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Beam_Slowing_v1.4_Ideal_Analytic(FAST).py
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#!/usr/bin/env python3
# -*- coding: utf-8 -*-
"""
Created on Wed Mar 6 10:37:32 2019
@author: Z0RA4
"""
#%%
import numpy as np
from numpy import pi,sin,cos,tan,sqrt
import matplotlib.pyplot as plt
from matplotlib import cm
from mpl_toolkits.mplot3d import Axes3D
from scipy.integrate import odeint
from scipy.interpolate import RegularGridInterpolator
# Units
mm = ms = 1e-3
um = us = 1e-6
# Magnetic field equations
# Functions defining the fields along each axis for the SFS and WFS states
def SFSy(y):
return (-6267)*y**2 - 0.106*y + 1.018
def SFSx(x):
return (2.518*10**4)*x**2 - 0.05364*x + 1.021
def WFSy(y):
return (1.081*10**10)*y**4 + (1.635*10**5)*y**3 - (1.133*10**4)*y**2 - 0.6312*y + 0.02394
def WFSx(x):
return (7.657*10**9)*x**4 - (1.166*10**5)*x**3 + (3.603*10**4)*x**2 + 0.2786*x + 0.03799
# Magnetic field from ZS-paper archived simulations; B field in [Teslas]
bz_mag_offset = 0.16 # approximately 0.16T if we assume only two stages in the numerical calculation of the Bfield; shift the minimum Bfield value
#stagescale == Scale the magnetic field such that two stages extend over the range of the numertor in mm
bz_scale = 0.97 # scale the B field so that the maximum value is approximately 1.16T
def zfield_sin(z,stagescale):
return sin(315.3*z/2/stagescale)
def zfield_cos(z,stagescale):
return cos(315.3*z/2/stagescale)
def fullfield(x,y,z,stagescale):
return (SFSx(x)*SFSy(y)*zfield_sin(z,stagescale)**2 + WFSx(x)*WFSy(y)*zfield_cos(z,stagescale)**2)*bz_scale+bz_mag_offset
def gSFSy(y):
return -0.106 - 12534*y
def gSFSx(x):
return -0.05364 + 50360*x
def gWFSy(y):
return -0.6312 - 22660*y + 490500*y**2 + (4.324*10**10)*y**3
def gWFSx(x):
return 0.2786 + 72060*x - 349800*x**2 + (3.0628*10**10)*x**3
def gxfullfield(x,y,z,stagescale):
return (gSFSx(x)*SFSy(y)*zfield_sin(z,stagescale)**2 + gWFSx(x)*WFSy(y)*zfield_cos(z,stagescale)**2)*bz_scale
def gyfullfield(x,y,z,stagescale):
return (SFSx(x)*gSFSy(y)*zfield_sin(z,stagescale)**2 + WFSx(x)*gWFSy(y)*zfield_cos(z,stagescale)**2)*bz_scale
def gzfullfield(x,y,z,stagescale):
return (315.3/stagescale)*(SFSx(x)*SFSy(y)*zfield_sin(z,stagescale)*zfield_cos(z,stagescale) - WFSx(x)*WFSy(y)*zfield_cos(z,stagescale)*zfield_sin(z,stagescale))*bz_scale
def gfullfield(x,y,z,stagescale):
return gxfullfield(x,y,z,stagescale)+gyfullfield(x,y,z,stagescale)+gzfullfield(x,y,z,stagescale)
# Creating arrays and meshgrids
xterms = yterms = zterms = 400 # ~400 is a good run time, but Simulations might be better with a higher number
radius = 2.5 # mm
zlen = 20 # mm
# Bore radius is actually x,y = 2.5mm & z = 40mm
x = np.linspace(-radius, radius, xterms)*mm
y = np.linspace(-radius, radius, yterms)*mm
z = np.linspace(0,zlen, zterms)*mm
#%% PLOT OF THE GRADIENTS OF THE MAGNETIC FIELD
scale = 20/20
zpos = 0*mm
fig = plt.figure()
ax = fig.add_subplot(111)
ax.grid()
ax.plot(z*1e3, gzfullfield(0,0,z,scale))
ax.set_title('GradB_z vs z')
ax.set_xlabel("z [mm]")
ax.set_ylabel("dB/dz [T/m]")
fig = plt.figure()
ax = fig.add_subplot(111)
ax.grid()
ax.plot(z*1e3, fullfield(0,0,z,scale))
ax.set_title('B_z vs z')
ax.set_xlabel("z [mm]")
ax.set_ylabel("B [T]")
fig = plt.figure()
ax = fig.add_subplot(111)
ax.grid()
ax.plot(x*1e3, gxfullfield(x,0,zpos,scale))
ax.set_title('GradB_x vs x at z = %.3f mm' %(zpos))
ax.set_xlabel("x [mm]")
ax.set_ylabel("dB/dx [T/m]")
fig = plt.figure()
ax = fig.add_subplot(111)
ax.grid()
ax.plot(y*1e3, gyfullfield(0,y,zpos,scale))
ax.set_title('GradB_y vs y at z = %.3f mm' %(zpos))
ax.set_xlabel("y [mm]")
ax.set_ylabel("dB/dy [T/m]")
#%% Physical quantities
amu = 1.660539040*10**-27 # amu to kg; NIST
hbar = 1.054571800*10**-34 #J*s, J/Hz ; NIST
e = 1.6021766208*10**-19 # C; NIST
kb = 1.38064852*10**-23 # J/K
me = 9.10938356*10**-31 # kg; NIST
mYb = 173.045 # amu, for 174Yb, which has an abundance of 31.896%; CIAAW 2015
mO = 15.999 # amu; google
mH =1.00794 #amu; google
# Properties of YbOH molecule
mYbOH = (mYb+mO+mH)
m = mYbOH * amu
mu_bohr = (-e*hbar)/(2*me) # J/T
# Let mu.B > 0 = WFS state
alpWFS = mu_bohr/m # * grad B = accel
# Let mu.B < 0 = SFS state
alpSFS = -mu_bohr/m # * grad B = accel
# Functions for generating gaussian distributions randomly
# output random values proportional to a gaussian distribution
def randG(mean,std):
return np.random.normal(mean, std)
# If the output is less than cutoff, repeat until output is greater than cutoff
def randG_trunc(mean,std,cutoff):
val = randG(mean,std)
while val <= cutoff:
val = randG(mean,std)
return val
# Solving for the velocity along one dimension from a temperature T
def velocity(T):
return sqrt(kb*T/m)
# Difference in kinetic energy equation
def KE(vf,v0):
dKE = (1/2)*m*((vf)**2-(v0)**2)
return dKE
#%% IDEAL CASE: Molecule state function, (position dependent)
init_state = alpWFS # Initial state of molecule
B_str = 1
def mol_state(x,y,z):
# if (abs(x) < radius*mm) and (abs(y) < radius*mm) and (z < 100*mm):
if (z < 90*mm):
if (0 <= z < 10*mm) \
or (20*mm <= z < 29*mm) \
or (38*mm <= z < 46*mm) \
or (54*mm <= z < 61*mm) \
or (68*mm <= z < 74*mm) \
or (80*mm <= z < 85*mm):
c_s = init_state
else:
c_s = alpSFS
else:
c_s = 0
return c_s
def stagescale(z):
if (0 <= z < 20*mm):
val = 20/20
elif (20*mm <= z < 38*mm):
val = 18/20
elif (38*mm <= z < 54*mm):
val = 16/20
elif (54*mm <= z < 68*mm):
val = 14/20
elif (68*mm <= z < 80*mm):
val = 12/20
elif (80*mm <= z < 90*mm):
val = 10/20
else:
val = 1 # Unless there is a better assumption to make
return val
def stagemod(z):
if (20*mm <= z < 38*mm):
val = (20)*mm
elif (38*mm <= z < 54*mm):
val = (20+18)*mm
elif (54*mm <= z < 68*mm):
val = (20+18+16)*mm
elif (68*mm <= z < 80*mm):
val = (20+18+16+14)*mm
elif (80*mm <= z < 90*mm):
val = (20+18+16+14+12)*mm
else:
val = 1 # Unless there is a better assumption to make
return val
tlast = 5*ms
t_terms = int(1e5)
t = np.linspace(0,tlast,t_terms) # s
def equation_system(b,t):
xt,yt,zt,vxt,vyt,vzt = b
db_over_dt = [vxt,vyt,vzt, B_str*mol_state(xt,yt,zt)*gxfullfield(xt,yt,zt,1), B_str*mol_state(xt,yt,zt)*gyfullfield(xt,yt,zt,1), B_str*mol_state(xt,yt,zt)*gzfullfield(xt,yt,zt%(stagemod(zt)),stagescale(zt))]
return db_over_dt
#%%
xt, yt, zt = [], [], []
vxt, vyt,vzt = [], [], []
particle_num = 100
xydev = 0.03*mm # deviation in the position in m
vxvydev = 0.11456 # deviation in the velocity in m/s
for n in range(particle_num):
r0 = np.array([randG(0,0.03*mm) ,randG(0,0.03*mm) ,0])# m
# v0 = np.array([randG(0,sigma_v(3)), randG(0,sigma_v(3)) ,randG_trunc(30,25,5)])
v0 = np.array([randG(0,0.11456), randG(0,0.11456) ,randG(30,2.5)])
s0 = np.concatenate([r0,v0])
solution = odeint(equation_system,s0,t)
xt.append(solution[:,0])
yt.append(solution[:,1])
zt.append(solution[:,2])
vxt.append(solution[:,3])
vyt.append(solution[:,4])
vzt.append(solution[:,5])
print(n)
#%% PHASE SPACE PLOTS along x and y directions
# Position at initial/final times
pos_i = 0
pos_f = -1
fig = plt.figure()
ax = fig.add_subplot(111)
ax.set_title('$V_x - x$ phase space')
ax.grid()
for n in range(len(xt)):
ax.scatter(xt[n][pos_i]/(3*mm),vxt[n][pos_i]/(velocity(3)), s = 10, color = 'red',alpha = 0.4)
ax.scatter(xt[n][pos_f]/(3*mm),vxt[n][pos_f]/(velocity(3)), s = 10, color = 'blue',alpha = 0.4)
ax.plot(xt[n]/(3*mm),vxt[n]/(velocity(3)), alpha = 0.4)
# ax.axhline(y = vxt[n][pos_i]/(sigma_v(3)), color = 'orange', linewidth = 0.75)
ax.axvline(x=(radius/3), color = 'salmon', linewidth = 0.75) # position of the bore
ax.axvline(x=(-radius/3), color = 'salmon', linewidth = 0.75) # position of the bore
ax.set_xlabel("$x/\sigma_x$ ")
ax.set_ylabel("$Vx/\sigma_{V_x}$ ")
fig = plt.figure()
ax = fig.add_subplot(111)
ax.set_title('$V_y - y$ phase space')
ax.grid()
for n in range(len(xt)):
ax.scatter(yt[n][pos_i]/(3*mm),vyt[n][pos_i]/(velocity(3)), s = 10, color = 'red',alpha = 0.4)
ax.scatter(yt[n][pos_f]/(3*mm),vyt[n][pos_f]/(velocity(3)), s = 10, color = 'blue',alpha = 0.4)
ax.plot(yt[n]/(3*mm),vyt[n]/(velocity(3)), alpha = 0.4)
# ax.axhline(y = vyt[n][pos_i]/(sigma_v(3)), color = 'orange', linewidth = 0.75)
ax.axvline(x=(radius/3), color = 'salmon', linewidth = 0.75)
ax.axvline(x=(-radius/3), color = 'salmon', linewidth = 0.75)
ax.set_xlabel("$y/\sigma_y$ ")
ax.set_ylabel("$Vy/\sigma_{V_y}$ ")
#%%
# Position at initial/final times
pos_i = 0
pos_f = -1
fig = plt.figure()
ax = fig.add_subplot(111)
ax.grid(True)
ax.set_title('$\Delta KE_{V_z}$ vs z')
for n in range(len(xt)):
ax.plot(zt[n]*1e3,KE(vzt[n],vzt[n][0])/kb)
ax.axvline(x=20)
ax.axvline(x=38)
ax.axvline(x=54)
ax.axvline(x=68)
ax.axvline(x=80)
ax.set_xlabel("z [mm]")
ax.set_ylabel("$\Delta KE$ [K]")
fig = plt.figure()
ax = fig.add_subplot(111)
ax.set_title('$V_z - z$ phase space')
ax.grid()
for n in range(len(xt)):
ax.scatter(zt[n][pos_i]*1e3,vzt[n][pos_i], s = 10, color = 'red',alpha = 0.4)
ax.scatter(zt[n][pos_f]*1e3,vzt[n][pos_f], s = 10, color = 'blue',alpha = 0.4)
ax.plot(zt[n]*1e3,vzt[n], alpha = 0.4)
ax.set_xlabel("$z$ [mm]")
ax.set_ylabel("$Vz$ [m/s]")
fig = plt.figure()
ax = fig.add_subplot(111)
ax.set_title('$KE_z - z$ phase space')
ax.grid()
for n in range(len(xt)):
ax.scatter(zt[n][pos_i]*1e3,(1/2)*m*(vzt[n][pos_i])**2/kb, s = 10, color = 'red',alpha = 0.4)
ax.scatter(zt[n][pos_f]*1e3,(1/2)*m*(vzt[n][pos_f])**2/kb, s = 10, color = 'blue',alpha = 0.4)
ax.plot(zt[n]*1e3,(1/2)*m*(vzt[n])**2/kb, alpha = 0.4)
ax.set_xlabel("$z$ [mm]")
ax.set_ylabel("$KE_z$ [K]")
#%%
# HISTOGRAM of Vz Distribution
vz_initial, vz_final = [], []
for n in range(len(xt)):
vz_initial.append(vzt[n][0])
vz_final.append(vzt[n][-1])
fig,ax = plt.subplots()
ax.set_title('Initial and final $V_z$ distrubution')
# plt.hist(vz_initial,bins=20,normed=True, alpha = 0.4)
# plt.hist(vz_final,bins=20,normed=True, alpha = 0.4)
plt.hist(vz_initial,bins=20, alpha = 0.4)
plt.hist(vz_final,bins=20, alpha = 0.4)
plt.xlim(0,60)
ax.set_xlabel("$Vz$ [m/s]")
ax.set_ylabel("Normalized counts")
plt.show()
#%% TRUNCATING PHASE SPACE VECTORS TO BE WITHIN AREA OF THE BORE
xtcut, ytcut, ztcut = [], [], []
vxtcut, vytcut,vztcut = [], [], []
for n in range(particle_num):
# Vectors such that plotting them will always be within the bore
xtcut.append(((xt[n]*1e9)[zt[n]*1e9 < 90e6])/1e9)
ytcut.append(((yt[n]*1e9)[zt[n]*1e9 < 90e6])/1e9)
ztcut.append(((zt[n]*1e9)[zt[n]*1e9 < 90e6])/1e9)
vxtcut.append(((vxt[n]*1e9)[zt[n]*1e9 < 90e6])/1e9)
vytcut.append(((vyt[n]*1e9)[zt[n]*1e9 < 90e6])/1e9)
vztcut.append(((vzt[n]*1e9)[zt[n]*1e9 < 90e6])/1e9)
#%% PHASE SPACE PLOTS along x and y directions with decreasing stage spacing
# Position at initial/final times
pos_i = 0
pos_f = -1
fig = plt.figure()
ax = fig.add_subplot(111)
ax.set_title('$V_x - x$ phase space with successively shorter stages')
ax.grid()
for n in range(particle_num):
ax.scatter(xtcut[n][pos_i]/(3*mm),vxtcut[n][pos_i]/(velocity(3)), s = 10, color = 'red',alpha = 0.4)
ax.scatter(xtcut[n][pos_f]/(3*mm),vxtcut[n][pos_f]/(velocity(3)), s = 10, color = 'blue',alpha = 0.4)
# ax.plot(xtcut[n]/(3*mm),vxtcut[n]/(velocity(3)), alpha = 0.4)
ax.axvline(x=(radius/3), color = 'salmon', linewidth = 0.75) # position of the bore
ax.axvline(x=(-radius/3), color = 'salmon', linewidth = 0.75) # position of the bore
ax.set_xlabel("$x/\sigma_x$ ")
ax.set_ylabel("$Vx/\sigma_{V_x}$ ")
fig = plt.figure()
ax = fig.add_subplot(111)
ax.set_title('$V_y - y$ phase space with successively shorter stages')
ax.grid()
for n in range(particle_num):
ax.scatter(ytcut[n][pos_i]/(3*mm),vytcut[n][pos_i]/(velocity(3)), s = 10, color = 'red',alpha = 0.4)
ax.scatter(ytcut[n][pos_f]/(3*mm),vytcut[n][pos_f]/(velocity(3)), s = 10, color = 'blue',alpha = 0.4)
# ax.plot(ytcut[n]/(3*mm),vytcut[n]/(velocity(3)), alpha = 0.4)
ax.axvline(x=(radius/3), color = 'salmon', linewidth = 0.75)
ax.axvline(x=(-radius/3), color = 'salmon', linewidth = 0.75)
ax.set_xlabel("$y/\sigma_y$ ")
ax.set_ylabel("$Vy/\sigma_{V_y}$ ")
#%%
# Position at initial/final times
pos_i = 0
pos_f = -1
fig = plt.figure()
ax = fig.add_subplot(111)
ax.grid(True)
# ax.set_title('$\Delta KE_{V_z}$ vs z with stage spacing of $\Delta z$ = %.1f mm' %(stage_offset))
ax.set_title('$\Delta KE_{V_z}$ vs z ' )
for n in range(len(xtcut)):
ax.plot(ztcut[n]*1e3,KE(vztcut[n],vztcut[n][0])/kb)
ax.axvline(x=20)
ax.axvline(x=38)
ax.axvline(x=54)
ax.axvline(x=68)
ax.axvline(x=80)
ax.axvline(x=90)
ax.set_xlabel("z [mm]")
ax.set_ylabel("$\Delta KE$ [K]")
fig = plt.figure()
ax = fig.add_subplot(111)
ax.set_title('$V_z - z$ phase space')
ax.grid()
for n in range(len(xtcut)):
ax.scatter(ztcut[n][pos_i]*1e3,vztcut[n][pos_i], s = 10, color = 'red',alpha = 0.4)
ax.scatter(ztcut[n][pos_f]*1e3,vztcut[n][pos_f], s = 10, color = 'blue',alpha = 0.4)
ax.plot(ztcut[n]*1e3,vztcut[n], alpha = 0.4)
ax.set_xlabel("$z$ [mm]")
ax.set_ylabel("$Vz$ [m/s]")
fig = plt.figure()
ax = fig.add_subplot(111)
# ax.set_title('$V_z - z$ phase space with stage spacing of $\Delta z$ = %.1f mm' %(stage_offset))
ax.set_title('$KE_z$ vs z')
ax.grid()
for n in range(len(xtcut)):
ax.scatter(ztcut[n][pos_i]*1e3,(1/2)*m*(vztcut[n][pos_i])**2/kb, s = 10, color = 'red',alpha = 0.4)
ax.scatter(ztcut[n][pos_f]*1e3,(1/2)*m*(vztcut[n][pos_f])**2/kb, s = 10, color = 'blue',alpha = 0.4)
ax.plot(ztcut[n]*1e3,(1/2)*m*(vztcut[n])**2/kb, alpha = 0.4)
ax.set_xlabel("$z$ [mm]")
ax.set_ylabel("$KE_z$ [K]")
#%%
# HISTOGRAM of Vz Distribution
vz_initial, vz_final = [], []
for n in range(len(xt)):
vz_initial.append(vztcut[n][0])
vz_final.append(vztcut[n][-1])
fig,ax = plt.subplots()
ax.set_title('Initial and final $V_z$ distrubution')
# plt.hist(vz_initial,bins=20,normed=True, alpha = 0.4)
# plt.hist(vz_final,bins=20,normed=True, alpha = 0.4)
plt.hist(vz_initial,bins=20, alpha = 0.4)
plt.hist(vz_final,bins=20, alpha = 0.4)
plt.xlim(0,60)
ax.set_xlabel("$Vz$ [m/s]")
ax.set_ylabel("Normalized counts")
plt.show()