1. BETA DECAY
Introduction:
β-particles emitted by radio active substances are very high energy electrons.
Madam Curie-
first to show that β-particles were a type of negatively charged particles.
much more penetrating than the α–particles.
2. Becquerel:
β-particles deflected much more than α–particles in a magnetic field.
This proved that β-particles are much lighter than α–particles.
Showed a rough estimate of the e/m of the β-particles which showed that they were electrons.
3. ENERGETICS OF β-DECAY:
Three different types of β-decay are observed:
1. β−
decay
2. β+
decay
3. Orbital electron capture.
In all the three processes,
the mass number A of the parent nucleus remains unchanged.
the atomic number Z changes by one unit.
4. β−
decay:
Since A = Z + N, where N is the neutron number.
As A remains constant and Z changes by one unit, the total number of protons and neutrons in
the nucleus remains unchanged after β – decay.
In β−
decay, Z is increased by one unit to (Z + 1) and N is decreased by one unit to (N – 1).
This happens because one neutron in the nucleus is transformed into a proton by β−
decay.
7. Consider the β−
decay of the nucleus of an atom X of atomic mass M(A,Z) into the nucleus of
the atom Y of atomic mass M(A,Z+1).
Process can be represented by the reaction:
ZXA
Z+1YA + β−
Using Einstein’s relation, E = 𝑚𝑐2 between the mass m and the energy E of a particle, we can
write the disintegration energy in β−
decay as the difference between the nuclear masses
M(A,Z) and M(A,Z+1) of X and Y respectively minus the electronic mass 𝑚𝑒 multiplied by 𝑐2
8. 𝑄β
− = [𝑀𝑛(A,Z) - 𝑀𝑛(A,Z+1) - 𝑚𝑒]𝑐2
Instead of the nuclear mass, if we take atomic masses:
𝑄β
− = [M(A,Z) - Z𝑚𝑒 - M(A,Z+1) + (Z+1) 𝑚𝑒 - 𝑚𝑒]𝑐2
= [M(A,Z) - M(A,Z+1)]𝑐2
If masses are all expressed in energy units, the above equation becomes:
𝑸β
− = [M(A,Z) - M(A,Z+1)]
9. Condition for β−
decay to take place:
𝑸β
− = [M(A,Z) - M(A,Z+1)]
from above equation we see that, 𝑸β
− > 0 if M(A,Z) > M(A,Z+1).
β−
- disintegration can take place if the mass of the parent atom ZXA is greater than the mass of
the daughter atom Z-1YA.
10. β+
decay:
Many artificially produced radioactive nuclei are found to undergo β+
- decay.
β+
or the positron is:
the anti-particle of the electron.
has same mass as the electron.
has equal and opposite charge.
11. When a nucleus undergoes β+
- decay it loses one unit of positive charge.
The atomic number Z is decreased to (Z – 1) and N is increased to (N + 1).
This happens because one proton in the nucleus is transformed into a neutron.
13. Process can be represented by the reaction:
ZXA
Z+1YA + β+
𝑄
β
+ = [𝑀𝑛(A,Z) - 𝑀𝑛(A,Z-1) - 𝑚𝑒]𝑐2
Instead of the nuclear mass, if we take atomic masses:
𝑄
β
+ = [M(A,Z) - Z𝑚𝑒 - M(A,Z-1) + (Z-1) 𝑚𝑒 - 𝑚𝑒]𝑐2
= [M(A,Z) - M(A,Z-1) - 2𝑚𝑒]𝑐2
14. If masses are all expressed in energy units, the above equation becomes:
𝑸
β
+ = [M(A,Z) - M(A,Z-1) - 2𝒎𝒆]
Condition for β+
decay to take place:
𝑸
β
+ = [M(A,Z) - M(A,Z-1) - 2𝒎𝒆]
from above equation we see that, 𝑸
β
+ > 0 if M(A,Z) > M(A,Z-1).
β+
disintegration is possible if the mass of the parent atom is greater than the sum of the
masses of the product atom and twice the electronic mass.
15. Rest mass energy of the electron is 0.511 MeV which implies that the mass of the parent atom
must be greater than that of daughter atom by an amount at least equal to 1.022 MeV.
16. Orbital Electron Capture:
Observed amongst many artificially produced radioactive nuclei.
Process: Proton within the nucleus captures an orbital electron [mostly K- electron, which is
nearest to the nucleus] and is transformed into a neutron.
Z is reduced to (Z – 1) and N is increased to (N + 1) in an orbital electron capture process type
of β – decay.
19. Process can be represented by the reaction:
ZXA + e−
Z-1YA
Binding energy of the electron in its orbit is 𝐵𝑒.
This energy must be subtracted from the energy released in the electron capture process, to
get the disintegration energy 𝑄𝑒
𝑄𝑒 = [𝑀𝑛(A,Z) + 𝑚𝑒 - 𝑀𝑛(A,Z-1)]𝑐2 - 𝐵𝑒
21. Condition for the orbital electron capture process to take place:
𝑸𝒆 = [M(A,Z) - M(A,Z-1)] - 𝑩𝒆
from above equation we see that, 𝑸𝒆 > 0 if M(A,Z) > M(A,Z-1).
This means that electron capture decay is possible if the mass of the parent atom is greater than the
mass of the daughter atom by at least the electron binding energy.
22. In the case of the lighter atoms, the electron binding energy is small [𝑩𝒆], this reduces M(A,Z)
being simply greater than M(A,Z-1).
23. Electron capture decay and β+
decay can occur in the same nuclei if:
𝑸
β
+ = [M(A,Z) - M(A,Z-1) - 2𝒎𝒆] ----------------------[1]
𝑸𝒆 = [M(A,Z) - M(A,Z-1)] - 𝑩𝒆 ---------------------[2]
are satisfied.
If equation [2] is satisfied and not [1] then only electron capture process is possible.
Example: 4Be7 + e−
3Li7
This decay cannot take place by β+
emission, because:
the difference in the atomic masses of the two nuclei is 0.864 eV,
the numerical value 0.864 MeV < 2𝑚𝑒𝑐2 = 1.022 MeV.
24. The decay of 80Br to 80Se can take place by both β+
emission and electron capture:
35Br80 + e−
34Se80
35Br80
34Se80 + β+
Note: The atomic mass difference between the two nuclei is 2.66 MeV which is greater than
2𝑚𝑒𝑐2 = 1.022 MeV.
25. DETERMINATION OF THE β ENERGY:
Experiments from Kaufmann and Bucherer showed that -
the β – particles from radioactive sources were emitted with a continuous distribution of
energy.
Energy distribution of the β – particles can be determined with the help of semi-circular
magnetic spectrograph.
Magnetic spectrograph was designed by Rutherford and Robinson.
26. Experimental Set-up:
S is a thin wire coated with a radioactive substance.
Acts as a source of β – particles.
β – particles emerge from the slit as a slightly divergent
beam and describe semi circular path.
The β – particles of a particular velocity in a slightly
divergent beam, after describing different semi-circular paths,
are all focused at a definite point on the photographic plate P.
27. β – particles --------- emitted from the source with a continuous distribution of velocities, the
different focal lines form continuous spectrum.
As the plate is developed, it is found to be continuously blackened from the end nearest to the
slit to definite farthest point.
β – particles are emitted with velocities and hence energies ranging from 0 up to a maximum.
The intensity of blackening is different at different points, which indicates that the intensities of
the β – particles of different velocities are different.
28. Use of Geiger – Muller
counter in measuring the
energy distribution of the β –
particles:
Geiger-Muller counter –
Used for detecting and
counting the number of β –
particles of different velocities.
Shifting the position of the
counter along horizontal path,
detects and counts β –
particles of different velocity.
Energy – distribution of β –
particles from different
radioactive substance can be
determined.
29. Equating the magnetic force due to magnetic induction with centripetal force:
𝑚𝑣2
𝑟
= 𝐵𝑒𝑟
p = mv = Ber ------------------------------------- [1]
where m = 𝑚0 1 − 𝛽2 ----- relativistic mass of the electron,
p = mv -------- momentum of the β – particle
r -------- the radius of curvature of its path from the source to the counter.
For particles of higher momenta ( i.e. higher energy ) travel along paths of larger radii of the
curvature.
Such particles are focussed further away from the slit.
30. For a given source, there is a maximum distance of the counter from slit up to which the β –
particle can be detected.
This confirms the fact that β – particles are emitted with velocities extending from 0 to a
maximum.
31. Magnetic spectrometer ---
Measures the momentum of the particle.
Total energy W can be determined by measuring the momentum of the particle.
𝑊2 = 𝑝2𝑐2 + 𝑚0
2
𝑐4
Kinetic energy of the β – particle-
𝐸𝛽 = W - 𝑚0𝑐2
= 𝑝2𝑐2 + 𝑚0
2
𝑐4 - 𝑚0𝑐2
From equation [1]-
𝐸𝛽 = 𝐵2𝑒2𝑟2𝑐2 + 𝑚0
2
𝑐4 - 𝑚0𝑐2
32. 𝐸𝛽 = 𝐵2𝑒2𝑟2𝑐2 + 𝑚0
2
𝑐4 - 𝑚0𝑐2
From the values of B and r, 𝐸𝛽 can be determined.
Corresponding number of β – particles for each 𝐸𝛽 are determined with the help of the G-M
counter.
33. Figure shows the energy distribution of the β – particles graphically with
N(𝐸𝛽) along the ordinate and 𝐸𝛽 along the abscissa.
Plot shows the energy distribution of the β – particles emitted by the
natural radio-active element RaE [210Bi] is plotted against 𝐸𝛽.
Shows a continuous distribution of the β – energy ranging from 0 to a
maximum 𝐸𝑚 = 1.17 MeV.
Number of β – particles with 𝐸𝛽 = 0 is finite.
34. Figure shows momentum distribution of the β – particles
emitted by artificially radioactive substance 198Au ( Z=79).
Abscissa is a measure of the momentum p of the β –
particles.
At some definite values of momentum sharp peak are
observed.
Sharp peaks –
Represent groups of mono – energetic electrons
superimposed on the continuous distribution.
These are called as internal conversion electrons or
conversion electrons.
36. ORIGIN OF CONTINUOUS β –
SPECTRUM: NEUTRINO HYPOTHESIS
Secondary electrons:
Give rise to discrete peak in β – spectrum.
Are not emitted from the β – disintegrating nuclei.
Area under the energy distribution curves is proportional to the number of
electrons emitted.
Areas under the peak:
are small compared with the area under the continuous distribution graph.
Indicates that the number of secondary electrons forming the peak is only
few percent of the total number of β – particles.
37. What does the experimental observations tell us??????
Total number of electrons [ in the peaks as well as in the continuous spectrum] > the number of
nuclei undergoing β – decay.
Number of electrons in the continuous spectrum = number of electrons emitted by the nuclei
undergoing β – decay.
Conclusion: Electrons emitted during β – decay form continuous spectrum [which excludes the
peak]
38.
39. Both β+
and β−
decay:
show continuous distribution of energies.
Velocities ranging from 0 to maximum.
Electron capture process:
No observable particle is emitted from the nucleus.
Only X-rays or Auger electrons, characteristic of the product atom are observed.
40. Conclusions from the experimental study of the α–disintegration:
α–particle spectra is discrete in nature.
which implies that the nuclei exist in discrete energy states [proved by theory of quantum
mechanics for a closed micro – system]
Transitions between these discrete levels in the parent and the product nuclei give rise to the
emission of mono – energetic groups of α–particles.
41. Study of β – energy: [ DISCUSSION OF THE DRAWBACKS OF β – DISINTEGRATION ENERGY]
1) Study of 𝛾 – ray spectra indicates: [Failure of conservation of energy]
β – particles with one or more definite energies are emitted.
Emission is due to transitions between the discrete energy levels in the parent and the product
nuclei.
Discrete spectra should be observed in β – decay as well.
The observation:
Continuous distribution of the β – energy.
Conclusion: Violates the principle of conservation of energy.
42. 2) Failure of principle of conservation of angular momentum:
Spin associated with a proton and neutron in the nucleus: s = 1
2
Two possibilities depending on the number of nucleons A present in the nucleus:
Case I: A = even.
Total spin angular momentum of the nucleus S = 𝑠𝑖 ---- either 0 or integral.
CASE II: A = odd.
Total spin angular momentum of the nucleus S = 𝑠𝑖 ---- half – odd integral.
43. Nucleons may have orbital angular momentum L:
Orbital angular momentum: can be only integral multiples of ℏ.
Total angular momentum (I) = L + S:
value of I is integral or half – odd integral in units of ℏ. [depends if A = even or odd]
44. Example:
if A is even I is integral or 0.
In β – disintegration process, A remains unchanged.
I --- 0 or integral
Implies I does not change or changes by one unit.
if A is odd I is half – integral, A remains unchanged.
I ----- half integral multiple.
Implies I changes by one unit.
45. The failure of previous assumptions:
electron intrinsic spin ½.
During emission of electron from the nucleus:
Electron carries away half – odd integral unit of angular momentum. [since the change in the
orbital angular momentum takes place in integral units of ℏ].
Emission of electron from the nucleus should change the angular momentum by half odd
integral multiple of ℏ.
This contradicts the statement that in β disintegration process I should change by integral
multiple of ℏ
Above we have discussed the two inconsistencies.
46. Wolfgang Pauli (1930) explained the discrepancies:
Assumptions by Pauli:
At the time of β – decay of a nucleus, an unobserved particle, in addition to electron, is
emitted.
This unobserved particle carries away the balance of energy 𝐸𝜐 = 𝐸𝑚 - 𝐸β.
The total energy of the two particles is equal to maximum β – energy 𝐸𝑚.
If electron is emitted with zero – kinetic energy, then second particle is emitted with maximum
energy 𝐸𝜐 = 𝐸𝑚
When electron is emitted with the energy 𝐸𝑚, the other particle has energy 𝐸𝜐 = 0.
47. THE NEW PARTICLE …………..
This new particle proposed by Pauli was named as neutrino.
One of the most important property neutrino must have…. Undetectable nature of neutrino.
48. Some of the properties of neutrino:
Neutrino must be electrically neutral.
Only change in the charge of the nucleus during β – decay is due to emission of the electron or
positron or due to the capture of an orbital electron.
This is in agreement with the experimental observations.
49. The mass of the neutrino should be zero or very nearly so.
This is due to the fact that the maximum energy 𝐸𝑚 of the emitted electron is equal to the mass
energy difference between the parent and the product nuclei less the rest energy of the
electron.
If neutrino has finite mass then its rest energy must also be subtracted to get 𝐸𝑚.
50. Intrinsic spin of the neutrino should be ½.
Since the electron spin is also ½, two spin ½ particles are emitted during the β – decay. Hence
the two together will take away an integral unit of angular momentum.
This is in agreement with the observation regarding the change in angular momentum in β –
decay.
The neutrino must obey Fermi – Dirac statistics like the electron since its spin is ½.
51. PHYSICAL PROPERTIES OF THE NEUTRINOS:
Difficult to detect.
Since they carry no charge, they cannot produce ionization in matter.
Usual methods of detection of charged particles cannot be applied for their detection.
Since the neutrinos are practically massless, they cannot transfer energy to any other particle by
elastic collision.
Hence the method applicable in the case of detection of an electrically neutral particle like
neutron cannot be employed in this case.
52. Neutrino has an anti – particle, known as the anti – neutrino 𝜈.
Neutrino is emitted at the time of 𝛽− decay.
Anti – neutrino is emitted during 𝛽+ decay and electron capture process.
We can represent the reaction as follows:
1) ZXA
Z+1YA + β−
+ 𝜈
(𝑙 = 0 0 1 -1)
2) ZXA
Z-1YA + β+
+ 𝜈
(𝑙 = 0 0 -1 1)
3) ZXA + e−
Z-1YA + 𝜈
(𝑙 = 0 1 0 1)
53. Neutrino and anti – neutrino are both massless and charge – less particles with the same
intrinsic spin (1/2).
Weakly interacting particles like electron, positron, neutrino and anti – neutrino belong to a
class of elementary particles called leptons.
Weakly interacting particles have associated with them lepton number 𝑙 with them.
Particle Lepton number
Electron, Neutrino +1
Positron, Anti – Neutrino -1
54. From equation (1),(2) and (3) we find that there is conservation of the lepton number after the
beta – disintegration process.
55. ORBITAL CAPTURE PROCESS:
Electron capture type of β – decay takes place when the positron emission is not energetically
possible.
Neutrino is emitted –
when nucleus captures an orbital electron
However, detection of neutrino is extremely difficult.
Only way of detecting an electron capture decay
Observing the characteristic X – rays of the daughter atom.
56. PROCESS:
An electron is captured from the K – shell.
Characteristic K X – rays are emitted.
Vacancy in the K – shell is filled up by the electrons from the
outer L, M ….shell.
Energies of the X- ray photons are given by: 𝐵𝐾 - 𝐵𝐿, 𝐵𝐿 -
𝐵𝑀,…….and so on.
𝐵𝐾, 𝐵𝐿, 𝐵𝑀 are the electron binding energies in the
corresponding shells.
57. Auger electron [ Auger or Radiation less transition ]:
Excess energy following the K – capture process directly
transferred to a L – electron.
The L – electron is emitted by the atom.
This known as Auger electron.
The process is called as Auger or radiation less transition.
58. FACTORS AFFECTING THE CAPTURE OF ORBITAL ELECTRON:
Electron capture probability depends on the two factors:
a) Probability of the electron being at the position of the nucleus.
b) Probability of capture of the electron by the nucleus.
Electron in the K – shell -
Has the largest probability of being at the position of the nucleus. [K – capture is observed]
59. Nucleus may also capture an L – electron –
Lower probability than K – electron.
Lower probability as the L – electron is farther away from the nucleus.
L – capture is followed by the emission of the characteristic L X rays of the daughter atom.
60. DETECTION OF
NEUTRINO:
Beta decay - a three body problem.
For the conservation of linear momentum to hold good:
Daughter nucleus recoils in a direction not exactly opposite
to the emitted β – particle.
If recoil of the daughter nucleus can be detected:
Indirect evidence of the existence of the neutrino.
Recoil energy is only about few eVs.
Experimental measurement on daughter recoil extremely
difficult.
65. The nuclear cross section of a nucleus is used
to describe the probability that
a nuclear reaction will occur.
The concept of a nuclear cross section can be
quantified physically in terms of
"characteristic area" where a larger area means
a larger probability of interaction.
