Model of multiple Dirac eikonal scattering of protons by nuclei
aa r X i v : . [ nu c l - t h ] D ec Model of multiple Dirac eikonal scattering of protons by nuclei
V. V. Pilipenko and V. I. Kuprikov
National Science Center “Kharkov Institute of Physics and Technology”,Kharkov 61108, [email protected]
The model of multiple Dirac eikonal scattering of incident proton by target-nucleusnucleons is developed, in which new expressions for the elastic pA -scattering amplitudesare obtained from the multiple scattering Watson series with employing the eikonalapproximation for the Dirac propagators of the free proton motion between successivescattering acts on nucleons. Basing on this model, calculations for the complete set ofobservables of the elastic p + Ca and p + Pb at 800 MeV have been performed withusing proton-nucleon amplitudes determined from the phase analysis and the nucleondensities obtained from describing the target-nucleus structure in the relativistic mean-field approximation. A comparison has been made of the results of these calculationswith analogous calculations on the basis of the Glauber multiple diffraction theory.
Keywords : Proton–nucleus scattering; multiple scattering; Dirac equation; eikonal ap-proximation.PACS numbers: 24.10.-i,25.40.Cm,11.80.La
1. Introduction
The multiple diffraction scattering theory (MDST), the groundwork of which waslaid by the initial paper of Glauber is a fairly successful and popular approach todescribing the processes of scattering of protons and other hadrons on atomic nucleiin the intermediate energy region. This approach has found its development andemployment in a great number of works of many authors and, in particular, differentrefinements and corrections to the initial MDST formulation were considered anddiscussed (see, for example, Refs. 2–14 and references therein). At the present time,another approach also often employed for describing the intermediate-energy pro-ton scattering by nuclei is the model of relativistic impulse approximation, which isbased on solving the Dirac equation with a relativistic microscopic optical potential(see, for example, Refs. 15–17 and references therein). Although the MDST doesnot lose its applicability and is succesfully employed to analyzing various scatter-ing processes of protons and nuclei, it seems interesting to consider a consistentrelativistic development of this approach basing also on the Dirac equation withsimultaneously employing a relativistic description of the structure of target nuclei.At present, in the literature a number of precise measurements are available fordifferential cross sections and spin observables of the proton-nucleus ( p – A ) scatter-ing at intermediate energies (namely, the order of several hundred MeV). Analysisof these experimental data can be used for obtaining valuable information aboutthe structure of nuclei. However, for performing a reliable analysis of these datasufficiently accurate theoretical methods are needed. In Refs. 11–14, we obtained V. V. Pilipenko, V. I. Kuprikov and tried out some rather sophisticated expressions for the p – A scattering am-plitudes in the framework of MDST basing on the realistic N N -amplitudes andnuclear densities and allowing for two-nucleon correlations by means of includingintermediate excitations of nuclei as well as taking account of noneikonal correc-tions. Notwithstanding the certain success of this consideration, some shortcomingsin the description of the analyzed spin observables may indicate that possible im-provements of the model should be sought for.In this article, we carry out the development of the MDST approach by meansof constructing a relativistic model of the multiple scattering of the incident pro-ton on the nucleons of the target nucleus basing on the eikonal approximation forthe Dirac equation, which we will call the multiple Dirac eikonal scattering model(MDES). The new expressions for the p – A scattering amplitude obtained by usin this approach are employed for calculations of the complete set of observablesfor the elastic p – A scattering with using the realistic nucleon densities calculatedon the basis of approximation of relativistic mean field (RMF) (see, for example,Refs. 18–24) in comparison with the analogous calculations by the MDST.
2. Description of the MDES model
We may suppose that for further improving this approach it would be probably ad-visable to employ a more consistent relativistic description of the process of multiplescattering of the incident proton on nucleons of the target nucleus, which we aregoing to develop basing on the eikonal approximation for the Dirac equation. Thisis analogous to considering MDST as the theory of the multiple eikonal scatteringof the incident particle on nucleons in the nucleus. Therefore, we will build ourmodel in the way close to the approach of Refs. 9, 10, which was formulated takingaccount of the results of the works of Refs. 7,8. This approach is based on a numberof assumptions, which allow for the mutual cancelation of different corrections tothe MDST, namely: one may neglect the motion of nucleons in the nucleus duringthe interaction with the incident proton, as well as the contribution of rescatteringsof the projectile on the same nucleon; the most essential matrix elements of the t -operator may be considered to be local and determined on the energy shell. Notethat in Ref. 25 the Dirac-eikonal amplitude was considered for the proton-nucleusscattering described by a phenomenological potentialWe will proceed from the representation of the p – A scattering T -matrix operatorin the form of the multiple scattering Watson series, neglecting here rescatteringson the same target nucleons and therefore restricting the series up to the terms oforder A : T = A X j =1 t j + A X j =1 A X k = j t j ˜ G (+) t k + A X j =1 A X k = j A X l = k = j t j ˜ G (+) t k ˜ G (+) t l + ... + term of order A , (1) odel of multiple Dirac eikonal scattering of protons by nuclei where t j are the t -matrix operators for the proton scattering on individual nucleonsof the target nucleus, which are considered to be the same as for scattering on freenucleons in the impulse approximation spirit. We assume that the quantities inEq. (1) are written in the p – A center-of-mass (c.m.) frame. The propagator of freemotion of the proton between the scattering acts on nucleons is taken in the form˜ G (+) = h E − H p (cid:16) ˆk (cid:17) − H t (cid:16) ˆk , { r j } (cid:17) + i i − , (2) H p (cid:16) ˆk (cid:17) = α ˆk + βm, H t (cid:16) ˆk , { x j } (cid:17) = q ˆk + h t ( { x j } ) . (3)Here, ˆk = ˆk p = − ˆk t = − i∂/∂ r is the momentum operator, r = r p − r t , r p and r t arethe coordinates of the proton and the nucleus as a whole (we assume ~ = c = 1).The Hamiltonian of the incident proton H p has the usual Dirac form, while theHamiltonian H t of the target in the case of scattering on a spinless nucleus ispresented in a model form, in which h t ( { x j } ) is the Hamiltonian of the internalmotion of the nucleus ( { x j } is the set of internal nucleon variables), which has toprovide relativistic kinematic relations when taking account of the target recoil.The system energy in the p – A c.m. frame is E = ε p + ε t , where the energies ofproton and nucleus are ε p = √ k + m and ε t = √ k + M , m and M being theirmasses; k = | k i | = | k f | is the magnitude of the initial and final momenta.The amplitude of the elastic p – A scattering is related to the matrix element ofthe operator (1) in the p – A c.m. frame by the following formula: F ( k f , k i ) = − π ε t ε p + ε t (cid:16) u µ ′ ( k f ) Ψ +0 h k f | T | k i i Ψ u µ ( k i ) (cid:17) . (4)Here, Ψ is the wave function of the ground state of the nucleus; | k i,f i are the vectorsof the initial and final plane waves normalized to the unit particle density; the protonbispinors u µ ( k ) are normalized according to the formula: ¯ u µ ( k ) · u µ ( k ) = 2 m .In Eq. (1) the j -th nucleon in the nucleus is displaced by the vector r j relativeto the coordinate r t of the nucleus as a whole. Therefore, if t ( j ) is the t -operatorfor the scattering on a nucleon of the sort j placed in the origin of coordinates,then for the t -matrix element for the scattering on the j -th nucleon in the nucleuswe have: h k ′ | t j ( r j ) | k i = exp (cid:0) i qr j (cid:1) h k ′ | t ( j ) | k i , where q = k − k ′ . It is necessaryto relate the matrix element h k ′ | t ( j ) | k i with the amplitude of the p – N scatteringdetermined in the p – N c.m. frame. As it was noted above we may assume that themain role is played by the matrix element values of t ( j ) , determined on the energyshell. In this case, we may consider these matrix elements to be local, i.e. such thatdepend only on the momentum transfer q and the fixed energy corresponding tothe energy shell (see Refs. 9, 10). For the off-energy-shell the matrix element valuesit will be acceptable to use the same approximation, as that determined in the p – N c.m. frame on the energy shell.In the noncovariant form, the pN -amplitude f j ( q c ) (here, q c = k c − k ′ c isthe momentum transfer in the p – N c.m. frame) has the following form (see, for V. V. Pilipenko, V. I. Kuprikov example, Ref. 15): f j ( q c ) = A j ( q c ) + B j ( q c ) σ p σ j + C j ( q c ) ( σ p + σ j ) n + D j ( q c ) ( σ p q c ) ( σ j q c ) + E j ( q c ) σ pz σ jz . (5)Here, σ p and σ j are the Pauli matrices of the incident proton and of the nucleonin the nucleus; the direction e z is chosen along the average momentum k ca = ( k c + k ′ c ) /
2, and the normal to the scattering plane is n c = k c × k ′ c / | k c × k ′ c | = ˆq c × e z ,where ˆq c = q c /q c , q c ⊥ k ca . The amplitude (5) is a spin operator, whose matrixelements are calculated between the spinors of proton χ p and nucleon χ j , which arenormalized to unity. In the noncovariant form the matrix element of t ( j ) is relatedto the amplitude f j ( q c ) by the formula: h k ′ c | t ( j ) | k c i n c = − πε pc ε Nc ( ε pc + ε Nc ) f j ( q c ) = − πε pc f j ( q c ) . (6)Let us introduce the invariant Dirac amplitude of the p – N scattering in theform (see, for example, Ref. 15):ˆ F (cid:0) s, q c (cid:1) = F S (cid:0) s, q c (cid:1) + F V (cid:0) s, q c (cid:1) γ µp γ jµ + F T (cid:0) s, q c (cid:1) σ µνp σ jµν + F P (cid:0) s, q c (cid:1) γ p γ j + F A (cid:0) s, q c (cid:1) γ p γ µp γ j γ jµ . (7)Here, γ µp and γ µj are the Dirac matrices of the proton and nucleon, γ = iγ γ γ γ ; σ µν = i [ γ µ , γ ν ] / F S , F V , F T , F P , and F A are thescalar, vector, tensor, pseudoscalar and pseudovector components of the amplitude,respectively; s = ( p µp + p µj ) = 4 ε pc is the invariant energy, q c = − t = − ( p µp − p ′ µp ) is the invariant momentum transfer. The amplitude (7) is a Lorentz invariant. Inthe p – N c.m. frame it can be related to the amplitude (5) in such a way that thefollowing relationship is satisfied: ¯ u µ ′ p ( k ′ c ) ¯ u µ ′ j ( − k ′ c ) ˆ F j (cid:0) s, q c (cid:1) u µ p ( k c ) u µ j ( − k c ) = χ µ ′ + p χ µ ′ + j f j ( q c ) χ µ p χ µ j . (8)The covariant matrix element h k ′ | t ( j ) | k i can be defined so that to have thefollowing correspondence:¯ u µ ′ p ( k ′ c ) ¯ u µ ′ j ( − k ′ c ) h k ′ c | t ( j ) | k c i u µ p ( k c ) u µ j ( − k c )= 4 ε pc χ µ ′ + p χ µ ′ + j h k ′ c | t ( j ) | k c i n c χ µ p χ µ j = − πε pc ¯ u µ ′ p ( k ′ c ) ¯ u µ ′ j ( − k ′ c ) ˆ F j (cid:0) s, q c (cid:1) u µ p ( k c ) u µ j ( − k c ) . (9)In Eq. (9) the matrix element in the left-hand side and the expression in the thirdline are invariant quantities. Thus, we may write the following covariant relation,which is valid in the p – A c.m. frame: h k ′ | t ( j ) | k i = − πε pc γ p γ j ˆ F j (cid:0) s, q (cid:1) . (10)Let us emphasize that the left and right sides of Eq. (10) are operators in thebispinor spaces of the proton and nucleon. Here, in the matrix element, we do notwrite explicitly the momenta of the nucleon. The matrix element of t ( j ) is obtained odel of multiple Dirac eikonal scattering of protons by nuclei on the energy shell in the p – A c.m. frame for the case of the elastic scattering, whichcorresponds to the situation | k ′ | = | k | = k for the proton momenta. In the caseof quasifree scattering, the nucleon in the nucleus must also have the same energybefore and after the collision. Owing to these on-shell conditions of the energyand momentum conservation, the initial and final momenta of the projectile andstruck nucleon should correspond to the situation of the Breit frame kinematics,with the nucleon initially moving relative to the c.m. of nucleus.
15, 28
Under theseconditions, the magnitudes of the momentum transfers in the p – N and p – A c.m.frames coincide: − t = q c = q . Taking into account Eq. (10) and supposing that theconcrete choice of off-shell extrapolation of the matrix element is not very importantwhen the matrix element decreases rapidly with the q increase,
7, 8, 10 we will takethe following approximation for the matrix element of the operator t j ( r j ): h k ′ | t j ( r j ) | k i = − πε pc exp (cid:0) i qr j (cid:1) γ p γ j ˆ F j (cid:0) s, q (cid:1) , q = k − k ′ . (11)In the coordinate representation we have a local matrix element h r ′ | t j ( r j ) | r i = δ ( r ′ − r ) t j ( r − r j ) where t j ( r ) = − π ε pc γ p γ j Z d q exp( − i qr ) ˆ F j (cid:0) s, q (cid:1) . (12)When neglecting correlations between nucleons in the nucleus, one may presentthe ground state wave function of the nucleus Ψ in the form of a product ofsingle-particle wave functions ϕ j ( r j ), which in the relativistic description of thenucleus are four-component bispinors ϕ T,αj = ( u Tj , w Tj ), where u j and w j aretwo-component spinors of the upper and lower (the latter being small in the stan-dard representation) components. Let us introduce the single-nucleon densities ofthe nucleus, for a spinless nucleus the nonzero ones being only the scalar density ρ Sj and the time component of the vector density ρ V j : ρ Sj ( r ) = (Ψ +0 · γ j δ ( r − r j )Ψ ) = 1 N j N j X i =1 h | u ( r i ) | − | w ( r i ) | i , (13) ρ V j ( r ) = (Ψ +0 · δ ( r − r j )Ψ ) = 1 N j N j X i =1 h | u ( r i ) | + | w ( r i ) | i , N j = N, Z. (14)Because there is no preferential direction, they depend on r = | r | , while the spacecomponents of the vector density as well as all other densities vanish. Neglectingthe nucleon correlations, we may insert the projection operator | i h | betweenneighbouring operators in Eq. (1). As a result, in the propagators ˜ G (+) we have H t (cid:16) ˆk , { r j } (cid:17) → p ˆk + M for the nucleus Hamiltonian, and every operator t j isaveraged over Ψ in Eq. (4):(Ψ +0 · t j ( r − r j ) Ψ ) = − π ε pc γ p Z d q e − i qr Z d r ′ e i qr ′ × (cid:2) ρ Sj ( r ′ ) F Sj (cid:0) s, q (cid:1) + γ p ρ V j ( r ′ ) F V j (cid:0) s, q (cid:1)(cid:3) . (15) V. V. Pilipenko, V. I. Kuprikov
Further, considering the distinction between densities (13) and (14) caused by thecontributions of the lower components as small (see, for example, Refs. 19, 20), atthis stage we will assume these densities to be equal to each other in the formulaefor the scattering amplitudes: ρ Sj ( r ) = ρ V j ( r ) ≡ ρ ( j )0 ( r ). As a result, we have:(Ψ +0 · h r ′ | t j ( r j ) | r i Ψ ) = δ ( r ′ − r ) γ p ˜ τ j ( r ) , (16)˜ τ j ( r ) = − π ε pc Z d q e − i qr (cid:2) F Sj (cid:0) s, q (cid:1) + γ p F V j (cid:0) s, q (cid:1)(cid:3) Q ( j )0 ( q ) , (17)where the elastic nucleon formfactor has been introduced: Q ( j )0 ( q ) = R d r e i qr ρ ( j )0 ( r ).Further we shall consider the free propagator of the following form (we willsimply denote γ p ≡ γ ): G (+) = ˜ G (+) γ = (cid:20) γ E − γ ˆk − m − γ q ˆk + M + i (cid:21) − . (18)For the propagator (18) we shall employ the eikonal expansion in powers of 1 /k ,choosing the z -direction e z k k a , where k a = ( k i + k f ) / p – A scattering. Thus, we present the free propagator in theform: G (+) ≈ ε t / ( ε p + ε t ) h G (+) e + δG (+) ne i , where the eikonal propagator and thenoneikonal corrections of the first order have the following form in the momentumrepresentation: G (+) e = − m Λ + ( k Λ ) h k a (cid:16) ˆk − k a (cid:17) − i i − , (19) δG (+)ne = 2 m Λ + ( k Λ ) (cid:16) ˆk − k a (cid:17) − q / h k a (cid:16) ˆk − k a (cid:17) − i i + γ ( ˆk − k Λ )2 k a (cid:16) ˆk − k a (cid:17) − i . (20)Here, Λ + ( k Λ ) = ( γ ε p − γ k Λ + m ) / (2 m ) is the projection operator onto the protonstates with the positive energy and the momentum k Λ . The direction of k Λ shouldbe close to the chosen z -direction for the eikonal approximation, however we have acertain freedom in choosing it: we may take k Λ = k a or k Λ = k i , or even k Λ = k f (we remind that k i = k a + q / k f = k a − q / q ⊥ k a , q = k i − k f ). Each of thesevariants is practically equivalent, although has its advantages when consideringthe eikonal approximation for the Dirac equation, and our choise of k Λ will beconctretized below from physical reasoning.The eikonal propagator in the coordinate representation takes the form: h r ′ | G (+) e | r i = − i k a m Λ + ( k Λ ) δ ( b ′ − b ) θ ( z ′ − z ) exp [ ik a ( z ′ − z )] , (21)where the transversal vector has been introduced: b = r ⊥ , b ⊥ e z . Neglecting thenoneikonal corrections and restricting to the eikonal expresson for the propaga-tor (21), with taking into account Eqs. (16) and (17), we find the amplitude of the odel of multiple Dirac eikonal scattering of protons by nuclei elastic p – A scattering (4) in the form: F ( k f , k i ) = ¯ u µ ′ ( k f ) ˜ F e ( q ) u µ ( k i ) where˜ F e ( q ) = − π Z d b e i qb Z dz A X j =1 ˆ τ j ( b , z )+ A X n =2 (cid:16) − i k a (cid:17) n − A X i =1 . . . A X i n = ... Z dz i . . . dz i n ˆ τ i ( b , z i ) × m Λ + ( k Λ ) ˆ τ i ( b , z i ) 2 m Λ + ( k Λ ) · · · ˆ τ i n − (cid:0) b , z i n − (cid:1) × m Λ + ( k Λ ) ˆ τ i n ( b , z i n ) θ ( z i − z i ) . . . θ (cid:0) z i n − − z i n (cid:1) , (22)ˆ τ j ( b , z ) = − ε t ε pc ε p + ε t π Z d q e − i ( qb + q z z ) × (cid:2) F Sj (cid:0) s, q (cid:1) + γ F V j (cid:0) s, q (cid:1)(cid:3) Q ( j )0 ( q ) . (23)In Eq. (22) all indices i . . . i n in the sums have different values. Further we shallpresent the amplitude in terms of the matrix elements with respect to the spinors χ in the initial and final rest frames of the proton. Introducing the projection operatoronto the upper components B + = (1 + γ ) / L ( k ) = r ε p + m m (cid:20) α k ε p + m (cid:21) , (24)we obtain the scattering amplitude in the form F ( k f , k i ) = χ µ ′ + ˆ F e ( q ) χ µ whereˆ F e ( q ) = 2 mB + L ( − k f ) ˜ F e ( q ) L ( k i ) B + = − π Z d b e i qb mB + L ( − k f ) × Z dz A X j =1 ˆ τ j ( b , z ) + A X n =2 (cid:16) − i k a (cid:17) n − A X i =1 . . . A X i n = ... Z dz i . . . dz i n × ˆ τ i ( b , z i ) 2 m Λ + ( k Λ ) ˆ τ i ( b , z i ) 2 m Λ + ( k Λ ) · · · ˆ τ i n − (cid:0) b , z i n − (cid:1) × m Λ + ( k Λ ) ˆ τ i n ( b , z i n ) θ ( z i − z i ) . . . θ (cid:0) z i n − − z i n (cid:1) L ( k i ) B + . (25)For considering further calculations we write down the following useful formulae:Λ + ( k i,f ) Λ + ( k i,f ) = Λ + ( k i,f ) , Λ + ( k i,f ) γ Λ + ( k i,f ) = ε p m Λ + ( k i,f ) , (26)Λ + ( k i,f ) = L ( k i,f ) B + L ( − k i,f ) , (27) B + L ( − k f ) L ( k i ) B + = 12 m [ ε p (1 − cos θ )+ m (1 + cos θ ) + i ( ε p − m ) σ n sin θ ] , (28) V. V. Pilipenko, V. I. Kuprikov B + L ( − k f ) γ L ( k i ) B + = 12 m [ ε p (1 + cos θ )+ m (1 − cos θ ) − i ( ε p − m ) σ n sin θ ] , (29) B + L ( − k i,f ) γ L ( k i,f ) B + = ε p m , L ( − k i,f ) L ( k i,f ) = 1 , (30)where n = k i × k f / | k i × k f | is the normal to the scattering plane and θ is thescattering angle in the p – A c.m. frame.Using the representation (27) for the projection operators Λ + in Eq. (25) andtaking into account formulae (28)–(30), we see that, when choosing k Λ = k a inall Λ + , we obtain a T -invariant expression for the amplitude but in the multiplescattering terms there are only two pN -interaction operators with spin rotation (i.e.those containing the operator σ n ), namely, at the first and last collisions. Taking k Λ = k i or k Λ = k f in all Λ + , we have only one operator with spin rotation atthe last or at the first collision, correspondingly, and besides, the model will notbe T -invariant. A more reasonable model would be the one in which the operatorwith the spin rotation could appear in each of the n successive collisions. In orderto obtain such a picture as well as the T -invariance of the model, in the expressionfor the amplitude (25) we shall simultaneously use the projection operators with k Λ = k i and k Λ = k f , writing Eq. (25) in the following symmetrized form:ˆ F e ( q ) = − π Z d b e i qb mB + L ( − k f ) Z dz A X j =1 ˆ τ j ( b , z ) + A X n =2 (cid:16) − i k a (cid:17) n − × n n X l =1 A X i =1 . . . A X i n = ... Z dz i . . . dz i n ˆ τ i ( b , z i ) 2 m Λ + ( k ( l )) × ˆ τ i ( b , z i ) 2 m Λ + ( k ( l )) · · · ˆ τ i n − (cid:0) b , z i n − (cid:1) m Λ + ( k n − ( l )) × ˆ τ i n ( b , z i n ) θ ( z i − z i ) . . . θ (cid:0) z i n − − z i n (cid:1) L ( k i ) B + , (31)where the momenta in the operators Λ + are as follows: k m ( l ) = k f for 1 ≤ m < l ,and k m ( l ) = k i for l ≤ m < n −
1, 1 ≤ l ≤ n . Using the representation (27) for theoperators Λ + and formulae (28)–(30), we obtain the following expression:ˆ F e ( q ) = − π Z d b e i qb A X n =1 (cid:16) − i k a (cid:17) n − n X { i ...i n } i n X i l = i ∞ Z −∞ dz . . . dz n × ˆ τ ′ i l ( b , z l ) ¯ τ i ( b , z ) ... ¯ τ i l − ( b , z l − ) ¯ τ i l +1 ( b , z l +1 ) · · · ¯ τ i n ( b , z n ) , (32)ˆ τ ′ i l ( b , z ) = B + L ( − k f ) ˆ τ i l ( b , z ) L ( k i ) B + , ¯ τ i m ( b , z ) = B + L ( − k i,f ) ˆ τ i m ( b , z ) L ( k i,f ) B + . (33)Here, P { i ...i n } is the sum over possible combinations of indices i ...i n . ObtainingEq. (32), we have taken into account the commutativity of all functions ˆ τ ′ i l ( b , z ) odel of multiple Dirac eikonal scattering of protons by nuclei and ¯ τ i m ( b , z ), which allowed us to sum up the θ -functions in Eq. (31). Integratingin Eq. (32) over all z -variables and collecting first the terms with the same values i l and i ...i n = i l and then the terms of the same sort, we find the following expressionfor the amplitude of the elastic p – A scattering:ˆ F e ( q ) = ik π Z d b e i qb Z ˆ E ′ p ( b ) Z − X n p =0 N X n n =0 ( − n p + n n n p + n n + 1 C Z − n p C Nn n ¯ E n p p ( b ) ¯ E n n n ( b )+ N ˆ E ′ n ( b ) Z X n p =0 N − X n n =0 ( − n p + n n n p + n n + 1 C Zn p C N − n n ¯ E n p p ( b ) ¯ E n n n ( b ) , (34)where C Zn p are the binomial coefficients and the following functions for nucleonshave been introduced:¯ E j ( b ) ≡ i k a ∞ Z −∞ dz ¯ τ j ( b , z ) = 2 ik a π ε t ε pc ( ε p + ε t ) m Z d q e − i qb × m (cid:2) mF Sj (cid:0) s, q (cid:1) + ε p F V j (cid:0) s, q (cid:1)(cid:3) Q ( j )0 ( q ) , (35)ˆ E ′ j ( b ) ≡ i k a ∞ Z −∞ dz ˆ τ ′ j ( b , z ) = 1 ik a π ε t ε pc ( ε p + ε t ) m Z d q e − i qb × m (cid:8) (1 − cos θ ) (cid:2) ε p F Sj (cid:0) s, q (cid:1) + mF V j (cid:0) s, q (cid:1)(cid:3) +(1 + cos θ ) (cid:2) mF Sj (cid:0) s, q (cid:1) + ε p F V j (cid:0) s, q (cid:1)(cid:3) + i ( ε p − m ) σ n sin θ (cid:2) F Sj (cid:0) s, q (cid:1) − F V j (cid:0) s, q (cid:1)(cid:3)(cid:9) Q ( j )0 ( q ) . (36)The expression (34) for the elastic p – A scattering amplitude can also be representedin the form:ˆ F e ( q ) = ik π Z d b e i qb Z dx n Z ˆ E ′ p ( b ) (cid:2) − x ¯ E p (cid:3) Z − (cid:2) − x ¯ E n (cid:3) N + N ˆ E ′ n ( b ) (cid:2) − x ¯ E p (cid:3) Z (cid:2) − x ¯ E n (cid:3) N − o . (37)Let us return to the definition of the pN -amplitudes (5)–(8). In calculationsby MDST, it is usual to take into account only the central and spin-orbit partsof the pN -amplitudes (see, for example, Refs. 6, 10). Further we will also restrictour consideration to these amplitudes A ( q ) and C ( q ) in Eq. (5). For this purpose,instead of the Eq. (8) we employ a simplified relation between the amplitudes,similarly to the work of Ref. 29, which may be written in the p – A c.m. frame withmaking use of the forward scattering conditions, as follows: B + L ( − k f ) (cid:2) F Sj (cid:0) s, q (cid:1) + γ p F V j (cid:0) s, q (cid:1)(cid:3) L ( k i ) B + = k c mk [ A j ( q ) + qC j ( q ) σ n ] . (38) V. V. Pilipenko, V. I. Kuprikov
Strictly speaking, the used pN -amplitudes should refer to the scattering in the Breitframe, but here we will neglect this distinction from the usual transition betweenthe p – N c.m. and p – A c.m. frames because it can yield considerable differencesin the calculated p – A scattering observables only for light target nuclei and forlarger scattering angles. From Eq. (38) we obtain the explicit formulae for therelationship between the pN -amplitudes:2 mF Sj (cid:0) s, q (cid:1) = kk c ( A j ( q )2( ε p + m ) − iC j ( q ) p k − q / (cid:20) ε p − q ε p + m ) (cid:21)) , (39)2 mF V j (cid:0) s, q (cid:1) = kk c ( A j ( q )2( ε p + m ) + iC j ( q ) p k − q / (cid:20) m + q ε p + m ) (cid:21)) . (40)We write the amplitudes in the p – N c.m. frame in the form: A ( q ) = ik c / (2 π ) f c ( q ), qC ( q ) = ik c / (2 π ) f s ( q ). Taking into account that 2 ε t ε pc / (( ε p + ε t ) m ) ≈ k/k c , thefunctions (35) and (36) can be written as follows:¯ E j ( b ) = E ( j )0 ( b ) + ¯ E ( j ) s ( b ) , ˆ E ′ j ( b ) = ¯ E j ( b ) + [(1 − cos θ ) + i σ n sin θ ] E ′ ( j ) s ( b ) , (41) E ( j )0 ( b ) = 12 π ∞ Z dqq J ( qb ) f ( j ) c ( q ) Q ( j )0 ( q ) , (42)¯ E ( j ) s ( b ) = i π ∞ Z dq J ( qb ) q p k − q / f ( j ) s ( q ) Q ( j )0 ( q ) , (43) E ′ ( j ) s ( b ) = − i π ∞ Z dq J ( qb ) k p k − q / f ( j ) s ( q ) Q ( j )0 ( q ) . (44)Thus, the pA -amplitude in Eq. (37) can be represented in the form: ˆ F e ( q ) = A ( q ) + σ n B ( q ) where A ( q ) = ik ∞ Z dbb J ( qb )Ω A ( b ) , B ( q ) = − k sin θ ∞ Z dbb J ( qb ) ˜Ω B ( b ) , (45)and the nuclear profile functions introduced in Eq. (45) are equal toΩ A ( b ) = Z dx n Z h ¯ E p ( b ) + (1 − cos θ ) E ′ ( p ) s ( b ) i (cid:2) − x ¯ E p ( b ) (cid:3) Z − (cid:2) − x ¯ E n ( b ) (cid:3) N + N h ¯ E n ( b ) + (1 − cos θ ) E ′ ( n ) s ( b ) i (cid:2) − x ¯ E p ( b ) (cid:3) Z (cid:2) − x ¯ E n ( b ) (cid:3) N − o , (46) odel of multiple Dirac eikonal scattering of protons by nuclei ˜Ω B ( b ) = Z dx n ZE ′ ( p ) s ( b ) (cid:2) − x ¯ E p (cid:3) Z − (cid:2) − x ¯ E n (cid:3) N + N E ′ ( n ) s ( b ) (cid:2) − x ¯ E p (cid:3) Z (cid:2) − x ¯ E n (cid:3) N − o . (47)If the proton and neutron E -functions may be put identical (for example, taken inthe averaged form), then the expressions (46) and (47) are simplified:Ω A ( b ) = (cid:20) − cos θ ) E ′ s ( b )¯ E ( b ) (cid:21) n − (cid:2) − ¯ E ( b ) (cid:3) A o , ˜Ω B ( b ) = E ′ s ( b )¯ E ( b ) n − (cid:2) − ¯ E ( b ) (cid:3) A o . (48)When taking account of the electromagnetic effects, the central A ( q ) and spin-orbit B ( q ) amplitudes of the elastic p – A scattering can be written as: A ( q ) = A C ( q ) + ik ∞ Z dbbJ ( qb ) n(cid:16) e iχ ( b ) − e iχ ( b ) (cid:17) + e iχ ( b ) [Ω A ( b ) − Ω B ( b ) χ s ( b )] o , (49) B ( q ) = B C ( q ) − ik ∞ Z dbbJ ( qb ) nh e iχ ( b ) χ s ( b ) − e iχ ( b ) χ s ( b ) i + e iχ ( b ) [Ω A ( b ) χ s ( b ) + Ω B ( b )] o . (50)where we have introduced the nuclear spin-orbit profile function Ω B ( b ) =( i/k ) d ˜Ω B ( b ) /db . In Eqs. (49) and (50), χ ( b ) = 2 ξln ( kb ) and χ s ( b ) = 2 ξκ / b are the central and spin-orbit scattering phases in the Coulomb field of two pointcharges in the eikonal approximation. The central and spin-orbit components of thescattering amplitude, corresponding to these phases, are given by the formulae A C ( q ) = − ξkq Γ (1 + iξ )Γ (1 − iξ ) exp (cid:16) − iξ ln q k (cid:17) , B C ( q ) = − iκqA C ( q ) . (51)Here, ξ = Ze (cid:14) ~ v is the Sommerfeld parameter for the p – A scattering, and theparameter κ characterizes the spin-orbit quantities χ s ( b ) and B C ( q ). The eikonalCoulomb phase χ ( b ) of the scattering on the volume charge of the target nucleushas the following form: χ ( b ) = χ ( b ) + 8 πξ ∞ Z b drr ρ ( p )0 ( r ) ln q − b (cid:14) r b / r − q − b (cid:14) r , (52)and the corresponding spin-orbit phase is χ s ( b ) = ( κk ) dχ ( b ) /db , which is analo-gous to the usual macroscopic allowance for the interaction of the proton magnetic V. V. Pilipenko, V. I. Kuprikov moment with the electromagnetic field of the target nucleus in MDST by introduc-ing a spin-orbit correction to the macroscopic proton–nucleus Coulomb phase shift χ ( b ). However, in contrast to this, in our previous works in the MDST frameworkwe considered this interaction microscopically through including the Coulomb spin-orbit term in the proton–nucleon amplitudes. In this case, the value of parameter κ was determined from the asymptotic behavior of the profile function Ω B ( b ) at b → ∞ (see, for example, Refs. 13, 14), which leads to the value κ = κ pp k c /k where κ pp is the corresponding spin-orbit parameter in the pp -amplitude. According toconcrete conventions of a used phase analysis, it may be κ pp = (3 + 4 µ a ) / m or κ pp = (3 / ( E c + m ) + 2 µ a /m ) / E c , where µ a = 1 .
79 is the anomalous magneticmoment of the proton. This κ value differs from that used in Ref. 6. In order toensure the correspondence with our microscopic consideration in Ref. 13, here weshall assume κ = κ pp k c /k with the latter variant of κ pp in accordance with the usedphase analysis for the pN -amplitude.
3. Results of calculations of the p – A scattering observables To perform the calculations of the p – A scattering amplitudes in the frameworkof the multiple scattering model, it is necessary to know the densities of nucleondistribution in the target nucleus. In this work, we have employed the nucleondensities obtained by us from the microscopic calculations of the nuclear structurein the approximation of the relativistic mean field (RMF). The RMF models knownfrom the literature provide the description of properties of the ground state of finitenuclei with a good accuracy (see Refs. 20, 24).The RMF model used in the present work is based on the nucleus Lagrangiandensity which has the following form:
23, 24 L = X j = n,p ¯Ψ j n iγ µ ∂ µ − m N + g σ σ − g ω γ µ ω µ − g ρ γ µ ρ µ · τ − e γ µ (1 − τ ) A µ o Ψ j ++ 12 ∂ µ σ∂ µ σ − m σ σ −
14 Ω µν Ω µν + 12 m ω ω µ ω µ − R µν R µν + 12 m ρ ρ µ · ρ µ − F µν F µν , (53)where σ , ω µ and ρ µ are the scalar, isoscalar-vector and isovector-vector mesonfields, respectively ( µ = 0 , , , A µ is the photon field ( e being the constantof electromagnetic interaction); Ψ n,p are the nucleon fields; τ are the isospin Paulimatrices ( τ = 1 for neutron, and τ = − γ µ are the Dirac matrices; g σ , g ω , and g ρ are the meson–nucleon couplings, which depend on the nucleon density inthe nucleus; m σ , m ω , m ρ , and m N are the masses of the mesons and nucleon; Ω µν , R µν , and F µν are the tensors of the fields of vector mesons and the electromagneticfield. All the meson–nucleon couplings and some of the meson masses are adjustableparameters of the model, whose values are determined in the literature from therequirement of the best description of properties of finite nuclei and nuclear matter.Owing to the stationarity of the problem and a number of symmetry requirements odel of multiple Dirac eikonal scattering of protons by nuclei imposed when deriving the RMF equations, they involve only the time componentsof the four-vectors of the nucleon and electromagnetic currents and vector-mesonfields. In the RMF models under consideration the antiparticle states are not takeninto account. The corresponding set of coupled equations, which is presented inRefs. 23, 24 and includes the Dirac equations for the spinor nucleon fields, thenonhomogeneous Klein–Gordon equations for the meson fields, and the equationfor the Coulomb field, was solved by us numerically by the iteration method. Inthe calculations for the Ca nucleus, after obtaining the self-consistent solution,we took into account the center-of-mass motion by recalculating the neutron- andproton-density distributions in the harmonic-oscillator approximation as it was donein Ref. 13.By means of the MDES model developed above, which bases on the eikonalapproximation for the Dirac equation and on employing realistic nucleon densitiescalculated in the RMF approximation and pN -amplitudes found from the phase-analysis solutions, we have developed an original numerical computer code andperformed the corresponding analysis of the differential cross sections, analyzingpowers, and spin rotation functions for the elastic p + Ca and p + Pb scatteringat the proton energy of 800 MeV with using different variants of the relativisticeffective
N N -interaction in the nuclear-structure calculations. The results obtainedby us on the basis of MDES are compared with the analogous results obtained inthe framework of the MDST approach under the same calculation conditions. InFig. 1, we present the results of such calculations with making use of the nucleondensities, obtained by us in the RMF approach with the DD-ME2 interaction (note that in Ref. 13 a comparison was performed of the results of the MDSTcalculations with using different nucleon densities in the target nuclei obtained inthe RMF and Skyrme–Hartree–Fock approaches). In these calculations we haveused the approximation from Ref. 31 for the pN -amplitudes, which was determinedfrom the phase analysis in Ref. 32.In our previous works (see Refs. 11, 13, 14), along with the usual variant ofMDST, we considered its more sophisticated formulation with taking account ofthe effects of two-nucleon correlations through including the most essential inter-mediate excitations (IE) of the target nuclei as well as of the effects of the noneikonalcorrections (NE) to the p – A amplitude. For this reason, in Fig. 1 we show a com-parison not only with the curves calculated by the usual MDST, which is analogousto the present variant of the MDES model, but also with the results obtained by theimproved MDST calculation allowing for IE and NE. As can be seen from Fig. 1,the calculations based on the MDES model and in the framework of the MDSTwithout taking into account IE and NE yield not so much differing results at theconsidered incident proton energies. The most significant distinctions are observedfor the scattering on Ca in the region of diffraction minima.For the analyzing power A y ( θ ), there are certain distinctions of the resultsobtained in these two approaches, in the case of the scattering on Ca the MDESmodel providing somewhat more encouraging results, however for the scattering V. V. Pilipenko, V. I. Kuprikov -2 (mb/sr) p+ Ca (deg) MDES MDST MDST+IE+NE Expt. (a) -3 -1 (deg)(mb/sr) (b) MDES MDST MDST+IE+NE Expt. p+ Pb (deg) (c) A y (deg) (d) A y (deg) (e) Q (deg) (f) Q Fig. 1. Differential cross sections σ ( θ ), analyzing powers A y ( θ ), and spin rotation functions Q ( θ )for the elastic p + Ca and p + Pb scattering at 800 MeV, calculated with the RMF nucleondensities for the DD-ME2 interaction on the basis of the new MDES model in comparison withcalculations by MDST without corrections as well as with the IE and NE corrections. The exper-imental data are taken from Refs. 33–36. odel of multiple Dirac eikonal scattering of protons by nuclei on Pb this curve rises above the experimental points with the scattering angleincrease, as it also does in the case of MDST. Certain distinctions between thesetwo models are also observed in the calculations of the spin rotation functions Q ( θ ). On the whole, we may say that the above-developed approach MDES inits present form without taking account of the IE and NE corrections does notyield a significant improvement in describing the analyzed experimental data in theconsidered energy region. In Fig. 1 the shown results of calculations on the basisof MDST with the allowance for the mentioned corrections clarify their role forthe data description. The contribution of IE of nuclei becomes essential with thescattering angle increase and the NE corrections manifest themselves in smoothingthe diffraction minima of the observables, and in general, the allowance for theIE and NE corrections improves the description of experimental data. From thiscomparison we can make a conclusion about the necessity of further developing theMDES model, in particular, in order to take into account the effects of IE of targetnuclei and NE corrections also in the approach basing on the Dirac equation. Asother possible refinements of this model, we can also mention the allowance forthe contributions to the p – A amplitude coming from spin-spin terms in the N N -amplitude as well as refining the formulae by retaining the contributions of the lowercomponents in the nucleon wave functions calculated in the RMF approximation.
4. Conclusion
By analogy with the Glauber multiple diffraction scattering theory (MDST), be-ing a highly effective and popular approach to analyzing processes of the nuclear-particle scattering on atomic nuclei at intermediate energies, in the present workan attempt has been made to provide a consistent relativistic description of theprocess of multiple scattering of the incident proton on nucleons of the target nu-cleus. For this purpose, a new model has been built, which is developed basingon the eikonal approximation for the Dirac equation and, therefore, can be calledas the model of multiple Dirac eikonal scattering (MDES). New expressions havebeen obtained for the amplitudes of the elastic p – A scattering on a spinless targetnucleus basing on the consideration of the multiple scattering Watson series withmaking use of the eikonal approximation for the Dirac propagators of free protonmovement between the acts of successive scattering on nucleons. In the frameworkof the developed MDES model an analysis of the complete set of the observablesfor the elastic p + Ca and p + Pb scattering has been performed at the incidentproton energy of 800 MeV. This analysis is based on using the realistic nucleon den-sities, calculated by means of modern models for describing the nucleus structure inthe approximation of relativistic mean field, and on using realistic
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