NEWTON LAWS OF MOTION
FRICTION
VELOCITY AND ACCELERATION
CENTRAL FORCES
UNIFORMLY ROTATING FRAME- CENTRIFUGAL AND CORIOLIS FORCES
CONSERVATION LAWS
CENTRE OF MASS AND VRIABLE MASS SYSTEMS
RIGID BODY DYNAMICS
FLUID DYNAMICS
COULOMB LAW AND ELECTRIC FIELD
GAUSS LAW OF ELECTROSTATICS AND APPLICATIONS
POLARIZATION OF DIELECTRICS
WORK AND ENERGY IN ELECTROSTATICS
BOUNDARY VALUE PROBLEMS
CURRENT ELECTRICITY
MAGNETOSTATICS
FARADAY LAW OF ELECTROMAGNETIC INDUCTION
MAGNETIC MATERIALS
DC CIRCUITS
AC CIRCUITS
MAXWELL EQUATIONS and poynting vector
ELECTROMAGNETIC WAVES
REFLECTION AND REFRACTION OF EM WAVES AT THE INTERFACE OF TWO DIELECTRICS
Section 3: MATHEMATICAL PHYSICS
MULTIPLE INTEGRAL
VECTOR CALCULUS
DIFFERENTIAL EQUATIONS
MATRICES
DIFFERENTIAL CALCULUS
FOURIER SERIES
PARTICLE NATURE OF WAVE
WAVE NATURE OF PARTICLE
H ATOM
POSTULATES OF QUANTUM MECHANICS
SCHRONDINGER WAVE EQUATION
NUCLEAR PHYSICS
SPECIAL THEORY OF RELATIVITY
SIMPLE HARMONIC OSCILLATION
DAMPED AND FORCED OSCILLATION
WAVES
GEOMETRICAL OPTICS
INTERFERENCE
DIFFRACTION
POLARIZATION OF LIGHT
THERMAL EXPANSION
CALORIMETRY
TRANSMISSION OF HEAT
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Basic Nuclear Properties

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(a) Charge : Charge of the nucleus = +Ze
Where e is the charge on the electron =  1.6 \times 10^{-19} coulomb
(b) Mass : Approximately equal to

 Z = (A- Z)m_n + Zm_p

Where  m_n &  m_p are mass of neutron & proton  m_n \approx 1840 m_e, \ m_p \approx 1836 m_e

(c) Size : The volume is proportional to total number of nucleons in it

 \dfrac{4}{3} \pi R^3 \propto A

 \Rightarrow \ \ \ \ R \propto A^{1/3}

So, radius of nuclear  R = R_\circ A^{1/3} where  R_\circ = 1.1 - 1.2 \mathrm{fm} (1 \mathrm{fm} = 10^{-15}m)

Nuclear surface is defined as the surface outside which there is negligible probability of finding any of the nuclear constituents. The electric charge distribution within the nucleus is approximately same as nuclear mass distribution. The electrical radius of the nucleus and nuclear matter radius are approximately the same.

Distriabution of charge =  \dfrac{\rho_{ch} (r)}{\rho_\circ}

 \rho_{ch}(r) = \dfrac{\rho_\circ}{1 + \exp\left [ \dfrac{(r - R)}{a} \right ]} where, a = 0.5 fm

(d) Density

Nuclear density,  \rho_N = \dfrac{\mathrm{Nuclear \ mass}}{\mathrm{Nuclear \ volume}}

Nuclear mass =  Am_n

Where A = mass number

 m_n = mass of nucleon

 = 1.67 \times 10^{-27} kg

Nuclear volume =  = \dfrac{4}{3} \pi R^3

 = \dfrac{4}{3} \pi (R_\circ A^{1/3})^3

 = \dfrac{4}{3} \pi R_{\circ}^{3} A

 \rho_N = \dfrac{Am_n}{\dfrac{4}{3} \pi R_{\circ}^{3} A} = \dfrac{m_n}{\dfrac{4}{3} \pi R_{\circ}^{3}} = 2 \times 10^{17} kg/m^3

As the density of the nucleus is independent of A, its value is almost same for all nuclei

(e) Angular momentum

(i) Spin angular momentum : It is due to particles spinning motion about its own axis through centre of mass magnitude of spin angular momentum  \left| {\vec S} \right| = \sqrt{{\not S(\not S + 1)}}\hbar where  \not S = \dfrac{1}{2} is the spin angular momentum quantum number.
Z-component of angular momentum is  S_Z = m_s \hbar , where  m_s is magnetic spin quantum number  m_s = \pm \dfrac{1}{2}
+ ve for spin axis parallel to Z-axis
– ve for anti parallel spin

 S_z = \pm \dfrac{1}{2} \hbar

(ii) Orbital Angular Momentum : Each individual nucleon may be pictured as having as angular momentum associated with orbital motion. This is called orbital angular momentum. Its magnitude is  |\vec l| = \sqrt{l(l + 1)\hbar}, where l is orbital angular momentum quantum number l = 0, 1, 2, ….

Z-component of orbital angular momentum

 L_z = m_l \hbar where  m_l is orbital magnetic quantum number  m_l = -l, -(l - 1),\cdots 1, -1, 0, 1, \cdots (l - 1), l

(iii) Total angular momentum : Total angular momentum of a nucleus J is vector sum of its orbital & spin angular momenta

 \vec j = \vec l + \vec s

 \left | \vec J \right | = \sqrt{j(j + 1) \hbar}

where j is total angular momentum quantum number and for both proton & neutron,  S = \dfrac{1}{2}

So, J is always half integral

Z component of total angular momentum  J_z = m_j \hbar where  m_j is total magnetic angular momentum quantum number  m_j = j,(j - 1), \cdots -(j - 1), -j

Total angular momentum of nucleus

Total angular momentum of the nucleus  \vec I is the resultant of the individual total angular momentum of all the constituent nucleons in the nucleus.

Its magnitude  \vec I = \sqrt{l(l + 1)\hbar} where I is total angular momentum quantum number for the nucleus. The value of I depends on the type of interaction or coupling in the nucleus.

(f) Nuclear Spin

Nuclear magnetic moment is proportional to angular momentum

 \vec \mu_I = \dfrac{g_I \mu_N}{\hbar} \vec I

Where g is called nuclear g factor

 \mu_I = g_I \mu_N \sqrt{I(I + 1)}

where  \mu_N is called the nuclear magneton

 \mu_B = 5.79 \times 10^{-5} eV/T

 = 9.27 \times 10^{-24} J/T

 \dfrac{\mu_n}{\mu_B} = \dfrac{m_e}{m_p} = \dfrac{1}{1836}

 \mu = 3.15 \times 10^{-8} ev/\tau

 = 5.05 \times 10^{-27} J/T

Nuclear g-factor varies from nucleus to nucleus

 \mu_{\mathrm{proton}} = \pm 2.792 \mu_N in Z-direction

 \mu_{\mathrm{neutron}} = \mp 1.913 \mu_N

 (\pm)  sign used for proton because  \mu_{pz} || \vec S

 (\mp) sign used for neutron because  \mu_{nz} is anti-parallel to  \vec S

Negative nuclear magnetic moment of the neutron indicates that negative charge on the average is father from the axis.

The total angular momentum of the whole atom

 \vec F = \vec L + \vec S + \vec I

 = \vec J + \vec I

Vector  \vec I &  \vec J process around their resultant  \vec F

(g) Quadrupole moment :

Quadrupole moment q is a measure of the departure from spherical symmetry of the nuclear charge distribution.
The electric quadrupole moment of a nuclear charge distribution which is symmetric about axis is given by

 q = \int_{v} (3z^2 - r^2) \rho(x, y, z)d \tau where  \rho is average nuclear charge density in terms of terms of proton charges &  r^2 = x^2 + y^2 + z^2 for uniformly charged ellipsoid of revolution defined by the equation

 \dfrac{x^2 + y^2}{a^2} + \dfrac{z^2}{b^2} = 1

The electric quadrupole moment reduces to  q = \dfrac{2Ze}{5} (b^2 - a^2) where Ze is total nuclear charge

q = 0 for spherically symmetric charge distribution i.e. a = b

q > 0 for charged distribution stretched in z-direction (b > a) i.e. Oblate spheroid.

q < 0 for charged distribution stretched in perpendicular to z-direction (b < a) i.e. Oblate spheroid.

(h) Wave Mechanical Properties

Nucleus has two have wave mechanical properties

(i) Statistics (ii) parity

(i) Statistics : Quantum mechanical description of system with number of particles like nucleus is given by either Bose – Einstein or Fermi – Dirac Statistics.

Bose Einstein Statistics : All the particles with integral spin (in unit of  \hbar ) or zero obey B.E. statistics & are called bosons e.g. Photon  \pi -meson, deuteron

All nuclei with even mass number A obey B.E. statistics

The wave function of system obeying B.E. statistics is symmetric.

Fermi-Dirac Statistics : The particle with half integral spin  (\hbar/2) obey Fermi – dirac (F.D.) statistics, are called fermions eg. electrons, proton & neutrons.

All the nuclei with odd mass number A obey F.D. statistics.

The wave functions of system obeying F.D. statistics are anti-symmetric. This means that if all the co-ordinates of any pair of identical particles are intercharged, the new system will be identical with the original except for the charge of sign in wave function  \psi(x_1, x_2,\cdots x_i, x_j, \cdots, x_n) = -(x_1, x_2 \cdots x_j, x_i, \cdots x_n)

All fermions obey Pauli-exclusion principle.

(ii) Parity : The parity of a system refers to the behaviour of the wave function  \psi under inversion of co-ordinates through the origin.

The particle is said to have been parity or positive parity

If  \psi (x, y, z) = \psi(-x, -y, -z)

& odd parity or negative parity

If  \psi(x, y, z) = P \psi (-x, -y, -z)

In general,  \psi(x, y, z) = P \psi(-x, -y, -z)

“P” can be taken as quantum number and property defined by it is called parity of the system

P = +1 even parity

P = –1 odd parity

In case of hydrogen like atoms.

 P = (-1)^l where l is the orbital quantum number.

The intrinsic parity of the proton, neutron, neutrino &  \mu meson is even where as intrinsic parity of  \pi meson is odd.

The parity of a whole system is the product of parities of individual particles. Parity remain conserved in nuclear reaction.

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