β
– Decay
The energy spectrum of β – particles is continuous (unlike that of α –
particles which is discrete)
β
– Decay
Difficulties in interpreting observed β - spectrum
(1) Nuclear energy states are discrete. However, the observed β–spectrum is
continuous
(2) The Q–value of a particular β–decay is constant & it determines the
maximum kinetic energy of the β–particle (Em). Thus for a β–particle
β
– Decay
Difficulties in interpreting observed β - spectrum
+
+
→
n
e
p
−
+
→
p
e
n
Since p, n, β–particles all have intrinsic spin 1/2, the above reactions
(3) As supposed, the β–emission is due to conversion of a proton into a
neutron and vice – versa
Since p, n, β–particles all have intrinsic spin 1/2, the above reactions
appears to be a violation of the law of conservation of angular momentum
In a famous letter written in 1930, Wolfgang Pauli attempted to resolve the beta-particle energy conundrum by suggesting that, in addition to electrons and protons, atomic nuclei also contained an extremely light neutral particle, which he called the neutron. He suggested that this "neutron" was also emitted during beta decay (thus accounting for the known missing energy, momentum,
β
– Decay
Pauli’s neutrino hypothesis
β
– Decay
Properties assigned to neutrino
1) It must be electrically neutral, so that the only change in the charge of the nucleus is due to emission of the β – particle.
β
– Decay
Properties assigned to neutrino
3) Intrinsic spin of the neutrino should be 1/2. Since the electron spin is also 1/2, two spin 1/2 particles are emitted during β – decay. Hence the two
together will take away an integral unit of angular momentum – in agreement with the observed change in angular momentum of the nuclei during β – decay
Interested in neutrino? Visit https://en.wikipedia.org/wiki/Neutrino
β
– Decay
How neutrino hypothesis eased the difficulties
Conservation of energy:
The maximum energy available for a particular β–decay is shared by the β–
particle & the neutrino. The sharing occurs in a variable manner that gives rise to observed continuous β–spectrum. The upper limit corresponds to the
case for lowest share (nearly zero) by the neutrino while the lower limit of the
β
– Decay
How neutrino hypothesis eased the difficulties
Conservation of angular momentum:
The spin of neutrino is assigned to be 1/2. This solves the conundrum in conservation of angular momentum.
e
All the above particles carry an intrinsic spin angular momentum 1/2 (in unit of ħ). The spins of the β–particle & the neutrino are oriented in parallel
How neutrino hypothesis eased the difficulties
Conservation of linear momentum:
The neutrino carries some energy and hence some linear momentum. However, it interacts with matter feebly and hence not detected easily unlike the β–particle. This explains why there is apparent violation in linear
momentum. This conundrum can be solved by taking the linear momentum of
β
– Decay
neutrino in account.
β
– Decay
Selection Rules: Allowed Transitions
Nuclear spin I = L + S
A transition occurs from an initial spin – parity state Ii to a final spin state If , change in I is
i
f
I
I
I
=
−
∆
∆
I
=
∆
L
+
∆
S
For allowed transitions
∆
L
=
0
∆
I
=
∆
S
β
– Decay
Selection Rules: Allowed Transitions
Both the β - particle & neutrino are spin – 1/2 particles
β
– Decay
Selection Rules: Allowed Transitions
For S = 0 (anti-parallel alignment of spin of β - particle & neutrino), the spin
change of the nuclear state will be
∆
I
=
∆
S
=
0
Fermi selection ruleFor S = 1 (parallel alignment of spin of β - particle & neutrino), the spin
change of the nuclear state will be
1
I
Gamow – Tellerβ
– Decay
Allowed Transitions: Examples
+
This is allowed Fermi transition (pure) Example 1
β
– Decay
Allowed Transitions: Examples
Spin and parity of 14Co is 5+ ; Spin – parity of 60Ni is 4+
No change in parity & Ii = 5, If = 4 hence ∆I = – 1
This is allowed Gamow - Teller transition (pure) Example 3 6027
Co
→
6028Ni
+
e
−Spin and parity of n is 1/2+ ; Spin – parity of 1H is 1/2+
No change in parity & Ii = 1/2, If = 1/2 hence ∆I = 0
β
– Decay
Selection Rules: Summary
Spin and parity of 3H is 1/2+ ; Spin – parity of 3He is 1/2+
No change in parity & Ii = 1/2, If = 1/2 hence ∆I = 0
This is both allowed Fermi & Gamow - Teller transition Example 5 31
H
→
23He
+
e
−Kurie plot: Fermi’s theory of β – decay based on Pauli’s neutrino hypothesis
yields the momentum distribution of the β – particles as follows
β
where pβand E β are respectively the momentum and kinetic energy of the β–
particles, Em is the maximum kinetic energy, A is a constant & F(Z, pβ) is
β
– Decay
called Fermi function
Kurie plot
A plot of this equation is known as Kurie plot or Fermi – Kurie plot. It is a straight line (for allowed transitions). It helps to find the limit on the effective mass of a neutrino
γ
– Decay
Gamma ray (or gamma radiation) is penetrating electromagnetic radiation of a kind arising from the radioactive decay of atomic nuclei.
It consists of photons in the highest observed range of photon energy.
Paul Villard, a French chemist and physicist, discovered gamma radiation in 1900 while studying radiation emitted by radium.
γ
– Decay
Wavelength:
<0.06 Å
Frequency:
> 5 ×10
19Hz
γ
+
→
∗
X
X
AZ A
Z
γ
– Decay
Interaction with matter
When a gamma ray passes through matter, the probability for absorption is proportional to the
(a) thickness of the layer (b) density of the material &
(c) absorption cross section of the material
The total absorption shows an exponential decrease of intensity with distance from the incident surface
where x is the thickness of the material from the incident surface, μ = nσ is
the absorption coefficient, measured in cm−1, n the number of atoms per
cm3 of the material (atomic density) and σ the absorption cross section in cm2
x
e
I
x
γ
– Decay
Interaction with matter
As it passes through matter, gamma radiation ionizes via three processes:
(a) Photoelectric effect
(b) Compton scattering
γ
– Decay
Photoelectric Effect
This describes the case in which a gamma photon interacts with and transfers its energy to an atomic electron, causing the ejection of that electron from the atom.
The kinetic energy of the resulting photoelectron is equal to the energy of the incident gamma photon minus the energy that originally bound the electron to the atom (binding energy).
γ
– Decay
Compton Effect
This is an interaction in which an incident gamma photon looses enough energy to an atomic electron to cause its ejection, with the remainder of the original photon's energy emitted as a new, lower energy gamma photon whose emission direction is different from that of the incident gamma photon (hence the term "scattering")
Probability of Compton scattering decreases with increasing photon energy Compton scattering is thought to be the principal absorption mechanism for gamma rays in the intermediate energy range 100 keV to 10 MeV.
γ
– Decay
Pair Production
Pair production is a phenomenon of paramount interest which involves the creation of a particle & its antiparticle from the interaction of a photon with mater
The phenomenon becomes significant for high energetic photons having energy greater than the combined rest mass of the particle – antiparticle pair
γ
– Decay
Pair Production
The reaction involves the production of particle – antiparticle pair from a photon and hence it satisfies the conservation of all types of charges (like electric charge, lepton charge etc.)
γ
– Decay
Pair Production
Dirac’s Theory
An electron can have both positive and negative energy states
In the physical world, the electron can only be found in the positive energy state. The electron in the negative energy state does not normally manifest
4
state. The electron in the negative energy state does not normally manifest their existence
By Dirac’s theory, all the negative energy states are completely filled by
γ
– Decay
Dirac’s Theory
A vacant state can be created if an electron, from the negative energy state, jumps into the positive energy state by absorption of sufficient electromagnetic energy
Such a vacant state in the negative energy state is interpreted as a positron, the antiparticle of electron. Thus an electron – positron pair can be formed
If the energy is gamma ray is greater than 1.022 MeV, the excess energy
γ
– Decay
The total absorption coefficient of Al13 for gamma rays, plotted versus gamma
energy, and the contributions by the three effects. As is usual, the photoelectric effect is largest at low energies, Compton scattering dominates at intermediate energies, and pair production dominates at high energies.
