Cross Sections

Nuclear Physics Lectures
Cross-Sections
Why concept is important
Learn about dynamics of interaction and/or constituents (cf Feynman’s watches).
Needed for practical calculations.
Experimental Definition
How to calculate σ
Fermi Golden Rule
Breit-Wigner Resonances
QM calculation of Rutherford Scattering

Nuclear Physics Lectures
Definition of σ
a+b?x
Effective area for reaction to occur is σ
Beam a
dx
Na
Na(0) particles type a/unit time hit target b
Nb atoms b/unit volume
Number /unit area= Nb dx
Probability interaction = σ Nbdx
dNa=-Na Nb dx σ
Na(x)=Na(0) exp(-x/λ) ; λ=1/(Nb σ)

Nuclear Physics Lectures
Reaction Rates
Na beam particles/unit volume, speed v
Flux F= Na v
Rate/target b atom R=Fσ
Thin target x<<λ: R=(NbT) F σTotal
This is total cross section. Can also define differential cross sections, as a function of reaction product, energy, transverse momentum, angle etc.
dR(a+b?c+d)/dE=(NbT) F dσ(a+b?c+d) /dE

Nuclear Physics Lectures
Breit-Wigner Line Shape
Start with NR Schrödinger equation:
X by φ*n and integrate
Start in state m ? exponential decay

Nuclear Physics Lectures
Breit-Wigner Line Shape - 2
For

Nuclear Physics Lectures
Breit-Wigner Line Shape -3
Normalised Breit-Wigner line shape
Q: where have you seen this shape before?
We will see this many times in NP and PP.

Nuclear Physics Lectures
Breit-Wigner Resonance
Important in atomic, nuclear and particle physics.
Uncertainty relationship
Determine lifetimes of states from width.
, Γ=FWHM;

Nuclear Physics Lectures
Fermi Golden Rule
Want to be able to calculate reaction rates in terms of matrix elements of H.
Warning: We will use this many times to calculate σ but derivation not required for exams, given here for completeness.

Nuclear Physics Lectures
Discrete ? Continuum
Decays to a channel i (range of states n). Density of states ni(E). Assume narrow resonance

Nuclear Physics Lectures
Cross Section
Breit Wigner cross section.
Definition of σ and flux F:

Nuclear Physics Lectures
Breit-Wigner Cross Section
Combine rate, flux & density states ?

Nuclear Physics Lectures
Breit-Wigner Cross Section
n + 16O? 17O

Nuclear Physics Lectures
Low Energy Resonances
n + Cd total cross section.
Cross section scales σ ~ 1/E1/2 at low E.
B-W: 1/k2 and Γ~n(E)~k

Nuclear Physics Lectures
Rutherford Scattering 1

Nuclear Physics Lectures
Rutherford Scattering 2

Nuclear Physics Lectures
Rutherford Scattering 3
Fermi Golden Rule:

Nuclear Physics Lectures
Rutherford Scattering 4
Compare with experimental data at low energy
Q: what changes at high energy ?
pi
pf

Nuclear Physics Lectures
Low Energy Experiment
Scattering of α on Au & Ag ? agree with calculation assuming point nucleus
Sin4(θ/2)
dN/dcosθ