A SHELL MODEL OF POSITIVE IONS IN 3He-RICH LIQUID HELIUM

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A SHELL MODEL OF POSITIVE IONS IN 3He-RICH

LIQUID HELIUM

H. Namaizawa

To cite this version:

JOURNAL ~ f i PHYSIQUE Colloque C6, suppl6ment au no 8 , Tome 39, aoiit 1978, page C6-72

A

SHELL

MODEL O F

POSITIVE IONS

IN

3 ~ e LIQUID HELIUM - ~ ~ ~ ~ H. Namaizawa

Department o f Physics and Astronoq, Northwestern University, Evmston, I l l i n o i s 60201, U.S.A and I n s t i t u t e o f Physics, CoZZege of Genera2 Education, University of Tokyo, Komaba, Meguro-ku, Tokyo 153, Japan t

Rdsum6.- Nous proposons un mod5le de coquille pour ions positifs dans 3 ~ e dilu6 dans 4 ~ e liquide. En raison de leur masse plus grande, les atomes 4 ~ e forment une coquille compllte autour de la boule de neige. En se basant sur ce mod6le on peut donner une explication de la mobilitd multiple des ions positif s rscemment ddcouverte dans 3 ~ e normal.

Abstract.- We propose a shell model for positive ions in 3 ~ e liquid containing dilute 4 ~ e . Due to their heavier mass 4 ~ e atoms should tend to form a closed shell structure around the snowball. Based on this model, an explanation is given for the recently discovered multiple positive-ionic mobility in normal 3 ~ e .

In pure helium the structure of a positiveion is conventionally understood in terms of a snowball. a compressed accumulation of helium atoms about the ion due to the electrostatic field /1,2/. When im- purity 4 ~ e atoms are added to 3 ~ e liquid 4 ~ e atoms are attracted to the snowball by the sum of two in- dependent potentials, the polarization potential from the ion (V ) and the excess Van der Waals

PO!

potential due to Increased density accumulation

In the aboveaand~respectively stand for the ato- mic polarizability and the dielectric constant, and v(r) is the helium-helium potential and n(r) repre- sents the number density function of liquid 3 ~ e .

The bound states of B e atom are found by sol- ving

$2

+ (- + v(r))$,j;> = eVJlv(r)

2m4 (4)

where v labels eigenstates. Since V has a strong attractive part, the low-lying bound states of (4) have well localized radial distributions so that the influence from the background 3 ~ e on the iner- tia, for example, is considered to be small.

It should be noted, however, that independent- ly bound 4 ~ e atoms cannot reside in the pressure field produced by compressed 3 ~ e background near f

Permanent address and address after April, 1978

the snowball. It is the many-body effect among bound 4 ~ e atoms which provides the resisting force against the pressure field to create a stable shell around the snowball. The many-body problem is quite analo- gous to the 4 ~ e adsorption problem upon a substrate except for the above constraint and the scale and shape. Solving the many-body problem from first principles is impractical/Z/ since there is current- ly no quantum mechanical description of positive ions even in pure helium. Instead we borrow the physical ideas and results /3,4/ on the physisorp- tion of 4 ~ e upon macroscopic, flat, and smooth substrates. Interacting '~e atoms on the v-th orbit of (4) are thus considered to be a monolayer having certain energy Ev, spreading pressure $v(=02a~ /ao) and chemical potential 5 (=EV+oaE /ao) as functions

v

of the areal density u. Let us suppose the film is confined between r = rA and rg. The film resides in the pressure field P(r) and the mechanical stabili- ty condition requires

P(rA)

-

P(rg) = 2t~~(o)/r, (5) where P(r) is the local pressure of the 3 ~ e back- ground and rM is, the mean radius of the 'He shell. At the density (say urn) where (5) is fulfilled the shell becomes mechanically stable against the pres- sure field.

In an actual calculation we have approxima- ted Vex by

where nS is the number density of solid 3 ~ e , x i s the number density of liquid 3 ~ e under the applied pressure Pa, and rS(Pm) is the corresponding radius

of the bare snowball. As for the helium interatomic potential the conventional Lennard-Jones form is applied, and the eigenvalue problem (4) is solved numerically. The snowball radius and the pressure profile are calculated as functions of the external pressure Pm according to the prescription presented in references /1,2/. For P,=O and 5 atm there are four bound states of angular momentum R = 0, 1, 2, 3 having no nodes in radial wave functions. We have chosen rM of a bound shell to be the peak position of the corresponding radial wave function of the atomic bound state. Since the origin of resistence against the pressure field is the atomic hard-core, the effective thickness of 4 ~ e shell can roughly

0

be given by the radius of the core S2.2A. Then r A and r are determined respectively by

B

r = r M + d/2 A,B

Calculating spreading pressure @vm from (5), an e s -

timate of the equilibrium densities o is done by vm

reading the

+

vs. a relation obtained from the results of reference 131. These are given in table I together with eV, rA, rB, rM2/rS2, and the num- ber of bound 4 ~ e atoms, N (=4rr *u ) . Notice

4m M vm o

that the bare snowball radius rS is 6.22 and 6.46A at Pm=O and 5 atm respectively.

Table I

Bound state energy e of 4 ~ e atom for angular momentum R= 0, 1 , 2, 3 under applied pressure P =O and 5 atm. Spreading pressure

+,

and areal density a, at the mechanical equilibrium are obtayned as described in the text. Nlm is the number of 4 ~ e atoms on the bound shell.

References

The present results predict distinct shell / I / Atkins,K.R., Phys. Rev.

116

(1959) 1339

I Bound State

4

=o

1

-1

1

-2 -4 =3 ?

structures of positive ions in 3~e-rich liquid

/ 2 / Fetter,A.L. in "The Physics of Liquid and Solid helium. The recent experiments /5,6/ on mobility Helium", edited by K.H. Bennemann and J.B. show a hierarchy of discrete positive-ionic struc- Ketterson (John Wiley,New York71976)PartI,p242 tures which can be interpreted by the present model. 131 Lee,M.A., Lowy,D.N, Woo,C.W,Phys.Rev.Bfi(l976) The detailed comparison with experiment will be 4874

p~r (atm) 0 5 0

.

5 0 5 0 5

discussed together with the self-consistent treat- /4/ Namaizawa,H.,J.Low.Temp.Phys.3J(1978)in print ment of the thermal equilibrium of the bound 4 ~ e

/5/ Alexander ,P .W., Barber ,C .N., McClintock,P.V.E.,

shell. Pickett,G.P.,Phys.Rev.Lett. 2(1977)1544 e (K) -0.769 -0.575 -0.663 -0.482 -0.456 -0.297 -0.161 -0.041