Ecole Joliot-Curie 2007

Transcription

1 Ecole Joliot-Curie 007 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 (

2 Ecole Joliot-Curie 007 J. CUGNON, ULg 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

3 1. Introduction Experiment Control parameters: a, A, Einc, ( E, polarisations,...) Detection asymptotic states only a+a b1+b*+...+[,..] - if only b1 is detected: inclusive reactions a+a b1+x - if all are detected: exclusive reactions - intermediate: semi-inclusive reactions

4 Information is encoded in (differential) cross sections Standard theoretical description c = p t e i k c. r c c ' = f f... e 1 i k c '. r c '... H =H 0 V T =V V E= ℏ kc m 1 E H i E i= V ℏ kc' m E f = ℏ kc' m E i Q interaction d c c '= ℏ vc 4 ℏ k c m E i ℏ k c ' m structure f k ' T i k d E f c c

5 calculation of cross sections = formidable task connection with structure is not obvious reaction mechanism d.o.f hypotheses approximations reaction model (mechanism) structure there exist time dependent approaches i = [ H, ] ℏ t [ ] p. U. f r, p, t =I [ f ] p coll t m INC, BUU, LV, QMD... macroscopic properties

6 symmetries: (a) time reversal invariance detailed balance i k c T f k c ' = f k c ' T i k c k c d cc ' d =k c' d c ' c d simplifies if angular momentum is disregarded: S c ' c = c ' c i 1/ m k c k c ' c cc ' = ƛ S c ' c c ' c f k ' T i k does not depend upon angle c c S c c ' =S c ' c (b) conservation of norm S S=1 c' S c ' c =1 Tc = ƛc 1 ℜ S cc (c) conservation of angular momentum, isospin, etc specificity of typical reactions Tc = 4 kc ℑ f cc

7 . Variation of cross sections with control parameters A. Characteristic variations Behaviour close to threshold: 1. channel without Coulomb interaction (incident neutrons) elastic scattering cc C endothermic reaction Q<0 cc ' C E E th exothermic reaction Q>0 cc ' C / k c '. reactions with Coulomb interaction elastic scattering may be dominated by Coulomb scattering reaction: multiplication with Gamow factor G c =exp Z1 Ze ℏ vc 3. total reaction cross section R high energy

8 Typical energy dependences n induced exothermic reaction

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11 Typical target dependences ( ) 104 MeV

12 B. Comparison of control parameters with typical lengths and energies : mixing of regimes due to the impact ( uncontrolled ) parameter

13 First rough classification 1. Quantum treatment is needed elastic &slightly inelastic collisions resonant reactions pick up & transfert reactions wave function related information. Quasi classical treatment is (perhaps) sufficient more global properties (moderately) inelastic collisions fragmentation (spallation) collisions

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15 C. Comparison nucleon nucleus nucleus nucleus 1. Differences linked with the shape of the colliding system deep inelastic collisions fusion reactions & partial fusion reactions. Differences linked with the number of particles thermodynamic properties of nuclear matter possible phase transition (multifragmentation)

16 3. Coherent, quasi coherent & resonant reactions A. Resonant reactions. The compound nucleus. resonance definite energy with lifetime =ℏ/ E >> tpass=r/v Bohr' hypothesis: formation of a complex CN state, followed by a statistical decay in open channels formation & decay are independent, except for conservation laws symmetric angular distributions (around 90 )

17 The Breit Wigner formula S matrix form S cc ' =e i c c ' cc ' i E E i / Bohr Hypothesis 1/ = 1/, c c ' c c CN cc ' = c P c ' with CN c = ƛ c c E E / 4, cc ' = ƛ S cc ' cc ' c =, Pc '= c ' Structure Information (SI): E, (J ),, c 1/ =real c

18 The shell model approach H =H 0 V = h 0 i V i H 0 i =E i i, H 0 c E =E c E c = bi E i a cc ' E c ' E i c' Q P

19 in absence of direct coupling < c V c'>=0: S c c ' =e i c c ' [ c c ' V V i E E i V c j j j c' j c j c ] Note that j are (complicated) eigenstates of [PHP + PVQ (E H0) 1 QVP] j = Ej j They are bound states in the continuum (E>0) Also: continuum component in loosely bound states c = c V represents the coupling of the resonant states to channel c

20 Average cross sections and the overlapping resonance region NB: no average for overlapping resonances 1 E I / = I cc' ƛ E I / 1 I E I / ƛ E I / c c = ƛ c 1/ c E E i / c c ' E E / 4 c c ' D 1/ c' de de Hauser Feshbach formula: = ƛ cc' c D c D D c' F accounts for width correlations F c c ' = ƛ c T c T c' Tc c Fcc'

21 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 C V LS W>0 accounts for the loss of flux (above the 1 st inelastic channel) Two theories: 1. elimination of the inelastic components E PHP P =PHQ Q, E QHQ Q =QHP P E PHP PHQ E i QHQ 1 QHP P =0. mass operator They predict Vopt to be energy dependent and non local

22 JLM microscopic potential V C E, r = d r ' G E ', r ' f r r ' 3 Phenomenological potentials G is the Brueckner G matrix

23 Parameters are smoothly dependent upon the target The crucial parameter is the volume integral of the central part J = V C r d 3 r Largely diffractive scattering krsin /=n Structure Information (SI): geometry, deformation relativistic formulation is better for spin orbit part OM > <Scc>=ei c(1 c/d) OMP Tc (HF) average cross-sections

24 C. Inelastic scattering. (p,p') SI: Energy levels Direct interactions: only one NN interaction Distorted wave Born approximation (DWBA) d d c ' r c ' v r r i c r c i SI: transition form factors >w.f. Born: T= V c ' i q. r i e i i q. r c = e cc ' r = c ' cc ' r r r i c i

25 Selectivity: l k R, l ' k ' R, d d l q R j qr SI: J & parity Special cases: Coulex, (e,e') = l l '

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28 @ higher energy transfer: excitation of giant resonances

29 selectivity of probes ( ') (p,p') (3He,t) SI: giant resonances structure, sum rules, exchange forces

30 D. Transfert reactions. Stripping, Pick up, etc Stripping: (d,p) T d, pn q 1 ℏ q / m n B, An q q should not be too large angular momentum selection rule SI: Spectroscopic factors A 1= S d d 1/ r A =C D S d r 0 3 d S = r A A 1 =nlj r u r p r S =1 N

31 90Zr(d,p) E=1 MeV

32 Knock out: (p,p) T T pp q A, A 1 p p R, p R = p q, p 1= p i q q d5 d E 1 d 1 d d pp d =C S p R q SI: momentum distribution of sp w.f. + spectroscopic factors NB: (e, e' p) is simpler

33 4. Incoherent high energy reactions A. Introduction above ~ 50 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

34 B. Glauber models tot P surv z =exp NN z z 0 R z0 R = b db P surv z 0 z 0 X = tot R R NN tot NN [ dz= R 1 X X 1 e R R E X e X ]

35 A simplified quantum model: Glauber formalism + eikonal basic assumptions: small momentum transfer at each interaction i k. r i r scattering introduces a phase shift only: e e frozen nucleus approximation (no Fermi motion) f fi q = ik d b e i q. b A f 1 1 j=1 1 i k i q '. b s d q ' e f j q ' i j fj is the individual amplitude, sj is the tranverse position of nucleon j 1 q f q... Expanding the product multiple scattering expansion f ij q = f 1 ij ij f ij q = f q ij q, 1 ij q = d s e f ij q = d q ' f q f q q ' ij q, i q. s s s SI: transition prob. f k k i q. s i q q '. s ' ij q = d s d s ' e e f f k l s s k s s l f SI: correlations

36 C. Models for collision regime For many particle emission and deep inelastic collisions, Glauber and quantum multiple scattering theories (KMT,..) are unpracticable Except for very peripheral and slightly inelastic collisions, no evidence for quantum effects Quasi classical tools have been devised, where quantum effects is 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

37 INC

38 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 self consistently parameter free SI: geometry & momentum distribution

39 D. Spallation and fragmentation collisions

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41 5. Reactions involving statistical coherent and incoherent features A. 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 (Hauser Feshbach) 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

42 HF for many exit channels d HF E c = ƛ ' c T c T c ' c ' E c de ' c' E Q c' ' 0 T c ' ' c ' E c de '' c' ' NB: HF is limited to low energy, but includes angular momentum

43 B. Evaporation fission and other de excitation models The Weisskopf Ewing evaporation model Let us assume A* B*+b in a volume V; E*=EB*+S+ Probability of emission per unit time: d b= ℏ A T B b E d b = CN b B A B V k dk 3 E B ℏ E A For (E*)= p exp( ae*) and E*=aT CN b B A d b e / T d A b B T A E A = ℏ ℏ k / mv d can be used for calculating X sections b = b B m 3 CN mt 3 ℏ e S /T d = CN c d b E S b 0 d b

44 fission everything is determined by phase space at the barrier d f ℏ = B E B de B dp dq / ℏ / dt d f= A E 1 A B E B de A E A A d and dq / dt= p/ M, Mpdp=d f= T e B /T must be supplemented by a fission partition model SI: level density & barriers

45 Other after burning models 1. evaporation: simulation of successive separated emissions time scales b =ℏ / b t emiss b. 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 3. above some temperature copious simultaneous emission multifragmentation usual models include any partition of the system final states = partitions of an equilibrated system (SMM,...) SI: thermodynamics of nuclear matter / possible phase transition

46 C. Pre equilibrium reactions For resonant reactions in the continuum, if energy increases (above ~ 0 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 Harp Miller Berne sp model exciton model hybrid model GDH model FKK theory...

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48 1. Harp Miller Berne approach initial state: ni=1 for ki < kf and ki = kc, ki =0 otherwise evolution: d ni dt { } = ijkl n k n l 1 n i 1 n j n i n j 1 n k 1 n l esc n i j k l NN v prediction of spectra dp d T = dt n i t i 0 i abandoned, due to the numerical task

49 . The exciton model (J. Griffin) whole distribution exciton states 1p, p 1h, 3p h,...(n=1,3,5,7...) density of (n=p+h) exciton states of given energy E (Ericson) n g E n E = n 1 p! h! n 1! probability of emission from a n exciton state per unit time P n d = m n 1 U ℏ spectrum 3 n E n P d = n P n d = n=n 0 d U = E S n m ℏ 3 n=n 0 n 1 U n E n d exciton hybrid model (M. Blann) d d n = R n=n 0 n { [ m n 1 U ℏ D n = 1 P n ' =n 0 3 n ' n E } d d, D n =1 0 ][ Pn E n P n E E ] D n = R n n=n 0 P n d

50 M. Blann

51 OMP n = ℏ M E or M =KA 3 E 1 SI: density of exciton states, mfp W n = =1/ NN v= ℏ v

52 further developments: cross sections (GDH): d d = b db P b angular distributions: d d d n = R P n n n 0 L 1 L 4 f L n P L cos with fl= parameters (Kalbach) emission of clusters, through phenomenological probabilities attached to every n exciton state

53 3. The Feshbach Kerman Koonin (FKK) theory P chain=msd=at least one particle is unbound Q chain=msc=all particles are bound FKK: no communication between P and Q chains chaining only between n and n±1 states never come back hypothesis (n > n+1)

54 MSC: n 1 MSC = ƛ c 1 D1 r n=1 n = n V n 1 ℏ k k =1 k n MSD: d MSD d du d M d N du N = d 1 d du 3 = N d k N 1 d W N, N 1 k N, k N 1 d N du N 3 1 = ℏ 3 { scatt = n V n 1 n 1 E ℏ } d 1 d du d N du N = r U N 1, d d k N 1, k N d N 1 du N 1 d r k N k N 1 N V N 1 3 r d 3 k d W N, N 1 k N, k N 1 d W N 3 n r, dr u 4 r d M d du n U n V n 1 V u r u r u r u 0 n mk N ℏ 3 d W,1 k, k 1 d 1 d du r U N d 1 du 1

55 C. Comparison INC preequilibrium 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 00 MeV (where INC shouldn't work!!) INC+evaporation= a theory from 40 MeV to 10 GeV?

56 6. Specific features linked to particular projectiles A. Slightly bound nuclei easy fragmentation p 1 =m v inc pinternal d 3 d p 1 f p 1 SI: momentum distribution

57 surrogate reactions d 3 d p = d 3 d p 3 He A 1 p A 1 n A f = NC n A A 1 f tot DWBA f tot

58 B. Proton or neutron rich nuclei study of isospin degree of freedom by inverse kinematics IAS seen by resonant scattering 07Pb(p,p) or by production 08Pb(p,n) ordinary spectroscopy by p(a,a*)p with recoil measurements

59 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

60 D. Mesons, antiprotons, etc pions SI: surface densities, S=1, transitions, pair correlations elastic inelastic absorption antiprotons annihilation: ~ GeV without p,ℓ transfer good for study of hot nuclei

61 7. Summary & outlook

62 D. The shell model approach H =H 0 V = h 0 i V i H 0 i =E i i, H 0 c E =E c E c = bi E i a cc ' E c ' E i c' Q ce P ce' i

63 A.Volya &V. Zelevinsky PRL94(005)05501

64 M.G. Mustafa et al PRC 88

65 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

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67 D. TALYS= a code system for E<150 MeV

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70 3. Heavy ion reactions Danielewicz

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