Chapter 12: The Atomic Nuleus

Example 12.1, Page 432

In [1]:
import math

#Variable declaration
h = 6.62e-034;    # Planck's constant, Js
h_bar = h/(2*math.pi);    # Reduced Planck's constant, Js
m = 1.67e-027;    # Rest mass of proton, kg
e = 1.602e-019;    # Energy equivalent of 1 eV, J
c = 3.00e+008;    # Speed of light, m/s
delta_x = 8.0e-015;    # Size of the nucleus, m

#Calculations
delta_p = h_bar/(2*delta_x*e);    # Uncertainty in momentum of proton from Heisenberg Uncertainty Principle, eV-s/m
p_min = delta_p;    # Minimum momentum of the proton inside the nucleus, eV-s/m
K = (p_min*c)**2*e/(2*m*c**2*1e+006);    # The minimum kinetic energy of the proton in a medium sized nucleus, MeV

#Result
print "The minimum kinetic energy of the proton in a medium sized nucleus = %4.2f MeV"%K

The minimum kinetic energy of the proton in a medium sized nucleus = 0.08 MeV

Example 12.2, Page 436

In [3]:
import math

#Variable declaration
c = 3.00e+008;    # Speed of light, m/s
e = 1.602e-019;    # Energy equivalent of 1 eV, J
m_e = 0.511;    # Rest mass energy of electron, MeV
m_p = 938.3;    # Rest mass energy of proton, MeV
h = 6.62e-034;    # Planck's constant, Js
A = 40;    # Mass number of Ca-40
r0 = 1.2;    # Nuclear radius constant, fm

#Calculations&Results
R = r0*A**(1./3);    # Radius of Ca-40 nucleus, fm
print "The radius of Ca-40 nucleus = %3.1f fm"%R
lamda = 2.0;    # de-Broglie wavelength to distinguish a distance at least half the radius, fm
# Electron energy
E = math.ceil(math.sqrt(m_e**2+(h*c/(lamda*e*1e+006*1e-015))**2));    # Total energy of the probing electron, MeV
K = E - m_e;    # Kinetic energy of probing electron, MeV
print "The kinetic energy of probing electron = %3d MeV"%math.ceil(K)
# Proton energy
E = math.ceil(math.sqrt(m_p**2+(h*c/(lamda*e*1e+006*1e-015))**2));    # Total energy of the probing electron, MeV
K = E - m_p;    # Kinetic energy of probing electron, MeV
print "The kinetic energy of probing proton = %3d MeV"%math.ceil(K)

#answers differ due to rounding errors

The radius of Ca-40 nucleus = 4.1 fm
The kinetic energy of probing electron = 620 MeV
The kinetic energy of probing proton = 187 MeV

Example 12.3, Page 437

In [4]:
#Variable declaration
A_U238 = 238;    # Mass number of U-238
A_He4 = 4;    # Mass number of He-4
r0 = 1.2;    # Nuclear radius constant, nm

#Calculations&Results
R_U238 = r0*A_U238**(1./3);    # Radius of U-238 nucleus, fm
R_He4 = r0*A_He4**(1./3);    # Radius of He-4 nucleus, fm
print "The radii of U-238 and He-4 nuclei are %3.1f fm and %3.1f fm repectively"%(R_U238, R_He4)
ratio = R_U238/R_He4;    # Ratio of the two radii
print "The ratio of radius to U-238 to that of He-4 = %3.1f"%ratio

The radii of U-238 and He-4 nuclei are 7.4 fm and 1.9 fm repectively
The ratio of radius to U-238 to that of He-4 = 3.9

Example 12.4, Page 438

In [5]:
#Variable declaration
h = 6.62e-034;    # Planck's constant, Js
c = 3.00e+008;    # Speed of light, m/s
e = 1.602e-019;    # Energy equivalent of 1 eV, J
B = 2.0;    # Applied magnetic field, T
mu_N = 3.15e-008;    # Nucleon magnetic moment, eV/T

#Calculations
mu_p = 2.79*mu_N;    # Proton magnetic moment, eV/T
delta_E = 2*mu_p*B;    # Energy difference between the up and down proton states, eV
f = delta_E*e/h;    # Frequency of electromagnetic radiation that flips the proton spins, Hz
lamda = c/f;    # Wavelength of electromagnetic radiation that flips the proton spins, m

#Results
print "The energy difference between the up and down proton states = %3.1e eV"%delta_E
print "The frequency of electromagnetic radiation that flips the proton spins = %2d MHz"%(f/1e+006)
print "The wavelength of electromagnetic radiation that flips the proton spins = %3.1f m"%lamda

The energy difference between the up and down proton states = 3.5e-07 eV
The frequency of electromagnetic radiation that flips the proton spins = 85 MHz
The wavelength of electromagnetic radiation that flips the proton spins = 3.5 m

Example 12.5, Page 443

In [6]:
#Variable declaration
c = 3.00e+008;    # Speed of light, m/s
e = 1.602e-019;    # Energy equivalent of 1 eV, J
u = 931.5;    # Energy equivalent of 1 amu, MeV
m_n = 1.008665;    # Mass of a neutron, u
M_H = 1.007825;    # Mass of a proton, u
M_He = 4.002603;    # Mass of helium nucleus
M_Be = 8.005305;    # Mass of Be-8, u

#Calculations&Results
B_Be = (4*m_n+4*M_H - M_Be)*u;
B_Be_2alpha = (2*M_He - M_Be)*u;
print "The binding energy of Be-8 = %4.1f MeV and is positive"%B_Be
if (B_Be_2alpha > 0):
    print "The Be-4 is stable w.r.t. decay into two alpha particles.";
else:
    print "The Be-4 is unstable w.r.t. decay into two alpha particles since the decay has binding energy of %5.3f MeV"%B_Be_2alpha     

The binding energy of Be-8 = 56.5 MeV and is positive
The Be-4 is unstable w.r.t. decay into two alpha particles since the decay has binding energy of -0.092 MeV

Example 12.6, Page 444

In [7]:
#Variable declaration
Z = 92;    # Atomic number of U-238
A = 238;    # Mass number of U-238

#Calculations
E_Coul = 0.72*Z*(Z-1)*A**(-1./3);    # Total Coulomb energy of U-238, MeV

#Result
print "The total Coulomb energy of U-238 = %3d MeV"%E_Coul

The total Coulomb energy of U-238 = 972 MeV

Example 12.8, Page 447

In [8]:
#Variable declaration
u = 931.5;    # Energy equivalent of 1 amu, MeV
m_p = 1.007825;    # Mass of proton, amu
m_n = 1.008665;     # Mass of neutron, amu
M_Ne = 19.992440; # Mass of Ne-20 nucleus, amu
M_Fe = 55.934942;    # Mass of Fe-56 nucleus, amu
M_U = 238.050783;    # Mass of U-238 nucleus, amu
A_Ne = 20;    # Mass number of Ne-20 nucleus
A_Fe = 56;    # Mass number of Fe-56 nucleus
A_U = 238;    # Mass number of U-238 nucleus

#Calculations
BE_Ne = (10*m_p+10*m_n-M_Ne)*u;    # Binding energy of Ne-20 nucleus, MeV
BE_Fe = (26*m_p+30*m_n-M_Fe)*u;    # Binding energy of Fe-56 nucleus, MeV
BE_U = (92*m_p+146*m_n-M_U)*u;    # Binding energy of U-238 nucleus, MeV

#Results
print "The binding energy per nucleon for Ne-20 : %4.2f MeV/nucleon"%(BE_Ne/A_Ne)
print "The binding energy per nucleon for Fe-56 : %4.2f MeV/nucleon"%(BE_Fe/A_Fe)
print "The binding energy per nucleon for U-238 : %4.2f MeV/nucleon"%(BE_U/A_U)

The binding energy per nucleon for Ne-20 : 8.03 MeV/nucleon
The binding energy per nucleon for Fe-56 : 8.79 MeV/nucleon
The binding energy per nucleon for U-238 : 7.57 MeV/nucleon

Example 12.10, Page 451

In [9]:
import math

#Variable declaration
N_A = 6.023e+023;    # Avogadro's number
T = 138*24*3600;    # Half life of Po-210, s
R = 2000;    # Activity of Po-210, disintegrations/s
M = 0.210;    # Gram molecular mass of Po-210, kg

#Calculations
f = 1/3.7e+010;    # Conversion factor to convert from decays/s to Ci
A = R*f/1e-006;    # Activity of Po-210, micro-Ci
N = R*T/math.log(2);    # Number of radioactive nuclei of Po-210, nuclei
m = N*M/N_A;    # Mass of Po-210 sample, kg

#Results
print "The activity of Po-210 = %5.3f micro-Ci"%A
print "The mass of Po-210 sample = %3.1e kg"%m

The activity of Po-210 = 0.054 micro-Ci
The mass of Po-210 sample = 1.2e-14 kg

Example 12.11, Page 452

In [10]:
import math

#Variable declaration
T = 110;    # Half life of F-18, min
f_remain = 0.01;    # Fraction of the F-18 sample remained

#Calculations
t = -math.log(0.01)/(math.log(2)*60)*T;    # Time taken by the F-18 sample to decay to 1 percent of its initial value, h

#Result
print "The time taken for 99 percent of the F-18 sample to decay = %4.1f h"%t

The time taken for 99 percent of the F-18 sample to decay = 12.2 h

Example 12.12, Page 452

In [11]:
import math

#Variable declaration
N_A = 6.023e+023;    # Avogadro's number
M = 10e+03;    # Mass of the U-235, g
M_U235 = 235;    # Molecular mass of U-235, g
t_half = 7.04e+008;    # Half life of U-235, y

#Calculations
N = M*N_A/M_U235;    # Number of U-235 atoms in 10 kg sample
R = math.log(2)*N/t_half;    # The alpha activity of 10 kg sample of U-235, decays/y

#Result
print "The alpha activity of 10 kg sample of U-235 = %3.1e Bq"%(R/(365.25*24*60*60))

The alpha activity of 10 kg sample of U-235 = 8.0e+08 Bq

Example 12.13, Page 453

In [14]:
#Variable declaration
u = 931.5;    # Energy equivalent of 1 u, MeV
M_U230 = 230.033927;    # Atomic mass of U-230, u
m_n = 1.008665;    # Mass of a neutron, u
M_H = 1.007825;    # Mass of hydrogen, u
M_U229 = 229.033496;    # Gram atomic mass of U-230

#Calculations&Results
Q = (M_U230 - M_U229 - m_n)*u;    # Q-value of the reaction emitting a neutron
if (Q < 0):
    print "The neutron decay in this reaction is not possible."
else:
    print "The neutron decay in this reaction is possible."

Q = (M_U230 - M_U229 - M_H)*u;    # Q-value of the reaction emitting a proton
if (Q < 0):
    print "The proton decay in this reaction is not possible."
else:
    print "The proton decay in this reaction is possible."

The neutron decay in this reaction is not possible.
The proton decay in this reaction is not possible.

Example 12.15, Page 461

In [13]:
#Variable declaration
u = 931.5;    # Energy equivalent of 1 u, MeV
M_Fe55 = 54.938298;    # Atomic mass of Fe-55, u
M_Mn55 = 54.938050;    # Atomic mass of Mn-55, u
m_e = 0.000549;    # Mass of the electron, u

#Calculations&Results
Q = (M_Fe55 - M_Mn55 - 2*m_e)*u;    # Q-value of the reaction undergoing beta+ decay, MeV
if (Q < 0):
    print "The beta+ decay is not allowed for Fe-55"
else:
    print "The beta+ decay is allowed for Fe-55"

Q = (M_Fe55 - M_Mn55)*u;    # Q-value of the reaction undergoing electron capture, MeV
if (Q < 0):
    print "Fe-55 may not undergo electron capture"
else:
    print "Fe-55 may undergo electron capture"

The beta+ decay is not allowed for Fe-55
Fe-55 may undergo electron capture

Example 12.16, Page 462

In [15]:
#Variable declaration
def check_allowance(Q, decay_type):
    if (Q < 0):
	print "The %s is not allowed for Ac-226"%decay_type
    else:
	print "The %s is allowed for Ac-226"%decay_type


u = 931.5;    # Energy equivalent of 1 u, MeV
M_Ac226 = 226.026090;    # Atomic mass of Ac-226, u
M_Fr222 = 222.017544;    # Atomic mass of Fr-222, u
M_He4 = 4.002603;    # Atomic mass of He-4, u
M_Th226 = 226.024891;    # Atomic mass of M_Th226, u
M_Ra226 = 226.025403;    # Atomic mass of M_Ra226, u
m_e = 0.000549;    # Mass of the electron, u

#Calculations
# Alpha Decay
Q = (M_Ac226 - M_Fr222 - M_He4)*u;    # Q-value of the reaction undergoing alpha decay, MeV
check_allowance(Q, "alpha decay");
# Beta- Decay
Q = (M_Ac226 - M_Th226)*u;    # Q-value of the reaction undergoing beta- decay, MeV
check_allowance(Q, "beta- decay");
# Beta+ Decay
Q = (M_Ac226 - M_Ra226 - 2*m_e)*u;    # Q-value of the reaction undergoing beta+ decay, MeV
check_allowance(Q, "beta+ decay");
# Electron Capture
Q = (M_Ac226 - M_Ra226)*u;    # Q-value of the reaction undergoing electron capture, MeV
check_allowance(Q, "electron capture");

The alpha decay is allowed for Ac-226
The beta- decay is allowed for Ac-226
The beta+ decay is not allowed for Ac-226
The electron capture is allowed for Ac-226

Example 12.17, Page 463

In [16]:
#Variable declaration
u = 931.5;    # Energy equivalent of 1 u, MeV
E_ex = 0.072;    # Energy of the excited state, MeV

#Calculations
M = 226*u*1e+0;    # Energy equivalent of atomic mass of Th-226, MeV
x = E_ex/(2*M);    # The error introduced in the gamma ray energy by approximation

#Result
print "The error introduced in the gamma ray energy by approximation = %3.1e"%x

The error introduced in the gamma ray energy by approximation = 1.7e-07

Example 12.18, Page 467

In [17]:
import math

#Variable declaration
t_half = 4.47e+009;    # The half life of uranium ore, years
R_prime = 0.60;    # The ratio of Pb206 abundance to that of U238

#Calculations
t = t_half/math.log(2)*math.log(R_prime + 1);    # Age of the uranuim ore, years

#Result
print "The age of U-238 ore = %3.1e years"%t

The age of U-238 ore = 3.0e+09 years

Example 12.19, Page 469

In [18]:
import math

#Variable declaration
t_half = 5730;    # The half life of uranium ore, years
R0 = 1.2e-012;    # The initial ratio of C14 to C12 at the time of death
R = 1.10e-012;    # The final ratio of C14 to C12 t years after death

#Calculations
# As R = R0*math.exp(-lambda*t), solving for t
t = -math.log(R/R0)*t_half/math.log(2);    # Age of the bone, years

#Result
print "The %3d years age of bone does not date from the Roman Empire."%math.ceil(t)

The 720 years age of bone does not date from the Roman Empire.