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Submitted on 1 Jan 1978
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DENSITY PROFILES FOR ELECTRON AND
POSITRONIUM BUBBLES IN HELIUM
P. Hautojärvi, M. Manninen, R. Nieminen, K. Rytsölä, I. Välimaa
To cite this version:
JOURNAL DE PHYSIQUE Colloque C6, supp/Pment au no 8, Tome 39, aoBt 1978, page C6-70
D E N S I T Y P R O F I L E S FOR ELECTRON AND P O S I T R O N I U M BUBBLES I N H E L I U M
P. ~autojzrvi, M.Manninen, R.Nieminen, K.Ryts61; and I.Vllimaa
Department o f TechnicaZ Physics, He Zsinki University o f TechnoZogy, SF-021 50 Espoo 25, Finland Rdsum6.- A partir des fonctionnelles de densitd nous calculons les propridtds des bulles dVClec-
trons et de positronium dans l'hdlium. Les profils de densitQ ont dtd testds par des mesures de temps de vie de positron.
Abstract.- Density functional calculations have been performed on electron and positronium bubbles in helium. Density profiles of positronium bubbles have been tested by positron lifetime measure- ments.
The mobility of ions is extensively used to investigate the superfluid phases of helium,but the internal structure of ions themselves ("snow ball" bubble) is quite poorly known and difficult to stu- dy experimentally. However, experimental information is easily obtainable from the density fluctuations induced by positrons and positronium, since the an- nihilation rate of positrons is proportional to the density of their immediate vicinity. In analogy with the behaviour of positive ions, positrons induce
clustering of He atoms /I/. A striking feature is that this clustering ("droplet formation") occurs only below a critical temperature which is about twice that of bulk liquid 121. On the other hand, the positronium atom causes cavity formation simi- lar to the electron bubble /I ,3/ and this is the subject of the present investigation.
Bubbles have usually been described by a square-well potential and a rectangular density profile. We go one step further and use the densi- ty functional method which has successfully been applied to clusters around positrons and positive ions /4,5/. The density profile n(r) is determined by minimizing the free energy fudctional.
n[n] = F [n] + jd;(V[n]
-
u)n(i) r y'I
d;lv~/! (I)where F is the free energy of the inhomogeneous fluid, p is its chemical potential and V [n] is the Ps-He or e-He interaction. The last term describes the kinecic energy associated with the localized state
of
the trapped particle moving in the po- tential V [nl.
Thus n(;) and I(;) have to be sol- ved self-consistently. The condition for the sta- bility of the bubble is that its free energy (1)is less than that for the homogeneous system with a delocalized particle. F and p are expressed local-
ly using the van der Waals equation of state. V
[d
has been constructed from Wigner-Seitz calculations usingahard-sphere potentia1,the radius of which represents the Ps-He or e-He scattering length aps or ae. Some density profiles for Ps and elec-0
tron bubbles at 10K with
aps
= 0.60 A and ae = 00.62 A/6/ are shown in figure 1.
-
ELECTRON BUBBLE - 1.0-
L-
C-
VI-
z Ps BUBBLE --
- - DISTANCE r( A )
Fig.1 : Self-consistent density profiles at T= 10K for electron and positronium bubbles. Curves (a) no = 1 . 5 ~ 1 0 ~ ~ c m - ~
,
curves (b) no = 0.5x10~~cm-~.Some density profiles for Ps and electron
0 0
bubbles at l0K with a = 0.60 A and ae= 0.62 A/6/
P s
are shown in figure 1. One can see that the posi- tronium bubbles are smaller in size. This is due to the double mass, which reduces both the kine- tic energy of the localization and the repulsion with helium atoms. Preliminary calculations on the bubble stability semm to indicate that, in contrast to positron-induced droplets, the bubble formation would be possible at any temperature if
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1978633
the density of the fluid is high enough. We are indepted to Asko Vehanen and Jari Vetten- ranta for their assistance in the measurements.
Ps BUBBLE A
ATOMIC DENSITY ( J ~ ' ~ c r n - ~ )
Fig.2 : The pick-off annihilation rate of orthopo- sitronium in free and bubble states as a function of helium density. The solid line represents results of the theoretical calculations.
Positron lifetime measurement have been car- ried out in a conventional way /3/ at various tem- peratures and pressures. The pick-off rate
A
ofPO the ortho-Ps has been deduced from the lifetime spectra and some results are shown in figure 2. At high temperatures
A
increases proportional to thePO
helium density (free Ps). When the bubble is formed at low temperatures the Ps overlap with the fluid is reduced leading to a considerable decrease of h
PO Thetheoretical threshold density of bubble formation at 10K is about 0 . 3 ~ 1 0 ~ ~ c m - ~ . The pick-off rate can be estimated from the calculated profiles accor- ding to
3 - 1
A
PO = 2.61~10 s /df I$(;) (2)-
where n (r) is expressed in a.u.. The solid line in figure 2 corresponds to the results of (2) by using
0
ap
= 0.60 A.
It shnuid be mentioned that the calcu- lations are extremely sensitive to the value of a Ps' A satisfactory fit is achieved only if aps lies0
within 0.55 A and 0.65 A. This is slightly lower than the most recent theoretical value 0.735 /7/ an than the earlier experimental result 0.77 A /I/ based on the square-well model with rectangular den- sity profile.
More detailed theoretical calculati'on with a semiempirical equation of state /8/ together with experimental investigations of the temperature de- pendence of the threshold density for bubble forma- tion are in progress and will be reported later.
References
/I/ Canter,K., Mcnutt,J.,and Roellig,L.,Phys. Rev. A
12
(1975) 375./2/ Hautojsrvi,P., RytsZl;i,~., Tuovinen,P.,
Vehanen,A. and Jauho,P., Phys Rev.Lett.E(l977) 842
131 ~autoj&vi,~., ~ytsGls,~., Tuovinen,P. and Jauho,P., Phys.Lett.
57A
(1976) 175/ 4 / Stott,M. and Zaremba,E., Phys.Rev.Lett.
38
(1977) 1493./ 5 / Manninen,M.and Hautoj:rvi,P., Phys.Rev.B (to be published)
/6/ Bransden, B. and Mcdowell,M., J.Phys. B
