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odedirac.py
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odedirac.py
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import math
import numpy as np
import scipy
from scipy import special
from scipy import interpolate
import util
from util import *
def _sgn(x):
if x > 0: return 1.0
if x < 0: return -1.0
return 0.0
sgn = np.vectorize(_sgn)
def psireg(r_i, Ex, m): ### Not normalised here
uu = special.jn(m, (np.abs(Ex)*r_i))
ud = special.jn(m+1, (np.abs(Ex)*r_i)) * sgn(Ex)
return uu, ud
def psising(r_i, Ex, m): ### Not normalised here
vu = special.yn(m, (np.abs(Ex)*r_i))
vd = sgn(Ex) * special.yn(m+1, (np.abs(Ex)*r_i))
return vu, vd
#
# We write the wave function as
#
# psi_u = (r/R)^mu chi_u
# psi_d = i (r/R)^mu chi_d
#
# where mu = m for m >= 0, and -m-1 otherwise. R == r[-1] here
# This way, we can start with finite solution at r=0 and integrate
# it to r = R. After that, we match it to wave function of free particle:
#
# psi = A * psi_reg(E, r, m) + B * psi_sing(E, r, m)
#
# We then normalise the wave function to unit intensity: A^2 + B^2 = 1.
#
def odepsi_m(E, r, U, m):
chi_u = np.zeros((len(r), len(E)))
chi_d = np.zeros(np.shape(chi_u))
chi_u[0, :], chi_d[0, :] = psireg(r[0], E - U[0], m)
nchi = np.abs(chi_u[0, :]) + np.abs(chi_d[0, :]) # this does not involve
# squares of small numbers
np.sqrt(chi_u[0, :]**2 + chi_d[0, :]**2) # unlike prev version
chi_u[0, :] /= nchi
chi_d[0, :] /= nchi
if m >= 0:
Ku = 0
Kd = -1.0 - 2.0 * m
mu = m
else:
mu = -m-1
Ku = 1.0 + 2.0 * m
Kd = 0
#
# Right-hand side of the ODE: chi'(r) = rhs(r)
#
def rhs(chi_u, chi_d, r, U):
f_u = Ku / r * chi_u - (E - U)*chi_d
f_d = (E - U)*chi_u + Kd / r * chi_d
#f_u = (m-mu)/r * chi_u - (E - U)*chi_d
#f_d = (E - U)*chi_u - (1 + mu + m)/r * chi_d
return f_u,f_d
Us = interpolate.splrep(r, U)
#
# RK4 fourth order method
#
def rk4step(chi_u, chi_d, r_p, r_n, h):
r1 = r_p
r2 = r_p + 0.5 * h
r3 = r2
r4 = r_n
#rs = np.array([r1, r2, r3, r4])
Ui = interpolate.splev(np.array([r1, r2, r3, r4]), Us)
U1 = Ui[0]
U2 = Ui[1]
U3 = Ui[2]
U4 = Ui[3]
k1u, k1d = rhs(chi_u, chi_d, r1, U1)
k2u, k2d = rhs(chi_u+k1u*0.5*h, chi_d+0.5*k1d*h, r2, U2)
k3u, k3d = rhs(chi_u+k2u*0.5*h, chi_d+0.5*k2d*h, r3, U3)
k4u, k4d = rhs(chi_u+k3u*h, chi_d+k3d*h, r4 ,U4)
chi_un = chi_u + h*(k1u + 2*k2u + 2*k3u + k4u)/6.0
chi_dn = chi_d + h*(k1d + 2*k2d + 2*k3d + k4d)/6.0
return chi_un, chi_dn
for i in range(1,len(r)):
chi_un, chi_dn = chi_u[i-1, :], chi_d[i-1, :]
h = r[i] - r[i-1]
dxi = max(abs(E)) * h
# dxi is the phase change over this grid step
# We split the step into smaller sub-steps if dxi
# is too large
dxi0 = 0.1
n_steps = int(dxi / dxi0) + 1
dr = h / n_steps
# Go over sub-steps
for i_step in range(n_steps):
r_p = r[i-1] + i_step * dr
r_n = r[i-1] + (i_step + 1) * dr
chi_un, chi_dn = rk4step(chi_un, chi_dn, r_p, r_n, dr)
chi_u[i, :], chi_d[i, :] = chi_un, chi_dn
uu, ud = psireg(r[-1], E, m)
vu, vd = psising(r[-1], E, m)
D = uu * vd - vu * ud
A = (chi_u[-1, :] * vd - chi_d[-1, :] * vu) / D
B = (chi_d[-1, :] * uu - chi_u[-1, :] * ud) / D
r_mu = (r/r[-1])**(mu)
#
# The goal of the crazy expression below is to multiply
# chi by the corresponding values of r^mu. In numpy, multiplying
# a matrix chi(r, E) by vector r would not work: we have to
# transpose the matrix first.
#
psi_u = (r_mu * chi_u.transpose()).transpose()
psi_d = (r_mu * chi_d.transpose()).transpose()
psi_u /= np.sqrt(A**2 + B**2)
psi_d /= np.sqrt(A**2 + B**2)
return psi_u, psi_d
def odedos_m(E, r, U, m):
psi_u, psi_d = odepsi_m(E, r, U, m)
dos_m = np.abs(psi_u)**2 + np.abs(psi_d)**2
dos_m *= np.abs(E) / 4.0 / np.pi
return dos_m;
def doscalc(E,r,U,mlist):
dos_tot = np.zeros((len(r), len(E)))
for m in mlist:
print m, E[0,], E[-1]
dosm = odedos_m(E, r, U, m)
dos_tot += dosm
return 2.0 * dos_tot # We assume only m>=0 are included
def rhocalc(Emin, Emax, r, U, mlist):
N_e = 2000
Es = np.linspace(Emin, Emax, N_e)
dos = doscalc(Es, r, U, mlist)
weight = np.zeros(len(Es))
weight[:] = 1.0
weight[0] = weight[-1] = 0.5
dE = Es[1] - Es[0]
rho = np.dot(dos, weight) * dE
return rho
def plotode(Emax, mlist):
r = np.arange(0.01, 40.0, 0.05)
U = 0.0 * r
E = 1.0
m = 0
Es = np.arange(-Emax, Emax, 0.01)
#rho = odedos_m(E,r,U,m)
dos = doscalc(Es,r,U,mlist)
for r_i in [0.1, 1.0, 2.0, 5.0, 10.0, 20.0]:
i = np.abs(r - r_i).argmin()
plot(Es, dos[i, :], label='r = %g' % r[i])
plot (Es, np.abs(Es) / 2.0/math.pi, '--k', label='Expected')
legend()
title ('DOS')
#rho_0 = special.jn(m, E*r - r*U)**2 + special.jn(m+1, E*r - r*U)**2
#rho_0 *= abs(E) / 4.0 / np.pi
#title('Charge Density')
#plot(r, rho, label='rho from ode m=%d' %m)
#plot(r, rhomatch, label='rhomatch m=%d' %m)
#plot(r, rho_0, 'r--', label='bessel')
#plot(r,rho/rhomatch)
#legend()
#figure()
#title('density of states')
#plot(r, dos, label='dos for E=&g')
show()
def test_rho():
# import pylab as pl
Emin = -1.0
Emax = -1e-4
mlist = np.arange(0, 10.0, 1.0)
r_0 = 1.0
r = np.linspace(0.1, 50.0, 500)
# pl.figure()
for Z in [0.0, 0.1, 0.2, 0.3]:
U = - Z / np.sqrt(r**2 + r_0**2)
rho_0 = - (Emax**2 - Emin**2) / 4.0 / math.pi
rho_rpa = -Z / 16.0 * r_0 / np.sqrt(r**2 + r_0**2)**3
rho = rhocalc(Emin, Emax, r, U, mlist)
# pl.plot (r, rho, label='Z = %g' % Z)
# pl.plot (r, rho_0 + rho_rpa, label='Z = %g, rpa' % Z)
# pl.legend()
# pl.show()
if __name__ == '__main__':
mlist = np.arange(0,10)
#plotode(1.0, mlist)
test_rho()