9Be(α, p)12B reaction and primordial nucleosynthesis

Nuclear Physics A542 ( 1992 ) 97-107 North-Holland

e(a, p)

YSICS

reaction and primordial nucleosynthesis R.N. Boyd'

Institute for Physical and Chemical Research, Wako, Saitama 351-01, Japan

S. Kubono, N. Ikeda, M.Y. Tanaka, T. Nomura, Y. Fuchi and H. Kawashima. Institute for Nuclear Study, University of Tokyo, Tanahsi, Tokyo 188, Japan

M. Ohura and H. Orihara

Cyclotron and Radio-Isotope Center, Tohoku University, Sendai 980, Japan

S . Yun, H. Tokyokawa, M. Yosoi and H. Ohnuma Tokyo Institute of Technology, Meguro, Tokyo 152, Japan Received 2 December 1991 p)12B reaction using 65 MeV Abstract: The high-lying levels of 12 B have been studied via the 9Be(a, a-particles . The reaction protons were analyzed with a quadrupole-dipole-dipole spectrograph, yielding a resolution of about 40 keV. Angular distributions were measured for 20 states ranging in excitation energy from 7.55 to 13.31 MeV. Results of DWBA calculations were compared to the angular distributions to attempt to set limits on the angular momentum transfers . The states seen at excitation eneraieq of 10.57 and 13.31. MeV have potential significance to models of primordial nucleosynthesis via their possible importance to the '~Li(a, n)"B and 9Be(t, p)"B reactions respectively. A new reaction rate for the latter reaction was obtained .

E

12 NUCLEAR REACTIONS 9Be(a, p), E = 65 MeV ; measured o,(O), B deduced levels, J, ir, widths, resonance strengths. DWBA calculations .

1. Introduction Recent models of primordial nucleosynthesis in which the universe is assumed 1-4 to be inhomogeneous at that time ) have shown the importance of two reactions which involve 12 B as the compound nucleus : "Li(a,n)"B and 'Be(t,n)"B . The 12 thresholds for these two reactions lie at excitation energies in B of 10-001 and 12 12.928 MeV respectively . Thus B levels slightly above those energies are of interest, as they might well produce resonant reaction contributions which could influence the predictions of these models of primordial nucleosynthesis .

Correspondence to: Prof. R.N. Boyd, Department of Physics, The Ohio State University, 174W 18th Avenue, Columbus, OH 43210-1106, USA . ' Permanent address : Dept. of Physics, Dept . of Astronomy, The Ohio State University, Columbus, OH 43210, USA .

0375-9474/92/$05 .00 @) 1992 - Elsevier Science Publishers B.V. All rights reserved

R. N. Boyd et aL / qBe(a, P) 12B

98

Li, 0' 2 13 reaction study') performed some years ago identified many levels in the energy regions of interest, and gave some information about their total widths. In the present study we have investigated the 'Be(a, p)'2B reaction to attempt to check those data, and to attempt to learn more about possible spin-parity assignments of the levels seen . While this experiment does not give all the information necessary to calculate reqonant contributions to the thermonuclear reaction rates, as that requires energieS, spin assignments, total widths, and appropriate partial widths, it does provide part of that information. In the case of the 13.31 MeV state, it does allow an estimate of the resonant reaction rate. In sect. 2 we describe the experimental setup used for the '9Be(a, P) ' 2 B reaction study and present the data. Sect. 3 describes basic features of the DWBA calculations, and presents the conclusions which can be drawn by comparing their results to the data. Sect . 4 discusses some astrophysical implications of the present results. Finally, sect. 5 presents our conclusions concerning the levels Of 12 B from the present study. e(7

2.

xperimental details

The "Be(a, p)' 2 B reaction study was performed with the sector-focussing cyclotron of the Institute for Nuclear Study of the University of Tokyo. The 65 MeV a-particle beam was momentum analyzed via the double magnetic analysis system, which provided a beam energy uncertainty of about 20 keV. The beam was brought to bear on a 500 ~Lg/cm2 self-supporting Be foil located in the scattering chamber in the I NS quadrupole-dipole-dipole magnetic spectrometer 6 ) . The reaction protons were detected with a position-sensitive proportional counter) configured to provide not only position but ,AE-signals from the proportional wire region and E-signals from a backing scintillator as well . Reaction products were further defined by the time-offlight between the detected particles and the accelerator reference pulse. Use of both IE-E and time-of-flight signals cleanly defined the various reaction products accepted by the spectrograph detector . A sample proton spectrum is shown in fig. 1 . The sharpest peaks seen, together with the anticipated peak widths from the thickness of target foil used, define the resolution to be about 40 keV FWHM ; many of the peaks seen exhibit characteristic widths well in excess of the inherent resolution of the system. The peaks for which we have significant data are listed in table 1, along with their widths, from which the 40 keV intrinsic peak width has been subtracted (in quadrature). Data were taken at angles of 6, 10, 15, 20, 27 .5, 35, 42.5, 50, 57.5, and 65 degrees; the resulting angular distributions are shown in figs. 2-4. The curves shown are results of DWBA calculations described in the next section. The energy calibration was performed using other reactions which produced peaks of well known energy in the 12 B energy region of interest at the same spectrometer settings as were used for the 9Be(a, p) reaction study. In particular, the 25 14 Mg(a, h ) 26 Mg, 24Mg( a, h )25Mg, 26Mg(a, h )27 Mg, N(a, h) 15 N, "Be(a, h)' OBe,

R. N. Boyd et aP. / "Be(a, p )' 2B

99

400

300

1000

w z z

a

v

ô

800 600

w a

1000 800 600 400 200 0

240

280

320

CHANNEL NUMBER

360

Fig. 1 . Spectrum for the "Be(a, p)`ZB reaction over the range of excitation energies studied in the present experiment . The solid curves represent the peak fits to the data. The extra solid curve in the upper figure represents the profile of the broad state centered at 8.271V1eV.

R.N. Boyd et aL /

100

e(a, p) 12B

TABLE I

Levels obsen,, ed in 12 B

,,Be(7 Li, a)

9 13e(a, p)

E, [MeV] 7 .61 7.70 7.87 7.99 8.15 8.17 8.41 9.42 9.46 9.60 10 .22 10 .42 10 .57 10.90 11 .35 11-58 12.23 12.33 12.76 13 .31

12

L,,

Width [keV]

E,, [MeV]

Width [keV]

3,4,5,6 3,4,5,6 3,4,5,6 3,4,5,6 3,4,5,6 3,4,5,6 3,4,5,6 0,1,2 0, 1, 2 3,4,5,6 3,4,5,6

30 ± V; 100± 30 80 ± 30 135 ±40 260± 80 45±15 40±15 35 ± 10 60± 20 45 ± 10 15 :1: 15 115 :±40 ME 10 30 :± 20 125 :E 60 90 :1: 30 155 ±60 45 ±20 160± 60 55± 15

7 .51*5

<30

7.836 7.937 8.100 8.240 8.376 9.430

60±40 <40 900±200

9.585 10.2 i 0 10.435 10.580 10.887 11 .310 11 .590

60±30 50 --L- 20 75 :±40 50 :1-- 30 40 :E 20 130:E 60 75 :±25

12 .330 12 .770 13 .330

100±30 85± 40 50±20

0,1,2 0,1,2 0,1,2 0, 1, 2 3,4,5,6 3,4,5,6 0,1,2 0,1,2

40±20 85±3(,

C(a, h) 13C , and `O(a, h)"O reactions all produced at least one state in the

excitation region of interest. At-ter correctio for target thICKnesses, i ese produceu a 'third-order equation in position along the etector which reproduced the energies except at the low excitation energy of the various calibration states to about 10 end of the spectrum. The energies of the levels in 12 which resulted from this equation are listed in the first column of table 1 . eaction calculations an

Iscussion

The reaction calculations were performed with the exact finite range DWBA code TW FNR 8 ) . They used optical model parameters of Gaillard et aL 9) for the e+ 9 13e channel, and of Kolata and Galonsky ") for the P+ 12 B channel . These parameters are listed in table 2. The transferred triton was bound in a well of radius and diffuseness 1 .25A 1/3 fm and 0.65 frn, respectively . At the excitation energies of 2 interest, the strongly excited states would be expected to be populated by a 7rpI/2vpI/2 transfer configuration ; the triton angular-momentum transfer is thus limited to L,, --:::: 3. However, the possibility that some states could be populated by d-sh%ell transfer led us also to consider Lt, as large asi. The results of the calculated angular istributions, assuming a variety of angular momentum transfers L,r, are shown

R. N. Boyd et aL / 98e(a, p) "2B

101

C~ b 'a

10 4

I

20

40

60

~j 12 .76 80

8CM

Fig . 2. Angular distributions for 'Be(a, p) to states in 12 B which appear to be characterized by L,, = 0, 1, or 2. The excitation energy of the states represented by the various symbols are indicated to the right. Experimental error bars are indicated where they ace larger than the symbols. The curves represent the results of DWBA calculations discussed in the text.

together with the data in figs. 2 and 3. There it can be seen that only the L,, = 0 to 4 curves represent any of the angular distributions . The L,r = 6 angular distribution peaked at an even larger angle than that for Ltr = 5, so is not shown. The calculated angular distributions were found to be rather featureless; thus they do not identify any particular angular momentum transfer very accurately. However, they do appear to provide some restrictions . The angular distributions shown in fig. 2 all appear to fall off more or less monotonically with angle. Comparison of these differential cross sections with the DWBA results suggests that they are characteristic of Lt, = 0, 1, or 2 distributions. Note that this applies to both the 10.57 MeV state and the 13.31 MeV state. Fig. 3 shows the differential cross sections which were either flat out to a fairly large angle, or which peaked at a fairly large angle, typically at 30' or so. These distributions are seen to be characterized by DWBA angular distributions for which L,,. = 3 or 4. Finally, fig. 4 shows the remaining states . The angular distribution for

M M DeV et aL /

102

Wa, p)'

100 40

0 à

0 &

0 A

Il A

e

0 A

0

a

10

0 A

8.15 ,41

T X

(n 1.-1

10 -0 ZL ~- 40

1

I

Xîî

c~ Z~

b

a60 10

W22 1233

10

Y

4

0

7.87

AM

Fig. 3. Angular distributions for "Be(a, p) to states in ' 2 13 which appear to be characterized by L,, = 3,4, or 5. Otherwise, the caption is the same as for fig . 2.

one of them (10.42 MeV) is fairly sharply forward peaked, but most of the other angular distributions are peaked around 20 0 , suggesting their Ltr to be about 2 units. 4.

stro ysical implications

The 10.57 MeV state may have important astrophysical consequences because it could correspond to a resonance which could contribute to the 'Li(a, n) ' 1 B reaction, which has a threshold at 10.001 MeV. Such a resonance clearly dominates the cross section for 'Li(a, n) " B(g.s.), inferred from the " B(n, a)'Li inverse reaction ") . The 10.57 MeV gate is ideally located in energy to have a major impact at the energies typical of primordial nucleosynthesis . Calculation of the contribution of a state to the thermonuclear reaction rate requires, at least in principle, its total width, alpha partial width, and neutron partial width; the total width can be determined from the present experimt:nt 12) and the neutron partial width from an experiment 13) specifically designed to measure that quantity . The 1157 MeV mate is the narrowest one observed in this experiment ; its width was taken to be 10 keV, with an uncertainty

p)

R.N. Boyd el aL / 'Be(a,

100

0

B

103

1

0

40

12

0

0

0

0

10 40

13

TT

-0 :J_ 10

[3

10 .42 7.61

13 13 X

X

C3

13

X

0 13 :

'a 1

0

1-1

0

b

-0

1

0

100T

0

0

7:

~

8.17 7.70

0

7.99

40

10

40

20

+ T

80

60

ecm

12-23

Fig . 4. Angular distributions for 'Be(a, p) to states in 12 B which are not well characterized by any L,, . Otherwise the caption is the same as for fig. 2 .

of the same value, consistent with our experimental resolution . However the resonance seen ") in the " B(n, a) experiment is about 100 keV wide. Thus, identification of the state seen at 10.57 MeV in the present experiment as the only state associated with the resonance which dominates the "B(n, a)"Li reaction has difficulties . One must consider the possibility either that another state which is unresolved from that we see at 10.57 MeV, but which is very weakly excited in 9 13e(a, p), is actually the state which corresponds to the resonance which dominates "B(n, a), or that the resonance" seen in the " B(n, a) experiment is actually composed of several TABLF_ 2

Optical model parameters Channel a+ 9 13e p+ 12 B

VtI

40 .0 43 .36

rR

aR

W

r,

a,

1 .75 1 .08

0 .495 0 .712

16 .3

1 .75

0 .615 0 .527

V(r)=-V(J(XR)-i

W-4W, :,

1 .21

dx,

WI:)

VI . .

6 .61

)f(xl)+( M,C)

1 .2

8 .16

VI ..

where f(xk ) = [I +exp (xk )] - ,

r,,,.

x, = (r - rjA

1/3

a, - I

r dr

)1a,, .

a,.,,

rc

0.531

1 .4 1 .20

f (XI.11 .)

I

Ref.

10 4

R. N. Bo3,d ei aL / '9Be(a, p)'2 8

overlapping resonances, with the 10.57 MeV state corresponding to one of these states . A weak, but persistent, tail was indeed observed in the present experiment which appeared to overlap the 10.57 MeV state, but which extended to its high excitation-energy side. Furthermore, the state seen -5 ) in 'Be( 7Li, a) at 10.58 MeV is broader, 50 ± 30 keV, than the state we see at 10.57 MeV in 'Be(a, p). Although it might be argued that the uncertainties from the two experiments do just overlap, that clouds the issue. The 10.57 MeV state was clearly the narrowest state seen in the present experiment, while several other states were seen in the 9Be( 7 Li, a) experiment which were narrower than the 10.58 MeV state. Thus it does appear that there may be at least two states at around 10.57 MeV which could be contributing to the "'resonance"' seen in " B(n, a). The results from the neutron partial width experiment are also interesting in that they show that decays of the 10.57 MeV state to the B excited states are important to 'Li(a, n) " B in the region of the resonance. Indeed, that experiment finds that the decays to the excited state are at least as strong as those to the ground state, suggesting a large enhancement for the "Li(a, n) " B(all states) cross section over that to the " B(ground state). This enhancement is in qualitative agreement with that observed in a recent direct measurement of the 8 Li(a, n) " B cross section using a 'Li radioactive ion beam. While the latter experiment was not able to go to as low energies as the indirect methods, it also observed a large enhancement of the 8Li(a, n)"B(all states) cross section over that to just the "B(ground state) . The present experiment also yields an important result for the 13.31 MeV state. The cross section for the 9&(c-, n)" B reaction has been observed ' 5 ), at 0', from about 0.25 to 1 .4 MeV in the laboratory system. Unfortunately, if there wem a resonance corresponding to the 13 .31 M-N state in the compound nucleus it would be difficult to infer such from those data. Thus this resonance could dominate the low energy cross section, but still might inot be seen in the existing 9 Be(t, n) data. Since the 13.31 MeV state is strongly excitled in the present experiment, its strong contrib~ition to the "Be(t, n) reaction cross section requires only that it exhibits neutron decay; it is most unlikely that it would not do so. The present experiment determines the total width of that state to be 55 :1-- 15 keV, and its strength to be a significant fraction of the maximum strength for such a reaction, since the state is strongly excited in 913e(a, p) . Since LIr can be limited to be 0, 1, or 21 from the angular distributions, the spin of the resonance could be any integer value from 0 to 4. However, since the most plausible configuration for the transferred triton to a strongly excited state at this excitation energy would be V2pl/2 7Tpl/2, this would favor a negative-parity transfer, which in turn, would support an Lt, = I assignment, or a J' assignment for the resonance of 1', 2', or 3' . The one obtained by Ajzenberg-Selove et al. 5) was (2'), consistent with the present assignment. While the actual strength factor is sensitive to the optical-model parameters, the 13 .31 MeV state is sufficiently strongly excited by triton transfer that we have assumed the value for the spectroscopic factor to be 0.3 of its maximum possible strength, an assignment

R.N. Boyd et A / 9Be(a, p) 1.),B

105

which is not likely to be in error by more than a factor of three. The triton partial width of a resonance is given ") in terms of its nuclear parameters by

r, = (2h1Rj(2E1jA )1/2 PL(ER, RJOL

(1)

9

where r, is the triton partial width, R. is the nuclear radius, ER is the energy of the resonance, jA is the reduced mass, PL is the penetrability of the barrier, and OL is the spectroscopic factor for the cluster to populate the nuclear state in question . Using this expression, the triton partial width for this state is found to be 4.4 kev, assuming L,, = 1 . Thus only the neutron partial width is needed to determine the contribution of this state to the 913e(t, n)" B thermonuclear reaction rate. In the absence of that measurement it is not unreasonable to assume that the sum of the neutron partial widths to all " B states is very nearly equal to the total width. Then the reaction rate becomes 16) (~27r ( U V) = kT)

3/2

h2

2JR + i F exp (-ER IkT), (2J, + 1)(2J2 +1) '

(2)

whereJR is the spin of the resonance, and J, (J2 ) is the spin of 9Be (3 H) . In the standard notation, the resonant part of the 9Be(t, n) reaction rate is therefore No (o-v),,, = 1 .25 x 10" Tg 3/2 exp (-4.43/ Tg) -M3 S -1 mole- ',

(3)

where T, is the temperature in 109K and No is Avagadro's number. The data for 9Be(t, n) at 0" can also be used to infer a nonresenant contribution to the reaction rate. Those data yield, assuming the cross section to be isotropic, an S-factor of about 4.5 x 105 keV - b, which declines slowly with increasing energy. If we assunle the S-factor to be constant at that value, we can calculate the reaction rate for the nonresonant component from the exp,.-ession 16) NO(av) = 4.34 X 105

1

-T exp (_,T )S CM 3 S- I mole-'

AZIZ2

where S is the astrophysical S-factor, and 7=3

I

(4)

"I

.,ue4Z21Z22 ,h 2 2kT

I

(5)

1/3

These expressions yield a nonresonant reaction rate for 'Be(t, n) of No(av) . ... .. .e, = 3.80 x 1012T~ 2/3 exp(- 14.02 T~

1/3) CM 3 S- I

mole-'

.

(6)

The total reaction rate for the "Beft, n) reaction is the sum of the two expressions given in eqs. (3) and (6), or NO(av) = 3.80 X 1012T~ 2/3 exp (- 14.02 Tg- 1/3)+ 1 .25 x 108T~ 3/2 exp (-4.43 T~') CM 3 S - I mole- '

(7)

106

R.N. Boyd et aL / 'Be(a,

p)'2B

Since the ~ Be(t, n) " B reaction has not been measured directly, this indirect determination of that reaction rate yields the best estimaie of it to date. Note, however, that the uncertainties on this rate are large; the dominant term is the nonresonant component, and it was determined based on the assumption of an isotropic differential cross section . The extent to which this assumption could be in error is not known, although it is not likely to be more than a factor of three, judging from typical low-energy angular distributions. In addition, the value for the spectroscopic factor for the resonant contribution was assumed to be 0.3 ; this could again be in error by as much as a factor of three. 5. Conclusions For the most part, the states identified in the present study confirm the identifications seen in the 9Be( 7 Li, a )12 B reaction study ,5) . Furthermore, the energy associated with each of those states compares well with those of that study. The largest discrepancy observed is at the lower excitation energy end of the spectrum. In doing our energy calibration we used states from well-known nuclides in wellknown reactions, as discussed in sect. 2. However, we did not actually have many such states near the low excitation end ofthe spectrum. Thus detector nonlinearities, coupled with few calibration states near that end of the spectrum, might explain the energy discrepancies (still only around 50 keV) which we did observe. The most notable difference in widths of states was seen for the level previously identified) as lying at 8.1 MeV and having a width of around 0.9 MeV. We observe a consistent dip in our spectra at a point which would have to be in the middle of that peak if it were indeed that wide; this implies that the 0.9 MeV wide level is actually at least two states. Indeed the two broad levels we see at 7 .99 and 8.15 MeV were almost certainly the main contributors to the "8.1 MeV level" observed in the "Be( 7 Li, a) study. The state seen in the present experiment at 8.15 MeV does apparently have a large width, 260 keV. Another broad level is claimed in the e(7 Li, a) study, this being at 9.43 MeV. We identify two states, at 9.42 and 9.46 MeV (identified because one side of the multiplet consistently was much sharper than the other, requiring a fairly broad state beside a narrow one), which together would give about the width observed in the 9Be( 7 Li, a) study. The DWBA fits to the angular distributions did not prove to be particularly definitive, primarily due to the lack of features seen in both the data and in the WBA results. Nonetheless they were able to provide some restrictions on the spin-parity assignments of the states observed. Finally, the astrophysical implications of two of the states observed, those at 10.57 and 13.31 MeV, could, in principle, be important. A recent direct measurement 14) of the 8 Li(a, n) " B cross section determined that cross section at energies above 1 .5 MeV, slightly above the region ofthe dominant resonance, but information at energies below 1 .5 MeV was not obtained in that experiment, so would be useful.

R.N. Boyd et aL

/

9Be(a,

P ) 12 B

107

Assuming all the experiments done to infer the 'Li(a, n) " B reaction rate are correct, the 10.57 MeV level cannot, by itself, be assumed to represent the complete contribution to the resonance which dominates 'he low energy 8 Li(a, n) " B cross section. However, the strong decay of the 10.57 MeV state to the "B excited state does confirm the large enhancement in the reaction rate due to contributions from the 11 B excited states observed in the 8Li(a, n) " B direct measurement 14) . The present study has produced an estimate for the reaction rate for 'Be(t, n) " B, including both a resonant and a nonresonant contribution. Since this reaction has not yet been studied directly with sufficient detail to obtain a rate, the present value gives the best estimate for this reaction rate yet available. This reaction may be an important one in the synthesis of "B in primordial nucleosynthesis, both in the standard model and in some regions of the parameter space of the inhomogeneous models. Since observations of the "B abundance in very metal poor stars are just now becoming available 17), the present 'Be(t, n) rate may be timely in producing the theoretical abundance predictions necessary for meaningful tests of such models . This work was partially supported by the National Science Foundation through grant PHY89-20606. References 1) 2) 3) 4) 5) 6) 7) 8) 9) 10) 11) 12) 13) 14) 15) 16) 17)

J.H. Applegate, C.J. Hogan and R.J. Scherrer, Astrophys . J. 329 (1988) 592 C. Alcock, G.M. Fuller and G.J. Mathews, Astrophys . J. 320 (1987) 439 R.A. Malaney and W.A. Fowler, Astrophys. J. 333 (1988) 14 T. Kajino and R.N. Boyd, Astrophys . J. 359 (1990) 267 F. Ajzenberg-Selove, R. Middleton and J.D. Garrett, Phys. Rev . C12 (1975) 1868 S. Kato, D. Hasegawa and M. Tanaka, Nucl. Instr. Meth. 154 (1978) 19 S. Kato, S. Kubono, T. Miyachi, S. Ohkawa, Y. Fuchi, M.H. Tanka and M. Yasue, Nucl. Instr. Meth. A287 (1990) 499 M. Igarashi, Code TWOFNR, 1973, unpublished P. Gaillard, R. Bonuhe, L. Feuvrais, M. Gaillard, A. Guichard, M. Gusakow, J.L. Leonhardt and J .R. Pizzi, Nucl . Phys. A131 (1969) 353 J .J. Kolata and A. Galonsky, Phys. Rev. 182 (1969) 1073 T. Paradellis, S. Kossionides, G. Doukellis, X. Aslanoglou, P. Assim akapoulos, A. Pakou, C . Rolfs, and K. Langanke, Z. Phys. A337 (1990) 211 S. Kubono, R.N. Boyd, N. Ikeda, T. Nomura, Y. Fuchi, H . Kawashima, M . Ohura, H. Orihara, S. Yun, H. Toyokawa, M . Yosoi, H. Ohnuma, 1. Tanihata and T. Kajino, Z. Phys. A338 (1990) 459 S. Kubono, N. Ikeda, M.H . Tanaka, T. Nomura, 1. Katayama, Y. Fuchi, H. Kawashima, M. Ohura, H. Orihara, C.C. Yun, Y. Tajima, M . Yosoi, H. Ohnuma, H . Toyokawa, H. Miyatake, T. Shimoda, R .N. Boyd, T. Kubo, 1. Tanihata and T. Kajino, Z. Phys. A341 (1991) 121 R .N . Boyd, 1. Tanihata, D. Hirata, N. Inabe, T. Kubo, T. Nakagawa, T. Suzuki, M. Yonokura, XX Bai, K. Kimura, S. Kubono, S. Shimoura and H .S. Xu, Phys. Rev. Letts . 68 (1992) 1283 V.I. Serov and B .Ya . Guzhovskii, Sov. J. Atom. Energy 12 (1962) 1 C.E. Rolfs and W .S. Rodney, Cauldrons in the cosmos (University of Chicago Press, Chicago, 1988) D. Lambert, private communication, 1991