Evidence that rotons in helium II are interstitial atoms
EEvidence that rotons in helium II are interstitial atoms
Robert M Brady , Edward T Samulski & David H P Turban University of Cambridge Computer laboratory, JJ Thomson Avenue, Cambridge CB3 0FD, UK Department of Chemistry CB University of Cambridge Cavendish Laboratory, JJ Thomson Ave, Cambridge CB3 0HE, UK
Superfluid helium II contains excitations known as rotons. Their properties have been stud-ied experimentally for more than 70 years but their structure is not fully understood. Feyn-man’s 1954 description, involving rotating flow patterns, does not fully explain later exper-imental data. Here we identify volumetric, thermodynamic, colloidal, excitation, x-ray andneutron scattering evidence that rotons are composed of interstitial helium atoms. We showin particular that they have the same mass, effective mass and activation energy within ex-perimental accuracy. They readily move through the substrate, and couple through latticevibrations to produce quantized, loss-free flow which corresponds to the observed super-flow. Our observations revive London’s 1936 conclusion that helium II has a relatively opencrystal-like lattice with enough free volume for atoms to move relative to one another, andreconcile it with London’s 1938 description of a quantum fluid.
Including supplementary material on page 19.1 a r X i v : . [ c ond - m a t . o t h e r] J un ondensed phases of helium Every known element except helium has a triple point where solid, liquid and vapour coexist . Helium (figure 1) has a flowing phase, helium II, where a solid would be expected .Figure 1:
The phase diagram of He on a logarithmic scale, with linear insets . The meltingcurves of neon, argon and krypton are superposed with pressures and temperatures scaled sotheir triple points fall at ‘TP’. The γ phase (right inset) is currently classified bcc and exhibits apretransition specific heat anomaly (dotted) . ‘L’ is the path discussed in the text. London suggested how to resolve this anomaly in 1936 by showing that cold helium II hasnegligible entropy, indicating a regular atomic arrangement. He proposed a crystal-like lattice withenough free volume to allow atoms to move relative to one another, which he associated with thelow viscosity flow . Compare Andreev and Lifshitz’s 1969 proposal that atoms may advancethrough crystals, making them “neither a solid nor a liquid” , and Leggett’s 1970 proposal for2upersolid flow .In general, solidification occurs when atoms or molecules cohere, but helium atoms repel,assisted by zero point motion. London argued that this accounts for the transition near ‘ L ’ infigure 1, which “depends essentially” on volume rather than temperature . Advancing along ‘ L ’,the distance between the atoms in solid helium increases until they lose cohesion and fluidize.Compare dry sand, which flows when the grains separate and lose cohesion.Experiments in another field provide quantitative evidence for this description. Spheri-cal colloidal particles in a fluid medium also repel, assisted by Brownian motion, and form ahexagonal solid which loses cohesion when the volume per particle is increased by diluting thesuspension
11, 12 . At intermediate dilutions the colloidal solid is dispersed in a flowing phase whichoccupies a factor of 1.103 more volume (the volume fractions being 0.545 and 0.494 respectively) .This mirrors He at constant volume and low temperature, where a hexagonal close packed (hcp)solid is dispersed in helium II with the same volume ratio (figure 2) .London’s 1936 description places the triple point of He at ‘ TP ’ in figure 1. This quanti-tatively agrees with the other noble gases, whose melting curves align in dimensionless units oftemperature and pressure relative to their triple points. Figure 1 superposes the melting curves ofneon, argon and krypton after scaling the temperatures and pressures so their triple points fall at‘ TP ’. They align with each other and with He at higher temperatures (the small differences dependon the square root of the atomic weight; see the supplementary material). At lower temperatures,the solid phases of the other noble gases are face centred cubic, a maximally efficient packing3igure 2:
Experimental molar volumes V (solid lines), plotted as V /V hex where V hex is the volumeof the hexagonal solid under the same conditions
4, 12 . 1.103: flowing phase of spherical parti-cles suspended in a fluid medium, 1.088: bcc lattice of hard spheres (packing efficiencies in thesupplementary material), 1.016: bcc lattice saturated with interstitial atoms, discussed in the text. arrangement with insufficient volume to fluidize; this may be favoured by the occupied p , d or f electron orbitals, vacant in He. The scaled melting curves in figure 1 indicate approximatelywhere London’s fluidized solid (shaded) melts into an ordinary liquid.London suggested a diamond lattice as one possibility . Fr¨ohlich then showed this is equiva-lent to a body centred cubic (bcc) lattice with 50% vacancies, giving ample free volume to supportthe flow . However, in 1938 London showed a diamond lattice is not viable because the vacanciesare not stable
8, 13 . He sidestepped the unsolved problem of the precise geometry in order to “directattention” to a momentum space representation in which the flow is quantized . We will show thathis two descriptions, in physical space (1936) and momentum space (1938) , are complementary.4ore specifically, helium atoms advance through interstitial positions in a locally bcc lattice, andinteract via lattice vibrations to produce coherent and quantized flow. γ phase of He We begin with γ helium, a crystalline solid (figure 1 inset). Figure 2 shows that it occupies ap-proximately 7% less molar volume than expected for bcc crystals in a hard sphere approximation.Nevertheless Schuch and Mills claimed in 1962 that this “provides evidence that the γ phase isbcc” . They noted that the hcp- γ volume ratio in figure 2 is approximately the same as for the hcp- β transition in zirconium
14, 15 , presumed that β zirconium is bcc, and cited Pauling’s speculationthat metals have bonds with covalent character whose lengths depend on the crystal geometry (co-ordination number) . But the original paper on the coincidence admitted that the bonding is notthe same in metals and helium , and therefore Pauling’s speculation about covalent bond lengthsdoes not fully explain the anomalous molar volume of γ helium.Schuch and Mills also reported x-ray measurements which likewise exhibit anomalies thatare not explained by classifying γ helium as ordinary bcc crystals. The hcp phase has eight visiblereflections while single crystals of the presumed bcc phases of both He and He have only three,whose intensities decline steeply with increasing angle, and the pattern for a coarse powder exhibitsonly one ill-defined reflection
5, 17 . Schuch et al indexed the reflection angles to those of bcc crystalsand dismissed the anomalous intensities as due to zero point motion. However, this does notaccount for the difference between a coarse powder and single crystals, or hcp helium having more5eflections even though the reflection planes are closer. Based on later measurements, zero pointmotion would attenuate the (200) line by approximately 50% relative to the (110) line, much lessthan observed (calculated in the supplementary material from the steep potential energy barrier toatomic displacements at approximately . s where s is the length of a bcc cell ).We interpret the anomalous molar volume and x-ray intensities as evidence that the bcccrystals of γ helium are saturated with interstitial atoms. Figure 3a shows an effectively infinitebcc lattice of helium atoms, with one extra atom added, that has been relaxed to equilibrium atlow temperature in numerical simulation. The resulting interstitial defect occupies seven bcc cells,with the eight numbered atoms displaced as in 3b. Figure 3c is a superlattice of defects, or bcclattice saturated with interstitial atoms.This proposed structure for γ helium agrees with experiment in the following ways.A1 Interstitial atoms provide a ready supply of helium atoms at grain boundaries, which helpsdevelop large crystals. Most samples formed single crystals, and a fine powder could not bemade
5, 17 .A2 An unperturbed bcc cell contains two atoms, so the idealised arrangement in figure 3c willoccupy approximately the molar volume of bcc crystals. The resulting hcp- γ volume ratiois near the experimental value (figure 2) .A3 Thermal agitation will increase the longitudinal separation between interstitial atoms in figure3c, so the hcp- γ volume ratio will increase with temperature, as observed . The associated6igure 3: (a) Numerical simulation of an extra (interstitial) atom in a bcc lattice of He atomsat 1 atmosphere and low temperature. Only significantly perturbed regions are shown. Programdetail in supplementary material. (b) Displacement of atoms in the central row (c) Superlattice ofinterstitial defects. randomness raises the entropy of the γ phase, which will form on heating, not cooling, hcphelium, also as observed.A4 The atoms in figure 3c are significantly perturbed from a bcc arrangement, so the intensities oftheir x-ray reflections will decline steeply with increasing angle, where the reflection planesare closer, giving fewer visible reflections than for hcp crystals. The horizontal lines of atomswill produce weakened (200) reflections from single crystals. However, for the vertical (200)7eflection planes (marked), the added atoms alter the periodicity of alternate rows. This willmask the (200) reflections in a coarse powder, where the orientations are random. All thesefeatures were observed
5, 17 .A5 Rearranging an atom into an interstitial position changes the Gibbs free energy by ∆ G = ∆ U + P ∆ V − T ∆ S (1)where P is the pressure, T temperature, ∆ U the change in internal energy, ∆ V volume andand ∆ S entropy. This becomes negative at high enough pressure, since ∆ V is negative. Thus γ helium forms on increasing the pressure, as observed (figure 1).A6 Reducing the pressure of γ helium until ∆ G in (1) vanishes will expel some interstitial defects,but those that remain will be more dilute, with more entropy, which will stabilise them. Thusthe transition will not be sharp. This is observed as a substantial rise in the specific heatcapacity within 20mK of the melting temperature, a previously unexplained ‘pretransitionanomaly’ (figure 1 inset) .See the supplementary material for further evidence. Rotons
When the Gibbs free energy (1) changes sign on reducing the pressure, interstitial atoms will beexpelled from γ helium. This suggests the new phase is locally bcc. Figure 1 identifies it ashelium II, and figure 2 shows it has approximately the expected molar volume. Interstitial atoms8ay advance through it as shown in figure 4 (animation in supplementary material). These mobileinterstitial atoms have the properties attributed to the excitations known as rotons , as follows.Figure 4: An interstitial atom advancing through an idealised bcc lattice. Numerical calculationat low temperature and 1 atmosphere pressure.
B1 In 1999, Tucker and Wyatt created excitations in cold helium II using a pulse of heat, whichadvanced linearly until reaching the surface and ejecting a helium atom into the space above .By directing them at an angle and observing the ejected atoms, they showed that the excitationshave positive effective mass. These observations are consistent with interstitial atoms as infigure 4.B2 Tucker and Wyatt argued that excitations with nonzero effective mass do not lose energy toacoustic waves, as the energy and momentum changes cannot match simultaneously. Intersti-tial atoms have the same property. They observed linear motion without noticeable losses .B3 Vacancy defects resemble figure 4 with a missing atom instead of an added one, and move ina similar way. They have negative effective mass, and more enthalpy since they increase thevolume against the pressure. By colliding excitations together, Tucker and Wyatt discovered9obile excitations which they showed have negative effective mass . They were not createdby the pulse of heat, suggesting they have greater enthalpy.B4 In the 1940’s, Landau concluded from phonon dispersion data that helium II contains excitations .There are two species, R + (rotons) and R − (maxons), with positive and negative effective massrespectively. Interstitial atoms and vacancies have these properties.B5 When negative ions are drawn through helium II under an electric field and scatter inelastically,they will create interstitial atoms and vacancies together, since atoms are conserved. In 1976,Allum, Bowley and McClintock discovered “hitherto unrecognized selection rules wherebyrotons are only created in pairs” in this experiment .B6 The conventional model of rotons does not predict their effective mass. If the i ’th numberedatom in figure 3a advances at velocity v i , the kinetic energy will be T ≈ m Σ v i where m isthe atomic mass. Our numerical model at atmospheric pressure and low temperature indicates T = m ∗ v where v the velocity of the defect and m ∗ ≈ . m (see the supplementarymaterial). This is within 5% of the observed value .B7 In 1954, Feynman suggested that a roton’s kinetic energy is due to rotating flow patterns .This suggests that its effective mass is greater than its gravimetric mass (if any), unlike aninterstitial atom which has the mass of a helium atom. This difference can be studied usingdensity data. When dilute, interstitial defects move freely in one dimension, and their concen-tration can be calculated from their quantum wavelength similarly to a particle in a box. At10emperature T , the expectation number (cid:104) N (cid:105) in a bcc lattice of N o atoms is given by (cid:104) N (cid:105) N = 3 (cid:18) πm ∗ s h β (cid:19) e − β ∆ H (2)where s is the length of a bcc cell, h is Planck’s constant, ∆ H = ∆ U + P ∆ V is the enthalpyof a defect and β = ( K B T ) − where K B is Boltzmann’s constant. See the supplementarymaterial for the proof and discussion of small terms we have neglected. These interstitialdefects increase the density of helium II, which is known from dielectric observations, andfigure 5 shows good agreement with experiment spanning six orders of magnitude .Figure 5: Concentration of interstitial atoms in helium II, from the measured density ρ ( T ) at 15atmospheres pressure after subtracting out ordinary (Debeye) expansion , compared to equation(2). Inset – activation temperature as a function of pressure. The roton energy gap is from neutronscattering data at the lowest temperature measured, approximately 1.25K . B8 The enthalpy ∆ H of interstitial atoms, obtained above, is within 5% of the roton energy gap11t all pressures (figure 5 inset) .B9 Hcp helium has insufficient volume for interstitial atoms, and vacancies have a large excitationenergy (see B3). This will suppress supersolid flow in this phase (unlike helium II, whichhas more room for interstitial atoms). Attempts to demonstrate supersolidity in hcp heliumwere unsuccessful but it may have been seen in other systems with looser crystal-likestructures
26, 27 .See the supplementary material for further evidence.
Fluidization mechanisms in ‘solid’ helium II
External stresses will locally raise the pressure at protrusions in the surface of the bcc crystalsof helium II. This reverses the sign of ∆ G in (1) so that interstitial atoms form, which advancethrough the solid and contribute to the flow. The flow will be amplified because interstitial atomsare conserved (see B5) and move without resistance between collisions (B2). There may also beother flow mechanisms, resembling the giant plasticity of hcp helium .This agrees with experiment as follows.C1 The density of helium II at its saturated vapour pressure and 1.1K is 27.84g/mole . Bcc crys-tals with this density would have a (110) peak in their structure factor at 19.7nm -1 . Neutronscattering measurements indicate the peak is at 20.3nm -1 .122 We saw (figure 2) that colloids have phases resembling helium. The crystalline phases ofcolloids are delicate; for example, they are perturbed by gravity and substantially damagedunder shear
11, 12, 30 , and we would expect the crystal structure in helium II to be similarly dam-aged, for example by vibrations or flow. Its structure factor has a half-width of approximately15% , indicating that the lattice maintains coherence over distances of order the length of theinterstitial defect in figure 3a, but not much longer.C3 The perturbations described above will impair the mean packing efficiency. Figure 2 showsthat helium II occupies 1.4% more volume than for perfect bcc crystals on a hard sphereapproximation .C4 The factors driving the pretransition anomaly in γ helium (see A6) also apply elsewhere on themelting curve. In particular, near ‘ A ’ in figure 1, helium II has few interstitial atoms, from (2),and a γ -like pretransition phase would have many more, giving it more entropy when warmed(see A3). Thus the solid will form on heating, as observed. On further heating, the exponentialrise in (2) will reverse the entropy balance and the solid will re-melt, observed at ‘ B ’.See the supplementary material for further evidence, and comparison with current models. Lattice vibrations
The equation of motion for a uniform line of atoms in one dimension has wave-like solutions(phonons) up to a frequency f o where neighbouring atoms oscillate antiphase . The supplemen-tary material shows that f o ≈ GHz in cold helium II at atmospheric pressure.13ear a discontinuity such as an interstitial defect, there is a so-called ‘optical’ solutionjust above f o where the unnormalised displacement of the n ’th atom at position x n in the one-dimensional lattice is U n ≈ ( − n e νt − µx n cos( kx n − ωt ) (3)See the supplementary material for the proof and extension to three dimensions involving sphericalharmonics. The amplitude e νt − µx n decays with distance from the defect, where both parametersreverse sign. By inspection, it advances at velocity v = ν/µ , which we associate with the velocityof the defect. The wavevector is k = − ωv/c s where c s is the speed of sound.This resonance will be excited by the energetic processes that create an interstitial atom, andit is long-lived since propagating waves do not exist at the resonant frequency and cannot carryenergy away. Thus, a newly created interstitial atom is an association between a particle and awave. The wave will guide the trajectory of the interstitial atom, since any velocity changes wouldrequire changes to the wave modes. This is consistent with the trajectories observed by Tucker andWyatt (B2) , which were ballistic on distances significantly longer than the coherence length ofthe lattice.A similar phenomenon occurs in another association between a particle and a wave, a dropletof oil bouncing on a vibrating oil tray. The bouncing creates surface waves which guide or ‘pilot’the droplet as it moves across the surface in a so-called ‘path memory’ effect, which producesballistic trajectories as with interstitial atoms . If barriers are present, such as a pair of slits,the trajectories exhibit statistical diffraction and interference patterns resembling those of a third14ave-particle association, a quantum particle .If there are multiple interstitial atoms, their resonant modes will overlap and weakly couple.Coupled resonators have normal modes with raised and lowered frequencies, and they sponta-neously synchronize when one of them is selected for, a phenomenon of nonlinear origin firstnoticed in pendulum clocks in 1665 and now studied in the field of Kuramoto theory
37, 38 . Thealignment can be maximised when the resonators are separated by a fixed number of wavelengths;this produces coherent motion which is described by a shared order parameter ∆( x , t ) . Such co-herence is also seen in videos of oil drops, which move coherently across the surface, separated bya fixed number of wavelengths .When interstitial atoms are synchronized in this way, they will not collide with each other,or even their images in the boundary. We saw (B2) that they move between collisions withoutnoticeable loss , and so the flow they carry will also be without noticeable loss. We associate thiswith the superflow in helium II .There is a fourth particle-wave association, which exhibits a similar order parameter. In1957, Bardeen, Cooper and Schrieffer (BCS) showed that conduction electrons in a metal areassociated with acoustic waves in the lattice, which they represented as virtual phonons. If thewave frequency is f , they showed that the electrons synchronize when their energy levels differby less than hf . Josephson showed how to measure the phase of the associated order parameter ∆( x , t ) in 1962 . When in this coherent state, the electrons also move without resistance.15n BCS theory, the wavelength of the order parameter for the electrons is λ = h/p where p is the momentum of a pair of electrons
40, 42 . Compare a pair of interstitial helium atoms, whosemomentum is m ∗ v . The solution (3), and its associated order parameter, have wavelength λ = 2 π | k | = c s f v = h s m ∗ v (4)where we have defined f = ω/ π and h s = 2 m ∗ c s /f . Figure 6 shows that h s is the same asPlanck’s constant within experimental accuracy at all pressures . The reason for this empiricalagreement with the BCS theory of superconductivity remains to be understood.Figure 6: The parameter h s = 2 m ∗ c s /f in (4) divided by Planck’s constant, from data for heliumII at low temperature . Error bars correspond to different crystal directions and do not includeexperimental error, the damage to the bcc lattice due to the flow, or our approximation f ≈ f o (seethe supplementary material). He differs from He in having magnetised nuclei. By inspection of figure 4 or the animationin the supplementary material, each atom in the path of a moving defect advances by one bcc cell’slength. This will disturb the correlations among the magnetised nuclei, giving dissipative flowas observed. This source of dissipation will be quenched at millikelvin temperatures, where thenuclear spins in a bcc lattice become aligned
4, 43 . He is observed to become superfluid at thesetemperatures. 16 esophases
Rod-shaped colloidal particles dispersed in a fluid medium form mesophases (liquid crystals),which are intermediate between a liquid and a solid. When dilute, the rods are randomly oriented(isotropic), and at greater concentration they align parallel to one another (nematic), which min-imizes repulsive inter-particle excluded volume interactions
44, 45 . At intermediate concentrationsthere is a biphase of isotropic and nematic domains .Similar considerations apply to the rod-shaped interstitial defects in helium II, which arelocally aligned with the bcc lattice. At low temperature they are dilute and populate all threelocal directions of the bcc lattice randomly (isotropically). When warmed, the concentration rises(figure 5), which forces more of them into nematic domains, where they are closely correlated,giving additional modes to shed energy and momentum, which introduces dissipation similar to anormal liquid. This agrees with the empirical two-fluid model of helium II, where it is superfluidat low temperature with an increasing proportion of normal fluid when warmed .At ‘ λ ’ in figure 1, the isotropic domains become isolated, quenching the superflow on longdistance scales. Just above this temperature, the isotropic domains disappear. It is more difficultto increase the concentration of repulsive rod-like defects in a nematic phase, where the packingis more efficient, accounting for the sudden fall in the specific heat capacity and the reversal in thethermal expansion coefficient at ‘ ρ max ’ in figure 1 inset. This line ends at a quadruple point (‘ Q ’)where four boundaries meet . Each boundary imposes a constraint on the variables pressure andtemperature, which would exceed the available degrees of freedom if there were one component17the Gibbs phase rule). ‘ Q ’ can exist because there are two components, the lattice and interstitialdefects. Further research
Helium is not the only system where an unusual flowing phase coexists under pressure with abnor-mally large crystals that are presumed to be bcc. The Earth’s core and neutron stars are thoughtto have these characteristics
48, 49 , raising the possibility that the crystal structures and flow mech-anisms are related. Helium II has a non-stochiometric bcc-like structure, due to the interstitialatoms, as do high temperature superconductors such as sulfur hydride at high pressure , raisingthe possibility that their flow mechanisms are similar.18 upplementary Material File list
The following files are found athttps://drive.google.com/drive/folders/0B-zmIlkqbDkZX0owZXJyajc4alE?usp=sharing.File Descriptioninterstitial-atom-movie.gif Animation of moving interstitial atomdensityhelium.ods † Molar volumes and forces between helium atomsinterstitial.ods † Renders and analyses the output of the programinterstitial2.ods † † spreadsheet for the animationmoving-defect.odg* flow mechanismstructure-factor.ods † Width of the structure factor of helium IIthermal-expansion.ods † The thermal expansion coefficientstriple-points.ods † The phase diagrams of the noble gases † LibreOffice version 5.2 spreadsheet. * LibreOffice version 5.2 drawingThe program to calculate the equilibrium positions of the atoms near a defect in figure 3 isin the ‘program’ folder. It uses Microsoft Visual Studio 2015.19 elting curves of the noble gases
Figure 1 shows that the melting pressures and temperatures of the noble gases, including He,approximately coincide at warmer temperatures when plotted in dimensionless units of temperatureand pressure relative to their triple point values.The small differences depend systematically on the atomic weights. Figure 7 shows thedimensionless melting pressures of helium, neon and argon, at a temperature of T triple , plottedagainst √ W where W is the atomic weight. Note the linear trend line, with which He is in goodagreement when its triple point is placed at ‘ TP ’ in figure 1.Figure 7: The reduced melting pressures
P/P triple of argon, neon and He at 4 times the triplepoint temperature. He is in good agreement with the trend when its triple point is placed at ‘TP’in figure 1 . acking efficiencies The following table summarises the packing efficiencies of the geometric arrangements discussedin the text, using a hard sphere approximation unless otherwise stated.Geometric arrangement efficiency Our classification Literatureface centred cubic lattice 0.74 fcc solid fcc solidhcp lattice 0.74 hexagonal solid hexagonal solidbcc saturated with interstitial atoms 0.729 † γ phase of He –bcc lattice 0.68 – γ phase of Hefluidized bcc crystals 0.67 * helium II –quantum fluid 0.67 * – helium II: London 1938diamond lattice 0.34 helium II: London 1936 † Idealised arrangement in figure 3c *Experimental value from helium II and spherical colloids.
X-ray attenuation due to zero point motion
We now estimate the attenuation of the intensities of the x-ray diffraction lines in hypotheticallyperfect bcc crystal of helium atoms, caused by zero point motion. We will use recent estimatesof the potential energy of a helium atom, which is in a shallow well with a steep potential energybarrier to displacements greater than d o ≈ . s where s is the side of a bcc cell
18, 51 .We begin with the approximation that the probability density of a helium atom is constant21nside a sphere of radius d o . The (200) reflection planes are d = 0 . s apart, so the amplitude ofthe diffraction line will be attenuated by a factor π (cid:82) − π (1 − x ) cos(2 π [ d o /d ] x ) dx ≈ . , andthe attenuation for the (110) line would be 0.8.The square of this ratio, approximately 40%, is an estimate for the relative attenuation of thetwo lines. This is an over-estimate since we have assumed a constant probability density, whereasthe ground state would have a density peak near the centre. This is much smaller than required toaccount for the observed attenuation. For example, in 1958 Schuch, Grilly and Mills found that thex-ray line intensities from single crystals of He ‘declined steeply with increasing angle’ so thatonly three lines could be observed, and in a coarse powder only a single (110) line was visible .The corresponding phase of He is similar . Numerical calculation
The net inter-atomic force as a function of the distance between atoms was estimated from thedensity of He as a function of pressure at 0.1K (see figure 8). This force was then used as aninput to the computer program, which iterates through the atoms, relaxing them in the direction ofany net forces, until the forces are small. The calculation can thus be classified as a mean-field orsemi-classical approximation.In the starting conditions for the program, a bcc arrangement was set up in accordance withthe observed density at the relevant pressure. An extra atom was added near the origin and thenearby atoms were displaced slightly to assist convergence. The atoms were represented by objects22igure 8: The net forces between neighbouring atoms, estimated from the density of He as afunction of pressure at 0.1K . The line shows a quadratic fit to the data, which was used as aninput to the calculation; the forces at larger distances were extrapolated as shown. S is thedistance between nearest neighbour atoms in an unperturbed bcc crystal at 1 atmosphere pressureused in figure 3. in a collection, indexed by their unperturbed positions, so it was only necessary to store the objectsthat had been perturbed, thereby allowing computation with an effectively infinite lattice.The file ‘positions.csv’ in the program folder contains the output of the program, includinga large number of atoms which are perturbed very little. It has columns for the nominal positions(x0, y0, z0) and relaxed positions (x, y, z) of the atoms, in descending order of the displacementfrom their nominal positions. The unperturbed bcc lattice is comprised of two intersecting cubiclattices, each of side 2 units; see the ‘coordinates’ structure for detail.The spreadsheet ‘interstitial.ods’ contains the coordinates for the defect plotted in figure 3.In our semi-classical or mean field approximation at low temperature, helium has the prop-23rty, known since at least 1936, that the atoms can reduce their potential energy by becomingdisplaced slightly from the central position of a lattice . The resulting Peierls-like distortion canbe seen by careful examination of the program output (see for example the small asymmetry infigure 9), but it is small at 1 atmosphere pressure. γ helium: further experimental evidence Further experimental evidence that γ helium has the structure shown in figure 3 is as follows.a1 We saw (A5) that interstitial atoms are forced into the bcc crystals in γ helium under pressure.At the transition pressure, we would expect the energy P ∆ V to be comparable with otherrearrangement energies involved. Experimentally, P ∆ V ∼ − − J, or 7K expressed as atemperature, comparable to the temperature required to melt the crystals in He and He.a2 Schuch and Mills reconciled the molar volume of γ helium with their x-ray measurements .However, they did not report the width of the reflection spots or the likely error in their estimate,even though they noted that the data were of poor quality. When can estimate the scatterfrom their measurements of the d distance, which corresponded to the brightest reflection.They averaged 11 photographs and reported an estimated error of 0.31%. Assuming they usedstandard statistics, we can infer a scatter corresponding to 1% in their estimates of the positionsof the centres of the individual spots, which translates into a scatter of 3% in molar volume ifone took the centres of the spots (which is itself an average). This implies that the width of thespots was substantially greater, consistent with the proposed arrangement in figure 3c.243 Schuch noted in 1961 that the volume increase on the hcp- β transition in zirconium is pro-portionally the same as for the hcp- γ transition in helium . Schuch and Mills later inferredthat they must have the same structure . If so, β zirconium, like γ helium, is composed of bcccrystals saturated with intersitial atoms. This is supported by several observations. The transi-tion occurs on heating, not cooling , as expected from the entropy in the interstitial zirconiumatoms (compare A3). Like γ helium, it only exists over a narrow range of temperatures, it hasan indistinct phase boundary on heating (compare A6), and near the transition the intensitiesof its x-ray reflections decline steeply with increasing angle . Heating produces a phase with-out sharp x-ray reflection peaks, resembling helium II (we would not expect it to fluidize aszirconium is not inert). Effective mass
When an interstitial defect advances the length of a bcc cell, s , then the i ’th atom advances by ( d i +1 − d i ) where d i is the deviation shown in figure 3b. Thus v i v ≈ d i +1 − d i s where v i is the velocity of the i ’th atom and v the velocity of the defect. This is plotted in figure 9.The kinetic energy of the atoms in the central row of the defect is T = Σ mv i . Equating thisto the kinetic energy of the defect moving at velocity v with effective mass m ∗ , namely T = m ∗ v ,gives m ∗ = m Σ( v i /v ) . This sum is tabulated below, giving m ∗ = 0 . m . The sum (cid:80) v i /v differs from 1, indicating the error bar due to the boundary of our calculation.25igure 9: The velocity v i of the i ’th atom of an advancing defect, divided by the velocity v of thedefect. We attribute the small asymmetry and negative end values to the Peierls-like distortiondiscussed above. The effective mass of the defect is m (cid:80) ( v i /v ) where m is the mass of an atom. i v i /v ( v i /v ) Expectation number of defects
The partition function of a system with energy levels E i in equilibrium at temperature T is definedby Z = (cid:88) e − βE i β = ( K B T ) − and K B is Boltzmann’s constant.Consider a single interstitial defect in one dimension, which has effective mass m ∗ and mo-mentum p . Its partition function is like that of a particle in a one-dimensional box of length L ,namely Z (1 d )1 = 1 h (cid:90) (cid:90) dq dp e − β p m ∗ = Lh (cid:114) πm ∗ β If the centre row of the defect forms part of a line which (without the defect) contains M atoms, separated by distance s , then substituting L = sM gives Z (1 d )1 = M (cid:115) πm ∗ s βh Consider a large bcc crystal containing N o = kM atoms in k identical lines of atoms likethe above. Neglecting edge effects, the defect could occupy any one of the lines and the partitionfunction for defects oriented parallel to the x direction will be Z (3 d )1 x = kZ (1 d )1 = N o (cid:115) πm ∗ s βh The defect could be oriented in the x , y or z directions, giving the total partition function Z (3 d )1 = 3 N o (cid:115) πm ∗ s βh (5)where s is the length of a bcc cell (the lattice parameter).If the crystal contains N i interstitial defects of chemical potential µ , the grand partition func-27ion is defined by Z = (cid:88) e β ( N i µ − E i ) which evaluates to Z = 1 + Z e βµ + Z e βµ + ... = exp( Z e βµ ) The expectation number of defects is (cid:104) N (cid:105) = (cid:80) N i e β ( N i µ − E i ) Z = 1 β Z ∂ Z ∂µ = 1 β ∂ ln( Z ) ∂µ = Z e βµ Substituting (5) and ∆ H = µ gives equation (2, text) in the text (cid:104) N (cid:105) N = 3 (cid:18) πm ∗ s h β (cid:19) e − β ∆ H We now consider small terms which we have neglected in (2, text), and possible sources ofsystematic error in our comparison of the activation energy of interstitial atoms and rotons (figure5 inset).An interstitial defect is resonant. We have already counted its zero point energy, since thechemical potential µ is the energy required to create it at low temperature, which includes zeropoint energy, but we have not counted its excited states. Taking the lowest energy spherical har-monics, there are three orientations of resonance which may be excited at higher temperature,whose frequency we estimate (below) to be f ≈ f o = 1 . Hz in superfluid helium at 1 atmo-sphere pressure. This corresponds to a temperature of hf /K B = 7 . K . The partition functionof an individual oscillator (excluding zero point energy) is given by (1 − e − βhf ) − . Therefore the28otal partition function is modified by a factor of (1 − e − βhf ) − . Taking f ∼ . Hz, this givesa correction of 6.7% at 2K, close to the highest temperatures plotted. This is small in relation tothe agreement over several orders of magnitude in the plot, and we find it has little or no effect onour estimate of the roton energy using a least squares fit.We have also neglected perturbations to the existing phonon frequencies. The speed of soundfalls by approximately 4% from 1K to 2K . We attribute this to the extra mass of the interstitialatoms. Again, this is small in relation to the several orders of magnitude in the plot.We used a constant value for the effective mass of an interstitial atom, m ∗ = 0 . m , whencalculating the theoretical line from (2, text). This is the effective mass discussed in the text, fromour numerical calculation at atmospheric pressure and low temperature. The effective mass of aroton, which we associate with interstitial atoms, reduces with pressure and temperature, both byof order 20% . The square root dependence on m ∗ in (2, text) indicates a possible systematic errorof order 10%. This is also likely to have little effect on the activation energy for the same reasonas above.The roton activation energy in figure 5 (inset) used the experimental values from from neu-tron scattering data at the lowest measured temperature, near 1.25K. This slightly under-estimatesthe low temperature value.In the main plot in figure 5 we subtracted out the ordinary expansion of helium II by ex-trapolating from the expansion at low temperature, where the concentration of interstitial atoms is29egligible due to the exponential decay. This improved the match at low temperature, but had littleeffect at higher temperatures where the concentration of interstitial atoms rises exponentially from(2, text). At 15 atmospheres and above there was a reasonable fit to the expected T dependenceof the thermal expansion coefficient at low temperature. At lower pressures, there were also T terms, which we speculate may be due to variations in the effective pressure associated with sur-face tension. As a consequence the data at lower pressures is less accurate. Details can be seen inthe spreadheet file.We saw that interstitial atoms become aligned nematically when they are more concentrated,at higher temperature. This is likely to introduce additional terms which we have not considered.The rise in the graph near the transition temperature may be associated with this. Further experimental evidence: rotons
The following experiments also suggest that interstitial atoms have the properties of rotons.b1 We would expect the effective mass of an interstitial defect to reduce with temperature, due tothermal agitation elongating it. The roton effective mass has a plateau below about 1K, andreduces sharply with temperature above this .b2 In most other liquids, the speed of sound rises with temperature due to the reduction in density.In helium II, the mass of the interstitial atoms in (2, text) raises the density and will reduce thespeed of longitudinal sound. It falls by approximately 4% from 1K to 2K at 15 atmospheres30ressure .b3 The mobile defects observed by Tucker and Wyatt (which we associate with interstitial atoms)transport heat energy, giving helium II its high thermal conductivity. If an interstitial atommoves through the lattice faster than the speed of transverse sound, we would expect it toradiate an analogue of Cherenkov radiation, thereby limiting its maximum speed. The velocityof heat transport (second sound) is limited to of order 20ms -1 , an order of magnitude less thanthe speed of longitudinal sound.b4 At warmer temperatures, interstitial atoms will exert a pressure due to their kinetic energy. Thisis observed in the fountain effect, a phenomenon also attributed to rotons
19, 52, 53 .b5 An inhomogeneous electric field will trap and stabilise interstitial atoms, which are denser andhave stronger dielectric interactions. In 2007, Moroshkin, Hofer, Ulzega and Weis produced adendritic solid by melting the γ phase of He with positively charged impurities sputtered ontoit
18, 54, 55 . See the photograph in figure 10. The dendrites are attracted to a cathode (to the right),indicating they are positively charged. They fall under gravity, indicating they are denser thanhelium II.
Further experimental evidence: fluidization
The following additional observations relate to the fluidization of He, He and colloids.c1 We noted (C4) that the factors driving the pretransition anomaly in γ helium also apply else-31igure 10: Dendritic crystals in helium II, formed around charged impurities
18, 54, 55 , which weinterpret as due to stabilised interstitial atoms. An alternative interpretation is that the photographshows ‘frozen rotons’. Reproduced by kind permission of Peter Moroshkin. where, suggesting the existence of a narrow γ -like pretransition phase very close to the meltingcurve. There are experimental reports of a bcc-like phase very close to the melting curve of He near room temperature and 15GPa, but Frenkel calculated that an ordinary bcc structurewould not be stable . He showed that quantum effects would not stabilise the phase, but didnot consider a pretransition γ phase, which is stabilised by the entropy of the interstitial atoms.See later studies calling the experimental observations into question .c2 We saw that helium II solidifies on heating near ‘ A ’ in figure 1, and attributed it to the greaterentropy of a pretransition γ -like phase (see C4). Our explanation requires that the depressionin the melting curve between ‘ A ’ and ‘ B ’ in the figure cannot be wider than the anomaly itself.The anomaly in the specific heat capacity of γ helium is approximately 20mK wide , or, basedon the slope of the curve, 20kPa, while the depression is only 1kPa.c3 The fluidized phase of He also solidifies on heating (see ‘ A ’ in figure 11). The solid in this32egion has the same x-ray characteristics as the γ phase of He, indicating they have the samestructure
5, 17 , namely a bcc lattice saturated with interstitial atoms, which may be favoured bythe interactions between the magnetised nuclei in He. The explanation for the solidificationon heating in C4 applies directly. The magnitude of the depression in the melting curve is notlimited by the width of a pretransition anomaly, and it is much larger than in He.Figure 11:
Phase diagram for He on a logarithmic scale, with linear inset. The melting curve forneon is superposed with its pressures and temperatures scaled so its triple point is at ‘TP’
1, 58 . c4 In 1959, Bernardes and Primakoff predicted that He would solidify on heating using differentassumptions, namely that the solid is ordinary bcc crystals, and it has more entropy than theliquid (which they assumed is a Fermi liquid) due to nuclear spins . However, it was laterfound that their model implies a spin ordering temperature much higher than observed . Theydid not consider the entropy associated with interstitial defects, an omission corrected above.335 In 2012, Besseling, Hermes, Fortini, Dijkstra, Imhof and Blaaderen applied low amplitudeoscillatory shear to a colloid of spherical particles in the fluidized phase near the solidificationconcentration, and observed the appearance of bcc-like order, both in numerical simulation andin experiments . The structure was body centred tetragonal (i.e. body centred cubic, slightlyelongated perpendicular to the shear planes) distorted into a hexagonal structure at the extremesof the shear. We expect the onset of fluidization in this structure to be similar to our proposalsfor helium II. One possibility is that this has already happened: the bcc order in the colloidalfluid has been damaged by the flow, and the gentle oscillation in this experiment has helped toheal the damage to reveal the underlying bcc-like order. One-dimensional lattice equation of motion
The equation of motion for a one-dimensional line of atoms is well known. Suppose that atoms ofmass m are distance d apart, with an elastic constant C s between neighbours. If the n ’th atom isdisplaced by U n ( t ) then the force on it from its nearest neighbours will be m d U n dt = C s ( U n +1 + U n − − U n ) (6)We can make a continuous approximation, which is valid for long wavelength waves, by defining asmooth function φ ( x, t ) so that φ ( nd, t ) = U n ( t ) . Substituting the second-order Taylor expansionaround x = 0 , namely φ ( x, t ) = φ (0 , t ) + x ∂φ (0 , t ) ∂t + 12 x ∂ φ (0 , t ) ∂t ∂ φ∂t − c s ∂ φ∂x = 0 which is the standard wave equation where the speed of sound is c s = d (cid:112) C s /m . This describeslow frequency sound waves in the lattice.There is another continuous approximation which is valid near the maximum frequency ofwaves, when adjacent atoms oscillate almost antiphase with one another. We again define a smoothfunction φ ( x, t ) so that φ ( nd, t ) = ( − n U n ( t ) . The same Taylor expansion gives ∂ φ∂t + c s ∂ φ∂x = − ω o φ (7)where ω o = 4 C s /m or ω o = 2 c s /d .We can describe the propagating waves just below the maximum frequency by substituting asolution of form φ ∝ cos( kx − ωt ) . This gives the dispersion relation ω = ω o − c s k (8)It follows immediately from (8) that propagating waves do not exist at angular frequencies above ω o . We can estimate ω o in He at 1 atmosphere pressure for a bcc crystal structure where d is theside of a primitive cell and using the data reported in Brooks for helium II at atmospheric pressureand low temperature , giving 35 .
49 10 − m c s
225 m s − ω o c s /d s − f o ω o / (2 π ) Hzwhere f o is the maximum frequency of propagating waves. This is a lower estimate for f o ,based on assuming d is the length of a bcc cell. An upper estimate is a factor √ / larger, basedon the shortest distance between atoms. Resonant interstitial atoms
Near an interstitial atom, in a one-dimensional idealisation, the equation of motion (7) has a lo-calised solution in which the atoms oscillate almost antiphase.In this solution, the interstitial atom itself is stationary at x = 0 and it provides the boundarycondition for the solution, which is φ = cos( ωt ) e − µ | x | This obeys (7) when ω = ω o + c s µ Expressed in terms of the atomic displacements, this solution is U n ≈ ( − i cos( ωt ) e − µ | x n | (9)where the displacements U n are mirror images in the origin, U − i = − U i , and the interstitial atomis stationary, U = 0 . 36he above solution was for a stationary defect. When the defect advances at velocity v , thecorresponding solution is φ = e νt − µx cos( kx − ωt ) which obeys (7) when ( ν − iω ) + c s ( ik − µ ) = − ω o (10)and writing the atomic displacements explicitly gives the equation used in the text U n ≈ ( − i e νt − µx n cos( kx n − ωt ) The amplitude e νt − µx advances with the defect at velocity v = ν/µ since its value remainsconstant when x = ( ν/µ ) t + constant. Equating the imaginary parts of (10) gives an the velocity v = νµ = − c s kω (11)Equating the real part of (10) gives the dispersion relation ν − ω + c s ( µ − k ) = − ω o from which we can check that the group velocity is the same as the velocity of the defect v g = ∂ω∂k = − c s kω = v The wavelength of this solution is λ = 2 π/ | k | . Substituting into (11) and approximating ω = 2 πf ≈ ω o gives an approximate relationship between the momentum of a pair of interstitialatoms p and the wavelength of the waves p = 2 m ∗ v ≈ m ∗ c s f λ = h s λ h s = 2 m ∗ c s /f . Approximating f ≈ f o = 2 c s /d gives h s = 2 πm ∗ c s d . This ‘acousticPlanck constant’ is calculated in the spreadsheet densityhelium.ods and the results displayed figure6. It uses data from Brooks – namely, the speed of sound, the effective mass of a roton fromneutron scattering measurements, and the inter-atomic distance calculated from the density on abcc arrangement of atoms in the (200) and (111) directions . Extension to thee dimensions
We have described the resonances of an interstitial atom using a one-dimensional simplification. Inthree dimensions, the atoms in adjacent rows will be displaced, and there are also solutions wherethe displacements are not parallel to the direction of motion. We outline these extensions in turn.One extension is based on a perturbation of the one-dimensional solution (9) for the atomsin the central row of the defect. This solution is perturbed because the displacements disturb theatoms in the adjacent rows due to transverse strains. The associated forces are much smaller thanfor longitudinal strains, and so we expect the perturbation to decay rapidly with distance from thecentral line. This suggests the perturbation is a small effect.There is some experimental support for this approach. If the velocity of the defect exceedsthe speed of transverse sound in the crystal, then we would expect it to lose energy to an analogueof Cerenkov radiation. The maximum velocity of a defect is indeed significantly less than thespeed of longitudinal sound (see b3 above). 38n alternative perspective is to note that a displaced atom exerts forces on the atoms nearit, not just those in the one-dimensional line. In this idealization we neglect the bcc lattice en-tirely, and it is necessary to extend the continuous approximation to the equation of motion (7) tothree dimensions. The solutions are likely to involve spherical Bessel functions like those for anunbaffled loudspeaker in the open air.There is also another class of solutions in which the displacements are perpendicular to thedirection of motion of the interstitial atom. Suppose the indexes of the atoms are ( i, j, k ) , corre-sponding to the ( x, y, z ) directions, so their coordinates are ( x i , y j , z k ) . There is an approximatesolution in which the displacements of the atoms in the ( x, y ) plane are parallel to the y directionand have magnitude U ( y ) ijk ≈ ( − j e − µ | y j | cos( kx i − ωt ) It is easily verified, using the perturbation approach described above, that the dispersionrelation of this solution is in approximately the same form as that for a relativistic particle, namely ω = ω o + c t k where c t is the speed of transverse sound.In the above solutions, the motion of adjacent atoms is approximately antiphase in one di-rection and approximately in phase in the other two directions. To complete the picture, there arealso more complicated solutions in which the motion is antiphase in two and three directions.39 chematic of the isotropic and nematic arrangements Figure 12 is a schematic illustration of the isotropic and nematic domains of the rod-like interstitialdefects in helium II, which are locally aligned with the bcc lattice. The illustration is idealised.We saw from neutron scattering data (C2) that the lattice loses correlation on distance scales largerthan the length of a defect; this loss of correlation is not illustrated.Figure 12:
Schematic illustration of (a) isotropic and (b) nematic domains of rod-like interstitialdefects in equilibrium in helium. Distortions to the lattice, due to the fluidization, are not shown.
Critical comments
Experts in solid helium, quantum fluids, colloids and soft matter have kindly offered critical com-ments which have greatly helped the manuscript. Some observations arising are listed below.Q1
London repudiated his 1936 model of helium II when he showed it is a quantum fluid.The manuscript fails to acknowledge this.
Contrary to some reports, London did not repudiate his original 1936 paper, which describedthe correlations among helium atoms in ordinary geometrical terms . The stated aim of his1938 paper was to reject a proposal by Fr¨ohlich that helium II has a diamond lattice , and40hen to “direct attention to an entirely different interpretation” in momentum space. Repre-sentations in physical and momentum space are often complementary, and the manuscript at-tempts to show this in detail. Afterwards, London continued to advocate his 1936 description.In particular, in 1939 he emphasized that the rheology “depends essentially” on volume .Q2 Helium II is a liquid. Liquids have insufficient positional order for the concept of aninterstitial atom to be defined.
There is no evidence that the atoms in helium II are disordered like in an ordinary liquid.London showed in 1936 that cold helium II has negligible entropy , and concluded that theatoms must be ordered positionally. The manuscript describes the flow mechanism, which issimilar to supersolid flow
9, 10 .Q3
The flowing phase of a colloid can be seen in a microscope, but the predicted bcc-likeorder is not observed.
To the contrary, Besseling et al recently photographed the appearance of bcc-like order in acolloid of spherical particles in the fluidized phase near the solidification concentration . Intheir experiment they applied low amplitude oscillatory shear to the phase. There are a numberof possible interpretations, but we suggest that the gentle oscillation accelerates the approachto equilibrium. See c5 (in the supplementary material below) for further discussion.In the absence of such stimulation, colloids approach equilibrium extremely slowly; forexample, they typically take days to settle . Even after a long time they do not reach equi-librium on Earth, as evidenced by the fact that they behave differently on the space shuttle .This can be understood by noting that the arrangement of particles is very weak mechanically,41articularly in the flowing phase, so that vibrations, gravity and convection impede or preventthe approach to equilibrium.Helium approaches equilibrium faster than a colloid, due to the smaller scale, but nev-ertheless may not reach it on Earth since it appears to behave differently in space shuttleexperiments .Q4 The γ phase of Helium has been classified body centred cubic for many years. If thisclassification were mistaken, as claimed, then it would have been discovered by specialistsin the field by now.
Specialists working on γ helium told us that the 7% discrepancy in molar volume (figure 2) iswell known. It is not the subject of active study because it is believed to have been explainedhistorically. We traced this supposed explanation to three papers, by Schuch et al in 1958, 1961and 1962
5, 14, 17 and noticed that the 1961 and 1962 papers contradict each other, as discussedin the text. These papers also reported unexplained anomalies in the x-ray patterns and crystalsize which led us directly to the structure in figure 3.Q5
Does the manuscript make observable predictions that differ from the conventional model?
Yes. The manuscript predicts that the lowest energy excitations in helium II (other than acous-tic phonons) are interstitial atoms, which have the gravimetric mass m of a helium atom andeffective mass (from the relationship between velocity and kinetic energy) of m ∗ ≈ . m (see B6). Both are close to the observed values – the effective mass is known from neutronscattering experiments and the mass can be inferred from its effect on density (see figure 5).42ccording to the conventional model, the lowest energy excitations are rotons. The theorydoes not predict their effective mass, but Feynman thought a roton’s kinetic energy is primarilyassociated with circular flow patterns , from which it follows that their gravimetric mass ismuch smaller than m ∗ , contrary to the negative thermal expansion measurements plotted infigure 5.Q6 The manuscript claims that the flow in helium II is carried by interstitial helium atoms,which are mobile. But at low temperature their concentration is vanishingly small, asshown in figure 5
Figure 5 is the concentration of interstitial atoms in thermal equilibrium, but flow is not anequilibrium phenomenon. Interstitial atoms can be formed near surface imperfections by me-chanical, rather then thermal, energy, as described in the text. They are metastable, sincehelium atoms cannot be destroyed, and continue to carry the flow for a considerable time.Q7
The manuscript describes helium using classical equations of motion. But helium is aquantum fluid. Is this an attempt to dethrone quantum mechanics?
No. Textbooks on quantum fluids typically begin with the wavelength postulate λ = h/p where p is the momentum . The wavelength in (4), λ = h s /p , was derived from ordinary clas-sical equations of motion in the same way as for the experiments on bouncing droplets
35, 36, 39 .It was a surprise to the authors that the parameter h s is empirically the same as Planck’s con-stant to experimental accuracy at all pressures (figure 6). This suggests that quantum processesultimately underly the result. In particular, the forces between helium atoms are quantum me-chanical in origin. We suggest this is an area for further research.438 Liquid He is a Bose-Einstein condensate and liquid He is a Fermi liquid. Does themanuscript argue otherwise?
Not necessarily. ‘Bose-Einstein condensate’ and ‘Fermi liquid’ are descriptions in momentumspace whereas the manuscript discusses a geometrical representation, in physical space.However, there is a difference between the models which might, in future, be measuredexperimentally. The wavelength (4) depends on the momentum of a pair of interstitial atoms,based on their effective mass, while London’s 1938 model of a Bose-Einstein condensateinvolves the gravimetric mass of a single helium atom. This is approximately a factor of3 larger. We are not aware of any experiments to date which are capable of distinguishingbetween them. Compare the corresponding electron mass in superconductors, according tothe conventional model .Q9 The phase diagram of He has the ‘ λ ’ line, below which superflow is observed on a macro-scopic scale, and the line of maximum density which is very close to it. Which of these isthe actual thermodynamic transition? The line of maximum density ( ρ max in figure 1 inset) is the thermodynamic transition. To theright of it, the interstitial atoms are arranged nematically, and to the left there is a biphase ofnematic and isotropic domains. Ordinary liquid crystals are similar .The nematic phase is an efficient packing arrangement which lacks room for more inter-stitial atoms, so it is difficult or impossible for their concentration to rise with temperaturein this region. This accounts for the sudden fall in the specific heat capacity and the thermalexpansion coefficient reversing sign. The isotropic domains (which we associate with super-44uid phenomena, as discussed in the text) are quenched to the right of ρ max and superfluidphenomena are quenched accordingly.Cooling the substance below ρ max , the next event occurs at the line λ , where the isotropicdomains join up and superflow is possible over a macroscopic sample. This is a geometricalphenomenon rather than an ordinary thermodynamic transition. We would expect superfluidbehaviour (other than macroscopic flow) to persist between the λ and ρ max lines. This isobserved and called the superfluid fluctuation regime.The approximately exponential rise in the specific heat capacity from of order 1K to thetransition temperature can be understood in the same way as in the conventional theory, asdue to the exponential rise in the number of excitations (equation (2, text)). Very near thetransition, there is a spike in the specific heat capacity. We conjecture that this is associatedwith nonlinear effects due to the vanishingly small size of the isotropic domains, an area forfurther research.1. Young, D. The phase diagrams of the elements (Lawrence Livermore Laboratory, 1975).2. Donnelly, R. & Barenghi, C. The observed properties of liquid helium at the saturated vaporpressure.
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We thank Ross Anderson, Sebastien Balibar, Robin Ball, Michael Cates, YorgosKatsikis, Daphne Klotsa, Anthony Leggett and Peter Moroshkin for helpful comments and materials.
Author contributions
The authors’ principal contributions were: γ helium, numerical simulations andflow paradigm, R.M.B.; colloids and mesophases, E.T.S.; thermodynamic calculations, D.H.P.T. All authorscommented on and approved all aspects of the paper.helium, numerical simulations andflow paradigm, R.M.B.; colloids and mesophases, E.T.S.; thermodynamic calculations, D.H.P.T. All authorscommented on and approved all aspects of the paper.