OPTICAL AND MECHANICAL PROPERTIES OF ELECTRON BUBBLES IN SUPERFLUID HELIUM-4
Z. Xie, W. Wei, Y. Yang, H. J. Maris*
Department of Physics, Brown University, Providence, Rhode Island 02912, USA
Received May 14, 2014
A series of experiments has revealed the existence of a large number (about 18) of different types of negative ions in superfluid helium-4. Despite much effort, the physical nature of these "exotic ions" has still not been determined. We discuss possible experiments which may be able to help determine the structure of these objects.
Cwitribvtiwi for the JETP special issue in honor of A. F. Andrew's 75th birthday
DOI: 10.7868/S0044451014120116
1. INTRODUCTION
At first sight, it appears that it should bo easy to understand the behavior of an electron immersed in liquid helium. Because a helium atom has a closed shell of electrons, there is a strong repulsion between a helium atom and an electron. As a result, in order to enter liquid helium, an electron has to overcome an energy barrier of approximately 1 oV [la]. The experiment performed earlier [lb] gave the result 1.3 oV. This barrier, together with the very low surface energy a: of the liquid (0.375 erg-cm-2) [2], makes it favorable for an electron to force open a cavity in the liquid and become trapped there, rather than moving freely through the bulk liquid. The size of this bubble can be estimated, to a reasonable accuracy, from the approximate expression for the energy
h2 . 4?r
Ebubbk: = gmR2 + ^-R2«' + — (!)
where R is the bubble radius, rn is the electron mass, and the last term represents the energy associated with forming the bubble when a pressure P is applied to the liquid. In the absence of an applied pressure, we find from Eq. (1) that the energy should be a minimum for the radius
/ h2 \1/4 E-mail: humphrey_ maris'fflbrown.edu
These "electron bubbles" have been studied in many experiments.
1) Measurements have been made of the photon energies required to excite the electron to a higher energy-state [3 5]. Since these energies are dependent on the bubble size (approximately proportional to the inverse square of the radius), the experiments provide information about the radius.
2) The mechanical properties of the bubble can be studied by applying a negative pressure [6]. If a negative pressure larger than a critical value Pc is applied, the bubble becomes unstable and grows rapidly. It can then be detected optically. From Eq. (1), the critical pressure is found to be [7]
3) Measurements have been made of the mobility //. of these bubbles [8 10]. The mobility is limited by the drag force exerted on a moving bubble by thermally excited phonons and rotons. In superfluid holium-4 above 1 Iv, the drag is primarily due to rotons and the mobility can be expected to vary as
// x exp(A//,-n. (4)
where A is the rot on energy gap. The results of the mobility experiments give a temperature dependence in reasonable agreement with this. If a sufficiently large electric field is applied, the velocity reaches a critical value vc at which a quantized vortex ring is nucleated. The bubble then becomes attached to this vortex ring [11].
Signal, pA 0
-0.5 -1.0 -1.5 -2.0
0 5 10 15 20 25
Transit time, ms
Fig. 1. Results from a time-of-flight mobility experiment performed at 0.991 K as reported in Ref. [22]. The solid curve shows the signal arriving at the collector as a function of time. The dashed curve is the signal after an algorithm has been used to remove the peaks. The length of the experimental cell is 6.15 cm and the drift field is 82.1 V crrT1
4) The effective IIlclSS of the bubbles has been measured under the saturated vapor pressure [12] and under elevated pressure [13]. The results of the measurements are in good agreement with the values predicted from the bubble model.
Surprisingly, the experiments have revealed that in addition to the "normal" electron bubbles (NEB), there are other negatively charged objects of unknown physical structure [14 21]. These are referred to as the "exotic ions". The solid curve in Fig. 1 shows data obtained in a recent time-of-flight mobility experiment at 0.991 Iv [22]. In this experiment, ions entered the liquid from a continuous electrical discharge in helium vapor above the surface of the liquid. The discharge was produced by a voltage applied between electrodes positioned in the vapor. After the ions entered the liquid, gate grids were used to allow a pulse of negative ions to enter the upper part of the experimental cell. These ions moved through the cell under the influence of a uniform drift field and the charge arriving at a collector at the bottom of the cell was recorded as a function of time. In Fig. 1, we can see a strong signal at a time of around 19 ms coming from the NEB. In addition, there is a series of peaks at earlier times coming from the exotic ions. Figure 1 clearly shows at least ten exotic ions; more recent experiments [23] have resolved 18 ions, each with a different mobility.
We can make a fit to each peak and then subtract
the peak from the measured total signal. When this is done, a smoothly varying background signal is revealed as shown by the dashed curve in Fig. 1. The continuous background has a cutoff at a time that is approximately one half of the arrival time of the NEB. The time at which the cutoff appears in the signal is inversely proportional to the drift field, indicating that the background arises from ions. These ions must have a continuous distribution of mobility, and therefore presumably a continuous distribution of size.
It is interesting that although the signal from each individual exotic ion is much smaller than the signal from the NEB, the total signal from the exotic ions (including the continuous background) is of a magnitude comparable to the NEB signal (typically 20 % to 50%).
At a critical velocity vc, each of the exotic ions (except the fastest ion F) nucleates a vortex ring and becomes trapped on it [18]. The critical velocity is larger than the critical velocity for the NEB, indicating that the ions are smaller than the NEB. Since vc increases progressively with an increase in ion mobility, each of the exotic ions appears to be singly charged.
A rough estimate of the ion size can be made from the measured mobility. Since the mean free path of a rot on at temperatures around 1 Iv is large compared to the bubble size, the drag exerted on a moving bubble should be proportional to the cross-sectional area of the bubble. Hence, the mobility should vary approximately as the inverse square of the radius. Based on this, the radius of the fastest ion is found to be around 8 A [17].
Presently, there is no accepted theory of the makeup of the exotic ions. Three ideas and their associated difficulties are as follows.
1) Impurity model. Impurity atoms that have acquired an extra electron could form bubbles with a size in the range of the exotic ions. However, an electron that is bound to an impurity with a high electron affinity (e. g., greater than 2 eV) would have a wave function that decreases very rapidly with distance. This would result in a snowball [24] or a bubble of a very small radius. Thus, in order for impurities to be the explanation of the exotic ions, the impurities have to have low electron affinity. It is also possible that there are impurities that do not form negative ions in the vacuum but which bind an electron in a bubble when in liquid helium. A serious difficulty with the impurity model is that the number of impurities that might be present in liquid helium is very small; it is difficult to believe that there can be 18 different impurities with the required electron affinity, and that the same im-
purities occur in different labs in different countries. Also, a theory based on impurities cannot explain the continuous background.
2) Helium ion model. Negative ions of a helium atom [25] or helium diiiior [26] have been studied and their lifetime measured in a number of experiments. A negative helium ion immersed in a liquid should form a bubble state if the binding of the electron to the atom (or diiiior) is sufficiently weak. One problem is that the lifetimes of the known ions of helium atoms or diniors are much less than the time to traverse the mobility cells used in the experiments where exotic ions have been detected. Thus, the ions should decay before reaching the collector. In addition, the number of different ions is not sufficient to explain the observation of 18 distinct species of exotic ions. Also this model would not provide an explanation of the continuous background.
3) Fission model [27]. An electron entering the liquid has a complicated wave function. We can ask whether all of this wave function ends up in a single bubble. If the wave function ends up divided between two or more bubbles, it is not clear what would happen. One possibility is that the helium would make a measurement and determine that the electron is in one of the bubbles (call this bubble A). Then according to the Copenhagen interpretation of quantum mechanics, the wave function will suddenly change such that it is nonzero only in bubble A. The other bubbles, which contain no wave function, will collapse. But if this does not happen and the bubbles containing only a fraction of the wave function are stable, these should be smaller than the NEB and could provide an explanation of the exotic ions. Since the fraction of the wave function ending up in a bubble could have any value, this theory could explain the continuous distribution of mobility. In addition, it has been pointed out [27] that there is a mechanism that could lead to bubbles containing particular discrete fractions of the wave function, and this might explain the 18 ions with discrete values of mobility.
In this paper, we discuss how the possible experiments may allow distinguishing between these three models.
2. THEORETICAL MODEL
Shikin has written an excellent revi
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