Superfluid helium-4

352: 240:, launched in January 1983 to gather infrared data was cooled by 73 kilograms of superfluid helium, maintaining a temperature of 1.6 K (−271.55 °C). When used in conjunction with helium-3, temperatures as low as 40 mK are routinely achieved in extreme low temperature experiments. The helium-3, in liquid state at 3.2 K, can be evaporated into the superfluid helium-4, where it acts as a gas due to the latter's properties as a Bose–Einstein condensate. This evaporation pulls energy from the overall system, which can be pumped out in a way completely analogous to normal refrigeration techniques. 960: 360: 275:

is a fairly high velocity so superfluid helium can flow relatively easily up the wall of containers, over the top, and down to the same level as the surface of the liquid inside the container, in a siphon effect. It was, however, observed, that the flow through nanoporous membrane becomes restricted if the pore diameter is less than 0.7 nm (i.e. roughly three times the classical diameter of helium atom), suggesting the unusual hydrodynamic properties of He arise at larger scale than in the classical liquid helium.

3134: 1028: 465: 344: 3404:

excitations if the flow velocity was less than the sound velocity. In this model, the sound velocity is the "critical velocity" above which superfluidity is destroyed. (Helium-4 actually has a lower flow velocity than the sound velocity, but this model is useful to illustrate the concept.) Landau also showed that the sound wave and other excitations could equilibrate with one another and flow separately from the rest of the helium-4, which is known as the "condensate".

218:. Referred to as superfluid helium droplet spectroscopy (SHeDS), it is of great interest in studies of gas molecules, as a single molecule solvated in a superfluid medium allows a molecule to have effective rotational freedom, allowing it to behave similarly to how it would in the "gas" phase. Droplets of superfluid helium also have a characteristic temperature of about 0.4 K which cools the solvated molecule(s) to its ground or nearly ground 5717: 1036: 453: 6437: 284:

remains perfectly stationary. Once the first critical angular velocity is reached, the superfluid will form a vortex. The vortex strength is quantized, that is, a superfluid can only spin at certain "allowed" values. Rotation in a normal fluid, like water, is not quantized. If the rotation speed is increased more and more quantized vortices will be formed which arrange in nice patterns similar to the

2678:). The vessels are connected by a so-called superleak. This is a tube, filled with a very fine powder, so the flow of the normal component is blocked. However, the superfluid component can flow through this superleak without any problem (below a critical velocity of about 20 cm/s). In the steady state 274:

Many ordinary liquids, like alcohol or petroleum, creep up solid walls, driven by their surface tension. Liquid helium also has this property, but, in the case of He-IV, the flow of the liquid in the layer is not restricted by its viscosity but by a critical velocity which is about 20 cm/s. This

3418:

The Landau theory does not elaborate on the microscopic structure of the superfluid component of liquid helium. The first attempts to create a microscopic theory of the superfluid component itself were done by London and subsequently, Tisza. Other microscopical models have been proposed by different

3407:

From the momentum and flow velocity of the excitations he could then define a "normal fluid" density, which is zero at zero temperature and increases with temperature. At the so-called Lambda temperature, where the normal fluid density equals the total density, the helium-4 is no longer superfluid.

283:

Another fundamental property becomes visible if a superfluid is placed in a rotating container. Instead of rotating uniformly with the container, the rotating state consists of quantized vortices. That is, when the container is rotated at speeds below the first critical angular velocity, the liquid

265:

Superfluids, such as helium-4 below the lambda point, exhibit many unusual properties. A superfluid acts as if it were a mixture of a normal component, with all the properties of a normal fluid, and a superfluid component. The superfluid component has zero viscosity and zero entropy. Application of

3374:

the volume flow. The normal flow is balanced by a flow of the superfluid component from the cold to the hot end. At the end sections a normal to superfluid conversion takes place and vice versa. So heat is transported, not by heat conduction, but by convection. This kind of heat transport is very

197:

time, thus increasing or decreasing the defect density respectively, it was shown, via torsional oscillator experiment, that the supersolid fraction could be made to range from 20% to completely non-existent. This suggested that the supersolid nature of helium-4 is not intrinsic to helium-4 but a

1694: 3403:

phenomenological and semi-microscopic theory of superfluidity of helium-4 earned him the Nobel Prize in physics, in 1962. Assuming that sound waves are the most important excitations in helium-4 at low temperatures, he showed that helium-4 flowing past a wall would not spontaneously create

402:

Below the lambda line the liquid can be described by the so-called two-fluid model. It behaves as if it consists of two components: a normal component, which behaves like a normal fluid, and a superfluid component with zero viscosity and zero entropy. The ratios of the respective densities

3456:

The models are based on the simplified form of the inter-particle potential between helium-4 atoms in the superfluid phase. Namely, the potential is assumed to be of the hard-sphere type. In these models the famous Landau (roton) spectrum of excitations is qualitatively reproduced.

301:

that they obey. Specifically, the superfluidity of helium-4 can be regarded as a consequence of Bose–Einstein condensation in an interacting system. On the other hand, helium-3 atoms are fermions, and the superfluid transition in this system is described by a generalization of the

69:. The substance, which resembles other liquids such as helium I (conventional, non-superfluid liquid helium), flows without friction past any surface, which allows it to continue to circulate over obstructions and through pores in containers which hold it, subject only to its own 468:

Fig. 5. The liquid helium is in the superfluid phase. As long as it remains superfluid, it creeps up the wall of the cup as a thin film. It comes down on the outside, forming a drop which will fall into the liquid below. Another drop will form – and so on – until the cup is

3481:-like behaviour of the element's interior density and interparticle interaction potential. The long-wavelength part is the quantum many-body theory of such elements which deals with their dynamics and interactions. The approach provides a unified description of the 2491: 630:

is the velocity of the superfluid component. The time derivative is the so-called hydrodynamic derivative, i.e. the rate of increase of the velocity when moving with the fluid. In the case of superfluid He in the gravitational field the force is given by

383:

of He. It is a pressure-temperature (p-T) diagram indicating the solid and liquid regions separated by the melting curve (between the liquid and solid state) and the liquid and gas region, separated by the vapor-pressure line. This latter ends in the

892: 296:

Although the phenomenologies of the superfluid states of helium-4 and helium-3 are very similar, the microscopic details of the transitions are very different. Helium-4 atoms are bosons, and their superfluidity can be understood in terms of the

151:

lines in superfluid helium. In the 1960s, Rayfield and Reif established the existence of quantized vortex rings. Packard has observed the intersection of vortex lines with the free surface of the fluid, and Avenel and Varoquaux have studied the

395:

Figure 1 also shows the λ-line. This is the line that separates two fluid regions in the phase diagram indicated by He-I and He-II. In the He-I region the helium behaves like a normal fluid; in the He-II region the helium is superfluid.

1510: 2134: 563: 3447:

around 1955, developed microscopic theories for the roton, which was shortly observed with inelastic neutron experiments by Palevsky. Later on, Feynman admitted that his model gives only qualitative agreement with experiment.

419:) the density of the normal (superfluid) component, and ρ (the total density), depends on temperature and is represented in figure 3. By lowering the temperature, the fraction of the superfluid density increases from zero at 266:

heat to a spot in superfluid helium results in a flow of the normal component which takes care of the heat transport at relatively high velocity (up to 20 cm/s) which leads to a very high effective thermal conductivity.

1860: 1039:

Fig. 8. Demonstration of the fountain effect. A capillary tube is "closed" at one end by a superleak and is placed into a bath of superfluid helium and then heated. The helium flows up through the tube and squirts like a

2844: 1128: 2367: 709: 2216:= 9.8 m/s this corresponds with a liquid-helium column of 56 meter height. So, in many experiments, the fountain pressure has a bigger effect on the motion of the superfluid helium than gravity. 3273: 2329: 3901:

Avenel, O.; Varoquaux, E. (1985). "Observation of Singly Quantized Dissipation Events Obeying the Josephson Frequency Relation in the Critical Flow of Superfluid ^{4}He through an Aperture".

781: 4698: 4590:

Van Alphen, W. M.; Van Haasteren, G. J.; De Bruyn Ouboter, R.; Taconis, K. W. (1966). "The dependence of the critical velocity of the superfluid on channel diameter and film thickness".

399:

The name lambda-line comes from the specific heat – temperature plot which has the shape of the Greek letter λ. See figure 2, which shows a peak at 2.172 K, the so-called λ-point of He.

773:

the vertical coordinate. Thus we get the equation which states that the thermodynamics of a certain constant will be amplified by the force of the natural gravitational acceleration

2565: 3096: 4626: 2576:

shows that the superfluid component is accelerated by gradients in the pressure and in the gravitational field, as usual, but also by a gradient in the fountain pressure.

2031: 1494: 1463: 3372: 2210: 628: 483: 193:. When helium-4 is cooled below about 200 mK under high pressures, a fraction (≈1%) of the solid appears to become superfluid. By quench cooling or lengthening the 2973: 4757:

Alonso, J. L.; Ares, F.; Brun, J. L. (October 5, 2018). "Unraveling the Landau's consistence criterion and the meaning of interpenetration in the "Two-Fluid" Model".

1767: 3316: 1238: 3009: 2021: 3042: 2933: 2709: 2676: 2643: 1994: 1906: 3198: 3171: 3125: 2610: 2177: 1757: 1432: 1400: 1342: 1310: 1198: 1171: 941: 592: 981: 731: 179:

which also exhibit superfluidity. This work with ultra-cold atomic gases has allowed scientists to study the region in between these two extremes, known as the

1958: 1932: 1368: 3379:

where heat is transported via gas–liquid conversion. The high thermal conductivity of He-II is applied for stabilizing superconducting magnets such as in the

5334:

A. V. Avdeenkov & K. G. Zloshchastiev (2011). "Quantum Bose liquids with logarithmic nonlinearity: Self-sustainability and emergence of spatial extent".

3336: 2900: 2880: 1730: 1278: 1258: 1063: 1021: 1001: 771: 751: 3130:

In an experiment, arranged as in figure 8, a fountain can be created. The fountain effect is used to drive the circulation of He in dilution refrigerators.

355:

Fig. 2. Heat capacity of liquid He at saturated vapor pressure as function of the temperature. The peak at T=2.17 K marks a (second-order) phase transition.

3839: 5281:

K. G. Zloshchastiev (2012). "Volume element structure and roton-maxon-phonon excitations in superfluid helium beyond the Gross-Pitaevskii approximation".

3493:

excitations, and has noteworthy agreement with experiment: with one essential parameter to fit one reproduces at high accuracy the Landau roton spectrum,

180: 456:

Fig. 4. Helium II will "creep" along surfaces in order to find its own level – after a short while, the levels in the two containers will equalize. The

198:

property of helium-4 and disorder. Some emerging theories posit that the supersolid signal observed in helium-4 was actually an observation of either a

5549:

Liquid Helium II, Superfluid: demonstrations of Lambda point transition/viscosity paradox /two fluid model/fountain effect/creeping film/ second sound.

3501:

of superfluid helium-4. This model utilizes the general theory of quantum Bose liquids with logarithmic nonlinearities which is based on introducing a

3423:. To date, a number of models of this kind have been proposed, including: models with vortex rings, hard-sphere models, and Gaussian cluster theories. 96:

particle, which can form bosons only by pairing with itself at much lower temperatures, in a weaker process that is similar to the electron pairing in

3419:

authors. Their main objective is to derive the form of the inter-particle potential between helium atoms in superfluid state from first principles of

1689:{\displaystyle \mu (p,T)=\mu (0,0)+\int _{0}^{p}V_{m}(p^{\prime },0)\mathrm {d} p^{\prime }-\int _{0}^{T}S_{m}(p,T^{\prime })\mathrm {d} T^{\prime }.} 5608: 5093:

T. D. Lee; K. Huang & C. N. Yang (1957). "Eigenvalues and Eigenfunctions of a Bose System of Hard Spheres and Its Low-Temperature Properties".

4323: 1031:

Fig. 7. Demonstration of the fountain pressure. The two vessels are connected by a superleak through which only the superfluid component can pass.

6171: 636: 5622: 2725: 4086:

Sophie, A; Rittner C (2006). "Observation of Classical Rotational Inertia and Nonclassical Supersolid Signals in Solid 4 He below 250 mK".

3712: 2179:

is called the fountain pressure. It can be calculated from the entropy of He which, in turn, can be calculated from the heat capacity. For

3760:

Hall, H. E.; Vinen, W. F. (1956). "The Rotation of Liquid Helium II. II. The Theory of Mutual Friction in Uniformly Rotating Helium II".

3214: 2245: 156:

in superfluid helium-4. In 2006, a group at the University of Maryland visualized quantized vortices by using small tracer particles of

5654: 2612:

can show up as a real pressure. Figure 7 shows two vessels both containing He-II. The left vessel is supposed to be at zero kelvins (

4065: 3431:

Landau thought that vorticity entered superfluid helium-4 by vortex sheets, but such sheets have since been shown to be unstable.

3411:

To explain the early specific heat data on superfluid helium-4, Landau posited the existence of a type of excitation he called a "

17: 6468: 3559: 3200:

connected by a tube filled with He-II. When heat is applied to the hot end a pressure builds up at the hot end according to Eq.

3375:

effective, so the thermal conductivity of He-II is very much better than the best materials. The situation is comparable with

5482: 4646: 4506: 388:

where the difference between gas and liquid disappears. The diagram shows the remarkable property that He is liquid even at

3435:

and, later independently, Feynman showed that vorticity enters by quantized vortex lines. They also developed the idea of

6164: 5553: 3474: 3415:", but as better data became available he considered that the "roton" was the same as a high momentum version of sound. 4989: 4662:

Staas, F. A.; Severijns, A. P.; Van Der Waerden, H. C.bM. (1975). "A dilution refrigerator with superfluid injection".

80:

of helium atoms. This condensation occurs in liquid helium-4 at a far higher temperature (2.17 K) than it does in

5463: 5455: 5430: 4577: 4539: 3049: 2486:{\displaystyle \rho _{0}{\frac {\mathrm {d} {\vec {v}}_{s}}{\mathrm {d} t}}=-{\vec {\nabla }}(p+\rho _{0}gz-p_{f}).} 141: 140:

on August 2, 1911, the same day that he observed superconductivity in mercury. It has since been described through

5128:

L. Liu; L. S. Liu & K. W. Wong (1964). "Hard-Sphere Approach to the Excitation Spectrum in Liquid Helium II".

460:

also covers the interior of the larger container; if it were not sealed, the helium II would creep out and escape.

5921: 385: 3876: 6157: 5991: 5916: 5647: 3597: 351: 172: 5442: 6441: 6103: 5931: 3465:

This is a two-scale approach which describes the superfluid component of liquid helium-4. It consists of two

478:

The equation of motion for the superfluid component, in a somewhat simplified form, is given by Newton's law

429:

It is possible to create density waves of the normal component (and hence of the superfluid component since ρ

327: 113: 6113: 5986: 5731: 5401: 3579: 298: 77: 5163:

A. P. Ivashin & Y. M. Poluektov (2011). "Short-wave excitations in non-local Gross-Pitaevskii model".

952:

shows that, in the case of the superfluid component, the force contains a term due to the gradient of the

6403: 5559:

Rousseau, V. G. (2014). "Superfluid density in continuous and discrete spaces: Avoiding misconceptions".

887:{\displaystyle M_{4}{\frac {\mathrm {d} {\vec {v}}_{s}}{\mathrm {d} t}}=-{\vec {\nabla }}(\mu +M_{4}gz).} 229:, which allow the measurement of some theoretically predicted gravitational effects (for an example, see 5605: 4489:

Buckingham, M.J.; Fairbank, W.M. (1961). "Chapter III The Nature of the λ-Transition in Liquid Helium".

2512: 6418: 6218: 5467: 5389:, III. The Many-Worlds Interpretation of Quantum Mechanics: the theory of the universal wave function. 247:

to lower temperatures. So far the limit is 1.19 K, but there is a potential to reach 0.7 K.

6248: 6233: 5640: 125: 3863: 943:

is below a certain critical value, which usually is determined by the diameter of the flow channel.

6391: 6386: 6315: 6290: 6275: 6270: 6265: 6138: 6037: 5667: 2212:

the fountain pressure is equal to 0.692 bar. With a density of liquid helium of 125 kg/m and

6473: 6032: 1468: 1437: 5390: 3341: 2182: 597: 6478: 6463: 6408: 6330: 6057: 6047: 5797: 5792: 4996: 3858: 194: 133: 6381: 6355: 6180: 4454:

Keesom, W.H.; Keesom, A.P. (1935). "New measurements on the specific heat of liquid helium".

3524: 3380: 2938: 190: 109: 6320: 6243: 5976: 5736: 5578: 5500: 5353: 5300: 5239: 5182: 5137: 5102: 5067: 5025: 4954: 4919: 4874: 4827: 4776: 4710: 4671: 4599: 4463: 4428: 4375: 4280: 4219: 4166: 4105: 4020: 3960: 3910: 3850: 3812: 3769: 3724: 3677: 3632: 3534: 3506: 3294: 3137:

Fig. 9. Transport of heat by a counterflow of the normal and superfluid components of He-II

1999: 1208: 164: 4745:

Critical velocities and mutual friction in He-He mixtures at low temperatures below 100 mK

3206:. This pressure drives the normal component from the hot end to the cold end according to 2978: 8: 6413: 6371: 6340: 6300: 6285: 6260: 5951: 5843: 5833: 5746: 5701: 5365: 5217: 5052: 5010: 3519: 3502: 3014: 2905: 2681: 2648: 2615: 219: 5582: 5504: 5357: 5304: 5243: 5186: 5141: 5106: 5071: 5029: 4958: 4923: 4878: 4831: 4818:

F. London (1938). "The λ-Phenomenon of Liquid Helium and the Bose-Einstein Degeneracy".

4780: 4714: 4675: 4603: 4467: 4432: 4379: 4284: 4223: 4170: 4109: 4062: 4024: 3964: 3914: 3854: 3816: 3773: 3728: 3681: 3636: 2902:

apply to the left and right side of the superleak respectively. In this particular case

1969: 1881: 956:. This is the origin of the remarkable properties of He-II such as the fountain effect. 326:.) A unified description of superconductivity and superfluidity is possible in terms of 6305: 6195: 6098: 6027: 5861: 5594: 5568: 5369: 5343: 5316: 5290: 5263: 5229: 5198: 5172: 4892: 4843: 4800: 4766: 4396: 4363: 4344: 4327: 4304: 4270: 4243: 4209: 4182: 4156: 4129: 4095: 4044: 3986: 3785: 3693: 3650: 3466: 3176: 3149: 3103: 2588: 2155: 1735: 1405: 1373: 1315: 1283: 1176: 1149: 953: 946:

In classical mechanics the force is often the gradient of a potential energy. Eq.

919: 570: 323: 167:

from pairs of ultra-cold fermionic atoms. Under certain conditions, fermion pairs form

4966: 4638: 4498: 4475: 3100:

This means that the pressure in the right vessel is equal to the fountain pressure at

966: 716: 210:

Recently in the field of chemistry, superfluid helium-4 has been successfully used in

6350: 6238: 6208: 6203: 6128: 6123: 6093: 6052: 5941: 5893: 5878: 5771: 5741: 5598: 5478: 5459: 5426: 5373: 5320: 5255: 5202: 4985: 4792: 4726: 4683: 4642: 4611: 4573: 4535: 4502: 4401: 4348: 4296: 4235: 4186: 4121: 4036: 3978: 3926: 3872: 3789: 3742: 3549: 3544: 3539: 3420: 2129:{\displaystyle \int _{0}^{T}S_{m}(p,T^{\prime })\mathrm {d} T^{\prime }=V_{m0}p_{f}.} 1937: 1911: 1347: 285: 168: 97: 5267: 4804: 4419:

Swenson, C. (1950). "The Liquid-Solid Transformation in Helium near Absolute Zero".

4364:"Limited Quantum Helium Transportation through Nano-channels by Quantum Fluctuation" 4308: 4247: 4133: 959: 558:{\displaystyle {\vec {F}}=M_{4}{\frac {\mathrm {d} {\vec {v}}_{s}}{\mathrm {d} t}}.} 426:

to one at zero kelvins. Below 1 K the helium is almost completely superfluid.

6255: 6083: 5706: 5586: 5508: 5361: 5308: 5247: 5190: 5145: 5110: 5075: 5033: 4984:. World Scientific Series in 20th century Physics. Vol. 27. World Scientific. 4962: 4927: 4896: 4882: 4847: 4835: 4784: 4718: 4679: 4634: 4607: 4521:

E.L. Andronikashvili Zh. Éksp. Teor. Fiz, Vol.16 p.780 (1946), Vol.18 p. 424 (1948)

4494: 4471: 4436: 4391: 4383: 4336: 4288: 4261:

Pollet, L; Boninsegni M (2007). "Superfuididty of Grain Boundaries in Solid 4 He".

4227: 4174: 4113: 4048: 4028: 4008: 3990: 3968: 3918: 3868: 3820: 3777: 3762:

Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences

3732: 3697: 3685: 3654: 3640: 3498: 3486: 3321: 2885: 2865: 1715: 1263: 1243: 1006: 986: 756: 736: 359: 153: 148: 137: 5251: 4292: 4231: 4178: 4117: 6325: 6310: 6280: 6228: 6223: 6073: 5926: 5663: 5612: 5488: 5312: 5218:"Roton-Maxon Spectrum and Stability of Trapped Dipolar Bose-Einstein Condensates" 4788: 4069: 3554: 3444: 363:

Fig. 3. Temperature dependence of the relative superfluid and normal components ρ

230: 215: 175:. At the other limit, the fermions (most notably superconducting electrons) form 147:

In the 1950s, Hall and Vinen performed experiments establishing the existence of

58: 5617: 4147:

Sophie, A; Rittner C (2007). "Disorder and the Supersolid State of Solid 4 He".

3922: 1855:{\displaystyle \int _{0}^{p}V_{m}(p^{\prime },0)\mathrm {d} p^{\prime }=V_{m0}p} 6345: 6335: 5871: 5866: 5823: 5756: 5751: 5590: 5512: 5447: 3494: 3436: 3133: 315: 157: 89: 5627: 5194: 5149: 4589: 3824: 3803:

Rayfield, G.; Reif, F. (1964). "Quantized Vortex Rings in Superfluid Helium".

6457: 6396: 6295: 6108: 6088: 6011: 5971: 5906: 5838: 5761: 5114: 5079: 4796: 4722: 3945: 3746: 3529: 3470: 437:= constant) which are similar to ordinary sound waves. This effect is called 389: 380: 260: 211: 121: 117: 1027: 464: 202:

state or intrinsically superfluid grain boundaries in the helium-4 crystal.

6376: 6133: 6006: 6001: 5996: 5961: 5911: 5828: 5386: 5259: 5037: 4931: 4440: 4405: 4300: 4239: 4125: 4040: 4004: 3982: 3930: 3781: 3432: 438: 176: 5632: 5524: 4945:

Bijl, A; de Boer, J; Michels, A (1941). "Properties of liquid helium II".

4730: 3944:

Bewley, Gregory P.; Lathrop, Daniel P.; Sreenivasan, Katepalli R. (2006).

343: 6213: 6042: 5936: 5848: 5234: 4275: 4214: 4161: 4100: 2839:{\displaystyle p_{l}+\rho _{0}gz_{l}-p_{fl}=p_{r}+\rho _{0}gz_{r}-p_{fr}} 457: 307: 256: 244: 129: 76:

The formation of the superfluid is a manifestation of the formation of a

6149: 4699:"He flow in dilute He-He mixtures at temperatures between 10 and 150 mK" 4032: 2585:

has only mathematical meaning, but in special experimental arrangements

243:

Superfluid-helium technology is used to extend the temperature range of

5981: 5956: 5883: 5853: 5787: 5766: 3400: 3146:

Figure 9 depicts a heat-conduction experiment between two temperatures

1123:{\displaystyle \mathrm {d} \mu =V_{m}\mathrm {d} p-S_{m}\mathrm {d} T.} 303: 199: 186: 42: 4387: 4340: 3737: 347:

Fig. 1. Phase diagram of He. In this diagram is also given the λ-line.

4887: 4862: 4839: 4552: 3689: 3645: 3620: 3440: 3376: 1035: 226: 66: 5716: 3973: 3469:. The short-wavelength part describes the interior structure of the 6118: 5946: 5543: 5538: 4771: 3478: 311: 81: 46: 5573: 5348: 5333: 5295: 5177: 4633:. Progress in Low Temperature Physics. Vol. 13. p. 167. 1434:, so with constant pressure (see figure 6). In the first integral 452: 6078: 5966: 5901: 5818: 5813: 3668:

Allen, J. F.; Misener, A. D. (1938). "Flow of Liquid Helium II".

93: 70: 50: 5548: 4696: 4493:. Progress in Low Temperature Physics. Vol. 3. p. 80. 5687: 4747:, thesis, Appendix A, Eindhoven University of Technology, 1991. 4697:

Castelijns, C.; Kuerten, J.; De Waele, A.; Gijsman, H. (1985).

4627:"Chapter 3: Thermodynamics and Hydrodynamics of He–He Mixtures" 3482: 319: 54: 5696: 5682: 4661: 4011:(2004). "Probable Observation of a Supersolid Helium Phase". 3490: 3412: 314:, and the attractive interaction between them is mediated by 85: 62: 5162: 4200:

Boninsegni, M; Prokofev (2006). "Superglass Phase of 4 He".

225:

Superfluids are also used in high-precision devices such as

5215: 5092: 4559:, Vol. 5, Academy of Sciences of the USSR, p. 71. 3384: 2567:

the density of liquid He at zero pressure and temperature.

704:{\displaystyle {\vec {F}}=-{\vec {\nabla }}(\mu +M_{4}gz).} 237: 5692: 5127: 5454:, Vol. I, "SUPERFLOW AND VORTEX LINES", pp. 1–742, 4944: 4003: 3946:"Superfluid helium: Visualization of quantized vortices" 3943: 3598:"The Nobel Prize in Physics 1996 - Advanced Information" 5216:

Santos, L.; Shlyapnikov, G. V.; Lewenstein, M. (2003).

5011:"Atomic Theory of the Two-Fluid Model of Liquid Helium" 189:

may also have been discovered in 2004 by physicists at

3344: 3324: 3297: 3179: 3152: 3106: 3017: 2981: 2941: 2908: 2888: 2868: 2684: 2651: 2618: 2591: 2515: 2185: 2158: 1972: 1940: 1914: 1884: 1738: 1718: 1471: 1440: 1408: 1376: 1350: 1318: 1286: 1266: 1246: 1211: 1179: 1152: 1009: 989: 969: 922: 759: 739: 719: 600: 573: 5443:

Department of Energy Office of Science: Superfluidity

5053:"Energy Spectrum of the Excitations in Liquid Helium" 5050: 3217: 3052: 2728: 2370: 2248: 2034: 2002: 1770: 1513: 1066: 784: 639: 486: 5628:

Video including superfluid helium's strange behavior

5408:. American Journal of Mathematics (1957) pp. 152–156 3505:-type contribution to energy related to the quantum 5280: 3268:{\displaystyle \Delta p=-\eta _{n}Z{\dot {V}}_{n}.} 2324:{\displaystyle \mu (p,T)=\mu _{0}+V_{m0}(p-p_{f}).} 4982:Selected papers of Richard Feynman with commentary 4631:Thermodynamics and hydrodynamics of He–He mixtures 4625:De Waele, A. Th. A. M.; Kuerten, J. G. M. (1992). 4488: 4260: 4199: 3366: 3330: 3310: 3267: 3192: 3165: 3119: 3090: 3036: 3003: 2967: 2927: 2894: 2874: 2838: 2703: 2670: 2637: 2604: 2559: 2485: 2323: 2204: 2171: 2128: 2015: 1988: 1952: 1926: 1900: 1854: 1751: 1724: 1688: 1488: 1457: 1426: 1394: 1362: 1336: 1304: 1272: 1252: 1232: 1192: 1165: 1122: 1015: 995: 975: 935: 886: 765: 745: 725: 703: 622: 586: 557: 3711:van Delft, Dirk; Kes, Peter (September 1, 2010). 1055:in more familiar form we use the general formula 84:(2.5 mK) because each atom of helium-4 is a 6455: 5475:Superconductivity, superfluids, and condensates. 4910:L. Tisza (1947). "The Theory of Liquid Helium". 4624: 4491:The nature of the λ-transition in liquid helium 4146: 4085: 3900: 3473:using a non-perturbative approach based on the 5544:http://web.mit.edu/newsoffice/2005/matter.html 5519:London, F. Superfluids (Wiley, New York, 1950) 4756: 4570:An introduction to the theory of superfluidity 3621:"Viscosity of Liquid Helium Below the λ-Point" 392:. He is only solid at pressures above 25 bar. 6165: 5648: 5008: 4979: 4453: 3802: 3710: 3667: 3460: 3395: 5662: 473: 291: 6436: 6172: 6158: 5655: 5641: 4553:"The theory of superfluidity of helium II" 3318:is the viscosity of the normal component, 1280:plane. First we integrate from the origin 1240:can be found by a line integration in the 6179: 5572: 5395: 5347: 5294: 5233: 5176: 4886: 4817: 4770: 4395: 4274: 4213: 4160: 4099: 3972: 3862: 3759: 3736: 3644: 3467:nested models linked via parametric space 963:Fig. 6. Integration path for calculating 163:In the early 2000s, physicists created a 5558: 4909: 4860: 3584:Encyclopedia of Condensed Matter Physics 3132: 1034: 1026: 958: 463: 451: 441:. Due to the temperature dependence of ρ 358: 350: 342: 5487: 4418: 4321: 3837: 3618: 108:Known as a major facet in the study of 27:Superfluid form of the helium-4 isotope 14: 6456: 5623:The Hindu article on superfluid states 5522:Philippe Lebrun & Laurent Tavian: 4335:. Vol. 710. pp. 034911–1/8. 3560:Timeline of low-temperature technology 1712:We are interested only in cases where 310:takes place between atoms rather than 6153: 5636: 5051:R. P. Feynman & M. Cohen (1956). 3840:"Vortex photography in liquid helium" 3451: 3390: 1908:is the molar volume of the liquid at 333: 5439:(IOP Publishing Ltd., Bristol, 1990) 5437:Superfluidity and Superconductivity, 4361: 3713:"The discovery of superconductivity" 3426: 3208: 2719: 2361: 2239: 2025: 2023:which has the dimension of pressure 1761: 1504: 1057: 1044: 775: 753:the gravitational acceleration, and 236:The Infrared Astronomical Satellite 5525:The technology of superfluid helium 5425:Taylor & Francis, London 2003, 3586:. Elsevier. 2005. pp. 128–133. 2560:{\textstyle \rho _{0}=M_{4}/V_{m0}} 24: 5456:World Scientific (Singapore, 1989) 5414: 4863:"Transport Phenomena in Helium II" 3507:Everett–Hirschman entropy function 3218: 2430: 2411: 2385: 2092: 2083: 2074: 1828: 1819: 1804: 1678: 1669: 1660: 1613: 1604: 1589: 1473: 1442: 1110: 1089: 1068: 844: 825: 799: 661: 542: 516: 25: 6490: 5532: 5477:Oxford Univ. Press, Oxford 2005, 4572:(New York: W. A. Benjamin), 3141: 733:is the molar chemical potential, 338: 136:possibly observed the superfluid 6435: 5715: 5452:Gauge Fields in Condensed Matter 3475:logarithmic Schrödinger equation 1966:is also written as a product of 88:particle, by virtue of its zero 5539:Helium-4 Interactive Properties 5380: 5327: 5274: 5209: 5156: 5121: 5086: 5044: 5002: 4973: 4938: 4903: 4854: 4811: 4759:The European Physical Journal B 4750: 4737: 4690: 4655: 4618: 4583: 4562: 4545: 4524: 4515: 4482: 4447: 4412: 4355: 4315: 4254: 4193: 4140: 4079: 4055: 3997: 3937: 3091:{\displaystyle 0=p_{r}-p_{fr}.} 205: 61:in which matter behaves like a 5366:10.1088/0953-4075/44/19/195303 5336:J. Phys. B: At. Mol. Opt. Phys 4061:Moses Chan's Research Group. " 3894: 3831: 3796: 3753: 3704: 3661: 3612: 3590: 3572: 2477: 2439: 2433: 2396: 2315: 2296: 2264: 2252: 2079: 2060: 1815: 1796: 1665: 1646: 1600: 1581: 1550: 1538: 1529: 1517: 1421: 1409: 1389: 1377: 1331: 1319: 1299: 1287: 1227: 1215: 878: 853: 847: 810: 695: 670: 664: 646: 608: 527: 493: 13: 1: 6104:Macroscopic quantum phenomena 5252:10.1103/PhysRevLett.90.250403 4997:Section IV (pages 313 to 414) 4967:10.1016/S0031-8914(41)90422-6 4639:10.1016/S0079-6417(08)60052-9 4530:S. J. Putterman (1974), 4499:10.1016/S0079-6417(08)60134-1 4476:10.1016/S0031-8914(35)90128-8 4293:10.1103/PhysRevLett.98.135301 4232:10.1103/PhysRevLett.96.135301 4179:10.1103/PhysRevLett.98.175302 4118:10.1103/PhysRevLett.97.165301 3566: 3338:some geometrical factor, and 594:is the molar mass of He, and 306:of superconductivity. In it, 250: 114:macroscopic quantum phenomena 6114:Order and disorder (physics) 4684:10.1016/0375-9601(75)90087-0 4612:10.1016/0031-9163(66)90958-9 3873:10.1016/0378-4363(82)90510-1 3838:Packard, Richard E. (1982). 1759:is practically constant. So 1489:{\textstyle \mathrm {d} p=0} 1458:{\textstyle \mathrm {d} T=0} 449:are also temperature waves. 269: 7: 5618:Superfluid phases of helium 5554:Physics Today February 2001 5435:D.R. Tilley and J. Tilley, 4568:Khalatnikov, I. M. (1965), 4534:(Amsterdam: North-Holland) 3923:10.1103/PhysRevLett.55.2704 3512: 3367:{\textstyle {\dot {V}}_{n}} 3281: 3202: 2852: 2713: 2581: 2572: 2499: 2355: 2349: 2337: 2233: 2227: 2221: 2205:{\textstyle T=T_{\lambda }} 2142: 1962: 1868: 1702: 1498: 1202: 1136: 1051: 948: 912: 900: 623:{\textstyle {\vec {v}}_{s}} 445:(figure 3) these waves in ρ 278: 10: 6495: 6419:Thermoacoustic heat engine 5591:10.1103/PhysRevB.90.134503 5513:10.1103/RevModPhys.71.S318 5313:10.1140/epjb/e2012-30344-3 4980:Braun, L. M., ed. (2000). 4789:10.1140/epjb/e2018-90105-x 4329:AIP Conference Proceedings 4324:"Superfluid Vortex Cooler" 1200:the molar volume. With Eq. 254: 173:Bose–Einstein condensation 144:and microscopic theories. 103: 92:. Helium-3, however, is a 6469:Bose–Einstein condensates 6431: 6404:Immersive virtual reality 6364: 6194: 6187: 6066: 6020: 5892: 5806: 5780: 5724: 5713: 5675: 5493:Reviews of Modern Physics 5491:(1999). "Superfluidity". 5195:10.2478/s11534-010-0124-7 5150:10.1103/PhysRev.135.A1166 4999:deals with liquid helium. 3825:10.1103/PhysRev.136.A1194 3461:Gaussian cluster approach 3396:Landau two-fluid approach 1370:. Next we integrate from 1173:is the molar entropy and 318:fluctuations rather than 120:effect was discovered by 6387:Digital scent technology 6139:Thermo-dielectric effect 6038:Enthalpy of vaporization 5732:Bose–Einstein condensate 5606:superfluid hydrodynamics 5115:10.1103/PhysRev.106.1135 5080:10.1103/PhysRev.102.1189 4723:10.1103/PhysRevB.32.2870 4532:Superfluid Hydrodynamics 2968:{\textstyle z_{l}=z_{r}} 474:Superfluid hydrodynamics 299:Bose–Einstein statistics 292:Comparison with helium-3 78:Bose–Einstein condensate 6033:Enthalpy of sublimation 5222:Physical Review Letters 4362:Ohba, Tomonori (2016). 4322:Tanaeva, I. A. (2004). 3903:Physical Review Letters 1960:. The other term in Eq. 1049:In order to rewrite Eq. 328:gauge symmetry breaking 257:Helium § Helium II 18:Thermomechanical effect 6409:Magnetic refrigeration 6048:Latent internal energy 5798:Color-glass condensate 5611:March 3, 2016, at the 5391:Everett's Dissertation 5038:10.1103/PhysRev.94.262 5009:R. P. Feynman (1954). 4932:10.1103/PhysRev.72.838 4551:Landau, L. D. (1941), 4441:10.1103/PhysRev.79.626 4074:Penn State University, 3849:. 109–110: 1474–1484. 3782:10.1098/rspa.1956.0215 3368: 3332: 3312: 3311:{\textstyle \eta _{n}} 3269: 3194: 3167: 3138: 3121: 3092: 3038: 3005: 2969: 2929: 2896: 2876: 2840: 2705: 2672: 2639: 2606: 2561: 2487: 2325: 2206: 2173: 2130: 2017: 1990: 1954: 1928: 1902: 1856: 1753: 1726: 1690: 1490: 1459: 1428: 1396: 1364: 1338: 1306: 1274: 1254: 1234: 1233:{\textstyle \mu (p,T)} 1194: 1167: 1124: 1041: 1032: 1024: 1017: 997: 977: 937: 888: 767: 747: 727: 705: 624: 588: 559: 470: 461: 376: 356: 348: 6382:Cloak of invisibility 6181:Emerging technologies 5858:Magnetically ordered 3525:Large Hadron Collider 3381:Large Hadron Collider 3369: 3333: 3313: 3270: 3195: 3168: 3136: 3122: 3093: 3039: 3006: 3004:{\textstyle p_{fl}=0} 2970: 2930: 2897: 2877: 2841: 2706: 2673: 2645:) and zero pressure ( 2640: 2607: 2562: 2488: 2326: 2207: 2174: 2131: 2018: 2016:{\displaystyle p_{f}} 1991: 1955: 1929: 1903: 1857: 1754: 1727: 1691: 1491: 1460: 1429: 1397: 1365: 1339: 1307: 1275: 1255: 1235: 1195: 1168: 1125: 1038: 1030: 1018: 998: 978: 962: 938: 889: 768: 748: 728: 706: 625: 589: 560: 467: 455: 362: 354: 346: 288:in a superconductor. 191:Penn State University 110:quantum hydrodynamics 5737:Fermionic condensate 5421:Antony M. Guénault: 3619:Kapitza, P. (1938). 3535:Polariton superfluid 3342: 3322: 3295: 3215: 3177: 3150: 3104: 3050: 3037:{\textstyle T_{l}=0} 3015: 2979: 2939: 2928:{\textstyle p_{l}=0} 2906: 2886: 2866: 2726: 2704:{\textstyle v_{s}=0} 2682: 2671:{\textstyle p_{l}=0} 2649: 2638:{\textstyle T_{l}=0} 2616: 2589: 2513: 2368: 2246: 2183: 2156: 2032: 2000: 1970: 1938: 1912: 1882: 1768: 1736: 1716: 1511: 1469: 1438: 1406: 1374: 1348: 1316: 1284: 1264: 1244: 1209: 1177: 1150: 1064: 1007: 987: 967: 920: 782: 757: 737: 717: 637: 598: 571: 484: 165:Fermionic condensate 57:. A superfluid is a 6414:Phased-array optics 6372:Acoustic levitation 5952:Chemical ionization 5844:Programmable matter 5834:Quantum spin liquid 5702:Supercritical fluid 5583:2014PhRvB..90m4503R 5505:1999RvMPS..71..318L 5402:I.I. Hirschman, Jr. 5358:2011JPhB...44s5303A 5305:2012EPJB...85..273Z 5244:2003PhRvL..90y0403S 5187:2011CEJPh...9..857I 5142:1964PhRv..135.1166L 5136:(5A): A1166–A1172. 5107:1957PhRv..106.1135L 5072:1956PhRv..102.1189F 5030:1954PhRv...94..262F 4959:1941Phy.....8..655B 4924:1947PhRv...72..838T 4879:1938Natur.141..913T 4832:1938Natur.141..643L 4781:2018EPJB...91..226A 4715:1985PhRvB..32.2870C 4676:1975PhLA...53..327S 4604:1966PhL....20..474V 4468:1935Phy.....2..557K 4433:1950PhRv...79..626S 4380:2016NatSR...628992O 4285:2007PhRvL..98m5301P 4224:2006PhRvL..96m5301W 4171:2007PhRvL..98q5302R 4110:2006PhRvL..97p5301R 4033:10.1038/nature02220 4025:2004Natur.427..225K 3965:2006Natur.441..588B 3915:1985PhRvL..55.2704A 3882:on November 7, 2017 3855:1982PhyBC.109.1474P 3817:1964PhRv..136.1194R 3774:1956RSPSA.238..215H 3729:2010PhT....63i..38V 3682:1938Natur.142..643A 3637:1938Natur.141...74K 3520:Douglas D. Osheroff 2347:Substitution of Eq. 2049: 1989:{\textstyle V_{m0}} 1901:{\textstyle V_{m0}} 1785: 1635: 1570: 713:In this expression 371:/ρ as functions of 31:Superfluid helium-4 6099:Leidenfrost effect 6028:Enthalpy of fusion 5793:Quark–gluon plasma 5423:Basic superfluids. 5165:Cent. Eur. J. Phys 4743:Zeegers, J. C. H. 4557:Journal of Physics 4368:Scientific Reports 4068:2013-04-08 at the 3600:. Nobel Foundation 3477:; it suggests the 3452:Hard-sphere models 3443:in the 1940s, and 3391:Microscopic theory 3364: 3328: 3308: 3265: 3193:{\textstyle T_{L}} 3190: 3166:{\textstyle T_{H}} 3163: 3139: 3120:{\textstyle T_{r}} 3117: 3088: 3034: 3001: 2965: 2925: 2892: 2872: 2862:where the indexes 2836: 2701: 2668: 2635: 2605:{\textstyle p_{f}} 2602: 2557: 2483: 2321: 2202: 2172:{\textstyle p_{f}} 2169: 2126: 2035: 2013: 1986: 1950: 1924: 1898: 1852: 1771: 1752:{\textstyle V_{m}} 1749: 1722: 1686: 1621: 1556: 1486: 1465:and in the second 1455: 1427:{\textstyle (p,T)} 1424: 1395:{\textstyle (p,0)} 1392: 1360: 1337:{\textstyle (p,0)} 1334: 1305:{\textstyle (0,0)} 1302: 1270: 1250: 1230: 1193:{\textstyle V_{m}} 1190: 1166:{\textstyle S_{m}} 1163: 1120: 1042: 1033: 1025: 1013: 993: 973: 954:chemical potential 936:{\textstyle v_{s}} 933: 884: 763: 743: 723: 701: 620: 587:{\textstyle M_{4}} 584: 555: 471: 462: 377: 357: 349: 334:Macroscopic theory 324:fermion condensate 169:diatomic molecules 6451: 6450: 6427: 6426: 6234:complexity theory 6219:cellular automata 6147: 6146: 6129:Superheated vapor 6124:Superconductivity 6094:Equation of state 5942:Flash evaporation 5894:Phase transitions 5879:String-net liquid 5772:Photonic molecule 5742:Degenerate matter 5561:Physical Review B 5483:978-0-19-850756-7 5473:James F. Annett: 5406:A note on entropy 4861:L. Tisza (1938). 4826:(3571): 643–644. 4703:Physical Review B 4664:Physics Letters A 4648:978-0-444-89109-9 4508:978-0-444-53309-8 4388:10.1038/srep28992 4341:10.1063/1.1774894 4019:(6971): 225–227. 3909:(24): 2704–2707. 3738:10.1063/1.3490499 3550:Superdiamagnetism 3545:Quantum gyroscope 3540:Quantum acoustics 3427:Vortex ring model 3421:quantum mechanics 3355: 3289: 3288: 3253: 3044:). Consequently, 2860: 2859: 2507: 2506: 2436: 2419: 2399: 2345: 2344: 2237:obtains the form 2150: 2149: 1876: 1875: 1732:is small so that 1710: 1709: 1144: 1143: 1045:Fountain pressure 976:{\textstyle \mu } 908: 907: 850: 833: 813: 726:{\textstyle \mu } 667: 649: 611: 550: 530: 496: 286:Abrikosov lattice 181:BEC-BCS crossover 98:superconductivity 16:(Redirected from 6486: 6439: 6438: 6316:machine learning 6291:key distribution 6276:image processing 6266:error correction 6192: 6191: 6174: 6167: 6160: 6151: 6150: 6084:Compressed fluid 5719: 5664:States of matter 5657: 5650: 5643: 5634: 5633: 5602: 5576: 5516: 5499:(2): S318–S323. 5466:(also available 5409: 5399: 5393: 5384: 5378: 5377: 5351: 5331: 5325: 5324: 5298: 5278: 5272: 5271: 5237: 5235:cond-mat/0301474 5213: 5207: 5206: 5180: 5160: 5154: 5153: 5125: 5119: 5118: 5101:(6): 1135–1145. 5090: 5084: 5083: 5066:(5): 1189–1204. 5057: 5048: 5042: 5041: 5015: 5006: 5000: 4995: 4977: 4971: 4970: 4942: 4936: 4935: 4907: 4901: 4900: 4890: 4888:10.1038/141913a0 4858: 4852: 4851: 4840:10.1038/141643a0 4815: 4809: 4808: 4774: 4754: 4748: 4741: 4735: 4734: 4709:(5): 2870–2886. 4694: 4688: 4687: 4659: 4653: 4652: 4622: 4616: 4615: 4587: 4581: 4566: 4560: 4549: 4543: 4528: 4522: 4519: 4513: 4512: 4486: 4480: 4479: 4451: 4445: 4444: 4416: 4410: 4409: 4399: 4359: 4353: 4352: 4334: 4319: 4313: 4312: 4278: 4276:cond-mat/0702159 4258: 4252: 4251: 4217: 4215:cond-mat/0603003 4197: 4191: 4190: 4164: 4162:cond-mat/0702665 4144: 4138: 4137: 4103: 4101:cond-mat/0604528 4083: 4077: 4059: 4053: 4052: 4001: 3995: 3994: 3976: 3950: 3941: 3935: 3934: 3898: 3892: 3891: 3889: 3887: 3881: 3875:. Archived from 3866: 3844: 3835: 3829: 3828: 3800: 3794: 3793: 3757: 3751: 3750: 3740: 3708: 3702: 3701: 3690:10.1038/142643a0 3665: 3659: 3658: 3648: 3646:10.1038/141074a0 3616: 3610: 3609: 3607: 3605: 3594: 3588: 3587: 3576: 3499:structure factor 3373: 3371: 3370: 3365: 3363: 3362: 3357: 3356: 3348: 3337: 3335: 3334: 3329: 3317: 3315: 3314: 3309: 3307: 3306: 3283: 3274: 3272: 3271: 3266: 3261: 3260: 3255: 3254: 3246: 3239: 3238: 3209: 3199: 3197: 3196: 3191: 3189: 3188: 3172: 3170: 3169: 3164: 3162: 3161: 3126: 3124: 3123: 3118: 3116: 3115: 3097: 3095: 3094: 3089: 3084: 3083: 3068: 3067: 3043: 3041: 3040: 3035: 3027: 3026: 3010: 3008: 3007: 3002: 2994: 2993: 2974: 2972: 2971: 2966: 2964: 2963: 2951: 2950: 2934: 2932: 2931: 2926: 2918: 2917: 2901: 2899: 2898: 2893: 2881: 2879: 2878: 2873: 2854: 2845: 2843: 2842: 2837: 2835: 2834: 2819: 2818: 2806: 2805: 2793: 2792: 2780: 2779: 2764: 2763: 2751: 2750: 2738: 2737: 2720: 2710: 2708: 2707: 2702: 2694: 2693: 2677: 2675: 2674: 2669: 2661: 2660: 2644: 2642: 2641: 2636: 2628: 2627: 2611: 2609: 2608: 2603: 2601: 2600: 2566: 2564: 2563: 2558: 2556: 2555: 2543: 2538: 2537: 2525: 2524: 2501: 2492: 2490: 2489: 2484: 2476: 2475: 2457: 2456: 2438: 2437: 2429: 2420: 2418: 2414: 2408: 2407: 2406: 2401: 2400: 2392: 2388: 2382: 2380: 2379: 2362: 2339: 2330: 2328: 2327: 2322: 2314: 2313: 2295: 2294: 2279: 2278: 2240: 2215: 2211: 2209: 2208: 2203: 2201: 2200: 2178: 2176: 2175: 2170: 2168: 2167: 2144: 2135: 2133: 2132: 2127: 2122: 2121: 2112: 2111: 2096: 2095: 2086: 2078: 2077: 2059: 2058: 2048: 2043: 2026: 2022: 2020: 2019: 2014: 2012: 2011: 1995: 1993: 1992: 1987: 1985: 1984: 1959: 1957: 1956: 1953:{\textstyle p=0} 1951: 1933: 1931: 1930: 1927:{\textstyle T=0} 1925: 1907: 1905: 1904: 1899: 1897: 1896: 1870: 1861: 1859: 1858: 1853: 1848: 1847: 1832: 1831: 1822: 1808: 1807: 1795: 1794: 1784: 1779: 1762: 1758: 1756: 1755: 1750: 1748: 1747: 1731: 1729: 1728: 1723: 1704: 1695: 1693: 1692: 1687: 1682: 1681: 1672: 1664: 1663: 1645: 1644: 1634: 1629: 1617: 1616: 1607: 1593: 1592: 1580: 1579: 1569: 1564: 1505: 1495: 1493: 1492: 1487: 1476: 1464: 1462: 1461: 1456: 1445: 1433: 1431: 1430: 1425: 1401: 1399: 1398: 1393: 1369: 1367: 1366: 1363:{\textstyle T=0} 1361: 1343: 1341: 1340: 1335: 1311: 1309: 1308: 1303: 1279: 1277: 1276: 1271: 1259: 1257: 1256: 1251: 1239: 1237: 1236: 1231: 1199: 1197: 1196: 1191: 1189: 1188: 1172: 1170: 1169: 1164: 1162: 1161: 1138: 1129: 1127: 1126: 1121: 1113: 1108: 1107: 1092: 1087: 1086: 1071: 1058: 1022: 1020: 1019: 1014: 1002: 1000: 999: 994: 982: 980: 979: 974: 942: 940: 939: 934: 932: 931: 902: 893: 891: 890: 885: 871: 870: 852: 851: 843: 834: 832: 828: 822: 821: 820: 815: 814: 806: 802: 796: 794: 793: 776: 772: 770: 769: 764: 752: 750: 749: 744: 732: 730: 729: 724: 710: 708: 707: 702: 688: 687: 669: 668: 660: 651: 650: 642: 629: 627: 626: 621: 619: 618: 613: 612: 604: 593: 591: 590: 585: 583: 582: 564: 562: 561: 556: 551: 549: 545: 539: 538: 537: 532: 531: 523: 519: 513: 511: 510: 498: 497: 489: 379:Figure 1 is the 214:techniques as a 154:Josephson effect 149:quantized vortex 142:phenomenological 138:phase transition 21: 6494: 6493: 6489: 6488: 6487: 6485: 6484: 6483: 6454: 6453: 6452: 6447: 6423: 6360: 6271:finite automata 6183: 6178: 6148: 6143: 6074:Baryonic matter 6062: 6016: 5987:Saturated fluid 5927:Crystallization 5888: 5862:Antiferromagnet 5802: 5776: 5720: 5711: 5671: 5661: 5613:Wayback Machine 5535: 5530: 5417: 5415:Further reading 5412: 5400: 5396: 5385: 5381: 5332: 5328: 5283:Eur. Phys. J. B 5279: 5275: 5214: 5210: 5161: 5157: 5126: 5122: 5091: 5087: 5055: 5049: 5045: 5013: 5007: 5003: 4992: 4978: 4974: 4943: 4939: 4908: 4904: 4859: 4855: 4816: 4812: 4755: 4751: 4742: 4738: 4695: 4691: 4660: 4656: 4649: 4623: 4619: 4592:Physics Letters 4588: 4584: 4567: 4563: 4550: 4546: 4529: 4525: 4520: 4516: 4509: 4487: 4483: 4452: 4448: 4421:Physical Review 4417: 4413: 4360: 4356: 4332: 4320: 4316: 4263:Phys. Rev. Lett 4259: 4255: 4202:Phys. Rev. Lett 4198: 4194: 4149:Phys. Rev. Lett 4145: 4141: 4088:Phys. Rev. Lett 4084: 4080: 4070:Wayback Machine 4060: 4056: 4002: 3998: 3974:10.1038/441588a 3948: 3942: 3938: 3899: 3895: 3885: 3883: 3879: 3864:10.1.1.210.8701 3842: 3836: 3832: 3805:Physical Review 3801: 3797: 3758: 3754: 3709: 3705: 3666: 3662: 3617: 3613: 3603: 3601: 3596: 3595: 3591: 3580:"Superfluidity" 3578: 3577: 3573: 3569: 3564: 3555:Superfluid film 3515: 3463: 3454: 3445:Richard Feynman 3429: 3398: 3393: 3358: 3347: 3346: 3345: 3343: 3340: 3339: 3323: 3320: 3319: 3302: 3298: 3296: 3293: 3292: 3256: 3245: 3244: 3243: 3234: 3230: 3216: 3213: 3212: 3184: 3180: 3178: 3175: 3174: 3157: 3153: 3151: 3148: 3147: 3144: 3111: 3107: 3105: 3102: 3101: 3076: 3072: 3063: 3059: 3051: 3048: 3047: 3022: 3018: 3016: 3013: 3012: 2986: 2982: 2980: 2977: 2976: 2959: 2955: 2946: 2942: 2940: 2937: 2936: 2913: 2909: 2907: 2904: 2903: 2887: 2884: 2883: 2867: 2864: 2863: 2827: 2823: 2814: 2810: 2801: 2797: 2788: 2784: 2772: 2768: 2759: 2755: 2746: 2742: 2733: 2729: 2727: 2724: 2723: 2689: 2685: 2683: 2680: 2679: 2656: 2652: 2650: 2647: 2646: 2623: 2619: 2617: 2614: 2613: 2596: 2592: 2590: 2587: 2586: 2548: 2544: 2539: 2533: 2529: 2520: 2516: 2514: 2511: 2510: 2471: 2467: 2452: 2448: 2428: 2427: 2410: 2409: 2402: 2391: 2390: 2389: 2384: 2383: 2381: 2375: 2371: 2369: 2366: 2365: 2309: 2305: 2287: 2283: 2274: 2270: 2247: 2244: 2243: 2213: 2196: 2192: 2184: 2181: 2180: 2163: 2159: 2157: 2154: 2153: 2117: 2113: 2104: 2100: 2091: 2087: 2082: 2073: 2069: 2054: 2050: 2044: 2039: 2033: 2030: 2029: 2007: 2003: 2001: 1998: 1997: 1996:and a quantity 1977: 1973: 1971: 1968: 1967: 1939: 1936: 1935: 1913: 1910: 1909: 1889: 1885: 1883: 1880: 1879: 1840: 1836: 1827: 1823: 1818: 1803: 1799: 1790: 1786: 1780: 1775: 1769: 1766: 1765: 1743: 1739: 1737: 1734: 1733: 1717: 1714: 1713: 1677: 1673: 1668: 1659: 1655: 1640: 1636: 1630: 1625: 1612: 1608: 1603: 1588: 1584: 1575: 1571: 1565: 1560: 1512: 1509: 1508: 1472: 1470: 1467: 1466: 1441: 1439: 1436: 1435: 1407: 1404: 1403: 1375: 1372: 1371: 1349: 1346: 1345: 1317: 1314: 1313: 1285: 1282: 1281: 1265: 1262: 1261: 1245: 1242: 1241: 1210: 1207: 1206: 1184: 1180: 1178: 1175: 1174: 1157: 1153: 1151: 1148: 1147: 1109: 1103: 1099: 1088: 1082: 1078: 1067: 1065: 1062: 1061: 1047: 1008: 1005: 1004: 988: 985: 984: 968: 965: 964: 927: 923: 921: 918: 917: 866: 862: 842: 841: 824: 823: 816: 805: 804: 803: 798: 797: 795: 789: 785: 783: 780: 779: 758: 755: 754: 738: 735: 734: 718: 715: 714: 683: 679: 659: 658: 641: 640: 638: 635: 634: 614: 603: 602: 601: 599: 596: 595: 578: 574: 572: 569: 568: 541: 540: 533: 522: 521: 520: 515: 514: 512: 506: 502: 488: 487: 485: 482: 481: 476: 448: 444: 436: 432: 425: 418: 414: 410: 406: 370: 366: 341: 336: 294: 281: 272: 263: 253: 231:Gravity Probe B 216:quantum solvent 208: 106: 59:state of matter 53:of the element 28: 23: 22: 15: 12: 11: 5: 6492: 6482: 6481: 6476: 6474:Fluid dynamics 6471: 6466: 6449: 6448: 6446: 6445: 6432: 6429: 6428: 6425: 6424: 6422: 6421: 6416: 6411: 6406: 6401: 6400: 6399: 6389: 6384: 6379: 6374: 6368: 6366: 6362: 6361: 6359: 6358: 6353: 6348: 6343: 6338: 6333: 6331:neural network 6328: 6323: 6318: 6313: 6308: 6303: 6298: 6293: 6288: 6283: 6278: 6273: 6268: 6263: 6258: 6253: 6252: 6251: 6241: 6236: 6231: 6226: 6221: 6216: 6211: 6206: 6200: 6198: 6189: 6185: 6184: 6177: 6176: 6169: 6162: 6154: 6145: 6144: 6142: 6141: 6136: 6131: 6126: 6121: 6116: 6111: 6106: 6101: 6096: 6091: 6086: 6081: 6076: 6070: 6068: 6064: 6063: 6061: 6060: 6055: 6053:Trouton's rule 6050: 6045: 6040: 6035: 6030: 6024: 6022: 6018: 6017: 6015: 6014: 6009: 6004: 5999: 5994: 5989: 5984: 5979: 5974: 5969: 5964: 5959: 5954: 5949: 5944: 5939: 5934: 5929: 5924: 5922:Critical point 5919: 5914: 5909: 5904: 5898: 5896: 5890: 5889: 5887: 5886: 5881: 5876: 5875: 5874: 5869: 5864: 5856: 5851: 5846: 5841: 5836: 5831: 5826: 5824:Liquid crystal 5821: 5816: 5810: 5808: 5804: 5803: 5801: 5800: 5795: 5790: 5784: 5782: 5778: 5777: 5775: 5774: 5769: 5764: 5759: 5757:Strange matter 5754: 5752:Rydberg matter 5749: 5744: 5739: 5734: 5728: 5726: 5722: 5721: 5714: 5712: 5710: 5709: 5704: 5699: 5690: 5685: 5679: 5677: 5673: 5672: 5660: 5659: 5652: 5645: 5637: 5631: 5630: 5625: 5620: 5615: 5603: 5567:(13): 134503. 5556: 5551: 5546: 5541: 5534: 5533:External links 5531: 5529: 5528: 5520: 5517: 5485: 5471: 5448:Hagen Kleinert 5445: 5440: 5433: 5418: 5416: 5413: 5411: 5410: 5394: 5379: 5342:(19): 195303. 5326: 5273: 5228:(25): 250403. 5208: 5171:(3): 857–864. 5155: 5120: 5085: 5043: 5001: 4991:978-9810241315 4990: 4972: 4953:(7): 655–675. 4937: 4918:(9): 838–854. 4902: 4853: 4810: 4749: 4736: 4689: 4654: 4647: 4617: 4582: 4561: 4544: 4523: 4514: 4507: 4481: 4446: 4411: 4354: 4314: 4269:(13): 135301. 4253: 4208:(13): 135301. 4192: 4155:(17): 175302. 4139: 4094:(16): 165301. 4078: 4054: 3996: 3936: 3893: 3830: 3795: 3752: 3703: 3660: 3611: 3589: 3570: 3568: 3565: 3563: 3562: 3557: 3552: 3547: 3542: 3537: 3532: 3527: 3522: 3516: 3514: 3511: 3495:sound velocity 3462: 3459: 3453: 3450: 3437:quantum vortex 3428: 3425: 3401:L. D. Landau's 3397: 3394: 3392: 3389: 3361: 3354: 3351: 3331:{\textstyle Z} 3327: 3305: 3301: 3287: 3286: 3277: 3275: 3264: 3259: 3252: 3249: 3242: 3237: 3233: 3229: 3226: 3223: 3220: 3187: 3183: 3160: 3156: 3143: 3142:Heat transport 3140: 3114: 3110: 3087: 3082: 3079: 3075: 3071: 3066: 3062: 3058: 3055: 3033: 3030: 3025: 3021: 3000: 2997: 2992: 2989: 2985: 2962: 2958: 2954: 2949: 2945: 2924: 2921: 2916: 2912: 2895:{\textstyle r} 2891: 2875:{\textstyle l} 2871: 2858: 2857: 2848: 2846: 2833: 2830: 2826: 2822: 2817: 2813: 2809: 2804: 2800: 2796: 2791: 2787: 2783: 2778: 2775: 2771: 2767: 2762: 2758: 2754: 2749: 2745: 2741: 2736: 2732: 2700: 2697: 2692: 2688: 2667: 2664: 2659: 2655: 2634: 2631: 2626: 2622: 2599: 2595: 2554: 2551: 2547: 2542: 2536: 2532: 2528: 2523: 2519: 2505: 2504: 2495: 2493: 2482: 2479: 2474: 2470: 2466: 2463: 2460: 2455: 2451: 2447: 2444: 2441: 2435: 2432: 2426: 2423: 2417: 2413: 2405: 2398: 2395: 2387: 2378: 2374: 2343: 2342: 2333: 2331: 2320: 2317: 2312: 2308: 2304: 2301: 2298: 2293: 2290: 2286: 2282: 2277: 2273: 2269: 2266: 2263: 2260: 2257: 2254: 2251: 2199: 2195: 2191: 2188: 2166: 2162: 2148: 2147: 2138: 2136: 2125: 2120: 2116: 2110: 2107: 2103: 2099: 2094: 2090: 2085: 2081: 2076: 2072: 2068: 2065: 2062: 2057: 2053: 2047: 2042: 2038: 2010: 2006: 1983: 1980: 1976: 1949: 1946: 1943: 1923: 1920: 1917: 1895: 1892: 1888: 1874: 1873: 1864: 1862: 1851: 1846: 1843: 1839: 1835: 1830: 1826: 1821: 1817: 1814: 1811: 1806: 1802: 1798: 1793: 1789: 1783: 1778: 1774: 1746: 1742: 1725:{\textstyle p} 1721: 1708: 1707: 1698: 1696: 1685: 1680: 1676: 1671: 1667: 1662: 1658: 1654: 1651: 1648: 1643: 1639: 1633: 1628: 1624: 1620: 1615: 1611: 1606: 1602: 1599: 1596: 1591: 1587: 1583: 1578: 1574: 1568: 1563: 1559: 1555: 1552: 1549: 1546: 1543: 1540: 1537: 1534: 1531: 1528: 1525: 1522: 1519: 1516: 1485: 1482: 1479: 1475: 1454: 1451: 1448: 1444: 1423: 1420: 1417: 1414: 1411: 1391: 1388: 1385: 1382: 1379: 1359: 1356: 1353: 1333: 1330: 1327: 1324: 1321: 1301: 1298: 1295: 1292: 1289: 1273:{\textstyle T} 1269: 1253:{\textstyle p} 1249: 1229: 1226: 1223: 1220: 1217: 1214: 1187: 1183: 1160: 1156: 1142: 1141: 1132: 1130: 1119: 1116: 1112: 1106: 1102: 1098: 1095: 1091: 1085: 1081: 1077: 1074: 1070: 1046: 1043: 1016:{\textstyle T} 1012: 996:{\textstyle p} 992: 972: 930: 926: 916:only holds if 906: 905: 896: 894: 883: 880: 877: 874: 869: 865: 861: 858: 855: 849: 846: 840: 837: 831: 827: 819: 812: 809: 801: 792: 788: 766:{\textstyle z} 762: 746:{\textstyle g} 742: 722: 700: 697: 694: 691: 686: 682: 678: 675: 672: 666: 663: 657: 654: 648: 645: 617: 610: 607: 581: 577: 554: 548: 544: 536: 529: 526: 518: 509: 505: 501: 495: 492: 475: 472: 446: 442: 434: 430: 423: 416: 412: 408: 404: 386:critical point 368: 364: 340: 339:Thermodynamics 337: 335: 332: 308:Cooper pairing 293: 290: 280: 277: 271: 268: 252: 249: 207: 204: 158:solid hydrogen 105: 102: 26: 9: 6: 4: 3: 2: 6491: 6480: 6479:Superfluidity 6477: 6475: 6472: 6470: 6467: 6465: 6464:Liquid helium 6462: 6461: 6459: 6444: 6443: 6434: 6433: 6430: 6420: 6417: 6415: 6412: 6410: 6407: 6405: 6402: 6398: 6397:Plasma window 6395: 6394: 6393: 6390: 6388: 6385: 6383: 6380: 6378: 6375: 6373: 6370: 6369: 6367: 6363: 6357: 6356:teleportation 6354: 6352: 6349: 6347: 6344: 6342: 6339: 6337: 6334: 6332: 6329: 6327: 6324: 6322: 6319: 6317: 6314: 6312: 6309: 6307: 6304: 6302: 6299: 6297: 6294: 6292: 6289: 6287: 6284: 6282: 6279: 6277: 6274: 6272: 6269: 6267: 6264: 6262: 6259: 6257: 6254: 6250: 6247: 6246: 6245: 6242: 6240: 6237: 6235: 6232: 6230: 6227: 6225: 6222: 6220: 6217: 6215: 6212: 6210: 6207: 6205: 6202: 6201: 6199: 6197: 6193: 6190: 6186: 6182: 6175: 6170: 6168: 6163: 6161: 6156: 6155: 6152: 6140: 6137: 6135: 6132: 6130: 6127: 6125: 6122: 6120: 6117: 6115: 6112: 6110: 6109:Mpemba effect 6107: 6105: 6102: 6100: 6097: 6095: 6092: 6090: 6089:Cooling curve 6087: 6085: 6082: 6080: 6077: 6075: 6072: 6071: 6069: 6065: 6059: 6056: 6054: 6051: 6049: 6046: 6044: 6041: 6039: 6036: 6034: 6031: 6029: 6026: 6025: 6023: 6019: 6013: 6012:Vitrification 6010: 6008: 6005: 6003: 6000: 5998: 5995: 5993: 5990: 5988: 5985: 5983: 5980: 5978: 5977:Recombination 5975: 5973: 5972:Melting point 5970: 5968: 5965: 5963: 5960: 5958: 5955: 5953: 5950: 5948: 5945: 5943: 5940: 5938: 5935: 5933: 5930: 5928: 5925: 5923: 5920: 5918: 5917:Critical line 5915: 5913: 5910: 5908: 5907:Boiling point 5905: 5903: 5900: 5899: 5897: 5895: 5891: 5885: 5882: 5880: 5877: 5873: 5870: 5868: 5865: 5863: 5860: 5859: 5857: 5855: 5852: 5850: 5847: 5845: 5842: 5840: 5839:Exotic matter 5837: 5835: 5832: 5830: 5827: 5825: 5822: 5820: 5817: 5815: 5812: 5811: 5809: 5805: 5799: 5796: 5794: 5791: 5789: 5786: 5785: 5783: 5779: 5773: 5770: 5768: 5765: 5763: 5760: 5758: 5755: 5753: 5750: 5748: 5745: 5743: 5740: 5738: 5735: 5733: 5730: 5729: 5727: 5723: 5718: 5708: 5705: 5703: 5700: 5698: 5694: 5691: 5689: 5686: 5684: 5681: 5680: 5678: 5674: 5669: 5665: 5658: 5653: 5651: 5646: 5644: 5639: 5638: 5635: 5629: 5626: 5624: 5621: 5619: 5616: 5614: 5610: 5607: 5604: 5600: 5596: 5592: 5588: 5584: 5580: 5575: 5570: 5566: 5562: 5557: 5555: 5552: 5550: 5547: 5545: 5542: 5540: 5537: 5536: 5527: 5526: 5521: 5518: 5514: 5510: 5506: 5502: 5498: 5494: 5490: 5486: 5484: 5480: 5476: 5472: 5469: 5465: 5464:9971-5-0210-0 5461: 5457: 5453: 5449: 5446: 5444: 5441: 5438: 5434: 5432: 5431:0-7484-0891-6 5428: 5424: 5420: 5419: 5407: 5403: 5398: 5392: 5388: 5383: 5375: 5371: 5367: 5363: 5359: 5355: 5350: 5345: 5341: 5337: 5330: 5322: 5318: 5314: 5310: 5306: 5302: 5297: 5292: 5288: 5284: 5277: 5269: 5265: 5261: 5257: 5253: 5249: 5245: 5241: 5236: 5231: 5227: 5223: 5219: 5212: 5204: 5200: 5196: 5192: 5188: 5184: 5179: 5174: 5170: 5166: 5159: 5151: 5147: 5143: 5139: 5135: 5131: 5124: 5116: 5112: 5108: 5104: 5100: 5096: 5089: 5081: 5077: 5073: 5069: 5065: 5061: 5054: 5047: 5039: 5035: 5031: 5027: 5023: 5019: 5012: 5005: 4998: 4993: 4987: 4983: 4976: 4968: 4964: 4960: 4956: 4952: 4948: 4941: 4933: 4929: 4925: 4921: 4917: 4913: 4906: 4898: 4894: 4889: 4884: 4880: 4876: 4873:(3577): 913. 4872: 4868: 4864: 4857: 4849: 4845: 4841: 4837: 4833: 4829: 4825: 4821: 4814: 4806: 4802: 4798: 4794: 4790: 4786: 4782: 4778: 4773: 4768: 4764: 4760: 4753: 4746: 4740: 4732: 4728: 4724: 4720: 4716: 4712: 4708: 4704: 4700: 4693: 4685: 4681: 4677: 4673: 4669: 4665: 4658: 4650: 4644: 4640: 4636: 4632: 4628: 4621: 4613: 4609: 4605: 4601: 4597: 4593: 4586: 4579: 4578:0-7382-0300-9 4575: 4571: 4565: 4558: 4554: 4548: 4541: 4540:0-444-10681-2 4537: 4533: 4527: 4518: 4510: 4504: 4500: 4496: 4492: 4485: 4477: 4473: 4469: 4465: 4461: 4457: 4450: 4442: 4438: 4434: 4430: 4426: 4422: 4415: 4407: 4403: 4398: 4393: 4389: 4385: 4381: 4377: 4373: 4369: 4365: 4358: 4350: 4346: 4342: 4338: 4331: 4330: 4325: 4318: 4310: 4306: 4302: 4298: 4294: 4290: 4286: 4282: 4277: 4272: 4268: 4264: 4257: 4249: 4245: 4241: 4237: 4233: 4229: 4225: 4221: 4216: 4211: 4207: 4203: 4196: 4188: 4184: 4180: 4176: 4172: 4168: 4163: 4158: 4154: 4150: 4143: 4135: 4131: 4127: 4123: 4119: 4115: 4111: 4107: 4102: 4097: 4093: 4089: 4082: 4075: 4071: 4067: 4064: 4058: 4050: 4046: 4042: 4038: 4034: 4030: 4026: 4022: 4018: 4014: 4010: 4009:M. H. W. Chan 4006: 4000: 3992: 3988: 3984: 3980: 3975: 3970: 3966: 3962: 3959:(7093): 588. 3958: 3954: 3947: 3940: 3932: 3928: 3924: 3920: 3916: 3912: 3908: 3904: 3897: 3878: 3874: 3870: 3865: 3860: 3856: 3852: 3848: 3841: 3834: 3826: 3822: 3818: 3814: 3811:(5A): A1194. 3810: 3806: 3799: 3791: 3787: 3783: 3779: 3775: 3771: 3768:(1213): 215. 3767: 3763: 3756: 3748: 3744: 3739: 3734: 3730: 3726: 3722: 3718: 3717:Physics Today 3714: 3707: 3699: 3695: 3691: 3687: 3683: 3679: 3676:(3597): 643. 3675: 3671: 3664: 3656: 3652: 3647: 3642: 3638: 3634: 3630: 3626: 3622: 3615: 3599: 3593: 3585: 3581: 3575: 3571: 3561: 3558: 3556: 3553: 3551: 3548: 3546: 3543: 3541: 3538: 3536: 3533: 3531: 3530:London moment 3528: 3526: 3523: 3521: 3518: 3517: 3510: 3508: 3504: 3500: 3496: 3492: 3488: 3484: 3480: 3476: 3472: 3471:fluid element 3468: 3458: 3449: 3446: 3442: 3438: 3434: 3424: 3422: 3416: 3414: 3409: 3405: 3402: 3388: 3386: 3382: 3378: 3359: 3352: 3349: 3325: 3303: 3299: 3285: 3278: 3276: 3262: 3257: 3250: 3247: 3240: 3235: 3231: 3227: 3224: 3221: 3211: 3210: 3207: 3205: 3204: 3185: 3181: 3158: 3154: 3135: 3131: 3128: 3112: 3108: 3098: 3085: 3080: 3077: 3073: 3069: 3064: 3060: 3056: 3053: 3045: 3031: 3028: 3023: 3019: 2998: 2995: 2990: 2987: 2983: 2960: 2956: 2952: 2947: 2943: 2922: 2919: 2914: 2910: 2889: 2869: 2856: 2849: 2847: 2831: 2828: 2824: 2820: 2815: 2811: 2807: 2802: 2798: 2794: 2789: 2785: 2781: 2776: 2773: 2769: 2765: 2760: 2756: 2752: 2747: 2743: 2739: 2734: 2730: 2722: 2721: 2718: 2716: 2715: 2698: 2695: 2690: 2686: 2665: 2662: 2657: 2653: 2632: 2629: 2624: 2620: 2597: 2593: 2584: 2583: 2577: 2575: 2574: 2568: 2552: 2549: 2545: 2540: 2534: 2530: 2526: 2521: 2517: 2503: 2496: 2494: 2480: 2472: 2468: 2464: 2461: 2458: 2453: 2449: 2445: 2442: 2424: 2421: 2415: 2403: 2393: 2376: 2372: 2364: 2363: 2360: 2358: 2357: 2352: 2351: 2341: 2334: 2332: 2318: 2310: 2306: 2302: 2299: 2291: 2288: 2284: 2280: 2275: 2271: 2267: 2261: 2258: 2255: 2249: 2242: 2241: 2238: 2236: 2235: 2230: 2229: 2224: 2223: 2217: 2197: 2193: 2189: 2186: 2164: 2160: 2152:The pressure 2146: 2139: 2137: 2123: 2118: 2114: 2108: 2105: 2101: 2097: 2088: 2070: 2066: 2063: 2055: 2051: 2045: 2040: 2036: 2028: 2027: 2024: 2008: 2004: 1981: 1978: 1974: 1965: 1964: 1947: 1944: 1941: 1921: 1918: 1915: 1893: 1890: 1886: 1872: 1865: 1863: 1849: 1844: 1841: 1837: 1833: 1824: 1812: 1809: 1800: 1791: 1787: 1781: 1776: 1772: 1764: 1763: 1760: 1744: 1740: 1719: 1706: 1699: 1697: 1683: 1674: 1656: 1652: 1649: 1641: 1637: 1631: 1626: 1622: 1618: 1609: 1597: 1594: 1585: 1576: 1572: 1566: 1561: 1557: 1553: 1547: 1544: 1541: 1535: 1532: 1526: 1523: 1520: 1514: 1507: 1506: 1503: 1501: 1500: 1483: 1480: 1477: 1452: 1449: 1446: 1418: 1415: 1412: 1386: 1383: 1380: 1357: 1354: 1351: 1328: 1325: 1322: 1296: 1293: 1290: 1267: 1247: 1224: 1221: 1218: 1212: 1205: 1204: 1185: 1181: 1158: 1154: 1140: 1133: 1131: 1117: 1114: 1104: 1100: 1096: 1093: 1083: 1079: 1075: 1072: 1060: 1059: 1056: 1054: 1053: 1037: 1029: 1010: 990: 983:at arbitrary 970: 961: 957: 955: 951: 950: 944: 928: 924: 915: 914: 904: 897: 895: 881: 875: 872: 867: 863: 859: 856: 838: 835: 829: 817: 807: 790: 786: 778: 777: 774: 760: 740: 720: 711: 698: 692: 689: 684: 680: 676: 673: 655: 652: 643: 632: 615: 605: 579: 575: 565: 552: 546: 534: 524: 507: 503: 499: 490: 479: 466: 459: 454: 450: 440: 427: 422: 400: 397: 393: 391: 390:absolute zero 387: 382: 381:phase diagram 374: 361: 353: 345: 331: 329: 325: 321: 317: 313: 309: 305: 300: 289: 287: 276: 267: 262: 261:superfluidity 258: 248: 246: 241: 239: 234: 232: 228: 223: 221: 217: 213: 212:spectroscopic 203: 201: 196: 192: 188: 184: 182: 178: 174: 170: 166: 161: 159: 155: 150: 145: 143: 139: 135: 131: 127: 126:John F. Allen 123: 122:Pyotr Kapitsa 119: 118:superfluidity 115: 111: 101: 99: 95: 91: 87: 83: 79: 74: 72: 68: 64: 60: 56: 52: 48: 44: 40: 36: 32: 19: 6440: 6377:Anti-gravity 6321:metamaterial 6249:post-quantum 6244:cryptography 6134:Superheating 6007:Vaporization 6002:Triple point 5997:Supercooling 5962:Lambda point 5912:Condensation 5829:Time crystal 5807:Other states 5747:Quantum Hall 5564: 5560: 5523: 5496: 5492: 5474: 5458:; Paperback 5451: 5436: 5422: 5405: 5397: 5387:Hugh Everett 5382: 5339: 5335: 5329: 5286: 5282: 5276: 5225: 5221: 5211: 5168: 5164: 5158: 5133: 5129: 5123: 5098: 5094: 5088: 5063: 5059: 5046: 5021: 5017: 5004: 4981: 4975: 4950: 4946: 4940: 4915: 4911: 4905: 4870: 4866: 4856: 4823: 4819: 4813: 4762: 4758: 4752: 4744: 4739: 4706: 4702: 4692: 4667: 4663: 4657: 4630: 4620: 4595: 4591: 4585: 4569: 4564: 4556: 4547: 4531: 4526: 4517: 4490: 4484: 4459: 4455: 4449: 4424: 4420: 4414: 4371: 4367: 4357: 4328: 4317: 4266: 4262: 4256: 4205: 4201: 4195: 4152: 4148: 4142: 4091: 4087: 4081: 4073: 4057: 4016: 4012: 3999: 3956: 3952: 3939: 3906: 3902: 3896: 3884:. Retrieved 3877:the original 3846: 3833: 3808: 3804: 3798: 3765: 3761: 3755: 3723:(9): 38–43. 3720: 3716: 3706: 3673: 3669: 3663: 3631:(3558): 74. 3628: 3624: 3614: 3604:February 10, 3602:. Retrieved 3592: 3583: 3574: 3464: 3455: 3433:Lars Onsager 3430: 3417: 3410: 3406: 3399: 3290: 3279: 3201: 3145: 3129: 3099: 3046: 2861: 2850: 2712: 2580: 2578: 2571: 2569: 2508: 2497: 2354: 2348: 2346: 2335: 2232: 2226: 2220: 2218: 2151: 2140: 1961: 1877: 1866: 1711: 1700: 1497: 1201: 1145: 1134: 1050: 1048: 947: 945: 911: 909: 898: 712: 633: 566: 480: 477: 439:second sound 428: 420: 401: 398: 394: 378: 372: 295: 282: 273: 264: 242: 235: 224: 209: 206:Applications 185: 177:Cooper pairs 171:and undergo 162: 146: 107: 75: 38: 34: 30: 29: 6392:Force field 6341:programming 6301:logic clock 6286:information 6261:electronics 6043:Latent heat 5992:Sublimation 5937:Evaporation 5872:Ferromagnet 5867:Ferrimagnet 5849:Dark matter 5781:High energy 5489:Leggett, A. 4765:(10): 226. 3886:November 7, 3503:dissipative 458:Rollin film 245:cryocoolers 187:Supersolids 130:Don Misener 6458:Categories 6306:logic gate 6204:algorithms 6058:Volatility 6021:Quantities 5982:Regelation 5957:Ionization 5932:Deposition 5884:Superglass 5854:Antimatter 5788:QCD matter 5767:Supersolid 5762:Superfluid 5725:Low energy 5289:(8): 273. 5024:(2): 262. 4772:1806.11034 4670:(4): 327. 4598:(5): 474. 4462:(1): 557. 4427:(4): 626. 4063:Supersolid 3567:References 3377:heat pipes 2579:So far Eq. 1502:we obtain 1496:. With Eq. 411:/ρ, with ρ 304:BCS theory 255:See also: 251:Properties 227:gyroscopes 220:rovibronic 200:superglass 65:with zero 43:superfluid 6351:simulator 6239:computing 6209:amplifier 5599:118518974 5574:1403.5472 5374:119248001 5349:1108.0847 5321:118545094 5296:1204.4652 5203:118633189 5178:1004.0442 5130:Phys. Rev 5095:Phys. Rev 5060:Phys. Rev 5018:Phys. Rev 4912:Phys. Rev 4797:1434-6028 4374:: 28992. 4349:109758743 4187:119469548 3859:CiteSeerX 3847:Physica B 3790:120738827 3747:0031-9228 3441:Arie Bijl 3353:˙ 3300:η 3251:˙ 3232:η 3228:− 3219:Δ 3070:− 2821:− 2799:ρ 2766:− 2744:ρ 2518:ρ 2465:− 2450:ρ 2434:→ 2431:∇ 2425:− 2397:→ 2373:ρ 2303:− 2272:μ 2250:μ 2219:With Eqs. 2198:λ 2093:′ 2075:′ 2037:∫ 1829:′ 1805:′ 1773:∫ 1679:′ 1661:′ 1623:∫ 1619:− 1614:′ 1590:′ 1558:∫ 1536:μ 1515:μ 1213:μ 1097:− 1073:μ 1040:fountain. 971:μ 910:Eq. 857:μ 848:→ 845:∇ 839:− 811:→ 721:μ 674:μ 665:→ 662:∇ 656:− 647:→ 609:→ 567:The mass 528:→ 494:→ 312:electrons 270:Film flow 195:annealing 132:in 1937. 67:viscosity 41:) is the 35:helium II 6256:dynamics 6119:Spinodal 6067:Concepts 5947:Freezing 5609:Archived 5268:25309672 5260:12857119 4805:53464405 4406:27363671 4309:20038102 4301:17501209 4248:41657202 4240:16711998 4134:45453420 4126:17155406 4066:Archived 4041:14724632 3983:16738652 3931:10032216 3513:See also 3479:Gaussian 2717:implies 1344:, so at 407:/ρ and ρ 367:/ρ and ρ 279:Rotation 82:helium-3 47:helium-4 45:form of 6346:sensing 6326:network 6311:machine 6281:imaging 6229:circuit 6224:channel 6196:Quantum 6079:Binodal 5967:Melting 5902:Boiling 5819:Crystal 5814:Colloid 5579:Bibcode 5501:Bibcode 5354:Bibcode 5301:Bibcode 5240:Bibcode 5183:Bibcode 5138:Bibcode 5103:Bibcode 5068:Bibcode 5026:Bibcode 4955:Bibcode 4947:Physica 4920:Bibcode 4897:4116542 4875:Bibcode 4848:4143290 4828:Bibcode 4777:Bibcode 4731:9937394 4711:Bibcode 4672:Bibcode 4600:Bibcode 4464:Bibcode 4456:Physica 4429:Bibcode 4397:4929499 4376:Bibcode 4281:Bibcode 4220:Bibcode 4167:Bibcode 4106:Bibcode 4049:3112651 4021:Bibcode 3991:4429923 3961:Bibcode 3911:Bibcode 3851:Bibcode 3813:Bibcode 3770:Bibcode 3725:Bibcode 3698:4135906 3678:Bibcode 3655:3997900 3633:Bibcode 3439:rings. 3011:(since 322:. (See 320:phonons 222:state. 104:History 94:fermion 71:inertia 51:isotope 6336:optics 6188:Fields 5707:Plasma 5688:Liquid 5597: 5481: 5468:online 5462: 5429: 5372: 5319: 5266: 5258: 5201: 4988: 4895: 4867:Nature 4846: 4820:Nature 4803: 4795: 4729: 4645: 4576: 4538: 4505: 4404: 4394: 4347: 4307: 4299: 4246: 4238: 4185: 4132: 4124: 4047: 4039: 4013:Nature 4005:E. Kim 3989: 3981: 3953:Nature 3929: 3861: 3788: 3745: 3696: 3670:Nature 3653: 3625:Nature 3483:phonon 2975:, and 2711:so Eq. 2359:gives 1878:where 469:empty. 259:, and 128:, and 116:, the 55:helium 6365:Other 6296:logic 5697:Vapor 5683:Solid 5676:State 5595:S2CID 5569:arXiv 5370:S2CID 5344:arXiv 5317:S2CID 5291:arXiv 5264:S2CID 5230:arXiv 5199:S2CID 5173:arXiv 5056:(PDF) 5014:(PDF) 4893:S2CID 4844:S2CID 4801:S2CID 4767:arXiv 4345:S2CID 4333:(PDF) 4305:S2CID 4271:arXiv 4244:S2CID 4210:arXiv 4183:S2CID 4157:arXiv 4130:S2CID 4096:arXiv 4076:2004. 4045:S2CID 3987:S2CID 3949:(PDF) 3880:(PDF) 3843:(PDF) 3786:S2CID 3694:S2CID 3651:S2CID 3491:roton 3487:maxon 3413:roton 3291:Here 2509:with 2231:, Eq. 1146:Here 134:Onnes 86:boson 63:fluid 49:, an 39:He-II 6442:List 5668:list 5479:ISBN 5460:ISBN 5427:ISBN 5256:PMID 4986:ISBN 4793:ISSN 4727:PMID 4643:ISBN 4574:ISBN 4536:ISBN 4503:ISBN 4402:PMID 4297:PMID 4236:PMID 4122:PMID 4037:PMID 4007:and 3979:PMID 3927:PMID 3888:2017 3743:ISSN 3606:2017 3497:and 3489:and 3385:CERN 3173:and 2882:and 2225:and 1934:and 1003:and 316:spin 238:IRAS 124:and 112:and 90:spin 6214:bus 5693:Gas 5587:doi 5509:doi 5362:doi 5309:doi 5248:doi 5191:doi 5146:doi 5134:135 5111:doi 5099:106 5076:doi 5064:102 5034:doi 4963:doi 4928:doi 4883:doi 4871:141 4836:doi 4824:141 4785:doi 4719:doi 4680:doi 4635:doi 4608:doi 4495:doi 4472:doi 4437:doi 4392:PMC 4384:doi 4337:doi 4289:doi 4228:doi 4175:doi 4114:doi 4072:." 4029:doi 4017:427 3969:doi 3957:441 3919:doi 3869:doi 3821:doi 3809:136 3778:doi 3766:238 3733:doi 3686:doi 3674:142 3641:doi 3629:141 3383:at 3203:(7) 2714:(7) 2582:(5) 2573:(7) 2570:Eq. 2356:(1) 2353:in 2350:(6) 2234:(3) 2228:(5) 2222:(4) 1963:(3) 1499:(2) 1402:to 1312:to 1203:(2) 1052:(1) 949:(1) 913:(1) 433:+ ρ 233:). 37:or 6460:: 5695:/ 5593:. 5585:. 5577:. 5565:90 5563:. 5507:. 5497:71 5495:. 5450:, 5404:, 5368:. 5360:. 5352:. 5340:44 5338:. 5315:. 5307:. 5299:. 5287:85 5285:. 5262:. 5254:. 5246:. 5238:. 5226:90 5224:. 5220:. 5197:. 5189:. 5181:. 5167:. 5144:. 5132:. 5109:. 5097:. 5074:. 5062:. 5058:. 5032:. 5022:94 5020:. 5016:. 4961:. 4949:. 4926:. 4916:72 4914:. 4891:. 4881:. 4869:. 4865:. 4842:. 4834:. 4822:. 4799:. 4791:. 4783:. 4775:. 4763:91 4761:. 4725:. 4717:. 4707:32 4705:. 4701:. 4678:. 4668:53 4666:. 4641:. 4629:. 4606:. 4596:20 4594:. 4555:, 4501:. 4470:. 4458:. 4435:. 4425:79 4423:. 4400:. 4390:. 4382:. 4370:. 4366:. 4343:. 4326:. 4303:. 4295:. 4287:. 4279:. 4267:98 4265:. 4242:. 4234:. 4226:. 4218:. 4206:96 4204:. 4181:. 4173:. 4165:. 4153:98 4151:. 4128:. 4120:. 4112:. 4104:. 4092:97 4090:. 4043:. 4035:. 4027:. 4015:. 3985:. 3977:. 3967:. 3955:. 3951:. 3925:. 3917:. 3907:55 3905:. 3867:. 3857:. 3845:. 3819:. 3807:. 3784:. 3776:. 3764:. 3741:. 3731:. 3721:63 3719:. 3715:. 3692:. 3684:. 3672:. 3649:. 3639:. 3627:. 3623:. 3582:. 3509:. 3485:, 3387:. 3127:. 2935:, 415:(ρ 330:. 183:. 160:. 100:. 73:. 6173:e 6166:t 6159:v 5670:) 5666:( 5656:e 5649:t 5642:v 5601:. 5589:: 5581:: 5571:: 5515:. 5511:: 5503:: 5470:) 5376:. 5364:: 5356:: 5346:: 5323:. 5311:: 5303:: 5293:: 5270:. 5250:: 5242:: 5232:: 5205:. 5193:: 5185:: 5175:: 5169:9 5152:. 5148:: 5140:: 5117:. 5113:: 5105:: 5082:. 5078:: 5070:: 5040:. 5036:: 5028:: 4994:. 4969:. 4965:: 4957:: 4951:8 4934:. 4930:: 4922:: 4899:. 4885:: 4877:: 4850:. 4838:: 4830:: 4807:. 4787:: 4779:: 4769:: 4733:. 4721:: 4713:: 4686:. 4682:: 4674:: 4651:. 4637:: 4614:. 4610:: 4602:: 4580:. 4542:. 4511:. 4497:: 4478:. 4474:: 4466:: 4460:2 4443:. 4439:: 4431:: 4408:. 4386:: 4378:: 4372:6 4351:. 4339:: 4311:. 4291:: 4283:: 4273:: 4250:. 4230:: 4222:: 4212:: 4189:. 4177:: 4169:: 4159:: 4136:. 4116:: 4108:: 4098:: 4051:. 4031:: 4023:: 3993:. 3971:: 3963:: 3933:. 3921:: 3913:: 3890:. 3871:: 3853:: 3827:. 3823:: 3815:: 3792:. 3780:: 3772:: 3749:. 3735:: 3727:: 3700:. 3688:: 3680:: 3657:. 3643:: 3635:: 3608:. 3360:n 3350:V 3326:Z 3304:n 3284:) 3282:9 3280:( 3263:. 3258:n 3248:V 3241:Z 3236:n 3225:= 3222:p 3186:L 3182:T 3159:H 3155:T 3113:r 3109:T 3086:. 3081:r 3078:f 3074:p 3065:r 3061:p 3057:= 3054:0 3032:0 3029:= 3024:l 3020:T 2999:0 2996:= 2991:l 2988:f 2984:p 2961:r 2957:z 2953:= 2948:l 2944:z 2923:0 2920:= 2915:l 2911:p 2890:r 2870:l 2855:) 2853:8 2851:( 2832:r 2829:f 2825:p 2816:r 2812:z 2808:g 2803:0 2795:+ 2790:r 2786:p 2782:= 2777:l 2774:f 2770:p 2761:l 2757:z 2753:g 2748:0 2740:+ 2735:l 2731:p 2699:0 2696:= 2691:s 2687:v 2666:0 2663:= 2658:l 2654:p 2633:0 2630:= 2625:l 2621:T 2598:f 2594:p 2553:0 2550:m 2546:V 2541:/ 2535:4 2531:M 2527:= 2522:0 2502:) 2500:7 2498:( 2481:. 2478:) 2473:f 2469:p 2462:z 2459:g 2454:0 2446:+ 2443:p 2440:( 2422:= 2416:t 2412:d 2404:s 2394:v 2386:d 2377:0 2340:) 2338:6 2336:( 2319:. 2316:) 2311:f 2307:p 2300:p 2297:( 2292:0 2289:m 2285:V 2281:+ 2276:0 2268:= 2265:) 2262:T 2259:, 2256:p 2253:( 2214:g 2194:T 2190:= 2187:T 2165:f 2161:p 2145:) 2143:5 2141:( 2124:. 2119:f 2115:p 2109:0 2106:m 2102:V 2098:= 2089:T 2084:d 2080:) 2071:T 2067:, 2064:p 2061:( 2056:m 2052:S 2046:T 2041:0 2009:f 2005:p 1982:0 1979:m 1975:V 1948:0 1945:= 1942:p 1922:0 1919:= 1916:T 1894:0 1891:m 1887:V 1871:) 1869:4 1867:( 1850:p 1845:0 1842:m 1838:V 1834:= 1825:p 1820:d 1816:) 1813:0 1810:, 1801:p 1797:( 1792:m 1788:V 1782:p 1777:0 1745:m 1741:V 1720:p 1705:) 1703:3 1701:( 1684:. 1675:T 1670:d 1666:) 1657:T 1653:, 1650:p 1647:( 1642:m 1638:S 1632:T 1627:0 1610:p 1605:d 1601:) 1598:0 1595:, 1586:p 1582:( 1577:m 1573:V 1567:p 1562:0 1554:+ 1551:) 1548:0 1545:, 1542:0 1539:( 1533:= 1530:) 1527:T 1524:, 1521:p 1518:( 1484:0 1481:= 1478:p 1474:d 1453:0 1450:= 1447:T 1443:d 1422:) 1419:T 1416:, 1413:p 1410:( 1390:) 1387:0 1384:, 1381:p 1378:( 1358:0 1355:= 1352:T 1332:) 1329:0 1326:, 1323:p 1320:( 1300:) 1297:0 1294:, 1291:0 1288:( 1268:T 1260:– 1248:p 1228:) 1225:T 1222:, 1219:p 1216:( 1186:m 1182:V 1159:m 1155:S 1139:) 1137:2 1135:( 1118:. 1115:T 1111:d 1105:m 1101:S 1094:p 1090:d 1084:m 1080:V 1076:= 1069:d 1023:. 1011:T 991:p 929:s 925:v 903:) 901:1 899:( 882:. 879:) 876:z 873:g 868:4 864:M 860:+ 854:( 836:= 830:t 826:d 818:s 808:v 800:d 791:4 787:M 761:z 741:g 699:. 696:) 693:z 690:g 685:4 681:M 677:+ 671:( 653:= 644:F 616:s 606:v 580:4 576:M 553:. 547:t 543:d 535:s 525:v 517:d 508:4 504:M 500:= 491:F 447:n 443:n 435:s 431:n 424:λ 421:T 417:s 413:n 409:s 405:n 403:ρ 375:. 373:T 369:s 365:n 33:( 20:)

Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.

Knowledge

Superfluid helium-4

Index