0 Se p 20 05 dE / dx to Subleading Order in the Plasma Density Lowell

Dimensional continuation is employed to compute the energy loss rate for a non-relativistic particle moving through a highly ionized plasma. No restriction is made on the charge, mass, or speed of this particle, but it is assumed that the plasma is not strongly coupled in that the dimensionless plasma coupling parameter g = e 2 κ D /4πT is small, where κ D is the Debye wave number. To leading order in this coupling, dE/dx is of the generic form g 2 ln[g 2 C]. The prefactor of the logarithm is well known. We compute the constant C under the logarithm exactly. Our result differs from approximations given in the literature, with differences in the range of about 20% for cases relevant to inertial confinement fusion experiments. The stopping power of plasma component b for projectile p is of the form dE b dx = e 2 p e 2 b 4π n b m b v 2 p ln Λ b = e 2 p 4π κ 2 b β b m b v 2 p ln Λ b , (1) where the Coulomb logarithm ln Λ b involves a ratio of short-and long-distance length scales. To compute ln Λ b , we employ the method of dimensional continuation [1]. To introduce this method, we consider the Coulomb potential φ ν (r) of a point source in ν spatial dimensions: φ ν (r) ∼ 1/r ν−2. Clearly the long and short distance behavior of φ ν depends on the spatial dimensionality ν. In high ν, short distance (hard) interactions are accentuated, while in low ν the large distance (soft) physics predominates. For Coulomb interactions, ν = 3 is special in that neither hard nor soft processes are dominant. For ν < 3, the soft physics is predominant, and for ν > 3 the hard processes are dominant. The energy loss for ν > 3, dE B /dx, is obtained from the Boltzmann (B) equation, and it contains a pole (ν − 3) −1 that reflects an infrared divergence in the scattering process when ν → 3 +. The energy loss for ν < 3, dE LB /dx, is obtained from the Lenard-Balescu (LB) kinetic equation, and it contains a pole (3 − ν) −1 that reflects an ultraviolet divergence when ν → 3 −. The stopping power to subleading order is therefore dE dx = lim ν→3 dE LB …