Elas%c  Sca)ering  Nuclear  Physics  Group  Seminar  

February  12,  2015  S.  Bültmann  

Scattering Cross-Sections •  Describe the interaction (rate ) of particles •  Consider a beam of cross-sectional area A and particle

density nA with particles having an average velocity va

•  The particle flux describes the number of particles hitting the target per unit area and unit time

•  Consider a target of thickness d and particle density nb

•  The geometric cross-section is defined as

with

Φa = na∙ va = A Na ∙

N ∙

σb = Φa ∙ Nb

N ∙ Nb = nb ∙ A ∙ d =

∙ nb ∙ d N ∙

Na ∙

Luminosity •  This geometric scattering cross-section can be re-written

•  Here the luminosity is defined by

L = Φa ∙ Nb = ∙ nb ∙ d = na ∙ va ∙ Nb Na ∙

σb = Φa ∙ Nb

N ∙ =

L N ∙

Luminosity •  Luminosity for colliding beams with N bunches is given

by

with –  beam velocities v –  collider ring circumference C –  beam cross-section at collision point 4π σx σy

L = 4π σx σy

Na ∙ Nb ∙ N ∙ v / C

Interaction Rate •  The interaction rate depends on the interaction potential

described by an operator H •  The operator H transforms the initial wave function ψi

into the final wave function ψf •  The transition matrix element (transition probability

amplitude) is given by

M = ⟨ ψf ∣ H ∣ψi ⟩

Interaction Rate •  Consider a particle scattered into a volume V and

momentum interval p’ and p’ + dp’ •  The interaction rate also depends on the number of final

states available •  Ignoring spin, the number of final states is given by

with the volume of phase space occupied by each particle

•  Energy and momentum are connected by

d n (p’) = d p’ V ∙ 4π p’ 2 ( 2π ħ ) 3

( 2π ħ ) 3

d E’ = v’ d p’

Interaction Rate •  The density of final states in the energy interval d E’ is

given by

•  Fermi’s Golden Rule connects the interaction rate per target particle and per beam particle with the transition matrix element and the density of final states

ρ (E’) = = d n (E’) d E’

V ∙ 4π p’ 2 v’ ∙ ( 2π ħ ) 3

Wfi = Na ∙ Nb N (E ) ∙

= ∣ M ∣2 ∙ ρ (E’) ħ

2π

Feynman Diagrams •  Describe particle scattering through interacting currents •  A pictorial way of describing fermion and boson

interactions •  Example electron-electron scattering

Transition Amplitudes •  Initial and final state particles have wave functions •  Vertices have dimensionless coupling constants

–  Electromagnetic interactions ⇒ √α ∝ e –  Strong interactions ⇒ √α ∝ √αs

•  Virtual particles have propagators –  Virtual photon has propagator ∝ 1/q 2 –  Virtual boson of mass m has propagator ∝ 1/(q 2 − m 2)

•  Transition amplitudes for electron-electron or electron-nucleon scattering have a virtual photon as propagator and two vertices, resulting in

with q 2 the 4-momentum transfer squared

M ∝ e 2 / q 2

Electron-Nucleon Scattering Kinematics

•  Electron with incident 4-momentum k = (E, 0, 0, E) and scattered 4-momentum k’ = (E’, E’ sin θ, 0, E’ cos θ)

•  Nucleon with incident 4-momentum P = (M, 0, 0, 0) and scattered 4-momentum P’ = (M + ν, −E’ sin θ, 0, E−E’ cos θ)

•  Exchanged virtual photon with 4-momentum q = k − k’ •  Invariant virtual photon mass squared is given by

with θ the scattering angle in the lab frame E the incident electron energy E’ the scattered electron energy ν = E – E’ the energy transfer M the nucleon rest mass

q 2 = − Q 2 = − 4 E E’ ∙ sin 2 ( θ/2 )

Electron-Nucleon Scattering Kinematics

•  Invariant mass squared of the final state nucleon is given by

•  In elastic scattering W = M , which yields Q 2 = 2 M ν

W 2 = M 2 + 2 M ν − Q 2

Scattering Cross-Sections •  Rutherford cross-section for scattering of point-like

particles (no recoil)

•  Mott cross-section accounting for spin s of electron

•  Helicity is conserved for relativistic particles

= = Z 2 α 2 ( ħ c) 2 4 E 2 ∙ sin 4 ( θ/2 )

d σ d Ω

⎛ ⎞ ⎝ ⎠ R

= ∙ (1 − β 2 sin 2 ( θ/2 ) ) d σ d Ω

⎛ ⎞ ⎝ ⎠ M

d σ d Ω

⎛ ⎞ ⎝ ⎠ R

with β = v / c  

h = ∣s∣ ∙ ∣p∣

s ∙ p

*

4 Z 2 α 2 ( ħ c) 2 E’ 2

Q 4

Nuclear Form Factors •  Experimental data only agree with Mott cross-section for

very low q •  The spatial extension of nuclei can be accounted for by

form factors F (q 2)

•  The form factors are the Fourier transform of the charge distribution f (x)

(Born approximation and no recoil)

= ∙ ∣F (q 2) ∣ 2 d σ d Ω

⎛ ⎞ ⎝ ⎠ exp

d σ d Ω

⎛ ⎞ ⎝ ⎠ M

F (q 2) = ∫ e i q x / ħ f (x) d3x

*

Nuclear Form Factors •  For spherical symmetric cases the charge distribution

only depends on the radius r

F (q 2) = 4π ∫ f (r) r 2 dr sin ∣q∣ r / ħ ∣q∣ r / ħ

Nuclear Form Factors

Nuclear Form Factors

Nucleon Form Factors •  Scattering of nucleons with mass of about 938 MeV

requires recoil effects to be accounted for

= ∙ d σ d Ω

⎛ ⎞ ⎝ ⎠ M

d σ d Ω

⎛ ⎞ ⎝ ⎠ M

* E’ E

E’ E

= ∙ (1 − β 2 sin 2 ( θ/2 ) ) 4 Z 2 α 2 ( ħ c) 2 E’ 2

Q 4

Nucleon Form Factors •  Electron interaction with the magnetic moment μ of the

nucleon needs to be included also •  The magnetic moment of a charged, point-like spin-1/2

particle is given by

•  We obtain for the scattering cross section

μ = g 2 M

e ħ 2 with g = 2

= ∙ ( 1 + 2 τ tan 2 ( θ/2 ) ) d σ d Ω

⎛ ⎞ ⎝ ⎠

d σ d Ω

⎛ ⎞ ⎝ ⎠ M

with τ = 4 M 2 c2

Q 2

Nucleon Form Factors •  The magnetic moments of nucleon deviates from 2

because of the composite structure •  Measured values are

μp = μN = +2.79 ∙ μN for the proton gp

2

μn = μN = −1.91 ∙ μN for the neutron gn

2

with μN = 2 Mp

e ħ

Nucleon Form Factors •  Charge and current distribution can be described by form

factors as for nuclei •  Two form factors are needed to describe the charge and

magnetic distributions •  The cross-section is given by the Rosenbluth formula

•  As Q2 → 0 also τ → 0 and only GE2 (0) remains above

= ∙ ( + 2 τ GM2 (Q2) tan 2 ( θ/2 ))

d σ d Ω

⎛ ⎞ ⎝ ⎠

d σ d Ω

⎛ ⎞ ⎝ ⎠ M

GE2 (Q2) + τ GM

2 (Q2)

1 + τ

with τ = 4 M 2 c2

Q 2

Nucleon Form Factors •  For the form factors we expect in the limit Q2 → 0

GE p (0) = 1  

GM p (0) = +2.79  

GE n (0) = 0  

GM n (0) = −1.91  

Measuring Nucleon Form Factors

•  To determine the two form factors independently, we need to measure the cross section at fixed values of Q2 and vary the beam energy (scattering angle)

= + 2 τ GM2 tan 2 ( θ/2 )

d σ d Ω

⎛ ⎞ ⎝ ⎠

d σ d Ω

⎛ ⎞ ⎝ ⎠ M

GE2 + τ GM

2

1 + τ

Measuring Nucleon Form Factors

•  Result of measurements revealed dipole structure

GE p (Q2) = = = G

dipole (Q2)  GM

p (Q2)   GM n (Q2)  

−1.91  2.79  

G dipole (Q2)  

= ( 1 + Q2 / (0.71 GeV2) ) −2

Nucleon Charge Distribution •  Spatial extend of nucleon charge can be found from

measured form factors •  Fourier transform only applicable at low Q2 •  Dipole form factor corresponds to a diffuse and

exponentially falling charge distribution

•  The yields for the proton charge radius

ρ (r) = ρ (0) e with a = 4.27 fm−1 − a r

√ ⟨ r 2⟩p = 0.86 fm