- Email: [email protected]
Vibrational frequency spectrum of random crystal lattices
ANNALS
OF PHYSICS:
15: 411436
Vibrational
(1961)
Frequency Crystal J. C. RI.lS,
Spectrum Lattices
of Random
BRADLEY*
Baltimore,
Maryland
A met,hod was developed to determine the moments of the vil)rat,ional frequency spectrum associated with a three-dimensional, two-component, random lattice. The frequency spect.rum was approximated from it,s zeroth through eighteenth order moments. A hand gap no longer appears in the spectrum. A.lso it is inferred from these calculations that t,he maximum frequency of the lattice is approximately that associated nit,h a lattice of light atoms. The clero point energy of the random lattice was calculated from an expression involving the moments of the frequency spectrum directly; it was found that this energy is lower than that associated \Cth the perfect latt,ire. I. IKTROI~UCTION
Recently a significant amount of work has heen devoted to the study of the density of vihratjionnl frequencies associated wit’h disordered crystal lattices. In particular, Dyson (1) and Schmidt (2) have presented theories for disordered linear chains. Later Domh et al. (3) made an exhaustive study of the vihrational spectra associated with “random” linear chains. They derived a gcnerating function from which they could compute the “configuration averages” of moments of the spectrum. This exposition is devoted to the description of a method that was developed to oktin a probability density function for lattice vibrations associated with a t,hret dimensional (:3-D 1 simple cubic “random lattice” containing two kiuds of atoms. It was desirable to develop an approach to this problem since t.he previous work on vibrations in “random lattices” has been confined to the 1-D case. To he more specific n randor~l crystal lattice (phrases being defined are italicized) of two components and X” atoms (as we shall consider) contains N”p =l-atoms and N”(l - p) B-atoms with the provision that a given lat,tice site contains an d-atom with probability p or n B-atom with probability 1 - p. This means that both short- and long-range order parameters vanieh or that the state is completely disordered. We *hall consider a simple cubic crystal lattice where only first nearest) neighthe
* Hased on a t.hesis submitted reqrlirrments for the degree
to the rniversit,y of Maryland of Doctor of Philosophy. 411
in part,ial
fulfillment
of
bors are assumed to interact, through central and noncent8ral forces. In particular, t,he force is assumed t’o ha1.e the same direction as an at.omic displacement,. If the displacement is in the direction of the line joining two neighboring atoms the force constant is denoted by y1 (cent,ral) and if it is orthogonal t,he constant is denot’ed by y2 (noncentral). Consequently, y1 > y.’ . Let h represent, some property of a lattice whose value is dependent, upon the arrangement, of it,s atoms. Suppose a set 1 indexes t,he different arrangements of ;2 and B atmoms (‘1 denotes the number of element8s in 1 where t,he corresponding set of A’s is (Ai)i,I’). Then the propert,y A associated with t)he random latt,ice is called the con$gu&ion average of the ( Ai) cc I and is defined by
The quantities of primary concern in this connection are the distribut)ion funct,ion, its moment8s and the thermal properties of the lat#tice. One of the most fruit,ful means of determining t,he shape of the frequency spectrum for random lat,tices, (g(w)), is the moment-trace method (4); it, depends only on the details of t,he crystal model wit#h essentially no limiting features such as the exist,ence of a unit, cell or the dimension of the crystal. We obtain an approximat,ion t,o (g(o)) by it,s first N + 1 nonzero moments and the assumption that it can be represent)ed by the series
where w,,,:~~is the maximum
normal mode of frequency
of vibration,
x = w/wIrI:,x ) and fn (x) mesh
is a simple function
defined in the following
fashion:
Consider
the
so = 0, x1 ) x2 ) . . . , x.,7+1 = 1
of [0, l] where s, - .rP1 = l/(N
+ 1) for i = 11, . . , N + I}. Then
f’ (s) = 1 if .c C Lri , x,+11 0 ot herwiee
.8
for i = (0, . . , -V). 31ult8iplying Eq. (2) by J? and integrating over the unit interval gives
s 1
.P(g
( xw,,,:,x
1)
dx
=
g
(I'
.i2%
Cx)
dx)
ci
0
forj = (0, . . f , NJ. In matrix notation these equations may be written as
VIBRATIOIi
SPECTRA
OF
KAKDOM
4 13
L.lTTICES
G = Fc, where
and
Since F is nonsingulw tained from
the coeficicnts
in the expnwion
in Ey. ( 2 ) are readily oh-
c = F-‘G. If the (2k)th order moment of g( w i corresponding is defined by’
to one arrangement. of atoms
then
Let II:: denote the (u, IH )th clement of F-’ hy F!,$‘; come
then the elements of c he-
N
(c,,) = c
Vi=”
The predominant
numhrr
of :trralIgements
1 Since g(wj is an even function chmge in the spectrum is cawed we used
instead of Eq. (2) where the mirror image of ,f,,(.r)
reduces to
11 = IO;..
F!-,‘(w,:;-‘A,);
,iV},
(3aj
of atoms in the latt.icc allox for large
it] w it follows by Eq. (1) t.hat. (y(w)) is dso. No essential I,- the representation of (g(w)) I)>- l’kl. 12). For suppose
F,, is defined un I- 1, 11, F,,(S) :rbollt the ~1 usis; then clearl?
= ,fSt(x)
+ ,f.( -.r~
and ,f,&(-x)
is
clust8ers of the two atomic species to occur. In so far as these clust,ers can be regarded as separate lat’tice struct’ures we shall assume that t,he maximum frequency of the entire aggregate is the maximum frequency of a similar lattice structure containing only light, atoms. Consequently Eq. (3a) reduces to n, =
(0,
...
) NJ.
(33)
The first ten nonzero moments ((&, (&, . . . , (pl8)} mere ralculated for the crystal model described above, and (g( 0)) was estimated from Eq. (2 i for different physical parameters such as the ratio of masses of t,he two at,omic species. Also from the moments a calculation of the zero point, energy and an est.imate of the validity of the assumphion concerning the maximum frequency of t,he lattice were obtained. The zero point1 energy for the random lat,tice is lower than that for t,he perfect lat8t.ice. II.
MOMENTS
OF
The moments of t#he distribution
THE
1118TRIBUTION
function
FUNCTIOS
are related to the dynamical
matrix,
D (the matrix derived from t,he classical equations of motion and whose eigenvalues are the squares of the normal mode frequencies; equation : (~21;) = A-” (Trace
D")
see (4))
by the following
(4)
Since the motion of atoms in one direction are not coupled t,o the motion of atoms in other directions the dynamical matrix appears in the block form. It is necessary to consider only one block as was done in Eq. (4) and in the following development where each block is an N3 X N3 submat,rix. It is not pos-
sible, however, to furt,her simplify the problem in this fashion as is often done with regard to periodic arrays of atoms.
VIBlUTION
Each matrix pose the (~1, sometimes he zero elements
SPECTRA
OF
HAiYDOM
115
LATTICES
element, in the dynamical mat,rix is ident,ified by sis indices. Supus, us ; ul’, ZL~‘, u3’)th element is d~,LI,U?,L(Q;IL,,,L(~,,Ua,) . This will written for hrevit#y as d,: , where u = (ul , us , Q). The only nonappearing in the uth row of D for t,he lat,ticc described above are cl(I‘*.Ut,~:~;“1,“.‘.,~~) = 2l-u + a9)l~JIL,,M?,IIa , d(!<,,“.‘.~a:,,,*l.r,,.rcl) = --rJJ/
(I ("~,"~.ic:~;lL,.Il~*l,IL:~) = d (Ic,,s?.?,:~;l,,,UL,lLq~l)
(5)
Il,.II?.lC3 , =
C(j) -Ysj~~~q,q,q
.
(7)
It is important, for later use t,o not,icc that only one random mass variable, JI, , occurs in the uth row of D. The moments of the distribution function may be expressed as
(llw)= m lljl~~~,llp c d,,“,cl”,“, ... d”,,,) and since t)he lat#tice is random the sum in Eq. (8) commutes with the configurat8ion average t 0 give
The elemrnk in the summand of Eq. (9) are defined as brackets. Since the brackets are independent, of the diagonal element of D” in which they are (‘ontained, Eq. (9) may be summed over the first index t,o give Gad =
c (d”,,, . . cl”,“,). ll2,.“.llg
,is 5~11 example of Eq. (lo), (pqjis computed from Eqs. (.5)-(7), the only nonzero elements are given by (4
(10) realizing that
== g &,u, hl,“,)
The configuration average of the mass terms using Eq. (1) become
where q = 1 - p. It, is apparent that the configuration averages are computed assuming that neighboring atoms do not, influrncc one another (property of randomness). This allows for all possible c,on~elltrat,iolls of il and B atoms to
occur in the set, of configurations. It is useful to picture this latt#ice as a system in an ensemble of lat,tices in which the concentration of .4 and B at,oms is preserved. As a consequence of this, Weiss (5 ) has shown t,he fluct8uations from the stoichiomet,ric ratios of 9 and B atoms in the lat’tice are arbitrarily small in t#he limit as t,he number of atmoms in t,he lattice tends to infinity. The importrant feature of Eqs. (11) and (12) that are used in computing the moments can he sunlmarized as follows: Notice that only part,itions of 2 appear as exponents of t,he masses in these equat,ions. Similarly, only a partiCon of li can appear as exponents of masses in an element of the sununand of Eq. (10) ; e. g., only hrackct,s of the following form occur in (pzr;): nr,“: . . JI&
(I/w:: where the {a,} are positive
integers and al + Uz +
. . . + al =
Ii.
(The subscript,s on t,hc massesare considered distinct in t,his bracket. j Then t,he population of nominally distinct, brackets, such as that above, must he determined for each partition of k. The central problem is to deduce the population of equal brackets which occur in a given moment. It is useful to identify the elements in the summand of Ey. (10) with a set of permutations of a prescribed set of digits. k’or example a bracket
(where some of the subscripts on massesmay he equal) can he made to corrcspond to a set of numbers to the base 7 (septcnary nrmihers) in t)he following fashion’ 0 if ui = uj = (.U~, ~1~, u3j
duiu, -
1 if
Uj
= (U1 + 1, U2,
2 if
Uj
= (U1 - 1, II? , U:j)
Us)
3 if uj = i 111 , u? + 1, us). 1 if
Uj
=
( 111 , ll2
-
(13)
1, lL3)
5if uj = (ZL~,UO,U~ + 1) B if
Uj
=
(ZL1 , Uf , ?La -
1)
The number 7 nat’urally arises as the base since it corresponds to the number of elements in a row of D given hy ICys. (T,)-(7). This is a manifestation of the interaction of an atom at a lattice site with its 6 nearest neighbors. If we utilize 2 The
symbol
“u”
means
“one-to-one
correspondence.”
VIBRATIO;“i
septenary numbers each bracket, (~4)
can
SPECTRA4
and ignore be writ,ten
OF
RrlNDOM
the expressions as
~4 = COO) + perm(l2)
LATTICES
417
containing
+ perm(34)
force
constams
+ perm(56),
(14)
where perm(cl . ck) is used to desiguate t,he set of pernnnations of rrary number cr . . . ck . Each septenary number c1 . . . ck corresponds and only one bracket. The sept~enary numbers arising in terms of the type appearing in are characterized by the r&rictions that, the number of I’S (say q) to the number of Z’s, etc., and the number of O’s is equal to JC- 2( 12~ + I:or example, let, nl l’s, la3 3’s, Q, 5’s occur in a septenary numhcr; then consider permutations of Cl,
\Ii
-
2
(i'll
. ’ . ,o +
It,:{
. . .
1 -.L,L.zL,L.L +
?Lj)
1
‘>
. . .
”
'
3 I
1 111
. . ...3 c-;-i’
the septeto one Eq. (14) he equal rt3 + ns). we must
J,...,-l
nzj
111
113
5, . . , 5 6, . . . 6 L--J. L-,L
%i in connection it is obvious &)
with computing &). As another t,hat computing (~(12) gives us
= ( 000000)
+ (zj
perm( OOOOab) + +
\zj
example
C C perm(OOahcd 1a.b) (cd) C
perm(aaabbb)
(15)
%
similar
to l~q.
(14)~
pcrm(OOaahb) ) + (5)
c”.b)fcr.d)
+
in
,z,
perm (aabbcd)
to.h)#(r:d)
+ perm(
123456),
(o.bl
and &
.t,) means
wnm~ation
where j(1,2),
the pair
(a, h) takes
on values
(3, 4), (5, B)}.
.Inother feature of these septenary uumhers is that, the product of force const,ant’s appearing in a hrwket reprrsrntcd hy any permutation conlposed of elements in ( 15) is
where
The number
of elements
in the permutation
set, of digits
ill ( 1.5) is
418
BKADLEY
7(k)
= [Ii - 2(n1 + a3 +?,I:
and t,he number of elements in the summand
(nI!)yn3!)yn6!)2 of Eq. (IO) is
where [k/2] is the largest integer in /i/2. To st,ate the problem in a more useful form let us consider perm(cl . . . ckj. We already understand how an element of this permut,ation is related to a partition of k (the partitions of k are designat,ed by jpl , . . . , P,,(Q}. Denote t,his functional relation by
.f: perm(c1 . . ck> - {PI , . . . , P,(k,I. Then S is single-valued and defines an equivalence relation on perm(cl it is illustrated in Fig. 1. If we let, 0 denote t,his equivalence then Cl . . c,ec,,
. . . ck) ;
. Cik
if and only if .f(CI . . . cl;) = f(c,,
. . . Ci,).
A particular equivalence class, f-‘(p,), appears as the dot#ted enclosure in Fig. 1. Consequently, the problem of computing moment,s of the distribution function amounts t,o comput#ing the orders of all the equivalence classes comprising the set perm(cl . . . ck). Essentially three dist,inct, sets of entit,ies are used in computing momentsthe septenary numbers just described, a graphical representation, and a matrix
FIG.
to the
1. An illustration set of part,itions
of the salient features I pL , , ~,~,n, I of k.
relating
a permutation
set perm(c,
. .ct)
VIBRATION
SPECTRA
OF
RANDOM
LATTICES
310
.
oh--L .
.
.
.
.
.
.
.
. FIG. 2. A graphical
representation
.
0
FIG. 3. A correspondence
.
.
of a septenary nurnber
.
0
*
2
l
4
l
6
. .
het,ween septenary numbers ttnd segment,s of a graph
represcntntion of these numbers. To symbolize these nwnbers graphically a grid is constructed containing 1~ + I columns like that, illwt,rated for k = 6 in Fig. 2. Each nonzero bracket, in Ey. (10) is represented by a continuous
path connecting 0 and 0’ which sat.isfy the following two restrictions: (1 ) Segments of paths must connect points in successive columns, and (3) only wg-
430 me& similar to those in Icig. 3 are allowed. Euvh segment, of a path on a grid corresponds to an element, of 0. Thrse scgment,s arc correlated wit,h scptrnary numbers in l’ig. 3. h segment of a graph ( or a digit c, ) has associated with it a “vector” whirh charact,crizes t,he number of horizontal lines in the grid it connects and its slope. The rwtor corresponding to a segment or digit c, , denoted by n(rij’, is n(c;) = y2 - 2/l )
(Ifi)
where u1 and u2 are the U-coordinates of its left and right end point,s, reapectivcly. In Fig. 4, for example, n(l)
= +1,
n(2) = -1. Suppose cl . . c/; cont)ains rhl digits I, ‘n3digits 3, and n5 digits 5. It is desirable t,o choose the lengt#hsof segments, or more precisely, the vectors corresponding to these digits in such a way tfhat a vector relating to the digit c = 3 cannot be represented as a sum of vect,ors relating to digits c = 1 and c = 5, etc. This affords a necessary independence among t#hesedigits which is manifested in random walk problems by the number of dimensions in which the walk occurs. Consequently, to impose this condition we shall choose n(l)
= +I,
i 17a)
Fro. 4. A diagram of t,he vector associated with a segment, e.g., the vector relating to the segment. labeled 1 is n(l) = yz - yl = -1 where ye and y? are the y coordinates of the left and right end points, respectively, of the segment.
VIBHATION
SPECTB.1
OF
RAKIJOM
LATTICES
n( 3) = IZl + 1,
(in))
n(5)
(ITC)
= (n1 + I)(tl;~ + 1).
The matrix representation of the set) perm( cl . . . ck) is derived vwtors associated with the /c;j in the fdlcJ\viilg tray: (:i\cn c1 . 1:
=
((r’,
4’1
from the ck define
1 by
ai,-
EE
lJ ZJ =
For es:1mp1r, S -
for
0
for
i
>
i
5
j,
(18a) (18h)
j.
where
1023344
r 1 01
-1 021I
--I Ii?:=
1
3
21
-1o-
1 3 2-l
1 2
-1 0
2
-20
-29 --4 -_
Cl!,!
Some special properties of there matrices ( called Y mniriccs) aw noteworthy. I;irstly, it is clear that “,j
=
2 (T,~, for
i 5 j,
(“0)
which means that thr matrix is tmiquely specified by its elements on the principal diagonal. Swondly, consider two rows i and j(i < j) where (T,* = (Tjs
(20a)
for somes in lj, j + 1, . . . , k/ ; then it foliows that. IQ. (20~~)holds for all s in Ij,j + 1, ... , X). If a row shares this relationship with any row above it, then we say it is Icdznld~nt-othermisc it, is distinct. It follows from these definit,ionr t,hat rows 1, o -, 5, 6 in Erl. (19) are distinct. The following theorem allows us to use s matrices rat,her independemly of perm (cl ... ck) to compute the orders of rcluivalrnce classes.Theorem: Consider 1)erm (cl . . ci.) and the associat’edgraphical and m&is representations which are 1P1, ... , P,I and 1S,i . . . ) 2, 1, respectively. Suppose h and h’ are, respectiwly, defined by the construction of graphs and S matrices. Then there exist. ftmct,ions f’ and f” such that, the diagram
422
BRADLEY
h perm(q
*.. ck) \
h’ 1
(21,. . . ,Z,} :
‘f fN
is commutative, i.e., ,f(a) = j’(h(a)) Also h and h’ are each 1 - 1. Proof. We must show that
iP1, . . . , p,,(k) 1
= .f”(h’(a))
for all a in perm (cl *. .
ck).
h(cl .‘. ck) = h(ci, . . c,,) implies that Cl
. . . Ck
=
C,[
. . . c,,
.
If this were not, true we could contradict that the value of h is the sameevaluated at these point’s by constructing two different graphs. Similarly h’(cl . .
Ck)
= h’(c,, . . .
Cik)
implies that Cl . . . cl; = c;, . . . Cik. Suppose t,his were not. so. c, . . . ck # ci, . . . cik =+” for somej = ( 1, . . . , k} that cj # ci, =+ ?L(c~) # n(ci), or t,he corresponding mat,rices are not equal, and h’(Cl * .
Ck)
# h’(c<, . .
CiJ.
Thus t = r = ‘perm (cl . . . ct). Next we must show that for any c1 . . . ck E perm (r, . . . ck), there exist functions f’ and j” such that, j”[h’(c,
. . . ce)] = ,f(cl . . . cn) = .f’[h(s . . . cl;)],
or y(z)
= ,f(Cl . . * Ck) = f,(P),
where s-c1 .” cL.H P. Consider first, cr . . . ck and P. The means of construct’ing P has been explained. By defilmion f(c1 . . . Ck) = a1 + a2 + . . . + a?, where du,llz . . . &,u, v Cl . . . Ck 3 The
symbol
“3”
means
“implies
that.”
VIBRdTION
SPECTRA
OF
R.iNI)OM
[assuming that perm (cl . . . ck) is in 1 - 1 correspondence elements in the summand of Ey. (8)] means that cr,,,, . . . cl,,,, h
apart from factors containing P F lP, ) . . . , PJ by f’(P)
LU,"'11/~"+:
force
433
LATTICES
with a subset of the
. . . M",E-l
constants.
Let
us define f’
at
= cl1 + cY2+ . . . + cl!, )
where CY~ is the number of left. end points of segments of P lying on a horizontal line in the grid (say horizontal line on which graph begins), etc. Sow it, remains t,o show t.hat, Ia1 ) QL’, ... ) a,,) = (a1 ) a? , . . . ) al) . Suppose not. Then let there he an ai 6 [ cyl, . . . , cu,). From the definition of .f’ we cannot have the factors clUrn” a,, 7 (~“,,t%2) ’ . . 7LLumai (for any 1~) in d,,,,
. . d,,,, ; this means that .f’(Cl . . . CA) # ccl + . . . + a; + . . . + al .
Hewe, by cont.radirtion ai E 1aI , . . , CY,,~. Similarly it is simple to show that Ia, ‘.. , at!) c {al) .‘. , al}. Kom.-consider Z * c1 . . . cJC aud define ,f” at Z F rvY1, ... ) S,) by y(r)
= Pl + P? + . . + A.,
where ,& - 1 zeros occur in a distiuct row of 2, say row 1, etc. We shall always esclude elk = 0 since it doesn’t contribute information to the analysis. It. must be shown that,
Suppow, for definiteness, that, there are (Yeleft endpoints of segmentsof P falling on the horizontal lines of the grid which contains t,hc left, endpoint of the first segment of P. Sumher these segments by s1, s?, . . . , s,, . Then t,he sum of lengt,hs of paths from the 1st up to, but not including the length of SC, is zero. If s?is the wth segment of the pat.h, then al,m--L= 0. Similarly, the sum of the lengt,hs of t,he next successionof segmentsfrom .s?up to, but, not including s3, is zero. If sais the rbth segment of the path then ~l,r,+-l = 0, etc.; thus we have a collection of 01~- 1 zeros in thr first row of S and crl F {P1,...,PU1.Itisclear by t.his construction that [PI , . . . , &) C la, , . . , a,,} as c-anbe show1 by coiltradiction. Hence the theorem follows. Q.E.D. As a conscquen(~eof the analysis to this point we shall use Z matrices directly
in computing orders septcnary numbers.
of equivalcnc~e clasps
l;or esample, we know by inspection
without
necessarily
reverting
to
that
y(Ey.
19j =1Z3.
Also from Eq. (20) it’ follows t’hat the new matrices obtained the diagonal elements in Eq. ( 19) subject t,o the conditions
011
+
033
=
0,
055
+
C66
=
0,
fl44
+
m7
=
0,
from permuting
is a subset’ of the equivalence class f-‘( 1Z3). In general, it is necessary to develop a systematic procedure for computing the number of permutations of diagonal elements for a given arrangement, of zeros in a matrix to find those differing arrangements of zeros in the matrix corresponding to t,he same partition. To accomplish this let us consider a matrix containing only one zero to the right, of the diagonal. This matrix corresponds to f(c1 *.* Ch) = l”, where the only zero in 2 on or t)o the right’ of the principal diagonal is nl% . We seek the number of permut,aGons of the diagonal elements in Z subject to the condit,ion t,hat only ulii = 0. To state this permut.ation problem more conveniently consider a set of elementIs containing N, al’s N1 -al’s, N, at’s, iV2 -Q’S, N, u3’s, and iL’, -us’s where al , a2 , and a3 are positive integers with Nlal < a2 and Nlal + .V2u2 < a3 . (Here we identify nl with -VI , n3 with N, , and 1~~with Na.) The problem, restat’ed, amounts to finding the total number of distinc&t arrangements of these element’ssuch that, in any given arrangement, no sum of proper subsequencesof consecutive elements is zero; call this number A (‘~1,N,;N?,N?:.N~.N3) . N1 is called the occupation number for al’s, etc. If we divide the arrangements into 6 mutually exclusive sets of arrangements where each set hegins wit,h each of t,he six elements fa, , &us , &a, then
VIBRATIOX
SPECTRA
OF
R.th-DOM
LATTICES
whwc..v = 2(,V1 + :V2 + ~1~::)and 1 = 1fl, f2, kX/. It should tw understood in I+. (21‘) that the 5” operator annihilates the operand, i.e., ?‘,,A!$
)“.--I, _ ‘&, = 0 if CL,, -+- . . . + (I;, = 0 for I; < 2( S1 + A7Tg+ Ma.) or if the owupation of ai, is zero. From symmetry 15~1.(21 ) can hc reduced to
=-r8+96+18
numtwr
= 192.
In other words, ill decomposing perm (12:34.X) we have found that yf-‘( lfi) = 19”. This rec’urr’ence method can be easily extended for the cast of a m&is containing several zero elements on and to the right’ of the diagonal. This is illustrated using IQ. i 19) in which case m ~ =11,1;?.2;o.n T-- ‘& A”‘1;‘2;o,” = 16; 0 .T.---7 7 pi ~_ --.! i L-J L--J , L--A
the line connecting two positions means the elements emering those posit,ions must8 sum to zero, and elemems not comiected must, not sum to zero. The process of computing “w” has just heen described for one arrangement, of zeros in a 2 matrix. In comput,ing the order of one equivalence class labeled by t,he partition P = (al + 1) + . . . + ias + l), any Z matrix helonging to that, class must’ contain al zeros in one distinct8 row, a2 in anot,hcr, etc. For a given arrangement, of zeros, say arrangement, 1, we compute ~11~ (as dcscrihed above). The zeros must then t,ake a different arrangement in the matrix, say arrangement L>,and the resulting 777? is computed. If there are Q different arrangements
There is no algorithm for get,ting these different arrangements of zeros. Fortunately, Q is small enough t,o allow all the arrangements t,o be explicitly listed without too much difficulty. The following check on t)he procedure developed for computing t,he order of an equivalence class ran he used. Consider perm (cl . . . cI;) which contains TABLE LIST b)
=
1,
. ..
, (p8)
~p2i=%[~+~~&q
numbers
ooo
3 12
4 13 112 22 1011
(ma),
Septenary
Partitions of 3
Partitions of 4
I
OF MOI\IENTS
012
1 6
Septenary oooo
0012
numbers 1122
1234
4 2
16
1 8 4
8
VIBRATIOS
SPECTRA
OF
TABLE
RANDOM
427
L.-ZTTTCES
II
hJ! Septenary
Partitions of 5
ooooo
number
00012
01122
01234
10
40 40 -40
1
5 14 113 1112 122 23
10
10 10
10
TABLE
III
(PI?) Partitions of 6
_____--000000
G 15 24 33 123 114 222 1122 11112 1113 111111 Total nlunber permutations
~~l’s,
1&<‘s,
.~ 000012
Septenary 001122
numbers
001234
111222
112234
2 12
24
123456
1 12 12 G
18 12 30 18 6
144 72 24 72
G
72 48 24 12
144 288 9G 192
20
180
720
48
of
and
diagonal elements t,he P,t,h partition
1
30
90
3G0
If Hi, rrprewnts the number of ways of arranging the in a matrix corresponding t,o the st,h arrangement of zeros and at, some stage of the rnlculation, t,hen we have
1~~;)‘s.
(23) Whe.11 the equality Fig11 in Ey. (23) holds, the calculations have heea complet.ed. It would he impossible, sinrc me are s~lmming orders of equivalence classes, fol the inequality sign to be reversed. Anot,her problem which has manifested itself in the course of this calrulat,ion is t,hat of determining the range of f (which maps a permutation set into par-
428
BRAI)LEY
Septenary
Partitions of 7 7 61 52 511 43 421 4111 331 322 3211 31111 2221 22111 211111
number of permutations
Total
0000000
000001i
numbers
0001122
0111222
28 28 42 5G
14 28
0001234-
0112234
0123456
1 14 14 14
28 28
1
42
210
112
28 28 28
224 56 112 112 168
14
56
140
840
84 84 5G 28 520 168 168 168 84 1260
33G
1008 672 336 1344 1344 5040
titions). In t,he majority of cases it is simple t#o determine whether or notI a partition is in t#he range off, but occasionally t,his decision is not obvious. In such cases it, is best, to use Z matrices to decide this qu&ion. For example, is f-‘(P) vacuous (,where P is a partition in question)? Suppose the partition is arranged such that a ;2 a;+l for i = 11, . . , 1 - 1). Since t#he first row of a Z matrix is always distinct, arrange al - 1 zeros in this row. This automat)ically specifies a nnmhcr of redundant rows. Place a2 - 1 zeros in the second distinct row (enumerating distinct, rows from the top to the hotltom of the matrix) such t,hat this arrangement is compaGble with the conditions specified by row 1. If however, t,he arrangement of zeros in row 1 are not, compatible with any arrangement, of zeros in t#he 2nd distinct row, then the zeros in row 1 are rearranged until the 1st t,wo distinct8 rows are compatible. This process is continued down to the (I - 1) th distinct row. In practice there are relatively few arrangements of zeros in any row so that this procedure does not become laborious. The largest number of zeros is placed in the first row because this numher of zeros will appear in the first row of some mat’rix although it might, not, appear in some other row if the part,it)ion is in t,he range of .f. Consequently, if this arrangement, of zeros can be realized then the corresponding part)ition is in the range off. A brief summary of the procedure developed for computing (pZk) is listed on the following page.
VIBR.ITIOS
SPECTRA
OF
RANDOM
420
LATTICES
step 1: List, the partitions of k explicitly Step 2: List, t-he different8 permutat.ion setIs which comprise Eq. (20) Step 3: TJsc the procedure developed in the preceding paragraph to compute the range of .f. Step 1: Compute 111for each arrangement of zeros in each rquivalence class. St,ep 5: Use Eq. (23) to check t.he sum of decomposed element’s in each permutation set.
Septenary
Partitions of 8
0
0
0
:: 0
i 0
i 0
i 0 0
8
1 1 : 2 2 2 2
1 2 2
: 2 2 2
10 40 6-l 80
a2 96
: ;
: 22
3 4 .i 6
3 3 4 4
128 128 lMi4 640
128 192 576 128 40 320 608 256 80
1
71 62 (ill 53 521 5111 -1-l X31 122 4211 41111 832 3311 3221 32111 311111 222“ 22211 221111 2111111 11111111 Total nrxml,~r of permut ations
00 1 2
0 0 1
numhers
16 l(i 16
8
30 80 10
-lo
!c,O
24 1-1-l 48 48 -lx 82 80
2 I6 12
l(i 11;
320 (i-l
192 1!J2
320 l(i0 l!U
288 !Mi 8&l I!)2 Oli -HO 105(i i(i8 96 ‘36 38-l 240
1 Ii0 96 I!,2
l(i
8 8
1
56
-120
5Ml
70
1680
50-M
768 32 06
2:w-l 1152
!Mi 96 320
15X 1020 -Hi08 15X 192 2304 3840
1Ii 160 224 Ii-l 16
1120
20160
832 2752 2560 ‘J!J2
10080
2520
Septenary 0
0 1 1 2’
0 1
3 3
2 2
::
34
RI“8 1 GO! V2 1‘2384 1872 (iO48 15!)84 11232 144 3741 N4
720 1728 3816 720 576 2376 43”O .I 504 1080 12!)(j
144 504 1008 288 72 100x 11144 252 1080 iX
2 1MI G480 864 12’Ni
432 2lGO 8&4 1008 2lti
Hi4 8G4
2lli X%2 216
0 0 0 0 0 1
i 1 1 1 2 2 2
Partitions of 9
numbers
i 4
2:
111111111 21111111 2211111 282111 22221 3111111 321111 32211 3222 33111 3321 333 411111 42111 4‘,21 d 4311 432 4-11 51111 5’211 622 53 1 54 (ill1 ti21 G3 711 72 81 9 Total numhel of permut~a tions
18
54 144 3ti
12(i 108 234 lci2
18
18 18 18
54 108 108 108 90 54 54
72 108 288 108
54 18 108
54 108 72 54
54 72 18
72 216 72
2l(i 21(i 432 21(i 21ci 216 432 72 432
180 GO
5760 1728
?AiO 288
X(i40 1OXli8 1440 345G 4320
540 1 i‘%I !bOO 57G 1728 72 108 11.52 18X 1728 468 324 288 1512 21(i 57fi
1728 G!ll2 4320 4320
1728 4320
1440
100x 504 828 36 72 47”.Y 181)
3GO x0
216
1 1
72
75G
1080
630
3024
GO480
15120
90720
22680
10080
VIBRATION
III.
LIST
OF
SF’E(‘TH.4
MOMENTS
OF
OF
THE
R.ih-DORI
431
I,ATTI(‘ES
I~ISTltIUUT1ON
FUNCTI0X
The moments (p,,), * . , (~1~) arc listed in this s&ion in Tahlw I through 1’1. To explain these tables the rradw is refrrred to Tahlc V for (p16). In particular, wnsider the second row labeled hy t’he part,ition { 7, I } and t,he second column lahcled hy the srptrnary numhrr 00000012. The elemcntj Iocat.ed at, the interswtioII of t.his row at~d column is 16. Thus from this c,lcmcnt \vc ohtail
which contributes to (~~6). WC affix thr c*oc+kkient colttaining force cwnstants from thr second c4umn. This cwlumn lists t,he poplllation of ecluivalcnce biasses of perm (00000012). From the prwwding section of t,his report we know that thr voefFic+t~t. of force coast.ant.s associated wit.h each clcnwnt. of this permutat.iou set, is 71’y”/a6.Similar columns relating to prrm (00000034) and pwm (0000005ti ) havr ken omitted since their cquivalcnw (4ass dwompositions arr identkal to those for perm (00000012). The 0111~difkrenw among t#hes:Pt,hrcc sets arc thv co&ic+nts of force caonst’ants which they imply. Thp coeffkirnt~ associated ait, perm (00000034) is yzL-y3’ and that for ptrm (0000005ti) is t hc same. COI~SCclllclrtly, the (2, 2) rlrmtnt of Tablr \T cont.rihut.cs l(j(y,’
+ 2y2’)yaR(l,li~/,,,,,,,,,,:,~lli,,+l,,,,,,:,)
t,o (c(,~). Each entry of Table T’ makes a similar contrihut~ion to (~~6) and (p16) is t,hr sum of all such krms. All thr mriments arc rrprewlted in this fashion. LTsiug ‘l’ahlw I through VI the reducttd moments -2i (/Qj); k!J = %,a.x were csalculakd ill Table VII.
.II,(
for diffcrrnt
physical paramctrrs.
Reduced moments for = 25rjj, p = ‘i, yI = 10?!. /Lo p2 fl, pe pq p,,, p1: p,r /.I,6 /.m
= = = = = = = = = =
1 .o 0.375 O.ZO(il 0.1326 0.0!,228 0. w734 o.mO71 0.039oi 0.03068 0.02143
j = 10, . . . , !)} A list of these moments appeal
Reduced moments for Al., = >lfB , ‘y, = IO-), p,, = /I> = p, = ,uf, = p, = p,c, = p,z = p,t = jl,6 = glfi =
I .o 0. .5 0.3385 0.2578 0.2075 (I. 1725 O.lW 0.1201 0.1100 0.09G30
II 0
0.5
1.0
0.5
0-
1.0
Wwmax.
W”max
FIG. 5. (ii) The histogram representing the distribution function for the random for 1If.t = 2!W, , p = ‘6, y1 = 10yz (h) The hist(ogram and exact, curve representing distribution function for a lnt,ticr in which M,t = MB , y1 = 10~~ . IV.
THE:
FREQUENCY
SPECTRUM
Ah’I)
SOME
RELhTEl)
lat,tice the
PHYSICAL
l’R( ~1’ERTIES
The kchniqut descrihrd in Scctiou I and the numerical values listed in Tal+ VII for the moments were used t,o estimate the frequency sp&rum by histograms which appear in Fig. 5. The c’aw cited in Fig. 5(a) indicaks that no apparent, separation of opt,icul and acoustical hands appear. Two peaks occur on this histogram which are situatjcd approximat,ely at the positions of infinite discontim&ies in the spcc:t8rum associated witfh the prrfect lattice; there is no rvidencc, however, that infinitr peaks should occur in the speckwn for the random lattice. Figure 5( 1)) is an illust,ration of the spectrum associated with the perfect, monat,omic lattice as calculated by this procedure. The smooth curve in this figure corresponds to the exact spectrum for t,his case. Since wIII:Iy = lim (m)/(m4, P-m Figure
6 reflects
some information
on thr
assumption
made
earlier
about,
the
II I
2
III 3
4
5
I 6
II 7
8
k
I 9
I IO
I
masimnm frequency of the lattiw. The momcwk in Talk T’II (first’ column) were used ill this cxlculation. The value 1 on the ordinat,e corresponds to the mnximum frcquenc’y of the lat,tiw of light atoms. The dot,ted lille refers to the masimnm freclucnvy for a perfect Iatt)ire ill which dl, = %I/, and y1 = IOr2 . Zvyagina and Ireronova (6 ) hare cxl~+ulntrclthe changesin 8”, the mean-square de\-iation of atoms from thrir tquilihrirm~ position for ordered and cwmpletcly disordered :<-l) dintomk lutticcs, in an attempt to study the behavior of the strrlqth CJf materials in going to high temperatures. They c~~ncl~~clc that for a aompleirly disordered, low c’ollc~c,llt,r:Ltiol1, dintomic: lnttiw, w,,,:,~(disorder) < w,,,;,~Corder), which t#endsto dwrensc k’. Our c~alrulatiolls slIpport the opposite effect sinw the inequality sign ahove shordd 1~ reversed if the maximum frrqwncy of the ralrdom Iat,tiw has t,he same signifiranw as the maximum frcquen~y of the perfwt lattjiw. 1~01~~1 J it al. (3) cloreloptd a method t’o c:alrulntc t#hrzero point energy per dcgrcc of freedom, co, for a random linear (*hain. We cxtcnd this c*alcv~lutiollto the c’ase of a rnr&m x-I> lattSicc. 11sI)onllJ ci nl. indicateq it is more ~~~Jllv?llk!llt to IISC
434
BRAOLET
t,he moments expanded 1/I~,,}. Then using
about .r = 0 t,han the pzk’s. Call these rnonwnt,s
the (24)
the { I’~,,) are defined by
f’z,, 3s0l(l-.r?)“j(.r)d.r =W&Z” 2(-])“I 0; UB”,,
( 2 5 ‘)
where Il.1m = w,z
(k,).
(20)
We use the one-sided delta fun&on (“7) and
(28) and .f( a) = (‘01 + (‘zlxy + . . .
(29)
to obtain (30) RIaradudin
and Weiss ( 7) obt,ained an exact cxlcwlation lim (g(w)) w-,I
of c’l. Une would expwt
= y(O) = Co’,
(31)
where g(w) represents t’he distribut,ion function for the perfecst, 3-D lattice. This implies t,hat Co = 0 iu our case. Onre Ca’is known, expressions for the (C:,} CMI be computed. For example,
Corresponding to t#he parameters listed in Table VIIa, (‘1 E + 1.18. Finally the zero point energy per degree of freedom is ( X? )
(‘ol~~;etI”(‘“l-ly,
thr wro point cncrgy prr dcgrw
of fwcclom
is
The zero point’ energy f’or thr r:mdom lattice (wing the data from Tat+ I’IIa j is lower than that for the awwiated perfect lattice by onr prr writ. This, however, is not iL st,ahilit,y critcrioll since \w never wilsidcrcd n c.ollfigrtrtltiollnl energy in our citlculat ions.
436
BRADLEY
REFEREXCES 1. F. J. I)YSON, Phys. Rev. 92, 1331 (1953). 2. H. SCHiNIUT, Phys. Rev. 106, 425 (1957). 3. C. I)ORIB, A. A. M~R.~I~UDIN, E. W. MONTROLL, AND G. WEISS, Php. /2ev. 116, 18 (1059) 116, 2-l (1959). 4. E. W. MONTROLL, J. (‘hem. Phys. 10, 219 (1942). 5. G. WEISS, Ph.1). thesis, Universit,y of Maryland, 1958. 6. A. 1’. ZVYAGINA AND V. I. IVERONOVA, SoGet Phys., Solid State (Translation of lf’iz Yverdoyo Tda) 2, 108 (1960). 7. A. MARADUDIN AND G. WEISS, .I. Chem. Phys. 29, 631 (1958).
OF PHYSICS:
15: 411436
Vibrational
(1961)
Frequency Crystal J. C. RI.lS,
Spectrum Lattices
of Random
BRADLEY*
Baltimore,
Maryland
A met,hod was developed to determine the moments of the vil)rat,ional frequency spectrum associated with a three-dimensional, two-component, random lattice. The frequency spect.rum was approximated from it,s zeroth through eighteenth order moments. A hand gap no longer appears in the spectrum. A.lso it is inferred from these calculations that t,he maximum frequency of the lattice is approximately that associated nit,h a lattice of light atoms. The clero point energy of the random lattice was calculated from an expression involving the moments of the frequency spectrum directly; it was found that this energy is lower than that associated \Cth the perfect latt,ire. I. IKTROI~UCTION
Recently a significant amount of work has heen devoted to the study of the density of vihratjionnl frequencies associated wit’h disordered crystal lattices. In particular, Dyson (1) and Schmidt (2) have presented theories for disordered linear chains. Later Domh et al. (3) made an exhaustive study of the vihrational spectra associated with “random” linear chains. They derived a gcnerating function from which they could compute the “configuration averages” of moments of the spectrum. This exposition is devoted to the description of a method that was developed to oktin a probability density function for lattice vibrations associated with a t,hret dimensional (:3-D 1 simple cubic “random lattice” containing two kiuds of atoms. It was desirable to develop an approach to this problem since t.he previous work on vibrations in “random lattices” has been confined to the 1-D case. To he more specific n randor~l crystal lattice (phrases being defined are italicized) of two components and X” atoms (as we shall consider) contains N”p =l-atoms and N”(l - p) B-atoms with the provision that a given lat,tice site contains an d-atom with probability p or n B-atom with probability 1 - p. This means that both short- and long-range order parameters vanieh or that the state is completely disordered. We *hall consider a simple cubic crystal lattice where only first nearest) neighthe
* Hased on a t.hesis submitted reqrlirrments for the degree
to the rniversit,y of Maryland of Doctor of Philosophy. 411
in part,ial
fulfillment
of
bors are assumed to interact, through central and noncent8ral forces. In particular, t,he force is assumed t’o ha1.e the same direction as an at.omic displacement,. If the displacement is in the direction of the line joining two neighboring atoms the force constant is denoted by y1 (cent,ral) and if it is orthogonal t,he constant is denot’ed by y2 (noncentral). Consequently, y1 > y.’ . Let h represent, some property of a lattice whose value is dependent, upon the arrangement, of it,s atoms. Suppose a set 1 indexes t,he different arrangements of ;2 and B atmoms (‘1 denotes the number of element8s in 1 where t,he corresponding set of A’s is (Ai)i,I’). Then the propert,y A associated with t)he random latt,ice is called the con$gu&ion average of the ( Ai) cc I and is defined by
The quantities of primary concern in this connection are the distribut)ion funct,ion, its moment8s and the thermal properties of the lat#tice. One of the most fruit,ful means of determining t,he shape of the frequency spectrum for random lat,tices, (g(w)), is the moment-trace method (4); it, depends only on the details of t,he crystal model wit#h essentially no limiting features such as the exist,ence of a unit, cell or the dimension of the crystal. We obtain an approximat,ion t,o (g(o)) by it,s first N + 1 nonzero moments and the assumption that it can be represent)ed by the series
where w,,,:~~is the maximum
normal mode of frequency
of vibration,
x = w/wIrI:,x ) and fn (x) mesh
is a simple function
defined in the following
fashion:
Consider
the
so = 0, x1 ) x2 ) . . . , x.,7+1 = 1
of [0, l] where s, - .rP1 = l/(N
+ 1) for i = 11, . . , N + I}. Then
f’ (s) = 1 if .c C Lri , x,+11 0 ot herwiee
.8
for i = (0, . . , -V). 31ult8iplying Eq. (2) by J? and integrating over the unit interval gives
s 1
.P(g
( xw,,,:,x
1)
dx
=
g
(I'
.i2%
Cx)
dx)
ci
0
forj = (0, . . f , NJ. In matrix notation these equations may be written as
VIBRATIOIi
SPECTRA
OF
KAKDOM
4 13
L.lTTICES
G = Fc, where
and
Since F is nonsingulw tained from
the coeficicnts
in the expnwion
in Ey. ( 2 ) are readily oh-
c = F-‘G. If the (2k)th order moment of g( w i corresponding is defined by’
to one arrangement. of atoms
then
Let II:: denote the (u, IH )th clement of F-’ hy F!,$‘; come
then the elements of c he-
N
(c,,) = c
Vi=”
The predominant
numhrr
of :trralIgements
1 Since g(wj is an even function chmge in the spectrum is cawed we used
instead of Eq. (2) where the mirror image of ,f,,(.r)
reduces to
11 = IO;..
F!-,‘(w,:;-‘A,);
,iV},
(3aj
of atoms in the latt.icc allox for large
it] w it follows by Eq. (1) t.hat. (y(w)) is dso. No essential I,- the representation of (g(w)) I)>- l’kl. 12). For suppose
F,, is defined un I- 1, 11, F,,(S) :rbollt the ~1 usis; then clearl?
= ,fSt(x)
+ ,f.( -.r~
and ,f,&(-x)
is
clust8ers of the two atomic species to occur. In so far as these clust,ers can be regarded as separate lat’tice struct’ures we shall assume that t,he maximum frequency of the entire aggregate is the maximum frequency of a similar lattice structure containing only light, atoms. Consequently Eq. (3a) reduces to n, =
(0,
...
) NJ.
(33)
The first ten nonzero moments ((&, (&, . . . , (pl8)} mere ralculated for the crystal model described above, and (g( 0)) was estimated from Eq. (2 i for different physical parameters such as the ratio of masses of t,he two at,omic species. Also from the moments a calculation of the zero point, energy and an est.imate of the validity of the assumphion concerning the maximum frequency of t,he lattice were obtained. The zero point1 energy for the random lat,tice is lower than that for t,he perfect lat8t.ice. II.
MOMENTS
OF
The moments of t#he distribution
THE
1118TRIBUTION
function
FUNCTIOS
are related to the dynamical
matrix,
D (the matrix derived from t,he classical equations of motion and whose eigenvalues are the squares of the normal mode frequencies; equation : (~21;) = A-” (Trace
D")
see (4))
by the following
(4)
Since the motion of atoms in one direction are not coupled t,o the motion of atoms in other directions the dynamical matrix appears in the block form. It is necessary to consider only one block as was done in Eq. (4) and in the following development where each block is an N3 X N3 submat,rix. It is not pos-
sible, however, to furt,her simplify the problem in this fashion as is often done with regard to periodic arrays of atoms.
VIBlUTION
Each matrix pose the (~1, sometimes he zero elements
SPECTRA
OF
HAiYDOM
115
LATTICES
element, in the dynamical mat,rix is ident,ified by sis indices. Supus, us ; ul’, ZL~‘, u3’)th element is d~,LI,U?,L(Q;IL,,,L(~,,Ua,) . This will written for hrevit#y as d,: , where u = (ul , us , Q). The only nonappearing in the uth row of D for t,he lat,ticc described above are cl(I‘*.Ut,~:~;“1,“.‘.,~~) = 2l-u + a9)l~JIL,,M?,IIa , d(!<,,“.‘.~a:,,,*l.r,,.rcl) = --rJJ/
(I ("~,"~.ic:~;lL,.Il~*l,IL:~) = d (Ic,,s?.?,:~;l,,,UL,lLq~l)
(5)
Il,.II?.lC3 , =
C(j) -Ysj~~~q,q,q
.
(7)
It is important, for later use t,o not,icc that only one random mass variable, JI, , occurs in the uth row of D. The moments of the distribution function may be expressed as
(llw)= m lljl~~~,llp c d,,“,cl”,“, ... d”,,,) and since t)he lat#tice is random the sum in Eq. (8) commutes with the configurat8ion average t 0 give
The elemrnk in the summand of Eq. (9) are defined as brackets. Since the brackets are independent, of the diagonal element of D” in which they are (‘ontained, Eq. (9) may be summed over the first index t,o give Gad =
c (d”,,, . . cl”,“,). ll2,.“.llg
,is 5~11 example of Eq. (lo), (pqjis computed from Eqs. (.5)-(7), the only nonzero elements are given by (4
(10) realizing that
== g &,u, hl,“,)
The configuration average of the mass terms using Eq. (1) become
where q = 1 - p. It, is apparent that the configuration averages are computed assuming that neighboring atoms do not, influrncc one another (property of randomness). This allows for all possible c,on~elltrat,iolls of il and B atoms to
occur in the set, of configurations. It is useful to picture this latt#ice as a system in an ensemble of lat,tices in which the concentration of .4 and B at,oms is preserved. As a consequence of this, Weiss (5 ) has shown t,he fluct8uations from the stoichiomet,ric ratios of 9 and B atoms in the lat’tice are arbitrarily small in t#he limit as t,he number of atmoms in t,he lattice tends to infinity. The importrant feature of Eqs. (11) and (12) that are used in computing the moments can he sunlmarized as follows: Notice that only part,itions of 2 appear as exponents of t,he masses in these equat,ions. Similarly, only a partiCon of li can appear as exponents of masses in an element of the sununand of Eq. (10) ; e. g., only hrackct,s of the following form occur in (pzr;): nr,“: . . JI&
(I/w:: where the {a,} are positive
integers and al + Uz +
. . . + al =
Ii.
(The subscript,s on t,hc massesare considered distinct in t,his bracket. j Then t,he population of nominally distinct, brackets, such as that above, must he determined for each partition of k. The central problem is to deduce the population of equal brackets which occur in a given moment. It is useful to identify the elements in the summand of Ey. (10) with a set of permutations of a prescribed set of digits. k’or example a bracket
(where some of the subscripts on massesmay he equal) can he made to corrcspond to a set of numbers to the base 7 (septcnary nrmihers) in t)he following fashion’ 0 if ui = uj = (.U~, ~1~, u3j
duiu, -
1 if
Uj
= (U1 + 1, U2,
2 if
Uj
= (U1 - 1, II? , U:j)
Us)
3 if uj = i 111 , u? + 1, us). 1 if
Uj
=
( 111 , ll2
-
(13)
1, lL3)
5if uj = (ZL~,UO,U~ + 1) B if
Uj
=
(ZL1 , Uf , ?La -
1)
The number 7 nat’urally arises as the base since it corresponds to the number of elements in a row of D given hy ICys. (T,)-(7). This is a manifestation of the interaction of an atom at a lattice site with its 6 nearest neighbors. If we utilize 2 The
symbol
“u”
means
“one-to-one
correspondence.”
VIBRATIO;“i
septenary numbers each bracket, (~4)
can
SPECTRA4
and ignore be writ,ten
OF
RrlNDOM
the expressions as
~4 = COO) + perm(l2)
LATTICES
417
containing
+ perm(34)
force
constams
+ perm(56),
(14)
where perm(cl . ck) is used to desiguate t,he set of pernnnations of rrary number cr . . . ck . Each septenary number c1 . . . ck corresponds and only one bracket. The sept~enary numbers arising in terms of the type appearing in are characterized by the r&rictions that, the number of I’S (say q) to the number of Z’s, etc., and the number of O’s is equal to JC- 2( 12~ + I:or example, let, nl l’s, la3 3’s, Q, 5’s occur in a septenary numhcr; then consider permutations of Cl,
\Ii
-
2
(i'll
. ’ . ,o +
It,:{
. . .
1 -.L,L.zL,L.L +
?Lj)
1
‘>
. . .
”
'
3 I
1 111
. . ...3 c-;-i’
the septeto one Eq. (14) he equal rt3 + ns). we must
J,...,-l
nzj
111
113
5, . . , 5 6, . . . 6 L--J. L-,L
%i in connection it is obvious &)
with computing &). As another t,hat computing (~(12) gives us
= ( 000000)
+ (zj
perm( OOOOab) + +
\zj
example
C C perm(OOahcd 1a.b) (cd) C
perm(aaabbb)
(15)
%
similar
to l~q.
(14)~
pcrm(OOaahb) ) + (5)
c”.b)fcr.d)
+
in
,z,
perm (aabbcd)
to.h)#(r:d)
+ perm(
123456),
(o.bl
and &
.t,) means
wnm~ation
where j(1,2),
the pair
(a, h) takes
on values
(3, 4), (5, B)}.
.Inother feature of these septenary uumhers is that, the product of force const,ant’s appearing in a hrwket reprrsrntcd hy any permutation conlposed of elements in ( 15) is
where
The number
of elements
in the permutation
set, of digits
ill ( 1.5) is
418
BKADLEY
7(k)
= [Ii - 2(n1 + a3 +?,I:
and t,he number of elements in the summand
(nI!)yn3!)yn6!)2 of Eq. (IO) is
where [k/2] is the largest integer in /i/2. To st,ate the problem in a more useful form let us consider perm(cl . . . ckj. We already understand how an element of this permut,ation is related to a partition of k (the partitions of k are designat,ed by jpl , . . . , P,,(Q}. Denote t,his functional relation by
.f: perm(c1 . . ck> - {PI , . . . , P,(k,I. Then S is single-valued and defines an equivalence relation on perm(cl it is illustrated in Fig. 1. If we let, 0 denote t,his equivalence then Cl . . c,ec,,
. . . ck) ;
. Cik
if and only if .f(CI . . . cl;) = f(c,,
. . . Ci,).
A particular equivalence class, f-‘(p,), appears as the dot#ted enclosure in Fig. 1. Consequently, the problem of computing moment,s of the distribution function amounts t,o comput#ing the orders of all the equivalence classes comprising the set perm(cl . . . ck). Essentially three dist,inct, sets of entit,ies are used in computing momentsthe septenary numbers just described, a graphical representation, and a matrix
FIG.
to the
1. An illustration set of part,itions
of the salient features I pL , , ~,~,n, I of k.
relating
a permutation
set perm(c,
. .ct)
VIBRATION
SPECTRA
OF
RANDOM
LATTICES
310
.
oh--L .
.
.
.
.
.
.
.
. FIG. 2. A graphical
representation
.
0
FIG. 3. A correspondence
.
.
of a septenary nurnber
.
0
*
2
l
4
l
6
. .
het,ween septenary numbers ttnd segment,s of a graph
represcntntion of these numbers. To symbolize these nwnbers graphically a grid is constructed containing 1~ + I columns like that, illwt,rated for k = 6 in Fig. 2. Each nonzero bracket, in Ey. (10) is represented by a continuous
path connecting 0 and 0’ which sat.isfy the following two restrictions: (1 ) Segments of paths must connect points in successive columns, and (3) only wg-
430 me& similar to those in Icig. 3 are allowed. Euvh segment, of a path on a grid corresponds to an element, of 0. Thrse scgment,s arc correlated wit,h scptrnary numbers in l’ig. 3. h segment of a graph ( or a digit c, ) has associated with it a “vector” whirh charact,crizes t,he number of horizontal lines in the grid it connects and its slope. The rwtor corresponding to a segment or digit c, , denoted by n(rij’, is n(c;) = y2 - 2/l )
(Ifi)
where u1 and u2 are the U-coordinates of its left and right end point,s, reapectivcly. In Fig. 4, for example, n(l)
= +1,
n(2) = -1. Suppose cl . . c/; cont)ains rhl digits I, ‘n3digits 3, and n5 digits 5. It is desirable t,o choose the lengt#hsof segments, or more precisely, the vectors corresponding to these digits in such a way tfhat a vector relating to the digit c = 3 cannot be represented as a sum of vect,ors relating to digits c = 1 and c = 5, etc. This affords a necessary independence among t#hesedigits which is manifested in random walk problems by the number of dimensions in which the walk occurs. Consequently, to impose this condition we shall choose n(l)
= +I,
i 17a)
Fro. 4. A diagram of t,he vector associated with a segment, e.g., the vector relating to the segment. labeled 1 is n(l) = yz - yl = -1 where ye and y? are the y coordinates of the left and right end points, respectively, of the segment.
VIBHATION
SPECTB.1
OF
RAKIJOM
LATTICES
n( 3) = IZl + 1,
(in))
n(5)
(ITC)
= (n1 + I)(tl;~ + 1).
The matrix representation of the set) perm( cl . . . ck) is derived vwtors associated with the /c;j in the fdlcJ\viilg tray: (:i\cn c1 . 1:
=
((r’,
4’1
from the ck define
1 by
ai,-
EE
lJ ZJ =
For es:1mp1r, S -
for
0
for
i
>
i
5
j,
(18a) (18h)
j.
where
1023344
r 1 01
-1 021I
--I Ii?:=
1
3
21
-1o-
1 3 2-l
1 2
-1 0
2
-20
-29 --4 -_
Cl!,!
Some special properties of there matrices ( called Y mniriccs) aw noteworthy. I;irstly, it is clear that “,j
=
2 (T,~, for
i 5 j,
(“0)
which means that thr matrix is tmiquely specified by its elements on the principal diagonal. Swondly, consider two rows i and j(i < j) where (T,* = (Tjs
(20a)
for somes in lj, j + 1, . . . , k/ ; then it foliows that. IQ. (20~~)holds for all s in Ij,j + 1, ... , X). If a row shares this relationship with any row above it, then we say it is Icdznld~nt-othermisc it, is distinct. It follows from these definit,ionr t,hat rows 1, o -, 5, 6 in Erl. (19) are distinct. The following theorem allows us to use s matrices rat,her independemly of perm (cl ... ck) to compute the orders of rcluivalrnce classes.Theorem: Consider 1)erm (cl . . ci.) and the associat’edgraphical and m&is representations which are 1P1, ... , P,I and 1S,i . . . ) 2, 1, respectively. Suppose h and h’ are, respectiwly, defined by the construction of graphs and S matrices. Then there exist. ftmct,ions f’ and f” such that, the diagram
422
BRADLEY
h perm(q
*.. ck) \
h’ 1
(21,. . . ,Z,} :
‘f fN
is commutative, i.e., ,f(a) = j’(h(a)) Also h and h’ are each 1 - 1. Proof. We must show that
iP1, . . . , p,,(k) 1
= .f”(h’(a))
for all a in perm (cl *. .
ck).
h(cl .‘. ck) = h(ci, . . c,,) implies that Cl
. . . Ck
=
C,[
. . . c,,
.
If this were not, true we could contradict that the value of h is the sameevaluated at these point’s by constructing two different graphs. Similarly h’(cl . .
Ck)
= h’(c,, . . .
Cik)
implies that Cl . . . cl; = c;, . . . Cik. Suppose t,his were not. so. c, . . . ck # ci, . . . cik =+” for somej = ( 1, . . . , k} that cj # ci, =+ ?L(c~) # n(ci), or t,he corresponding mat,rices are not equal, and h’(Cl * .
Ck)
# h’(c<, . .
CiJ.
Thus t = r = ‘perm (cl . . . ct). Next we must show that for any c1 . . . ck E perm (r, . . . ck), there exist functions f’ and j” such that, j”[h’(c,
. . . ce)] = ,f(cl . . . cn) = .f’[h(s . . . cl;)],
or y(z)
= ,f(Cl . . * Ck) = f,(P),
where s-c1 .” cL.H P. Consider first, cr . . . ck and P. The means of construct’ing P has been explained. By defilmion f(c1 . . . Ck) = a1 + a2 + . . . + a?, where du,llz . . . &,u, v Cl . . . Ck 3 The
symbol
“3”
means
“implies
that.”
VIBRdTION
SPECTRA
OF
R.iNI)OM
[assuming that perm (cl . . . ck) is in 1 - 1 correspondence elements in the summand of Ey. (8)] means that cr,,,, . . . cl,,,, h
apart from factors containing P F lP, ) . . . , PJ by f’(P)
LU,"'11/~"+:
force
433
LATTICES
with a subset of the
. . . M",E-l
constants.
Let
us define f’
at
= cl1 + cY2+ . . . + cl!, )
where CY~ is the number of left. end points of segments of P lying on a horizontal line in the grid (say horizontal line on which graph begins), etc. Sow it, remains t,o show t.hat, Ia1 ) QL’, ... ) a,,) = (a1 ) a? , . . . ) al) . Suppose not. Then let there he an ai 6 [ cyl, . . . , cu,). From the definition of .f’ we cannot have the factors clUrn” a,, 7 (~“,,t%2) ’ . . 7LLumai (for any 1~) in d,,,,
. . d,,,, ; this means that .f’(Cl . . . CA) # ccl + . . . + a; + . . . + al .
Hewe, by cont.radirtion ai E 1aI , . . , CY,,~. Similarly it is simple to show that Ia, ‘.. , at!) c {al) .‘. , al}. Kom.-consider Z * c1 . . . cJC aud define ,f” at Z F rvY1, ... ) S,) by y(r)
= Pl + P? + . . + A.,
where ,& - 1 zeros occur in a distiuct row of 2, say row 1, etc. We shall always esclude elk = 0 since it doesn’t contribute information to the analysis. It. must be shown that,
Suppow, for definiteness, that, there are (Yeleft endpoints of segmentsof P falling on the horizontal lines of the grid which contains t,hc left, endpoint of the first segment of P. Sumher these segments by s1, s?, . . . , s,, . Then t,he sum of lengt,hs of paths from the 1st up to, but not including the length of SC, is zero. If s?is the wth segment of the pat.h, then al,m--L= 0. Similarly, the sum of the lengt,hs of t,he next successionof segmentsfrom .s?up to, but, not including s3, is zero. If sais the rbth segment of the path then ~l,r,+-l = 0, etc.; thus we have a collection of 01~- 1 zeros in thr first row of S and crl F {P1,...,PU1.Itisclear by t.his construction that [PI , . . . , &) C la, , . . , a,,} as c-anbe show1 by coiltradiction. Hence the theorem follows. Q.E.D. As a conscquen(~eof the analysis to this point we shall use Z matrices directly
in computing orders septcnary numbers.
of equivalcnc~e clasps
l;or esample, we know by inspection
without
necessarily
reverting
to
that
y(Ey.
19j =1Z3.
Also from Eq. (20) it’ follows t’hat the new matrices obtained the diagonal elements in Eq. ( 19) subject t,o the conditions
011
+
033
=
0,
055
+
C66
=
0,
fl44
+
m7
=
0,
from permuting
is a subset’ of the equivalence class f-‘( 1Z3). In general, it is necessary to develop a systematic procedure for computing the number of permutations of diagonal elements for a given arrangement, of zeros in a matrix to find those differing arrangements of zeros in the matrix corresponding to t,he same partition. To accomplish this let us consider a matrix containing only one zero to the right, of the diagonal. This matrix corresponds to f(c1 *.* Ch) = l”, where the only zero in 2 on or t)o the right’ of the principal diagonal is nl% . We seek the number of permut,aGons of the diagonal elements in Z subject to the condit,ion t,hat only ulii = 0. To state this permut.ation problem more conveniently consider a set of elementIs containing N, al’s N1 -al’s, N, at’s, iV2 -Q’S, N, u3’s, and iL’, -us’s where al , a2 , and a3 are positive integers with Nlal < a2 and Nlal + .V2u2 < a3 . (Here we identify nl with -VI , n3 with N, , and 1~~with Na.) The problem, restat’ed, amounts to finding the total number of distinc&t arrangements of these element’ssuch that, in any given arrangement, no sum of proper subsequencesof consecutive elements is zero; call this number A (‘~1,N,;N?,N?:.N~.N3) . N1 is called the occupation number for al’s, etc. If we divide the arrangements into 6 mutually exclusive sets of arrangements where each set hegins wit,h each of t,he six elements fa, , &us , &a, then
VIBRATIOX
SPECTRA
OF
R.th-DOM
LATTICES
whwc..v = 2(,V1 + :V2 + ~1~::)and 1 = 1fl, f2, kX/. It should tw understood in I+. (21‘) that the 5” operator annihilates the operand, i.e., ?‘,,A!$
)“.--I, _ ‘&, = 0 if CL,, -+- . . . + (I;, = 0 for I; < 2( S1 + A7Tg+ Ma.) or if the owupation of ai, is zero. From symmetry 15~1.(21 ) can hc reduced to
=-r8+96+18
numtwr
= 192.
In other words, ill decomposing perm (12:34.X) we have found that yf-‘( lfi) = 19”. This rec’urr’ence method can be easily extended for the cast of a m&is containing several zero elements on and to the right’ of the diagonal. This is illustrated using IQ. i 19) in which case m ~ =11,1;?.2;o.n T-- ‘& A”‘1;‘2;o,” = 16; 0 .T.---7 7 pi ~_ --.! i L-J L--J , L--A
the line connecting two positions means the elements emering those posit,ions must8 sum to zero, and elemems not comiected must, not sum to zero. The process of computing “w” has just heen described for one arrangement, of zeros in a 2 matrix. In comput,ing the order of one equivalence class labeled by t,he partition P = (al + 1) + . . . + ias + l), any Z matrix helonging to that, class must’ contain al zeros in one distinct8 row, a2 in anot,hcr, etc. For a given arrangement, of zeros, say arrangement, 1, we compute ~11~ (as dcscrihed above). The zeros must then t,ake a different arrangement in the matrix, say arrangement L>,and the resulting 777? is computed. If there are Q different arrangements
There is no algorithm for get,ting these different arrangements of zeros. Fortunately, Q is small enough t,o allow all the arrangements t,o be explicitly listed without too much difficulty. The following check on t)he procedure developed for computing t,he order of an equivalence class ran he used. Consider perm (cl . . . cI;) which contains TABLE LIST b)
=
1,
. ..
, (p8)
~p2i=%[~+~~&q
numbers
ooo
3 12
4 13 112 22 1011
(ma),
Septenary
Partitions of 3
Partitions of 4
I
OF MOI\IENTS
012
1 6
Septenary oooo
0012
numbers 1122
1234
4 2
16
1 8 4
8
VIBRATIOS
SPECTRA
OF
TABLE
RANDOM
427
L.-ZTTTCES
II
hJ! Septenary
Partitions of 5
ooooo
number
00012
01122
01234
10
40 40 -40
1
5 14 113 1112 122 23
10
10 10
10
TABLE
III
(PI?) Partitions of 6
_____--000000
G 15 24 33 123 114 222 1122 11112 1113 111111 Total nlunber permutations
~~l’s,
1&<‘s,
.~ 000012
Septenary 001122
numbers
001234
111222
112234
2 12
24
123456
1 12 12 G
18 12 30 18 6
144 72 24 72
G
72 48 24 12
144 288 9G 192
20
180
720
48
of
and
diagonal elements t,he P,t,h partition
1
30
90
3G0
If Hi, rrprewnts the number of ways of arranging the in a matrix corresponding t,o the st,h arrangement of zeros and at, some stage of the rnlculation, t,hen we have
1~~;)‘s.
(23) Whe.11 the equality Fig11 in Ey. (23) holds, the calculations have heea complet.ed. It would he impossible, sinrc me are s~lmming orders of equivalence classes, fol the inequality sign to be reversed. Anot,her problem which has manifested itself in the course of this calrulat,ion is t,hat of determining the range of f (which maps a permutation set into par-
428
BRAI)LEY
Septenary
Partitions of 7 7 61 52 511 43 421 4111 331 322 3211 31111 2221 22111 211111
number of permutations
Total
0000000
000001i
numbers
0001122
0111222
28 28 42 5G
14 28
0001234-
0112234
0123456
1 14 14 14
28 28
1
42
210
112
28 28 28
224 56 112 112 168
14
56
140
840
84 84 5G 28 520 168 168 168 84 1260
33G
1008 672 336 1344 1344 5040
titions). In t,he majority of cases it is simple t#o determine whether or notI a partition is in t#he range off, but occasionally t,his decision is not obvious. In such cases it, is best, to use Z matrices to decide this qu&ion. For example, is f-‘(P) vacuous (,where P is a partition in question)? Suppose the partition is arranged such that a ;2 a;+l for i = 11, . . , 1 - 1). Since t#he first row of a Z matrix is always distinct, arrange al - 1 zeros in this row. This automat)ically specifies a nnmhcr of redundant rows. Place a2 - 1 zeros in the second distinct row (enumerating distinct, rows from the top to the hotltom of the matrix) such t,hat this arrangement is compaGble with the conditions specified by row 1. If however, t,he arrangement of zeros in row 1 are not, compatible with any arrangement, of zeros in t#he 2nd distinct row, then the zeros in row 1 are rearranged until the 1st t,wo distinct8 rows are compatible. This process is continued down to the (I - 1) th distinct row. In practice there are relatively few arrangements of zeros in any row so that this procedure does not become laborious. The largest number of zeros is placed in the first row because this numher of zeros will appear in the first row of some mat’rix although it might, not, appear in some other row if the part,it)ion is in t,he range of .f. Consequently, if this arrangement, of zeros can be realized then the corresponding part)ition is in the range off. A brief summary of the procedure developed for computing (pZk) is listed on the following page.
VIBR.ITIOS
SPECTRA
OF
RANDOM
420
LATTICES
step 1: List, the partitions of k explicitly Step 2: List, t-he different8 permutat.ion setIs which comprise Eq. (20) Step 3: TJsc the procedure developed in the preceding paragraph to compute the range of .f. Step 1: Compute 111for each arrangement of zeros in each rquivalence class. St,ep 5: Use Eq. (23) to check t.he sum of decomposed element’s in each permutation set.
Septenary
Partitions of 8
0
0
0
:: 0
i 0
i 0
i 0 0
8
1 1 : 2 2 2 2
1 2 2
: 2 2 2
10 40 6-l 80
a2 96
: ;
: 22
3 4 .i 6
3 3 4 4
128 128 lMi4 640
128 192 576 128 40 320 608 256 80
1
71 62 (ill 53 521 5111 -1-l X31 122 4211 41111 832 3311 3221 32111 311111 222“ 22211 221111 2111111 11111111 Total nrxml,~r of permut ations
00 1 2
0 0 1
numhers
16 l(i 16
8
30 80 10
-lo
!c,O
24 1-1-l 48 48 -lx 82 80
2 I6 12
l(i 11;
320 (i-l
192 1!J2
320 l(i0 l!U
288 !Mi 8&l I!)2 Oli -HO 105(i i(i8 96 ‘36 38-l 240
1 Ii0 96 I!,2
l(i
8 8
1
56
-120
5Ml
70
1680
50-M
768 32 06
2:w-l 1152
!Mi 96 320
15X 1020 -Hi08 15X 192 2304 3840
1Ii 160 224 Ii-l 16
1120
20160
832 2752 2560 ‘J!J2
10080
2520
Septenary 0
0 1 1 2’
0 1
3 3
2 2
::
34
RI“8 1 GO! V2 1‘2384 1872 (iO48 15!)84 11232 144 3741 N4
720 1728 3816 720 576 2376 43”O .I 504 1080 12!)(j
144 504 1008 288 72 100x 11144 252 1080 iX
2 1MI G480 864 12’Ni
432 2lGO 8&4 1008 2lti
Hi4 8G4
2lli X%2 216
0 0 0 0 0 1
i 1 1 1 2 2 2
Partitions of 9
numbers
i 4
2:
111111111 21111111 2211111 282111 22221 3111111 321111 32211 3222 33111 3321 333 411111 42111 4‘,21 d 4311 432 4-11 51111 5’211 622 53 1 54 (ill1 ti21 G3 711 72 81 9 Total numhel of permut~a tions
18
54 144 3ti
12(i 108 234 lci2
18
18 18 18
54 108 108 108 90 54 54
72 108 288 108
54 18 108
54 108 72 54
54 72 18
72 216 72
2l(i 21(i 432 21(i 21ci 216 432 72 432
180 GO
5760 1728
?AiO 288
X(i40 1OXli8 1440 345G 4320
540 1 i‘%I !bOO 57G 1728 72 108 11.52 18X 1728 468 324 288 1512 21(i 57fi
1728 G!ll2 4320 4320
1728 4320
1440
100x 504 828 36 72 47”.Y 181)
3GO x0
216
1 1
72
75G
1080
630
3024
GO480
15120
90720
22680
10080
VIBRATION
III.
LIST
OF
SF’E(‘TH.4
MOMENTS
OF
OF
THE
R.ih-DORI
431
I,ATTI(‘ES
I~ISTltIUUT1ON
FUNCTI0X
The moments (p,,), * . , (~1~) arc listed in this s&ion in Tahlw I through 1’1. To explain these tables the rradw is refrrred to Tahlc V for (p16). In particular, wnsider the second row labeled hy t’he part,ition { 7, I } and t,he second column lahcled hy the srptrnary numhrr 00000012. The elemcntj Iocat.ed at, the interswtioII of t.his row at~d column is 16. Thus from this c,lcmcnt \vc ohtail
which contributes to (~~6). WC affix thr c*oc+kkient colttaining force cwnstants from thr second c4umn. This cwlumn lists t,he poplllation of ecluivalcnce biasses of perm (00000012). From the prwwding section of t,his report we know that thr voefFic+t~t. of force coast.ant.s associated wit.h each clcnwnt. of this permutat.iou set, is 71’y”/a6.Similar columns relating to prrm (00000034) and pwm (0000005ti ) havr ken omitted since their cquivalcnw (4ass dwompositions arr identkal to those for perm (00000012). The 0111~difkrenw among t#hes:Pt,hrcc sets arc thv co&ic+nts of force caonst’ants which they imply. Thp coeffkirnt~ associated ait, perm (00000034) is yzL-y3’ and that for ptrm (0000005ti) is t hc same. COI~SCclllclrtly, the (2, 2) rlrmtnt of Tablr \T cont.rihut.cs l(j(y,’
+ 2y2’)yaR(l,li~/,,,,,,,,,,:,~lli,,+l,,,,,,:,)
t,o (c(,~). Each entry of Table T’ makes a similar contrihut~ion to (~~6) and (p16) is t,hr sum of all such krms. All thr mriments arc rrprewlted in this fashion. LTsiug ‘l’ahlw I through VI the reducttd moments -2i (/Qj); k!J = %,a.x were csalculakd ill Table VII.
.II,(
for diffcrrnt
physical paramctrrs.
Reduced moments for = 25rjj, p = ‘i, yI = 10?!. /Lo p2 fl, pe pq p,,, p1: p,r /.I,6 /.m
= = = = = = = = = =
1 .o 0.375 O.ZO(il 0.1326 0.0!,228 0. w734 o.mO71 0.039oi 0.03068 0.02143
j = 10, . . . , !)} A list of these moments appeal
Reduced moments for Al., = >lfB , ‘y, = IO-), p,, = /I> = p, = ,uf, = p, = p,c, = p,z = p,t = jl,6 = glfi =
I .o 0. .5 0.3385 0.2578 0.2075 (I. 1725 O.lW 0.1201 0.1100 0.09G30
II 0
0.5
1.0
0.5
0-
1.0
Wwmax.
W”max
FIG. 5. (ii) The histogram representing the distribution function for the random for 1If.t = 2!W, , p = ‘6, y1 = 10yz (h) The hist(ogram and exact, curve representing distribution function for a lnt,ticr in which M,t = MB , y1 = 10~~ . IV.
THE:
FREQUENCY
SPECTRUM
Ah’I)
SOME
RELhTEl)
lat,tice the
PHYSICAL
l’R( ~1’ERTIES
The kchniqut descrihrd in Scctiou I and the numerical values listed in Tal+ VII for the moments were used t,o estimate the frequency sp&rum by histograms which appear in Fig. 5. The c’aw cited in Fig. 5(a) indicaks that no apparent, separation of opt,icul and acoustical hands appear. Two peaks occur on this histogram which are situatjcd approximat,ely at the positions of infinite discontim&ies in the spcc:t8rum associated witfh the prrfect lattice; there is no rvidencc, however, that infinitr peaks should occur in the speckwn for the random lattice. Figure 5( 1)) is an illust,ration of the spectrum associated with the perfect, monat,omic lattice as calculated by this procedure. The smooth curve in this figure corresponds to the exact spectrum for t,his case. Since wIII:Iy = lim (m)/(m4, P-m Figure
6 reflects
some information
on thr
assumption
made
earlier
about,
the
II I
2
III 3
4
5
I 6
II 7
8
k
I 9
I IO
I
masimnm frequency of the lattiw. The momcwk in Talk T’II (first’ column) were used ill this cxlculation. The value 1 on the ordinat,e corresponds to the mnximum frcquenc’y of the lat,tiw of light atoms. The dot,ted lille refers to the masimnm freclucnvy for a perfect Iatt)ire ill which dl, = %I/, and y1 = IOr2 . Zvyagina and Ireronova (6 ) hare cxl~+ulntrclthe changesin 8”, the mean-square de\-iation of atoms from thrir tquilihrirm~ position for ordered and cwmpletcly disordered :<-l) dintomk lutticcs, in an attempt to study the behavior of the strrlqth CJf materials in going to high temperatures. They c~~ncl~~clc that for a aompleirly disordered, low c’ollc~c,llt,r:Ltiol1, dintomic: lnttiw, w,,,:,~(disorder) < w,,,;,~Corder), which t#endsto dwrensc k’. Our c~alrulatiolls slIpport the opposite effect sinw the inequality sign ahove shordd 1~ reversed if the maximum frrqwncy of the ralrdom Iat,tiw has t,he same signifiranw as the maximum frcquen~y of the perfwt lattjiw. 1~01~~1 J it al. (3) cloreloptd a method t’o c:alrulntc t#hrzero point energy per dcgrcc of freedom, co, for a random linear (*hain. We cxtcnd this c*alcv~lutiollto the c’ase of a rnr&m x-I> lattSicc. 11sI)onllJ ci nl. indicateq it is more ~~~Jllv?llk!llt to IISC
434
BRAOLET
t,he moments expanded 1/I~,,}. Then using
about .r = 0 t,han the pzk’s. Call these rnonwnt,s
the (24)
the { I’~,,) are defined by
f’z,, 3s0l(l-.r?)“j(.r)d.r =W&Z” 2(-])“I 0; UB”,,
( 2 5 ‘)
where Il.1m = w,z
(k,).
(20)
We use the one-sided delta fun&on (“7) and
(28) and .f( a) = (‘01 + (‘zlxy + . . .
(29)
to obtain (30) RIaradudin
and Weiss ( 7) obt,ained an exact cxlcwlation lim (g(w)) w-,I
of c’l. Une would expwt
= y(O) = Co’,
(31)
where g(w) represents t’he distribut,ion function for the perfecst, 3-D lattice. This implies t,hat Co = 0 iu our case. Onre Ca’is known, expressions for the (C:,} CMI be computed. For example,
Corresponding to t#he parameters listed in Table VIIa, (‘1 E + 1.18. Finally the zero point energy per degree of freedom is ( X? )
(‘ol~~;etI”(‘“l-ly,
thr wro point cncrgy prr dcgrw
of fwcclom
is
The zero point’ energy f’or thr r:mdom lattice (wing the data from Tat+ I’IIa j is lower than that for the awwiated perfect lattice by onr prr writ. This, however, is not iL st,ahilit,y critcrioll since \w never wilsidcrcd n c.ollfigrtrtltiollnl energy in our citlculat ions.
436
BRADLEY
REFEREXCES 1. F. J. I)YSON, Phys. Rev. 92, 1331 (1953). 2. H. SCHiNIUT, Phys. Rev. 106, 425 (1957). 3. C. I)ORIB, A. A. M~R.~I~UDIN, E. W. MONTROLL, AND G. WEISS, Php. /2ev. 116, 18 (1059) 116, 2-l (1959). 4. E. W. MONTROLL, J. (‘hem. Phys. 10, 219 (1942). 5. G. WEISS, Ph.1). thesis, Universit,y of Maryland, 1958. 6. A. 1’. ZVYAGINA AND V. I. IVERONOVA, SoGet Phys., Solid State (Translation of lf’iz Yverdoyo Tda) 2, 108 (1960). 7. A. MARADUDIN AND G. WEISS, .I. Chem. Phys. 29, 631 (1958).