D.I. Pushkarov- On the Defecton Statistics in Quantum Crystals

8/3/2019 D.I. Pushkarov- On the Defecton Statistics in Quantum Crystals

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526 D. I. PUSHKAROVOn the other hand, the energy related to the excitation statistics (e.g. the Fermi

energy, El') is proportional toEl ' - To - Llx 2 j3

Consequently, at low enough concentrations (x < (Ll/VO3) the interaction energyE < EF ' The critical concentration fo r a gas of vacancions is Xc - 10-1. For the 3Heimpurities in 4He Vo - 10- 2 K, and Xc < 10- 5

2. Defecton Statistics in 3d CrystalsFirst of all note that the statistics of defecton excitations may not coincide, in general,with the Fermi-Dirac or Bose-Einstein statistics. Defectons are created not in vacuum,and the behaviour of their wave function with defecton permutations depends not onlyon the defect atom characteristics (e.g. spin), but also on the crystal matrix, and on thedefect positions in the lattice. Nevertheless, there are some cases when the defectonstatistics is well defined.Consider a simplified model of a crystal, in which the host a toms and defects differonly in spin. In such a manner the interatomic interaction is supposed to be independent of the kind of atoms. Hence,we may formally assume that each lattice site n can beformed in one of two possible states indicated by the subscript 8. The values 8 = 0,8 = 1 correspond to a lattice site occupied by a Bose or Fermi particle, respectively.

The state 8 = 1 in a Bose crystal can be treated as an excited state, whereas 8 = 0has to be accepted as a normal one. The presence of a Bose impurity (e.g. vacancy) ina Bose crystal is described by 8 = 2. In the same manner Bose and Fermi defects ina Fermi crystal are desc1 ibed by the values 8 = 0 and 8 = 3, respectively.

Since the number of particles in each state is equal to 0 or 1, the eigenvalues of theparticle number operators Nns are N ns = 0, 1. Obviously,L N ns = 1 .s (1)

Further, introduce creation and annihilation operators, a ~ s , ans of particles in thestate 8 in the site n. Then(2)

The wave function in the second quantization representation will be deno ted byI .. N n ., . . . ) .

The operators a+, a can be constructed in two steps. First, we introduce new creation and annihilation operators bt., bns with the following properties:

b;,bnR = Nns ,b ~ s l .. N ns ) = ( l - N ns ) I.. N ns + 1 ... ) ,bn.I .. N ns ) = Ne I.. Nne - 1 ... ) .

(3)

Consequently, b ~ s , bns satisfy the commutation relations typical of Pauli operators,bIlHb;,a - b;wbnH = bmn b,a(1 - 2Nn.) ,bmab nR - b"sbma = 0 , (4)


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On the Defecton Statistics in Quantum Crystals 527The cOl1lmutation relations (4) take into account only the fact that every lattice site

cannot be occupied by more than one particle. Operators b+, b do not take account 01'the correct wave function symmetry. So, relations (4) for two dif ferent lattice site!>coincide always with those for Bose operators, whereas these sites can be occupied byFermi atoms. To construct the correct operators a ~ . " an.', we renumber all lattices sitesattaching to every of them a scalar number n. The numeration manner may be arbitrary but must be fixed. Then a + and a can be written in tbe form

i : r . ~ .:..vl Ia ~ = e I


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528 D.1. PUSHKAROV

(i l)

I t is easy to see that the operator a ~ ( s , a) an(s, a) = a ~ S a n 8 = Nns coincides with thedefect number N ns . In spite of the complicated form of expressions (6) to (8), the commutation relations of operators iX +, iX are simple in form in the cases on interest. So;taking into account that iXn(S, a) a ~ ( s , a) = Nna and the normalization condition (1),one obtains

an(S, a) a;'(s, a) + a;'(s, a) an(s, a) = 1 .I f m =f= n, the direct commutation, using (6) to (8), yields

[iX l1I (S, a), ai;(s, a)] 'f = (ei:t(S-a) =j= 1) a ~ ( s , a) am(s, a) . (12)For definiteness consider a Bose crystal (e.g. 4He) with Fermi-type impurities (3He).Then s = 1, a = 0, and operators a; ' = a;'la"O, an = a;;Oa"l obey the exact Fermi commutation relations

( X m a ~ + a ~ a m = ~ m ,ana n = a;'a;' = .Substituting (13) into (9) yields

H = Amna;;.an m,""'n

(13)

(14)As the matrix elements A mn depend on the difference Rm - R n only, the Hamiltonian(14) can be easily diagonalized by means of the Fourier transformation,

N+ _ 1 '" + - i kR m 1 '" N e'ikRn (15)'-"m - '/R t a k e , an = VN t '-"k ,where N is the number of lat,tice sites. Then

H = e(k) atak .kOperators a ~ ak are creation and annihilation operators of defectons with the quasiwave vector k and dispersion law

c(k) = A m - n eik(Rm-R n ) nSince the Fourier transformation does not change the wave function symmetry,

operators a ~ a k are also of the Fermi type.In the case of vacancies or Bose-type impurities in Fermi crystals a; ' = a;'lanO playsthe role of an annihilation operator restoring the "host" - an atom in the lattice site n.

Hence, such defectons are also Fermi quasiparticles.I f defects obey the same statistics as the crystal atoms (e.g. vacancies in 4He),

then s - a = 2 and (11), (12) give(16)

where Nn = a ~ a n 'Since the defect concentration is always small, relations (16) may be replaced bythose of Bose operators.

The results obtained in this section are-presented in Table 1 where F and B denoteFermi- or Bose-type excitations, respectively.

The statist ics of defectons in complex lattices as well as the vacancion statistics insolid solutions consisting of Bose and Fermi atoms may be considered in the same way.


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On the Defecton Statistics in Quantum CrystalsTa b le 1defect in Fermi crystals1. vacancy F2. impurity of

Btype FFtype B

3. interstitiaIs obeytheir own statistics

in Bose crystalsBBF

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3. Kinematic InteractionIn spite of the fact that Pauli statistics at small concentration is very close to the BoseEinstein one, the difference between these two statistics may be of interest, mainly,due to the problem of Bose-Einstein condensation. The problem is whether the macroscopic number of defectons can be found in a quantum state.

The commutation relations in k-space can be obtained from (15), (16),2 "'-[CXk, cxp]_ == ~ k , p - N N p -k , (17)

where Np _ k == c x ~ c x n ei(p -k)Rnnand

No == C X ~ C X n == Nn . n

A second commutation yields2 v 2[cx q, [CXk' c x ~ ] _ ] _ == N [cx q , N p _ q]_ == N- CXq+k-p,

(18)

(19)

It is seen that the s ta te vector in the right-hand side is determined by the quasimomentum conservation law. Statist ics with such commutation relat ions has beenconsidered by Kaplan [6]. It has been shown that no more than N quasiparticles canbe found in a k-state.A state with nk quasiparticles in a k-state has the form

Ink) = Onk(CXk)n k 10) ,where

Naturally, ON+l == 0.The distribution function of an ideal gas with a maximum occupation number Nin a quantum state k coincides with the Gentile function [7]G ( Ck - !J; ) -1 ((Ck - !J;) (N + 1) )-1nk == exp -T-- - 1 - (N + 1) exp T - 1 34 physica (b ) 133/2


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530 D. I . PUSHKAROVHowever, for nk 1 defectons cannot be considered as noninteracting quasiparlicles,as seen from the following. The action of the operators (Xt and (Xk on t he st at e Ink)can be written in the form

Hence,


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On th e Defecton Statistics in Quantum Crystalscan be rewritten by means of new Fermi operators,

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in L NzF n == e l< n bn , - in L NlF == e l< n b ~ in the form

H == A L ( F ~ F n + l + F ~ + l F n ) nThe Fourier transformation then yields

H == L e(k) FtF k ,kwhere e(k) == 2A cos (k) is the defecton dispersion law in a Id lat tice . Note that thisresult is due to restricting to nearest-neighbour interaction not supposed in the 3dcase consideration.

References[I ] D. I. PUSHKAROV, Zh. eksper. teor. Fiz., Pisma 19, 386 (1974).[2] V. A. MIKHEEV, B. N. ESELSON, V. N. GRIGORIEV, and N. P. MIKHIN, Fiz. nizk. Temp. 3, 385

(1977).[3] A. R. ALLEN, M. G. RICHARDS, and J. SCHRATTER, J. low-Temp. Phys. 47" 289 (1982).[4J D. I. PUSHKAROV, Zh. eksper. teor. Fiz. 59,1755 (1970); Soviet Phys. - J. exper. theor . Phys.

32,954 (1971).[5] D. I. PUSHKAROV, Zh. eksper. teor. Fiz . 68, 147 (1975); Soviet Phys. -- J. exper. theor. Phys.

41, 735 (1976).[6] I. G. KAPLAN, theor . mat . Fiz. 27, 254 (1976).[7] G. GENTILE, Nuovo Cimento 17,493 (1940).[8] D. B. CHESNUT and A. SUNA, J. chem. Phys. 39, 146 (1963).

(Received July 12,1985)

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D.I. Pushkarov- On the Defecton Statistics in Quantum Crystals

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