Elas%c’Scaering’skuhn/NucPhys/Elastic.pdf · – Virtual photon has propagator ∝ 1/q 2! –...
Transcript of Elas%c’Scaering’skuhn/NucPhys/Elastic.pdf · – Virtual photon has propagator ∝ 1/q 2! –...
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