Quantum Momentum Distributions

Transcription

1 Journal of Low Temperature Physics, Vol. 147, Nos. 5/6, June 2007 ( 2007) DOI: /s Quantum Momentum Distributions Benjamin Withers and Henry R. Glyde Department of Physics and Astronomy, University of Delaware, Newark, DE 19716, USA glyde@udel.edu (Received October 29, 2006; revised February 22, 2007) The atomic momentum distribution in quantum fluids and solids differs significantly from the classical, Gaussian Maxwell Boltzmann distribution. There are more low momentum atoms than in a Gaussian. Using simple models, we show that deviation from Gaussian of the observed magnitude and sign can arise from anharmonicity in the potential or from the introduction of Bose statistics. KEY WORDS: quantum liquids; Bose statistics; quantum effects; anharmonic potentials; momentume distributions; neutron scattering. 1. INTRODUCTION In classical systems, the atomic momentum distribution (AMD) is always the renowned Maxwell Boltzmann distribution. 1,2 Essentially, the atomic momentum p = k and atomic position r are independent, canonical variables and the momentum distribution n(k) is always a Gaussian independent of the interatomic or any external potential. Neutron scattering measurements 3 10 have shown that n(k) for atoms in liquid 4 He, solid 4 He and liquid neon 5,6 differs significantly from a Gaussian. In these quantum systems there is a larger percentage of low-momentum atoms leading to an n(k) that is more sharply peaked around k = 0 than in a Gaussian. The deviation from a Gaussian is particularly pronounced in liquid and solid helium because these are highly quantum systems. The n(k) in liquid neon is nearly Gaussian because liquid neon is nearly a classical liquid. The measurements were chiefly carried out on the MARI instrument at the ISIS neutron scattering facility, UK, which has a high energy resolution option. This makes measurement of the intrinsic shape of n(k) of helium possible independent of the instrument resolution /07/ / Springer Science+Business Media, LLC

2 634 B. Withers and H. R. Glyde function. Deviations from a Gaussian have also been seen in hydrogenic systems. 11 The observed, quantum n(k) are well reproduced in path integral Monte Carlo (PIMC) calculations. 12,13 Although there is good agreement between PIMC calculations 12,13 and experiment, 3,4,9,10 the physical origin of the deviation of n(k) from a Gaussian is not clear. For example, in a highly quantum, but harmonic solid, n(k) is again a Gaussian at all temperatures. Does the deviation from a Gaussian arise from anharmonic effects, and what sign of anharmonic effect is needed to obtain a sharper n(k)? Similarly, can the introduction of quantum effects through symmetrization of the one-body density matrix (OBDM) lead to deviation of n(k) from a Gaussian of the observed sign and magnitude? The goal of this paper is to consider some very simple models to explore these questions. The aim is qualitative understanding rather than quantitative agreement with experiment. Specifically, the spherically symmetric, classical, Gaussian n(k) is, following the notation in the experimental papers, [ ] n G (k) = (2πα 2 ) 3/2 exp 1 k 2, (1) 2 where, α 2 = kq 2 =(mkt / 2 ) = 2 π ( λ 2 and λ 2 = 2πmkT/h 2) is the thermal wavelength. The kq 2 is the mean square wave vector projected along an axis Q, k Q = k. ˆQ. In general, n(k) is the Fourier transform of the OBDM, ρ 1 (r, r ) (r ) (r) dr2...dr N r, r 2...r N e βh r (2),r 2...r N given by, = N Z(β) n(k) = 1 (2π) 3 N dr α 2 dr e ik (r r ) ρ 1 (r, r ). For a translationally invariant system, ρ 1 (r, r ) ρ 1 (r r ) = ρ 1 (r) this reduces to, ρ 1 (r) = n dk n(k)e i k.r = n e i k.r, (3) where n = N/V is the number density, Z is the partition function, β = 1/kT and H is the Hamiltonian. In the quantum case, ρ 1 (r) should be made symmetric with respect to interchange of the atoms. The process of symmetrization of ρ 1 (r) will broaden ρ 1 (r) and sharpen n(k). Again it is not

3 Quantum Momentum Distributions 635 clear what degree of symmetrization or temperature is needed to reproduce the observed n(k). We examine both these effects to investigate qualitatively what determines n(k) in these systems. To characterize the deviation of n(k) from a Gaussian, we expand the OBDM in cumulants and note the magnitude of the higher cumulants. Specifically, the intermediate scattering function observed in neutron scattering experiments in the Impulse Approximation (IA), J IA (s), is the OBDM. That is, J IA (s) = 1 n ρ 1(s) = e ik Qs, (4) where ρ 1 (s) is the OBDM (3) for displacements s = r. ˆQ projected along the scattering vector Q. The neutron data are analyzed by expanding the OBDM as, [ ] e ikqs = exp 2 1 α 2s 2 + 4! 1 α 4s 4 6! 1 α 6s 6 + (5) e 2 1 α 2s 2 [ 1 + δ 4! (α 2 ) 2 s 4 ] and fitting [ the expanded form to the data. In the fit, the cumulants α 2 = kq 2, α 4 = kq 4 3 k2 Q 2],α 6 are treated as free parameters to be determined from the data. Three cumulants could be empirically determined for liquid 4 He with α 6 small. Only α 2 and α 4 in solid 4 He and liquid neon could be determined. In a Gaussian, random system all cumulants vanish except α 2. We use the kurtosis or excess δ α 4 /(α 2 ) 2 to characterize the deviation of the OBDM from a Gaussian. The observed values are δ = 0.60 ± 0.07 in normal liquid 4 He at SVP (T = 2.3K) 9, δ = 0.40 ± 0.10 in solid 4 He on the melting line (V = 21.0 cm 3 /mole) 10 and δ = 0.10 ± 0.07 in liquid neon. 6 The corresponding 3D momentum distribution is 7,14 [ ( )] n(k) = n G (k) 1 + δ 5 10 k 2 + k4 8 3 α 2 3(α 2 ) 2, (6) where n G (k) is the Gaussian component. The observed n(k) in solid helium 10 is reproduced in Fig. 1. In Sect. 2, we evaluate the momentum distribution n(k) of a single particle in an anharmonic well. The goal is to explore whether anharmonic effects can lead to a non-gaussian n(k) with the observed kurtosis. Similarly, we evaluate n(k) for non-interacting Bosons in a box. The aim is to introduce symmetrization (quantum effects) by degrees and evaluate the deviation of n(k) from a Gaussian, i.e., the kurtosis, that results.

4 636 B. Withers and H. R. Glyde Fig. 1. Momentum Distribution of Solid 4 He at T = 1.6 K and V = 21.0 cm 3 /mol compared with its Gaussian component, from Ref. 10. The momentum distribution n(k) is observed to be non-gaussian and is characterized by a larger occupation of the low-momentum states. To evaluate n(k), we write the density matrix as ρ(r, r i, r 2,...,r N ) = φ i(r, r 2,...,r N ) e βe i φ i (r, r 2,...,r N ), (7) Z(β) where φ i is the ith energy eigenstate of the system, and E i its corresponding energy. The expectation value is performed in momentum space. The OBDM is then obtained by integrating over r 2,...,r N, leaving a function of r and r only. 2. ANHARMONIC POTENTIAL ( x ) We consider a single particle oscillating in a 1D potential V(x)= ɛf a where ɛ = ω has energy units, f is dimensionless and a is a characteristic length which we take as a 2 = ( ω/mω 2 ). Expanding the potential about x = 0 and assuming a potential symmetric in x, wehave, V(x) ɛ = f(0) + f (0) ( x ) 2 f (4) (0) ( x ) 4 + 2! a 4! a + f (6) (0) ( x ) 6 ( x ) 8 +. (8) 6! a a

5 Quantum Momentum Distributions 637 We choose f(0) = 0,f (0) = 1,f (4) (0) = γ 4 and f (6) (0) = γ 6 and introduce a 2 = ( /mω) so that ( ) ( ) 2 V(x)= 1 2 mω2 x 2 + γ 4 mω 2 mω 2 x 4 + γ 6 mω 2 mω 2 x 6. (9) 4! ω 6! ω With γ 4 and γ 6 considered as perturbations to the harmonic term, we evaluate the perturbed harmonic energies E i and wave functions φ i as needed in Eq. (7) to evaluate the OBDM for temperatures up to kt (1.5) ω. Figure 2 shows the kurtosis of the 1D n(k) when only γ 4 is retained and γ 6 = 0. To get a positive δ as observed, a negative γ 4 or a shallower well than harmonic is needed. A δ can be obtained. However, a larger δ cannot be obtained since for γ the ground state energy decreases rapidly and becomes unbound as a function of γ 4. When both fourth and sixth order terms are retained γ 4 can be decreased provided γ 6 is positive and comparable to γ 4. Figure 3 shows the well, V(x), and the corresponding wave function (x) when γ 4 = 2.5 and γ 6 = 2.05 giving a kurtosis of δ = 0.4, the value observed in solid helium. However, in the model n(k), while the δ (fourth order) is large, there is a significant sixth order contribution. The fourth and sixth orders tend to cancel so that the full model n(k) is similar to the Gaussian component of n(k) as shown in Fig. 3, RHS. This anharmonic model shows that a shallower well than harmonic creates a sharper n(k) than a Gaussian as observed. In this simple E Z k B T= 1 E Z 1.5 E Z Kurtosis γ 4 Fig. 2. Kurtosis values for γ 6 = 0, γ 4 0 and m = 1K 1 Å 2,ω= 1 K. Temperatures are given as multiples of the zero-point energy.

6 638 B. Withers and H. R. Glyde Ψ(x) 2 [Angstroms -1 ], V(x) [Kelvin] Perturbed V(x) Harmonic V(x) Ψ(x) 2 Harm. Comp. of Ψ(x) 2 n(k) n(k) Gaus. Comp of n(k) x [Angstroms] k [Angstrom -1 ] Fig. 3. Left: Perturbed potential and (x) 2. Right: Associated AMD for kurtosis 0.4 at γ 4 = 2.5, γ 6 = 2.05 for m = 1K 1 Å 2,ω= 1 K and k B T = 0.25E Z. model, it is difficult to get δ as large as observed and still have a stable oscillator. Clearly, self-consistent methods, as is well known, 14 are required for stability. However, this simple model demonstrates that a large kurtosis can be obtained for a shallow well potential. Also, a large kurtosis is obtained provided k B T is much less than the zero point energy. 3. SYMMETRIZATION OF THE OBDM In this section, we evaluate the impact on n(k) of symmetrizing the wave function φ(r, r 2...,r N ) in the OBDM. We do this for non-interacting Bosons confined in a cube of side length, L. At high temperatures we anticipate that making φ(r, r 2...,r N ) symmetric with respect to pair exchange will be most important and ( sufficient. As ) temperature 1/2 is lowered and the thermal wavelength λ = 2π 2 /mk B T lengthens, exchanges involving a larger number of atoms will become important and should be included. Since there are no interactions, the effect of statistics only will be displayed in the symmetrization. Our goal is to calculate the kurtosis of n(k) arising from symmetrization Method of Symmetrization All symmetries can be reached by symmetrizing one wave function only in Eq. (7). We symmetrize this by degrees beginning with pair exchanges, then tripled exchanges and so on up to exchanges of p max

7 Quantum Momentum Distributions 639 Bosons. Including up p max exchanges gives a total of N terms in the wavefunction. The symmetrized OBDM in Eq. (7) is, ρ N N ( ) L 3N Z(β)N d 3N k e γ(k2 1 +k k2 N ), (10)! 2π where γ = λ 2 /4π, = φ 1 (r)φ 2 (r 2 ) φ N (r N ), and = φ 1 (r )φ 2 (r 2 ) φ N (r N ) + Pair Exchanges + Triplet Exchanges + +p max Exchanges. These exchanges can be split into two types. Type I exchanges involve particle 1 (unique since r r ), and Type II do not. There is also the no exchange term (direct). Using the standard notation for permutation of labels, where the top line corresponds to the original ordering (the ordering in the wave function in (10) to the left of the exponential), and the bottom line corresponds to the new arrangements, two examples of these permutations are: [ ] Type I 3421 order = 4, [ ] 1234 Type II order = 3, (11) Direct: [ The number of exchanged labels gives the order of the exchange. A single cycle is always formed for Type II exchange, and a single cycle with one break between r and r in the Type I exchange. To evaluate (10) we use the results, ( ) 1 3 2π ( d 3 r j e W n(r k r j ) 2 e W m(r j r i ) 2 = ]. d 3 ke ık r i e γ k2 e ık r j = 1 λ 3 e (r i r j )2 4γ, (12) π W n + W m ) 3/2 e WnWm Wn+Wm (r k r i ) 2. (13) Equation (12) is used for calculating the integral over momentum space in Equation (10), and (13) is used to integrate out the particle coordinates (r i, i 1) in the OBDM. The particle exchanges lead to terms of the type in (12). Each type of exchange term (I or II) is calculated separately for a general order, by repeated use of the expression in (13). Each exchange term is then multiplied by the number of possible permutations for its order and its type.

8 640 B. Withers and H. R. Glyde Each type (I or II) of order p contributes to the OBDM: [ ] 1 ρ 1,I (p) = (N 1)! N ( Lλ ) 3(N p+1) ( ) 3 1 1L Z (N p)! N! p e (r r )2 3/2 4γp, [ ] ρ 1,II (p) = Z 1 (N 1)! N p N ( Lλ ) 3(N p+1) ( ) 3 1 1L (N p)! p N! p e (r r )2 3/2 4γ, (14) 1 ρ 1,DIRECT = Z [1] N ( Lλ ) 3N ( ) 3 1L N! e (r r ) 2 4γ, where the terms in square parentheses are the number of permutations of that type of order p. Summing (14) over all orders up to p max, the OBDM is given by p max ρ 1 (r, r [ ) = ρ 1,DIRECT + ρ1,i (p) + ρ 1,II (p) ]. (15) p=2 The partition function is then given by the expression d 3 rρ 1 (r, r) = 1, (16) which gives Z = N N! ( ) L 3N λ pmax 1 + p=2 (N 1)! (N p)! ( ) λ 3(p 1) N L p 5/2, (17) where N is now a function of p max, so that when p max = 0, Z = (L/λ) 3N. Fourier transforming the OBDM yields the momentum distribution: ( ) λ 3 γ k2 n(k) = e 2π (18) { 1 + p max (N 1)! ( λl ) 3(p 1) p=2 (N p)! [(N p)p e (p 1)γ k2]} 1 + p max p=2 (N 1)! ( λl ) 3(p 1) (N p)! Np Impact of Symmetrization Figure 4 shows the kurtosis, δ, ofn(k) when the OBDM is symmetrized with respect to exchanges involving up to 2, 4, 6, 8, and 10 Bosons. The δ is shown as a function of the quantum parameter, ζ = ( ) ( ) 1/3 λ Nλ 3 =, l L 3.

9 Quantum Momentum Distributions # of exchanges 8 included 6 (p max ) 4 2 Kurtosis Fig. 4. Kurtosis of n(k) for various p max values, parameterized by ζ where ζ = (nλ 3 ) 1/3,λ is the thermal wavelength and n is the density. ζ 0.5 p max =2 Gaussian Component p max =6 Gaussian Component p max =10 Gaussian Component n(k) k [Angstrom -1 ] k [Angstrom -1 ] k [Angstrom -1 ] Fig. 5. Momentum distributions and associated gaussian components at ζ = 1.5, and p max = 2, 6, and10 with corresponding values of the kurtosis 0.015, 0.100, and 0.236, respectively. where λ is the period wavelength and l is a mean distance between Bosons. In the classical limit, ζ 0, the n(k) is a Gaussian and δ = 0. At ζ =0.5, δ is small and symmetrization with respect to pairs (p max =2) captures all the effects of exchange. At ζ = 1.0,δ 0.05 and exchanges of up to 6 10 particles are important. For ζ 1.5,δ can be large, comparable to observed values in liquid 4 He, and exchanges of a large number of particles are needed to capture the full effects of exchange. The shape of n(k) at ζ = 1.5 and exchanges of up to 2, 6, and 10 Bosons is shown in Fig. 5.

10 642 B. Withers and H. R. Glyde The n(k) differs from a Gaussian qualitatively as is observed in liquid and solid 4 He. Exchanges of 10 or more Bosons are needed to obtain the full impact of quantum effects on n(k). 4. DISCUSSION AND CONCLUSION The present results show that a deviation of n(k) from a Gaussian can arise from an anharmonic potential well. Specifically, a positive kurtosis in n(k) of the observed magnitude can be obtained by incorporating quartic anharmonic terms that lead to a well shallower than harmonic. However, a single oscillator in this well is not stable without adding higher anharmonic terms. To obtain stability in a shallow well, self-consistent methods such as the self-consistent phonon theory 14 are needed or direct Monte Carlo evaluation of n(k). A deviation in n(k) similar to that observed can be obtained in a Bose gas by introducing Bose symmetrization. A temperature low enough that the thermal wavelength is comparable to or greater than the inter-particle spacing is needed. When symmetrization is fully incorporated at low temperature we anticipate ( that ) 1 n(k) would become the Bose distribution function n(k) = e βɛ(k) 1 which is sharply peaked at low k. We expect symmetrization to be most applicable to liquid 4 He. The exchange integrals governing inter-particle exchange are three orders of magnitude larger in liquid helium than in solid helium. Liquid helium is also in the highly quantum range at all temperatures (T 3 K) investigated. The thermal wavelength is λ = 9 and 5 Å at T = 1 and 3 K (ζ = 2 and 1.4), respectively. The anharmonic interaction model applies to both solid and liquid helium. Anharmonic effects are most likely to be the origin of a non-gaussian n(k) in solid helium. Both anharmonic interaction and Bose symmetrization are incorporated exactly in quantum Monte Carlo calculations. The deviation from a Gaussian is small in liquid neon probably because the liquid temperature is high and the liquid is nearly classical. REFERENCES 1. J. C. Maxwell, Phil. Trans. 157, 49 (1867). 2. L. Boltzmann, Wien. Ber. 53, 195 (1866). 3. K. H. Andersen, W. G. Stirling, H. R. Glyde, R. T. Azuah, S. M. Bennington, A. D. Taylor, Z. A. Bowden, and I. Bailey, Physica B-197, 198 (1994). 4. H. R. Glyde, Phys. Rev. B 50, 6726 (1994). 5. R. T. Azuah, W. G. Stirling, H. R. Glyde, P. E. Sokol, and S. M. Bennington, Phys. Rev. B 51, 605 (1995).

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