Volume 51A, number 3 PHYSICS LETTERS 24 February 1975 ENERGY LOSSES OF CHANNELED PARTICLES AT HIGH ENERGIES M.A. KUMAKHOV and R. WEDELL Institute of Nuclear Physics, Moscow State University, Moscow, USSR and F.F. KOMAROV University of Minsk, Minsk, USSR Received 6 January 1975 The ratio of the energy loss of channeled particles to the energy loss in the amorphous medium has been calculated including relativistic corrections. In the limit of high velocities, the ratio has been shown not to tend to unity. In the various problems associated with the use of the directional effects, it is necessary to know the energy losses of the channeled particles [1]. It is inte- resting to consider the ratio of energy losses of chan- neled particles to the losses in an amorphous medium at high energies. At low velocities, as has been shown theoretically [2, 3] and experimentally [4] also the ratio has a maximum. The calculation is made in two ways. First, we make use of the equipartition rule for the energy losses in close and distant collisions [5,6], the we resort to the classical dipole approximation. In the first case, the ratio turns out to be 7 = 1 - ½Zeor/Z 2 , (i) where Zeor = Z 2 - Zou t, Z 2 is the atomic number of the target, Zou t the number of the outer electrons. In the derivation of (1) it has been assumed that the stopping of channeled particles in close collisions takes place on the outer electrons. Now we make a more detailed calculation assuming that the distribution of valence electrons in the channel is uniform. In this case n = 7t + 72, (2) where 71 and 72 are the ratios of losses to the valence and inner electrons to those of the random beam. For instance, when the relativistic effects are not taken into account, we have Zou t lnl2mo2/hwp I Zou t (1+ lnll/h~pl) 71 = Z~ lnl2mo2/ii = Z2 \ ln~2rn°2/----~1 ('3) where hoop is the plasmon energy, o the particle velo- city, I the atomic ionization potential. As is seen from (3), 71 decreases slowly with velocity. The decrease is somewhat compensated by an increase in r/2 as with the increasing velocity greater number of inner electrons in the neighbouring channels take part in the stopping. Indeed, according to the adiabatic criterion, the impact parameter b, at which an inner electron may effecti- vely participate in the stopping, is equal to b = oh/1 i where I i is the proper ionization potential. The calcu- lations were carried out for protons perfectly channeled in Si along the (111 ) axis. The energy losses of the random beam were calculated using the Bloch relati- vistic formula [7], and the plasma losses were found by the Could formula [8]. The results of the calculations are given in table 1. Clearly, the value of 7 little changes with energy. At energies of the order of several MeV, a small increase the L-shell electrons begin to take an effective part in the stopping of channeled particles. From the fact that at relativistic velocities 7 < 1 also one may draw a conclusion that the channeling effect can be detected even at these energies by measuring the energy losses. It would be of interest to verify this statement expe- rimentally. When the equipartition rule is used (i.e., formula (2)), we have 7 ~ 0.642. This value, as can be seen from table 1, is slightly different from those values of 7 given by a detailed calculation. At low velocities, a maximum of 7 is quite pronounced. 151