Introduction to Quantum Collision Theory Pierre Capel 16 July 2015 - - PowerPoint PPT Presentation

Introduction to Quantum Collision Theory

Pierre Capel 16 July 2015

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Quantum Collisions

Quantum Collisions

Quantum collisions used to study the interaction between particles/nuclei/atoms. . . analyse the structure of particles/nuclei/atoms. . . measure reaction rates of particular interest (stars, nuclear reactors, production of radioactive isotopes. . . ) Measurement scheme :

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Quantum Collisions

Reaction types

Various reactions can happen :

1

a + b → a + b

(elastic scattering)

2

→ a + b∗

(inelastic scattering)

3

→ c + f + b (breakup)

4

→ d + e

(rearrangement or transfer) Examples :

1

11Be + 208Pb → 11Be + 208Pb

(elastic scattering)

2

→ 11Be∗(1/2−) + 208Pb

(inelastic scattering)

3

→ 10Be + n + 208Pb

(breakup)

4

→ 10Be + 209Pb

(transfer)

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Quantum Collisions

Energy conservation

Total energy is conserved :

ma c2 + mb c2 + incident kinetic energy

= mass of products c2 + kinetic energy The Q value of a reaction is

Q = ma c2 + mb c2 − mass of products c2

A channel will be open if the incident kinetic energy > −Q

• therwise the channel is closed

Q > 0 : exoenergetic, always open Q < 0 : endoenergetic, requires a minimal incident kinetic energy

The elastic channel is always open (Q = 0)

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Notion of cross section

Cross section

For an incident flux Fi on N particles in the target, if ∆n of particle/events detected in direction Ω = (θ, ϕ) per unit time within the solid angle ∆Ω

∆n = Fi N ∆σ [∆σ] = surface ; unit : barn 1 b = 10−24cm2

The differential cross section

dσ dΩ = lim

∆Ω→0

∆σ ∆Ω = ∆n FiN∆Ω

Taking Z as the beam axis, dσ/dΩ depends only on θ by symmetry

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Notion of cross section

Theoretical framework

Let us consider particles a and b of mass ma and mb interacting through potential V(R), where R = Ra − Rb is the a-b relative coordinate The Hamiltonian reads

H = Ta + Tb + V(R),

where

Ta = p2

a

2ma = −2∆Ra 2ma Tb = p2

b

2mb = −2∆Rb 2mb

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Notion of cross section

Change to cm and a-b relative motion

The coordinate change

Rcm = (maRa + mbRb)/M R = Ra − Rb

with M = ma + mb, the total mass, leads to

H = Tcm + TR + V(R),

where

Tcm = P2

cm

2M = −2∆Rcm 2M TR = P2 2µ = −2∆R 2µ ,

with µ = mamb/M the reduced mass of a and b

H is then the sum of two Hamiltonians : Hcm(Rcm) + H(R)

Hence the two-body wave function factorises

Ψtot(Ra, Rb) = Ψcm(Rcm) Ψ(R)

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Notion of cross section

cm motion

The cm wave function is solution of

Hcm Ψcm(Rcm) = Ecm Ψcm(Rcm),

where

Hcm = P2

cm

2M

is the Hamiltonian of a free particle of mass M The cm motion is described by a plane wave

ΨKcm(Rcm) = (2π)−3/2eiKcm·Rcm

with Ecm = 2K2

cm/2M

The factor (2π)−3/2 is chosen such that

ΨK′

cm|ΨKcm = δ(Kcm − K′cm) 8 / 30

Stationary Scattering States

Stationary Scattering States

The Hamiltonian of the a-b relative motion reads

H = TR + V(R)

A stationary scattering state ΨK(R) is solution of

H ΨK = E ΨK

where E = 2K2/2µ with the asymptotic behaviour

Ψ

K ˆ

Z(R) −→

R→∞(2π)−3/2

• eiKZ + fK(θ) eiKR

R

• ,

with Z chosen as the beam axis

fK is the scattering amplitude

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Stationary Scattering States

Physical interpretation

To interpret the stationary scattering state

Ψ

K ˆ

Z(R) −→

R→∞(2π)−3/2

• eiKZ + fK(θ) eiKR

R

• ,

let us recall the probability current

J(R) = 1 µℜ[Ψ∗(R) P Ψ(R)]

The plane wave describes the incoming current

Ji(R) = (2π)−3/2K µ ˆ Z = (2π)−3/2 v ˆ Z

where v is the a-b relative velocity The spherical wave fK(θ) eiKR

R describes the scattered current

J s(R) = (2π)−3/2 v |fK(θ)|2 1 R2 ˆ R + O( 1 R3)

is purely radial at R → ∞ ; directed outwards ; ∝ v but varies with θ

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Stationary Scattering States

Physical interpretation

Incoming wave : Scattered wave :

eiKZ → Ji(R) ∝ v ˆ Z fK(θ) eiKR R → J s(R) ∝ v | fK(θ)|2 1 R2 ˆ R

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Stationary Scattering States

Theoretical scattering cross section

We can assume the incoming flux

Fi = C Ji

The scattered flux in direction Ω is then

Fs = C Js

For one scattering nucleus, the number of event per unit time detected in direction Ω reads

dn = Fs dS = C Js R2dΩ ⇒ dσ dΩ = dn Fi dΩ = R2Js Ji = |fK(θ)|2

The scattering amplitude fK contains all information about V

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Partial-wave expansion and phaseshift

Partial-wave expansion

If the potential does not depend on Ω, i.e. V(R),

[H, L2] = [H, LZ] =

the angular motion is described by spherical harmonics

ψKLM(R) = 1 R uKL(R) Y M

L (Ω),

where uKL is solution of the radial equation

d2 dR2 − L(L + 1) R2 − 2µ 2 V(R) + K2

• uKL(R) = 0

This can be solved using numerical techniques

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Partial-wave expansion and phaseshift

Phase shift δL

The scattering amplitude fK is obtained from the asymptotics of ΨK If we assume R2V(R) −→

R→∞ 0, uKL(R) −→ R→∞ uas KL(R), which is solution of

d2 dR2 − L(L + 1) R2 + K2

• uas

KL(R) = 0

whose solutions

uas

KL(R)

= A KR jL(KR) + B KR nL(KR) −→

R→∞

A sin(KR − Lπ/2) + B cos(KR − Lπ/2)

where jL and nL are regular and irregular spherical Bessel functions Posing A = C cos δL and B = C sin δL

uas

KL(R) −→ R→∞C sin(KR − Lπ/2 + δL)

C is just a normalisation factor ; δL is the phaseshift

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Partial-wave expansion and phaseshift

Scattering matrix S L

uas

KL(R) −→ R→∞C sin(KR − Lπ/2 + δL)

better interpreted in terms of incoming and outgoing waves :

uKL(R) −→

R→∞

iCe−iδL 2

• e−i(KR−Lπ/2) − S L ei(KR−Lπ/2)

where

S L = e2iδL

is the scattering matrix The outgoing wave is shifted from the incoming wave by 2δL due to the effect of V ⇒ used to compute fK

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Partial-wave expansion and phaseshift

Scattering amplitude

The scattering wave function can be expanded in partial waves

(2π)3/2Ψ

K ˆ

Z(R) = 1 KR

∞

• L=0

cL uKL(R) Y0

L(Ω)

−→

R→∞

eiKZ + fK(θ)eiKR R

Since eiKZ

−→

R→∞ ∞

• L=0

(2L + 1)iLPL(cos θ) i 2KR

• e−i(KR−Lπ/2) − ei(KR−Lπ/2)

and

uKL −→

R→∞

ie−iδL 2

• e−i(KR−Lπ/2) − S L ei(KR−Lπ/2)

comparing the incoming waves we obtain cL =

√ 4π √ 2L + 1iLeiδL

and deduce the scattering amplitude from S L = e2iδL

fK(θ) = 1 2iK

∞

• L=0

(2L + 1)(S L − 1)PL(cos θ)

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Partial-wave expansion and phaseshift

Scattering cross section

dσ dΩ = | fK(θ)|2 =

• 1

2iK

∞

• L=0

(2L + 1)(S L − 1)PL(cos θ)

• 2

After integration over Ω the total scattering cross section reads

σ = 4π K2

∞

• L=0

(2L + 1) sin2 δL

Each partial wave contributes to σ but with variable importance

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Partial-wave expansion and phaseshift

Contribution of partial waves

σ = 4π K2

∞

• L=0

(2L + 1) sin2 δL

Centrifugal barrier L(L+1)

R2

ensures δL −→

L→∞ 0

⇒ limited sum

At very low E, only L = 0 contributes and σ = 4π

K2 sin2 δ0

At large E many partial waves must be included to reach convergence (⇒ low-energy method)

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Partial-wave expansion and phaseshift

Resonance

Resonance ≡ significant variation of a cross section on a short energy range In elastic scattering, contribution of partial wave L

σL = 4π K2 sin2 δL

small if δL ∼ nπ (n ∈ Z) large if δL ∼ π/2 If δL goes quickly from 0 to π → rapid increase and decrease of σL i.e. resonance structure Definite L ⇒ quantum numbers and parity similar to bound state

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Optical model

Reaction cross section

So far we have described only elastic scattering Other channels can be open, like transfer : a + b → d + e We can define a differential cross section for these other channels

dσ dΩ(a + b → d + e) = lim

R→∞

R2Jd+e Ji

The sum of all channels but elastic scattering (inelastic, transfer, breakup,. . . ) gives the reaction cross section

σr =

• channel\a+b

σ(a + b → channel)

The interaction cross section corresponds to all channels but elastic and inelastic scattering

σI =

• channel\(a+b)∪(a+b∗)∪(a∗+b)∪(a∗+b∗)

σ(a + b → channel)

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Optical model

Optical Model

Using real scattering potential V implies that ∇J = 0

⇔ flux stays in elastic channel

To simulate other channels, use complex potential

Uopt(R) = V(R) + iW(R) ⇒ − 2

2µ∆Ψ + UoptΨ = EΨ

∇J = ∇1 µℜ{[Ψ∗ P Ψ]} = −i 2µ∇[Ψ∗ ∇Ψ − Ψ ∇Ψ∗] = −i 2µ[∇Ψ∗ · ∇Ψ − ∇Ψ · ∇Ψ∗ + Ψ∗ ∆Ψ − Ψ ∆Ψ∗] = i 2µ[Ψ∗ 2µ 2 UoptΨ − Ψ 2µ 2 U∗

• ptΨ∗]

= 2 W(R) |Ψ(R)|2

To have absorption from elastic channel W(R) ≤ 0

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Optical model

Partial-wave expansion

The optical potential Uopt leads to a complex phaseshift :

δL = ℜ(δL) + iℑ(δL)

with ℑ(δL) ≥ 0 because W ≤ 0

⇒ S L = ηL e2iℜ(δL)

where

ηL = e−2ℑ(δL) < 1

simulates the absorption from the elastic channel in

fK(θ) = 1 2iK

∞

• L=0

(2L + 1)(ηL e2iℜ(δL) − 1)PL(cos θ)

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Optical model

Absorption cross section

The absorption cross section σa corresponds to all other channels simulated by Uopt :

σa = −

• ∇JdR

Ji = − limR→∞

• J · ˆ

R R2dΩ Ji = π K2

∞

• L=0

(2L + 1)

• 1 − η2

L

• It can be compared to the reaction cross section σr

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Optical model

Summary

Collisions used to study interaction and structure of “particles” Notion of cross section used to characterise a process in quantum collision theory

dσ dΩ = lim

∆Ω→0

∆n FiN∆Ω

Computing stationary scattering state

Ψ

K ˆ

Z(R) −→

R→∞(2π)−3/2

• eiKZ + fK(θ)eiKR

R

• ,

gives the scattering cross section

dσ dΩ = |fK(θ)|2

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Optical model

Summary (2)

In partial-wave expansion, defining phaseshifts

uas

KL(R) −→ R→∞C sin(KR − Lπ/2 + δL)

the scattering cross section reads

dσ dΩ =

• 1

K

∞

• L=0

(2L + 1)(S L − 1)PL(cos θ)

• 2

Optical model uses complex potentials to simulate other channels in scattering theory

⇒ absorption cross section σa = π K2

∞

• L=0

(2L + 1)

• 1 − η2

L

• The Coulomb case is similar (though a bit more complicated)

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Coulomb force

Coulomb Interaction

We assumed R2V(R) −→

R→∞ 0, which excludes Coulomb VC(R) = ZaZbe2 4πǫ0R

Coulomb requires special treatment, but similar results are obtained Defining the Sommerfeld parameter η = ZaZbe2

4πǫ0v,

Schr¨

• dinger equation for a and b scattered by Coulomb reads
• ∆R − 2ηK

R + K2

• ΨC(R) = 0,

which can be solved exactly and

ΨC(R) −→

R→∞

(2π)−3/2      ei[KZ+η ln K(R−Z)] + fC(θ)ei[KR−η ln 2KR] R       ,

with

fC(θ) = − η 2k sin2(θ/2) e2i[σ0−η ln sin(θ/2)] σ0 = arg Γ(1 + iη)

the Coulomb scattering amplitude

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Coulomb force

Rutherford cross section

The same analysis can be done defining Ji and J s to define the Coulomb elastic scattering cross section

• r Rutherford cross section :

dσR dΩ = |fC(θ)|2 = ZaZbe2 4πǫ0 2 1 16E2 sin4(θ/2)

Note that it diverges at θ = 0

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Coulomb force

Partial-wave analysis

We can again separate the angular from the radial part solution of

d2 dR2 − L(L + 1) R2 − 2ηK R − 2µ 2 VN(R) + K2

• uKL(R) = 0

If additional (nuclear) term R2VN(R) −→

R→∞ 0, uKL(R) −→ R→∞ uas KL(R) :

uas

KL(R)

= A FL(η, KR) + B GL(η, KR) −→

R→∞

A sin(KR − Lπ/2 − η ln KR + σL) +B cos(KR − Lπ/2 − η ln KR + σL)

where FL and GL are regular and irregular Coulomb functions and σL = arg Γ(l + 1 + iη) is the Coulomb phaseshift

⇒ uas

KL(R)

−→

R→∞

C sin(KR − Lπ/2 − η ln KR + σL + δL) δL is an additional phaseshift,

which contains all information about the nuclear interaction VN

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Coulomb force

(Additional) scattering amplitude

The stationary scattering states have now the asymptotic behaviour

Ψ(R) −→

R→∞

ΨC(R) + (2π)−3/2 fadd(θ)ei(KR−η ln KR) R

with

fadd(θ) = 1 2iK

∞

• L=0

(2L + 1)e2iσL(e2iδL − 1)PL(cos θ)

the additional scattering amplitude The total scattering amplitude f(θ) = fC(θ) + fadd(θ) gives the elastic-scattering cross section

dσ dΩ = | fC(θ) + fadd(θ)|2

At forward angles (θ ≪ 1), fC ≫ fadd, and dσ/dΩ ≈ dσR/dΩ

⇒ usually (dσ/dΩ)/(dσR/dΩ) is plotted

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Coulomb force

Example : 6He + 64Zn @ 14MeV

[Rodr` ıguez-Gallardo et al. PRC 77, 064609 (2008)]

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