Today’s lecture: Radioactivity
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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) + 4He
Beta decay
(Z,N) => (Z+1,N-1) + e- +
Gamma decay (Z,N)* => (Z,N) + γ
Particle decays
n => p + e- + ν
K+ => μ+ + νμ
π0 => 2γ
Radioactive decay
• Intensity of decays:
N ~ # of nuclei (Z,N)
ω ~ probability of a decay
t ~ 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 Ra
Becquerel [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)=0.36
β -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+x
2→3+y
Real example:
Measurement of transition rate
• If half-life is reasonably short:
because:
measure I(t) or N(t) a.k.a. “decay curve”
(no need to know N(0) or I(0) !!!)
• If half-life is very long:
• 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:
Note: in reality it may not be trivial to deduce N(t) and N0 (or I(t) and I0)
2nd Note: limits of a particular dating method are given by τ and the
accuracy of determining N and N0 (or I 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 energy
Γ 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
• What is the probability that projectile hits something ?
Beam particle
(projectile)
Target particles
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 nucleus
THEN: • 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 target
2) Inelastic scattering
3) 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
• Let’s put detector with cross‐section area A at angle Θ and distance r
from the target
THEN: A
where: r
Θ
target
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 number
A’ is atomic weight of the target material (NOT mass 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 m
and standard deviation of √m.