ЯДЕРНАЯ ФИЗИКА, 2014, том 77, № 3, с. 395-398
= ЯДРА
METHOD OF INTEGRAL TRANSFORMS TO CALCULATE REACTIONS
©2014 V. D. Efros*
National Research Centre "Kurchatov Institute", Moscow, Russia Received January 29, 2013
A method to calculate reactions in quantum mechanics is outlined. It is advantageous, in particular, in problems with many open channels of various nature, i.e., when energy is not low. In the difference with more conventional approaches the dynamics calculations to be performed are bound-state type-calculations. Continuum spectrum states never enter the game. In the course of calculations there is no need to consider reaction channels, as well as reaction thresholds. Reaction channels and thresholds come into play at merely the kinematics level and only after a dynamics calculation is done.
DOI: 10.7868/S0044002714030064
The approach reviewed in the paper is advantageous, in particular, in problems with many open channels of various nature, i.e., when energy is not low. It was successfully applied in the range of nuclei with 3 < A < 7 proceeding from NN or NN + + NNN forces. Many cases of reactions induced by perturbation, i.e. electromagnetic orweak interaction, were considered. Some of the results can be found in the review paper [1]. Both inclusive and exclusive processes were studied. Reactions induced by strong interaction were never treated in this way although the method is similar in this case, see below.
In the presentation below the interaction is assumed to be of a short range. In the case of inclusive reactions, the Coulomb interaction is dealt within the same way as a short-range interaction and does not cause a problem. For exclusive reactions there are peculiarities as to the Coulomb interaction, and the problem is not completely elaborated. Also the problem of the efficient description of narrow resonances in the present method is to be studied in future. To this aim, additional information on such resonances from independent calculations or from experiment would be useful in the framework of the present approach. At the same time, such resonances are normally located at low energy and in the present approach the quality of their description does not seem to noticeably influence the quality of results emerging at higher energy.
The present method was created as a "non-conventional" one. Among various contributions of
E-mail: v.efros@mererand.com
V.B. Belyaev to few-body theory his work on a non-conventional approach to calculate reactions [2] is to be mentioned.
The main features of the present approach are the following. The dynamics calculations to be performed are bound-state-type calculations. In the course of calculations there is no need to consider reaction channels, as well as reaction thresholds. Reaction channels and thresholds come into play at merely the kinematics level only after a dynamics calculation is done.
Continuum spectrum states never enter the game. In place of them, "response-like" quantities of the type
R(E) = J2(Q'\^n)(^n\Q)5(E - En) + (1)
n
+ ^dYQ'^Y\Q)S(E - Ey)
are basic ingredients of the approach. Here ^n are bound states and are continuum-spectrum states. They represent a complete set of eigenstates of the Hamiltonian H of a problem. The subscript y denotes collectively a set of continuous and discrete variables labeling the states which is symbolized in the summation over integration notation. The normalizations <^n\^n') = $n,n> and (^ ) = 5(y - i) are assumed so that
E\*n)(*n \ + rfdj \ ^y )<*7 \ = I, (2)
n
I being the identity operator.
In the method discussed the quantities R(E) of Eq. (1) are obtained not in terms of the complicated
395
7*
396
EFROS
states entering their definition but via a bound-state-type calculation. And reaction observables are expressed in terms of R(E) as quadratures.
Let us first explain the latter of these points. Consider strong-interaction-induced reactions. Denote Afci(E) and Afcf (E) the antisymmetrized "channel free-motion states". Here, the subscript i refers to the initial state of a reaction, the subscript f refers to final states of a reaction, fciff (E) are products of fragment bound states and of factors describing their free motion [3], and A denotes the operator realizing antisymmetrization with respect to identical particles [3]. Denote ^(E) = A(H - E)fci(E) and fcf (E) = A(H - E)f (E). One has fc = AV^fc and fcf = AV^fcf, where Vf are interactions between fragments in the initial and final states. Here, it will be assumed that these interactions are of a short range so that the long-range inter-fragment Coulomb interaction is disregarded1). The T matrix determining the reaction rates is [3]
Tfi = T?r + (faf(E)\(E — H + ie)~1ME)), (3) e ^ +0. Here Tfiorn is the simple Born contribution,
Tfr = (f ) = fa \ fa),
and the main problem consists in calculating the second contribution in (3) that includes the Green function (E — H + ie)-1. This contribution may be represented as
J dE'Re(E)(E — E' + ie)
-1
where
+
Re (E') =
J2(<Pf (E№n)(Vn\fa (E))S(E' — En ) +
n
^dY (faf (E)\^( №(E))S(E' — Ey ).
(4)
(5)
The quantity (5) is just of Eq. (1) structure (with the E ^ E' replacement). Thus, indeed, to calculate matrix elements of the T matrix it is sufficient to have quantities of this structure. Once they are available, the integrations (4) are readily done.
In order to calculate a perturbation-induced reaction amplitude (%-\O\%0), where O is a perturbation and %0 is an unperturbed initial state; the same is to be done with the ^ O%0 replacement in the above relations.
!)This restriction can be weakened or removed. This is done in part in [4].
Now let us explain the above-mentioned point on calculating the Eq. (1)-type quantities. It should also be noted that such quantities may be of interest themselves representing observable response functions for inclusive perturbation-induced reactions. Let us rewrite Eq. (1) as
R(E) = ^ RnS(E - En) + f (E), (6)
n
Rn = (Q' \ %n)(%n \ Q),
f (E) = ^dY(Q'\%)%\Q)5(E - Ey). (7)
Calculation of the bound-state contributions Rn can be done separately, see also below. The contribution (7) includes an integral over few- or many-body continuum states that are very complicated except for low energies, and the problem just lies in calculating this contribution. If Ethr denotes the threshold value for continuum state energies, then f (E) is different from zero at Ethr < E <<x>.
An easy task is the sum-rule calculation. Using Eq. (2) one gets
X
I f (E)dE = (Q'\Q) — Y, Rn.
Ethr n
(8)
Obviously, the quantity (8) does not allow reconstruction of f (E) itself. To achieve this goal, let us consider "generalized sums" of the form
K(a, E)f (E)dE.
(9)
Eth
Using Eq. (2) one obtains "continuous sum rules"
X
i K(a,E)f(E)dE = (10)
Eth
= ^ dY (Q' \ % )K (a,Ey \ Q) = = (Q'\K(a, H)\Q) -J2 RnK(a, En),
n
where as above H is the Hamiltonian of the problem and Rn are defined in Eq. (6). If one is able to calculate the quantity (Q'\K(a,H) \Q) entering Eq. (10) then one comes to the integral equation for f (E)
<x
J K(a, E)f (E)dE = $(a), ai < a < a2, (11)
Ethr
HŒPHAfl OH3HKA tom 77 № 3 2014
METHOD OF INTEGRAL TRANSFORMS
397
with
$(a) = <Q'\K (a, H )\Q) - E RnK (a, En). (12)
n
And at proper choices of the kernel K one can completely reconstruct f (E) from this equation.
The presented approach to calculate reactions has been introduced in [5]2). The transforms with the kernels K(a, E) = (E - a)-p where p = 1 and 2 were employed. These are the Stieltjes transform and the generalized Stieltjes transform. Here, a is chosen taking real values lower than the continuum spectrum threshold and ranging outside neighborhoods of the discrete spectrum values En. In accordance with Eq. (12) the input in the integral equation is
$(a) = ^ Q'
~ E - n)P
(H - a)P Rn
{En ~
Q-
(13)
Denoting (H - a)-lQ = § and (H - a)-lQ' = §' this can also be written in the two respective cases as
Rn
$(a) = Q§) -
where (H - a)§ = Q, and
n En - a
*(a) = <§ '\§ )-Y
Rn
(En - a)2:
(14)
(15)
where (H - a)§' = Q' and (H - a)§ = Q.
The states § and §' are localized. Therefore the inputs $(a) are indeed calculable with bound-state-type methods.
The transform with the "Lorentz kernel" [9] was intensively used. The kernel can be written as
K(a, E) = [(E - a*)(E - a)]
(16)
where a = aR + iai is now complex. The quantity $(a) obtained in this case is of Eq. (15) form with the
replacement (En - a)-2 [(En - a*)(En - a)]-1. One can also obtain the Lorentz input $(a) from the dynamics equations that, like the Stieltjes case (14), involve only the initial-state source term Q. For this purpose one rewrites the above-mentioned $(a) in the form [10]
$(a) = (2ai)-1 <Q'\§ - ^2)-_Rn_
(En — a*)(En — a)'
(17)
E
where §1 = (H - a)-1Q and ^2 = (H - a*)-1Q.
Both §§1 and §§2 are calculated from the initial state of a reaction. Final states enter here via the known quantity Q' i.e., as quadratures.
In the cases (14) and (15) it is convenient to calculate Rn as the limits of the expression (En - a)<Q'\§(a)) and of the expression (En -- a)2<§'(a)\§(a)) respectively at a tending to En. Here § and §' are the solutions to the corresponding inhomogeneous equations. In the Lorentz case one can use a similar relation with both a and a* tending to En, i.e. with aR — En and aI — 0.
Choosing the kernel K as that of the Laplace transform, one gets
$(a) = <Q'\e-^H\Q)-J2 Rne
-Ena
2)For observable responses R(E), i.e. in case of inclusive perturbation-induced reactions, a bound-state-type calculation of their integral transforms has been suggested in [6] in case of the Stieltjes transform and in [7] in case of the Laplace transform. Inversions of the transforms were not considered in those works. An alternative approach [8] was also developed in which R(E) is reconstructed from its moments of the type (E-n), n = 0,...,N. The quantity of Eq. (8) represents then the zero moment. Subsequent
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