A quantum mechanical description of channeling under the influence of ultrasound (US) excited in a single crystal has been developed [1-2]. It was shown that a resonant interaction between a longitudinal standing ultrasonic wave and planar or axially channeled relativistic charged particles can take place if the frequency of US approaches some critical value. This value corresponds to the energy difference between two neighbouring states a and b of transverse motion. The quantum numbers N of these states obey the selection rules | N_a-N_b | = 1, 2, 3... for axial and | N_a-N_b | = 2, 4, 6... for planar channeling.
The frequency of US necessary for resonance reads

nn* = vs c | ea-eb | n h

where e denotes the corresponding energy level of the channeled particle and v_s the velocity of US inside the crystal. Since the effect resembles a parametric resonance, an integer n = 1, 2... appears in the denominator of the formula. This is of great practical importance because it opens the possibility of resonant interaction even at frequencies n_s lower than typically 10 GHz for n = 1. At the same amplitude of crystal potential modulation, higher values of n, of course, diminish the effect of US.
Probability distributions of a planar channeled electron are shown in Figs. 1 and 2 versus the dimensionless coordinates X  =  ax and Z  =  k_s z where the spatial coordinate z is chosen along the channeling direction, x is the transversal one, a denotes a scaling parameter of the crystal potential and k_s is the wave number of US. The only difference between the two figures consists in the US-frequency. At resonance (Fig. 1) US strongly mixes the corresponding channeling states a and b. Even a slight deviation of n_s from the resonant value n₁^* (Fig. 2) results in an abrupt reduction of the influence of US on the state of the channeled particle.

Fig. 1 Probability distribution P(X,Z) of a planar channeled electron at resonance (n_s = n₁^*).

Fig. 2 The same as in Fig. 1 but slightly out of resonance (n_s = 0.95 n₁^*). ¹ Institute of Applied Problems in Physics, NAS Yerevan, Armenia

References

[1] L.Sh. Grigoryan et al., Rad. Eff. & Def. in Sol. 152 (2000) 225; ibid. p. 269

[2] L.Sh. Grigoryan et al., Rad. Eff. & Def. in Sol. 153 (2000) 13  IKH 05/30/01 © W. Wagner