３. 振動板実験A. De Troyer et al., Physica 17, 50 (1951)

Ø 液体中で円盤を一点振幅で振動させるときのトルクを測定

Ø 粘性(h)はTl以下で徐々に減少し、T → 0でゼロに向かう

回転トルク :

§3. Superfluidity of Liquid 4He1. キャピラリーフロー実験

流速 :

P0 P1P1 - P0 = DP

2r

L

He-II

Ø 差圧なしに流れる(粘性ゼロ)Ø 管径が小さいほど流速が大きい(0.1 ≤

2r ≤1 µm)Ø 粘性(h)はTl直下でゼロに落ちる

K.R. Atkins, Philos. Mag. Supp. 1, 169 (1952)2. 回転粘性計実験

r1r0

h

wHe-II

固定円筒にかかるトルク :

A.C. Hollis-Hallett, Proc. Cambridge Phil. Soc., 49, 717 (1953)

Ø 液体中で外筒を回転させたときに内筒が回転しようとするトルクを測定

Ø hはTl以下で減少するがゼロには落ちず、T < 1 Kで急激に上昇

• vn ≈ 0, measuring only vs• measuring h = hn (µ l)

l: mean free path• measuring rh (= rnhn µ rn)

Two fluid model

: total density (const.): total mass current

rs: superfluid density (∝ No. of particles at ground state)dissipationless, zero entropy

rn: normal fluid density (∝ No. of particles at excited states)frictional, finite entropy

L. Tisza, Nature 141, 913 (1938)

No mutual friction between rs and rn.

Thermal counter flow (j = 0)huge thermal conductivity

>>>>T1 T2

ideal thermal counter flow (T1 ≈ T2)

heater

vs vn

Second sound:

ρS

ρn

T

x

temperature (entropy) wave

2nd sound in ultra cold Fermi gasL. A. Sidorenkov, et al. Nature 498, 78 (2013)

液体4Heの超流動転移 (T = 2.17 K)

熱伝導が異常に高く、突沸が止む

温度のゆらぎによる突沸

粘性がないので、熱対流がすみやかに温度のゆらぎを除去するため

4

ヘリウムガスを液体窒素中にパイプで吹き込むと、窒素の突沸 (泡立ち)が数10分間近くも止まる。.... まるで液体ヘリウムの超流動状態のように！

• ヘリウムガスを注入した内壁近くが特に突沸が停止しやすい？

• ガラス内壁の沸騰核がヘリウムガスのせいで働かなくなる？

液体窒素の超流動？

水素ガスを注入しても同様の現象が見られる。

我々も試してみたが、再現する

液温が下がることが突沸停止の原因だった。

意外な原因

確かに、水素・ネオンガスを注入しても同様の現象が見られるが、窒素ガスでは見られない。

高吉慎太郎、穀山 渉, 固体物理 42, 91 (2007)S. Takayoshi et al., Cryogenics 49, 221 (2009)

液温が低下するときに液体窒素の蒸発が早い 。

窒素の泡中に内部蒸発して、蒸発潜熱が奪われて冷却するに違いない！

内部蒸発

ヘリウムガス＋

窒素ガス

ヘリウムガス↓

液体窒素

§4 Elementary Excitations in superfluid 4HeLandau theory (phonon-roton dispersion)

L.D. Landau, Phys. Rev. 60, 356 (1941); J. Phys. USSR 5, 71 (1941); ivid. 8, 1 (1944); ivid. 11, 91 (1947)

phonon: density waves due to repulsive interactions

roton： "vortex motions (rot vs ≠ 0) of a whole quantum liquid"

c =283 m/sDr/kB = 8.65 K,p0/ℏ = 19.1 nm-1,mr = 0.16 m4

Dr/kB = 9.6 K,p0/ℏ = 19.5 nm-1,mr = 0.77 m4

in 1947 latest exp.

Rot v commutes with p and v and, therefore, also with the Hamiltonian,

Elementary excitations in He-II

Phonon (T ≤ 0.6 K): density waves (only longitudinal mode)due to repulsive interactions

D.S. Greywall, PRB 18, 2127 (1978)

specific heat

Roton (1 K ≤ T ≤ Tl): short range correlation

Dr/kB = 8.65 K, p0/ℏ = 19.1 nm-1, mr = 0.16 m4 D. de Klerk et al., PR 89, 662 (1953)

2nd sound

rn µ T4

rn µ T5.6

L.D. Landau, J. Phys. USSR 11, 91 (1947)

Dilute Bose gases of phonons and rotons at low T: (rn)

Inelastic neutron scattering of superfluid 4He

D.G. Henshaw and A.D.B. Woods, Phys. Rev. 121, 1266 (1961)

T = 1.12 Kln = 0.404 nm

free particles

critical velocity(vL≈ 50 m/s)

ener

gy e

(K)

momentum p (Å-1)

phonon

maxon

roton

roton gap ≈ 9 K

W.G. Stirling and H.R. Glyde, PRB 41, 4224 (1990)

superfluid

Q = 0.4 A-1T = 1.35 K

Q = 0.4 A-1T = 2.24 -

2.96 K

normal

phonon

2.17 K

1.79 K

2.02 K

Q = 1.925 A-1

roton

Feynman theory for He-IIR.P. Feynman, PR 94, 262 (1954)Excited-state wave fn. Yk will be given by

: variational fn.

By minimizing the total energy with respect to f (ri), we have the following three eqs.

: density correlation fn.(probability to find an atom both at r and r')

: static structure factorS(k)S(k) µ k

Experimentally, S(k) µ k for k→ 0,which is consistent with e(k) µ kfor k → 0 (phonons).

exp.

Feynman-CohenFeynman

•

•

•

T-dependence of phonon-roton dispersion in liq. 4He through Tl W. Dmowski et al., Nat.

Commun. 8, 15294 (2017)

dynamic structure factor: S (q, w)

T = 1.83 K T = 2.85 K

S (q,w, T = 1.83 K)− S (q,w, T = 2.85 K)

What is Roton?

crystal

e

p0

0

2p/a

D p

e

0

energy gap in superconductors

Remnant crystalline periodicity?

superfluid 4He classical liquids

Why no viscosity？

Landau critical velocity:vL = 50 m/s

• energy conservation：

• momentum conservation：

In order to decelerate the initial velocity (vi) of moving tube to vf by emitting an elementary excitation with an energy e and a momentum p into liq. He, the excitation should satisfy:

L.D. Landau (1941)

momentum pEn

ergy

e

e = p2/2m (ideal Bose gas)

slope vi

0

vL

4He-II

BEC3D ideal gas

3D interactinggas & liquid

superfluid

interacting 2D (& 1D) gas & liquid

Condensation fraction in superfluid 4He

n0 = N0/V : number density of condensate ⇔ rs

N0/N ≈ (na3)1/2 ≈ 10 % : large depletion of condensatedue to interactions in He-II

: one-particle wave fn. of condensate

N0/V = lim r(r - r')|r - r'|→∞ Green fn. M.C.

variationalnormal state

T = 0

P..E. Sokol, Bose Eisnstein Condensation, p.51 (Cambridge University Press, 1995)

N0/N ≈ 10%

H.R. Glyde, R.T. Azuah, W.G. Stirling, Phys. Rev. B 62, 14337 (2000)

Pressure (density) dependencies of Dr and n0 determined from neutron experiments

• Roton gap (Dr) in metastable liquid 4He seems to vanish at a very high pressure, P ≈ 160 bars (r ≈ 0.22 g/cm3).

J.V. Pearce et al., PRL 93, 145303 (2004)

Data taken from metastable liquidin porous material

liquid

melting pressure

solid

r = 0.182 g/cm3

H. R. Glyde et al., PRB 83, 100507(R) (2011)

(phaseseparated)

P-r relation: L. Vrangeš et al., PRL 95, 145302 (2005)

• Condensation fraction (n0) seems to vanish at a relatively low density, r = 0.19−0.20 g/cm3, which is close to the lowest density of hcp 4He. → Metastable superfluid 4He with n0 = 0 can exist

in a wide pressure (density) range.