PHY401: Nuclear and Particle Physics

Lecture 13, Monday, September 21, 2020 Dr. Anosh Joseph IISER Mohali Radioactivity

PHY401: Nuclear and Particle Physics Dr. Anosh Joseph, IISER Mohali Radioactivity

A nuclide with a low binding energy(“parent nucleus”) would prefer to decay...

... into one with a higher binding energy(“daughter nucleus”).

Since it is energetically favorable.

PHY401: Nuclear and Particle Physics Dr. Anosh Joseph, IISER Mohali Radioactivity

This decay process can happen by 1. α-decay: the nuclide releases an an α-particle (helium nucleus). 2. β− decay: an electron and an anti-neutrino are emitted. 3. β+ decay: a positron and a neutrino are emitted. 4. γ-decay: a nucleus in a metastable excited state (“isomer”) decays directly or indirectly to its ground state emitting one or more high energy photons (γ-rays).

PHY401: Nuclear and Particle Physics Dr. Anosh Joseph, IISER Mohali Radioactivity

In the last lecture we saw the decay equation

−λt N(t) = N0e . (1)

N0: number of parent nuclei at time t = 0.

τ: mean lifetime.

That is, time taken for the number of parent nuclei to fall to1 /e of its initial value.

PHY401: Nuclear and Particle Physics Dr. Anosh Joseph, IISER Mohali Radioactivity

From Eq. (1) we see that

1 τ = . (2) λ

The “half-life” of a radioactive nucleus, τ 1 is the time 2 taken for the number of parent nuclei to fall to one-half of its initial value.

From Eq. (1) we see that

ln 2 τ 1 = = τ ln 2. (3) 2 λ

PHY401: Nuclear and Particle Physics Dr. Anosh Joseph, IISER Mohali Carbon Dating

Life on earth is carbon based.

14 Living organisms absorb the carbon isotope 6 C.

This isotope is created in the atmosphere by cosmic ray activity.

14 The production of 6 C from cosmic ray bombardment roughly cancels the rate at which that isotope decays.

PHY401: Nuclear and Particle Physics Dr. Anosh Joseph, IISER Mohali Carbon Dating

14 Thus the global concentration of 6 C remains almost constant.

The totality of atmosphere, biosphere, and oceans is known as the carbon exchange reservoir.

The concentration ratio between C-14 and C-12 is approximately the same throughout the reservoir.

PHY401: Nuclear and Particle Physics Dr. Anosh Joseph, IISER Mohali Carbon Dating

A sample of carbon taken from a living organism will have a C-14 concentration of one part in 1.3 × 1012.

The living organism is being continually rejuvenated, by exchanging carbon with the environment - by photosynthesis, or by eating plants or by eating other animals.

On the other hand, a sample of carbon from a dead object cannot exchange its carbon with the environment and therefore cannot rejuvenate its concentration of C-14.

PHY401: Nuclear and Particle Physics Dr. Anosh Joseph, IISER Mohali Carbon Dating

14 C-14 decays radioactively into 7 N (nitrogen), via β-decay with a half-life of 5730 years.

Thus, by measuring the concentration of the isotope C-14 in a fossil sample,...

... using techniques of mass spectroscopy, the age of the fossil can be determined.

PHY401: Nuclear and Particle Physics Dr. Anosh Joseph, IISER Mohali Multi-modal Decays

There are situations in which a radioactive nucleus can sometimes decay into more than one channel, each of which has its own decay constant.

When a nuclide can decay by more than one pathway, it is said to exhibit multi-modal decay.

In this situation, the total decay constant is given by the sum of decay constant λ1 along branch 1 and decay constant λ2 along branch 2.

PHY401: Nuclear and Particle Physics Dr. Anosh Joseph, IISER Mohali Multi-modal Decays

Parent nuclei decays according to

dN(t) = −λ N(t) − λ N(t). (4) dt 1 2

Solution is −(λ1+λ2)t N(t) = N0e . (5)

PHY401: Nuclear and Particle Physics Dr. Anosh Joseph, IISER Mohali Multi-modal Decays

Let us consider an example.

212 Bismuth-212 ( 83 Bi) can decay as

212 208 83 Bi → 81 Tl + α, (6) 212 212 − 83 Bi → 84 Po + e + ν¯, (7) with a total mean lifetime of 536 seconds. (Tl: Tellurium, Po: Polonium.)

208 212 Ratio of 81 Tl to 84 Po from these decays is found to be 9:16.

PHY401: Nuclear and Particle Physics Dr. Anosh Joseph, IISER Mohali Multi-modal Decays

What are the decay constants λ1 and λ2, for each of these decay modes?

We have 1 λ + λ = = 1.86 × 10−3s−1. (8) 1 2 536

Ratio of the number of decay products is equal to the ratio of the decay constants.

λ 9 1 = . (9) λ2 16

PHY401: Nuclear and Particle Physics Dr. Anosh Joseph, IISER Mohali Multi-modal Decays

This gives

−4 −1 λ1 = 6.8 × 10 s , (10) −3 −1 λ2 = 11.8 × 10 s . (11)

PHY401: Nuclear and Particle Physics Dr. Anosh Joseph, IISER Mohali Decay Chains

There are cases in which a parent nucleus decays, with decay constant λ1 into a daughter nucleus,...

... which is itself radioactive, and decays (either into a stable nuclide or into another radioactive nuclide) with decay constant λ2.

An example of this is

210 β 210 α 206 83 Bi −→ 84 Po −→ 82 Pb (12)

PHY401: Nuclear and Particle Physics Dr. Anosh Joseph, IISER Mohali Decay Chains

Mean lifetime for the ﬁrst stage of decay is 7.2 days.

Mean lifetime for the second stage is 200 days.

Suppose at time t we have N1(t) nuclei of the parent nuclide and N2(t) nuclei of the daughter nuclide.

Then for N1(t) we have dN (t) 1 = −λ N (t). (13) dt 1 1

PHY401: Nuclear and Particle Physics Dr. Anosh Joseph, IISER Mohali Decay Chains

This gives −λ1t N1(t) = N1(0)e . (14)

For N2 there is a production mechanism,...

... which contributes a rate of increase of N2 equal to the rate of decrease of N1.

In addition, there is a contribution to the rate of decrease of N2 from its decay process.

PHY401: Nuclear and Particle Physics Dr. Anosh Joseph, IISER Mohali Decay Chains

Thus we have dN (t) 2 = λ N (t) − λ N (t). (15) dt 1 1 2 2

That is,

dN2(t) = λ N (0)e−λ1t − λ N (t). (16) dt 1 1 2 2

PHY401: Nuclear and Particle Physics Dr. Anosh Joseph, IISER Mohali Decay Chains

This is an inhomogeneous differential equation whose solution with N2(0) = 0 is given by λ 1 −λ1t −λ2t N2(t) = N1(0) e − e . (17) (λ2 − λ1)

This equation is known as the Bateman equation.

(It is in fact a simple form of the set of Bateman equations. Here this equation is for a chain of three isotopes.)

PHY401: Nuclear and Particle Physics Dr. Anosh Joseph, IISER Mohali Decay Chains

Initially, as the parent decays, the quantity of the daughter nuclide grows faster than it decays.

However, after some time, the available quantity of the parent nuclide is depleted...

... so the production rate decreases.

This in turn allow the decay rate of the daughter nuclide to dominate so that the quantity of the daughter nuclide also decreases.

PHY401: Nuclear and Particle Physics Dr. Anosh Joseph, IISER Mohali Decay Chains

Some heavy nuclides have a very long decay chain, decaying at each stage to another unstable nuclide before eventually reaching a stable nulcide.

An example of this is U-238.

It decays in no fewer than 14 stages - eight by α-decay and six by β-decay before reaching a stable isotope of Pb.

The lifetimes for the individual stages vary from around 10−4 s to 109 years.

PHY401: Nuclear and Particle Physics Dr. Anosh Joseph, IISER Mohali Decay Chains

Note: Since uranium compounds are common in granite, uranium and its radioactive daughters are a part of the stone walls of buildings.

They therefore contribute to the environmental radiation background.

This is particularly true of the inert gas Rn-222 (radon), which escapes from the walls and is inhaled into the lungs.

PHY401: Nuclear and Particle Physics Dr. Anosh Joseph, IISER Mohali Decay Chains

The α-decay of Rn-222 is responsible for about 40% of the average natural human radiation exposure.

Figure below: Illustration of the U-238 decay chain in the N-Z plane. The half-life of each of the nuclides is given together with its decay mode.

PHY401: Nuclear and Particle Physics Dr. Anosh Joseph, IISER Mohali Decay Chains

PHY401: Nuclear and Particle Physics Dr. Anosh Joseph, IISER Mohali Induced Radioactivity

It is possible to convert a nuclide which is not radioactive into a radioactive one by bombarding it with neutrons or other particles.

The stable nuclide (sometimes) absorbs the projectile in order to become an unstable, radioactive nucleus.

23 For example, bombarding 11Na (sodium) with 24 neutrons can convert the nuclide to 11Na.

PHY401: Nuclear and Particle Physics Dr. Anosh Joseph, IISER Mohali Induced Radioactivity

24 It is radioactive and it decays to 12Mg (magnesium) through β-decay.

Let us assume that the rate at which the radioactive nuclide (with decay constant λ) is being generated is R.

Then the number of such nuclei is given by

dN(t) = R − λN. (18) dt

PHY401: Nuclear and Particle Physics Dr. Anosh Joseph, IISER Mohali Induced Radioactivity

If we take the number of these nuclei as zero at time t = 0 (that is, we start the bombardment at t = 0) then the above equation has the solution R N(t) = (1 − e−λt ). (19) λ

At large t we have R = λN. (20)

That is, we achieve the equilibrium state in which the production rate R is equal to the decay rate λN.

PHY401: Nuclear and Particle Physics Dr. Anosh Joseph, IISER Mohali Beta Decay

Consider nuclei with equal mass number A (isobars).

We can write the semi-empirical mass formula as

2 − 1 M(A, Z) = αA − βZ + γZ + δA 2 , (21)

PHY401: Nuclear and Particle Physics Dr. Anosh Joseph, IISER Mohali Beta Decay

− 1 aa α = M − a + a A 3 + , (22) n v s 4 β = aa + (Mn − Mp − me), (23) −1 − 1 γ = aaA + acA 3 , (24)

δ = 0 or ± δ0. (25)

PHY401: Nuclear and Particle Physics Dr. Anosh Joseph, IISER Mohali Beta Decay

In the above form, nuclear mass M(A, Z) is a quadratic function of Z.

A plot of such nuclear masses, for constant mass number A, as a function of Z yields a parabola for odd A.

For even A, the masses of the even-even and the odd-odd nuclei are found to lie on two vertically shifted parabolas.

PHY401: Nuclear and Particle Physics Dr. Anosh Joseph, IISER Mohali Beta Decay

The odd-odd parabola lies at twice the pairing energy − 1 2δA 2 above the even-even one.

The minimum of the parabolas is found at Z = β/2γ.

The nucleus with the smallest mass in an isobaric spectrum is stable with respect to β-decay.

PHY401: Nuclear and Particle Physics Dr. Anosh Joseph, IISER Mohali Beta decay in odd mass nuclei

Let us look at different kinds of β-decay.

We will use the example of the A = 101 isobars.

For this mass number, the parabola minimum is at the isobar Ru-101 (ruthenium) which has Z = 44.

PHY401: Nuclear and Particle Physics Dr. Anosh Joseph, IISER Mohali Beta decay in odd mass nuclei

101 Isobars with more neutrons, such as 42 Mo 101 (molybdenum)and 43 Tc (technetium), decay through the conversion − n → p + e + ν¯e. (26)

The charge number of the daughter nucleus is one unit larger than that of the parent nucleus.

An electron and an electron-antineutrino are also produced

PHY401: Nuclear and Particle Physics Dr. Anosh Joseph, IISER Mohali Beta decay in odd mass nuclei

See ﬁgure below.

Mass parabola of the A = 101 isobars. Possible β-decays are shown by arrows. Z is on the x-axis. Zero point of the mass scale was chosen arbitrarily.

PHY401: Nuclear and Particle Physics Dr. Anosh Joseph, IISER Mohali Beta decay in odd mass nuclei

101 101 − 42 Mo → 43 Tc + e + ν¯e, (27) 101 101 − 43 Tc → 44 Ru + e + ν¯e. (28)

(Tc-43: technetium, Ru-44: Ruthenium.)

PHY401: Nuclear and Particle Physics Dr. Anosh Joseph, IISER Mohali Beta decay in odd mass nuclei

Historically such decays where a negative electron is emitted are called β−-decays.

Energetically, β−-decay is possible whenever the mass of the daughter atom M(A, Z + 1) is smaller than the mass of its isobaric neighbor

M(A, Z) > M(A, Z + 1). (29)

PHY401: Nuclear and Particle Physics Dr. Anosh Joseph, IISER Mohali Beta decay in odd mass nuclei

We consider here the mass of the whole atom...

... and not just that of the nucleus alone.

Rest mass of the electron created in the decay is automatically taken into account.

The tiny mass of the (anti-)neutrino (< 2 eV) is negligible in the mass balance.

101 Isobars with a proton excess, compared to 44 Ru, decay through proton conversion

+ p → n + e + νe. (30)

PHY401: Nuclear and Particle Physics Dr. Anosh Joseph, IISER Mohali Beta decay in odd mass nuclei

101 The stable isobar 44 Ru (rhodium) is eventually produced via

101 101 + 46 Pd → 45 Rh + e + νe, and (31) 101 101 + 45 Rh → 44 Ru + e + νe. (32)

(Rh-45: Rhodium, Pd-45: Palladium.)

Such decays are called β+-decays.

PHY401: Nuclear and Particle Physics Dr. Anosh Joseph, IISER Mohali Beta decay in odd mass nuclei

Since the mass of a free neutron is larger than the proton mass, the process Eq. (30) is only possible inside a nucleus.

By contrast, neutrons outside nuclei can and do decay via Eq. (26).

Energetically, β+-decay is possible whenever the following relationship between the masses M(A, Z) and M(A, Z − 1) (of the parent and daughter atoms respectively) is satisﬁed

M(A, Z) > M(A, Z − 1) + 2me. (33)

PHY401: Nuclear and Particle Physics Dr. Anosh Joseph, IISER Mohali Beta decay in odd mass nuclei

This relationship takes into account the creation of a positron...

... and the existence of an excess electron in the parent atom.

PHY401: Nuclear and Particle Physics Dr. Anosh Joseph, IISER Mohali Beta decay in odd mass nuclei

Note: We now know that neutrinos do have a tiny mass.

The ﬁrst hint of this was during the observation of the Supernova in 1987, when a burst of neutrinos were observed a few seconds after the burst of γ-rays, implying that the neutrinos had not travelled from the supernova with exactly the speed of light.

This was conﬁrmed by the Super-Kamiokande Collaboration. In 1998 they announced the ﬁrst evidence of neutrino oscillation.

PHY401: Nuclear and Particle Physics Dr. Anosh Joseph, IISER Mohali Beta decay in even nuclei

Even mass-number isobars form, as we described above, two separate (one for even-even and one for odd-odd nuclei) parabolas,...

... which are split by an amount equal to twice the pairing energy.

Often there is more than one β-stable isobar, especially in the range A > 70.

Let us consider the example of the nuclides with A = 106. (See ﬁgure below.)

PHY401: Nuclear and Particle Physics Dr. Anosh Joseph, IISER Mohali Beta decay in even nuclei

Figure: Mass parabolas of the A = 106-isobars. Possible β-decays are indicated by arrows. Z is on the x-axis. Zero point of the mass scale was chosen arbitrarily.

PHY401: Nuclear and Particle Physics Dr. Anosh Joseph, IISER Mohali Beta decay in even nuclei

106 106 The even-even 46 Pd (palladium) and 48 Cd (cadmium) isobars are on the lower parabola,...

106 ... and 46 Pd is the stablest.

106 48 Cd is β-stable, since its two odd-odd neighbors both lie above it.

PHY401: Nuclear and Particle Physics Dr. Anosh Joseph, IISER Mohali Beta decay in even nuclei

106 Conversion of 48 Cd is thus only possible through a 106 double β-decay into 46 Pd

106 106 + 48 Cd → 46 Pd + 2e + 2νe. (34)

106 Probability for such a process is so small that 48 Cd may be considered to be a stable nuclide.

PHY401: Nuclear and Particle Physics Dr. Anosh Joseph, IISER Mohali Beta decay in even nuclei

Odd-odd nuclei always have at least one more strongly bound even-even neighbor nucleus in the isobaric spectrum.

They are therefore unstable.

The only exceptions to this rule are the very light 2 6 10 14 nuclei 1H, 3Li, 5 B and 7 N, which are stable to β-decay, since the increase in the asymmetry energy would exceed the decrease in pairing energy.

PHY401: Nuclear and Particle Physics Dr. Anosh Joseph, IISER Mohali Electron capture

Another possible decay process is the capture of an electron from the cloud surrounding the atom.

Quantum mechanics tells us that there is a ﬁnite probability of ﬁnding such an electron inside the nucleus.

In such circumstances it can combine with a proton to form a neutron and a neutrino in the following way

− p + e → n + νe. (35)

PHY401: Nuclear and Particle Physics Dr. Anosh Joseph, IISER Mohali Electron capture

Figure: Electron capture at the quark level.

PHY401: Nuclear and Particle Physics Dr. Anosh Joseph, IISER Mohali Electron capture

This reaction occurs mainly in heavy nuclei where the nuclear radii are larger and the electronic orbits are more compact.

Usually the electrons that are captured are from the innermost (the “K”) shell since such K-electrons are closest to the nucleus and their radial wave function has a maximum at the centre of the nucleus.

Since an electron is missing from the K-shell after such a K-capture, electrons from higher energy levels will successively cascade downwards and in so doing they emit characteristic X-rays.

PHY401: Nuclear and Particle Physics Dr. Anosh Joseph, IISER Mohali Electron capture

Figure below: the decay of K-40.

In this nuclear conversion, β−- and β+-decay as well as electron capture compete with each other.

PHY401: Nuclear and Particle Physics Dr. Anosh Joseph, IISER Mohali Electron capture

PHY401: Nuclear and Particle Physics Dr. Anosh Joseph, IISER Mohali Lifetimes

Lifetimes τ of β-unstable nuclei vary between a few ms and 1016 years.

They strongly depend upon both the energy E which is released and upon the nuclear properties of the mother and daughter nuclei.

Decay of a free neutron into a proton, an electron and an antineutrino releases 0.78 MeV...

... and this particle has a lifetime of τ ≈ 880 s.

PHY401: Nuclear and Particle Physics Dr. Anosh Joseph, IISER Mohali Lifetimes

A well-known example of a long-lived β-emitter is the nuclide K-40.

It transforms into other isobars by both β−-and β+-decay.

Electron capture in K-40 also competes here with β+-decay.

PHY401: Nuclear and Particle Physics Dr. Anosh Joseph, IISER Mohali Lifetimes

The stable daughter nuclei are Ar-40 and Ca-40 respectively, which is a case of two stable nuclei having the same mass number A.

Note: K-40 contributes considerably to the radiation exposure of human beings and other biological systems.

Potassium is an essential element: for example, signal transmission in the nervous system functions by an exchange of potassium ions.

PHY401: Nuclear and Particle Physics Dr. Anosh Joseph, IISER Mohali Lifetimes

The fraction of radioactive K-40 in natural potassium is 0.01%, ...

...and the decay of K-40 in the human body contributes about 16% of the total natural radiation which we are exposed to.

PHY401: Nuclear and Particle Physics Dr. Anosh Joseph, IISER Mohali References

I B. Povh, K. Rith, C. Scholz and F. Zetsche, Particles and Nuclei: An Introduction to the Physical Concepts, Springer, 6th edition (2008).

I A. Das and T. Ferbel, Introduction To Nuclear And Particle Physics, World Scientiﬁc (2003).

PHY401: Nuclear and Particle Physics Dr. Anosh Joseph, IISER Mohali Quiz 3

1. Is it possible to describe the atomic nuclei by incorporating aspects of both the shell nuclear model and the liquid-drop model? (Yes/No).

If the answer is yes, describe why one should do that.

If the answer is no, describe why one should not do that.

PHY401: Nuclear and Particle Physics Dr. Anosh Joseph, IISER Mohali End

PHY401: Nuclear and Particle Physics Dr. Anosh Joseph, IISER Mohali