ЯДЕРНАЯ ФИЗИКА, 2014, том 77, № 3, с. 369-375
= ЯДРА ^^
BINARY SCATTERING AND BREAKUP IN THE THREE-NUCLEON SYSTEM
©2014 P. A. Belov*, S. L. Yakovlev**
Department of Computational Physics, St. Petersburg State University, Russia
Received January 29, 2013
We present the further development of the three-particle formalism for differential Faddeev equations. The asymptotic boundary conditions in the hyperspherical adiabatic representation have been constructed. We prove that these conditions are asymptotically equivalent to the standard Merkuriev boundary conditions. With these boundary conditions we have formulated the boundary-value problem for Faddeev equations which has the property that the binary channel and the breakup channel are explicitly orthogonal. The effective numerical scheme for solving the formulated boundary-value problem is given.
DOI: 10.7868/S0044002714030039
We cordially dedicate this work to Vladimir B. Belyaev on the occasion of his 80th birthday
1. INTRODUCTION
One of the reliable approaches to the three-particle scattering above the breakup threshold is the formalism of differential Faddeev equations [1]. In this formalism the special asymptotic boundary conditions should be implemented in order to determine the solution which corresponds to the particular scattering process. The Merkuriev boundary conditions [2] are used as the standard in applications [1, 3]. Recently, we have developed a new approach [4] for solving the three-particle scattering problem on the basis of Faddeev equations in the configuration space. The key ingredient of this approach is the new form of the asymptotic boundary conditions. While the applicability of the approach has been already demonstrated in [4] for neutron—deuteron scattering, in this paper we give the necessary proof of these new boundary conditions. Although these conditions are different in their form from the Merkuriev conditions, they are equivalent in the sense that they match each other when the separation of particles tends to infinity. The asymptotic boundary conditions are represented in the form of the hyperspherical adiabatic expansion. This expansion is constructed in such a way that the binary and breakup channels are orthogonal at any values of the hyper-radius. This property allows using the asymptotic value of the Faddeev components as
E-mail: pavelbelov@gmail.com
E-mail: yakovlev@cph10.phys.spbu.ru
the boundary value [4] for the Faddeev equations. This approach makes it possible to calculate the scattering parameters at the asymptotic region through the solution of the boundary-value problem with the in-homogeneous boundary conditions in the asymptotic region, i.e. without reconstruction of the solution over the entire configuration space.
For the transparency of the exposition we deal with the simplest s-wave Faddeev differential equation formulation of the three-body scattering problem with short-range two-body interaction between particles. The generalization of the formalism described in this note on the general case (without partial wave decomposition) is straightforward and will be given in further notes.
The plan of the paper is as follows. In Section 2 the conventional formulation of the three-particle scattering problem on the basis of Faddeev equations with Merkuriev boundary conditions is presented. In Section 3 we develop, firstly, the hyperspherical adiabatic representation for the Faddeev equations and, secondly, by making the transformation to the dia-batic representation we find the asymptotic boundary conditions for diabatic representation. Using the property of the geometric connection matrix we find the boundary conditions for the adiabatic representation. In Section 4 we apply the obtained asymptotic representations for constructing the asymptote of the solution of the original Faddeev equations and, as a result, we obtain the desired asymptotic approach to the Faddeev equations. Section 5 contains the description of the numerical scheme for solving the Faddeev equations. Section 6 concludes the paper.
2. FADDEEV EQUATIONS WITH MERKURIEV BOUNDARY CONDITIONS
In the present paper we consider the neutron— deuteron (nd) system in the quartet state (total spin 3/2) as a typical example of the three-body system with equal masses. The nd system under consideration is described by the differential Faddeev equation of the form [ 1]
(-A + V(x) - E) U(X) = (1)
= -V(x) (P + + P-) U(X)
for the Faddeev component U of the wave function Here A = Ax + Ay is the Laplace operator and V(x) is a two-body pair potential. The center-of-mass frame of properly scaled Jacobi coordinates {x, y} = X [1] is used throughout. The expansion of the wave function into components is written as
tf(X) = (I + P+ + P-) U(X),
where P± are the cycling and anti-cycling permutation operators of three particles and I is the unit operator. The s-wave equation for the radial part of the Faddeev wave function component appears from Eq. (1) after projection onto the states with the zero orbital momentum in all pairs of the three-body system. This s-wave Faddeev equation is given by [5]
d2 d2 \ -ft?+ «(*■»)
1
1
where
x
Vs
2\ 1/2 3y2\
,--xyu + , ,
4 2 4 /
3x
y2
+ — xyß + —
1/2
ß = cos (X, y).
q2 = E — e. The deuteron ground-state wave function satisfies the equation
hLP= ("¿2 <fi(x) (3)
with the zero boundary conditions at zero and infinity.
The solution of the s-wave Faddeev equation (2) for nd scattering above the breakup threshold (E > 0) should satisfy the Merkuriev boundary condition [2] as p
U (x, y) ~ ^>(x)[sin qy + a(q) exp iqy] + (4)
+ A(d,E)
exp l^
VP
where p = \/x2 + y2, tg 9 = y/x. The conditions U(x, 0) = U(0, y) = 0 guarantee the regularity of the solution at the origin.
Taking into account the change of the unknown function U(p, 9) = y/pU(x, y), the transformation to the hyperspherical coordinates {p, 9} leads to the following equation
dp2
1
v
p2 892
+
(5)
+ V(p cos 9) - E^jU(p,9)
e+(e)
(2) =^=V(pcoS9) J U{p,9')d9'.
0-(0)
The integration limits are defined, in turn, as 9-(9) = = 3 — 9|, 9+(9) = n/2 — \n/6 — 9|. The boundary condition (4) takes the form
U(p, 9) ~ y/~p<f{p cos 9) [sin{gp sin 9} +
+ a{q) exp{iqp sin 9}] + A(9, E)exp i\[Ëp.
Functions a(q) and A(9, E) are the binary scattering amplitude and the Faddeev component of the breakup amplitude, respectively. The s-wave total breakup amplitude ^.(9, E) can be assembled from A(9, e) by the formula
Here and throughout x, y represent the radial values of Jacobi vectors, i.e. x = |x|, y = |y|. By X, y the unit vectors x/x, y/y are denoted. The potential V(x) stands for the triplet component of NN interaction [6]. It is assumed that V is acting only in the state with the zero orbital momentum of the pair of nucleons.
The center-of-mass energy E and the relative neutron momentum q are associated with the deuteron ground-state energy e < 0 by the equation
1
sin 9 cos 9
A(9,E) =
0+(0)
A(9,E)-^= J d9'A(9',E)
0-(0)
3. HYPERSPHERICAL ADIABATIC AND DIABATIC REPRESENTATIONS OF FADDEEV EQUATIONS 3.1. Adiabatic Representation By the hyperspherical adiabatic representation one means an expansion of the solution along an angular
x
y
MEPHA^ OH3HKA TOM 77 № 3 2014
basis of functions depending not only on the angular variable e but also on the hyper-radius p as a parameter [7]. As the basis to expand the solution we choose the eigenfunction set of the operator h(p)
h(p)fa (e\p) =
(6)
1
-fdo + v(Pcos^(Olp) = \k(p)M0\p)
considered on the interval [0,n/2] with zero boundary conditions fa(0\p) = fa(n/2\p) = 0. Here, one should realize that p is a parameter for the operator h(p) and, as a consequence, its eigenfunctions and eigenvalues inherit parametric dependence on p. The spectral properties of the operator h(p) as p -^to are related to the two-body Hamiltonian h from (3) defined on [0, to). The main properties, which play the key role in the constructions given bellow, can be formulated as follows
A = lim A0(p) = £,
p^X
M0\p) = Vptp(pCOS 0)(1 + 0(p~n)
with some p > 0. For exited states (6\p), k > 1, the following asymptotics are valid as p ^to
\k(P)
(2k ~ v7
--7= sin(2A;0).
Jn
Due to this asymptotics, it is convenient to represent the eigenvalues Ak (p) for k > 1 as
Vk (p)
\k (P) =
p2
where the vk(p) quantities are the eigenvalues of the operator from which the 1/p2 factor is factored out, namely
+ p2V(pcos 6)) fak(6\p) = Vk(p)fa(9\p). (7)
It is clear that eigenfunctions in (6) and (7) are the same. It is worth repeating that the asymptotics of eigenvalues vk(p) as p ^to are of the form
Vk (p) - (2k)2, k = 1,2,3,...
It is important to point out here that for the operator from (7) we can get the asymptotics as p ^ 0
-d2e + p2V(pcos 6) - -6$ + O(p2-a), (8)
where a corresponds to the behavior of the potential V(x) at the origin, i.e.
V(x) -O(\x\-a).
Let us recall that due to the self-ajointness the a parameter should obey the inequality
a < 2
and, as a consequence, in (8) we have (2 - a) > 0. Now it is clear from (8) that the asymptotics for eigenvalues vk(p) as p ^ 0 have the form
Vk(p) - (2k)2.
The operator h(p) is self-adjoint (Hermitian) on [0,n/2] interval with zero boundary conditions and its eigenfunction set {fa(6\p)}g° is orthonormal and complete. This set can be used to expand the solution of Eq. (5)
U(p, 6) = fa(6\p)Fo(p) + ^ fa(6\p)Fi(p). (9)
i=1
Introducing expansion (9) into (5) and projecting onto basis functions (6\p) lead us to the following set of the coupled-channel equations for Fk(p), k = = 0,1,...,
-92p-^ + Xk(p)-E^jFk(p)= (10)
= {2Aki(p)dpFi(p) + [Bki(p) + Wki(p)]Fi(p)} .
i=0
Here, the nonadiabatic matrix elements Aki(p), Bki(p) and potential coupling matrix Wki(p) are given by integrals
n/2
Aki(p) = J d6fa(6\p)dpfa(6\p), 0
n/2
Bki(p) = i d6fa(6\p)d2pfa(6\p),
n/2
Wik(p) = I ddfa(d\p)V(pcosd) x 0
0+(0)
x J de'fa(e'\p).
2—(2)
Now we need to study the asymptotics of functi
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