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Cluster Integral Theory of Distribution Functions of Spatially Nonuniform Systems
The theory of the pair distribution function of quantum and classical systems developed by Fujita, Isihara, and Montroll is extended to apply to spatially nonuniform systems. Cluster integral expansio...

Motion of an Impurity Particle in a Boson Superfluid

Phys. Fluids 4, 279 (1961); doi:10.1063/1.1706323

Issue Date: March 1961

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M. Girardeau
Boeing Scientific Research Laboratories, Seattle, Washington
The problem of the low-lying spectrum of a system of many bosons and one impurity particle interacting by repulsive, central, two-body forces is treated by first eliminating the dynamical variables of the impurity in terms of an added effective boson-boson interaction, and then treating the boson system by an extension of Bogolubov's method. In this way various physical properties of the clothed impurity particle, including its dispersion relation and effective mass, are obtained; singularities associated with the onset of acoustical wave drag as the impurity speed reaches the speed of sound in the boson medium are predicted. The parameters measuring the smallness of the corrections to the above results are shown to be the dimensionless boson-boson interaction, the dimensionless boson-impurity interaction, and the boson-impurity mass ratio. The treatment is then adapted to the case of hard spheres at low density by employing the Huang-Yang-Lee pseudopotential method; the results are valid if the dimensionless boson density and the boson-impurity mass ratio are small. A few remarks are made concerning the connection of the models treated with experimentally accessible results. ©1961 The American Institute of Physics
History: Received October 17, 1960
Permalink: http://dx.doi.org/10.1063/1.1706323
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0031-9171 (print)   1089-7666 (online)
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REFERENCES (31)
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  1. For a bibliography covering the work done prior to 1954 see H. Fröhlich, Advances in Phys. 3, 325 (1954).
  2. N. N. Bogolubov, J. Phys. (U.S.S.R.) 11, 23 (1947).
  3. T. D. Lee, F. E. Low, and D. Pines, Phys. Rev. 90, 297 (1953).
  4. See, e.g., K. A. Brueckner, and K. Sawada, Phys. Rev. 106, 1117 (1957), Sec. II.
  5. See Eq. (9) and the associated discussion; we shall find later that for weak coupling (1−n0/n) is so small that n0 can be replaced by n in most calculations.
  6. The summation

    [dformula -Omega[sup -1][summation][sub k][sup [prime]]((rho[sub 0] nu[sub bi][sup 2](k)m[sup -1][bold q] [center-dot] [bold k])/((1/4)k[sup 4] + k[sup 2] rho[sub 0] nu[sub b](k) - (m[sup -1][bold q] [center-dot] [bold k])[sup 2]))]

    vanishes identically because nub and nubi are even functions of k; this result has been used in simplifying the expression for the constant term in Eq. (18).

  7. This expression for gk is identical with what one would obtain in the absence of the impurity particle; the reason is that the off-diagonal part of (−[summation]k[prime]m−1q·kU<sub>2</sub><sup>-1</sup>NkU2) vanishes identically because gk = g−k.
  8. This relationship is easily proved using Eq. (9) and the fact that in the Schrödinger representation Phi<sub>[bold q]</sub><sup>i</sup> = Omega<sup>-(1/2)</sup>eiq·r, so that the integral over the impurity coordinate r merely yields a factor unity.
  9. M. Girardeau, Phys. Rev. 115, 1090 (1959).
  10. Reference 9, Eq. (2.1); the function Lk appearing therein differs from gk [our Eq. (20)] only in sign and through the replacement of rho by rho0. The error term O(rho*<sup>(1/2)</sup>lambda<sub>b</sub><sup>(1/2)</sup>) in Eq. (2.1) of reference 9 is incorrect, and should be replaced by the (smaller) term O(rho*lambdab) as in Eq. (37).
  11. We assume that vb(x) is sufficiently well behaved that its Fourier transform nub(k) falls off at least as rapidly as (const/k2) for kalphab>>1 and does not differ appreciably from nub(0) for kalphab<<1.
  12. We assume that the boson-impurity interaction vbi(x) is well behaved in the same sense that vb(x) is, i.e., that nubi(k) falls off at least as rapidly as (const/k2) for kalphabi>>1 and does not differ appreciably from nubi(0) for kalphabi<<1.
  13. Denoting the ground state of [fraktur H]q by Phiq, one has

    [dformula [del][sub q]E[sub q] = [del][sub q](Phi[sub q],[fraktur H][sub q] Phi[sub q]) = (Phi[sub q],([del][sub q][fraktur H][sub q])Phi[sub q]) + [[del][sub q[prime]](Phi[sub q[prime]],[fraktur H][sub q] Phi[sub q[prime]])][sub q[prime] = q] [equivalent] <[del][sub q][fraktur H][sub q]> + [[del][sub q[prime]],(Phi[sub q[prime]],[fraktur H][sub q] Phi[sub q[prime]])][sub q[prime] = ]]

    . The expression in square brackets vanishes at q[prime] = q as a consequence of the variational theorem.

  14. The boson-boson pair correlation function C(x) is related to the boson-boson pair distribution function D(x) (normalized to unity at large x) by D(x) = 1−C(x). The asymptotic behavior of C(x) for xi [equivalent] rho<sup>(1/2)</sup>nu<sub>b</sub><sup>(1/2)</sup>(0)|x|>>|1 is

    [dformula C(x) [approximate] (1/2)pi[sup -2] rho[sup (1/2)] nu[sub b][sup (3/2)](0)xi[sup -4]]

    . This result can be obtained by substituting the weak-coupling approximation phi0(k) = −Lk [Eq. (1) of reference 9] into the asymptotic form of D(x) given in Eq. (29) of reference 15. On the other hand, according to Eqs. (52) and (53)

    [dformula C[sub bi](x - r) [approximate] pi[sup -1] rho[sup (1/2)] nu[sub b][sup (1/2)](0)nu[sub bi](0)xi[sup -1]e[sup -2 xi]]

    .

  15. M. Girardeau and R. Arnowitt, Phys. Rev. 113, 755 (1959).
  16. K. Huang and C. N. Yang, Phys. Rev. 105, 767 (1957).
  17. T. D. Lee, K. Huang, and C. N. Yang, Phys. Rev. 106, 1135 (1957).
  18. This is Eq. (33) of reference (16) with the following changes of notation: a-->ab, [h-bar]-->1, m-->1, rj-->xj.
  19. Since the lowest-order pseudopotential takes into account only the effects of two-body collisions, it can depend only upon the parameters entering into the center-of-mass wave equation describing the scattering of two hard spheres, and in the case where one of these hard spheres is a boson and the other the impurity, the only such parameters are the effective mass µ = (1+m−1)−1 and the distance (1/2)(ab+ai) between the hard spheres at collision (we recall that m is the impurity mass in units chosen so that the boson mass is unity). Thus Eq. (55) is obtained from the Huang-Yang-Lee pseudopotential by the replacement therein of the boson mass by 2µ, the boson diameter by (1/2)(ab+ai), and the boson-boson relative coordinate by the boson-impurity coordinate (r−xj).
  20. Of course, the corresponding dimensionless factors might well turn out to be either zero of infinite. In the former case this would imply that the corrections actually are of higher order than those indicated above, whereas in the latter case they would be of lower order than the dimensional analysis suggests but still of higher order than the terms appearing explicitly in Eqs. (58) and (59). The former alternative occurs in the case of the ground-state energy of the hard-sphere Bose gas, where the next correction beyond the Lee-Huang-Yang result

    [dformula (E[sub 0]/n) = 2 pi rho a[1 + (128/15)(rho a[sup 3]/ pi)[sup (1/2)]]]

    turns out to be of order rho2a4 In (rhoa3), rather than the order rho5/3a3 suggested by dimensional analysis [see, e.g., T. T. Wu, Phys. Rev. 115, 1390 (1959)]. It seems likely that the same thing happens in the impurity-particle problem; then the corrections to Eqs. (58) and (59) would be of orders rho2a4 In (rhoa3) and rho3/2a7/2 In (rhoa3), respectively.

  21. This low-density limit of the boson-impurity distribution function can also be obtained by an elementary physical argument as follows. The low-density limit of rho(x−r)/rho is clearly independent of rho, hence of the number of bosons. It must therefore be obtainable by solving a two-body problem involving the impurity and only one boson. Since Eq. (63) is restricted to the case of zero total linear momentum (q = 0), we want the ground state psi0 of this two-body system, which is an s state: psi0 = psi0(|x−r|). This ground state is the zero-energy scattering state satisfying the wave equation

    [dformula -(1/(2 mu))(1/r[sub bi][sup 2])(([partial-derivative])/([partial-derivative]r[sub bi]))(r[sub bi][sup 2](([partial-derivative] psi[sub 0])/([partial-derivative]r[sub bi]))) = 0]

    ,

    [dformula r[sub bi] > (1/2)(a[sub b] + a[sub i])]

    , where rbi = |x−r| and µ = m/(1+m); the boundary conditions on psi0 are

    [dformula psi[sub 0][(1/2)(a[sub b] + a[sub i])] = 0]

    ,

    [dformula psi[sub 0] --> [subformula [underaccent r[sub bi] --> [infinity] [below] ]1]

    . The solution is

    [dformula psi[sub 0] = 1-((a[sub b] + a[sub i])/(2r[sub bi])) = 1-((a[sub b] + a[sub i])/(2|x - r|))]

    . The distribution function is the probability density

    [dformula | psi[sub 0]|[sup 2] = [1-((a[sub b] + a[sub i])/(2|x - r|))][sup 2]]

    , which agrees with the low-density limit of rho(x−r)/rho.

  22. Note added in proof. The abscissa is the boson-impurity separation in units of 2|x−r|/(ab+ai).
  23. In order that the impurity component of neutron scattering be determined by the dispersion relation of a single impurity particle, it is necessary that the impurity density be so low that impurity-impurity interactions are negligible.
  24. H. Palevsky, K. Otnes, and K. E. Larsson, Phys. Rev. 112, 11 (1958).
  25. D. G. Henshaw, Phys. Rev. Letters 1, 127 (1958);
    26. Bull. Am. Phys. Soc. Ser. II, 5, 12 (1960).
  26. J. L. Yarnell, G. P. Arnold, P. J. Bendt, and E. C. Kerr, Phys. Rev. 113, 1379 (1959).
  27. Note added in proof. The impurity effective mass might also be determined from the velocity of second sound at low temperatures. See K. R. Atkins, Phys. Rev. 116, 1339 (1959), Eq. (23).
  28. The Bogolubov replacement b0-->n<sub>0</sub><sup>(1/2)</sup>, b<sub>0</sub><sup>[dagger]</sup>-->n<sub>0</sub><sup>(1/2)</sup> previously carried out in obtaining Eq. (14) has also been employed in obtaining the triad terms (b†bb+b†b†b) in Eq. (A.l); the precise conditions that must be satisfied in order that such a replacement be rigorously valid in the limit n-->[infinity] are not known at present, although it is clearly necessary that n0-->[infinity] as n-->[infinity], a condition which is indeed implied by Eqs. (36)–(39) in the case of weak coupling [in fact, Eqs. (36)–(39) imply that n0[proportional]n as n-->[infinity]].
  29. According to Eq. (17), fk is of order n<sup>-(1/2)</sup>.
  30. An additional contribution is made to w<sub>i</sub><sup>(1)</sup> by the fact that there is a correction to rho0 due to the perturbation ([fraktur H]q−[fraktur H]<sub>q</sub><sup>(0)</sup>). This correction is, however, certainly smaller than (rho<sub>0</sub><sup>(0)</sup>−rho), the correction to the totally unperturbed value, rho, of rho0 arising from that part of the interparticle interaction contained in [fraktur H]<sub>q</sub><sup>(0)</sup>. Since we have already found that replacement of rho0 by rho evaluating Eq. (25) leads to errors of higher order than the terms appearing explicitly in Eq, (43), it is clear that the even smaller correction due to the modification of rho0 due to ([fraktur H]q−[fraktur H]<sub>q</sub><sup>(0)</sup>) is also of higher order than the terms appearing explicitly in Eq. (43).
  31. These single-particle states of U<sub>2</sub><sup>-1</sup>U<sub>1</sub><sup>-1</sup>[fraktur H]<sub>q</sub><sup>(0)</sup>U1U2 correspond to single-phonon states of [fraktur H]<sub>q</sub><sup>(0)</sup>.

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