Sound velocity measurements of nuclear-spin ordered BCC solid 3He

We measured sound velocity, v(T, B), in the two nuclear-spin ordered phases (the low field phase (U2D2) and the high field phase (CNAF)) of single crystal of BCC solid He-3 below 1 mK along the melting curve (P-m similar or equal to 34 bar). The change in sound velocity, Delta v(T, B)(equivalent to v(T, B) - v(0, 0)), arises from the nuclear-spin parts of the internal energy. In the U2D2, the measured Delta v(T, 0.06 T) was proportional to T-4 and Delta v(0.5 mK, B) was proportional to B-2. In the CNAF, the measured Delta v(T, 0.50 T) was proportional to T-4 and Delta v(0.5 mK, B) was proportional to B - B-c1(T), where B-c1 (T) is the lower critical field. From Delta v for different crystal orientations, we determined the nuclear-spin elastic-stiffness tensors Delta c(ij)(l,h)(T, B), six components for U2D2 and three components for CNAF as a function of T and B. We obtained the generalized Gruneisen constants Gamma(1/chi)(ij) for the magnetic susceptibility chi from Delta c(ij)(l) (0.5 mK, B) and Gamma(c)(ij) for the spin wave velocity c from Delta c(ij)(l)(T, 0.06 T) in the U2D2 by using the nuclear-spin part of the internal energy U(T, B). In particular, the obtained Gamma(c)(ij) and F-ij(1/chi) are compared with the theoretical estimation based on the multiple-spin exchange model (MSE) with three parameters. The validity of the MSE with three parameters is quantitatively discussed. (c) 2006 Elsevier B.V. All rights reserved.