4.3.2 Solid Nitrogen in an Electric Field

An electric field was applied to a sample of solid nitrogen (s-N₂) by the addition to the sample cell of two fine wire grids outside the sample cell spaced 3.0 mm from the mylar cell windows. High positive and negative voltages (with respect to ground) were then applied to the grids to establish an electric field through the sample. Conventional time-differential spectra were measured by TF $\mu{\cal SR}$ with the electric field either along or opposed to the direction of the incoming muon beam, producing the asymmetries shown in Fig. 4.13. A positive E denotes an electric field parallel to the muon's momentum (the conventional forward direction); negative E in the opposite direction. The magnitude E is simply the potential difference between the grids divided by the separation of 1.2 cm.

The results unambiguously indicate directional asymmetry in the effect of an electric field. With $\vec{E}$ in the $+\hat{z}$ direction the muon fraction was increased and muonium decreased. With the direction of the electric field reversed, the asymmetries change in the opposite way. This is consistent with a mechanism for muonium formation in which free electrons generated in the muon's radiolysis track neutralize the positive muon. Applying an external electric field drives these electrons either toward or away from the stopped muon, increasing or decreasing the likelihood of muonium formation by putting more or fewer electrons in the vicinity of the thermal muon. Muonium formation in this way is always exothermic and therefore can proceed after the muon has come to thermal energies. Assuming this model, we can conclude that the electrons originate predominantly behind the muon. This model implies that the muon stops downstream from the end of its ionization track, more or less preserving its direction of travel while slowing down.

In order for the external field to have much effect on the trajectory of electrons in the Coulomb fields of nearby ions and the muon, the electron that eventually reaches the muon must be far enough away from them, at least for part of the time-of-flight, One way for this to happen may result from the end of the cyclic charge exchange part of the muon track. A hot muonium atom can undergo a collision with a neutral atom and ionize itself, leaving the electron in the lattice while the bare muon can continue on with the last few eV and eventually come to rest further down stream. In this picture neither the electron or muon are in the immediate vicinity of any other ion. Most importantly, this electron that has been carried in a muonium atom away from the ion from which it came is the closest one to the muon and therefore is the electron that reaches the thermal muon first. This model implies that the direction of travel of the muon, at the very end of the track, is still on average in the direction of the muon beam.

  

Figure 4.13: Slowly relaxing diamagnetic asymmetry $A_{\rm S}$and muonium asymmetry $A_{\rm Mu}$ (boxes and circles respectively) measured in a sample of solid nitrogen at T=20 K with an external electric field applied either along (E>0) or opposite (E<0) to the incoming muon beam direction.

In order for an external electric field to have an appreciable effect on the motion of electrons in the vicinity of the muon, the magnitude of the external electric field would have to be comparable to (or larger than) the electric field due to the muon's charge. One can then estimate the typical initial distance between the electron and muon from the electric field dependence of the asymmetries. An electric field of about 3 kv/cm falls midway on the curve between E=0 and saturation of the effect. Taking the electric field of the muon at the electron to be E₀=3.0 kV/cm, one may conclude that the typical electron-muon distance is approximately $R=\sqrt{e/\epsilon E_0}$ = 60 nm, in which $\epsilon = 1.3$ is the dielectric constant of solid nitrogen.