NUCLEAR AND PARTICLE PHYSICS -2
Properties of Nucleus_- I
I-Nuclear angular momentum
L- Nuclear orbital angular moment
S- Nuclear spin angular momentum
Angular momentum- angular momentum of a nucleus is a constant
quantity because nucleus being an isolated system i.e. the external torque is
equal to zero.
I represent angular momentum quantum number. The eigen
values of the square of the angular momentum operator are I(I+1) in the unit of
ℏ
Î2
Ψ = I(I+1) ℏ2 Ψ
Iz is the Z- component of total angular momentum
ÎzΨ = m ℏ Ψ, m
is the eigen values of Îz,
m ℏ is the Z component of angular momentum.
both have common eigen function [
Total angular momentum (I) = ℏ✓{I(I+1)}
For a given value of I, m can take 2I+1
values from +I to –I
I is not only integer but half integer
i.e. I= 0, 1/2, 1, 3/2 …..
Half integer value of I is due to the
Pauli’s proposed intrinsic spin.
Total angular momentum is the sum of
orbital angular momentum and spin angular momentum, I=L + S
Intrinsic spins are integers if the mass
number A is even and half integer when A is odd.
Intrinsic spin ranges from zero as in 4He
and 12C up to 7 as in 176Lu.
The total, orbital and spin angular
momentum of the nuclus,
PI2= I(I+1)
ℏ2
PL2
= L(L+1) ℏ2
PS2=
= S(S+1) ℏ2
Parity:
This refers to the behaviour of a
wave function at the symmetric position. the wave function of a physical system
at (X, Y, Z) is Ψ (X, Y, Z) that of Ψ (-X, -Y, -Z) at (-X, -Y, -Z). The Hamiltonian of
the system is invariant under space invariance then
Ψ
(-X, -Y, -Z) = Ψ (X, Y, Z)
Ψ (-X, -Y,
-Z) = - Ψ (X, Y, Z)
In the first case, the wave
function has even parity, and in the second case, the wave function has odd
parity. Elementary particles may possess intrinsic parity which is related to
the inversion of some internal axis of the particle. The law of conservation of
parity, states that the parity is conserved in a process if the mirror image of
process is also a process which can occur in nature. By convention, the parity
of nucleus is taken as even.
By interchanging of
the coordinates of two particles, if the sign of the wave function does not
change, then the wave function is symmetric and the resulting Quantum
statistics is Bose-Einstein statistics. When the sign of the wave function
changes as a result of interchange of the coordinates, then we get anti-
symmetric wave function and the statistics is Fermi-Dirac statistics.
P Ψ
(X1, X2) = ±
Ψ (-X1, -X2)
P is the parity operator whose
eigen values are +1 and -1.
P= +1 for even parity and P= -1
for odd parity.
Magnetic
dipole moment
The magnetic moment 𝜇
is related to the angular momentum by,
𝜇 = ( e/2m) L and more generalised relation is 𝜇 = g ( e/2m)L
For nucleus, the magnetic moment 𝜇 = g (e/2Mp) I, where I is the total angular momentum.
𝜇= g 𝜇N (I/ℏ), where 𝜇N= ( eℏ/2Mp) is called nuclear magneton.
The measured values of the magnetic moment of proton and
neutron are
𝜇p =
2.7927 𝜇N 𝜇B = 9.27 × 10-24 J/T
𝜇n =
-1.9131 𝜇N 𝜇N = 5.05 × 10-27 J/T
The above values so that the proton and neutron magnetic
moment are of the order 10-3 times the electronic magnetic moment.
The values are unexpected. The expected ones: The protons should have 1 𝜇N and the
neutron should have 0 because neutrons have no charge.
This indicates that both neutron
and proton have non-uniform charge distribution and in the case of neutron the
direction of the magnetic moment is opposite to that of the spin angular
momentum vector. This can be understood from meson's theory. It means that the
sign of electric charge constantly exchanged between nuclear particles. This
helps to contribute to non-zero orbital magnetic moments.
The magnetic moments of protons and neutrons
are related to their intrinsic spin angular momentum as follows.
Spin angular momentum pp = sp ℏ pn=sn ℏ where sp = sn
= ½
For electrons, 𝜇e / pe= ge(e/2Me ) where ge = -2, Lande’s factor
For protons and neutrons,
μp/ pp= gp(e/2Mp ) ; gp =2μp /μN = 2x 2.7927=5.5855
μn/ pn= gn(e/2Mn ) ; gn =2μn /μN = -3.8263
𝜇p = gp(e/2Mp )sp ℏ ; 𝜇p = (gp/2)𝜇N
𝜇n = gn(e/2Mn )sn ℏ ; 𝜇n = (gn/2)𝜇N
These results show that proton and neutron
have a complex structure.
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