Transcript ppt - RHIP
V. Nuclear Reactions Topics to be covered include: Two-potential formula Distorted Wave Born Approximation Coupled-Channels Applications Eikonal Theory General References: 1) Rodberg and Thaler, “The Quantum Theory of Scattering,” Chapter 12. 2) Roy Glauber, in “Lectures in Theoretical Physics,” Vol. 1, Interscience, Boulder 1958 series, starting on page 337. 1 In previous chapters we have covered elastic scattering in detail. Here I focus on methods used to calculate inelastic scattering from nuclei and single-nucleon transfer reactions. I will not get bogged down in the intense amount of angular momentum algebra required in order to do real calculations. My goal is to show the structure of the scattering equations and give you a flavor of how the reaction calculations are done. I will start with the two-potential formula in formal scattering theory which provides the basis for many applications. Assume the total Hamiltonia n can be written as H H 0 V0 V1 H 0 V H 0 H P H A TPA , where H P is the internal Hamiltonia n of the projectile and H A is the Hamiltonia n of the nucleus, TPA is the KE operator for the relative P A motion. Potential V0 depends only on the coordinate between P and A and is the dominant interactio n in the system. Potential V1 is a perturbati on that depends on projectile and the internal nuclear coordinate s. P A projectile Target nucleus 2 Two-potential formula: Let (f ) be the scattering solution of H 0 V0 , where subscript f indicates the final reaction channel in the P T scattering . 1 1 (f ) f V0 (f ) f V0 f E H 0 i E H 0 V0 i and for brevity define the propagator s 1 1 ~ G() and G ( ) E H 0 i E H 0 V0 i where the transitio n matrix for H 0 V0 is T fi( 0 ) f V0 i( ) (f ) V0 i The t - matrix for the full Hamiltoni an is T fi f V i( ) f V0 V1 i( ) f V0 i( ) f V1 i( ) , where ~ i( ) i G ( ) V0 V1 i( ) We will assume that V1 induces the reaction of interest and not V0 . What we must do then is to separate the t - matrix into two parts correspond ing to the two potentials where the wave functions used in the matrix elements account for the dominant V0 . This requires some formal operator manipulati ons. From the definition of (f ) we obtain (f ) f f V0G ( ) and substitute this into the above eqn. for T fi T fi f V0 i( ) (f ) V1 i( ) f V0G ( )V1 i( ) 3 Two-potential formula: 1 1 1 1 ( A B) to get A B B A 1 ~ ~ ~ 1 ~ G ( ) G ( ) G ( ) G ( ) G ( ) G ( ) G ( )V0G ( ) ~ ~ ~ G ( ) G ( ) G ( )V0G ( ) G ( ) G ( )V0G ( ) Next, use the operator identity : Expand the wave function i( ) ~ i( ) i G ( ) V0 V1 i( ) ~ i G ( ) V0 V1 i( ) G ( )V0G ( ) V0 V1 i( ) i G ( ) V0 V1 i( ) G ( )V0 i( ) i i G ( )V0i G ( )V1 i( ) i( ) i( ) G ( )V1 i( ) and i( ) i( ) G ( )V1 i( ) where i( ) is in the incident channel and perturbati on interactio n V1 induces scattering out of the incident channel into final reaction channel i( ) . The t - matrix in the two - potential formula is : T fi f V0 i( ) (f ) V1 i( ) f V0 i( ) (f ) V1 i( ) T fi( 0) (f ) V1 i( ) For reactions generated by interactio n V1 (where V0 does not), the approximat e t - matrix is : T fiDW BA (f ) V1 i( ) , the Distorted Wave Born Approximat ion (DWBA) 4 DWBA: T fiDW BA (f ) V1 i( ) The distorted waves i( ) may be approximat ed by the optical model wave functions discussed in Chapter III using the KMT, Watson, Fesbach, g - matrix or phenomenol ogical models. The interactio n V1 can be estimated as follows. Starting with the P A Hamiltoni an : A A i 1 i 1 H TP H P H A vPi H 0 vPi Add/subtra ct the optical potential H H0 U opt A vPi U opt i 1 We identify V0 U A opt and V1 vPi U opt in the two - potential formula wh ere i( ) is the elastic i 1 channel wave function solution for the optical potential discussed in Chapter III. Using the methods in Chpt. III V1 can be approximat ed as A V1 vPi U i 1 A opt tPiKMT U opt i 1 5 DWBA: Apply the DWBA to inelastic scattering where the target nucleus is excited from the g.s. to some excited state F* and the projectile remains in its g.s. The nuclear physics notation for this reaction is A(p,p’)A*. For a single particle excitation: 18O P + 18O 1d3/2 2s1/2 1d5/2 1d3/2 2s1/2 1d5/2 1p1/2 1p 3/2 1p1/2 1p 3/2 1s1/2 p 1s1/2 n P p n T fiDW BA k( ) F * V1 F gs k( ) f i where k( ) is the wave function for the relative P A motion in the initial channel i and F gs is the nuclear g.s. Product F gs k( ) correspond s to i( ) in the previous slides; i the projectile state function is not relevant in this applicatio n. 6 DWBA: The transitio n matrix element is evaluated as A F * V1 F gs F * v i 1 A eff Pi F gs F * t U i 1 W Pi A opt F gs or F * tPiKMT U opt F gs i 1 where the Watson or KMT effective interactio n t - matrices discussed in Chapter III may be used as the effective interactio n. For single - nucleon excitation the core nucleon portion of the A overlap F * tPiKMT U opt F gs cancels, leaving only the active, valence nucleon overlap given by i 1 A F * t i 1 U KMT Pi opt F gs njF core A t i 1 KMT i U opt nj F core nj tPKMT val nj where the active, valence nucleon does not contribute to the U opt portion of the overlap. The inelastic transitio n is evaluated from the integral ( ) () T fiDW BA k( )nj tPKMT d 3rP d 3rN k( )* ( rP )n*j ( rN )tPKMT val nj k val rP rN nj ( rN ) k ( rP ) f i f i Actual calculatio n of this integral requires a ton of angular momentum algebra wh ich I will not bother wit h. A For collective state excitation s F * tPiKMT U opt F gs i 1 d opt U (r) dr 7 DWBA: 16O(p,p’)16O* at 800 MeV fixed target PRL 43, 421 (1979). Angular position of first peak indicates the L transfer to the nucleus. For 0+ ground-states, this gives the state’s Jp. 8 DWBA: 116,124Sn(p,p’) at 800 MeV PRL 42, 363 (1979) 9 DWBA: 12C,208Pb(p,p’) at 800 MeV Phys. Rev. C 18, 1436 (1978) 10 DWBA for single nucleon transfer: Here we consider reactions where the incoming projectile either sheds a nucleon which then binds to the target or picks up a nucleon from the target. The first is called a stripping reaction and the second a pick-up reaction. Examples of stripping reactions: AZ(d,p)A+1Z*, AZ(3He,d)A+1(Z+1)* Examples of pickup reactions: AZ(p,d)A-1Z*, AZ(3He,a)A-1Z* where (*) indicates that the final-state nucleus can be in an excited energy level. The DWBA amplitude for a pick-up reaction (see cartoon) is obtained as follows: Entrance channel p Exit channel p n C n C The full Hamiltonia n is : A1 A1 i 1 i 1 H Tp Tn H C v pi vni v pn 11 DWBA for single nucleon transfer: For the entrance channel the terms can be grouped as : A1 A1 A1 opt H Tp Tn vni H C v pi v pn H 0 U i v pi v pn U iopt i 1 i 1 i 1 where the last group of terms represents perturbati on term V1 in the two - potential formula and in the DWBA applicatio n. For g.s. matrix elements we may replace the sum over interactio ns with the sum over effective interactio ns, A1 V1 v pi v pn U iopt ( A 1) t KMT U iopt v pn v pn i 1 A 1 opt U i U iopt v pn A The DWBA integral for this reaction has the form T fiDW BA S n1/ j2 d 3 rp d 3 rn k( )* (rpnC )d* (rp rn ) F C v pn (rp rn ) F C nj (rn ) k( ) (rp ) f i where d is the deuteron w ave function, F C is the nuclear core w.f., nj (rn ) is the neutron bound state w.f., and rpnC is the deuteron core relative c.m. coordinate m r m r n n p p rpnC mn m p S is the spectrosco pic factor defined as the overlap between th e initial and final nuclear state where for a (p, d) pick - up reaction A 1, Z A, Z S n1/ j2 nj , or nj A 1, Z A, Z S n1/ j2 where A, Z and A 1, Z are the true state functions and nj is the normalized single - particle shell - model wave function. 12 DWBA for single nucleon transfer: For light - ion projectile s it is usually sufficient to neglect its size relative to that of the target nucleus, or we assume that s rp rn r0 A1/ 3 , resulting in the Zero - Range Approximat ion : T fiZR DW BA S n1/ j2 d 3 sd* ( s )v pn ( s ) d 3 r k( )* (r )nj (r ) k( ) (r ) f (amplitude ) i (shape) Spectrosco pic factors are roughly of order 1, but vary w idely from ~ 0.1 to ~ 10 depending on the type of reaction, the excitation energy of the final state, how close the nuclei are to having closed shells, etc. Advanced shell model calculatio ns attempt to explain th ese factors. See : " Ground state neutron spectrosco pic factors for Z = 3 - 24 isotopes, ' ' Jenny Lee, M.B. Tsang, W.G. Lynch, (2005), e - Print : nucl - ex/0511024 In the next few slides I show some examples of single-nucleon transfer reactions with DWBA shape predictions, and amplitude fits using the spectroscopic factor. These are (d,p) stripping reactions where a neutron is injected into the target nucleus into one of its states. The DWBA t-matrix has the same form as above where the single particle shell model state is the one receiving the transferred neutron. 13 DWBA for single nucleon transfer: Proton stripping reaction to various final states in 29P at beam energy of 35 MeV Phys. Rev. C 13, 1367 (1976) DWBA CC p 28Si 14 DWBA for single nucleon transfer: Proton dropped into d5/2 levels 15 DWBA for single nucleon transfer: Proton dropped into f7/2 and p3/2 levels 16 Coupled-Channels: An alternate reaction model is obtained by expanding the total P A wave function in terms of the nuclear states Fa with coefficien ts representi ng the relative P A motion, i.e. the P A relative wave function. This is given by i( ) k(,a) Fa a i and the Sch. eq. is TP H P H A V0 V1 a k(,a) Fa E k(,a) Fa i a i Project from the left - side with arbitrary target nucleus state F F TP H P H A V0 E V1 k(,a) Fa 0 i a All but the V1 term are diagonal. Substituti ng the above forms for V0 and V1 yields T P ( E ) F V0 F () ki , F V1 Fa k(,a) 0 a i which are a set of coupled, differenti al equations which can be solved for all k(,a) . i The diagonal m.e. of V0 is approximat ed with the g.s. expectatio n value, where F V0 F F A t i 1 KMT Pi F F gs U opt F gs and F V1 Fa , referred to as the coupling potential is given by F V1 Fa A KMT F t Pi U opt Fa which va nishes if a . i 1 17 Coupled-Channels: 12C(p,p’) at 800 Mev PRL 40, 1547 (1978) 4+ 2+ 0+ Coupled-Channels Coupled-channels allows multiple pathways to get to the final-state. These become important when the coupling strength increases, e.g. highly deformed nuclei or with strong vibrational excitations. 18 Coupled-Channels: DWBA Coupled-channels 176Yb(p,p’) at 800 MeV Phys. Rev. C 22, 1168 (1980) The multiple pathways, or scattering amplitudes interfere which can significantly shift the diffractive patterns. 19 Coupled-Channels: DWBA Coupled-channels 154Sm(p,p’) at 800 MeV Phys. Rev. C 22, 1168 (1980) 20 Eikonal approximation – the Glauber model: The eikonal approximation is a high energy, forward scattering limit solution of the Schrodinger equation, which provides an intuitive connection between the scattering potential and the cross section. Consider a plane wave scattering from a spherical potential V: a Roy Glauber z V ( x, y , z ) Incident plane wave d2 d2 d2 2 Sch. Eq. 2 2 2 k 2 ( x, y, z ) 2 V ( x, y, z ) ( x, y, z ) dy dz dx and assume the limits E V and ka 1 which is the high energy limit where the scattered wave is forward peaked. The asymptotic limit of the w.f. is eikr ( x, y, z ) r e f ( , ) r ikz where r x 2 y 2 z 2 . At very forward angles z x or y and therefore r z . 21 Eikonal approximation – the Glauber model: In this limit the asymptotic wave function can be approximat ed by ( x, y, z ) r eikz 1 f ( , ) / z where the w.f. has the form of a modulated plane wave along the z - axis. Glauber' s assumption is that the general wave function in the high energy, forward scattering limit should have the form of a plane wave along z with a slowly var ying modulation , or ( x, y, z ) e ikz ( x, y, z ) The Schrodinge r equation is d2 d2 d2 2 2 2 2 k 2 eikz ( x, y, z ) 2 Veikz ( x, y, z ) dy dz dx The z derivative is : 2 d 2 ikz d ikz d d ikz d ikz e ( x, y , z ) e ( x, y, z ) ike ( x, y, z ) e 2 2ik k 2 2 dz dz dz dz dz and assuming ( x, y, z ) is slowly var ying we neglect all second derivative s and get d 2 eikz 2ik k 2 k 2 eikz ( x, y, z ) 2 Veikz ( x, y, z ) dz d 2 2ik ( x, y , z ) 2 V ( x, y , z ) ( x, y , z ) dz d i i ( x, y , z ) 2 V ( x, y , z ) ( x, y , z ) V ( x, y , z ) ( x, y , z ) dz k v where v k / is the velocity (in the c.m.) 22 Eikonal approximation – the Glauber model: The solution t o d i ( x, y , z ) V ( x, y , z ) ( x, y , z ) dz v is ( x, y , z ) e i V ( x , y , z ) dz c v The b.c. at z is built into the wave function' s form; the b.c. at z is ( x, y, z ) z e ikz and therefore ( x, y, z ) z 1. This enables the integratio n constant c to be determined where ( x, y , z ) e i v z V ( x , y , z) dz ikz i v z V ( x , y , z) dz and the wave function is ( x, y, z ) e Now switch to cylindrica l coordinate s and introduce the impact parameter. ˆ r b kz k r b b z Incident plane wave V b kˆz k̂z 23 Eikonal approximation – the Glauber model: The scattering amplitude is given by ik r f k , k e V (r ) k (r )d 3r 2 2p and substituti ng the eikonal approximat ions gives z ikz iv V (b kˆz) dz ik r f k , k e V ( r )e dzd 2b 2 2p i z V ( b kˆz ) dz i ( kz k ( b kˆz )) v ˆ e V b kz e dzd 2b 2 2p where for elastic scattering k k and at forward angles, 1, the momentum transfer q k k is approximat ely perpendicu lar to the z - axis. k Therefore the leading phase factor simplifies to e i ( kz k ( b kˆz )) e i ( kz kz k b ) e iq b k The error in this approximat ion to leading order in is (see diagram) kz k (b kˆz ) kz k (sin )qˆ b (k q sin( / 2)) z where k sin q cos( / 2) k q to leading order in . The above simplifies to kz q b kz qz / 2 q b kz 2 / 2 So we must require that kz 2 / 2 1, or 2 as the range ka q / 2 k q k k k q z q sin( / 2) q / 2 of applicabil ity of the eikonal approximat ion. 24 Eikonal approximation – the Glauber model: Continuing with the calculatio n of the eikonal t - matrix i z V ( b kˆz ) dz f ( k , k ) e iqb d 2b dzV (b kˆz )e v 2 2p a Vdz 1 a Vdz The z - integratio n gives : dzVe e c where if V 0, then the integral must a vanish and c 1 / a. The above z - integral reduces to i v v V ( b kˆz) dz e 1 i v iqb iv V ( b kˆz) dz 2 f ( k , k ) 1d b e e 2p 2 i i V ( b kˆz ) dz 2 k iqb v f ( k , k ) e e 1 d b 2pi This has the form of an incident plane wave which passes along a straight line through the potential at some impact parameter and undergoes a phase shift which depends on the “profile” of the scattering potential. 25 Eikonal approximation – the Glauber model: If V (b kˆz ) is azimuthall y symmetric then the integratio n over the impact parameter can be simplified to the form, d 2 2p 0 0 b bdb d The azimuth integratio n can be done analytical ly where 2p de iqbcos 2pJ 0 (qb) (Bessel function) 0 and we finally get f (k , ) ik J 0 (qb) e i (b ) 1 bdb 0 1 (b) V (b kˆz )dz , the nuclear potential profile function v 2 King Glauber I ka Coulomb potentials , spin - orbit potentials and multiple scattering have also been included with the restrictio ns : E V , ka 1, in the eikonal approximat ion. For 800 MeV p A scattering , k ~ 7 fm -1 and a ~ 5 fm, so 14o 26 Eikonal approximation – the Glauber model: First-order p + 16O optical potential obtained by folding the free-space N+N t-matrix with the nuclear matter density estimated from the measured charge density; including Coulomb and spin-orbit potentials in the Glauber eikonal approximation. From Varma and Zamick, Phys. Rev. C 16, 308 (1977). 27 Eikonal approximation – the Glauber model: Varma and Zamick, Phys. Rev. C 16, 308 (1977). 28 This concludes Chapter V: Nuclear Reactions 29