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 3 He below 1 mK along the melting curve ( P m ≃ 34 bar ) . The change in sound velocity, Δ v ( T , B ) ( ≡ v ( T , B ) - v ( 0 , 0 ) ) , arises from the nuclear-spin parts of the internal energy. In the U2D2, the measured Δ v ( T , 0.06 T ) was proportional to T 4 and Δ v ( 0.5 mK , B ) was proportional to B 2 . In the CNAF, the measured Δ v ( T , 0.50 T ) was proportional to T 4 and Δ v ( 0.5 mK , B ) was proportional to B - B c 1 ( T ) , where B c 1 ( T ) is the lower critical field. From Δ v for different crystal orientations, we determined the nuclear-spin elastic-stiffness tensors Δ 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 Γ ij 1 / χ for the magnetic susceptibility χ from Δ c ij l (0.5 mK, B ) and Γ ij c for the spin wave velocity c from Δ 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 Γ ij c and Γ ij 1 / χ 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.