- Email: [email protected]
Dynamics of a crystal lattice containing isotopes
Pirenne, J e a n 1958
P)lysica X X l V 73-92
DYNAMICS OF A CRYSTAL LATTICE CONTAINING ISOTOPES by
JEAN
PIRENNE
The Institute for advanced Study, Princeton, N. J. and Universit~ de Liege, Belgique Synopsis B y means of a m o m e n t method, t h e v i b r a t i o n s p e c t r u m of a m o n o a t o m i c crystal c o n t a i n i n g several isotopes, whose a t o m s are distributed at r a n d o m t h r o u g h o u t the lattice points, is o b t a i n e d in t e r m s of the solution of t h e conventional v i b r a t i o n p r o b l e m of a singIe isotope crystal and in terms of the quantities
where c, is t h e c o n c e n t r a t i o n of the i - t h isotope and A/~l the deviation of its inverse a t o m i c mass/~f from t h e average value (/*)av = ~ , ci/~f. T h e f r e q u e n c y distribution is represented b y a series d e v e l o p m e n t following t h e increasing powers of a h e t e r o g e n e i t y p a r a m e t e r to which all the A/~ are assumed to be proportional. The first t e r m is the f r e q u e n c y distribution of a fictious crystal whose a t o m s would all h a v e the mass ~1. The n e x t two t e r m s are proportional toav and Av; t h e f o u r t h t e r m brings contributions in av and in ~, etc. These three additional t e r m s h a v e been written down explicitly, b u t the following ones could be easily obtained. E a c h t e r m b u t t h e first one brings contributions of t h e f o r m f d ' ( 2 -- t') K(2') d)d, where 2 is the square of t h e angular frequency. These functionals m a y n o t be simply p u t equal to K'(2), as /f(2) has singularities which m u s t be carefully considered. I n t h e r m o d y n a m i c applications, however, this difficulty is i r r e l e v a n t for t h e n t h e c5' f u n c t i o n is simply replaced b y the 2-derivative of the desired t h e r m o d y n a m i c function of the h a r m o n i c oscillator and the remaining integral is trivial.
Introduction. Let us consider a crystal formed b y a single chemical element possessing several isotopes and let us assume these isotopes to be distributed at random at the various lattice points. Then arises the problem how to compute the frequency distribution or spectrum from which all the thermodynamical properties of this crystal m a y be immediately derived. The solution of the eigen-vibration problem is elementary and well known when all the masses are equal, but here we wish to take account explicitly of the fact that different isotopes have different masses, although the forces remain the same. In other words, the kinetic energy of the crystal is no more invariant under the operations of the symmetry group of the crystal, but the potential energy still is. This problem was suggested to us by G. P l a c z e k in connection with --
Physica XXIV
73
--
74
JEAN
PIRENNE
the scattering of neutrons or X-rays b y such a crystal when one wishes to investigate the influence of the mass differences between the isotopes which have always been neglected hitherto. It turns out however that the well known relations between scattering cross section, correlation functions and frequency spectrum are destroyed b y the introduction of mass differences. The scattering problem is therefore quite distinct from the present one the solution of which m a y however clarify the situation. During the early development of this work, two papers of I. P r i g o g i n e and collaborators Eli, [2] appeared in which the actual problem was considered from the point of view of t h e p e r t u r b a t i o n theory. In fact, the main object of these papers was not to compute the frequency spectrum itself but rather to obtain an integral value representing the zero-point energy of the crystal. It is however much simpler to derive the sum of the perturbations of all the eigenvalues than the eigenvalues themselves. The first quantity m a y be deduced from the trace of the second order perturbation operator, which can be evaluated in any basis, in particular in the conventional one defined by the usual system of plane waves. On the contrary, the computation of the individual perturbations requires the explicit construction of a system of unperturbed waves adapted to the perturbation, a much more complicated problem, at least in the three-dimensional case. It is therefore not yet clear whether the perturbation method would easily lead to the derivation of the frequency spectrum itself, which, anyway, is not indicated in these papers. For the linear chain, D y s o n E3] has developed a rigorous method of deriving the frequency spectrum; its calculations have been somewhat simplified b y B e l l m a n E4] and, more recently, another presentation has been given by D e s C l o i z e a u x E51, w h o introduces the concept of phase shift. All these calculations only apply to the particular case where the interatomic interaction is confined to pairs of nearest neighbours only. As far as we know, it has not been possible either to lift this restriction or to extend the theory to the three-dimensional case. • In the present paper, we use an apparently easier method based on the computation of the moments of the frequency spectrum b y means of traces of certain matrices. A somewhat similar starting point has been used b y M on t r o l l E61 to derive the frequency spectrum of a lattice without isotopes, b u t here the aim and the treatment are entirely different. We assume indeeed that the frequencies and polarisations have already been computed for each wave vector for the crystal without isotopes, and that the frequency distribution of the latter has also been obtained; no special assumption is therefore necessary concerning the potential energy which m a y be quite general. These results are then used to express the frequency distribution of the crystal with isotopes in terms of the moments of the spectrum of the inverse masses, #~ = m~-1 or more precisely in terms of the moments of the deviations A#, = t L, -- Av.
DYNAMICS OF A CRYSTAL LATTICE C O N T A I N I N G ISOTOPES
75
Definitions. Let us represent by ml < m2 < . s.. u~,a,,
(1)
the first sommation being carried over all lattice points x' and x'. Introducing te hyper-vector u = (u,~ 1, u ~ . . . . . .
)
(2)
in a 3N-dinensional linear space, W can be written in the more condensed matrix form : w --k(u,
(3)
Vu)
where (A, B) denotes the scalar product of the hyper-vectors A and B. The matrix element is of course invariant under the operations of the translation and symmetry group of the lattice; it depends therefore on x and x' b y the distance x -- x' only: = v~,,,(x -
x')
(4)
The equation of motion of the system m a y be written: Mii = -- Vu
(5)
= m. 6.,.. Oa.,,,
(6)
with The characteristic (angular) frequencies wj and the corresponding eigenamplitudes uj then satisfy the equation: Vuj : 2jMuj
(7)
aj = ~ .
(8)
with Our problem is to determine the number of proper vibrations Ng(to) do) = N9(2) d2
(9)
for which to ~ toj ~ t o + d t o , ,~ ~ 2 j ~ 2 + d ; t As will be later seen, the frequency distribution g(to), or the function 9(2), becomes independant of N for an infinitely large crystal, provided the
76
JEAN
PIRENNE
isotopes are distributed at random so that their presence probabilities at the various lattice points are independent of each other *).
Limits o/ the [requency spectrum. From (7), we derive ]tj = (uj, Vuj)/(uj, Mul)
(10)
Suppose now, for simplicity, that ul is normalised to unity, then: (UJ, V u ] ) ~
-max](1) . m l
-m&x~ll21 .
=
m2 . . . . . .
(11)
where A~ax is the square of the highest frequency of a crystal containing the l isotope only. Fl~rther
(uj, Muj)
(12)
Therefore, In other words, the spectrum of a mixed crystal is extended no more than the spectrum of the pure crystal containing the lightest isotope only.
Normal coordilmtes.
The eigenamplitudes are solutions of the equation =
M-1
juj
(I 4)
leading to a familiar secular equation. However, as the matrix M - I V is not symmetric, they are not orthogonal. We introduce therefore the normal coordinates ~j = M1/2uj (15) which are solution of the equation H~j = ~j~j
(16)
H = M-112VM -1/9"
(17)
where As H is a symmetrical matrix, the vectors ~j are orthogonal and we shall normalise them to unity: =
(18)
The unperturbed problem. In the particular case of a single isotope of mass m, equation (17) reduces to: m_lV~O)= 21.0,~o) (19) *) T h i s b e c o m e s o n l y t r u e w h e n N - + co. S u p p o s e it w e r e t r u e for a fixed v a l u e of N a n d d e n o t e b y Pl, Pa,...Pr t h e a p r i o r i i n d e p e n d a n t p r o b a b i l i t i e s of f i n d i n g i s o t o p e s 1, 2, . . . , r a t an), l a t t i c e p o i n t . T h e n , t h e t o t a l n u m b e r of a t o m s of t h e v a r i o u s i s o t o p e s w o u l d n o m o r e b e well d e t e r m i n e d , b u t t h e i r a v e r a g e c o n c e n t r a t i o n s w o u l d be e q u a l to Pt, . . . , Pr a n d the s t a n d a r d d e v i a t i o n s f r o m t h e s e a v e r a g e c o n c e n t r a t i o n w o u l d t e n d to zero as N - ½ . T h e r e f o r e , for N -+ oo. t h e r e is no d i f f e r e n c e w h e t h e r t h e c o n c e n t r a t i o n s are a priori fixed or w h e t h e r t h e p r o b a b i l i t i e s Pl, po . . . . , Pr are independant.
DYNAMICS OF A CRYSTAL LATTICE C O N T A I N I N G ISOTOPES
77
whose solutions are
•101 k,$ ;x =
e (k, s)
ei~'/v'~
(2o~)
or, in Dirac notations,
a[k, s> = e~(k, s) . eikx/v'~
(2Oh)
where N is the number of atoms of the crystal and e(k, s), the polarisation vector whose index s runs from 1 to 3 *) . k is subjected to the usual BornK~rman conditions. We assume that this "unperturbed" problem has been completely solved; in other words e(k, s) and 20(k, s) are well known functions of the wave vector k and the polarisation index s. We shall further assume that the number of eigenvibrations N.90(2 ) d l , for which 2 E 2j E 2 + d2, has also been computed. Our task now is to derive the distribution function ~0(2) for the general case of several isotopes distributed at random, which we shall refer to as the perturbed problem. It is however not our intention to consider the mass differences as a small perturbation and to use standard perturbation theory, but rather to derive 9(2) from its moments for arbitrary mass differences. It turns out however that our result appears as a series whose terms are easily expressible by means of the solutions of the unperturbed problem with an appropriate fictious mass. For cubic crystals, the three first terms depend on ~oo(2) only, but for the computation of the next ones, the knowledge of the polarisation is also necessary. For other crystals, the same is already true from the second term. For the following, it will be useful to express the matrix element of any power of the potential matrix in terms of its eigenvalue as follows (x, a
IVpl x', a'> = 2Ek (x, a Is, k>(s, k I x', a'>
= FAe,,(s, k) ,~(s, k) e~,,(s, k).~'*'-'~lN
(21)
(21)
On the otherthand, by taking the trace of Vao we find: N-1
• m-
1k'~Y'x,a r (x, a [V~[ x, a) : N-1 . m-1. Trace V h = f 2~ 90(2) d2 (22)
or, as Vv is invariant under the translations of the lattice, m -1. ~ a (x, a IVY[ x, a) = f 2 ~ ° 90(2) d2
(22b)
Further, as (x, a IV[ x, a'> is also invariant under the symmetries of the lattice, it is necessarily diagonal with respect to a, a'. For cubic crystals, it follows therefore that m -1. (x, a IVY[ x, a') = ~5~.,f2~oo(2) d2
(23)
*) For simplicity we consider only Bravais lattices, such as the simple cubic, face-centred and body-centred cubic lattices.
78
JEAN PIRENNE
T h e / r e q u e n c y d i s t r i b u t i o n / o r the perturbed problem. The n u m b e r of eigenvibrations 2~ of the crystal whose 4j lie in in the interval
dR = 2o~ dw
(24a)
Ng(o~) d¢o = N9(4 ) dR = ~{d~} (2t, 2~)
(24b)
m a y be expressed as
where the s u m m a t i o n extends over all eigenvectors of the interval (24a); (U, ~") is the scalar product of the hyper-vectors 2' and 2". Now, the norm and the various m o m e n t s of the frequency distribution (or r a t h e r of the square angular frequency density) 9(4) m a y be written in similar w a y as: dr'0 = f0~ 9(4) dR = N -1. ]~j (~j, 2t)
= N -1. Trace 1 = 3
J t ' l = f0~= 9(4)4 dR = N -1 . ~ j (~j;H2j) = N -1. Trace H .
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
,
(25) .
.At'n = f ~ 9(4)t n dR = N -1 . Y,j (21,Hn2j) = N -1. Trace H n If we can evaluate these moments, we deduce from t h e m the Fourier transform of 9(),), ~p(s)----f0~= 9(4)e ~s~ dR = N - 1 . T r a c e ei*H = Y~°=0 ((is)n/n!).'¢[n,
(26a)
and finally, the distribution 9(4) itself: 9(4) = (1/2=) f+o~ ~0(s) e-~SAds.
(26b)
I t m a y be mentioned here t h a t the series (26a) is absolutely and uniformly convergent as H is bounded. T h e m o m e n t s o/ the /requency distribution.
B y introducing the inverse
mass
#(x)
=
m-l(x),
(27)
the m o m e n t ~ ' n is explicitely given by J t ' n = N -1. Trace ( M - l / 2 V M - 1 / 2 ) n = N -1. Trace (M -1 V) n =
= N - 1 . Z{~,~}/~(x') #(x") . . . . . . . . . . ;,(x(~))
= N -1. 2~{~}#(x') #(x") . . . . #(x(n)) . $/" (x', x", . . . . x(n))
(28a)
with
~"(x', x", . . . , x('O) = Z{~, . . . . . ....
x', a'> (28b)
where the subscript {x} indicates a s u m m a t i o n over all lattice points, {a}
DYNAMICS OF A CRYSTAL LATTICE C O N T A I N I N G ISOTOPES
79
a s u m m a t i o n over all indices a', a", . . . . and (x, a} a s u m m a t i o n over both lattice points and a-indices. To visualise these summations, it will be useful to represent a set of points x', x", . . . . x(n) b y a closed graph obtained b y joining b y a line each point to the following one and x~m to x ' . It m a y happen, of course, t h a t the various points of the sequence are not all distinct one from each other; then, if r of them coincide, the graph will pass r times through the same point, which will therefore be a multiple point with multiplicity r. R e t u r n i n g now to (28b), we see t h a t the function $/" (x', x", . . . . x(n)) a t t a c h e d to a graph is invariant under the translations of the graph. Therefore, in (28b), the product # ( x ' ) . # ( x " ) . . . # ( x ( m m a y be replaced b y its average over all translations keeping the graph inside the crystal. As the n u m b e r of these translations is practically equal to the number of atoms, if the crystal is sufficiently large, the s u m m a t i o n over t h e m m a y be dropped, if we simultaneously leave out the factor N-1. It is further clear that, for an infinite crystal and a r a n d o m distribution of isotopes, the above average of the inverse mass product only depends of the multiplicities rl, r2 . . . . . r v of the p different points Xl, x2 . . . . . x~ which constitute the graph, but is otherwise independent of its shape: <#(x') #(x')
...
#,(x(n)>Av =
(#r'>Av . (#r2)Av . . . . . .
(#~'>Av
(29)
Therefore J C n ---- ~Irl (#rl)Av • (#~2)Av . . . . . . .
(/z~)Av . Vr2...r ~
(30a)
with vn...,., = Xla} v(G),
v(G) = Z{:~} ~ ( x ' ,
x", . . . . x(n))
(30b, c)
In (30a), the s u m m a t i o n is carried out over all values of rl, rs . . . . . rv compatible with rl + r2 + . . . + r v = n and p = 1, 2, 3 . . . . , n. In (30b) the s u m m a t i o n is carried out over all possible graphs having p distinct points of multiplicity rl, rz . . . . . . rp (subscript G) and, in (30c), it is extended, for each particular graph G, over all possible positions of p -- 1 of its points, one of t h e m being kept fixed as the translations of the graphs are now excluded. Let us briefly note here t h a t the m o m e n t s (#r)A v of the inverse mass spectrum are immediately expressible in terms of the inverse masses and concentrations of the various isotopes b y the s t a n d a r d formula: (#r)Av = ~ c i#ir
(31a)
L a t e r on we shall also need the r-st order average deviations (A#~>a~ = X c,(#, - (#)A~) ~
(31 b)
D e c o m p o s i t i o n o / t h e m o m e n t s a n d new expression o / t h e / r e q u e n c y distribution. As will be later seen, the c o m p u t a t i o n of Vrl. ..rp b y means of (28b)
B0
JEAN PIRENNE
is greatly simplified by dropping the above mentioned condition t h a t xl, x~, . . . , x~ be always different from each other. B y doing so, however, we add terms truly corresponding to other graphs having points of higher multiplicity, but we count them with a wrong mass coefficient (29). Therefore, when we shall come to these other graphs, we shall have to substract from the corresponding exact term, with its exact mass coefficient, the previously added term with its wrong mass coefficient. To apply this method, the various graphs must appear in such an order t h a t no such correction should affect a term which has not yet been met. As will be seen, this requires the graphs to be classed be order of increasing value of the number r = rl + r2 + . . . r~, which we shall call the order of the graph. All graphs of the same order should further be considered by order of increasing value of the highest of the multiplicities rl, r2 . . . . . r~. Each moment will then appear as a sum:
~n
= ~oo.~,~,)
(32)
We shall then build the functions
~or(s) = Y,~°=0 ( (is) n/n !) Md(~)
(33a)
9r(~) = l[2~r f + ~ ~or(s) e~8~ ds
(33b)
and we finally obtain the frequency distribution 9(~) as a new series 9(~) = X~'=1 9r(}t)
(34)
The r-st term of this series will be refered to in the following as the r-st order term. It arises from all graphs of multiplicity r and it is related with the average r-st order deviationAv and eventually to other deviations whose sum of the orders is also equal to r.
First order term. Ler us first consider a graph without any multiple point. According to (30), the contribution of all such graphs to the moment Mt'n is obtained b y summing up the corresponding q/'~(x', x", . . . . x(n)), given by (28b), over all possible positions of n - 1 of these points under the only restriction that x', x", . . . . x(n) be all different one from each other. If, however, we drop this restriction we add new terms that should be later substracted, but the summation (28b) greatly simplifies as it now reduces to a simple matrix product. Inserting the result so obtained in (30) and taking account of (22b), we find the following contribution to ~ ' n : ~ ' ~ ' = (#>3~ Z a (x, a IV~l x, a> = f o ~" 2 n 90(2) d2,
(35)
where 9o(2) is unperturbed the frequency distribution for a fictitious inverse mass #* = <#>Av. Introducing (35) in (33a) and using then (33b) we obtain the first term of
DYNAMICS OF A CRYSTAL LATTICE CONTAINING ISOTOPES
8|
the series (34) representing the perturbed frequency distribution: y,l(S) ---- Z~'=o ((is)hi n!) "£~) = fo~" e~sa ¢o0.) d).
(36a)
fol(2) = ( I / 2 u ) f + ~ 91(s) e -iSa ds = 90(2).
(366)
This result coincides with that of the perturbation method used by P r i g o g i n e et al. ; however, the fictitious mass used by these authors was the average mass (m)Av whereas we are dealing here with a fictitious mass ~q¢* = (m - 1 )av. - 1 Both methods lead therefore to different results; in fact, the two series representing ~(~l) differ from the second term already. Second order term. We now have to consider graphs containing multiple points. If we want to represent such a graph on a sheet of paper, it will immediately be realised that a mere polygonal ]ine does not provide a complete description. In fact, the label of each point should be indicated, or at least their succession order, as any circular permutation of x', . . . . . x ("~ does not change the value of the corresponding term of (28). The orientation of each loop should therefore be marked in some way, for instance by an arrow.
A perhaps simpler way of drawing such graphs is however to represent the sequence x', . . . x(n) on the sheet of paper by a set of n distinct points connected in their prescribed order by a full fine and to indicate those points, which are indeed identical in the crystal, by connecting them by dotted lines• Examples will be found in fig. 1, 2, 3, 4.
X~.j )~ X'P'"g \
X'
X" iX"' I = x'"
........
Xc~"~ ~
^
Fig. 1.
A simple notation for each type of graph is easily obtained by writing down only those points of the sequence which are multiple and by denoting each of them by a separate letter; for instance, the graphs represented in fig. 1 to 4 will be denoted by (xx), ( x x x ) , ( x x y y ) , (xyxy) and ( x x x x ) . Let us now consider the simplest graph containing multiple points, namely (xx), which has but one double point (fig. 1). Its contribution to ~ ' n is, according to (30), ( / t 2 ) A v • /\ / 2 /~'~-~ Av
. v(xx)
(37a)
82
JEAN
PIRENNE
where v(xx) is given b y (30c). Let us remenber, however, that, in computing the first order term ~/t'C1) we carried out an unrestricted x-summation, introducing thereby additional contributions which, in fact, corresponded to more complicated graphs b u t were nevertheless counted with the wrong factor <#>],. In particular, the added contribution, relative to (xx) graphs, was
•
<#>~v. v(xx).
(37b)
Substracting now this from (37a), we obtain the total contribution: (<~2>A~
(~o~)
• <~>~72 •
v(xx)
(37c)
Now, to compute v(xx), we have again to sum up ~F'(x', x", . . . . x(n)), given b y (28b), over all possible positions of all points but one, which can always be taken as the double point. As this summation should correspond to (xx) graphs only, the summation points a n d the double point should always be kept distinct one from each other, but, as we have already done before, we shall again drop this restriction, introducing thereby additional terms which correspond to more complicated graphs that will later have to be substracted. The summation however greatly simplifies and reduces to simple matrix products and we obtain in this w a y the second order term
"£~) = (<#2>Av -- (tt>]v) • <#>Av'~-2-W(XX)
(38a)
w(xx) == ½,, Z~=I .-1 Za, a'
(38b)
with : (38b) is obtained as follows. Suppose first that the double point x is x'; then, b y unrestricted summation over x", . . . , x(n), we immediately find the above term w(xx), b u t without the factor n/2. The double point, however, m a y also be x", . . . . x(m, and, after eventually changing the name of the variables, we arrive to the same result in each case. Thus, for any fixed value p' of p, there are n equivalent graphs leading to the same term, but in summing up over p, we find again these same graphs for p = n -- p'. Therefrom follows the factor n/2. We shall now compute this sum for a cubic crystal. Using (23), we find:
...£~, =Av
-
<~>5-----~
n
1 ~ ; ~ " ~ ;~"
Y
3
a0 a0 X $ - I 2 'n-~R'~00(R')90(U)d2'dR" (39a)
6 ~ o ~,
a, _
a.
a'z" ~o(z') ~o(r') da' d~"
(39b)
Introducing this expression in (33a) and noting that the contribution M/(~), being related to (xx) graphs, only starts from n = 2, we get: v,2(s) = Z --
((is),,/n ! ) . . ~ )
1
Av
6
<#>~
. (is)
(4o~)
|
fa" F~= e*S;~'--etS;~" .l';t"
dO
|
JO
2 , _ 2,,
~o(~t') ¢o(;(')d).' dit" (40b)
DYNAMICS OF A CRYSTAL LATTICE CONTAINING ISOTOPES
83
In (40a), it has been possible to sum from n = t, instead of n = 2 as was announced, because expression (39b) vanishes identically for n = 1. Inserting this in (33b), we obtain, for the second order term of the frequency distribution,
1 (A/z2)Av 1 [.a=f~,,, eiS(~'-~)_e~SC~"-~) Jo (is) 2' -- 2# 2'Y'~o(~t')9o(2")d2'd2"ds(41a)
92(2)-- 6 Qe2)Av ~ J o or, finally,
I
-- 2) -- S(Y'
2'--2"
-- 2)
2'2"9o(2')qoo(2")d2'd2" (42a)
Applications of this formula will be later discussed. For non cubic crystals, we would obtain in a similar way, b y using (21b) instead of (23):
92(2) - 2,
<~>~
Z,.,.
ff
20(k', s') - 20(k", s") -
-
s,,)
v0(dk') 3 • 2o(k', s') 20 (k", s") (e(k', s') . e(k", s")) 2 . - -
(2~)3
-
vo(dk")3 (42b) (2~)3
where vo is the atomic volume, i.e. the crystal volume per atom.
X'=~X
~
X=X c~
Fig. 2.
Third order term. We now turn to the next simple graph after (xx), namely xxx), which has one triple point only (fig. 2). It brings to the m o m e n t ~ ' n the contribution Q*3>Av • (#) Av'~-3 . v(xxx),
(43a)
where v(xxx) is given b y (30b). But we must substract from this the contribution (#)'~v" v(xxx), (43b) introduced in the second order term b y dropping any restriction on the x-summation, and three times the contribution
(([.12)A v
-
-
(#t)~v) (,")Av n--3 v(xxx)
(43c)
84
JEAN PIRENNE
introduced in the second order t e r m ~'~), a first time when the double point x was x ' ---- x(~ +1) a n d we s u m m e d over x(~+q+z) w i t h o u t omitting x(~+e +1) ---= x', a second time, when the double point x was x(~+l) = x(~+q+l) and we failed to o m i t this point in the x ' - s u m m a t i o n , a third time when x was x(~+a+l) __= x' and we did not omit this point. Substracting therefore (43b) and three times (43c) from (43a), we find the following contribution to ~ ' n : {A~ _
<~>a
_ 3<~>A~ (<,,2>~
_
<~>~)}
<~O~3. =
v(xxx)
Av~ a .
=
v(xxx).
(44)
v ( x x x ) is obtained b y s u m m a t i o n of ¢/'(x', . . . . x(n)), given b y (28b), over all possible ( x x x ) graphs having a fixed point which m a y always be choosen to coincide with the triple point. If we however drop the restriction t h a t the s u m m a t i o n points and x be all distinct, as we have already done before, the s u n m m t i o n again simplifies and we obtain in this w a y the third order term of ~ ' n : ~tt~) = (Atza>A v . <#>~ga . w(xxx) (45a) w(xxx) =
with
--½ _
~
~-I
~-I Z~=~Z~=IZ~,,,,~,~,,~,,.(45b)
The factors would give n counted three L e t us now
~n appears because the triple point m a y be x', . . . . x(n) which equal contributions, b u t in fact, each graph woald t h e n be times. evaluate w ( x x x ) for cubic crystals. Using again (23), we find
w ( x x x ) ---- (n/3)l~vfoa'foa'foX'Fna(it', it", it')~oo(2')qoo(it")~oo(it')dit'dit"dit" (46) with Fn3 = it,it,,it,, £~=o . - a £~=o it,n-a-~ it"~-~ it,,,q =
=-
it'a"it"
it'it"it", Ds(a',
~", a')
1
2'
it, n - z
1
2"
it" n-Z
l
2"
it"n-i
(47a) (47b)
~'.-~ (it" - it") + a " . - * (it" - a') + z ' . - ~ (it' -
where
Dn(it', it", . . . . it(n)) =
it")(it" -
it")(it" -
(it' - 2'3
it')
I
2'
it'2
. . . . it' n-1
1
it"
it"2
. . . . it"n-1
1
2(m
it(n) 9' . . . . 2{n)n-x
(47c)
(48)
is the V a n d e r M o n d e determinant. If we insert (47b or c) into (46) and the latter into (45a), t h e n we can easily build the series
~oz (s) = ~'~=1 ((is)hi n .J~~'~nu(a)
(49a)
DYNAMICS OF A CRYSTAL LATTICE CONTAINING ISOTOPES
85
The s u m m a t i o n should start at n = 3, beca~lse there cannot be a n y triple point before this; b u t it can nevertheless be performed from n = 1, because expression (48), and therefore also (47), (46) and (45a), identically vanish for n = 1, 2. We therefore have: 1 J.' d "s)"
~ps(s) --
-,,--V777,,,-,,, 1 a" e*ha" 9o(2')9o(2") q~o(k")dR'da"d2" (49b) (#>a 27333 Da(2 ,a ,~ ) 1 2 " e ~sa''
Inserting this in (33b), and introducing the notation 1
An(.a', ,V, . . . . ~(n); ~) =
~'
1 .
.
l
~t'2
~" .
.
.
.
.
....
~"~ .
).(n)
.
.
.
.
.
,t'n-2
~(~' - - ,l)
. . . . ~"~-~ ,
.
.
.
.
.
.
)(n)2 . . . .
.
.
.
.
.
~(~" -- ,l) .
.
* *
.
.
.
.
,
, (5o) •
).(n)n-2 (~(2(n) __ ~)
we find, for the third order term of the frequency distribution:
1 (Att3)A v [[(~,~,,j.,,, As(J.',2",,Tt";2) ,]J.] D3(2',2",,t") 9o(it')~oo(Z")9o(2")dk'd2"d,l" (51a)
(p(~t) = 27 ~
27 <#>av Jd,J
(a --2 )(X --,~ )(3. --,~ ) • ,o(a') ~o(Z') vo(a") da' d~" da" (Sib)
Fourth order term. After (xxx), the next simple graphs are (xxyy) and (xyxy), having both two distinct points (fig. 3a, b and 3c), and (xxxx) having one quadruple point only (fig. 4). T h e y bring to dgn the following contributions: gt2) ~ gt)~-~l . v(xxyy) (52a)
q,~>]~ <~>~;~. v(xyxy)
(52b)
~;*. v(xxxx)
" (52c)
where the v-factors are again given b y (30b). B u t v(xxyy) and v(xyxy) contributions were already introduced, once in a(1) with the " w r o n g " factor mass factor (/~5]~, and twice in Me'cs), with the wrong mass factor ((#2}Av-- (l~)~v) (#)av n-2 , a first time when the double point of t h a t graph was x and a second time when it was y. Substracting these terms we obtain for both (xxyy) and (xyxy) graphs, the following mass factor" {qts>2 _ (~54 _ 2(qt25 -- <#>2) <#>2}<~>n-4 = = {(kt2)2 __ 2(/~2) ( # ) 2 _~_ ( / ~ ) 4 } ( ~ ) n - 4 :
(/]/~2)2 ( # ) n - 4
(53)
JEAN PIRENNE
86
Similarily, v(xxxx) contributions have been introduced: a) once in J/~) with the wrong factor <#>2,; b) six times in d / ~ ) with the wrong factorAv<#)~7~, i.e. each time the double point of the (xx) graph coincided with any of the six distinct pairs which can be selected from the four points x% x#, x~, x ~ actually taken to coincide in the present (xxxx) graph;
7 .......... ,,%,,__...._.----,' a
X,=,=X~
=Xc~)
Xc~')X ~
/J'y X(~
b X--°=X ~ X(~")-'~X
=X(~) C
Fig. 3.
c) four times in .~(3) with the wrong factorAv~-~3, when the triple point of the (xxx) graph coincided with any of the above considered four points. Substracting all these terms, we obtain for the (xxxx) term, the corrected mass factor: -- 4(Av
--~ < / ~ > ~ i {Av- -
=
--
3AvAv + 3AvQ~>2av --
~v)
• Av} =
4AvAv-~- 6Av2Xv-- 4Av3v-}- ~v}--~
~4.a,.
(54)
DYNAMICS OF A CRYSTAL LATTICE CONTAINING ISOTOPES
87
We now have to compute v(xxyy), v(xyxy) apd v(xxxx) b y summing the corresponding ~//'(x', . . . . x(n)), given b y (28a), over all possible (xxyy), (xyxy) and (xxxx) graphs having one fixed point which can always be taken as coinciding with the multiple point x. This operation will be however simplified in the same way as before by dropping any restriction on the summation-points. The additional contributions thereby introduced Will be substracted during the computation of higher order terms, except that particular contribution corresponding to x = y, which is to be substracted right now as it gives rise to an (xxxx) term with the wrong mass factor 3~v 4vn-4,
(55)
the factor 3 coming from the three situations pictured in fig. 3. CoLlecting these results, the fourth order term of ~ ' n finally writes:
~ ' ~ ' = <~>'~'~ {av -- 3)~ w(xxxx)}
(56)
with n
w(xxyy) =
•-i 1o-1 q--I X p = 3 ~-~,/=2 ~-'~r=l ~x',a.~',a'"
a'> .
•
w(xyxy) =
In
(57a)
n--1 I o - - I q--1 X~-3 Xq=2 Er=~ E~',.,.',." .
(57b)
~9--I Zr=1 q--1 Z x ' , a , a ' , a " . n--X Xq=2 w ( x x x x ) = ~ n ~-~/~=3
•
a" IVq-r I x, a'>
(57c)
The factor n correspond to the fact that, in the summation (28b), equal contributions are found if the multiple point x is any of the n points x', x", . . . . x(n); however, as the same graph would then be counted twice in the first case and four times in the others, the result must be divided b y 2 and 4 respectively *). We shall now carry on the calculations in the case of a cubic lattice. *) The factors n/2. n/4, n/4 may easily be checked by counting tile total ntunber of terms of the sums occurring in (57a, b, c), namely: ~'~)~--1 p = l ~ pq-=- 1l ~;'~r~--~ -- 1 = (n - - l ) ( n - 2)(n - - 3 ) / 6 . Therefore, the total numbers of contributing graphs are respectively equal to 2C, C, and C, with C = n(n -- l)(n -- 2)(,i -- 3)/24 = nll4!(n -- 4)! O n the other hand, there are just C different w a y s of chosing four distinct points from the n points x', x", . .., x(n) and it is clear from fig. 3 and 4 that each choice leads to two distrinct (xxyy) graphs, one (xyxy) and one (xxxx) graph. T h e total n u m b e r s of graphs are therefore respectively 2C, C
and C, as before.
88
JEAN PIRENNE
a) The (xxyy) contribution. As has been noticed before, is, by s y m m e t r y reason, the product of the unit matrix 6=a' by a const .a~lt (cf. equ. 23). Therefore, (57) becomes: zo(xxyy) = ½nX=,¢ E,=m ~-~ Z ~~-I = l
IV~-ql
x,
• (n - - p - -
a>
.
1) (58)
By using (23), we get: zo(xxyy)
1 1
,=-s
• ~o(a') ~o(~")
= ~nfff
(~' a/o~' -
= ~nfff
(;U alaa'
~oo(~") d X ~ " ~ "
=
])F.~(;v;~";~') ~o(;~') ~o(;~") ~o(;~") da' d~" d;~"
+ a" a/aa" + a" a m " - 3). • 2%~(~'~"~"
,o(~')
~o(~")
~o(~') d~'
d~" d~"
(sg)
where Fn8 is given by (47a). As this function is homogeneous, we have, by
a~X= X ~
X ,X
Fig. 4.
Euler theorem: (~' a/a~' + ~" ala~." + #." a/a~."~) Fn3 ----n~n3
(60)
Therefore:
~(xxyy) = A~(n - a ) f f f F , ~ ( Z ' ~ " ~ " ) ~o(~') ~o(Z") ~o(~') d~' d~" dZ" (61) This expression, when inserted in (56) brings to ~#~1 a contribution
~)
(xxyy) =
(~l~,,).w(xxyy)
(62)
which, introduced in (33b), gives the function"
~o~y(s) = Z°~=I ( ( i s ) n / n ! ) ~ ) ( x x y y )
(63a)
where the sum only starts at n = 4 as there is no ~ 1 (xxyy) for lower n. However, it m a y be performed, by convenience, from n = 1, as, by (47b)
DYNAMICS OF A CRYSTAL LATTICE CONTAINING ISOTOPES
89
a n d (61), w ( x x y y ) = 0 for n = 1, 2, 3, Therefore:, ,
~%(s) =
54
<~>L
;{"~"~""
[~'(is) ~'--2is] e~', (Y--~") + [~"(is) ~--2is] e ~ "' (~ "--M') + [~" (is) 2--2is] eta'' (;t'--;t") (~, _ ~-)(~- _ ~,,,)(~,,, _ ~,) • 9o(~t') 90(2") ~oo(~t') cUt' dg" d Jr'. (63b) Inserting this in (33b), we find the c o n t r i b u t i o n of the (xxyy) graphs to the f o u r t h order t e r m of the f r e q u e n c y distribution, n a m e l y :
¢~.,,(,,t.) = 27 < # > ~ (~' -
z')[~'(~"
-
"
~) -
~
~'i(~" -
(~,
_
~)]
~-)(~-
+ (~,,, _ ~,)[~,(~- _ ~} - ~(~(~' -
x")(x"
-
x')(~"
-
+ _
(z' -
~,,,)(~,,,
~")[~'(~'
-
~) -
~(~'
-
~)].+
~,)
_
_ ~}] ~')
• ~'z'z'.
vo(Z)
~o(r')
~o(~")
(64)
b) The (xyxy) contribution. The coefficient zo(xyxy), given b y (57b) m a y be simplified b y m e a n s of (2 l) ; we thus o b t a i n :
w ( x y x y ) ---- ¼n Yqk,4 C(/z', s . .,. k. . . , s ; /z", s ' , &IV, siV)
Fn4(~', ;t", ~t', 2IV) .
• N - 4 • X eiV(x'-~')+v'{x'-x)+k"(x-x'}+v(~-x)
(65a)
with . . . , s ; /~", s'~; kIV, sIV) = C ( h ' , s . .,&
= (e(k', s'). e(k", s")) (e(&", s"). e(~', s")) (e(&', s"). • e(/z Iv, sir)) (e(/z Iv, s Iv) . e(/z', s'))
(655)
F n 4 ( t ' , ~[", t " , ~.IV) = ~O~=3n-I~'~'q =2v-1 Zr=l~/-1 j~'n-l~ X"~-~/~'aq -'r XI v r ,. (65C)
where we h a v e written for simplicity )l' = 2o(k', s'); ;t" = 2o(/Z", s"); 2" = 2o(I¢", s"); 2IV ~ ~o(/zlV, sir) (65d) It m a y easily be shown that
n4(A, t ,
Physica XXIV
=
2'2"2"2Iv D4(;t', ~.', ~', ;tIv)
1
2'
2 '2
2'~
1
2"
2"~
2"n
1
~"
$'2
~"n
1
2Iv
2 IV2
2Iyn
(66)
90
PIRENNE
JEAN
On the other hand, owing to the well known completeness of an orthogo: system of plane waves subjected to Born-Karman periodicity conditio the x'-summation of (65a) simply gives: N - 1 Z ei(~'-k"+k . . . .
~'~)(~-~')
= ~(k' + k", k" + k Iv)
(
Introducing (66) and (67) in (65) and replacing further the k-summati, b y integrations, we get:
w(xyxy)
=
~n (Vo/(2~-~)3)zf f f C(k' s', k" s", k"s', klVs Iv) . • F n 4 ( a , ' ~',, ~,,, ~iv)
(dk')3(dk")3 (dk,,,)3 (
'where
vo = V / N is the atomic volume and k Iv = k' + k" -- k"q Following (56), this term brings to ~g~) the contribution
which, by (33b),gives the function (4} ~Oxxyy(S) ~"
Zn=l
((is)n~ n!) "41~ ) (xxyy),
where the summation m a y again formally be started at n = 1 as, b y ( (68) and (71), w(xyxy) = 0 for n = 1, 2, 3. This function is immedia obtained b y replacing in (71) the factors A,n, Un, T"n ~zvn arising f: (66), by
eiSl~, e~SA't, e~s~Ut, eis~IV and the factor n b y is. Inserting the result in (33b), we obtain, for (xyxy)-contribution to the fourth order term of the frequency distribut
~,,
(4) -
4
<~>~
\ 8~a /
"
• C(k',s';k"s";k's" ;klVs Iv) ~'4k^ ,~ ,A ,A , ~ TUT.2I v (dk,)a(dk.)3(dk.,) 3 4~A ,A ,~. jA
]
with k Iv given b y (70). c) The (xxxx)-contribution. Transforming (57c) b y means of (23), we
Zg)(XX.XX)
=
¼n . ~7 " F n 4 ( 2'' ha' 2'tt' 2IV)
which, b y (56), contributes to J d ~ ) the term:
./.t~)(XXXX) = ( ( A # 4 > _ 3(A#2>2) . w(xxxx)/Q04
DYNAMICS OF A CRYSTAL L E T T I C E C O N T A I N I N G ISOTOPES
Using again (33a and b) as before, we find, for tim ( x x x x ) - c o n t r i b u t i o n the frequency distribution 1
9! to
v f f f A4(Z, Z', a", alV; a)
q)'~=''" (A) = 10----8
<#>~v
• A'A"A"A
D4(it', ~", ~", ~11v)
TM
~oo(A')9o(A")9o(,V')9o(AIV).clA'd~" dA" ciaw (76)
Conclusions. B y means of a m o m e n t method, rather different from Montroll's one, and b y the use of a Fourier transformation, we have been able to derive a formal series development of the frequency distribution of the mixed crystal. If we write the deviations of the inverse mass #~ of the i-th isotope with respect to its average <#>Av under the form A#,=#i--<#>Av---pKf, where p is a parameter characterising the degree of heterogeneity of the crystal, our series is nothing but the development of the frequency distribution following the increasing powers of p. The first term represents the frequency distribution of a fictitious pure crystal whose atoms would have a mass equal to ~. The following terms involve the standard deviationsAv of higher and higher order as one proceeds further in the development. The first three of these following terms have been explicitely computed and there is no difficulty to push further although the expressions of higher order terms become increasingly more complicated. Each term from the second one appears as a linear functional f S ' ( l -- 4') K(~') d~' of the derivative of Dirac's delta function, t being the square of the angular frequency. K(,I) is expressed in terms of the solutions of the classical eigen-vibration problem presented b y the above mentioned fictitious pure crystal. In general, the polarisations and frequencies must have been determined for each wave vector, but in the case of cubic crystals, the knowledge of the frequency distribution alone is required to obtain the second and third order term. The application of this development to the derivation of any thermodynamic function of the randomly mixed crystal does not seem to raise any difficulty, the 6' function being simply replaced b y the k-derivative of the corresponding thermodynamic function of the harmonic oscillator. The consistent introduction of the entropy of mixing seems however to deserve some more detailed consideration and it would be also desirable to see in what domain of temperature physical effects are to be expected. Concerning the frequency distribution itself, several points still remain t o be investigated: the behaviour of the Van Hove's singularities when we pass from the single isotope to the several isotopes crystal, the numerical
92
DYNAMICS OF A CRYSTAL LATI'ICE CONTAINING ISOTOPES
evaluation of our terms, which contain several singularities (6' functions and poles), the convergence of our development. We intend to discuss these various points in a following paper.
Acknowledgements. I wish to express here m y deepest gratitude to professor R. J. O p p e n h e i m e r for his hospitality and support at the Institute for advanced Study, Princeton, N. J., where most of this work was completed during the winter 1954-55. I would like also to pay here a last tribute to the m e m o r y of the late G e o r g e P l a c z e k who inspired me the subject of this paper and followed these developments with his invaluable criticism and his unforgetable kindness. Received 17-9-57 REFERENCES I) 2) 3) 4) 5) 6)
P r i g o g i n e , I., B i n g e n , R. et J e e n e r , J., Physica 20 (1954) 383. P r i g o g i n e , I. et J e e n e r , J., Physica 20 (1954) 516. Dyson, F. J., Phys. Rev. 92 (1953) 1331. B e l l m a n , R., Phys. Rev. 1 0 1 (1955) 19. Des C o i z e a u x , J., J. Phys. Radium, 18 (1957) 131. M o n t r o l l , E. W., J. chem. Phys. 11 (1943) 481; 12 (1944) 98.
P)lysica X X l V 73-92
DYNAMICS OF A CRYSTAL LATTICE CONTAINING ISOTOPES by
JEAN
PIRENNE
The Institute for advanced Study, Princeton, N. J. and Universit~ de Liege, Belgique Synopsis B y means of a m o m e n t method, t h e v i b r a t i o n s p e c t r u m of a m o n o a t o m i c crystal c o n t a i n i n g several isotopes, whose a t o m s are distributed at r a n d o m t h r o u g h o u t the lattice points, is o b t a i n e d in t e r m s of the solution of t h e conventional v i b r a t i o n p r o b l e m of a singIe isotope crystal and in terms of the quantities
where c, is t h e c o n c e n t r a t i o n of the i - t h isotope and A/~l the deviation of its inverse a t o m i c mass/~f from t h e average value (/*)av = ~ , ci/~f. T h e f r e q u e n c y distribution is represented b y a series d e v e l o p m e n t following t h e increasing powers of a h e t e r o g e n e i t y p a r a m e t e r to which all the A/~ are assumed to be proportional. The first t e r m is the f r e q u e n c y distribution of a fictious crystal whose a t o m s would all h a v e the mass ~1. The n e x t two t e r m s are proportional to
Introduction. Let us consider a crystal formed b y a single chemical element possessing several isotopes and let us assume these isotopes to be distributed at random at the various lattice points. Then arises the problem how to compute the frequency distribution or spectrum from which all the thermodynamical properties of this crystal m a y be immediately derived. The solution of the eigen-vibration problem is elementary and well known when all the masses are equal, but here we wish to take account explicitly of the fact that different isotopes have different masses, although the forces remain the same. In other words, the kinetic energy of the crystal is no more invariant under the operations of the symmetry group of the crystal, but the potential energy still is. This problem was suggested to us by G. P l a c z e k in connection with --
Physica XXIV
73
--
74
JEAN
PIRENNE
the scattering of neutrons or X-rays b y such a crystal when one wishes to investigate the influence of the mass differences between the isotopes which have always been neglected hitherto. It turns out however that the well known relations between scattering cross section, correlation functions and frequency spectrum are destroyed b y the introduction of mass differences. The scattering problem is therefore quite distinct from the present one the solution of which m a y however clarify the situation. During the early development of this work, two papers of I. P r i g o g i n e and collaborators Eli, [2] appeared in which the actual problem was considered from the point of view of t h e p e r t u r b a t i o n theory. In fact, the main object of these papers was not to compute the frequency spectrum itself but rather to obtain an integral value representing the zero-point energy of the crystal. It is however much simpler to derive the sum of the perturbations of all the eigenvalues than the eigenvalues themselves. The first quantity m a y be deduced from the trace of the second order perturbation operator, which can be evaluated in any basis, in particular in the conventional one defined by the usual system of plane waves. On the contrary, the computation of the individual perturbations requires the explicit construction of a system of unperturbed waves adapted to the perturbation, a much more complicated problem, at least in the three-dimensional case. It is therefore not yet clear whether the perturbation method would easily lead to the derivation of the frequency spectrum itself, which, anyway, is not indicated in these papers. For the linear chain, D y s o n E3] has developed a rigorous method of deriving the frequency spectrum; its calculations have been somewhat simplified b y B e l l m a n E4] and, more recently, another presentation has been given by D e s C l o i z e a u x E51, w h o introduces the concept of phase shift. All these calculations only apply to the particular case where the interatomic interaction is confined to pairs of nearest neighbours only. As far as we know, it has not been possible either to lift this restriction or to extend the theory to the three-dimensional case. • In the present paper, we use an apparently easier method based on the computation of the moments of the frequency spectrum b y means of traces of certain matrices. A somewhat similar starting point has been used b y M on t r o l l E61 to derive the frequency spectrum of a lattice without isotopes, b u t here the aim and the treatment are entirely different. We assume indeeed that the frequencies and polarisations have already been computed for each wave vector for the crystal without isotopes, and that the frequency distribution of the latter has also been obtained; no special assumption is therefore necessary concerning the potential energy which m a y be quite general. These results are then used to express the frequency distribution of the crystal with isotopes in terms of the moments of the spectrum of the inverse masses, #~ = m~-1 or more precisely in terms of the moments of the deviations A#, = t L, -- Av.
DYNAMICS OF A CRYSTAL LATTICE C O N T A I N I N G ISOTOPES
75
Definitions. Let us represent by ml < m2 < . s..
(1)
the first sommation being carried over all lattice points x' and x'. Introducing te hyper-vector u = (u,~ 1, u ~ . . . . . .
)
(2)
in a 3N-dinensional linear space, W can be written in the more condensed matrix form : w --k(u,
(3)
Vu)
where (A, B) denotes the scalar product of the hyper-vectors A and B. The matrix element
x')
(4)
The equation of motion of the system m a y be written: Mii = -- Vu
(5)
(6)
with The characteristic (angular) frequencies wj and the corresponding eigenamplitudes uj then satisfy the equation: Vuj : 2jMuj
(7)
aj = ~ .
(8)
with Our problem is to determine the number of proper vibrations Ng(to) do) = N9(2) d2
(9)
for which to ~ toj ~ t o + d t o , ,~ ~ 2 j ~ 2 + d ; t As will be later seen, the frequency distribution g(to), or the function 9(2), becomes independant of N for an infinitely large crystal, provided the
76
JEAN
PIRENNE
isotopes are distributed at random so that their presence probabilities at the various lattice points are independent of each other *).
Limits o/ the [requency spectrum. From (7), we derive ]tj = (uj, Vuj)/(uj, Mul)
(10)
Suppose now, for simplicity, that ul is normalised to unity, then: (UJ, V u ] ) ~
-max](1) . m l
-m&x~ll21 .
=
m2 . . . . . .
(11)
where A~ax is the square of the highest frequency of a crystal containing the l isotope only. Fl~rther
(uj, Muj)
(12)
Therefore, In other words, the spectrum of a mixed crystal is extended no more than the spectrum of the pure crystal containing the lightest isotope only.
Normal coordilmtes.
The eigenamplitudes are solutions of the equation =
M-1
juj
(I 4)
leading to a familiar secular equation. However, as the matrix M - I V is not symmetric, they are not orthogonal. We introduce therefore the normal coordinates ~j = M1/2uj (15) which are solution of the equation H~j = ~j~j
(16)
H = M-112VM -1/9"
(17)
where As H is a symmetrical matrix, the vectors ~j are orthogonal and we shall normalise them to unity: =
(18)
The unperturbed problem. In the particular case of a single isotope of mass m, equation (17) reduces to: m_lV~O)= 21.0,~o) (19) *) T h i s b e c o m e s o n l y t r u e w h e n N - + co. S u p p o s e it w e r e t r u e for a fixed v a l u e of N a n d d e n o t e b y Pl, Pa,...Pr t h e a p r i o r i i n d e p e n d a n t p r o b a b i l i t i e s of f i n d i n g i s o t o p e s 1, 2, . . . , r a t an), l a t t i c e p o i n t . T h e n , t h e t o t a l n u m b e r of a t o m s of t h e v a r i o u s i s o t o p e s w o u l d n o m o r e b e well d e t e r m i n e d , b u t t h e i r a v e r a g e c o n c e n t r a t i o n s w o u l d be e q u a l to Pt, . . . , Pr a n d the s t a n d a r d d e v i a t i o n s f r o m t h e s e a v e r a g e c o n c e n t r a t i o n w o u l d t e n d to zero as N - ½ . T h e r e f o r e , for N -+ oo. t h e r e is no d i f f e r e n c e w h e t h e r t h e c o n c e n t r a t i o n s are a priori fixed or w h e t h e r t h e p r o b a b i l i t i e s Pl, po . . . . , Pr are independant.
DYNAMICS OF A CRYSTAL LATTICE C O N T A I N I N G ISOTOPES
77
whose solutions are
•101 k,$ ;x =
e (k, s)
ei~'/v'~
(2o~)
or, in Dirac notations,
a[k, s> = e~(k, s) . eikx/v'~
(2Oh)
where N is the number of atoms of the crystal and e(k, s), the polarisation vector whose index s runs from 1 to 3 *) . k is subjected to the usual BornK~rman conditions. We assume that this "unperturbed" problem has been completely solved; in other words e(k, s) and 20(k, s) are well known functions of the wave vector k and the polarisation index s. We shall further assume that the number of eigenvibrations N.90(2 ) d l , for which 2 E 2j E 2 + d2, has also been computed. Our task now is to derive the distribution function ~0(2) for the general case of several isotopes distributed at random, which we shall refer to as the perturbed problem. It is however not our intention to consider the mass differences as a small perturbation and to use standard perturbation theory, but rather to derive 9(2) from its moments for arbitrary mass differences. It turns out however that our result appears as a series whose terms are easily expressible by means of the solutions of the unperturbed problem with an appropriate fictious mass. For cubic crystals, the three first terms depend on ~oo(2) only, but for the computation of the next ones, the knowledge of the polarisation is also necessary. For other crystals, the same is already true from the second term. For the following, it will be useful to express the matrix element of any power of the potential matrix in terms of its eigenvalue as follows (x, a
IVpl x', a'> = 2Ek (x, a Is, k>
= FAe,,(s, k) ,~(s, k) e~,,(s, k).~'*'-'~lN
(21)
(21)
On the otherthand, by taking the trace of Vao we find: N-1
• m-
1k'~Y'x,a r (x, a [V~[ x, a) : N-1 . m-1. Trace V h = f 2~ 90(2) d2 (22)
or, as Vv is invariant under the translations of the lattice, m -1. ~ a (x, a IVY[ x, a) = f 2 ~ ° 90(2) d2
(22b)
Further, as (x, a IV[ x, a'> is also invariant under the symmetries of the lattice, it is necessarily diagonal with respect to a, a'. For cubic crystals, it follows therefore that m -1. (x, a IVY[ x, a') = ~5~.,f2~oo(2) d2
(23)
*) For simplicity we consider only Bravais lattices, such as the simple cubic, face-centred and body-centred cubic lattices.
78
JEAN PIRENNE
T h e / r e q u e n c y d i s t r i b u t i o n / o r the perturbed problem. The n u m b e r of eigenvibrations 2~ of the crystal whose 4j lie in in the interval
dR = 2o~ dw
(24a)
Ng(o~) d¢o = N9(4 ) dR = ~{d~} (2t, 2~)
(24b)
m a y be expressed as
where the s u m m a t i o n extends over all eigenvectors of the interval (24a); (U, ~") is the scalar product of the hyper-vectors 2' and 2". Now, the norm and the various m o m e n t s of the frequency distribution (or r a t h e r of the square angular frequency density) 9(4) m a y be written in similar w a y as: dr'0 = f0~ 9(4) dR = N -1. ]~j (~j, 2t)
= N -1. Trace 1 = 3
J t ' l = f0~= 9(4)4 dR = N -1 . ~ j (~j;H2j) = N -1. Trace H .
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
,
(25) .
.At'n = f ~ 9(4)t n dR = N -1 . Y,j (21,Hn2j) = N -1. Trace H n If we can evaluate these moments, we deduce from t h e m the Fourier transform of 9(),), ~p(s)----f0~= 9(4)e ~s~ dR = N - 1 . T r a c e ei*H = Y~°=0 ((is)n/n!).'¢[n,
(26a)
and finally, the distribution 9(4) itself: 9(4) = (1/2=) f+o~ ~0(s) e-~SAds.
(26b)
I t m a y be mentioned here t h a t the series (26a) is absolutely and uniformly convergent as H is bounded. T h e m o m e n t s o/ the /requency distribution.
B y introducing the inverse
mass
#(x)
=
m-l(x),
(27)
the m o m e n t ~ ' n is explicitely given by J t ' n = N -1. Trace ( M - l / 2 V M - 1 / 2 ) n = N -1. Trace (M -1 V) n =
= N - 1 . Z{~,~}/~(x')
= N -1. 2~{~}#(x') #(x") . . . . #(x(n)) . $/" (x', x", . . . . x(n))
(28a)
with
~"(x', x", . . . , x('O) = Z{~,
x', a'> (28b)
where the subscript {x} indicates a s u m m a t i o n over all lattice points, {a}
DYNAMICS OF A CRYSTAL LATTICE C O N T A I N I N G ISOTOPES
79
a s u m m a t i o n over all indices a', a", . . . . and (x, a} a s u m m a t i o n over both lattice points and a-indices. To visualise these summations, it will be useful to represent a set of points x', x", . . . . x(n) b y a closed graph obtained b y joining b y a line each point to the following one and x~m to x ' . It m a y happen, of course, t h a t the various points of the sequence are not all distinct one from each other; then, if r of them coincide, the graph will pass r times through the same point, which will therefore be a multiple point with multiplicity r. R e t u r n i n g now to (28b), we see t h a t the function $/" (x', x", . . . . x(n)) a t t a c h e d to a graph is invariant under the translations of the graph. Therefore, in (28b), the product # ( x ' ) . # ( x " ) . . . # ( x ( m m a y be replaced b y its average over all translations keeping the graph inside the crystal. As the n u m b e r of these translations is practically equal to the number of atoms, if the crystal is sufficiently large, the s u m m a t i o n over t h e m m a y be dropped, if we simultaneously leave out the factor N-1. It is further clear that, for an infinite crystal and a r a n d o m distribution of isotopes, the above average of the inverse mass product only depends of the multiplicities rl, r2 . . . . . r v of the p different points Xl, x2 . . . . . x~ which constitute the graph, but is otherwise independent of its shape: <#(x') #(x')
...
#,(x(n)>Av =
(#r'>Av . (#r2)Av . . . . . .
(#~'>Av
(29)
Therefore J C n ---- ~Irl (#rl)Av • (#~2)Av . . . . . . .
(/z~)Av . Vr2...r ~
(30a)
with vn...,., = Xla} v(G),
v(G) = Z{:~} ~ ( x ' ,
x", . . . . x(n))
(30b, c)
In (30a), the s u m m a t i o n is carried out over all values of rl, rs . . . . . rv compatible with rl + r2 + . . . + r v = n and p = 1, 2, 3 . . . . , n. In (30b) the s u m m a t i o n is carried out over all possible graphs having p distinct points of multiplicity rl, rz . . . . . . rp (subscript G) and, in (30c), it is extended, for each particular graph G, over all possible positions of p -- 1 of its points, one of t h e m being kept fixed as the translations of the graphs are now excluded. Let us briefly note here t h a t the m o m e n t s (#r)A v of the inverse mass spectrum are immediately expressible in terms of the inverse masses and concentrations of the various isotopes b y the s t a n d a r d formula: (#r)Av = ~ c i#ir
(31a)
L a t e r on we shall also need the r-st order average deviations (A#~>a~ = X c,(#, - (#)A~) ~
(31 b)
D e c o m p o s i t i o n o / t h e m o m e n t s a n d new expression o / t h e / r e q u e n c y distribution. As will be later seen, the c o m p u t a t i o n of Vrl. ..rp b y means of (28b)
B0
JEAN PIRENNE
is greatly simplified by dropping the above mentioned condition t h a t xl, x~, . . . , x~ be always different from each other. B y doing so, however, we add terms truly corresponding to other graphs having points of higher multiplicity, but we count them with a wrong mass coefficient (29). Therefore, when we shall come to these other graphs, we shall have to substract from the corresponding exact term, with its exact mass coefficient, the previously added term with its wrong mass coefficient. To apply this method, the various graphs must appear in such an order t h a t no such correction should affect a term which has not yet been met. As will be seen, this requires the graphs to be classed be order of increasing value of the number r = rl + r2 + . . . r~, which we shall call the order of the graph. All graphs of the same order should further be considered by order of increasing value of the highest of the multiplicities rl, r2 . . . . . r~. Each moment will then appear as a sum:
~n
= ~oo.~,~,)
(32)
We shall then build the functions
~or(s) = Y,~°=0 ( (is) n/n !) Md(~)
(33a)
9r(~) = l[2~r f + ~ ~or(s) e~8~ ds
(33b)
and we finally obtain the frequency distribution 9(~) as a new series 9(~) = X~'=1 9r(}t)
(34)
The r-st term of this series will be refered to in the following as the r-st order term. It arises from all graphs of multiplicity r and it is related with the average r-st order deviation
First order term. Ler us first consider a graph without any multiple point. According to (30), the contribution of all such graphs to the moment Mt'n is obtained b y summing up the corresponding q/'~(x', x", . . . . x(n)), given by (28b), over all possible positions of n - 1 of these points under the only restriction that x', x", . . . . x(n) be all different one from each other. If, however, we drop this restriction we add new terms that should be later substracted, but the summation (28b) greatly simplifies as it now reduces to a simple matrix product. Inserting the result so obtained in (30) and taking account of (22b), we find the following contribution to ~ ' n : ~ ' ~ ' = (#>3~ Z a (x, a IV~l x, a> = f o ~" 2 n 90(2) d2,
(35)
where 9o(2) is unperturbed the frequency distribution for a fictitious inverse mass #* = <#>Av. Introducing (35) in (33a) and using then (33b) we obtain the first term of
DYNAMICS OF A CRYSTAL LATTICE CONTAINING ISOTOPES
8|
the series (34) representing the perturbed frequency distribution: y,l(S) ---- Z~'=o ((is)hi n!) "£~) = fo~" e~sa ¢o0.) d).
(36a)
fol(2) = ( I / 2 u ) f + ~ 91(s) e -iSa ds = 90(2).
(366)
This result coincides with that of the perturbation method used by P r i g o g i n e et al. ; however, the fictitious mass used by these authors was the average mass (m)Av whereas we are dealing here with a fictitious mass ~q¢* = (m - 1 )av. - 1 Both methods lead therefore to different results; in fact, the two series representing ~(~l) differ from the second term already. Second order term. We now have to consider graphs containing multiple points. If we want to represent such a graph on a sheet of paper, it will immediately be realised that a mere polygonal ]ine does not provide a complete description. In fact, the label of each point should be indicated, or at least their succession order, as any circular permutation of x', . . . . . x ("~ does not change the value of the corresponding term of (28). The orientation of each loop should therefore be marked in some way, for instance by an arrow.
A perhaps simpler way of drawing such graphs is however to represent the sequence x', . . . x(n) on the sheet of paper by a set of n distinct points connected in their prescribed order by a full fine and to indicate those points, which are indeed identical in the crystal, by connecting them by dotted lines• Examples will be found in fig. 1, 2, 3, 4.
X~.j )~ X'P'"g \
X'
X" iX"' I = x'"
........
Xc~"~ ~
^
Fig. 1.
A simple notation for each type of graph is easily obtained by writing down only those points of the sequence which are multiple and by denoting each of them by a separate letter; for instance, the graphs represented in fig. 1 to 4 will be denoted by (xx), ( x x x ) , ( x x y y ) , (xyxy) and ( x x x x ) . Let us now consider the simplest graph containing multiple points, namely (xx), which has but one double point (fig. 1). Its contribution to ~ ' n is, according to (30), ( / t 2 ) A v • /\ / 2 /~'~-~ Av
. v(xx)
(37a)
82
JEAN
PIRENNE
where v(xx) is given b y (30c). Let us remenber, however, that, in computing the first order term ~/t'C1) we carried out an unrestricted x-summation, introducing thereby additional contributions which, in fact, corresponded to more complicated graphs b u t were nevertheless counted with the wrong factor <#>],. In particular, the added contribution, relative to (xx) graphs, was
•
<#>~v. v(xx).
(37b)
Substracting now this from (37a), we obtain the total contribution: (<~2>A~
(~o~)
• <~>~72 •
v(xx)
(37c)
Now, to compute v(xx), we have again to sum up ~F'(x', x", . . . . x(n)), given b y (28b), over all possible positions of all points but one, which can always be taken as the double point. As this summation should correspond to (xx) graphs only, the summation points a n d the double point should always be kept distinct one from each other, but, as we have already done before, we shall again drop this restriction, introducing thereby additional terms which correspond to more complicated graphs that will later have to be substracted. The summation however greatly simplifies and reduces to simple matrix products and we obtain in this w a y the second order term
"£~) = (<#2>Av -- (tt>]v) • <#>Av'~-2-W(XX)
(38a)
w(xx) == ½,, Z~=I .-1 Za, a'
(38b)
with : (38b) is obtained as follows. Suppose first that the double point x is x'; then, b y unrestricted summation over x", . . . , x(n), we immediately find the above term w(xx), b u t without the factor n/2. The double point, however, m a y also be x", . . . . x(m, and, after eventually changing the name of the variables, we arrive to the same result in each case. Thus, for any fixed value p' of p, there are n equivalent graphs leading to the same term, but in summing up over p, we find again these same graphs for p = n -- p'. Therefrom follows the factor n/2. We shall now compute this sum for a cubic crystal. Using (23), we find:
...£~, =
-
<~>5-----~
n
1 ~ ; ~ " ~ ;~"
Y
3
a0 a0 X $ - I 2 'n-~R'~00(R')90(U)d2'dR" (39a)
6 ~ o ~,
a, _
a.
a'z" ~o(z') ~o(r') da' d~"
(39b)
Introducing this expression in (33a) and noting that the contribution M/(~), being related to (xx) graphs, only starts from n = 2, we get: v,2(s) = Z --
((is),,/n ! ) . . ~ )
1
6
<#>~
. (is)
(4o~)
|
fa" F~= e*S;~'--etS;~" .l';t"
dO
|
JO
2 , _ 2,,
~o(~t') ¢o(;(')d).' dit" (40b)
DYNAMICS OF A CRYSTAL LATTICE CONTAINING ISOTOPES
83
In (40a), it has been possible to sum from n = t, instead of n = 2 as was announced, because expression (39b) vanishes identically for n = 1. Inserting this in (33b), we obtain, for the second order term of the frequency distribution,
1 (A/z2)Av 1 [.a=f~,,, eiS(~'-~)_e~SC~"-~) Jo (is) 2' -- 2# 2'Y'~o(~t')9o(2")d2'd2"ds(41a)
92(2)-- 6 Qe2)Av ~ J o or, finally,
I
-- 2) -- S(Y'
2'--2"
-- 2)
2'2"9o(2')qoo(2")d2'd2" (42a)
Applications of this formula will be later discussed. For non cubic crystals, we would obtain in a similar way, b y using (21b) instead of (23):
92(2) - 2,
<~>~
Z,.,.
ff
20(k', s') - 20(k", s") -
-
s,,)
v0(dk') 3 • 2o(k', s') 20 (k", s") (e(k', s') . e(k", s")) 2 . - -
(2~)3
-
vo(dk")3 (42b) (2~)3
where vo is the atomic volume, i.e. the crystal volume per atom.
X'=~X
~
X=X c~
Fig. 2.
Third order term. We now turn to the next simple graph after (xx), namely xxx), which has one triple point only (fig. 2). It brings to the m o m e n t ~ ' n the contribution Q*3>Av • (#) Av'~-3 . v(xxx),
(43a)
where v(xxx) is given b y (30b). But we must substract from this the contribution (#)'~v" v(xxx), (43b) introduced in the second order term b y dropping any restriction on the x-summation, and three times the contribution
(([.12)A v
-
-
(#t)~v) (,")Av n--3 v(xxx)
(43c)
84
JEAN PIRENNE
introduced in the second order t e r m ~'~), a first time when the double point x was x ' ---- x(~ +1) a n d we s u m m e d over x(~+q+z) w i t h o u t omitting x(~+e +1) ---= x', a second time, when the double point x was x(~+l) = x(~+q+l) and we failed to o m i t this point in the x ' - s u m m a t i o n , a third time when x was x(~+a+l) __= x' and we did not omit this point. Substracting therefore (43b) and three times (43c) from (43a), we find the following contribution to ~ ' n : {
<~>a
_ 3<~>A~ (<,,2>~
_
<~>~)}
<~O~3. =
v(xxx)
Av~ a .
=
v(xxx).
(44)
v ( x x x ) is obtained b y s u m m a t i o n of ¢/'(x', . . . . x(n)), given b y (28b), over all possible ( x x x ) graphs having a fixed point which m a y always be choosen to coincide with the triple point. If we however drop the restriction t h a t the s u m m a t i o n points and x be all distinct, as we have already done before, the s u n m m t i o n again simplifies and we obtain in this w a y the third order term of ~ ' n : ~tt~) = (Atza>A v . <#>~ga . w(xxx) (45a) w(xxx) =
with
--½ _
~
~-I
~-I Z~=~Z~=IZ~,,,,~,~,,~,,
The factors would give n counted three L e t us now
~n appears because the triple point m a y be x', . . . . x(n) which equal contributions, b u t in fact, each graph woald t h e n be times. evaluate w ( x x x ) for cubic crystals. Using again (23), we find
w ( x x x ) ---- (n/3)l
=-
it'a"it"
it'it"it", Ds(a',
~", a')
1
2'
it, n - z
1
2"
it" n-Z
l
2"
it"n-i
(47a) (47b)
~'.-~ (it" - it") + a " . - * (it" - a') + z ' . - ~ (it' -
where
Dn(it', it", . . . . it(n)) =
it")(it" -
it")(it" -
(it' - 2'3
it')
I
2'
it'2
. . . . it' n-1
1
it"
it"2
. . . . it"n-1
1
2(m
it(n) 9' . . . . 2{n)n-x
(47c)
(48)
is the V a n d e r M o n d e determinant. If we insert (47b or c) into (46) and the latter into (45a), t h e n we can easily build the series
~oz (s) = ~'~=1 ((is)hi n .J~~'~nu(a)
(49a)
DYNAMICS OF A CRYSTAL LATTICE CONTAINING ISOTOPES
85
The s u m m a t i o n should start at n = 3, beca~lse there cannot be a n y triple point before this; b u t it can nevertheless be performed from n = 1, because expression (48), and therefore also (47), (46) and (45a), identically vanish for n = 1, 2. We therefore have: 1 J.' d "s)"
~ps(s) --
-,,--V777,,,-,,, 1 a" e*ha" 9o(2')9o(2") q~o(k")dR'da"d2" (49b) (#>a 27333 Da(2 ,a ,~ ) 1 2 " e ~sa''
Inserting this in (33b), and introducing the notation 1
An(.a', ,V, . . . . ~(n); ~) =
~'
1 .
.
l
~t'2
~" .
.
.
.
.
....
~"~ .
).(n)
.
.
.
.
.
,t'n-2
~(~' - - ,l)
. . . . ~"~-~ ,
.
.
.
.
.
.
)(n)2 . . . .
.
.
.
.
.
~(~" -- ,l) .
.
* *
.
.
.
.
,
, (5o) •
).(n)n-2 (~(2(n) __ ~)
we find, for the third order term of the frequency distribution:
1 (Att3)A v [[(~,~,,j.,,, As(J.',2",,Tt";2) ,]J.] D3(2',2",,t") 9o(it')~oo(Z")9o(2")dk'd2"d,l" (51a)
(p(~t) = 27 ~
27 <#>av Jd,J
(a --2 )(X --,~ )(3. --,~ ) • ,o(a') ~o(Z') vo(a") da' d~" da" (Sib)
Fourth order term. After (xxx), the next simple graphs are (xxyy) and (xyxy), having both two distinct points (fig. 3a, b and 3c), and (xxxx) having one quadruple point only (fig. 4). T h e y bring to dgn the following contributions: gt2) ~ gt)~-~l . v(xxyy) (52a)
q,~>]~ <~>~;~. v(xyxy)
(52b)
~;*. v(xxxx)
" (52c)
where the v-factors are again given b y (30b). B u t v(xxyy) and v(xyxy) contributions were already introduced, once in a(1) with the " w r o n g " factor mass factor (/~5]~, and twice in Me'cs), with the wrong mass factor ((#2}Av-- (l~)~v) (#)av n-2 , a first time when the double point of t h a t graph was x and a second time when it was y. Substracting these terms we obtain for both (xxyy) and (xyxy) graphs, the following mass factor" {qts>2 _ (~54 _ 2(qt25 -- <#>2) <#>2}<~>n-4 = = {(kt2)2 __ 2(/~2) ( # ) 2 _~_ ( / ~ ) 4 } ( ~ ) n - 4 :
(/]/~2)2 ( # ) n - 4
(53)
JEAN PIRENNE
86
Similarily, v(xxxx) contributions have been introduced: a) once in J/~) with the wrong factor <#>2,; b) six times in d / ~ ) with the wrong factor
7 .......... ,,%,,__...._.----,' a
X,=,=X~
=Xc~)
Xc~')X ~
/J'y X(~
b X--°=X ~ X(~")-'~X
=X(~) C
Fig. 3.
c) four times in .~(3) with the wrong factor
--~ < / ~ > ~ i {Av- -
=
--
3AvAv + 3
~v)
• Av} =
4AvAv-~- 6Av
~4.
(54)
DYNAMICS OF A CRYSTAL LATTICE CONTAINING ISOTOPES
87
We now have to compute v(xxyy), v(xyxy) apd v(xxxx) b y summing the corresponding ~//'(x', . . . . x(n)), given b y (28a), over all possible (xxyy), (xyxy) and (xxxx) graphs having one fixed point which can always be taken as coinciding with the multiple point x. This operation will be however simplified in the same way as before by dropping any restriction on the summation-points. The additional contributions thereby introduced Will be substracted during the computation of higher order terms, except that particular contribution corresponding to x = y, which is to be substracted right now as it gives rise to an (xxxx) term with the wrong mass factor 3
(55)
the factor 3 coming from the three situations pictured in fig. 3. CoLlecting these results, the fourth order term of ~ ' n finally writes:
~ ' ~ ' = <~>'~'~ {av -- 3
(56)
with n
w(xxyy) =
•-i 1o-1 q--I X p = 3 ~-~,/=2 ~-'~r=l ~x',a.~',a'"
a'> .
•
w(xyxy) =
In
(57a)
n--1 I o - - I q--1 X~-3 Xq=2 Er=~ E~',.,.',."
(57b)
~9--I Zr=1 q--1 Z x ' , a , a ' , a "
•
a" IVq-r I x, a'>
(57c)
The factor n correspond to the fact that, in the summation (28b), equal contributions are found if the multiple point x is any of the n points x', x", . . . . x(n); however, as the same graph would then be counted twice in the first case and four times in the others, the result must be divided b y 2 and 4 respectively *). We shall now carry on the calculations in the case of a cubic lattice. *) The factors n/2. n/4, n/4 may easily be checked by counting tile total ntunber of terms of the sums occurring in (57a, b, c), namely: ~'~)~--1 p = l ~ pq-=- 1l ~;'~r~--~ -- 1 = (n - - l ) ( n - 2)(n - - 3 ) / 6 . Therefore, the total numbers of contributing graphs are respectively equal to 2C, C, and C, with C = n(n -- l)(n -- 2)(,i -- 3)/24 = nll4!(n -- 4)! O n the other hand, there are just C different w a y s of chosing four distinct points from the n points x', x", . .., x(n) and it is clear from fig. 3 and 4 that each choice leads to two distrinct (xxyy) graphs, one (xyxy) and one (xxxx) graph. T h e total n u m b e r s of graphs are therefore respectively 2C, C
and C, as before.
88
JEAN PIRENNE
a) The (xxyy) contribution. As has been noticed before,
IV~-ql
x,
•
a>
.
1) (58)
By using (23), we get: zo(xxyy)
1 1
,=-s
• ~o(a') ~o(~")
= ~nfff
(~' a/o~' -
= ~nfff
(;U alaa'
~oo(~") d X ~ " ~ "
=
])F.~(;v;~";~') ~o(;~') ~o(;~") ~o(;~") da' d~" d;~"
+ a" a/aa" + a" a m " - 3). • 2%~(~'~"~"
,o(~')
~o(~")
~o(~') d~'
d~" d~"
(sg)
where Fn8 is given by (47a). As this function is homogeneous, we have, by
a~X= X ~
X ,X
Fig. 4.
Euler theorem: (~' a/a~' + ~" ala~." + #." a/a~."~) Fn3 ----n~n3
(60)
Therefore:
~(xxyy) = A~(n - a ) f f f F , ~ ( Z ' ~ " ~ " ) ~o(~') ~o(Z") ~o(~') d~' d~" dZ" (61) This expression, when inserted in (56) brings to ~#~1 a contribution
~)
(xxyy) =
(
(62)
which, introduced in (33b), gives the function"
~o~y(s) = Z°~=I ( ( i s ) n / n ! ) ~ ) ( x x y y )
(63a)
where the sum only starts at n = 4 as there is no ~ 1 (xxyy) for lower n. However, it m a y be performed, by convenience, from n = 1, as, by (47b)
DYNAMICS OF A CRYSTAL LATTICE CONTAINING ISOTOPES
89
a n d (61), w ( x x y y ) = 0 for n = 1, 2, 3, Therefore:, ,
~%(s) =
54
<~>L
;{"~"~""
[~'(is) ~'--2is] e~', (Y--~") + [~"(is) ~--2is] e ~ "' (~ "--M') + [~" (is) 2--2is] eta'' (;t'--;t") (~, _ ~-)(~- _ ~,,,)(~,,, _ ~,) • 9o(~t') 90(2") ~oo(~t') cUt' dg" d Jr'. (63b) Inserting this in (33b), we find the c o n t r i b u t i o n of the (xxyy) graphs to the f o u r t h order t e r m of the f r e q u e n c y distribution, n a m e l y :
¢~.,,(,,t.) = 27 < # > ~ (~' -
z')[~'(~"
-
"
~) -
~
~'i(~" -
(~,
_
~)]
~-)(~-
+ (~,,, _ ~,)[~,(~- _ ~} - ~(~(~' -
x")(x"
-
x')(~"
-
+ _
(z' -
~,,,)(~,,,
~")[~'(~'
-
~) -
~(~'
-
~)].+
~,)
_
_ ~}] ~')
• ~'z'z'.
vo(Z)
~o(r')
~o(~")
(64)
b) The (xyxy) contribution. The coefficient zo(xyxy), given b y (57b) m a y be simplified b y m e a n s of (2 l) ; we thus o b t a i n :
w ( x y x y ) ---- ¼n Yqk,4 C(/z', s . .,. k. . . , s ; /z", s ' , &IV, siV)
Fn4(~', ;t", ~t', 2IV) .
• N - 4 • X eiV(x'-~')+v'{x'-x)+k"(x-x'}+v(~-x)
(65a)
with . . . , s ; /~", s'~; kIV, sIV) = C ( h ' , s . .,&
= (e(k', s'). e(k", s")) (e(&", s"). e(~', s")) (e(&', s"). • e(/z Iv, sir)) (e(/z Iv, s Iv) . e(/z', s'))
(655)
F n 4 ( t ' , ~[", t " , ~.IV) = ~O~=3n-I~'~'q =2v-1 Zr=l~/-1 j~'n-l~ X"~-~/~'aq -'r XI v r ,. (65C)
where we h a v e written for simplicity )l' = 2o(k', s'); ;t" = 2o(/Z", s"); 2" = 2o(I¢", s"); 2IV ~ ~o(/zlV, sir) (65d) It m a y easily be shown that
n4(A, t ,
Physica XXIV
=
2'2"2"2Iv D4(;t', ~.', ~', ;tIv)
1
2'
2 '2
2'~
1
2"
2"~
2"n
1
~"
$'2
~"n
1
2Iv
2 IV2
2Iyn
(66)
90
PIRENNE
JEAN
On the other hand, owing to the well known completeness of an orthogo: system of plane waves subjected to Born-Karman periodicity conditio the x'-summation of (65a) simply gives: N - 1 Z ei(~'-k"+k . . . .
~'~)(~-~')
= ~(k' + k", k" + k Iv)
(
Introducing (66) and (67) in (65) and replacing further the k-summati, b y integrations, we get:
w(xyxy)
=
~n (Vo/(2~-~)3)zf f f C(k' s', k" s", k"s', klVs Iv) . • F n 4 ( a , ' ~',, ~,,, ~iv)
(dk')3(dk")3 (dk,,,)3 (
'where
vo = V / N is the atomic volume and k Iv = k' + k" -- k"q Following (56), this term brings to ~g~) the contribution
which, by (33b),gives the function (4} ~Oxxyy(S) ~"
Zn=l
((is)n~ n!) "41~ ) (xxyy),
where the summation m a y again formally be started at n = 1 as, b y ( (68) and (71), w(xyxy) = 0 for n = 1, 2, 3. This function is immedia obtained b y replacing in (71) the factors A,n, Un, T"n ~zvn arising f: (66), by
eiSl~, e~SA't, e~s~Ut, eis~IV and the factor n b y is. Inserting the result in (33b), we obtain, for (xyxy)-contribution to the fourth order term of the frequency distribut
~,,
(4) -
4
<~>~
\ 8~a /
"
• C(k',s';k"s";k's" ;klVs Iv) ~'4k^ ,~ ,A ,A , ~ TUT.2I v (dk,)a(dk.)3(dk.,) 3 4~A ,A ,~. jA
]
with k Iv given b y (70). c) The (xxxx)-contribution. Transforming (57c) b y means of (23), we
Zg)(XX.XX)
=
¼n . ~7 " F n 4 ( 2'' ha' 2'tt' 2IV)
which, b y (56), contributes to J d ~ ) the term:
./.t~)(XXXX) = ( ( A # 4 > _ 3(A#2>2) . w(xxxx)/Q04
DYNAMICS OF A CRYSTAL L E T T I C E C O N T A I N I N G ISOTOPES
Using again (33a and b) as before, we find, for tim ( x x x x ) - c o n t r i b u t i o n the frequency distribution 1
9! to
q)'~=''" (A) = 10----8
<#>~v
• A'A"A"A
D4(it', ~", ~", ~11v)
TM
~oo(A')9o(A")9o(,V')9o(AIV).clA'd~" dA" ciaw (76)
Conclusions. B y means of a m o m e n t method, rather different from Montroll's one, and b y the use of a Fourier transformation, we have been able to derive a formal series development of the frequency distribution of the mixed crystal. If we write the deviations of the inverse mass #~ of the i-th isotope with respect to its average <#>Av under the form A#,=#i--<#>Av---pKf, where p is a parameter characterising the degree of heterogeneity of the crystal, our series is nothing but the development of the frequency distribution following the increasing powers of p. The first term represents the frequency distribution of a fictitious pure crystal whose atoms would have a mass equal to ~. The following terms involve the standard deviations
92
DYNAMICS OF A CRYSTAL LATI'ICE CONTAINING ISOTOPES
evaluation of our terms, which contain several singularities (6' functions and poles), the convergence of our development. We intend to discuss these various points in a following paper.
Acknowledgements. I wish to express here m y deepest gratitude to professor R. J. O p p e n h e i m e r for his hospitality and support at the Institute for advanced Study, Princeton, N. J., where most of this work was completed during the winter 1954-55. I would like also to pay here a last tribute to the m e m o r y of the late G e o r g e P l a c z e k who inspired me the subject of this paper and followed these developments with his invaluable criticism and his unforgetable kindness. Received 17-9-57 REFERENCES I) 2) 3) 4) 5) 6)
P r i g o g i n e , I., B i n g e n , R. et J e e n e r , J., Physica 20 (1954) 383. P r i g o g i n e , I. et J e e n e r , J., Physica 20 (1954) 516. Dyson, F. J., Phys. Rev. 92 (1953) 1331. B e l l m a n , R., Phys. Rev. 1 0 1 (1955) 19. Des C o i z e a u x , J., J. Phys. Radium, 18 (1957) 131. M o n t r o l l , E. W., J. chem. Phys. 11 (1943) 481; 12 (1944) 98.