def get_lawson_parameters(ni, Ti, settings, reaction='D_T_to_n_alpha'):
"""
Lawson minimal confinement time [s], and maximal flux [s^-1] from each end assuming some fusion volume.
T in [keV], ni in [m^-3].
"""
sigma_v_fusion = get_sigma_v_fusion(Ti, reaction=reaction)
MeV_to_J = define_electron_charge() * 1e6
E_charged = get_E_charged(reaction=reaction) * MeV_to_J # J
kB_keV = define_electron_charge() * 1e3
n_tau_lawson = 12 * kB_keV * Ti / (E_charged * sigma_v_fusion)
tau_lawson = n_tau_lawson / ni
flux_lawson = 0.5 / n_tau_lawson * settings['volume_main_cell'] * ni**2
return tau_lawson, flux_lawson
def get_sigma_v_fusion_approx(T, reaction='D_T_to_n_alpha', n=None):
"""
Approximation of <sigma*v> (Maxwell averaged reactivity), applicable for non-resonant reactions.
Source:"2011 - Adelberger et al - Solar fusion cross sections. II. The pp chain and CNO cycles", eq. 8.
T in [keV], n is number density in [cm^3], reactivity sigma*v in [m^3/s].
"""
if reaction == 'D_D_to_p_T_n_He3':
reactions = ['D_D_to_p_T', 'D_D_to_n_He3']
branching_ratios = [0.5, 0.5]
return sum([
branching_ratio *
get_sigma_v_fusion_approx(T, reaction=reaction_branch, n=n)
for reaction_branch, branching_ratio in zip(
reactions, branching_ratios)
])
else:
# constants
A_1, A_2 = get_As_for_reaction(reaction=reaction)
Z_1, Z_2 = get_Zs_for_reaction(reaction=reaction)
m_p = define_proton_mass() # kg
m_r = m_p * A_1 * A_2 / (A_1 + A_2) # reduced mass
alpha = define_fine_structure_constant()
e = define_electron_charge()
c = define_speed_of_light() # m/s
c_cm = c * 1e2
m_r_keV_over_c2 = m_r * c**2 / e / 1e3
m_r_keV_cm = m_r_keV_over_c2 / c_cm**2
# effective S factor
E0 = T * (np.pi * alpha * Z_1 * Z_2 /
np.sqrt(2))**(2 / 3) * (m_r_keV_over_c2 / T)**(1 / 3) # keV
delta_E0 = T * 4 * np.sqrt(E0 / (3 * T)) # keV
S0, S0_der, S0_der2 = get_astrophysical_S_factor(reaction=reaction)
S_eff = S0 * (1 + 5 * T / (36 * E0)) + S0_der * E0 * (1 + 35 * T / (36 * E0)) \
+ 0.5 * S0_der2 * E0 ** 2 * (1 + 89 * T / (36 * E0)) # [keV * barn]
barn = define_barn()
barn_cm = barn * 1e4
S_eff_cm2 = S_eff * barn_cm
# electron screening factor
if n is None:
f0 = 1 # assume no electron screening
else:
zeta = ((0.5 * Z_1**2 / A_1 + 0.5 * Z_2**2 / A_2) + 0.92 *
(0.5 * Z_1 / A_1 + 0.5 * Z_2 / A_2))**0.5
rho_kg_over_cm3 = n * m_r # specific density [kg/cm^3] since n in [cm^3]
rho0 = rho_kg_over_cm3 * 1e3 # [g/cm^3]
T_eV = T * 1e3
T_K = T_eV * define_factor_eV_to_K()
T_6 = T_K / 1e6 # in 10^6 Kelvin
f0 = np.exp(0.188 * Z_1 * Z_2 * zeta * rho0**0.5 * T_6**(-3 / 2))
sigma_v_cm3_over_s = np.sqrt(
2 / (m_r_keV_cm * T)) * delta_E0 / T * f0 * S_eff_cm2 * np.exp(
-3 * E0 / T)
sigma_v_m3_over_s = 1e-6 * sigma_v_cm3_over_s
return sigma_v_m3_over_s
def get_gamow_energy(reaction='D_T_to_n_alpha'):
"""
Return Gamow energy (tunneling barrier) in [keV].
Source: "Atzeni, Meyer-ter-Vehn - The Physics of Inertial Fusion", chapter 1.
"""
# alpha_fine = 1 / 137.035999084 # fine structure constant
A_1, A_2 = get_As_for_reaction(reaction=reaction)
Z_1, Z_2 = get_Zs_for_reaction(reaction=reaction)
A_r = A_1 * A_2 / (A_1 + A_2) # reduced A
# E_g = 986.1 * (Z_1 * Z_2) ** 2 * A_r # shortened form
m_p = define_proton_mass() # kg
m_r = m_p * A_1 * A_2 / (A_1 + A_2) # reduced mass
alpha = define_fine_structure_constant()
e = define_electron_charge()
c = define_speed_of_light() # m/s
m_r_keV = m_r * c**2 / e / 1e3
E_g = (np.pi * alpha * Z_1 * Z_2)**2 * 2 * m_r_keV
return E_g
def get_fusion_sigma_v_E_reaction(T, reaction='D_T_to_n_alpha'):
"""
Multiply sigma*v by the reaction energy and return in [J].
T in [keV].
"""
if reaction == 'D_D_to_p_T_n_He3':
reactions = ['D_D_to_p_T', 'D_D_to_n_He3']
branching_ratios = [0.5, 0.5]
return sum([
branching_ratio * get_sigma_v_fusion(T, reaction=reaction_branch) *
get_E_reaction(reaction=reaction_branch)
for reaction_branch, branching_ratio in zip(
reactions, branching_ratios)
])
else:
sigma_v_E = get_sigma_v_fusion(
T, reaction=reaction) * get_E_reaction(reaction=reaction)
e = define_electron_charge()
return sigma_v_E * 1e6 * e # MeV to J
def get_specific_coulomb_scattering_rate(ne,
Te,
ni,
Ti,
settings,
impact_specie='e',
target_specie='i'):
"""
Coulomb scattering rate based on the general formula of
"2007 - Fundamenski et al - Comparison of Coulomb Collision Rates in the
Plasma Physics and Magnetically Confined Fusion Literature"
densities in m^-3 and temperatures in eV
"""
eps0 = define_vacuum_permittivity()
e = define_electron_charge()
coulomb_log_dict = calculate_coulomb_logarithm(
ne,
Te,
ni,
Ti,
Z=settings['Z_ion'],
A=settings['A_atomic_weight'],
coulomb_log_min=settings['coulomb_log_min'])
if impact_specie == 'e':
m_s1 = settings['me']
Z_s1 = 1
T_s1 = Te
else:
m_s1 = settings['mi']
Z_s1 = settings['Z_ion']
T_s1 = Ti
if target_specie == 'e':
m_s2 = settings['me']
Z_s2 = 1
T_s2 = Te
n_s2 = ne
else:
m_s2 = settings['mi']
Z_s2 = settings['Z_ion']
T_s2 = Ti
n_s2 = ni
if impact_specie == 'e' and target_specie == 'e':
coulomb_log = coulomb_log_dict['ee']
elif impact_specie == 'i' and target_specie == 'i':
coulomb_log = coulomb_log_dict['ii']
elif (impact_specie == 'e'
and target_specie == 'i') or (impact_specie == 'i'
and target_specie == 'e'):
# as a shortcut, check only the first cell temperatures to pick the correct temperature regine
Ti_0 = settings['Ti_0']
Te_0 = settings['Te_0']
if Ti_0 / Te_0 > settings['mi'] / settings['me']:
coulomb_log = coulomb_log_dict['ie_overheated_ions']
else:
if Te_0 > 10 * settings['Z_ion']**2.0:
coulomb_log = coulomb_log_dict['ie_hot_electrons']
else:
coulomb_log = coulomb_log_dict['ie_cold_electrons']
else:
raise ValueError('invalid option for impact/target species.')
scat_rate = 2 ** 0.5 / (12 * np.pi ** 1.5) * n_s2 * Z_s1 ** 2 * Z_s2 ** 2 * e ** 4 * coulomb_log / \
(eps0 ** 2 * m_s1 ** 0.5 * (e * T_s1) ** 1.5) \
* (1 + m_s1 / m_s2) / (1 + m_s1 / m_s2 * T_s1 / T_s2) ** 1.5
return scat_rate