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

Introduction to Quantum Collision Theory Pierre Capel 16 July 2015 1 / 30

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 : 2 / 30

Quantum Collisions Reaction types Various reactions can happen : a + b → a + b (elastic scattering) 1 → a + b ∗ (inelastic scattering) 2 → c + f + b (breakup) 3 → d + e (rearrangement or transfer) 4 Examples : 11 Be + 208 Pb → 11 Be + 208 Pb (elastic scattering) 1 → 11 Be ∗ (1 / 2 − ) + 208 Pb (inelastic scattering) 2 → 10 Be + n + 208 Pb (breakup) 3 → 10 Be + 209 Pb (transfer) 4 3 / 30

Quantum Collisions Energy conservation Total energy is conserved : m a c 2 + m b c 2 + incident kinetic energy = mass of products c 2 + kinetic energy The Q value of a reaction is Q = m a c 2 + m b c 2 − mass of products c 2 A channel will be open if the incident kinetic energy > − Q otherwise 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 ) 4 / 30

Notion of cross section Cross section For an incident flux F i 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 = F i N ∆ σ [ ∆ σ ] = surface ; unit : barn 1 b = 10 − 24 cm 2 The differential cross section d σ ∆ σ ∆ n d Ω = lim ∆Ω = F i N ∆Ω ∆Ω → 0 Taking Z as the beam axis, d σ/ d Ω depends only on θ by symmetry 5 / 30

Notion of cross section Theoretical framework Let us consider particles a and b of mass m a and m b interacting through potential V ( R ) , where R = R a − R b is the a - b relative coordinate The Hamiltonian reads H = T a + T b + V ( R ) , where p 2 = − � 2 ∆ R a a T a = 2 m a 2 m a p 2 = − � 2 ∆ R b b T b = 2 m b 2 m b 6 / 30

Notion of cross section Change to cm and a - b relative motion The coordinate change R cm = ( m a R a + m b R b ) / M R = R a − R b with M = m a + m b , the total mass, leads to H = T cm + T R + V ( R ) , 2 M = − � 2 ∆ R cm P 2 cm where T cm = 2 M P 2 2 µ = − � 2 ∆ R T R = 2 µ , with µ = m a m b / M the reduced mass of a and b H is then the sum of two Hamiltonians : H cm ( R cm ) + H ( R ) Hence the two-body wave function factorises Ψ tot ( R a , R b ) = Ψ cm ( R cm ) Ψ ( R ) 7 / 30

Notion of cross section cm motion The cm wave function is solution of H cm Ψ cm ( R cm ) E cm Ψ cm ( R cm ) , = P 2 cm where H cm = 2 M is the Hamiltonian of a free particle of mass M The cm motion is described by a plane wave Ψ K cm ( R cm ) = (2 π ) − 3 / 2 e i K cm · R cm with E cm = � 2 K 2 cm / 2 M The factor (2 π ) − 3 / 2 is chosen such that � Ψ K ′ cm | Ψ K cm � = δ ( K cm − K ′ cm ) 8 / 30

Stationary Scattering States Stationary Scattering States The Hamiltonian of the a - b relative motion reads H = T R + V ( R ) A stationary scattering state Ψ K ( R ) is solution of H Ψ K = E Ψ K where E = � 2 K 2 / 2 µ with the asymptotic behaviour � e iKZ + f K ( θ ) e iKR � R →∞ (2 π ) − 3 / 2 Z ( R ) −→ Ψ , K ˆ R with Z chosen as the beam axis f K is the scattering amplitude 9 / 30

Stationary Scattering States Physical interpretation To interpret the stationary scattering state � e iKZ + f K ( θ ) e iKR � R →∞ (2 π ) − 3 / 2 Z ( R ) −→ Ψ , K ˆ R let us recall the probability current J ( R ) = 1 µ ℜ [ Ψ ∗ ( R ) P Ψ ( R )] The plane wave describes the incoming current J i ( R ) = (2 π ) − 3 / 2 � K Z = (2 π ) − 3 / 2 v ˆ ˆ Z µ where v is the a - b relative velocity The spherical wave f K ( θ ) e iKR R describes the scattered current J s ( R ) = (2 π ) − 3 / 2 v | f K ( θ ) | 2 1 R + O ( 1 R 2 ˆ R 3 ) is purely radial at R → ∞ ; directed outwards ; ∝ v but varies with θ 10 / 30

Stationary Scattering States Physical interpretation Incoming wave : Scattered wave : f K ( θ ) e iKR → J s ( R ) ∝ v | f K ( θ ) | 2 1 e iKZ → J i ( R ) ∝ v ˆ R 2 ˆ Z R R 11 / 30

Stationary Scattering States Theoretical scattering cross section We can assume the incoming flux F i = C J i The scattered flux in direction Ω is then F s = C J s For one scattering nucleus, the number of event per unit time detected in direction Ω reads C J s R 2 d Ω dn = F s dS = d σ dn ⇒ = d Ω F i d Ω R 2 J s = J i | f K ( θ ) | 2 = The scattering amplitude f K contains all information about V 12 / 30

Partial-wave expansion and phaseshift Partial-wave expansion If the potential does not depend on Ω , i.e. V ( R ) , [ H , L 2 ] = 0 [ H , L Z ] 0 = the angular motion is described by spherical harmonics ψ KLM ( R ) = 1 R u KL ( R ) Y M L ( Ω ) , where u KL is solution of the radial equation � d 2 � dR 2 − L ( L + 1) − 2 µ � 2 V ( R ) + K 2 u KL ( R ) = 0 R 2 This can be solved using numerical techniques 13 / 30

Partial-wave expansion and phaseshift Phase shift δ L The scattering amplitude f K is obtained from the asymptotics of Ψ K If we assume R 2 V ( R ) −→ R →∞ u as R →∞ 0 , u KL ( R ) −→ KL ( R ) , which is solution of � d 2 � dR 2 − L ( L + 1) + K 2 u as KL ( R ) = 0 R 2 whose solutions u as KL ( R ) = A KR j L ( KR ) + B KR n L ( KR ) −→ A sin( KR − L π/ 2) + B cos( KR − L π/ 2) R →∞ where j L and n L are regular and irregular spherical Bessel functions Posing A = C cos δ L and B = C sin δ L u as KL ( R ) −→ R →∞ C sin( KR − L π/ 2 + δ L ) C is just a normalisation factor ; δ L is the phaseshift 14 / 30

Partial-wave expansion and phaseshift Scattering matrix S L u as KL ( R ) −→ R →∞ C sin( KR − L π/ 2 + δ L ) better interpreted in terms of incoming and outgoing waves : iCe − i δ L e − i ( KR − L π/ 2) − S L e i ( KR − L π/ 2) � � −→ u KL ( R ) 2 R →∞ where e 2 i δ L S L = is the scattering matrix The outgoing wave is shifted from the incoming wave by 2 δ L due to the effect of V ⇒ used to compute f K 15 / 30

Partial-wave expansion and phaseshift Scattering amplitude The scattering wave function can be expanded in partial waves ∞ 1 � (2 π ) 3 / 2 Ψ c L u KL ( R ) Y 0 Z ( R ) = L ( Ω ) K ˆ KR L = 0 e iKZ + f K ( θ ) e iKR −→ R R →∞ ∞ i e − i ( KR − L π/ 2) − e i ( KR − L π/ 2) � � � Since e iKZ (2 L + 1) i L P L (cos θ ) −→ 2 KR R →∞ L = 0 ie − i δ L e − i ( KR − L π/ 2) − S L e i ( KR − L π/ 2) � � −→ and u KL 2 R →∞ √ √ 2 L + 1 i L e i δ L comparing the incoming waves we obtain c L = 4 π and deduce the scattering amplitude from S L = e 2 i δ L ∞ 1 � (2 L + 1)( S L − 1) P L (cos θ ) f K ( θ ) = 2 iK L = 0 16 / 30

Partial-wave expansion and phaseshift Scattering cross section d σ | f K ( θ ) | 2 = d Ω 2 � � ∞ � 1 � � � � (2 L + 1)( S L − 1) P L (cos θ ) = � � � � 2 iK � � L = 0 � � After integration over Ω the total scattering cross section reads ∞ σ = 4 π � (2 L + 1) sin 2 δ L K 2 L = 0 Each partial wave contributes to σ but with variable importance 17 / 30

Partial-wave expansion and phaseshift Contribution of partial waves ∞ σ = 4 π � (2 L + 1) sin 2 δ L K 2 L = 0 Centrifugal barrier L ( L + 1) R 2 ensures δ L −→ L →∞ 0 ⇒ limited sum K 2 sin 2 δ 0 At very low E , only L = 0 contributes and σ = 4 π At large E many partial waves must be included to reach convergence ( ⇒ low-energy method) 18 / 30

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 σ L = 4 π small if δ L ∼ n π ( n ∈ Z ) K 2 sin 2 δ L large if δ L ∼ π/ 2 If δ L goes quickly from 0 to π → rapid increase and decrease of σ L i.e. resonance structure Definite L ⇒ quantum numbers and parity similar to bound state 19 / 30

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 R 2 J d + e d σ d Ω ( a + b → d + e ) = lim J i R →∞ The sum of all channels but elastic scattering (inelastic, transfer, breakup,. . . ) gives the reaction cross section � σ r = σ ( a + b → channel ) channel \ a + b The interaction cross section corresponds to all channels but elastic and inelastic scattering � σ ( a + b → channel ) σ I = channel \ ( a + b ) ∪ ( a + b ∗ ) ∪ ( a ∗ + b ) ∪ ( a ∗ + b ∗ ) 20 / 30

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 − � 2 ⇒ U opt ( R ) = V ( R ) + iW ( R ) 2 µ ∆Ψ + U opt Ψ = E Ψ ∇ 1 µ ℜ{ [ Ψ ∗ P Ψ ] } ∇ J = − i � 2 µ ∇ [ Ψ ∗ ∇ Ψ − Ψ ∇ Ψ ∗ ] = − i � 2 µ [ ∇ Ψ ∗ · ∇ Ψ − ∇ Ψ · ∇ Ψ ∗ + Ψ ∗ ∆Ψ − Ψ ∆Ψ ∗ ] = i � 2 µ [ Ψ ∗ 2 µ � 2 U opt Ψ − Ψ 2 µ � 2 U ∗ opt Ψ ∗ ] = 2 � W ( R ) | Ψ ( R ) | 2 = To have absorption from elastic channel W ( R ) ≤ 0 21 / 30