See also the source file: he3_grad.f. $$ F_\nabla = \frac12 \Delta^2 \left[ K_1 (\nabla_j R_{ak})(\nabla_j R_{ak}) + K_2 (\nabla_j R_{ak})(\nabla_k R_{aj}) + K_3 (\nabla_j R_{aj})(\nabla_k R_{ak}) \right] $$
| he3_grad_K0(ttc,p) | $K_1=K_2=K_3$ without fermi-liquid corrections, see VW7.23m |
| he3_grad_c(ttc,p)
he3_grad_delta(ttc,p) |
$c$ and $\delta$ paramters calculated with fermi-liquid
corrections (Cross-1975), see VW7.25. These values are
used to calculate all other things:
$c=-\frac{\rho_s}{10}\ \frac{3+F_1^a}{3+F_1^s} \ \frac{1}{1+F_1^a(5-3\rho_s/\rho)/15},\qquad \delta = \frac{F_1^a \rho_s/\rho}{3+F_1^s(1-\rho_s/\rho)}$ |
| he3_grad_K12(ttc, p)
he3_grad_K3(ttc, p) |
$K_1 = K_2 = -\frac{2}{\Delta^2}\left(\frac{\hbar}{2m}\right)^2 c,\qquad K_3 = -\frac{2}{\Delta^2}\left(\frac{\hbar}{2m}\right)^2 (1+\delta)c $ |
| he3_grad_K(ttc, p)
he3_grad_Kp(ttc, p) |
$K = 2K_1 + K_2 + K_3,\qquad K' = K_2 + K_3$ |
| he3_grad_lg1(ttc, p)
he3_grad_lg2(ttc, p) he3_grad_lsgb(ttc, p) |
Thuneberg's $\lambda_{G1}$, $\lambda_{G2}$ and $\lambda_{SG}^b$:
$\lambda_{G1} = \frac12\Delta^2 (K_2+K_3),\qquad \lambda_{G2} = \frac12\Delta^2 K_1,\qquad \lambda_{SG}^b = \Delta^2 K_2$ |
| he3_cpar(ttc,p)
he3_cperp(ttc,p) |
Velocity of transverse spin waves parallel
and perpendicular to the $l$ direction [cm/s]:
$c_\parallel^2 = \frac{\gamma^2\Delta^2}{\chi_B} K,\qquad c_\perp^2 = \frac{\gamma^2\Delta^2}{\chi_B}(K-K'/2),\qquad c_\parallel^2/c_\perp^2\approx 4/3.$ |
| he3_clpar(ttc, p)
he3_clperp(ttc, p) |
Same for longitudinal waves [cm/s]:
$C_\parallel^2 = \frac{\gamma^2\Delta^2}{\chi_B}(K-K'),\qquad C_\perp^2 = \frac{\gamma^2\Delta^2}{\chi_B} K,\qquad C_\parallel^2/C_\perp^2\approx 1/2.$ |
Transverse spin-wave velocities at $T/T_c\approx0$ was measured in Zajalov-2015. This measurements was used to extract value of $F^1_a$. The functions for spin-wave velocities use this value, and thus agree with the experiment.