D - Alpha Decay

Question 1

How to calculate Q and what does the value of Q indicate?

Answer 1

Q = Reactants - Products

Q > 0 Exothermic — Reaction can occur

Q < 0 Endothermic — Reaction requires external energy to occur

Question 2

Calculate B(4,2) with helium having an atomic rest mass of 3728.337MeV.

Answer 2

B(4,2) is the binding energy of (A,Z)
M_He•C2 = 3728.337 MeV

E_B = ((2•939.550) + (2•938.256) + (2•0.511)) - (3728.337) = 28.3MeV

Positive BE meaning mass is lost when constituents combined meaning nucleons are more stable in this form than by themselves

Question 3

What is the Geiger-Nuttal law and what is is implying?

Answer 3

Geiger-Nuttal law shows that there is a extreme variation in half-life for alpha particles for a small range in alpha energies

Implies that isotopes with higher Z tend to have shorter half-lives for alpha decay — as the energy of the emitted alpha particles increases, the decay constant also increases.

Decay constant - probability of a nucleus decaying per unit time

Question 4

What is the energetic of alpha decay? KE recoil energy etc

Answer 4

Qa = KE_d + KE_alpha

Conservation of momentum p_a = -p_d | KE = p^2/2m

KE_a = Q_a•(1-4/A)

Recoil energy of daughter is:
KE_d ~~ Q_a•(4/A)

A is for the parent nucleus, hence (4/A) is approx 0. — Hence a majority of the energy is going towards the alpha particle

**Newtonian mechanics works well for alpha (relatively low speeds) — need QM for beta decay

Question 5

What does a +ve Q_a indicate and what is the equation?

Answer 5

Q_a > 0 is indicative of nuclei that undergo spontaneous alpha decay

Q_a = B(A-4, Z-2) + B(4,2) - B(A,Z)

Question 6

When is alpha emission most dominant? What is the approximate range of MeV for alpha particles?

Answer 6

Alpha decay dominates when A > 200 | alpha emission increases the BE of the whole system

Approximate MeV range 3-7

Coulomb repulsion term significantly decreased as reduction in charge of parent nucleus

Question 7

Working out calculations involving a potential barrier

Answer 7

r = r0(A1^1/3 + A2^1/3)
{r0 = 1.2fm
• tells you separation of daughter nuclei at point of decay
• This may be used to calculate the theoretical Coulomb barrier potential height, CP

CP = (k•Q1•Q2)/r
{r = distance above; Q charges of each nuclei•e; k is Coulomb’s constant 8.99x10^9Nm2
•this formula can also be rearranged to find r ,if given the KE of the alpha particle, which would tell you width of the barrier (minus the first r value obtained above)

Question 8

Wave-particle solutions in reference to QT in alpha decay

Answer 8

QM states that particles can behave like a wave. Hence a particle can have a wave function explain its behaviour

In quantum tunnelling for alpha decay, there are 3 regions:
1) inside the parent nucleus
2) the potential well
3) the daughter and alpha particles

We can use the TISE Schrödinger equation to find wave like solutions in regions 1 and 3 and decaying exponential solutions in region 2

Question 9

How does one compute the half-life of alpha decay? What assumptions are made?

Answer 9

1) need to know the probability of getting through the barrier, T.

2) need to know the frequency the alpha particle is trying to escape

Time between collisions = distance/speed = 2•R/V
Frequency, f = 1/period(T) = V/2•R

Decay rate = f•T = ln(2)/ t1/2

Assumptions
1) v&raquo_space; E (not great assumption)
2) modelled barrier as rectangle (assuming v has same value across entire width, which in reality it tails off)