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The angular dependence of a broad peripheral resonance state
N~&ur Physics A222 (1974) 365-316; Not to be reproduced by photoprint
THEl ANGULAR
@ North-HoliandPu62i~~
Co., Amsterdam
or microfilm without written permission from the publisher
~EPEN~ENC~
OF A BROAD
RESONANCE
PE~~~L
STATE
R. C. FULLER t Institut fiir Theoretische Phisik der Universitiit Heidelberg, Germany Y. AVISHAI tt ~a~-Planck-Instit~~t fiir ker~pbysik, Heidelberg, Germnay Received
17 December
1973
Abstract: We discuss the angular dependence oftbe shape-elastic resonance wave function for a state so broad decay occurs before the ions orbit the interaction region. In the shadow of the interaction region the non-stationa~ angular dependence is described by a Legendre function for complex angular momentum, P, (cos@. In the lit region the same state has the angrdar dependence e-in=P,(--cos @). After presenting theoretical arguments which show the necessity of using these different analytic continuations in the shadow and lit regions to states of complex angular momentum we show, by way of illustration of the theoretical argument, results for a strong absorption model introduced several years ago by Austern.
1.lntrodu~tion It has been found that the optical model potentials which fit heavy-ion elastic scattering above the Coulomb barrier must be sufficiently absorptive to damp shape-elastic resonances. On the other hand it has been suggested that a Regge-pole parameterization of the nuclear S-matrix can be used to fit elastic scattering data [see McVoy “) and references contained therein]. The Regge pole describes a peripheral shape-elastic resonance; however, as discussed in a previous paper “) by one of the present authors, the resonance lifetime is so short that the ions do not rotate about one another; i.e. the resonance lifetime is commensurate with the direct reaction time scale. This implies that the decay of such a state is fore-aft asymmetric in angle. Such an angular distribution cannot be described by a single Legendre polynomial. As shown in ref. “) the wave packet in angular momentum space necessary to describe such a state is provided by a Regge pole whose imaginary part gives the rate of decay in angle of the state. Strong volume absorption gives rise to a shadow on the downstream side of the reaction region. The full wave function in this region of conjuration space is dominated by peripheral resonances which live su~cie~tly long to propagate into the shadow. As one moves out of the shadow it is necessary to subtract from
t Supported by a grant from the Gesellschaft ftir Schwerionenforschung, ti Supported by a Minerva Fellowship. Present address: University of the Negev, Beer Sheba, Israel. 365
Darmstadt,
Germany.
366
R. C. FULLER
AND
Y. AVISHAI
the full wave function the contribution which has not traversed the shadow in order to obtain the part of the wave function dominated by resonant contributions. This leads to the characteristic signature of a Regge pole at back angles in the scattering distribution although this same signature appears in the full wave function just downstream from the interaction region at forward angles. In the next section we give a general discussion of the geometrical interpretation of the broad peripheral resonance. In the last section we discuss a semiclassical model which Austern “) introduced several years ago and which he describes as “perhaps as consistently ‘black’ as any model can be”. This model generates a broad peripheral resonance and the model is sufficiently simple to allow its detailed study. In order to retain this simplicity we neglect the Coulomb interaction. 2. Geometrical
interpretation
of a broad peripheral resonance
Since we intend to investigate broad peripheral states using a schematic model, the following analytic discussion will stress physical interpretation with little pretense of mathematical rigor. Our discussion of the wave function near the surface will follow previous work “) of one of the present authors while the discussion of the relation of the surface wave in the shadow to the asymptotic angular distribution follows Nussenzveig “). The wave function for the scattering of two spinless particles has the partial-wave expansion,
Y(k, r) =
k C eit'(21
+ l)$j+)(k, r)~,(cos $),
(1)
where $i+‘(k, r) may conveniently be decomposed into incoming and outgoing contributions as $I”
= $iei4”r(f,f-)(k,
r)-e-‘“‘,Sl(k)f,(+)(k,
r)>.
(2)
The incoming and outgoing waves,fiF)(k, r) satisfy the radial Schriidinger equation and the angular momentum independent boundary condition f/‘)(k,
at large r. To write the partial wave transformation, it is necessary integer values of the angular convenient for obtaining the eq. (2) the SW transformation
WC r>=
r) w ecikr,
(3)
sum of eq. (1) using a Sommerfeld-Watson (SW) to choose an interpolation of the summand between momentum. We choose an interpolation which is resonance contribution to the wave function. Using for the wave function may be written as
k/c-$$ (f’-‘(A
k,
r)-e-‘“(“-%(A,
k)f”‘(A,
k, r)}P,_,(cos
S),
ANGULAR
DEPENDENCE
OF RESONANCE
STATE
367
where c is a contour which encircles the real axis in the counter-clockwise direction. The partial wave series is recovered as the residue series arising from the zeros of cos nl. Since f”)(A, k, r) satisfy the radial Schrddinger equation which depends on the angular momentum through an entire function, Z(Z+ l), and further satisfy A (or 1) independent boundary conditions, they are entire functions of 1, at fixed k and r [ref. “)I. Therefore at fixed k and I the only poles in 2 in the radial wavefunction arise from the S-matrix S(A, k). We now assume the integration contour is distorted to pick up at least a subset of these pole contributions to give the residue series
Pae+za-f(+)(&, k, r)P,l_*(cos 9), where p, is the S-matrix residue lim,,,= (A-,$$(A). If this approach is to be useful, the first few terms of this series must approximate the wave function in the shadow. It is essentially for this reason that we chose the continuation of eq. (4) which led to the residue series involving P,(cos 9) which is singular at 9 = rc and not at 9 = 0. It is not our purpose to discuss the general problem of convergence which involves discussion of the residua /3,. We shall only discuss those aspects of the problemwhich do not depend on the details of the interaction. For this reason our considerations apply to the region of configuration space where the interaction may be assumed weak, i.e. near and beyond the interaction surface. Heavy ion scattering above the Coulomb barrier is characterized by strong absorption which implies that the leading Regge pole contribution to eq. (5) corresponds to a peripheral configuration. The angular momenta associated with such a configuration are generally sufficiently large to justify the use of asymptotic expressions for the functions appearing in eq. (5). For values of the radius corresponding to configurations with the ions sufficiently separated to ignore their interaction, we have
where h,‘l’(k, r) is the outgoing spherical Hankel function. Following Nussenzveig [ref. “)] we assume in addition that Yis sufficiently large to satisfy kr-cc,
>> LX?,
where a, = Rei, -3 and 1, is the leading Regge pole. We can then use the asymptotic approximation for the Hankel function Ii:’
w P-+($---T)+eexp
(i [(p’---a:)+--~,
cos-r ‘3 P
-tn]),
p = kr.
(7)
368
R. C. FULLER
AND
Y. AVISHAI
Substituting eq. (6) into eq. (5) gives for a single Regge-pole contribution
Rl(r)=
n eFiSa
c-
e -i3nalh!~)t(p)P1,_~(cos 9). 1
Unitarity restricts 1, to the upper-half &plane “) where we may write
For large angular momenta the asymptotic expression for the Legendre function is P,_*(cos 9)
M
+
(-&-Jcosp],
p>
Substituting eqs. (lo), (9), and (7) into eq. (8) gives for a single Regge-pole contribution
R,(k,r) = e-i*(2zal)3 8 ats,i-n/4l+,ilnta,-n/41) x x_e 1)mievc -
)
(11)
where 6, = 9,-9+2nln, 0, = 9,+9+2nm, 9, = gn-cos-l
5 = sin-.% P
P
(12)
This approach, which follows Nussenzveig 4), leads to a geometrical interpretation of the Regge-pole contribution. If we define the radius a by ka = [xl,
03)
then for the leading Regge pole a can be expected to approximate the radius of the interaction surface since lower partial waves are absorbed, i.e. weakly coupled to the elastic channel. Then $, =t sin-l 5 P
(141
is the cylindrical boundary, 0 < v < v0 6 $n, of the geometrical shadow of a sphere of radius a. The contribution from the leading Regge pole to the wave function in the geometrical shadow described by eq. (11) is shown schematically in fig. 1. For the angular momentum a,, the particles are incident on a trajectory corresponding to the impact parameter a. At TX and T, the system forms a resonance which is attenuated both by decay into the elastic channel and by absorption.
ANGULAR
DEPENDENCE
OF RESONANCE T*
Fig. 1. A schematic
representation
7/2-
(cps-1
369
STATE
alr+&=%-9rb.
of the contribution of a broad peripheral shadow region given by eq. (11).
resonance
in the
At the point P(r, 31, within the geometrical shadow, one has the two contributions emanating tangentially from T; and Ti which for v # 0 have unequal amplitudes because they have propagated through different angles. Contributions from the m # 0 terms correspond to encircling the sphere m times before decay and therefore are generally strongly attenuated unless a2 is quite small. Obviously the rate of convergence of the series in eq. (9) is determined by cc2. In this connection we refer the reader to the discussion in ref. “> on the relationship between a2 and the “life-angle” of the resonance. As one moves to larger angles the m = 0 contribution from the &mode increases in amplitude. As one passes through the boundary of the geometrical shadow the magnitude increases exponentially in an,gle. Higher order Regge poles, corresponding to larger a*, increase even faster form f 0, leading to slow convergence or even divergence of the Regge-pole series. This is to be expected since outside the shadow the wave function must include at least the incident plane wave in addition to the resonance contribution. Obviously slow convergence or divergence can only arise from the m # 0 contributions which we now proceed to isolate in order to move into the lit region. Following Nussenzveig “) we introduce the functions Q!” (cos 3) and Q$” (cos 3) defined by Q$l’ ‘)(cos 3) = ; [P&OS 3)t
2 Qv(cos 3)] , n
,
(15)
where P, and Q, are Legendre functions of the first and second kind respectively. In the same angular range where the asymptotic expression for P, (cos9) given by eq. (10) is valid, we have Q$~~)(cos 3) M (2xR sin 3)-*eri(h’9-&n)_ (16) From eq. (15) we have P,(cos 3) = Q$“(cos 3) + Q!~)(cos 3).
(17)
R. C. FULLER
370
AND
Y. AVISNAf
From eqs. (17), {16), (1.1) and (8) it follows that the only divergent terms in the Regge-pole series arise from the IM= 0 co~~rib~~io~s to Qcf (cos 9). Referring to eqs. (8) and (9) we are therefore led to write e -i&n&
J
P,,_*(cos
8)
=
2
ei*‘%J6$+(cos S)+d,_Jcos
$),
cos na,
(181
where the term which gives rise to divergence has been explicitly isolated and
d,*_&os 8) = 2&‘-
=
2&(-
l)nreix~,(2m+s)Q!~,~(cos 9)
f)meixl,f2tnf~))(Q~~~~$(COS 9) -e-2z~2*Q~L,(cos9)).
However from Nussenzveig “) we have - i~?‘V’++(-cos
9) = Q&(cos
@--e2”““*rQ~‘,‘-;(cos3),
and therefore eq. (18) may be written
Upon deleting the troublesome in eq. (51,
first term in eq, (21) we have for the residue series
B,f’“‘(x,,
k +&;(-cos
91,
(221
which is valid in the lit region. Since this equation results from deleting the 6e term from eq. (X.1>,the contribution of a single Regge pole in the lit region can be represented by the obvious extension of fig. 1 shown in fig. 2. In moving from the shadow to the lit region, we have chosen to follow an argument which at each step stressed the physical interpretation of the variables, Let us now show what the ar~ment has accomplished. Eq. (21) may be rewritten as P,(cos 23)= e?Pa( - cos 23)f 2i cos z(ct + ~~e~~~Q~~~(~o~ 3).
(2W
This expression provides an analytic continuation of the relation P,(cos 3) = (- 1)” P,( - cos 9) which holds at integer values of the angular momentum. The necessity of such a continuation can be understood by recalling P,(cos 9) is singular at 9 = n for non-integer angular momentum while I>a(-cos 9) is singular at 9 =: 0. In the shadow, which includes the angle 9 = 0, the Regge-pole contribution given by eq. (5) has the angular dependence Pz(cos 9). Upon moving into the lit region which includes the angle 9 = z, the angular dependence of the Regge-pole contribution changes to e’“”P,(-cos 9) which is non-singular in the lit region. As stressed by
ANGULAR
DEPENDENCE
OF RESONANCE
STATE
371
Fig. 2. A schematic representation of the contribution of a broad peripheral resonance in the lit region showing the contribution which arises from components of the wave which have traversed the shadow.
Nussensveig “) the use of different continuations of the Legendre function in different angular regions does not imply that the Regge-pole contribution is discontinuous. Obviously one must move into the lit region to determine the scattering angular distributioll. We have just shown that the contribution of a single Regge pole to the scattered wave in the lit region has the angular dependence I’,( -cos 9). This contribution may be continued to large Ywhere it leads to the same angular dependence in the scattering amplitude. As discussed in ref. 2), states with sufficiently large CI~, decay before rotating about the interaction region and give rise to angular distributions which decrease expol~entially toward oscillation at back angles. We suggest that such states be called “Ay-off resonances”. The strong valume absorption which characterizes heavy ion applications leads to the dominance of “peripheral fly-off resonances” (PFOR) at back angles. This interpretation of the exponential decrease of the angular distribution is to be contrasted with the eikonal explanation offered by Broglia et al. “). Referring to fig. 2 it is apparent that the Regge-pole contributions to the scattering amplitude arise from components of the wave function which have traversed the semi-classically forbidden shadow region. In this connection it is interesting to recall Goldberg and Smith’s observation ‘) that by fitting data to angles slightly beyond the maximal semi-classical deflection angle into the angular region where the angular distribution starts its exponential decrease, they removed the depth ambiguity in the optical model fit. On the basis of the previous discussion we are led to conjecture that this ambiguity is removed by fitting into the angular region where the angular distribution is largely determined by the imaginary part of the leading Regge pole which is a sufficiently sensitive function of the potential depth to remove the ambiguity.
372
R. C. FULLER
AND
Y. AVISEIAI
3. Asstem model Several years ago Austern “> suggested a model which incorporates strong absorption for the full three-dimensional wave function. The lower partial waves are assumed to be completely absorbed in the usual sense that each partial wave has no outgoing component; i.e., the reflection coefficients vanish,
s, =
0,
l6
(23)
L*in*
The high partial waves are unscattered,
S-matrix elements for peripheral partial waves are determined by requiring that the three-dimensional wave function on the illuminated side of the reaction surface, specifically at r ;5! R and 3 = x, satisfy the boundary condition
-FL. Xc++-,7c)= - pyr, n>,
dp
appropriate for no reflection. In this region, neglecting the Coulomb interaction, the wave function is well approximated by x’“‘(k, r) = 4 c (- l)I(ZI f l)[@)(p) -l-& ~~l~~P)~~~(COS zJ)* Eqs. (23~(26)
(26)
may be combined to give LmHX c (I+ l)d,pjlyP)+ Lli,
4:)&l]
= 0,
(27)
where A, = (-i)‘(Si++$
(28)
which, upon combining eqs. (23) and (24), satisfy the sum rule Lnar
C ?A, = 1.
(2%
I‘min
TheMax-Ami,+ 1 unknowns are obtained by requiring eq. (27) to hold to order ~ = L,,, - Lmin- 1 in the Taylor expansion of ~~i)(p~ about p = K-R = 8. The resulting Mf 1 equations plus eq. (29) are then solved for the M+2 unknowns, AI, and the S-matrix elements, S,, are obtained from eq. (28). A model this simple which neglects all the subtilities of analyticity cannot be expected to give a really clean manifestation of a Regge pole. The model includes strong absorption by way of a physically reasonable boundary condition. It does not require additional assumptions concerning the interaction and therefore the following results may be rather general. We were led to the study of the model by Austern’s calculation “) of the wave function in the shadow shown in the lower half of fig. 3. The regular oscillation apparent at small angles is the signature of the
00
0
/ 30”
I
I
I
60”
90”
120*
i
I
I
I
I
I
I 60”
I
I
i
90”
120”
30”
I
1
1500
1E
I
I
Eon 9
Fig. 3. The angular dependence of the wave function at p = 25, showing the resonance ~ntr~bntjon in the shadow. For the upper plot Lmax= 22, Li, = 17, 8 = 19; for the lower pIot Lax = 12, Lmin= 7, p = 10. *.
‘I-
l
.
i-
El
t
018
_;_
2. --ReS
------c 1.22 1
-
l***
iA
I
ImS
l9
Fig. 4. Argand plots and reelection coefficients for parameters corresponding to fig. 3. Note the slight violation of unit+rity and the absence of a kink in pIot of reflection coef’ficients,
374
R. C. FULLER
00
30”
60”
AND Y. AVISHAI
90”
120”
150”
I . 180”
Fig. 5. The scattering angular distributions do/d@ for the same parameters as used in figs. 3 and 4. Note the smooth fall toward slight oscillation at back angles.
presence of a Regge pole since it implies that a component of the wave function has traveled into the shadow “). The regularity of the oscillation indicates that this component is an eigenstate of the angular momentum, generally not a physical integer value, however. The upper portion of figs. 3-5 show calculations for L max = 22, Lmin= 17 and p = 19 while for the lower part Austern’s parameters, L max= 12, Lmin= 7 and B = 10 were used. The wave function is normalized to an incident plane-wave of unit magnitude eikZ.In fig. 3 it is apparent that the magnitude of the wave approaches that of the plane wave as one moves around the limb into the illuminated region. Austern shows, and we find, that the phase of the full wave
ANGULAR
DEPENDENCE
OF RESONANCE
STATE
375
a 3 -1 K. * r: 3
-2
2
-3
30”
60”
90”
Fig. 6. A plot of the 6 and a-modes for I&,, = 22, Lmi,
=
17,p = 19.
Fig. 7. The upper plot shows an Argand plot of the wave function in the deep shadow for 22, L,i, = 17, /I = 19 and p = 25. The plot has been rotated as indicated to compare with the lower plot of the Legendre function for cc = 19.5~+O.Si. The lower plot has been normalized to the upper plot at 11”.
L Inax=
fun~t~oll also approximates that of the plane wave in this region. This behaviour is characteristic of the wave function for absorbing potentials as suggested by McCarthy and Pursey “). This can be seen in the calculations of Amos “). Fig. 4 shows the reflection coefficients and Argand plots of the S-matrix. We do not find the %ink” in the reflection coefficients found by Austern. Obviously the model slightly violates unitarity. Fig. 5 shows plots of the two scattering angular distributions. One does not observe the expected exponential fall-off although, as expected, the angular distribution falls smoothly toward slight oscillation at extreme back angles. Eqs. (17) and (I 5) can be used to write the partial wave expansion of the wave function, eq. (25), in the region external to the interaction as a sum of the 5- and 6-
376
R. C. FULLER
AND Y. AVISHAI
modes shown in fig. 1 and discussed following eq. (11). The magnitudes of the two components, multiplied by (sin 9)“, are shown in fig. 6. As discussed in the previous section the b-mode is the portion of the surface wave which has traveled through the lesser angle to arrive at a n.on-zero angle 9. One observes its exponential fall-off with decreasing angle as expected on the basis of eq. (16). The z-mode, already having traveled to 9 = 0 starts with a magnitude equal to that of the o-mode at 9 = 0 and then decreases as one moves out of the deep shadow. Obviously the two modes interfere less as the angle increases. This is apparent in fig. 1. The upper part of fig. 7 shows an Argand plot of the full wave function. This plot has been rotated in order to more easily compare with the angular dependence of P,(cos$), GI= 19.5 +0.8i which has been normalized to agree in magnitude with the wave function plot at 11” and is shown in the lower half of the figure. The similarity of the two plots is apparent and indicates that the angular dependence of the wave function in the shadow is well approximated by _&(r)P,(cos 8) where J;(r) is generally complex which accounts for the rotation. We are grateful to H. A. Weidenm~ller for helpful discussions and one of the authors (Y.A.) would like to thank him for his hospitality during his stay at the Max Planck Institut fur Kerphysik. We thank H. M. Hofmann for his help with computational problems. References 1) IL W. McVoy, Phys. Rev. C3 (1971) 1104 2) 3) 4) 5) 6) 7) 8)
R. C. Fuller, Nucl. Phys. A216 (1973) 199 N. Austern, Ann. of Phys. 15 (1961) 299 N. M. Nussenzveig, Ann. of Phys. 34 (1965) 23 V. de Alfaro and T. Regge, Potential scattering (North-Holland, Amsterdam, 1965) R. A. Broglia, S. Landowne and A. Winther, Phys. Lett. 4OB (1972) 293 D. A. Goldberg and S. M. Smith, Phys. Rev. Lett. 29 (1972) 500 I. E. McCarthy and P. L. Pursey, Proc. Int. Conf. on nuclear structure, Kingston, Canada, 1960, ed. D. A. Bromley and E. Vogt (Univ. of Toronto Press, Toronto, 1960) 9) K. A. Amos, Nucl. Phys. 77 (1966) 225
THEl ANGULAR
@ North-HoliandPu62i~~
Co., Amsterdam
or microfilm without written permission from the publisher
~EPEN~ENC~
OF A BROAD
RESONANCE
PE~~~L
STATE
R. C. FULLER t Institut fiir Theoretische Phisik der Universitiit Heidelberg, Germany Y. AVISHAI tt ~a~-Planck-Instit~~t fiir ker~pbysik, Heidelberg, Germnay Received
17 December
1973
Abstract: We discuss the angular dependence oftbe shape-elastic resonance wave function for a state so broad decay occurs before the ions orbit the interaction region. In the shadow of the interaction region the non-stationa~ angular dependence is described by a Legendre function for complex angular momentum, P, (cos@. In the lit region the same state has the angrdar dependence e-in=P,(--cos @). After presenting theoretical arguments which show the necessity of using these different analytic continuations in the shadow and lit regions to states of complex angular momentum we show, by way of illustration of the theoretical argument, results for a strong absorption model introduced several years ago by Austern.
1.lntrodu~tion It has been found that the optical model potentials which fit heavy-ion elastic scattering above the Coulomb barrier must be sufficiently absorptive to damp shape-elastic resonances. On the other hand it has been suggested that a Regge-pole parameterization of the nuclear S-matrix can be used to fit elastic scattering data [see McVoy “) and references contained therein]. The Regge pole describes a peripheral shape-elastic resonance; however, as discussed in a previous paper “) by one of the present authors, the resonance lifetime is so short that the ions do not rotate about one another; i.e. the resonance lifetime is commensurate with the direct reaction time scale. This implies that the decay of such a state is fore-aft asymmetric in angle. Such an angular distribution cannot be described by a single Legendre polynomial. As shown in ref. “) the wave packet in angular momentum space necessary to describe such a state is provided by a Regge pole whose imaginary part gives the rate of decay in angle of the state. Strong volume absorption gives rise to a shadow on the downstream side of the reaction region. The full wave function in this region of conjuration space is dominated by peripheral resonances which live su~cie~tly long to propagate into the shadow. As one moves out of the shadow it is necessary to subtract from
t Supported by a grant from the Gesellschaft ftir Schwerionenforschung, ti Supported by a Minerva Fellowship. Present address: University of the Negev, Beer Sheba, Israel. 365
Darmstadt,
Germany.
366
R. C. FULLER
AND
Y. AVISHAI
the full wave function the contribution which has not traversed the shadow in order to obtain the part of the wave function dominated by resonant contributions. This leads to the characteristic signature of a Regge pole at back angles in the scattering distribution although this same signature appears in the full wave function just downstream from the interaction region at forward angles. In the next section we give a general discussion of the geometrical interpretation of the broad peripheral resonance. In the last section we discuss a semiclassical model which Austern “) introduced several years ago and which he describes as “perhaps as consistently ‘black’ as any model can be”. This model generates a broad peripheral resonance and the model is sufficiently simple to allow its detailed study. In order to retain this simplicity we neglect the Coulomb interaction. 2. Geometrical
interpretation
of a broad peripheral resonance
Since we intend to investigate broad peripheral states using a schematic model, the following analytic discussion will stress physical interpretation with little pretense of mathematical rigor. Our discussion of the wave function near the surface will follow previous work “) of one of the present authors while the discussion of the relation of the surface wave in the shadow to the asymptotic angular distribution follows Nussenzveig “). The wave function for the scattering of two spinless particles has the partial-wave expansion,
Y(k, r) =
k C eit'(21
+ l)$j+)(k, r)~,(cos $),
(1)
where $i+‘(k, r) may conveniently be decomposed into incoming and outgoing contributions as $I”
= $iei4”r(f,f-)(k,
r)-e-‘“‘,Sl(k)f,(+)(k,
r)>.
(2)
The incoming and outgoing waves,fiF)(k, r) satisfy the radial Schriidinger equation and the angular momentum independent boundary condition f/‘)(k,
at large r. To write the partial wave transformation, it is necessary integer values of the angular convenient for obtaining the eq. (2) the SW transformation
WC r>=
r) w ecikr,
(3)
sum of eq. (1) using a Sommerfeld-Watson (SW) to choose an interpolation of the summand between momentum. We choose an interpolation which is resonance contribution to the wave function. Using for the wave function may be written as
k/c-$$ (f’-‘(A
k,
r)-e-‘“(“-%(A,
k)f”‘(A,
k, r)}P,_,(cos
S),
ANGULAR
DEPENDENCE
OF RESONANCE
STATE
367
where c is a contour which encircles the real axis in the counter-clockwise direction. The partial wave series is recovered as the residue series arising from the zeros of cos nl. Since f”)(A, k, r) satisfy the radial Schrddinger equation which depends on the angular momentum through an entire function, Z(Z+ l), and further satisfy A (or 1) independent boundary conditions, they are entire functions of 1, at fixed k and r [ref. “)I. Therefore at fixed k and I the only poles in 2 in the radial wavefunction arise from the S-matrix S(A, k). We now assume the integration contour is distorted to pick up at least a subset of these pole contributions to give the residue series
Pae+za-f(+)(&, k, r)P,l_*(cos 9), where p, is the S-matrix residue lim,,,= (A-,$$(A). If this approach is to be useful, the first few terms of this series must approximate the wave function in the shadow. It is essentially for this reason that we chose the continuation of eq. (4) which led to the residue series involving P,(cos 9) which is singular at 9 = rc and not at 9 = 0. It is not our purpose to discuss the general problem of convergence which involves discussion of the residua /3,. We shall only discuss those aspects of the problemwhich do not depend on the details of the interaction. For this reason our considerations apply to the region of configuration space where the interaction may be assumed weak, i.e. near and beyond the interaction surface. Heavy ion scattering above the Coulomb barrier is characterized by strong absorption which implies that the leading Regge pole contribution to eq. (5) corresponds to a peripheral configuration. The angular momenta associated with such a configuration are generally sufficiently large to justify the use of asymptotic expressions for the functions appearing in eq. (5). For values of the radius corresponding to configurations with the ions sufficiently separated to ignore their interaction, we have
where h,‘l’(k, r) is the outgoing spherical Hankel function. Following Nussenzveig [ref. “)] we assume in addition that Yis sufficiently large to satisfy kr-cc,
>> LX?,
where a, = Rei, -3 and 1, is the leading Regge pole. We can then use the asymptotic approximation for the Hankel function Ii:’
w P-+($---T)+eexp
(i [(p’---a:)+--~,
cos-r ‘3 P
-tn]),
p = kr.
(7)
368
R. C. FULLER
AND
Y. AVISHAI
Substituting eq. (6) into eq. (5) gives for a single Regge-pole contribution
Rl(r)=
n eFiSa
c-
e -i3nalh!~)t(p)P1,_~(cos 9). 1
Unitarity restricts 1, to the upper-half &plane “) where we may write
For large angular momenta the asymptotic expression for the Legendre function is P,_*(cos 9)
M
+
(-&-Jcosp],
p>
Substituting eqs. (lo), (9), and (7) into eq. (8) gives for a single Regge-pole contribution
R,(k,r) = e-i*(2zal)3 8 ats,i-n/4l+,ilnta,-n/41) x x_e 1)mievc -
)
(11)
where 6, = 9,-9+2nln, 0, = 9,+9+2nm, 9, = gn-cos-l
5 = sin-.% P
P
(12)
This approach, which follows Nussenzveig 4), leads to a geometrical interpretation of the Regge-pole contribution. If we define the radius a by ka = [xl,
03)
then for the leading Regge pole a can be expected to approximate the radius of the interaction surface since lower partial waves are absorbed, i.e. weakly coupled to the elastic channel. Then $, =t sin-l 5 P
(141
is the cylindrical boundary, 0 < v < v0 6 $n, of the geometrical shadow of a sphere of radius a. The contribution from the leading Regge pole to the wave function in the geometrical shadow described by eq. (11) is shown schematically in fig. 1. For the angular momentum a,, the particles are incident on a trajectory corresponding to the impact parameter a. At TX and T, the system forms a resonance which is attenuated both by decay into the elastic channel and by absorption.
ANGULAR
DEPENDENCE
OF RESONANCE T*
Fig. 1. A schematic
representation
7/2-
(cps-1
369
STATE
alr+&=%-9rb.
of the contribution of a broad peripheral shadow region given by eq. (11).
resonance
in the
At the point P(r, 31, within the geometrical shadow, one has the two contributions emanating tangentially from T; and Ti which for v # 0 have unequal amplitudes because they have propagated through different angles. Contributions from the m # 0 terms correspond to encircling the sphere m times before decay and therefore are generally strongly attenuated unless a2 is quite small. Obviously the rate of convergence of the series in eq. (9) is determined by cc2. In this connection we refer the reader to the discussion in ref. “> on the relationship between a2 and the “life-angle” of the resonance. As one moves to larger angles the m = 0 contribution from the &mode increases in amplitude. As one passes through the boundary of the geometrical shadow the magnitude increases exponentially in an,gle. Higher order Regge poles, corresponding to larger a*, increase even faster form f 0, leading to slow convergence or even divergence of the Regge-pole series. This is to be expected since outside the shadow the wave function must include at least the incident plane wave in addition to the resonance contribution. Obviously slow convergence or divergence can only arise from the m # 0 contributions which we now proceed to isolate in order to move into the lit region. Following Nussenzveig “) we introduce the functions Q!” (cos 3) and Q$” (cos 3) defined by Q$l’ ‘)(cos 3) = ; [P&OS 3)t
2 Qv(cos 3)] , n
,
(15)
where P, and Q, are Legendre functions of the first and second kind respectively. In the same angular range where the asymptotic expression for P, (cos9) given by eq. (10) is valid, we have Q$~~)(cos 3) M (2xR sin 3)-*eri(h’9-&n)_ (16) From eq. (15) we have P,(cos 3) = Q$“(cos 3) + Q!~)(cos 3).
(17)
R. C. FULLER
370
AND
Y. AVISNAf
From eqs. (17), {16), (1.1) and (8) it follows that the only divergent terms in the Regge-pole series arise from the IM= 0 co~~rib~~io~s to Qcf (cos 9). Referring to eqs. (8) and (9) we are therefore led to write e -i&n&
J
P,,_*(cos
8)
=
2
ei*‘%J6$+(cos S)+d,_Jcos
$),
cos na,
(181
where the term which gives rise to divergence has been explicitly isolated and
d,*_&os 8) = 2&‘-
=
2&(-
l)nreix~,(2m+s)Q!~,~(cos 9)
f)meixl,f2tnf~))(Q~~~~$(COS 9) -e-2z~2*Q~L,(cos9)).
However from Nussenzveig “) we have - i~?‘V’++(-cos
9) = Q&(cos
@--e2”““*rQ~‘,‘-;(cos3),
and therefore eq. (18) may be written
Upon deleting the troublesome in eq. (51,
first term in eq, (21) we have for the residue series
B,f’“‘(x,,
k +&;(-cos
91,
(221
which is valid in the lit region. Since this equation results from deleting the 6e term from eq. (X.1>,the contribution of a single Regge pole in the lit region can be represented by the obvious extension of fig. 1 shown in fig. 2. In moving from the shadow to the lit region, we have chosen to follow an argument which at each step stressed the physical interpretation of the variables, Let us now show what the ar~ment has accomplished. Eq. (21) may be rewritten as P,(cos 23)= e?Pa( - cos 23)f 2i cos z(ct + ~~e~~~Q~~~(~o~ 3).
(2W
This expression provides an analytic continuation of the relation P,(cos 3) = (- 1)” P,( - cos 9) which holds at integer values of the angular momentum. The necessity of such a continuation can be understood by recalling P,(cos 9) is singular at 9 = n for non-integer angular momentum while I>a(-cos 9) is singular at 9 =: 0. In the shadow, which includes the angle 9 = 0, the Regge-pole contribution given by eq. (5) has the angular dependence Pz(cos 9). Upon moving into the lit region which includes the angle 9 = z, the angular dependence of the Regge-pole contribution changes to e’“”P,(-cos 9) which is non-singular in the lit region. As stressed by
ANGULAR
DEPENDENCE
OF RESONANCE
STATE
371
Fig. 2. A schematic representation of the contribution of a broad peripheral resonance in the lit region showing the contribution which arises from components of the wave which have traversed the shadow.
Nussensveig “) the use of different continuations of the Legendre function in different angular regions does not imply that the Regge-pole contribution is discontinuous. Obviously one must move into the lit region to determine the scattering angular distributioll. We have just shown that the contribution of a single Regge pole to the scattered wave in the lit region has the angular dependence I’,( -cos 9). This contribution may be continued to large Ywhere it leads to the same angular dependence in the scattering amplitude. As discussed in ref. 2), states with sufficiently large CI~, decay before rotating about the interaction region and give rise to angular distributions which decrease expol~entially toward oscillation at back angles. We suggest that such states be called “Ay-off resonances”. The strong valume absorption which characterizes heavy ion applications leads to the dominance of “peripheral fly-off resonances” (PFOR) at back angles. This interpretation of the exponential decrease of the angular distribution is to be contrasted with the eikonal explanation offered by Broglia et al. “). Referring to fig. 2 it is apparent that the Regge-pole contributions to the scattering amplitude arise from components of the wave function which have traversed the semi-classically forbidden shadow region. In this connection it is interesting to recall Goldberg and Smith’s observation ‘) that by fitting data to angles slightly beyond the maximal semi-classical deflection angle into the angular region where the angular distribution starts its exponential decrease, they removed the depth ambiguity in the optical model fit. On the basis of the previous discussion we are led to conjecture that this ambiguity is removed by fitting into the angular region where the angular distribution is largely determined by the imaginary part of the leading Regge pole which is a sufficiently sensitive function of the potential depth to remove the ambiguity.
372
R. C. FULLER
AND
Y. AVISEIAI
3. Asstem model Several years ago Austern “> suggested a model which incorporates strong absorption for the full three-dimensional wave function. The lower partial waves are assumed to be completely absorbed in the usual sense that each partial wave has no outgoing component; i.e., the reflection coefficients vanish,
s, =
0,
l6
(23)
L*in*
The high partial waves are unscattered,
S-matrix elements for peripheral partial waves are determined by requiring that the three-dimensional wave function on the illuminated side of the reaction surface, specifically at r ;5! R and 3 = x, satisfy the boundary condition
-FL. Xc++-,7c)= - pyr, n>,
dp
appropriate for no reflection. In this region, neglecting the Coulomb interaction, the wave function is well approximated by x’“‘(k, r) = 4 c (- l)I(ZI f l)[@)(p) -l-& ~~l~~P)~~~(COS zJ)* Eqs. (23~(26)
(26)
may be combined to give LmHX c (I+ l)d,pjlyP)+ Lli,
4:)&l]
= 0,
(27)
where A, = (-i)‘(Si++$
(28)
which, upon combining eqs. (23) and (24), satisfy the sum rule Lnar
C ?A, = 1.
(2%
I‘min
TheMax-Ami,+ 1 unknowns are obtained by requiring eq. (27) to hold to order ~ = L,,, - Lmin- 1 in the Taylor expansion of ~~i)(p~ about p = K-R = 8. The resulting Mf 1 equations plus eq. (29) are then solved for the M+2 unknowns, AI, and the S-matrix elements, S,, are obtained from eq. (28). A model this simple which neglects all the subtilities of analyticity cannot be expected to give a really clean manifestation of a Regge pole. The model includes strong absorption by way of a physically reasonable boundary condition. It does not require additional assumptions concerning the interaction and therefore the following results may be rather general. We were led to the study of the model by Austern’s calculation “) of the wave function in the shadow shown in the lower half of fig. 3. The regular oscillation apparent at small angles is the signature of the
00
0
/ 30”
I
I
I
60”
90”
120*
i
I
I
I
I
I
I 60”
I
I
i
90”
120”
30”
I
1
1500
1E
I
I
Eon 9
Fig. 3. The angular dependence of the wave function at p = 25, showing the resonance ~ntr~bntjon in the shadow. For the upper plot Lmax= 22, Li, = 17, 8 = 19; for the lower pIot Lax = 12, Lmin= 7, p = 10. *.
‘I-
l
.
i-
El
t
018
_;_
2. --ReS
------c 1.22 1
-
l***
iA
I
ImS
l9
Fig. 4. Argand plots and reelection coefficients for parameters corresponding to fig. 3. Note the slight violation of unit+rity and the absence of a kink in pIot of reflection coef’ficients,
374
R. C. FULLER
00
30”
60”
AND Y. AVISHAI
90”
120”
150”
I . 180”
Fig. 5. The scattering angular distributions do/d@ for the same parameters as used in figs. 3 and 4. Note the smooth fall toward slight oscillation at back angles.
presence of a Regge pole since it implies that a component of the wave function has traveled into the shadow “). The regularity of the oscillation indicates that this component is an eigenstate of the angular momentum, generally not a physical integer value, however. The upper portion of figs. 3-5 show calculations for L max = 22, Lmin= 17 and p = 19 while for the lower part Austern’s parameters, L max= 12, Lmin= 7 and B = 10 were used. The wave function is normalized to an incident plane-wave of unit magnitude eikZ.In fig. 3 it is apparent that the magnitude of the wave approaches that of the plane wave as one moves around the limb into the illuminated region. Austern shows, and we find, that the phase of the full wave
ANGULAR
DEPENDENCE
OF RESONANCE
STATE
375
a 3 -1 K. * r: 3
-2
2
-3
30”
60”
90”
Fig. 6. A plot of the 6 and a-modes for I&,, = 22, Lmi,
=
17,p = 19.
Fig. 7. The upper plot shows an Argand plot of the wave function in the deep shadow for 22, L,i, = 17, /I = 19 and p = 25. The plot has been rotated as indicated to compare with the lower plot of the Legendre function for cc = 19.5~+O.Si. The lower plot has been normalized to the upper plot at 11”.
L Inax=
fun~t~oll also approximates that of the plane wave in this region. This behaviour is characteristic of the wave function for absorbing potentials as suggested by McCarthy and Pursey “). This can be seen in the calculations of Amos “). Fig. 4 shows the reflection coefficients and Argand plots of the S-matrix. We do not find the %ink” in the reflection coefficients found by Austern. Obviously the model slightly violates unitarity. Fig. 5 shows plots of the two scattering angular distributions. One does not observe the expected exponential fall-off although, as expected, the angular distribution falls smoothly toward slight oscillation at extreme back angles. Eqs. (17) and (I 5) can be used to write the partial wave expansion of the wave function, eq. (25), in the region external to the interaction as a sum of the 5- and 6-
376
R. C. FULLER
AND Y. AVISHAI
modes shown in fig. 1 and discussed following eq. (11). The magnitudes of the two components, multiplied by (sin 9)“, are shown in fig. 6. As discussed in the previous section the b-mode is the portion of the surface wave which has traveled through the lesser angle to arrive at a n.on-zero angle 9. One observes its exponential fall-off with decreasing angle as expected on the basis of eq. (16). The z-mode, already having traveled to 9 = 0 starts with a magnitude equal to that of the o-mode at 9 = 0 and then decreases as one moves out of the deep shadow. Obviously the two modes interfere less as the angle increases. This is apparent in fig. 1. The upper part of fig. 7 shows an Argand plot of the full wave function. This plot has been rotated in order to more easily compare with the angular dependence of P,(cos$), GI= 19.5 +0.8i which has been normalized to agree in magnitude with the wave function plot at 11” and is shown in the lower half of the figure. The similarity of the two plots is apparent and indicates that the angular dependence of the wave function in the shadow is well approximated by _&(r)P,(cos 8) where J;(r) is generally complex which accounts for the rotation. We are grateful to H. A. Weidenm~ller for helpful discussions and one of the authors (Y.A.) would like to thank him for his hospitality during his stay at the Max Planck Institut fur Kerphysik. We thank H. M. Hofmann for his help with computational problems. References 1) IL W. McVoy, Phys. Rev. C3 (1971) 1104 2) 3) 4) 5) 6) 7) 8)
R. C. Fuller, Nucl. Phys. A216 (1973) 199 N. Austern, Ann. of Phys. 15 (1961) 299 N. M. Nussenzveig, Ann. of Phys. 34 (1965) 23 V. de Alfaro and T. Regge, Potential scattering (North-Holland, Amsterdam, 1965) R. A. Broglia, S. Landowne and A. Winther, Phys. Lett. 4OB (1972) 293 D. A. Goldberg and S. M. Smith, Phys. Rev. Lett. 29 (1972) 500 I. E. McCarthy and P. L. Pursey, Proc. Int. Conf. on nuclear structure, Kingston, Canada, 1960, ed. D. A. Bromley and E. Vogt (Univ. of Toronto Press, Toronto, 1960) 9) K. A. Amos, Nucl. Phys. 77 (1966) 225