i n t e n s i t y c a u s e d b y t h e d i s t o r t i o n , w h i c h is k n o w n as t h e f r o z e n h e a t m o t i o n . A t h e o r e m , v a l i d for all i n h o m o g e n e o u s d i s t o r t i o n s , is p r o v e d w i t h t h e h e l p of t h i s e x p r e s s i o n . T h i s t h e o r e m c a n b e u s e d for t h e d i s c u s s i o n of t h e b r o a d e n i n g or d e c r e a s e of i n t e n s i t y of t h e reflect i o n s for d i s t o r t e d c r y s t a l s .
§ I. Introduction. The broadening of D e b y e-S c h e r r e r lines, caused b y cold-working on metals, has been explained in several ways. One of t h e m represents the t h e o r y of the inhomogeneous distortion, which has been considered b y D e h I i n g e r *), B o a s 5), B r i n d 1 e y and R i d 1 e y 6) a.o. In these theories it is assumed t h a t the a t o m s are displaced from their ideal positions over small distances. This distortion must be distinguished from the homogeneous distortion, e.g. a single stretch. Here the displacements cannot be small for all the atoms, as t h e y are linear functions of the ideal coordinates of the atoms, the crystal retaining after distortion an ideal crystal s t r u c t u r e with new parameters. Now the papers referred to deal with special inhomogeneous distortions. These distortions m a y cause a broadening of the reflection, or a decrease of intensity, or both. I t is h o w e v e r not quite clear how this broadening, or the decrease of i n t e n s i t y can be derived from the chosen t y p e of distortion. A general t h e o r y does not exist at present. In this paper an a t t e m p t will be made 1° to formulate a general theory, from which special theories m a y be deduced, and 2 ° to p u t the several m e t h o d s of evaluating the broadening of reflections on a c o m m o n basis. T h o u g h it seems likely t h a t not much can be said --
29
j. BOUMAN
30
a b o u t a general distortion, at least one t h e o r e m can be p r o v e d (though a p p r o x i m a t e l y ) , a n d this in t u r n m a y be used to elucidate t h e prob l e m of t h e b r o a d e n i n g of reflections•
§ 2. Preliminary remarks. L e t us consider first a perfect crystal, m e a s u r i n g N I p r i m i t i v e t r a n s l a t i o n s along t h e a-axis, N2 along the b-axis, Na along the c-axis. F o r the sake of s i m p l i c i t y we shall a s s u m e t h a t e a c h cell c o n t a i n s one a t o m only. A parallel, m o n o c h r o m a t i c b e a m of X - r a y s falls on t h e crystal. I t s direction-cosines are %, ~p, 3% F o r the a m p l i t u d e of the d i f f r a c t e d b e a m h a v i n g the direction ct,, [~,, y~ we find Nt--I Nt--I
F = Z,,, 0
Ns--I
Z,,, Z., e x p 2~ri [A. ix'i' + A2x ~ + A3x~]. 0
(I)
0
F is p r o p o r t i o n a l to the a m p l i t u d e , we can h ~ w e v e r leave out the several factors, as a t o m f a c t o r etc. x"i, x"2, x~ are the c o o r d i n a t e s of the a t o m nln2n3, r e l a t i v e l y to the e l e m e n t a r y axes a, b, c. In the case of an u n d i s t o r t e d c r y s t a l x7 =
nl,
x~ =
n 2 , x~ =
ns.
At, A2, Aa a r e . t h e c o m p o n e n t s in t h e reciprocal lattice of the vect o r w i t h C a r t e s i a n c o m p o n e n t s 7) ~,
'
X
'
X
W r i t i n g instead of Na--I N t - - I N , - - I
X-,
)2-, ~ . ,
0
0
------E-
0
the f o r m u l a ( 1) m a y be t r a n s f o r m e d for an u n d i s t o r t e d c r y s t a l into F = Z - e x p 27ti [Alnl + A 2 ~ + A3n3] e 21riAtNt - -
=
I
e 2~iAtNt --
e2*rla' - - 1
]
e 2~'iAtN* - -
e2=*~, - - 1
l
e2=~a' - - 1
(2)
T h e i n t e n s i t y I will t h e n be I -----F •
F * ---- sin2 =A lNl sin 2 ~A 1
sin 2 x A 2 N 2 sin 2 r~A2
sin 2 rcA3N s sin 2 rcA3
(2a)
N o w this i n t e n s i t y will h a v e a v e r y s t e e p m a x i m u m for
A i : h, A 2 = k, As = l (h, k, I being integral n u m b e r s )
A GENERAL THEORY
OF LATTICE-DISTORTIONS
31
where We can visualize this function b y ascribing to each point A~, A2, As in the reciprocal lattice the value of I. T h e m a x i m u m values are to be found at the lattice points h, k, l, b u t there will be a d o m a i n r o u n d each point h, k, l, where I has a finite intensity. F o r the b o u n d a r i e s of this d o m a i n we shall take the surfaces, where I a t t a i n s for the first time the value zero, 1
1
A t=h4-~,
A 2=k
4- N 2 '
1
A3=14-
N3
As is well known, we can find the t o t a l e n e r g y of a reflection b y integrating I over the domain r o u n d h, k, l, and m u l t i p l y i n g the integral with the L o r e n t z factor. F o r our purpose we m a y disregard the L o r e n t z factor too, the total energy then being
E = fffI
dA1 dA2 dA3.
We have to integrate over all parts of the reciprocal space, where I has a finite value. Of course we have to take care not to include those parts of the energy, t h a t belong to a n o t h e r line, e.g. h + 1, k, I. We have therefore to take as boundaries h+
1
"o,
"+"T T I
E =f
k +. - .L .
.
.
l+½.
dA, dA2dA3 --=-N ~ N 2 N 3 * ).
(3)
h---4t k--~ ~ - 4
Let us call this d o m a i n the domain, connected to hkl, and the domain h + 1/Nl, k -~: 1IN 2, l + l/N3 the domain o/intensity. The extension of this d o m a i n of i n t e n s i t y is also one of the determining factors for the b r e a d t h of the reflection. In order to deduce a formula for this b r e a d t h of the reflection, we have to refer to a formula for a cubical crystal derived b y v o n L a u e. He finds for the angular breadth ofa Debye-Scherrer line~) b=
v'=-z N . a cos 8 "
*) T h i s f o r m u l a differs m s o m e r e s p e c t s f r o m the o r d i n a r y one. H e r e the i n t e g r a l s e x t e n d f r o m - - oo to 0% b u t t h e d e n o m i n a t o r sin t ~ A 1 is r e p l a c e d b y sin t ~r (A~ - - h) a n d this in t u r n b y ~rI (AI - - h) t. O u r f o r m u l a is r i g o r o u s l y t r u e , a n d can be p r o v e d b y p u t t i n g I = Y.n Y.n' exp 2~i [Altt t + .42n 2 + Astt J -- Atn't a n d b y i n t e g r a t i n g before s u m m i n g .
--
.4~n'~_ - -
.-Isn't]
32
1. B O U M A N
N being a kind of average of NtN2N3, a the primitive translation, 2~ the angle between the primary and the secondary beam, X the wave-length. (The angular breadth is b y definition the total energy of the li.ne, divided b y the radius of the film and b y the energy in the m .aximum, which is supposed~"to be i n t h e centre.) If we introduce the theoretical breadth bt, i.e. an average dimension of the domain of intensity, 2 b, = - ~ we can write,
b=
½v/n.X a ~-o-~- b,
(4)
expressing the breadth of the line with the aid .of the domain of intensity. The problem is now reduced to the question : How are the function I, and the domain of intensity affected b y a distortion of the lattice ?
§ 3. The general distortion. We will first state some results. For a distorted crystal we have to subdivide the reciprocal space dividing a* into Nx parts, b* into N2 parts and c* into N3 parts. The domain, connected with hkl, has now the following lattice-points h+
, k+-~-~2,
l+
N--~'
where q lies in the range s
in
the range
in the range
--
N l -2
l
N2 - -
1
N32
1
+
N 1- - -
1
N2 --
I
N3-
1
-
2
~
(taking for N l, N 2, N3 odd numbers. For even numbers we shall find the same results, but the description of the results is less simple). Other values of q, s, t will lead into another domain. Each point has its own domain of intensity 1
s
1
t
I
A GENERAL THEORY OF LATTICE-DISTORTIONS
33
It will be seen that the domains of intensity overlap in such a manner that each new lattice point lies on the boundary of a neighbouring domain. For a perfect crystal the intensity in a point hkl will have the value N~t. N~. N~ and the total energy of the domain will be NI . N2. N3. Now the intensities of the new points will be
A ¢st A *a N~ N~2 N] and the intensity of the original line
A ooo A *oo N~t N~2 N~
Aqa a r e
complex numbers with the property [ Aqa I < 1. This holds also for Aooo. We can now integrate ~he energy of a single domain of intensity, the result being A~,, A*a Nt Nz N3. In general this value will have no special significance, because we cannot neglect the energies of the other domains in the neighbourhood of this domain. It will be shown, however, that the sum o] the energies A qs~A *n N tN2N3 will be the total energy o[ the domain belonging to hkl, and that this energy will have the value NINaNa, i.e. the energy o[ the reflection o] the undistorted crystal. The first part will be ipso facto true, if each point is surrounded by domains, for which Aqst = 0. The second part of the proposition is approximately true, the approximation being all the better, if only the domains in the neighbourhood of hkl have finite values of Aq,t. We shall now derive the expression for the amplitude of the diffracted beam for a distorted crystal, in order to prove the results, mentioned above. The displacement of the atom ntn2n ~ may have the components u . , v., w., relatively to the axes a, b, c. Then X~ =
n l + Un,
X~=
n 2 + v,,,
x] =
n a + wn,
and F = Z- exp 2~i [At(nt + u.) + A2(n~ + v.) + Aa(n3 + w.)]. Physica IX
3
34
j. BOUMAN
I n this f o r m u l a nln2n3 assume v e r y great values, u, v, w are b o u n d ed, a n d are small for all values of nln2n3. We can separate the two factors of the s u m m a n d b y the following device. NI--I
Nt '--I
2
2
F = Y." Z-' Y.q
N.--I 2
Z,
Nx~l 2
~t .
Nl--I 2
Nt--] 2
l
×
NI N2 N3
xexp2~i[Alnt+A2*h+A3n3+ x
2hi [A lu: +
exp
,
.
,
nl + ~-~2n2 + ~-~3n3] × . q
,
s
,
l
A 2 v . + A 3 w n - - ~ ] n, - - N-2 ~ - - -~3 n~].
This t r a n s f o r m a t i o n can be ;¢erified with the aid of the following relation N--I 2
Z, exp 2hi ~s- ( n -
n') = 0 . . . . n . ~ n',
N--I 2
~-N
....
~=~'
(sa)
(n < N, n' < N) We shall also m a k e f r e q u e n t use of the relation NZ, -I
s (n - - n ' ) = 0 . . . . n : # n ' exp 2~i -~-
0
(sb) I n t r o d u c i n g the symbol N~.~I 2 S -Y.~ Nt--] 2
Nt--I 2 Z, Mr--| 2
Na--I 2 Zt Nt--I 2
we can write
1 3 X,, exp A,a=.Ntl~2N
2~i
[Alu,,+A2v,,+A3w,,--~ nl---~2s ha--N-3 t n3]
+
+ (A2+
+ (A3+
(6)
,7o,
A GENERAL
THEORY
OF LATTICE-DISTORTIONS
35
The summation Y.- can be effected ; we find in this way
F = SAq~
'exp[2~i(A, + ~)Nl]--I
exp[2rd(A2 + N2)N2]--I
oxp[~!S,+ k)]--~
exp[~(S~+ ~)]--,
§ 4. Discussion o~the results. Evaluating the intensity for a given direction At. A2, A3, we find sinn(A~+~-~)N~, sinTr(A2+~2) N2 F ----- S A a,t
sinTr(Ai+Nfi~),
(
sinn(A2+~)
')
t' sin =(A3+ ~3 )
× expTri Al+
(Nt--l) +
s
+ (A2+ ~-=)(N2--1) + (A3+ N--a)(N3--1) ] and
I ~ FF* -=
sin~(A,+~,) ~, ~in~(A. ~) ~ ~in~(~+~)~ ~A,.,A.*~ s)" ( ~) + sin2~(A,+N~ ) sin2~(A2+~-~2
sin[=(A'+ ~)N']sin[=(A'+~)N'I
sin2~ A3+~-~3
.(q___q')
~-SS'Av,,A**vv sin[~r(Al+~)]sin[Tr( A'------fr~+~Nll/j ]| expa~----XY--× ~ (N'--I) ' 1 x terms with the index 2 x terms with the index 3.. (8) *) T h e n u m e r a t o r s in (7b) a n d in t h e f o l l o w i n g f o r m u l a s m a y be s i m p l i f i e d , as e x p
[2~i ( A t N L + q)] = e x p 2rri A vNl. We h a v e r e f r a i n e d f r o m t h i s s i m p l i f i c a t i o n , in o r d e r t o m a k e t h e s t r u c t u r e of the f o r m u l a s m o r e clear.
36
J . BOUMAN
Each term of the first summation can be Compared with formula (2a). If we could neglect the second summation SS', the intensity would be the sum o f a series of functions, each of ~:hem having a steep m a x i m u m in a point
A~+
=h, A 2 + ~ = k ,
A3+~=l
with the value * 2 Aq. Aq,, NI/~2 N]3.
The total energy in the domain of intensity belonging to At = h
q NI'
A 2 ---- k
s --N--~'
Az = l
t --N--~
A3 = I
t --N-3"
would be , Aqa Aqa NI N2 N3,
if we take for Aqn the value for At = h
q Nl'
A2 = k
s --~'
The total energy of the domain, connected with hkl, would be
S Aqa A*a NI N2 N3. As we have already remarked, this reasoning is not quite sound, for we cannot, in general, neglect the second term of I. Now we turn to Aqn. It is evident from formula (6) that 1
IAq,t I < NtN2N3 Z" I = 1.
Aq,t is a function of AtAzA3. The variation with respect to these variables is but small: aA qn I exp 2rd [At u . + A2 v. + aAt = N IN 2N 3 2r~i Y.- u. s t + A~w.--~n~ -- ~n~ -- ~n~]
~A qst I 2~ I -~'AI < NtN2N3 E,, I u. I
'2= I u. I
We shall now evaluate the total energy in the domain, connected with hkl, making use of the formula (7a).
A G E N E R A L T H E O R Y OF L A T T I C E - D I S T O R T I O N S h+½ k+½ t+½
E = S Z - S ' Z,; f
f
t
f dA, dA2dA~AqnA~,vv
h--~ k--t ~--t
x exp2rd
[(,) A, +-~l
37
n, +
(s) A2+-~2
'
n2+
×
(A 3 + ~ ') n3l x
~).'
s', '--(A~
The i n t e g r a t i o n can be effected, if we t r e a t the t e r m s Aqn as constants. Now we h a v e seen t h a t the values Aqn v a r y slowly w i t h A 1A2A 3. W e can also infer f r o m f o r m u l a (8) t h a t the terms, in which Aqn occurs, have a m a x i m u m for
q A2=k A, = h - - N--~l,
s t - - N----2' A 3 = l - - N--33" /
The error m a d e is, therefore, not a serious one, if we s u b s t i t u t e for the function Aoa its value for the values of A 1, A2, .43, m e n t i o n e d above. Aqa --
N,N2N~I
Z exp 27ti [(h _ _ ~ ) l
u,, + (k ___~2) v. +
t
__ s
T h e integrals being zero, if n v~ n', and equal to 1 if n = n', we can write
E : SS' ~,,, Aqa Aq,* s,t, × x e,,p 2,~i 7 ,
(q-
q') +
-~
-~
•
The s u m m a t i o n Z,, will also lead to the value zero, if q :/: q', s :/: s', t -7s t', and to NIN2N3 if q = q', s -----s', t = t'. E = S Aqs t
A*:iN1 N2 Na
(9)
This is the first p a r t of the proposition, which has been s t a t e d in § 3. To prove the second part, we h a v e to show t h a t S Aq, t Aq*t =
i,
or
1
q
s
t
S Y., Z,,'.N~tN~N~ exp 2rd [A t u . + A2v.+ A3wn'-'--XT-nt--~-~--n'z---~n3]'~.t "'2 3
exp2~[--A,u,',.---A2v,',--A3w:,+
+ x /r n 3 3,_ = N~I n,' + xv-n6 s 1v2
.LV3
Z.
38
J. BOUMAN Now Al, A2, As have in Aq,t the values h
q
N, '
k- s N2'
1
t N3"
We shall neglect the second terms, and put
At=h,
Az=k,
A3=l.
This means that we take the values of Aqs, for the point hkl. This "approximation will be justified, if only the lattice-points in the neighbourhood of hk/ have. finite Aqs,, i.e. if for great values of qst Ava = 0. We m a y remark that in the speci~ theories this approxi'mation always occurs. In this case the truth of the proposition can be seen immediately by summing over qst. We shall retain this last approximation through out the following part of this paper.
§ 5. Some examples. 'Frozen heat motion'. The c o n s t a n t s Aqa , and not the displacements u,, v,, w,, define the broadenings of reflections. Let us consider a-distorted crystal, for which A qa = 0, except for--2
39
A G E N E R A L T H E O R Y OF L A T T I C E - D I S T O R T I O N S
Now an analogous effect c a n n o t be e x p e c t e d tbr a single distorted crystal. T h e crystal will have one distinct configuration. If we take into account, t h a t the D e b y e-S c h e r r e r lines are due to a cong l o m e r a t e of m a n y small crystals, we can e x p e c t the frozen heat m o t i o n to be represented b y a r a n d o m distribution of several types of d i s t o r t i o n o v e r the various crystals. To test this hypothesis, we will assume t h a t the displacements u,,, v., w. are v e r y small. The formula for Aqn can be developed into a series. Before giving this expression, it will be well to consider a n o t h e r m e t h o d of dealing with u . . v., w,,, which is always valid. These n u m b e r s can be developed into finite F o u r i e r series.
u. -----S(U)o.,exp2"m;
q n, + N-~n2 + ~-~3n3
~. = s(v),., exp. 2=i [q ~ , . , + ~s. ~ + ~ 1 .~
s
(,o~)
,]
As the sum S is t a k e n over the range - - ( N t 1)/2 -+ ( N t - 1)/2 etc., there are NIN2N3 n u m b e r s (U)qa etc. These F o u r i e r components m a y be used to define the displacements. With the aid of (5a) we find 1 £,, u,, exp - - 2~i [NI q nl (U)qst = .NIN2Na and
+
~S n2 + h na [(10b) *'a J
1 (U)ooo -- NIN=Na Z,, u. =- u.
In the same m a n n e r we can express u2., u . v . , etc.
If
'
u~ =-- S(U2),,,exp 2rd -~ln, + ~ n 2
lq
u,,v,, = S ( U V ) o a e x p 2rd - ~ l " '
,4.1 I
+ *,a
+ ~s; n 2 + . ,, , , n a l *'3
J
( U2)ooo = u'~. " (UV)ooo-- u.v. Now, from (6), a n d replacing A1, A=, Aa b y hkl, A qa ~
1 £,, exp 2rff [ - - q--s t NIN2N3 Ni nt - - N2 n2 - - -N-an3] x × I1 + 2 n i ( h u . + kv. + l w . ) -
2re2 (hu. + kv. + lw.) 2 ...].
40
j. BOUMAN Les us consider Aooo first.
Aooo = 1 + ~
[ h ( U ) ~ + k(V)ooo + Z(W)ooo] - --27,2 W(O)ooo + ~k(UV)ooo + . . . .
+ t2(w2)ooo].
Now (00ooo, (V)ooo, (W)ooo, being u,,, v., w,,, can always be put e q u a l to zero. And, as (U2)ooo = u--~.etc. Aooo ---- ! - - 2 ~ : 2 [(hu, + kv,, + lw,,)2]. Now hkl are the c o m p o n e n t s in the reciprocal lattice of the v e c t o r p ] d ~ , if dh,~ is the distance between two lattice planes (hkl) and p the c o m m o n d e n o m i n a t o r of h, k, and l. Its direction is the normal to the plane (hkl). I f u~ denotes the c o m p o n e n t of the displacement of the n ~ a t o m along the n o r m a l to (hkl), it follows t h a t :
hu,, + kv, + lw,,
Pu~" -
Aooo = 1
8~2 X2 sin2 ~ ~
d~
2 sin = ~
~
~-~ exp [ - - 8 ~
u~
sin2 ~ 2 -~ (-~7~] •
We will write ~ = u~, the m e a n square of the d i s p l a c e m e n t along the normal to the plane (hkl), t h e n : sin2 $ 2Aooo ---- exp [ - - 8~ ~ ~ uv]. N o w we t u r n to Aqa.
Aq,, = 27ri [h(U)qn + k(V)on + l(W)o,,j + . . . . The form between brackets will be the F o u r i e r c o m p o n e n t qst of 2 sin -
Unv '
-
The distribution of e n e r g y for a single crystal will be defined by 16n 2 sin 2 I Aooo 12 -----exp [ - X2 . u~ (l la) [A,,, 12 = 4re2 I h(O),,,
+ k(V)q,, + Z(W)o,, 12
(l lb)
If we take At = h, A2 = k, A 3 = l, as we have done before, then •
S [aqal2+ IAoool 2--- 1 qst~O
or, denoting S b y S ' q,s,t~O
S' ) Aqa 12
--
167ta s i n 2 ~
x~
u~.
(12)
A GENERAL THEORY
OF LATTICE-DISTORTIONS
41
We m a y add the following remark. The energy of the original line, given b y (11 a), is decreased i n t h e same m a n n e r as in the case of the heat effect. B u t as the neighbouring domains of i n t e n s i t y m a y have high values of Aqa, and so are to be reckoned to the original line, the energy m a y be greater t h a n in the case of the heat-effect. The expression is only valid" for small values t)f u, v, w, or r a t h e r of hu,, + kv, + lw,,. If we turn now to an aggregate with a r a n d o m distribution of deformations over the individual crystals, we m u s t take the average values of [ Aoo0 ]2 and [ Aqa ]2. We m a y assume in this case t h a t [ (V)qn [~,
] (V)q,, (V)*a ],~ etc. *)
will not depend on qst. (In general the second form will not be zero). Therefore: J A,,, 1,2~= 492 [h.2O 2, + 2hk (UV),, . . . .
+ 12W2v]
The formula for hA00o [2 will not bc altered, but u 2 will now stand for the mean square displacement over the atoms of one crystal, taking then the average over all crystals. F r o m formula (12) 1
1671:2 s i n 2
',A [~,. = [.Aq,, J2,v -- N , N 2 N 3 __ 1 l 16re2 sin 2 -- N I N 2 N 3 . X2
X2
(uv2)"o
(U~)av.
(12a)
Though this relation is not strictly true, as I Aq,t [2 is a function of A t, A2, As, and we have replaced these variable numbers b y hkl, w e m a y infer t h a t [Aqs t I~v will be very small. The original line will s t a n d out against a weak, continuous background. It is not difficult to calculate the intensity of this background. For this calculation it will not be necessary to assume t h a t the displacements are very small. The average of [Aot [2 will still be independent of qst, but the formula (12a) must now be written I
[Aqst 12v - - I A I~ -- N~N2N3 [1 - - ] A0oo 1,2~]
(12b)
*) !We d e n o t e b y a h o r i z o n t a l b a r an a v e r a g e over the a t o m s of one c r y s t a l , and b y av. the average o v e r the crystals of a n aggregate.
42
J. BOUMAN
We may assume too that the average of Aq,t Acvv * will be zero. Making use of (8), we find I,° = IAooo12,o sin2 ~ A tNi sin 2 n A2N2 sin 2 ~ AaN3 • sin2r~At sin2~A2 sin2nAz + sin2n(A2+ ~ ) N 2
sinZ~(Aa+ ~--~a)Na
sin2~(A2+ ~ )
sin2n (A3+ ~-~3)
+ I A t,vS 2 , The sum can be calculated easily• For sinZx(A,+ k ) N l = Y,- exp 2rri (A (cf. (2), (2a,)), and sinZx(A2 + ~ ) N 2
(
+ A2+
s
s!n2.n(A3 + ~ ) N 3
(
(n2--n~) + A a +
'F
(na--n3) .
The summation over qst will make each term zero, if not nt = n;, n2 ---- n2, na ---- n3. In this case the value of the exponential function 2 ~dll be 1 and the total sum will be N~tN2N 32 . e
t
t N3 sin2~ (A l + --~t)Nl sinZn (A2 + N~)N2 sin27r (Aa +. N~) S, S
sin2n (AI + ~ ) __
--
We find I,v
=
sin 2 rc A iNt sin2x At
AT2~T2AT2
,t v l , t , 2 ~ . 3
o sin2~ AtN1
I A o o o I~v
s i n 2 rc A i
+ IA 12,v[N~tN~2N]
sin 27:A2N2 sin 2n AaN 3 sin2~ A2 sin27r As
sin2~r N2A2 sin2~ AsN3 + sin 27r A2 sin 27r As sin2~AtNl sin27rA2N2 sin2nAaNa] sin2~A| sin2rcA2 " s-~r~A-~ ]"
A GENERAL
THEORY
OF LATTICE-DISTORTIONS
43
As I A [~ can be neglected, c o m p a r e d with [ Aooo I~, we find the final expression I~=
tAoool.2v sin2 ~ A t N ' • sin 2 ~ A t
sin 2 ~ A 2 N 2 sin 2 h A 2
sin 27:AaN3 sin 2 n A a +
+ NtN2N 3 [I - - 1Aoool2].
(13~
T h e first p a r t of I,v is the expression for the i n t e n s i t y of the original line, the second p a r t t h a t for the continuous scattering• R e t u r n i n g to our assumption of small values of u, v, w and introducing M --
8n 2 sin 2 X2
(u2~).v
we can write I . . = e -2M sin2~ A iN, sin2r~ A2N2 sin2~ AaNa • sin2~A I sin2~A 2 s i n 2 ~ A a +2MNIN2Na.(13a) This r a t h e r l e n g t h y c a l c u l a t i o n , leading to the well-known result
(13a), m a y serve as a check on our theory, and m a y m a k e clear, how the c o n t i n u o u s scattering m a y be conceived and which assumptions had to be made.
§6. Comparison with the work o/ D e h l i n g e r and B o a s . F o r the sake of simplicity, the previous papers on this subject assume t h a t the displacement has b u t one c o m p o n e n t u . . and f u r t h e r t h a t the displacement is a function of nl only. We find 1 Z . exp 2hi [A I u., --A¢a = NtN2N3
n, - - Nt n2 - - ~ n3]
This is in general = 0, but Aqoo = ~
Y'", exp 2m' [A, u. - -
n,].
We will suppress the index 1, and replace A, b y h. The formula
(7b) will then be t r a n s f o r m e d into exp [2rci (Al + ~ t ) N I ] - - I V
::
S Ago0
exp (2~i A2N2) - - 1 exp (2hi A2) - - l
exp (2hi A3Na) - - 1 exp (2~:i Aa) - - 1
44
J. BOUMAN
The t w o l a t t e r factors of the p r o d u c t are the same as for an und i s t o r t e d crystal ; we m a y leave t h e m out too. T h e t h i r d c o m m o n feature of the papers, m e n t i o n e d above, is t h a t the displacement is p_eriodicaJ with a periotl of r i d t e r a t o m i c distances (Nit beiiag a integer). This mekns t h a t Aq0o has o n l y then a finite value," when q is a multiple of N/r. To prove this proposition, we r e m a r k t h a t the F o u r i e r series (10a) m a y now be written as
u,, --= S(u)~ exp q
V
2~i -- n, r
N
~
V m
T h e t r u t h of the proposition is evident, from the expansion of Ag into a series (pag. 012). T h e formula's becor0e therefore A~oo = --1 Z,, , - i exp 2hi 0
(hu,, --7 n)
(l 4)
exp [2ni (A, + r) N , ] - - I F = SA,,oo This m e a n s t h a t the domains of i n t e n s i t y are s e p a r a t e d b y e m p t y domain.s, and a crystal of this kind will show separate lines in the n e i g h b o u r h o o d of the original line. These are the so-called ghosts. W e can now calculate A for the several t y p e s of distortion, and we shall find the intensities of the ghosts, as c o m p u t e d b y D e h 1 i ng e r a n d B o a s. We will, however, t r e a t one distortion in particular, as t h e r e is a discrepancy b e t w e e n our results and the results of D e h 1 i n g e r 2) 5). This d e p l a c e m e n t is for a p a r t (l - - p) of the period r S ~t~ =
a COS 2 ~ n - -
The o t h e r p a r t is not distorted. We will assume at once t h a t r and s are large numbers; e.g. ~ ~ I0', s = 102, and t h a t (1 - - p ) r is a multiple of the period of the cosine-distortion r/s. It follows t h a t (1 -- p)s a n d therefore ps are integers.
A GENERAL
THEORY
OF
LATTICE-DISTORTIONS
45
N o w . f r o m (14) A ~oo = .2
Z e x p 2~:i h a
r
r
r
r
+
2
+ -=
e x p - - 27ri
r
o
Z~ Z* - -
o
o
.
o
(l--k)!
2; e x p ~ 2 r c i
----
(l~p)r
Es(/--2k)--v)]
k!exp
+
+ --1 e x p [ - - 2~i(1 - - p)v] e x p [ - - 2rri pv] - - 1
I n t h e first s u m we r e p l a c e v b y ~s +
A,oo = rl ~1o Eko - -
" (l--k)!kt. ×
x
e x p 2~i (1 - - p ) [ s ( l - - 2 k - - ~ ) - - -r] - -
1
+
e x p 2hi s ( l - - 2 k - - tL) - - "r _ _ l ¥
+ - - I -exp [ - - 27riv] - - e x p [ - - 2hi (1 - - p)v] r
exp [--2~irl--1
N o w t a k i n g , i n s t e a d of l a n d k k andre = l--2k--~, a n d r e a r r a n g i n g t h e d o u b l e s u m m a t i o n , we c a n w r i t e A~oo = 1r +Z~ i~'+" Jo~+..) (27rah) e x p [2ni (1 - - p ) ( s r n ---¢o S~ -- T exp 27ri--
z)] - - 1
1
1 e x p [ - - 2~:i (1 - - p) (s~ + ~)] - - e x p [ - - 2~i (s~ + z)] r
exp[ -2nis~+r
*]--I (v = s~ + .).
W e will t r e a t s e p a r a t e l y t h e cases v = 0, v = Fs, v = ~, v = Fs + . 1°. t L = T = 0 , ~=0 A v0o =
-
1 ~
-
r
--0o
i " J . , (2nah) e x p 2 h i (1 - - p ) ( s i n ) - e x p 27ri
$~ 1"
--
1
1 + p
.
46
J. BOUMAN
As (! - - p ) s is an integer, the t e r m s of the sum will be zero, e x c e p t when the d e n o m i n a t o r is zero too. This will occur for m -----0, m = ± r (or 7"Is if this n u m b e r is a integer). B u t the B e s s e 1 functions with a large i n d e x h a v e v e r y small values for small a r g u m e n t s . We retain m = 0 only, and find Ao0o = (I - - p) J o (2nab) + p. In the o t h e r cases (v :# 0) we m a y write • A~o
o o i v + " j(~,+,,) (27rah) e x p [ - - 2rci(1 - - p) x] - - 1 _ _ = --1 +Zm /'
"
--oo
exp 2~i
S?H, - -
T
1
7,
1 exp [ - - 2~i(1 - - p) x] - - 1 exp[--2~isV'+r ~]--1 2 ° .
T ~--- O ,
v ~
~l.S.
The secofid p a r t will be zero, in the first p a r t we retain m = 0 only Am,o,o = (1 - - p) iv I v (2nah). These results h a v e also been o b t a i n e d b y D e h I i n g e r. It is clear, however, t h a t also for o t h e r values of v, finite values for A~oo m a y be found. W e replace again the series b y the t e r m m = 0. 3 °. t L = 0 , v = x . As x m a y be a small number, we will not omit the second p a r t of A~oo; we find I A~oo 12 ---- 7,21 sin 2 n px [Jo (2~ah) - - l ]2. sin2 ~rx 7,
4 °. v = ~s + x. N o w we can omit the second p a r t J A~00 12 --
I sin27rp, -2 (2nab).
r ~ s~n2~ Jv r
If we now take X I A~0o j2 we m u s t find 1, in accordance with our general result: B o a s 6) calculates Y. lAv.,.o.o js and finds a value for this sum, which is less t h a n I. As this sum is used for calculating the b r e a d t h of the line, the i m p o r t a n c e of our general proposition is clear (vid § 7).
A G E N E R A L T H E O R Y OF L A T T I C E - D I S T O R T I O N S
47
§ 7. Conclusion. In the a b o v e the problem has been solved as to how the distribution of i n t e n s i t y in the reciprocal lattice is affected b y distortion. The following step should be to evaluate the distribution of e n e r g y on the film b u t this problem presents considerable difficulties. W i t h the aid of simplifying assumptions, however, it can be solved for an u n d i s t o r t e d crystal. This leads to the f o r m u l a for the b r e a d t h of a line (4) as a function of the size of the particles. Now the size of the d o m a i n of i n t e n s i t y is inversely p r o p o r t i o n a l to the size of the particle. W e have seen t h a t in general the d o m a i n of i n t e n s i t y will increase for a d i s t o r t e d crystal, so t h a t the effect of distortion will in some respects be the same as t h a t of a breaking u p of the particles. B u t the distribution of energy in the reciprocal lattice will bear in general no resemblance to the distribution for a small, und i s t o r t e d particle. This will occur only if the Aqa, considered as a function of qsl, have a m a x i m u m for q = 0, s = 0, t = 0, and decrease with increasing values of q, s, t, to zero. In this case B o a s 5) uses the following m e t h o d . We consider a one-dimensional reciprocal space. F o r an u n d i s t o r t e d crystal we find for the distribution of energy sin 2 ~ A N sin 2 ~ A
I-As h+½
f
I.dA--N,
I(A = h ) = N 2 = I m ~
h--i
so we can write for the theoretical b r e a d t h b,
2 bt-
N
f I . dA -- 2
Imp,
In our case the domain of the distorted crystal will have a distrib u t i o n of energy, which resembles the distribution for an u n d i s t o r t e d crystal. We can take the integral over the range h - - ½ ~ h + ½, as all Aq,t values are zero, e x c e p t in the neighbourhood of 0, 0, 0. As this integral is N, and Ima~ = N 2 [ Aooo 12
2N bi --
N2 I Aoool 2
and b--
1
acos~
2
" IAooo[2"X"
48
J. BOUMAN
tn general it will be uncertain, which d o m a ~ s of intensity' must be reckoned to the original line. This means thai the range of the' ivitegr~d is uncertain. But, if we integrate over the domain, connected with hkl, the breadth must be inversely proportional to [ A0oo [2. We have mentioned this calculation to elucidate our remark in § 6, concerning f I . dA, but do not intend to discuss the breadth of the line for. all possible cases. The final problem must be the determination of the distortion u., 'v., w. from the intensity and the breadth of the D e b y e-S c h e rr e r fines, or from the distribution of energy over these lines. The intermediate step will be the determination of the values of Aq,,. If ~these were known, then u . , v., q. could be determined. This is easy for small displacements. Then the formula (I I b) can be used, and (U)q;,, (V)qst, (Wqa) can be found from the reflections, and from these the distortion. If only [Aqst [ is k~own, the phases of the F o u r i e r components of the distortion still remain unknown, and we know only whether the components with short or with long wavelength are preponderant. Without determining the connection between the Aq,t values and the properties of the reflection, we can formulate some rules (cf. § 5). A reflection, with the original breadth, b u t diminished intensity, appearing on a continuous background, will indicate a frozen heat motion. All terms of the F o u r i e r series will occur, with the same average amplitude. A broadened llne, with the original intensity, can only be caused b y a distortion of the type, already mentioned. In the F o u r i e r series~only the components with long wavelength appear, and in the reciprocal space we shall find energy only in the neighbourhood of hkl. The total energy will be found in the line, and there will be no scattered energy at all. An example is the cosine-distortion of D e hlinger. If the broadened line has a diminished intensity, then energy must b e found between the lines, and the F o u r i e r series will have at teast a few terms with short wavelength, as there will be factors Aq. with large values of q, s, t. We have shown some consequences of the proposition, that the total energy in a domain, connected to hkl, remains approximately constant. The proposition will be a guiding principle for determining the deformation of a crystal. Received November 3rd, 1941.
A G E N E R A L T H E O R Y OF L A T T I C E - D I S T O R T I O N S
49
REFF_.RENCES 1) 2) 3) 4) 5) 61
M. y o n L a u e , Z. Kristallogr. 64, 115, 1926. U. D e h l i n g e r , Z. Kristallogr. 65,615, 1927. J. H e n g s t e n b e r g a n d H . M a r k , Z. Phys. 61, 435,1930. W. B o a s, Z. Kristallogr. 1t6, 214~ 1937. W. B o a s, Z. Kristallogr. 97,345, 1937. G.W. Brindley and P. R i d l e y , Proc. phys. Soe. (London) 50, 501, 1938; 51,432, 1939: P) J. B o u m a n a n d W . F. d e J o n g , P h y s i c a S , 817, 1938.
Physica IX