P H P 4 8 0 : Nu c l e a r P h ys i c s
C o u r s e I n c h a r g e : Dr B a s k a r a n R a n g a s a m y
Study Material for Week 6 (27.04.2020 to 01.05.2020)
Beta Decay
Beta Decay: Introduction
The emission of ordinary negative electrons from the nucleus was among the earliest observed
radioactive decay phenomena. The inverse process, capture by a nucleus of an electron from its atomic
orbital was not observed until 1938 when Alvarez detected the characteristic X rays emitted in the filling of
the vacancy left by the captured electron. The Joliot-Curies in 1934 first observed the related process of
positive electron (positron) emission in radioactive decay, only two years after the positron had been
discovered in cosmic rays. These three nuclear processes are closely related and are grouped under the
common name beta (β) decay.
Radioactive nuclides have been found to exhibit three types of β-radioactivity viz, electron
emission, positron emission and electron capture. The first type i.e., electron emission is very much
common in comparison to the other two. Bucherer in 1909 was the first to prove that β-particles are
nothing but fast moving electrons.
The most basic β decay process is the conversion of a proton to a neutron or of a neutron into a
proton. In a nucleus, β decay changes both Z and N by one unit: 𝑍 → 𝑍 ± 1. 𝑁 → 𝑁 ∓ 1 so that A = Z + N
remains constant. Thus β decay provides a convenient way for an unstable nucleus to “slide down” the
mass parabola of constant A and to approach the stable isobar.
In contrast with α decay, progress in understanding β decay has been achieved at an extremely slow
pace, and often the experimental results have created new puzzles that challenged existing theories. Just as
Rutherford's early experiments showed particles to be identical with 4He nuclei, other early experiments
showed the negative β particles to have the same electric charge and charge to-mass ratio as ordinary
electrons. As evidenced, the presence of electrons is nuclear constituents and so we must regard the β
decay process as “creating” an electron from the available decay energy at the instant of decay; this
electron is then immediately ejected from the nucleus. This situation contrasts with α decay, in which the α
particle may be regarded as having a previous existence in the nucleus. The basic decay processes are thus:
𝑛 →𝑝+𝑒
negative beta decay (β–)
𝑝 →𝑛+𝑒
positive beta decay (β+)
𝑝+𝑒 →𝑛
orbital electron capture (𝜀)
These processes are not complete, for there is yet another particle (a neutrino or antineutrino)
involved in each. The latter two processes occur only for protons bound in nuclei; they are energetically
forbidden for free protons or for protons in hydrogen atoms.
Pauli's Neutrino Hypothesis.
Pauli in 1930 forwarded another explanation which is known as Neutrino hypothesis. Pauli assumed
the emission of a β-particle from the nucleus as due to the transition of the nuclear heavy particle from the
neutron quantum state to proton quantum state with the simultaneous creation of electron-neutrino pair.
This pair of electron- neutrino then escapes with the constant discrete total energy equal to the difference of
the initial neutron state to the final proton state.
The continuous spectrum represents the way in which the energy is shared between the electron and
the neutrino. The upper limit of the continuous β- spectrum corresponds to the fact that the neutrino
escapes with zero energy and thus the total energy is shared by the electron. The lower limit, similarly,
represents that a greater amount of energy is shared by the neutrino than the electron.
The new hypothetical particle suggested by Pauli must be neutral to conserve charge and must have
a rest mass small compared with that of an electron. All discrepancies in the β- decay process can therefore,
be explained if the neutrino possesses the following properties:
mass, m = 0, charge = 0, and spin =
ℏ
Recent researches have proved that a particle of these properties does exist. This led to Pauli's hypothesis
on a firm footing.
Fermi's Theory of β- Decay
In 1934, Fermi developed a theory to explain the continuous β- ray spectrum. This theory is called
neutrino theory of β- decay. According to this theory, a β-particle and a neutrino are created in the nucleus
and both are emitted simultaneously. The total energy of these two particles is a constant which is equal to
the end-point energy observed in the β- ray spectrum. This maximum energy is shared by the β-particle, the
neutrino and also by the recoiling nucleus. The electron will carry the maximum energy when the energy of
the neutrino is zero. In all other cases, electron will carry an energy less than the maximum. The sum of the
energies carried by the electron and the neutrino will always be the same. This energy may be shared by the
two particles in any proportion. Hence it explains the continuous β-ray spectrum. When a nucleus emits a
β-particle its charge changes by one unit while its mass practically remains unchanged. When the ejected βparticle is an electron, the number of protons in the nucleus is increased by one and the number of neutrons
is decreased by one. In positron emission, the reverse process takes place i.e. the number of protons
decreases by one and the number of neutrons increases by one, β-transformations may then be represented
by the following processes:
𝑛 →𝑝+𝛽 +𝜈
(1)
𝑝 →𝑛+𝛽 +𝜈
(2)
where v and 𝜈 represent neutrino and antineutrino.
The n-p process can be performed with the help of a catalyst. Fe56 is bombarded with neutrons of
few MeV energy and Mn56 is produced by an (n-p) reaction.
56
26Fe
+0n1 → 1H1 + 25Mn56
Mn56 then decays as
56
25Mn
∴ The net result is
1
0n
→ 26Fe56 + 𝛽 + 𝜈̅
→ 1H1 + 𝛽 + 𝜈̅
In the above two equations, [i.e., equ (1) and (2)] neutron is not to be considered as composed of a
proton, electron and neutrino but is considered to be transformed into the three particles at the instant of 𝛽
emission. Similarly a proton is transformed into a neutron, positron and neutrino at the time of 𝛽
emission. The neutron or proton that is transformed is not a free particle but is bound to the nucleus by the
nuclear forces.
The most important thing in the theory of β- decay is to calculate the probability of the above
process. This is done as follows.
We know a large number of experimental evidences that electrons do not exist inside the nucleus, it
must be formed at the moment of its emission just as a proton is formed at the moment of its emission from
an atom. The neutrino is also created at the moment of its emission. These particles are created into states
represented by the wave functions 𝜓 and 𝜓 . Let us assume that these are functions for plane waves with
momenta 𝑝 𝑎𝑛𝑑 𝑝 respectively.
𝜓 = 𝑁 .𝑒
(3)
𝜓 = 𝑁 .𝑒
ℏ
where 𝑘 = and r represents the space co-ordinate and N is a normalization factor.
𝜓 is actually more complicated than shown here because it is affected by the nuclear charge Ze.
The probability of emission can be assumed to depend upon the expectation value for the electron and
neutrino to be at the nucleus i.e. on the factor.
(4)
𝜓 (0) . |𝜓 (0)|
It also depends on other factors, whose nature is uncertain.
One such factor is the square modulus of a matrix element M taken between the initial and final states of
the nucleus. This matrix element in its simplest form can be taken as,
𝑀 = ∫ 𝜓 ∗ 𝜓 𝑑𝜏
(5)
assuming that only one nucleon participates. 𝜓 represents the initial state of the nucleon and 𝜓 the final
state i.e. the proton state. We can also take M to be a vector having x-component given by,
𝑀 = ∫ 𝜓 ∗ 𝜎 𝜓 𝑑𝜏
(6)
where 𝜎 is the x-component of a (relativistic) spin operator.
Then
|𝑀| = |𝑀 | + 𝑀
(7)
+ |𝑀 |
The choice of M, i.e. whether given by (5) or (6) depends upon the selection rules.
The expression for the probability of emission depends also upon a constant factor g2 which represents the
strength of the coupling giving rise to emission and is found to have value,
g = 10-48 to 10-49 g cm5 sec-2 = 9 x 10-5 MeV. fm3
(8)
Thus the probability of emission per unit time is
=
Where
ℏ
(10)
𝜓 (0) . |𝜓 (0)|. |𝑀|. |𝑔|
represents the energy density of the final states and 0 refers to the location of the nucleus.
Let Ω be the volume of a big box in which we enclose the system for normalization purposes, then
∫ 𝜓 ∗ . 𝜓 𝑑𝜏 = 1
∴𝑁=
√
(11)
Therefore
where 𝑘 =
ℏ
,𝑘 =
𝜓 =
/
𝑒
𝜓 =
/
𝑒
.
.
(12)
ℏ
when the nucleus is at r = 0
𝜓 (0) =
(13)
= 𝜓 (0)
√
The number of plane wave states having magnitude of momentum between p and p + dp with the particle
anywhere in the volume Ω is given by
(14)
ℏ
∴
𝑑𝑛 =
×
ℏ
.
(15)
×Ω
ℏ
𝑑𝑝 𝑑𝑝
(16)
𝑑𝑝 𝑑𝑝 = 𝐽𝑑𝑝 𝑑𝐸
(17)
=Ω
where J is the Jacobian. 𝑖. 𝑒. ,
ℏ
𝐽=
=
1
=
0
Using the relation 𝐸 = 𝐸 + 𝐸 = 𝑐𝑝 + 𝐸 , J is found to be equal to
(mass of neutrino v is assumed to
be zero for this derivation) then,
=Ω
.
(18)
𝑑𝑝
ℏ
Substituting the value in equation (10), the probability of emission per unit time 𝑃(𝑝 , 𝑝 )𝑑𝑝 is,
𝑃 𝑝 , 𝑝 𝑑𝑝 =
|𝑀|𝑔
ℏ
(19)
ℏ
Using the relation
𝑝 𝑐=𝐸 =𝐸
−𝐸
to eliminate 𝑝 and replacing 𝑝 by simply 𝑝, we get
𝑃(𝑝)𝑑𝑝 =
Using the equation
We get
𝐸
𝑃(𝑝)𝑑𝑝 =
=
| |
ℏ
| |
ℏ
𝐸
−𝐸
(20)
𝑝 𝑑𝑝
(21)
𝑚 𝑐 +𝑐 𝑝
𝑚 𝑐 +𝑐 𝑝
− 𝑚 𝑐 +𝑐 𝑝
𝑝 𝑑𝑝
(22)
β-Decay Life Time.
The decay constant λ which is equal to the reciprocal of the mean decay life time τ is obtained by
calculating the probability per unit time for emission of a β-particle with momentum values between 0 and
pmax. Thus
| |
𝜆= =
ℏ
∫
𝑚 𝑐 +𝑐 𝑝
− 𝑚 𝑐 +𝑐 𝑝
𝑝 𝑑𝑝
(23)
Let us define
𝑚 𝑐 𝜂 = 𝑝 𝑎𝑛𝑑 𝑚 𝑐 𝜂 = 𝑝
.
Expressing the integral in terms of 𝜂 𝑎𝑛𝑑 𝜂 and calling it 𝐹(𝜂 ), we get
=
and
| | .
ℏ
𝐹(𝜂 ) = ∫
. 𝐹(𝜂 )
(24)
(1 + 𝜂 ) − (1 + 𝜂 ) 𝜂 𝑑𝜂
(25)
If the distortion of the wave function by coulomb field of the nucleus is taken into account, the factor
𝐹(𝜂 ) should be written as 𝐹(𝑧, 𝜂 ).. But for small z,
𝐹(𝑧, 𝜂 ) ≈ 𝐹(𝜂 )
The final expression for 𝜏 is therefore
therefore,
=
| | .
ℏ
𝐹(𝑧, 𝜂 )
(26)
In the above expression |𝑀| is uncertain, but for aallowed
llowed transitions (governed by selection rules), it is of
the order of unity. From eq. (26)) it is seen that if |𝑀| does not change, then
𝐹(𝑧, 𝜂 ) = 𝐶𝑜𝑛𝑠𝑡𝑎𝑛𝑡
(27)
The product 𝐹. 𝜏 is a measure of the forbiddenness of a transition. The lowest 𝐹. 𝜏 values correspond to the
transitions which are permitted by the selection rules.
It is also seen that if the integration of eq. ((23) is carried out in terms of end-point
point energy 𝐸
, then the
decay constant λ is found to be proportional to the fifth power of the end point energy i.e.
𝜆=𝑘 𝐸
or
𝑙𝑜𝑔 𝜆 = 𝑙𝑜𝑔 𝑘 + 5 𝑙𝑜𝑔 𝐸
(28)
This equation is found to hold good fairly with Sargent's Cu
Curves as shown below (Fig.1).
(
This confirms the
validity of Fermi's theory.
Fig.1:
1: Sargent
Sargent’s curve- 𝒍𝒐𝒈𝒆 𝝀 𝒗𝒆𝒓𝒔𝒖𝒔 𝒍𝒐𝒈𝒆 𝑬𝒎𝒂𝒙
𝜷
Gamma-Rays
Introduction
While studying α-decay and ββ-processes,
processes, we have seen that under certain circumstances γ-rays are
also emitted. The emission of γ-rays
rays thus speaks of the variou
variouss internal intricacies of the nuclear
phenomena. Unlike α-rays and β-rays,
rays, γ-rays
rays are electromagnetic radiation and therefore have no electric
charge, they cannot be deflected by electric and magnetic fields. The measurement of γ-ray energies is
therefore not possible by the usual magnetic spectrographs and special methods have to be used as we shall
see later on for the precise measurement of γ-ray
ray energies. Similarly the absorption of γ-rays is also
different from those of charged particles like α and β-rays
ays as these radiations are very mucb penetrating.
The absorption of γ-rays
rays obeys exponential law as is evident from the following treatment.
Origin of γ-Rays
The large values of energy associated with γ-rays
rays show that they must be of nuclear origin. γ-ray
spectrum consists of sharp lines and this indicates the existence of a number of energy lev
levels in the
nucleus. Stable nuclides are usually in th
thee state of least energy or ground state, but they can be excited by
particle or photon bombardment. Hence nuclei can exist in states of definite energies, just as atoms can. An
excited nucleus is denoted by an asterisk (*) after its usual symbol. Thus 38Srr87* refers to 38Sr87 in an
excited state. One way an excited nucleus can return to the ground state is by the emission of γ-rays. γ-ray
decay is represented schematically by
by,
(ZXA)* → ZXA + γ.
The star (*) indicates an excited nucleus, and both the daught
daughter
er and the parent have the same
structure of nuclear particles. If E* is the energy associated with the excited state and E is the energy of the
ground state, then the γ-rays
rays have an energy
energy,
hv = E* – E.
Here v is the frequency of the emitted γ-ray.
Fig.1: Decay scheme
A simple example of the relationship between energy levels and decay schemes is shown in Fig-1
which pictures the β-decay of 12Mg277 to 13Al27. The half-life
life of the decay is 9.5 minutes, and it may take
27
place to either of the two excited states of 13Al . The resulting 13Al27* nucleus then undergoes one or two
gamma decays to reach the ground state. Most excited nuclei have very short half
half--lives against γ-decay, but
a few remain excited for as long
ong as several hours. A long lived excited nucleus is called an isomer of the
same nucleus in its ground state. The excited nucleus 38Sr87* has a half-life of 2.8 hours and is an isomer of
87
38Sr .
The Absorption of γ-rays by Matter.
Let us consider a beam of γ-rays of intensity I, incident on a sheet of thickness t. The emergent beam will
be found to have decreased intensity. The change in intensity of the beam is proportional to the thickness of
the sheet and the intensity of the incident beam, i.e.
𝑑𝐼 ∝ −𝐼. 𝑑𝑡
or
(1)
𝑑𝐼 = −𝜇 𝐼. 𝑑𝑡
= −𝜇. 𝑑𝑡
where 𝜇 is the constant of proportionality and is known as “absorption coefficient”. Integrating above
equation, we get
𝑙𝑜𝑔 𝐼 = −𝜇𝑡 + 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡 𝐶
At t = 0, I = I0, the initial incident intensity, then
𝐶 = 𝑙𝑜𝑔 𝐼
𝐼 = 𝐼 .𝑒
(2)
Equation (2) gives the intensity of the γ-rays after passing a thin sheet of thickness t. The intensity I may
also be written as
𝐼 = 𝐵ℎ𝜈
(3)
where B is “the number of photons crossing unit area in unit time” and hv is the energy per photon. B is
often called the “flux” and is defined as the number of photons per sq. m. per sec and I is the corresponding
energy flux. Eq. (2) can also be written as
𝐵 =𝐵 ×𝑒
(4)
The above equations hold good only, when
(i)
(ii)
(iii)
the γ-rays are mono-energetic i.e. the beam is homogeneous
the beam is collimated and of small solid angle
the absorber is thin.
Interaction of γ-Rays with Matter.
When a beam of γ-ray photons is incident on a thin absorber, each photon that is removed from the
beam is removed individually in a single event. Two possibilities arise:
(a) absorption, in which the photon disappears — the photon energy being converted into kinetic energy of
absorbed particles, and
(b) scattering, in which the photon is deflected out of the beam.
The removal of γ-ray photon in one single event is responsible for the exponential absorption. There are
three processes which are mainly responsible for the absorptions of γ-rays. These are (1) Photo-electric
absorption (2) Compton scattering by the electrons in the atom (3) Production of electron-positron pairs as
a result of the interaction between γ-rays and the electric field of atomic nuclei.
It is possible to calculate the probability of each event taking place and this can be expressed in
terms of the absorption coefficient or cross-section. The total absorption coefficient referred in eqns. (1)
and (2) is the sum of the absorption coefficients for the three processes and may be expressed as,
𝜇 =𝜇 +𝜇 +𝜇
(5)
where 𝜇 = photo-electric absorption coefficient
𝜇 = compton-effect absorption coefficient
𝜇 = pair production absorption coefficient
Each absorption coefficient is related to the corresponding cross-section by the following equation:
𝜇=
𝜇
(6)
Where 𝜌 = density in gm/cm3
N = Avogadro's number
A = Atomic weight
aμ
= cross-section per atom.
Internal Conversion.
While studying the theory of α and β-decay, we have seen that in either case, the nucleus is left in
an excited state. The transition from an excited state is accomplished by the emission of γ-rays. When a
nucleus passes from a higher excited state to the ground state, the difference in energy of the two states is
emitted as a γ-ray. Thus the emission of an α-particle or a β-particle is mostly accompanied by a γ-ray
photon. But the transition from an excited level of a nucleus to a lower level of the same nucleus can also
be achieved without the emission of a photon. As an alternative to γ-decay, an excited nucleus, in some
cases, may return to its ground state by giving up its excitation energy to one of the orbital electrons around
it. The energy of excitation is directly transferred to a bound electron of the same atom. The nuclear energy
difference W (Difference between the excited state and the ground state) is converted to energy of an
atomic electron which is ejected from the atom with a kinetic energy EK.
The emitted electron has a K.E. equal to the lost nuclear excitation energy minus the binding energy
of the electron in the atom. i.e.,
K.E. of the ejected electron = EK = W – Ei
(1)
Here, W = the available excitation energy (or)
energy difference between the excited state and the ground state
Ei = binding energy of the ejected electron in its shell of origin (or)
the original atomic binding energy of the electron
This process is called internal conversion. The emitted electron is called decay a conversion electron. Thus
internal conversion and emission of a γ-ray from the nucleus are two alternate ways of accomplishing the
same nuclear transition. The internal conversion is not a two step process in which a γ-ray photon is first
emitted and then it knocks out an orbital electron. It is in better accord with experiment to regard internal
conversion as representing a direct transfer of excitation energy from a nucleus to an orbital electron.
Hence, internal conversion is a single step process in which the excited nucleus inter
interacts directly with the
orbital electron. The energy of the ejected electron ((β-particle)
particle) has discrete values.
values Therefore the
corresponding β-particle energy spectrum is a line spectrum having discrete energies
energies.
Fig.1:
1: Disintegration processes of a radioactive nuclei
Fig-1 illustrates the various kinds of disintegration processes that radioactive nuclei may undergo.
und
The
nucleus is represented as an assembly of protons and neutrons. A proton is in
indicated
dicated by a cross, and a
neutron by an open circle.
The Fig-2 shows
hows the spectrum of conversion electrons which are ejected from the K, L and M shells of
indium by internal conversion of thee 392 KeV transition in In113.
Fig.2: Spectrum of conversion electrons ejected from Indium by Internal Conversion
These conversion electrons carry away the spins and parity changes of the nuclear states. The
kinetic energy of conversion electron escaping from an atom is given as
as,
Ei = W – BK
or
W – BL
or
W – BM etc.
(2)
where BK, BL and BM represent the binding energy of an electron in the K, L, M etc.
Internal Conversion Coefficient. Let the decay constant λγ represent the probability per unit time for the
emission of a photon whose energy is W = hv. Let the decay constant λe represent the probability per unit
time for the internal conversion phenomenon to take place: then excluding other possible modes of decay
(e.g., β-decay) we can write,
λ = λγ+ λe
(3)
The internal conversion coefficient is defined as
𝛼=
+
where Ne and Nγ are the numbers of conversion electrons and of photons emitted in the same interval from
the same sample, in which identical nuclei are undergoing the same nuclear transformation characterised
by the energy W. The total transition probability λ is thus given by,
λ = λγ (1+α)
(4)
and total number of nuclei transforming is Nγ + Ne. The theoretical value of internal conversion coefficient
is found to depend on,
W = the energy of the transition;
Z = atomic number of the transforming nucleus;
I = the multipole order of the transition atomic shell (K, L, M etc.) in which conversion takes place,
and electric multipole or magnetic multipole.
Nuclear Isomerism.
The life times of most of the γ-ray transitions have been found to be very small but in few cases
(about 250) it has been found that life time for γ-decay is measurable with observed half lives varying from
10-10 sec to many years. These delayed transitions are known as isomeric transitions and the states from
which such transitions take place are known as isomeric states or isomeric levels. On account of these
delayed transitions, such pairs of nuclides exist which have the same atomic number and mass numbers i.e.
they are isotropic and isobaric but have different radioactive properties. Such nuclides are called nuclear
isomers and their existence is termed as Nuclear Isomerism.
UX2 and UZ both having the same atomic number and same mass number but having different halflives and emitting different radiations were the first to be discovered demonstrating nuclear isomerism.
Both UX2 and UZ are formed by β– decay of UX1 (90Th234) and thus have the same mass number 234 and
both have the same atomic number 90 + 1 = 91. The newly formed nuclide UX2 has a half life of 118
minutes and emits three groups of electrons with end point energies of 2.31 MeV (90%), 1.50 MeV (9%)
and 0.58 MeV (1%). On the other hand UZ emits four groups of β-particles with end point energies of 0.16
MeV (28%), 0.32 MeV (32%), 0.53 MeV (27%) and 1.13 MeV (13%) and has a half life of 6.7 hours. UX2 is
more highly excited than UZ and about 0.15% of the UX2 nuclei decay to the ground state UZ by emitting a
γ-ray with an energy of 0.394 MeV. UZ which is nothing but ground state of 91Pa234 then decays to UII
(92U234) by β-decay. About 100% of UX2 atoms decay directly to UII by β-decay process.
Reference Books:
1. Modern Physics, R Murugesan, Kiruthiga Sivaprasath, 18th edition, (2016) S Chand and company limited.
2. Elements of Nuclear Physics, M.L.Pandya, R.P.S. Yadav, 7th edition, (1996) Kedar Nath Ram Nath publications, Delhi.
3. Introductory Nuclear Physics, Kenneth S. Krane, (1988) John Wiley & Sons Inc, New York, USA.
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