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{{Short description|State of matter at low temperatures}}
{{Use American English|date=January 2019}}
{{Use mdy dates|date=January 2019}}
'''Superfluid helium-4''' ('''helium II''' or '''He-II''') is the [[superfluid]] form of [[helium-4]], the most common [[isotope]] of the element [[helium]]. 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 [[inertia]].<ref>{{cite book | chapter-url=https://www.sciencedirect.com/topics/physics-and-astronomy/superfluidity | title=Encyclopedia of Condensed Matter Physics | chapter=Superfluidity | year=2005 | pages=128–133 | publisher=Elsevier }}</ref>

The formation of the superfluid is a manifestation of the formation of a [[Bose–Einstein condensate]] of helium atoms. This condensation occurs in liquid helium-4 at a far higher temperature (2.17&nbsp;K) than it does in [[helium-3]] (2.5&nbsp;mK) because each atom of helium-4 is a [[boson]] particle, by virtue of its zero [[Spin (physics)|spin]]. Helium-3, however, is a [[fermion]] 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 [[superconductivity]].<ref>{{Cite web|url=https://www.nobelprize.org/nobel_prizes/physics/laureates/1996/advanced.html|title=The Nobel Prize in Physics 1996 - Advanced Information|publisher=Nobel Foundation|access-date=2017-02-10}}</ref>

== History ==
Known as a major facet in the study of [[quantum hydrodynamics]] and [[macroscopic quantum phenomena]], the [[superfluidity]] effect was discovered by [[Pyotr Kapitsa]]<ref>{{cite journal|last1=Kapitza|first1=P.|year=1938|title=Viscosity of Liquid Helium Below the λ-Point|journal=Nature|volume=141|issue=3558|page=74|bibcode=1938Natur.141...74K|doi=10.1038/141074a0|s2cid=3997900|doi-access=free}}</ref> and [[John F. Allen (physicist)|John F. Allen]], and [[Don Misener]]<ref>{{cite journal|last1=Allen|first1=J. F.|last2=Misener|first2=A. D.|year=1938|title=Flow of Liquid Helium II|journal=Nature|volume=142|issue=3597|page=643|bibcode=1938Natur.142..643A|doi=10.1038/142643a0|s2cid=4135906}}</ref> in 1937. [[Heike Kamerlingh Onnes|Onnes]] possibly observed the superfluid [[phase transition]] on August 2, 1911, the same day that he observed superconductivity in mercury.<ref>{{Cite journal |last1=van Delft |first1=Dirk |last2=Kes |first2=Peter |date=2010-09-01 |title=The discovery of superconductivity |journal=Physics Today |language=en |volume=63 |issue=9 |pages=38–43 |doi=10.1063/1.3490499 |bibcode=2010PhT....63i..38V |issn=0031-9228|doi-access=free }}</ref> It has since been described through [[Phenomenology (particle physics)|phenomenological]] and microscopic theories.

In the 1950s, Hall and Vinen performed experiments establishing the existence of [[quantized vortex]] lines in superfluid helium.<ref>{{cite journal|last1=Hall|first1=H. E.|last2=Vinen|first2=W. F.|year=1956|title=The Rotation of Liquid Helium II. II. The Theory of Mutual Friction in Uniformly Rotating Helium II|journal=Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences|volume=238|issue=1213|page=215|bibcode=1956RSPSA.238..215H|doi=10.1098/rspa.1956.0215|s2cid=120738827}}</ref> In the 1960s, Rayfield and Reif established the existence of quantized vortex rings.<ref>{{cite journal|last1=Rayfield|first1=G.|last2=Reif|first2=F.|year=1964|title=Quantized Vortex Rings in Superfluid Helium|journal=Physical Review|volume=136|issue=5A|article-number=A1194|bibcode=1964PhRv..136.1194R|doi=10.1103/PhysRev.136.A1194}}</ref> Packard has observed the intersection of vortex lines with the free surface of the fluid,<ref>{{cite journal|last1=Packard|first1=Richard E.|year=1982|title=Vortex photography in liquid helium|url=http://physics.berkeley.edu/sites/default/files/_/vortexphotography1982_0.pdf|journal=Physica B|volume=109–110|pages=1474–1484|bibcode=1982PhyBC.109.1474P|citeseerx=10.1.1.210.8701|doi=10.1016/0378-4363(82)90510-1|access-date=November 7, 2017|archive-date=November 7, 2017|archive-url=https://web.archive.org/web/20171107221820/http://physics.berkeley.edu/sites/default/files/_/vortexphotography1982_0.pdf}}</ref>
and Avenel and Varoquaux have studied the [[Josephson effect]] in superfluid helium-4.<ref>{{cite journal|last1=Avenel|first1=O.|last2=Varoquaux|first2=E.|year=1985|title=Observation of Singly Quantized Dissipation Events Obeying the Josephson Frequency Relation in the Critical Flow of Superfluid ^{4}He through an Aperture|journal=Physical Review Letters|volume=55|issue=24|pages=2704–2707|bibcode=1985PhRvL..55.2704A|doi=10.1103/PhysRevLett.55.2704|pmid=10032216}}{{Dead link|date=June 2018|bot=InternetArchiveBot|fix-attempted=no}}</ref> In 2006, a group at the University of Maryland visualized quantized vortices by using small tracer particles of [[solid hydrogen]].<ref>{{cite journal|last1=Bewley|first1=Gregory P.|last2=Lathrop|first2=Daniel P.|last3=Sreenivasan|first3=Katepalli R.|year=2006|title=Superfluid helium: Visualization of quantized vortices|url=http://users.ictp.it/~krs/pdf/2006_001.pdf|journal=Nature|volume=441|issue=7093|page=588|bibcode=2006Natur.441..588B|doi=10.1038/441588a|pmid=16738652|s2cid=4429923|doi-access=free}}</ref>

In the early 2000s, physicists created a [[Fermionic condensate]] from pairs of ultra-cold fermionic atoms. Under certain conditions, fermion pairs form [[diatomic molecule]]s and undergo [[Bose–Einstein condensation]]. At the other limit, the fermions (most notably superconducting electrons) form [[Cooper pairs]] 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 BEC-BCS crossover.

[[Supersolid]]s may also have been discovered in 2004 by physicists at [[Penn State University]]. When helium-4 is cooled below about 200&nbsp;mK under high pressures, a fraction (≈1%) of the solid appears to become superfluid.<ref>{{Cite journal|author=[[Kim Eunseong|E. Kim]] and [[Moses H. W. Chan|M. H. W. Chan]]|year=2004|title=Probable Observation of a Supersolid Helium Phase|journal=Nature|volume=427|issue=6971|pages=225–227|bibcode=2004Natur.427..225K|doi=10.1038/nature02220|pmid=14724632|s2cid=3112651}}</ref><ref>Moses Chan's Research Group. "[http://www.phys.psu.edu/~chan/index_files/Page526.htm Supersolid] {{webarchive|url=https://web.archive.org/web/20130408065008/http://www.phys.psu.edu/~chan/index_files/Page526.htm|date=2013-04-08}}." ''Penn State University,'' 2004.</ref> By quench cooling or lengthening the [[annealing (metallurgy)|annealing]] 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 property of helium-4 and disorder.<ref>{{Cite journal|last=Sophie|first=A|author2=Rittner C|year=2006|title=Observation of Classical Rotational Inertia and Nonclassical Supersolid Signals in Solid 4 He below 250 mK|journal=Phys. Rev. Lett.|volume=97|issue=16|article-number=165301|arxiv=cond-mat/0604528|bibcode=2006PhRvL..97p5301R|doi=10.1103/PhysRevLett.97.165301|pmid=17155406|s2cid=45453420}}</ref><ref>{{Cite journal|last=Sophie|first=A|author2=Rittner C|year=2007|title=Disorder and the Supersolid State of Solid 4 He|journal=Phys. Rev. Lett.|volume=98|issue=17|article-number=175302|arxiv=cond-mat/0702665|bibcode=2007PhRvL..98q5302R|doi=10.1103/PhysRevLett.98.175302|s2cid=119469548}}</ref> Some emerging theories posit that the supersolid signal observed in helium-4 was actually an observation of either a [[superglass]] state<ref>{{Cite journal|last=Boninsegni|first=M|author2=Prokofev|year=2006|title=Superglass Phase of 4 He|journal=Phys. Rev. Lett.|volume=96|issue=13|article-number=135301|arxiv=cond-mat/0603003|bibcode=2006PhRvL..96m5301W|doi=10.1103/PhysRevLett.96.135301|pmid=16711998|s2cid=41657202}}</ref> or intrinsically superfluid grain boundaries in the helium-4 crystal.<ref>{{Cite journal|last=Pollet|first=L|author2=Boninsegni M|year=2007|title=Superfuididty of Grain Boundaries in Solid 4 He|journal=Phys. Rev. Lett.|volume=98|issue=13|article-number=135301|arxiv=cond-mat/0702159|bibcode=2007PhRvL..98m5301P|doi=10.1103/PhysRevLett.98.135301|pmid=17501209|s2cid=20038102}}</ref>

== Applications ==
Recently{{timeframe|date=September 2024}} in the field of chemistry, superfluid helium-4 has been successfully used in [[spectroscopic]] techniques as a [[quantum solvent]]. 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&nbsp;K which cools the solvated molecule(s) to its ground or nearly ground [[Rovibronic coupling|rovibronic]] state.

Superfluids are also used in high-precision devices such as [[gyroscope]]s, which allow the measurement of some theoretically predicted gravitational effects (for an example, see [[Gravity Probe B]]).

The Infrared Astronomical Satellite [[IRAS]], launched in January 1983 to gather infrared data was cooled by 73 kilograms of superfluid helium, maintaining a temperature of {{convert|1.6|K|C}}. When used in conjunction with helium-3, temperatures as low as 40&nbsp;mK are routinely achieved in extreme low temperature experiments. The helium-3, in liquid state at 3.2&nbsp;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. (See [[dilution refrigerator]])

Superfluid-helium technology is used to extend the temperature range of [[cryocoolers]] to lower temperatures. So far, the limit is 1.19&nbsp;K, but there is a potential to reach 0.7&nbsp;K.<ref>{{cite book|last1=Tanaeva|first1=I. A.|title=AIP Conference Proceedings|date=2004|volume=710|chapter=Superfluid Vortex Cooler|issue=3 |pages=034911–1/8 |doi=10.1063/1.1774894|s2cid=109758743 |url=https://pure.tue.nl/ws/files/1719607/Metis192474.pdf |chapter-url=https://research.tue.nl/nl/publications/the-superfluid-vortex-cooler(422e0405-c46f-4eb5-8088-2fc2689972b4).html}}</ref>

== Properties ==
{{see also|Helium#Helium II|superfluidity}}
Superfluids, such as helium-4 below the lambda point (known, for simplicity, as '''helium II'''), 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 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&nbsp;cm/s) which leads to a very high effective thermal conductivity.

=== Film flow ===

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-II, the flow of the liquid in the layer is not restricted by its viscosity but by a critical velocity which is about 20&nbsp;cm/s. This 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&nbsp;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.<ref>{{Cite journal | doi=10.1038/srep28992| pmid=27363671| pmc=4929499| title=Limited Quantum Helium Transportation through Nano-channels by Quantum Fluctuation| journal=Scientific Reports| volume=6| article-number=28992| year=2016| last1=Ohba| first1=Tomonori| bibcode=2016NatSR...628992O}}</ref>

=== Rotation ===
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 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 [[Abrikosov lattice]] in a superconductor.

=== Comparison with helium-3 ===
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 [[Bose–Einstein statistics]] 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 [[BCS theory]] of superconductivity. In it, [[Cooper pair]]ing takes place between atoms rather than [[electron]]s, and the attractive interaction between them is mediated by [[Spin (physics)|spin]] fluctuations rather than [[phonon]]s. (See [[fermion condensate]].) A unified description of superconductivity and superfluidity is possible in terms of [[spontaneous symmetry breaking|gauge symmetry breaking]].

== Macroscopic theory ==

=== Thermodynamics ===
[[File:Phase diagram of 4He 01.svg|thumb|Fig. 1. Phase diagram of <sup>4</sup>He. In this diagram is also given the λ-line.]]
[[File:He liquid tbotevadobis damokidebuleba TemperaTurastan 350-en.svg|thumb|Fig. 2. Heat capacity of liquid <sup>4</sup>He at saturated vapor pressure as function of the temperature. The peak at T=2.17 K marks a (second-order) phase transition.]]
[[File:Normal and superfluid density 01.jpg|thumb|Fig. 3. Temperature dependence of the relative superfluid and normal components ρ<sub>n</sub>/ρ and ρ<sub>s</sub>/ρ as functions of ''T''.]]

Figure 1 is the [[phase diagram]] of <sup>4</sup>He.<ref>{{cite journal|last1=Swenson|first1=C.|year=1950|title=The Liquid-Solid Transformation in Helium near Absolute Zero|journal=Physical Review|volume=79|issue=4|page=626|bibcode=1950PhRv...79..626S|doi=10.1103/PhysRev.79.626}}</ref> 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 [[critical point (thermodynamics)|critical point]] where the difference between gas and liquid disappears. The diagram shows the remarkable property that <sup>4</sup>He is liquid even at [[absolute zero]]. <sup>4</sup>He is only solid at pressures above 25 [[bar (unit)|bar]].

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.

The name lambda-line comes from the specific heat – temperature plot which has the shape of the Greek letter λ.<ref>{{cite journal|last1=Keesom|first1=W.H.|last2=Keesom|first2=A.P.|year=1935|title=New measurements on the specific heat of liquid helium|journal=Physica|volume=2|issue=1|page=557|bibcode=1935Phy.....2..557K|doi=10.1016/S0031-8914(35)90128-8}}</ref><ref>{{cite book|last1=Buckingham|first1=M.J.|title=The nature of the λ-transition in liquid helium|last2=Fairbank|first2=W.M.|date=1961|isbn=978-0-444-53309-8|series=Progress in Low Temperature Physics|volume=3|page=80|chapter=Chapter III The Nature of the λ-Transition in Liquid Helium|doi=10.1016/S0079-6417(08)60134-1}}</ref> See figure 2, which shows a peak at 2.172 K, the so-called λ-point of <sup>4</sup>He.

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 ρ<sub>n</sub>/ρ and ρ<sub>s</sub>/ρ, with ρ<sub>n</sub> (ρ<sub>s</sub>) the density of the normal (superfluid) component, and ρ (the total density), depends on temperature and is represented in figure 3.<ref>E.L. Andronikashvili Zh. Éksp. Teor. Fiz, Vol.16 p.780 (1946), Vol.18 p. 424 (1948)</ref> By lowering the temperature, the fraction of the superfluid density increases from zero at ''T''<sub>λ</sub> to one at zero kelvins.  Below 1&nbsp;K the helium is almost completely superfluid.

It is possible to create density waves of the normal component (and hence of the superfluid component since ρ<sub>n</sub> + ρ<sub>s</sub> = constant) which are similar to ordinary sound waves. This effect is called [[second sound]]. Due to the temperature dependence of ρ<sub>n</sub> (figure 3) these waves in ρ<sub>n</sub> are also temperature waves.

[[File:helium-II-creep.svg|thumb|right|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 [[Rollin film]] also covers the interior of the larger container; if it were not sealed, the helium II would creep out and escape.]]

[[File:Liquid helium Rollin film.jpg|thumb|right|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 empty.]]

=== Superfluid hydrodynamics  ===
The equation of motion for the superfluid component, in a somewhat simplified form,<ref>S. J. Putterman (1974), ''Superfluid Hydrodynamics'' (Amsterdam: North-Holland) {{ISBN|0-444-10681-2}}.</ref> is given by Newton's law

<math display="block">\vec F = M_4\frac{\mathrm{d}\vec v_s}{\mathrm{d}t}.</math>

The mass <math display="inline">M_4</math> is the molar mass of <sup>4</sup>He, and <math display="inline">\vec v_s</math> 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 <sup>4</sup>He in the gravitational field the force is given by<ref>Landau, L. D. (1941), [http://e-heritage.ru/Book/10088665 "The theory of superfluidity of helium II"], ''Journal of Physics'', Vol. 5, Academy of Sciences of the USSR, p. 71.</ref><ref>Khalatnikov, I. M. (1965), ''An introduction to the theory of superfluidity'' (New York: W. A. Benjamin), {{ISBN|0-7382-0300-9}}.</ref>

<math display="block">\vec F = -\vec \nabla (\mu + M_4 gz).</math>

In this expression <math display="inline">\mu</math> is the molar chemical potential, <math display="inline">g</math> the gravitational acceleration, and <math display="inline">z</math> 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

{{NumBlk2|:|<math display="block">M_4\frac{\mathrm{d}\vec v_s}{\mathrm{d}t}  = -\vec \nabla (\mu + M_4 gz).</math>|1}}

Eq.&nbsp;{{EquationNote|1|(1)}} only holds if <math display="inline">v_s</math> is below a certain critical value, which usually is determined by the diameter of the flow channel.<ref>{{cite journal |doi=10.1016/0031-9163(66)90958-9 |title=The dependence of the critical velocity of the superfluid on channel diameter and film thickness |year=1966 |last1=Van Alphen |first1=W. M. |last2=Van Haasteren |first2=G. J. |last3=De Bruyn Ouboter |first3=R. |last4=Taconis |first4=K. W. |journal=Physics Letters |volume=20 |issue=5 |page=474 |bibcode = 1966PhL....20..474V}}</ref><ref>{{cite book |doi=10.1016/S0079-6417(08)60052-9 |title=Thermodynamics and hydrodynamics of <sup>3</sup>He–<sup>4</sup>He mixtures |chapter=Chapter 3: Thermodynamics and Hydrodynamics of <sup>3</sup>He–<sup>4</sup>He Mixtures |series=Progress in Low Temperature Physics |date=1992 |last1=De Waele |first1=A. Th. A. M. |last2=Kuerten |first2=J. G. M. |isbn=978-0-444-89109-9 |volume=13 |page=167|chapter-url=https://research.utwente.nl/en/publications/thermodynamics-and-hydrodynamics-of-hehe-mixtures(17c67fdc-75ba-4c0f-9e13-c247b78ccf97).html}}</ref>

In classical mechanics the force is often the gradient of a potential energy. Eq.&nbsp;{{EquationNote|1|(1)}} shows that, in the case of the superfluid component, the force contains a term due to the gradient of the [[chemical potential]]. This is the origin of the remarkable properties of He-II such as the fountain effect.

[[File:Integration path in pT diagram 01.jpg|thumb|Fig. 6. Integration path for calculating <math display="inline">\mu</math> at arbitrary <math display="inline">p</math> and <math display="inline">T</math>.]]
[[File:Demo fountain pressure 01.jpg|thumb|Fig. 7. Demonstration of the fountain pressure. The two vessels are connected by a superleak through which only the superfluid component can pass.]]
[[File:Helium fountain 01.jpg|thumb|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 fountain.]]

=== Fountain pressure ===
In order to rewrite Eq.{{EquationNote|1|(1)}} in more familiar form we use the general formula

{{NumBlk2|:|<math display="block">\mathrm{d} \mu = V_m\mathrm{d}p - S_m\mathrm{d}T.</math>|2}}

Here <math display="inline">S_m</math> is the molar entropy and <math display="inline">V_m</math> the molar volume. With Eq.{{EquationNote|2|(2)}} <math display="inline">\mu(p,T)</math> can be found by a line integration in the <math display="inline">p</math>–<math display="inline">T</math> plane. First, we integrate from the origin <math display="inline">(0,0)</math> to <math display="inline">(p,0)</math>, so at <math display="inline">T=0</math>. Next, we integrate from <math display="inline">(p,0)</math> to <math display="inline">(p,T)</math>, so with constant pressure (see figure 6). In the first integral <math display="inline">\mathrm{d}T=0</math> and in the second <math display="inline">\mathrm{d}p=0</math>. With Eq.{{EquationNote|2|(2)}} we obtain

{{NumBlk2|:|<math display="block">\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.</math>|3}}

We are interested only in cases where <math display="inline">p</math> is small so that <math display="inline">V_m</math> is practically constant. So

{{NumBlk2|:|<math display="block">\int_{0}^{p} V_{m}(p^\prime,0)\mathrm{d}p^\prime = V_{m0}p</math>|4}}

where <math display="inline">V_{m0}</math> is the molar volume of the liquid at <math display="inline">T=0</math> and <math display="inline">p=0</math>. The other term in Eq.{{EquationNote|3|(3)}} is also written as a product of <math display="inline">V_{m0}</math> and a quantity <math>p_f</math> which has the dimension of pressure

{{NumBlk2|:|<math display="block">\int_{0}^T S_{m}(p,T^\prime)\mathrm{d}T^\prime=V_{m0}p_{f}.</math>|5}}

The pressure <math display="inline">p_f</math> is called the fountain pressure. It can be calculated from the entropy of <sup>4</sup>He which, in turn, can be calculated from the heat capacity. For <math display="inline">T=T_\lambda</math> the fountain pressure is equal to 0.692 bar. With a density of liquid helium of 125&nbsp;kg/m<sup>3</sup> and {{mvar|g}} = 9.8&nbsp;m/s<sup>2</sup> 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.

With Eqs.{{EquationNote|4|(4)}} and {{EquationNote|5|(5)}}, Eq.{{EquationNote|3|(3)}} obtains the form

{{NumBlk2|:|<math display="block">\mu(p,T) = \mu_0 + V_{m0}(p-p_{f}).</math>|6}}

Substitution of Eq.{{EquationNote|6|(6)}} in {{EquationNote|1|(1)}} gives

{{NumBlk2|:|<math display="block">\rho_0 \frac{\mathrm{d} \vec v_s}{\mathrm{d}t} = - \vec\nabla (p + \rho_0gz-p_{f}).</math>|7}}

with <math display="inline">\rho_0 = M_4/V_{m0}</math> the density of liquid <sup>4</sup>He at zero pressure and temperature.

Eq.{{EquationNote|7|(7)}} 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.

So far Eq.{{EquationNote|5|(5)}} has only mathematical meaning, but in special experimental arrangements <math display="inline">p_f</math> 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 (<math display="inline">T_l=0</math>) and zero pressure (<math display="inline">p_l = 0</math>). 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&nbsp;cm/s). In the steady state <math display="inline">v_s=0</math> so Eq.{{EquationNote|7|(7)}} implies

{{NumBlk2|:|<math>p_{l}+\rho_0gz_{l}-p_{fl}=p_{r}+ \rho_0gz_{r}-p_{fr}</math>|8}}

where the indexes <math display="inline">l</math> and <math display="inline">r</math> apply to the left and right side of the superleak respectively. In this particular case <math display="inline">p_l = 0</math>, <math display="inline">z_l = z_r</math>, and <math display="inline">p_{fl} = 0</math> (since <math display="inline">T_l = 0</math>). Consequently,

<math display="block">0=p_{r}-p_{fr}.</math>

This means that the pressure in the right vessel is equal to the fountain pressure at <math display="inline">T_r</math>.

In an experiment, arranged as in figure 8, a fountain can be created. The fountain effect is used to drive the circulation of <sup>3</sup>He in dilution refrigerators.<ref>{{cite journal|doi=10.1016/0375-9601(75)90087-0|title=A dilution refrigerator with superfluid injection|year=1975|last1=Staas|first1=F. A.|last2=Severijns|first2=A. P.|last3=Van Der Waerden|first3=H. C.bM.|journal=Physics Letters A|volume=53|issue=4|page=327|bibcode = 1975PhLA...53..327S }}</ref><ref>{{cite journal|title=<sup>3</sup>He flow in dilute <sup>3</sup>He-<sup>4</sup>He mixtures at temperatures between 10 and 150 mK|doi=10.1103/PhysRevB.32.2870|pmid=9937394|year=1985|last1=Castelijns|first1=C.|last2=Kuerten|first2=J.|last3=De Waele|first3=A.|last4=Gijsman|first4=H.|journal=Physical Review B|volume=32|issue=5|pages=2870–2886|bibcode = 1985PhRvB..32.2870C |url=https://research.tue.nl/nl/publications/3he-flow-in-dilute-3he4he-mixtures-at-temperatures-between-10-and-150-mk(d7aefa27-5cff-4379-9b28-e0623ec7de38).html}}</ref>
[[File:Counterflow heat exchange 01.jpg|thumb|Fig. 9. Transport of heat by a counterflow of the normal and superfluid components of He-II]]

=== Heat transport ===
Figure 9 depicts a heat-conduction experiment between two temperatures <math display="inline">T_H</math> and <math display="inline">T_L</math> 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.{{EquationNote|7|(7)}}. This pressure drives the normal component from the hot end to the cold end according to

{{NumBlk2|:|<math display="block">\Delta p = -\eta_nZ\dot V_n.</math>|9}}

Here <math display="inline">\eta_n</math> is the viscosity of the normal component,<ref>Zeegers, J. C. H. ''Critical velocities and mutual friction in <sup>3</sup>He-<sup>4</sup>He mixtures at low temperatures below 100 mK'', thesis, Appendix A, Eindhoven University of Technology, 1991.</ref> <math display="inline">Z</math> some geometrical factor, and <math display="inline">\dot V_n</math> 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 effective, so the thermal conductivity of He-II is very much better than the best materials. The situation is comparable with [[heat pipe]]s 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 [[Large Hadron Collider]] at [[CERN]].

== Microscopic theory ==

=== Landau two-fluid approach ===
{{Main|Two-fluid model}}
[[Lev Landau|L. D. Landau's]] 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 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".

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.

To explain the early specific heat data on superfluid helium-4, Landau posited the existence of a type of excitation he called a "[[roton]]", but as better data became available, he considered that the "roton" was the same as a high momentum version of sound.

The Landau theory does not elaborate on the microscopic structure of the superfluid component of liquid helium.<ref>{{Cite journal|last1=Alonso|first1=J. L.|last2=Ares|first2=F.|last3=Brun|first3=J. L.|date=2018-10-05|title=Unraveling the Landau's consistence criterion and the meaning of interpenetration in the "Two-Fluid" Model|journal=The European Physical Journal B|language=en|volume=91|issue=10|page=226|doi=10.1140/epjb/e2018-90105-x|issn=1434-6028|arxiv=1806.11034|bibcode=2018EPJB...91..226A|s2cid=53464405}}</ref> The first attempts to create a microscopic theory of the superfluid component itself were done by London<ref>{{cite journal|author=F. London|journal= Nature |volume=141|pages= 643–644 |year=1938|doi=10.1038/141643a0|title=The λ-Phenomenon of Liquid Helium and the Bose-Einstein Degeneracy|issue=3571|bibcode = 1938Natur.141..643L |s2cid= 4143290 }}</ref> and subsequently, Tisza.<ref>{{cite journal|author=L. Tisza|journal= Nature |volume= 141|doi=10.1038/141913a0|title=Transport Phenomena in Helium II|year=1938|issue=3577|page=913|bibcode = 1938Natur.141..913T |s2cid= 4116542 |doi-access=free}}</ref><ref>{{cite journal|author=L. Tisza|journal=Phys. Rev. |volume=72|pages= 838–854 |year=1947|doi=10.1103/PhysRev.72.838|title=The Theory of Liquid Helium|issue=9|bibcode = 1947PhRv...72..838T }}</ref>
Other microscopical models have been proposed by different authors. Their main objective is to derive the form of the inter-particle potential between helium atoms in superfluid state from first principles of [[quantum mechanics]].
To date, a number of models of this kind have been proposed, including: models with vortex rings, hard-sphere models, and Gaussian cluster theories.

=== Vortex ring model ===
Landau thought that vorticity entered superfluid helium-4 by vortex sheets, but such sheets have since been shown to be unstable.
[[Lars Onsager]] and, later independently, Feynman showed that vorticity enters by quantized vortex lines. They also developed the idea of [[quantum vortex]] rings.
[[Arie Bijl]] in the 1940s,<ref>{{Cite journal
| last = Bijl
| first = A
|author2=de Boer, J
|author3=Michels, A
 | title = Properties of liquid helium II
| journal = Physica
| volume = 8
| issue = 7
| pages = 655–675
|year = 1941
| doi = 10.1016/S0031-8914(41)90422-6
| bibcode=1941Phy.....8..655B}}</ref>
and [[Richard Feynman]] around 1955,<ref>{{Cite book
| editor = Braun, L. M.
| title = Selected papers of Richard Feynman with commentary
| publisher = World Scientific
| series = World Scientific Series in 20th century Physics
| volume = 27
| date = 2000
| isbn = 978-981-02-4131-5 }}
[https://books.google.com/books?id=qnwkqcVixucC&q=feynman Section IV (pages 313 to 414)] deals with liquid helium.</ref> 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.<ref>{{cite journal|author=R. P. Feynman|journal= Phys. Rev.|volume= 94|page= 262 |year=1954|doi=10.1103/PhysRev.94.262|title=Atomic Theory of the Two-Fluid Model of Liquid Helium|issue=2|bibcode = 1954PhRv...94..262F |url=http://authors.library.caltech.edu/3539/1/FEYpr54a.pdf}}</ref><ref>{{cite journal|author=R. P. Feynman|author2=M. Cohen|name-list-style=amp|journal= Phys. Rev. |volume=102|pages= 1189–1204 |year=1956|doi=10.1103/PhysRev.102.1189|title=Energy Spectrum of the Excitations in Liquid Helium|issue=5|bibcode = 1956PhRv..102.1189F |url=https://authors.library.caltech.edu/3549/1/FEYpr56.pdf}}</ref>

=== Hard-sphere models ===
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.<ref>{{cite journal|author=T. D. Lee|author2= K. Huang|author3=C. N. Yang|name-list-style=amp|journal= Phys. Rev. |volume=106|pages= 1135–1145 |year=1957|doi=10.1103/PhysRev.106.1135|title=Eigenvalues and Eigenfunctions of a Bose System of Hard Spheres and Its Low-Temperature Properties|issue=6|bibcode = 1957PhRv..106.1135L }}</ref><ref>{{cite journal|author=L. Liu|author2=L. S. Liu|author3=K. W. Wong|name-list-style=amp|journal= Phys. Rev. |volume=135|pages= A1166–A1172 |year=1964|doi=10.1103/PhysRev.135.A1166|title=Hard-Sphere Approach to the Excitation Spectrum in Liquid Helium II|issue=5A|bibcode = 1964PhRv..135.1166L }}</ref><ref>{{cite journal|author=A. P. Ivashin|author2=Y. M. Poluektov|name-list-style=amp|journal= Cent. Eur. J. Phys. |volume= 9|pages= 857–864 |year=2011|doi=10.2478/s11534-010-0124-7|title=Short-wave excitations in non-local Gross-Pitaevskii model|issue=3|bibcode = 2011CEJPh...9..857I |arxiv=1004.0442|s2cid=118633189}}</ref>
In these models the famous Landau (roton) spectrum of excitations is qualitatively reproduced.

=== Gaussian cluster approach ===
This is a two-scale approach which describes the superfluid component of liquid helium-4. It consists of two [[Critical phenomena|nested models linked via parametric space]]. The short-wavelength part describes the interior structure of the [[fluid element]] using a non-perturbative approach based on the [[logarithmic Schrödinger equation]]; it suggests the [[Gaussian]]-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.<ref>{{cite journal | url=https://link.aps.org/doi/10.1103/PhysRevLett.90.250403 | doi=10.1103/PhysRevLett.90.250403 | title=Roton-Maxon Spectrum and Stability of Trapped Dipolar Bose-Einstein Condensates | year=2003 | last1=Santos | first1=L. | last2=Shlyapnikov | first2=G. V. | last3=Lewenstein | first3=M. | journal=Physical Review Letters | volume=90 | issue=25 | article-number=250403 | pmid=12857119 | arxiv=cond-mat/0301474 | bibcode=2003PhRvL..90y0403S | s2cid=25309672 }}</ref> The approach provides a unified description of the [[phonon]], [[maxon excitation|maxon]] and [[roton]] excitations, and has noteworthy agreement with experiment: with one essential parameter to fit one reproduces at high accuracy the Landau roton spectrum, [[sound velocity]] and [[structure factor]] of superfluid helium-4.<ref>{{cite journal|author=K. G. Zloshchastiev|journal= Eur. Phys. J. B |volume= 85|article-number= 273 |year=2012|doi=10.1140/epjb/e2012-30344-3|title=Volume element structure and roton-maxon-phonon excitations in superfluid helium beyond the Gross-Pitaevskii approximation|issue=8|bibcode = 2012EPJB...85..273Z|arxiv = 1204.4652 |s2cid= 118545094 }}</ref> This model utilizes the general theory of quantum Bose liquids with logarithmic nonlinearities<ref>{{cite journal|author=A. V. Avdeenkov|author2=K. G. Zloshchastiev|name-list-style=amp|journal= J. Phys. B: At. Mol. Opt. Phys. |volume= 44|article-number= 195303|year=2011|doi=10.1088/0953-4075/44/19/195303|title=Quantum Bose liquids with logarithmic nonlinearity: Self-sustainability and emergence of spatial extent|issue=19|bibcode = 2011JPhB...44s5303A|arxiv = 1108.0847 |s2cid=119248001}}</ref> which is based on introducing a [[Open quantum system|dissipative]]-type contribution to energy related to the quantum [[Entropic uncertainty|Everett–Hirschman entropy function]].<ref>[[Hugh Everett]], III. The Many-Worlds Interpretation of Quantum Mechanics: the theory of the universal wave function. [https://www.pbs.org/wgbh/nova/manyworlds/pdf/dissertation.pdf Everett's Dissertation]</ref><ref name="Hirschman">[[Isidore Isaac Hirschman, Jr.|I.I. Hirschman, Jr.]], ''A note on entropy''. American Journal of Mathematics (1957) pp. 152–156</ref>

== See also ==
{{colbegin|colwidth=22em}}
* [[Douglas D. Osheroff]]
* [[Large Hadron Collider]]
* [[London moment]]
* [[Polariton superfluid]]
* [[Quantum acoustics]]
* [[Quantum gyroscope]]
* [[Superdiamagnetism]]
* [[Superfluid film]]
* [[Timeline of low-temperature technology]]
{{colend}}

== References ==
{{Reflist|30em}}

== Further reading ==
{{refbegin|30em}}
* Antony M. Guénault: ''Basic superfluids.'' Taylor & Francis, London 2003, {{ISBN|0-7484-0891-6}}
* D.R. Tilley and J. Tilley, ''Superfluidity and Superconductivity,'' (IOP Publishing Ltd., Bristol, 1990)
*[https://www.energy.gov/science/bes/articles/discovering-secrets-superfluids Department of Energy Office of Science: Superfluidity]
* [[Hagen Kleinert]], ''Gauge Fields in Condensed Matter'', Vol. I, "SUPERFLOW AND VORTEX LINES", pp.&nbsp;1–742, [https://archive.today/20060514143926/http://www.worldscibooks.com/physics/0356.htm World Scientific (Singapore, 1989)]; Paperback {{ISBN|9971-5-0210-0}} (also available [http://www.physik.fu-berlin.de/~kleinert/kleiner_reb1/contents1.html online] {{Webarchive|url=https://web.archive.org/web/20080527174206/http://www.physik.fu-berlin.de/~kleinert/kleiner_reb1/contents1.html |date=May 27, 2008 }})
* James F. Annett: ''Superconductivity, superfluids, and condensates.'' Oxford Univ. Press, Oxford 2005, {{ISBN|978-0-19-850756-7}}
* {{Cite journal | last1 = Leggett | first1 = A. | author-link1= Anthony Leggett| title = Superfluidity | doi = 10.1103/RevModPhys.71.S318 | journal = Reviews of Modern Physics | volume = 71 | issue = 2 | pages = S318–S323 | year = 1999 |bibcode = 1999RvMPS..71..318L }} 
* London, F. Superfluids (Wiley, New York, 1950)
* Philippe Lebrun & Laurent Tavian: ''[https://cds.cern.ch/record/808569?ln=en The technology of superfluid helium]''
{{refend}}

== External links ==
* [http://www.mas.ncl.ac.uk/helium/ Helium-4 Interactive Properties]
* [https://news.mit.edu/2005/matter http://web.mit.edu/newsoffice/2005/matter.html]
* [http://www.alfredleitner.com/ Liquid Helium II, Superfluid: demonstrations of Lambda point transition/viscosity paradox /two fluid model/fountain effect/creeping film/ second sound.]
* [https://web.archive.org/web/20060408121158/http://www.aip.org/pt/vol-54/iss-2/p31.html Physics Today February 2001]
* {{Cite journal|doi=10.1103/PhysRevB.90.134503|title=Superfluid density in continuous and discrete spaces: Avoiding misconceptions|journal=Physical Review B|volume=90|issue=13|article-number=134503|year=2014|last1=Rousseau|first1=V. G.|arxiv=1403.5472|bibcode=2014PhRvB..90m4503R|s2cid=118518974}}
* [http://www.emis.de/journals/LRG/Articles/lrr-2008-10/articlesu23.html superfluid hydrodynamics] {{Webarchive|url=https://web.archive.org/web/20160303171517/http://www.emis.de/journals/LRG/Articles/lrr-2008-10/articlesu23.html |date=March 3, 2016 }}
* [http://ltl.tkk.fi/research/theory/helium.html Superfluid phases of helium]
* [https://web.archive.org/web/20031206032515/http://www.hindu.com/seta/2003/10/09/stories/2003100900030200.htm The Hindu article on superfluid states]
* [https://www.youtube.com/watch?v=2Z6UJbwxBZI Video including superfluid helium's strange behavior]

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