Today’s lecture: Radioactivity

• radioactive decay• mean‐life• half‐life• decay modes• branching ratio• sequential decay• measurement of the transition rate• radioactive dating techniques

Radioactive decay

• spontaneous transition of particle or atomic nucleus into a more energetically favorable state

Examples:Alpha decay (Z,N) => (Z-2,N-2) + 4HeBeta decay (Z,N) => (Z+1,N-1) + e- +Gamma decay (Z,N)* => (Z,N) + γParticle decays n => p + e- + ν

K+ => μ+ + νμπ0 => 2γ

Radioactive decay

• Intensity of decays: N ~ # of nuclei (Z,N)ω ~ probability of a decayt ~ time

Note: Decay probability ω constant in time and equal for all mother nuclei => # of mother nuclei decreases exponentially within statistical limits

Units:Curie [Ci] = number of decays equal to disintegrations in 1 sec in 1g of pure RaBecquerel [Bq] = 1 decay in 1 second

1 Ci = 3.7 x 1010 Bq

Mean-life vs. Half-life

•Let’s define τ= 1/ω

•Let’s consider average decay time ‹t› :

where u=t/τ

τ is the “mean life”

•Let’s consider time t1/2 when half of the nuclei decay :

=> therefore N=N0/2 and

Decay modes

• Unstable particle or nucleus may have several different decay modes

Example:

τ = 87min

In general:

Branching ratios

BR = fraction of particles which decay by an individual decay mode with respect to the total number of particles which decay

Problem: What are the half-lives of individual decay modes for 212Bi ?

α -decay : BR(α)=0.36β -decay : BR(β)=0.64

Production of radioactive material

• Stable nuclides undergo such nuclear reaction that the product is unstable(e.g. accelerators, nuclear reactors, cosmic rays)

14C produced in upper layers of atmosphere

18F produced by bombarding oxygen target with accelerated protons (for PET scanning)

In general:

where p is the production rate

If N=0 at t=0

Sequential decay

• If daughter nucleus is also unstable it can undergo another decay => decay chain

General example:1 → 2 + x2 → 3 + y

Real example:

Measurement of transition rate

• If half-life is reasonably short:

• If half-life is very long:

measure I(t) or N(t) a.k.a. “decay curve”(no need to know N(0) or I(0) !!!)

because:

• then ω→0 and decay curve is almost flat• we have to determine N(0) independently and measure I

Radioactivity & nuclear collisions

• radioactive dating techniques

• estimation of the age of the Earth 

• decay and the uncertainty principle

• collisions and cross‐sections

• partial and differential cross‐section

• probabilities, expectations, fluctuations

Radioactive dating

solving “survival equation” for t:

2nd Note: limits of a particular dating method are given by τ and the accuracy of determining N and N0 (or I and I0).

Note: in reality it may not be trivial to deduce N(t) and N0 (or I(t) and I0)

Examples:• radiocarbon dating (t1/2(14C)=5730 y) => age of organic remains• potassium-argon dating (t1/2(40K)=1.248x109 y) => sediments & lava• uranium-lead dating => age of mineral zircon (ZrSiO4), age of Earth

Estimation of the age of the Earth

Problem: How old is the Earth ?

Solution:

• let’s assume that initial concentrations of 235U and 238U were equal at the time of earth creation• let’s consider current natural uranium composition(i.e. 0.72% of 235U and 99.28% of 238U)

Earth is 6 GY old !!!

Decay and the uncertainty principle

• decaying state is a system with uncertainty of lifetime (i.e. mean life τ) that is bound to the uncertainty of total energy given by:

• uncertainty in energy of an excited state is reflected in the line shape  of the radiation emitted in decay to the ground state.

probability of measuring energy E given by Lorentzian distribution:

E0 is the central energyΓ is full width in half height (FWHM)

• because Γ is FWHM then  and thus

If mean‐life τ of some state is too short, we still can measure Γ and then determine τ.

Decay and the uncertainty principle

• main peak consistent with production of particle of mass 770 MeV => ρ0

ρ0 decays into to pions π++ π‐

FWHM(=153 MeV) of the peak indicates mean‐life of ρ0 to be τ ~4x10‐24s

Collisions and cross-sections

Macroscopic shooting

Microscopic shooting

Collisions and cross-sections

Beam particle(projectile)

Target particles

• What is the probability that projectile hits something ?

Note:Hit = something happens to projectile (i.e. scattered or absorbed)

Simplest interpretation: if πb2≤σTOT then we have a reaction

!!! σTOT is the effective (cross section) area of the target

Collisions and cross-sections

• let’s suppose we have n atoms per cm3 in a target of thickness x• let’s assume that σ is the cross section area of a single target nucleusTHEN: 

• mean free path for a projectile moving through target is 

AND: 

• # of unaffected particles of beam decreases deeper in the target: 

beam is exponentially attenuated as it traverses through the target

• # of collisions in the target within the thickness x:

• for thin targets when            :

Total and partial cross-sections

• individual nuclear collisions can have generally different outcome

1) Elastic scattering by the target2) Inelastic scattering3) Absorption by the target

• total collision cross‐section is a sum of individual probabilities of all the possible reaction outcomes 

• σe, σi and σa are called partial cross-sections and represent probability of reaction progressing through a particular reaction channels

Differential cross-sections

• Let’s consider elastic  scattering: – projectile can scatter at any angle with respect to incident direction

cross‐section per unit solid angle with respect to Θ

target

Θ

r

A

• Let’s put detector with cross‐section area A at angle Θ and distance rfrom the target

where: 

THEN: 

Differential elastic scattering cross-sections (e-+Fr)

Cross-sections (one last time)

• Cross sections have dimensions of an effective area (but small compared to human scale) 

Usual units “fermis squared” : 1fm2 = 10‐30m2 = 10‐26

cm2

OR: “barns” : 1b = 10+2fm2 = 10‐24cm2

• target thicknesses expressed in “mass per unit area” : [g∙cm‐2]

• how to calculate n (atoms per cm3 in a target)?:

ρ is target density;NA is Avogadro’s numberA’ is atomic weight of the target material (NOTmass number)

Probabilities, expectations, and fluctuations

• let’s suppose we measure for 10s decay of isotope with τ»10s

THEN: according to decay law we “expect”m decays:

N is # of atoms in the measured sample

• Problem: How often will we measure exactly m?

In general:• Decay is a random process, with each measurement the observed number of decays will fluctuate.• Probability P that we observe n decays when m decays are expected is described by Poisson distribution:

• for large m distribution approximates to Gaussian with mean of mand standard deviation of √m.