def __init__(
self,
initial=None,
final=None,
r0=[0, 0, 0], # center of mass vector
form='L',
ngauss=8):
self.initial = initial
self.r0 = np.array(r0) / Bohr
self.form = form
if final is None:
self.final = PlaneWave(initial.gd)
else:
self.final = final
self.Ekin = None
# Gauss-Legendre weights and abscissas
self.gl = {}
self.gl['x'] = GaussLegendre(-1., 1., ngauss)
self.gl['phi'] = GaussLegendre(0, 2 * pi, ngauss)
self.gl['psi'] = self.gl['phi']
self.angle = {}
# sin and cos of the magic angle (54.7 deg)
self.costhm = 1. / sqrt(3)
self.sinthm = sqrt(2. / 3.)
def __init__(self,
initial=None,
final=None,
r0=[0, 0, 0], # center of mass vector
form='L',
ngauss=8):
self.initial = initial
self.r0 = np.array(r0) / Bohr
self.form = form
if final is None:
self.final = PlaneWave(initial.gd)
else:
self.final = final
self.Ekin = None
# Gauss-Legendre weights and abscissas
self.gl = {}
self.gl['x'] = GaussLegendre(-1., 1., ngauss)
self.gl['phi'] = GaussLegendre(0, 2 * pi, ngauss)
self.gl['psi'] = self.gl['phi']
self.angle = {}
# sin and cos of the magic angle (54.7 deg)
self.costhm = 1. / sqrt(3)
self.sinthm = sqrt(2. / 3.)
class CrossSectionBeta:
def __init__(
self,
initial=None,
final=None,
r0=[0, 0, 0], # center of mass vector
form='L',
ngauss=8):
self.initial = initial
self.r0 = np.array(r0) / Bohr
self.form = form
if final is None:
self.final = PlaneWave(initial.gd)
else:
self.final = final
self.Ekin = None
# Gauss-Legendre weights and abscissas
self.gl = {}
self.gl['x'] = GaussLegendre(-1., 1., ngauss)
self.gl['phi'] = GaussLegendre(0, 2 * pi, ngauss)
self.gl['psi'] = self.gl['phi']
self.angle = {}
# sin and cos of the magic angle (54.7 deg)
self.costhm = 1. / sqrt(3)
self.sinthm = sqrt(2. / 3.)
def calculate(self, Ekin):
"""Calculate the necessary overlaps."""
Ekin = Ekin / Hartree
if self.Ekin == Ekin:
return
self.Ekin = Ekin
# photoelectron momentum
self.k = sqrt(2 * self.Ekin)
for angle in ['x', 'phi', 'psi']:
self.angle[angle] = self.gl[angle].get_x()[0]
self.T20, self.T2m = self.gi_x()
# we need the average
self.T20 /= 8 * pi**2
self.T2m /= 8 * pi**2
def get_omega(self):
"""Return the necessary photon energy."""
return self.Ekin - self.initial.get_energy() / Hartree
def get_beta(self, Ekin=None):
"""Return the asymmetry parameter.
E: photoelectron kinetic energy [eV]
"""
if Ekin is not None:
self.calculate(Ekin)
return self.T20 / self.T2m - 1.
def get_ds(self, Ekin=None, units='Mb'):
"""Return the total cross section.
Ekin: photoelectron kinetic energy [eV]
units:
'Mb', 1 Mb = 1.e-22 m**2
'Ang', 1 A**2 = 1.e-20 m**2
'a.u.', 1 a_0**2 = 2.8e-21 m**2
as output units
"""
if Ekin is not None:
self.calculate(Ekin)
try:
pre = ds_prefactor[units]
except KeyError:
raise NotImplementedError, 'Unknown units: >' + units + '<'
# me_c = self.initial.get_me_c(np.array([0., 0., self.k]), self.form)
# T2mana = np.abs(np.dot(me_c,me_c)) / 3.
# print "T2m:", T2mana, self.T2m
omega = self.get_omega()
# integration over momentum agles
pre *= self.k * 4 * pi
# print omega, self.initial.get_ds(self.k, omega, self.form), \
# (self.k * 4 * pi * (2 * pi)**2 / 137.0359895 * self.T2m / omega)
return pre * ((2 * pi)**2 * alpha * self.T2m / omega)
def gauss_integrate(self, angle, function):
T20 = 0.
T2m = 0.
gl = self.gl[angle]
for x, w in zip(gl.get_x(), gl.get_w()):
# print angle, x, w
self.angle[angle] = x
t20, t2m = function()
T20 += w * t20
T2m += w * t2m
# print "<gauss_integrate> angle=", angle, T2m
return T20, T2m
def gi_x(self):
"""Gauss integrate over x=cos(theta)"""
return self.gauss_integrate('x', self.gi_phi)
def gi_phi(self):
"""Gauss integrate over phi"""
return self.gauss_integrate('phi', self.gi_psi)
def gi_psi(self):
"""Gauss integrate over psi"""
return self.gauss_integrate('psi', self.integrand)
def integrand(self):
# polarisation in the direction of vk
costh = self.angle['x']
sinth = sqrt(1. - costh**2)
sinphi = sin(self.angle['phi'])
cosphi = cos(self.angle['phi'])
eps0 = np.array([sinth * cosphi, sinth * sinphi, costh])
vk = self.k * eps0
# polarisation at the magic angle
costhm = self.costhm
sinthm = self.sinthm
sinpsi = sin(self.angle['psi'])
cospsi = cos(self.angle['psi'])
epsm = np.array([
sinthm * (cosphi * sinpsi * costh + sinphi * cospsi) +
costhm * cosphi * sinth,
sinthm * (sinphi * sinpsi * costh - cosphi * cospsi) +
costhm * sinphi * sinth, costhm * costh - sinthm * sinth * sinpsi
])
# initial and final state on the grid
initial_G = self.initial.get_grid()
final_G = self.final.get_grid(vk, self.r0)
ini_analyt = H1s(self.initial.gd, self.r0)
gd = self.initial.gd
if self.form == 'L':
if_G = initial_G * final_G
omega = self.get_omega()
if 0:
me_c = []
for c in range(3):
xyz_G = ((np.arange(gd.n_c[c]) + gd.beg_c[c]) * gd.h_c[c] -
self.r0[c])
shape = [1, 1, 1]
shape[c] = -1
xyz_G.shape = tuple(shape)
np.resize(xyz_G, gd.n_c)
me_c.append(gd.integrate(if_G * xyz_G))
me_c = np.array(me_c) * omega
else:
me_c = gd.calculate_dipole_moment(if_G)
me_c += self.r0 * gd.integrate(if_G)
me_c *= -omega
elif self.form == 'V':
dtype = final_G.dtype
phase_cd = np.ones((3, 2), dtype)
if not hasattr(gd, 'ddr'):
gd.ddr = [Gradient(gd, c, dtype=dtype).apply for c in range(3)]
dfinal_G = gd.empty(dtype=dtype)
me_c = np.empty(3, dtype=dtype)
for c in range(3):
# print "rank, c, apply", rank, c, dtype, final_G.shape, dfinal_G.shape
gd.ddr[c](final_G, dfinal_G, phase_cd)
me_c[c] = gd.integrate(initial_G * dfinal_G)
else:
raise NotImplementedError
# print self.k, self.initial.get_me_c(vk, self.form)[0].imag, me_c[0].imag
if 0:
omega = self.get_omega()
me_analyt = ini_analyt.get_me_c(vk, self.form)[0].imag
me = me_c[0].imag
def ds(me):
return self.k / omega * me**2
print omega, ds(me_analyt), ds(me), me_analyt, me
# print 'analyt', self.initial.get_me_c(vk, self.form)
# print 'num', me_c
# print 'analyt/num', self.initial.get_me_c(vk, self.form) / me_c
# return the squared matrix elements
T2 = []
for eps in [eps0, epsm]:
me = np.dot(eps, me_c)
# print "eps, T2:", eps, (me * me.conj()).real
T2.append((me * me.conj()).real)
# print vk, T2
return T2[0], T2[1]
class CrossSectionBeta:
def __init__(self,
initial=None,
final=None,
r0=[0, 0, 0], # center of mass vector
form='L',
ngauss=8):
self.initial = initial
self.r0 = np.array(r0) / Bohr
self.form = form
if final is None:
self.final = PlaneWave(initial.gd)
else:
self.final = final
self.Ekin = None
# Gauss-Legendre weights and abscissas
self.gl = {}
self.gl['x'] = GaussLegendre(-1., 1., ngauss)
self.gl['phi'] = GaussLegendre(0, 2 * pi, ngauss)
self.gl['psi'] = self.gl['phi']
self.angle = {}
# sin and cos of the magic angle (54.7 deg)
self.costhm = 1. / sqrt(3)
self.sinthm = sqrt(2. / 3.)
def calculate(self, Ekin):
"""Calculate the necessary overlaps."""
Ekin = Ekin / Hartree
if self.Ekin == Ekin:
return
self.Ekin = Ekin
# photoelectron momentum
self.k = sqrt(2 * self.Ekin)
for angle in ['x', 'phi', 'psi']:
self.angle[angle] = self.gl[angle].get_x()[0]
self.T20, self.T2m = self.gi_x()
# we need the average
self.T20 /= 8 * pi ** 2
self.T2m /= 8 * pi ** 2
def get_omega(self):
"""Return the necessary photon energy."""
return self.Ekin - self.initial.get_energy() / Hartree
def get_beta(self, Ekin=None):
"""Return the asymmetry parameter.
E: photoelectron kinetic energy [eV]
"""
if Ekin is not None:
self.calculate(Ekin)
return self.T20 / self.T2m - 1.
def get_ds(self, Ekin=None, units='Mb'):
"""Return the total cross section.
Ekin: photoelectron kinetic energy [eV]
units:
'Mb', 1 Mb = 1.e-22 m**2
'Ang', 1 A**2 = 1.e-20 m**2
'a.u.', 1 a_0**2 = 2.8e-21 m**2
as output units
"""
if Ekin is not None:
self.calculate(Ekin)
try:
pre = ds_prefactor[units]
except KeyError:
print('Unknown units: >' + units + '<')
raise
# me_c = self.initial.get_me_c(np.array([0., 0., self.k]), self.form)
# T2mana = np.abs(np.dot(me_c,me_c)) / 3.
# print "T2m:", T2mana, self.T2m
omega = self.get_omega()
# integration over momentum agles
pre *= self.k * 4 * pi
# print omega, self.initial.get_ds(self.k, omega, self.form), \
# (self.k * 4 * pi * (2 * pi)**2 / 137.0359895 * self.T2m / omega)
return pre * ((2 * pi) ** 2 * alpha * self.T2m / omega)
def gauss_integrate(self, angle, function):
T20 = 0.
T2m = 0.
gl = self.gl[angle]
for x, w in zip(gl.get_x(), gl.get_w()):
# print angle, x, w
self.angle[angle] = x
t20, t2m = function()
T20 += w * t20
T2m += w * t2m
# print "<gauss_integrate> angle=", angle, T2m
return T20, T2m
def gi_x(self):
"""Gauss integrate over x=cos(theta)"""
return self.gauss_integrate('x', self.gi_phi)
def gi_phi(self):
"""Gauss integrate over phi"""
return self.gauss_integrate('phi', self.gi_psi)
def gi_psi(self):
"""Gauss integrate over psi"""
return self.gauss_integrate('psi', self.integrand)
def integrand(self):
# polarisation in the direction of vk
costh = self.angle['x']
sinth = sqrt(1. - costh ** 2)
sinphi = sin(self.angle['phi'])
cosphi = cos(self.angle['phi'])
eps0 = np.array([sinth * cosphi,
sinth * sinphi,
costh])
vk = self.k * eps0
# polarisation at the magic angle
costhm = self.costhm
sinthm = self.sinthm
sinpsi = sin(self.angle['psi'])
cospsi = cos(self.angle['psi'])
epsm = np.array([sinthm * (cosphi * sinpsi * costh +
sinphi * cospsi) +
costhm * cosphi * sinth,
sinthm * (sinphi * sinpsi * costh -
cosphi * cospsi) +
costhm * sinphi * sinth,
costhm * costh - sinthm * sinth * sinpsi])
# initial and final state on the grid
initial_G = self.initial.get_grid()
final_G = self.final.get_grid(vk, self.r0)
ini_analyt = H1s(self.initial.gd, self.r0)
gd = self.initial.gd
if self.form == 'L':
if_G = initial_G * final_G
omega = self.get_omega()
if 0:
me_c = []
for c in range(3):
xyz_G = ((np.arange(gd.n_c[c]) + gd.beg_c[c]) * gd.h_c[c]
- self.r0[c])
shape = [1, 1, 1]
shape[c] = -1
xyz_G.shape = tuple(shape)
np.resize(xyz_G, gd.n_c)
me_c.append(gd.integrate(if_G * xyz_G))
me_c = np.array(me_c) * omega
else:
me_c = gd.calculate_dipole_moment(if_G)
me_c += self.r0 * gd.integrate(if_G)
me_c *= -omega
elif self.form == 'V':
dtype = final_G.dtype
phase_cd = np.ones((3, 2), dtype)
if not hasattr(gd, 'ddr'):
gd.ddr = [Gradient(gd, c, dtype=dtype).apply for c in range(3)]
dfinal_G = gd.empty(dtype=dtype)
me_c = np.empty(3, dtype=dtype)
for c in range(3):
gd.ddr[c](final_G, dfinal_G, phase_cd)
me_c[c] = gd.integrate(initial_G * dfinal_G)
else:
raise NotImplementedError
if 0:
omega = self.get_omega()
me_analyt = ini_analyt.get_me_c(vk, self.form)[0].imag
me = me_c[0].imag
def ds(me):
return self.k / omega * me ** 2
print(omega, ds(me_analyt), ds(me), me_analyt, me)
# print 'analyt', self.initial.get_me_c(vk, self.form)
# print 'num', me_c
# print 'analyt/num', self.initial.get_me_c(vk, self.form) / me_c
# return the squared matrix elements
T2 = []
for eps in [eps0, epsm]:
me = np.dot(eps, me_c)
# print "eps, T2:", eps, (me * me.conj()).real
T2.append((me * me.conj()).real)
# print vk, T2
return T2[0], T2[1]