def Analatic_sol(R, n=0, De=10, a=1, Re=1):
e = exp(1)
X = R * a
Xe = Re * a
lam = ((2 * m * De)**.5) / (a * h)
eps = -(lam - n - .5)**2
alp = (2 * lam - 2 * n - 1) #alpha
En = (eps * (a * h)**2 / (2 * m)) + 10
Z = 2 * lam * e**(-(X - Xe))
Nn = (fac(n) * (alp) / (gm(2 * lam - n)))**.5
L = lag(n, alp, Z) #lagragian
Psi = Nn * Z**(alp / 2) * e**(-.5 * Z) * L
Psi_N, A = Normalize(R, Psi)
if n % 2 == 1:
Psi_N = -Psi_N
return Psi_N, En
from scipy.special import gamma as gm import numpy as np from math import exp from math import factorial as fac import matplotlib.pyplot as plt De = 10 a = .8 Re = 1 e = exp(1) R = np.linspace(-1, 5, 100) X = R * a Xe = Re * a m, h = 1, 1 lam = ((2 * m * De)**.5) / (a * h) n = 5 #n<=[lamda-1/2] eps = -(lam - n - .5)**2 alp = (2 * lam - 2 * n - 1) #alpha En = eps * (a * h)**2 / (2 * m) + 10 Z = 2 * lam * e**(-(X - Xe)) Nn = (fac(n) * (alp) / (gm(2 * lam - n)))**.5 L = lag(n, alp, Z) #lagragian Psi = Nn * Z**(alp / 2) * e**(-.5 * Z) * L plt.plot(X, Psi) plt.show()
def beta_function(a, b, u):
# function to calculate beta distribution value for given a, b, and u
beta = (gm(a + b) / (gm(a) * gm(b))) * (u**(a - 1)) * ((1 - u)**(b - 1))
return beta
def Deuteron(B, minimized=False):
#Constants
#---------------------------------------
hbarC = 197.327054 #MeV fm
m1 = 938.28 #Proton Mass
m2 = 939.57 #Neutron Mass
mu = m1 * m2 / (m1 + m2)
R = 20 #fm
dx = 1 / 128
xi = 2 * mu / hbarC**2
gamma = np.sqrt(xi * B)
#---------------------------------------
#Long Range Solutions VNN = 0
# s-wave l=0
#---------------------------------------
f_s = np.exp(-gamma * R)
f_s_prime = -gamma * f_s
#---------------------------------------
# d-wave l=2
#---------------------------------------
f_d = (1 + 3 / (gamma * R) + 3 / (gamma**2 * R**2)) * f_s
f_d_prime = -f_s * (gamma + 3 / R + 6 / (gamma * R**2) + 6 /
(gamma**2 * R**3))
#---------------------------------------
#Starting Value of Independent Solutions ~0
#---------------------------------------
c = 10**-6
U_zero = np.array([c, 0])
U_two = np.array([0, c])
# Run Numerov
#--------------------------------------
r, u0, w0 = Numerov(U_zero, B, xi, dx, R)
r, u2, w2 = Numerov(U_two, B, xi, dx, R)
#---------------------------------------
#Compute Derivatives -- Central Difference
#---------------------------------------
phi = [u0[-2],\
w0[-2],\
(u0[-1] - u0[-3])/(2*dx),\
((w0[-1] - w0[-3]))/(2*dx)]
psi = [u2[-2],\
w2[-2],\
(u2[-1] - u2[-3])/(2*dx),\
((w2[-1] - w2[-3]))/(2*dx)]
#---------------------------------------
#Generate Matrix (4x4)
#---------------------------------------
FS = np.array([-f_s, 0, -f_s_prime, 0])
FD = np.array([0, -f_d, 0, -f_d_prime])
M = np.array([phi, psi, FS, FD])
#---------------------------------------
#Compute Determanint -- Should be zero
#---------------------------------------
det = abs(np.linalg.det(M))
#---------------------------------------
########################################
# Deuteron Properties #
########################################
if minimized == True:
print("Binding Energy: " + str(-B))
phi_c, phi_t, phi_c_prime, phi_t_prime = phi
psi_c, psi_t, psi_c_prime, psi_t_prime = psi
# a = (f_s*psi_c_prime-f_s_prime*psi_c)/(phi_c*psi_c_prime-phi_c_prime*psi_c)
# b = (f_s*phi_c_prime-f_s_prime*phi_c)/(psi_c*phi_c_prime-psi_c_prime*phi_c)
b = (f_s * phi_c_prime - f_s_prime * phi_c) / (psi_c * phi_c_prime -
psi_c_prime * phi_c)
a = b * (f_d * psi_t_prime - psi_t * f_d_prime) / (phi_t * f_d_prime -
phi_t_prime * f_d)
u = a * u0 + b * u2 #s-wave Solution
w = a * w0 + b * w2 #d-wave Solution
#Proportion of Deuteron in d-wave
eta = (a * phi_t + b * psi_t) / f_d
print("Eta: " + str(eta))
#---------------------------------------
Iu = simps(u**2, r)
Iw = simps(w**2, r)
fs_I = (np.exp(-2 * gamma * R)) / (2 * gamma)
fd_I = fs_I * (1 + 6 / (gamma * R) + 12 / (gamma * R)**2 + 6 /
(gamma * R)**3)
inf = fs_I + eta**2 * fd_I
As = Iu + Iw + inf
As2 = 1 / (As)
print("As: " + str(np.sqrt(As2)))
#---------------------------------------
#Useful Calculation -- Total Probabilty of d-wave
Pd = Iw + eta**2 * fd_I
Pd *= As2
print("Pd: " + str(Pd))
#---------------------------------------
muP = 2.793 #Magnetic Moment of Proton (Exp)
muN = -1.913 #Magnetic Moment of Neutron (Exp)
#Deuteron Magnetic Moment
muD = (muN + muP) * (1 - 1.5 * Pd) + .75 * Pd
print("Magnetic Moment: " + str(muD))
#---------------------------------------
#Useful Constants -- infinite integration of fs -- fd -- *r^2
# integrand = exp(-2*gamma*R)*r^(2)/(gamma*r)^(pow)
#factored out exp(-2*gamma*R)/(4*gamma**3)
I0 = 2 * gamma * R * (gamma * R + 1) + 1 #pow = 0
I1 = 2 * gamma * R + 1 #pow = 1
I2 = 2 #pow = 2
#----
#Incomplete Gamma Function -- Scipy Uses Regularized -- Fixed
a = 10**(-8)
ginc = gm(a) * gammaincc(a, 2 * R * gamma) / np.exp(-2 * gamma * R)
#----
I3 = 4 * ginc
I4 = 4 / (gamma * R) - 8 * ginc
#---------------------------------------
#Deteron Radius:
integrand = (r**2) * (u**2 + w**2)
rd_inf = I0 + eta**2 * (I0 + 6 * I1 + 15 * I2 + 18 * I3 + 9 * I4)
rd_inf *= np.exp(-2 * gamma * R) / (4 * gamma**3)
rd = simps(integrand, r) + rd_inf
rd = np.sqrt(As2 * rd / 4)
print("Deteron Radius: " + str(rd))
#---------------------------------------
#Quadrapole Moment:
integrand = (u * w * np.sqrt(8) - w**2) * r**2
Qd_inf = eta * np.sqrt(8) * (I0 + 3 * I1 + 3 * I2) - eta**2 * (
I0 + 6 * I1 + 15 * I2 + 18 * I3 + 9 * I4)
Qd_inf *= np.exp(-2 * gamma * R) / (4 * gamma**3)
Qd = simps(integrand, r) + Qd_inf
Qd *= As2 / 20
print("Quadrapole Moment: " + str(Qd))
#---------------------------------------
########################################
else:
print("Evaluated at B = " + str(B))
return det