QUASIPARTICLE THEORY OF THE SURFACE TENSION OF LIQUID 3He AND DILUTE SOLUTION OF 4He IN 3He+

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QUASIPARTICLE THEORY OF THE SURFACE TENSION OF LIQUID 3He AND DILUTE SOLUTION

OF 4He IN 3He+

W. Saam

To cite this version:

W. Saam. QUASIPARTICLE THEORY OF THE SURFACE TENSION OF LIQUID 3He AND

DILUTE SOLUTION OF 4He IN 3He+. Journal de Physique Colloques, 1978, 39 (C6), pp.C6-201-

C6-202. �10.1051/jphyscol:1978689�. �jpa-00218369�

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JOURNAL DE PHYSIQUE Colloque C6, supplkment au no 8, T o m 39, aorit 1978, page C6-201

QUASI PARTICLE THEORY OF THE SURFACE TENS ION OF LIQUID

3 ~ e

3 4-

AND DILUTE SOLUTION OF

4 ~ e

IN

He

W.F. Saam

Physics Department, The Ohio S t a t e University, CoZombus, Ohio, 43210, U.S.A

Rlsumd.- Une thlorie du type quasiparticule est donnle pour la tension superficielle du liqui'de 3 ~ e pur et pour des mglanges diluls de 4 ~ e dans 3 ~ e . Seuls des systsmes non-superfluides sent consid6rls.

En utilisant l'approximation des longueurs de diffusion pour deri ddphasages 2 la surface, des com- paraisons entre th6orie et explrience donnent une longueur

-

³

¹

pour les quasiparticules 4 ~ e 3 la surface libre. Le signe de la longueur indique que les quasiparticules sont repoussles de la surface.

Abstract.- A quasiparticle theory of the free surface tensions of pure liquid 3 ~ e and dilute solutions of 4 ~ e in 3 ~ e is given. Only normal, i.e., non-superfluid, systems are considered. Within the scattering length approximation forquasilgarticlephase shifts at the surface, comparison of theory with experiment yields a length -3 for 4 ~ e quasiparticles at the 3 ~ e free surface. The sign of the length indicates that the 4 ~ e quasiparticles are repelled from the surface.

In recent years a considerable amount of attention / I / has been focused on the properties of the free surfaces of superfluid 4 ~ e and dilute superfluid solutions of 3 ~ e in %e. This work has led to a ra- ther clear understanding of the surface tension of these systems and considerable clarification of the nature of interactions both between free 3 ~ e and 4 ~ e atoms and the surfaces and bulk quasiparticles and the surfaces. Less attention has been paid to the properties of free surfaces of pure liquid. 3 ~ e and dilute solutions of 4 ~ e i n 3 ~ e . This paper attempts to remedy this situation in part by developing a ra- ther general theory of the surface tension of such systems, in the process pointing out that surface tension measurements can yield quite interesting information about the interactions between bulk quasiparticles and the free surface. Only normal, i.e.

non-superfluid, systems are considered.

We begin by addressing the case of pure 3 ~ e in the low temperature regime where the excitations of the system are describable in terms of a quasiparticle picture. The temperature-dependent part of

the surface tension a(T) is given /2/ by the part of the total grand canonical potential,

Q

-

^kgT^+C ^an ⁺^exp^(-@(E+_{k s n}

-

^F13)^jJ ⁽¹⁾

k s n

(planar) free surface, n is a quantum index for de- grees of freedom for motion normal to the surface, p3 is the 3 ~ e chemical potential, and B = Ilk T is

B the inverse temperature in energy units. In contrast to the case of 4 ~ e , there will be no ripplon contribution, ripplons not being expected to be valid ele- mentary excitations in this case 131. At sufficien- tly low T only quasiparticles bound to the liquid will contribute, vapor contributions being negligi- ble.

Measured relative to the binding energy per atoms,

where q(n) is a wavevector to the z-direction (normal to the surface) whose allowed values remain to be determined. The quasiparticle wavefunctions will

take on the form

for z more than a few atomic diameters away from the surface. Here 6(q,2) is the phase shift incurred by the particle upon reflection from an interior surface of the liquid. We now suppose that the li- which is proportional to the surface area A . In equa- quid possesses two parallel surfaces separated b~ a

tion (1) the

w -

are the independent quasiparticle macroscopic distance L. The sum in equation (1) will

K&ll

energy levels, k is a wavevoctor parallel to the then possess a bulk term plus twice the relevant + Work suuuorted in Dart by U.S. National Science surface tension contribution. Note now thatsvmmetr~

~oundatibn Grant ~Lmber ~MR-757-21866

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1978689

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(parity) requirements on @I (r) require that + k,n

which determines the allowed values of q(n). Appli- cation of the Euler-Maclaurin sum formula /4/ pro- duces

valid through orders AL and A, but neglecting contributions of order A/L. The intergration over n is converted to one over q via equation (4). There remain three terms on the right-hand side of equation (5), one a bulk term -AL, and two

-

A. The latter two terms divided by 2A (there being 2 surfaces) gi- ve the desired surface tension contribution. With a(0) = 0 the first of these is

the second being

The result a l is easily evaluated in the low temperature limit applicable here, giving

Here AT3 is a thermal wavelength. To evaluate a2 we make the simplifying assumption that

where b33 is a (scattering) length expected to be of the order of the surface width (i.e., -several

i)

Then,

where ng is the 3 ~ e bulk atomic number density.

to the 3 ~ e . For the temperature range of interest (roughly 0.001 s T s 0.3 K) the 4 ~ e contribution should be that of a classical ideal quasiparticle gas with binding energy E

-

6.61 K and effective

40

mass m4" = 4.5m4(5). The resulting temperature dependent part of the surface tension, to be added to a l + a2 is

where n,, is the 4 ~ e number density and bk3 is a scattering length for 4 ~ e quasiparticles at the free surf ace.

The available surface tension data 161 allow

0

the estimate bb3 3A, the sign of which indicates that 4 ~ e quasiparticles are repelled from the 3He surface (i.e. there is negative adsorbtion of 4He atoms at the surface).

The author thanks Professor D.O. Edwards for very useful discussions and for the estimate of b43 from the experimental data.

Ref erencea

/I/ For a review of this subject, see the article by D.O., Edwards and W.F., Samm in Progress in Low Temperature Physics, Volume VII, edited by D.F. Brewer (North-Holland) to be published in 1978

/2/ See, e.g., Landau, L.D. and Lifshitz, E.M., Statistical Physics (Pergamon, London) 1958, Chapter XV

/3/ Fomin, I.A., Zh. Eksp. Teor. Fiz.

61

(1971) 2562 (Sov. Phys. JETP (1972) 1371)

/4/ Margenau, H. and Murphy, G.M., The Mathematics of Physics and Chemistry (Van Nostrand, Prince- ton, New Jersey) 1956 p. 474

/5/ See, e.g. Saam, W.F. and Laheurte, J.P., Phys.

Rev. Af? (1971) 1170

/6/ Esel'son, B.N., Ivantsov, V.G. and Shvets, A.D., Zh. Eksp. Teor. Fiz. 44 (1963) 483, (Sov. Phys.

JETP

17

(1 963) 330)

The result (eq. 10) is easily generalized to include the effects of adding a small amout, of 4 ~ e