A Model Study of the Possible Effect of Beryllium Grain Sound Speed Anisotropy on ICF Capsule Implosions

The sound speed in a Be grain is markedly different in orthogonal directions due to an anisotropic Young’s modulus. The impact of this fact on ICF capsules machined from multi-crystalline Be is not clear, but is of concern if the shock velocity is likewise grain orientation dependent. In this paper the expected inner wall break out profile due to grain affected shock velocity variations is calculated for a Be capsule, as a function of the grain size and effective shock velocity anisotropy factor factor p = v_‖ / v_⊥, where v_‖ and v_⊥ are the effective maximum and minimum orthogonal shock speeds in a grain. In this simple model it is assumed that grain boundaries have no effect other than to mark the location where the shock speed changes as it moves from one grain to another. The grain structure of bulk beryllium is modeled by randomly placing N points in a volume V to define Wigner-Seitz cells (grains) of average volume V/N. Each grain is given a random orientation. The spherical shell wall is modeled by a 150 µm thick planar slab of this multi-crystalline material, 2πR in length where R is the capsule radius, taken to be 1000 pm. The slab is sampled at 3600 points along its 2πR length, at each point the average shock velocity through the sample is determined based on the model slab grain structure at that point. This data is used to create the expected spatial breakout profile, which is then Fourier transformed to give a power spectral representation that is compared to the current outside surface design specification. In order to match the design specification, grain diameters less than 10 pm and an effective shock velocity anisotropy, p, of less than 1.001 are necessary.