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Ecole Joliot-Curie 2007
“One has to realize that the experimental and theoretical understanding of nuclear reactions is one of the major achievements of Nuclear Physics of the last half century, largely unrecognized or celebrated, even by nuclear physicists themselves” (H. Feshbach)
Nuclear reactions: development of collision theory conservation of mass
(http://www.theo.phys.ulg.ac.be)
Nuclear Reactions. A short digest.● Generalities● Typical variations & first classification● Coherent reactions● Incoherent (high energy) reactions● Reactions involving coherent & incoherent effects● Specific features linked to particular projectiles● Outlook
Ecole Joliot-Curie 2007
J. CUGNON, ULg
1. Introduction● Experiment
● Control parameters: a, A, Einc, (E, polarisations, ...)
● Detection asymptotic states only
+ a A b1+b2*+...+[,..]
- if only b 1 : is detected inclusive + reactions a A b1+X- : if all are detected exclusive reactions- : intermediate -semi inclusive reactions
● Information is encoded in (differential) crosssections
● Standard theoretical description
c=pt ei k c . r c c '= f 1
f 2...ei k c ' . r c ' ...
T=VV 1E−Hi
V
E=ℏ2 k c
2
2 mE i=
ℏ2 k c '2
2 mE f=
ℏ2 k c '2
2 mE i−Q
d c c '=24
ℏ vc
ℏ2 k c
2
2 mE i−
ℏ2 k c '2
2 m−E f ∣⟨ f k c '∣T∣i k c ⟩∣
2d
H=H 0V
interaction structure
● calculation of crosssections = formidable task● connection with structure is not obvious● reaction mechanism
hypotheses
approximationsd.o.f reaction model
(mechanism)structure
● there exist timedependent approaches
∂
∂ t=−
iℏ
[H , ]
[∂∂ tp
m. ∇− ∇U . ∇ p] f r ,p , t =I coll [ f ]
macroscopic properties
INC, BUU, LV, QMD...
symmetries:
⟨ i−k c∣T∣ f − k c ' ⟩=⟨ f k c '∣T∣i k c ⟩
(a) timereversal invariance ⇒ detailed balance
k c2 dcc '
d=k c '
2 d c ' c
d−
simplifies if angular momentum is disregarded:
S c ' c=c ' c−2 i m2 k c k c '
2 1/2
⟨ f k c '∣T∣i k c ⟩ does not depend upon angle
cc '=ƛc2∣Sc ' c−c ' c∣
2S c c '=S c ' c
∗
(b) conservation of norm
S S=1 ∑c '∣Sc ' c∣
2=1 c
T=2ƛc21−ℜScc c
T=−4k c
ℑ f cc
(c) conservation of angular momentum, isospin, etc specificity of typical reactions
2. Variation of crosssections with control parameters
Behaviour close to threshold:1. channel without Coulomb interaction (incident neutrons)
elastic scattering cc≈C
endothermic reaction Q<0
exothermic reaction Q>0
cc '≈C E−E th
cc '≈C /k c '
2. reactions with Coulomb interaction
multiplication with Gamow factor Gc=exp −Z 1 Z 2 e2
ℏ vc
elastic scattering may be dominated by Coulomb scattering
reaction:
3. total reaction crosssection≈R2≈ 1b @ high energy
A. Characteristic variations
ninduced exothermic reactionTypical energy dependences
Typical target dependences
() 104 MeV
B. Comparison of control parameters with typical lengths and energies
: mixing of regimes due to the impact (“uncontrolled”) parameter
First “rough” classification
1. Quantum treatment is needed elastic &slightly inelastic collisions resonant reactions pickup & transfert reactions
2. Quasiclassical treatment is (perhaps) sufficient (moderately) inelastic collisions fragmentation (spallation) collisions
wave functionrelated information
more global properties
C. Comparison nucleonnucleus ⇔nucleusnucleus
1. Differences linked with the shape of the colliding system deep inelastic collisions fusion reactions & partial fusion reactions
2. Differences linked with the number of particles thermodynamic properties of nuclear matter possible phase transition (multifragmentation)
3. Coherent, quasicoherent & resonant reactionsA. Resonant reactions. The compound nucleus.
resonance ⇒”definite” energy with lifetime =ℏ/∆E >> tpass=R/v
:Bohr' hypothesis formation of a complex
,CN statefollowed by a statistical decay in open channels
formation & decay are independent, except for conservation laws
⇒ symmetric angular distributions (around 90°)
The BreitWigner formula
Smatrix form
Scc '=eicc 'cc '−i∑
E−Ei
/2
Bohr Hypothesis =
c1 /2
c '1 /2 , ∑
c
c=,
c1/2=real
cc '=cCN Pc '
withc
CN=ƛc2
c
E−E2
2 /4
, Pc '=
c '
cc '=ƛc2∣Scc '−cc '∣
2
Structure Information (SI): E, (J), , c
The shellmodel approach
H=H 0V=∑i
h0i V H 0i=E ii , H 0c E =EcE
c =∑
ibiE i∑
c 'ac
c ' E c ' E
PQ
in absence of direct coupling <c|V|c'>=0:
Sc c '=ei cc '[c c '−i 2∑j
⟨c∣V∣ j ⟩ ⟨ j∣V∣c ' ⟩E−E j−i∑
c∣⟨ j∣V∣c ⟩∣
2 ]Note that j are (complicated) eigenstates of [PHP + PVQ (EH0)1 QVP]j = Ej j
They are bound states in the continuum (E>0)Also: continuum component in loosely bound states
c=∣⟨c∣V∣⟩∣
2represents the coupling of the resonant states to channel c
HauserFeshbach formula:
Average crosssections and the overlapping resonance regionNB: no average for overlapping resonances
⟨c c ' ⟩=1I∫
E−I /2
EI /2
ƛc2∣∑
c1/2
c '1/2
E−Ei
/2∣
2
dE
≈1I∫
E−I /2
EI /2
ƛc2∑
cc '
E−E2
2 /4
dE
=ƛc2 2
D c c '
⟨c c ' ⟩=ƛc2 2
D
c 2D
c '
2D
F c c '=ƛc2 T c T c '
∑c
T c
F c c '
F accounts for width correlations
Two theories:1. elimination of the inelastic components
2. mass operator
They predict Vopt to be energydependent and nonlocal
B. Elastic scattering. The optical model
The optical model: elastic scattering can be reduced to potential scattering
V opt r ≈−V 0
r 0
V LS−i W r =V CV LS
W>0 accounts for the loss of flux (above the 1st inelastic channel)
E−PHPP=PHQQ , E−QHQQ=QHP PE−PHP−PHQEi −QHQ−1QHP P=0
V C E ,r =∫d 3r ' G E ' ,r ' f r−r '
JLM microscopic potential
Phenomenological potentials
G is the Brueckner Gmatrix
Parameters are smoothly dependent upon the target The crucial “parameter” is the volume integral of the central part
Largely diffractive scattering
Structure Information (SI): geometry, deformation relativistic formulation is better for spinorbit part OM > <Scc>=ei2c(1c/D)
J=∫V C r d3r
maxima @ 2kRsin /2=n
OMP ⇒ Tc⇒ ( ) -HF average cross sections
C. Inelastic scattering. (p,p')
SI: Energy levels
Direct interactions: only one NN interactionDistorted wave Born approximation (DWBA)
dd
∝∣⟨c ' r c '∣∑iv r−r i∣c r c⟩∣
2
∝∣⟨c '∣∑ie−iq . r i∣c⟩∣
2
=∣e−iq .r cc ' r ∣2
SI: transition form factors >w.f.
Born: T=”V”
cc ' r =⟨c '∣∑ir−r i∣c⟩
Selectivity:
l≈k R , l '≈ k ' R , l≈q R
dd
∝∣ jqR∣2
SI: J & parity
Special cases: Coulex, (e,e')
=∣l−l '∣
@ higher energy transfer:excitation of giant resonances
SI: giant resonances structure, sum rules, exchange forces
selectivity of probes
(') (p,p') (3He,t)
D. Transfert reactions. Stripping, Pickup, etc
Stripping: (d,p)
T ∝d , pnq1
−ℏ2 q2 /2 mn
B , Anq
q should not be too large angular momentum selection rule
SI: Spectroscopic factorsA1=∑
S1 /2
r A =nlj
dd
=C D02 S
∣∫d3r d∗ r u
r pr ∣2
S=∣⟨
r A∣A1 ⟩∣2
∑ S=1−N
90Zr(d,p)
E=12 MeV
Knockout: (p,2p)
T ∝T ppq A ,A−1 p pR , pR= p2−q , p1=pi−q
d5
d E1 d1 d2
=C S∣
pR∣2 d pp
dq
SI: momentum distribution of sp w.f. + spectroscopic factors
NB: (e, e' p) is simpler
q
4. Incoherent highenergy reactionsA. Introduction
above ~ 250 MeV: dominance of NN collisions elastic scattering is limited to extreme forward angles, even though the
cross section remains important (see later) angular momentum > impact parameter as the relevant parameter coherent inelastic scattering is limited to very peripheral collisions (and
to very forward angles) opening of huge number of channels: many ejected particles
> fragmentation
B. Glauber models
R=∫0
R
2b db∫−z0
z0
PsurvzNNtot dz=R2[1−2
X 21−e−X
2X
e−X ]
Psurvz=exp −NNtot zz0
X=2NNtot R≈R R≈0.8−0.9R2≈E
A simplified quantum model: Glauber formalism + eikonal
basic assumptions: small momentum transfer at each interaction scattering introduces a phase shift only: frozen nucleus approximation (no Fermi motion)
eik .r eir
f fiq=ik
2∫d2b eiq .b ⟨ f∣1−∏j=1
A 1− 12 i k
∫d 2 q ' eiq ' .b−s j f j q ' ∣i⟩fj is the individual amplitude, sj is the tranverse position of nucleon j
Expanding the product multiple scattering expansion
f ij1q= f qij q , ij q=∫d2s eiq .s ⟨ f∣∑k
s−sk∣ f ⟩f ij2q=∫d 2q ' f q f q−q ' ij
2q , ij2q=∫d 2s d 2s ' eiq .s ei q−q ' .s ' ⟨ f∣∑k≠l
s−sks−sl∣ f ⟩SI: correlations
f ij q= f ij1q f ij
1q...
SI: transition prob.
C. Models for collision regime
For many particle emission and deep inelastic collisions,Glauber and quantum multiple scattering theories (KMT,..)are unpracticableExcept for very peripheral and slightly inelastic collisions,no evidence for quantum effects
Quasiclassical tools have been devised, where quantum effectsis restricted to small binary collision regions and translated in using cross sections.
INC, QMD, BUU, LV,...
They are based on simulations and/or transport equations
INC
brief description of INCL:
ordered & separated NN collisions elastic or inelastic subject to Pauli blocking potential well transmission, reflection, (refraction) stochasticity relativistic kinematics isospin degree of freedom accomodates p, n, d, t, He3 & He4 as projectiles
must be supplemented by an evaporation model stopping time is determined selfconsistently
“parameterfree” SI: geometry & momentum distribution
D. Spallation and fragmentation collisions
5. Reactions involving statistical coherent and incoherent featuresA. Similarity between resonant reactions in the “overlapping regime” and the ultimate stage of spallation/fragmentation collisions
In the former, the reaction can be described by the formation of CN followed by an “independent” decay governed by average decay widths (HauserFeshbach)
Note that the formation of CN involves average over many different states, i.e. many degrees of freedom ⇒ equilibrated system
At the end of “hard collision” stage of spallation reactions, the system is also expected to be in equilibrium
HF for many exit channels
dHF Ec '∗ =ƛc
2 T c T c 'c ' Ec '∗ dE c '
∗
∑c ' '∫0
EQ
T c ' 'c ' Ec ' '∗ dE c ' '
∗
NB: HF is limited to low energy, but includes angular momentum
The WeisskopfEwing evaporation modelLet us assume A*→B*+b in a volume V; E*=EB*+S+Probability of emission per unit time:
d b=2ℏ
∣⟨ A∣T∣B b ⟩∣2E B∗
V k2 dk23
CN b B ∗ A ∗ =2ℏ
∣⟨b B∣T∣A ⟩∣2E A∗
ℏ k /mV
d b=CN b B ∗ A ∗ 2 m
23ℏ2
E B∗
E A∗
d
can be used for calculating Xsections d=cCN d b
∑b∫0
E−S
d bFor (E*)= p exp(2aE*) and E*=aT2
b=CN b B ∗ A ∗ 2 mT 2
23ℏ2e−S /T
B. Evaporationfission and other deexcitation models
d b∝e−/T d
fission
everything is determined byphase space at the barrier
d f
ℏ=BE B
∗ dE B∗ dp dq /2ℏ/dt
AE A∗ dE A
∗and dq /dt=p /M , Mpdp=d
d f=1
2
BE B∗
AE A∗
d ⇨ f=T
2e−B /T
must be supplemented by a fission partition model
SI: level density & barriers
Other “afterburning” models
1. evaporation: simulation of successive separated emissionstime scales b=ℏ/b≫temiss b
3. above some “temperature” ►► copious simultaneous emission multifragmentation
usual models include any partition of the system final states = partitions of an equilibrated system (SMM, ...)
2. for increasing temperature n may become smaller than temiss (fission)neutrons may be emitted from the system on its way to fission=fission delayed or friction in fissioning motion
SI: thermodynamics of nuclear matter / possible phase transition
C. Preequilibrium reactions
For resonant reactions in the “continuum”, if energy increases (above ~ 20 MeV), the spectra are no longer thermal &emission no longer isotropic
Idea = addition of fast +/ coherent emission and slow evaporation
Many special tools: HF+DR HarpMillerBerne sp model exciton model hybrid model GDH model FKK theory ...
1. HarpMillerBerne approach
initial state: ni=1 for ki < kF and ki = kc , ki =0 otherwiseevolution:
d ni
dt=∑
j∑
k∑
lijkl {nk nl 1−ni1−n j−ni n j 1−nk1−nl}−esc ni
≈NN ⟨v ⟩
prediction of spectradP
d =∫
0
T
dt∑ini t −i
abandoned, due to the numerical task
2. The exciton model (J. Griffin)
whole distribution exciton states 1p, 2p1h, 3p2h,...(n=1,3,5,7...) density of (n=p+h) exciton states of given energy E (Ericson)
nE =gn E n−1
p ! h !n−1!
probability of emission from a nexciton state per unit time
Pnd =m
2ℏ3
n−1U
n E d U=E−S−
spectrum
P d =∑n=n0
n
n Pnd =m
2ℏ3 ∑n=n0
n n−1U
nE n d
exciton hybrid model (M. Blann)dd
=R ∑n=n0
n [ m
2ℏ3
n−1U
n E d ][ Pn E
Pn E n2E ]Dn=R ∑
n=n0
n
Pnd
Dn=∏n '=n0
n−2
{1−∫Pn ' d } , Dn0=1
M. Blann
n2=
2ℏ∣M∣2E ∗ or
n2=
v=1/NN v=2 W
ℏ
OMP
∣M∣2=KA−3 E−1
SI: density of exciton states, mfp
further developments:
crosssections (GDH):
angular distributions:
dd
=∫2b db Pb
dd d
=R ∑nn0
n
Pn∑L
2 L14
f L nPL cos
with fL= parameters (Kalbach)
emission of clusters, through phenomenological probabilities attached to every nexciton state
3. The FeshbachKermanKoonin (FKK) theory
Pchain=MSD=at least one particle is unboundQchain=MSC=all particles are bound
FKK:no communication between P and Qchainschaining only between n and n±1 statesnevercomeback hypothesis (n > n+1)
MSC:
MSC=ƛc2 21
D1
∑n=1
r ∏k=1
n−1
k
k n
n
n=
2ℏ
∣⟨n∣V∣n1 ⟩∣2nU n=
2ℏ
∣⟨n∣V∣n1 ⟩∣2n1E
⟨n∣V∣n1 ⟩≈V 0∫u1r u2r u3r {uscatt r ,u4r }dr
r2
MSD: d 2MSD
ddU=
d 21
ddU
d 2M
ddUd21
ddU=r U ⟨ d
d ⟩d 2M
dN dU N
=∑N∫
d3 k N−1
23⋯∫
d3 k2
23d 2 W N , N−1k N ,k N−1
d N dU N
d 2 W N 1, N−2 k N−1 ,k N−2
dN−1 dU N−1
⋯d2 W 2,1 k2,
k1
d 2 dU 2
d 21
d1 dU 1
d 2 W N , N−1k N ,k N−1
dN dU N
=2ℏ ∣∫d3r k N
− k N−1 ⟨N∣V∣N−1⟩∣2 mk N
23ℏ2r U N
C. Comparison INCpreequilibrium models All have to be supplemented by evaporation
Phase space is classical in INC, discrete in PE models
Pauli blocking is more naturel in INC
same common physics in all models: interaction in a Fermi gas
mediated by binary collisions
INC is a more consistent model (only the stopping is “by hand”)
more or less equivalent results below 200 MeV (where INC
shouldn't work!!)
INC+evaporation= a theory from 40 MeV to 10 GeV?
6. Specific features linked to particular projectiles
A. Slightly bound nuclei
easy fragmentation
p1=m vincpinternal
dd3 p1
∝ f p1
SI: momentum distribution
surrogate reactions
dd3p
=dd3 p
3 HeA−1 pA1 ∗ DWBA
f
tot
nA f =NC nAA1 ∗ f
tot
B. Proton or neutron rich nuclei
study of isospin degree of freedom by inverse kinematics
IAS
seen by resonant scattering 207Pb(p,p) or by production 208Pb(p,n)
ordinary spectroscopy by p(A,A*)p with recoil measurements
C. Photons
Real photons probe the nucleus with an elm field: very good for electric giant resonances (T=1) photodesintegration (,n)
Virtual photon (e,e) (e,e') charge density transition densities parton structure
D. Mesons, antiprotons, etc
pions
SI: surface densities, S=1, transitions, pair correlations
elastic inelastic absorption
antiprotons
annihilation: ~2 GeV without p,ℓ transfer
good for study of hot nuclei
7. Summary & outlook
D. The shellmodel approach
H=H 0V=∑i
h0i V H 0i=E ii , H 0c E =EcE
c =∑
ibiE i∑
c 'ac
c ' E c ' E
EcEcEc iE
c '
PQ
A.Volya &V. ZelevinskyPRL94(2005)052501
M.G. Mustafa et alPRC 88
special features:
quantum effects are simulated (stochasticity, Pauli blocking, mean field, transmission and reflection)
predictive power for (almost) all channels
substantiated by nuclear transport theory
D. TALYS= a code system for E<150 MeV
3. Heavy ion reactions
Danielewicz
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