научная статья по теме EFFECT OF THREE-BODY CLUSTERS IN THE GROUND-STATE PROPERTIES OF SPIN-POLARIZED LIQUID [1]HE Физика

EFFECT OF THREE-BODY CLUSTERS IN THE GROUND-STATE PROPERTIES OF SPIN-POLARIZED LIQUID 3He

Z. Razavifar* A. Rajabi

Physics Department. Shahid Rajaee Teacher Training University 167SS, Lavizan. Tehran. Iran

Received March 15, 2014

The ground-state energy of polarized and unpolarized liquid JHo is calculated using the variational theory. A variational wave function is constrained to be normalized appropriately by including the three-body terms in the cluster expansion of the two-body radial distribution function. The higher-order terms have been found to be important to obtain an equation of state which is in agreement with experimental data. The saturation density of unpolarized liquid JHo was found to be 0.267<r-J, which decreases by enhancing the polarization. For all relevant densities, the ground-state energy of the spin-polarized system is higher than that in the unpolarized case.

DOI: 10.7868/S0044451014100034

1. INTRODUCTION

Liquid 3He is an interesting system in which many-body correlations play an important role in determining its properties [1]. This system obeys Fermi Di-rac statistics the same as neutron stars, whereas describing 3He is easer than neutron stars because of the simplicity of inter-particle interaction. Moreover, we have accumulated a huge amount of experimental information about 3He. Hence, for theoreticians, liquid 3He can be considered an excellent laboratory to test many-body theories applied to neutron stars.

Experimentally, the zero-temperature equation of state of liquid 3He is known and the density of equilibrium is p0 = 0.277(T-3 with a = 2.556 A [2]. Theoretically, most of the available many-body methods have been applied for investigating the properties of liquid 3He, two successful approaches are Fermi hypernet-ted-chain (FHNC) and quantum Monte Carlo (QMC) methods. Viviani et al. used a variational wave function which includes pair, triplet, backflow, and spin dependent correlations in the FHNC method to obtain an equation of state which is in very close agreement with the experimental data [3]. Casulleras and Boronat in 2000, using optimized backflow correlations, applied the diffusion Monte Carlo (DMC) method and generated an equation of state of liquid 3He which is in

E-mail: Zahrarazavifar62(fflp;mail.com

excellent agreement with experimental data from equilibrium up to freezing [4]. In 2003, this computation was revisited by using exactly the same potential, wave function, and number of particles as used by Casulleras and Boronat, but their results were not confirmed [5].

In addition, in 1979 liquid 3He was polarized by a rapid melting of a highly polarized solid 3He [6]. In this state, nuclear spins of 3He aligned and because of nuclear magnetic interaction, the intrinsic relaxation time of partially polarized 3He is long, which allows using it for magnetic resonance imaging [7]. Most of the theoretical investigations based on QMC predict the fully polarized state with a lower energy than for the unpolarized state [8, 9]. By considering back-flow and three-body wave functions and twist-averaged boundary conditions in the QMC approach, it was found that the energy of the polarized state was higher than the unpolarized one, but the obtained susceptibility had discrepancy with extrapolated experimental data [5, 10]. Manosuki et al., by using the FHNC technique, found that the energy of spin-polarized phase was above that of the normal phase [11]. They conclude that the three-body and backflow correlations are very important for their variational wave function. They predicted that this system could exhibit new phase transitions to ferromagnetism, while no such new phase has been discovered so far.

The lowest-order constrained variational (LOCV) method is a many-body approach which has been developed to study the bulk properties of the quantum

fluids [12 14]. In recent years, this method has been applied to study homogeneous normal liquid 3He [15 18]. In this variational approach, as we see in the next section, we use a cluster expansion to calculate the energy and other properties of system. Convergence of the expansion and the effect of higher-order cluster terms in the energy of unpolarized liquid 3He was studied and it has been shown that higher-order cluster terms in the normalization constraint improve the equation of state [14]. The LOCV method has several advantages with respect to the other many body methods which go beyond the lowest order [14]. Two of them are: (i) the LOCV method is fully self-consistent, i. e., there are no free parameters in this variational approach, (ii) It considers a particular form for the long-range part of the correlation function in order to perform an exact functional minimization of the energy. It is shown that correlation functions obtained from the extended LOCV (ELOCV) lead to more accurate results for the momentum distribution [19, 20], 3He droplets [21], and 3He atoms in nanotubc [22]. In a series of papers, Bordbar et al. applied the LOCV method to the polarized case and calculated some properties of this system [18,23 25]. In their recent work, they considered the ground-state properties with the three-body cluster contributions [26]. But they did not consider the effect of three-body cluster expansion of the two-body radial distribution function (ELOCV). We expect that the same as in QMC and FHNC methods, three-body correlations are very important in spin-polarized systems. Hence, in this paper, we intend to consider the effect of higher-order terms in the cluster expansion of the radial distribution function and calculate the energy by the extended LOCV approach with the three-body cluster contributions.

2. SCHEME OF CALCULATIONS

function (<!)) for single-particle states to find the variational wave function of the interacting system:

= F<f>, (2.2)

where F is a correlation function which incorporates the correlations induced by interactions. To calculate the energy expectation value, we use the variational principle and a cluster expansion developed in Ref. [27],

The one-body term E\ is just the familiar Fermi-gas kinetic energy, i.e.,

1 N /

¿=1 x

The two-body energy E-2 is

£'2 = ¿F&'.;|W"(12)|ij)Q (2-5)

ij

and the "effective interaction operator" W"(12) is given

by

M'(12) = ^(V/(12))2 + /2(12)F(12), (2.6)

where /(12) and V'(12) are the two-body correlation and inter-atomic potential.

Higher-order correlations are considered in terms of statistically irreducible two-body correlations. So, the three-body energy is written as

E3 = Eah + Eahh + E31., (2.7)

where

E3h = ^^[(umwv(i2)ma -

ijk

- (ik\h(13)\ik)a(ij\W(12)\U)a], (2-8)

trkf 2 m

(2.4)

2.1. Cluster expansion of energy

The Hamiltonian of the normal liquid 3He consists of AT atoms interacting with each other and is usually-writ ten as

N

i=i

pi_

2 m

1

El'to).

iri

(2.1)

where I '(ij) is the two-body inter-atomic potential. In this work, we use the Lcnnard Jones potential. In the LOCV method, we use an ideal Fermi gas type wave

E3hh = — ^^{ijk\h(13)h(23)W(12)\ijk)a, (2.9)

ijk

hjr>f — — x 31 2 N

x Y, (ijk

ijk ^

and

—/2(31)V2/i(12)V2/i(23)

hdj) = f(tj) - 1.

(2.10) (2.11)

Wo note that in Eq. (2.3), to collect all contributions which are conventionally assigned to the first order in the sniallnoss parameter, we have to compute a special portion of the four-body terms, like the three-body cluster terms [27]:

= EM/>(34)|W)a<W|H'(12)|u)a. (2.12) ijkl

In the LOCV formalism we constrain the two-body correlation function to normalize the wave function of the system. We hope this constraint makes the cluster expansion converge very rapidly.

2.2. Spin polarized calculations

We now specialize the above cluster expansion to the spin-polarized system including Ar atoms with Ar(+) spins up and Ar(_) spins down, with

p=- = p<+>

P

(2.13)

being the total number density and the spin asymmetry parameter Q defined as

N(+) _ N(-)

c =

N

(2.14)

By considering the single-particle states |/) as plane waves, we can calculate the energy terms introduced in the last section. The one-body energy term E\ is

El = T0 |W3[(l+0^ + (W)n (2.15) The two-body energy E2 introduced in Eq. (5) is

E-2 = 2ttp / rl-jdri-2

l-|(l + c )2i2(4+)r12)

1(1-0 2i2(HrV12)

H'(i-i2), (2.16)

where

i'(x) = 3:—— = — (sin(x) — xcos(x)) (2.17)

is called the statistical correlation function or the Slater factor [28]; = (6?rV+))1/3 and k<T} =

are the Fermi momenta of spin up and spin down states, respectively.

The three-body cluster energies in Eqs. (8), (9), and (10) are

Eu ~ 8iV X

x j iiriiir2iir3/i(ri3)W"(ri2)ri(ri,r2,r3), (2.18)

p3 f

E?,h.h = 7^7 / dr1dr2dr3h(r13) x

X M"(j'12)/»(»'23)r2(ri, r2, r3), (2.19)

()3 r ff

E?>I = 2N J dTldT2(lT?'^~, x

X f (r31)■V2h(j'12)V2h(r23 )T2 (n, r2, r3). (2.20)

Here, the three-body energy terms Fi(ri,r2,r3) and F2(ri,r2,r3) are defined as follows:

Ti(ri,r2,r3) =

= (1 + OH(k£)r12U(k£)r23miIt)r31) +

>,,.)<(*) 'r23)l(k) >3I) -[(1 + O3 + (1 + 02(1 - 0]i2(^F+),''23) -

• [(1 - O3 + (1 - 02(1 + 0]i2(A-irV23) (2.21)

and

r2 (n, r2, r3) = l - - [(l + O3 + (l + O2 (1 - 0] x

X [£2(^+Vi2) + 2i2(fr<+V23)] -4[(l-03 + (l-C)2(l+C)][£2(*-ir)ri2)+2£2(i-ir)r23)]-

(1 + 03i(4+)'-12)i(4+)'-23)i(4+)r31)

1

f4

i(l - ü3m-; lrl2)l(k) lr23)l(k) 'r3,). (2.22)

And finally,

E±h. = J ^riiir2iir3iir4/i(r34)W"(ri2) x

x {(1 + 04[i2(^+)r13)i2(4+)r24) - i(^+)r31) x x i(4+)r23)i(4+)r14)i(4+)r24)] +

+ (1 - 04[i2(Ä-iT)ri3)i2(Ä-ir)r24) - W^rn) X

x l(k) :r23)l(k) :rn)l(k) '' r21) J. (2.23)

The above terms can be simplified by putting particle 1 at the origin and replacing p J dr± = AT.

2.3. Normalization constraint and the Euler—Lagrange equation

Now, we minimize the energy with respect to /(»'), whereas in LOCV formalism we are interested in obtaining a more physical correlation function that satisfies the normalization constraint (ipvH'v) = 1- In

the lowest-order approximation, this constraint is given by [28]

(> / (.92 ('''12) - l)d3

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