Derivation of K-matrix reaction theory in a discrete basis formalism
DDerivation of K -matrix reaction theory in a discrete basisformalism Y. Alhassid, G.F. Bertsch, and P. Fanto Center for Theoretical Physics, Sloane Physics Laboratory,Yale University, New Haven, Connecticut 06520, USA Department of Physics and Institute for Nuclear Theory, Box 351560University of Washington, Seattle, Washington 98195, USA
Abstract
The usual derivations of the S and K matrices for two-particle reactions proceed through theLippmann-Schwinger equation with formal definitions of the incoming and outgoing scatteringstates. Here we present a simpler alternative derivation that is carried out completely in theHamiltonian representation, using a discrete basis of configurations for the scattering channels aswell as the quasi-bound configurations of the combined fragments. We use matrix algebra to derivean explicit expression for the K matrix in terms of the Hamiltonian of the internal states of thecompound system and the coupling between the channels and the internal states. The formula forthe K matrix includes explicitly a real dispersive shift matrix to the internal Hamiltonian that iseasily computed in the formalism. That expression is applied to derive the usual form of the S matrix as a sum over poles in the complex energy plane. Some extensions and limitations of thediscrete-basis Hamiltonian formalism are discussed in the concluding remarks and in the Appendix. a r X i v : . [ phy s i c s . a t m - c l u s ] A p r . INTRODUCTION The K -matrix formalism for reactions between particles with an internal structure iswidely used in many domains of physics, including molecular collisions [1], mesoscopicphysics [2, 3], hadronic spectra [4], nuclear reactions [5], and statistical reaction theoryin general [6]. Its advantage over the competing R -matrix theory [7–9] is a simplified con-nection between internal states of the compound system and the channel wave functions ofthe incoming or outgoing particles. In particular, in the K -matrix approach, the Hamil-tonian dynamics within the internal states can be treated by well-known configuration-interaction (CI) methods [10]. However, the derivation of equations relating the K matrixto the Hamiltonian can be rather obscure in the literature. The derivations often start fromthe Lippmann-Schwinger equation and its associated T matrix, which is already severalsteps removed from the Hamiltonian equation expressed in a computationally transparentbasis [6, 11–14]. Here we carry out the derivations starting from a representation of theHamiltonian H in a discrete basis. As a benefit, we find an expression for the dispersivecouplings of the internal states to the continuum that is computationally quite simple. Incontrast, many derivations in the literature suppress these terms in the final formulas.A simple version of our formalism has been applied in nuclear reaction theory [15, 16].In the Mazama code introduced in Ref. [15], the diagonal S -matrix element is computed forone specific channel, providing the elastic cross section in that channel and the total reactioncross section. Here we consider a general scattering problem of any number of two-particlechannels. II. DISCRETE-BASIS FORMULATION OF THE SCATTERING PROBLEMA. Discretized two-particle Hilbert space
The Hilbert space of the two-particle scattering system consists of two subspaces. The firstcontains configurations, labeled by λ , that are used to construct internal wave functions ofthe compound system; the scattering wave function amplitude for each internal configuration We use the term ‘particles’ both for the elementary constituents and the (possibly composite) reactantsin the initial or final states of the reaction. will be denoted ψ λ . The second subspace contains all the scattering channels. Eachchannel c is defined by the set of configurations having the same internal structures for thetwo particles and differing only in the relative coordinate between the particles’ centers ofmass. We introduce a discretized mesh of separation distances r n = R + n ∆ r ( n = 0 , , . . . )with finite spacing ∆ r . The channel wave function in channel c then consists of the set ofamplitudes ϕ c ( n ) of the configurations on the mesh points, (cid:126)ϕ c = { ϕ c (0) , ϕ c (1) , ... } . (1) R is assumed to be sufficiently large such that potential interactions between the reactantsat larger distances r > R can be ignored. The first configuration ϕ c (1) is connecting to theinternal states, either directly or through some extension of the chain into the interactingregion. A less restrictive definition of the channel wave function that allows for a potentialinteraction V c ( r ) in each channel c is given in Appendix A. B. Hamiltonian matrix elements
1. Channel Hamiltonian
The Hamiltonian in the channel space is taken to be the kinetic energy operator of therelative motion of the two particles. It is approximated by the second-order difference for-mula on neighboring mesh points. Following nomenclature from condensed-matter physics,we denote the Hamiltonian matrix element between adjacent states in the channel by t c (here t c = (cid:126) / M c (∆ r ) where M c is the reduced mass of the two fragments). Then theHamiltonian matrix H c describing the relative motion of the fragments in channel c has thematrix elements H cn,n (cid:48) = − t c δ n,n (cid:48) +1 + (2 t c + E c ) δ n,n (cid:48) − t c δ n,n (cid:48) − , (2)where E c is the summed energy of the two reactants at rest. In the region r n > R theHamiltonian is invariant under translations, so its eigenfunctions at energy E can be ex-pressed as a superposition of an incoming wave and an outgoing wave with wave number k c and amplitudes a ( − ) c and a ( − ) c , respectively ϕ c ( n ) = a ( − ) c e − ik c r n − a (+) c e ik c r n . (3)3 IG. 1: Connectivity of the discretized Hamiltonian. The internal states are enclosed in thelarge dashed circle. Small open circles represent states of the internal Hamiltonian and the solidlines indicate off-diagonal matrix elements of the internal Hamiltonian. Solid circles represent thediscretized channel configurations. They are coupled to each other through the dotted lines togenerate the channel Hamiltonian. The dashed lines denote matrix elements v λ,c connecting thechannels to the internal states. Using [ H c (cid:126)ϕ c ]( n ) = Eϕ c ( n ) for n > E − E c = 2 t c (1 − cos κ c ) . (4)where κ c = k c ∆ r . In the continuum limit, Eq. (4) reduces to the usual quadratic dispersion E − E c = ( (cid:126) / M c ) k c .
2. Interaction with internal states
The Hamiltonian matrix elements involving states in the interaction region r ≤ R areof two kinds: those strictly between internal states and those that connect with the channelwave functions at the n = 1 site. We denote the latter matrix elements connecting theinternal state λ with channel c by v λ,c . Fig. 1 demonstrates the states and the Hamiltonianmatrix elements that connect them.We consider N i internal states and N c channels, and assume that the internal state4amiltonian is diagonal with energies E λ . For each channel c , the wave function is regularat n = 0, i.e., ϕ c (0) = 0. At radial site n = 1 the scattering wave function satisfies theHamiltonian equation − t c ϕ c (2) + (2 t c + E c ) ϕ c (1) + N i (cid:88) λ =1 v λ,c ψ λ = Eϕ c (1) , c = 1 , ..., N c , (5)while the corresponding equations for the internal-state amplitudes are N c (cid:88) c =1 v λ,c ϕ c (1) + E λ ψ λ = Eψ λ , λ = 1 , ..., N i . (6)Eliminating the internal state amplitudes ψ λ from Eqs. (6) and substituting in Eqs. (5), wefind − t c ϕ c (2) + (2 t c cos κ c ) ϕ c (1) + (cid:88) λ,c (cid:48) v λc v λc (cid:48) E − E λ ϕ c (cid:48) (1) = 0 . (7) III. D MATRIX
Substituting the channel wave function form (3) in Eqs. (7) yields a set of coupled linearequations relating the vector of outgoing amplitudes (cid:126)a (+) = ( a (+)1 , a (+)2 , ..., a (+) N c ) to the vectorof incoming amplitudes (cid:126)a ( − ) = ( a ( − )1 , a ( − )2 , ..., a ( − ) N c ) t c a ( − ) c + (cid:88) λc (cid:48) v λc v λc (cid:48) E − E λ e − iκ c (cid:48) a ( − ) c (cid:48) = t c a (+) c + (cid:88) λc (cid:48) v λc v λc (cid:48) E − E λ e iκ c (cid:48) a (+) c (cid:48) , (8)where we have absorbed a factor of e − ik c R in a ( − ) c and a factor of e ik c R in a (+) c .In principle, we could define an N c × N c matrix that transforms (cid:126)a ( − ) to (cid:126)a (+) but this is notthe S -matrix. The S -matrix preserves the total probability flux and requires the amplitudes a ( ± ) c to be normalized to the unit flux. To change to flux-normalized variables, we note that,for a tridiagonal channel Hamiltonian, the probability current J c ( n → n + 1) from a site n to the neighboring site n + 1 is given by J c ( n → n + 1) = iH cn,n +1 [ ϕ c ( n ) ϕ ∗ c ( n + 1) − ϕ ∗ c ( n ) ϕ c ( n + 1))] (9)up to a channel-independent constant. Applying (9) to the wave functions a ( ± ) c e ± ik c r n forthe Hamiltonian (2), we find for the current J c in channel cJ c = ± t c sin κ c | a ( ± ) c | , (10)5hich is independent of n . The flux-normalized amplitudes are thus b ( ± ) c = a ( ± ) c /d c , (11)where d c = (2 t c sin κ c ) − / . (12)Eqs. (8) can be rewritten for the flux-normalized amplitudes b ( ± ) c d c t c b ( − ) c + (cid:88) λc (cid:48) v λc v λc (cid:48) E − E λ d c (cid:48) e − iκ c (cid:48) b ( − ) c (cid:48) = d c t c b (+) c + (cid:88) λc (cid:48) v λc v λc (cid:48) E − E λ d c (cid:48) e iκ c (cid:48) b (+) c (cid:48) . (13)Dividing both sides of the equation by d c t c , we obtain D(cid:126)b ( − ) = D ∗ (cid:126)b (+) , (14)where the matrix D is defined by . D c,c (cid:48) = δ c,c (cid:48) + 2 π N i (cid:88) λ =1 W λc W λ,c (cid:48) E − E λ d c (cid:48) t c (cid:48) e − iκ c (cid:48) (15)with W λc = 1 √ π v λc d c t c . (16) D ∗ is obtained from D by simply replacing e − iκ c (cid:48) → e iκ c (cid:48) . The S matrix is defined by (cid:126)b (+) = S(cid:126)b ( − ) and, using Eq. (14), is given by S = D ∗− D . (17)
IV. K MATRIX
The K matrix is defined from the S matrix by the implicit relation S = 1 + iK − iK . (18)Substituting Eq. (17) in Eq. (18), and solving for K , we express K in terms of D and D ∗ K = − i ( D + D ∗ ) − ( D − D ∗ ) . (19) The definitions include factors of 2 π and (2 π ) − / following the convention in the literature [6]
6n the following we derive an explicit expression for the matrix elements of K . UsingEq. (15), we have D + D ∗ W T ( E − H ) − V , D − D ∗ − iπW T ( E − H ) − W , (20)where H = (cid:88) λ | λ (cid:105) E λ (cid:104) λ | (21)is the internal state Hamiltonian of the compound system. The matrix V is defined by V λc = πW λc cot κ c , (22)where we have used d c t c = (2 sin κ c ) − .Substituting Eq. (20) in (19), we find K = − (1 + X ) − X tan κ = − [1 − (1 + X ) − ] tan κ , (23)where the matrix X is defined by X = W T ( E − H ) − V , (24)and tan κ is a diagonal matrix with elements tan κ c along its diagonal.To invert 1+ X we use the operator identity B − ( B − A ) A − = A − − B − with A = E − H and B = E − H + V W T to find( E − H + V W T ) − V W T ( E − H ) − = ( E − H ) − − ( E − H + V W T ) − . (25)Multiplying by W T on the left and by V on the right, we obtain Y X = X − Y , (26)where Y = W T ( E − H + V W T ) − V . (27)Solving (26), we find (1 + X ) − = 1 − Y . Substituting in (23), we find K = − Y tan κ = − W T ( E − H + V W T ) − V tan κ = − πW T ( E − H + V W T ) − W , (28)where we have used Eq. (22). 7he final expression for K is thus K = πW T ( H + ∆ − E ) − W , (29)where W is given in Eq. (16) and describes the coupling matrix of the channels to theinternal states, while ∆ = − V W T is the real shift matrix∆ λλ (cid:48) = − π (cid:88) c W λc W λ (cid:48) c cot κ c . (30)The above expression for K has the same form as the usual K matrix, c.f. Eq. (18)of Ref. [3]. However, our term includes the real shift matrix ∆ that is usually ignored inexpressions for the K matrix. In other derivations of the S matrix, this shift arises fromoff-shell couplings to the channels; see, e.g., Eqs. (28-30) of Ref. [6]. In our approach, thisshift arises naturally from the matrix algebra.The K matrix in (29) is real symmetric, which guarantees that the S matrix in (18) issymmetric and unitary. V. S MATRIX
To find an explicit expression for the S matrix, we use again the operator identity B − ( B − A ) A − = A − − B − but now for A = E − ( H +∆) and B = E − ( H +∆ − iπW W T ).We obtain iπ [ E − ( H + ∆ − iπW W T )] − W W T [ E − ( H + ∆)] − = [ E − ( H + ∆)] − − [ E − ( H + ∆ − iπW W T )] − . (31)Multiplying by πW T on the left and by W on the right, we find iπZK = K + πZ , (32)where Z = W T [ E − ( H + ∆ − iπW W T )] − W , (33)and we have used the expression (29) for the K matrix. Relation (32) can be rewritten inthe form K − iK = − πZ . (34)8sing the relation (18) between the S matrix and the K matrix, we find S = 1 + 2 i K − iK = 1 − πiZ (35)where we have used (34) to obtain second equality. We thus find an explicit expression for S S = 1 − πiW T [ E − ( H + ∆ − iπW W T )] − W , (36)which includes both a real shift ∆ and an imaginary shift − iπW W T to the Hamiltonian H . This expression coincides formally with Eqs. (28-30) in Ref. [6] for the S matrix in theabsence of background scattering. VI. CONCLUDING REMARKS
We have described an alternative derivation of the K matrix of scattering theory, asshown for the Hamiltonian specified by Eqs. (2), (5) and (6). Using this derivation, weare able to avoid imposing formal structures such as the continuum Green’s functions ofLippmann-Schwinger reaction theory. It practice, it is well-suited to many-body Hamiltoni-ans of equal-mass particles, in which case it may be difficult to identify a relative coordinate.This includes nuclei and atomic condensates where common practice follows the Hartree-Fock or Hartree-Fock-Bogoliubov approximations and their extensions in the CI framework.For large systems, this approach needs much less computational effort than other reactionformalisms, which rely on explicit antisymmetrization and/or the use of a Jacobi coordinaterepresentation to separate out a channel wave function ϕ rc ( r ) in the relative coordinate r ofthe two particles.In the above derivation, we left unspecified the exact relationship of the usual channelwave function ϕ rc to the discrete-basis wave function (cid:126)ϕ c . These quantities have differentdimensions: the components of (cid:126)ϕ c are dimensionless amplitudes in the CI formalism while ϕ rc has dimension [length] − / , the same as ordinary coordinate-space wave functions. Theformal connection between the two is not obvious, since it is difficult to separate out arelative coordinate wave function unless it is already defined in the CI basis. Our approachonly involves the role of the relative coordinate at large separations, where the absence ofinteractions leads to the simplified Hamiltonian approximation in Eq. (2).9he present formalism might be applicable to problems in nuclear reaction theory suchas fission [17]. It should also simplify the treatment of the interaction between droplets ofatomic condensates, such as the fusion reaction described in Ref. [18]. Acknowledgements
We thank J.J. Rehr for discussion on possible applications to molecular reactions. Thework of Y.A. and P.F. was supported in part by the U.S. DOE grant No. de-sc0019521,and by the U.S. DOE NNSA Stewardship Science Graduate Fellowship under cooperativeagreement No. de-na0003864.
Appendix A: Potential interactions in the channels
Our definition of the channel Hamiltonian requires that the starting point at R be beyondthe range of the interaction in the given channel. This is obviously inefficient if there arelong-range potential interactions between the reactants. As in other formulations of reactiontheory, the present framework can include elastic scattering potentials V c ( r ) in the channelsto reduce the size of the interaction zone.We define a mesh for which n = 1 is the point where the channel configurations interactwith the internal ones. First we solve the one-dimensional Schr¨odinger equation for thechannel wave function U c ( n ) in the absence of all coupling terms v λ,c − t c U c ( n −
1) + [ V c ( n ) + 2 t c + E c − E ] U c ( n ) − t c U c ( n + 1) = 0 (1 ≤ n ≤ N ) , (A1)where V c ( n ) is the channel potential at site n . The wave function U c ( n ) is assumed to bereal and can be written in terms of incoming and outgoing wave functions, I c ( n ) and I ∗ c ( n ),respectively, U c ( n ) = i [ I c ( n ) − I ∗ c ( n )] . (A2)At the upper mesh points where the potential V c ( n ) can be ignored, the incoming wave I c ( n )has the following asymptotic form I c ( n ) = e − iδ c e − iκ c n , (A3)where δ c is the phase shift for scattering in a potential V c .10t n = 1, the real wave function U c satisfies the Hamiltonian equation[2 t c cos κ c + V c (1)] U c (1) − t c U c (2) = 0 , (A4)where we have used U c (0) = 0 and the dispersion relation (4).After including the interaction v λc with the internal wave function amplitudes, the channelwave function acquires a different mixture of incoming and outgoing waves ϕ c ( n ) = a ( − ) c I c ( n ) − a (+) c I ∗ c ( n ) . (A5)Eliminating the internal state amplitudes, the Hamiltonian equation acting at site n = 1has the form (7) but with the additional contribution of the channel potential[2 t c cos κ c + V c (1)] ϕ c (1) − t c ϕ c (2) + N c (cid:88) c (cid:48) =1 N i (cid:88) λ =1 v λc v λc (cid:48) E − E λ ϕ c (cid:48) (1) = 0 . (A6)Multiplying Eq. (A4) by ϕ c (1), and subtracting it from Eq. (A6) multiplied by U c (1), weobtain − t c [ ϕ c (2) U c (1) − U c (2) ϕ c (1)] + (cid:88) c (cid:48) ,λ U c (1) v λc v λc (cid:48) E − E λ ϕ c (cid:48) (1) = 0 . (A7)Inserting Eqs. (A2) and (A5) in Eq. (A7) and simplifying, yields − it c W c ( a ( − ) c − a (+) c ) + (cid:88) c (cid:48) ,λ U c (1) v λc v λc (cid:48) E − E λ ( a ( − ) c (cid:48) I c (cid:48) (1) − a (+) c (cid:48) I ∗ c (cid:48) (1)) = 0 , (A8)where W c = I c (1) I ∗ c (2) − I ∗ c (1) I c (2) . (A9)In analogy with the Wronskian of a second-order differential operator, W c ( n ) = I c ( n ) I ∗ c ( n + 1) − I ∗ c ( n ) I c ( n + 1) is independent of the mesh position n andthus can be evaluated in the asymptotic regime to give W c = 2 i sin κ c . (A10)The current J c ( ∓ ) n,n +1 is given an expression similar to Eq. (9) but with the wave functions a ( − ) c I c and a (+) c I ∗ c . It is proportional to W c and is thus independent of the mesh position,leading to the same result (10) as in the case without potential interactions. As in the maintext, we introduce the flux-normalized variables b ( ± ) c = a ( ± ) c /d c . (A11)11nserting Eq. (A11) into Eq. (A8), multiplying both sides by d c , and separating the termsin that are proportional to b ( − ) c and b (+) c yields (cid:88) c (cid:48) (cid:18) δ cc (cid:48) + (cid:88) λ U c (1) d c v λc v λc (cid:48) d c (cid:48) E − E λ I c (cid:48) (1) (cid:19) b ( − ) c (cid:48) = (cid:88) c (cid:48) (cid:18) δ cc (cid:48) + (cid:88) λ U c (1) d c v λc v λc (cid:48) d c (cid:48) E − E λ I ∗ c (cid:48) (1) (cid:19) b (+) c (cid:48) . (A12)Thus, the S matrix is given by S = ( D ∗ ) − D , where the D matrix is now defined by D cc (cid:48) = δ cc (cid:48) + (cid:88) λ U c (1) d c v λc v λc (cid:48) d c (cid:48) E − E λ I c (cid:48) (1) . (A13)The corresponding S matrix can be shown to be unitary and symmetric.Finally, in the absence of a channel potential, I c (1) = e − iκ c , U c (1) = 2 sin κ c , and the D matrix in Eq. (A13) reduces to Eq. (15). [1] X.H. Lin, Y.G. Peng, Y. Wu, et al., Chem. Phys. , 10 (2019).[2] Y.V. Fyodorov and H.-J. Sommers, Phys. Rev. Lett. (1996).[3] Y. Alhassid, Rev. Mod. Phys. , 895 (2000).[4] R.S. Longacre, A. Etkin, K.J. Foley, et al., Phys. Lett. B , 223 (1986).[5] T. Kawano, P. Talou, and H.A. Weidenm¨uller, Phys. Rev. C , 044617 (2015).[6] G.E. Mitchell, A. Richter, and H.A. Weidenm¨uller, Rev. Mod. Phys. , 2845 (2010).[7] E.P. Wigner and L. Eisenbud, Phys. Rev. , 29 (1947).[8] A. M. Lane and R. G. Thomas, Rev. Mod. Phys. , 257 (1958).[9] P. Descouvemont and D. Baye, Rep. Prog. Phys. , 036301 (2010).[10] P.O. L¨owdin, Phys. Rev. , 1474 (1955).[11] R.H. Dalitz, Rev. Mod. Phys. , 471 (1961).[12] C. Mahaux and H.A. Weidenm¨uller, Shell-Model Approach to Nuclear Reactions (North-Holland, Amsterdam, 1969).[13] S.U. Chung, J. Brose, R. Hackmann, et al., Ann. Physik , 404 (1995).[14] John R. Taylor, Scattering Theory of Non-relativistic Collisions (Dover Publications, Mineola,2006)[15] P. Fanto, G.F. Bertsch, and Y. Alhassid, Phys. Rev. C , 014604 (2018); see its SupplementalMaterial for the Mazama code.
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