New techniques, both for generating and detecting turbulence in the helium superfluids 3He-B and 4He, have recently given insight in how turbulence is started, what the dissipation mechanisms are, and how turbulence decays when it appears as a transient state or when externally applied turbulent pumping is switched off. Important simplifications are obtained by using 3He-B as working fluid, where the highly viscous normal component is practically always in a state of laminar flow, or by cooling 4He to low temperatures where the normal fraction becomes vanishingly small. We describe recent studies from the low temperature regime, where mutual friction becomes small or practically vanishes. This allows us to elucidate the mechanisms at work in quantum turbulence on approaching the zero temperature limit.
Superfluid $T_c$ of liquid helium-3 and its pressure dependence are calculated by using a relation obtained from our macro-orbital microscopic theory. The results agree closely with experiments. This underlines the accuracy of our relation and its potential to provide superfluid $T_c$ of electron fluid in widely different superconductors and renders experimental foundation to our conclusion related to the basic factors responsible for the formation of (q, -q) bound pairs of fermions and the onset of superfluidity in a fermionic system.
We present measurements of how a single vortex line in superfluid helium interacts with a macroscopic bump on the chamber wall. At a general level our measurements confirm computational work on vortex pinning by a hemispherical bump, but not all the details agree. Rather than observing a unique pin location, we find that a given applied velocity field can support pinning at multiple sites along the bump, both near its apex and near its edge. We also find that pinning is less favorable than expected. A vortex can pass near or even traverse the bump itself with or without pinning, depending on i…
When two vortices cross, each of them breaks into two parts and exchanges part of itself for part of the other. This process, called vortex reconnection, occurs in classical as well as superfluids, and in magnetized plasmas and superconductors. We present the first experimental observations of reconnection between quantized vortices in superfluid helium. We do so by imaging micron-sized solid hydrogen particles trapped on quantized vortex cores (Bewley GP, Lathrop DP, Sreenivasan KR, 2006, Nature, 441:588), and by inferring the occurrence of reconnection from the motions of groups of recoiling…
We present measurements on both energy loss and pinning for a vortex terminating on the curved surface of a cylindrical container. We vary surface roughness, cell diameter, fluid velocity, and temperature. Although energy loss and pinning both arise from interactions between the vortex and the surface, their dependences on the experimental parameters differ, suggesting that different mechanisms govern the two effects. We propose that the energy loss stems from reconnections with a mesh of microscopic vortices that covers the cell wall, while pinning is dominated by other influences such as the…
An equation previously proposed to describe the evolution of vortex line density in rotating counterflow turbulent tangles in superfluid helium is generalized to incorporate nonvanishing barycentric velocity and velocity gradients. Our generalization is compared with an analogous approach proposed by Lipniacki, and with experimental results by Swanson et al. in rotating counterflow, and it is used to evaluate the vortex density in plane Couette and Poiseuille flows of superfluid helium.
Formation of fourth-sound shock waves in narrow channels filled with superfluid helium is studied. Physical and mathematical conditions at the surface of discontinuity are established. These conditions differ somewhat from those in case of first- and second-sound waves. The velocity of discontinuity coincides with that of fourth sound. The jumps of temperature and the superfluid velocity are shown to be of the first order as to the pressure jumps.
A new method of obtaining the interaction Hamiltonian of phonons at superfluid helium-solid interface is proposed in the work. Equations of hydrodynamic variables are obtained in terms of second quantization if helium occupies a half-space. The contributions of all processes to the heat flux from solid to superfluid helium are calculated based on the obtained Hamiltonian. The angular distribution of phonons emitted by a solid is found in different processes. It is shown that all the exit angles of superfuild helium phonons are allowed. The obtained results are compared with experimental data a…
The mechanism of the roughening induced partial depinning of gliding dislocations from Helium-3 impurities is proposed as an alternative to the standard "boiling off". We give a strong argument that Helium-3 remains bound to dislocations even at large temperatures due to very long equilibration times. A scenario leading to the similarity between elastic and superfluid responses of solid Helium-4 is also discussed. Its main ingredient is a strong suppression of the superfluidity along dislocation cores by dislocation kinks (D. Aleinikava, et. al., arXiv:0812.0983). These kinks, on one hand, det…
It is shown that the critical velocity of a small sphere oscillating in superfluid helium increases with the square root of the oscillation frequency. This behavior can be described by a simple dimensional argument. The size of the sphere and the temperature of the superfluid are found to have no or only very little effect. Surface properties of the sphere and remanent vorticity may have an influence but have not been under systematic investigation in these measurements
We proceed from the premise that the spectrum of elementary excitations in the normal component in Landau's theory of superfluidity should depend on the superfluid helium temperature. This leads to generalization of the Landau superfluidity criterion. On this basis, taking into account available experimental data on inelastic neutron scattering, it is shown that, in addition to phonon--roton excitations, there is one more type of elementary excitations in superfluid helium, which we called helons. The energy spectrum with such a momentum dependence was first proposed by Landau. The helon energ…
Generation of a quasi-stationary flow of the superfluid helium normal part in the presence of intense first- and second-sound waves is studied. Relevant equations are obtained. The contribution to the process of energy dissipation at the shock front layer and of fluid viscosity is analysed in detail for the case of a second-sound wave. An estimate concerning possible experimental observation of the process is made.
Tungsten filaments used as sources of electrons in a low temperature liquid or gaseous helium environment have remarkable properties of operating at thousands of degrees Kelvin in surroundings at temperatures of order 1 K. We provide an explanation of this performance in terms of important changes in the thermal transport mechanisms. The behavior can be cast as a first-order phase transition.
An analysis of experimental data shows that, in addition to phonon--roton excitations in superfluid helium, there necessarily exist at least one branch of elementary excitations whose energy spectrum strongly depends on temperature. On this basis, the Landau superfluidity criterion is generalized for several branches of elementary excitations, taking into account that the critical velocity should vanish during the phase transition of liquid helium from the superfluid state to the normal state. We suppose that a new spectrum of excitations with a gap, depending on interparticle interaction, cor…
The superfluid Reynolds number $Re_s = (v - v_c) \,D/ κ$ can be expressed simply by the number of vortex rings that are shed during a half-period of the oscillation.
Nearly five decades have elapsed since the seminal 1966 paper of P.W. Anderson on the flow of superfluid helium, $^4$He at that time. Some of his "Considerations" -- the role of the quantum phase as a dynamical variable, the interplay between the motion of quantised vortices and potential superflow, its incidence on dissipation in the superfluid and the appearance of critical velocities, the quest for the hydrodynamic analogues of the Josephson effects in helium -- and the way they have evolved over the past half-century are recounted below. But it is due to key advances on the experimental fr…
We explore the possibility that the fast and exotic negative ions in superfluid helium are electrons bound to quantized vortex structures, the simplest being a ring. In the states we consider, the electron energy is only slightly below the conduction band minimum of bulk helium. To support our proposal we present two calculations. In the first, we show that the electron pressure on the vortex core is insufficient to cavitate the helium and form an electron bubble. In the second, we estimate the equilibrium radius of the vortex ring that would bind an electron and find it is much smaller than t…
System of two-fluid hydrodynamics of superfluid helium with the account of electric field is obtained. These equations are obtained in kinetic approach using quasi-equilibrium distribution function of quasi-particles, which vanishs collision integral of quasi-particles, and contains dependence on electric field by means of phenomenological parameter α. Using experimental data at temperature range of 1,4 - 2 K, where basic role plays roton hydrodynamics, the value of phenomenological parameter, is obtained.
A complete study of the propagation of waves (namely longitudinal density and temperature waves, longitudinal and transversal velocity waves and heat waves) in turbulent superfluid helium is made in three situations: a rotating frame, a thermal counterflow, and the simultaneous combination of thermal counterflow and rotation. Our analysis aims to obtain as much as possible information on the tangle of quantized vortices from the wave speed and attenuation factor of these different waves, depending on their relative direction of propagation with respect to the rotation vector.
The fact that superfluid helium always leaks out of an open container is usually explained by the phenomenon of wetting. In the present paper it is demonstrated that this explanation is unconvincing. The fact can be readily explained from the viewpoint of the interpretation of superfluidity proposed earlier by the author according to which superfluidity is an equilibrium state of liquid helium where the symmetry is spontaneously broken because of an intrinsic superflow. Experiments on the thickness of moving helium films that have given rise to much controversy are discussed as well. Some expe…
Using an optical interferomertric method, the homogeneous cavitation density of superfluid helium at $T=0.96\,$K is measured and found to be $ρ_{cav}=0.1338\pm0.0002\rm\,g.cm^{-3}$. A well established equation of state for liquid helium at negative pressures converts this to the cavitation pressure $P_{cav}=-5.1\pm0.1\,$bar. This cavitation pressure is consistent with a model taking into account the presence of quantized vortices, but disagrees with previously published experimental values of $P_{cav}$.
Recent experiments on quantum turbulence in superfluid helium made use of small tracer particles to track the motion of quantized vortices, determine velocity statistics and visualize vortex reconnections. A problem with this visualization technique is that it may change the turbulent flow which it visualizes. To address this problem, we derive and solve the equations of motion of a quantized vortex ring which contains a given number of particles trapped on the vortex core, hence derive a quantitative criterion to determine in which measure small particles trapped in quantized vortices affect …
This article reviews recent developments in quantum fluid dynamics and quantum turbulence (QT) for superfluid helium and atomic Bose-Einstein condensates. Quantum turbulence was discovered in superfluid $^4$He in the 1950s, but the field moved in a new direction starting around the mid 1990s. Quantum turbulence is comprised of quantized vortices that are definite topological defects arising from the order parameter appearing in Bose-Einstein condensation. Hence QT is expected to yield a simpler model of turbulence than does conventional turbulence. A general introduction to this issue and a br…
The flow phase diagram for He II and $^3$He-B is established and discussed based on available experimental data and the theory of Volovik [JETP Letters {\bf{78}} (2003) 553]. The effective temperature - dependent but scale - independent Reynolds number $Re_{eff}=1/q=(1+α')/α$, where $α$ and $α'$ are the mutual friction parameters and the superfluid Reynolds number characterizing the circulation of the superfluid component in units of the circulation quantum are used as the dynamic parameters. In particular, the flow diagram allows identification of experimentally observed turbulent states I an…
In this paper we propose a thermodynamically consistent model for superfluid-normal phase transition in liquid helium, accounting for variations of temperature and density. The phase transition is described by means of an order parameter, according to the Ginzburg-Landau theory, emphasizing the analogies between superfluidity and superconductivity. The normal component of the velocity is assumed to be compressible and the usual phase diagram of liquid helium is recovered. Moreover, the continuity equation leads to a dependence between density and temperature in agreement with the experimental …
Unprecedented transport efficiency is demonstrated for electrons on the surface of micron-scale superfluid helium filled channels by co-opting silicon processing technology to construct the equivalent of a charge-coupled device (CCD). Strong fringing fields lead to undetectably rare transfer failures after over a billion cycles in two dimensions. This extremely efficient transport is measured in 120 channels simultaneously with packets of up to 20 electrons, and down to singly occupied pixels. These results point the way towards the large scale transport of either computational qubits or elect…
The irrotational nature of superfluid helium was discovered through its decoupling from the container under rotation. Similarly, the resonant period drop of a torsional oscillator (TO) containing solid helium was first interpreted as the decoupling of solid from the TO and appearance of supersolid. However, the resonant period can be changed by mechanisms other than supersolid, such as the elastic stiffening of solid helium that is widely accepted as the reason for the TO response. To demonstrate the irrotational nature more directly, the previous experiments superimposed the dc rotation onto …
Electrons floating on the surface of liquid helium are possible spin-qubits for quantum information processing. Varying electric potentials are not expected to modify spin states, which allows their transport on helium using a charge-coupled device (CCD)-like array of underlying gates. This approach depends upon efficient inter-gate transfer of individual electrons. Measurements are presented here of the charge transfer efficiency (CTE) of few electrons clocked back and forth above a short microscopic CCD-like structure. A charge transfer efficiency of 0.99999992 is obtained for a clocking fre…
We find the pressure, due to the thermal excitations of superfluid helium, at the interface with a solid. The separate contributions of phonons, $R^-$ rotons and $R^+$ rotons are derived. The pressure due to $R^-$ rotons is shown to be negative and partially compensates the positive contribution of $R^+$ rotons, so the total roton pressure is positive but several times less than the separate $R^-$ and $R^+$ roton contributions. The pressure of the quasiparticle gas is shown to account for the fountain effect in $HeI I$. An experiment is proposed to observe the negative pressure due to $R^-$ ro…
By large-scale quantum Monte Carlo simulations we show that grain boundaries in Helium-4 crystals are generically superfluid at low temperature, with a transition temperature of the order of ~0.5K at the melting pressure; insulating grain boundaries are found only for special orientations of the grains. We also find that close vicinity to the melting line is not a necessary condition for superfluid grain boundaries, and a grain boundary in direct contact with the superfluid liquid at the melting curve is found to be mechanically stable and the grain boundary superfluidity observed by Sasaki et…
We demonstrate the trapping of millimeter-scale superfluid Helium drops in high vacuum. The drops are sufficiently isolated that they remain trapped indefinitely, cool by evaporation to 330 mK, and exhibit mechanical damping that is limited by internal processes. The drops are also shown to host optical whispering gallery modes. The approach described here combines the advantages of multiple techniques, and should offer access to new experimental regimes of cold chemistry, superfluid physics, and optomechanics.
We perform fully coupled two--dimensional numerical simulations of plane channel helium II counterflows with vortex--line density typical of experiments. The main features of our approach are the inclusion of the back reaction of the superfluid vortices on the normal fluid and the presence of solid boundaries. Despite the reduced dimensionality, our model is realistic enough to reproduce vortex density distributions across the channel recently calculated in three--dimensions. We focus on the coarse--grained superfluid and normal fluid velocity profiles, recovering the normal fluid profile rece…
We argue that classical transitions can be the key to explaining the long standing puzzle of the fast A-B phase transition observed in superfluid Helium 3 while standard theory expects it to be unobservably slow. Collisions between domain walls are shown to be capable of reaching phases inaccessible through homogenous nucleation on the measured timescales. We demonstrate qualitative agreements with prior observations and provide a definite, distinctive prediction that could be verified through future experiments or, perhaps, a specific analysis of existing data.
We have developed a two-dimensional model of quantised vortices in helium II moving under the influence of applied normal fluid and superfluid in a counterflow channel. We predict superfluid and vortex-line density profiles which could be experimentally tested using recently developed visualization techniques.
The quasi-one-dimensional system of surface electrons over superfluid helium with negative charge on substrate is investigated. The temperature dependence of conductivity is ladder-like which is observed at temperature lower 1.3 K. The parameters of ladder have been different in different experiments. The observed dependence isn't explained by classical models of carry with account to the usual scatters in channel. It is supposed the dependence have been connected with quantum character of electron transport in narrow channels.
We report the results of high frequency acoustic shear impedance measurement on superfluid helium-3 confined in 98% porosity silica aerogel. Using 8.69 MHz continuous wave excitation, we measured the acoustic shear impedance as a function of temperature for the sample pressure of 28.4 and 33.5 bar. We observed the A-B transition on warming in zero magnetic field. Our observations show that the A and B phases in aerogel coexist in a temperature range of about 100 micro-Kelvin in width. We propose that the difference in the relative stability of the A and B phases, arising from anisotropic scatt…
We present an experimental and theoretical study of the 2D dynamics of electrically charged nanoparticles trapped under a free surface of superfluid helium in a static vertical electric field. We focus on the dynamics of particles driven by the interaction with quantized vortices terminating at the free surface. We identify two types of particle trajectories and the associated vortex structures: vertical linear vortices pinned at the bottom of the container and half-ring vortices travelling along the free surface of the liquid.
We comment on the paper by M. Sýkora, M. Pavelka, M. La Mantia, D. Jou, and M. Grmela "On the relations between large-scale models of superfluid helium-4," Physics of Fluids, 33(12):127124(2021), where the authors have developed a formalism for describing a coarse-grained flow of superfluid helium. This formalism is greatly based on the Hall-Vinen-Bekarevich-Khalatnikov (HVBK) model. We strongly disagree with the use of the HVBK equation approach for the case of the three-dimensional quantum turbulence and expose our objections in this comment. We discuss the HVBK method and also criticize the…
Electrons floating on the surface of liquid helium are possible qubits for quantum information processing. Varying electric potentials do not modify spin states, which allows their transport on helium using a charge-coupled device (CCD)-like array of underlying gates. This scheme depends upon efficient inter-gate electron transfer and on the absence of electron traps. We will present a measurement of the charge transfer efficiency (CTE) of electrons clocked back and forth above a short CCD-like structure. The CTE obtained at low clocking frequencies is 0.999 with an electron density of about 4…
We have studied spinning superfluid $^4$He nanodroplets at zero temperature using Density Functional theory. Due to the irrotational character of the superfluid flow, the shapes of the spinning nanodroplets are very different from those of a viscous normal fluid drop in steady rotation. We show that when vortices are nucleated inside the superfluid droplets, their morphology, which evolves from axisymmetric oblate to triaxial prolate to two-lobed shapes, is in good agreement with experiments. The presence of vortex arrays confers to the superfluid droplets the rigid-body behavior of a normal f…
Preparing and detecting spatially extended quantum superpositions of a massive object comprises an important fundamental test of quantum theory. These quantum states are extremely fragile and tend to quickly decay into incoherent mixtures due to the environmental decoherence. Experimental setups considered up to date address this threat in a conceptually straightforward way -- by eliminating the environment, i.e. by isolating an object in a sufficiently high vacuum. We show that another option exists: decoherence is suppressed in the presence of a strongly interacting environment if this envir…
Building on a general variational framework for multi-fluid dynamics, we discuss finite temperature effects in superfluids. The main aim is to provide insight into the modelling of more complex finite temperature superfluid systems, like the mixed neutron superfluid/proton superconductor that is expected in the outer core of a neutron star. Our final results can also (to a certain extent) be used to describe colour-flavour locked quark superconductors that may be present at the extreme densities in the deep neutron star core. As a demonstration of the validity of the model, which is based on t…
We solve the problem of beams of phonons and rotons incident on, and interacting with, solid surfaces. Phonons and rotons are the quasiparticles of superfluid helium and have a unique dispersion curve. The dispersion curve controls the transmission, reflection and mode change of these quasiparticles at the interface with another medium. We develop a non-local hydrodynamic theory in a consistent and unified way. The structure of the solutions in the quantum fluid is discussed. The creation probabilities of all quasiparticles are derived when any one of them is incident on the interface. The dep…
Based on electronic structure calculations, the structure of intrinsic positive charge solvated in superfluid helium is identified as triatomic He3^+ ion, which is bound to the surrounding ground state helium atoms through the charge - charge induced dipole interaction in a pairwise additive manner. Bosonic density functional theory calculations show that this ion forms the well-known Atkins' snowball solvation structure where the first rigid helium shell is effectively disconnected from the rest of the liquid. Evaluation of the total energy vs. helium droplet size N shows distinct regions rel…
We present a systematic study on the excited electron-bubble states in superfluid helium-4 using a time-dependent density functional approach. For the evolution of the 1P bubble state, two different functionals accompanied with two different time-development schemes are used, namely an accurate finite-range functional for helium with an adiabatic approximation for electron versus an efficient zero-range functional for helium with a real-time evolution for electron. We make a detailed comparison between the quantitative results obtained from the two methods, which allows us to employ with confi…
The thermo-mechanical effect in superfluid helium is used to create an initial chemical potential difference, $Δμ_0$, across a solid $^4$He sample. This $Δμ_0$ causes a flow of helium atoms from one reservoir filled with superfluid helium, through a sample cell filled with solid helium, to another superfluid-filled reservoir until chemical potential equilibrium is restored. The solid helium sample is separated from each of the reservoirs by Vycor rods that allow only the superfluid component to flow. With an improved technique, measurements of the flow, $F$, at several fixed solid helium tempe…
An equilibrium multielectron bubble in liquid helium is a fascinating object with a spherical two-dimensional electron gas on its surface. We describe two ways of creating them. MEBs have been observed in the dome of a cylindrical cell with an unexpectedly short lifetime; we show analytically why these MEBs can discharge by tunneling. Using a novel method, MEBs have been extracted from a vapor sheath around a hot filament in superfluid helium by applying electric fields up to 15 kV/cm, and photographed with high-speed video. Charges as high as 1.6x10-9 C (~1010 electrons) have been measured. T…
In the previous paper (Attard 2022d)on the fountain pressure in superfluid helium, it was shown that the experimentally confirmed expression of H. London (1939) was thermodynamically equivalent to equality of chemical potential. However this theoretical equivalence was not reflected in the experimental data. The problem has now been traced to errors in the enthalpy and entropy derived from the measured heat capacity by Donnelly and Barenghi (1998). In this paper the corrected thermodynamic data yields almost exact agreement between the two expressions and the measured fountain pressure. A phys…
Heat is carried in superfluid He-4 by the motion of the normal fluid$^{1}$, a counterflowing superfluid component serving to eliminate any net mass flow. It has been known for many years that above a critical heat current the superfluid component in this counterflow becomes turbulent. This turbulence takes the form of a disorganized tangle of quantized vortex lines and is maintained by the relative motion of the two fluids$^{2-3}$. It has been suspected that the normal fluid may also become turbulent$^{4}$, but experimental verification is difficult without a technique for visualizing the flow…
The flow of liquid helium through a single nanohole with radius smaller than 25 nm was studied for the first time. Mass flow was induced by applying a pressure difference of up to 1.4 bar across a 50 nm thick Si3N4 membrane and was measured directly by means of mass spectrometry. In liquid He I, we experimentally show that the fluid is not clamped by the short pipe with diameter-to-length ratio D/L~1, despite the small diameter of the nanohole. This viscous flow is quantitatively understood by making use of a model of flow in short pipes. In liquid He II, a two-fluid model for mass flow is use…
We show that a two-excitation process in superfluid helium, combined with sensitivity to meV energy depositions, can probe dark matter down to the ~keV warm dark matter mass limit. This mass reach is three orders of magnitude below what can be probed with ordinary nuclear recoils in helium at the same energy resolution. For dark matter lighter than $\sim 100$ keV, the kinematics of the process requires the two athermal excitations to have nearly equal and opposite momentum, potentially providing a built-in coincidence mechanism for controlling backgrounds.
An energetic electron passing through liquid helium causes ionization along its track. The ionized electrons quickly recombine with the resulting positive ions, which leads to the production of prompt scintillation light. By applying appropriate electric fields, some of the ionized electrons can be separated from their parent ions. The fraction of the ionized electrons extracted in a given applied field depends on the separation distance between the electrons and the ions. We report the determination of the mean electron-ion separation distance for charge pairs produced along the tracks of bet…
We derive an analytical expression for a critical temperature of spontaneous symmetry breaking in a repulsive hard-core interacting Bose system. We show that the critical temperature of spontaneous symmetry breaking in a hard-core interacting Bose system is determined by the three physical parameters: the density of Bose liquid at absolute zero (rho_0), the mass (m) and the hard sphere diameter (sigma) of a boson. The formula that we have derived is Tc = rho_0 pi hbar^2 sigma / m^2 kB. We report Tc of liquid helium-4 is 2.194 K, which is significantly close to the lambda temperature of 2.1768 …
We report a counter-intuitive self-heating effect of helium-4 superflow. This fundamentally unusual heating effect bears a phenomenological resemblance to the Peltier effect of electric current across two different conductors. It reveals that helium-4 superflow carries thermal energy and entropy, which is in contrast to the two-fluid model of superfluid helium-4. A natural understanding of this heating effect is provided by a recently developed microscopic quantum theory of superfluid helium-4.
Two different microscopic phases have been observed via inelastic neutron scattering for superfluid helium confined within highly monodisperse, smooth, and unidimensional silica pores. When the helium forms a thin film on the pore walls, it supports a dramatically modified phonon-roton spectrum as well as a 2D surface roton. The energies of these modified phonon-roton modes are consistent with predictions for a dilute, low-density film, while the energy of the 2D surface roton corresponds to that of a dense film. The relatively smooth and weak surface potential of the substrate permits the for…
Immersed mechanical resonators are well suited for probing the properties of fluids, since the surrounding environment influences the resonant characteristics of such oscillators in several ways. Quartz tuning forks have gained much popularity in recent years as the resonators of choice for studies of liquid helium. They have many superior properties when compared to other oscillating bodies conventionally used for this purpose, such as vibrating wires. However, the intricate geometry of a tuning fork represents a challenge for analyzing their behavior in a fluid environment - analytical appro…
The specific heat of liquid helium was calculated theoretically in the Landau theory. The results deviate from experimental data in the temperature region of 1.3 - 2.1 K. Many theorists subsequently improved the results of the Landau theory by applying temperature dependence of the elementary excitation energy. As well known, many-body system has a total energy of Galilean covariant form. Therefore, the total energy of liquid helium has a nonlinear form for the number distribution function. The function form can be determined using the excitation energy at zero temperature and the latent heat …
