Properties of Few-Electron Bubbles in Superfluid Helium-4
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Properties of Few‑Electron Bubbles in Superfluid Helium‑4 Yiming Xing1 · Humphrey J. Maris1 Received: 7 July 2020 / Accepted: 8 August 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract We present calculations of the properties of bubbles in liquid helium-4 containing 3 or 8 electrons. The pressure range in which these bubbles are stable is determined. For Z =3, we find that the bubbles are unstable except in a pressure range between − 0.78 and − 0.90 bars. For Z = 8, the bubbles are stable provided the pressure is in the range from zero to – 0.42 bars. At the upper end of the pressure range, the bubbles break into smaller bubbles each containing a fraction of the electrons; at the lower end the bubbles become unstable against unlimited expansion. Keywords Electrons · Liquid helium-4 · Critical pressures
1 Introduction An electron entering liquid helium repels the helium atoms and forms a bubble. A bubble containing a single electron is referred to as a normal electron bubble or NEB. The energy E of this bubble is given by the simple expression
E=
h2 4𝜋 3 R P, + 4𝜋R2 𝛼 + 3 8mR2
(1)
where h2 ∕8mR2 is the zero point energy of the electron ( m is the mass of the electron, R is the bubble radius), 4𝜋R2 𝛼 is the surface energy of the bubble, and the last term is the work done against the applied pressure P in forming the bubble. There are a number of other effects that have not been allowed for in Eq. 1, but these are small corrections [1]. From Eq. 1, we can find the value of the radius that makes the energy a minimum. For zero pressure and temperatures below 1 K this comes out to be 19 Å. It is also possible to produce bubbles that contain a large number Z of electrons, e.g., 105–107; these are referred to as multi- electron bubbles or MEB [2–5]. These bubbles are formed when a high density of electrons is produced just above the * Yiming Xing [email protected] 1
Department of Physics, Brown University, Providence, RI 02912, USA
13
Vol.:(0123456789)
Journal of Low Temperature Physics
surface of the liquid. For these objects, the zero point energy makes a contribution to the energy which is small compared to both the surface energy and the energy associated with the Coulomb repulsion between electrons. To minimize the Coulomb energy, the electrons are distributed within a thin layer just inside the bubble surface. The energy of an MEB is then given by the approximate expression
E=
4𝜋 3 Z 2 e2 + 4𝜋R2 𝛼 + R P, 2𝜀R 3
(2)
where 𝜀 is the dielectric constant of liquid helium. At a critical negative pressure Pc , the bubble will grow without limit. This pressure for MEB is given by
)1∕3 ( 27𝜋𝜀𝛼 4 Pc = − 2Z 2 e2
(3)
For more detailed calculations of the energy of the structure and energy of an MEB see the papers by Shikin [6], and Salomaa et al. [7–9]. When a NEB contains an electron in the ground state (as is assumed in writing Eq. 1), the NEB is spherically symmetric. The MEB also have spherical symmetry, apart from small dimples at the location of each electron
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