""" he3.py 3He parameter table 26 Jun 2009 Lev - Greywall superfluid 3He phase diagram changed on 20-21 Jun 2010 by Lev References: [1] Greywall PRB 33 No.11, 7250 (1986) [2] Lifshitz and Pitaevskii Stat. Phys pt 2 (vol. IX on L&L theor. phys course) [3] Wheatley RMP 47, 415 (1975) """ # # Changelog # 19 Jul 2010: chiN added, from [3] # import numpy, flib def P_AB_melt(): "melting pressure [bar] at AB line"; return 34.358 def Pc_melt(): "melting pressure [bar] at Tc line (P_A)"; return 34.338 def P_tri(): "pressure [bar] of the tri-critical point"; return 21.22 def Pc_solid(): "melting pressure [bar] at solid phase transition"; return Pc_melt() + 0.05252 def Tc_solid(): "melting temperature [mK] at solid phase transition"; return 0.931 def Tc(P): "return a Tc [mK] at a given pressure P [bar]" scalar = numpy.isscalar(P) if scalar: P = numpy.array([P]) else: P = numpy.asanyarray(P) # the expression for Tc(P) is a polynomial in and after Eq. (5), page 7527 in Ref. 1 T = numpy.polyval([0.53010918e-7, -0.57248644e-5, 0.25685169e-3, -0.69302185e-2, 0.13867188, 0.92938375], P) T[(P < 0) | (P > Pc_melt())] = numpy.nan if scalar: T = T[0] return T __Pc__ = numpy.concatenate([numpy.arange(0.0, Pc_melt(), 0.1), [Pc_melt()]]) __Tc__ = Tc(__Pc__) __T_tri__ = Tc(P_tri()) def T_tri(): "temperature [mK] of the tri-critical point"; return __T_tri__ def Pc(T): "return a P [bar] at which the Tc takes a given value T [mK]" return numpy.interp(T, __Tc__, __Pc__, left=numpy.nan, right=numpy.nan) def T_AB(P): "return a T_AB [mK] at a given pressure P [bar]" scalar = numpy.isscalar(P) if scalar: P = numpy.array([P]) else: P = numpy.asanyarray(P) # the expression for T_AB - T_tri(P - P_tri) is a polynomial in and after Eq. (15), page 7529 in Ref. 1 T = numpy.polyval([0.17038992e-5, -0.61709783e-4, 0.83437032e-3, -0.53633181e-2, -0.10322623e-1, T_tri()], P - P_tri()) T[(P < P_tri()) | (P > P_AB_melt())] = numpy.nan if scalar: T = T[0] return T __P_AB__ = numpy.concatenate([numpy.arange(P_tri(), P_AB_melt(), 0.1), [P_AB_melt()]]) __T_AB__ = T_AB(__P_AB__) def P_AB(T): "return a P [bar] at which the T_AB takes a given value T [mK]" return numpy.interp(T, __T_AB__, __P_AB__, left=numpy.nan, right=numpy.nan) def Aphase_contour(): """ return a bundle (T,P) of arrays defining the outline of the A phase, the melting curve is considered flat """ above_tri = (__Tc__ >= T_tri()) return (numpy.concatenate([__T_AB__[-1::-1], __Tc__[above_tri]]), numpy.concatenate([__P_AB__[-1::-1], __Pc__[above_tri]])) def Bphase_contour(): """ return a bundle (T,P) of arrays defining the outline of the B phase, the melting curve is considered flat """ below_tri = (__Tc__ <= T_tri()) return (numpy.concatenate([__Tc__[below_tri], __T_AB__, [0, 0]]), numpy.concatenate([__Pc__[below_tri], __P_AB__, [Pc_solid(), 0]])) # table VI, page 7532 in [1] # column 1: Pressure [bar] # column 2: V [cm^3/mol] # column 3: gamma [1/K] # column 4: m*/m # column 5: F_1^S (Landau Fermi-liquid parameter) __P_V_gamma_mstar_F1s__ = numpy.array([ [0, 36.84, 2.78, 2.80, 5.39], [1, 35.74, 2.85, 2.93, 5.78], [2, 34.78, 2.92, 3.05, 6.14], [3, 33.95, 2.98, 3.16, 6.49], [4, 33.23, 3.04, 3.27, 6.82], [5, 32.59, 3.10, 3.38, 7.14], [6, 32.03, 3.16, 3.48, 7.45], [7, 31.54, 3.21, 3.58, 7.75], [8, 31.10, 3.27, 3.68, 8.03], [9, 30.71, 3.32, 3.77, 8.31], [10, 30.35, 3.37, 3.86, 8.57], [11, 30.02, 3.42, 3.95, 8.84], [12, 29.71, 3.48, 4.03, 9.09], [13, 29.42, 3.53, 4.12, 9.35], [14, 29.15, 3.58, 4.20, 9.60], [15, 28.89, 3.62, 4.28, 9.85], [16, 28.64, 3.67, 4.37, 10.10], [17, 28.41, 3.72, 4.45, 10.35], [18, 28.18, 3.77, 4.53, 10.60], [19, 27.96, 3.82, 4.61, 10.84], [20, 27.75, 3.87, 4.70, 11.09], [21, 27.55, 3.92, 4.78, 11.34], [22, 27.36, 3.97, 4.86, 11.58], [23, 27.18, 4.02, 4.94, 11.83], [24, 27.01, 4.06, 5.02, 12.07], [25, 26.85, 4.11, 5.10, 12.31], [26, 26.70, 4.16, 5.18, 12.55], [27, 26.56, 4.21, 5.26, 12.79], [28, 26.42, 4.26, 5.34, 13.03], [29, 26.29, 4.31, 5.42, 13.26], [30, 26.17, 4.36, 5.50, 13.50], [31, 26.04, 4.40, 5.58, 13.73], [32, 25.90, 4.45, 5.66, 13.97], [33, 25.75, 4.50, 5.74, 14.21], [34, 25.58, 4.54, 5.82, 14.46], [34.39, 25.50, 4.56, 5.85, 14.56] ]) def V_mol(P): "molar volume of liquid 3He [cm^3/mol] at a given pressure [bar]" return numpy.interp(P, __P_V_gamma_mstar_F1s__[:,0], __P_V_gamma_mstar_F1s__[:,1], left=numpy.nan, right=numpy.nan) def gamma(P): "normal 3He heat capacity slope gamma [1/K] at a given pressure [bar]" return numpy.interp(P, __P_V_gamma_mstar_F1s__[:,0], __P_V_gamma_mstar_F1s__[:,2], left=numpy.nan, right=numpy.nan) def m_star_over_m(P): "Fermi liquid mass enchancement m*/m for normal 3He at a given pressure [bar]" return numpy.interp(P, __P_V_gamma_mstar_F1s__[:,0], __P_V_gamma_mstar_F1s__[:,3], left=numpy.nan, right=numpy.nan) def F1s(P): "Landau Fermi liquid parameter F_1^s at a given pressure [bar]" return numpy.interp(P, __P_V_gamma_mstar_F1s__[:,0], __P_V_gamma_mstar_F1s__[:,4], left=numpy.nan, right=numpy.nan) __hbar__ = 1.05457148e-34 # Plank's constant [kg m^2/s] __NA__ = 6.0221415e23 # Avogadro's number __k_B__ = 1.3806503e-23 # Boltzmann's constant: [m^2 kg / (s^2 K)] def p_Fermi(P): "liquid 3He Fermi momentum p_F [kg m/s] at a given pressure [bar]" # ref: [2] Eq. (1.1) return (3 * numpy.pi**2 * __hbar__**3 * __NA__ / (1e-6 * V_mol(P)))**(1./3) def m_star(P): "3He quasiparticle mass [kg] at a given pressure [bar]" # bare 3He atom mass is 3.016 a.m.u., 1 a.m.u = 1.66e-27 kg return m_star_over_m(P) * 3.0160293 * 1.66053878e-27 def v_Fermi(P): "liquid 3He Fermi velocity v_F [m/s] at a given pressure [bar]" # ref: [2] Eq. (1.14) return p_Fermi(P) / m_star(P) # chiN(P) from table V of [3] __chi__ = {0: 3.81, 3: 4.88, 6: 5.67, 9: 6.38, 12: 7.09, 15: 7.75, 18: 8.38, 21: 8.92, 24: 9.42, 27: 9.91, 30: 10.42, 33: 10.94, 34.36: 11.17} def chiN(P): "abs zero normal state susceptibility [3], table V" pp = sorted(__chi__.keys()) return numpy.interp(P, pp, [__chi__[p] for p in pp], left=numpy.nan, right=numpy.nan) * 1e-8 def xi0(P): "return zero temperature coherence length [nm]" # xi_0 = \sqrt{\frac{7 \zeta(3)}{20}} \frac{\hbar v_F}{2\pi k_B T_c} return (7 * 1.2021 / 20)**0.5 * __hbar__ * v_Fermi(P) / (2*numpy.pi * __k_B__ * 1e-3*Tc(P)) * 1e9 def xi_GL(P, T, reducedT = True): "return Ginzburg-Landau coherence length [nm]" if not reducedT: T /= Tc(P) return xi0(P) / (1 - T)**0.5