Liquid Helium

The element helium comes in to isotopes. He3 with two protons and a single neutron in its nucleus is a fermion which does not form a Bose-Einstein condensate at low temperature. He4 has an extra neutron, making the nucleus a boson, and does make the condensate in the liquid state. It does not freeze to a solid under normal pressure.

He4 liquifies from gas to liquid at 5.2 K above absolute zero temeprature at normal pressure. The superfluid phase transition happens at 2.2 K. The density-temperature function has a slope discontinuity there. This means a discontinuous thermal expansion coefficient. There is also a sudden change in the dielectric constant. The specific heat at constant volume has a singularity at 2.19 K that looks like the Greek letter lambda. Hence the name "lambda point". He4 liquid called "HeII" below the lambda point creeps along thin films and flows through narrow tubes with zero viscosity below a critical speed. The "thermomechanical effect" is the flow of HeII through a superleak from low to high temperature at constant pressure until a pressure difference equal to the temperature difference multiplied by the specific entropy and the density is established. The mechanocaloric "fountain effect" is the flow of HeII through a superleak at common temperature with a pressure difference from high to low pressure. What is surprising is that the low pressure region cools down! This is explained by the phenomenological "two-fluid model" consisting of a superfluid of zero viscosity and a normal fluid. The ratio of superfluid to the total approaches 1 as the temperature approaches absolute zero.

Fritz London noted the possible connection to the Bose-Einstein condensation of an ideal gas which predicts a lambda point at 3.13 K. Landau (1941) made a theory good near absolute zero, but not near the lambda point. The normal fluid is a gas of excitations where the vacuum is the superfluid. One can explain some of the properties of HeII by imposing a special energy-momentum relation on these excitations or quanta. At low momenta, i.e., long wavelengths, the energy is proportional to the momentum like an acoustic phonon in a solid crystal lattice. Landau also put in a second "roton" spectrum with an energy gap plus the square of the momentum divided by twice an effective mass like an ordinary slow moving particle. Further exerimental work caused Landau to modify his roton spectrum in 1947 to delta + (p - po)^2/2 mu, where delta/Boltzmann's constant = 9.6 K, po/hbar = 1.95 inverse Angstroms, and mu/mass of helium = 0.77 to get a good fit to thermodynamic data. Note that this spectrum has a region where the energy decreases with increasing momentum. The group speed dE/dp is opposite in direction to the phase speed E/p in this region. There are also critical points, a max and a min, where the group speed is zero for finite phase speed. Again this is also phenomenology with no derivation from quantum mechanics. That is what Feynman did. Bogoliubov did model the weakly interacting boson gas with a hard core potential using perturbation theory in quantum mechanics. But the scheme did not fit actual HeII well at all. Bogoliubov did introduce his famous transformation which is useful in several different, seemingly unrelated, fundamental problems of physics like the Unruh radiation seen by a constantly accelerating particle in a vacuum. The latter is related to the Hawking radiation in the explosion of mini-blackholes. The Bogoliubov transformation in second quantization is from a set of interacting particles to non-interacting quasiparticles. This model did show the that Bose-Einstein condensation is not 100% at absolute zero and it gave a formula for the ground state energy, pressure and speed of sound. The excitations did not have the final phonon roton spectrum with po in it. It did have Landau's original spectrum of 1941 with po = 0 of a phonon and an excitation with a gap.

Feynman's idea.

How can we understand the Tisza, London, Landau, Bogoliubov phenomenological models in terms of atoms described by quantum mechanics? Landau's theory of quantum hydrodynamics left atoms and the role of nonlocal Bose-Einstein quantum statistics out of the picture. Feynman argued that Bose-Einstein statistics created the energy gap in the roton excitations which were NOT large scale coordinated motions of many atoms. These rotons are not the same as quantized vortices seen in HeII which do involve such large-scale coordinated motions of many atoms.

"[Feynman] argued that, unless Bose statistics intervened, excitations involving large-scale motions of the atoms could be created with as little energy as one wants; it was only because the situation was governed by Bose statistics that large-scale motions became redundant and one was forced to invoke motions on an atomic scale, which in turn required a finite minimum energy to get excited." Mehra, p. 364

Feynman's first superfluid paper argued that in spite of the inter-atomic interactions, superfluid helium was a Bose-Einstein condensate similar to what happens in the ideal gas where the interactions are absent.

Note that there are theories today that the human mind is a room-temperature Bose-Einstein condensate of bosonic phonons forming Frohlich modes of collective electric dipole oscillations in the microtubules inside the cells of our body. The Bose-Einstein condensate is robust against thermal perturbations. Robert Worden has a model of how such a Bose-Einstein condensate makes an internal representation of the outer 3D world.

Feynman used his path integral to write the partition function of statistical mechanics for liquid helium. One can do this because the real time of classical mechanicsis formally analogous to an imaginary inverse absolute temperature. That is t -> i/T. Infinite time then corresponds to aboslute zero. The Lagrangian of classical mechanics corresponds to a free-energy in thermodynamics. Therefore, the action is like a free energy divided by temperature. In other words, action in imaginary time is like thermodynamic entropy. The Feynman amplitude is essentially e^iaction/hbar which is analogous to the partition function (i.e., "the sum-over-states") e^-entropy/k, where k is Boltzmann's quantum of entropy. Note, that any erasing of one bit of information requires an increase of entropy of order k. This is a consequence of the second law of thermodynamics.

"[Feynman] expressed the partition function of the liquid in the form of a path-integral which took into account the fact that the sum-over-states was to be carried only over those states that were symmetric with respect to the exchange of the particles, and went on to determine the class of trajectories that contributed most to this integral. He showed that the most important trajectories corresponded to situations in which the displacements of the individual atoms went hardly beyond their nearest neighbors; in other words, only those permutations of the coordinates were important in which the atoms were either left in their original positions or were moved to a neighboring location."

Mehra, p.364

Feynman then develops a clever geometric model of nonintersecting polygons on a 3D lattice to compute his path integral which has a third order phase transition with a continuous specific heat and discontininuous slope corresponding to a singularity in the partition function when the "fugacity" = 1. The basic math here is very simple. The partition function has a factor which is a geometric series of powers of the fugacity that sums to 1/(1 - fugacity) when fugacity < 1.

This, unfortunately, is not the observed second-order the lambda point transition. However, adding more polygons did give the actual second-order transition thirty years later in work by Ceperly and Pollock. There were no Landau phonons and rotons in Feynman's first paper on superfluid helium.

"The above conclusions rested very much on the argument that, despite interparticle interactions, the motion of one atom through the others is not opposed by a potential barrier because the others simply move out of the way." Mehra, p. 365

Feynman's second superfluid paper in 1953 looked at the ground state many-atom wave function that dominates the behavior of the liquid near absolute zero. The problem is to construct an excited state. He argued, like Landau in 1941, that the lowest energy excitations were compressional sound waves whose quanta are longitudinal phonons with no energy gap. Additional particle-like excitations must have an energy gap delta' and an effective mass m'.

"The requirement of [Bose-Einstein permutational] symmetry of the wave functions ruled out any large-scale motions of the atoms (which simply permute the atoms and do not yield anything truly different from the gound state." Mehra, p. 366

That is, the ground state is the equally weighted sum of all possible permutations of the atoms corresponding to a single row Young pattern with a huge number ( 10^23) of boxes.

In contrast, for a fermi liquid and a classical liquid with distinguishable atoms, "one could have large-scale motions with excitation energy as small as one likes; these excitations would be more than a match for the phonons in determining the low-temperature properties". Mehra, p. 366

Feynman's third superfluid helium paper appeared in 1954 on the excited many-atom wave-functions. He considered (quotes from Feynman): 1. rotation of a small ring of atoms; 2. "excitation of an atom in a local cage formed around it by its neighbors"; 3. "motion of a single atom with wave number k = 2pi/d, where d is the atomic spacing, the other atoms moving about to get out of the way in front and to close in from behind". The form of the excited state for all three cases had to be the ground state multiplied by the sum of single particle functions f(ri) from i = 1 to N. For excitations of type 3, f(ri) = e^ik.ri, but f(ri) for 1 and 2 was not clear. Feynman used a variational principle with constraints to minimize the expectation value of the Hamiltonian of the N atoms. This gave him an integro-partial differential equation for f(ri) whose kernel was the conditional probability to find a second atom at a distance from a first atom whose position is given. One possible solution was precisely f(r) = e^ik.r like in case 3 above.

This solution of the variational minimization of the total energy of the superfluid gave a single-quantum excited state which was the product of the ground state multiplied by the Fourier transform of the density of the superfluid for momentum hbar multiplied by wave number k. The energy of this single excitation was E(k) = hbar^2 k^2/2m S(k), where the "structure factor" S(k) was the ground state expectation value of the product of the Fourier transform of the density with its Hermitian conjugate divided by N. This, in turn, is equal to the spatial Fourier transform of the above conditional probability for a pair of atoms. S(k) is directly measured from the scattering of neutrons and x-rays from the superfluid. Indeed measuring S(k) for x-ray scattering from the DNA molecule led Crick and Watson to the double-helix picture of the universal genetic code at about the same time that Feynman was thinking about superfluid helium. Feynman's great "Aha!" here was that f(r) was the same for all three kinds of excitations. Feynman told Mehra that "I cannot remember exactly how it happened. I was walking along the street ... and zing! I understood it!..."

Feynman knew that lim S(k) -> k for k ->0 , and that lim S(k) -> 1 for k -> infinity. The problem was what happens in between? Feynman at first thought that Landau's 1947 phonon-roton spectrum "was nutty". He had no idea, at first, on what should happen in between. He then realized that S(k) must give something like what Landau proposed in 1947 where the energy of the excitation quanta rises to a max with increasing momentum and then goes through a min. Feynman's reasoning, that he told Merha, was:

"But, of course, that is just what liquids must do. If you measure X-ray diffraction, then because of the spatial structure of the liquid, which is almost like a solid, there will be a maximum corrresponding to the first diffraction ring of the X-ray pattern! That was a terrific moment! It was most interesting. I say this because it illustrates a moment when discovery was made. In a terrific flash of a few seconds, I saw that, since S(k) appeared in the denominator, this peak in S(k) would make a notch in the curve of excitation energy, so that the curve of excitation energy would be linear for low momenta, which would correspond to phonons, and at higher momenta there would be a minimum around some po. I suddenly understood this thing that Landau was talking about." Mehra p. 368

What is the excitation curve for the alleged Frohlich Bose-Einstein condensate in the microtubule infra-structure of the cells in our body?

Is the "terrific moment" of discovery a phase-transition in the brain triggered by an advanced quantum fluctuation from the future modulating the retarded memory quantum information pattern from the past? This is what Fred Hoyle suggests in his book, The Intelligent Universe.

Neutron scattering experiments done a year earlier than Feynman's terrific moment showed the roton energy gap was 18K about twice as big as Landau's original number. The phonon-roton spectrum was a continuous whole. There was no rotational motion in the "roton" which was misnamed. The roton is roughly "a vortex ring of such small radius that only one atoms could pass through its center ... the energy ... is considerable smaller than p^2/2m because there is a correlated motion of many atoms together, sharing the total momentum p .. the effective inertia is close to 2.5m..." (p. 372, Mehra).

The Feynman-Cohen "back-flow".

"The roton picture emerging ... failed to meet the requirement of particle conservation as one attempted to describe the excitation through a wave packet that drifts with a group velocity v = dE/dp but carries a current p/m; the roton at the minimum of the E(p) curve was the extreme example of this, for it drifted with a vanishing group velocity but carried a sizable current with it. Feynman suggested that this difficulty could be resolved if one allowed for a "backflow" of the fluid surrounding the excitation, for that would reduce the current overall and, with proper adjustments, would enable one to satisfy the requirement of conservation. ... this backflow would be bipolar at larger distances from the excitation and ... the coupling of the roton with the backflow would not only lower its energy, thus bringing the parameter delta closer to the Landau value, but would also provide a mechanism for the roton-roton interaction." Mehra p. 371

Feynman and Cohen used a more sophisticated trial wave function for their variational algorithm involving two-atom spatial correlations. But the improvement with experiment was not significant.

The energy spectrum E(p) is an input which allows the computation of thermodynamic properties. Feynman showed how all of this can be derived from a single structure factor S(p). Therefore, it should be possible to measure these excitations independently from estimating them as best fits to macroscopic thermodynamic data. The phonon part of the spectrum is measured directly from the energy distribution of very slow neutrons inelastically, but coherently, scattered from the liquid. If only a single phonon is excited in the inelastic collision, then neutrons scattered at any given angle could only have definite energies.

Note: The angle of scattering for the neutron is @. Conservation of energy and momentum give

E = (pi^2 - pf^2)/2mn

p^2 = pi^2 + pf^2 - 2pipfcos@

So keep pi fixed, vary @, measure pf to compute E(p) directly.

The single phonon excitation will look like a sharp line in the neutron spectrum. Multiple excitations will be in the continuous background. This background is small for neutron waves longer than 4 angstroms at temperatures less than 2K. If T < 1K, Feynman and Cohem found that linewidths of scattered neutrons waere small so one could get good practical resolution of the E(p) spectrum. Let pi and pf be the initial and final momenta of the neutron of mass mn. E and p are the energy and momentum of the phonon excitation in the liquid.

Neutrons from nuclear reactors were used. For example, in the Yarnell data at 1.1K: the speed of sound in superfluid helium liquid was measured at 239 meters/sec (+-5m/s) for p/hbar = 0.55inverse Angstroms. The max of E(p) measured at 13.92K for p/hbar = 1.11 inverse Angstroms (+- 0.02). The min (roton gap) measured to be 8.65K (+- 0.04) at po/hbr = 1.92 +- ).01 inverse Angstroms. The effective mass at the roton min was measured to be only mu = 0.16 m, instead of the expected much larger 2.5m from the back-flow picture. It is odd that Mehra does not comment on this discrepancy between his pages 372 (where 2.5m is cited) and 375 (where 0.16m is cited) for the same physical property. Above p/hbar = 2.18 inverse Angstroms, the spectrum is phonon-like rising linearly with same speed of sound slope at 240 meters/sec. Only the roton min gets a bit smaller at 1.6 and 1.8 K. Also at the high momentum end of 2.4 inverse Angstroms, d^2E/dp^2 changes sign suggesting a secomnd possible max. "The last two features of the spectrum had, in fact been predicted theoretically." (Mehra p. 376). The slope of the roton curve beyond the min at po cannot exceed the speed of sound because the roton would then decay to a roton and a phonon. The spectrum should end at pc with Ec = 2delta and dE/dp|c = 0 because that is where a single roton decays to two lower energy rotons.

Superflow.

Below the lambda point of 2.19K, there are two different viscosities (only one above lambda point). The viscosity measured from a pair of coaxial cylinders in relative rotation varies smoothly wit temperature T right through the lamba point. The viscosity of flow through narrow tubes depend sensitively on both T and the flow speed v below the lambda point. While above the lambda point this flow viscosity is the same as the one for the coaxial cylinders. The critical velocity is larger for tubes of smaller diameter. For diameters 10^-3 - 10^-5 cm the critical speed, above which there is viscosity, below which there is frictionless superflow, in in the range 2 - 20 cm/sec. The two fluid model is a non-fundamental phenomemological explanation. The rotating cyclinders measure the normal fluid, the thin tubes measure the superfluid flow. The critical speed in thin tubes (capillaries) marks the onset of breakdown of the superfluid. What is causing this breakdown?

Superflow is irrotational potential flow with vanishing curl of the velocity field. The superfluid slips past the walls of the thin tube. Landau showed that there is a minimum critical speed needed to make an excitation of the E(p) = p^2/2mS(p) spectrum derived from first principles by Feynman. Landau's argument is that the mass M of superfluid at velocity v must slow down as a whole to maintain the irrotational curl-free flow. The change in energy dE = Mv.dv = vdp. This creates an excitation of energy e and momentum p. Conservation of energy and momentum gives e = vpcos@ < vp, where @ is the angle between p of the excitation and v of the mass M of superfluid. That is, to create an excitation of any type, the velocity of the superfluid must be at least equal to the minimum phase speed e/p of the excitation spectrum. But these numbers were way too large to account for the experimental data, e.g., 240 m/s for phonons, 60-70 m/s for rotons compared to observed 2 - 20 cm/s. So something new was happening! Also Landau's idea could not explain the dependence of the critical speed on the diameter of the tube.

Feynman (1955) explained the critical speed for superflow as the creation of at least mesoscopic if not macroscopic quantized vortices. Onsager had already conjectured quantized vortex sheets not lines. Small parts of the fluid do in fact slow down. Landau's model does not work. The flow cannot remain curl-free everywhere. A state of uniform flow is the ground state multiplies by a phase factor e^i& where & = mv sum of ri/hbar. Feynman generalized this to nonuniform flow. He then considered a ring of He atoms in the liquid. The atoms moved over to their successive neigbors on the ring. By Bose-Einstein statistics the wavefunction must return to its original value which means a phase change of 2piN where N is an integer. The result is a quantized vortex flow of circulation Nh. The vortex lines are singularities in the velocity field. The energy per unit length of a vortex line is mean mass density multiplied by [(Nh/m)^2/4pi] ln(b/a) where b is of the order of the diameter of the tube and a is the size of the vortex core where there is nonvanishing curl in the flow.

Feynman modeled the creation of vortices when superfluid comes out of an orifice. The vortex lines are perpendiculat to the flow lines. Feynman computed a critical speed of order vc = (h/md)ln(d/a), where d is the diameter of the orifice. But this was 1 m/s still much to high compared to experiment. But it was lower than Landau's number based on the phonon-roton E(p) spectrum. The better picture is the creation of vortex rings rather than vortex lines.

Paradox of rotating buckets of superfluid.

Rotate an ordinary liquid uniformly in a bucket. The shape of the free liquid surface is parabolic from the gravity and the centrifugal force with curvature omega^2/2g, where omega is the rotational rate and g is the acceleration of gravity. The two-fluid model predicted a different reult that would depend on temperature since only the normal component should have any viscosity like the ordinary liquid. In fact, the rotating bucket of superfluid has the same parabolic profile for the free surface as does the ordinary liquid! Why is the zero viscosity superfluid also rotating? Feynman suggested a uniform distribution of quantized vortex lines parallel to the axis of rotation of the bucket like the magnetic field inside a long straight coil. The number density of lines was 2m omega/h per unit area of the free surface. The mean spacing between the lines was b = (h/m omega)^1/2 -- a fraction of a milimeter in real experiments. The energy needed to generate these vortices was only 1% of the energy of the equivalent rotating solid. Different experiments on second sound attenuation and dc second sound pulses designed before Feynman's theory confirmed Feynman's picture here. Vinen in a third experiment used vibrating wires to actually detect the Feynman vortices in rotating superfluid helium. Later Whitmore and Zimmerman observed stable Feynman line vortices in rotating "buckets" for N = 1, 2, 3.

The velocity field of a vortex ring is very different from that of a vortex line. The ring of radius r moves in a direction perpendicular to the plane of its ring with speed v = h/mr. The energy e of the ring is

e = 2pi^2hbar^2(number density/mass of atom)r ln(r/a)

the momentum is

p = 2pi^2hbar (number density)r^2

Note that the e(p) curve is qualitatively p^1/2 ln(p) so that the group speed of energy propagation is de/dp or order 1/p^1/2, so that the ring slows down as the ring radius gets larger.

The speed of the ring is v = (h/2mr)ln(r/a).

Note the vortex-ring critical speed (h/mR)ln(R/a) still gives too big a number of 1 m/s. But Fetter shows how image vortex rings form for the large rings at the inner walls of the tube. Fetter's image correction gives vc = (1/8)(h/mR)ln(R/a) which agrees much closer with actual experiment! In 1964 Rayfield and Reif showed the reality of Feynman vortex rings in narrow tubes using by putting an electric charge on them with He ions.

Feynman uses quantum probability amplitudes A rather than wave functions psi. Both are complex numbers of the form z = re^i&. The amplitude is more general than the wave function. There is always an amplitude. There is not always a wave function. Bohm's quantum force pilot-wave theory is for wave functions and it must be generalized to amplitudes. Amplitudes are global defined in terms of the world lines of particles in spacetime. Wave functions are defined on constant time (i.e., space-like slices) of spacetime. One can generalize the ideas of amplitudes and wave functions to other spaces like configuration space. The wave function psi(x,t) at a given point event (x,t) in configuration space x at time t is a sum of all contributions of retarded amplitudes from all paths that come from the past and end at the given point event. Note that x = (r1, r2, ... rN) for N "atoms". The complex-conjugate wave function psi*(x,t) is a sum of all contributions of advanced amplitudes from those same paths that go backward in time starting from the same given point. The Born probability density psi*(x,t)psi(x,t) is the set of loops in time formed by the modulation of the retarded amplitudes by their advanced time-reversed amplitudes. Imagine a second set of advanced paths from the future ending at (x,t). This second set of advanced paths multiplies the original set of retarded paths in a sum over all x for the same t to get the Feynman "transition amplitude" to quantum leap from a state psi in the past, to a new state chi* in the future. That is, <chi|psi> = integral chi*(x,t)psi(x,t)dx = <psi|chi>*.

Quantum statistics, whether Fermi-Dirac, for spin ½ "atoms", or Bose-Einstein, for spin 0 and spin 1 "atoms" (He4 has spin 0, He3 has spin 1/2), come about because of special kinds of Einstein-Podolsky-Rosen nonlocal quantum correlations in entangled many-atom amplitudes that form irreducible representations of the finite group of permutations of the N atoms. These group representations are called Young patterns which can be pictured as arrays of square boxes drawn on a sheet of paper.

Fermi-Dirac quantum statistics for N fermions has a quantum amplitude that is totally antisymmetric under the permutation exchange of any two atoms. This means that the many-atom amplitude changes its sign under each pair permutation. Lets take the simple case for N = 2 "atoms". The Feynman amplitude for two identical fermion is FD(1,2), where FD(2,1) = - FD(1,2). In particular, if x1 = x2 = x at the same time t, then FD(x,x,t) = - FD(x,x,t) = 0, which is the Pauli exclusion principle that goes a very long way in explaining the stability and diversity of ordinary matter. The Young pattern in this case is the single column of two boxes. Similarly, the Young pattern for two bosons is a single row of two boxes which has a symmetric amplitude in which BE(1,2) = BE(2,1). The formal exchange in the entangled many-atom amplitude results in a nonlocal repulsive context-dependent quantum force between two identical fermions, and an attractive quantum force between two identical bosons even when their wave packets do not overlap in space. This Bohm quantum force is not exactly the same as the Coulomb exchange energy in ferromagnets, for example, which does require a spatial overlap of the wave functions of the different identical particles.

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