This library provides constant and functions for various $^3$He properties. Code can be found in https://github.com/slazav/he3lib
Old documentation (last updated 2016): http://slazav.github.io/he3lib/index_2016.html
The library can be used with different programming languages and via a command-line interface. Example programs and scripts can be found in the examples/ folder.
There are a few functions in the library with complex return value (he3_lancwire_*, math_stokes). This interface is experimental, you can always use separate functions for real/imag component, in this case you will call the code twice. Support for complex functions in the library:
Example with command line interface:
$ he3 he3_lancwire_n 0.011 0 16.5 124 1700 1.687029e+03 2.546024e+01 $ he3 he3_lancwire_n_f 0.011 0 16.5 124 1700 1.687029e+03 $ he3 he3_lancwire_n_w 0.011 0 16.5 124 1700 2.546024e+01
const_e = 2.7182818284590452 -- e
const_pi = 3.1415926535897932 -- pi
const_2pi = 6.2831853071795864 -- 2*pi
const_euler = 0.5772156649015329 -- Euler's constant
const_z2 = 1.6449340668482264 -- zeta(2)
const_z3 = 1.2020569031595943 -- zeta(3)
const_z4 = 1.0823232337111382 -- zeta(4)
const_z5 = 1.0369277551433699 -- zeta(5)
const_na = 6.02214129e+23 -- Avogadro constant, [1/mole]
const_kb = 1.3806488e-16 -- Boltzmann constant, [erg/K]
const_r = 8.314472e+7 -- R-gas constant, kb*na, [erg/K/mol]
const_h = 6.62606957e-27 -- Planck constant, [g*cm2/s]
const_hbar = 1.054571726e-27 -- reduced Planck constant, [g*cm2/s]
const_mu0 = 1.2566370614 -- vacuum permeability [G*cm/A]
const_ev = 1.602176634e-12 -- electronvolt [erg]
h1_gyro = 26752.218744 -- H1 g-factor, [rad/s/G]
h2_gyro = 4106.5 -- H2 g-factor, [rad/s/G]
h_amass = 1.67355755e-24 -- H molar mass, [g/mol]
h_mmass = 1.00784 -- H molar mass, [g/mol]
he3_gyro = 20378.9 -- He3 g-factor, [rad/s/G]
he3_amass = 5.00789994e-24 -- He3 atom mass, [g]
he3_mmass = 3.0158281 -- He3 molar mass, [g/mol]
he4_amass = 6.64647641e-24 -- He4 atom mass, [g]
he4_mmass = 4.002602 -- He4 molar mass, [g/mol]
he3_pcr = 1.16317 -- He3 critical pressure [bar]
he3_tcr = 3.324 -- He3 critical temperature [K]
he3_pm = 29.3113 -- He3 Melting curve minimum [bar] (PLTS2000)
he3_tm = 0.31524 -- He3 Melting curve minimum [K] (PLTS2000)
he3_pa = 34.3380 -- He3 A-N-Solid crit.pt [bar] (Greywall-86)
he3_ta = 2.491 -- He3 A-N-Solid crit.pt [mK] (Greywall-86)
he3_pb = 34.3580 -- He3 A-B-Solid crit.pt [bar] (Greywall-86)
he3_tb = 1.932 -- He3 A-B-Solid crit.pt [mK] (Greywall-86)
he3_ps = 34.3905 -- Solid He3 AFM transition at melting curve [bar] (Greywall-86)
he3_ts = 0.9291 -- Solid He3 AFM transition at melting curve [mK] (Greywall-86)
he3_pabn = 21.22 -- He3 A-B-N crit.pt. [bar] (Greywall-86)
he3_tabn = 2.273 -- He3 A-B-N crit.pt. [mK] (Greywall-86)
he3_pa_plts = 34.3407 -- He3 A-N-Solid crit.pt [bar] (PLTS2000)
he3_ta_plts = 2.444 -- He3 A-N-Solid crit.pt [mK] (PLTS2000)
he3_pb_plts = 34.3609 -- He3 A-B-Solid crit.pt [bar] (PLTS2000)
he3_tb_plts = 1.896 -- He3 A-B-Solid crit.pt [mK] (PLTS2000)
he3_ps_plts = 34.3934 -- Solid He3 AFM transition at melting curve [bar] (PLTS2000)
he3_ts_plts = 0.902 -- Solid He3 AFM transition at melting curve [mK] (PLTS2000)
he3_pabn_plts = 21.222 -- He3 A-B-N crit.pt. [bar] (PLTS2000, converted from Greywall scale)
he3_tabn_plts = 2.2315 -- He3 A-B-N crit.pt. [mK] (PLTS2000, converted from Greywall scale)
he3_pvap(T) -- Vapor pressure [bar] vs T [K] (1962 temperature scale)
[Sherman-1964]
he3_pmelt_greywall_org(T) -- Melting pressure [bar] vs T [K], Greywall-1986, T = 0.0009 - 0.25 K
Greywall. PRB33 7520 (1986) f.A1
he3_pmelt_plts_org(T) -- Melting pressure [bars] vs T [K], PLTS-2000, T = 0.0009 - 1 K
he3_pmelt_osborne_org(T) -- Melting pressure [bars] vs T [K], Osborne-1952, 0.5-1.5 K
Osborne, Abraham, Weinstock, 1951, 1952
he3_pmelt_mills_org(T) -- Melting pressure [bars] vs T [K], Mills-1955, 2-31K
Mills, Grilly, Phys. Rev. 99, 480486 (1955)
he3_pmelt(T) -- He3 melting pressure [bar] vs T[K], T = 0 .. 31 K, Greywall-86 scale
he3_pmelt_plts(T) -- He3 melting pressure [bar] vs T[K],T = 0 .. 31 K, PLTS-2000 scale
he3_tc(P) -- T_c [mK] vs P [bar], Greywall-86 scale
Greywall. PRB33 (1986) f.5.
Alvesalo scale should be multiplied by 0.893 to convert temperature.
he3_tab(P) -- T_ab [mK] vs P [bar], Greywall-86 scale
Greywall. PRB33 (1986) f.15
he3_tc_plts(P) -- T_c [mK] vs P [bar], PLTS-2000 scale
he3_tab_plts(P) -- T_ab [mK] vs P [bar], PLTS-2000 scale
Names of functions and constants on the plot:
See Wheatley, Rev.Mod.Phys. 47, 415(1975), tables at p.467
he3_vm(P) -- He3 molar volume [cm^3/mole] vs P [bar] (exp data, Greywall-86)
he3_gammaf(P) -- He3 specific heat Cv/RT [1/K] vs P [bar], (exp data, Greywall-86)
see also Alvesalo PRL44 1076 (1980) - they have different values!
he3_c_n(T,P) -- He3 heat capacity [C/R] vs T [K] and P [bar]
he3_rho(P) -- He3 density [g/cm^3] vs P [bar]
he3_2n0(P) -- 2N0 [1/erg/cm^3] vs P [bar]
he3_pf(P) -- He3 Fermi momentum [g cm/s] vs P [bar]
he3_vf(P) -- He3 Fermi velocity [cm/s] vs P [bar]
he3_meff(P) -- He3 effective mass [g] vs P [bar]
he3_mm(P) -- He3 effective mass ratio, m_eff/m_3 vs P [bar]
he3_f1s(P) -- He3 F1s fermi-liquid parameter vs P [bar]
he3_a(P) -- He3 average atomic spacing [Å] vs P [bar]
he3_gdk(P) -- He3 average dipolar coupling energy [K] vs P [bar]
he3_tfeff(P) -- He3 effective Fermi temperature [K] vs P [bar]
he3_c1(P) -- He3 first sound velocity c1 [m/s] vs P [bar] (exp data, Wheatley-75)
he3_f0s(P) -- He3 F0s fermi-liquid parameter vs P [bar]
he3_f0a(P) -- He3 F0a fermi-liquid parameter (same as Z0/4) vs P [bar] (Hensley-1993)
Hensley, JLTP89 501 (1992), JLTP90 149 (1993)
See also: Wheatley-75; Ramm, JLTP 2 539 (1970);
he3_chi_n(P) -- Susceptibility [sgs] vs P [bar]
see Einzel-1991 f.10
he3_f1a(P) -- He3 F1a fermi-liquid parameter vs P [bar] (Zavjalov-2015, from spin-wave velocities);
Zavjalov-2015 -- Spin-wave velocity in 3He-B;
See also:
Corruccini PRL27 650 (1971) -- Leggett-Rice effect in 3He-N, not accurate;
Osheroff PhB90 20 (1977) -- Spin-wave velocity in 3He-B, not accurate;
Greywall-1983 -- high temperature Cv;
theory, spin waves: Dorfle PRB23 3267 (1981) + F3s;
theory, spin waves: Cross JLTP 21 525 (1975);
he3_f2a(P) -- He3 F2a fermi-liquid parameter vs P [bar] (zero at the moment)
Halperin???
he3_f2s(P) -- He3 F2s fermi-liquid parameter vs P [bar] (zero at the moment)
See also:
Engel, Ihas, Phys. Rev. Lett. 55, 955958 (1985);
Hamot, Lee, ... Halperin, JLTP 99 p651 (1995);
Mastumoto et al. JLTP 102 p227 (1996);
he3_bcsgap(ttc) -- BCS energy gap, $\Delta/k_BT_c$ vs $T/T_c$
Newton iteration based on a note by E.Thuneberg and R.Hanninen.
Taken from ROTA texture library.
See: [1]
[2]
he3_bcsgap_fast(ttc) -- BCS energy gap, $\Delta/k_BT_c$ vs $T/T_c$, Einzel approximation
D.Einzel JLTP 84 (1991) f.68.
<0.5% accuracy in the whole temperature range,
70 times faster then he3_bcsgap
he3_bcsdgap2(ttc) -- Derivative of BCS energy gap, $d\Delta^2/d(T/Tc)$ vs $T/T_c$
Same method as in he3_bcsgap calculation, V.Zavjalov, 2020
he3_dcbcn(p) -- Heat capacity jump for He3-B, $\Delta C_b/C_n$ vs P [bar], (exp.data, Greywall-1986)
Greywall-1986, Fig.19
he3_dcacn(p) -- Heat capacity jump for He3-A, $Delta C_a/C_n$ vs P [bar], (exp.data, Greywall-1986)
Greywall-1986, Fig.19
he3_trivgap(ttc,p) -- Trivial strong-coupling correction (WCP) to the BCS energy gap. $\Delta/k_BT_c$ vs $T/T_c$, P[bar]
Approximation of Serene,Rainer-1983 corrections (Phys.Rep. 101, 221), table 4.
Note that derivative of
the $\Delta^2$ in $T_c$ is not strictly proportional to the heat capacity jump.
This shows that exact heat capacity calculation requires WCP energy terms,
not just BCS calculations with modified gap...
he3_trivdgap2(ttc,p) -- Derivative of the trivial strong-coupling (WCP) gap: d(Delta^2)/d(T/Tc)
he3_todogap(ttc,p) -- Gap corrected to Todoschenko's value 1.99 at T=0,P=Pmelt, delta/Tc vs T/Tc, P[bar]
Linear interpolation in density between BCS value at zero bar
and Todoschenko's value 1.99 at melting pressure
he3_gap(ttc,p) -- Wrapper function which should be used everywhere in the lib, same as he3_trivgap
he3_egap(ttc,p) -- he3_gap expressed in energy units [erg] rather then $T_c$
he3_yosida(ttc,gap,n) -- Yosida function of order n vs T/Tc, gap
See D.Einzel JLTP 84
$Y_n = 2\int_0^\infty \left(\frac{\xi}{E}\right)^n\left(-\frac{\partial f^0}{\partial E}\right)$
At T -> 0:
$Y_n = 2\Gamma\left(\frac{n+1}{2}\right)\left(\frac{2k_BT}{\Delta}\right)^{(n-1)/2} \exp\left(-\frac{\Delta}{k_BT}\right)$
he3_yosida_s(ttc,gap) -- Entropy Yosida function vs T/Tc, gap, see Einzel-2004
he3_yosida_c(ttc,gap,dgap2) -- Heat Capacity Yosida function vs T/Tc, gap, dgap2, see D.Einzel-2003
he3_yosida_par(ttc,gap) -- $Y_\parallel = 2/5 Y_0 + 3/5 Y_2$, see Eizel-1991 f.90
he3_yosida_perp(ttc,gap) -- $Y_\perp = 4/5 Y_0 + 1/5 Y_2$, see Eizel-1991 f.90
Code from http://ltl.tkk.fi/research/theory/qc/bcsgap.html Original nsplit=10 is too small for (z3 - 0.9*z5 + 0.9*z5.^2./z3 - 1.5*z7) combination in he3_text_lhv
he3_z3(ttc,gap) -- Z3 function
he3_z5(ttc,gap) -- Z5 function
he3_z7(ttc,gap) -- Z7 function
he3_lambda(ttc,gap) -- Lambda function
he3_rho_nb(ttc,p) -- B-phase Normal component density \rho_n^B/\rho_0
VW book f.3.92
he3_chi_b(ttc,p) -- He3-B susceptibility chi_b/chi_0
see VW book ch.10 p.449, ch2 p.90;
see Wheatley-75 f 3.7;
There is also additional term to 3*chi0: + 2/5 F2a (1-Y0)^2
he3_chi_bp(ttc,p) -- He3-B Cooper pair susceptibility ratio chi_bp/chi_b
see Leggett-Takagi 1975, f.12
he3_c_b(ttc,P) -- He3-B heat capacity (C/R)
Some gap-related functions on the plot:
he3_gd(p) -- Dipole coefficient $g_D$, [1/(erg cm^3)] vs P [bar]
restored from experimental data by E.Thuneberg.
From ROTA texture library.
he3_ld(ttc,p) -- Dipole coefficient lambda_d [erg/cm^3] vs T/Tc, P [bar]
See Thuneberg-2001 f.5 and f.24.
he3_nu_b(ttc,p) -- B-phase Leggett frequency [Hz] vs T/Tc, P[bar]
See Thuneberg-2001 f.47.
he3_nu_b1(ttc,p) -- B-phase Leggett freq, Hz (less accurate formula without use of g_d)
he3_grad_k0(ttc,p) -- K1=K2=K3 in a simple approximation, see VW 7.23m
he3_grad_c(ttc,p) -- gradient energy c parameter, see VW 7.25
he3_grad_delta(ttc,p) -- gradient energy delta parameter, see VW 7.25
he3_grad_k12(ttc,p) -- K1=K2 with Fermi-liquid corrections
he3_grad_k3(ttc,p) -- K3 with Fermi-liquid corrections
he3_grad_k(ttc,p) -- K = 2K1+K2+K3
he3_grad_kp(ttc,p) -- K' = K2+K3
he3_grad_lg1(ttc,p) -- lambda_G1 = Delta^2/2 (K2+K3)
he3_grad_lg2(ttc,p) -- lambda_G2 = Delta^2/2 K1
he3_grad_lsgb(ttc,p) -- lambda_SG^b = Delta^2 K2
he3_cpar(ttc,p) -- Fomin's spin wave velocity c_par (Fomin-1980 f.51)
he3_cperp(ttc,p) -- Fomin's spin wave velocity c_perp (Fomin-1980 f.51)
he3_clpar(ttc,p) -- Leggett's spin wave velocity c_par (Leggett-1975 XII.B)
he3_clperp(ttc,p) -- Leggett's spin wave velocity c_perp (Leggett-1975 XII.B)
he3_dorfle_rl(ttc,p,f1a,f3a) -- Test function Rl for F1a+F3a gradient energy parameters Dorfle, PRB23 3267 (1981)
K1 = K2 = rho_s/40m* Rl
K3 = rho_s/40m* (4Rt-3Rl)
he3_dorfle_rt(ttc,p,f1a,f3a) -- Test function Rt for F1a+F3a gradient energy parameters Dorfle, PRB23 3267 (1981)
K1 = K2 = rho_s/40m* Rl
K3 = rho_s/40m* (4Rt-3Rl)
he3_text_a(ttc,p) -- Textural parameter a, erg/cm^3 1/G^2
See Thuneberg-2001 f.25 and f.6
he3_text_ldv(ttc,p) -- Textural parameter lambda_{DV}, erg/cm^3 1/(cm/s)^2
See Thuneberg-2001 f.26 and f.7
he3_text_lhv(ttc,p) -- Textural parameter lambda_{HV}, g/(cm^3 Gauss^2)
See Thuneberg-2001 f.27 and f.8
he3_text_d(ttc,p) -- Surface energy coefficient d, erg/(cm^2 Gauss^2)
Came from ROTA texture library.
Some G-L extrapolation is used
he3_text_llh(ttc,p,omega) -- Vortex energy coefficient \lambda_{LH}
Came from ROTA texture library. (difference: 5/2a)
See Thuneberg-2001 f.30 and Kopu-2007 f.5
Some G-L extrapolation is used
he3_text_lo(ttc,p,omega) -- lambda/omega value used in texture library
he3_text_xih(ttc,p,h) -- Magnetic length, cm
see Thuneberg-2001, p.662
he3_text_xid(ttc,p) -- Dipole length, cm
see Thuneberg-2001, p.662
he3_text_vd(ttc,p) -- Dipole velocity vd in cm/s
he3_crsect_w(P) -- Scattering crossection he3_crsect_wi(P) -- Scattering crossection he3_crsect_wd(P) -- Scattering crossection Scattering parameters
$\lambda_n^+$ ($\lambda_n$, $\lambda_n^s$), $\lambda_n^-$ ($\lambda_n^a$),
$\delta_n^+$, $\delta_n^-$,
$\gamma_n^+$, $\gamma_n^-$,
he3_scatt_l1a(P) -- Scattering parameter $\lambda_1^-$ ($\lambda_1^a$)
he3_scatt_l2(P) -- Scattering parameter $\lambda_2$ ($\lambda_2^+$)
he3_scatt_g0(P) -- Scattering parameter $\gamma_0$,
he3_scatt_d0(P) -- Scattering parameter $\delta_0$,
he3_sykes_c(l) -- Normal phase viscosity correction c(\lambda), (Sykes-1970)
he3n_visc(p) -- Normal phase viscosity * T^2 (Sykes-1970)
he3_sykes_h(l) -- Normal phase thermal conductivity correction H(\lambda), (Sykes-1970)
he3_tau_n0(ttc,p) -- Normal state quasiparticle lifetime at the Fermi level $\tau_N(0,T)$, s
he3_tau_n_av(ttc,p) -- Thermal average of normal state quasiparticle lifetime $3/4\tau_N(0,T)$, s
he3_tau_nd(ttc,p) -- Thermal average of normal state spin diffusion transport time, s
he3_tau_nv(ttc,p) -- Thermal average ot normal state viscous transport time, s
he3_diffn_hydr(ttc,p) -- Hydrodynamic spin diffusion in normal liquid, cm2/s
he3_diffn_perp(ttc,p,nu0) -- Frequency-dependent spin diffusion D_perp in normal liquid, cm2/s
he3_coll_int(xi,ttc,gap,g0,d0) -- Collision integral in Einzel approximation
he3_coll_int_lt(xi,ttc,gap,g0,d0) -- Collision integral for low temp (good for < 0.7Tc)
he3_coll_int_ht(xi,ttc,gap,g0,d0) -- Collision integral for high temp (good above 0.95 Tc)
he3_tau0lt(ttc,p) -- He3-B quasiparticle lifetime, low temp limit (no Ek dep)
he3_tau0(ttc,p) -- He3-B quasiparticle lifetime at Fermi level
he3_tau_av(ttc,p) -- Averaged quasiparticle lifetime
he3_fpath(ttc,p) -- Mean free path of Bogoliubov quasiparticles [cm]
he3_rmsv(ttc,p) -- RMS group velocity of Bogoliubov quasiparticles [cm/s]
he3_visc_fpath(ttc,p) -- Viscous free path of Bogoliubov quasiparticles [cm]
he3_hvisc(ttc,p) -- Hydrodinamic (freq=0) viscosity of Bogoliubov quasiparticles [g/cm/s]
he3b_slip_length(ttc,p) -- Slip length to mean free path ratio
he3_tau_dperp(ttc,p) -- Spin diffusion perpendicular transport time, s
he3_tau_dpar(ttc,p) -- Spin diffusion parallel transport time, s
he3_diff_hperp_zz(ttc,p) -- Hydrodynamic spin diffusion D_perp, cm2/s
he3_diff_hpar_zz(ttc,p) -- Hydrodynamic spin diffusion D_par, cm2/s
he3_diff_perp_xx(ttc,p,nu0) -- Spin diffusion coefficient D_perp_xx [cm2/s]
he3_diff_perp_xx_im(ttc,p,nu0) -- Spin diffusion coefficient D_perp_xx_im [cm2/s]
he3_diff_perp_zz(ttc,p,nu0) -- Spin diffusion coefficient D_perp_zz [cm2/s]
he3_diff_perp_zz_im(ttc,p,nu0) -- Spin diffusion coefficient D_perp_zz_im [cm2/s]
he3_diff_par_xx(ttc,p,nu0) -- Spin diffusion coefficient D_par_xx [cm2/s]
he3_diff_par_zz(ttc,p,nu0) -- Spin diffusion coefficient D_perp_zz [cm2/s]
he3_b2hcr(ttc,P) -- critical field B_ab [mk] vs ttc, P [bar]
he3_b2tab(P,H) -- inverse function: find Tab [mK] with known P [bar], H [G]
Based on Ashida and Nagai paper (Progr.Theor.Phys. 74 949 (1985)). he3_b2gap1(ttc,p,H) -- gap distortion
he3_b2gap2(ttc,p,H) -- gap distortion
he3_b2heff(ttc,p,H) -- effective field
he3_b2mag(ttc,p,H) -- magnetization
he3_b2rho_npar(ttc,p,H) -- He3-B normal fluid density vs T/Tc, p, H
he3_b2rho_nper(ttc,p,H) -- He3-B normal fluid density vs T/Tc, p, H
he3_b2rhoab_npar(ttc,p) -- He3-B normal fluid density at the A-B boundary vs T/Tc, p
he3_b2rhoab_nper(ttc,p) -- He3-B normal fluid density at the A-B boundary vs T/Tc, p
he3_b2magab(ttc,p) -- magnetization at the A-B boundary
he3_xigl(ttc,p) -- Extrapolated GL coherence length, cm
he3_vneq(ttc,p,omega,r) -- Equilibrium vortex number
he3_tau_r(ttc) -- Leggett-Takagi tau_r [s] vs T/Tc, 20bar
he3_tau_f(ttc) -- Leggett-Takagi tau_f [s] vs T/Tc, 20bar
he3_cv_n(t,v) -- Heat capacity, Cv/R vs T [K], Vm [cm^3/mol]
he3_tcond_n(t,p) -- He3-n thermal conductivity, K [erg/s cm K] vs T [K] and P [bar].
he3_emery_factor(ttc,p) -- Emery factor vs T/Tc and P [bar].
In normal He3 transport properties (viscosity, spin diffusion, thermal conductivity?)
are suppressed just above Tc because of some fluctuation effects (Emery-1976).
This is clearly seen at viscosity measurements (Parpia-1978, Carless-1983, Nakagawa-1996).
The factor has form 1 - $G(1 - \theta/\alpha\mbox{atan}(\alpha/\theta))$, where $\theta = \sqrt{T/T_c - 1}$.
Values of $G$ (pressure independent) and $\alpha$ were obtained by fitting
viscosity data from Carless-1983 and Nakagawa-1996 in assumption that at high
temperatures they should follow Dyugaev-1985 model.
Note that in Carless-1983 temperature scale Alvesalo-80 is used. To convert
it to Greywall-86 scale one should multiply temperature by $k=0.893$.
he3_visc_n0(t,p) -- He3-n viscosity, eta [poise] vs T [K] and P [bar].
he3_visc_n(t,p) -- He3-n viscosity, eta [poise] vs T [K] and P [bar].
There is also a complete viscosity model in Huang-2012, but for me it does
not look as good as this one.
he3_nu_a(ttc,p) -- Legget frequency nu_a [Hz] vs P, ttc
he3p_chi_par(ttc,p) -- susceptibility component along d vector chi_par / chi_n
he3p_xid_perp(ttc,p) -- dipolar length perpendicular to the l vector, 4g_d/K1
he3p_xid_par(ttc,p) -- dipolar length parallel to the l vector, 4g_d/(K1+K2+K3)
he3p_xih_perp(ttc,p,h) -- magnetic length perpendicular to the l vector (4g_d/K1)
he3p_xih_par(ttc,p,h) -- magnetic length parallel to the l vector (4g_d/(K1+K2+K3))
rota_rcell = 0.2925 -- ROTA: cell radius (2011-2014)
rota_nmra = 96.69139692 -- ROTA: field/current in nmrA solenoid [G/A] (normal phase measurements 2014-10-10: f0=832803.65Hz I0=2655.663813mA)
rota_nmrb = 136.6058277 -- ROTA: field/current in nmrB solenoid [G/A] (normal phase measurements 2014-10-29, 2014-11-03: f0=1176586.91Hz I0=1907.345mA)
rota_hmina_r = 1.032 -- ROTA: effective radius of the HminA coil [cm]
rota_hmina_n = 4 -- ROTA: number of turns of the HminA coil
rota_hmina = 2.2305 -- ROTA: field/current in the center of HminA coil [G/A]
rota_hmina_mr = 1.652 -- ROTA: quadratic radial term of the HminA field, [G/A/cm^2]
rota_hmina_i0i = -0.02918 -- ROTA: effectve HminA coil current divided by NMR current
rota_hmina_i0f = -9.3050e-08 -- ROTA: effectve HminA coil current divided by NMR frequency, rota_hmina_i0i = rota_hmina_i0f * he3_gyro/2/pi * rota_nmra
rota_rrda = 1.01e-4 -- ROTA: radiation damping constant for the nmrA spectrometer
rota_c_ns(t,i) -- Nuclear stage heat capacity [J/K] vs T[K] and I[A]
rota_fork_cal(w,p,n) -- Calibration of fork N, T/Tc, vs width (Hz) and P (bar)
rota_nmra_q(f0) -- Q value of the nmrA spectrometer vs frequency (measured)
rota_nmra_f(n) -- ROTA: frequencies of nmrA spectrometer,kHz for n=1..8 (use real*8 n!)
rota_bza(I,Imin,r,z) -- ROTA: Bz field profile of the A spectrometer
rota_nspeca(f0,I,Imin) -- ROTA: normal phase spectrum
rota_qball_dbetan(p,f0) -- ROTA: Derivative of the textural angle beta_N in the center of the cell (rota-specific, measured), [rad/cm]
rota_qball_fz0(p,f0,imin) -- ROTA: nu_z (1/2 of distance between visible axial levels) (no rotation, rota-specific, measured), [Hz]
rota_qball_fr0(p,f0,imin) -- ROTA: nu_r (1/2 of distance between visible radial levels) (no rotation, rota-specific, measured), [Hz]
rota_qball_az0(P,f0,imin) -- ROTA: z size of the magnon condensate (no rotation, rota-specific, measured), [cm]
rota_qball_ar0(P,f0,imin) -- ROTA: r size of the magnon condensate (no rotation, rota-specific, measured), [cm]
rota_qball_trd(P,f0,fr,fz) -- ROTA: tau_RD for the magnon condensate with given radial and axial frequencies (rota-specific, measured) [s]
rota_qball_trd0(P,f0,imin) -- ROTA: tau_RD for the magnon condensate (no rotation, rota-specific, measured) [s]
he3b_spec1s(ttc,P,H,kv,ak,bk,an,bn) -- He3-B acoustic magnon spectrum, simple formula
he3b_spec2s(ttc,P,H,kv,ak,bk,an,bn) -- He3-B optical magnon spectrum, simple formula
he3b_spec3s(ttc,P,H,kv,ak,bk,an,bn) -- He3-B longitudinal magnon spectrum, simple formula
he3b_spec1(ttc,P,H,kv,ak,bk,an,bn) -- He3-B magnon spectrum, full formula, lowest mode
he3b_spec2(ttc,P,H,kv,ak,bk,an,bn) -- He3-B magnon spectrum, full formula, middle mode
he3b_spec3(ttc,P,H,kv,ak,bk,an,bn) -- He3-B magnon spectrum, full formula, highest mode
he3b_spec_kx2a(ttc,P,H,w,an,bn) -- He3-B magnon spectrum, kx^2(w), lowest mode
he3b_spec_kx2b(ttc,P,H,w,an,bn) -- He3-B magnon spectrum, kx^2(w), middle mode
he3b_spec_kx2c(ttc,P,H,w,an,bn) -- He3-B magnon spectrum, kx^2(w), highest mode
This file contains mathematical functions and subroutines.
Some of them are not included in the library public interface, but still
can be used in fortran/C programs: 1D and 2D real and complex integration with
Gauss-2pt and Gauss-7pt+Kronrod-15pt quadratures, error estimation,
adaptive breakpoints; 2D integration of delta-function (f1*delta(f2));
solving qubic equations. math_ele(x) -- complete elliptic integral E(x)
math_elk(x) -- complete elliptic integral K(x)
loop_bz(Rl,r,z) -- magnetic field Bz of a current loop [G/A] vs Rloop, r, z [cm]
loop_br(Rl,r,z) -- magnetic field Br of a current loop [G/A] vs Rloop, r, z [cm]
math_stokes(g) -- Evaluates complex Stokes function K + 1i*K' (complex)
math_stokes_k(g) -- Stokes K function, real component of math_stokes(g).
math_stokes_kp(g) -- Stokes K' function, imag component of math_stokes(g).
he4_tcv = 2.1720 -- He4 superfluid transition temperature at vapor pressure [K] (1958 temperature scale)
he4_pcv = 0.050396 -- He4 vapor pressure at superfluid transition [bar] (1958 temperature scale)
he4_tcm = 1.750 -- He4 superfluid transition temperature at melting curve [K] (Swenson-1952, shifted)
he4_pcm = 30.033 -- He4 superfluid transition pressure at melting curve [bar] (Swenson-1952)
he4_tcr = 5.1994 -- He4 critical temperature [K] (1958 temperature scale)
he4_pcr = 2.2905 -- He4 critical pressure [bar] (1958 temperature scale)
he4_tc(p) -- He4 superfluid transition temperature [K] vs pressure [bar].
he4_pmelt(t) -- He4 Melting pressure [bar] vs temperature [K], 0..4K
he4_pvap(t) -- He4 Vapor pressure [bar] vs temperature [K], 0..tcr
he4_vm(t) -- He4 molar volume at saturated vapor pressure [cm^3/mol] vs T [K]
he34_xcr = 0.674 -- critical point at saturated pressure, concentration of He3
he34_tcr = 0.867 -- critical point at saturated pressure, temperature [K]
Phase diagram functions are from Chaudhry PhD thesis (Massachusetts, 2009) he34_xdil(t) -- Dilute phase-separation curve, concentration vs temperature [K]
he34_xcon(t) -- Concentrated phase-separation curve, concentration vs temperature [K]
he34_tlambda(x) -- Lambda curve, temperature [K] vs concentration
he34_tcr_p(p) -- Tricritical line vs pressure [bar], 0..10 bar
he34_xcr_p(p) -- Concentration along the tricritical line vs pressure [bar]:
he34_xdil_p(t,p) -- Dilute phase-separation curve, x vs t[K] and p[bar]
he34_xcon_p(t,p) -- Concentrated phase-separation curve, x vs t[K] and p[bar]
he34_tlambda_p(x,p) -- Lambda curve, t[K] vs x and p[bar]
Phase diagram functions on the plot:
These functions produce same results as old Lancaster code, I do not plan to change them.
I plan to do a separate version using functions from the library (Yosida functions, viscosity, etc.) he3_lancwire_d(t,rho,diam,fre) -- Calibration of vibrating wire in mixing chamber (diluted phase), original Lancaster version, complex: freq + 1i*width (complex)
he3_lancwire_d_f(t,rho,diam,fre) -- Calibration of vibrating wire in mixing chamber (diluted phase), frequency
he3_lancwire_d_w(t,rho,diam,fre) -- Calibration of vibrating wire in mixing chamber (diluted phase), width
he3_lancwire_b(t,p,rho,diam,fre) -- Calibration of vibrating wire in superfluid He3-B (original Lancaster version), complex: freq + 1i*width (complex)
he3_lancwire_b_f(t,p,rho,diam,fre) -- Calibration of vibrating wire in superfluid He3-B, frequency
he3_lancwire_b_w(t,p,rho,diam,fre) -- Calibration of vibrating wire in superfluid He3-B, width
he3_lancwire_n(t,p,rho,diam,fre) -- Calibration of vibrating wire in normal He3 (original Lancaster version), complex: freq + 1i*width (complex)
he3_lancwire_n_f(t,p,rho,diam,fre) -- Calibration of vibrating wire in normal He3 (original Lancaster version), freq
he3_lancwire_n_w(t,p,rho,diam,fre) -- Calibration of vibrating wire in normal He3 (original Lancaster version), width
TODO: it's better to re-chack magnetization units. It's checked that magnetization
is consistent with susceptibility and entropy is consistent with heat capacity and
demagnetization cooling effect. See [Kochmansky]. magn_cw_y(ttc,btc) -- y-function, dimensionless magnetization of S=1/2 Curie-Weiss magnet, M/mu vs T/Tc and muB/kTc
magn_cw_m(T,B,Tc,gyro) -- Molar magnetization of S=1/2 Curie-Weiss magnet, M[J/T/mole] vs T[K], B[T], Tc[K], gyro[rad/s/T]
magn_cw_chi(T,B,Tc,gyro) -- Molar magnetic susceptibility of S=1/2 Curie-Weiss magnet, chi [J/T^2/mole] vs T[K], B[T], Tc[K], gyro[rad/s/T]
magn_cw_s(T,B,Tc,gyro) -- Entropy of S=1/2 Curie-Weiss magnet, S/R vs T[K], B[T], Tc[K], gyro[rad/s/T]
magn_cw_c(T,B,Tc,gyro) -- Heat capacity of S=1/2 Curie-Weiss magnet, C/R vs T[K], B[T], Tc[K], gyro[rad/s/T]
magn_cw_d(T,B,Tc,gyro) -- Cooling effect of demagnetization D[K/T] vs T[K], B[T], Tc[K], gyro[rad/s/T]
Example for Curie-Weiss material with Curie temperature $T_c=0.5$ mK and
gyromagnetic ratio $\gamma=203.789\cdot10^6$ rad/s/T: See Pobell book f9.15 magn_par_m(T,B,Bi,gyro,spin) -- Molar magnetization of paramagnetic material, M[J/T/mole] vs T[K], B[T], Bi[T], gyro[rad/s/T], spin[half-int]
magn_par_chi(T,B,Bi,gyro,spin) -- Molar magnetic susceptibility of paramagnetic material, chi[J/T^2/mole] vs T[K], B[T], Bi[T], gyro[rad/s/T], spin[half-int]
magn_par_s(T,B,Bi,gyro,spin) -- Entropy of paramagnetic material, S/R vs T[K], B[T], Bi[T], gyro[rad/s/T], spin[half-int]
magn_par_c(T,B,Bi,gyro,spin) -- Heat capacity of paramagnetic material, C/R vs T[K], B[T], Bi[T], gyro[rad/s/T], spin[half-int]
magn_par_d(T,B,Bi,gyro,spin) -- Cooling effect of demagnetization D[K/T] vs T[K], B[T], Bi[T], gyro[rad/s/T], spin[half-int]
Example for copper nuclei. Internal field $B_i = 0.36\cdot 10^{-3}$ T,
gyromagnetic ratio $\gamma = 71.118\cdot10^6$ rad/s/T, spin $J$ = 3/2:
$\lambda_0^+ = \lambda_1^+ = 1$, $\lambda_0^- = 3$,
$\delta_0^+ = \delta_1^+ = \delta_0^-/3 = \delta_0$
See Sykes-1970 f.26-29, Einzel-1978 f66,67,71,74, Einzel-1984 f.24
l1a = 1 + 2 <W*cos(th)>/<W>
l2 = 1 - 3 <W*sin^4(th/2)sin^2(phi)>/<W>
Quasiparticle lifetimes
Einzel JLTP32 (1978) p.28,34
Einzel JLTP84 (1991) f.4
In VW2.38. tau_n0 is different by pi/4 factor!
Einzel JLTP84 (1991) f.5
Einzel JLTP84 (1991) p.328
See Einzel JLTP84 (1990) p.41
Einzel JLTP84 (1991) f.23
1/3 * vf^2 * tau_n0 * (1+f0a) * 3/4 1/(1-L1) # Einzel-1991
1/3 * vf^2 * tau_n0 * (1+f0a) * f_e(L1) # VW 2.40 + 2.71
Result is same if tau_n0 is different by pi/4 factor
Einzel JLTP84 (1991) f.22, Bunkov PRL65
He3-B transport properties
[he3_transp_b.f]
Collision integral for Bogoliubov quasiparticles
Einzel, Wolfle, Hirschfeld, JLTP80 (1990), Appendix, p.66
+ my small fixes
Einzel, JLTP84 (1991), p.345
Einzel, JLTP84 (1991), p.345
Einzel, JLTP32 (1978), f.80 - first (gap/ttc)^2 term
Quasiparticle lifetime, mean free path
Einzel-1978 f.79
Einzel-1984 f.60,60a
Einzel JLTP84 (1991) p.344
Einzel JLTP84 (1991) p.345
Einzel JLTP32 (1978) f.84
Einzel JLTP32 (1990) f.28 and below
Viscosity
Einzel 1990 Eq.26
Einzel 1990 Eq.28
"improved lower bound" for diffuse scattering,
[Hojgaard-1980]
$ \frac{\zeta}{l} = \frac12\left(\frac{8}{15}\sqrt{\frac{Y_2 Y_0}{Y_1^2}} + \frac{5}{8}\sqrt{\frac{Y_3^2 Y_0}{Y_2^3}}\right)$
TODO: in Einzel-1983 exact calculation is done (no big difference). There is also a calculation for arbitrary diffusive-specular
scattering and Andreev scattering. In Einzel-1990 effect of surface curvature
and roughness is calculated. Good review Einzel-Parpia-1997!
Spin diffusion
Einzel JLTP84 (1991) f.90,96
Einzel JLTP84 (1991) f.90,96
Einzel JLTP84 (1991) f.102
Einzel JLTP84 (1991) f.102
B phase in strong magnetic field (B2 phase).
[he3_b2.f]
Inseob Hahn PhD thesis, p79
see also https://doi.org/10.1016/0921-4526(94)90737-4
see also code at http://spindry.phys.northwestern.edu/he3.htm
Calculation of B-phase gap distortion and spin polarization.
Misc. functions for superfluid He3-B
[he3_other.f]
see Thuneberg-2001, p.667
No strong coupling corrections are needed!
Ref: WV pic.10.5 20bar
Normal 3He liquid parameters beyond zero-temperature limit
[he3_normal.f]
Original formula from Greywall-1983.
Note that Cv = Cp up to terms (T/Tf)^3.
Dyugaev-1985. Measurements from Greywall-1984 (7mK-1K) and
Kerrisk,Keller-1969 (1.5K-Tcr) are used to obtain some semi-theoretical
model for thermal conductivity and viscosity (see below).
Pure Dyugaev-1985 model without Emery effect. See function he3_visc_n below.
Model from Dyugaev-1985, it uses thermal conductivity experimental data to get viscosity.
At low temperature Emery effect, reduction of viscosity close to $T_c$ due to fluctuation
effects, is taken into account.
Very good agreement with Betts-1963,1965 at high temperatures, and with Carless-1983, Nakagawa-1996
at low tempeatures (temperature scale correction for Carless-1983 is needed).
A phase
[he3_a.f]
Interpolation formula by A.Yudin based on Halperin and ROTA data
There is no check that t > t_ab
He3 Polar phase
[he3_polar.f]
chi_perp/chi_0 = 1
see Leggett-1975, VIIID f.7.53 and f.7.54
ROTA-specific functions
[he3_rota.f]
ROTA-specific constants
Q-balls in the zero temperature limit
B phase magnon spectra.
[he3_bspec.f]
Mathematics
[he3_math.f]
from IMSL library
from IMSL library
Uses the methods outlined in STOKES - Mathematical and physical papers Vol III.
For G>=3 use eq.113, for G<3 eq.103-105.
Code is taken from Lancaster ULT wire calibration program.
Helium-4 parameters
[he4.f]
Constants
Data from [[Swenson-1952]], shifted by -0.014K to
have Tc(Pvap) = 2.1720K (according to 1958 temperature scale).
[[Swenson-1950,1951]]
Fit of 1958 temperature scale (~1.5% accuracy)
[[Kerr,Taylor-1964]]
3He-4He mixtures
[he34.f]
Constants
Phase diagram at saturated vapor pressure
(0.08014 < x < 0.674; 0.15 K < T < 0.867 K):
Chaudhry PhD thesis (Massachusetts, 2009)
Below 0.15 K: Edwards, Ifft, Sarwinski, Phys.Rev. 177, 380 (1969) Eq.23
(0.674 < x < 1; 0.15 K < T < 0.867 K).
Chaudhry PhD thesis (Massachusetts, 2009)
Below 0.15 K: Edwards, Ifft, Sarwinski, Phys.Rev. 177, 380 (1969) Eq.25.
Chaudhry PhD thesis (Massachusetts, 2009)
Phase diagram at pressures 0..10bar (T>0.15K)
Chaudhry PhD thesis (Massachusetts, 2009)
Chaudhry PhD thesis (Massachusetts, 2009)
Chaudhry PhD thesis (Massachusetts, 2009)
Chaudhry PhD thesis (Massachusetts, 2009)
Chaudhry PhD thesis (Massachusetts, 2009)
Vibrating wire calibration programs from Lancaster ULT (original version)
[he3_wire_orig.f]
Arguments: temperature [K], rho wire [g/cm^3], wire diameter [um], frequency [Hz]
arguments: temperature [K], rho wire [g/cm^3], wire diameter [um], frequency [Hz]
arguments: temperature [K], rho wire [g/cm^3], wire diameter [um], frequency [Hz]
arguments: temperature [K], pressure [bar], rho wire [g/cm^3], wire diameter [um], frequency [Hz].
arguments: temperature [K], pressure [bar], rho wire [g/cm^3], wire diameter [um], frequency [Hz]
arguments: temperature [K], pressure [bar], rho wire [g/cm^3], wire diameter [um], frequency [Hz]
Cylinder programme using wide line treatment and slip fudge.
arguments: temperature [K], pressure [bar], rho wire [g/cm^3], wire diameter [um], frequency [Hz].
Cylinder programme using wide line treatment and slip fudge.
arguments: temperature [K], pressure [bar], rho wire [g/cm^3], wire diameter [um], frequency [Hz]
Cylinder programme using wide line treatment and slip fudge.
arguments: temperature [K], pressure [bar], rho wire [g/cm^3], wire diameter [um], frequency [Hz]
Magnetic models
[magn.f]
Curie-Weiss magnet with spin 1/2
Solving equation m = \tanh((m+btc)/ttc) by Newton method.
Works for positive and negative field.
In the demagnetization process $dQ = T\,dS = T(dS/dT)\,dT + T(dS/dB)\,dB$.
Then $dT = dQ/C - D\,dB$, where $D = (dS/dB)/(dS/dT)$
Paramegnetetic material with internal field
In the demagnetization process $dQ = T\,dS = T(dS/dT)\,dT + T(dS/dB)\,dB$.
Then $dT = dQ/C - D\,dB$, where $D = (dS/dB)/(dS/dT)$
