# 6. Radioactivity Flashcards

Radioactivity

Definition

-spontaneous change of unstable nuclei to a more stable state

Valley of Stability

- the stable nuclides lie on a stepped line on the nuclide map (in the N, Z plane), this is the valley of stability
- the further we go from the valley, the less stable the nuclides
- this related to the nuclides binding energy: less tightly bound nuclides are less stable and will tend to decay to nuclides with a higher binding energy

Decay Constant

- since nuclear decay is governed by quantum mechanics, there is an element of randomness involved
- we cannot know in advance when a particular nucleus will decay
- we can only measure the probability of decay in a given time interval
- this is the decay constant λ

Radioactivity Equations

dN/dt = -λN

=>

N(t) = N(0)*e^(-λt)

Lifetime

Definition

-we define the lifetime of the nuclide as:

τ = 1/λ

-this is the time taken for N to drop by a factor of e

Half Life

t1/2 = τ*ln2

-this is the time taken for N to drop by a factor of 2

Spontaneous Deacy

-since the decay is spontaneous with no input of energy it can only occur if energetically favoured, i.e if energy is released during the process

-so we require the mass of the initial (parent) nuclide to be greater than the sum of the masses of the final (daughter) nuclides:

M(A,Z) > Σ M(Ai,Zi)

Half Life and the Valley of Stability

-half-lives/lifetimes decrease the further we go from the valley of stability

Beta Minus Deacy

n -> p + e- + ant ie- neutrino

-with a half-life of around 10 mins

Beta Plus Decay

p -> n + e+ + (e- neutrino)

How is beta plus decay possible spontaneously?

-for a free proton, beta plus decay will never happen, however in some cases it can be energetically favourable

-consider the semi-empirical mass formula, rewrite to eliminate N, N=A-Z

-collect like terms in Z:

M(A,Z) = c1*A + c2*Z + c3*Z² + c4

-since beta decay does not alter A, we can see that the relevant nuclide masses for this decay mode are quadratic in Z, there are two cases to consider

Beta Decay

Semi-Empirical Mass Formula

-consider the semi-empirical mass formula, rewrite to eliminate N, N=A-Z

-collect like terms in Z:

M(A,Z) = c1*A + c2*Z + c3*Z² + c4c1 = mn - av + as*A^(-1/3) + aa/4

c2 = mp + me - mn - aa

c3 = aa/A + ac/A^(1/3)

c4 = 𝛿/A^(1/2)

-with 𝛿=±11.2MeV/c²

-since beta decay doesn’t alter A, we can see that the relevant nuclide masses for this decay mode are quadratic in Z

Beta Decay

Case 1) A is odd

-the nuclear masses of isobars fall on a parabola

-the minimum of this parabola lies at:

Z’stab = -c2/2c3

-since Z can only take integer values, the actual minimum Zstab, is the integer closest to Z’stab

-those nuclides with non-minimal mass for a given A will under go β-decay:

β- decay if Z < Zstab

β+ decay if Z > Zstab

Beta Decay

A is even

- if A is even, then either N and Z are both even or they are both odd
- but β+ or β- decay in this case switches between the two possibilities
- so there are two parabolas to consider, the lower-mass parabola has N, Z even
- each decay jumps from one to the other which can lead to more than one stable nuclide on the same curve since it is each jump that must be individually energetically favoured rather than a whole chain of such jumps

Discovery of the Neutrino

- during β decay the nucleus changes atomic number and a β± particle is emitted, these are the only two products detected
- two-body decays have a very well defined momentum Ek, e.g. for α decay, but β decay was found to have a much broader spectrum of possible momenta
- this is what would be expected from a three body decay and observations showed that β decay conserved mass number and charge but not spin or momentum
- there must be an unobserved third particle produced, neutral, with small mass and spin 1/2
- this is how the existence of the neutrino was predicted