Radioactivity

Question 1

Why do nuclei decay?

Answer 1

Nuclei that are unstable due to high numbers of neutrons (and therefore lower binding energy) decay into a nucleus higher binding energy to become more stable. The change in binding energy means that the process is energetically favourable.

Question 2

What is produced in alpha decay (other than the daughter nucleus)?

Answer 2

alpha particle/He nucleus

Question 3

What is produced in beta decay (other than the daughter nucleus)?

Answer 3

electron + anti electron neutrino [b- decay]
OR
positron + electron neutrino [b+ decay]

Question 4

In nuclear decay, what is the difference in binding energy equal to?

Answer 4

kinetic energy of decay products

Question 5

What is an isomer?

Answer 5

Nucleus in a metastable excited state which will decay to its ground state and produce gamma rays.

Question 6

Give the decay rate equation and its solution

Answer 6

No of nuclei per second, dN(t)/dt = -lambda N(t)

solution: N(t) = N_0 e^{-lambda*t}

Question 7

what does lambda*N(t) represent in the decay equation?

Answer 7

Activity

Question 8

What does lambda represent in decay rate equations?

Answer 8

Decay constant/ decay rate

Question 9

Mean lifetime

Answer 9

t = 1/lambda

Question 10

Half-life

Answer 10

ln2/lambda

Question 11

Generic equation for t

Answer 11

t = -1/lambda * ln(N(t)/N(0))

Question 12

Units for decay per second

Answer 12

Becquerel (Bq)

Question 13

What is a decay chain?

Answer 13

When the daughter nucleus decays as well.

Question 14

What is “secular equilibrium” and when does it occur?

Answer 14

It is when quantities of various daughter nuclei in a decay chain remain unchanged. It occurs when the first parent has a v. long lifetime and the no.s of nuclei in the chain N_A, N_B, N_C, etc are in the ratio lambda_AN_A = lambda_BN_B, etc.

Question 15

Why does secular equilibrium occur?

Answer 15

Rate of decay = rate of decay of B so quantity of B remains constant (as production of B = decay of B)

Question 16

Explain signs in decay chain equations

Answer 16

See notes (pg 20)

Question 17

What is Induced Radioactivity?

Answer 17

Process of making an unradioactive nuclide radioactive by bombarding it w/ neutrons and other particles to make it unstable when it absorbs these particles.

Question 18

If R is the decay rate at which radioactive nuclei are GENERATED through induced radioactivity, what does the equation for the number of nuclei per second change to?

What is its new solution?

How does the solution look graphically (N(t) plotted against t)?

Answer 18

dN(t)/dt = R - lambda*N

solution (w/ initial condition R=0 when t=0 as bombardment starts at t = 0): N(t) = R/lambda * (1-e^{-lambda*t})

Graphically, this starts at 0 and grows until it asymptotically reaches the equilibrium state N = R/lambda (production rate R = decay rate lambda*N)

Question 19

What is multi-modal decay?

Answer 19

When a nuclide can decay in different ways (each with different products)

Question 20

Each pathway in multi-modal decay has a different decay constant. What would be the new dN(t)/dt equation in the case of multi-modal decays for a nuclei type? What is its solution? What would the mean lifetime be in this case?

Answer 20

dN(t)/dt = -lambda1N(t) - lambda2N(t)

solution: N(t) = N_0 * e^{ - (lambda1 + lambda2) * t }

mean lifetime: t = 1/lambda where lambda = lambda1 + lambda2

Question 21

What does the Q value of a decay process represent?

Answer 21

Energy produced

Question 22

Why do alpha particles emerge with a kinetic energy less than Q_alpha?

Answer 22

Because if the parent nucleus was at rest before decay, the daughter nucleus must have some recoil to conserve momentum (which requires energy)

Question 23

In some cases, alpha particles produced have less energy than expected from Q value calcs. What is going on in these cases?

Answer 23

The daughter nucleus is in an excited state, so there is less energy for the alpha particle (or the recoil of the daughter nucleus). This exited nucleus can de-excite, producing gamma rays.

Question 24

In some cases, alpha particles produced have more energy than expected from Q value calcs. What is going on in these cases?

Answer 24

When the parent nucleus was the product of another decay so it may have been produced in an excited state and decay to ground state (producing gamma rays) BEFORE undergoing alpha decay.

Question 25

What happens when the excited state of a nucleus is long-lived but the decay constant for the nucleus is large?

Answer 25

The excited state nucleus can directly alpha decay, but the Q-value will be larger by an amount = excitation energy.

Question 26

Describe the mechanism for alpha decay.

Answer 26

2 protons from highest p+ energy levels and 2 neutrons from highest neutron energy levels combine to form an alpha particle inside the nucleus -> “quasi-bound state” [energy: Q_alpha]

There are strong, short range forces and Coulomb repulsion b/w quasi alpha particle and rest of nucleus. Together, these form a potential barrier equal to Coulomb potential at radius R of nucleus: V=2Ze^2/4piepsilon0*R

Beyond R, the alpha particle has enough energy to escape. It needs to overcome this potential to be emitted (alpha decay), however it doesn’t have enough energy so the only was this is possible is by Quantum Tunnelling.

Question 27

Isobars

Answer 27

Atoms of same mass

Question 28

Why is beta decay not energetically allowed?

Answer 28

Because of the difference in the binding energies of the parent and daughter nuclei. When a neutron is converted into a proton this increases the Coulomb repulsion, so increases the Coulomb term in the binding energy equation.
There is also a pairing term in the semi-empirical mass formula that favours even numbers of p+s and ns and a symmetry term that tells us that the numbers of p+s and ns should be roughly equal.

Question 29

When is beta decay energetically permitted?

Answer 29

When the mass of the parent exceeds the mass of the daughter + mass of electron.

Question 30

When will beta decay occur in nuclei with even A?

Answer 30

If number of p+s and ns are both odd, due to the pairing term in binding energy).

If they are both even, it will only decay if number of neutrons is too large or too small for stability.

Question 31

When will beta decay occur in nuclei with odd A?

Answer 31

Here you will have either even numbers of p+s or ns, and pairing term does not change b/w isobars. Therefore, defining factor for stability would be the symmetry term (prefers equal numbers of both) and Coulomb term (prefers fewer p+s). This give us only one stable isobar: Z_A [see notes for equation to determine Z_A]

Therefore, instances that beta decay would occur:
If Z > Z_A: too many p+s -> beta +ve decay
If Z < Z_A: too many ns -> beta -ve decay

Question 32

Discuss the evidence leading to the prediction of the existence of the neutrino.

Answer 32

The kinetic energy of the electron released was expected to equal Q_beta, however experimental results showed a spectrum of energies w/ a maximum energy of Q_beta, which could be explained by a neutrino (v. low mass particle that is also neutral leading it to evade detection) taking some of the energy.

Question 33

Explain why the neutrino has spin 1/2.

Answer 33

Spin of parent and daughter nuclei are integers and as electrons have spin 1/2, neutrinos must also have spin 1/2, as adding to the spin of the electron gives an integer and therefore conserves the total angular momentum in the beta decay process.

Question 34

What is another mechanism of beta decay?

Answer 34

Electron capture where an electron from an inner shell is absorbed by the nucleus, releasing a daughter nucleus with Z-1 and and electron neutrino.

Question 35

Which produces more energy per nucleon, nuclear fusion or nuclear fission?

Answer 35

Nuclear fusion

Question 36

What is the minimum energy required for nuclear fusion to occur?

Answer 36

Both nuclei are +vely charged so need enough energy to overcome Coulomb repulsion between them, so minimum energy required = e^2/4piepsilon0*R

Question 37

Why is as temperature such that E = 1/2 * k * T required in nuclear fusion?

Answer 37

For p+s to have an avg energy associated to the degrees of freedom corresponding to motion in direction of target proton.

Question 38

What condition must be satisfied for quantum tunnelling for nuclear fusion to occur?

Answer 38

barrier height is NOT&raquo_space; KE of incoming particle

Question 39

Explain thermalisation process in Sun.

Answer 39

Gamma rays produced from nuclear fusion/proton cycle will scatter against other charged particles in the Sun, losing energy. They eventually “thermalise”, i.e. settle at an energy (wavelength) distribution which will be the black body distribution at surface temp of Sun -> this peak lies int he visible light range, hence we are able to see it.

Question 40

Why does nuclear fission occur?

Answer 40

Heavier nuclei have smaller binding energy per nucleon, therefore it is energetically favourable for it to split into 2 smaller nuclei, thereby releasing energy which goes into KE of the fragments.

Question 41

During nuclear fission, how can further stability be achieved?

Answer 41

A few ‘spare’ neutrons also being released which also take some of the energy released. These neutrons can induce further fission reactions (chain reaction).

Question 42

Spontaneous fission is very rare. Name one method that can be used to induce fission and explain what is involved.

Answer 42

Neutron capture: parent absorbs neutron, neutron binds to parent, releasing energy (binding energy) in the form of vibrational energy. This energy can be enough to overcome the potential barrier (= surface energy + Coulomb repulsion b/w fission fragments before nucleus has split)

Question 43

In cases where binding energy of neutron is not enough to overcome potential barrier and induce fission, what condition needs to be satisfied to induce it?

Answer 43

Incident neutron needs to have some minimum KE to help overcome the potential barrier.

Question 44

Suggest one way extra binding energy is released during neutron capture (that will help overcome potential barrier).

Answer 44

When nucleus w/ an unpaired neutron absorbs extra neutron and they pair up.

Question 45

What value is used to determine if a chain reaction in nuclear fission is going to occur?

Answer 45

k = no. of neutrons (J) / no.of neutrons (J-1) where J is the stage of the reaction.

If k < 1: stops
If k > 1: uncontrolled
If k = 1: controlled