- Email: [email protected]
An exactly soluble three-body model of resonance scattering
ANNALS
OF PHYSICS:
An Exactly
61, 57-77 (1970)
Soluble
Three-Body
Model
of Resonance
Scattering
P. BEREGI AND I. LOVAS Central Research Institute for Physics, Budapest, Hungary AND
J. REVA? Service de Physique Thdorique, Centre d’Etudes NuclGaires de Saclay BP no 2-91, Gif-sur- Yvette, France Received February 23, 1970
A simple model is constructed consisting of two particles and a potential having two bound states. The model is solved exactly, applying Faddeev’s method, and approximately, using the basic assumptions of the microscopic nuclear reaction theories. In the elastic scatterjng channel a compound-type resonance is produced and the properties of this resonance are studied in order to make comparison between exact and approximate solutions.
I. INTRODUCTION In the last few years a great deal of effort was concentrated on the development of microscopic nuclear reaction theories [l-6]. The main aim of these theories is to calculate the S matrix by means of the detailed dynamical treatment of the nuclear many-body problem. This ambitious goal can be achieved, of course, only at the price of a number of different simplifying assumptions and approximations which can be divided into two groups. To the first group belong the specific assumptions characteristic of the scattering problem, which are introduced to overcome the difficulties connected with the continuous spectrum. The second group contains more general assumptions and approximations, such as the assumptions concerning the two-body interactions, the average field, the residual interaction, the truncation of .the configurational space, etc. The aim of this paper is to investigate the effect of the approximations belonging to the first group [17, 18,191. For this purpose we have constructed a simple model + On leave of absence from the Central Research Institute for Physics, Budapest. 57
58
BEREGI, LOVAS, AND REVAI
which has the following general properties: (i) It incorporates all the essential features of the nuclear reactions; (ii) It can be solved exactly; (iii) It can be treated applying the most frequently used approximations. This model consists of two particles and a potential having two bound states; consequently it is appropriate for the study of elastic and inelastic scattering, scattering of composite particle, stripping, and pick-up and break-up reactions. In elastic scattering a “compound’‘-type resonance is produced and in this paper we shall focus our attention on this resonance, studying the effect of different approximations on the resonance parameters. Since the potential can be considered as a third particle having infinite mass, the exact solution of our model can be worked out by Faddeev’s method [7,8]. Although Faddeev has eliminated all the principal difficulties associated with the quantum mechanical theory of the threebody problem, the practical applicability of his method is somewhat limited, since in the general case it leads to a system of integral equations containing two variables. There exists, however, a special class of potentials, namely the class of the separable nonlocal potentials [9], with the help of which Faddeev’s equations can be reduced to simple one-variable integral equations [8]. Therefore it is very convenient to represent all the interactions of the model by separable nonlocal potentials. For the sake of further simplicity we use potentials acting only in relative s states. The interaction between the infinitely heavy particle and the light ones is represented by a sum of two separable nonlocal terms. In this way, as is well known [lo], two bound states can be produced, which is the most salient feature of our model. It is necessary to say a few words about the two light particles of the model. We assume that they are identical fermions. In this way we shall be able to study the effect of the antisymmetrization, which is frequently ignored in nuclear reaction calculations. Since all interactions of our model are spin independent, the total spin is a good quantum number, in other words, the S = 0, and S = 1 states are not coupled. In the spin triplet state, associated with an antisymmetric wave function, there is no interaction between the two particles since the potential acts only in a relative s state; consequently it is enough to treat the nontrivial S = 0 state associated with a symmetric wave function.
II. THE MODEL The Hamiltonian
of our model is given by w,
2) = t(l) + t(2) + u(l) + U(2) + u(l, 2),
(1)
RESONANCE
59
SCATTERING
where t(i) and u(i) are the operators of the kinetic and potential energy, respectively, and v(l,2) is the interaction between the two particles. It is convenient to define these operators with the help of their matrix elements in momentum representation. Using the fi = 1 and m = l/2 units they read (ki 1 t(i) 1k;)
= 6(ki - k,l) ki2,
(2)
(ki I ~0) I ki’) = - i g,W g&i’),
(3)
5=1
g2(ki)
(k, , k, I ~62)
=
I kl’, k,‘)
(p22
= -S@
P = k, + k, , g(k)
'2
;
p2
:
-
y22
2
- p’) g(k) g(k’),
k = i& =
j&2
- k,), k2
&2
'
(3b)
(4) (9
WI
3
where A, , h2, 6,) h and P1 , B2 , y2, P are the strength and range parameters, respectively. The Schrodinger equation for the i-th particle reads ki2y(ki)
+ 1 (ki 1u(i) 1k,‘)
&ki’)
dki’ = Ey(ki’).
(5)
By an appropriate choice of parameters, this equation has two bound state solutions, which are easily obtained in the form
Q)dkd) =
ki2
-’
E,
i s=l
N,Yg,(ki),
(u
=
0,
b),
where the index v = a labels the lower lying state. The coefficients N,” are determined by the following system of linear equations:
where M,,(z) is given by
60
BEREGI,
LOVAS,
AND
REVAI
The two-particle system can have a bound state with the eigenvalue E~ = -2~~2 if the equation M(Q) = 0 (9) is satisfied, where M(z) is defined by M(z) = 1 + 47r j”
;(y ;($
k2 dk.
0
The wave function of this bound state can be expressed as
The essential features of the model can be seen easily by looking at the spectrum shown in Fig. 1. ENERGY 0
THRESHOLO OF THE TWO-PARTICLE CONTINUUM THRESHOLD OF THE P/CK-UP THRESHOLD OF THE INELASTIC SCATTERING
6 Lb Eb+Eb ‘lb b EO
k-b+&*& &+Eb*&b
REsONAhcE
IN THE ELASTIC
THRESHOLO
OF THE ONE-PARTICLE
TWO EXCITED
GROUND
6*6*&a
1.
FIG.
The
SCATTERIM CONTlNiJlJtl
STATES
STATE
spectrum of
the model.
In this paper we study only a part of this spectrum lying between the elastic and break-up thresholds.
III.
THE
EXACT
SOLUTION
As was mentioned in the Introduction, we investigate only the spin singlet state, when the wave function must be symmetric. In the treatment of the scattering processes, however, it is more convenient to consider the two light particles to be distinguishable and to derive the symmetric scattering amplitude in terms of the ordinary and exchange scattering amplitudes. Applying Faddeev’s method [7] the wave function of the system lu,(+) is written as e+’
= *, + $4 + *n )
RESONANCE
61
SCATTERING
where #I , & , and $I2 are the solutions of the following system of integral equations: (13)
Here @, is the eigenfunction of the operator momentum representation it is given by
H1 = r(1) + t(2) + u(l).
In
@a(k, 7k,) = S(k, - ka) dkd. (14) The Green operator for the noninteracting system is denoted by G, . Its matrix elements in momentum representation read (k, >k, I Go(z) I kl’, k,‘) = @,’ The T operators of the “two-particle” by the following equations:
- W W,’
- k2) z _ kl; _ k 2 . 2
(15)
subsystems tl(z), tz(z), and tlz(z) are defined
h(z) = 4) + 41) Go(z) h(z), k44 = 42) + ~(2) Go(z) G4 (16) d4 = 4L2) + 41,2) Go(z) h(z). As a first step these equations have to be solved. Because of the separability of the potentials the solutions can be obtained very easily by algebraic methods in the following explicit form: (h >k, I h(z) I kl’, k,‘) = --6&z - k2’) i
g,(h) gs&‘> K,l(z
- h2),
g,(k,)g&,')
-
r,s=1 (k,,
(k,
k,
3 k,
I t2(4
I h2t.4
I kl',
k,')
=
-W,
-
k,')
2 7, s=l
-JG;(z
k2),
(17)
I kl’, k,‘) = -S@ - P’) g(k) g(k’) M-l@ - 4~‘).
Using these solutions of the two-body problems, the solutions of the Faddeev equations can be written as follows: A&
Y W = @a&
3 k,) +
E _ k 2:
1
‘2(k1 ’ k2) = Mk,
2W =
1
k 2 + iv 2
2
c g&) E - /cl2 - k,2 + iq i,j=l
1 E - k12 - k22 -I- iq
g(k) M-’
&‘(E
- k12 + id $‘&,
(E - i p2 + iv) F(O)@, k,),
k,),
(18)
62
BEREGI,
LOVAS,
AND
REVAI
where the functions &‘j*)(x, k,) = Fi(l)(x, k,) & Fj2)(x, ka) and F(O)(x, k,) defined as the solutions of the following system of equations:
Fj+)(x,k,) = i j- 4,(x,
Y) Mi?(E
- y2 + id @)(Y,
Y) M-ICE
- Sy2 + id F”‘(Y,
are
k,) dy
s.i=l
2 1 &(x,
+
$‘(x, ka) = - .S.i=li j &(x, F(O)(x,k,) =
Y) Ki’(E
2 &(y, x) M;‘(E
k,) dy - g,(kJ
- y2 + h) 4-ky,
ka) 4
wdx),
+ gjk)
cp,(x),
- y2 + iv) F$+‘(y, k,) dy
S.&-l -
g(b
-
W
dx
-
(19)
&A
where 44%
&( Y) g,(x)
Y) = -
(20)
E - x2 - y2 + iv
and &(Y - xl &Y - x> 4(x, Y) = - E - x2 - (x - y)” + iq *
(21)
In order to obtain integral equations depending only on a single variable we make a partial wave expansion,
Bj(X, y) = c “j$ Z
9%(x
z”$
Y) g (; x - Y) = T
-
y, j,
&T&-&J YZrnG-u~
(23)
y,
Y&(&J
(24)
5
YZl,(Q,).
W&=-Z
Finally we obtain
z~j+)(x, kJ= 5 j-m Aj,(x, y) xy MS’@ - y2 + iv> 4&,
s.i=1
+ 2 SD z&(x,Y>M-‘(E 0 -
zd+)(y,
0
47r~zovaW g&J
xka
9
BY" + iq> zF'O'(y, kJ dy
k,) dy
63
RESONANCE SCA-ITERING
&)(x,
k,) = - 47r8~oi
Srn&(x,
S.&l
y) xy M;yE
- y2 + iq) &)(y,
k,) dy
0
As is seen, the equations with different I are not coupled because of the special choice of the potentials.
IV.
THE TRANSITION
AMPLITUDES
The cross sections of all possible reactions can be calculated from the transition matrix elements given by qi
= (@, I v, I W’h
(26)
where cD~ describes the “final” state and is a solution of the Hamiltonian Hf , while V, is defined as H = Hf + V, . In our model, below the break-up threshold, the following processes occur: (i)
Ordinary
scattering, @f = @2 - kv) ~v&.l, vr = 43 + 4,2),
(v = a, b), (27)
E = k,2 + E, .
(ii)
Exchange scattering,
@t = %k, - kv) vdk,), v, = 4) + u&2),
(v = a, b), (28)
E = k,2 + E, .
(iii)
Pick-up, @f = &P - w rpdw, v, = 41) + u(2), E = *pd2 f cd.
The pick-up reaction does not exist, of course, if there is no bound state in the
64
BEREGI,
LOVAS,
AND
REVAI
two-particle system. The transition amplitudes (26) for these reactions can be expressed in terms of the functions Fi (l), Fjzl, and F(O) as follows: (i)
Ordinary
scattering, r::
= (v, k, 1 To’ 1a, k,)
= i
NdVF2’(k, , k,),
i=l
(ii)
Exchange scattering, 9-f:
= (v, k, I Tex 1a, k,)
= ;
NivF~‘(ky
, k,),
(30
i=l
(iii)
Pick-up Y--,p,u= (d, pa 1 TPU 1 a, k,)
= 2NdFco)(p,, k,),
(32)
where NiV and Nd are defined by (7) and (1 l), respectively. It is worthwhile to note that usually the Faddeev equations are formulated directly for the transition operators, because their matrix elements between the states Qf and Qi give immediately the transition amplitudes. If the initial fragmentation is given, then the number of transition operators is equal to the number of final fragmentations. In our case there are three independent operators which can be labeled according to the above-discussed processes. Their off-the-energy-shell matrix elements are connected with the quantities F(l) z 9F!2) z 3 and F(O) as follows* .
g&W M& - k,4 F%
,U
i,i=l
(33) +
g(W
+
(z
(k, , k, I T”“(z) I a, k,) = i
M-‘(z -
h2
MM
-
k,')
+g2) @a&,
F”‘(P,
kJ W,
M;(z - k,3 F,(“(kz , k,)
i.j=l
+ g&h) MYj’(z - k,2) F?(kl +
(z
-
ha
-
k22)
@a@,
, W
, k,)]
RESONANCE
65
SCATTERING
To get the scattering amplitude for the case of indistinguishable have to sum the ordinary and exchange scattering amplitudes,
=
particles we
+ (v, k, I TeX I a, k,),
(34)
which can be expressed readily as (v, k, 1 Tsc I a, k,)
= i
NJ++‘(ky
, k,).
(35)
V. THE APPROXIMATESOLUTIONS The starting point for the majority of microscopic nuclear reaction theories is the exact wave function expanded in terms of the target eigenstates. In the case of our model this expansion reads v=a.b
where the bound states and the continuum states of the target defined by (5) are denoted by y,(kd and Q)&), respectively. The functions &(kJ and #,&) are considered as unknown coefficients of the expansion to be calculated from the equation (H - E)Y, = 0. The determination of the functions & and $, is equivalent, of course, to the exact solution of the problem. We can get different approximations of the microscopic nuclear reaction theories by neglecting or approximating some parts of the expansion (36). In this paper we investigate three kinds of approximations. V. 1. The Coupled Channels Approximation Without Exchange If we neglect the second part of the expansion (36) containing the continuum states of the target we get the usual “coupled channels” approximation. In this case the approximate wave function contains only the discrete states of the target, (37) Equations can be obtained for $a and 8,&by the requirement I 44W
- E) Y&G 3 k,) dk, = 0,
(38)
which have the following form: Wz2 + Ev - El M4 595/61/1-s
+ c 1
&’
= 0,
(3%
66
BEREGI,
LOVAS,
AND
REVAI
where
6 I ww I k,‘) = L(k, I 49 I k,‘) + j- dk&k,
3 k, I 4 192) I kl’, k,‘) v,(k,‘) dk, db’.
To get a solution for Y&r , k,) which corresponds to the “initial I,& has to be written as follows: Z/I”(X) = &,6(x - k,) - E -“,“” Performing
Y
>‘-1- iq 3
condition”
(40) (14),
(41)
a partial wave expansion fy&
ka) = c &(x, I‘
ka)
(42)
we obtain a system of coupled integral equations,
where
and t~,(p, X) is defined by (24). In this approximation only ordinary scattering can take place since the wave function does not contain continuum states for the first particle. It is easy to see that the ordinary scattering amplitude can be expressed as
(v, k, I Tar I a, kJ =L&
, k,).
(45)
V.2. The Coupled-Channels Approximation with Exchange The method of coupled channels can be easily generalized for including the possibility of exchange processes. In this way we can take into account the indistinguishability of the particles. For this purpose we write the approximate wave function in the following form:
RESONANCE
67
SCATTERING
where $yd and #ye are unknown functions to be determined from the following requirements: I dWW
- El ya@, , W 4
= 0,
I dk,W
- El Ya@, 3 kd &
= 0.
(47)
Apparently in this way we get four coupled equations for the four unknown functions. If we introduce, however, the linear combinations 9’*‘(k) Y
= # YW
f 9 Y“00 9
(48)
we obtain two uncoupled systems of equations
(k2 + Ev - E) $!+)(k) + 1 j- (k I d:’ I k’) Sl”’ or’> A’ ” IL c y,(kW, + Ev - E) j- q4.0 $?‘(k’) dk’ = 0, u
(4%
x {(k, kl I v I k’, k,‘) f
(50)
where
Writing
down the wave function in terms of 4!*)(k),
we see that the two terms in squared brackets are the symmetric and antisymmetric components of the wave function. In contrast to the previous approximation Eqs. (49) do not determine uniquely the functions #j,+‘(k). This difficulty is connected with the fact that the wave function given by (46) cannot be considered as a part of an orthogonal expansion. This problem was studied in detail by Friedman [12] who has shown that the transition amplitudes of the physical processes are not affected by this ambiguity. As far as the mathematics of the problem is concerned, the uniqueness of the functions #!*)(k) can be ensured by appropriate subsidiary conditions. In our case these
68
BEREGI,
LOVAS,
subsidiary conditions can be incorporated we get the following equations: (k2 + E, - E) #i*‘(k)
AND
REVAI
into the equations and instead of (49)
+ c j- (k / NJ::’ j k’) &‘(k’) u
dk’
f + C y,(k)(E, + Ev- E) j [v,,(W #?‘(k’) f q#‘) d? @‘)I dk’ = 0. (52) II Because of the separability of the potentials the following relation holds: @, , k, I u&2)
I kl’, k,) = 0% , k, I v&2)
I kz’, k,‘);
(53)
therefore the equations can be simplified to a great extent. The equations for $L-‘(k) do not contain the interaction ~(1, 2). Therefore their solutions can be expressed simply by the solutions of Eq. (5). The equations for #J+‘(k) are very similar to the previously-discussed ones given under (39); consequently we may follow the same procedure. Introducing the analogues of (41) and (42) we get again the integral equations (43) except for the kernel, which is given by
The transition matrix elements are also very similar,
(55)
where
fyd,(x,k,) = Hfk%
kc4+ fkb,
kz)),
fvBa(x, k,)
kJ - f:iI’(x,
W.
(56) = Hfi%
It is easy to see again that the scattering amplitudes for the case of indistinguishable particles can be obtained directly from the solution fY(O+),
(57)
RESONANCE
69
SCATTERING
V.3. The Isolated Resonance Approximation If we assume that in the neighbourhood of an isolated resonance the wave function can be approximated by the linear combination of a continuum state and a bound state “embedded into the continuum,” then we get the so-called isolated resonance approximation. In our case this assumption corresponds to the following approximate wave function: (57) Ydk, , k,) = w&) v+dk,) + cpdkd M4, where 4, is an unknown function and y is an unknown number. It is obvious that this approximation is a restricted version of the coupledchannels approximation. Equations (39) in this case can be written as follows:
W + Ecz- E) $a&) + j (k, I wasI k,‘) A&‘) & E
-Y
j
1 wab
I k,‘)
dk,‘)
dk,‘,
(58) (kz2
+
&
-
=--s
El
Y’?‘a@,)
& I wba
+
I k,‘)
y j
#a(k,‘)
1 Wbb
I k,‘)
yb(k,‘)
&
&‘.
With the help of the second equation we can eliminate the coefficient y and obtain
(b2 + Em- E) #a@4 + j (k2 I was I k,‘) &&‘) dk,’ =-
I s
I W
Yb(kd
P)b&‘)&’
1 wba
1 k,‘)
,J
(k,‘)
&
a
E - %??b- (b,b 1v I b,b)
where the expectation value of the two-body interaction embedded into the continuum is defined as
I dk
2
&,‘,
c59j
’
for the bound state
(b, b I 0 l b, b) = j 9)&) $%Ckz)&2kz I v&2) I h’, k2’) x P)b(kl’) Q’&‘)
&
dk, dk,’ dk,‘.
(60)
Writing again the solution in the form of (41) and repeating the same procedures as previously we get the scattering amplitude in a straightforward way, (a, k, I To’ / a, k,)
= f(ka , k,).
(61)
70
BEREGI,
LOVAS,
AND
VI. R~su~-rs AND
REVAI
DISCUSSION
As a first step of the analysis we have studied the properties of the scattering amplitude given by (62)
We have calculated the various trajectories of its poles in the complex k, plane, varying the parameters A1 , h, , 6, , /?1 , p, , and y2 . In this way we could select out a family of parameters which ensures the existence of two poles on the positive imaginary axis, in other words, the existence of two bound states. In such a case the position of the poles in the complex k plane is indicated in Fig. 2. Assuming short range interaction, the wave function of a bound Im k
Re k
FIG.
2.
The poles
of the I~ matrix.
state must behave in the asymptotic region as e-+” where E, = (i~4~)~[13]. This requirement is equivalent to the condition 0 < 01, < mWV2p ~~1. (63) It is well known that a pair of complex poles situated in the lower half-plane at resonance if p > a. %P - io produces a “potential”
RESONANCE
71
SCATTERING
For sets of parameters satisfying condition (63) no poles giving rise to potential resonances were found. The values of the parameters used throughout this paper, together with the “coordinates” of the poles, are given in Table I. At real positive values of k1 we TABLE I The Parameters of the Potential and the Poles of the I Matrix. 4 A2 82 61 Be Y2
0.725 0.900 4.178 1.200 1.000 3.000
or, al, a.3’ “a’
0.928 0.372 -8.298 -0.916 0.159 1.243
P
D
have analysed the scattering amplitude (62) in terms of the phase shift. The dependence of the phase shift 6, on k, is shown in Fig. 3 and it can be easily checked that the Levinson theorem [ 14, 151 6(k = 0) - 6(k = co) = mr
(64)
is fulfilled since the number of bound states denoted by IZ is actually equal to two.
FIG.
3. The phase shift of the “potential
scattering.”
72
BEREGI,
LOVAS,
AND
REVAI
As a next step we have solved the integral equations given by formulas (25) and (43) for the quantities Fj*), F(O),f,, , f$‘, andfat several values of the parameter h, taking only the value 1 = 0. The integral equations were approximated by systems of linear equations using the method of Gaussian quadratures. Special care was exercised in the neighbourhood of the singularities of the kernels. 6
(a) 0
1
ldJGS4
’ :
I 1
#I’
I 1 I I I I I
/ /‘I --__
..’
---
/’
EXACT ---SYMMETRK .-.-ORDIMRY -
EXCMME
a76
076
(b)
r; f I
a=L.2854
I : : : I 1’
_
--*_
---_
-- -----r/
,
I I i I ; I I I I
COIJPLEO EXcHAa __-
CH4h’NEi.S
SYFlflETRlC
-.-.DRDMRY _
EXCMNGE
ka
WIM
RESONANCE
73
SCATTERING
15 ---COWLED CWISFLS WITHOUT EXWWGE -ISOLATED RESON+VKE APPROXIMATION
10 -
a5 -
al
I a68
I 070
I 072
I a74
I a76
I U78
ka
FIG. 4. The I = 0 elastic scattering cross sections around to resonance at X = 0.954. The axes of q, and k, are scaled in units of (47r/122)-1and & , respectively.
This numerical procedure for the exact solution was checked by the method of Hetherington and Schick [ 161, viz., the integral equations were solved along a line 1k I e-iQ(O < CJI< 7r/6) and the solutions were analytically continued back to the real k axis. To check the approximate solutions we have calculated them also by two independent methods, viz., the integral equations were solved with boundary conditions for “outgoing wave” and for “standing wave,” as well. In the first case the solutions are complex-valued and from the solutions we have obtained immediately the T matrix elements as was described in the previous section. In the second case the solutions are real and after having obtained the K matrix we calculated the T matrix elements with the help of the K matrix. We performed these calculations with the parameters j3 = 2.5, h = 0.954, h = 1.350, and X = 1.909. At the value X = 1.909 the two light particles have a bound state and therefore the pick-up reaction is possible, while at the other two values of h the bound state does not exist. The elastic scattering cross sections around the resonance for I = 0 at two values of h, viz., at A = 1.350 and h = 0.954, are shown in Figs. 4 and 5, respectively. As is indicated in the figures, together with the “symmetric” cross section corresponding to indistinguishable particles, we computed the ordinary and exchange cross sections, too. In addition to the elastic scattering cross section we calculated also the inelastic
74
BEREGI,
LOVAS,
AND
REVAI
and pick-up cross sections just for checking the consistency of our methods. The optical theorem, the unitarity, and the detailed balance were used as tools for testing. In order to draw some conclusions it is worth while to enumerate the main assumptions of the studied approximations. The coupled-channels approximation with exchange differs from the exact solution only in one respect, viz., in this approximation two particles cannot be simultaneously in the continuum. This is the basic assumption of the theories proposed by Bloch and Gillet [3] and Weidenmiiller [5].
COOPLED EXCMMGE
‘\ 2.0 -
15 y
1. -
‘\
1.
\
\
‘\
‘\
i-7 ‘.
‘\. ‘A
‘l ‘...
‘.t
\
WITH
_ _ _ SYNMETRC
/ i
l-/’
ChMNNELS
1 I I I I I I I I I I I
_.-_ _
ORDINARY EXCHANGE
0.5 /----A,L.-w-. -.-.__ 0
060
a65
am
a75
atw
--__ -.-.__
--__ a&i
--__ 090
-‘k,
RESONANCE
75
SCATTERING
~~~COIJPLED CHANNELS WIThyUT EXCHMGE -ISOLATED RESONANCE APPROXIMATKWV
0
060
0.65
070
075
080
085
090-
ia
FIG. 5. The 1 = 0 elastic scattering cross sections around the resonance at X = 1.350. The axes of O, and k, are scaled in units of (4&3-’ and j2, respectively.
The coupled-channels approximation without exchange has an additional restriction compared to the previous one, viz., only the bombarding particle can be in the continuum. This restriction is the characteristic feature of an enormous number of nuclear reaction calculations in which exchange effects are neglected. The isolated resonance approximation, introduced by Feshbach [l] and applied, e.g., by Balashov et al. [4], contains a further restriction, viz., the bombarding particle can be in the continuum only if the target is in its ground state. .
,titititiHHHHH 1
L_cg~$R~ EXACT
FIG. 6. The allowed configurations is indicated by black points.
in the various approximations.
The bombarding
particle
The essence of the approximations is summarized in Fig. 6, which shows the allowed configurations. We have analysed the exact and the various approximate
76
BEREGI, LOVAS, AND REVAI
cross sections of the elastic scattering in terms of resonance parameters by fitting to them a simple Breit-Wigner formula,
(a, k, I T I a, k,) = A + BkG2+ (ka2 - Ez + (i/2)r ’
(651
where A, B, and C are complex numbers. The resonance energy E, , the width r, and the level shift Ebbgiven in Table 11 can serve as tools for quantitative comparison between the exact and approximate solutions. The level shift Ebbis defined as the difference of the resonance energy and the energy of the unperturbed quasibound state associated with the resonance, Ebb= E, - (2E, - E,).
(66)
TABLE II The Energy and Width of the Resonance
E, Exact Coupled channels with exchange Coupled channels without exchange Isolated resonance approximation
x = 0.954 r Ebb
E,
h = 1.350 r %b
-5
h = 1.909 r %a
0.543 0.0117 0.551 0.0094
-0.042 -0.034
0.499 0.525
0.0330 0.0253
-0.086 -0.060
0.333 0.478
0.0709 0.0609
-0.252 -0.107
0.553
0.0042
-0.032
0.528 0.0104
-0.057
0.486
0.0223
-0.099
0.548
0.0054
-0.037
0.519 0.0146
-0.066
0.477 0.0343
-0.108
Looking at Table II it is seen immediately that the resonance width and the level shift decrease together with the strength of the twoparticle interaction and the agreement between exact and approximate results improves, as is expected. It is also quite natural that the results of the coupled-channels approximation with exchange are closest to the exact ones; the deviations, however, both in the resonance width and in the level shift are rather large. As far as the energy of the resonance is concerned, the various approximations give nearly the same results; the resonance width, however, is very strongly affected by the exchange. Of course this effect would be reduced if the number of particles were higher. It is interesting to note that the result of the poorest approximation, viz., that of the isolated resonance approximation, is not the worst. The numbers exhibited in Table II would suggest a somewhat pessimistic conclusion: The reliability of the approximate methods seems to be rather limited. These results, however, cannot be taken too seriously until similar tests have been done with local potentials, which is more difficult but in principle can be done
77
RESONANCE SCATTERING
[17]. Nevertheless, it seems to us that the special choice of the potential cannot be the source of such a large deviation between exact and approximate solutions and we are convinced that the error caused by an approximate method can be counterbalanced only by the use of appropriate effective interactions. ACKNOWLEDGMENTS The authors
are grateful
to Dr. Gy. Bencze
and Dr.
P. Doleschall
for several
helpful
discussions.
REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. Il.
12. 13. 14. 15. 16. 17. 18. 19.
FESHBACH, Ann. Phys. (New York), 5 (1958), 357; 19 (1962), 287. M. MACDONALD, N&. Phys. 54 (1954), 393. BL~CH AND V. GILLET, Phys. Left. 16 (1965), 62. V. BALASHOV, P. DOLESCHALL, G. YA. KORENMAN, V. L. KOROTKZH, AND V. N. FET~SOV, Sov. J. Nucl. Phys. 2 (1965), 461. H. A. WEWENM~~LLER, Nucl. Phys. 75 (1966), 189. C. MAHAUX AND H. A. WEIDEN~~LLER, “Shell-Model Approach to Nuclear Reactions,” North-Holland, Amsterdam, 1969. L. D. FADDEEV, Sov. Phys.-JETP 12 (1961), 1014; Sov. Phys.-Dokl. 6 (1961), 384. C. LQVELACE, Phys. Rev. B 135 (1964), 1225. Y. YAMAGUCHI, Phys. Rev. 95 (1954), 1628. G. C. GHIRARDI AND A. RIMINI, J. Math. Phys. 6 (1965), 40. W. A. FRTEDMAN AND H. FESHBACH, “Spectroscopic and Group Theoretical Methods in Physics,” The Racah Memorial Volume (F. Bloch, S. G. Cohen, Eds.) North-Holland, Amsterdam, 1968. W. A. FRIEDMAN, Ann. Phys. (New York), 45 (1967), 265. Y. YAMAGUCH~ AND Y. YAMAGLJCHI, Phys. Rev. 95 (1954), 1635. N. LEVINSON, Kgl. Danske Videnskab. Selskab. Mat. Fys. Medd. 25 (1949), No. 9. F. CALOGERO AND G. JAGANNATHAN, Nuovo Cimenfo A 47 (1967), 178. J. H. HETHER~NGTON AND L. H. SCHICK, Phys. Rev. B 137 (1965), 935. A. I. BAZ, V. F. DEMIN, AND I. I. KUZMIN, Yud. Fiz. 4 (1966), 1131. A. S. REINER AND A. I. JAFFE, Phys. Rev. 161 (1967), 9, 35. P. E. SHANLEY AND R. AARON, Ann. Phys. (New York), 44 (1967), 363.
H. V. C. V.
OF PHYSICS:
An Exactly
61, 57-77 (1970)
Soluble
Three-Body
Model
of Resonance
Scattering
P. BEREGI AND I. LOVAS Central Research Institute for Physics, Budapest, Hungary AND
J. REVA? Service de Physique Thdorique, Centre d’Etudes NuclGaires de Saclay BP no 2-91, Gif-sur- Yvette, France Received February 23, 1970
A simple model is constructed consisting of two particles and a potential having two bound states. The model is solved exactly, applying Faddeev’s method, and approximately, using the basic assumptions of the microscopic nuclear reaction theories. In the elastic scatterjng channel a compound-type resonance is produced and the properties of this resonance are studied in order to make comparison between exact and approximate solutions.
I. INTRODUCTION In the last few years a great deal of effort was concentrated on the development of microscopic nuclear reaction theories [l-6]. The main aim of these theories is to calculate the S matrix by means of the detailed dynamical treatment of the nuclear many-body problem. This ambitious goal can be achieved, of course, only at the price of a number of different simplifying assumptions and approximations which can be divided into two groups. To the first group belong the specific assumptions characteristic of the scattering problem, which are introduced to overcome the difficulties connected with the continuous spectrum. The second group contains more general assumptions and approximations, such as the assumptions concerning the two-body interactions, the average field, the residual interaction, the truncation of .the configurational space, etc. The aim of this paper is to investigate the effect of the approximations belonging to the first group [17, 18,191. For this purpose we have constructed a simple model + On leave of absence from the Central Research Institute for Physics, Budapest. 57
58
BEREGI, LOVAS, AND REVAI
which has the following general properties: (i) It incorporates all the essential features of the nuclear reactions; (ii) It can be solved exactly; (iii) It can be treated applying the most frequently used approximations. This model consists of two particles and a potential having two bound states; consequently it is appropriate for the study of elastic and inelastic scattering, scattering of composite particle, stripping, and pick-up and break-up reactions. In elastic scattering a “compound’‘-type resonance is produced and in this paper we shall focus our attention on this resonance, studying the effect of different approximations on the resonance parameters. Since the potential can be considered as a third particle having infinite mass, the exact solution of our model can be worked out by Faddeev’s method [7,8]. Although Faddeev has eliminated all the principal difficulties associated with the quantum mechanical theory of the threebody problem, the practical applicability of his method is somewhat limited, since in the general case it leads to a system of integral equations containing two variables. There exists, however, a special class of potentials, namely the class of the separable nonlocal potentials [9], with the help of which Faddeev’s equations can be reduced to simple one-variable integral equations [8]. Therefore it is very convenient to represent all the interactions of the model by separable nonlocal potentials. For the sake of further simplicity we use potentials acting only in relative s states. The interaction between the infinitely heavy particle and the light ones is represented by a sum of two separable nonlocal terms. In this way, as is well known [lo], two bound states can be produced, which is the most salient feature of our model. It is necessary to say a few words about the two light particles of the model. We assume that they are identical fermions. In this way we shall be able to study the effect of the antisymmetrization, which is frequently ignored in nuclear reaction calculations. Since all interactions of our model are spin independent, the total spin is a good quantum number, in other words, the S = 0, and S = 1 states are not coupled. In the spin triplet state, associated with an antisymmetric wave function, there is no interaction between the two particles since the potential acts only in a relative s state; consequently it is enough to treat the nontrivial S = 0 state associated with a symmetric wave function.
II. THE MODEL The Hamiltonian
of our model is given by w,
2) = t(l) + t(2) + u(l) + U(2) + u(l, 2),
(1)
RESONANCE
59
SCATTERING
where t(i) and u(i) are the operators of the kinetic and potential energy, respectively, and v(l,2) is the interaction between the two particles. It is convenient to define these operators with the help of their matrix elements in momentum representation. Using the fi = 1 and m = l/2 units they read (ki 1 t(i) 1k;)
= 6(ki - k,l) ki2,
(2)
(ki I ~0) I ki’) = - i g,W g&i’),
(3)
5=1
g2(ki)
(k, , k, I ~62)
=
I kl’, k,‘)
(p22
= -S@
P = k, + k, , g(k)
'2
;
p2
:
-
y22
2
- p’) g(k) g(k’),
k = i& =
j&2
- k,), k2
&2
'
(3b)
(4) (9
WI
3
where A, , h2, 6,) h and P1 , B2 , y2, P are the strength and range parameters, respectively. The Schrodinger equation for the i-th particle reads ki2y(ki)
+ 1 (ki 1u(i) 1k,‘)
&ki’)
dki’ = Ey(ki’).
(5)
By an appropriate choice of parameters, this equation has two bound state solutions, which are easily obtained in the form
Q)dkd) =
ki2
-’
E,
i s=l
N,Yg,(ki),
(u
=
0,
b),
where the index v = a labels the lower lying state. The coefficients N,” are determined by the following system of linear equations:
where M,,(z) is given by
60
BEREGI,
LOVAS,
AND
REVAI
The two-particle system can have a bound state with the eigenvalue E~ = -2~~2 if the equation M(Q) = 0 (9) is satisfied, where M(z) is defined by M(z) = 1 + 47r j”
;(y ;($
k2 dk.
0
The wave function of this bound state can be expressed as
The essential features of the model can be seen easily by looking at the spectrum shown in Fig. 1. ENERGY 0
THRESHOLO OF THE TWO-PARTICLE CONTINUUM THRESHOLD OF THE P/CK-UP THRESHOLD OF THE INELASTIC SCATTERING
6 Lb Eb+Eb ‘lb b EO
k-b+&*& &+Eb*&b
REsONAhcE
IN THE ELASTIC
THRESHOLO
OF THE ONE-PARTICLE
TWO EXCITED
GROUND
6*6*&a
1.
FIG.
The
SCATTERIM CONTlNiJlJtl
STATES
STATE
spectrum of
the model.
In this paper we study only a part of this spectrum lying between the elastic and break-up thresholds.
III.
THE
EXACT
SOLUTION
As was mentioned in the Introduction, we investigate only the spin singlet state, when the wave function must be symmetric. In the treatment of the scattering processes, however, it is more convenient to consider the two light particles to be distinguishable and to derive the symmetric scattering amplitude in terms of the ordinary and exchange scattering amplitudes. Applying Faddeev’s method [7] the wave function of the system lu,(+) is written as e+’
= *, + $4 + *n )
RESONANCE
61
SCATTERING
where #I , & , and $I2 are the solutions of the following system of integral equations: (13)
Here @, is the eigenfunction of the operator momentum representation it is given by
H1 = r(1) + t(2) + u(l).
In
@a(k, 7k,) = S(k, - ka) dkd. (14) The Green operator for the noninteracting system is denoted by G, . Its matrix elements in momentum representation read (k, >k, I Go(z) I kl’, k,‘) = @,’ The T operators of the “two-particle” by the following equations:
- W W,’
- k2) z _ kl; _ k 2 . 2
(15)
subsystems tl(z), tz(z), and tlz(z) are defined
h(z) = 4) + 41) Go(z) h(z), k44 = 42) + ~(2) Go(z) G4 (16) d4 = 4L2) + 41,2) Go(z) h(z). As a first step these equations have to be solved. Because of the separability of the potentials the solutions can be obtained very easily by algebraic methods in the following explicit form: (h >k, I h(z) I kl’, k,‘) = --6&z - k2’) i
g,(h) gs&‘> K,l(z
- h2),
g,(k,)g&,')
-
r,s=1 (k,,
(k,
k,
3 k,
I t2(4
I h2t.4
I kl',
k,')
=
-W,
-
k,')
2 7, s=l
-JG;(z
k2),
(17)
I kl’, k,‘) = -S@ - P’) g(k) g(k’) M-l@ - 4~‘).
Using these solutions of the two-body problems, the solutions of the Faddeev equations can be written as follows: A&
Y W = @a&
3 k,) +
E _ k 2:
1
‘2(k1 ’ k2) = Mk,
2W =
1
k 2 + iv 2
2
c g&) E - /cl2 - k,2 + iq i,j=l
1 E - k12 - k22 -I- iq
g(k) M-’
&‘(E
- k12 + id $‘&,
(E - i p2 + iv) F(O)@, k,),
k,),
(18)
62
BEREGI,
LOVAS,
AND
REVAI
where the functions &‘j*)(x, k,) = Fi(l)(x, k,) & Fj2)(x, ka) and F(O)(x, k,) defined as the solutions of the following system of equations:
Fj+)(x,k,) = i j- 4,(x,
Y) Mi?(E
- y2 + id @)(Y,
Y) M-ICE
- Sy2 + id F”‘(Y,
are
k,) dy
s.i=l
2 1 &(x,
+
$‘(x, ka) = - .S.i=li j &(x, F(O)(x,k,) =
Y) Ki’(E
2 &(y, x) M;‘(E
k,) dy - g,(kJ
- y2 + h) 4-ky,
ka) 4
wdx),
+ gjk)
cp,(x),
- y2 + iv) F$+‘(y, k,) dy
S.&-l -
g(b
-
W
dx
-
(19)
&A
where 44%
&( Y) g,(x)
Y) = -
(20)
E - x2 - y2 + iv
and &(Y - xl &Y - x> 4(x, Y) = - E - x2 - (x - y)” + iq *
(21)
In order to obtain integral equations depending only on a single variable we make a partial wave expansion,
Bj(X, y) = c “j$ Z
9%(x
z”$
Y) g (; x - Y) = T
-
y, j,
&T&-&J YZrnG-u~
(23)
y,
Y&(&J
(24)
5
YZl,(Q,).
W&=-Z
Finally we obtain
z~j+)(x, kJ= 5 j-m Aj,(x, y) xy MS’@ - y2 + iv> 4&,
s.i=1
+ 2 SD z&(x,Y>M-‘(E 0 -
zd+)(y,
0
47r~zovaW g&J
xka
9
BY" + iq> zF'O'(y, kJ dy
k,) dy
63
RESONANCE SCA-ITERING
&)(x,
k,) = - 47r8~oi
Srn&(x,
S.&l
y) xy M;yE
- y2 + iq) &)(y,
k,) dy
0
As is seen, the equations with different I are not coupled because of the special choice of the potentials.
IV.
THE TRANSITION
AMPLITUDES
The cross sections of all possible reactions can be calculated from the transition matrix elements given by qi
= (@, I v, I W’h
(26)
where cD~ describes the “final” state and is a solution of the Hamiltonian Hf , while V, is defined as H = Hf + V, . In our model, below the break-up threshold, the following processes occur: (i)
Ordinary
scattering, @f = @2 - kv) ~v&.l, vr = 43 + 4,2),
(v = a, b), (27)
E = k,2 + E, .
(ii)
Exchange scattering,
@t = %k, - kv) vdk,), v, = 4) + u&2),
(v = a, b), (28)
E = k,2 + E, .
(iii)
Pick-up, @f = &P - w rpdw, v, = 41) + u(2), E = *pd2 f cd.
The pick-up reaction does not exist, of course, if there is no bound state in the
64
BEREGI,
LOVAS,
AND
REVAI
two-particle system. The transition amplitudes (26) for these reactions can be expressed in terms of the functions Fi (l), Fjzl, and F(O) as follows: (i)
Ordinary
scattering, r::
= (v, k, 1 To’ 1a, k,)
= i
NdVF2’(k, , k,),
i=l
(ii)
Exchange scattering, 9-f:
= (v, k, I Tex 1a, k,)
= ;
NivF~‘(ky
, k,),
(30
i=l
(iii)
Pick-up Y--,p,u= (d, pa 1 TPU 1 a, k,)
= 2NdFco)(p,, k,),
(32)
where NiV and Nd are defined by (7) and (1 l), respectively. It is worthwhile to note that usually the Faddeev equations are formulated directly for the transition operators, because their matrix elements between the states Qf and Qi give immediately the transition amplitudes. If the initial fragmentation is given, then the number of transition operators is equal to the number of final fragmentations. In our case there are three independent operators which can be labeled according to the above-discussed processes. Their off-the-energy-shell matrix elements are connected with the quantities F(l) z 9F!2) z 3 and F(O) as follows* .
g&W M& - k,4 F%
,U
i,i=l
(33) +
g(W
+
(z
(k, , k, I T”“(z) I a, k,) = i
M-‘(z -
h2
MM
-
k,')
+g2) @a&,
F”‘(P,
kJ W,
M;(z - k,3 F,(“(kz , k,)
i.j=l
+ g&h) MYj’(z - k,2) F?(kl +
(z
-
ha
-
k22)
@a@,
, W
, k,)]
RESONANCE
65
SCATTERING
To get the scattering amplitude for the case of indistinguishable have to sum the ordinary and exchange scattering amplitudes,
=
particles we
+ (v, k, I TeX I a, k,),
(34)
which can be expressed readily as (v, k, 1 Tsc I a, k,)
= i
NJ++‘(ky
, k,).
(35)
V. THE APPROXIMATESOLUTIONS The starting point for the majority of microscopic nuclear reaction theories is the exact wave function expanded in terms of the target eigenstates. In the case of our model this expansion reads v=a.b
where the bound states and the continuum states of the target defined by (5) are denoted by y,(kd and Q)&), respectively. The functions &(kJ and #,&) are considered as unknown coefficients of the expansion to be calculated from the equation (H - E)Y, = 0. The determination of the functions & and $, is equivalent, of course, to the exact solution of the problem. We can get different approximations of the microscopic nuclear reaction theories by neglecting or approximating some parts of the expansion (36). In this paper we investigate three kinds of approximations. V. 1. The Coupled Channels Approximation Without Exchange If we neglect the second part of the expansion (36) containing the continuum states of the target we get the usual “coupled channels” approximation. In this case the approximate wave function contains only the discrete states of the target, (37) Equations can be obtained for $a and 8,&by the requirement I 44W
- E) Y&G 3 k,) dk, = 0,
(38)
which have the following form: Wz2 + Ev - El M4 595/61/1-s
+ c 1
&’
= 0,
(3%
66
BEREGI,
LOVAS,
AND
REVAI
where
6 I ww I k,‘) = L(k, I 49 I k,‘) + j- dk&k,
3 k, I 4 192) I kl’, k,‘) v,(k,‘) dk, db’.
To get a solution for Y&r , k,) which corresponds to the “initial I,& has to be written as follows: Z/I”(X) = &,6(x - k,) - E -“,“” Performing
Y
>‘-1- iq 3
condition”
(40) (14),
(41)
a partial wave expansion fy&
ka) = c &(x, I‘
ka)
(42)
we obtain a system of coupled integral equations,
where
and t~,(p, X) is defined by (24). In this approximation only ordinary scattering can take place since the wave function does not contain continuum states for the first particle. It is easy to see that the ordinary scattering amplitude can be expressed as
(v, k, I Tar I a, kJ =L&
, k,).
(45)
V.2. The Coupled-Channels Approximation with Exchange The method of coupled channels can be easily generalized for including the possibility of exchange processes. In this way we can take into account the indistinguishability of the particles. For this purpose we write the approximate wave function in the following form:
RESONANCE
67
SCATTERING
where $yd and #ye are unknown functions to be determined from the following requirements: I dWW
- El ya@, , W 4
= 0,
I dk,W
- El Ya@, 3 kd &
= 0.
(47)
Apparently in this way we get four coupled equations for the four unknown functions. If we introduce, however, the linear combinations 9’*‘(k) Y
= # YW
f 9 Y“00 9
(48)
we obtain two uncoupled systems of equations
(k2 + Ev - E) $!+)(k) + 1 j- (k I d:’ I k’) Sl”’ or’> A’ ” IL c y,(kW, + Ev - E) j- q4.0 $?‘(k’) dk’ = 0, u
(4%
x {(k, kl I v I k’, k,‘) f
(50)
where
Writing
down the wave function in terms of 4!*)(k),
we see that the two terms in squared brackets are the symmetric and antisymmetric components of the wave function. In contrast to the previous approximation Eqs. (49) do not determine uniquely the functions #j,+‘(k). This difficulty is connected with the fact that the wave function given by (46) cannot be considered as a part of an orthogonal expansion. This problem was studied in detail by Friedman [12] who has shown that the transition amplitudes of the physical processes are not affected by this ambiguity. As far as the mathematics of the problem is concerned, the uniqueness of the functions #!*)(k) can be ensured by appropriate subsidiary conditions. In our case these
68
BEREGI,
LOVAS,
subsidiary conditions can be incorporated we get the following equations: (k2 + E, - E) #i*‘(k)
AND
REVAI
into the equations and instead of (49)
+ c j- (k / NJ::’ j k’) &‘(k’) u
dk’
f + C y,(k)(E, + Ev- E) j [v,,(W #?‘(k’) f q#‘) d? @‘)I dk’ = 0. (52) II Because of the separability of the potentials the following relation holds: @, , k, I u&2)
I kl’, k,) = 0% , k, I v&2)
I kz’, k,‘);
(53)
therefore the equations can be simplified to a great extent. The equations for $L-‘(k) do not contain the interaction ~(1, 2). Therefore their solutions can be expressed simply by the solutions of Eq. (5). The equations for #J+‘(k) are very similar to the previously-discussed ones given under (39); consequently we may follow the same procedure. Introducing the analogues of (41) and (42) we get again the integral equations (43) except for the kernel, which is given by
The transition matrix elements are also very similar,
(55)
where
fyd,(x,k,) = Hfk%
kc4+ fkb,
kz)),
fvBa(x, k,)
kJ - f:iI’(x,
W.
(56) = Hfi%
It is easy to see again that the scattering amplitudes for the case of indistinguishable particles can be obtained directly from the solution fY(O+),
(57)
RESONANCE
69
SCATTERING
V.3. The Isolated Resonance Approximation If we assume that in the neighbourhood of an isolated resonance the wave function can be approximated by the linear combination of a continuum state and a bound state “embedded into the continuum,” then we get the so-called isolated resonance approximation. In our case this assumption corresponds to the following approximate wave function: (57) Ydk, , k,) = w&) v+dk,) + cpdkd M4, where 4, is an unknown function and y is an unknown number. It is obvious that this approximation is a restricted version of the coupledchannels approximation. Equations (39) in this case can be written as follows:
W + Ecz- E) $a&) + j (k, I wasI k,‘) A&‘) & E
-Y
j
1 wab
I k,‘)
dk,‘)
dk,‘,
(58) (kz2
+
&
-
=--s
El
Y’?‘a@,)
& I wba
+
I k,‘)
y j
#a(k,‘)
1 Wbb
I k,‘)
yb(k,‘)
&
&‘.
With the help of the second equation we can eliminate the coefficient y and obtain
(b2 + Em- E) #a@4 + j (k2 I was I k,‘) &&‘) dk,’ =-
I s
I W
Yb(kd
P)b&‘)&’
1 wba
1 k,‘)
,J
(k,‘)
&
a
E - %??b- (b,b 1v I b,b)
where the expectation value of the two-body interaction embedded into the continuum is defined as
I dk
2
&,‘,
c59j
’
for the bound state
(b, b I 0 l b, b) = j 9)&) $%Ckz)&2kz I v&2) I h’, k2’) x P)b(kl’) Q’&‘)
&
dk, dk,’ dk,‘.
(60)
Writing again the solution in the form of (41) and repeating the same procedures as previously we get the scattering amplitude in a straightforward way, (a, k, I To’ / a, k,)
= f(ka , k,).
(61)
70
BEREGI,
LOVAS,
AND
VI. R~su~-rs AND
REVAI
DISCUSSION
As a first step of the analysis we have studied the properties of the scattering amplitude given by (62)
We have calculated the various trajectories of its poles in the complex k, plane, varying the parameters A1 , h, , 6, , /?1 , p, , and y2 . In this way we could select out a family of parameters which ensures the existence of two poles on the positive imaginary axis, in other words, the existence of two bound states. In such a case the position of the poles in the complex k plane is indicated in Fig. 2. Assuming short range interaction, the wave function of a bound Im k
Re k
FIG.
2.
The poles
of the I~ matrix.
state must behave in the asymptotic region as e-+” where E, = (i~4~)~[13]. This requirement is equivalent to the condition 0 < 01, < mWV2p ~~1. (63) It is well known that a pair of complex poles situated in the lower half-plane at resonance if p > a. %P - io produces a “potential”
RESONANCE
71
SCATTERING
For sets of parameters satisfying condition (63) no poles giving rise to potential resonances were found. The values of the parameters used throughout this paper, together with the “coordinates” of the poles, are given in Table I. At real positive values of k1 we TABLE I The Parameters of the Potential and the Poles of the I Matrix. 4 A2 82 61 Be Y2
0.725 0.900 4.178 1.200 1.000 3.000
or, al, a.3’ “a’
0.928 0.372 -8.298 -0.916 0.159 1.243
P
D
have analysed the scattering amplitude (62) in terms of the phase shift. The dependence of the phase shift 6, on k, is shown in Fig. 3 and it can be easily checked that the Levinson theorem [ 14, 151 6(k = 0) - 6(k = co) = mr
(64)
is fulfilled since the number of bound states denoted by IZ is actually equal to two.
FIG.
3. The phase shift of the “potential
scattering.”
72
BEREGI,
LOVAS,
AND
REVAI
As a next step we have solved the integral equations given by formulas (25) and (43) for the quantities Fj*), F(O),f,, , f$‘, andfat several values of the parameter h, taking only the value 1 = 0. The integral equations were approximated by systems of linear equations using the method of Gaussian quadratures. Special care was exercised in the neighbourhood of the singularities of the kernels. 6
(a) 0
1
ldJGS4
’ :
I 1
#I’
I 1 I I I I I
/ /‘I --__
..’
---
/’
EXACT ---SYMMETRK .-.-ORDIMRY -
EXCMME
a76
076
(b)
r; f I
a=L.2854
I : : : I 1’
_
--*_
---_
-- -----r/
,
I I i I ; I I I I
COIJPLEO EXcHAa __-
CH4h’NEi.S
SYFlflETRlC
-.-.DRDMRY _
EXCMNGE
ka
WIM
RESONANCE
73
SCATTERING
15 ---COWLED CWISFLS WITHOUT EXWWGE -ISOLATED RESON+VKE APPROXIMATION
10 -
a5 -
al
I a68
I 070
I 072
I a74
I a76
I U78
ka
FIG. 4. The I = 0 elastic scattering cross sections around to resonance at X = 0.954. The axes of q, and k, are scaled in units of (47r/122)-1and & , respectively.
This numerical procedure for the exact solution was checked by the method of Hetherington and Schick [ 161, viz., the integral equations were solved along a line 1k I e-iQ(O < CJI< 7r/6) and the solutions were analytically continued back to the real k axis. To check the approximate solutions we have calculated them also by two independent methods, viz., the integral equations were solved with boundary conditions for “outgoing wave” and for “standing wave,” as well. In the first case the solutions are complex-valued and from the solutions we have obtained immediately the T matrix elements as was described in the previous section. In the second case the solutions are real and after having obtained the K matrix we calculated the T matrix elements with the help of the K matrix. We performed these calculations with the parameters j3 = 2.5, h = 0.954, h = 1.350, and X = 1.909. At the value X = 1.909 the two light particles have a bound state and therefore the pick-up reaction is possible, while at the other two values of h the bound state does not exist. The elastic scattering cross sections around the resonance for I = 0 at two values of h, viz., at A = 1.350 and h = 0.954, are shown in Figs. 4 and 5, respectively. As is indicated in the figures, together with the “symmetric” cross section corresponding to indistinguishable particles, we computed the ordinary and exchange cross sections, too. In addition to the elastic scattering cross section we calculated also the inelastic
74
BEREGI,
LOVAS,
AND
REVAI
and pick-up cross sections just for checking the consistency of our methods. The optical theorem, the unitarity, and the detailed balance were used as tools for testing. In order to draw some conclusions it is worth while to enumerate the main assumptions of the studied approximations. The coupled-channels approximation with exchange differs from the exact solution only in one respect, viz., in this approximation two particles cannot be simultaneously in the continuum. This is the basic assumption of the theories proposed by Bloch and Gillet [3] and Weidenmiiller [5].
COOPLED EXCMMGE
‘\ 2.0 -
15 y
1. -
‘\
1.
\
\
‘\
‘\
i-7 ‘.
‘\. ‘A
‘l ‘...
‘.t
\
WITH
_ _ _ SYNMETRC
/ i
l-/’
ChMNNELS
1 I I I I I I I I I I I
_.-_ _
ORDINARY EXCHANGE
0.5 /----A,L.-w-. -.-.__ 0
060
a65
am
a75
atw
--__ -.-.__
--__ a&i
--__ 090
-‘k,
RESONANCE
75
SCATTERING
~~~COIJPLED CHANNELS WIThyUT EXCHMGE -ISOLATED RESONANCE APPROXIMATKWV
0
060
0.65
070
075
080
085
090-
ia
FIG. 5. The 1 = 0 elastic scattering cross sections around the resonance at X = 1.350. The axes of O, and k, are scaled in units of (4&3-’ and j2, respectively.
The coupled-channels approximation without exchange has an additional restriction compared to the previous one, viz., only the bombarding particle can be in the continuum. This restriction is the characteristic feature of an enormous number of nuclear reaction calculations in which exchange effects are neglected. The isolated resonance approximation, introduced by Feshbach [l] and applied, e.g., by Balashov et al. [4], contains a further restriction, viz., the bombarding particle can be in the continuum only if the target is in its ground state. .
,titititiHHHHH 1
L_cg~$R~ EXACT
FIG. 6. The allowed configurations is indicated by black points.
in the various approximations.
The bombarding
particle
The essence of the approximations is summarized in Fig. 6, which shows the allowed configurations. We have analysed the exact and the various approximate
76
BEREGI, LOVAS, AND REVAI
cross sections of the elastic scattering in terms of resonance parameters by fitting to them a simple Breit-Wigner formula,
(a, k, I T I a, k,) = A + BkG2+ (ka2 - Ez + (i/2)r ’
(651
where A, B, and C are complex numbers. The resonance energy E, , the width r, and the level shift Ebbgiven in Table 11 can serve as tools for quantitative comparison between the exact and approximate solutions. The level shift Ebbis defined as the difference of the resonance energy and the energy of the unperturbed quasibound state associated with the resonance, Ebb= E, - (2E, - E,).
(66)
TABLE II The Energy and Width of the Resonance
E, Exact Coupled channels with exchange Coupled channels without exchange Isolated resonance approximation
x = 0.954 r Ebb
E,
h = 1.350 r %b
-5
h = 1.909 r %a
0.543 0.0117 0.551 0.0094
-0.042 -0.034
0.499 0.525
0.0330 0.0253
-0.086 -0.060
0.333 0.478
0.0709 0.0609
-0.252 -0.107
0.553
0.0042
-0.032
0.528 0.0104
-0.057
0.486
0.0223
-0.099
0.548
0.0054
-0.037
0.519 0.0146
-0.066
0.477 0.0343
-0.108
Looking at Table II it is seen immediately that the resonance width and the level shift decrease together with the strength of the twoparticle interaction and the agreement between exact and approximate results improves, as is expected. It is also quite natural that the results of the coupled-channels approximation with exchange are closest to the exact ones; the deviations, however, both in the resonance width and in the level shift are rather large. As far as the energy of the resonance is concerned, the various approximations give nearly the same results; the resonance width, however, is very strongly affected by the exchange. Of course this effect would be reduced if the number of particles were higher. It is interesting to note that the result of the poorest approximation, viz., that of the isolated resonance approximation, is not the worst. The numbers exhibited in Table II would suggest a somewhat pessimistic conclusion: The reliability of the approximate methods seems to be rather limited. These results, however, cannot be taken too seriously until similar tests have been done with local potentials, which is more difficult but in principle can be done
77
RESONANCE SCATTERING
[17]. Nevertheless, it seems to us that the special choice of the potential cannot be the source of such a large deviation between exact and approximate solutions and we are convinced that the error caused by an approximate method can be counterbalanced only by the use of appropriate effective interactions. ACKNOWLEDGMENTS The authors
are grateful
to Dr. Gy. Bencze
and Dr.
P. Doleschall
for several
helpful
discussions.
REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. Il.
12. 13. 14. 15. 16. 17. 18. 19.
FESHBACH, Ann. Phys. (New York), 5 (1958), 357; 19 (1962), 287. M. MACDONALD, N&. Phys. 54 (1954), 393. BL~CH AND V. GILLET, Phys. Left. 16 (1965), 62. V. BALASHOV, P. DOLESCHALL, G. YA. KORENMAN, V. L. KOROTKZH, AND V. N. FET~SOV, Sov. J. Nucl. Phys. 2 (1965), 461. H. A. WEWENM~~LLER, Nucl. Phys. 75 (1966), 189. C. MAHAUX AND H. A. WEIDEN~~LLER, “Shell-Model Approach to Nuclear Reactions,” North-Holland, Amsterdam, 1969. L. D. FADDEEV, Sov. Phys.-JETP 12 (1961), 1014; Sov. Phys.-Dokl. 6 (1961), 384. C. LQVELACE, Phys. Rev. B 135 (1964), 1225. Y. YAMAGUCHI, Phys. Rev. 95 (1954), 1628. G. C. GHIRARDI AND A. RIMINI, J. Math. Phys. 6 (1965), 40. W. A. FRTEDMAN AND H. FESHBACH, “Spectroscopic and Group Theoretical Methods in Physics,” The Racah Memorial Volume (F. Bloch, S. G. Cohen, Eds.) North-Holland, Amsterdam, 1968. W. A. FRIEDMAN, Ann. Phys. (New York), 45 (1967), 265. Y. YAMAGUCH~ AND Y. YAMAGLJCHI, Phys. Rev. 95 (1954), 1635. N. LEVINSON, Kgl. Danske Videnskab. Selskab. Mat. Fys. Medd. 25 (1949), No. 9. F. CALOGERO AND G. JAGANNATHAN, Nuovo Cimenfo A 47 (1967), 178. J. H. HETHER~NGTON AND L. H. SCHICK, Phys. Rev. B 137 (1965), 935. A. I. BAZ, V. F. DEMIN, AND I. I. KUZMIN, Yud. Fiz. 4 (1966), 1131. A. S. REINER AND A. I. JAFFE, Phys. Rev. 161 (1967), 9, 35. P. E. SHANLEY AND R. AARON, Ann. Phys. (New York), 44 (1967), 363.
H. V. C. V.