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THE DISORDERED LATTICE PROBLEM: A REVIEW

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The Disordered Lattice Problem: A Review
Author(s): Alexei Maradudin and George H. Weiss
Source: Journal of the Society for Industrial and Applied Mathematics, Vol. 6, No. 3 (Sep.,
1958), pp. 302-319
Published by: Society for Industrial and Applied Mathematics
Stable URL: https://www.jstor.org/stable/2098700
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J. SOC. INDUST. APPL. MATH.
Vol. 6, No. 3, September, 1958
Printed in U.S.A.
THE DISORDERED LATTICE PROBLEM: A REVIEW*t
ALEXEI MARADUDIN AND GEORGE H. WEISS
Introduction. In the eighteenth century a mathematician would probably
have been familiar with the latest advances and problems in physics.
In the twentieth century, due to the increasing tendency to specialize in
both physics and mathematics the problem of communication is a real
one, notwithstanding the obvious mutual benefits which result from exchanges between the two sciences.
It is the purpose of this paper to describe a mathematical problem of
some interest in current investigations of solid state physics which does not
seem to have received any previous publicity in mathematical circles.
This problem, that of determining the vibrational properties of a disordered
lattice of masses and springs, has not been solved in a satisfactory fashion.
Since the main difficulties are mathematical it is our hope that this paper
will interest mathematicians who would not ordinarily be acquainted with
solid state physics.
We begin by giving some of the preliminaries necessary to a discussion
of the problem. The heat capacity at constant volume, Cv, is defined as
the amount of energy needed to raise the temperature of a solid by one
degree centigrade. If E is the energy, C, equals ( 3E/T)X , where T is the
absolute temperature and the subscript v refers to the fact that the volume
of the solid is held constant during this process. A modern theory of the
heat capacity of solids was originated by Einstein [1] in 1907 through the
application of the Planck quantum theory. Einstein approximated a crystal
by an assembly of 3N independent harmonic oscillators each vibrating with
the same frequency P. The energy of such an ensemble is given by
(1) E(T) = 3Nhv [ + exp (hV/lT)-1]'
where h is Planck's constant, and k is Boltzmann's constant. The simple
Einstein theory described the qualitative aspects of the heat capacity of
solids quite accurately. However, even though the quantitative agreement
was good there were discrepancies which argued for a more sophisticated
* This research was supported by the United States Air Force through the Air
Force Office of Scientific Research, Air Research and Development Command, under
Contract No. AF18(600)1315 and Contract No. AF18(600)1015. Reproduction in whole
or in part is permitted for any purpose of the United States Government.
t Received by the Editors April 27, 1958 and in revised form May 16, 1958.
302
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THE DISORDERED LATTICE PROBLEM 303
model. Subsequent more refined investigations were not long in appearing.
It was obvious at first that experimental deviations from the Einstein
formula were due to the fact that the atoms in a solid do not vibrate inde-
pendently of each other. In particular, if the approximation (valid at low
temperatures) of linear restoring forces between atoms is made, one cannot
characterize a solid by a single frequency v, but rather by an ensemble of
frequencies {Iv }. If we consider a model with a large number of normal
mode frequencies which are more or less continuously distributed throughout some interval, then it is convenient to define a frequency distribution
function g(v) such that g(v) dpv measures the fraction of normal modes with
frequencies in the interval (v, v + dv) as dp -* 0. Any thermodynamic or
vibrational property of the solid can then be expressed as an average over
g(v). For example, the internal energy is expressed in the form
(2) E(T) = h Ei z i [ + exp (h'i/kT)-11
Nh] g) [2+ exp (hv/kT) -1] d
where VL is the maximum frequency and must be computed from
ticular model.
One of the first models investigated was due to Born and von Kairmain
[2]. This consisted of a lattice of masses connected by springs. Born and
von Karman considered two cases; in the first the masses were identical
and in the second atoms of two different masses appeared in regular alternation in the lattice. They were able to solve the equations of motion ex-
plicitly and thus find the eigenfrequencies. Just at the time of this study a
paper by Debye [3] appeared which contained a calculation of g(v) from a
continuum model. Debye's model proved to be quite accurate in making
quantitative predictions so that interest in the Born-von Karman model
lapsed for approximately twenty years. In 1935, Blackman [4] pointed out
that there still remained differences between the predictions of the Debye
theory and experiment which arose from the continuum approximation.
Hence emphasis again shifted back to the discrete Born-von Ka'rman
model. Excellent reviews of the modern work on the evaluation of g(v) f
monatomic lattices are to be found in [5] and [6].
In 1953, Dyson [71 proposed as a model for glass a one-dimensional chain
of masses and springs in which the mass at any given lattice point is a ran*dom variable as is the spring constant. Prigogine, Bingen, and Jeener
[8] studied the same model in connection with the separation into separate
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304 ALEXEI MARADUDIN AND GEORGE H. WEISS
phases of an isotopic mixture at the absolute zero of temperature. It is this
model with which we are concerned. We will consider only the simplest type
of lattice since even in this case no satisfactory solutions exist. We discuss
an n-dimensional cubic lattice with nearest neighbor interactions only (i.e.,
each integer point in a Cartesian coordinate system is occupied by an atom
which is connected by a spring to each of its nearest neighbor atoms).
In three dimensions the equations of motion for the x-component of the
atom located at (x, y, z) = (il. i2, i3) can be written
Mjjii3t~ili2i3 = YT( -1),2i3(Xi-li2i3 -Xili23) + 'Y1) -,i2i3(Xi1+li2i3-Xil
(3) + yTjti2-l,i3(Xi1,i2-l,i3 - Xili2i3) + -YT i2+li3(Xi1,i2+l?i3 - Xili2iO
+ -itl2.i3-l(Xi-i2,i3-l Xili2i3 + +1jt2,i3+l(Xili2,i3+l - Xili2i3)
In this equation Mili2i3 is a random mass and the y's are ran
constants. Now we let xili2i3 uilii23 exp(icot) and we are left with a set of
homogeneous equations in the u's containing v2. We may complete the
specification of the system by assuming that' uili923 = Uil+N i~i3
Uj1,i2+N, 3 = *.. . In order to find a nonzero solution to the equations
for the u's we must solve a certain determinantal equation. In one dimension, for example, we require
M1,W2 - _y, - _y2 72 0
'Y2 M2 W2 - 'Y2 - 73 73 0
(4)
0
AI33(2-y3-34
74
=0.
If the lattice is a regular one, say M1 = M2= = MN and
Y1 = 72 = ... = 'YN, the solution for the {wj} is not too di
the M's and -y's are not regularly distributed the problem red
for the roots of a very high order algebraic equation, obviously a problem
of a different order of magnitude. For reference we shall list some results
for the frequency spectra of monatomic lattices [6]:
1 It can be shown that the frequency distribution of an N-particle lattice tends to a
fixed distribution independent of the boundary conditions as N --> 0o The proof
follows similar theorems by Weyl [19] on the spectra of eigenvalues for differential
operators.
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THE DISORDERED LATTICE PROBLEM 305
1-D: g() = 2N (2 2)- (V < PL),
7r
=0
(V
PL=
>
PL),
27r
(4y/
2-D: 9(v) - N K (71 _2))
7r(PL 2 - p2) \12 2P(PL2 -21
(5) ( 2[L2 - V2] > 'Y1 Y2/ir)
4N
2v' K 2r V(L -V y
- K
(71 72)2 (71 Y2) /
(Y -Y2/ir > v [VL2- _2] > 0)
=
VL
=
0
-
otherwise,
?
where K(x) is a complete elliptic integral of the first kind. For a discussion
of the three-dimensional case the reader is referred to [6]. Notice that
there are infinities in g(i) in both the one and two-dimensional cases. This
is no longer the case in three dimensions. Some plots of g(v) are shown in
Fig. 1. The distribution of frequencies for a regular diatomic lattice (one
in which, say, the atoms are arranged ABABA ... ) splits into branches but
retains infinities [9]. This is illustrated in Fig. 2.
With the preceding facts as background we may now proceed to a discussion of the disorder problem and some of the efforts that have been
made to obtain solutions or approximate solutions. We have stated that the
solutions for the normal mode frequencies are not too difficult when the
lattice is a regular one. For any configuration of atoms in the lattice there
exists a frequency distribution g(i). We require (g(q)) where ( ) refers
to an average over all configurations. An equivalent formulation that we
will find useful later is, given any vibration property 4(v) find its average
value,
rPL
rPL
(6) V)= K] i(v)g(v) dv) = J +(v) (P(v)) dv.
What can we say quite generally about each g(v)? The distri
frequencies for a regular lattice is nonzero for a finite range (cf. Fig. 1).
The same must also be true for any lattice for which the masses are not
arbitrarily light and for which the springs are not arbitrarily stiff (i.e.,
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2.5
2.0
z
-~1.5
1:5L.0
0.5
1
1
I
0 .2 .4 .6 .8 1.0
f - //L
FIG. la. One-dimensional frequency spectrum
8
o
z
N
Y TT
2
0
0 .2 .4 .6 .8 LO
t-V/ ZL
FIG. lb. Two-dimensional frequency spectrum
306
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THE DISORDERED LATTICE PROBLEM 307
2.0
1.6
1I.2
0.8
0.4
02 0.4 0.6 Q8 1.0
f =v
VL
FIG. 2. Frequency function for a diatomic linear chain in the form ABAB
e(2j3 < y < o and M > M > 0 for all j, ii, i2 and i3). For, by a
theorem of Rayleigh2 [10], a decrease in a mass of a vibrating system or the
stiffening of a spring constant can only decrease any particular frequency
or leave it unchanged. If we now consider a lattice made up of the lightest
masses allowable in the lattice and the stiffest springs, this will have a maximum frequency PL. Any other lattice can be formed from this one by
either adding masses to particles or loosening springs; hence its spectrum
of frequencies is nonzero in a finite interval which is bounded above by
VL . Unfortunately little more of a general nature can be stated about this
part of the problem.
Special methods for one-dimensional problems. At the present time a
closed form solution to the disordered lattice problem exists only for a particular one-dimensional chain [7]. There exist only partial results for the
binomial isotope chain, that is, a chain wherein an atom of mass Mi occupies
a given lattice site with probability pi (where Xi pi = 1) [12, 13]. The two
and three-dimensional problems remain virtually uncharted territory
although some approximate methods exist for their solution, [8, 13, 14].
We will now describe some of the methods of Dyson and others for the
linear chain.
2More generally [111 if Ai and A2 are completely continuous symmetric transformations with A = Ai +A2 , and if A2 > 0, each characteristic value of A is greater
than or equal to the characteristic value of Ai with the same index.
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308 ALEXEI MARADUDIN AND GEORGE H. WEISS
We start with the equations of motion in the form
(7) m = - ) -j(xj+l -Xj) --jl(xj-Xj-1).
We can symmetrize the matrix of the coefficients of the x's by introducing
new dependent variables by
(8)
Yj
=mjXj
and new force constants by
(9) =2j-1 'Yj/m, X2j = 'Yj/Mj+l This puts the equations of motion into the form
(10) Y (X2jX2j-1)2Yj+1 + (X2j-2X2j-3) 2Y- -(X2j-2 + X2j-l)Yj
We now introduce new variables {zig given by
(11) Zj = 2j Yj+- x2j-lYj
which puts (10) in the form
(12) j= 2j-1Zj-2j-2Zj-1
Equations (11) and (12) can be reduced to a single set by introducing the
variables u2j- = yj and U2j = zj . The u's then satisfy
I
I
(13) ?j = XMuj+1 M-juj-
If we assume solutions of the form vj = uj exp (iwt) then the normal mode
frequencies can be found from the (2N - 1) X (2N - 1) determinantal
equation
A2N-1(C, X1 X * *2N-1) = WIT + H2N-1
C) iV-X 0 .
(14)
=
i-
1
co
iVX
*
*
=..
0 -iV\2 W ...
In order to find the distribution of normal mode frequencies Dyson
introduces the function
1
N2
Q(x) = lim 2 E In (1 + xc())
1
N
= lim In fl (1 + ix\-wcj) (1 -i\C-j)
(15)
(15) ~N--oo 2N -1 j=j
= lim 2N I 1 ln II + ix/IH2N-l(Xl, X2, ...)
= f In (1 + xA)D(G) dA,
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THE DISORDERED LATTICE PROBLEM 309
where D(g) is the distribution of the squared frequencies and the logarithm
is defined as the branch of the function that is real for positive x. That is to
say
(16) D(y) = dM(4)1d,
where M(y) is the fraction of the root
assumes without proof that M(y) is differentiable.
It is not too difficult to show that
(17) Re [(iir)' lim Q(-z + iE)] = 1 - M(1/z)
or
3
(18) D(1/z) = -z Re [(fir) lim Q'(-z + iE)].
We consider now the determinant I + i-\xH2N-l 1. This
1 -\/Xxi 0 ..
i\IHN(19) A2N-1 = |I| +
iVXH2l | = 0V/Xx
- 1e
1 - V
This is the determinant of a Jacobi matrix which can be evaluated as a
continued fraction by noting that /X2N-1 satisfies the relation
/A2N _1(1, X Al, -XX2 X * - * , XX2N-1)
(20) = A2N-2(1, -XX2, -XX3, ... XX2N-1)
+ xX1A2N-3(1, -XX3 XX4, XX2N-1),
or
(21) \A2N-1 + XA 1+XAl XX2
a2N-2 A2N-2/A2N-3 1 ? 1 +
Hence we have
(22) A2N-1 [1 + ( { X})]X
where
(23)
,=
+
1
+
1
+
1+ 1+ 1+
3 This relation follows from the known inversion formula for the Stieltjes transform [21].
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310 ALEXEI MARADUDIN AND GEORGE H. WEISS
Since
(24) in A2N-1 = En In [1 + An,
we have
1 2N-1
(25) Q(x) = rn 1E in [1 + AJ.
N-ixo 2N - 1n=i
If one knows U(x), then one can determine D(x) through the inversion
formula
(26) D(x) = 212 f cos h7ra da f j cos [a In (xy)]Q'(y) dy
which can be derived by Fourier transform methods.
At this point we are able to discuss statistical questions with the aid of
(25) and (26). We assume that each of the parameters defined by (9) is
an independent random variable with the probability distribution G(X).
Each of the quantities P(j) will have (in the limit as N --> X ) a distribution
F(h), the same for all j. However we also have
(27) U x
(27) ?(i) ~~~1 + P(j - 1)
Since X and t(j - 1) are independent random variables, F(v) must be a
solution of the following integral equation:
(28) F() f F(~')G [d( + )] (1 + ) di'.
When we have found the solution of this equation which satisfies the
normalization condition
00
(29) f F() d- = 1,
the characteristic equation of the c
(30) (Q(x)) = 2 f F(v) In (1 + A) do,
where the brackets indicate an ensemble average. Dyson
in closed form only for the distribution functions
n
(31)
Jn(')
n n1 _nx
=(n)
In the physically more interesting case where the chain is composed of
two kinds of atoms with masses m, M distributed at random in the propor-
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THE DISORDERED LATTICE PROBLEM 311
tion p: (1 - p) the integral equation of (28) reduces to the functional
equation
F(q) = p(l - n + m/xK)-2F[(l - t + m/xK)- - 1]
(32) + (1 - p)(1 - n + MjxK)-2F[(1 - + MjxK)1 - 1],
where X2j-1 = j-2 = K/mj . There have been no theoretical or numerical
investigations of this equation. It would be of some interest to discover
whether the distribution of normal mode frequencies has infinities in it as
does the one-dimensional monatomic spectrum of (5) or the ordered
diatomic spectrum given in [9].
The treatment that we have outlined above is applicable to one-dimensional lattices only since it depends on the matrix H2N-1 of (14) being a
Jacobi matrix. In more than one dimension other off-diagonal terms appear,
hence the determinant I cI + H2N-1 I does not satisfy the simple second
order difference equation (21), which in turn implies that the A2N-1 cannot
be expressed in terms of continued fractions as in (21).
There are two other treatments of the one-dimensional disordered chain
that cannot be generalized to higher dimensions [12, 15]. These use the socalled transfer matrix. For example, Schmidt studies the case of isotope
defects only. The treatment in Schmidt's paper starts from the equations
of motion of the chain in the form
(33)
(33) ~~~Mnvn = kC(Vn+l + Vn-1 - 2Vn) (n = 1, 2, .. i, N
V0 = VN+1 = ?,
where Mn is the mass at the nth lattice site and k is the force constant.
Letting Vn = Un exp (iwt) and Mn = mn/k Schmidt is able to rewrite (33)
in the form
un(2 -Mn C2) = un+1 + un-1 (n = 1, 2, .., N)
(34)
Uo = UV1= ?
or, equivalently
(35) (n0M (n+)
/ku(-1
2 _-M"2 ( n
At this point Schmidt studies the properties of
(36) V ((w2) (Un )
where un,- is regarded as an x-comp
(Pn/2 is defined as the angle betwee
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312 ALEXEI MARADUDIN AND GEORGE H. WEISS
Schmidt shows that Sp(.,2) increases mono
Schmidt's proof we let
(37) zn( 2) = un,1/un = tan (Pn/2),
which, with (34) implies
(38) Zn+1 =2M 2 (n = 1, 2, *,N),
with the boundary conditions z1 = 0, ZN
require that sPn = 27rm + ir with m an int
It is now easily verified that
(39) (89n+l) > O (,Pn+l > 0,
(3)(n constant w2 &,2 constant n =
so that IN is indeed a monotonic function. The reason for stu
is that it is very simply related to the integrated frequency spectrum
M(c,2), which is defined by
NM(W2) = {number of eigenvalues W,2 with W,2 < (,2
In fact, if c2 increases from c to b then ON+l(c2) increases monotonically
from 'N+?(cWa2) to CON+1(Wb2) and whenever in this region (N+1(WV 2) = 7r + 2irm,
2.-
cWV is an eigenvalue since UN+1 = 0. Hence, we find
(40) M(c,2) = (27rN) -1N+l(cW2) + constant.
To study the disordered chain Schmidt defines
(41) wj[Z] = no. of chains for which z < z.(w 2) < z + dz
total no. of chains
and
(42) Wn[z] H f w.z V dz'
where Wn[z] is a multiple valued function of
specified by Wn[z]. If the nth lattice site ca
mass Mj with probability pj, then a simple
satisfies
Wn+1[Z] = z2 pj~w2 - 2 - z']
(43)
W1[Z] = 6(Z).
Furthermore, defining W[z] by
1
(44) W[z] = lim - I W1 z] + W2 z] + * + WnHV[z }
n-,oof n
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THE DISORDERED LATTICE PROBLEM 313
Schmidt shows that W[z] is a continuous nond
satisfies
(45a) W[zJ = pj W[2 -Mj z-1] - WV-[ ]o
(45b) W[ool - W[- ool = 1, V[OJ = 0.
This is similar to the functional equation given by Dyson. Furthermor
for a randomly disordered chain, he obtains the important relation for t
mean value (averaged over all chain configurations) of M(W2),
(46) M~o) WI-s 00
This result, together with the functional equation given in (45) constitutes Schmidt's formal solution to the problem.
Asahi and Hori [15] have also used the transfer matrix method to study
disordered lattices. It has been shown in [14] and [16] that if one expands
any thermodynamic function in powers of the concentration c, the coefficient of c' depends only on lattices with m defects where m < n. The
main contribution of Asahi and Hori was to give a reasonably simple
method of writing down the eigenvalue equation for a chain with a finite
number of different isotope atoms. However these authors do not give as
complete a treatment of the disordered lattice as does Schmidt; hence we
do not reproduce it.
Perturbation methods for higher dimensions. The disordered lattice prob-
lem in two and three dimensions is further from a solution than the onedimensional problem. The elegant considerations of Dyson depend upon
the fact that the secular determinant is that for a Jacobi matrix. The studies
of Schmidt and Asahi and Hori depend on the fact that the transfer matrix
is 2 X 2 matrix. In higher dimensions there are additional off-diagonal
terms in the secular determinant and a transfer matrix cannot even be
constructed. There have been no investigations in the higher dimension
cases that come as close to a solution as Dyson's does in one dimension.
The only results known so far are all found by some variant of perturbation
theory. We will briefly describe a method which we have developed since
through its use one can ensure the convergence of the perturbation series,
a feature which is absent in all other discussions of this problem.
We may express any additive function of the normal mode frequencies
as a contour integral [17],
(47) H Ejof)(itj) s l dt a(n)d In a conu) wI
where JH(cw) If is the secular determinant and C is a contour which encloses
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314 ALEXEI MARADUDIN AND GEORGE H. WEISS
all of the zeros of I H(w) I but none of the p
ordering of masses only but not of spring constants although more general
cases can be treated by this method. Let us decompose the matrix H(w) in
the form
H (w)= Ho(w.) + 5H(w.)
(48) = Ho(w)[I + H '(wo)6H(wo)] = Ho(w)A(w)
where Ho(X) is the matrix corresponding to a regular lattice and 6H(X)
contains the contributions from defect atoms. We may therefore write
27 JC f(w)d In I Ho(X) I + 2iJf(w) d In I (X)I
(493
= SO + AS,
where presumably SO is known. We may therefore concentrate our attention
on AS. Notice that I A(W) I generally must have poles at the zeros of
Ho(X) I in order to shift the frequencies. Let us denote the eigenvalues of
Ho'(cw)6H(cw) by {AXs. This yields the representation for I(c) !,
(50) 1 A('v) I = IfL (1 + Xj).
The logarithm of | A(cw) I can be expanded in the
X (_ 1) n,+l
(51) In 1A(c,) I = E (1 j )j'
n=l n
provided that I Xj < 1 for all co. W
Ho1 (w)5H(w) by D(cw) then we no
X0 (_ 1n+l
(52) In I A ) I En=l
TrD
T()
n
Henceforth we confine ourselves to the simple two isotope case where A
atoms with mass M1 occupy a lattice site with probability p and B atoms
of mass M2 occupy it with probability 1 - p. We may write the mass at
lattice site m = (ml, M2, m3) as
(53) M(m) =, M + crmAM
where am is a random variable with the property
-M (with probability p)
(54) am = l -
lM2-M (with p
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THE DISORDERED LATTICE PROBLEM 315
The choice of M and AM is arbitrary and can be made in several ways,
all of which are consistent with the conditions I Xi I < 1. Now we choose t
matrix Ho(wo) to be the secular matrix corresponding to a lattice consisting
only of atoms of mass M. For a cubic lattice the elements of Ho(co) and
Ho1(co) are known [17]. In particular we have for the elements of Ho'(W),
(55) a M, = N Es [Xo(s)K' exp [21ris * (m - m')/N]
where N is the number of atoms on an edge, n is the dimensionality of the
lattice, and
(56) Xo(s) = fMW2 - 2 E j (1 - cos 2r)
The question of whether the X, are less than one in absolute value can
be investigated in some detail with the aid of a theorem due to Gershgorin
[18]. We will restrict ourselves to the one-dimensional isotope case for sim-
plicity and find the conditions under which the eigenvalues Xj71 of D-1 are
greater than one in absolute value. The theorem of Gershgorin states:
Let B = (BkX) be an arbitrary square matrix of order N and set
N
(57) Pk = I iBkxj|
X-1
Then each eigenvalue of B lies in the interior or on the boundary of at least
of the N circles
(58) I z - Bkk I < Pk
Since D-1(w) = [6H(wo)]-'Ho(wo) we find t
circles as there are isotopes and these are specified by
(59)
z
-2
<
2'y
=M2-
AMc*.2U
which implies that the circles can be described by
(60) center: M2(j _ M), radius: 2 M
in the complex co-plane. The most convenient contour C for the integral
of (49) is a semicircle in the right half plane with the diameter on the
imaginary axis and a radius which goes to infinity. For all additive func-
tions of interest the only contribution comes from the integration down the
imaginary axis. It is easily seen from (60) that in order for I XT' I to b
greater than one on both portions of the contour we must have
(61)
M>
IMj-MJ,
which imposes a restriction only on the choice of M but not on AM.
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316 ALEXEI MARADUDIN AND GEORGE H. WEISS
When there is disorder in the lattice we are interested in (AS) the value
of AS averaged over all possible configurations. If M1- M2 is so small that
higher powers of (M1 -M2)/ are negligible in comparison to the first,
we may write for AS
(62) AS 2 (Zm om)f g(z) d{z amm }.
The element a(-') is independent of m. The randomness
scribed by the term Zm 0m . If N is large this term has a Gaussian distribution with a mean given by
(63) KErm nm) = M [pMl + (1 - p)M2-M]
and a variance
(64) (Z _m2)m2~~~(AM)2
N P(l -p)(M -M2)2.
Em(m
UM
In general it is not possible to neglect higher powers of AM so that one
must find an expansion using (49) and (52). Hence we must find
(AS) = 21ri X g(z) d(ln I A(z) l)
(65) 27rs 1 n~~E g 9(Z) d (Tr (D n))
where the various interchanges of summation and differentiation can be
justified [13]. There remains now the task of evaluating terms of the form
(Tr (D')) or
(66) (Tr (Dn)) = a . .. Zamlr~a3 KO'ml Om2 ... aMn)
As can be seen, a combinatorial problem now arises in sorting out the terms
since, for example,
(67)
(2
)
()
2.
It is, in fact, not too difficult to sort out all of the terms appearing in
( Tr (Dn)) together with the proper combinatorial factors, but for values
of n greater than four the detailed calculations become quite tedious.
In the limit as N oo the amm, are given by
(68) amfm'(wd) = Jam J exp [2ri i( (m - m')/N]
he sumsM an 2Zof 2 (l -c osl ) it n
The sums appearing in the expansion of ( Tr (D n)
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THE DISORDERED LATTICE PROBLEM 317
terms of certain "irreducible sums" such as
ak(C) (AMfc2)" r1- a (-1) (-) (1
ak~u), _ ( LM ) E.M ... EMk iM1m2am2m3 * a**
(AMW2)k Es [o(S)
(69)
A
s
k
(AMw2)" pr fT dfp1 .2... d(Pn
man Jo JO [MW + 2 Ejy j(l c j)]
bk (icO) m2 EM1 Em2 (ai mI2)k.
We have used a pure imaginary argument for ak since all contributions
from the semicircular part of the D-shaped contour vanish as its radius
becomes infinite.
When the detailed calculations are made the terms up to order ju2
(= (AM/M)2) are
1 doe (In I L(ico) 2 - 2(of) [a, (i)- j-a2(ico)]
(70) - 2((_2) _ ()2) al(iC)[ai(iw)-
-2 (0f) [a2(iw) 1a3(i)] +
C)
It is not our purpose to give a detailed account of the theory but only of
the techniques; hence, we will not carry this discussion further except to
point out that like all perturbation theories, this one cannot yield any
qualitative information. For example we cannot find out whether the averaged frequency spectrum of a disordered lattice contains any infinities
as does the spectrum of the ordered lattice (5). The only other approach to
the disordered lattice problem has been by the use of the method of moments
[9]. This method assumes that the frequency spectrum can be expanded in
a series of Legendre polynomials
(71) (g(co)) = anPn (P )
n=O (WL
where (0L is the maximum possible frequency. It is not known whether
(g(co)) can in fact be expanded in such a series, hence even the beginning
stages of this method are of dubious validity. The coefficients in the expansion of (71) are given by
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318 ALEXEI MARADUDIN AND GEORGE H. WEISS
an 2 f (g(XWL))P,(X) dx
(72) 2n +1 [n(21) (2n - 2r) !n2
2 r=O 2nr!(n - -_2r _ L))x dx.
Thus the an can be found in terms of the moments of (g(co)). However,
the even moments of (g (w)) can also be written as
( _1 E 2k - Tr (D2k)
(73)
/12k
where
=
D
-
is
the
secular
de
symmetry of (g(co)). Hence
an iS to find the traces of
tional average. The an can
(72). The calculation of even Tr (D10) is a rather complicated problem.
However, some recent work by Professor Cyril Domb of the University of
London indicates that by the use of random walk methods one can find an
explicit expression for the moments of any order in one dimension. It remains for the future to show whether the series of (71) can be summed
using these results to find a closed form expression for (g(w)).
We have thus concluded our survey of the disordered lattice problem.
As we have tried to indicate, it is still in a comparatively unsatisfactory
state, although many attacks have been made on it. It is hoped that this
article will serve to stimulate interest in the problem on the part of people
not familiar with the physical aspects of crystal lattices but who might
bring to bear on it fresh mathematical techniques that will result in a more
satisfactory solution.
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THE DISORDERED LATTICE PROBLEM 319
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DEPARTMENT OF PHYSICS AND INSTITUTE FOR FLUID DYNAMICS
AND APPLIED MATHEMATICS, UNIVERSITY OF MARYLAND, COLLEGE PARK,
AND WEIZMANN INSTITUTE OF SCIENCE, REHOVOTH, ISRAEL
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