

In [263]:

    
import pandas as pd
import numpy as np
import matplotlib.pyplot as plt
from scipy.optimize import curve_fit  # import the curve fitting function
%matplotlib inline

Lead

1



In [264]:

    
N_net = np.array([10802,6943,5828,4180,3115,2062,1514,1114,805])



In [265]:

    
thickness = np.array([0,5.35,10.78,16.23,21.56,26.87,32.07,36.78,41.88])*1e-3



In [266]:

    
Ln = np.array([np.log(entry) for entry in N_net])



In [267]:

    
dy = np.array([1/np.sqrt(entry) for entry in N_net])



In [268]:

    
plt.figure(figsize=(12,6))
plt.scatter(thickness,Ln);
plt.xlabel('Thickness',size=20);
plt.ylabel('Ln($N_{net}$)',size=20);
plt.xticks(size = 13);
plt.yticks(size = 13);

2



In [269]:

    
def myfun(x,N_o,alpha):
    ans = np.log(N_o) - alpha*x  # this is y, "the function to be fit"
    return ans



In [270]:

    
p0 = [20000,3/(0.04)]



In [271]:

    
xlots = np.linspace(0,.05,100)  # need lots of data points for smooth curve
yfit = np.zeros((len(N_net),xlots.size))

plsq, pcov = curve_fit(myfun, thickness, Ln, p0,dy)  # curve fit returns p and covariance matrix
# these give the parameters and the uncertainties
N_o = plsq[0]
eN_o = np.sqrt(pcov[0,0])
alpha = plsq[1]
ealpha = np.sqrt(pcov[1,1])

yfit = myfun(xlots,plsq[0],plsq[1])  # use fit results for a, b, c
    
print('N_o = %.0f +/- %.0f' % (plsq[0], np.sqrt(pcov[0,0])))
print('alpha = %.0f +/- %.0f' % (plsq[1], np.sqrt(pcov[1,1])))









    



N_o = 10584 +/- 326
alpha = 60 +/- 2

3



In [272]:

    
plt.figure(figsize=(12,6));
plt.xticks(size = 13);
plt.yticks(size = 13);
plt.scatter(thickness*1e2,Ln);
plt.xlabel('x (mm)');
plt.ylabel('y (mm)');
plt.plot(xlots*1e2,yfit);
plt.xlim(0,.05*1e2)
plt.legend(['data','Fit'],loc='best');
#plt.text(0.03,8.5,'$N_o$ = %.0f +/- %.0f' % (plsq[0], np.sqrt(pcov[0,0])),size=17)
#plt.text(0.03,8,'alpha = %.0f +/- %.0f' % (plsq[1], np.sqrt(pcov[1,1])),size=17)
plt.xlabel('Plate Thickness (cm)',size=20);
plt.ylabel('Ln($N_{net}$)',size=20);

plt.savefig('Lead')

4



In [273]:

    
#range lambda = 1/alpha
ry = 1/alpha
#uncertainty dlambda = alpha^-2 dalpha
dry = 1/alpha**2 *ealpha



In [274]:

    
ry*1e2 #This is in cm









    Out[274]:





1.6694328164558021



In [275]:

    
dry*1e2 #cm









    Out[275]:





0.053400174792917345

5



In [276]:

    
np.log(2)/alpha * 1e2 #cm









    Out[276]:





1.1571626498605878



In [277]:

    
dry*1e2*np.log(2) #cm









    Out[277]:





0.037014180599118915

6



In [278]:

    
rho = 11.3*1e3 #kg/m^3



In [279]:

    
mu = alpha/rho
mu









    Out[279]:





0.0053009366024752416



In [280]:

    
dmu = ealpha / rho
dmu









    Out[280]:





0.00016956114576644637



In [281]:

    
#From chart 0.51

Aluminum



In [282]:

    
N_net = np.array([8313,7145,5867,4845,4060,3072,2799])



In [283]:

    
thickness = np.arange(1,8)*12.93*1e-3



In [284]:

    
Ln = np.array([np.log(entry) for entry in N_net])



In [285]:

    
dy = np.array([1/np.sqrt(entry) for entry in N_net])



In [286]:

    
plt.figure(figsize=(12,6))
plt.scatter(thickness,Ln);
plt.xlabel('Thickness',size=20);
plt.ylabel('Ln($N_{net}$)',size=20);
plt.xticks(size = 13);
plt.yticks(size = 13);



In [287]:

    
xlots = np.linspace(0,.1,100)  # need lots of data points for smooth curve
yfit = np.zeros((len(N_net),xlots.size))

plsq, pcov = curve_fit(myfun, thickness, Ln, p0,dy)  # curve fit returns p and covariance matrix
# these give the parameters and the uncertainties
N_o = plsq[0]
eN_o = np.sqrt(pcov[0,0])
alpha = plsq[1]
ealpha = np.sqrt(pcov[1,1])

yfit = myfun(xlots,plsq[0],plsq[1])  # use fit results for a, b, c
    
print('N_o = %.0f +/- %.0f' % (plsq[0], np.sqrt(pcov[0,0])))
print('alpha = %.0f +/- %.0f' % (plsq[1], np.sqrt(pcov[1,1])))









    



N_o = 10231 +/- 238
alpha = 15 +/- 0



In [288]:

    
plt.figure(figsize=(12,6));
plt.xticks(size = 15);
plt.yticks(size = 15);
plt.scatter(thickness*1e2,Ln);
plt.plot(xlots*1e2,yfit);
plt.xlim(0,.1*1e2)
plt.legend(['data','Fit'],loc='best');
#plt.text(0.01,8.2,'$N_o$ = %.0f +/- %.0f' % (plsq[0], np.sqrt(pcov[0,0])),size=17)
#plt.text(0.01,8,'alpha = %.1f +/- %.1f' % (plsq[1], np.sqrt(pcov[1,1])),size=17)
plt.xlabel('Plate Thickness (cm)',size=20);
plt.ylabel('Ln($N_{net}$)',size=20);

plt.savefig('Aluminum')



In [289]:

    
#range lambda = 1/alpha
ry = 1/alpha
#uncertainty dlambda = alpha^-2 dalpha
dry = 1/alpha**2 *ealpha



In [290]:

    
ry*1e2 #This is in cm









    Out[290]:





6.8518298052926623



In [291]:

    
dry*1e2 #cm









    Out[291]:





0.22353858326810663



In [292]:

    
np.log(2)/alpha * 1e2 #cm









    Out[292]:





4.7493265112152088



In [293]:

    
dry*1e2*np.log(2) #cm









    Out[293]:





0.15494513873865268



In [294]:

    
rho = 2.7*1e3 #kg/m^3



In [295]:

    
mu = alpha/rho
mu*1e1









    Out[295]:





0.054054227979258851



In [296]:

    
dmu = ealpha / rho
dmu*1e1









    Out[296]:





0.001763500537155952



In [297]:

    
#from chart 0.53



In [ ]: