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HELIUM, SOLID

1

Helium, Solid

Henry R. Glyde

Introduction

Helium was first solidified at the famous Kamerlingh Onnes low-temperature physics laboratories in Leiden by W. H. Keesom [1] on June 25, 1926. The initial experiments by Sir Francis Simon at Oxford University and by Keesom and their collaborators focused on the melting curve, the specific heat, and the thermal conductivity of solid helium as a test of our early understanding of solids. These measurements showed, for example, that the Lindemann criterion of melting does not hold in solid helium. This pioneering work up to 1957 is elegantly and beautifully reviewed by Domb and Dugdale [2], a review that stands today as an excellent introduction to solid helium along with the books and reviews by Wilks [3], Keller [4], Wilks and Betts [5], Glyde [6], Dobbs [7] and Roger et al. [8]. The pair potential v(r) between helium atoms is precisely known [9,10]. It is weakly attractive at large separation, r 3 °-1 , with a maximum well depth A = 10.95 K. At close approach r = 2.63 °, where hard-core radius A defined by v() = 0, v(r) becomes steeply repulsive. The potential parameters and of the rare gases are compared in Table 1. The potential seen by a helium atom lying between two atoms in a linear lattice is depicted in Fig. 1. The well shape, which is wide and anharmonic, is clearly dominated by the repulsive core of v(r). Table 1: Comparison of solid 3 He (at V = 24 cm3 /mole) and solid 4 He (at V = 21.1 cm3 /mole) with the heavier rare-gas crystals. The interatomic potential parameters are the core radius [v() = 0] and the well depth for the following potentials: He [9,10]; Ne, HFD-C2 [9]; Ar, HFD-C [9]; Kr, HFD-C (HFGKK) [9]. For Xe we quote and from Barker et al. [11]. Debye Melting Debye zero Lindemann Potential parameters Rare-gas temperature temperature point energy parameter = u2 1/2 /R (°) A (K) crystal D (K) TM (K) EZD = 9 D 8 3 He(bcc) 19 0.65 21 0.368 2.637 10.95 4 He(bcc) 25 1.6 28 0.292 2.637 10.95 Ne 66 24.6 74 0.091 2.758 42.25 Ar 84 83.8 95 0.048 3.357 143.22 Kr 64 161.4 72 0.036 3.579 199.9 Xe 55 202.0 62 0.028 3.892 282.35 Since helium is light, its thermal wavelength, T , is long, e. g., at T = 1.0 K, T 10 ° for 4 He. Helium is therefore difficult to localize. Attempts to localize A it lead to a high kinetic or zero point energy. Since v(r) is weak, helium does not solidify under attraction via v(r). Rather, it solidifies only under pressure

de Boer parameter 0.325 0.282 0.061 0.019 0.011 0.0065

2

Fig. 1: The potential well seen by a helium atom in a linear solid arising from its nearest neighbors (solid line). R is the interatom space of the linear lattice. Dashed line is the interatomic potential, v(r). Note that the second derivative of the potential well is negative at the lattice point R.

Fig. 2: Phase diagram of helium.

and then solidifies because of the hard core of v(r), much as billiard balls form a lattice under pressure. The total energy E is small and positive except at the lowest pressures. The phases of helium are sketched in Fig. 2. At p 35 bar, the lighter isotope 3 He solidifies into an expanded body-centered-cubic (bcc) structure, having V = 24.4 cm3 /mole with interatom spacing R = 3.75 °. At higher A pressure, both 3 He and 4 He are compressed into the close-packed (fcc and hcp) phases (e. g., at p = 4.9 kbar, fcc 4 He has V = 9.03 cm3 /mole, R = 2.11 °). A A discussion of phases can be found in the books by Keller [4], Wilks [3], and Wilks and Betts [5].

HELIUM, SOLID

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Helium, the Quantum Solid

p The basic character of solid helium is revealed in the specific heat, Cv , arising from vibration of the atoms about their lattice points, the phonons (p). At low p temperature (T ), Cv is well described in all phases by the traditional Debye expression 3 T 4 4 p . (1) Cv = 3R 5 D

Here R is the gas constant and D is a free parameter, the Debye temperature, p obtained by fitting to the observed Cv . D is the characteristic energy of the phonons. From Table 1, D is clearly large compared to the melting temperature (TM ) in solid helium. In Debye's model the zero point vibrational energy is (EZ )D = 2 KE

D

=

9 D . 8

(2)

This energy, a purely quantum effect, clearly dominates thermal energies (TM ) so that solid helium may be called a quantum solid. From D we may also evaluate the mean square vibrational amplitude u2 of the atoms about their lattice points. At T = 0 K this is u2 =

qj

2M qj

= 109.2

1 M D

°2 , A

(3)

where is Planck's constant, M is the atomic mass and qj are the phonon frequencies. From Table 1 we see that u2 and the Lindemann [12] ratio u2 1/2 /R are large in solid helium. Simply from D we see that the atoms are not well localized. There is a trade-off between reducing the (energy), e.g. EZ in Eq. 2 and localizing the particle u2 in Eq. 3. To keep the zero point energy manageable each nucleus has a large vibrational distribution about its lattice point. The atoms therefore explore a wide range of the potential well depicted in Fig. 1. For the dynamics, the hard core is the important part of the potential and all force constants will be the derivatives of v(r) averaged over large-amplitude vibration. The vibrational distribution may be described by a nuclear wave function with wavelength given by the famous de Broglie wavelength, . The degree of quantumness of a solid can be characterized by the de Boer [13] parameter, 1 h = 2 (2m)1/2 (4)

which is the ratio of the de Broglie wavelength, = h/p [with momentum p = (2/m)1/2 ], to the minimum separation of atoms in the crystal, . If is comparable to , the solid is highly quantum (see Table 1). Also, is clearly mimicked by the Lindemann ratio (see Table 1). A large therefore means quantum and very anharmonic vibration.

4 The large nuclear wave function of atoms means that wave functions of atoms on adjacent lattice sites overlap. This leads to direct nuclear exchange integrals and tunnelling between sites. In solid 3 He, which has a nuclear spin 1 , this 2 exchange can be traced in nuclear magnetic resonance experiments and in thermodynamic properties. The characteristic energies in helium are the phonon energies (D ) or kinetic energy KE 25 K, the total energy E 1 K, and the exchange energy, J 10-3 K.

Ground State and Dynamics

A key to understanding helium is finding a suitable wave function (r1 . . . rN ) to describe the vibrational distribution of the atomic nuclei. Nobel laureate Max Born and his student D. J. Hooten first recognized that this vibrational distribution and the phonon dynamics must be determined consistently [2]. Nosanow [14] proposed a Gaussian function, the wave function for a harmonic crystal, for the distribution with corrections to account for short-range correlations between pairs of atoms induced by the hard core of v(r). The short-range correlations can be described by a Jastrow pair function or by use of a Brueckner T -matrix method, both developed initially to describe short-range correlations between nucleons in nuclei. The self-consistent determination of the phonon frequencies and lifetimes and the vibrational distribution constitutes the selfconsistent phonon (SCP) theory. This theory is exhaustively reviewed [6,15-17]. Phonons in solid helium are most directly observed by inelastic neutron scattering. Measurements on all phases show that, although solid helium is highly anharmonic, most phonons are well defined and have long lifetimes [6]. The expanded bcc 4 He phase is so anharmonic that phonon energies can be determined at low wave vector (Q) only (see Fig. 3). At higher Q, the scattering intensity contains fascinating interference effects between the one-phonon and multiphonon scattering contributions that broadens the response and can be well described using the SCP theory [6]. The more compressed fcc phase of helium is significantly less anharmonic and can be quite well described without short-range corrections. Recent measurements [20] identify an anomaly in the intensity suggesting a new mode. The static properties such as the ground state energy, E = KE + V , and the atomic momentum distribution, n(k), are most accurately calculated using Monte Carlo (MC) methods [21], Diffusion Monte Carlo (DMC) [22,23] at T = 0 K and Path Integral Monte Carlo (PIMC) [21] at T > 0 K. Typically KE 25 - 50 K depending on the density. The KE and V nearly cancel to give E = ±1 K in solid 3 He. In spite of this, DMC values of E for both solid 3 He and 4 He agree remarkably well with experiments over a wide density range [23]. The atomic kinetic energy KE and the atomic momentum distribution n(k) can be measured directly by high energy transfer neutron scattering [6]. As seen from Fig. 4, the observed KE and those calculated by Monte Carlo methods agree well for solid 4 He. It is interesting that the observed and M C KE M C = 25 K at V = 21 cm3 /mole in Fig. 4 is twice the Debye model

HELIUM, SOLID

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Fig. 3: The phonon-frequency dispersion curves observed in bcc 4 He by Minkiewicz et al. [18] and calculated by Glyde and Goldman [19] in the selfconsistent harmonic (SCH) and complete first-order self-consistent (SC1) approximations. value KE D = (9/16) D = 14 K obtained using the observed D in Table 1. This difference is a direct manifestation of the anharmonic hard core of v(r). Interactions via the hard core introduce high-energy anharmonic tails in the phonon response functions. These high-energy components raise the average KE above that predictable by an adjusted Debye model which has no tails. High-energy tails are common to all phases of helium. Recent measurements Diallo [24] show that n(k) in solid 4 He differs significantly from a Gaussian (see Fig. 5). There are more low momemtum (low n(k)) atoms than in a classical, Maxwell-Boltzmann n(k). It is not clear whether this arises from anharmonic efforts or from the onset of Bose statistics at low temperature since the shape of n(k) is similar [24,25] in liquid and solid 4 He. This deviation from a Gaussian n(k) is also found in PIMC calculations [21,26] in both liquid and solid 4 He. In addition, the Debye-Waller factor is found to deviate from a Gaussian in PIMC calculations [27].

Exchange and Magnetic Properties

Solid 3 He, a nuclear spin 1/2 Fermi solid, displays a rich spectrum of nuclear spin phases at low temperature (T 10-3 K = 1 mK). This is in both bulk, 3D bcc solid 3 He close to the melting line [8,28] and in 2D triangular lattice layers that form on grafoil surfaces. In bcc 3 He there is a transition from the paramagnetic to an antiferromagnetic phase at T = 1 mK at low magnetic field (B 0.46 T). Nuclear magnetic resonance [29] and neutron-diffraction data [30] are consistent with this being a tetragonal [29] U2D2 ordering displayed in Fig. 6. The antiferromagnetic resonant frequency, 0 , of the U2D2 ordering agrees

6

45

Previous Exp't This Exp't PIMC

Atomic Kinetic Energy (K)

GFMC

40

35

30

25

20 14 16 18 20 22

V (cm /mole)

3

Fig. 4: Atomic kinetic energy in solid 4 He. The solid symbols are experimental values. The open symbols are GFMC [22] and PIMC [26] calculations (from Ref. [24]).

0.06

4

Solid

He

Observed n(k) Gaussian Comp. 0.04

n(k)

(Å )

0.02 0.00 0 2 4

-1

-3

k (Å )

Fig. 5: Momentum distribution of solid 4 He at T = 1.6 K and V = 20.87 cm3 /mol compared with its Gaussian component. The momentum distribution n(k) is observed to be non-Gaussian and is characterized by a larger occupation of the low momentum states [24].

HELIUM, SOLID

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Fig. 6: The tetragonal up­up down­down (U2D2) and cubic canted normal antiferromagnetic (CNAF) structures. Exchange processes involving two, three, and four atoms in HM SE , J2 = -J1N , J3 = -T1 , J4 = -Kp .

better with experiment [31] than do other orderings not excluded by neutron and NMR data. At higher field, B 0.46 T, there is a transition to a second ordered phase believed to be a canted normal antiferromagnetic (CNAF) phase (see Fig. 6). As B is increased, the canting angle decreases until it goes to zero at an upper critical field [31], BC2 = 21.7 ± 1 T. For B 10 T, the phase diagram is not well known. The physical origin of the rich magnetic behaviour is the large vibrational amplitudes of the nuclei noted above. The nuclear wave functions are broad and the wave functions of nuclei on adjacent sites overlap leading to large nuclear exchange integrals, J 1 mK. This is true only at large molar volumes where u2 is large. J decreases exponentially with decreasing molar volume [32,33]. Also, since the pair potential has a repulsive hard core, simple pair exchange is suppressed relative to exchange integrals involving ring exchanges of three or more atoms [34]. Indeed in 2D layers the magnitudes [33] of the Jn involving exchange of n atoms are in the order n = 3, 2, 4, 6, 5. Also exchanges of an even number of atoms (odd parity 4 exchanges) favour antiferromagnetism while exchanges of an odd number (even parity) favours ferromagnetism [34]. This leads to a multiple spin exchange (MSE) Hamilitonian [35,36] with competing antiferromagnetic and ferromagnetic terms,

8

HM SE

=

n

(-1)n Jn

(n)

Pn P3 + J4

(3) (4)

= J2

(2)

P2 - J3

P4 - J5

P5 + J6

(5) (6)

P6 , (5)

where Pn is a permutation involving n atoms. The sign convention in HM SE is chosen so the exchange energies Jn are positive. The first term in HM SE is the Heisenberg Hamiltonian, HH = J2 ij P2 (ij) with P2 (ij) = 2Si · Sj + 1/2. In pioneering work, Roger, Hetherington and Delrieu [8] and Roger [28] were able to reproduce the magnetic phase diagram of bulk bcc 3 He as well as the magnetic specific heat keeping up to four particle exchanges (n = 2 - 4) in HM SE , The exchange constants J2 = -J1N , J3 = -T1 and J4 = Kp were treated as fitting parameters where J1N , T1 and Kp is the notation used by Roger et al [8]. Ceperley and Jacucci [32] in a remarkable application of path integrals have evaluated the Jn . They confirm that the first four terms in HM SE are the most important and find J2 = 0.46 mK, J3 = 0.19 mK and J4 = 0.27 mK at V = 24.12 cm3 /mole within 10 % of the Roger et al [8] values. Exchanges n > 4 are non-negligible and including these Godfrin and [37] obtain impressive agreement with experiment, especially the Curie-Weiss constant. The magnetization and specific heat of 2D solid 3 He (on grafoil) also displays remarkable behaviour [38,39]. For example, there is a cross-over from antiferromagnetic to ferromagnetic response in the second 2D layer on grafoil with increasing density. This magnetic behaviour and the cross-over can be reproduced using HM SE in (5) keeping terms up to n = 6 with the antiferromagnetic (J4 , J6 ) and the ferromagnetic (J3 ) terms having different density dependencies [35,40]. Spin liquids are suggested [41]. Indeed exotic spin states having magnetic short range order, a spin gap but no Neel long range order are predicted [36] using HM SE . These novel spin orderings serve as realizations of spin states, such as valence-bond spin liquids, of broad interest in condensed matter [36]. The direct energies such as KE , V and the ground state E are 104 times larger than the exchange energies, Jn . Thus statistics (Fermi or Bose) play no role in determining the the ground static wave function. The magnitude of the exchange integrals will therefore be similar in solid 3 He and 4 He.

Supersolid Helium

Kim and Chan [42-44]have observed a nonclassical, superfluid component in solid 4 He. This is in both bulk [42] solid 4 He and in solid 4 He immersed in porous media [43,44]. The moment of inertia of solid or liquid 4 He can be accurately measured using a torsional oscillator. In a normal solid or liquid, the full moment of the solid is observed. When solid 4 He is cooled below 0.23 K, the moment of inertia is observed to decrease indicating that a fraction of the solid has decoupled from the walls of the oscillator exactly as if it were

HELIUM, SOLID

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a superfluid. The "superfluid" fraction, s /, of solid 4 He is approximately 2 % at low temperature (as against 100 % in liquid 4 He). The temperature dependence of s (T )/ is very different in solid and liquid 4 He. Superfluidity can be suppressed by adding a few hundred parts per million of 3 He. The "supersolid" phase extends up to pressures of 65 bar (compressed 4 He) which is surprising. Possible mechanisms of superflow are based on the large exchange integrals in quantum solids (tunnelling) and often involve a quantum defect (e.g. vacancies) as the superfluid component. This observation of "supersolidity" opens up an exciting new dimension to quantum solids. See also: Crystal Binding; Crystal Defects; Interatomic and Intermolecular Forces; Rare Gases and Rare-Gas Compounds; Tunneling. References [1] Keesom W. H. Keesom, Comm. Kamerlingh Onnes Lab., Leiden No. 184b (1926); Helium, p. 180. Elsevier, Amsterdam, 1942. [2] Domb:57 C. Domb and J. S. Dugdale, Prog. Low Temp. Phys. 2, 338 (1957). [3] Wilks:67 J. Wilks, The Properties of Liquid and Solid Helium. Clarendon Press, Oxford, 1967. [4] Keller:69 W. E. Keller, Helium-3 and Helium-4. Plenum, New York. 1969. [5] Wilks:87 J. Wilks and D. S. Betts, Liquid and Solid Helium. Clarendon Press, Oxford, 1987. [6] Glyde:94 H. R. Glyde, Excitations in Liquid and Solid Helium. Clarendon Press, Oxford, 1994. [7] Dobbs:94 E. R. Dobbs, Solid Helium Three. Clarendon Press, Oxford, 1994. [8] Roger:83 M. Roger, J. H. Hetherington, and J. M. Delrieu, Rev. Mod. Phys. 55, 1 (1983). [9] Aziz:84 R. A. Aziz, in Inert Gases (M. L. Klein, ed.). Springer-Verlag, Berlin, Heidelberg, 1984. [10] Korona:97 T. Korona, H. L. Williams, R. Bukowski, B. Jeziorski and K. Szalewicz, J. Chem. Phys. 106, 5109 (1997). [11] Barker:76 J. A. Barker, M. L. Klein, and M. V. Bobetic, IBM J. Res. Dev. 20, 222 (1976). [12] Lindemann:11 F. A. Lindemann, Phys. Z. 11, 609 (1911).

10 [13] deBoer:57 J. de Boer, in Progress in Low Temperature Physics, Vol. 2 (J.C. Gorter, ed.). North-Holland, Amsterdam, 1957. We use the definition introduced by Newton Bernardes, Phys. Rev. 120, 807 (1960), which is (2 2)-1 times de Boer's definition. [14] Nosanow:66 L. H. Nosanow, Phys. Rev. 146, 120 (1966). [15] Koehler:75 T. R. Koehler, in Dynamical Properties of Solids, Vol. II (G. K. Horton and A. A. Maradudin, eds.). North-Holland, Amsterdam, 1975. [16] Glyde:76 H. R. Glyde, in Rare Gas Solids, Vol. I (M. L. Klein and J. A. Venables, eds.). Academic Press, New York, 1976. [17] Horner:72 H. Horner, in Dynamical Properties of Solids, Vol. I (G. K. Horton and A. A. Maradudin, eds.). North-Holland, Amsterdam, 1974. [18] Minkiewicz:73 V. J. Minkiewicz, T. A. Kitchens, G. Shirane, and E. B. Osgood, Phys. Rev. A 8, 1513 (1973). [19] Glyde:76a H. R. Glyde and V. V. Goldman, J. Low Temp. Phys. 25, 601 (1976). [20] Markovich:02 T. Markovich, E. Polturak, J. Bossy, and E. Farhi, Phys. Rev. Lett. 88, 195301 (2002). [21] Ceperley:95 D. M. Ceperley, Rev. Mod. Phys. 67, 279 (1995). [22] Whitlock:87 P. A. Whitlock and R. M. Panoff, Can. J. Phys. 65, 1409 (1987). [23] Moroni:00 S. Moroni, F. Pederiva, S. Fantoni, and M. Boninsegni, Phys. Rev. Lett. 84, 2650 (2000). [24] Diallo:04 S. O. Diallo, J. V. Pearce, R. T. Azuah and H. R. Glyde, Phys. Rev. Lett. 93, 075301 (2004). [25] Glyde:00 H. R. Glyde, R. T. Azuah and W. G. Stirling, Phys. Rev. B 62, 14337 (2000). [26] Ceperley:89 D. M. Ceperley, in Momentum Distributions, (R. N. Silver and P. E. Sokol, eds). Plenum, New York, 1989. [27] Draeger:00 E. W. Draeger and D. M. Ceperley, Phys. Rev. B 61, 12094 (2000). [28] Roger:84 M. Roger, Phys. Rev. B 30, 6432 (1984). [29] Osheroff:80 D. D. Osheroff, M. C. Cross, and D. S. Fisher, Phys. Rev. Lett. 44, 792 (1980). [30] Benoit:85 A. Benoit, J. Bossy, J. Flouquet, and J. Schweizer, J. Phys. Lett. 46, L923 (1985).

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[31] Osheroff:87 D. D. Osheroff, H. Godfrin, and R. R. Ruel, Phys. Rev. Lett. 58, 2458 (1987). [32] Ceperley:87 D.M. Ceperley and G. Jacucci, Phys. Rev. Lett. 58, 1648 (1987). [33] Bernu:02 B. Bernu and D. M. Ceperley, J. Phys. Condens. Mat. 14, 9099 (2002). [34] Thouless:65 D. J. Thouless, Proc. Phys. Soc. 86, 893 (1965). [35] Roger:97 M. Roger, Phys. Rev. B 56, R2928 (1997). [36] Misguich:99 G. Misguich, C. Lhuillier, B. Bernu, and C. Waldmann, Phys. Rev. B 60, 1064 (1999). [37] Godfrin:88 H. Godfrin and D. D. Osheroff, Phys. Rev. B 38, 4492 (1988). [38] Godfrin:95 H. Godfrin and R. E. Rapp, Adv. Phys. 44, 113 (1995). [39] Collin:04 E. Collin and Yu. M. Bunkov and H. Godfrin, J. Phys. Condens. Mat. 16, S691 (2004). [40] Roger:90 M. Roger, Phys. Rev. Lett. 64, 297 (1990). [41] Ishida:97 K. Ishida, M. Morishita, K. Yawata and H. Fukuyama, Phys. Rev. Lett. 79, 3451 (1997). [42] Kim:04 E. Kim and M. H. W. Chan, Science 305, 1941 (2004). [43] Kim:04a E. Kim and M. H. W. Chan, Nature 427, 227 (2004). [44] Kim:05 E. Kim and M. H. W. Chan, J. Low Temp. Phys. 138, 859 (2005).

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