ANNALS
OF PHYSICS:
53, 115l
32 (1969)
Tests of the Extended
RMatrix
Theory
of Nuclear
Reactions*
L. GARSIDE Lawrence Radiation Laboratory, University of California, Berkeley, California 94720 AND
W. TOBOCMAN Physics Department, Case Western Reserve University, Cleveland, Ohio 44106
The extended Rmatrix theory of nuclear reactions is a generalization of the Rmatrix formalism that makes it possible to calculate the collision matrix from the compound nucleus eigenstates provided by any nuclear structure calculation. In addition, optical potentials for each channel may be used to represent the nonresonant part of the collision matrix. To test the extended Rmatrix formalism (ERF) it was applied to simple, soluble, two channel, Swave scattering models. The ERF was found to be superior to the standard Rmatrix formalism (SRF). Like the SRF, the ERF results are very sensiiive to the choice of the channel radius parameters. In addition the ERF is plagued by the appearance of narrow “false resonances”. A modification of the ERF called the Xmatrix formalism (XF) completely eliminates the dependence on the matching radius. The XF is found to give fairly good results in our model calculations and reduces the false resonance difficulty. However, the XF has the drawback of being sensitive to a set of boundary condition parameters whose values will be somewhat ambiguous in practical calculations. This shortcoming is eliminated in the modified Xmatrix formalism (MXF) which is found to give excellent results for our two channel model and shows no false resonances. We conclude that the MXF is the version of choice among the various versions of the ERF. I. INTRODUCTION
There has been considerable successin the last few years in describing the properties
of nuclear
bound
states
by means of nuclear
structure
calculations
based
on the shell model and generalizations thereof. Unified nuclear reaction theory is an attempt to capitalize on this successby utilizing the results and methods of the nuclear structure work to provide a true manybody basis for multichannel nuclear reaction calculations. The Rmatrix formalism (I) appears at first sight to be the ideal vehicle for such a program. This formalism usesthe compound nucleus eigenstatesto provide a basis for calculating the nuclear reaction collision matrix. However, the Rmatrix * Supported by the U.S. Atomic Energy Commission. 115
116
GARSIDE
AND
TOBOCMAN
formalism has two drawbacks which reduce its usefulness for unified nuclear reaction theory. First of all, the compound nucleus eigenstates used in Rmatrix theory are required to fulfill homogeneous boundary conditions at the hypersurface in configuration space separating the “inside” or “compound nucleus” region of configuration space from the “outside” or “asymptotic” region. It is very difficult to comply with these boundary conditions in all but the simplest type of nuclear structurecalculation. Secondly, in any practical calculation it is necessary to truncate the sums over the infinite set of compound nucleus eigenstates to just a few terms. The neglected terms may not be important for determining rapid resonance variation of the collision matrix over a narrow range of energy. However, the coherent effect of these neglected terms is necessary to alter the nonresonance background part of the collision matrix from the hard sphere scattering form it would otherwise have in the standard Rmatrix formalism. The extended Rmatrix formalism (2) (ERF) is a generalization of the standard &matrix formalism (SRF) that seeks to remedy the two difficulties described above. The ERF makes it possible to calculate the collision matrix from compound nucleus wave functions which fulfill homogeneous boundary conditions on a hypersurface located outside of the boundary between the “inside” and “asymptotic” regions. By allowing the boundary condition hypersurface to be located at infinity, one can be free to use conventional nuclear structure wave functions. In addition, the ERF makes it possible to use optical potential scattering phase shifts in place of the hard sphere phase shifts that provide the lowest order contribution to the background scattering. Thus a judicious choice of the optical potential for each channel should render the final result for the collision matrix less sensitive to truncation of the compound nucleus eigenstate sum. Within the framework of the ERF there is a variety of ways of relating the collision matrix to the compound nucleus eigenstates and the channel optical potentials. Each way has some special advantage to recommend it. In order to test the efficacy of the ERF in the few level approximation and in order to compare the merits of some the various options provided by the ERF we have applied the formalism to a simple, two channel, Swave potential scattering model. The simplicity of the model makes it possible for us to compare the elastic and inelastic scattering cross sections calculated on the basis of the ERF with the exact values. We performed three series of calculations. In these calculations we considered a scattering model consisting of two coupled Swave nucleon channels. For the first series of calculations (3) each channel had a square potential well providing an elastic interaction, and the two channels were coupled together by a square well. The results of SRF and ERF calculations in the N level approximation were compared with the exact elastic and inelastic cross sections. The ERF was found
EXTENDED
RMATRIX
THEORY
OF
NUCLE.~R
REACTIONS
117
to be a considerable improvement over the SRF. In this calculation the ERF revealed one shortcoming; a narrow “false resonance” appeared in the elastic cross section. However, this resonance was very narrow and tended to disappear as the number of levels N was increased. In a second series of calculations (4) we used zero range potential wells in our two channel scattering model. In this series we limited ourselves to the two level approximation. In the limit of zero coupling between channels, there would be one bound eigenstate in each channel. The two compound nucleus eigenstates that were used in the ERF were the result of using the two unperturbed bound states to diagonalize the exact Hamiltonian. Thus our compound nucleus wave functions were only approximate eigenstates of the Hamiltonian, and they had tails extending to infinity. This aspect added a new element of realism to our calculations. The ERF calculations were found to give a good fit to the elastic cross section provided that the matching radius was appropriately adjusted. False resonances appeared, but they disappeared when a normalization matrix was included in the definition of the Rmatrix. However, the inelastic cross section always turned out much too small. An alternativeversionof theERFwasalso triedinthe second series of calculations. We call this version the Xmatrix formalism (XF). In the XF the matching radius has been eliminated, but certain boundary condition parameters Is,> must be used. The (.sJ are related to the asymptotic boundary conditions fulfilled by the compound nucleus eigenstates. For approximate shell model type eigenstates, such as were used in our zero range model calculations, the values to be assigned to the fs,; are not well determined. However, we found it was possible to find reasonable values for the boundary condition parameters (s,f for which the XF gave fairly good results for both the elastic and inelastic cross sections. No false resonances appeared. There is a modification of the XF, which we will call the modified Xmatrix formalism (MXF), in which the boundary condition parameters (sJ enter as arbitrary input parameters unreIated to the compound nucleus states used. Since the MXF cannot be applied to zero range potentials, we returned to the coupled square wells for our third series of calculations (5) which tested the MXF. The MXF was found to give excellent results in the few level approximation. Several different choices for the {se} were tried, but changing the {s,) was found to have very little effect on the calculated cross section. Our model calculations indicate that the modified Xmatrix formalism (MXF) is probably the most effective way to use the extended Rmatrix formalism (ERF) in the few level approximation. The MXF would appear to be an excellent vehicle for unified nuclear reaction theory calculations. In Section II we review the results of the ERF. Section III is devoted to the application of the ERF to the two channel scattering model. We present our conclusions in Section IV.
118
GAFWDE AND TOBOCMAN II. THE EXTENDED
Let the Schroedinger nucleons be
equation
RMATRIX
FORMALISM
for the relative motion
of a system of A
(E  H)ul, = 0.
(1)
We consider the case where the energy E is small enough so that only two body channels are open. Associated with each channel 01there will be a total channel wave function $a = 5ULl ,5,, , QJ. q& is constructed by first vector coupling the internal motion wave functions of the two nuclides 01~and 01~that constitute channel 01to form a channel spin wave function. The channel spin wave function is then vector coupled to a spherical harmonic for the relative angular motion of the two nuclides to form the total angular mementum eigenfunction & . The subscript iy that appears on the relative motion wave function ?Pais there to indicate that this wave function describes the state where there is unit incident current in channel ~1and purely outgoing flux in all other channels. The collision matrix U serves to characterize the asymptotic behavior of the radial wave functions z+,Jr):
In the asymptotic region u,,(r)
=
k’(r)
6,~

$+‘(d
UB,
>
for
r > a6 .
(3)
The boundary separating the inside region from the asymptotic region is defined by the channel radii (a,}. In Eq. (3) &’ and $+) are the incoming and outgoing unit current radial wave functions for channel /3. Our objective is the calculation of the elements of the collision matrix. The relationship of the collision matrix to the Rmatrix may be expressed in the following manner: ua6 = e”““{(l + iw(+))’ e”‘* = { 5~)(u,)/5~+)(u,))‘/2
(1  i?!f))},6
eivB
W (4b) (4c)
EXTENDED RMATRIX
119
THEORY OF NUCLEAR REACTIONS
The quantity v, is the hard sphere phase shift, mBis the reduced mass in channel j?, the (LZ:}are a second set of channel radii such that {LZ: 3 a,}. The quantities {g&u, , u;)} are the elements of the Rmatrix.
The Da) are the compound
nucleus eigenstates. (E,  H)E(“’
= 0.
(6)
They fulfill homogeneous boundary conditions in each channel
where ui 2 ui 3 uB . When the three sets of channel radii{a:}, {a:>, and {a,)are set equal to each other, the relationships outlined above constitute the standard Rmatrix formalism (SRF). For each channel 01we introduce an optical model Hamiltonian H, and a radial wave function x, , regular at the origin, such that (E  H,) +,~;~x,(r,)
= 0.
(8)
The optical model Hamiltonian is such that the residual interaction vanishes in the asymptotic region. <@&)I V& I y> = 0,
for
V, = H  H,
r > a,.
(9)
The optical model phase shift 6, characterizes the asymptotic behavior of the optical model radial wave function.
x,(r)=
s
sin(k,r  I,,42 + S,),
OL
(To simplify matters we have neglected the Coulomb model Rmatrix has as elements
for
r 2 a, .
interaction.)
The optical
120
GARSIDE
AND
TOBOCMAN
where 4~;‘y~(rJ is a solution of Eq. (8) which is independent of &r;‘xol . Specifically, y. is chosen so that it has the asymptotic form
Y&>=
$$
e
 Z,~r/2 + 8,)  s, sin&r,
{cos(k,r,
 l,rr/2 + S,)}, for r, 3 a,.
(12)
The boundary condition parameters {s,} in Eq. (12) are related to the log. derivatives A, of Eq. (7) by the requirement A, = a; $ 8
=
ln Y,&,“)
k,&[l + s, cot(k&rs”  l&2 + S,)] [cot(k,a,  &Jr/2 + S,)  $1
(13)
Eq. (13) expresses the requirement that the optical model Rmatrix fulfill the same asymptotic boundary conditions as the Rmatrix itself, The optical model is injected into the Rmatrix formalism by using the substitution R = R, $ RaVaR
(14)
for the Green’s function R in the definition of the Rmatrix. When this substitution is carried out, Eq. (4) is replaced by Urns= &{(l
+ iX’+))1
(1  iX())},,
~w&, ,44 = <@,@,>IR, VaR I @BW>.
eiBB
(1W
(15c)
We see that the hard sphere phase shifts have been replaced by the optical model phase shifts and that the Rmatrix has been replaced by the Ymatrix. Eq. (15) constitutes what we will refer to as the extended Rmatrix formalism (ERF). This is to be compared with Eq. (4) when {a, = ai = a:> which is the standard Rmatrix formalism (SRF). In order to make the formalism less sensitive to the values assigned to the matching radii {a,} and {ul} we make the substitution R = RO+ RVBRs
into the expression for the Ymatrix, Eq. (15~).
(16)
EXTENDED
RMATRIX
This results in the following
THEORY
expression
OF NUCLEAR
REACTIONS
121
for the Kmatrix.
*x$$ = X&( 1 ? iSB)
(1W U’b) (17c)
S,fi = ,’ Ar2x,(r,)/ x,, I 4dx,h3)) x,, = v, + V,RV,
Eq. (17) together with Eq. (15a) constitute the Xmatrix formalism (XF). Note that the matching radii (a,) and {a:: have been eliminated. The boundary radii {a:) are implicit in our expressions inasmuch as they define the extent of the configuration space integrations required to evaluate the various matrix elements. The (a”; are also implicit in the definition of the boundary condition parameters. However, there is nothing to prevent us from setting all the boundary radii equal to infinity. The modified Xmatrix formalism (MXF) results when Eq. (17~) is replaced by (6) x,, = V,( v, 
V&V&l
= v, + VJ v, The Green’s function
v, V~R/jvpy
V,R,VB .
(18a) (18b)
in Eq. (17~) has the representation R z
c
) E(n)
The asymptotic boundary conditions other hand, the related quantity (V, 
)(E  E&l
(Ecn)* I.
(19)
fulfilled by R are implicit in the LP). On the
VBR,VJ1
= RV;lR,l
(20)
which appears in Eq. (18) has the representation (Va 
V~R,V&l (Ml)nm
= 1 c I W’) = ;ijyn)* / V, 
Mnm(@(m’* I I/,R,V,
/ cPcm’).
(214 @lb)
The {@,)} are any complete set of compound nucleus states; they need not be eigenstates of H. When Eq. (21) is used in the (MXF). the boundary condition parameters to be inserted in Eq. (17a) must be the same as those used in the definition of the optical model Green’s function. The (8,) used do not depend on the {CD(“)} and can be chosen arbitrarily. Thus the asymptotic boundary conditions for the Xmatrix never become ambiguous in the MXF as they do in the XF when approximate eigenfunctions are used for the {Bn)> in Eq. (19). When all the S, are chosen equal to i the Xmatrix becomesidentical with the conventional Tmatrix. When all the S, are chosen equal to zero the Xmatrix becomes identical with the conventional Kmatrix.
122
GARSIDE
III. APPLICATION
TO THE
TWO
AND
TOBOCMAN
CHANNEL
POTENTIAL
SCATTERING
MODEL
The scattering model we used consists of a zero angular momentum particle of mass m = 1 amu being scattered by an infinite mass target. The target can exist in its ground state .$1 or in its single excited state I&. The wave equation for this system is (
T + V,(r)  E V12(r)
V,,(r) %1(r) = 0 T + V,(r) + Q  E I( u2dr) 1
where T = (fi2/2m) d2/dr2 and Q is the excitation energy of q& . We considered the case where V, , V2 , and VI2 were all square well potentials of radius a, = 6F. We also considered the case where they were zero range potentials: V, = (@/2m) lii(rl
+ 6,) 6(r  c>
VI2 = (fi2i2u,/2m) liior‘6(r
The optical equations:
model
regular
functions
(2%
 7).
fulfilled
the following
Schroedinger
(T + V,(r)  E) xl(r) = 0
CW
(T + V,(r) + Q  E) x2(r) = 0. Thus the residual interaction
(25b)
is just
For the square well calculations use was made of the following compound nucleus states: ’ @)’
(2/f~“)l/~ 1sin(n77/2)) (sin(n?rr/2a”))/r 1) 7rr/2a”))/r *
= ((2/031’2 1sin((n  1)57/2)1(sin((n n = 1, 2, 3, ..*
(27)
The Pn) are seen to be normalized in the interval 0 < r < a” and to have zero slope at r = u”. For the zero range potential calculations we employed the following compound nucleus wave functions: 1O(n)) = (2b,)1/2r1 exp(b,r)
(t 1 y),
n = 1,2.
PO
EXTENDED
RMATRIX
THEORY OF NUCLEAR REACTIONS
A 0 0
123
EXACT 2 LEVELS 6 LEVELS IO LEVELS 20 LEVELS
(b:
b 0 0
020
EXACT 2 LEVELS 6 LEVELS IO LEVELS 20 LEVELS
,015 “7 In /1
010
‘\
I
\ \
j
005
\
;
I
0’ 35
1 4
\
‘. _ _
6
a
I IO
MC’J
FIG. 1. /~,~(47r)~ X elastic cross section (a). kIe(4?r)1 x inelastic cross section (b). Standard Rmatrix formalism (SRF) calculation for the coupled square wells model: a, = 6F, VI = 32 MeV, V, = 39 MeV, V,, = 1 MeV, a = a' = a" = SF, Q = 3.5 MeV.
124
GARSIDE AND TOBOCMAN
(a)
a

2 LEVELS EXACT A
6 LEVELS

,020
EXACT 2 LEVELS A
6 LEVELS
.o15/T\\ \ “7 ur 4
\ \ \
,010
\ \ \
,005 1
\
‘1
\
\
\ ‘. .w
0'3.5
41
5
6 MS
7
I 8
9
1 IO
FIG. 2. k,‘(hml X elastic cross section (a). /~,~(4.rr)~ X inelastic cross s&ion (b). Extended Rmatrix formalism (ERF) calculation for the coupled square wells ‘model: a, = 6~, VI = 32 MeV, V, = 39 MeV, VI, = 1 MeV, a = 6F, a’ = a” = 23F, Q = 3.5 MeV.
EXTENDED
RMATRIX
THEORY
OF
NUCLEAR
RE.~CTIONS
125
These states are bound eigenstates of the optical model Hamiltonian normalized in the interval 0 < r < a” = co. In our first series of calculations (3) we applied the SRF and the ERF to the square well case. We chose Q = 3.5 MeV, V, = 32 MeV, Vz = 39 MeV. and V,, = 1 MeV. With these parameters we find a resonance in the elastic channel of about 1.6 MeV width. The SRF calculations were performed using a = a’ := a” = 8F. for the channel radii. The ERF calculations used a = 6F. and a’ = a” ~~ 8F. The compound nucleus eigenstates EC”) used in Eq. (5) to calculate the Rmatrix were found by diagonalizing the Hamiltonian in the space spanned by the first twentyfive compound nucleus wave functions @cn). Then the Nlevel approximation SRF and ERF collision matrices were calculated by using N of the .Yn) in Eq. (5). We calculated the quantities 0.25 j 1  U,, I2 = (k12/4n) ull = 0.25
1  SI1 ,B 0.25 ( U,, I2 = (k12/4x) u‘gl = 0.25 1S,, I2 k12 = 2mE/fi2
(29~1) (29b) (29~)
where cI1 is the elastic cross section and uzl is the inelastic cross section. The SRF results are displayed in Fig. (1) while the ERF results appear in Fig. (2). Both formalisms give cross sections that are seen to be converging on the exact cross sections, but the convergence of the ERF is much more rapid. The ERF gives a nearly perfect fit to the exact cross section over a 10 MeV interval in the &level approximation while the SRF result is only fair in the 20level approximation. However, the ERF does show a narrow “false resonance” which tends to disappear with increasing N. The calculations described above employed compound nucleus states E@) which fulfilled homogeneous boundary conditions at the boundary radius a” and which, to a good approximation, were Hamiltonian eigenstates. The approximate compound nucleus eigenstates that would be provided by a nuclear structure calculation would have tails extending to infinity in each channel. They would not fulfill homogeneous boundary conditions at the surface of any finite region of configuration space. In our second series of calculations (4) we employed compound nucleus eigenstates of this type. The zero range, two channel scattering model was considered in our second series of calculations (4). We choose Q = 4 MeV, (fi2b12/2nz) = 3 MeV, (fi2b22/2mj =: 2 MeV, and L’,, = 0.05 Fl. This choice of parameters leads to a resonance in the elastic channel of about 0.3 MeV width. In all our calculations we used a two level approximation based on the approximate compound nucleus eigenstates that result from using the two states displayed in Eq. (28) to diagonalize the Hamiltonian. The results of an ERF calculation are shown in Fig. (3) labelled U for three values of the matching radius a = a’. (The boundary radius a” was set equal to
126
GAFWDE
AND
TOBOCMAN
ENERGY &M’)
46.01 (b)
ENERGY IMeV)
FIG. 3. Elastic cross section (a). Inelastic cross section (b). Extended Rmatrix @RF) calculation for the coupled zero range potential wells model: (fie/2m)bla = 3 MeV, (tia/2m)6za = 2 MeV, u,, = 0.05 F1, Q = 4 MeV. The calculations were performed for various values of the matchmg radius a using the usual (IJ) and normalized (N) representation of the Rmatrix in the two level approximation.
EXTENDED
RMATRIX
THEORY
OF
NUCLEAR
REACTIONS
127
infinity.) We find a) the inelastic cross section is much too small, b) the cross sections are very sensitive to the choice of matching radius, and c) a false resonance appears in the elastic cross section. Also displayed in Fig. (3) and labelled N are a set of cross sections calculated using the “normalized” Rmatrix Green’s function (2)
in place of the R defined by Eq. (19) in the ERF. The subscript a on the matrix element in Eq. (30b) serves to indicate that the configuration space integration used to evaluate the matrix element is confined to the inside region bounded by the matching radius a. Use of the normalized Rmatrix is seen to bring about some improvement. The false resonances are eliminated, and the inelastic cross section is increased. The sensitivity to a remains. The results of applying the XF to the zero range two channel scattering model are shown in Fig. (4). Sensitivity to a has been replaced by sensitivity to {s,}. The quantities in the parentheses identify the values assigned to the boundary condition parameters (s,} for calculation of the cross section. When a pair of numbers appears in the parentheses, the first is used in the inelastic channel below threshold and the second is used for open channels. In reference (2) it was indicated that in calculations like this, a reasonable guess would be to set (sur= i> for closed channels and {s, = real number} for open channels. This choice is seen to give pretty good results in our XF calculations. The sensitivity of our XF calculated cross sections to the {s,} suggests that it will be worth the extra complication of the MXF to eliminate ambiguity in the values to be assigned to the boundary condition parameters. The MXF cannot be used when the residual interaction V is separable as it is for the zero range case. So to test the MXF we applied it to the square well case in our third series of calculations (5). The Nlevel approximation in the context of the MXF consisted in limiting the sums shown in Eq. (21) to N terms. The compound nucleus states {CD(“)}used were those that appear in Eq. (27). XF calculations were also done, but instead of using a large number of the @Cn)‘sto create high quality compound nucleus eigenstates 8@), we replaced the definition of R shown in Eq. (19) by the following one R = c c 1@n)) Lnm<@(m) i
(314
nm
(Ll),,
The Nlevel approximation on n and m to N terms.
= ;@(,)
1 E  H 1 @cm)).
for these XF calculations consisted in limiting
@lb)
the sums
128
GARSIDE AND TOBOCMAN
ENERGY (MeV)
40.90 NE 35.76 Y 0 30.67

8 t; 25.56 w
‘. \\
..\
g2045
“\
; 15.34 
\
‘\ ‘. ’ \Y ..,‘‘\
lOt.W.lO.U ICd h.0) ..,‘.xcl,,cl,.;
% g 10.22 5.112 
011”“““““” 360 4.80
6.00
7.20
8.40
9.60
IO 80
l2.00
ENERGY t Me”1
FIG. 4. Elastic cross section (a). Inelastic cross section (b). Xmatrix formalism (Xl?) calculation for the coupled zero range potential wells model: (fi2/2m)ble = 3 MeV, (fiz/2m)bze = 2 MeV, v0 = 0.05 Fl, Q = 4 MeV. The calculations were performed for various values of the boundary condition parameters {se}. The number that appears in the parentheses has been assigned to both s1 and sz . When two numbers appear in the parentheses, the first is used when the channel is closed and the second when it is open. The two level approximation was used. a = 3.8F.
EXTENDED RMATRIX THEORY OF NUCLEAR REACTIONS
129
When we applied the MXF to the case used in our first series of calculations, the two level fits were essentially perfect. So we turned to a case where the channel coupling is much stronger. For our third series of calculations we used the following parameters: Q = 3.5 MeV, VI = 10 MeV, V, = 15 MeV, and V,, = 4 MeV. Here VI2 is so large that the elastic scattering resonance is broadened beyond recognition. However there is a prominent threshold cusp in the elastic cross section. The results of MXF calculations using the boundary condition parameters s, = (K,2  k,2) sin 2K,a” 2K&, k,2 = (2m/h2)[E  Q(1  S,3]
Wb)
K,2 = k,2  (2m/A2)V,
(32~)
are shown in Fig. (5). These boundary conditions result from the choice y,Jr) = const. x cos KJ in Eq. (13). Those calculations based on Eq. (18a) are identified with a C while those based on Eq. (18b) are identified with a D. The D option appears to converge a little more rapidly than the C option. Both give essentially perfect results in the six level approximation. Other nonsingular choices for the {s,} give cross sections that are almost identical with the MXF results shown in Fig. (5). Presented in Fig. (6) are the results of MXF calculations using the boundary condition parameters sa = tan cotl(k;lKm cot Kea”) (33) which derive from zero slope (A, = 0) boundary conditions. The s, become singular at certain energies. The C option of the MXF seems to be immune to these singularities, but the D option is strongly effected below the inelastic threshold. However, in the ten level approximation the D option gives essentially exact results. The cross sections in Fig. (6) labelled with a B are the results of XF calculations. In these calculations there is no ambiguity in the choice of the (s,); those provided by Eq. (33) are the correct choice. We see that the XF suffers from false resonances in the vicinity of 2 MeV. These resonances become narrower as N is increased, but even for N = 20 they are not completely eliminated.
IV. CONCLUSION
We conclude that the modified Xmatrix formalism (MXF) is the most suitable version of the extended Rmatrix formalism (ERF) for use in unified nuclear reaction calculations. It can be constructed from the results of existing nuclear
130
GARSIDE
.
AND
TOBOCMAN
I :
.761
N” .652

E.
d N* 
,543.
3= L .435, ,326
‘.
.i?l7
r
,109 
.oool
0.0
’
’
1.0
’
’
2.0
’
’
3.0
1
’
4.0
’
5.0
ENERGY
’
’
’
6.0 (MeV
’
’
’
’
’
7.0
8.0
9.0
7.0
8.0
9.0
’
I
1.528
x N 
1.019 z 3 
,764 t
00
1.0
2.0
30
4.0
50 ENERGY
6.0
10.0
(Me’,)
FIG. 5. Elastic (a) and inelastic (b) scattering cross sections (multiplied by k12/4r) for the coupled square wells model calculated by the modified Xmatrix formalism (MXF): a, = 6F, V, = 10 MeV, V, = 15 MeV, V,, = 4 MeV, a” = 8F, Q = 3.5 MeV. E identifies the exact cross sections. The MXF calculations are labelled CN for cases where Eq. (21a) was used and DN for cases where Eq. (21 b) was used. N identifies the number of levels used. “Y = cos Kr” boundary conditions were used in the MXF.
I.
.
,
132
GARSIDE AND TOBOCMAN
structure and optical model calculations. It contains no channel radii and the values of the boundary condition parameters can be assigned in a fairly arbitrary manner. The MXF appears to be free of the false resonances that appear in some of the other versions of the ERF. RECEIVED: October 2, 1968
REFERENCES 1. A. M. LANE 2. L. GARSIDE 3. L. GAR.SIDE Research
AND R. G. THOMAS, AND W. TOBOCMAN, AND W. TOBOCMAN,
Rev. Mod. Phys. 30,257 (1958). Phys. Rev. 173, 1047 (1968).
A test of the extended RMatrix theory of nuclear reactions I. Report, Physics Department, Case Western Reserve University. 4. W. TOBOCMAN AND L. GARSIDE, A test of the extended RMatrix theory of nuclear reactions II. Research Report, Physics Department, Case Western Reserve University. 5. L. GARSIDE AND W. TOBOCMAN, A generalized KMatrix method for nuclear reaction calculations. Research Report, Physics Department, Case Western Reserve University. 6. J. H~~NER AND R. H. LEMMER, Phys. Rev. 175, 1394 (1968). J. H~FNER AND C. M. SHAKIN, Phys. Rev. 175, 1350 (1968).