aa r X i v : . [ nu c l - t h ] A ug Neutron capture cross sections of radioactive nuclei
C. A. Bertulani
Department of Physics and Astronomy, Texas A&M University-Commerce, Commerce, TX, USA ∗ Abstract:
Alternative methods to calculate neutron capture cross sections on radioactive nucleiare reported using the theory of Inclusive Non-Elastic Breakup (INEB) developed by Hussein andMcVoy [1]. The statistical coupled-channels theory proposed in Ref. [2] is further extended inthe realm of random matrices. The case of reactions with the projectile and the target being two-cluster nuclei is also analyzed and applications are made for scattering from a deuteron target [3]. Anextension of the theory to a three-cluster projectile incident on a two-cluster target is also discussed.The theoretical developments described here should open new possibilities to obtain information onthe neutron capture cross sections of radioactive nuclei using indirect methods. keywords: neutron capture, radioactive nuclei, inclusive reactions
To the Memory of Mahir Hussein
I. INTRODUCTION
Research on reactions involving radioactive nuclei has provided invaluable information about the properties of nucleiclose to the drip line [4, 5, 7]. Further, indirect methods with rare isotopes yield cross sections of neutron capture andof other fusion reactions needed to fill the knowledge gaps in reaction chains such as the r- and s-processes of interestfor astrophysics [8]. The experimental information on neutron capture reactions is mostly constrained by captureon stable nuclei. The neutron capture on radioactive nuclei, in particular those near the drip nuclei, are difficult toobtain and are not available or have to be inferred indirectly [5, 6]. Another way to obtain neutron capture crosssections on radioactive nuclei is by means of indirect hybrid reactions. Among others, a few methods used frequentlyby experimentalists are the surrogate method [9–11], the Trojan horse method [12–16], or the ANC method [17–19].The surrogate method is most often used to infer neutron capture cross sections for fast neutrons on actinide nucleiwith the purpose to study fast breeder reactors. In the Trojan horse method one is interested to extract informationon the reaction x + A → y + B by studying a reaction of the form a + A → b + y + B . The ANC method uses transferreactions to extract asymptotic normalization coefficients useful to calculate radiative capture reactions of the form A + a → B + γ . All these indirect methods have one thing in common: the many-cluster feature of nuclear reactions.The method employed in this article is the Inclusive Nonelastic Breakup (INEB) reaction theory initially developedin Refs. [1, 20–22] and widely popularized by Hussein and collaborators (see, e.g., Ref. [23]). In this article wealso discuss the use of the INEB theory to study the case of breakup reactions of a radioactive projectile on adeuteron target, including a further extension of the method employed in Ref. [24]. The INEB theory can be easilyapplied to the case of a long lived radioactive target, such as Xe. The theory is also adaptable to study transferin reactions involving multi-clusters both in the projectile and the target, starting with the simpler (d,p) case (amodern example is Ref. [25]). This is often the case in transfer reactions with rare nuclear isotopes. For the simplercases of a weakly-bound projectile or for the coupling to compound nucleus states too many channels might beinvolved, with computations costing prohibitive large CPU times. An statistical theory to tackle these cases havebeen proposed in Ref. [2]. In this article we show that another path to simplify the theoretical description of thereaction mechanism can be achieved by using statistical concepts involving random matrices. The INEB theory isrelatively simple since reaction mechanisms can be quite difficult to describe theoretically. It is interesting to notethat one of the first publications concerning reactions involving unstable targets was done by Chew and Low backin 1959 [26]. Nowadays, with the construction of radioactive beam facilities, the most intense experimental studiesinvolve radioactive projectiles reacting with stable nuclear targets.For the extension of the INEB theory to multi-cluster reaction cases, we consider first the case of a three-bodynon-cluster radioactive projectile reacting with a two-body cluster target, using the deuteron as a surrogate. Thereaction in this case is a neutron pickup reaction. The main idea of the surrogate method is that a measurementof the inclusive proton recoil spectrum allows the extraction of quantities involved in neutron capture reactions [9–11]. But, the capture reaction is not the free neutron capture. Different reaction mechanisms are involved in the ∗ Electronic address: [email protected] surrogate reaction owing to the fact that the neutron within the deuteron is bound and carries the details of its internalwavefunction. We also consider a second case in which a four-body reaction involves a two-fragment projectile anda two-fragment target. As examples of these reactions we mention the one proton halo nucleus B, involved in thereactions B + d → p + B or p + ( Be + d). Thus, two features will be exhibited in the inclusive proton spectrum:(a) at low proton energies the incomplete fusion of Be + d will play a role and (b) the capture neutron reactionwill be influenced by higher proton energies. The case of one-neutron halo nuclei are then considered. Be and Care good projectile examples. As an example of an outcome of such studies, we anticipate that the inclusive protonspectrum would show a low energy peak at 2.22 MeV, and a weaker peak at a higher energy of ≈ Xe where the
Xe has a lifetime of 9.8 hours.
Xe is a notorious nuclear reactor poisonand it is widely known that its thermal neutron capture cross section is incredibly large, of the order of 2 . × barns. One of the goals of this theoretical exercise is to use Xe as a benchmark to assess how the proton spectrumemerges in, e.g., a reaction of the type d + Xe → p + Xe. Further, our multi-cluster discussion should constitutean important extension of the four-body formulation involving a three fragment projectile on a “structureless” targetdeveloped recently in Refs. [24, 27] to two-fragment on two-fragment nuclear reactions [3]. It would be more convenientto use B as a target as it is stable. The thermal neutron capture in this case is about 4000.00 barns, very largeconsidering the size of this nucleus and the fact that other nuclei in its vicinity have much smaller cross sections. Thisnucleus, is used in radiotherapy under the name Boron Neutron Capture Therapy. It would interesting to extract themodified capture cross section ˆ σ (n + B) and check wether it is feasible to use the inclusive proton spectrum in thebreakup reaction d + B → p + (n + B) to obtain the capture cross section. This would supply a check on theconsistency of the theory. The surrogate method has also been benchmarked using the INEB theory in Refs. [28–32].
II. INCLUSIVE NON-ELASTIC BREAKUP REACTIONS OF RADIOACTIVE PROJECTILES WITHTWO CLUSTER TARGETSA. Radioactive projectiles and neutron poisons
Let us take a radioactive nuclear poison projectile, e.g.,
Xe incident on a two-cluster target such as the deuteron.The lifetime of
Xe is very long, about 9.6 hours, easily allowing the production of a radioactive beam with presenttechniques. We would like to describe a reaction of the type
Xe + d → p + Xe, which is known as a pickupreaction. We will also assume that the proton energy spectrum is possible to be measured. We give a short summaryof the theoretical equations developed in Refs. [1, 20–22] which are pertinent to the present discussion. These theorieswould be appropriate to extract the neutron capture cross section n + Xe → Xe from the experimental analysisof the
Xe + d → p + Xe reaction. This is a general reaction of the type a + A → p + (n + A) and the INEBformulation of Ref. [1] yields for the differential cross section d σ INEB p dE p d Ω p = ˆ σ nR ρ p ( E p , Ω p ) , (1)where ρ b ( E p , Ω p ) is the density of states of the spectator fragment, p , as measured in the experiment and where ˆ σ nR is the total reaction cross section of the process n + A including medium corrections, ˆ σ nR = ˆ σ R ( n + A ) . Therefore, in this theory, the breakup cross section is directly proportional to the reaction cross section ˆ σ nR obtainedfrom ˆ σ nR = − k n E n h ˆ ρ n ( r n ) | W n ( r n ) | ˆ ρ n ( r n ) i , (2)where a complex optical potential U n = V n + iW n is assumed for the interaction between the neutron and the targetA. Note that we avoid the practical discussion of how to theoretically obtain a consistent optical potential from firstprinciples. Recent work based on the concept of self-energy [33] has recently been the focus of reaching a self-contenttheory for optical potentials including modern approaches of nuclear forces, such as chiral effective field theories [34].For a recent work in this direction, see Ref. [35].The most difficult part of the reaction formalism is to calculate the “source” function ˆ ρ n ( r n ) which is the overlapof the neutron distorted wave and the total wave function of the surrogate nucleus d in the incident channel. Eq. (2)is exact, but readily amenable to approximations. Using the DWBA limit and the post-form of the interaction, V pn ,one can show that [1] ˆ ρ n ( r n ) = ( χ ( − ) p | χ (+) d Φ d > ( r n ) , (3)where Φ d ( r p , r n ) is the internal wavefunction of the d = p + n system. In Ref. [20] a different approach was taken,based on the post form of the interaction V np . In their case, the source function includes a Green’s function accountingfor the neutron propagator, namely, ˆ ρ n ( r n ) = 1 E n − U n + iε ( χ ( − ) p | V np | χ (+) d Φ d i . (4)As shown in Ref. [24], if one additionally assumes that the three-body distorted waves χ i depend only on therelative coordinates between the fragment and the target, the cross section in Eq. (2) can be further decomposed into d σ p dE p d Ω b = E n k n Z d r n | ˆ S p ( r n ) | W ( r n ) | χ (+) n ( r n ) | , (5)where ˆ S p ( r n ) ≡ Z d r p h χ ( − ) p | χ (+) p i ( r p )Φ d ( r p , r n ) . (6)This formalism was the starting point of an analysis performed in Ref. [24] To study reactions involving the deuteronas a projectile and neutron poisons, such as Xe as a target or vice-versa. In any case, this surrogate reaction willcertainly yield useful information on the total reaction cross section for n +
Xe, as displayed in the equationsabove. To obtain the capture cross section one needs to take the difference between the reaction cross section and theother direct reaction contributions, as for example the inelastic excitation of
Xe.Several neutron poisons exist, even stable nuclei such as B and
Gd, for which the cross sections for captureof epithermal neutrons with E n < ∼ . . × barns and 2 . × barns, respectively. A very large(8 . × barns) thermal neutron absorption cross section was also observed for Zr [36]. Medical applications,such as the the Gadolinium Neutron Capture Therapy (GNCT) [37] and Boron Neutron Capture Therapy (BNCT)[38] are also based on large neutron absorption by the nuclei. Other examples of neutron poisons are
Cd withneutron absorption cross section of 2 × barns and Xe with 3 × barns, the largest known cross section forneutron induced reactions. Cd, a cadmium isotope is often used in reactors as a neutron absorber-moderator. Theneutron absorption on
Xe has a cross section reaching atomic values, but for other neighboring isotopes the crosssections are inexplicably much smaller. A list of empirical neutron capture cross sections on several nuclei is shownin Table I, with data extracted from Refs. [39–41].No satisfactory explanation for the very large cross sections in some of the isotopes is available [42]. In Ref. [43]it was suggested that the reaction proceeds via the population of a single 1p-2h doorway in the compound nucleus,connected with an intermediate structure [44–46]. But a detailed calculation is sill missing in the literature. Thephenomenon is most likely of statistical nature, based on a fortuitous isolated compound nucleus resonance. Theproblem with this idea is that such a resonance requires very narrow conditions on its position and strength, whichare seldom met. The probability that the reaction hits this resonance is given by [43] P ( η ) = 12 π
11 + η , (7)with η ≡ Γ D,n / Γ q,n as a measure of the enhancement due to the doorway state, where Γ D,n is the doorway widthand Γ q,n is the compound neutron width. But, while Γ
D,n is in the keV region, Γ q,n is in the eV region. Therefore, η ≫ P ( η ) ≪
1, and the probability for the occurrence of the doorway enhancement is very small. Since most ofthe neutron capture cross sections are inhibited by statistical fluctuations, the probable cause for the large neutronpoison cross sections remain within the domain of random phenomena.In Ref. [24] the INEB formalism was further developed to relate the neutron capture cross sections with (d,p)reactions in inverse kinematics. It is readily noticed that the zero point motion of the neutron inside the deuteronhas a large effect in reducing the neutron absorption cross section from the free neutron values by several orders ofmagnitudes. It is also found out that the best energies for such studies is about 30 MeV/nucleon ions incident ondeuteron targets. In Table II we show a numerical evaluation for the contribution of the s and d deuteron boundstates to the neutron capture cross sections [24]. The purpose of these calculations is to motivate experiments withneutron poisons. In fact, for projectiles such as Xe and
Gd large cross sections are also obtained for deuterontargets. These cross sections are amenable to experimental studies using radioactive beams of
Xe and
Gd.
B. Statistical coupled-channels theory
Numerical calculations for elastic and inelastic scattering of a system involving a very large number of channels isoften a prohibitive computational task. Reactions with radioactive beams often involves this feature, and in particular,
Nucleus Cross section (barn) Be [8.77 ± . × − B 0.5 ± N [79.8 ± . × − N [0.024 ± . × − O [0.19 ± . × − Ne [37 ± × − Ne 0.666 ± . Si [177 ± × − Ar 0.660 ± Ca 0.41 ± Fe 2.59 ± Co 37.18 ± Ni 4.5 ± Cu 4.52 ± Kr 0.111 ± Zr [8.61 ± × Rh 145 ± Cd [2.06 ± . × Cd 0.34 ± Xe 2 . × [39] Xe ∼ × − [41] Sm [4.014 ± . × Gd [2.54 ± . × Tb 23.3 ± Pb [0.23 ± . × − Bi 0.0338 ± Th 7.35 ± U 2.68 ± E n < ∼ . σ s (mb) σ d (mb) Co 2 . × − . × − Ni 1 . × − . × − Cu 2 . × − . × − Zr 1 . × . × Xe 7 . × . × Nucleus σ s (mb) σ d (mb) Sm 11 . Gd 7 . × . Tb 7 . × − . × − Th 2 . × − . × − U 4 . × − . × − TABLE II: Neutron absorption cross section for (d,p) reactions for several target nuclei. Effective deuteron energies of 30 MeVin inverse kinematics were considered. Note the large cross sections, within experimental reach, for Zr,
Xe and
Gd. for weakly bound nuclei the continuum might need a special treatment with a continuum discretization, often knownas continuum-discretized-coupled-channels (CDCC) [47, 48, 51]. For a comprehensive review of the CDCC method,see [50]. Practical continuum discretization methods have been discussed in Ref. [52] that can also be used insemiclassical calculations.A few works have pursued the challenging task to implement a full numerical implementation of reaction crosssections for nucleon-nucleus scattering including the coupling of the elastic channel with a large number of particle-hole excitations in the nucleus. The particle-hole states are regarded as doorway states through which the reactionflows to more complicated nuclear configurations, followed by a flux to long-lived compound nucleus resonances. InRef. [53] the many channels were microscopically obtained from a random-phase calculation with a Skyrme force. Agood agreement was obtained for proton and neutron scattering at 10-40 MeV incident energies. In Ref. [2] alternativestatistical methods were proposed to mitigate the problem of coupling to an intractable large number of channels.Two relevant theoretical approaches have been developed based on the average of a large number of couplings tobackground channels while keeping the couplings of main doorway channels fully in the coupled-channels procedure.But, despite the appealing feature of simplifying continuum-discretized-coupled-channels calculations, and of manydoorway states, the statistical CDCC equations developed in Ref. [2] have not been implemented numerically.To complement the techniques developed in Ref. [2], we introduce another method based on random matrices totreat coupled channels with a very large number of states. We will describe reactions with many couplings to break-upchannels. We take the CDCC Hamiltonian to be H = H + T R + V tc (cid:18) R + A f A p r (cid:19) + V tf (cid:18) R − A c A p r (cid:19) , (8)where H is the nuclear intrinsic Hamiltonian, c and f are the two fragments of the projectile. We expand the partialwave components of the wave function asΨ JMπ ( R , r ) = 1 rR X ljLi u JπljLi ( R ) φ jli ( r ) Y JMljL (Ω R, Ω r ) , (9)with Y JMljL (Ω R, Ω r ) = i l + L h ( Y l (Ω r ) ⊗ χ s ) j ⊗ Y L (Ω R ) i JM . (10)In the equations above, φ jli ( r ) are the wave functions describing the relative motion of the projectile fragments c and f , satisfying H φ jli = ε jli φ jli , (11)and the u JπljLi ( R ) the wave functions describing the relative motion between the projectile center of mass and thetarget. These satisfy the equation (cid:20) − ¯ h µ (cid:18) d dR − L ( L + 1) R (cid:19) + U c ( R ) + ε c − E (cid:21) u Jπc ( R ) = X c ′ v cc ′ ( R ) u Jπc ′ ( R ) , (12)where we have now substituted the index c for the set ( ljLi ), with the understanding that ε c = ε jli . We have dividedthe interaction into a diagonal part U c ( R ), which contains the monopole nuclear and Coulomb potentials, and a term v cc ′ ( R ), which describes the coupling among the channels.To solve these equations we expand the wave function in the internal region (0 ≤ R ≤ a ) in a basis ϕ n ( R ), as u Jπc,int ( R ) = X n A Jπcn ϕ n ( R ) , R ≤ a, (13)and match at R = a to the Coulomb wave functions of the external wave function, u Jπc,ext ( R ) = 1 √ v C (cid:0) I c ( k c R ) δ cc − O c ( k c R ) U Jπcc (cid:1) , (14)where v c and k c are the relative velocity and wave number in the channel, O c = I ∗ c are the outgoing and incomingCoulomb waves and U Jπcc ′ is the scattering matrix.We use the Bloch operator [44–46] L c = ¯ h µ δ ( R − a ) (cid:18) ddR − B c (cid:19) , (15)where we will take B c = 0. Then we define a matrix C as C Jπcn,c ′ n ′ = (cid:10) ϕ n (cid:12)(cid:12) ( T c + U c + ε c + L c − E ) δ cc ′ + V Jπcc ′ (cid:12)(cid:12) ϕ n ′ (cid:11) (16)and the R-matrix as R Jπcc ′ = X dnd ′ n ′ γ c,dn (cid:20) C Jπ (cid:21) dn,d ′ n ′ γ c ′ ,d ′ n ′ , (17)where γ c,dn = s ¯ h µa ϕ n ( a ) δ cd . (18)Suppressing the indices, we can write R Jπ = γ C Jπ γ T . (19)We can now write the scattering matrix U Jπ in terms of R Jπ as U Jπ = ρ / O (cid:0) − R Jπ D (cid:1) − (cid:0) − R Jπ D ∗ (cid:1) Iρ − / , (20)where ρ cc ′ = k c aδ cc ′ , O cc ′ = O c ( k c a ) δ cc ′ , I cc ′ = I c ( k c a ) δ cc ′ (21)and D cc ′ = k c a O ′ c ( k c a ) O c ( k c a ) δ cc ′ = k c a (cid:20) I ′ c ( k c a ) I c ( k c a ) (cid:21) ∗ δ cc ′ . (22)We now write (cid:0) − R Jπ D (cid:1) − = 1 + R Jπ D + R Jπ DR Jπ D + R Jπ DR Jπ DR Jπ D + . . . (23)= 1 + γ C Jπ γ T D + γ C Jπ γ T Dγ C Jπ γ T D + γ C Jπ γ T Dγ C Jπ γ T Dγ C Jπ γ T D + . . . = 1 + γ C Jπ − γ T Dγ γ T D. Substituting, we may reduce (cid:0) − R Jπ D (cid:1) − (cid:0) − R Jπ D ∗ (cid:1) = 1 + γ C Jπ − γ T Dγ γ T ( D − D ∗ ) (24)and write the scattering matrix as U Jπ = ρ / O (cid:18) γ C Jπ − γ T Dγ γ T ( D − D ∗ ) (cid:19) Iρ − / . (25)Now let us assume that we are only interested in a strongly-coupled subset of the states c . We can the write thematrix as C Jπ − γ T Dγ = C Jπ − γ T D γ VV † C Jπx − γ T D x γ ! , (26)where the sub-matrix containing the set of interest is denoted by 0 and the remaining set by x . Since the matrixelements of γ T Dγ are diagonal in c (but not in n ), the only matrix elements coupling the two sets of states are the(weak) interaction terms v cn,c ′ n ′ , which we denote here by V. Its inverse is given by (cid:16) C Jπ − γ T D γ − V C Jπx − γ T D x γ V † (cid:17) − (cid:16) C Jπ − γ T D γ − V C Jπx − γ T D x γ V † (cid:17) − V C Jπx − γ T D x γ C Jπx − γ T D x γ V † (cid:16) C Jπ − γ T D γ − V C Jπx − γ T D x γ V † (cid:17) − (cid:18) C Jπx − γ T D x γ − V C Jπ − γ T D γ V † (cid:19) − (27) Up to this point, the development has been exact. To simplify the calculation of the component of interest, wewould like to approximate the matrix V C Jπx − γ T D x γ V † . (28)We would also like to approximate the term V C Jπx − γ T D ∗ x γ γ T πk x ρ x | O x | γ C Jπx − γ T D x γ V † , (29)which enters the expression for the summed cross section of the weakly-coupled states. Weak DWBA
If the coupling among the channels weakly-coupled to the entrance channel is also weak, the simplest approximationwould be to neglect the coupling term in C Jπx , a type of DWBA approximation. In that case, we would take (cid:0) C Jπx (cid:1) cn,c ′ n ′ → h ϕ n | T c + U c + ε c + L c − E | ϕ n ′ i δ cc ′ (30)and (cid:0) C Jπx − γ T D x γ (cid:1) cn,c ′ n ′ → (cid:18) h ϕ n | T c + U c + ε c + L c − E | ϕ n ′ i − ¯ h µa D c (cid:19) δ cc ′ . (31)We would then have (cid:18) V C Jπx − γ T D x γ V † (cid:19) c n,c ′ n ′ → X c x mm ′ v c n,c x m (cid:18) C Jπx − γ T D x γ (cid:19) c x m,c x m ′ v c x m ′ ,c ′ n ′ . (32)The coupling through the weakly-coupled terms would then include only their optical propagation part - the c x submatrices would not be coupled. The R-matrix solution would still require solution of each c x channel in theexpansion set { ϕ n } but would not require calculation and diagonalization of the c x n matrix. Statistical CDCC
Alternatively, if the coupling among the channels weakly-coupled to the entrance channel is sufficiently strong tomix them, we can consider using statistical hypotheses to in approximations to the cross sections. If the expansionfunctions φ jli ( r ) and ϕ n ( R ) are real, the matrix C Jπx is real symmetric and the matrix C Jπx − γ T D x γ complex symmetric.There is thus a unitary transformation U for which U (cid:0) C Jπx − γ T D x γ (cid:1) U T is diagonal. This does not look to be thatuseful an expression, since we cannot expect special behavior of the matrix elements of U U T .On the other hand, whether the basis functions are real or not, the fact that C Jπx is Hermitian implies that thereexist a U for which U C
Jπx U † is diagonal. In this case we have V C Jπx V † = V U E x − E U † V † , (33)where E x is the diagonalized Hamiltonian in the x subspace. Rewriting the matrix elements V U as(
V U ) c n,λ = X c ′ x n ′ v c n,c ′ x n ′ U c ′ x n ′ ,λ , (34)we expect that we might find (cid:18) V U E x − E U † V † (cid:19) c n,c ′ n ′ = X λ ( V U ) c n,λ ǫ λ − E (cid:0) U † V † (cid:1) λ,c ′ n ′ ≈ X λ (cid:12)(cid:12)(cid:12) ( V U ) c n,λ (cid:12)(cid:12)(cid:12) ǫ λ − E δ c c ′ δ nn ′ (35)due to the fact that matrix elements with different values of c and n will have different phases and will tend to averageto zero.To perform calculations, a stronger version of this hypothesis is needed. First, we require the number of states N Λ to be large in intervals ∆ ǫ λ in which optical quantities, such as k , ρ, and D vary slowly and denote these intervals byΛ . We then assume that X λ ∈ Λ ( V U ) c n,λ (cid:0) U † V † (cid:1) λ,c ′ n ′ = N Λ (cid:12)(cid:12)(cid:12) ( V U ) c n, Λ (cid:12)(cid:12)(cid:12) δ c c ′ δ nn ′ , (36)so that (cid:18) V U E x − E U † V † (cid:19) c n,c ′ n ′ = X λ ∈ Λ (cid:12)(cid:12)(cid:12) ( V U ) c n, Λ (cid:12)(cid:12)(cid:12) N Λ ǫ Λ − E δ c c ′ δ nn ′ . (37)However, the matrix element we want to simplify is (cid:18) V C Jπx − γ T D x γ V † (cid:19) c n,c ′ n ′ = (cid:18) V U E x − E − U † γ T D x γU U † V † (cid:19) c n,c ′ n ′ . (38)This will be easy to calculate if (cid:0) U † γ T D x γU (cid:1) λλ ′ = X c x nn ′ (cid:0) U † γ T (cid:1) λ,c x n D c x ( γU ) c x n ′ λ ′ → (cid:16) γ T D x γ (cid:17) Λ δ λλ ′ for λ ∈ Λ ≡ i Γ Λ / δ λλ ′ (39)which we will assume to be the case. We then have (cid:18) V C Jπx − γ T D x γ V † (cid:19) c n,c ′ n ′ = (cid:18) V U E x − i Γ x / − E U † V † (cid:19) c n,c ′ n ′ → X Λ (cid:12)(cid:12)(cid:12) ( V U ) c n, Λ (cid:12)(cid:12)(cid:12) N Λ ǫ Λ − i Γ Λ / − E δ c c ′ δ nn ′ . (40)If the average matrix elements vary smoothly with Λ, we can write ( N Λ → ρ x ( ǫ Λ ) dǫ Λ ) X Λ (cid:12)(cid:12)(cid:12) ( V U ) c n, Λ (cid:12)(cid:12)(cid:12) N Λ ǫ Λ − i Γ Λ / − E = Z (cid:12)(cid:12)(cid:12) ( V U ) c n, Λ (cid:12)(cid:12)(cid:12) ρ x ( ǫ Λ ) dǫ Λ ǫ Λ − i Γ Λ / − E ≡ ∆ c n ( E ) + i Γ ↓ c n ( E ) / ≈ πi (cid:12)(cid:12)(cid:12) ( V U ) c n, Λ (cid:12)(cid:12)(cid:12) ρ x ( ǫ Λ = E ) . (41)If the statistical hypotheses are consistent with the physics of the problem, this is equivalent to (cid:18) V C Jπx − γ T D x γ V † (cid:19) c n,c ′ n ′ = (cid:0) ∆ c n ( E ) + i Γ ↓ c n ( E ) / (cid:1) δ c c ′ δ nn ′ . (42)We can then approximate the S-matrix of the strongly-coupled states as U Jπ = ρ / O γ C Jπ − γ T D γ − ∆ − i Γ ↓ / γ T ( D − D ∗ ) ! I ρ − / (43)and can calculate the cross sections accordingly. Note that these cross sections only take into account the loss offlux to the weakly-coupled states. They do not include the fluctuation contribution due to coupling through theweakly-coupling states back to the strongly-coupled ones. The lowest order fluctuation contribution will come fromthe term U Jπ ,fl = ρ / x O x γ C Jπ − γ T D γ − ∆ − i Γ ↓ / V C Jπx − γ T D x γ V † (44) × C Jπ − γ T D γ − ∆ − i Γ ↓ / γ T ( D − D ∗ ) I ρ − / , (45)where the couplings V and V † must be averaged with those of the conjugate contribution, U Jπ † ,fl . The fluctuationcontribution is thus of fourth-order in the coupling V .The lowest order contribution to the S-matrix elements of the channels weakly coupled to the entrance channel aregiven by U Jπx = ρ / x O x γ C Jπx − γ T D x γ V † C Jπ − γ T D γ − ∆ − i Γ ↓ / γ T ( D − D ∗ ) I ρ − / , (46)with cross sections given by σ x = πk x (cid:12)(cid:12) U Jπx (cid:12)(cid:12) . (47)The inclusive cross section to the weakly-coupled states can be written as X x σ x = S † V C Jπx − γ T D ∗ x γ γ T πk x ρ x | O x | γ C Jπx − γ T D x γ V † S , (48)where S = 1 C Jπ − γ T D γ − ∆ − i Γ ↓ / γ T ( D − D ∗ ) I ρ − / . (49)We write (cid:18) V C Jπx − γ T D ∗ x γ γ T πk x ρ x | O x | γ C Jπx − γ T D x γ V † (cid:19) c n,c ′ n ′ == (cid:18) V U E x + i Γ x / − E U † γ T πk x ρ x | O x | γU E x − i Γ x / − E U † V † (cid:19) c n,c ′ n ′ (50)= X Λ (cid:12)(cid:12)(cid:12) ( V U ) c n, Λ (cid:12)(cid:12)(cid:12) N Λ Γ ↑ Λ / | ǫ Λ − i Γ Λ / − E | δ c c ′ δ nn ′ where we have assumed that (cid:18) U † γ T πk x ρ x | O x | γU (cid:19) λλ ′ = (cid:18) γ T π ρ x k x | O x | γ (cid:19) Λ δ λλ ′ ≡ Γ ↑ Λ / δ λλ ′ . (51)Transforming the sum over states to an integral, as before, we find X Λ (cid:12)(cid:12)(cid:12) ( V U ) c n, Λ (cid:12)(cid:12)(cid:12) N Λ Γ ↑ Λ / | ǫ Λ − i Γ Λ / − E | → Z (cid:12)(cid:12)(cid:12) ( V U ) c n, Λ (cid:12)(cid:12)(cid:12) ρ x ( ǫ Λ ) Γ ↑ Λ / | ǫ Λ − i Γ Λ / − E | dǫ Λ ≈ π Γ ↑ Λ Γ Λ ρ x ( ǫ Λ = E ) (cid:12)(cid:12)(cid:12) ( V U ) c n, Λ (cid:12)(cid:12)(cid:12) . (52)The inclusive cross section to the channels weakly coupled to the entrance channel can then be written as X x σ x = π Γ ↑ Λ Γ Λ ρ x ( ǫ Λ = E ) X c ′ n ′ n (cid:12)(cid:12)(cid:12) ( V U ) c n, Λ (cid:12)(cid:12)(cid:12) (cid:12)(cid:12) S c ′ n ′ ,c n (cid:12)(cid:12) . (53)These cross sections also have higher-order corrections corresponding to coupling through the weakly-coupled statesto the strongly-coupled ones and then back to the weakly-coupled ones.We note that the loss terms also have higher-order corrections coming from averages over different pairs of interac-tions, such as the following one, in which the middle two instances and the outer two instances of the interaction areaveraged V C Jπx − γ T D x γ V † C Jπ − γ T D γ V C Jπx − γ T D x γ V † . (54)When the coupling is weak, this term can usually be neglected in comparison to the term V C Jπx − γ T D x γ V † , (55)for which it would serve as a correction in the development above. (The average over the first pair of V ’s and thesecond pair of V ’s furnishes the second-order iteration of the latter term in the expansion rather than a correctionto it.) When the number of weakly-coupled states is large, it can be shown that the other possible combination ofaverages - the first and third instances and the second and fourth instances of V - is much smaller than the othertwo. This method has not been implemented numerically, but work in this direction is in progress. It will simplifycalculations of reaction cross sections involving too many states, beyond the reach of present computing capabilities. III. INCLUSIVE NON-ELASTIC BREAKUP REACTIONS OF TWO-FRAGMENT PROJECTILE ON ATWO-FRAGMENT TARGET
We now consider a four-body breakup problem for the reaction of a two-fragment projectile with a two-fragmenttarget. Both projectile and target can dissociate into their two clusters, forming a genuine four-body system. Theformalism of Ref. [3] can be applied by describing the target as a = b + x cluster, and the projectile as anotherA = x + B cluster. Therefore, the inclusive measurement of b will include the breakup of the projectile with x reacting with the target a = x + d , and also the breakup of the target with x reacting with the projectile, formedby the x + A cluster. This reaction is a complicated high-energy four-body problem. To simplify it to a manageableproblem we treat the reaction as a two three-body process. The breakup of the projectile is assumed to proceedwithout affecting the target and the breakup of the target is also assumed to proceed without affecting the projectile.One then obtains two separate groups of detected spectator fragments, one belonging to the target and the otherbelonging to the projectile. This approach can become valuable to treat reactions involving a neutron or proton-richprojectile with targets such as the deuteron.Let us consider the reaction B + d, yielding p + (n + B) → p + B, or p + ( Be + d). Note that B is a oneproton halo with proton separation energy of 0.137 MeV. The reaction p + (n + B) leads to a neutron capture by0a one-proton halo nucleus, whereas the reaction p + ( Be + d) yields an incomplete fusion of the core of the halonucleus with the deuteron target. Denoting the proton originating from the radioactive projectile by p and that fromthe deuteron target breakup by p , we can write the inclusive non-elastic proton spectrum as d σ p dE p d Ω p = ρ ( E p )ˆ σ R (n + 8B) + ρ ( E p )ˆ σ R (d + 7Be) + · · · (56)The first term on the right-hand side contains the neutron capture cross section of the halo nucleus and is peakedat the higher proton energy since the proton separation energy in the deuteron is 2.22 MeV. In contrast, the secondterm corresponds to the incomplete fusion in the form of Be + d which leads to the emission of the halo proton in B first, followed by the collision of its core, Be, with the deuteron. The last reaction mechanism will dominate thelow energy part of the inclusive proton spectrum.For a one-neutron halo projectile such as Be or C, the same type of reaction yields an inclusive proton spectrumthat will exhibit a low energy peak associated with the deuteron breakup at 2.22 MeV, and a weaker higher energypeak related to the removal of a proton from the tightly bound cores, Be, C. That is, d σ p dE p d Ω p = ρ ( E p )ˆ σ R (n + Be) + ρ ( E p )ˆ σ R (d + Be) + · · · (57) d σ p dE p d Ω p = ρ ( E p )ˆ σ R (n + C) + ρ ( E p )ˆ σ R (d + B) + · · · (58)The cross sections, ˆ σ R ( n + Be), ˆ σ R ( n + C), ˆ σ R ( d + Be), ˆ σ R ( d + B), are obtained from expressions similar to Eq.(5). One needs the S-matrix elements, ˆ S p ( r p ) and ˆ S p ( r p ) to evaluate the cross sections. The matrix elements canbe calculated once the optical potentials appropriate for the scattering of protons on deuterons and on the differenthalo projectiles are known. Optical potentials for the projectile and target systems are also needed, as well as thoseto obtain distorted waves of the participant fragments. These are n + Be, n + C, d + Be, d + B. In the case of theproton halo nucleus B, we need similar ingredients: ˆ S p ( r p ) for p + d elastic scattering and ˆ S p ( r p ) for p + B.Similarly one needs the optical potentials for d + B, n + B and d + Be. These potentials could be extracted fora phenomenological fit to elastic scattering data.Once the incomplete fusion cross sections are calculated from fusion theory [55], the neutron capture cross sectionscan be obtained from the general form of the breakup cross sections, Eqs. (56, 57, 58). Thus the Inclusive Non-ElasticBreakup is a potentially powerful method to extract the neutron capture cross section of short-lived radioactive nuclei.
IV. INCLUSIVE NON-ELASTIC BREAKUP REACTIONS OF THREE CLUSTER PROJECTILESWITH TWO CLUSTER TARGETS
Let us consider three-cluster projectiles, a = b + x + x , such as Be = He + He + n and Borromean nuclei suchas Li = Li + n + n. The cross section for the four-body process, b + x + x + A , where b is the observed spectatorfragment and x and x are the interacting participants fragments, is given by [54] d σ INEBb dE b d Ω b = ρ b ( E b ) σ BR , (59) σ BR = k a E a (cid:20) E x k x σ x R + E x k x σ x R + E CM ( x , x )( k x + k x ) σ BR (cid:21) , (60)where, σ x R = k x E x h ˆ ρ x ,x | W x | ˆ ρ x ,x i , σ x R = k x E x h ˆ ρ x ,x | W x | ˆ ρ x ,x i , (61)and, σ BR = ( k x + k x ) E CM ( x , x ) h ˆ ρ x ,x | W B | ˆ ρ x ,x i , (62)1is a three-body, x + x + A , reaction cross section. The different energies of the fragments are determined by thebeam energy for weakly bound projectiles with the binding energy being marginally relevant for the energies of thethree outgoing fragments. In this case, we can use e.g., E x ,Lab = E a,Lab ( M x /M a ), with M a and M x being themass numbers of the projectile and of the fragment x , respectively. The three-body source function, ˆ ρ x ,x , is ageneralization of the two-body source function defined in Eqs. (3,4), i.e.,ˆ ρ x ,x ( r x , r x ) = ( χ ( − ) b ( r b ) | χ (+) a ( r b , r x , r x )Φ a ( r b , r x , r x ) i . (63)The cross sections σ x R and σ x R are the reaction cross sections of x + A and of x + A whereas the other clusters, x and x are scattered and not observed. In analogy with the developments presented in the previous section, onegets [54] E x k x σ x R = Z d r x d r x | ˆ S b ( r x , r x ) | | χ (+) x ( r x ) | W ( r x ) | χ (+) x ( r x ) | , (64) E x k x σ x R = Z d r x d r x | ˆ S b ( r x , r x ) | | χ (+) x ( r x ) | W ( r x ) | χ (+) x ( r x ) | . (65)The three-body reaction cross section σ BR corresponds to a new type of absorption where both fragments interactswith the target simultaneously ( x + x + A). To our knowledge the formulation for such a process has not yet beenexplored in the literature for the calculation of reaction cross sections. V. CONCLUSIONS
We summarized a few works aimed at extending the formalism developed by Hussein and McVoy [1] on InclusiveNon-Elastic Breakup (INEB) reactions. The main application of the formalism is to obtain neutron absorption crosssections by radioactive nuclei in inverse kinematics. First, we have considered the case of a radioactive projectileincident on a two-cluster target, such as the deuteron. Applications have been done to extract neutron capture crosssections on neutron poisons based on the results of Ref. [24]. Further, an assessment of the statistical CDCC theorydeveloped in Ref. [2] has lead to an additional method using random matrix theory.The case of INEB reactions two-projectile cluster (or fragment) reacting with a two-target cluster has also beendiscussed [3]. A schematic formulation was also introduced for the case of INEB reactions of three cluster projectileswith two cluster targets. All these ideas and theoretical methods have been elaborated in collaboration with M.S.Hussein along many years and decades. His enthusiasm and openness to solve problems in physics was a motivationfor many of his collaborators, including myself.
Acknowledgements
This work was partly supported by the U.S. DOE Grant No. DE-FG02-08ER41533. Funding contributed throughthe LANL Collaborative Research Program by the Texas A&M System National Laboratory Office and Los AlamosNational Laboratory.
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