The solid dashed curve shows the calculation with without spin-orbit term, respectively.. The linepoints curve shows the calculation with the full effective SLy5 Skyrme interaction.. The
OPTICAL POTENTIAL 4
Microscopic optical potentlal cà 6
In literature, there are several microscopic approaches to calculate the MOP: the ab initio methods [26-38], the semi-microscopic approaches [39, 40], the methods based on self-energy [41, 42], the nuclear matter approaches [43-45],the nuclear structure approaches |4, 5, 15-18, 46-48], a method which combines nuclear matter and nuclear structure approaches [49-51], and so on.
Ab initio approaches 2 0.0.0.0 000 een 6
Recently, the ab initio approaches have impressive progresses in many as- pects: mass number, precision, and accuracy These models have demonstrated their ability to describe the microscopic optical potential for neutron elastic scattering of 4°48Ca [52] Within the high performance computer and modern calculation technique, these models can now reach to the reliable predictions for medium nuclei by using the modern nucleon-nucleon (NN) and three-nucleon forces (3NFs) from chiral effective field theory.
The intrinsic Hamiltonian of the scattering system reads
‡= with p; being the momentum of the nucleon i of mass m, P; = ye p; being the momentum associated with the center-of-mass motion, W; and Vj;, are the two- body (NN) and three-body (3NFs) interaction, respectively Using the Green’s function approach [52], the microscopic optical potential is defined as y= D*+U, (1.6) where %*(7,6,F) is the self-energy which can be calculated by inverting the matrix
S1) = (GO (BE) — G"(B), (1.7) and the one-body potential U is the HF potential, and the Green’s function G satisfies the Dyson equation
7,0 where œ,,+,ð denote single-particle states, and G is the first-order Green’s function ©! in Eq.(1.6) is the nuclear optical potential for E > Egy while Xƒ is the mean-field potential for the bound states (A+1) nucleon system for E < Egs. ĩ
To get the optical potential, the Dyson equation has been inverted by using the coupled-cluster approach [37] for the Green’s function GC° as
1 - cc E) = (®1\daz at G'* (a, B, E) (®o,r|d oe BÀ) +ủn 0Ì 0)
BH (BA — H) —ùn where (®g r| = (®o|(1 + A) is the left ground state with reference state |®o) and
A a linear combination particle-hole de-excitation operators, EA is the energy of the ground state, œ and 6 denote single-particle states, the parameter 7 > 0 is such that 7 — 0 at the end of the calculation The operators a„ = e age? and al =e Tạ Ver are the similarity-transformed annihilation and creation operators, respectively, and the Baker-Campbell-Hausdorff expansion yields the relations
Ga = dạ + (aa, TÌ, (1.10) al, =a, + [ab, TÌ, (1.11) where the operators aq and ah annihilate and create a fermion in the single- particle states a and đ, respectively, and are shorthand for the quantum numbers a=(n,l,j,jz,Tz) Here, n,l,j,j2,7, label the radial quantum number, the orbital angular momentum, the total orbital momentum, its projection on the z axis, and the isospin projection, respectively a cluster operator T denotes
T'—T\ + Tạ = date ta; + — 3 a balajai free, (1.12) where 7¡ and 7; induce lạnh and 2p-2h excitations of the HF reference, re- ijab spectively The single-particle states i,7, refer to hole states occupied in the reference state |®o) while a,b, denote valence states above the reference state.
This approach has been successfully applied to calculate the differential elastic cross section for elastic scattering 4°Ca(n,n)*°Ca, and #ŠSCa(n,n)'®Ca at some incident energies lower than 15 MeV The obtained results have been compared with the experimental data and the phenomenological KD potential However, due to the complicated structure of the equation (1.7), the absorption from the non-elastic channel of the optical potential has been dropped Therefore, the very important contributions at low-energy from the collective excitations (giant resonances and low-lying states) are missing.
Another work uses the self-consistent Green’s function theory within the three-nucleon (3W) interaction [53] The Hamiltonian of the system is
H(A) =T ~ Tem.(A + 1) + Vo + V3, (1.13) where Tean (A+ 1) is the center of mass kinetic energy operator for the A-nucleon target plus the projectile, and V2, V3 are the two-body (NN) and three-body
(3N) interactions The partial wave (1,7) decomposition of the self-energy which is the optical potential reads
Del (hk BLT) = So Raa()S(E,D)Rua(E), (1.14) Xa nt ( nn! which is energy-dependent, nonlocal and in a separable form, R,, i(k) are the ra- dial harmonic oscillator wave functions in momentum space And the irreducible self-energy >*(œ) is the solution of the Dyson equation g(w) = g°(w) + 9° (w)E*(w)9(w), (1.15) where g°(w) is the free particle propagator Then, the irreducible self-energy
&*(w) is in the general spectral representation ằ*¿(E,I) = 5% Mi M; 2a) cot (ge rere) 7 38 1
Rp), ` (1.16) 1 i where X4; is the first order and energy independent mean field, a(đ) is single- particle quantum numbers of the harmonic oscillator basis, is the boundary condition parameter, matrices M(N) are the coupling between the single particle states to the intermediate p-h configuration, C(D) are the interaction matrices, and K are the unperturbed energies.
This model has been applied to describe the neutron elastic scattering off !©O (*°Ca) at 3.286 (3.2) MeV respectively [53] To show some important features of optical potential at low-energy, the obtained results are quantitatively com- pared with experimental data This encouraging results showing the self-energy generated from the self-consistent Green function is a promising candidate to link the theories of structure and reactions for medium mass nuclei.
Finally, although ab initio methods have made progress in handling light and medium nuclei, they are still not suited for heavy targets due to the time- consuming and complex numerical calculations Also, the main drawback of this model is that the important collective states of the targets such as low-lying states and giant resonances at low-energy can not be described.
Another way to calculate the nucleon self-energy is to use the Brueckner- Hartree-Fock potential in nuclear matter by using a complex reaction matrix analogous to the (real) Brueckner G-matrix [54] These methods are quite suc-
9 cessful to produce globally satisfactory results over a wide energy range = 50
MeV Since this method is built for the infinite nuclear matter, the local density approximation has been used to describe the OP in finite nuclei To get the lo- cal optical potential, the solution of the self-consistent equation has been folded with the resulting density-dependent mean-field with a realistic point-nucleus density distribution We report below one of the most modern nuclear matter models which takes into account the two-body and three-body interactions from chiral effective field theory In this approach [38, 55], the first order of 3 is the
Hartree-Fock potential wi (4,4 kp) = S > (ahissitti|Vonlabissitti)ni, (1.17)
1 which is real, non-local, and energy-independent as the external momentum q crosses the Fermi surface The sum runs over the momentum h, spin s, and isospin £ of the intermediate hole state |hy,s1,t:) The Von is the anti- symmetrization of Voy, n1 = ỉ(k; — |hạ|) is the zero-temperature occupation probability, w is the incident energy, ky is the Fermi momentum.
The second order has the direct and exchange terms which are calculated from the particle states above the Fermi level
1 ằ [(hịhas1s3f1f2|V2Ä |qpassaffa) |? w+ €2 — €1 — €3 — †fỊ where 7, = 1 — ng selects particle states lying above the Fermi momentum, ng = ỉ(kr— kj) is the occupation probability, the summation is over intermediate- state momenta for particles p„ and holes hạ, their spins s„, and isospins t, with k = 1,2,3, and c are the single-particle energies The second orders are non-local, energy-dependent, and complex The resulting optical potentials for proton and neutron are
Up( Es RP, kt) = Vp(Es KP, kit) + iWy(Es RP, kt), (1.20)
Un (Es, BP) = Vn (EB; ke, KP) + 1Wy (Esk? KP), (1.21) with
Vi(E; kỳ, kƑ) = Re ¥i(q, E(4); ky, KƑ), (1.22) Wi(B: kỳ, kip) = Se Im Sị(g, Mr E(4): kes FP) (1.23) where the subscript i denotes a propagating proton or neutron.
Finally, the nucleon-nucleus optical potential has been defined by using the improved local density approximation (ILDA)
WLDA(E;r) = (a [verre 2 dr’, (1.24) with the optical potential in the standard local density approximation
V(E;r) +iW(B;r) = V(E: ki), ke (r)) + 7W (Bs ki), ke (r)), (1.25) where ¢ is an adjustable length scale, k#(z) and ki (r) are local neutron and proton
SELF-CONSISTENT MEAN FIELD AND BEYOND 18
Self-consistent mean-field 00.0 18
As many other theories in nuclear physics, the Hartree-Fock (HF) approach was originally implemented in another branch of many-body theory, namely atomic physics In nuclei, nucleons are bound solely by their mutual interac- tion and they do not experience any central field, unlike the electrons in atom. Therefore, straightforward applications of many-body theories (from atomic or condensed matter physics) to nuclei typically present difficulties as the nuclei has its own distinctive characteristics such as the short-range repulsive character of the nuclear force or the finite number of nucleons However, the Hartree-Fock approximation used by Kelson for the first time in nuclear physics in 1963 [61] remains a great success To describe the structure of nuclei, we use a microscopic approach which is based on the following hypothesis:
- the nuclei is a many-body quantum system of A nucleons (fermions) without structure;
- the motion of nucleons in the ground states and/or in the excited states at low-energy is non-relativistic;
- the wave function of the system W is a static solution of the non-relativistic Schrodinger equation
AW = EV, (2.1) where the Hamiltonian contains the kinetic operator K and the two-body interaction V
It is well known that we could not solve exactly the Eq (2.2) for A > 4 even there are many progress of few-body physics in the last decade Also, the exact analytic form of the nuclear force is unknown The first macroscopic model which remains the great success is a classical liquid drop model This approach considers the nucleons are confined in an imcompressive finite volume by the strong interaction At the same time, the nucleon mean free path is much larger than the size of nuclei due to the Pauli principle The co-existence of two types of properties of the fluid (classics and quantum) shows the mesoscopic character of nuclei.
Another approach assumes that nucleons are independent particles moving in an average potential created by the interaction of nucleons with each other. The experimental manifestation of this mean-field is the shell effects which are well-known in atomic physics In nuclear physics, the shell model was built to describe the nuclei which have the magic number The main problem of nuclear physics is to generate this mean-field from the nucleon-nucleon interactions. Historically, the most widely used empirical potentials are:
* The Nilsson potential [62] consists an axially-deformed harmonic oscillator with a correction proportional to l? allowing to lower the states with high angular momenta It is also the first single-particle (sp) potential consider- ing the deformations of nuclei.
Vnilsson = 2 la: (x? y”) œ0„Z2] 2K hwoo (78+ u(P ~~ (P))) ) (2.3) m where wo is harmonic oscillator constant of spherical nuclei „go = 41.A1⁄3
MeV that reproduces the nuclear radius, w;(w 1) is harmonic oscillator con- stant for z (L) respectively, ô, are potential parameters adjusted separately for protons and neutrons. ằ The Woods-Saxon potential [63] represents the Hartree part of the mean- field potential For the spherical nuclei, its expression in coordinate space reads
VWws(r) = “T1 cứ=RJja" Vo where Vp is the depth of potential well, the radius R and surface diffusion a. With this form, the magic numbers of nuclei can not be explained except the first three numbers (2, 8, 20) In 1949, by adding a crucial spin-orbit interaction to the central field, Jensen and Goeppert-Mayer have succeeded
20 to explain the shell closures of all double-closed shell nuclei The spin-orbit term reads
= I.8 2.5 r dr ? (25) which meets the requirements in description of shell structure of nuclei.
These above phenomenological potentials can be used to calculate the single particle wave functions and energies With the progress of computer in the last half of the last century, the one-body self-consistent mean-field can be derived from the effective two-body interaction by using the variational principle in the framework of Hartree-Fock theory The phenomenological potentials are usually used to initiate the self-consistent process in order to obtain the microscopic mean-field.
The fundamental assumption of the HF theory is to use the Slater determinant to describe the nuclear wave function VY This determinant is an asymmetric product of N independent (independent probability) particles wave functions ®.
Since the nucleons are fermions, the system must be antisymmetrized to satisfy the Pauli principle The asymmetrization operation leads to the normalization of the Slater determinant: on(t1) ÓA(f2) ÓN(N) where VN! is the renormaliazation factor, ở¡(r;) is the wave function of parti- cle 7 in the one-body state i All the correlations between nucleons are neglected in Ứng This is a great simplification, but also a crude approximation, which neglects a significant part of the nucleon-nucleon interaction such as pairing correlations, and three-body interaction For example, the treatment of the pairing correlation is still a main dilemma of the nuclear structure after more than 60 years The Slater determinant is the mathematical expression of the independent particle model To describe the nuclear system, we need to find the single-particle wave function ¢; together with the mean-field potential from the nucleon-nucleon interaction To do it, the variational principle is the best choice.
Consider an A-body system described by the Hamilton ƒŸ and its eigenfunction
|) which satisfy the static Schrédinger equation
The variational principle states that the solving Eq (2.7) is equivalent to the minimization of the function of energy defined by ọE[W 4] = 0, (2.8) with
This method is especially well suited to determine the ground states One can show for any trial wave function |¿) that with the values Ep are always lower than variational ones The Hartree-Fock approximation is a variational method which is restricted in the subspace of Hilbert space containing only the Slater determinants.
In second quantification, the Slater determinant (2.6) can be expressed as
(eur) = | [+i10): (2.11) i=l where creation operators al corresponding to single-particle wave functions ¢;, and |0) is the particle vacuum To find the Hartree-Fock equations, one expresses the energy of the nuclear system E[®yp] by a function of the one-body matrix density associated with the Slater determinant pig = (1|ô|j) = (®nr|a}a¡|®np) (2.12)
First, we want to minimize the expectation value of the many-body Hamilto- nian
H=K+YV, (2.13) where # is the kinetic energy operator and V an effective two-body interaction.
In the second quantization formalism it mm
=> hija ly + 1 j tonal alajar, (2.14) ij where k¿; = (i|k|j), and ÕjjkI = (ij|6|kl) The HF energy is the mean value of the
Hamiltonian of the state @yp
Using of the Wick’s theorem, the energy of the nuclear system can be given as a function of the matrix dong
Emrl2 cà, k¡j0j¡ T 3 3 = Tr(kp )+ gr Tr(pip) (2.16) 1 ~
To get the HF equations, the variational "- has been applied to Eq (2.16) ij ỗửEnr = Ehr|ứ + dp] — Enele i> hij dpij, (2.17) where ỉEhnr[0] hij = : 2.18 3 Opi; ( ) is related to the single-particle HF Hamiltonian hup = k + Ông (2.19)
In Eq (2.16), the one-body potential typ = Tr(pi) is a self-consistent mean- field obtained by folding the two-body potential with a density distribution The condition d6Eyr = 0 is equivalent to
By using the mean-field approach, one can reduce the many-body problem to one-body problem The HF equation is a nonlinear equation since the Hamilto- nian hyp depends on the density p Also, the operators hyp and p have common eigenstates and they can be diagonalized simultaneously The eigenvalues of p are
1 (occupied states) and/or 0 (unoccupied states) One usually uses the occupied states to define the HF basis and to diagonalize the Hartree-Fock Hamiltonian. This converts (2.20) into the system of HF equations hp|ởi) = eildi), (2.21) where e; are single-particle energies.
Nuclear implementation of density functional theory
In the Hartree-Fock approach with the extended Skyrme interaction, the total energy density of a nucleus can be expressed as
H=K+ Ho + H3 + Hest + Hein + Hgo + TÍsg + Hout, where K = 5-7 is the kinetic-energy term and Ho, H3, ?(etr, Hin, Hso, Hsg are given by
Ho = TM [(2 + zo)ứ” — (2zo + 1) (ứạ + ỉ0) ] › (2.27)
H3 = tse" [(2 + zs)ứ” — (2x3 + 1) (p, + ứ2)] (2.28) Huy = : [t1(2 + 21) + to(2+ z;)| pr
[/4(2z4 3 1)ứ — tgs(2zs 4 1)p"] (Pn Tn + PpTp); (2.29)
32 t x5 [(28 + 3)f4(2 + z4)ứ” — fs(2 + 25)07] (Vứ)Ÿ t( + 1)p°'Vp S~ ng Vu, (2.30)
Here, ứ = ứn + ỉp,T = Tạ + Tp, and J = Jy+ Jn are the particle number density, kinetic-energy density, and spin density, with p and n denoting the protons and neutrons, respectively The Coulomb energy density can be expressed as
Heoul = Heo + Hour (2.33) where #8? 1 is the direct term of the form ¡ 1 pp(r)dềr! dir _ ˆ 2 P
Hooul — 2° pp(r) lr — r| h (2.34) and He, is the exhange term
HES = Fete a) (2.35) where the densities are defined by e the nucleon density p(r) = ằ Joi(r, 0)”, (2.36)
28 e the kinetic energy density rr) =o [Voi(r,0)| , (2.37) ứ the so-called spin-orbit currents
F(r) = 3` lof (r,0)) |Ÿo.ơ') x (ơlZlz)] (2.38) where we sum over all occupied single-particle states Here, the total densities are p = ỉn+ỉp,7 = Tn+Tp (kinetic energy density) and J = J,+J, (spin densities), where n, p corresponds to neutrons and protons, respectively.
To find the Hartree-Fock equations, one uses the variational principle on the energy-density function.
The procedure for deriving the Skyrme-Hartree-Fock (SHF) equations can be easily found in the textbook We will thus simply outline here, the major points of this equation For the imposed spherical symmetry, the first assumption is that the ground state wave function is a Slater determinant built on the single- particle (s.p.) wave functions of the occupied states {¿4(r)} The total energy must be stationary
5(E — So eo / len) Pa°r) = 0, (2.39) which leads to the non-linear Skyrme-Hartree-Fock equation: h2
2m>(r) + Ug(r) + qVo(r) — 7W¿(r) - (V x ứ) |yÄŒ) = cay2(r), (2.40) where the index a refers to occupied states Since the potential is local (due to the zero-range interaction), these equations can be solved in coordinate space or in a discrete basis (harmonic oscillator) In the case of finite-range Gogny interaction, the HF equations must be solved in the harmonic oscillator basis since the corresponding potential is non-local To conserve the particle number, one introduces in Eq (2.40) the eigenvalues €, (usually called the single particle energies) are just the Lagrange multipliers The SHF equation is non-linear since the effective masses mj, central potentials U, and spin-orbit potentials W,are function of the local densities ứ„, the kinetic energy densities 7,, and the spin-orbit densities Jg For even-even nuclei, one usually diagonalizes the HF
29 matrix by blocks (Q") for the occupied states due the time-reversal symmetry.
In order to keep the local properties of all the potentials in the SHF equation, the local density approximation has been used for the exchange term of the long- range Coulomb interaction (unlike the Skyrme force, has a non-zero range) This approximation is called the Slater approximation which has been tested many years ago for the ground state of eight light nuclei (from !°O to °°Ni) [73].
Finally, the one-body local Coulomb potential reads e 0p) 37 (3 3
Within the spherical symmetry, the single particle wave functions can be factor- ized into
PA 0) = —* Di) 8 X()] malt) (2.43) where y is a two-component (iso-)spinor One can easily deduce the the three- dimensional HF equations (2.40) into a set of simple one-dimensional differential equations in coordinate space h? „(+ 1) ủˆ \'¿
2m) Ug + yz Ua | 1 W(r)uủa — (sm) Ug = €atla; (2.44) where the HF potential is a sum of central, Coulomb and spin-orbit terms:
Vg(r) = VicemTM"l(r) + dg aVo(t) + VÈ“(r) (2.45)
After solving the HF equation, we get all necessary ingredients for the RPA calculations such as: single-particle states, and mean-field potential.
Beyond mean-field approach 00 29
The static self-consistent mean-field approaches have successfully described the nuclear systems in their ground state There are many excited states beyond the mean-field which can be explained by the particle-hole excitations in the framework of shell model for example But there is also a variety of excitation spectra with features that cannot be understood from the elementary excitations of single-particle nature such as rotational motion and vibrational collective modes These modes can only be explained by a constructive superposition of elementary excitations participation by many nucleons resulting in a collective excitation of the nuclei At low-energy, the well-known examples are the low- lying states and the high-lying giant resonances which are important to describe
30 the imaginary part of the optical potential Such excitations can be thought as small-amplitude collective motions of nuclei after the action of an external field.
There is now a well-established microscopic framework called the Random
Phase Approximation (RPA) which is the well-known small amplitude limit of the time-dependent Hartree-Fock (TDHF) approximation In 1953, Bohm and Pines [74] developed the RPA to describe the plasma oscillations of the electron gas The first application of RPA in nuclear physics is to study the monnopole vibration of ©O by Ferrel [75] Then, this approach was intensively exploited in the early 1970 within the Skyrme and Gogny interaction The RPA, starting from a HF calculation, goes beyond a simple independent-particle picture to take into account the correlations in the ground state as well as the excited states Fully self-consistent RPA code using the effective Skyrme interaction was published several years ago Nowadays, RPA is one of the most widely used approaches to describe the small-amplitude collective motions of nuclei at low-energy Also, there are many extensions from the original RPA with more refinements: deformations, finite temperature, pairing correlations, etc.
The two-body Hamiltonian reads
H = Ho + Vies = So each, Ca — 3 Uœw8a8 + 1 ằ Vapys : cheesey 1 (2.46) 1 1 a a8 apy6 which corresponds to the Schrodinger equation
(|) = Eu|), (2.47) where {|v)}] is a set of exact eigenstates, and {„} is the associated eigenvalues.
We can write the Schrédinger equation as an equation of motion
[4.01] |0) = (Eu — Ea)T}|0), and multiply from the left with (0/6T
(0|[ðT, [H,Tjÿ]]I0) = (2, — Ea)(0|[ọT, TỶ |I0) (2.48) where TỶ, is defined by rh = > XY chen — Yinchep, (2.49) which leads to the definition of the RPA ground state
From Eq (2.48), we obtain two sets of equations
(RPA|[ch ep, [H,Tÿ]]IRPA) = hwy(RPA|[cl cp, Pf ]|RPA), (2.50a)
(RPA|[chen, [H,Tj]]IRPA) = hwy (RPA| [chen Tf] |RPA), (2.50b) where fw, is the excitation energy hw, = E, — Eo of the state |v) It is very complicated to solve these equations since the state |RPA) is not known yet. Therefore, the quasi-boson approximation has been used to simplify these equa- tions In this approximation, the operators of particle-hole pairs behave like ideal bosons leading to the quasi-boson commutator
Therefore, one can calculate all expectation values in the HF approximations since the correlated |RPA) ground state does not differ very much from the HF ground state Note that, this approximation works well when the RPA states are composed from many coherent particle-hole excitations such as the giant resonances or low-lying states Within the QBA, the absolute square of the amplitudes X%, (Y,},) gives the probability of finding the state cen |0) (ch ep|0)) in the excited state |v), respectively
(0lej|eplz) ~ (HE|[cje;.Tj]|IHF) = X72, (2.52a)
(0leje„lz) ~ (HF | [chen, F}]|HF) = Y2, (2.52b)
The Eqs (2.50a) and (2.50b) could be expressed in the matrix form called the RPA secular matrix
A B x X where the two amplitudes X and Y are eigenvectors, and the symmetric matrices
Apnp = (HEI[elep, LH, chen] | HF) = (sp — €n)5pp San + ủpyn: (2.54a) Bohpw = —(HE|[ch ep, [H, c],ey]]|HE) = ủgyin, (2.54b) with the particle-hole coupled matrix elements of the residual interaction being defined from the HF energy functional
The RPA states are excited under the action of an external field, which can be electromagnetic or hadronic To describe the transition from the ground state to excited states, the reduced transition probability is given by
BUPA > f) = SIMA, (2.56) where (f||F\||i) is the reduced matrix element of F\„ and 7 stands for E (elec- tromagnetic), IS (isoscalar) or IV (isovector) For the RPA state, one can write
(2.55)VaBys 32 in details the matrix element of the operator
(2.57) where App, is the square of the so-called transition amplitude.
It is of interest to define the transition densities since their spatial shape show the type of excitations: volume or surface, isoscalar or isovector, etc Moreover, together with optical potential, they can be used as input in calculations of inelastic scattering cross section Using the spherical harmonic expansion, the transition density reads
Inserting the single-particle wave functions defined by Eq (2.43), the radial part of the transition density can be written as a function of the X” and Y”
Spr) $Y") x (pll¥y iin), (2,59) ~ V97J+I 5_ (Xn r2 1 ph
The isoscalar (IS) and isovector (IV) transition densities can be defined as ðpp (r) =ðpz(r)+ðpJ(n) — and — ðpp(r) = bpp (r) — 5pR(r) (2.60)
One of the most important tests for the validity of the RPA calculation is to use the Energy Weighted Sum Rule (EWSR) Given an external field represented by the operator F\,, its strength function is defined as
From the definition (2.56), the strength function can be rewritten
S(E) = À ` (Eu— Eu)°B(TA;0 — v), (2.62) which establishes a relation between the excitation probability B(7^) and the
EWSR Among them, the most important linear moment 5¡ (known as Energy Weighted Sum Rule) is referred to the double commutator sum rule
51 = S7(Ey — En)|(0|Fl0)lÊ = š 9 OLE: (H, F]JI0) (2.63)
2.4 Schrodinger equations for microscopic optical poten- tial
In order to describe the elastic scattering of the nucleon by the target, we have to solve the Schrédinger equation:
V(r) = EV(r), (2.64) where / is the reduced mass of the system, F is the energy in the center of mass frame and V(r) is the potential which describes the interaction between the incident nucleon and the target When the nucleon is very far from the target, the scattering wave function V(r) satisfies the asymptotical condition eikz
U(r, 0,6) — e** + Ƒ(0,ó - ) where the first term is the incoming plane waves, and f(0,¢) is the scattering amplitude which represents the fraction of the incident waves scattered at angles (0,ứ) Then, the differential cross section are defined by dơ(0 ử)
This formula establishes a very important link between scattering theory and experiment data In our work, the microscopic optical potential is qualified by comparing the theoretical calculations with the experimental angular distribu- tions and analyzing powers.
Within ¥(r,r’; E), the Schrodinger equations reads h2 9 h2 = " ơ
== fier u(r ober where m* is the effective mass, Uso, is the spin-orbit potential, and the central potential Up is defined by
To avoid the first derivative in Eq (2.67), we use the approximation proposed by Dover and Giai [76] To do it, the scattering wave function is rewritten m m*(r) 2 ~
W(r,o;E) = ( ) W(r,ơ; E) = ƒ(r)W(r,ơ; E), (2.69) where two wave-functions W and W have the same asymptotic behavior since mr) _, 1 when r + oo This approximation allows us to decompose the first
34 two terms on the left hand side of Eq › 67)
"an = pews) = Fy ?(ƒV?Ù + ÙV?ƒ +2V/.VừÙI, (2.70) 2 2 and
V he - VỮ = h (Vf?) -V(f¥) = c VỀ (WV f+ fVW) (271)
The sum of the last term in Eq (2.70) and o one of Eq (2.71) can simplify the first derivative ry +za = VW[-2f ?Vƒ~ f(-2f ”Vƒ)] = 0 (2.72)
Therefore, the three dimensional Schrédinger equation for V reads
"ơ (fev? f+ view + [Up + Uso.(r)l-s — Elf
Finally, we obtain the integro-differential equation h” cog Ba
(2.74) which contains ® a local, real, and energy-dependent one-body potential The locality is due to the zero-range character of the effective Skyrme interaction.
Uur(r;E) = _ 'V?f 2 = 2f7|VF)’) — /?Ua + Eq = 8) + PUso (rls
= Uị+L2[7(j+ 1) — l(+ 1) aly (2.75) where f(r),Upo(r),Us.o.(r) can be calculated by using the Skyrme-Hartree-
Fock code [12]. ằ anon-local, complex, energy-dependent one-body potential The non-locality arises from the channel couplings as shown in theory of Feschbach
Using the partial wave expansion within scattering wave function, the first term of the Eq.(2.74) reads he fid 12? ity; h? 1đ2 iy tứ DI
2m (2 3 ằ r 3m (ẫ.2) = 3m 22 yr dr? r2 Vijm(" 0); ljm ljm
39 and the central term is
The spin-orbit is decomposed by
The integral term of Eq (2.74) reads apy rr B)fUdr = f S° [inl ` l?m;Ù!?'m! ll — S QS 3 >> Q —, 4 M ô = ae J3 = e ơ CS xy 5 Q xX Lm `5 œ ơ — and the last term is
After summing up all the above terms, the radial integro-differential equation in symmetric form reads
-[ [Kư.r r)+Kứr)T+ẹ (v0) | ua") =0, where V(r) = Dmg(r), K(r.r? = rƒ(r)>(r.r E)r' f(r’), Vi) = 0, Va(r) = 0, Kf(r,r)) 0, K"(r,r') = 0.
To get the nuclear reactions observables, this equation has been solved by using the standard DWBA98 code written by J Raynal [77].
RESULTS 36
Numerical aspects 2 0.0.0 ee 36
As the first step, we perform a Hartree-Fock calculation to obtain the single particle states and the lowest-order term of the self-energy (HF potential) For all considered nuclei, the radial HF equations are solved in a box of 15 fm radius with a radial mesh of 0.1 fm The spherical symmetry has been imposed. The NN effective phenomenological interaction SLy5 has been chosen since this interaction can well describe the spectroscopic properties of the exotic nuclei
[66] In order to describe the particle states, the continuum has been discretized by adopting box boundary conditions To build the matrix for RPA calculations, all the hole states are considered to build the particle-hole configurations while we limit the lowest eight unoccupied states for the particle states This choice is showed to be enough to describe single-particle energies for “°Ca and 28Pb by using the PVC calculations as in Ref.[72].
Using the outputs of the HF code, the ground states and excited states are then described within a fully self-consistent RPA calculation [12] The same effective SLy5 interaction has been used for the residual interaction Note that, all the terms of the interaction are taken into account except the tensor term in both HF and RPA calculations To prepare the inputs for the next PVC calculations, we must set two cut-offs to avoid the divergence due to the using of zero-range forces in the beyond-mean-field approaches (the PVC approach in our case) First, only natural parity phonon with the multi-polarity Z from 0 to 5 whose energies are smaller than 50 MeV are taken into account Second,the model space for the unoccupied states À is restricted to ca < 50 MeV These
37 restrictions are justified in the PVC calculations for both structure and reactions
[4] Note that, the cut-off must be taken into account even with the finite-range effective interaction (Gogny interaction) since it has the 6 function in the density- dependent term To qualify each multi-pole response, we must carefully check the energy-weighted sum rules (EWSR) For almost excited states, the EWSRs satisfy the double commutator value by more than 90% For the most strong collective states such as 3— and 4T, the exhaustion reaches 99.50%.
As well known, the QBA has been used to simplify the RPA equations This approximation works well for the collective states composed by the coherent particle-hole configurations and/or for medium and heavy nuclei To eliminate the non-collective states, only the states whose total isocalar or isovector strength are larger than 5% are taken into account The second order optical potential is added to reinforce the treatment of the spurious states due to the violation of Pauli principle of the RPA approach Also, we remove by hand the first dipole excited state which is a spurious mode associated with the center-of-mass motion.
The imaginary part of the optical potential is responsible for the absorption part of the optical potential To study the magnitude, the shape, and the non- locality of imaginary part, it is convenient to define the quantity
W(R,s) = ImAX(r,r’,w), (3.1) where R = j(r +1’) corresponds to the radius and the shape, and s = r — r' shows its non-locality N Vinh Mau and A Bouyssy practically showed that
[46] W(R,s) is independent from the angle 6 between the R and s Therefore, to a very good approximation it can be rewritten
Using the partial wave decomposition for AX);(r,r’,w), we obtain w(R,s)= ` 2 = ImASiy(,rh,6), (3.3) lj where r= + § and r'= R- §.
Systematic calculations (neutron scattering)
To construct a new generation of optical potential in the near future, we ex- plore as much as possible the influence of nuclear structure on n — A scattering observables, particularly in the stable region first The microscopic optical po- tential based on a fully self-consistent particle vibration coupling approach has
38 been performed only for neutron elastic scattering off some specific targets (e.g., for !8O and *4O in Refs [14, 15], “Ca and *8Ca in Refs [16-18], and ?°8Pb in
Ref [5] In this thesis, a systematic calculation has been performed for neu- tron elastic scattering off a series of doubly closed-shell targets !°O, 4°Ca, 48Ca, 208Pb at different incident energies below 50 MeV It is worth noting that all the parameters are fixed for all nuclei at all incident energies Therefore, the theoretical results can be judged by comparing with the experimental data for nuclei ranging from the light to the heavy region.
16Q(n,n) a *Ca(nn) ail do/dQ[b/sr] # 5 sod = in a
= na= do/dQ[b/sr] = = dơ/dO[b/sr] = = ca =
Figure 3.1: Angular distributions of neutron elastic scattering by !8O, “Ca, Ca, and 208Pb at different incident energies below 50 MeV The solid curves show the results of the MOP calculations using the SLy5 interaction The calculated results for 7°°Pb using the same MOPs are adopted from Fig 7 of Ref [5] The experimental data are the tabulated cross-sections taken from Ref [78] These figures are extracted from Ref.[58].
First, we are interested in the dynamical effect caused by the PVC on the cross sections at low incident energies In Fig 3.1, angular distributions have been systematically compared with experimental data given by the National Nuclear Data Center, Brookhaven National Laboratory Online Data Service
[78] It is surprising that the experimental data are overall well reproduced In the region of medium and heavy nuclei, the diffraction patterns are more visible than the ones obtained in the light nuclei region at all incident energies This result shows that the real part of the optical potential are better described in the medium and heavy nuclei where the underlying nuclear mean-field works very well For all targets at all incident energies, the angular distributions at small-scattering-angle region match very well with the experimental data This result, indicates that the absorption part on the surface of microscopic optical potential has been satisfactorily described As well known, it is very hard to have a global agreement for microscopic optical potential due to the richness of the excitations of nuclei from the light to heavy region Therefore, the obtained results are very encouraging.
However, at the large scattering angles, there is a systematic disagreement with experimental data for all nuclei at all incident energies This is an intricate problem for both phenomenological [22] and semi-microscopic optical potential (39, 41] For example, the authors in Refs [79, 80] showed that this disagree- ment is due to the parity- or angular-momentum-dependent components of the phenomenological optical potential In the recent calculation, even these com- ponents are taken into account the results are not better We suggest that the reason is due to a lack of the absorption in the interior part of the imaginary part of MOPs This is the consequence of several limits of our models: the weak effective Skyrme interaction, the too simple Iplh configurations in the RPA states, the neglected unnatural parity states, and the treatment of the contin- uum Also, the missing compound elastic contributions are the responsible of the lack of absorption for incident energies below 10 MeV Finally, the inclusion of the pick-up and knock-out reactions, or multiple excitations could improve the description of the imaginary part of the MOP.
As well known, the effective interactions are initially designed for the nuclear structure at low-energy Therefore, these interactions are a priori adapted for nuclear reactions at incident energies lower than 30 MeV Beyond 30 MeV, the saturation of the HF potential leads to the overestimate of the differential cross sections at backward angles as shown in Ref[l7] In our calculations, it is surprising to have the good agreement with experimental data up to 45 MeV which shows the performance of modern mean-field HF code The obtained results for '°O are coherent with ones of Ref [15] Also, for “Ca and for48Ca, the angular distributions are in nice agreement with ones of Refs [16-18].
Mainly, the obtained results show that our MOP works well for light, medium and heavy nuclei However, the obtained precision is not high compared with the phenomenological optical potential, especially at large scattering angles.
The systematic calculations show the deficiencies of the microscopic optical potential based on the Skyrme energy-density-functional theory This is a major step forward in the applications of the nuclear structure model to build up the global MOPs at low-energy.
Proton elastic scattering 0000004 40
So ° dứơ/d@[mb/sr] 5 bh dứ/dO[mb/sr] S cn ° Ss
10° 208 b(p,p) zB Sz do/dQ[mb/sr] do/dQ[mb/sr] °
Figure 3.2: Angular distributions of proton elastic scattering by !®O, “Ca, “Ca, and 208Pb at different incident energies below 50 MeV The solid curves show the results of the MOP calculations using the SLyð interaction The experimental data are the tabulated cross-sections taken from Ref [78] These figures are extracted from Ref.PHẾ
For the next step, it is naturally to extend our calculations to the proton elastic scattering These optical potentials are the most important inputs for many nuclear reaction codes To do it, the direct and exchange terms of the Coulomb interaction have been included into the residual interaction of the PVC. The direct matrix element reads ơ Are? ; ; re
Vi(ihjp) = c1 09/l000l3/llĐ) f đang cặp) leap)
(3.4) where r>(r