def shear_modulus(T,V,params):
# low T:
# C_v = 234. * n * Av * boltzmann_constant (T/theta_D) ^ 3. (3.43 from Poirier/EarthInterior)
# high T:
# C_v = 3. * n * gas_constant
# n = number of moles
#C_v = 3. * n * gas_constant # in J * mol / K
P_th = lambda x: mgd.mgd_thermal_pressure(T,x, params)
P_th_ref = lambda x: mgd.mgd_thermal_pressure(300.,x, params)
f =.5*(pow(params['ref_V']/V,2./3.)-1.) # EQ 24
a2_s = -2.*params['ref_grueneisen'] - 2.*params['eta_0s'] # EQ 47
a1_ii = 6. * params['ref_grueneisen'] # EQ 47
a2_iikk = -12.*params['ref_grueneisen']+36.*pow(params['ref_grueneisen'],2.) - 18.*params['q0']*params['ref_grueneisen'] # EQ 47
nu_o_nu0_sq = 1.+ a1_ii*f + (1./2.)*a2_iikk * pow(f,2.) # EQ 41
#gamma = params['ref_grueneisen'] * (pow(params['ref_V']/V,-params['q0'])) # EQ 49
gamma = 1./6.*pow(nu_o_nu0_sq,-1.)*(2*f+1.)*(a1_ii+(a2_iikk*f))
eta_s = - gamma - (1./2. * pow(nu_o_nu0_sq,-1.) * pow((2.*f)+1.,2.)*a2_s) # EQ 46 NOTE the typo from Stixrude 2005
delta_U = (P_th(V) - P_th_ref(V))*(V/gamma)
G_twoterms = pow(1.+2.*f, 5./2.) * (params['ref_mu'] + (3.*(params['ref_K']*params['mu_prime']) - 5.*params['ref_mu'])*f) \
- (eta_s*(1./V)*delta_U) # EQ 33 up to the second order
G_threeterms = pow(1.+2.*f, 5./2.) * (params['ref_mu'] + (3.*(params['ref_K']*params['mu_prime']) - 5.*params['ref_mu'])*f \
+ ((6.*params['ref_K']*params['mu_prime']) - 24.*params['ref_K'] - 14.*params['ref_mu']+ 9./2.*params['ref_K']*params['K_prime']) * pow(f,2)) \
- (eta_s*(1./V)*delta_U) # EQ 33 up to the second order
return G_twoterms
def volume(p,T,params):
P_th = lambda x: mgd.mgd_thermal_pressure(T,x, params) #From Stixrude 2005: delta U (T)*gamma*ro (1/V)
P_th_ref = lambda x: mgd.mgd_thermal_pressure(300.,x, params)#From Stixrude 2005: delta U (300 K)*gamma*ro (1/V)
b_iikk= 9.*params['ref_K'] # EQ 28
b_iikkmm= 27.*params['ref_K']*(params['K_prime']-4.) # EQ 29
f = lambda x: 0.5*(pow(params['ref_V']/x,2./3.)-1.) # EQ 24
func = lambda x: (1./3.)*(pow(1.+2.*f(x),5./2.))*((b_iikk*f(x)) \
+(0.5*b_iikkmm*pow(f(x),2.))) + P_th(x) - P_th_ref(x) - p #EQ 21
V = opt.brentq(func, 0.09*params['ref_V'], 3.*params['ref_V'])
return V
def bulk_modulus(T,V,params):
P_th = lambda x: mgd.mgd_thermal_pressure(T,x, params)
P_th_ref = lambda x: mgd.mgd_thermal_pressure(300.,x, params)
func_int_cv = lambda t: math.exp(t)*pow(t,4.)/pow(math.exp(t)-1.,2.)
Dcv = integrate.quad(func_int_cv,0.,params['ref_Debye']/T)
Dcv_300 = integrate.quad(func_int_cv,0.,params['ref_Debye']/300.)
cv_300 = 9.*params['n']*gas_constant*(pow(300./params['ref_Debye'],3.))*Dcv_300[0]
Cv = (9.*params['n']*gas_constant*(pow(T/params['ref_Debye'],3.))*Dcv[0])-cv_300 #why substract this? Sanne #units of R
#Kth_0 = params['ref_K'] - (pow(params['ref_grueneisen'],2.) * density_0 * delta_U_300/params['molar_mass'])
f =.5*(pow(params['ref_V']/V,2./3.)-1) # EQ 24
gamma = params['ref_grueneisen'] * (pow(params['ref_V']/V,-params['q0'])) # EQ 49
delta_U = (P_th(V) - P_th_ref(V))*(V/gamma) #convert into Stixrudian units
K = pow(1.+2.*f, 5./2.) * ( params['ref_K'] + (3*params['ref_K']*params['K_prime'] -5.*params['ref_K'])*f) \
+ ((gamma+1.-params['q0'])*(gamma/V) *delta_U ) \
- ((pow(gamma,2.) / V )*(Cv*T - cv_300*300.) ) # EQ 32 up to the second order (the third order is basically zero when K'~4
#+27./2.*(params['ref_K']*params['K_prime']-4.*params['ref_K'])*pow(f,2.)) \
return K
