Chapter 4: Structure of the Atom

Example 4.1, Page 129

In [2]:
import math

#Variable declaration
m_e = 0.000549;    # Rest mass of an electron, u
m_He = 4.002603;    # Rest mass of a helium, u

#Calculations
M_alpha = m_He - 2*m_e;    # Mass of alpha particle, u
theta_max = 2*m_e/M_alpha;    # Maximum scttering angle for aplha particle, rad

#Result
print "The maximum scattering angle for alpha particle = %5.3f degrees"%(theta_max*180/math.pi)

The maximum scattering angle for alpha particle = 0.016 degrees

Example 4.2, Page 137

In [3]:
import math

#Variable declaration
rho = 19.3;    # Density of gold, g/cc
N_A = 6.02e+023;    # Avogadro's number
N_M = 1;    # Number of atoms per molecule
M_g = 197;    # Gram atomic mass of gold, g/mol
n = rho*N_A*N_M/(M_g*1e-006);    # Number density of gold atoms, atoms/metre-cube
Z1 = 79;    # Atomic number of gold
Z2 = 2;    # Atomic number of He nucleus
t = 1e-006;    # Thickness of the gold foil, m
e = 1.602e-019;    # Charge on an electron, C
k = 9e+009;    # Coulomb constant, N-Sq.m/C^2
theta = 90.;    # Angle of deflection of alpha particle, degrees
K = 7.7;    # Kinetic energy of alpha particles, MeV

#Calculations
f = math.pi*n*t*(Z1*Z2*e**2*k/(2*1.6e-013*K))**2*(1./math.tan(theta*math.pi/180/2))**2; # The fraction of alpha particles deflected

#Result
print "The fraction of alpha particles deflected = %1.0e"%f

The fraction of alpha particles deflected = 4e-05

Example 4.3, Page 138

In [4]:
import math

#Variable declaration
rho = 19.3;    # Density of gold, g/cc
N_A = 6.02e+023;    # Avogadro's number
N_M = 1;    # Number of atoms per molecule
M_g = 197;    # Gram atomic mass of gold, g/mol
Z1 = 79;    # Atomic number of gold
Z2 = 2;    # Atomic number of He nucleus
t = 2.1e-007;    # Thickness of the gold foil, m
e = 1.602e-019;    # Charge on an electron, C
k = 9e+009;    # Coulomb constant, N-Sq.m/C^2
r = 1e-002;    # Distance of the alpha particles from the target, m
theta = 45;    # Angle of deflection of alpha particle, degrees
K = 7.7;    # Kinetic energy of alpha particles, MeV

#Calculations
n = rho*N_A*N_M/(M_g*1e-006);    # Number density of gold atoms, atoms/metre-cube
f = n*t*(Z1*Z2*e**2*k)**2/((r*1.6e-013*K)**2*math.sin(theta*math.pi/180/2)**4*16); # The fraction of alpha particles deflected

#Result
print "The fraction of alpha particles deflected at %d degrees = %3.1e per mm square"%(theta, f/1e+006)

The fraction of alpha particles deflected at 45 degrees = 3.2e-07 per mm square

Example 4.5, Page 139

In [5]:
#Variable declaration
Z1 = 2;    # Atomic number of He nucleus
Z2 = 13;    # Atomic number of aluminium
e = 1.602e-019;    # Charge on an electron, C
k = 9e+009;    # Coulomb constant, N-Sq.m/C^2
K = 7.7;    # Kinetic energy of alpha particles, MeV

#Calculations
r_min = Z1*Z2*e**2*k/(K*1.6e-013);    # Size of the aluminium nucleus, m

#Result
print "The size of the aluminium nucleus = %3.1e m"%r_min

The size of the aluminium nucleus = 4.9e-15 m

Example 4.6, Page 140

In [6]:
#Variable declaration
c = 3.00e+008;    # Speed of light, m/s
r = 0.5e-010;    # Radius of the atom, m
e = 1.6e-019;    # Charge on an electron, C
m_e = 9.11e-031;    # Mass of the electron, kg
k = 9e+009;    # Coulomb constant, N-Sq.m/C^2

#Calculations
v = e*k**(1./2)/(m_e*r)**(1./2);    # Speed of the electron, m/s

#Result
if v < 0.01*c:
    print "The nonrelativistic treatment for calculating speed of the electron = %3.1e m/s is justified"%v

The nonrelativistic treatment for calculating speed of the electron = 2.2e+06 m/s is justified

Example 4.7, Page 146

In [7]:
import numpy 
import math

#Variable declaration
def check_region(lamda):
    if lamda >= 400. and lamda < 700.:
        return  "visible";
    else:
        return "infrared";
    

n_l = 3.;    # Lower electron orbit in Paschen series
n_u = [4, numpy.inf];    # First and limiting upper orbits in Paschen series
R_inf = 1.0974e+007;    # Rydberg constant, per metre

#Calculations&Results
lambda_max = 1./(R_inf*(1./n_l**2-1./n_u[0]**2)*1e-009);    # The longest wavelength in Paschen series, nm
region = check_region(lambda_max);    # Check for the region
print "The maximum wavelength is %d nm and is in the %s region"%(math.ceil(lambda_max), region)
lambda_min = 1./(R_inf*(1./n_l**2-1./n_u[1]**2)*1e-009);    # The shortest wavelength in Paschen series, nm
region = check_region(lambda_min);    # Check for the region
print "The minimum wavelength is %d nm and is also in the %s region"%(lambda_min, region)

The maximum wavelength is 1875 nm and is in the infrared region
The minimum wavelength is 820 nm and is also in the infrared region

Example 4.8, Page 149

In [8]:
#Variable declaration
m_e = 0.0005486;    # Mass of an electron u
m_p = 1.007276;    # Mass of a proton, u
m_d = 2.013553;    # Mass of a deutron, u
m_t = 3.015500;    # Mass of a triton, u
R_inf = 1.0974e+007;    # Rydberg constant, per metre

#Calculations
R_H = 1./(1+m_e/m_p)*R_inf;    # Rydberg constant for hydrogen
R_D = 1./(1+m_e/m_d)*R_inf;    # Rydberg constant for deuterium
R_T = 1./(1+m_e/m_t)*R_inf;    # Rydberg constant for tritium
lambda_H = 1./(R_H*(1./2**2-1./3**2)*1e-009);    # Wavelength of H_alpha line for hydrogen, nm
lambda_D = 1./(R_D*(1./2**2-1./3**2)*1e-009);    # Wavelength of H_alpha line for deuterium, nm
lambda_T = 1./(R_T*(1./2**2-1./3**2)*1e-009);    # Wavelength of H_alpha line for tritium, nm

#Results
print "The wavelength of H_alpha line for hydrogen = %6.2f nm"%lambda_H
print "The wavelength of H_alpha line for deutruim = %6.2f nm"%lambda_D
print "The wavelength of H_alpha line for tritium = %6.2f nm"%lambda_T

The wavelength of H_alpha line for hydrogen = 656.45 nm
The wavelength of H_alpha line for deutruim = 656.27 nm
The wavelength of H_alpha line for tritium = 656.22 nm

Example 4.9, Page 150

In [9]:
import numpy

#Variable declaration
R = 1.0974e+007;    # Rydberg constant, per metre
Z = 3;    # Atomic number of Li
n_l = 1;    # Lower orbit of Li++ ion
n_u = numpy.inf;    # Limiting orbit of Li++ ion

#Calculations
lamda = 1./(Z**2*R*(1/n_l**2-1/n_u**2)*1e-009);    # The shortest wavelength emitted by Li++ ion, nm

#Result
print "The shortest wavelength emitted by Li++ ion = %4.1f nm"%lamda

The shortest wavelength emitted by Li++ ion = 10.1 nm