# Chapter4-Detection and Measurement of Nuclear Reactions¶

## Ex1-pg178¶

In :
##  Exa4.1 : : Page 178 (2011)
#find The resultant pulse height recorded in the fission chamber
import math
N = 200e+006/35.;    ## Total number of ion-pairs
e = 1.60218e-019;     ## Charge of an ion, coulomb
Q = N*e;    ## Total charge produced in the chamber, coulomb
C = 25e-012;     ## Capacity of the collector, farad
V = Q/C;     ## Resultant pulse height, volt
print'%s %.2e %s'%("\nThe resultant pulse height recorded in the fission chamber = ",V," volt");

## Result
## The resultant pulse height recorded in the fission chamber = 3.66e-002 volt

The resultant pulse height recorded in the fission chamber =  3.66e-02  volt


## Ex2-pg178¶

In :
##  Exa4.2 : : Page 178 (2011)
#find The energy of the alpha particles
import math;
V = 0.8/4.;    ## Pulse height, volt
e = 1.60218e-019;     ## Charge of an ion, coulomb
C = 0.5e-012;     ## Capacity of the collector, farad
Q = V*C;    ## Total charge produced, coulomb
N = Q/e;     ## Number of ion pairs
E_1 = 35.;    ## Energy of one ion pair, electron volt
E = N*E_1/10**6;    ## Energy of the alpha particles, mega electron volt
print'%s %.2f %s'%("\nThe energy of the alpha particles = ",E," MeV");

## Result
## The energy of the alpha particles = 21.845 MeV (The answer is wrong in the textbook)

The energy of the alpha particles =  21.85  MeV


## Ex3-pg178¶

In :
##  Exa4.3 : : Page 178 (2011)
#find Total number of ion pairs produced and Total charge flow in the counter
import math
E = 10e+06;    ## Energy produced by the ion pairs, electron volts
N = E/35.;        ## Number of ion pair produced
m = 10**3;        ## Multiplication factor
N_t = N*m;        ## Total number of ion pairs produced
e = 1.60218e-019;  ## Charge of an ion, coulomb
Q = N_t*e;         ## Total charge flow in the counter, coulomb
t = 10**-3;        ## Pulse time, sec
R = 10**4;        ## Resistance , ohm
I = Q/t;        ## Current passes through the resistor, ampere
V = I*R;        ## Height of the voltage pulse, volt
print'%s %.2f %s %.2e %s %.2e %s '%("\nTotal number of ion pairs produced: ",N_t,""and " \nTotal charge flow in the counter : ",Q," coulomb"and "\nHeight of the voltage pulse :",V," volt")

## Result
## Total number of ion pairs produced: 2.857e+008
## Total charge flow in the counter : 4.578e-011 coulomb
## Height of the voltage pulse : 4.578e-004 volt

Total number of ion pairs produced:  285714285.71  4.58e-11
Height of the voltage pulse : 4.58e-04  volt


## Ex4-pg178¶

In :
##  Exa4.4 : : Page 178 (2011)
#find The radial electric field and The life of the G.M. Counter
import math;
V = 1000.;    ## Operating voltage of Counter, volt
x = 1e-004;        ## Time taken, sec
b = 2.;            ## Radius of the cathode, cm
a = 0.01;        ## Diameter of the wire, cm
E_r = V/(x*math.log(b/a)); ## Radial electric field, V/m
C = 1e+009;              ## Total counts in the GM counter
T = C/(50.*60.*60.*2000.); ## Life of the G.M. Counter, year
print'%s %.2f %s %.2f %s '%("\nThe radial electric field: ",E_r,"V/m"and"\nThe life of the G.M. Counter : ",T," years");

## Result
## The radial electric field: 1.89e+006V/m
## The life of the G.M. Counter : 2.778 years

The radial electric field:  1887391.66
The life of the G.M. Counter :  2.78  years


## Ex5-pg178¶

In :
##  Exa4.5 : : Page 178 (2011)
import math;
#find The avalanche voltage in G.M. tube
I = 15.7;    ## Ionisation potential of argon, eV
b = 0.025;            ## Radius of the cathode, metre
a = 0.006e-02;        ## Radius of the wire, metre
L = 7.8e-06;     ## Mean free path, metre
V = round(I*a*math.log(b/a)/L);    ## Avalanche voltage in G.M. tube, volt
print'%s %.2f %s'%("\nThe avalanche voltage in G.M. tube = ",V," volt");

## Result
## The avalanche voltage in G.M. tube = 729 volt

The avalanche voltage in G.M. tube =  729.00  volt


## Ex6-pg179¶

In :
##  Exa4.6 :  : Page 179 (2011)
#find The voltage fluctuation GM tube
import math;
C_r = 0.1e-02;     ## Counting rate of GM tube
S = 3.;             ## Slope of the curve
V = C_r*100*100/S;       ## Voltage fluctuation, volt
print'%s %.2f %s'%("\nThe voltage fluctuation GM tube = ",V," volt");

## Result
## The voltage fluctuation GM tube = 3.33 volt

The voltage fluctuation GM tube =  3.33  volt


## Ex7-pg179¶

In :
##  Exa4.7 : : Page-179 (2011)
#find Time measurement for actual count and Time measurement for backward count
import math;
R_t = 100.;        ## Actual count rate, per sec
R_B = 25.;         ## Backward count rate, per sec
V_S = 0.03;        ## Coefficient of variation
R_S = R_t-R_B;        ## Source counting rate,per sec
T_t = (R_t+math.sqrt(R_t*R_B))/(V_S**2*R_S**2);   ## Time measurement for actual count, sec
T_B = T_t*math.sqrt(R_B/R_t);       ## Time measurement for backward count, sec
print'%s %.2f %s  %.2f %s '%("\nTime measurement for actual count : ",T_t," sec"and " \nTime measurement for backward count : ",T_B," sec");

## Result
##  Time measurement for actual count : 29.630 sec
##  Time measurement for backward count : 14.8 sec

Time measurement for actual count :  29.63
Time measurement for backward count :   14.81  sec


## Ex8-pg179¶

In :
##  Exa4.8 : : Page-179 (2011)
#find The capacitance of the detector and The capacitance of the detector
import math;
A = 1.5e-4;           ## Area of capacitor plates, square metre
K = 12.;               ## Dielectric constant
D = K*8.8542e-012;   ## Electrical permittivity of the medium, per newton-metre-square coulomb square
x = 50e-06;           ## Width of depletion layer, metre
C = A*D/x*10**12;      ## Capacitance of the silicon detector, pF
E = 4.5e+06;          ## Energy produced by the ion pairs, eV
N = E/3.5;            ## Number of ion pairs
e = 1.60218e-019;     ## Charge of each ion, coulomb
Q = N*e;              ## Total charge, coulomb
V = Q/C*10**12;        ##  Potential applied across the capacitor, volt
print'%s %.2f %s %.2e %s '%("\nThe capacitance of the detector : ",C," pF"and "\nThe capacitance of the detector : ",V," volt");

## Result
## The capacitance of the detector : 318.75 pF
## The potential applied across the capacitor : 6.46e-004 volt

The capacitance of the detector :  318.75
The potential applied across the capacitor :  6.46e-04  volt


## Ex9-pg180¶

In :
##  Exa4.9 : : Page-180 (2011)
#find The statistical error of the measured ratio
import math
N_A = 1000.;        ## Number of count observed for radiation A
N_B = 2000.;        ## Number of count observed for radiation B
r = N_A/N_B;        ## Ratio of count A to the count B
E_r = math.sqrt(1./N_A+1./N_B);        ## Statistical error
print'%s %.2f %s'%("\nThe statistical error of the measured ratio = ", E_r*r,"");

## Result
## The statistical error of the measured ratio = 0.02 (Wrong answer in the textbook)

The statistical error of the measured ratio =  0.02


## Ex10-pg180¶

In :
##  Exa4.10 : : Page 180 (2011)
#find The charge collected at the anode of photo multiplier tube
import math;
E = 4e+006;        ## Energy lost in the scintillator, eV
N_pe = E/10**2*0.5*0.1;        ## Number of photoelectrons emitted
G = 10**6;                     ## Gain of photomultiplier tube
e = 1.6e-019;                 ## Charge of the electron, C
Q = N_pe*G*e;                 ## Charge collected at the anode of photo multiplier tube, C
print'%s %.4e %s'%("\nThe charge collected at the anode of photo multiplier tube : ",Q," C");

## Result
## The charge collected at the anode of photo multiplier tube : 3.2000e-010 C

The charge collected at the anode of photo multiplier tube :  3.2000e-10  C


## Ex11-pg180¶

In :
##  Exa11 : : Page 180 (2011)
#find Charge collected at the anode of photo multiplier tube
E = 4e+06;        ## Energy lost in the scintillator, eV
N_pe = E/10**2*0.5*0.1;        ## Number of photoelectrons emitted
G = 10**6;                     ## Gain
e = 1.6e-019;                 ## Charge of the electron, C
Q = N_pe*G*e;                    ## Charge collected at the anode of photo multiplier tube, C
print'%s %.4e %s'%("\nCharge collected at the anode of photo multiplier tube : ",Q," C");
## Result
##  Charge collected at the anode of photo multiplier tube : 3.2000e-010 C

Charge collected at the anode of photo multiplier tube :  3.2000e-10  C


## Ex12-pg181¶

In :
##  Exa4.12 : : Page 181 (2011)
#find Standard deviation of the reading
## Defining an array
import math;
import numpy
n = numpy.zeros((6,1)); ## Declare the cell matrix of 1X6
n[0,0] = 10000;
n[1,0]= 10200;
n[2,0] = 10400;
n[3,0] = 10600;
n[4,0] = 10800;
n[5,0] = 11000;
g = 0.; ##
k = 6;
H = 0.;
for i in (0,k-1):
g = g + n[i,0]

N = g/k; ## Mean of the count
D = math.sqrt(N);
for i in range (0,k-1):
H = H+((n[i,0]-N)*(n[i,0]-N))

S_D = round(math.sqrt(H/(k-1)));
print'%s %.2f %s'%("\nStandard deviation of the reading  : ", S_D,"");
delta_N = math.sqrt(N);
if (S_D > delta_N) :
print("\nThe foil cannot be considered uniform..!");
else:
print("\nThe foil can be considered uniform.");

## Result
## Standard deviation of the reading  : 374
## The foil cannot be considered uniform..!

Standard deviation of the reading  :  6906.00

The foil cannot be considered uniform..!


## Ex13-pg181¶

In :
##  Exa4.13 : : Page 181 (2011)
#findNo. of electrons in the output
import math
V = 2e-03;         ## Voltage impulse, volt
C = 120e-012;    ## Capacitance of the capacitor, F
e = 1.6e-019;    ## Charge of the electron, C
n = C*V/(15.*e);     ## No. of electons
N = n**(1/10.);     ## No. of electrons in the output
print'%s %.2f %s'%("\nNo. of electrons in the output : ",N," (approx)");

## Result
## No. of electrons in the output : 3.16 (approx)

No. of electrons in the output :  3.16  (approx)


## Ex14-pg181¶

In :
##  Exa4.14 : : Page 181 (2011)
#find Time of flight of proton and Time of flight of electron
import math
m_p = 0.938;    ## Mass of the proton, GeV
E = 1.4;        ## Total energy of proton, GeV
gama = E/m_p;     ## Boost parameter
bta = math.sqrt(1-1/gama**2);    ## Relativistic factor
d = 10.;        ## Distance between two counters,m
C = 3e+08;     ## Velocity of light ,m/s
t_p = d/(bta*C);    ## Time of flight of proton ,sec
T_e = d/C;        ## Time of flight of electron, sec
print'%s %.2f %s %.2f %s '%("\nTime of flight of proton: ",t_p/1e-009," ns"and " \nTime of flight of electron : ",T_e/1e-009," ns ");

## Result
## Time of flight of proton: 44.90 ns
## Time of flight of electron : 33.33 ns

Time of flight of proton:  44.90
Time of flight of electron :  33.33  ns


## Ex15-pg182¶

In :
##  Exa4.15 : : Page 182 (2011)
#find The fractional error in rest mass of the particle
import math;
p = 100.;        ## Momentum of the particle, GeV
n = 1+1.35e-04;    ## Refractive index of the gas
m_0 = 1.;         ## Mass, GeV per square coulomb
gama = math.sqrt((p**2+m_0**2)/m_0);     ## Boost parameter
bta = math.sqrt (1-1/gama**2);        ## Relativistic parameter
d_theta = 1e-003;    ## Error in the emission angle, radian
theta = math.acos(1/(n*bta));         ## Emision angle of photon, radian
F_err = (p**2*n**2*2*theta*10**-3)/(2*m_0**2);        ## Fractional error
print'%s %.2f %s'%("\nThe fractional error in rest mass of the particle =  ", F_err,"");

## Result
## The fractional error in rest mass of the particle =  0.13

The fractional error in rest mass of the particle =   0.13


## Ex16-pg182¶

In :
##  Exa4.16 : : Page 182 (2011)
#find The total number of quantas during emission of visible light
import math;
u = 1.49;        ## Refractive index
E = 20*1.60218e-019;        ## Energy of the electron, joule
m_e = 9.1e-031;            ## Mass of the electron, Kg
C = 3e-08;                ## Velocity of the light, m/s
bta = (1 + (1/(E/(m_e*C**2)+1))**2 );   ## Boost parameter
z = 1.;         ##
L_1 = 4000e-010;            ## Initial wavelength, metre
L_2 = 7000e-010;            ## Final wavelength, metre
N = 2*math.pi*z**2/137.*(1./L_1-1./L_2)*(1-1./(bta**2*u**2));        ## Number of quanta of visible light, quanta per centimetre
print'%s %.2f %s'%("\nThe total number of quantas during emission of visible light = ", round(N/100),"quanta/cm");

## Result
## The total number of quantas during emission of visible light = 270 quanta/cm

The total number of quantas during emission of visible light =  270.00 quanta/cm