Today s lecture: Radioactivity

Transcription

1

2 Today s lecture: Radioactivity radioactive decay mean life half life decay modes branching ratio sequential decay measurement of the transition rate radioactive dating techniques

3 Radioactive decay spontaneous transition of particle or atomic nucleus into a more energetically favorable state Examples: Alpha decay (Z,N) => (Z-2,N-2) + 4 He Beta decay (Z,N) => (Z+1,N-1) + e - + Gamma decay (Z,N)* => (Z,N) + γ Particle decays n => p + e - + ν K + => μ + + ν μ π 0 => 2γ

4 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 Bq

5 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 t 1/2 when half of the nuclei decay : => therefore N=N 0 /2 and

6 Decay modes Unstable particle or nucleus may have several different decay modes Example: τ = 87min In general:

7 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 212 Bi? α -decay : BR(α)=0.36 β -decay : BR(β)=0.64

8 Production of radioactive material Stable nuclides undergo such nuclear reaction that the product is unstable (e.g. accelerators, nuclear reactors, cosmic rays) 14 C produced in upper layers of atmosphere 18 F 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

9 Sequential decay If daughter nucleus is also unstable it can undergo another decay => decay chain General example: x y Real example:

10 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

11 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

12 Radioactive dating solving survival equation for t: Note: in reality it may not be trivial to deduce N(t) and N 0 (or I(t) and I 0 ) 2 nd Note: limits of a particular dating method are given by τ and the accuracy of determining N and N 0 (or I and I 0 ). Examples: radiocarbon dating (t 1/2 ( 14 C)=5730 y) => age of organic remains potassium-argon dating (t 1/2 ( 40 K)=1.248x10 9 y) => sediments & lava uranium-lead dating => age of mineral zircon (ZrSiO 4 ), age of Earth

13 Estimation of the age of the Earth Problem: How old is the Earth? Solution: let s assume that initial concentrations of 235 U and 238 U were equal at the time of earth creation let s consider current natural uranium composition (i.e. 0.72% of 235 U and 99.28% of 238 U) Earth is 6 GY old!!!

14 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: E 0 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.

15 Decay and the uncertainty principle main peak consistent with production of particle of mass 770 MeV => ρ 0 ρ 0 decays into to pions π + + π FWHM(=153 MeV) of the peak indicates mean life of ρ 0 to be τ ~4x10 24 s

16 Collisions and cross-sections Macroscopic shooting Microscopic shooting

17 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 πb 2 σ TOT then we have a reaction!!! σ TOT is the effective (cross section) area of the target

18 Collisions and cross-sections let s suppose we have n atoms per cm 3 in a target of thickness x let s assume that σ is the cross section area of a single target nucleus THEN: AND: mean free path for a projectile moving through target is # 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 :

19 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 σ e, σ i and σ a are called partial cross-sections and represent probability of reaction progressing through a particular reaction channels

20 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

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

22 Cross-sections (one last time) Cross sections have dimensions of an effective area (but small compared to human scale) Usual units fermis squared : 1fm 2 = m 2 = cm 2 OR: barns : 1b = fm 2 = cm 2 target thicknesses expressed in mass per unit area : [g cm 2 ] how to calculate n (atoms per cm 3 in a target)?: ρ is target density; N A is Avogadro s number A is atomic weight of the target material (NOT mass number)

23 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.