Birefringence of hexagonal single crystals of helium-4 at various pressures

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Birefringence of

hexagonal single crystals

of helium-4

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»

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Birefringence of

hexagonal single crystals

of helium-4

at various pressures

Proefschrift

ter verkrijging van de graad van doctor in de technische wetenschappen aan de Technische Hogeschool Delft op gezag vian de rector magnificus dr. ir. C. J. D. M. Verhagen, hoogleraar in de Afdeling der Technische Natuurkunde, voor een commissie uit de senaat te verdedigen op woensdag 29 mei 1968 te 16 uur

door

Johan Eeuwe Vos natuurkundig ingenieur geboren te Rotterdam

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Dit proefschrift is goedgekeurd door de promotor prof. dr. B. S. Blaisse.

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Voor hen die hebben bijgedragen aan het tot stand komen van dit proefschrift.

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o Introduction

Due to advanced experimental techniques and increased impetus from theoretical physics much work has been done in the last decade on solid inert gases, and especially on solid heUum. The latter is the most pronounced 'quantum solid'. Only in the last years experimental results are obtained from more or less perfect single crystals of helium; measurements have previously been carried out on polycrystalline material, the grain size of which was unknown.

The most obvious way to check the size and the quality of a crystal that belongs to any non-cubic class is an optical observation in polarized light. As helium crystallizes in a wide range of pressures and temperatures in the hexagonal close-packed struc-ture, this type of optical control of single crystals is ideal for the the hexagonal helium crystals. These are uniaxial; a complete set of observations yields not only infor-mation regarding the quality of the crystal, but the direction of the optic and crystallo-graphic c-axis as well.

In this thesis we will discuss the following subjects: the phase diagram of helium-4, the stability of the three crystalline phases, some properties of hexagonal helium crystals, the microscopic theory of birefringence, the apparatus used by us and the results of our experiments.

Originally the reason to set up our experiments on hexagonal helium crystals was the challenge to observe a possible optical anisotropy and to measure the value of the birefringence n^—n^. Recently we succeeded in measuring this quantity even as a function of density. Another result of our optical work was, that we were able to observe visually the solidification and recrystallization processes, and the phase transitions from the hexagonal to each of the cubic phases.

Probably the most important practical result of the present work is that in the future it will be possible to combine optical measurements with the determination of other anisotropic properties of hexagonal helium crystals. It will not be very diflScult to measure such properties in differently orientated single crystals.

Finally, in this thesis our results of the birefringence of hexagonal helium-4 crystals will be interpreted in terms of the theory of birefringence originally due to Ewald. It will be shown that the axial ratio cja of the hexagonal crystal increases with density; at the highest densities at which we carried out measurements this ratio approaches the value ^ ^ 6 which holds for the closest packed structure.

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1 The phase diagram of helium-4

Introduction

Before discussing the properties of single crystals in the hexagonal close-packed structure, we wish to indicate under which conditions of tempera-ture and pressure this structempera-ture can exist, i.e. in what part of the phase diagram it is situated. It seems appropriate to consider the whole phase diagram of helium-4 in the beginning of this work, and to show several particular properties of this phase diagram.

1.1 The pT diagram

The/»r diagram of helium-4 is depicted in figs. 1.1 and 1.2. In the first figure we see two fluid phases and two solid phases, all at relatively low pressures and temperatures. In the second diagram the melting line is extended to high pressures and high tem-peratures; there is a third solid phase as well. For purposes of comparison the phase diagram of heUum-3 is also shown (fig. 1.3).

50 40 30 20 10 0 1 2 3

Fig. 1.1 The phase diagram of helium-4 at low temperatures and pressures. The X-line, the melting line and the transition line between h.c.p. and b.c.c. solid are shown. The vapour pressure line almost coincides with the horizontal axis. A tangent has been drawn from the origin to the melting line.

One of the most obvious consequences of the quantum character of helium is evident from the fact that helium, being the only exception to the rule, cannot be solidified under its own vapour pressure. The melting line of helium-4 comes to zero temperature

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at a pressure of 24.96 atm; the vapour pressure line drops to zero pressure at zero temperature. The liquid exists everywhere between the vapour pressure line and the melting line. It can exist in two phases, normal liquid He I and superfluid He II, separated by the A-line. The transition between these two liquid phases is called a A-transition because of the A-shaped curve for the specific heat at this second-order transition.

The melting line has been determined, with increasing accuracy, by various investig-ators, e.g. Keesom (1926, blocked capillary method), Domb & Dugdale (1957), Grilly & Mills (1959), and Langer (1961, up to 14 140 atm at 77.3 °K).

2000 1000 800 600 400 200 100' 80 60 40 20. ,1 2 4 6 8 10 20

Fig. 1.2 The phase diagram of helium-4 extended to elevated pressures and temperatures, so that the transition line between h.c.p. and f.c.c. solid can be seen. Also some approximate isochores in the solid phase have been drawn.

1.2 The structures of the solid phases

Keesom & Taconis (1938) were the first to analyse the structure of solidified helium by means of X-ray diffraction. They found a hexagonal close-packed structure (h.c.p.), with nearest neighbour distance 0.357 nm at 1.45 °K and 37 atm. The h.c.p. structure

1 1 1 1 1 1 K„ = 10 cm'/mol flatm 12 -14 -/ I« / - / h.c.p. / - 1 8 / — — — •—— -'J / / / 20 / . 2 0 . 7 _ / " ' " \ l 1 1 t 1 1 1 1 1 J / / / fluid -1 -1 -1 1 ' _., /f.c.c./ T.'K 1 / •

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-is shown in fig. 1.4, where the quantities a and c are the nearest neighbour d-istance, and twice the layer spacing, respectively.

An important parameter of an h.c.p. lattice is the value c/a, which could not yet be calculated from the results of Keesom & Taconis. We shall refer to this quantity often, calling it the axial ratio.

We wish to adhere to convention by referring to all those lattices that can be con-structed out of closest packed layers (with hexagonal symmetry) in the sequence ABABAB . . . as hexagonal close-packed or h.c.p. In the exceptional case of hexagonal closest packing c/a has the value 1.^6 = 1.63299 .. ., but actual monatomic h.c.p.

2000 1000 800 600 400 200 100 80 60 40 20 1 2 4 '6 8 10 20

Fig. 1.3 The phase diagram of helium-3 on the same scale as fig. 1.2. The position of the h.c.p. to b.c.c. transition line in the solid is clearly different from the one for helium-4.

lattices have values of c/a varying in a range as broad as from 1.58 to 1.85.

More recent determinations of the structure in the hexagonal phase of helium-4 using neutron difl'raction have been performed by Henshaw (1958) who found lattice con-stants from which follows c/a = 1.63 ± 0.03 at a molar volume of 18.57 cm^. A re-calculation of his data by Donohue (1959) resulted in the value c/a = 1.612 ± 0.004.

T I I I I T^ r I T

iï atm

T,°K

J I I I I 1111 i n

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Fig. 1.4 The position of atoms in an h.c.p. structure, which consists of close-packed layers in an alternating arrangement ABABA . .. The nearest neighbour distance within each layer is a; the layer spacing is Jc.

Lipschultz et al. (1967) found for a large single crystal at 27 atm c/a = 1.638 ± 0.001. The latter experiments are further described in sec. 3.4d.

On the other hand, the following results from experiments with X-ray diifraction can be mentioned: Mills & Schuch (1962) found lattice constants from which c/a = = 1.628 lb 0.009 at a molar volume of 20.66 cm^; Schuch & Mills (1963) report values for which c/a = 1.627 at 17.4 cm^/mol. The importance of the value of c/a for optical properties will become clear in the discussion in chap. 4.

A first-order phase transition within the solid phase has been found by Dugdale & Simon (1953). They observed a slight latent heat in their measurements of the heat capacity of sohd helium at pressures above 1200 atm; they attributed this fact to a first-order transition from hexagonal close-packed to cubic close-packed or face-centered cubic (f.c.c), as the AVin this transition is extremely small. A confirmation of their supposition was presented by the X-ray diffraction results of Mills & Schuch (1961). (This face-centered cubic phase is usually called the p phase, as distinct from the hexagonal close-packed one which is usually called the a phase. We shall not use this designation with Greek letters, because it leads to confusion as the nomenclatures for helium-4 and helium-3 are not consistent).

A second phase transition in the soUd was found by Vignos & Fairbank (1961) who were measuring longitudinal sound velocities in crystalline helium-4 at relatively low pressures.

A discontinuity in their results at temperatures several hundredths of a degree below the melting line between 1.448 °K and 1.770 °K led them to the conclusion of an additional first-order phase transition. The structure of this new-found phase has been determined by Schuch & Mills (1962), again using X-ray techniques. The struc-ture appeared to be body-centered cubic, b.c.c.; (generally called the y phase). The existence of three different crystalline structures needs some further considera-tion. In the first place, the h.c.p. structure which exists in the major part of the phase diagram is not the structure we would expect if we compare helium with the other sohd inert gases. All other solid inert gases have the f.c.c. structure, which helium has only at elevated temperatures, above about 15 °K. The relative stabilities of both close-packed structures will be discussed separately in chap. 2. In the second place

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we remark that the existence of the less dense b.c.c. phase is limited to a very small temperature interval, its maximum width being 0.08 deg. An explanation for its stabiUty in just this range of the phase diagram has never been given; a qualitative argument lies in the fact that the b.c.c. phase occurs in the part of the pT diagram that has relatively the lowest pressure at the highest temperature.

Nosanow (1966) has calculated the ground state energy for h.c.p. and b.c.c. heüum-4 and helium-3 in the range of molar volumes between 17 and 21 cm^. Using correlated wave functions (see sec. 2.8) he finds that the b.c.c. structure has the lowest ground state energy in this complete range. His methods being less appropriate for higher densities and hexagonal symmetry, it is not surprising that the stability problem cannot be settled in this way.

Some particular properties of the phase diagram

The point where the A-Hne cuts the melting line (the 'upper A point', or A' point) is very near the upper triple point of h.c.p., b.c.c. and liquid, see fig. 1.5. It would be purely accidental if these points should coincide. The experimental work of Kierstead (1966) showed that the interval between triple point and upper A point is Ti—T^. = = (10.00 ± 0.10)X 10"^ deg and p^-p^ = 0.305 ± 0.003 atm. These differences increase if some percents of helium-3 are added to the helium-4.

Fig. 1.5 The position of the b.c.c. phase of helium-4, as determined by Vignos & Fairbank (1961). Also the upper part of the X-line and the tangent to the melting line through the origin are shown in greater detail than in fig. 1.1.

As Simon (1934) has observed, the A-line meets the melting line very near to the point where the tangent to the melting line passes through the origin (see again fig. 1.5).

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In this point is / , \

Po_ ^ (dp\ To \dTjo

From the phase diagram we find TQ = 1.70 °K.

This gives an indication of the difference in internal energy between the liquid and the solid phase at either side of the 'upper A point', as can be seen from the following. We consider Clapeyron's equation:

dp^^AS_ dT ~ AV

For T> To is AS/AV > p/T or TAS > pAV. Therefore AU = TAS-pAV > 0 and the Uquid has the higher internal energy. But for T < TQ is JC/ < 0: the solid has now a higher internal energy than the liquid, which is superfluid in almost the same region, namely below 1.764 °K.

HeUum is the only substance for which the melting curve does not begin at zero pressure; therefore, it is not possible to draw a tangent to the melting curve as we have done above, for any other substance except helium. Consequently helium is the only substance for which the liquid state can be the state of lowest energy.

A minimum in the melting curve occurs at T = 0.77 °K, and is 7.5 x 10"^ atm below the melting pressure at zero temperature, po = 24.96 atm; see Wiebes & Kramers

10 8 6 4 2 0 ' 0.4 0.5 06 0.7 0.8 09

Fig. 1.6 The minimum in the melting line of hellum-4, as determined by Straty & Adams (1966).

(1963) and also Straty & Adams (1966), from which fig. 1.6 has been taken. The occurrence of this minimum proves, again via Clapeyron's equation, that below 0.77 °K also the entropy of the soHd is larger than that of the liquid. This unusual fact can be understood qualitatively in the following way: at a fixed low temperature the thermal disorder of the solid arising from the excitations of phonons of all possible polarizations may become larger than that of the liquid arising from longitudinally polarized phonons alone; the liquid cannot propagate transverse elastic waves (see e.g. Goldstein & Mills, 1967). Baum, Brewer, Daunt & Edwards (1959) found a much more pronounced minimum in the melting line of helium-3. The minimum melting pressure is here 29.3 atm at T = 0.32 °K. The curve rises to 31.1 atm at 0.1 and 0.58 °K.

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2 The phase transitions

in solid hehum

Introduction

In the preceding chapter we have mentioned the existence of three lattice types, h.c.p., f.c.c, and b.c.c, in solid helium. In this respect, helium has an exceptional position between the inert gases, becauses all others exist only in the f.c.c. structure. In this chapter we wish to discuss briefly the phase transition between the close-packed structures in the case of helium, and the stability of the b.c.c. structure. Much attention has been paid by various authors to the problem of the stability of both close-packed phases. Mostly the goal has been to explain the h.c.p. structure observed for helium and the f e e . structure for the other inert gases. (A similar problem is the explanation of the structures observed in the case of the alkali halides). But even if this comparison of the struc-tures of various inert gases was the goal, the same arguments may be relevant to the phase transition between h.c.p. and f.c.c. in helium. We shall now give some of the arguments found in the literature, without making a definite choice between the various approaches.

2.1 The interatomic potential

The attractive interaction between two atoms of an inert gas is due to the well-known Van der Waals forces between induced dipoles and multipoles (see e.g. Mar-genau, 1939). A simple discussion of the interaction between induced dipoles leads to an attractive potential proportional to the inverse sixth power of the interatomic distance.

Dipole-quadrupole forces, quadrupole-quadrupole forces and contributions of the higher multipoles give rise to attractive potentials proportional to the inverse eighth-, tenth power, etc. of the interatomic distance. The effects become rapidly smaller, even at distances comparable to the interatomic distance in an inert-gas crystal. At very smaU interatomic distance a repulsion becomes important that is caused by the net effect of overlap forces. Quantum mechanical reasoning leads to an exponential repulsive potential. We shall not discuss the above effects in any detail, because we are only interested in finding a shape of the potential function that is suitable for calculations. The aforementioned effects would give rise to a potential of the form used for instance by Slater and Kirkwood (1931):

(p{r) = C e x p ( - A r ) - j ; r ~ ^

Because of greater ease in mathematical handling another potential function is adopted by most workers. This function is due to Jones and to Mie (see e.g. Lennard-Jones and Ingham, 1925), and has the form

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We shall use the generally accepted value s = 12 in our considerations mentioned here-under. If one wishes to derive the potential energy of a complete lattice by summation over two-body terms, numerical results are obtained more easily with a power function than with an exponential function. The values used for e and <T are such that the Lennard-Jones potential describes the repulsive interaction equally well as the Slater-Kirkwood potential. In fact, a large number of experimental results can be described consistently with an interaction gouvemed by a Lennard-Jones function. The parameters e and a can be determined for each species of atoms from various experimental data, as for example second virial coefficients at low temperatures. The proper set of values should explain different properties of the species in question. Usually the Lennard-Jones potential is supposed to be additive, i.e. one supposes that the total potential energy of a set of three atoms is equal to the sum of the potential energies of the three pairs of atoms. We shall mention in sec. 2.7 the calculations of Jansen and co-workers (1964-1965) who find the total Van der Waals interaction in a crystal to be non-additive; the correction due to this effect will not be considered in the next sections.

The total energy of the crystal

Apart from the static lattice potential, which can be calculated as a sum over pairs of atoms of contributions like those given in the preceding section, the total energy of a crystal includes thermal (vibrational) energy and zero-point energy. If the frequen-cies Vj of the normal modes of vibration of a crystal have been calculated, the vibra-tional quantum numbers «j give both contributions: ^ , «j Avj is the thermal energy and YJ, i^Vj is the zero-point energy. It is important to note that the normal modes are dependent on volume (and thus on temperature and pressure); consequently the zero-point energy is not independent of the temperature of a given sample!

If one wishes to determine the most stable configuration for the crystal, the total free energy must be minimized. Even if the stability at zero temperature must be found, the normal modes have to be calculated in order to find the zero-point energy. In former publications (see e.g. Prins, Dumoré and Lie, 1952), only the static lattice energy arising from Van der Waals-forces was considered.

The work of Barron and Domb (1955), which had been stimulated directly by the discovery of the phase transition in solid heUum by Dugdale and Simon (1953), took into account the static lattice energy plus lattice dynamics. These authors also estim-ated the zero-point energy, but they did not believe the latter result to be correct in the case of helium on account of another exceptional effect in this case, namely the large anharmonicity of the lattice vibrations, even at very low temperatures. We shall consider the calculations of the static lattice energy first in more detail, and discuss the zero-point energy problem in sec. 2.6.

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2.3 The static lattice energy

To find the static lattice energy, one must calculate lattice sums of the shape X*"* ''* ^ where we take s = 6 and 12 (see fig. 2.1). Because the numbers of neigh-bours «t at a distance r^ are different in the h.c.p. and f.c.c. lattices (see table 2.1), the static lattice energies are different. In these calculations one assumes that the axial

Table 2.1 Number of neighbours in the first shells in a closest packed h.c.p. and in an f.c.c. crystal; the last two columns give the cumulative numbers up to and including the shell concerned.

fry

[")

1 2 2f 3 3Ï 4 5 5f 6 no. of neighbours in h.c.p. 12 6 2 18 12 6 12 12 6 f.cc. 12 6 -24 -12 24 -8 cumulative no. of atoms in h.c.p. 12 18 20 38 50 56 68 80 86 f.c.c. 12 18 18 42 42 54 78 78 86

ratio of the h.c.p. crystal is equal to the ideal value |>/6, and that the molar volumes of both lattices are the same. We shall see in sec. 7.8 that the axial ratio in h.c.p. heHum-4 is very near this ideal value.

Calculations of such lattice sums for various lattices and various values of s have been performed by Lennard-Jones & Ingham (1925), Kane & Goeppert-Mayer (1940), Prins, Dumoré & Lie (1952), Kihara & Koba (1952), Barron & Domb (1955) and Wallace & Patrick (1965).

Generally speaking, the result of the calculations of static lattice energies is that, for vanishing pressure, the h.c.p. lattice has a lower energy than the f.cc. lattice. The difierence in energy is about 0.01% if one uses the Lennard-Jones (12; 6) potential. Prins et al. (1952) also calculated the change in the energy difference if the lattices are compressed or elongated along the trigonal direction. They find that the difference in-creases with compression and dein-creases with elongation.

Kihara & Koba (1952) studied the Lennard-Jones (s; 6) potentials for s = 7, 8, 9, 10, 12, 14, 16 and 18, and the exponential potential

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for various values of a'. They found that a very wide potential well, i.e. a' < 8.675, stabilizes the f.cc lattice (see fig. 2.1). For the usually adopted (12; 6) potential and for the (13; 6) potential the h.c.p. lattice is favoured to the extent of 0.01%; also for all other values of s studied the h.c.p. lattice is stable, but the differences between h.c.p. and f.c.c. are smaller in these cases.

Fig. 2.1 The potential wells for the interaction of two inert gas atoms. The (12;6) Lennard-Jones potential is drawn; the exponential function used by Kihara & Koba (1952), eq. 2.3.1, is represented by the dashed line. For the latter fimction the well is exactly so wide that the f.c.c. lattice is stable.

Barron & Domb (1955) calculated the energy difference also as a function of molar volume and found that, at a volume of about half the static equiUbrium volume at vanishing pressure, the difference in energy changes sign for the Lennard-Jones (12; 6) potential function. Therefore, at high pressures a transition can be expected from h.c.p. to f.c.c.

This does not yet explain the phase transition observed in solid helium, because this transition is a thermal eïïect: the f.c.c. lattice exists at temperatures above circa 15 °K, the transition line being almost vertical in the pT diagram.

Barron & Domb predicted for heUum a transition to the f.cc. lattice at a molar volume of less than 5 cm^, corresponding to a pressure of about 25 000 atm at zero temperature. 2.4 Harmonic, quasiharmomc and anharmonic approximations

We consider a system of vibrating atoms and expand its potential energy in a Taylor series in the displacements of the atoms from their equilibrium positions. If the poten-tial well in which each atom moves can be approximated accurately enough in the region of its minimum by a paraboUc shape, the expansion can be restricted to the

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zero- and second-order terms. (The first-order term is zero because the force on the atom vanishes in the equilibrium position.)

These terms are not sufficient, however, if the shape of the potential well near its minimum is not parabolic, or if the amplitude of the vibration of the atom is so large that the shape of the potential well far away from its minimum is also needed. In these cases the most straightforward technique is to take also the next two terms in the Taylor series into account in the complete calculations. In this way one arrives at the first anharmonic approximation, consisting of zeroth, second, third and fourth order contributions.

A somewhat less complicated technique consists of adapting the second-order term in the expansion such that it also describes the third- and fourth-order effects, and then taking into account only this modified second-order term in the remainder of the cal-culations. This is the quasiharmomc approximation; in this approximation the motion of the atoms is harmonic, but the frequencies are shifted with respect to those in the harmonic approximation (see Hooton, 1955a, b, c for the application to solid helium). In the case of the inert gases it is necessary to use the anharmonic or the quasi-harmonic approximation. For sohd helium even this description does not give the

Fig. 2.2 The potential well seen by a helium atom between two neighbours. The interatomic distance is greater than the value for which the two-body potential has an inflexion point; there-fore the total potential function has two minima.

right predictions with regard to the stability of the h.c.p. and the f.c.c. lattice (see sec. 2.5). This is due to the large deviations of the atoms from their equilibrium positions, and to the irregular shape of the potential well. As an indication of the first effect we may well note that the amplitude of the zero point motion at zero

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perature is one third of the interatomic distance! (Domb, 1952) The shape of the effective potential well for each atom can be far from parabolic in helium, because the average interatomic distance is larger than the value for which the pair-potential has its inflexion point as a result of the large zero-point energy which 'blows up' the crystal (see fig. 2.2).

Moreover, each of the neighbours of an atom performs a motion with an amplitude so large that the potential well of the atom considered changes continually and in an asymmetric way. The resulting short-range correlations in these motions will be discussed in sec. 2.8.

The frequency distribution and the Debye temperature

If one wishes to distinguish between the energies of both close-packed lattices, it is of course not sufficient to assume for instance a Debye-spectrum for the lattice vibra-tions. One has to calculate the actual frequency distributions, using the dynamical matrices in either case. The general theory for this procedure has been given by Born (1923); a more recent description can be found in Born & Huang (1954). The dynam-ical matrix D(q) is derived from the equations of motion of the atoms in the lattice; it is a 3n x 3n matrix, if there are n atoms per unit cell. The eigenvalues of the dynam-ical matrix D(q) are the squares of the angular frequencies co{q) of the 3« normal modes, each as a function of the wave number q.

To be able to set up D(q) one must have specified the interaction potential between pairs of atoms, and one has to perform the summation over all contributing neigh-bours.

It is only practicable to establish the dynamical matrix in the case of additive two-body interactions, and in the (quasi-)harmonic approximation; the latter because an angular frequency m is not defined in the anharmonic approximation. One can calculate the 'anharmonic free energy', however, (see e.g. Wallace, 1963, 1964) by treating the third and fourth order terms in the energy as perturbations to the solution found with the 'harmonic' values for a>{q).

Barron & Domb (1955) derived the velocities of long waves in the h.c.p. and f.c.c. lattices from the dynamical matrix, using the Lennard-Jones (12; 6) potential and taking only nearest neighbour forces into account.

They found a relative difference between the Debye temperatures at zero temperature of 0.77%, the value of ©o for the f e e lattice being the smaller one.

The importance of this will be obvious from the following: The Helmholtz free energy determines the stability in the case of a sample at a given temperature and volume. Assuming C„ = b(T ^/©o^). we find for the Helmholtz free energy

F = u-Ts = t/o-h j c „ d r - T J—yr

0 0 T '

-U -^Tl

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Therefore if ©o is smaller for the f.c.c. lattice, the f e e . lattice can become stable at some elevated temperature, when the difference in U(T)~TS has become greater than the difference in the static lattice energy UQ. (We still have not taken the zero-point energy into account.)

In an attempt to explain the occurrence of the h.c.p. structure observed in thin films of neon (Goringe & Valdre, 1964) and in bulk solid argon (Meyer et al., 1964), also Feldman (1965) has calculated the differences of 0Q for the f.c.c. and h.c.p. lattices of inert gases. He used the quasi-harmonic approximation and an (s; 6) Lennard-Jones all neighbour force model, with the zero-point energy taken into account. He finds a relative difference in ©o of 2%, the value for f.c.c. again being the smaller. In this way the results of Barron «fe Domb have been confirmed by results of a some-what more realistic model.

Calculations of the frequency spectrum of close-packed monatomic lattices with a Lennard-Jones force-law have been performed by Nijboer & De Wette (1965), De Wette & Nijboer (1965), and Isenberg & Domb (1964). The work of Nijboer & De Wette showed that even the quasi-harmonic approximation gives incorrect predictions for the stability, because at an interatomic distance as exists in sohd helium the squares of the eigenfrequencies as found from the dynamical matrix are negative. Thus the usual stability criterion - that all vibrational frequencies have to be real for a crystal to be stable - cannot be apphed in the case of a 'quantum crystal'.

The zero-point energy

In the Debye (continuum) model a frequency distribution g(v) = av^ is assumed, where a is determined by putting the total number of vibrational states equal to three times the number of particles, 3nN (if there are N cells, and n particles per cell). The total zero-point energy is then:

ƒ (xv^-ihvdv = ^/snNhv„^^ 0

The zero-point energy per oscillator can be written, with the usual substitution

Let us investigate how we have to modify this result in the case of the atomistic theory of lattice vibrations. In the first place we remark that ©^ itself, as found from specific heat measurements, is a function of temperature, varying about 15% at low temperatures. Furthermore, the zero-point energy is determined for the greater part by the high-frequency part of the frequency distribution, whilst the Debye-spectrum is at most an accurate approximation of the real spectrum at low frequencies. Domb & Salter (1952) have shown that the zero-point energy is actually approximated very well (within about 1%) by

where ©c» is the characteristic temperature in the limit of high temperatures.

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The rigourous relation

k0„ ^ J5ii2

h V 3

holds. Here /ij is the second moment of the frequency distribution Vm

A*2 = I v^g(y)dv 0

which, unhke the first moment fi^, can be calculated from the dynamical matrix D(q) without knowing the shape of the frequency spectrum g(y); see Maradudin, Montroll & Weiss (1963).

v^ is the true maximum frequency in the lattice, defined in terms of the force con-stants. In the nearest neighbour model the values for fi2 of f e e and h.c.p. lattices are the same, so that then the zero-point energies are equal.

In the model with interactions between all neighbours £,(f.ec.) - £,(h.c.p.) « lO-^E,

(see Barron & Domb, 1955); this would indicate that at low temperatures the h.c.p. lattice is stabilized by the zero-point energy. However, this result is in contrast to the values obtained by Feldman (1965) for ©Q; see sec. 2.5. No conclusive results are known as yet.

2.7 Non-additivity of the interaction between atoms

Much attention has been paid by Jansen and co-workers (1965, where further ref-erences are given) to the question as to whether the Van der Waals-interaction (which is described by one or more terms in a Lennard-Jones formula) is additive. Three-body interactions were first studied by Axilrod & Teller (1943), giving the 'triple-dipole' effect resulting from third-order perturbation calculations for non-overlapping atoms. Rosen (1953) and Shostak (1955) calculated the first-order effect due to overlap or exchange forces between three helium atoms.

Both types of calculations yielded changes in the energy that were negative for an equilateral triangle and positive for a linear array of atoms. Jansen (1964) performed calculations for the solid inert gases of first- and second-order three-body exchange interactions, assuming a Gaussian charge distribution on each atom. The vaHdity of this assumption has recently be questioned by Swenberg (1967).

Jansen took into account all 66 triplets of atoms formed by a central atom and any two of its 12 nearest neighbours; the resulting isosceles triangles include 9 specimens of different shape for the h.c.p. and the f.c.c. lattice.

He found that the first-order effect stabilizes the h.c.p. lattice by a few percent of the first-order energy; the second-order effect has the opposite sign. The net effect for sohd argon is calculated by Jansen to be 4% of the cohesive pair-interaction energy, favouring the f.c.c. lattice. But the total cohesive energy of the f.c.c. and h.c.p. lattices for argon decreases in absolute value by 21% and 25%, respectively, as a result of the non-additivity of the interactions.

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gases except helium. The magnitude of the difference between the f.c.c. and the h.c.p. lattice depends on the dimensions of the atoms concerned and on the interatomic distance; it decreases for lighter atoms and for larger distances in such a way that the difference vanishes for solid helium (it is not reported at what density exactly). It seems possible that the non-additivity gives rise to the stability of the f e e lattice for helium at densities much greater than the density for which the difference vanishes as described above; Jansen does not mention this possibility.

Short-range correlations

We have remarked earlier that the amplitudes of the vibrations of helium atoms in a crystal are very large. In fact, it is necessary to take into account correlations between the motions of neighbouring atoms, because the 'hard core' of the Lennard-Jones potential function prohibits a close approach of any two atoms. The work of Nosanow (1966, where further references are given) has been devoted to a description of the necessary short-range correlations in b.c.c. and h.c.p. heUum-4 (and helium-3) at low densities.

In the Hartree approximation for a crystal the wave function for its ground state is written as

m<n m

where

(Pm = <P(km-«ml) = (PiOm) = ^Xpi-^Ag^)

are the single particle functions, localized around the average position R„ of each atom; and ƒ„„ =/(|f„—»-„|) =/(r,„„) is a Jastrow-factor (introduced by Jastrow, 1955, in nuclear physics) with the properties that it gets the value unity for r„„ > a, and rapidly approaches zero for r„„ < a. (The Lennard-Jones potential function changes sign for r„„ = a). Nosanow uses the analytical form (see fig. 2.3):

f(r„„) = exp(-XnO/4e)

where .^is a variational parameter and V(r„„) is the (12; 6) Lennard-Jones function.

a

Fig. 2.3 The Jastrow-factor as used by Nosanow (1966), as a function of interatomic distance, a is the value for which the potential function has the value zero; i.e. the value of r below which the 'hard core' exists.

He then makes a cluster expansion of the ground-state energy (see e.g. Fowler and Guggenheim, 1939 and Van Kampen, 1961), including only the first three terms. It is

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supposed to be highly improbable that more than three particles will be very close to each other, and it is supposed that the cluster expansion converges rapidly after the third term.

The ground-state energy is minimized with respect to the wave function (p„ and the value K in the Jastrow-factor, both for h.c.p. and b.c.c. helium-4 (and in either case for helium-3 as well). A differential equation has been established for the wave function; this equation is solved by iteration. The numerical result for cp„ can be remarkably well approximated by a Gaussian function, which already has been assumed earlier. It is found that, at all molar volumes studied, i.e. between 17 and 21 cm^, the b.c.c. structure has the lowest ground-state energy; the values for the energy are about 33.5 J/mol higher than the experimental ones. Experiment shows that for helium-4 at T = 0 the b.c.c. phase does not exist; at higher temperatures it exists only at about 21 cm^/mol. For helium-3 the b.c.c. phase is stable a.tT = 0 and at molar volumes between 20 and 25 cm'/mol.

The values derived for the ground-state pressure and ground-state compressibility as a function of molar volume agree with experimental values within 10%.

The discrepancy between calculations and experiment with regard to the structure may be due to two reasons: In the first place, the wave function (p{r) = exp (—^Ar^) is supposed to be spherically symmetric, which is a good approximation of the situation in a b.c.c. crystal, but which is a bad approximation in an h.c.p. crystal. Nosanow advocates for h.c.p. helium the use of a function (p(r) = exp(—iy4r^—i5z^), where z is the component of r along the c-axis, and B is another variational parameter. In the second place, as the density increases, the truncation of the cluster expansion after the third term may become increasingly illegitimate (see Brueckner & Froberg, 1965), so that the results obtained for the stability of the b.c.c. phase at higher den-sities may be erroneous.

It is not very probable that Nosanow's technique will soon lead to an accurate stabiU-ty criterion for the phase transition at higher densities, viz between h.c.p. and f.c.c. lattices. In principle, however, interesting results could be expected if the proper symmetry conditions were included in the calculations of short range correlations in this scheme, Hke the inclusion of symmetry considerations in the calculation of the frequency spectrum of the lattices yields interesting results for the long range corre-lations in crystals.

Concluding remarks

This chapter gives a brief description of several different aspects which are important for the question as to which one of the two close-packed lattices is stable. The in-teresting thing is, that all aspects together determine the final conclusion; one should know all that happens inside the crystal in order to be able to calculate the criterion for the transition point from h.c.p. to f.c.c. in solid heHum-4.

As we have stated in sec. 2.1, and seen in sees. 2.2, 2.5 and 2.8, the generally accepted manner of describing the interatomic forces is the use of a Lennard-Jones (12; 6)

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potential function. This function may be accurate enough for not very refined cal-culations, like that of the static lattice energy. But if one wishes to calculate very small differences between the two lattices, for instance in the frequency spectrum, this approximation may be too crude. As Jansen has shown (see sec. 2.7), the assumption of additive forces between induced dipoles is illegitimate; it introduces serious errors.

Before the question can be definitively settled, one should probably first solve the 13-body problem of a central atom and its twelve nearest neighbours, derive a poten-tial function from this solution, and calculate its dynamical behaviour, which will include per se the actual short range correlations. Only then can the long range corre-lations in the crystal (its lattice dynamics) be properly described in terms of normal modes, from which the partition function and the Helmholtz free energy (working at constant volume) can be computed.

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3 The hexagonal close-packed

structure

Introduction

In this chapter we discuss some of the crystallographic properties of the hexagonal close packing of identical atoms. Next we show how the symmetry of a crystal restricts the occurrence of anisotropy in several of its physical properties. The results are appHed to the case of h.c.p. helium, and a brief review is given of the measurements of its anisotropic prop-erties that have been reported in the literature.

3.1 The relation between space group and point group

We have seen in chap. 1 and in fig. 1.4 how the atoms are arranged in an h.c.p. structure of identical atoms: above and below each hexagonal close-packed layer of atoms two other close-packed layers are found. The atoms of the latter layers have positions above and below the centres of half the number of equilateral triangles of the first layer.

The unit cell of the h.c.p. structure is a rhombic parallelepiped containing two atoms: the positions of these atoms can be described by the orthogonal coordinates (0, 0, 0) and (ia, \a^3, \c); see fig. 3.1 for the choice of the coordinate system.

^^%^

/lie-Fig. 3.1 The unit cell of the h.c.p. structure, containing two atoms.

An infinite array can be formed by regular repetition in space of identical unit cells. In this way a crystal is built up. The complete set of the operations on an infinite, three-dimensional periodic array (the crystal) which bring it into self-coincidence is called the 'space group' of the crystal. These operations are called the symmetry operations; they are the elements of a group.

The symmetry operations of a specific crystal are generated by its 'symmetry elements'. These symmetry elements can be of the following types:

a. centre of symmetry (or inversion point) b. mirror plane

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c. n-fold rotation axis

d. n-fold inversion axis ( = rotation followed by an inversion) e. translation

f. glide plane ( = reflection followed by a translation parallel to the mirror plane) g. n-fold screw axis ( = rotation followed by a translation parallel to the rotation

axis).

In crystals only 1-, 2-, 3-, 4- and 6-fold axes are possible.

If we study on a crystal the relation between two physical quantities, e.g. electric field and electric polarization, the relation found can be called a physical property of the crystal; this property is the polarizabiUty in the case considered. Such physical properties are often tensors, T. The symmetry of the physical property, as char-acterized by the relations which hold between the elements T^ of such a tensor, is determined by the symmetry of the crystal under consideration.

Normally the physical properties which we measure are macroscopic properties, because the characteristic dimension of the 'probes' we use (electrodes, thermometers, electromagnetic waves) is generally large as compared to the interatomic spacing. The use of X-rays as a probe is, of course, one of the exceptions. The symmetry of an X-ray pattern is indeed determined by the macroscopic symmetry of the crystal; its detailed features, however, give information about the positions of the atoms, which are not a macroscopic property.

We may conclude that, when we wish to find those symmetry elements which are responsible for the symmetry of any (macroscopic) physical property, the complete set of symmetry elements of the crystal under consideration must be deprived of all translational components.

Therefore, to obtain these symmetry elements when we know the space group of the crystal, we have to substitute a mirror plane for each glide plane and an n-fold rotation axis for each n-fold screw axis. The resulting set of symmetry elements may be imagined as being 'collapsed' so as to pass through one point, which has, of course, no definite location in the lattice.

The set of operations associated with these elements is again a group; it is called the 'point group' of the crystal: there is one point that is left in its original position if any symmetry operation of the point group is applied.

There is an alternative description of the way in which one can find the point group after the space group has been established: put one arrow in an arbitrary direction in space and apply all possible symmetry operations of the space group of the crystal on this arrow. This will result in an infinite number of arrows, but only a limited number of different directions of these arrows will occur. Now bring one arrow of each different direction with its tail into one single point. The group of symmetry operations of this finite set of arrows is the point group associated with the space group.

There are 230 space groups but only 32 crystallographic point groups.

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The space group and the point group of the h.c.p. structure

Now we return to the h.c.p. crystal. In its structure the following symmetry elements can be found:

a. 6-fold inversion axes (along the c-axis) through each atom, with a centre in each atom;

b. 6-fold screw axes in the same direction, having a translation of |c, through the centres of the set of non-adjacent triangles of each layer above which no atom is positioned;

e twofold screw axes in the same direction, having a translation of ^c, half-way between neighbouring elements of set (a);

d. three sets of mirror planes parallel to the c-axis, intersecting in the 6-fold inversion axes and in the 6-fold screw axes.

• atoms in a layer of type A ® atoms in a layer of type B A 6-fold screw axis

^ 6-füld inversion axis

£ 2-fold screw axis

mirror plane

Fig. 3.2 A projection on the two triangular nets of the symmetry elements of the h.c.p. structure, from which the space group can be determined to be P6Jmmc.

Fig. 3.3 Stereogram of the symmetry elements of the point group 6/mmm.

These elements are shown in fig. 3.2. They constitute sufficient information to con-clude that the space group of the h.c.p. structure is Pó^/mmc (see International Tables, 1952 for the exact meaning of this notation and for the full enumeration of all sym-metry elements). As we have described above, the crystallographic point group can be found by disregarding all translational elements of the space group; the sixfold screw axis '63' is replaced by a sixfold axis '6'; the glide plane 'c' is replaced by a

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mirror plane 'm'; the fact that the structure is primitive ' P ' is irrelevant. In this way the point group is found to be 6/mmm (in International notation; the Schoenflies symbol is D^y).

The symmetry elements corresponding to this point group are: a sixfold rotation axis with a mirror plane perpendicular thereto; six twofold rotation axes perpendicular to the sixfold axis; six mirror planes through the sixfold axis; and a centre of symmetry (see fig. 3.3 for the symmetry elements in stereographic projection).

The operations of this point group determine the symmetry of the macroscopic physical properties of the crystal. The position symmetry of the atoms in the h.c.p. structure, however, is only 6m2 (or D^f). This is a lower symmetry; we use it in the calculation of lattice sums (see chap. 4).

Symmetry and physical properties of an h.c.p. crystal

To find the symmetry of the physical properties of an h.c.p. crystal, appearing as a reduction in the number of independent constants in any tensor representing such a property, we make use of Neumann's principle, which can be stated in the following way: 'every physical property of a crystal must possess at least the symmetry of the point group of the crystal'.

This means, once a relation between a set of physical quantities has been set up, e.g. the three equations describing the relation between two vector quantities, we can apply the symmetry operations of the point group of the crystal to this relation and demand that it remains invariant.

We take as a specific example the relation between electric polarization and electric field in an orthogonal coordinate system:

P = a£, or

P; = o-ijEj (summation convention).

Because of the inherent symmetry of the polarizability (see e.g. Bhagavantam, 1966 and NiggU, 1955), we have a^ = a^;.

To reduce the number of independent variables further, we apply one of the operations of the point group 6/mmm. Let us rotate the crystal and the coordinate system, for instance through an angle of 120° around the z-axis. For £•,- and for P; we may then write Xi = P y Z / or:

Xi = Xicosl20° - X'iSm 120° X2 = X\ sin 120° + X'2 cos 120°

x^ = x^

where X^ means either Ei or P,-, and R is the rotation matrix.

We substitute these transformation equations into the relations P; = OLIJEJ to get Pj' = a'ljE'j and find in this way the elements of the transformed matrix a' = R~^eiR:

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K = (iaii-W3ai2-|-|a22)£i-|-(iV3ai,-iai2-iV3a22)£i-l-(-iai3-l-iV3a23)£'3 P'i = (W3aii-iai2-iV3°'22)£i+ (iaii+iV3ai2+ia22)£2+(-W3ai3-ia23)£'3

P3 = (-iai3+W3a23)£'i+ (-W3ai3-ia23)Ê2+ «33^3 Because the equations have to be invariant, we may estabUsh the identities

i a i i - i \ / ^ " i 2 + J«22 =«11. etcetera, from which we find

«11 = « 2 2 .

«12 = «13 = «23 = 0.

None of the other symmetry operations yields a further reduction in the number of independent constants.

Thus the matrix at, for crystals belonging to the class 6/mmm has the following simple form:

/ « u 0 0 a = j 0 «11 0

\ 0 0 «33

In an analogous way one can determine the maximum number of independent con-stants for various other types of physical properties. Bhagavantam (1966) gives a table with the results for crystals of all classes. As an illustration of the limitation which symmetry sets to the number of independent constants we show some of these results in table 3.1.

Table 3.1 Number of constants necessary to describe some physical properties in a crystal of the class / and in a crystal of the class 6/mmm (according to Bhagavantam, 1966).

physical property represents relation between

scalar and scalar scalar and vector vector and vector, same as

scalar and symmetric tensor vector and symmetric tensor

symmetric tensor and symmetric tensor

physical property for example density pyro electricity optical polarization, thermal conductivity, thermal expansion piezo electricity elasticity max. no. of constants in any class 1 3 6 18 21 no. of constants in ójmmm 1 0 2 0 5

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3.4 Anisotropic properties of h.c.p. helium-4

We shall now proceed to describe the results of those measurements on hexagonal single crystals of helium-4, in which the symmetry of the crystal plays a role. These measurements are concerned with (i) index of refraction, (ii) velocity of sound and (iii) thermal conductivity. The phonon dispersion curves have up till now only been determined in one direction, perpendicular to the c-axis. Further one could for in-stance devise experiments to measure anisotropics in the thermal expansion coeffi-cient and in the compressibility.

3.4a Birefringence

In all aforementioned types of experiments, the realized as well as the future ones, probably the most straightforward way to determine the orientation of a crystal is a measurement of the phase retardation produced by it in polarized light. Then, at the same time, one can visually check the quality of the crystal on which the other measurements are going to be performed.

Apart from our results, which are shown in chap. 7 of the present work, (see also Vos, Veenenga Kingma, Van der Gaag & Blaisse, 1967), Heybey & Lee (1967) reported measurements at Brookhaven National Laboratory only at 26 atm. They have grown crystals, 5.8 cm long, between flat windows and observed the state of polarization of the transmitted light. Under the assumption that a number of their crystals had been single crystals with axes in arbitrary directions, they calculated the birefringence in a way analogous to our statistical calculations (see sec. 6.7). They suppose the biggest measured value of the retardation to result from a single crystal viewed per-pendicular to its optic and crystallographic axis, and they can neglect the error in this value thanks to the large dimensions of the crystals.

The result they quote is |«e—«d = (2-6 ± 0.1)x 10"*, which agrees rather well with our result at comparable density. It should be understood that they are only able to determine the absolute value of n^—n„, in contrast to the possibility provided by our method of 'double' measurements, in two perpendicular directions on one crystal.

3.4b Velocity of sound

Lipschultz & Lee (1965) found velocities of 10 MHz transverse sound in h.c.p. helium-4 at about 25.6 atm and 1.3 °K to vary between 230 and 315 m/s, indicating a high de-gree of anisotropy. (The errors in their results are estimated to be smaller than 1%.) The sound pulse received was less sharp in the h.c.p. phase than in the b.c.c. phase. This is possibly due to the occurrence of a few large crystallites in the 2 cm long chamber.

Vignos & Fairbank (1961, 1962, 1966) reported measurements of the velocity of longitudinal 10 MHz sound on samples, 2 cm long, 0.5 cm in diameter, of liquid and solid helium-4, heUum-3 and He^-He* mixtures. In this survey we will only discuss their results on h.c.p. helium-4.

The velocity of longitudinal sound increases roughly from 480 m/s at 25 atm to about

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790 m/s at 140 atm. But there is a scatter of up to 12% in the experimental points for different samples, while the estimated error in the results is ^%. This scatter can again be attributed to an anisotropy in the velocity of sound; because the composition of the sample (one or more crystallites) is unknown, no definitive information can be extracted from these results.

Nosanow & Werthamer (1965) have given a theoretical calculation of sound velocities for different modes in crystaUine helium-4 and helium-3 at zero temperature. They used the time-dependent Hartree approximation, together with the results of varia-tional calculations of the ground state energy using correlated trial wave functions; see also sec. 2.8. For h.c.p. helium-4 they found theoretical values for the longitudinal sound velocity in the [0001] and [01Ï1] directions to differ up to 100 m/s, and those for the transverse sound velocity in the [0001] and [0111] directions to differ up to 170 m/s. The experimental results quoted above agree reasonably well with this theoretical range of values.

3.4c Thermal conductivity

The first record of an anisotropy in the thermal conductivity of h.c.p. helium-4 crystals was by Bertman, Fairbank, White & Crooks (1966). They used samples 3 cm long and 0.2 cm in diameter, with three temperature sensors along the length. In some cases, even after annealing the sample, different conductivities were measured between different sets of temperature sensors. Then, after reannealing the same sample, both conductivities had risen to the same value. The authors attribute this to a change in orientation of two (or more) crystallites to give one favourably oriented crystal. Apart from these cases, their results from both sets of temperature sensors were always consistent and reproducible. Therefore, the authors suppose that there is a preferred orientation during crystal growth in their sample tube. A more conclusive exper-imental result is described by Guyer & Hogan (1967). These authors grew crystals of h.c.p. helium-4 at a pressure of 85 atm in a stainless steel sample chamber 5 cm long and 0.18 cm in diameter.

Six out of thirteen crystals showed no difference in conductivity between two sets of temperature sensors. The temperature dependence of the thermal conductivity be-tween 0.4 and 1.6 °K did in all six cases agree exactly with the theoretical dependence for a single crystal. But the six values of the conductivity at its maximum differ by a factor 5. This fact is attributed to differences in orientation of the c-axes, the orien-tations making angles between 0° and 75° with the direction of the heat current. Com-parison with the thermal conductivity of a polycrystalline sample showed that the conductivity parallel to the c-axis is smaller than that perpendicular to it.

It is indicated in the same paper that the anisotropy of the thermal conductivity arises primarily from an anisotropy of the phonon-phonon scattering rate, and, to a smaller extent, also from the anisotropy of the velocity of sound. Actually, both anisotropics must have their origin in the five different elastic constants for an h.c.p. crystal.

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3.4d Phonon dispersion curve

By means of triple-axis neutron spectrometry, Lipschultz et al. (1967) have recently determined the first phonon dispersion curves for a hehum single crystal at 27 atm and 0.99 °K. This determination was done for wave vectors along the [1010] direction, see fig. 3.4 for their results. The crystal had been grown at a constant temperature of 1.30 °K by supplying helium in the superfluid phase through an open capillary. A temperature gradient existed over the cylindrical sample container (5 cm long and 2.4 cm in diameter) in such a way that the solidification rate was 1.0-1.5 mm/min. The sample was annealed near the b.c.c. phase boundary and afterwards cooled to 0.99 °K. The orientation of the crystal used was determined by elastic neutron

scat-1 scat-1

phonon energy, meV

\ \

\ \ LO

LA l\ \ i • \

, , \

1 1

__

TO

44f

,^r

-1/ wave vector. A"'

f

1 1

1 0.5. 0 0.5 1

Fig. 3.4 The first dispersion curves to be found in an h.c.p. single crystal of helium-4 by Lip-schultz et al. (1967).

tering; the crystal could be tilted ±5°. By means of beam masking techniques its dimensions were found to be approximately 12xl2xl0mm^.

The theoretical values of Nosanow & Werthamer (1965) for longitudinal and trans-verse sound velocities are in good agreement with the initial slopes of the dispersion curves (see again fig. 3.4). Experimental values for the velocities of sound at 10 MHz also agree to a certain extent with these slopes, indicating that some of the crystals used had the same orientation as the one for which the phonon dispersion curve was measured.

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3.5 Concluding remarks

For several types of experiments on h.c.p. helium such as the last three mentioned above, it is important to know the orientation of the crystal. It is preferable that this orientation be determined independently of the property to be measured. For this purpose a determination of the optic axis, by measuring the birefringence in two perpendicular directions, wiU often be less difficult than the use of neutron or X-ray diffraction techniques.

Therefore it is advisable in such cases to combine in one sample cell the facihties for measuring the property desired and the birefringence. The latter can be done in a way analogous to the procedure described in chaps. 5, 6 and 7 of this work.

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4 Calculation of/2e—^o

from the structure of the crystal

Introduction

After the qualitative descriptions of various measurements in chap. 3 we turn now to the central problem of this thesis: the relation between the macroscopic property birefringence and the microscopic structure and properties of the h.c.p. phase of helium.

As is well known the theory of Lorentz and Lorenz leads from the atomic polarizability to the refractive index of matter. The theory originally developed by Ewald (1912, 1916, 1921) succeeded in calculating the different indices of refraction for different directions of polarization in a rhombic crystal. Kronig & Sonnen (1958) applied Ewald's theory (1921) to the case of hexagonal helium crystals, the hexagonal close-packed structure being represented by two interpenetrating rhombic lattices. As a parameter for the calculations they used c/a, the ratio of the hexagonal crystaUine axes (see fig. 1.4). The result of their work, which has been done without a computer, is given for reference in table 4.1; the error is stated to be 2 units in the last decimal.

Table 4.1 Values of «e—«o as calculated by Kronig & Sonnen.

da tie-to 1.528 +0,000119 1.633 +0.000002 1.745 -0.000143

We have calculated more numerous and more accurate results with the aid of Kronig and Sonnen's formulae and verified these results by applying a straightforward Lorentz-summation over a very large number of lattice points.

4.1 The atomic polarizability

As the term 'polarizability of an atom' will play an important role in the following discussions, we shaU first examine its meaning in the classical, and in the quantum mechanical case.

In the classical case we consider an atom exposed to a harmonically varying electric field E = Ej^'^'; one electron is bound elasticaUy to the nucleus, the natural circular frequency is (OQ. Then the electronic polarizability is defined as the induced dipole moment divided by the electric field:

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E m(o}l — o)^)

Ultraviolet spectroscopy has shown that the mean resonance circular frequency for hehum is in the order of OJQ = 3.6 x 10'* rad/s (A^s « 0.5 nm).

In the visible spectrum co is a factor 10 smaller; therefore we may ignore co^ in the above expression for a, and thus find

Substituting the constants, we find for helium in this approximation a w 2.5 X 10"*' C V / k g

In the quantum mechanical case we can give a more satisfactory description of the influence of an external electric field on the electrons bound to an atom.

Suppose we have solved Schrödinger's equation for a free atom, the Hamiltonian HQ consisting of the kinetic energies of the electrons, their potential energies in the field of the nucleus and their mutual Coulomb repulsions. We apply a perturbing Hamil-tonian Hi, representing the action on the electrons of a static external field E^ along the z-axis.

To write down this Hamiltonian we calculate the potential energy of the electrons i with charge -e and z-coordinates Zj in the field E^:

Hi = X eE^z, = -E,M, i

where M^ is the dipole moment operator in the z-direction. If we could solve the new Schrödiger equation

iHo + H,)il,„ = ^ A

we would find an average induced dipole moment proportional to the field E^ as long as E, is small. For larger E^ the average dipole moment should be written as

( M X = «£z + j8E' + ...

In such cases the polarizability a is defined as the coefficient of the first term in the expansion for (M^X^; it is also the coefficient in the first term in the expansion for the energy:

^ = - i a £ | - i ) 8 £ | - . . .

We use this definition of a in the present work, and ignore higher-order effects. In the case we are dealing with, the interaction of dielectric and light, only the first term in these expansions needs to be taken into account.

Now second-order perturbation methods can give the energy correctly to terms in the square of the perturbation. Therefore, it is not necessary to go further than second-order perturbation theory for our purpose. In the appUcation of this theory, the matrix elements of the perturbing Hamiltonian

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play an important role. These are exactly the same as those occurring in the expression for the optical transition probabilities between the ground state and the n-th excited state. We know that only the matrix elements connecting the ground state with excited P states are different from zero.

Moreover, as all diagonal matrix elements of M^ are zero, there is no term in the average energy in the first order of the perturbing field; the second-order terms being the first nonvanishing ones. Keeping this in mind, we can write down the foUowing expressions for the energy and the perturbed wave function of the ground state:

and

^?' = ^.+i:^-^^^Ei+.

„ 0 i - < 5 „

The average dipole moment can be found in the usual way

(MX, = f ip['^'M,il,Y^dx = - 2 £ , X ^-^^^^ + terms of higher order in £, The first term should by definition be equal to aE^, thus

« = 2i:

m,)

|2

Inl

•1

• 2

which is also exactly equal to minus two times the coefficient of E/ in the expan-sion for the energy, as we stated earlier that it should be.

Thus far we have only considered a static appUed field E^. If the applied field is har-monic with a frequency v, the result for an atom in its ground state becomes:

«(v) = 2 E J ^ ^ % i ^

^(hv„,r-(hvr

where hv„i = ^„—^i. A more common notation is

«(v) =

4n^m n v^i —v^ with the so-called oscillator strengths/„i given by

871 m 2

fni = -^—v„i\iM,)i„\ eh

For these osciUator strengths the Kuhn-Thomas sum rule holds: X ƒ„„ = N for aU m

n

where A'^ is the number of electrons in the atom. This sum rule is very useful when it comes to calculating actual atomic polarizabilities.

Birefringence of hexagonal single crystals of helium-4 at various pressures

Birefringence of

hexagonal single crystals

of helium-4

Birefringence of

hexagonal single crystals

of helium-4

at various pressures

Contents

o Introduction

1 The phase diagram of helium-4

2 The phase transitions

in solid hehum

fry

[")

F = u-Ts = t/o-h j c „ d r - T J—yr

-U -^Tl

f(r„„) = exp(-XnO/4e)

3 The hexagonal close-packed

structure

^^%^

x^ = x^

\ \

\ \ LO

, , \

1 1

TO

44f

,^r

f

1 1

4 Calculation of/2e—^o

from the structure of the crystal

^?' = ^.+i:^-^^^Ei+.

« = 2i:

m,)

«(v) = 2 E J ^ ^ % i ^

^(hv„,r-(hvr