Critical angular velocity for vortex lines formation
aa r X i v : . [ c ond - m a t . o t h e r] J un Critical angular velocity for vortex lines formation
Enore Guadagnini
Dipartimento di Fisica
E. Fermi dell’Universit`a di Pisa,and INFN Sezione di Pisa,Largo B. Pontecorvo 3, 56127 Pisa, Italy.
Abstract
For helium II inside a rotating cylinder, it is proposed that the formation of vortex linesof the frictionless superfluid component of the liquid is caused by the presence of the ro-tating quasi-particles gas. By minimising the free energy of the system, the critical value Ω of the angular velocity for the formation of the first vortex line is determined. Thisvalue nontrivially depends on the temperature, and numerical estimations of its tempera-ture behaviour are produced. It is shown that the latent heat for a vortex formation andthe associated discontinuous change in the angular momentum of the quasi-particles gasdetermine the slope of Ω ( T ) via some kind of Clapeyron equation. The formation of vortex lines, with quantized vorticity, in helium II can be understood as amacroscopic quantum mechanics effect [1, 2]. Vortex lines in helium II have been observed inrotating containers [3, 4, 5, 6, 7] and, by means of an analysis on friction and drag on quantisedvortices [8, 9], some of their phenomenological coefficients have been deduced. The forma-tion of vortices has been studied [10] also in the case of freely rotating fluid drops of heliumII. Discussions on the possible connections of the quantised vortex dynamics with superfluidturbulence can be found for instance in [11] and in the articles collection of Ref.[12].The emergence of vortex lines is a common feature of quantum liquids. Vortices have animportant impact on high-temperature superconductors [13] and have been observed [14, 15,16] in rotating superfluid He. Arrays of vortex lines have also been observed [17, 18, 19, 20,21] and studied [22, 23, 24, 25, 26, 27, 28] in Bose-Einstein condensates of cold atoms.In order to investigate the mechanism of vortex formation, in the present article the cominginto being of a single vortex line in helium superfluid, which is contained in a rotating cylinder,is considered. The determination of the critical value Ω of angular velocity at which the first1ortex line appears is discussed. Since the superfluid component of liquid helium has no viscos-ity and its dynamics is unaffected by the moving walls of the container, it is proposed that thecreation of a vortex line is induced by the rotating quasi-particles gas. Minimization of the freeenergy of the system is used to derive the value of Ω , which turns out to depend nontrivially onthe temperature. The angular momentum and the thermodynamic variables which characterisethe formation of the vortex line are examined, and their statistical mechanics expressions aredetermined. It is shown that the latent heat for a vortex formation and the associated discontin-uous change in the angular momentum of the quasi-particles gas determine the slope of Ω ( T ) via some kind of Clapeyron equation.In order to make this article self-contained, a few basic definitions of helium superfluid andthe main properties of the quasi-particles are briefly recalled in Section 2, where a derivationof the densities of the thermodynamic potentials energy, free energy and entropy is presented.The deduction of the critical value of the angular velocity for the formation of a vortex line iscontained in Section 3. In Section 4 it is shown that the latent heat of vortex formation and thediscontinuous change in the angular momentum of the quasi-particles gas determine the slopeof the curve Ω( T ) by means of an equation of the Clapeyron type. During the vortex formation,the changes in entropy and in angular momentum are computed. Finally, numerical estimationsof the temperature dependence of Ω( T ) are reported. The main conclusions are collected inSection 5. When the value of the temperature is below the critical value T ≃ . K corresponding to the λ -point, at ordinary pressure the behaviour of helium He is similar (but it is not equal) to thebehaviour of a two-components liquid in which • one component, which has velocity v s and mass density ρ s , corresponds to the so-calledsuperfluid motion; this fluid component has no viscosity and carries zero entropy; • the second component, with velocity v n and mass density ρ n , corresponds to the normalmotion and behaves as a normal viscous fluid.This peculiar quantum liquid can be described [29, 30, 31, 32, 33] by means of a gas of quasi-particles, which represent the localized energy fluctuations of the system above its ground state,and by means of additional degrees of freedom which are related with the global (zero entropy)motion of the ground state wave function, that will simply be called the global motion of thecondensate with macroscopic velocity v s . The mass density of the liquid helium is given by ρ = ρ s + ρ n , (2.1)and the momentum density is written as P /V = ρ s v s + ρ n v n . (2.2)Let us consider an inertial reference system. When the condensate is at rest ( v s = 0 ) and when v n = 0 , the dependence of the energy ε of a single quasi-particle on its momentum p is given2y the energy spectrum ε ( p ) , where p = | p | . For small momenta, the function ε ( p ) has a typicallinear behaviour, ε ≃ up , where u denotes the speed of first sound. In a neighbourhood of p ≃ × ~ / cm, the function ε ( p ) has a deep local minimum and it can be approximatedas ε ( p ) ≃ ∆ + ( p − p ) / m ∗ . Quasi-particles obey the Bose-Einstein statistics and the quasi-particles gas has vanishing chemical potential. The hydrodynamic motions of the superfluid component and of the normal component of theliquid appear to be essentially independent, apart from a modification of the energy spectrum ofthe quasi-particles which takes place when the relative velocity v = v n − v s is not vanishing. Letus consider a small portion of the liquid with well-defined thermodynamics variables and givenvelocities v s and v n . When v = v n − v s = 0 , the energy spectrum of a single quasi-particle(which belongs to this part of the liquid) with momentum p is given by E v ( p ) = ε ( p ) − vp = ε ( p ) − ( v n − v s ) p . (2.3)This equation is a consequence [29] of the nonrelativistic transformation properties of energyand momentum under a change of an inertial reference system into another inertial referencesystem. The peculiar form (2.3) of the energy spectrum is responsible [29] of the absence ofviscosity of the superfluid motion.Equation (2.3) also determines the values of the local densities of the thermodynamic poten-tials which are specified by the rules of statistical mechanics. For a portion of liquid, in thermalequilibrium with temperature T and with well defined velocities v n and v s , the relevant thermo-dynamic potentials —that will be useful for the following discussion— will now be computed.Let us concentrate on the case in which the values v n = | v n | and v s = | v s | are smaller thanany intrinsic velocity scale of helium liquid (speeds of the first and second sound,...), so that thethermodynamic potentials can be approximated by their Taylor expansion up to second orderin powers of v n and v s . The low density approximation for the quasi-particles gas will also beconsidered, and thus the interactions between the quasi-particles will be neglected. When the condensate is at rest ( v s = 0 ), a small portion of the liquid, in which v n = 0 , hasa momentum density P /V which is associated to the normal component of the fluid exclu-sively, P /V = ρ n v n . This momentum density coincides with the momentum density of thequasi-particles gas, P /V = [ P /V ] q.p. . Let n ( ε ) denote the quasi-particles (Bose-Einstein)distribution function, at a given temperature T ; then [ P /V ] q.p. = Z dτ p n ( ε − v n p ) ≃ v n Z dτ ( p / (cid:20) − ∂n ( ε ) ∂ε (cid:21) = v n ρ n , (2.4)where dτ = d p/h , and a first order expansion of n ( ε − v n p ) in powers of the velocity v n hasbeen considered. In the integration function, the multiplicative factor p i p j has been replacedby δ ij p / . Equation (2.4) determines [29] the value ρ n of the mass density of the normal3omponent of the fluid ρ n = Z dτ ( p / (cid:20) − ∂n ( ε ) ∂ε (cid:21) . (2.5)Let us now consider another small part of the liquid in which both velocities v n and v s arenonvanishing. In this case, the total momentum density is the sum of the momentum density [ P /V ] q.p. due to the quasi-particles and the momentum density [ P /V ] s which is associated withthe condensate motion with velocity v s , P /V = [ P /V ] q.p. + [ P /V ] s . (2.6)In agreement with expressions (2.3), the quasi-particles contribution is given by [ P /V ] q.p. = Z dτ p n ( ε − ( v n − v s ) p ) ≃ ( v n − v s ) Z dτ ( p /
3) [ − ∂n ( ε ) /∂ε ]= v n ρ n − v s ρ n = v n ρ n − v s ( ρ − ρ s )= v n ρ n + v s ρ s − v s ρ . (2.7)Therefore, by comparing expression (2.2) with equations (2.6) and (2.7), one gets [ P /V ] s = v s ρ . (2.8) Let us now derive the value of the energy density
U/V for a portion of liquid in which bothvelocities v n and v s are nonvanishing. From equation (2.3) it follows that the energy density [ U/V ] q.p. of the quasi-particles gas is given by [ U/V ] q.p. = Z dτ ( ε − vp ) n ( ε − vp ) , (2.9)where v = v n − v s . A second order expansion in powers of v gives [ U/V ] q.p. ≃ Z dτ (cid:20) ε n ( ε ) + ( vp ) ∂n ( ǫ ) ∂ε + ( vp ) ε ∂ n∂ε (cid:21) . (2.10)By means of the replacement p i p j → δ ij p / , one obtains [ U/V ] q.p. ≃ U /V + v Z dτ (cid:26) ( ε p / ∂ n ( ε ) ∂ε − ( p / (cid:20) − ∂n ( ε ) ∂ε (cid:21)(cid:27) , (2.11)where U /V = Z dτ ε n ( ε ) . (2.12)Finally, in agreement with the expression (2.8), the energy density [ U/V ] s which is related tothe condensate zero entropy motion takes the form (in an inertial reference system) [ U/V ] s = ρ v s . (2.13)4hus the total energy density is given by U/V = U /V + ρ v s + ( ρ ∗ n − ρ n ) ( v n − v s ) , (2.14)in which ρ n is defined in equation (2.5) and ρ ∗ n = Z dτ ( ε p / ∂ n ( ε ) ∂ε . (2.15)Expression (2.14) can also be obtained by transforming the energy density of the liquid heliumfrom the inertial reference system in which v s = 0 to the inertial reference system in which v s = 0 . Equation (2.3) implies that the contribution [ F/V ] q.p. of the quasi-particles gas to the free energydensity is given by [ F/V ] q.p. = kT Z dτ ln (cid:0) − e − ( ε − vp ) /kT (cid:1) , (2.16)and the expansion up to second order in powers of the fluid velocities, kT Z dτ ln (cid:0) − e − ( ε − vp ) /kT (cid:1) ≃ Z dτ n kT ln (cid:0) − e − ε/kT (cid:1) + ( vp ) ∂n ( ε ) ∂ε o , gives [ F/V ] q.p. = F /V − ρ n ( v n − v s ) , (2.17)where F /V = kT Z dτ ln (cid:0) − e − ε/kT (cid:1) . (2.18)The free energy density [ F/V ] c which is associated with the zero entropy motion of the con-densate coincides with [ U/V ] s = ρv s . Therefore, the density F/V of free energy of the liquidis given by
F/V = F /V + ρ v s − ρ n ( v n − v s ) . (2.19)By means of the thermodynamic relation F = U − T S , the density of entropy turns out to be
T S/V = ( U − F ) /V + (2 ρ ∗ n − ρ n ) ( v n − v s ) . (2.20) This section contains the theoretical determination of the mass density ρ ∗ n . For completeness,the computation [29, 34] of ρ n is also reported. In a neighbourhood of p = 0 , the Bose-Einsteindistribution for quasi-particles n ( ε ) = 1 e ε ( p ) /kT − , (2.21)can be approximated by n ( ε ) ph ≃ e up/kT − , (2.22)5nd describes the phonons distribution. Whereas, in a neighbourhood of p = p , the distributionfor the quasi-particles (rotons) can be approximated by the Maxwell-Boltzmann distribution n ( ε ) r ≃ e − ∆ /kT e − ( p − p ) / m ∗ kT , (2.23)because rotons constitute a low density gas. One can write ρ n = ρ n,ph + ρ n,r , (2.24)where ρ n,ph reads ρ n,ph ≃ Z d ph p kT e up/kT ( e up/kT − = 2 π ( kT ) ~ u , (2.25)and ρ n,r is given by ρ n,r ≃ Z d ph p kT e − ∆ /kT e − ( p − p ) / m ∗ kT ≃ p π ) / ~ (cid:18) m ∗ kT (cid:19) / e − ∆ /kT . (2.26)Let us now concentrate on ρ ∗ n shown in equation (2.15); one can put ρ ∗ n = ρ ∗ n,ph + ρ ∗ n,r , (2.27)in which the phonons contribution is given by ρ ∗ n,ph ≃ Z dτ ( p p / u ) ∂ n ( ε ) ph ∂p = 52 ρ n,ph , (2.28)and the rotons part ρ ∗ n,r turns out to be ρ ∗ n,r ≃ e − ∆ /kT kT ) Z dτ p (cid:20) ∆ + ( p − p ) m ∗ (cid:21) e ( p − p ) / m ∗ kT ≃ ρ n,r (cid:20) ∆2 kT + 14 (cid:21) . (2.29) Let us consider the case in which the container of the helium fluid is a cylinder which is rotatingaround its axis with a constant angular velocity Ω . The laboratory system is assumed to be aninertial reference system. The normal component of the liquid helium, which has nonvanishingviscosity and interacts with the container walls, has perception of the motion of the container.Whereas the superfluid component of liquid helium, with vanishing viscosity, is insensitive tothe rotation of the vessel. As a result, after a transient period, the whole system reaches thestable condition in which, for small values of Ω , the viscous component of the fluid is rotatingwith the same angular velocity of the container ( v n = 0 ), whereas the superfluid componentremains at rest ( v s = 0 ).In order to determine the precise motion of the quasi-particles gas which is induced by therotation of the container, one can use the Landau reasoning [29]. In the coordinate system which6s rotating with the same angular velocity Ω , the container is at rest, and the boundary condi-tions for the normal component of the liquid coincide with the stationary conditions of a staticcontainer. Therefore, in this reference system, the motion is determined by the standard actionprinciple and the statistical distribution is expressed in terms of the Gibbs factor exp( − E ′ /kT ) ,where E ′ denotes the energy of a quasi-particle in the rotating system E ′ = ε − Ω · ( r ∧ p ) . (3.1)Thus, in order to find the macroscopic motion of the quasi-particle gas, one can minimise thethermodynamic potentials which are obtained by means of the energy (3.1), and this implies[29] that the quasi-particle gas is rotating as a whole with angular velocity Ω .The same conclusion can also be obtained by considering the laboratory point of view, wherethe equilibrium boundary condition is determined by the requirement that the part of the viscousliquid in contact with the walls of the container must have the same velocity of the walls. Thisimplies that, in the equilibrium state, the viscous fluid must rotate as a solid body with the sameangular velocity of the cylinder, so that there is no energy dissipation caused by friction.To sum up, because of the nontrivial interactions between the normal viscous component ofthe fluid with the moving walls of the container, in the laboratory system the velocity v n takesthe value v n = v n ( r ) = Ω ∧ r , (3.2)and, in agreement with Landau argument, the thermodynamic potentials can be computed bymeans of the standard rules of statistical mechanics in which the energy spectrum of the quasi-particles is given in equation (2.3), with v n shown in equation (3.2) and v s = 0 .As the value of Ω increases, a critical value Ω is reached in which the condensate also startsmoving ( v s = 0 ). In order to proceed —as much as possible— according to an irrotationalmotion, which means ∇ ∧ v s = 0 , the best solution consists in concentrating the vorticity in asingle line (with a quantized vorticity). This line must be closed, or it must have its end-pointson the boundaries of the superfluid region. The stable configuration is obtained when a vortexline is created along the axis of the container. In the presence of a vortex line, the velocity v s of the condensate is directed as the tangent to concentric circles belonging to a plane whichis orthogonal to the axis of the cylinder and, for the minimum nontrivial value of the vorticity πχ = h/m , it has magnitude | v s | = v s ( r ⊥ ) = ~ m r ⊥ , (3.3)where r ⊥ denotes the distance from the central axis. Now the main issue to be discussed is thededuction of the critical value Ω .As a first possibility, one could try to extend the Landau reasoning, which is valid for themotion of the quasi-particles gas, to the condensate motion also. According to this hypothesis,one should consider the energy U ′ vor = U vor − Ω M vor , where U vor and M vor denote the energyand the angular momentum —in the laboratory system— of the motion of the liquid in thepresence of a vortex line. The minimisation of U ′ vor leads [34, 35, 36] to the results: • the critical value of the angular velocity is given by Ω = ~ mR ln (cid:18) Ra (cid:19) , (3.4)7here R represents the radius of the cylinder and a denotes the size of the core of thevortex; • the condensate starts moving in the same direction of the viscous normal component ofthe fluid, i.e. the velocity v s is directed as v n defined in equation (3.2).This procedure appears to be not completely established because the condensate displays noviscosity and is uninfluenced by the rotation of the walls of the container. As a consequence,differently from the case of the normal viscous component of the fluid, the boundary conditionsfor the condensate remain the same in any rotating coordinate system independently of the spe-cific value of the angular velocity. So, as far as the motion of the condensate is concerned, itseems that the thermodynamic potential to be minimised cannot be of the form ( U vor − Ω M vor ) ,because Ω appearing in this expression is totally undetermined since it is not fixed by the con-densate boundary conditions. Also, expression (2.19) shows that, if v n and v s have the samedirection then, as a consequence of the formation of a vortex line, the free energy of the systemwould increase; this seems rather odd. Let us consider then a second possibility, in which it is supposed that the formation of the vortexline is induced by the motion of the quasi-particles gas.It is assumed that some external equipment is acting on the system in order to maintainthe temperature and the value Ω of the angular velocity fixed. The velocity v n of the normalcomponent of the fluid is specified in equation (3.2). In this way, the equilibrium boundaryconditions between the viscous component of the fluid and the walls of the rotating cylinder aresatisfied, and the quasi-particle gas is indeed in thermal equilibrium. The velocity v s is the onlyvariable we are interested in; this variable specifies the (zero entropy) motion of the frictionlesssuperfluid component of helium. In agreement with the laws of thermodynamics, it is assumedthat the vortex line formation is determined by the minimisation condition of the free energy ofthe system.The free energy F of the helium liquid is obtained by integrating the density (2.19) in thevolume, F = Z d r (cid:8) F /V + ρ v s − ρ n ( v n − v s ) (cid:9) . (3.5)The result is the sum of three terms, F = e F + F I + F II . (3.6)The first term e F does not depend on v s , e F = F − Z d r ρ n | v n | , (3.7)and then it is not involved in the computation of Ω . The contribution F I is linear in v s , F I = Z d r ρ n v n v s , (3.8)8hereas F II is quadratic in v s , F II = Z d r ρ s | v s | . (3.9)The formation of the vortex line takes place when F I + F II < . For small velocities one canassume that the mass densities are constant; one finds F I = ± ρ n πLR ~ m Ω , (3.10) F II = ρ s πL ~ m ln (cid:18) Ra (cid:19) , (3.11)where πR L is the volume of the cylinder. The formation of the meniscus has been neglectedbecause, for small velocities, it gives rise to minor effects. The sign in expression (3.10) ispositive if the directions of v n and v s coincide, and it is negative when v n and v s have oppositedirections. Therefore the condition F I + F II < is satisfied when • Ω > Ω , in which the critical value Ω of the angular velocity is given by Ω = (cid:18) ρ s ρ n (cid:19) ~ mR ln (cid:18) Ra (cid:19) ; (3.12) • the condensate starts moving in the opposite direction of the viscous normal componentof the fluid ( i.e. v n v s = −| v n | | v s | < ).Expression (3.12) looks similar to equation (3.4) but predicts a nontrivial dependence of thecritical angular velocity on the temperature. In particular, Ω vanishes in the T → T limit, andtends to diverge when T → . Perhaps, the result that v n and v s must have opposite directionsmay appear unexpected; in any case, this conclusion is also confirmed by the requirement ofthermodynamic stability, as it is shown in the next section. The idea that the motion of the condensate is caused by the presence of the rotating gas ofquasi-particles, and that the emergence of the vortex line is related to the minimisation of thefree energy, seems to be quite reasonable. But of course only the comparison of the prediction(3.12) with the experiments will determine the actual reliability of this approach.In addition to the measure of the critical angular velocity Ω , one could also examine thebehaviour of certain thermodynamic variables which are involved in the formation of the vortexline. For fixed volume, the equilibrium thermal states of helium II inside a rotating containercan be characterised by the variables Ω and T . It is assumed that the quasi-particles gas isrotating with velocity v n shown in equation (3.2). Let us consider the critical curve Ω ( T ) inthe cartesian plane (Ω , T ) shown in Figure 1. The curve Ω ( T ) describes states of coexistenceof two different types of liquid motions; the points in the region (1) of the plane correspond tostates without vortex lines, and the points in the region (2) refer to states in which one vortex9ine is present. The transition from region (1) to region (2) corresponds to the formation of onevortex line. Ω TT no vortex line (1) one vortex line (2) Figure 1. Critical curve Ω ( T ) in the (Ω , T ) -plane.In crossing the critical curve, the latent heat of vortex formation and the discontinuos changein the angular momentum of the quasi-particles gas determine the slope of the curve Ω ( T ) bymeans of some kind of Clapeyron equation.In the differential of the free energy, dF = − SdT − J d Ω , the variable J corresponds tothe vertical component of the angular momentum of the quasi-particles gas. Along the criticalcurve, the free energies F (1) and F (2) of the two types of states are equal; therefore from equation dF (1) = dF (2) , − S (1) dT − J (1) d Ω = − S (2) dT − J (2) d Ω , (4.1)one obtains d Ω dT = − S (2) − S (1) J (2) − J (1) = − λT ( J (2) − J (1) ) , (4.2)where λ = T ( S (2) − S (1) ) denotes the latent heat for the vortex formation. The total angular momentum of the liquid helium is the sum of the angular momentum J of thequasi-particles gas, that for generic values of the velocities v = v n − v s is given by J = Z d r dτ n ( ε − vp ) r ∧ p ≃ Z d r dτ r ∧ p ( vp ) (cid:20) − ∂n ( ε ) ∂ε (cid:21) ≃ Z d r ρ n r ∧ ( v n − v s ) , (4.3)and the angular momentum M due to the motion of the condensate, M = Z d r ρ r ∧ v s . (4.4)10he resulting total angular momentum is J + M = Z d r ( ρ s r ∧ v s + ρ n r ∧ v n ) . (4.5)When v n = Ω ∧ r , with the angular velocity directed as the vertical axis Ω = Ω b z , from theexpression (3.5) of the free energy one gets ∂F∂ Ω = − b z (cid:18)Z d r ρ n r ∧ ( v n − v s ) (cid:19) = − J z ≡ − J . (4.6)The discountinuous change of J , which is due to the formation of a vortex line, is given by ∆ J = J (2) − J (1) = − b z (cid:18)Z d r ρ n r ∧ v s (cid:19) = ρ n πLR ~ m . (4.7)It should be noted that, as a result of the formation of one vortex line, the vertical component ofthe angular momentum of the quasi-particle gas increases, whereas the total angular momentumof helium II decreases ∆ ( J z + M z ) = − ρ s πLR ~ m . (4.8) The change in entropy due to the formation of a vortex line can be obtained by integrating thechange of entropy density (2.20) in the volume, λ = T ( S (2) − S (1) ) = Z d r (2 ρ ∗ n − ρ n )( v s − v n v s )= (2 ρ ∗ n − ρ n ) (cid:18) ρρ n (cid:19) πL ~ m ln (cid:18) Ra (cid:19) . (4.9)Equations (2.28) and (2.29) imply that the mass density (2 ρ ∗ n − ρ n ) which appears in equation(4.9) is positive, therefore after the formation of a vortex line the value of the entropy is in-creased. The mass density (2 ρ ∗ n − ρ n ) is also related with the rate of increment of ρ n with thetemperature. Indeed, in the approximation in which the total mass density ρ is constant, fromequation (3.12) it follows d Ω dT ≃ − ρρ n ∂ρ n ∂T ~ mR ln (cid:18) Ra (cid:19) . (4.10)By comparing equation (4.10) with equations (4.2), (4.7) and (4.9), one derives T ∂ρ n ∂T = 2 ρ ∗ n − ρ n . (4.11)Equation (4.11) can also be obtained form expressions (2.19) and (2.20) by means of the relation S = − ( ∂F/∂T ) V , or it can be derived directly from the definitions (2.5) and (2.15).The macroscopic motions of the liquid which are associated with the two velocities v n and v s represent “ordered” motions of the elementary constituents of the fluid, as opposed to the11haotic thermal motion of the atoms. In the case of a rotating container, a possible measureof the ordered motion of the fluid is given by the magnitude of its total angular momentum.By keeping the value of Ω fixed, during the formation process of a vortex line the amount ofmacroscopic ordered motion reduces, and the amount of disordered microscopic motion (valueof entropy) grows. Precisely because v n and v s have opposite orientations, the discontinuouschange of the entropy is positive and the change in the total angular momentum is negative. Since Ω is obtained by minimising a quadratic function of the macroscopic velocities of theliquid —in which the value of v s is specified in equation (3.3) and v n is described in equation(3.2)—, Ω is proportional to Ω shown in equation (3.4). The proportionality coefficient D ,given by D = ρ s ρ n = ρρ n − , (4.12)nontrivially depends on the temperature T . In the range . ≤ T ≤ . where the temperatureis expressed in Kelvin, the rotons contribution to the mass density turns out to be dominant[34, 37, 38]. By using the experimental data [37, 38] of the normal fluid ratio ρ n /ρ , the resultingvalues of D are shown in Table 1.Table 1: Values of D and of the normal fluid ratio at different temperatures. D . × . × . × ρ n /ρ . × − . × − . × − . × − T (K) 0.6 0.8 1 1.2 D
12 4 .
88 2 .
12 0 .
78 0 . ρ n /ρ . × − .
17 0 .
32 0 .
56 0 . T (K) 1.4 1.6 1.8 2.0 2.1In the interval from . to . Kelvin, equation (3.12) predicts a variation of Ω of three ordersof magnitude. This effect becomes even more important at lower temperatures. For T < . K,as the temperature become smaller the value of D is rapidly increasing with the approximatebehaviour ∝ T − . In the T → limit, the asymptotic value of D is given by D −→ ρρ n ≃ ρ ~ u π ( kT ) , (4.13)where ρ denotes the helium mass density, ρ ≃ . g/cm . The asymptotic behaviour (4.13) isa consequence of the fact that the density of quasi-particles vanishes in the T → limit. On theother hand, in a neighbourhood of the transition temperature T = T , the low density approxi-mation for the quasi-particles gas cannot be adopted. When T ≈ T , in order to determine thevalue of the free energy of the system, the interactions between quasi-particles should be takeninto account. 12 .4 Vortices array Finally, when Ω ≫ Ω several vortices are formed. The experimental data can be described as Provided that the angular velocity Ω is not too small, the vortex lines in uniformlyrotating helium are straight and parallel to the axis of rotation, and they form anarray with uniform density ... [9]Computations on the formation of vortex patterns in rotating superfluid, which are based on theminimisation of ( U vor − Ω M vor ) , can be found for instance in Ref.[39].Differently from the case of a single vortex, the presence (and the time evolution) of a vortexarray in the fluid is not described by stationary velocity fields for the fluid components. Atany fixed time, the spatial positions of the the vortex filaments break the continuous rotationalsymmetry around the vertical axis of the cylinder. The cores of the vortices posses nontrivialvelocities that, combined with the localized positions of the filaments, give rise to a nontrivialspace and time dependence of the velocity field v s , and then of the quasi-particles energy E v ( p ) .Consequently, in order to describe the details of the emergence of a vortex array, one needs toconsider the full set of thermodynamic and hydrodynamic equations (containing the relevantfriction and drag parameters) for the dynamics of liquid helium II.In addition to v s and v n , the positions and the velocities of the vortex filaments must bespecified. If the position of one filament is parametrised as s ( ξ, t ) , where ξ represents thearclength, then the time evolution of s can be approximated [9, 40, 41] by d s dt ≃ v s + α s ′ ∧ ( v n − v s ) − α ′ s ′ ∧ [ s ′ ∧ ( v n − v s )] , (4.14)where s ′ = d s /dξ and α and α ′ denote the mutual friction coefficients. In the case of n vortices, the velocities { v s , v n , d s j /dt } , for j = 1 , , ...n , display nontrivial space and timedependence and intricate couplings. The cores of the vortices can be understood as defects inthe condensate; they interact with the quasi-particles gas and are dragged by the rotating viscousfluid component. As a result, the superfluid vortex filaments tend to align with the normal fluidvorticity. This effect has been observed, for instance, in the case of evolving turbulent flows,where the driven motion of the quantum vortex filaments by the normal fluid velocity has beendetermined and computed [41] by means of numerical simulations.It should be noted that, in the case of a rotating container, the formation of a single quantumvortex corresponding to a counter-rotating superfluid flow —which is proposed in the presentarticle— is not in contradiction with the behaviour of the motion of the vortex filaments. Theemergence of one quantum straight vortex line, which is placed on the axis of the rotatingcylinder, differs from the dragged motion of the core of the vortex filaments of an array because,unlike the dynamics of the quantum filaments, the nucleation of the first vortex line is specifiedby the minimisation condition of the free energy (3.6), as discussed in Section 3. In this article it has been proposed that, with a rotating container, the formation of the firstvortex lines of the superfluid component of helium II is caused by the presence of the rotating13uasi-particles gas, and that the critical angular velocity Ω for the formation of one vortex linecan be obtained by minimising the free energy of the system. During the emergence of the firstvortex line, the condensate starts moving in the opposite direction with respect to the motionof the rotating viscous component of the liquid, the entropy of the system increases and thetotal angular momentum decreases. The value of Ω that has been derived displays a nontrivialdependence on the temperature; Ω ( T ) vanishes in the T → T limit, and tends to diverge when T → . Numerical estimations of the behaviour of Ω ( T ) as a function of the temperaturehave been presented. It has been shown that the latent heat for the formation of one vortex lineand the corresponding discontinuos change in the angular momentum of the quasi-particles gasdetermine the slope of the curve Ω ( T ) through a sort of Clapeyron equation. The incrementin the entropy and the reduction of the total angular momentum of the liquid during the vortexformation have been determined.As far as the experimental side is concerned, the direct determination of the condensatecirculation of the first nucleated vortex appears to be difficult to implement. 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