Difficulties in interpreting observed β - spectrum
The energy spectrum of β – particles is continuous (unlike that of α – particles which is discrete)
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 (E ). Thus for a β–particle
m
emitted with kinetic energy E emitted with kinetic energy E (<E (<E ), the difference E ), the difference E – E – E can not be can not be
k k m m m m k k
accounted for
Difficulties in interpreting observed β - spectrum
- →
e n p −
e p 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 (4) Experiments shows that the principle of conservation of linear momentum is not obeyed in β–emission
Pauli’s neutrino hypothesis
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, and angular momentum). In 1931, Enrico Fermi renamed Pauli's "neutron" to
neutrino . In 1934, Fermi published his landmark theory for beta decay, where
he applied the principles of relativity to matter particles, supposing that they can be created and annihilated, just as the light quanta in atomic transitions.
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.
2) The mass of neutrino should be zero or very nearly zero. This follows from the fact that the maximum energy E of the emitted electron is
m
equal to the mass energy difference between the parent & daughter equal to the mass energy difference between the parent & daughter nuclei less the rest mass energy of the β – particle.
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 4) Neutrino interacts feebly with matter 5) It obeys Fermi – Dirac statistics
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 –spectrum corresponds to greatest share of the total available energy
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. −- + + + n → p e υ e
- p → n e υ e e
14
- Example 2
- Example 4
- 1
- No change in parity & I = 1/2, I = 1/2 hence ∆I = 0
- Z Z
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 or anti-parallel to each other such that total angular momentum remains conserved
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 neutrino in account.
p A Z p A Z p p ( , ) ( , = + 1 ) + + −
ν A A − β
X Y e Z Z +1 e
→ υ
+ A A + + X → Y e υ + +Selection Rules: Allowed Transitions
Nuclear spin I = L + S A transition occurs from an initial spin – parity state I to a final spin state I , i f change in I is
∆ I = I −
I f i ∆ = ∆ +
I L ∆ S
For allowed transitions
∆ L = ∆ I = ∆ S
Hence no change in parity occurs
Selection Rules: Allowed Transitions
Both the β - particle & neutrino are spin – 1/2 particles Their spins can be aligned either parallel or anti-parallel to each other , so that total spin angular momentum carried by them S = 0 (anti-parallel) or S = 1 (parallel)
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
Fermi selection rule ∆ I = ∆ S =
For S = 1 (parallel alignment of spin of β - particle & neutrino), the spin change of the nuclear state will be
Gamow – Teller
I S ∆ = ∆ =
1 selection rule
I ∆ ∆I = = 0 ± , , ±
1 I = → i f I =
1
Allowed Transitions: Examples
14
14
∗ + Example 1O → N + e
8
7
O is 0 ; Spin – parity of N is 0
14 Spin and parity of
No change in parity & I = 0, I = 0 hence ∆I = 0
i f
This is allowed Fermi transition (pure)
6
6 −
He → Li e
2
3
6
He is 0 ; Spin – parity of Li is 1
6 Spin and parity of
No change in parity & I = 0, I = 1 hence ∆I = +1
i f
Allowed Transitions: Examples
60
60
− Example 3Co → Ni + e
27
28
14
; Spin – parity of Ni is 4
60 Spin and parity of Co is 5
No change in parity & I = 5, I = 4 hence ∆I = – 1
i f
This is allowed Gamow - Teller transition (pure)
1
1
−
n → H e
1
Spin and parity of n is 1/2 ; Spin – parity of H is 1/2
i f
3
3 − Example 5
H → He + e
1
2
3 +
3 +
Spin and parity of H is 1/2 ; Spin – parity of He is 1/2 No change in parity & I = 1/2, I = 1/2 hence ∆I = 0
i f
This is both allowed Fermi & Gamow - Teller transition
Selection Rules: Summary Selection Rules: Summary
Kurie plot: Fermi’s theory of β – decay based on Pauli’s neutrino hypothesis
yields the momentum distribution of the β – particles as follows 2 2 N ( p ) dp AF ( Z , p ) p ( E E ) dE
β β β β β β
= − m where p and E are respectively the momentum and kinetic energy of the β– β β particles, E is the maximum kinetic energy, A is a constant & F(Z, p ) is m β called Fermi function1 / 2 N ( p ) β k = constant k ( E E ) 2 β = − m F ( Z , p ) p β β
Kurie plot 1 / 2 N ( p ) β k = constant k E E 2 = ( − ) m β
F Z p p ( , ) β β
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
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.
In 1903, Ernest Rutherford named this radiation gamma rays. In 1903, Ernest Rutherford named this radiation gamma rays.
A ∗ A X →
X γ
Wavelength: <0.06 Å
19 > 5 ×10 Hz
Frequency: Energy: > 30 keV
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
µ x −
from the incident surface
I ( x ) = I e
where x is the thickness of the material from the incident surface, μ = nσ is
−1
the absorption coefficient, measured in cm , n the number of atoms per
Interaction with matter
As it passes through matter, gamma radiation ionizes via three processes: (a) Photoelectric effect (b) Compton scattering (c) Pair production
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).
The photoelectric effect is the dominant energy transfer mechanism for X- ray and gamma ray photons with energies below 50 keV, but it is much less important at higher energies.
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.
Compton scattering is relatively independent of the atomic number of the
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 The threshold energy of the photon for the production of electron – positron pair is twice the rest mass of electron i.e. 1.022 MeV. Thus the gamma ray having energy more than 1.022 MeV can energetically allow the production of electron – positron pair after interaction with the matter
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.)
The pair creation occurs usually in the immediate vicinity of the nucleus where the Coulomb field is very strong, such that the momentum and energy of the gamma ray is balanced. That is why the pair production can not take place in vacuum
Pair Production 2 2 2 2 4 Dirac’s Theory 2 + E = p c m c 2 2 4 = + ± E p c m c
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 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 electrons
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 will imparted to the pair in the form of kinetic energy energy, and the contributions by the three effects. As is usual, the photoelectric