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Essential singularities of the specific heat and of the density of states in a model of a random harmonic chain
Volume 62A, number 1
PHYSICS LETTERS
11 July 1977
ESSENTIAL SINGULARITIES OF THE SPECIFIC HEAT AND OF THE DENSITY OF STATES IN A MODEL OF A RANDOM HARMONIC CHAIN Walter F. WRESZINSKI* lnstitut de Physique, Universitd de Neuchdtel, CH - 2000 Neuchdtel, Switzerland Received 10 March 1977 Revised manuscript received 25 May 1977 We prove that the specific heat of a two component random harmonic chain in the limit of infinite greatest mass has an essential singularity at zero temperature, which implies that the average density of states has an essential singularity at zero energy of type consistent with the one proposed by Lifschitz. The phenomenon is shown to have essentially the same origin as the Griffiths' singularities of random Ising systems.
Griffiths [ 1 ] showed that the magnetization of random Ising systems [2] exhibits some remarkable singularities (now called Griffiths singularities) which are quite different in nature from those displayed by pure systems: in particular, the magnetization is singular in a range of temperatures above the critical temperature. The origin of these singularities was shown in greater detail by Wortis [3,4] in the one-dimensional Ising model. He proved that they arise from the fact that for 0 < p < 1 (p being the concentration of magnetic sites), there is a finite probability for the existence of intact (that is, free of nonmagnetic impurities) chains o f n sites, with n arbitrarily large but not infinite, and which behaves as e - n for large n. Further, the susceptibility has an essential singularity and is infinitely differentiable at the critical point [3]. For an electron in a crystal with randomly distributed impurities, there exists an attractive physical argument by Lifschitz [5,7] which implies that the (electronic) density of states g(e) has the following behaviour at small energies e (v being the number of space dimensions)
cently, Luttinger [6,7] obtained eq. (1) - as well as the first correction to it - from a new variational principle. The main connection between this conjecture and the structure of Griffiths' singularities (already noted in ref. [4]) is the crucial role played in Lifschitz's argument by the fact that the probability that a large region of volume V~ is free of impurities is proportional to exp [ - const. V~]. In this note, we consider the effects (and interrelation) of both phenomena (the Griffiths singularities and an essential singularity of type (1)) for the simplest soluble model of lattice vibrations, namely a two-component random harmonic chain with nearest-neighbour interactions in the limit where the greatest mass tends to infinity [8,9]. We prove that, as a consequence of the Griffiths singularities, the specific heat has an anomalous (i.e., as compared to the pure system) behaviour at low temperatures, more precisely, an essential singularity of type exp(--Cl/X/--T) as TJ~0 (e 1 > 0). We also show that this behaviour is consistent with an essential singularity at zero energy co of the average density of states ~+,(co)) of the following type:
g(e) ~ cexp[--d/e v/2] e¢0
(g(co))
c,d>O
(1)
where d is a given function of the electron density. Re* On leave from the Inst. de Fi'sica USP, S~l'oPaulo, Brazil. Supported in part by the Eidgen/Jssische Kommission fiir Auslffndische Studierende, 8044 Ziirich, and in part by the Swiss National Science Foundation.
~ exp [ - e / w v] to40
c > O.
(2)
This is the Lifschitz form of the singularity for lattice vibrations, because the phonon energy behaves as 1IV llv, for small energies, where Vis the volume [5,7]. We consider a harmonic chain (with nearest neighbour interactions) formed of two species of atoms of masses m 1 and m2, with m 1 .( m2, such that the mass 15
Volume 62A, number 1
PHYSICS LETTERS
at each site is an independent random variable. Let p be the probability of occupation of a given lattice site by a "light" atom (i.e., of mass m). In the limit m 2 -~oo, the linear chain is divided into "islands" of light masses bounded by "walls" of rigid atoms. The model in this limit was treated in refs. [8,9]: the probability of having an "island" or run of n light atoms is [pn(1 - p ) 2 ] , and the normal mode frequencies of a chain of n particles ef mass m enclosed between rigid walls are
cos,n = cOL
I1
as -]1/2
-- COS ~ - ) J
~rs = coL sin 2(n + 1-~)
11 July 1977
Proof. The function h(x) =- ( x / s i n h x ) 2
(8)
is monotone decreasing in x eR+, with lira x ~0 h(x) = 1 and limx t,~ h(x) = 0. Using this and (3), we obtain from (6) (notice that by (3) COs,n >~COl,n) c(T)~<(l-p)2[
max
(npn/2h(~col,n)}]
Lne[ 1, '~ ) oo
x ~ (pl/2),.
(9)
n=l
s = 1 ..... n
(3)
where CO2L= 4 k/m 1' k being the coupling constant between neighbour sites. In refs. [8,9] the probability associated with the frequency cos,n was evaluated, and a plot of the average density of states (g(.)) was given. We are, however, interested in the analytic form of (g(.)) for coa,0 which is difficult to obtain directly. However, due to the division into "islands", the model is precisely analogous to the one-dimensional Ising model studied in [3]. Accordingly, let fn be the free energy of an open chain o f n sites, i.e.,
Let o~(p) =- (1 -p)2pl/2/(1 _ p l / 2 ) . We note that for n of the order of x/-ff, both pn/2 and h(18COl,n/2 ) decrease as e-',/~. We therefore choose M to be the greatest integer smaller or equal to x/~. From (9), we get, assuming also M~> 1/log (1/Vp): c(T)~
(npn/2h(18col, n/2)}
L n e [ 1 ,M)
+ n emax[M,~) (np n/2 h~3 col,n/2)}/ <~c~(p) (Mth(f3 COl,M/2 ) +pM/21 }.
(lO)
n
fn@)= kTlog
l-I (1 - e-~C°s,n).
s=l
(4)
It may then be proven, as in ref. [3], that the free energy per site f ~ ) in the thermodynamic limit is equal to f ~ ) = (1 - p)2 ~ n=l
pn fn(18)
(5)
and, correspondingly, the specific heat per site c(T) in the thermodynamic limit is equal to
c(T)=( 1 _ p ) 2 ~ pn n=l
[. 13cos,n/2 7 2 (flcos, n/2)J "
s=lLsinh
d/(n + 1) ~< COl,n ~< c/(n + 1).
(11)
By (10) and (11) and using M + 1 < x/~ + 2, we obtain:
c(T) ~< c@) M[( [18d/2 (M + 1)]/sinh [13d/2(M + 1)] }2 +pM/21
<. r e-a/"/-T/T
which is (7a). (7b) follows similarly using the lower bound
(6)
Proposition 1. There exist constants ot1 ) O, r > O,
c(T) ~> (1 - p)2 max ne[1,~)
{pn h ~ COl, n/2)} (12)
~> (1 - p)2 pM {[/3c/2(M + 1)1 / sinh [18e/2 (M + 1)] }2
s > O, a > 0 and b ;> 0 such that for all T < a 1 c(T) ~< r e - a / " / T / T .
(7a)
c(T) >~s e-b/x/T/T.
(7b)
16
Now, there exist c > O, d > 0 such that
and choosing M as before. As a corollary of proposition 1, we have:
Volume 62A, number 1
PHYSICS LETTERS
lira [c(T) exp(p/T a) T -r] = 0 V 0 ~ < a < 1/2, T¢O (13a) VreZ, VO~
11 July 1977 60 1
I(a, T) =- f
d¢o exp(-dco -a) h(co/2kT)
and
and
lim [c(T) exp(p/T a) T -r] = oo V a > 1/2, T+0
J(ot, T) -~ h(¢Ol/2k T ) .
VreZ,
V0 ~
This anomalous behaviour for T~, 0 must be compared to the pure one-dimensional crystal's [8], where
c(T) =k/lr f0 ~
dco
8k 2
h(co/2kV) ~ 0 T
(19)
Then, it follows from (17) that: (13b)
2
(18)
0
T " (14)
We see therefore that the asymptotic form g(w) "~w +0 1 / ~ for the density of states is not compatible with (7). We introduce the average density of states by the Definition. An average density of states is, if it exists, a real positive locally integrable function (g(.)) on R+, with the properties
c03 t> c2 I(~, T)
(20)
and
c(r) • c I [(a, T)+ J(ot, r ) .
(21)
Suppose, firstly, that a > 1, and let p -- c~/(a + 1). It follows that 1/2 < p < 1.
(22)
Let
eTfy) =-exp [--d/(2kTy) ~]
(23)
We have: co
0 <
J ~ d w (g(co))= 1 0
(15)
= 2kT
dy eT(y) h(v) +
j
dy eT(v) h(v)
T-O
c(T) = ; de (g(co)) h(w/2kT) 0
(16) oo
where h is given by (8). Note that eq. (15) is the conventional normalization and (16) is natural, [8, pp. 46, 191]. Given eq. (16), it is easy to see that no power-law asymptotic behaviour: ~g(o~))~ ~0 c°r, r eZ+, is consistent with (7) (or (13)). Further, the following proposition shows that (13) is consistent with an essential singularity of (g(.)) of Lifschitz's form (2) (with v = 1): Proposition 2. Suppose that 36o I > 0, c 1 > 0, c 2 ~> 0, d > 0 and a > 0 such that
<<.2kT[exp{-d/[(2k) c~T ° ] ) f dy h(y) 0
I
+ 2T-P [exp(T -o - I1-1 f dy (v/sinhy) .(24) T-P From (22) and (24) it follows that we may choose p' such that p > p' > 1/2 and such that t
lim [exp(1/TP )I(~, r)] = 0. T*0 Similarly, since, for T sufficiently small:
(25)
J(a, T) ~
(26)
c 2 exp(-dw - a ) ~<~(w)) ~
(17)
Then (13) implies that a = 1. Conversely, (17), with a = 1, implies (13). Proof. Let
(al' a2 > 0)
and P' < 1, it follows that i
lim [e 1/rp J(~, 7)1 =0 T*0
(27) 17
Volume 62A, number 1
PHYSICS LETTERS
11 July 1977
i
(25) and (27) in (21) imply that limT+ 0 [e 1/Tp c(T)] = 0, which contradicts (13b). Suppose, now, that a < 1. We have, by (20),
wl/2kT
c(T) >~2c2kT
f
dy eT(y) h(y).
(28)
0 It is easy to see that the integrand in (28) has one and only one maximum at a point y m ~ ct. xT -p for T sufficiently small, and converges to zero both as y 4 0 andyl"~. Since a < 1 implies a < 1/2, we may write for T sufficiently small,
514T-P
c(T)>12c2kTf
dy er(y) h0,) >t
3/4T-O 1/2T-O eT(3/4T-P)h(3/4T --o) if 5/4T-P <~Ym 1/2T-P er ( 5 / 4 T - 0 ) h(5/4T --p) >~2c2kT X' if 3 / 4 T - ° >~Ym
(29)
1/2T-P eT(1/4T--O) h(1/4T-P) if 3/4 T - ° <'Ym <"5/4T-0 From (29), it follows that for T sufficiently small,
c(T)>/rT 1-3p e-s/Tp
r,s>O.
(30)
Hence, choosing p
18
only if ct = 1, which corresponds to Lifschitz's form. Our conclusions would not be affected, for instance, by replacing (17) by a more complicated behaviour of type (g(w)) ~o~ ~0 ~ - r e x p ( - d ~ - a ) ' but our arguments would again determine only the exponent a. In fact, a nonrigorous but probably correct argument suggests (g(c@ ~ ~0 w - 2 e x p ( - d w - 1). We thank Prof. R. Mills for this and other useful remarks. I am greatly indebted to Dr. J. Hammerberg for calling Lifschitz's conjecture to my attention and for numerous helpful discussions. I should also like to thank Prof. M. Romerio for several helpful discussions, a critical reading of the manuscript and for his warm hospitality at Neuch~tel, Dr. S.R. Salinas for referring me to ref. [3], and Prof. J. Lebowitz for an interesting discussion in Lausanne.
References [1] R.B. Griffiths, Phys. Rev. Lett. 23 (1969) 17. [2] R.B. Griffiths and LL. Lebowitz, J. Math. Phys. 9 (1968) 1284. [3] M. Wortis, Phys. Rev. B10 (1974) 4665. See also F. Matsurbara et al., Can. J. Phys. 51 (1973) 1053. [4] A.B. Harris, Phys. Rev. B12 (1975) 203. [5] I.M. Lifschitz, Adv. Phys. 13 (1964) 483. [6] J.M. Luttinger, Phys. Rev. Lett. 37 (1976) 609. [7] J.M. Luttinger and R. Friedberg, Phys. Rev. B12 (1975) 4460. [8] A.A. Maradudin, E.W. Montroll and G. Weiss, in Solid State Phys., Suppl. 3 (1963). [9] C. Domb, et al. Phys. Rev. 115 (1959) 24. [10] H. Matsuda and K. Ishii, Progr. Theor. Phys. Suppl. 45 (1970) 56. [11] M.S. Lu, M. Netkin and M. Arita, Phys. Rev. B10 (1974) 2315. [12] M.V. Romerio, J. Math. Phys. 12 (1971) 552.
PHYSICS LETTERS
11 July 1977
ESSENTIAL SINGULARITIES OF THE SPECIFIC HEAT AND OF THE DENSITY OF STATES IN A MODEL OF A RANDOM HARMONIC CHAIN Walter F. WRESZINSKI* lnstitut de Physique, Universitd de Neuchdtel, CH - 2000 Neuchdtel, Switzerland Received 10 March 1977 Revised manuscript received 25 May 1977 We prove that the specific heat of a two component random harmonic chain in the limit of infinite greatest mass has an essential singularity at zero temperature, which implies that the average density of states has an essential singularity at zero energy of type consistent with the one proposed by Lifschitz. The phenomenon is shown to have essentially the same origin as the Griffiths' singularities of random Ising systems.
Griffiths [ 1 ] showed that the magnetization of random Ising systems [2] exhibits some remarkable singularities (now called Griffiths singularities) which are quite different in nature from those displayed by pure systems: in particular, the magnetization is singular in a range of temperatures above the critical temperature. The origin of these singularities was shown in greater detail by Wortis [3,4] in the one-dimensional Ising model. He proved that they arise from the fact that for 0 < p < 1 (p being the concentration of magnetic sites), there is a finite probability for the existence of intact (that is, free of nonmagnetic impurities) chains o f n sites, with n arbitrarily large but not infinite, and which behaves as e - n for large n. Further, the susceptibility has an essential singularity and is infinitely differentiable at the critical point [3]. For an electron in a crystal with randomly distributed impurities, there exists an attractive physical argument by Lifschitz [5,7] which implies that the (electronic) density of states g(e) has the following behaviour at small energies e (v being the number of space dimensions)
cently, Luttinger [6,7] obtained eq. (1) - as well as the first correction to it - from a new variational principle. The main connection between this conjecture and the structure of Griffiths' singularities (already noted in ref. [4]) is the crucial role played in Lifschitz's argument by the fact that the probability that a large region of volume V~ is free of impurities is proportional to exp [ - const. V~]. In this note, we consider the effects (and interrelation) of both phenomena (the Griffiths singularities and an essential singularity of type (1)) for the simplest soluble model of lattice vibrations, namely a two-component random harmonic chain with nearest-neighbour interactions in the limit where the greatest mass tends to infinity [8,9]. We prove that, as a consequence of the Griffiths singularities, the specific heat has an anomalous (i.e., as compared to the pure system) behaviour at low temperatures, more precisely, an essential singularity of type exp(--Cl/X/--T) as TJ~0 (e 1 > 0). We also show that this behaviour is consistent with an essential singularity at zero energy co of the average density of states ~+,(co)) of the following type:
g(e) ~ cexp[--d/e v/2] e¢0
(g(co))
c,d>O
(1)
where d is a given function of the electron density. Re* On leave from the Inst. de Fi'sica USP, S~l'oPaulo, Brazil. Supported in part by the Eidgen/Jssische Kommission fiir Auslffndische Studierende, 8044 Ziirich, and in part by the Swiss National Science Foundation.
~ exp [ - e / w v] to40
c > O.
(2)
This is the Lifschitz form of the singularity for lattice vibrations, because the phonon energy behaves as 1IV llv, for small energies, where Vis the volume [5,7]. We consider a harmonic chain (with nearest neighbour interactions) formed of two species of atoms of masses m 1 and m2, with m 1 .( m2, such that the mass 15
Volume 62A, number 1
PHYSICS LETTERS
at each site is an independent random variable. Let p be the probability of occupation of a given lattice site by a "light" atom (i.e., of mass m). In the limit m 2 -~oo, the linear chain is divided into "islands" of light masses bounded by "walls" of rigid atoms. The model in this limit was treated in refs. [8,9]: the probability of having an "island" or run of n light atoms is [pn(1 - p ) 2 ] , and the normal mode frequencies of a chain of n particles ef mass m enclosed between rigid walls are
cos,n = cOL
I1
as -]1/2
-- COS ~ - ) J
~rs = coL sin 2(n + 1-~)
11 July 1977
Proof. The function h(x) =- ( x / s i n h x ) 2
(8)
is monotone decreasing in x eR+, with lira x ~0 h(x) = 1 and limx t,~ h(x) = 0. Using this and (3), we obtain from (6) (notice that by (3) COs,n >~COl,n) c(T)~<(l-p)2[
max
(npn/2h(~col,n)}]
Lne[ 1, '~ ) oo
x ~ (pl/2),.
(9)
n=l
s = 1 ..... n
(3)
where CO2L= 4 k/m 1' k being the coupling constant between neighbour sites. In refs. [8,9] the probability associated with the frequency cos,n was evaluated, and a plot of the average density of states (g(.)) was given. We are, however, interested in the analytic form of (g(.)) for coa,0 which is difficult to obtain directly. However, due to the division into "islands", the model is precisely analogous to the one-dimensional Ising model studied in [3]. Accordingly, let fn be the free energy of an open chain o f n sites, i.e.,
Let o~(p) =- (1 -p)2pl/2/(1 _ p l / 2 ) . We note that for n of the order of x/-ff, both pn/2 and h(18COl,n/2 ) decrease as e-',/~. We therefore choose M to be the greatest integer smaller or equal to x/~. From (9), we get, assuming also M~> 1/log (1/Vp): c(T)~
(npn/2h(18col, n/2)}
L n e [ 1 ,M)
+ n emax[M,~) (np n/2 h~3 col,n/2)}/ <~c~(p) (Mth(f3 COl,M/2 ) +pM/21 }.
(lO)
n
fn@)= kTlog
l-I (1 - e-~C°s,n).
s=l
(4)
It may then be proven, as in ref. [3], that the free energy per site f ~ ) in the thermodynamic limit is equal to f ~ ) = (1 - p)2 ~ n=l
pn fn(18)
(5)
and, correspondingly, the specific heat per site c(T) in the thermodynamic limit is equal to
c(T)=( 1 _ p ) 2 ~ pn n=l
[. 13cos,n/2 7 2 (flcos, n/2)J "
s=lLsinh
d/(n + 1) ~< COl,n ~< c/(n + 1).
(11)
By (10) and (11) and using M + 1 < x/~ + 2, we obtain:
c(T) ~< c@) M[( [18d/2 (M + 1)]/sinh [13d/2(M + 1)] }2 +pM/21
<. r e-a/"/-T/T
which is (7a). (7b) follows similarly using the lower bound
(6)
Proposition 1. There exist constants ot1 ) O, r > O,
c(T) ~> (1 - p)2 max ne[1,~)
{pn h ~ COl, n/2)} (12)
~> (1 - p)2 pM {[/3c/2(M + 1)1 / sinh [18e/2 (M + 1)] }2
s > O, a > 0 and b ;> 0 such that for all T < a 1 c(T) ~< r e - a / " / T / T .
(7a)
c(T) >~s e-b/x/T/T.
(7b)
16
Now, there exist c > O, d > 0 such that
and choosing M as before. As a corollary of proposition 1, we have:
Volume 62A, number 1
PHYSICS LETTERS
lira [c(T) exp(p/T a) T -r] = 0 V 0 ~ < a < 1/2, T¢O (13a) VreZ, VO~
11 July 1977 60 1
I(a, T) =- f
d¢o exp(-dco -a) h(co/2kT)
and
and
lim [c(T) exp(p/T a) T -r] = oo V a > 1/2, T+0
J(ot, T) -~ h(¢Ol/2k T ) .
VreZ,
V0 ~
This anomalous behaviour for T~, 0 must be compared to the pure one-dimensional crystal's [8], where
c(T) =k/lr f0 ~
dco
8k 2
h(co/2kV) ~ 0 T
(19)
Then, it follows from (17) that: (13b)
2
(18)
0
T " (14)
We see therefore that the asymptotic form g(w) "~w +0 1 / ~ for the density of states is not compatible with (7). We introduce the average density of states by the Definition. An average density of states is, if it exists, a real positive locally integrable function (g(.)) on R+, with the properties
c03 t> c2 I(~, T)
(20)
and
c(r) • c I [(a, T)+ J(ot, r ) .
(21)
Suppose, firstly, that a > 1, and let p -- c~/(a + 1). It follows that 1/2 < p < 1.
(22)
Let
eTfy) =-exp [--d/(2kTy) ~]
(23)
We have: co
0 <
J ~ d w (g(co))= 1 0
(15)
= 2kT
dy eT(y) h(v) +
j
dy eT(v) h(v)
T-O
c(T) = ; de (g(co)) h(w/2kT) 0
(16) oo
where h is given by (8). Note that eq. (15) is the conventional normalization and (16) is natural, [8, pp. 46, 191]. Given eq. (16), it is easy to see that no power-law asymptotic behaviour: ~g(o~))~ ~0 c°r, r eZ+, is consistent with (7) (or (13)). Further, the following proposition shows that (13) is consistent with an essential singularity of (g(.)) of Lifschitz's form (2) (with v = 1): Proposition 2. Suppose that 36o I > 0, c 1 > 0, c 2 ~> 0, d > 0 and a > 0 such that
<<.2kT[exp{-d/[(2k) c~T ° ] ) f dy h(y) 0
I
+ 2T-P [exp(T -o - I1-1 f dy (v/sinhy) .(24) T-P From (22) and (24) it follows that we may choose p' such that p > p' > 1/2 and such that t
lim [exp(1/TP )I(~, r)] = 0. T*0 Similarly, since, for T sufficiently small:
(25)
J(a, T) ~
(26)
c 2 exp(-dw - a ) ~<~(w)) ~
(17)
Then (13) implies that a = 1. Conversely, (17), with a = 1, implies (13). Proof. Let
(al' a2 > 0)
and P' < 1, it follows that i
lim [e 1/rp J(~, 7)1 =0 T*0
(27) 17
Volume 62A, number 1
PHYSICS LETTERS
11 July 1977
i
(25) and (27) in (21) imply that limT+ 0 [e 1/Tp c(T)] = 0, which contradicts (13b). Suppose, now, that a < 1. We have, by (20),
wl/2kT
c(T) >~2c2kT
f
dy eT(y) h(y).
(28)
0 It is easy to see that the integrand in (28) has one and only one maximum at a point y m ~ ct. xT -p for T sufficiently small, and converges to zero both as y 4 0 andyl"~. Since a < 1 implies a < 1/2, we may write for T sufficiently small,
514T-P
c(T)>12c2kTf
dy er(y) h0,) >t
3/4T-O 1/2T-O eT(3/4T-P)h(3/4T --o) if 5/4T-P <~Ym 1/2T-P er ( 5 / 4 T - 0 ) h(5/4T --p) >~2c2kT X' if 3 / 4 T - ° >~Ym
(29)
1/2T-P eT(1/4T--O) h(1/4T-P) if 3/4 T - ° <'Ym <"5/4T-0 From (29), it follows that for T sufficiently small,
c(T)>/rT 1-3p e-s/Tp
r,s>O.
(30)
Hence, choosing p
18
only if ct = 1, which corresponds to Lifschitz's form. Our conclusions would not be affected, for instance, by replacing (17) by a more complicated behaviour of type (g(w)) ~o~ ~0 ~ - r e x p ( - d ~ - a ) ' but our arguments would again determine only the exponent a. In fact, a nonrigorous but probably correct argument suggests (g(c@ ~ ~0 w - 2 e x p ( - d w - 1). We thank Prof. R. Mills for this and other useful remarks. I am greatly indebted to Dr. J. Hammerberg for calling Lifschitz's conjecture to my attention and for numerous helpful discussions. I should also like to thank Prof. M. Romerio for several helpful discussions, a critical reading of the manuscript and for his warm hospitality at Neuch~tel, Dr. S.R. Salinas for referring me to ref. [3], and Prof. J. Lebowitz for an interesting discussion in Lausanne.
References [1] R.B. Griffiths, Phys. Rev. Lett. 23 (1969) 17. [2] R.B. Griffiths and LL. Lebowitz, J. Math. Phys. 9 (1968) 1284. [3] M. Wortis, Phys. Rev. B10 (1974) 4665. See also F. Matsurbara et al., Can. J. Phys. 51 (1973) 1053. [4] A.B. Harris, Phys. Rev. B12 (1975) 203. [5] I.M. Lifschitz, Adv. Phys. 13 (1964) 483. [6] J.M. Luttinger, Phys. Rev. Lett. 37 (1976) 609. [7] J.M. Luttinger and R. Friedberg, Phys. Rev. B12 (1975) 4460. [8] A.A. Maradudin, E.W. Montroll and G. Weiss, in Solid State Phys., Suppl. 3 (1963). [9] C. Domb, et al. Phys. Rev. 115 (1959) 24. [10] H. Matsuda and K. Ishii, Progr. Theor. Phys. Suppl. 45 (1970) 56. [11] M.S. Lu, M. Netkin and M. Arita, Phys. Rev. B10 (1974) 2315. [12] M.V. Romerio, J. Math. Phys. 12 (1971) 552.