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Strength functions for fragmented doorway states
ANNALS
OF PHYSICS
125, 253-275
Strength
(1980)
Functions
for Fragmented
WILLIAM Department qf Physics and Astronomy,
Doorway
States*
M. MACDONALD
University of Maryland,
CoIlege Park, Maryland
20742
Received June 13, 1979
Coupling a strongly excited “doorway state” to weak “hallway states” distributes its strength into micro-resonances seen in differential cross sections taken with very good energy resolution. The distribution of strength is shown to be revealed by reduced widths of the K-matrix rather than by the imaginary part of poles of the S-matrix. Different strength functions (SF) constructed by averaging the K-matrix widths are then investigated to determine their dependences on energy and on parameters related to averages of microscopic matrix elements. A new sum rule on the integrated strength of these SF is derived and used to show that different averaging procedures actually distribute the strength differently. Finally, it is shown that the discontinuous summed strength defines spreading parameters for the doorway state only in strong coupling, where it approximates the indefinite integral of the continuous SF of MacDonald-Mekjian-Kerman-De Toledo Piza. A new method of “parametric continuation” is used to relate a discontinuous sliding box-average, or a finite sum, of discrete terms to a continuous function.
1. INTRODUCTION The concept of a doorway state, introduced by Feshbach and Block, has proved to be the key to understanding many features of nuclear reactions. As emphasized by Feshbach, Kerman, and Lemmer [I] the doorway state is an eigenstate of an approximate or model Hamiltonian so strongly coupled to certain reaction channels that its properties determine the essential feature of reactions proceeding through these channels. More complicated states “hallway states” [2] of this Hamiltonian may be weakly coupled to these channels but their participation in the reaction process is dominated by the strength which they acquire by coupling to the doorway state. Thus the strength of the doorway is spread over some energy range and the doorway state acquires a spreading width in addition to its decay width. In some cases, the density of hallway states is so large that their participation can be observed in cross sections only as a broadening of the doorway resonance. Other doorways, most notably the isobaric analog states (IAS), may be observed in high resolution experiments as a large number of distinct resonances whose widths increase near the energy of the original doorway state. The most beautiful examples of this latter case are to be found in the high resolution data of the Duke University experimental group as was first pointed out by D. Robson [3]. * Supported by the U. S. Department
of Energy.
253 0003-4916/80/040253-23$05.00/O Copyright Q 1980 by Academic Press, Inc. All rights of reproduction in any form reserved.
254
WILLIAM
M. MACDONALD
The central problem in the analysis of fine structure associated with a fragmented doorway state (FDS) is the extraction of its properties and the determination of average parameters which characterize its interaction with the hallway states as well as the coupling of the latter to the open channels. The spin and parity of the doorway state is, of course, the same as that of the fine structure, which can be determined from the angular distributions on resonance. Other information must be extracted from an analysis of the resonance pattern. When the fine structure is resolved, a resonance energy and one or more partial widths can be found for each of the resonances. Together with the background phase shift these resonance parameters summarize all the information available from the differential cross sections. The first determinations of the rms value of the charge dependent matrix element coupling an isobaric analog state (IAS) to hallway states of different isospin attempted to use the envelope of Wigner-Eisenbud reduced widths [4]. Fluctuations in the magnitude of the coupling matrix elements severely limited the accuracy of this approach. MacDonald and Mekjian [5] and, independently, Kerman and De Toledo Piza [6] showed that a Lorentz-weighted average of the reduced widths defines a “strength function” (SF) which has the characteristic shape of an asymmetric Lorentzian whenever a fragmented doorway state (FDS) is present.1 The shape of this SF is completely determined by only five parameters: (1) the energy of the doorway state, (2) its particle emission width, (3) a “spreading width” which measures the coupling of the doorway to the hallway states, (4) the average intrinsic particle emission width of the hallway states (before coupling to the doorway state), and (5) the average phase angle between matrix element coupling a hallway state to the doorway state and that coupling it to the continuum. M. Di Toro successfully used this SF to analyze the fine structure in several IAS to deduce these parameters [7]. Recently it has also been thoroughly tested [S] by a model study on a large number of FDS. Other SF can be constructed in various ways. First, different resonance expansions can be used to represent the differential cross sections and these parameters used to construct an SF. For example, the micro-resonances can be represented by a pole expansion of the S-matrix and the resulting real and imaginary parts of the poles used to construct an SF. Second, different averaging procedures can be used to construct different SF from the same set of resonance parameters [9]. The MMKP-SF was constructed using Lorentz-weighted averaging, but A. M. Lane favored an SF which is a sliding box-average of reduced widths [IO]. This latter is a common method for treating statistical data, but it yields a discontinuous histogram not well suited to analyzing the data on fragmented IAS. In a preliminary presentation of some of the results of this paper, I have given a “parametric continuation” of this box-averaged SF which provides a continuous transition from this highly discontinuous histogram to a smooth SF. The shape of this new SF also is determined by only five parameters when an FDS is present [l 11. Each of these SF has a different dependence on energy. Difficulties in using the box-averaged SF led Lane, Lynn, and Moses (LLM) [12] to suggest use of a “summed strength function” (SSF). This SSF was used by Bilpuch, 1 This will be referred to as the MMKP-SF.
255
STRENGTH FUNCTIONS
Lane, Mitchell, and Moses (BLMM) in their analysis of the high resolution data on IAS [13]. II: suffers from the same deficiency of the box-averaged SF of being a discontinuous function whose shape is poorly defined in weak coupling. But the same procedure I used on the box-averaged SF can be used to construct a’parametric continuation of the SSF. This new SSF can be used to analyze an FDS even in weak coupling. Tn fact, it will be found to be identical to the indefinite integral of the MMKP-SF. This paper presents a complete discussion of the above-mentioned SF and SSF. Section 2 begins with a brief review of different resonanceexpansions of the S-matrix and a discussion of the relations between their respective resonance energies and widths. The picket-fence model is used to obtain equations for the widths of a reactance matrix and for the imaginary part of the S-matrix poles in order to make explicit the.ir differences. Section 3 presents a complete discussion of these SF which can be constructed from the same set of resonance parameters for the reactance matrix. In Sect. 4a sum rule is derived for each SF in order to establish its normalization. Section 5 is devoted to the subject of SSF and their inter-relations. Section 6 summarizes the findings of this investigation of strength functions and sum rules for the analysis:of FDS.
2. RESONANCE PARAMETERS FOR FINE STRUCTURE 2.1. Comparison of Different ResonanceExpansions In principle, the fine structure in high resolution measurementsof differential cross sections can be used to determine poles and residuesof the S-matrix in the expansion
(1) Unfortunately the quantities (E,,s, rAs, g,J f or d’ff 1 erent levels are not independent becausethey must satisfy complicated unitarity constraint equations [14]. For this reason crosssectionscontaining numerous resonancesare more conveniently analyzed by using a reactance matrix K in place of the S-matrix. S,,,(E) = ei(‘c+‘c’)[(l + iK)(l - iK)-l],,,
.
(2)
The K-matrix [2, 5, 61 can also be expanded to display resonance energies I& and partial widths fi,l/’ (3) The resonance parameters (.8$, F;,‘,/3 of the K-matrix are both real and independent; unitarity is guaranteed by the form of Eq. (2). The Wigner-Eisenbud (WE) R-matrix theory [15] is a reactance matrix formalism based on the division of the 3A-dimensional configuration space into an internal or
256
WILLIAM
“interaction” the s-matrix
M.
MACDONALD
region and an external or “asymptotic” in the form SC,, = eidcPE”[(l - RL,)-l
region. This formulation
(1 - RL,*)],,, ei*c’P~l’z
gives (4)
where (L,), = S, - B, + iP, contains a slowly energy dependent level shift S, and a penetrability factor P, . The &matrix is given by R,,, = - c -?!d!& a E-6
with energy independent partial reduced widths Y,,~and resonance energies E,, . A very similar expression for the S-matrix emerges from both the K-matrix formulation of the shell model approach to reaction theory (SMART) [5,16] and the more general projection operator formalism of Feshbach [ 171. These approaches divide the Hilbert space into (1) a portion spanned by quasi-bound or closed-channel components and (2) a portion spanned by scattering states or open-channel components. In SMART the K-matrix has a direct interaction term as shown in Eq. (3). K:,
= 4xcW
I f’e I x&D.
(f-5)
The resonance term is the second term in Eq. (3) with p;:” = (2741’2 Wa I ve I xc(E)>
(7)
4 =.
(8)
and It should be noted that both in Eq. (5) and Eq. (6) the xc(E) are distorted waves of a real optical potential. This is a consequence of the fact that these are matrix elements of
a reactance matrix and not of the S-matrix. In the literature on IAS there has been considerable confusion on this point because of a general failure to appreciate the differences between the K-matrix and the s-matrix. The partial reduced width p,1,/” has an energy dependence arising from the “penetrability” of xc(E) through a real potential barrier. As suggested by Weidenmuller and Mahaux [ 181 this can be removed as an energy dependent factor. l-t’,‘” = (2fe(E))1’2 j& .
This gives an equation for the S-matrix similar to that displayed SC,, = e”““ft’“[(l Although
+ iRDf + iRRf)--l (1 - iRDf-
in Eq. (4).
&Rf)]ec, f~1'zei60'.
(10)
the resonant part of the “reduced” K-matrix
(11)
STRENGTH
FUNCTIONS
257
has the form of the WE R-matrix (except for the sign), the sum is only over aJinite number of internal states ul, instead of the infinite number of states of Eq. (5). As shown by C. Bloch [19], the sum over “distant levels” of the WE R-matrix provides the direct :interaction term which is exhibited explicitly in Eq. (10) as RD. These “distant levels” also correct the physically unreasonable hard-sphere shifts CJ$= 2ka, and square-well penetrability factors P, which appear in Eq. (4). Indeed, unless these “distant level” terms of the WE R-matrix are properly included by modifying & and P, of Eq. (3) to the potential phase shifts 6, and penetrability for a diffuse well, the reduced widths and resonance energies obtained for Eq. (4) from an R-matrix analysis of differential cross sections will not be correct [20]. In the analysis of FDS an incorrect distribution of reduced widths with energy will result from a failure to include these corrections. It is essential to note that the “R-matrix” used by A. M. Lane and D. Robson [21] to discuss line-broadening of IAS is not related to the R-matrix of Wigner-Eisenbud but to the resonance expansion of the S-matrix introduced by Kapur-Peierls [22]. S,,, = ei”c[l + 2iP~‘2R,‘~?P,t] ei”“‘. The Kapur-Peierls
expansion
has the properties of the expansion in Eq. (l), viz. the energies &?A= E,“’ - i(I’fp/2) are complex and the resonance parameters for different levels are connected by unitarity constraints. The 8A are not really poles of the S-matrix, however, but eigenvalues of a complete set of internal states which satisfy a complex boundary condition. Although the Kapur-Peierls “R-matrix” is also based on an expansion of the solution of the SchrGdinger equation in an interaction volume in (ri ,..., rA) space, it is not a reactance matrix and does not share the properties of the R-matrix of Wigner-Eisenbud. The resonance parameters of this “R-matrix” can not properly be identified with the (EA , yiC) obtained by the Duke University group from their multilevel analyses of experimental differential cross sections using the WignerEisenbud R-matrix. In fact, the entire discussion of the qualitative behavior of fine structure widths for analog resonances presented by Lane in his review does not apply to these resonance parameters, as will be evident in the next section. 2.2. Resonance Energies and Widths of the K-Matrix for an FDS The effects of distant levels (R,) must be included in Eq. (4) to obtain partial reduced widths and resonance energies which correctly reflect the fragmentation of a doorway state. As noted above the inclusion of these terms leads to Eq. (10) with the resonant part of the reduced K-matrix of Eq. (11). The strength functions and sum rules of this paper are therefore based on the resonance parameters $ and +& of Eq. (11). Tn using the results to be presented one must hope that corrections due to
258
WILLIAM
M.
MACDONALD
R, have been included in existing analyses of the experimental data, or are small, so that quoted E,, and yACcan be identified with i?,, and j& . For this reason (and simplifi-
cation of notation) the tilde will be omitted from all K-matrix quantities in the remainder of this paper. Before fragmentation a doorway state at E, has reduced partial widths yDCand, in general, the hallway state at .Q has reduced partial width yic . If Mi denotes the coupling matrix element of the doorway to the hallways, the resonant part of the reduced K-matrix is given explicitly by
where
The poles and residues of RR give the resonance energies and poles of Eq. (II).
(17)
The qualitative features of the distribution of widths can best be seen in a simple model [5] called the picket-fence wherein only one channel is open and Mi = A4 = constant; ci = co i
yic = 0 n = 0, 1, 2,..., co.
nD;
(18)
For this case, Eq. (17) can be evaluated to give a distribution rtc = Y?Jc
(4
-
r”D/271 Ed2 + W’M2
determined by r’” = 2nM2/D. This quantity (or its average) is usually called the “spreading width,” but the width of the distribution of reduced widths is actually given by W = ((rL)2 + M2)lp. The envelope of theseyt” is Lorentzian for all values of yiC , FL, M, and D. No distinction exists between casesof strong coupling (r= > D) and casesof weak coupling (P
E, = (r”/2)cot[n(EA
- q,)/D];
(20)
they are also independent of yiC . In the picket-fence model W is approximately equal to the spreading width of Feshbach, Kerman, and Lemmer [1] for D/277 M < 1. However, for a physical
STRENGTH
259
FUNCTIONS
FDS the fluctuations in the magnitudes of Mi2, exhibited by the & for IAS [13], for example, prevent an accurate determination of W by fitting to Eq. (19) [4, 51. This fact led to the introduction of strength functions constructed by averaging the reduced widths [5, 61. 2.3. S-Matrix
Poles of an FDS
Although not determined directly by the high resolution experiments on fragmented doorways, the S-matrix poles can be found from Eq. (10) by using the results just found for the K-matrix. The complex poles 6, = Evs - i(rVs/2) satisfy the determinantal equation Det(l + iK) = 0. (21) For the case of more than one channel this equation involves both diagonal and nondiagonal elements of the K-matrix. Therefore partial reduced widths for both elastic and inelastic channels participate in determining the S-matrix poles. If there is only one open channel, Eq. (21) simplifies somewhat. The explicit form of irir* given by Eq. (14) provides the equation for the poles of a fragmented doorway.
From this equation it can be shown that 6, = Evs - iTus/ with rys > 0. A comparison of Eqs. (16)-(17) with Eq. (22) reveals the gulf between the resonance energies and widths of the K-matrix and those of the S-matrix. For the picket-fence model, with now an additional assumption that KD -= 0, Eq. (22) becomes ’ + 8, - ED - (r&/2) cot[7r(b,
-
Q)/D~.
(23)
Two coupled equations for Evs and rys result from setting the real and imaginary parts of this equation equal to zero. No independent equation for the S-matrix resonance energies Evs, analogous to Eq. (16), can be disentangled. From Eq. (23) Lane derives [lo] the following equation for the widths.
dAs
tanh __ D
=
wDc (ED -
- m/2
EAs)” + [(r&y + (r,,
- r,y/4
(24)
From this equation it can easily be seen that even that shape of the distribution of S-matrix widths depends on the relative magnitudes of r,, , P, and D. In contrast to the distribution of K-matrix widths, the distribution of S-matrix widths is not even approximately Lorentzian except for special cases [lo]. Even in this case the width of the distribution of S-matrix widths is not FL, as for the K-matrix widths, but [(rl)2 + (rDC)2]1/2 [lo]. Unfortunately this crucial difference between the S-matrix parameters and the K-matrix parameters has not been generally realized and such
260
WILLIAM M. MACDONALD
concepts as “weak mixing due to rapid decay,” a valid notion for S-matrix poles, have been used in discussions of K-matrix or R-matrix widths (particularly from the Duke data), to which it does not apply.
3. STRENGTH FUNCTIONS FOR K-MATRIX
WIDTHS
Fluctuations in the magnitude of the matrix element coupling the doorway to hallway states compel us to smooth the distribution of widths. The resulting function of energy is called a strength function (SF). In this section it will be seen that different averaging procedures produce SF with different energy dependences. Each SF also depends on different spreading parameters. 3.1. MMKP
Strength Function
Averaging the widths with Lorentz-weighting produces a stength function with parameters that are themselves the Lorentz-weighted averages of the microscopic parameters [5, 61. First define the Lorentz-weighted average of a resonance parameter FA for the states Y,, .
(FAID,) = ; 7 (E _ 2)2 + 12 Then define the Lorentz-weighted hallway states.
average of the microscopic
parameter fi for the
= ; 1i (E - $2 + 12 The K-matrix strength function defined by Mekjian and MacDonald and De Toledo Piza uses the average given by Eq. (25).
and by Kerman
A very important relation for constructing the strength function of a fragmented doorway state follows directly from Eqs. (1 l), (27). S(E; I) = - + Im RP,(E + ir>
(28)
Substitution of the explicit form for RR given by Eq. (14) into Eq. (28) gives directly the parametric form of this SF. SD(E.
r) 7
=
s
I
3’k,
0 T
(~d4
~0s
24~ -
@
-
2~)
(E - ED), + (T,/2)2
sin
24
*
(29)
261
STRENGTH FUNCTIONS
The parameters in this equation characterize the spreading. Miyic qDcei* = YDC + 2 i E-cEi+iZ 8, = ED + A,
(W
A, 3 (n/Z)(Mt(E
- cJ/Di)
@lb)
r, f r,J + 21
(324
rIJ E 2~(MJDJ
Wb)
(33)
So =
Averages containing (E - ci) in Eqs. (30), (31) can be neglected in comparison with other terms because contributions from hallways above and below energy E tend to cancel. Neglect of these terms given 8, = E. and the simpler expressions tan 4 = -4MiyiclDi)lyDc h
=
yDc
set
(34) (35)
6
The Eq. (29) is suggestive but all the parameters are actually functions of E and I. The expectation is that the microscopic averages A, , rlL, and S, can be taken as constant for some reasonable choice of I. The SF constructed from experimental data by using Eq. (27) show that this expectation is well fulfilled for many IAS. A (relatively poor) example is the SF for the .Z* = l/2+ fragmented IAS observed in 40Ar (p, p) l/2+ shown in Fig. 1. Nevertheless calculations based on models are useful for establishing the magnitude of fluctuations expected in r,L and the proper choice of Z [S].
5
sFi!
01 2300
2350
2400
2450
2500
2550
2600
E, (keV)
FIG. 1. The Lorentz-weighted or MMKP strength function for a fragmented IAS in 4oAr(p,p) constructed from experimentally determined reduced widths by using Eq. (27) (dashed line) is compared with the doorway SF of Eq. (29) (solid line) obtained by a least-squares fit for Z = 35 keV to fix EA = 2458.5 keV, 4 = 0.18, y& = 7.5 keV, YS = 13.7 keV, and S, = 0.011. 595/125/2-3
262
WILLIAM
M.
MACDONALD
Some insight into this matter is also provided by the picket-fence molel. For this model r,qE)
1 - e-4nIlD 2TrA!P -= D 1 + e-4n1/D - 2e-2n11D cos 2(E - q,) n/D
(36)
e-2n11Dsin 2(E - q,) n/D 1 + e-4n11D - 2e-2rIlD cos 2(E - q,) n-/D
(37)
A, =E
D
For I/D it is an excellent approximation to take I’,$ = 2nM2/D and A, = 0. This model does not, however, include the effects of statistical fluctuations on the magnitude of the matrix elements and in the level spacings. These cause fluctuations in I’,& which are larger than those in Eqs. (36), (37). These fluctuations have been evaluated both numerically and analytically in a model study of 400 fragmented doorway states [8]. There it has been found that on average I’,& shows fractional fluctuations of order (D/nI)l12. Nevertheless, fitting Eq. (27) to Eq. (29) with constant parameters determines I’,i with an error of less than ten percent when I is chosen of order 2~(.Mi2)/(D,)
C+ I’,‘.
3.2. Box-Averaged
Strength Function
Statistical data is often “smoothed” by averaging with a sliding box [9]. This method has been favored by A. M. Lane in defining an SF [lo].
This SF is actually a discontinuous function of energy. In the case of very strong coupling (r,J > D) there will be many levels within the width of the maximum of the pattern, and one can hope to approximate S, by a continuous function with parameters related to microscopic averages like those in Eqs. (30)-(33). Lane has asserted that S’,(Ej I) has the same form as that given in Eq. (29). This can be shown not to be true even for the picket-fence model wherein all effects of fluctuating matrix elements and level spacings are absent (see Fig. 3). Use Eq. (19) for the widths to find S, for a fragmented doorway state. For very strong coupling (rJ > D) this distribution well approximates a continuous function in which E,, assumes all values. This will enable us to replace summation by integration in finding the SF defined by Eq. (38). As a check, the integral on E,, gives the correct sum rule value in this same limit.
Therefore in this strong coupling limit the continuous function which best approximates S, for an FDS is (40)
STRENGTH
263
FUNCTIONS
The result is S/(E;
I) cz &
(tan-’
The multiple branches of the arctangent result. Another form is SBD(E; I) fg 6
tan-l
(E - E#
ED + tan-’
E +;,;
function
;7W,2)z
2 i
n - tan-l
1.
call for care in transforming
(41) this
- I” (E -
z= &
ED-E+I w/2
ED)2 + (;j”
- I” =; 0
(42)
IW I2 - (W/2)2 - (E - ED)2 1 (E -
ED)2 + (;j’
- I3 < 0.
This continuous approximation to the box-averaged SF for the fragmented doorway must be compared with the MMKP-SF given by Eq. (29) in this limit (S, = C$= 0). (43) The box-averaged SF is more sharply peaked than the Lorentzian form conjectured by Lane. In the wings the box-averaged SF is approximately Lorentzian but the “width” is (W/2)2 - Z2. However, only if I< W/2 does Eq. (42) approximate a Lorentzian throughout its range. ‘All the known cases of fragmented IAS fall into the category of weak to intermediate coupling. The fractional discontinuities in the box-averaged SF are then large and can not be reduced below a certain level. To see this, consider the discontinuity at ITA introduced as the front of the box slides past this level. The S,(E; I) increases by &2Z. The sum rule x,, & = &, shows that the maximum value of S, prior to this jump is &,/2Z. The fractional jump in S, must therefore exceed (&&,). Over most of the pattern this must be multiplied by a factor equal to the reciprocal of the fraction of the summed width in the box at any energy. Thus S,(E; I) does not even approximately define a continuous function in the case of intermediate to weak coupling. Figure 2 shows plots of S,(E; I) for the IAS at E, = 2459 keV in 40Ar + p with J= = l/2+ for two values of the averaging interval. 3.3. Parametric A function imates S,(& parameter”’ l This function
Continuation
of S,(E; I)
continuous in E and Zcan be constructed which arbitrarilyclosely approxI) [l I]. This new SF is also a continuous function of a “smoothing , and it approaches S,(E; I) and becomes equal to it in the limit of E + 0. will be denoted by SPCB(E; I, l ) and called the “parametric continua-
40
2300
2350
2400
2400
2500
Ar(p, P) I/Z’
2550
x00
E, (keV) 0.125 40Ar(p,p)
l/2’
0.100 F
FIG. 2. (a) and (b): The box-averaged strength function for Eq. (38) is shown for Z = 35 keV
(Fig. 2a) and for Z = 70 keV (Fig. 2b). Note that the larger averaging interval does not reduce the fractional discontinuities.
0.500 0400 1 0.200
40
Artp,
P) l/2+
_
Ic_I 2300
2350
HO0
2450
2500
2550
2600
E, N.eW
FIG. 3. The box-averaged SF of Eq. (38) is compared with the asymmetric Lorentzian (29) for Z = 0 using the parameters of Fig. 1 for this IAS.
of Eq.
265
STRENGTH FUNCTIONS
tion” of S&Y; Z). Unlike S, , but like S(E; Z) of Eq. (27), this strength function can be used to analyze fragmented doorway states even in weak to intermediate coupling. To derive this SF begin by representing S, as a contour integral in the complex energy plane. The contour is illustrated in Fig. 4 as a rectangle with two sides parallel E-I+is
-,
E+l+ir
E+I-ic
E-1-k
Cl
FIG. 4. Contour for the box-averaged strength function S, of Eq. (44).
to the real axis and a distance E above and below it. The ends are perpendicular to the real axis and located at E - Z and E + I. Now use Eq. (11) to write the contour integral representation S,(E; I) = (47rZi)-l $ dz Izp,(z).
(44
The integral receives contributions from the sides, C, and C, , as well as from the ends, C, and C, . Use the contributions from C, and C, to define the parametric continuation of S, . The equation for S,, can be written 1
dE’ Im &fc(E’
+ k).
(45)
The pole expansion for 92 given by Eq. (11) then leads directly to an explicit result. - tan-l from C, and C, define a fluctuating
The contributions
The definitions
E
1
in Eqs. (45), (47) define a decomposition
- Z + iv))
of the box-averaged
S,(E; I) = S,,,(E; Z, 4 + S&T A l ) Recognition
(46)
function.
1 4 e) = 4nli I -~’ dy(Zt:c(E + Z + iy) - &f-G
ME;
E-Z-EA
(47) SF. (48)
that E i
Z - E., _ tan-l E
E - I - E,, E
(E - EAjz < I2 otherwise
(49)
266
WILLIAM
M.
MACDONALD
leads to
G-l Sp,(E; z, 6) = S,(E; Z). The same result follows
w-0
less directly from the limit Ii?
S,,(E; Z, c) = 0.
The decomposition has a unique feature which follows in the next section.
(51) from the sum rule on S derived /
m dE S,,(E; Z, E) = 0. s -m
(521
This equation justifies the notational designation of S,, as the fluctuating part of the box-averaged SF. It remains to find the form of S,,, for a fragmented doorway. This is done by using Eq. (14) to evaluate the integral in Eq. (45). A comparison of Eq. (45) with Eq. (28) gives the relation 1 SPCB(E;
4
4
=
z
s
ET1dE’ ELI
S(E); E).
(53)
Thus, SP,, is the average over the interval (E - Z, E i Z) of the MMKP-SF! Evaluation of S,,, for a fragmented doorway simply requires evaluation of this integral with SD@: c) derived in Eq. (29). The integral can only be evaluated explicitly, however, by assuming that the parametric quantities in Eq. (29) are independent of energy. SpDCR(E; Z, c) = S, + $$ (% cos 24 -
Y sin 2+)
(54)
The energy dependent quantities C, S are defined by the equations +? = -!- (tan-l 7r y _
E + Z - ED _ tan-l rS(4/2
E - Z - ED 1 J-d42
1 ,n (E + Z - ED)~ + (r~(4/2)~ 2~ (E - Z - ED)2 + (TS(E)/~)~ ’
(55)
The parameters in both Eqs. (54) and Eq. (55) are the same Lorentz-weighted averages defined in Eqs. (30)-(35), but with the smoothing parameter E replacing I. Thus Z’&E) = Z’,i + 2~ by replacing Z by E in Eq. (32a). As shown in Eq. (48), S,,, differs from S, by a fluctuating term of zero integral. Thus SD pcB gives the parametric form of S, for a fragmented doorway when the discontinuities are smoothed into a continuous function. This result can be checked against the form of S, derived for the picket-fence model in very strong coupling, as given in Eq. (41). In the picket-fence model I$ = 0 and W ‘v 2rrM2/D. As Eq. (36) shows, the choice Z 5 D gives r,L = 2rM2/D = W to very good approximation.
STRENGTH FUNCTIONS
267
of Eq. (54) has exactly the sameform as Eq. (41) and differs in the replaceThus SpDCB ment of W !Z r,i by rS(c) = r,J + 2~. This difference merely reflects the fact that the undefined smoothing operation used to derive Eq. (41) did not properly include the additional spreading of the strength due to smoothing. The Eq. (54) showsthat when S, is usedto analyze the fine structure of a fragmented doorway, the correct parametric form is not Eq. (29) as conjectured by Lane and collaborators. This result has also been demonstrated by direct application to the .I= = 3/2-- isobaric analog state in 40Ar(P, p)40Ar at E, = 1875 keV [II]. Finally, it should be noted that while SpcB provides a satisfactory replacement for the discontinuous box-averaged S, , there is no reason to prefer it to the MMKP strength function. Rather to the contrary, S,,-, is merely the average of the MMKP-SF, as shown by Eq. (53). This averaging is entirely unnecessary since the MMKP-SF is smooth.
4. SUM RULES FOR SF The SF discussedin the previous section differ from one another in their energy dependenaes.To compare them one must first determine how each is normalized. The sum rules provide this information. A sum rule will first be established for the general form of each strength function, and then the parametric form of each SF will be studied. There are some surprisesin the latter cases. All sum rules relate back to the sum rule on the K-matrix widths. This is easily extracted by noting that Eq. (11) is identically equal to Eq. (14) when the microresonancesarise from an FDS. Equating the coefficients of l/E in these two equations for KzCgives the desired sum rule. (56) In the following sections it will be shown that every SF integrates to this same sum rule value. 4.1. Sum Rule for MMKP-SF The sum rules on SF relate to the total area under the SF, i.e., to the infinite integral. The sum rule for the MMKF-SF requires evaluation of
Straightforward integration combined with Eq. (56) gives
s
m dE S(E; I> = 1 yfc + &, . -03
(58)
268
WILLIAM
M.
MACDONALD
The identity of Eq. (27) with Eq. (29) assures us that SD@; Z) satisfies the same sum rule when the energy dependence of allparameters is included. This is not true of the parametric form when the parameters are taken to be constant. A straightforward evaluation of the infinite integral of S”(E; Z) does not even converge. The indefinite integral is
4 I El
dE SD(E; Z) = (E, - El) S,, +
cos 24 - S sin 24)
(59)
where the constants C and S are C = +- (tan-l s
-
1
ln
277
(4 (ED
Ez
-
ED
_
tan-l
EI
Ts!2 -
-
rs/2
ED)~
+
(r.4)”
Q2
+
(r.s/2)’
ED
)
’
(60)
Examining the separate terms in Eq. (59) we see that the first term corresponds to C yi, of Eq. (56). No limit has been derived for this sum, so none can be applied to the first term of Eq. (59). Nevertheless, identification of this term implies that the sum rule value for the second term of Eq. (54) is &, . BLMM have evaluated this term by taking d = E, - ED = ED - El so that S = 0 according to Eqs. (60). If (E, - ED), (ED - El) > r,/2 then the constant C = 1 and
s
ED+A
dE (SD(E; Z) - S,,) = &, cos 24/cos2 4.
ED-d
This result can not be correct because it disagrees with Eq. (58)! The origin of the difficulty in getting the correct sum rule appears to be that the assumption of a constant 4 is not compatible with a normalizable interaction, i.e.,
A finite value for the right-hand side implies that Mi must become zero far from the analog. According to Eq. (34) this implies that $ also approaches zero at energies far from the analog. The result in Eq. (59) can still be obtained as an approximate result by integrating by parts and neglecting terms in derivatives of 4. Extension of the range of integration and using the limiting values of 4 = 0 and C = 1 then gives the expected result * dE (SD(E; Z) s-m
SO) = y;, .
(631
Both Eqs. (58), (59) state that the integrals of the strength functions S(E; Z) and SD(E; Z) exceed the intrinsic background strength xi y& by the quantity ytc contributed by the doorway state.
STRENGTH
269
FUNCTIONS
4.2. Sum Rule for S,
The sum rule on SB(E; I) is most easily evaluated by writing a more explicit form of Eq. (38). SB(E; Z) = &c
A
y;@(EA
+ Z - E) - O(E, - Z -
EN.
(38’)
This equation uses the step function. (64)
Straightforward
integration
and Eq. (56) give the expected result. m dE SdE; I) = c y:c + y;e s--m 2
(65)
4.3. Sum Rule for S,,,
The integrated strength of S,,(E; Z, l ) is most easily evaluated by inserting Eq. (11) into Eq. (45), the defining equation, and interchanging orders of integration. Thus one has first
m 2z7TI=--mdElFy dE’ ; Y:C(E’ _ &e + Ez. s--adES,,:,(E; 1, 4 = -!-
(66)
Use the variable x = E’ - E in the second integral
and then interchange the order of integrations to obtain easily the final result.
m dE Sm(E; s-02
1, l 1 = c Y%+ Y;,
(68)
z
This same device can be used in combination with the result Eq. (63) to derive the sum rule on S&(E; Z, e), the parametric form of S,, for an FDS. The result in Eq. (54) came from substituting Eq. (29) into Eq. (53). Return to this form j-m dE SpD,,(E; Z, c) = & J-= dE j-E: dE’ SD(E’; m --m
and change from E’ to x = E’ - E as an integration orders of integration then gives Irn dE S&(E; -m
l)
variable. Interchanging
I, c) = & SI, dx Ja dE S”(E - OL
t .u;l ).
the
(70)
270
WILLIAM
M. MACDONALD
Now it is only necessary to note that the result of the inner integration on E will be independent of x and therefore that the outer average on x gives unity.
j z dE S&(E; --ic Thus S&
I, e) = j” dE SD@; e) -?)
(71)
satisfiesthe same sum rule as SD.
4.4. Implication
of the Sum Rules
Each of the above SF is normalized to the samearea, the sum rule value given in Eq. (56). Therefore these SF can be compared directly and the different shapesreally represent different distributions of the same total strength. The lesson to be learned is that each averaging processgives a different strength function which has a different energy dependence with different parameters. These parameters can be determined only by fitting a particular SF, calculated from a given set of (I?,,, rf,) for the Kmatrix, to the correct parametric form of that same SF for an FDS. At this point note that Eq. (65) and Eq. (68) lead directly to the important Eq. (51). Since S, and S,,, satisfy the same sum rule, the quantity S,, has zero total strength.
5. SUMMED
STRENGTH
FUNCTION
The difficulties encountered in attempts to usethe box-averaged SF to analyze data on TAS led LLM to advocate the use of the “summed strength function” (SSF).
This SSF was used by Bilpuch, Lane, Mitchell, and Moses (BLMM) to analyze the high resolution data on LAS. The analysis was based on the conjecture that the parametric form of this SSF for an FDS is the indefinite integral of the strength function SD given by Eq. (29), i.e.,
$ (Es, E) = j” dE SD(E; I). ES The IAS parameters were determined by fitting Ce , calculated from Eq. (72) to the right side of Eq. (73). However, no account was taken of Z and the spreading width I’,J was taken equal to I’, (hence Z = 0). Tn investigating the properties of & it is interesting to note that
(74) This is easily verified by using Eq. (38), given in Sec. 4, to evaluate the indefinite
STRENGTH
integral. Therefore, Eq. (73) does not flow Indeed, since SBD # SD Eq. (73) would not The parametric form of &, is easily found coupling. Use Eq. (19) and approximate the
; (Es, E) ,/E; dE’ G
271
FUNCTIONS
from a relation between x,B and SBD. be compatible with equality in Eq. (74). for the picket-fence model in very strong sum by an integral.
= j-” dE’ ES
&S/2?r (If? - ED)’ + (w/2)’
In very strong coupling r” ‘v W = 2rrM2/D. 5 (Es, E) cz $L (tan-’
E w/f
+ tan-l
ED - Es 1 WI2
(76)
Setting Es ~= -cc gives the result of Eq. (28) in LLM. The resullt provides some justification for the use of Eq. (73) in strong coupling, but most 1A.S are cases of intermediate to weak coupling, to which this derivation does not apply. lln such cases Ce is really no more satisfactory than S, . Both are highly discontinuous functions on which the points to be fitted must be selected arbitrarily. (LLM chose to fit the midpoints of steps.) However, as we shall see the eye is less able to judge the significance of a fit to CB than to S, . What is needed at this point is a continuous SSF whose shape is well determined for an FDS. This is provided by a new SSF.
5.1. MMK,P-SSF A continuous SSF can be constructed M MKP-SF. From Eq. (27) follows
simply by using the indefinite integral of the
C (Es, E; I) = [E: dE’ S(E’; I) = + T & (tan-1 E r EA + tan-1 Ed r E~~). (77) The quantity on the right is a continuous by the choice of I. It is easily seen that
This equation
shows
continuatiotr of &(E,
function
whose smoothness
is regulated
that the MMKP-SSF defined by Eg. (77) is the parametric , E) with the smoothing parameter I. This result can be verified
by using thfe contour representation
for CB
c (Es, E) = &
f dz IZR(z)
B
where the contour
is a rectangle with sides parallel to the real axis running from Es
272
WILLIAM M. MACDONALD
Es-iI-
E-i1 Cl
FIG. 5. Contourfor the summedstrengthfunction & of Eq. (79). to Eat a distance I above and below, as shown in Fig. 5. The parametric continuation of & is defined Es,
E; I) = -
+ s,” CtE’ Im RR(E’ -f iZ).
s
From Eqs. (28), (77), and (80) follows directly the desired result. & (Es, E; 0 = c (Es, E; 0
(81)
The parametric form of C (ES , E; I) for an FDS is easily found by inserting Eq. (29) into Eq. (77). f
(Es,
E; I) = ~0s” “’
(@%Y ES, ED) ~0s 273- p(E, ES, ED) sin 24)
t &(E - Es) g(E,
Es,
ED) = $
*9(E7 Es ' ED) = &
(tan-l
(82) EFs,f
(E - Ed2 In (ED _ Es)2
+ tan-l ED - Es r&Y/2 1 + V.Y/~>~ + (r&)2
(83)
The parameters in these equations are the same as those defined by Eqs. (30)-(33). Thus x (Es, E; Z) is a continuous, smooth function whose energy dependence for an FDS is determined by five parameters which are the Lorentz-weighted averagesof microscopic matrix elements. These parameters can be determined by a least-squares of C (Es, E; Z) to 1:” (Es, E; Z) without any arbitrariness in the selection of the fitting points as shown by Fig. 6. For the picket-fence model wherein S, = 4 = 0 and Mi is constant, the SSF in Eqs. (82)-(83) reduces to the form of the result in Eq. (76). Tt is important to note, however, that r, = W + 21. Despite the fact that C (Es , E; I) is the parametric continuation of CR (Es , E) little theoretical or practical sensecan be made of Eq. (73) as it has been used in the analysis of TAS by BLMM. The introduction of a finite smoothing or averaging parameter Z is an absolutely essential step in obtaining a five-parameter strength function for a fragmented doorway state. There is no sensibleway to let Z go to zero
STRENGTH
2300
2350
2400
273
FUNCTIONS
2450
2500
2550
2600
E,(keV)
6. The MMKP-SSF is constructed by using Eq. (77) (dashed line) and compared with the doorway form of this SSF calculated from Eq. (83) with the same parameters as in Fig. 1. FIG.
12.5 _
40Artp,
P) l/2+ .r
2xX,
2350
2400
2450
2500
2560
2600
E, (keV)
FIG. 7. The discontinuous SSF of Lane, Lynn, and Moses [12] given by Eq. (72) is shown with the parametric form of Eq. (83) using the same parameters as in Fig. 1 but with Z = 0.
in Eq. (73). From a practical point of view, the discontinuous function CB (Es , E) can not be used reliably to determine the parameters of the right-hand size in cases of intermediate to weak coupling. This is not apparent from the fits exhibited by BLMM because the eye cannot properly assessthe quality of these fits. The Eq. (83) shows that .the spreading width rlJ is determined by the slopeor derivative of the SSF rather than by its value, as in the case of the SF. Thus fits presented by BLMM must be judged by the accuracy with which the slope of their SSF is determined. It is not sufficient to note whether a continuous curve has been run through most of the steps in the discontinuous function & . This can be seen from Fig. 7 which shows CB with x (Es, E; I = 0). The IAS parameters have the same values shown in Figs. 1 and 6 (although the parameters would not exist for I = O! Cf., Eq. (26)). Note that the fit also depends on Es!
214
WILLIAM
M.
6.
MACDONALD
SUMMARY
Strength functions (SF) and summed strength functions (SSF) are used to extract directly from micros-resonance parameters for high resolution differential cross sections the averages of microscopic coupling matrix elements between a fragmented doorway state (FDS) and “hallway states.” A distinction must be made between resonance parameters of the S-matrix and those of a reactance or K-matrix because their respective SF will have very different energy dependences and will depend in different ways on spreading parameters. Tn the picket-fence model for an FDS the K-matrix widths have a Lorentzian envelope of widths equal to the spreading width rk g 2~ikP/D. In the same model the distribution of S-matrix widths is not Lorentzian except in special cases, and the width of the distribution is the sum of the spreading width and the escape width. Different averaging procedures applied to the same K-matrix parameters also give different SF. The MMKP-SF, which is the Lorentz-weighted average of the reduced widths, has a characteristic asymmetric Lorentzian shape determined by five parameters that are themselves Lorentz-weighted averages of microscopic matrix elements. On the other hand, the sliding-box weighted average favored by A.M. Lane and collaborators is a discontinuous function which only approximates a continuous function in the limit of weak coupling between the doorway and the hallway states. This function is the sum of arctangent functions rather than the asymmetric Lorentzian established for the MMKP-SF. Although the box-weithted SF does not really determine a continuous function in intermediate to weak coupling (which obtains in the IAS, for example), a parametric continuation of this SF can be constructed which is defined for this regime. For an FDS this parametric continuation confirms the arctangent dependence on energy found in extremely strong coupling. These results explain the failure of attempts [23] to use the box-weighted SF to determine the spreading parameters for IAS by equating it to an incorrect parametric form, viz., the asymmetric Lorentzian derived for the Lorentz-weighted MMKP-SF. The apparent failure of the box-weighted SF led BLMM [13] to use the summed strength function (SSF) to analyze the high resolution data on IAS. For an FDS, this SSF was assumed by BLMM to be equal to the integral of the asymmetric Lorentzian found for the MMKP-SF. This assumption can, in fact, be very easily verified for the strong coupling limit of the picket-fence model. However, as with the box-averaged SF, the SSF of BLMM also does not really serve to define a continuous function in the intermediate to weak coupling regime which applies to most IAS. Therefore it has been again necessary to construct a parametric continuation of this SSF which is continuous and does define a parametric form even in intermediate to weak coupling. This parametric continuation proves to be identical to the integral of the continuous MMKP-SF, and it is therefore indeed equal to the integral of the parametric form found for MMKP-SF. Thus, although the BLMM-SSF is a discontinuous function which does not accurately determine the spreading parameters, the form assumed by BLMM is at least not inconsistent. Therefore the parameters found by BLMM contain considerable uncertainty but this arises from the discontinuous nature of
STRENGTH FUNCTIONS
275
their SSF a:nd not from the use of an incorrect form. Nevertheless, the deficiencies of the BLMM analysis of IAS call for a new analysis of this data with a continuous SF which more accurately determines the spreading parameters of the IAS. The results will be presented in another publication [24].
ACKNOWLEDGMENTS The author wishes to acknowledge the support of the University of Maryland Center.
Computer Science
REFERENCES 1. H. FESHBACH, A. KERMAN, AND R. H. LEMMER, Ann. Phys. 41 (1967), 230. 2. R. A. FERRELL AND W. M. MACDONALD, P/y. Rev. Lett. 16 (1966), 187. 3. D. ROBSOI~,Phys. Rev. 137 (1965), B535. 4. W. M. MACDONALD AND A. Z. MEKJIAN, Phys. Rev. Lett. 18 (1967), 706; C. MAHAUX AND H. A. WEIDENMULLER, Nucl. Phys. A 94 (1967), 1. 5. W. M. MACDONALD AND A. Z. MEKJIAN, Phys. Rev. 160 (1967), 730. 6. A. K. KEIXMAN AND A. F. R. DE TOLEDO PIZA, Ann. Phys. 48 (1968), 173. 7. M. DI TOKO, Nucl. Phys. A 155 (1970), 285. 8. W. M. MACDONALD, to be published in Phys. Rev. C. 9. H. FESHBPKH, in “Reaction Dynamics,” pp. 186-191, Gordon & Breach, New York, 1973.
10. A. M. LANE, in “Isospin in Nuclear Physics” (D. N. Wilkinson, Ed.), pp. 509-590, NorthHolland, Amsterdam, 1969. 11. W. MACDONALD, Phys. Rev. Lett. 40 (1978), 1066. 12. A. M. LAPSE, J. E. LYNN, AND J. MOSES, Nucl. Phys. A 232 (1974), 189. 13. E. G. BILPUCH, A. M. LANE, G. E. MITCHELL, AND J. D. MOSES, Phys. Rep. 28C (1976), 145. 14. L. ROSENFELD, Acta Phys. Pol. A 38 (1970), 603. 15. E. P. WIGNER AND L. ELSENBUD, Phys. Rev. 72 (1947), 29. 16. C. MAHA~X AND H. A. WEIDENMULLER, “Shell Model Approach to Nuclear Reactions,” NorthHolland, Amsterdam, 1969. 17. H. FESHBPCH, Ann. Phys. (N.Y.) 5 (1958), 357; ibid, 19 (1962), 287. 18. C. MAHACJX AND H. A. WEIDENMULLER, Nucl. Phys. A 97 (1967), 378. 19. C. BLOCH:, Nucl. Phys. 3 (1957), 137. 20. E. VOGT, Rev. Mod. Phys. 34 (1962), 723. 21. D. ROBSO~J,in “Isospin in Nuclear Physics” (D. H. Wilkinson, Ed.), pp. 461-508, North-Holland, Amsterdam, 1969. 22. P. L. KAPUR AND R. E. PEIERLS, Proc. Roy. Sot. London Ser. A 166 (1938), 277. 23. G. E. MITCHELL, Bull. Amer. Phys. Sac. 22 (1977), 573. 24. W. M. MACDONALD, Strength function analysis of high resolution data on isobaric analog states, to be published.
OF PHYSICS
125, 253-275
Strength
(1980)
Functions
for Fragmented
WILLIAM Department qf Physics and Astronomy,
Doorway
States*
M. MACDONALD
University of Maryland,
CoIlege Park, Maryland
20742
Received June 13, 1979
Coupling a strongly excited “doorway state” to weak “hallway states” distributes its strength into micro-resonances seen in differential cross sections taken with very good energy resolution. The distribution of strength is shown to be revealed by reduced widths of the K-matrix rather than by the imaginary part of poles of the S-matrix. Different strength functions (SF) constructed by averaging the K-matrix widths are then investigated to determine their dependences on energy and on parameters related to averages of microscopic matrix elements. A new sum rule on the integrated strength of these SF is derived and used to show that different averaging procedures actually distribute the strength differently. Finally, it is shown that the discontinuous summed strength defines spreading parameters for the doorway state only in strong coupling, where it approximates the indefinite integral of the continuous SF of MacDonald-Mekjian-Kerman-De Toledo Piza. A new method of “parametric continuation” is used to relate a discontinuous sliding box-average, or a finite sum, of discrete terms to a continuous function.
1. INTRODUCTION The concept of a doorway state, introduced by Feshbach and Block, has proved to be the key to understanding many features of nuclear reactions. As emphasized by Feshbach, Kerman, and Lemmer [I] the doorway state is an eigenstate of an approximate or model Hamiltonian so strongly coupled to certain reaction channels that its properties determine the essential feature of reactions proceeding through these channels. More complicated states “hallway states” [2] of this Hamiltonian may be weakly coupled to these channels but their participation in the reaction process is dominated by the strength which they acquire by coupling to the doorway state. Thus the strength of the doorway is spread over some energy range and the doorway state acquires a spreading width in addition to its decay width. In some cases, the density of hallway states is so large that their participation can be observed in cross sections only as a broadening of the doorway resonance. Other doorways, most notably the isobaric analog states (IAS), may be observed in high resolution experiments as a large number of distinct resonances whose widths increase near the energy of the original doorway state. The most beautiful examples of this latter case are to be found in the high resolution data of the Duke University experimental group as was first pointed out by D. Robson [3]. * Supported by the U. S. Department
of Energy.
253 0003-4916/80/040253-23$05.00/O Copyright Q 1980 by Academic Press, Inc. All rights of reproduction in any form reserved.
254
WILLIAM
M. MACDONALD
The central problem in the analysis of fine structure associated with a fragmented doorway state (FDS) is the extraction of its properties and the determination of average parameters which characterize its interaction with the hallway states as well as the coupling of the latter to the open channels. The spin and parity of the doorway state is, of course, the same as that of the fine structure, which can be determined from the angular distributions on resonance. Other information must be extracted from an analysis of the resonance pattern. When the fine structure is resolved, a resonance energy and one or more partial widths can be found for each of the resonances. Together with the background phase shift these resonance parameters summarize all the information available from the differential cross sections. The first determinations of the rms value of the charge dependent matrix element coupling an isobaric analog state (IAS) to hallway states of different isospin attempted to use the envelope of Wigner-Eisenbud reduced widths [4]. Fluctuations in the magnitude of the coupling matrix elements severely limited the accuracy of this approach. MacDonald and Mekjian [5] and, independently, Kerman and De Toledo Piza [6] showed that a Lorentz-weighted average of the reduced widths defines a “strength function” (SF) which has the characteristic shape of an asymmetric Lorentzian whenever a fragmented doorway state (FDS) is present.1 The shape of this SF is completely determined by only five parameters: (1) the energy of the doorway state, (2) its particle emission width, (3) a “spreading width” which measures the coupling of the doorway to the hallway states, (4) the average intrinsic particle emission width of the hallway states (before coupling to the doorway state), and (5) the average phase angle between matrix element coupling a hallway state to the doorway state and that coupling it to the continuum. M. Di Toro successfully used this SF to analyze the fine structure in several IAS to deduce these parameters [7]. Recently it has also been thoroughly tested [S] by a model study on a large number of FDS. Other SF can be constructed in various ways. First, different resonance expansions can be used to represent the differential cross sections and these parameters used to construct an SF. For example, the micro-resonances can be represented by a pole expansion of the S-matrix and the resulting real and imaginary parts of the poles used to construct an SF. Second, different averaging procedures can be used to construct different SF from the same set of resonance parameters [9]. The MMKP-SF was constructed using Lorentz-weighted averaging, but A. M. Lane favored an SF which is a sliding box-average of reduced widths [IO]. This latter is a common method for treating statistical data, but it yields a discontinuous histogram not well suited to analyzing the data on fragmented IAS. In a preliminary presentation of some of the results of this paper, I have given a “parametric continuation” of this box-averaged SF which provides a continuous transition from this highly discontinuous histogram to a smooth SF. The shape of this new SF also is determined by only five parameters when an FDS is present [l 11. Each of these SF has a different dependence on energy. Difficulties in using the box-averaged SF led Lane, Lynn, and Moses (LLM) [12] to suggest use of a “summed strength function” (SSF). This SSF was used by Bilpuch, 1 This will be referred to as the MMKP-SF.
255
STRENGTH FUNCTIONS
Lane, Mitchell, and Moses (BLMM) in their analysis of the high resolution data on IAS [13]. II: suffers from the same deficiency of the box-averaged SF of being a discontinuous function whose shape is poorly defined in weak coupling. But the same procedure I used on the box-averaged SF can be used to construct a’parametric continuation of the SSF. This new SSF can be used to analyze an FDS even in weak coupling. Tn fact, it will be found to be identical to the indefinite integral of the MMKP-SF. This paper presents a complete discussion of the above-mentioned SF and SSF. Section 2 begins with a brief review of different resonanceexpansions of the S-matrix and a discussion of the relations between their respective resonance energies and widths. The picket-fence model is used to obtain equations for the widths of a reactance matrix and for the imaginary part of the S-matrix poles in order to make explicit the.ir differences. Section 3 presents a complete discussion of these SF which can be constructed from the same set of resonance parameters for the reactance matrix. In Sect. 4a sum rule is derived for each SF in order to establish its normalization. Section 5 is devoted to the subject of SSF and their inter-relations. Section 6 summarizes the findings of this investigation of strength functions and sum rules for the analysis:of FDS.
2. RESONANCE PARAMETERS FOR FINE STRUCTURE 2.1. Comparison of Different ResonanceExpansions In principle, the fine structure in high resolution measurementsof differential cross sections can be used to determine poles and residuesof the S-matrix in the expansion
(1) Unfortunately the quantities (E,,s, rAs, g,J f or d’ff 1 erent levels are not independent becausethey must satisfy complicated unitarity constraint equations [14]. For this reason crosssectionscontaining numerous resonancesare more conveniently analyzed by using a reactance matrix K in place of the S-matrix. S,,,(E) = ei(‘c+‘c’)[(l + iK)(l - iK)-l],,,
.
(2)
The K-matrix [2, 5, 61 can also be expanded to display resonance energies I& and partial widths fi,l/’ (3) The resonance parameters (.8$, F;,‘,/3 of the K-matrix are both real and independent; unitarity is guaranteed by the form of Eq. (2). The Wigner-Eisenbud (WE) R-matrix theory [15] is a reactance matrix formalism based on the division of the 3A-dimensional configuration space into an internal or
256
WILLIAM
“interaction” the s-matrix
M.
MACDONALD
region and an external or “asymptotic” in the form SC,, = eidcPE”[(l - RL,)-l
region. This formulation
(1 - RL,*)],,, ei*c’P~l’z
gives (4)
where (L,), = S, - B, + iP, contains a slowly energy dependent level shift S, and a penetrability factor P, . The &matrix is given by R,,, = - c -?!d!& a E-6
with energy independent partial reduced widths Y,,~and resonance energies E,, . A very similar expression for the S-matrix emerges from both the K-matrix formulation of the shell model approach to reaction theory (SMART) [5,16] and the more general projection operator formalism of Feshbach [ 171. These approaches divide the Hilbert space into (1) a portion spanned by quasi-bound or closed-channel components and (2) a portion spanned by scattering states or open-channel components. In SMART the K-matrix has a direct interaction term as shown in Eq. (3). K:,
= 4xcW
I f’e I x&D.
(f-5)
The resonance term is the second term in Eq. (3) with p;:” = (2741’2 Wa I ve I xc(E)>
(7)
4 =
(8)
and It should be noted that both in Eq. (5) and Eq. (6) the xc(E) are distorted waves of a real optical potential. This is a consequence of the fact that these are matrix elements of
a reactance matrix and not of the S-matrix. In the literature on IAS there has been considerable confusion on this point because of a general failure to appreciate the differences between the K-matrix and the s-matrix. The partial reduced width p,1,/” has an energy dependence arising from the “penetrability” of xc(E) through a real potential barrier. As suggested by Weidenmuller and Mahaux [ 181 this can be removed as an energy dependent factor. l-t’,‘” = (2fe(E))1’2 j& .
This gives an equation for the S-matrix similar to that displayed SC,, = e”““ft’“[(l Although
+ iRDf + iRRf)--l (1 - iRDf-
in Eq. (4).
&Rf)]ec, f~1'zei60'.
(10)
the resonant part of the “reduced” K-matrix
(11)
STRENGTH
FUNCTIONS
257
has the form of the WE R-matrix (except for the sign), the sum is only over aJinite number of internal states ul, instead of the infinite number of states of Eq. (5). As shown by C. Bloch [19], the sum over “distant levels” of the WE R-matrix provides the direct :interaction term which is exhibited explicitly in Eq. (10) as RD. These “distant levels” also correct the physically unreasonable hard-sphere shifts CJ$= 2ka, and square-well penetrability factors P, which appear in Eq. (4). Indeed, unless these “distant level” terms of the WE R-matrix are properly included by modifying & and P, of Eq. (3) to the potential phase shifts 6, and penetrability for a diffuse well, the reduced widths and resonance energies obtained for Eq. (4) from an R-matrix analysis of differential cross sections will not be correct [20]. In the analysis of FDS an incorrect distribution of reduced widths with energy will result from a failure to include these corrections. It is essential to note that the “R-matrix” used by A. M. Lane and D. Robson [21] to discuss line-broadening of IAS is not related to the R-matrix of Wigner-Eisenbud but to the resonance expansion of the S-matrix introduced by Kapur-Peierls [22]. S,,, = ei”c[l + 2iP~‘2R,‘~?P,t] ei”“‘. The Kapur-Peierls
expansion
has the properties of the expansion in Eq. (l), viz. the energies &?A= E,“’ - i(I’fp/2) are complex and the resonance parameters for different levels are connected by unitarity constraints. The 8A are not really poles of the S-matrix, however, but eigenvalues of a complete set of internal states which satisfy a complex boundary condition. Although the Kapur-Peierls “R-matrix” is also based on an expansion of the solution of the SchrGdinger equation in an interaction volume in (ri ,..., rA) space, it is not a reactance matrix and does not share the properties of the R-matrix of Wigner-Eisenbud. The resonance parameters of this “R-matrix” can not properly be identified with the (EA , yiC) obtained by the Duke University group from their multilevel analyses of experimental differential cross sections using the WignerEisenbud R-matrix. In fact, the entire discussion of the qualitative behavior of fine structure widths for analog resonances presented by Lane in his review does not apply to these resonance parameters, as will be evident in the next section. 2.2. Resonance Energies and Widths of the K-Matrix for an FDS The effects of distant levels (R,) must be included in Eq. (4) to obtain partial reduced widths and resonance energies which correctly reflect the fragmentation of a doorway state. As noted above the inclusion of these terms leads to Eq. (10) with the resonant part of the reduced K-matrix of Eq. (11). The strength functions and sum rules of this paper are therefore based on the resonance parameters $ and +& of Eq. (11). Tn using the results to be presented one must hope that corrections due to
258
WILLIAM
M.
MACDONALD
R, have been included in existing analyses of the experimental data, or are small, so that quoted E,, and yACcan be identified with i?,, and j& . For this reason (and simplifi-
cation of notation) the tilde will be omitted from all K-matrix quantities in the remainder of this paper. Before fragmentation a doorway state at E, has reduced partial widths yDCand, in general, the hallway state at .Q has reduced partial width yic . If Mi denotes the coupling matrix element of the doorway to the hallways, the resonant part of the reduced K-matrix is given explicitly by
where
The poles and residues of RR give the resonance energies and poles of Eq. (II).
(17)
The qualitative features of the distribution of widths can best be seen in a simple model [5] called the picket-fence wherein only one channel is open and Mi = A4 = constant; ci = co i
yic = 0 n = 0, 1, 2,..., co.
nD;
(18)
For this case, Eq. (17) can be evaluated to give a distribution rtc = Y?Jc
(4
-
r”D/271 Ed2 + W’M2
determined by r’” = 2nM2/D. This quantity (or its average) is usually called the “spreading width,” but the width of the distribution of reduced widths is actually given by W = ((rL)2 + M2)lp. The envelope of theseyt” is Lorentzian for all values of yiC , FL, M, and D. No distinction exists between casesof strong coupling (r= > D) and casesof weak coupling (P
E, = (r”/2)cot[n(EA
- q,)/D];
(20)
they are also independent of yiC . In the picket-fence model W is approximately equal to the spreading width of Feshbach, Kerman, and Lemmer [1] for D/277 M < 1. However, for a physical
STRENGTH
259
FUNCTIONS
FDS the fluctuations in the magnitudes of Mi2, exhibited by the & for IAS [13], for example, prevent an accurate determination of W by fitting to Eq. (19) [4, 51. This fact led to the introduction of strength functions constructed by averaging the reduced widths [5, 61. 2.3. S-Matrix
Poles of an FDS
Although not determined directly by the high resolution experiments on fragmented doorways, the S-matrix poles can be found from Eq. (10) by using the results just found for the K-matrix. The complex poles 6, = Evs - i(rVs/2) satisfy the determinantal equation Det(l + iK) = 0. (21) For the case of more than one channel this equation involves both diagonal and nondiagonal elements of the K-matrix. Therefore partial reduced widths for both elastic and inelastic channels participate in determining the S-matrix poles. If there is only one open channel, Eq. (21) simplifies somewhat. The explicit form of irir* given by Eq. (14) provides the equation for the poles of a fragmented doorway.
From this equation it can be shown that 6, = Evs - iTus/ with rys > 0. A comparison of Eqs. (16)-(17) with Eq. (22) reveals the gulf between the resonance energies and widths of the K-matrix and those of the S-matrix. For the picket-fence model, with now an additional assumption that KD -= 0, Eq. (22) becomes ’ + 8, - ED - (r&/2) cot[7r(b,
-
Q)/D~.
(23)
Two coupled equations for Evs and rys result from setting the real and imaginary parts of this equation equal to zero. No independent equation for the S-matrix resonance energies Evs, analogous to Eq. (16), can be disentangled. From Eq. (23) Lane derives [lo] the following equation for the widths.
dAs
tanh __ D
=
wDc (ED -
- m/2
EAs)” + [(r&y + (r,,
- r,y/4
(24)
From this equation it can easily be seen that even that shape of the distribution of S-matrix widths depends on the relative magnitudes of r,, , P, and D. In contrast to the distribution of K-matrix widths, the distribution of S-matrix widths is not even approximately Lorentzian except for special cases [lo]. Even in this case the width of the distribution of S-matrix widths is not FL, as for the K-matrix widths, but [(rl)2 + (rDC)2]1/2 [lo]. Unfortunately this crucial difference between the S-matrix parameters and the K-matrix parameters has not been generally realized and such
260
WILLIAM M. MACDONALD
concepts as “weak mixing due to rapid decay,” a valid notion for S-matrix poles, have been used in discussions of K-matrix or R-matrix widths (particularly from the Duke data), to which it does not apply.
3. STRENGTH FUNCTIONS FOR K-MATRIX
WIDTHS
Fluctuations in the magnitude of the matrix element coupling the doorway to hallway states compel us to smooth the distribution of widths. The resulting function of energy is called a strength function (SF). In this section it will be seen that different averaging procedures produce SF with different energy dependences. Each SF also depends on different spreading parameters. 3.1. MMKP
Strength Function
Averaging the widths with Lorentz-weighting produces a stength function with parameters that are themselves the Lorentz-weighted averages of the microscopic parameters [5, 61. First define the Lorentz-weighted average of a resonance parameter FA for the states Y,, .
(FAID,) = ; 7 (E _ 2)2 + 12 Then define the Lorentz-weighted hallway states.
average of the microscopic
parameter fi for the
and by Kerman
A very important relation for constructing the strength function of a fragmented doorway state follows directly from Eqs. (1 l), (27). S(E; I) = - + Im RP,(E + ir>
(28)
Substitution of the explicit form for RR given by Eq. (14) into Eq. (28) gives directly the parametric form of this SF. SD(E.
r) 7
=
s
I
3’k,
0 T
(~d4
~0s
24~ -
@
-
2~)
(E - ED), + (T,/2)2
sin
24
*
(29)
261
STRENGTH FUNCTIONS
The parameters in this equation characterize the spreading. Miyic qDcei* = YDC + 2 i E-cEi+iZ 8, = ED + A,
(W
A, 3 (n/Z)(Mt(E
- cJ/Di)
@lb)
r, f r,J + 21
(324
rIJ E 2~(MJDJ
Wb)
(33)
So =
Averages containing (E - ci) in Eqs. (30), (31) can be neglected in comparison with other terms because contributions from hallways above and below energy E tend to cancel. Neglect of these terms given 8, = E. and the simpler expressions tan 4 = -4MiyiclDi)lyDc h
=
yDc
set
(34) (35)
6
The Eq. (29) is suggestive but all the parameters are actually functions of E and I. The expectation is that the microscopic averages A, , rlL, and S, can be taken as constant for some reasonable choice of I. The SF constructed from experimental data by using Eq. (27) show that this expectation is well fulfilled for many IAS. A (relatively poor) example is the SF for the .Z* = l/2+ fragmented IAS observed in 40Ar (p, p) l/2+ shown in Fig. 1. Nevertheless calculations based on models are useful for establishing the magnitude of fluctuations expected in r,L and the proper choice of Z [S].
5
sFi!
01 2300
2350
2400
2450
2500
2550
2600
E, (keV)
FIG. 1. The Lorentz-weighted or MMKP strength function for a fragmented IAS in 4oAr(p,p) constructed from experimentally determined reduced widths by using Eq. (27) (dashed line) is compared with the doorway SF of Eq. (29) (solid line) obtained by a least-squares fit for Z = 35 keV to fix EA = 2458.5 keV, 4 = 0.18, y& = 7.5 keV, YS = 13.7 keV, and S, = 0.011. 595/125/2-3
262
WILLIAM
M.
MACDONALD
Some insight into this matter is also provided by the picket-fence molel. For this model r,qE)
1 - e-4nIlD 2TrA!P -= D 1 + e-4n1/D - 2e-2n11D cos 2(E - q,) n/D
(36)
e-2n11Dsin 2(E - q,) n/D 1 + e-4n11D - 2e-2rIlD cos 2(E - q,) n-/D
(37)
A, =E
D
For I/D it is an excellent approximation to take I’,$ = 2nM2/D and A, = 0. This model does not, however, include the effects of statistical fluctuations on the magnitude of the matrix elements and in the level spacings. These cause fluctuations in I’,& which are larger than those in Eqs. (36), (37). These fluctuations have been evaluated both numerically and analytically in a model study of 400 fragmented doorway states [8]. There it has been found that on average I’,& shows fractional fluctuations of order (D/nI)l12. Nevertheless, fitting Eq. (27) to Eq. (29) with constant parameters determines I’,i with an error of less than ten percent when I is chosen of order 2~(.Mi2)/(D,)
C+ I’,‘.
3.2. Box-Averaged
Strength Function
Statistical data is often “smoothed” by averaging with a sliding box [9]. This method has been favored by A. M. Lane in defining an SF [lo].
This SF is actually a discontinuous function of energy. In the case of very strong coupling (r,J > D) there will be many levels within the width of the maximum of the pattern, and one can hope to approximate S, by a continuous function with parameters related to microscopic averages like those in Eqs. (30)-(33). Lane has asserted that S’,(Ej I) has the same form as that given in Eq. (29). This can be shown not to be true even for the picket-fence model wherein all effects of fluctuating matrix elements and level spacings are absent (see Fig. 3). Use Eq. (19) for the widths to find S, for a fragmented doorway state. For very strong coupling (rJ > D) this distribution well approximates a continuous function in which E,, assumes all values. This will enable us to replace summation by integration in finding the SF defined by Eq. (38). As a check, the integral on E,, gives the correct sum rule value in this same limit.
Therefore in this strong coupling limit the continuous function which best approximates S, for an FDS is (40)
STRENGTH
263
FUNCTIONS
The result is S/(E;
I) cz &
(tan-’
The multiple branches of the arctangent result. Another form is SBD(E; I) fg 6
tan-l
(E - E#
ED + tan-’
E +;,;
function
;7W,2)z
2 i
n - tan-l
1.
call for care in transforming
(41) this
- I” (E -
z= &
ED-E+I w/2
ED)2 + (;j”
- I” =; 0
(42)
IW I2 - (W/2)2 - (E - ED)2 1 (E -
ED)2 + (;j’
- I3 < 0.
This continuous approximation to the box-averaged SF for the fragmented doorway must be compared with the MMKP-SF given by Eq. (29) in this limit (S, = C$= 0). (43) The box-averaged SF is more sharply peaked than the Lorentzian form conjectured by Lane. In the wings the box-averaged SF is approximately Lorentzian but the “width” is (W/2)2 - Z2. However, only if I< W/2 does Eq. (42) approximate a Lorentzian throughout its range. ‘All the known cases of fragmented IAS fall into the category of weak to intermediate coupling. The fractional discontinuities in the box-averaged SF are then large and can not be reduced below a certain level. To see this, consider the discontinuity at ITA introduced as the front of the box slides past this level. The S,(E; I) increases by &2Z. The sum rule x,, & = &, shows that the maximum value of S, prior to this jump is &,/2Z. The fractional jump in S, must therefore exceed (&&,). Over most of the pattern this must be multiplied by a factor equal to the reciprocal of the fraction of the summed width in the box at any energy. Thus S,(E; I) does not even approximately define a continuous function in the case of intermediate to weak coupling. Figure 2 shows plots of S,(E; I) for the IAS at E, = 2459 keV in 40Ar + p with J= = l/2+ for two values of the averaging interval. 3.3. Parametric A function imates S,(& parameter”’ l This function
Continuation
of S,(E; I)
continuous in E and Zcan be constructed which arbitrarilyclosely approxI) [l I]. This new SF is also a continuous function of a “smoothing , and it approaches S,(E; I) and becomes equal to it in the limit of E + 0. will be denoted by SPCB(E; I, l ) and called the “parametric continua-
40
2300
2350
2400
2400
2500
Ar(p, P) I/Z’
2550
x00
E, (keV) 0.125 40Ar(p,p)
l/2’
0.100 F
FIG. 2. (a) and (b): The box-averaged strength function for Eq. (38) is shown for Z = 35 keV
(Fig. 2a) and for Z = 70 keV (Fig. 2b). Note that the larger averaging interval does not reduce the fractional discontinuities.
0.500 0400 1 0.200
40
Artp,
P) l/2+
_
Ic_I 2300
2350
HO0
2450
2500
2550
2600
E, N.eW
FIG. 3. The box-averaged SF of Eq. (38) is compared with the asymmetric Lorentzian (29) for Z = 0 using the parameters of Fig. 1 for this IAS.
of Eq.
265
STRENGTH FUNCTIONS
tion” of S&Y; Z). Unlike S, , but like S(E; Z) of Eq. (27), this strength function can be used to analyze fragmented doorway states even in weak to intermediate coupling. To derive this SF begin by representing S, as a contour integral in the complex energy plane. The contour is illustrated in Fig. 4 as a rectangle with two sides parallel E-I+is
-,
E+l+ir
E+I-ic
E-1-k
Cl
FIG. 4. Contour for the box-averaged strength function S, of Eq. (44).
to the real axis and a distance E above and below it. The ends are perpendicular to the real axis and located at E - Z and E + I. Now use Eq. (11) to write the contour integral representation S,(E; I) = (47rZi)-l $ dz Izp,(z).
(44
The integral receives contributions from the sides, C, and C, , as well as from the ends, C, and C, . Use the contributions from C, and C, to define the parametric continuation of S, . The equation for S,, can be written 1
dE’ Im &fc(E’
+ k).
(45)
The pole expansion for 92 given by Eq. (11) then leads directly to an explicit result. - tan-l from C, and C, define a fluctuating
The contributions
The definitions
E
1
in Eqs. (45), (47) define a decomposition
- Z + iv))
of the box-averaged
S,(E; I) = S,,,(E; Z, 4 + S&T A l ) Recognition
(46)
function.
1 4 e) = 4nli I -~’ dy(Zt:c(E + Z + iy) - &f-G
ME;
E-Z-EA
(47) SF. (48)
that E i
Z - E., _ tan-l E
E - I - E,, E
(E - EAjz < I2 otherwise
(49)
266
WILLIAM
M.
MACDONALD
leads to
G-l Sp,(E; z, 6) = S,(E; Z). The same result follows
w-0
less directly from the limit Ii?
S,,(E; Z, c) = 0.
The decomposition has a unique feature which follows in the next section.
(51) from the sum rule on S derived /
m dE S,,(E; Z, E) = 0. s -m
(521
This equation justifies the notational designation of S,, as the fluctuating part of the box-averaged SF. It remains to find the form of S,,, for a fragmented doorway. This is done by using Eq. (14) to evaluate the integral in Eq. (45). A comparison of Eq. (45) with Eq. (28) gives the relation 1 SPCB(E;
4
4
=
z
s
ET1dE’ ELI
S(E); E).
(53)
Thus, SP,, is the average over the interval (E - Z, E i Z) of the MMKP-SF! Evaluation of S,,, for a fragmented doorway simply requires evaluation of this integral with SD@: c) derived in Eq. (29). The integral can only be evaluated explicitly, however, by assuming that the parametric quantities in Eq. (29) are independent of energy. SpDCR(E; Z, c) = S, + $$ (% cos 24 -
Y sin 2+)
(54)
The energy dependent quantities C, S are defined by the equations +? = -!- (tan-l 7r y _
E + Z - ED _ tan-l rS(4/2
E - Z - ED 1 J-d42
1 ,n (E + Z - ED)~ + (r~(4/2)~ 2~ (E - Z - ED)2 + (TS(E)/~)~ ’
(55)
The parameters in both Eqs. (54) and Eq. (55) are the same Lorentz-weighted averages defined in Eqs. (30)-(35), but with the smoothing parameter E replacing I. Thus Z’&E) = Z’,i + 2~ by replacing Z by E in Eq. (32a). As shown in Eq. (48), S,,, differs from S, by a fluctuating term of zero integral. Thus SD pcB gives the parametric form of S, for a fragmented doorway when the discontinuities are smoothed into a continuous function. This result can be checked against the form of S, derived for the picket-fence model in very strong coupling, as given in Eq. (41). In the picket-fence model I$ = 0 and W ‘v 2rrM2/D. As Eq. (36) shows, the choice Z 5 D gives r,L = 2rM2/D = W to very good approximation.
STRENGTH FUNCTIONS
267
of Eq. (54) has exactly the sameform as Eq. (41) and differs in the replaceThus SpDCB ment of W !Z r,i by rS(c) = r,J + 2~. This difference merely reflects the fact that the undefined smoothing operation used to derive Eq. (41) did not properly include the additional spreading of the strength due to smoothing. The Eq. (54) showsthat when S, is usedto analyze the fine structure of a fragmented doorway, the correct parametric form is not Eq. (29) as conjectured by Lane and collaborators. This result has also been demonstrated by direct application to the .I= = 3/2-- isobaric analog state in 40Ar(P, p)40Ar at E, = 1875 keV [II]. Finally, it should be noted that while SpcB provides a satisfactory replacement for the discontinuous box-averaged S, , there is no reason to prefer it to the MMKP strength function. Rather to the contrary, S,,-, is merely the average of the MMKP-SF, as shown by Eq. (53). This averaging is entirely unnecessary since the MMKP-SF is smooth.
4. SUM RULES FOR SF The SF discussedin the previous section differ from one another in their energy dependenaes.To compare them one must first determine how each is normalized. The sum rules provide this information. A sum rule will first be established for the general form of each strength function, and then the parametric form of each SF will be studied. There are some surprisesin the latter cases. All sum rules relate back to the sum rule on the K-matrix widths. This is easily extracted by noting that Eq. (11) is identically equal to Eq. (14) when the microresonancesarise from an FDS. Equating the coefficients of l/E in these two equations for KzCgives the desired sum rule. (56) In the following sections it will be shown that every SF integrates to this same sum rule value. 4.1. Sum Rule for MMKP-SF The sum rules on SF relate to the total area under the SF, i.e., to the infinite integral. The sum rule for the MMKF-SF requires evaluation of
Straightforward integration combined with Eq. (56) gives
s
m dE S(E; I> = 1 yfc + &, . -03
(58)
268
WILLIAM
M.
MACDONALD
The identity of Eq. (27) with Eq. (29) assures us that SD@; Z) satisfies the same sum rule when the energy dependence of allparameters is included. This is not true of the parametric form when the parameters are taken to be constant. A straightforward evaluation of the infinite integral of S”(E; Z) does not even converge. The indefinite integral is
4 I El
dE SD(E; Z) = (E, - El) S,, +
cos 24 - S sin 24)
(59)
where the constants C and S are C = +- (tan-l s
-
1
ln
277
(4 (ED
Ez
-
ED
_
tan-l
EI
Ts!2 -
-
rs/2
ED)~
+
(r.4)”
Q2
+
(r.s/2)’
ED
)
’
(60)
Examining the separate terms in Eq. (59) we see that the first term corresponds to C yi, of Eq. (56). No limit has been derived for this sum, so none can be applied to the first term of Eq. (59). Nevertheless, identification of this term implies that the sum rule value for the second term of Eq. (54) is &, . BLMM have evaluated this term by taking d = E, - ED = ED - El so that S = 0 according to Eqs. (60). If (E, - ED), (ED - El) > r,/2 then the constant C = 1 and
s
ED+A
dE (SD(E; Z) - S,,) = &, cos 24/cos2 4.
ED-d
This result can not be correct because it disagrees with Eq. (58)! The origin of the difficulty in getting the correct sum rule appears to be that the assumption of a constant 4 is not compatible with a normalizable interaction, i.e.,
A finite value for the right-hand side implies that Mi must become zero far from the analog. According to Eq. (34) this implies that $ also approaches zero at energies far from the analog. The result in Eq. (59) can still be obtained as an approximate result by integrating by parts and neglecting terms in derivatives of 4. Extension of the range of integration and using the limiting values of 4 = 0 and C = 1 then gives the expected result * dE (SD(E; Z) s-m
SO) = y;, .
(631
Both Eqs. (58), (59) state that the integrals of the strength functions S(E; Z) and SD(E; Z) exceed the intrinsic background strength xi y& by the quantity ytc contributed by the doorway state.
STRENGTH
269
FUNCTIONS
4.2. Sum Rule for S,
The sum rule on SB(E; I) is most easily evaluated by writing a more explicit form of Eq. (38). SB(E; Z) = &c
A
y;@(EA
+ Z - E) - O(E, - Z -
EN.
(38’)
This equation uses the step function. (64)
Straightforward
integration
and Eq. (56) give the expected result. m dE SdE; I) = c y:c + y;e s--m 2
(65)
4.3. Sum Rule for S,,,
The integrated strength of S,,(E; Z, l ) is most easily evaluated by inserting Eq. (11) into Eq. (45), the defining equation, and interchanging orders of integration. Thus one has first
m 2z7TI=--mdElFy dE’ ; Y:C(E’ _ &e + Ez. s--adES,,:,(E; 1, 4 = -!-
(66)
Use the variable x = E’ - E in the second integral
and then interchange the order of integrations to obtain easily the final result.
m dE Sm(E; s-02
1, l 1 = c Y%+ Y;,
(68)
z
This same device can be used in combination with the result Eq. (63) to derive the sum rule on S&(E; Z, e), the parametric form of S,, for an FDS. The result in Eq. (54) came from substituting Eq. (29) into Eq. (53). Return to this form j-m dE SpD,,(E; Z, c) = & J-= dE j-E: dE’ SD(E’; m --m
and change from E’ to x = E’ - E as an integration orders of integration then gives Irn dE S&(E; -m
l)
variable. Interchanging
I, c) = & SI, dx Ja dE S”(E - OL
t .u;l ).
the
(70)
270
WILLIAM
M. MACDONALD
Now it is only necessary to note that the result of the inner integration on E will be independent of x and therefore that the outer average on x gives unity.
j z dE S&(E; --ic Thus S&
I, e) = j” dE SD@; e) -?)
(71)
satisfiesthe same sum rule as SD.
4.4. Implication
of the Sum Rules
Each of the above SF is normalized to the samearea, the sum rule value given in Eq. (56). Therefore these SF can be compared directly and the different shapesreally represent different distributions of the same total strength. The lesson to be learned is that each averaging processgives a different strength function which has a different energy dependence with different parameters. These parameters can be determined only by fitting a particular SF, calculated from a given set of (I?,,, rf,) for the Kmatrix, to the correct parametric form of that same SF for an FDS. At this point note that Eq. (65) and Eq. (68) lead directly to the important Eq. (51). Since S, and S,,, satisfy the same sum rule, the quantity S,, has zero total strength.
5. SUMMED
STRENGTH
FUNCTION
The difficulties encountered in attempts to usethe box-averaged SF to analyze data on TAS led LLM to advocate the use of the “summed strength function” (SSF).
This SSF was used by Bilpuch, Lane, Mitchell, and Moses (BLMM) to analyze the high resolution data on LAS. The analysis was based on the conjecture that the parametric form of this SSF for an FDS is the indefinite integral of the strength function SD given by Eq. (29), i.e.,
$ (Es, E) = j” dE SD(E; I). ES The IAS parameters were determined by fitting Ce , calculated from Eq. (72) to the right side of Eq. (73). However, no account was taken of Z and the spreading width I’,J was taken equal to I’, (hence Z = 0). Tn investigating the properties of & it is interesting to note that
(74) This is easily verified by using Eq. (38), given in Sec. 4, to evaluate the indefinite
STRENGTH
integral. Therefore, Eq. (73) does not flow Indeed, since SBD # SD Eq. (73) would not The parametric form of &, is easily found coupling. Use Eq. (19) and approximate the
; (Es, E) ,/E; dE’ G
271
FUNCTIONS
from a relation between x,B and SBD. be compatible with equality in Eq. (74). for the picket-fence model in very strong sum by an integral.
= j-” dE’ ES
&S/2?r (If? - ED)’ + (w/2)’
In very strong coupling r” ‘v W = 2rrM2/D. 5 (Es, E) cz $L (tan-’
E w/f
+ tan-l
ED - Es 1 WI2
(76)
Setting Es ~= -cc gives the result of Eq. (28) in LLM. The resullt provides some justification for the use of Eq. (73) in strong coupling, but most 1A.S are cases of intermediate to weak coupling, to which this derivation does not apply. lln such cases Ce is really no more satisfactory than S, . Both are highly discontinuous functions on which the points to be fitted must be selected arbitrarily. (LLM chose to fit the midpoints of steps.) However, as we shall see the eye is less able to judge the significance of a fit to CB than to S, . What is needed at this point is a continuous SSF whose shape is well determined for an FDS. This is provided by a new SSF.
5.1. MMK,P-SSF A continuous SSF can be constructed M MKP-SF. From Eq. (27) follows
simply by using the indefinite integral of the
C (Es, E; I) = [E: dE’ S(E’; I) = + T & (tan-1 E r EA + tan-1 Ed r E~~). (77) The quantity on the right is a continuous by the choice of I. It is easily seen that
This equation
shows
continuatiotr of &(E,
function
whose smoothness
is regulated
that the MMKP-SSF defined by Eg. (77) is the parametric , E) with the smoothing parameter I. This result can be verified
by using thfe contour representation
for CB
c (Es, E) = &
f dz IZR(z)
B
where the contour
is a rectangle with sides parallel to the real axis running from Es
272
WILLIAM M. MACDONALD
Es-iI-
E-i1 Cl
FIG. 5. Contourfor the summedstrengthfunction & of Eq. (79). to Eat a distance I above and below, as shown in Fig. 5. The parametric continuation of & is defined Es,
E; I) = -
+ s,” CtE’ Im RR(E’ -f iZ).
s
From Eqs. (28), (77), and (80) follows directly the desired result. & (Es, E; 0 = c (Es, E; 0
(81)
The parametric form of C (ES , E; I) for an FDS is easily found by inserting Eq. (29) into Eq. (77). f
(Es,
E; I) = ~0s” “’
(@%Y ES, ED) ~0s 273- p(E, ES, ED) sin 24)
t &(E - Es) g(E,
Es,
ED) = $
*9(E7 Es ' ED) = &
(tan-l
(82) EFs,f
(E - Ed2 In (ED _ Es)2
+ tan-l ED - Es r&Y/2 1 + V.Y/~>~ + (r&)2
(83)
The parameters in these equations are the same as those defined by Eqs. (30)-(33). Thus x (Es, E; Z) is a continuous, smooth function whose energy dependence for an FDS is determined by five parameters which are the Lorentz-weighted averagesof microscopic matrix elements. These parameters can be determined by a least-squares of C (Es, E; Z) to 1:” (Es, E; Z) without any arbitrariness in the selection of the fitting points as shown by Fig. 6. For the picket-fence model wherein S, = 4 = 0 and Mi is constant, the SSF in Eqs. (82)-(83) reduces to the form of the result in Eq. (76). Tt is important to note, however, that r, = W + 21. Despite the fact that C (Es , E; I) is the parametric continuation of CR (Es , E) little theoretical or practical sensecan be made of Eq. (73) as it has been used in the analysis of TAS by BLMM. The introduction of a finite smoothing or averaging parameter Z is an absolutely essential step in obtaining a five-parameter strength function for a fragmented doorway state. There is no sensibleway to let Z go to zero
STRENGTH
2300
2350
2400
273
FUNCTIONS
2450
2500
2550
2600
E,(keV)
6. The MMKP-SSF is constructed by using Eq. (77) (dashed line) and compared with the doorway form of this SSF calculated from Eq. (83) with the same parameters as in Fig. 1. FIG.
12.5 _
40Artp,
P) l/2+ .r
2xX,
2350
2400
2450
2500
2560
2600
E, (keV)
FIG. 7. The discontinuous SSF of Lane, Lynn, and Moses [12] given by Eq. (72) is shown with the parametric form of Eq. (83) using the same parameters as in Fig. 1 but with Z = 0.
in Eq. (73). From a practical point of view, the discontinuous function CB (Es , E) can not be used reliably to determine the parameters of the right-hand size in cases of intermediate to weak coupling. This is not apparent from the fits exhibited by BLMM because the eye cannot properly assessthe quality of these fits. The Eq. (83) shows that .the spreading width rlJ is determined by the slopeor derivative of the SSF rather than by its value, as in the case of the SF. Thus fits presented by BLMM must be judged by the accuracy with which the slope of their SSF is determined. It is not sufficient to note whether a continuous curve has been run through most of the steps in the discontinuous function & . This can be seen from Fig. 7 which shows CB with x (Es, E; I = 0). The IAS parameters have the same values shown in Figs. 1 and 6 (although the parameters would not exist for I = O! Cf., Eq. (26)). Note that the fit also depends on Es!
214
WILLIAM
M.
6.
MACDONALD
SUMMARY
Strength functions (SF) and summed strength functions (SSF) are used to extract directly from micros-resonance parameters for high resolution differential cross sections the averages of microscopic coupling matrix elements between a fragmented doorway state (FDS) and “hallway states.” A distinction must be made between resonance parameters of the S-matrix and those of a reactance or K-matrix because their respective SF will have very different energy dependences and will depend in different ways on spreading parameters. Tn the picket-fence model for an FDS the K-matrix widths have a Lorentzian envelope of widths equal to the spreading width rk g 2~ikP/D. In the same model the distribution of S-matrix widths is not Lorentzian except in special cases, and the width of the distribution is the sum of the spreading width and the escape width. Different averaging procedures applied to the same K-matrix parameters also give different SF. The MMKP-SF, which is the Lorentz-weighted average of the reduced widths, has a characteristic asymmetric Lorentzian shape determined by five parameters that are themselves Lorentz-weighted averages of microscopic matrix elements. On the other hand, the sliding-box weighted average favored by A.M. Lane and collaborators is a discontinuous function which only approximates a continuous function in the limit of weak coupling between the doorway and the hallway states. This function is the sum of arctangent functions rather than the asymmetric Lorentzian established for the MMKP-SF. Although the box-weithted SF does not really determine a continuous function in intermediate to weak coupling (which obtains in the IAS, for example), a parametric continuation of this SF can be constructed which is defined for this regime. For an FDS this parametric continuation confirms the arctangent dependence on energy found in extremely strong coupling. These results explain the failure of attempts [23] to use the box-weighted SF to determine the spreading parameters for IAS by equating it to an incorrect parametric form, viz., the asymmetric Lorentzian derived for the Lorentz-weighted MMKP-SF. The apparent failure of the box-weighted SF led BLMM [13] to use the summed strength function (SSF) to analyze the high resolution data on IAS. For an FDS, this SSF was assumed by BLMM to be equal to the integral of the asymmetric Lorentzian found for the MMKP-SF. This assumption can, in fact, be very easily verified for the strong coupling limit of the picket-fence model. However, as with the box-averaged SF, the SSF of BLMM also does not really serve to define a continuous function in the intermediate to weak coupling regime which applies to most IAS. Therefore it has been again necessary to construct a parametric continuation of this SSF which is continuous and does define a parametric form even in intermediate to weak coupling. This parametric continuation proves to be identical to the integral of the continuous MMKP-SF, and it is therefore indeed equal to the integral of the parametric form found for MMKP-SF. Thus, although the BLMM-SSF is a discontinuous function which does not accurately determine the spreading parameters, the form assumed by BLMM is at least not inconsistent. Therefore the parameters found by BLMM contain considerable uncertainty but this arises from the discontinuous nature of
STRENGTH FUNCTIONS
275
their SSF a:nd not from the use of an incorrect form. Nevertheless, the deficiencies of the BLMM analysis of IAS call for a new analysis of this data with a continuous SF which more accurately determines the spreading parameters of the IAS. The results will be presented in another publication [24].
ACKNOWLEDGMENTS The author wishes to acknowledge the support of the University of Maryland Center.
Computer Science
REFERENCES 1. H. FESHBACH, A. KERMAN, AND R. H. LEMMER, Ann. Phys. 41 (1967), 230. 2. R. A. FERRELL AND W. M. MACDONALD, P/y. Rev. Lett. 16 (1966), 187. 3. D. ROBSOI~,Phys. Rev. 137 (1965), B535. 4. W. M. MACDONALD AND A. Z. MEKJIAN, Phys. Rev. Lett. 18 (1967), 706; C. MAHAUX AND H. A. WEIDENMULLER, Nucl. Phys. A 94 (1967), 1. 5. W. M. MACDONALD AND A. Z. MEKJIAN, Phys. Rev. 160 (1967), 730. 6. A. K. KEIXMAN AND A. F. R. DE TOLEDO PIZA, Ann. Phys. 48 (1968), 173. 7. M. DI TOKO, Nucl. Phys. A 155 (1970), 285. 8. W. M. MACDONALD, to be published in Phys. Rev. C. 9. H. FESHBPKH, in “Reaction Dynamics,” pp. 186-191, Gordon & Breach, New York, 1973.
10. A. M. LANE, in “Isospin in Nuclear Physics” (D. N. Wilkinson, Ed.), pp. 509-590, NorthHolland, Amsterdam, 1969. 11. W. MACDONALD, Phys. Rev. Lett. 40 (1978), 1066. 12. A. M. LAPSE, J. E. LYNN, AND J. MOSES, Nucl. Phys. A 232 (1974), 189. 13. E. G. BILPUCH, A. M. LANE, G. E. MITCHELL, AND J. D. MOSES, Phys. Rep. 28C (1976), 145. 14. L. ROSENFELD, Acta Phys. Pol. A 38 (1970), 603. 15. E. P. WIGNER AND L. ELSENBUD, Phys. Rev. 72 (1947), 29. 16. C. MAHA~X AND H. A. WEIDENMULLER, “Shell Model Approach to Nuclear Reactions,” NorthHolland, Amsterdam, 1969. 17. H. FESHBPCH, Ann. Phys. (N.Y.) 5 (1958), 357; ibid, 19 (1962), 287. 18. C. MAHACJX AND H. A. WEIDENMULLER, Nucl. Phys. A 97 (1967), 378. 19. C. BLOCH:, Nucl. Phys. 3 (1957), 137. 20. E. VOGT, Rev. Mod. Phys. 34 (1962), 723. 21. D. ROBSO~J,in “Isospin in Nuclear Physics” (D. H. Wilkinson, Ed.), pp. 461-508, North-Holland, Amsterdam, 1969. 22. P. L. KAPUR AND R. E. PEIERLS, Proc. Roy. Sot. London Ser. A 166 (1938), 277. 23. G. E. MITCHELL, Bull. Amer. Phys. Sac. 22 (1977), 573. 24. W. M. MACDONALD, Strength function analysis of high resolution data on isobaric analog states, to be published.