LA-7828-T Thesis UC-34a Issued: May 1979

Neutron- Radius Differences and Isovector Deformations from n+ and ir Inelasiic Scattering from 18O

Steven G. Iversen*

- NOTICE- This report was prepared as an accounl of work sponsored by the United Slates Government. Neither the United Stales noi the United States Depart men t of Eivrgy. nor any of their employees, nor any of their contractors, su be oni factors, or their employees, makes any warranty, express or implied, or assumes ai:y legal liability or responsibility for the accuracy, complett-wss or usefulness of any information, apparatus, product ur process disclosed, or represents thai its use would noi *Visiting Staff Member. infringe privately owned rights.





Chap ter



A. The EPICS Facility 16 B. The 18O Targets 24 C. The Spectrometer 30 D. Detection System 33


A. Angular Binning 43 B. Peak Fitting 44 C. Normalized Yield 45 D. Cross Sections 45 E. Corrections 47 F. Beam Contaminants 48 G. Experimental Results 50 H. Overall Errors 58


A. Experimental Information 78

B. Theoretical Description 93


A. 108 B. Inelastic Scattering 121 C. Discussion of Inelastic Results 140

iv Appendix



I. The EVENT ANALYZER 161 II. The TEST FILE 165 III. The DISPLAY PACKAGE 169 IV. Run Analysis I70 V. Calculation of Missing 171 C. CHAMBER CALIBRATION 173

I. Front Chamber Calibration I73

II. Drift Corrections for Rear Chambers 176


I. Angular Binning IS3 II. Peak Fitting I88 III. Yield Calculation 191 IV. Calculation 194 V. Total Errors 200 E. THE OPTICAL MODEL 202

I. Multiple Scattering Theory 202 II. Optical Model Potential 205 III. Corrections to the Optical Potential 208 IV. The Program DWPI 211





1-1. Characteristics of in the -nudeon

system. 4

1-2. Fitted values for IT p and TT p total cross sections. 6

II-l. Characteristics of the EPICS pion beam. 21

II-2. Straggling AE due to substances in the EPICS beam. 21

II-3. 180 target content and target window impurities. 26

II-4. Characteristics of detectors used in these experiments. 32 II-5. Sealer, TDC, ADC quantities. 41

III-l. Summary of integrated cross section results for states 55 in 180.

III-2. Experimental energies for ail states observed in the 57 present experiment.

III-3. Experimental energies for the 4j,02,22 unresolved 57 triplet of states at 3.56,3.63,3.92 MeV.

IV-1. Energy levels of i80. 80

IV-2. (a) Spectroscopic factors for the single trans- 82 fer reactions 170(d,p)180 and 19F(d,3He)180. (b) Spectroscopic factors for the pick-up 83 reactions 18O(p,d)170 and 18O(d,t)17O.

IV-3. States in 18O from 16O(t,p)18O. 84

IV-4. Experimentally determined percentages of components of 85 wave functions for states of 180.

IV-5. (a) Transition strengths for the 2* and 37 states in 90 180 by inelastic scattering. (b) Transition strengths for states in 180 by lifetime 91 measurements. (c) Measurements of the quadrupole moment of the 2\ 91 state of 1&0.

IV-6. (a) Theoretical predictions for electromagnetic 94 transition rates for states of 180. (b) Theoretical presictions for the quadrupole moment 94 of the 2i state of 180.

VI Table

IV-7. Theoretical predictions for the wave functions of the 95 Ot and 2f states of 180.

V-l. Potential amplitudes used in DWPI for our analysis. 106

V-2. Deformation lengths calculated from the 164 MeV data. 124

V-3. Deformation lengths calculated from the 230 MeV data. 125

V-4. Potential parameters bo.bi used in calculations for 147 the 230 MeV data of this experiment.

V-5. Calculations of transition matrix elements M and M 153 for the 2"f state of 180 using electromagnetic tran-^ sition rates.

D-l. Resolution widths observed in the present experiments. 192

D-2. Ca) ir+p data fit by Dodder. 196 (b) ir~p data fit by Dodder. 197

D-3. Predicted HCrr1,^1)!! cross sections according to 198 analysis by Dodder.

D-4. normalization factors, N, for all cases. 199

D-5. Total percentage errors calculated for the ratios p2 201 of summed cross sections.



1-1. IT p and IT p total cross sections. 5

II-l. Experimental areas at LAMPF. 17

II-2. A diagram of EPICS. 19

II-3. A diagram of the EPICS spectrometer. 22

II-4. The spectrometer coordinate system. 23

II-5. An energy loss spectrum taken from a run during this 25 experiment.

II-6. Exploded diagram of the arrangement used to maintain 29 18 the frozen H2O .

II-7. Schematic diagram of EPICS showing the relative posi- 31 tions of the detectors and magnetic elements.

II-8. Diagram of the fast electronics used at EPICS for 39 this experiment.

III-l. Composite summed spectrum for 164 MeV IT scattering 51 from 180.

III-2. High excitation region in ir1 scattering from 180. 52

III-3. Angular distributions for elastic scattering of 164 MeV 60 T\- from 180, for 2° binning of the spectrometer angular acceptance.

III-4. Angular distributions for excitation of the 2j state of 61 180 by 164 MeV ir1, for 2° bins.

III--5. Angular distributions for excitation of the unresolved 62 triplet of states in 180 - the 4f at 3.56 MeV, o£ at 3.63 MeV and l\ at 3.92 MeV - by 164 MeV TT1, for 2° bins.

III-6. Angular distributions for excitation of the 3a state of 63 180 by 164 MeV •n±, for 2° bins.

III-7. Angular distributions for elastic scattering of 164 MeV 64 fr~ from 180, for 1° binning of the spectrometer angular acceptance.

III-8. Angular distributions for excitation of the 2j state of 65 180 by 164 MeV IT*, for 1° bins.

viii Figure

III-9. Angular distributions for excitation of the 3~ state 66 of 180 by 164 MeV TT1, for 1° bins.

III--10. Angular distributions for elastic scattering of 230 67 MeV TT1 from 180, for 2° bins.

III-ll. Angular distributions for excitation of the 2i state 68 of 180 by 230 MeV TT1, for 2° bins.

111-12. Angular distributions for excitation of the unresolved 69 triplet of states in 180 - the 4f at 3.56 MeV, o£ at 3.63 MeV and it at 3.92 MeV - by 230 MeV TT±, for 2° bins.

111-13. Angular distributions for excitation of the 3i state of 70 180 by 230 MeV TT*, for 2° bins.

111-14. Angular distributions for elastic scattering of 230 MeV 71 TT* from 180, for 1° bins.

111-15. Angular distributions for excitation of the 2i state oi 72 180 by 230 MeV ff*, for 1° bins.

111-16. Angular distributions for excitation of the 3i state of 73 180 by 230 MeV IT*, for 1° bins.

111-17. A comparison of spectra for IT scattering on 180 from 74 SIN and EPICS.

111-18. Angular distributions taken at SIN for elastic scattering 75 of 163 MeV TT1 from 180.

111-19. Angular distributions taken at SIN for excitation of the 76 2t state of 180 by 163 MeV IT*.

111-20. Angular distributions taken at SIN for excitation of the 77 37 state of 180 by 163 MeV IT1.

IV-1. Energy levels and decay branches for low-lying states 79 of 180. + V-l. Experimental points for 164 MeV TT elastic scattering 113 in the vicinity of the minima.

V-2. Our 164 MeV elastic scattering data with curves calcu- 115 lated with DWPI.

V-3. Our 164 Me7 elastic scattering data with curves calcu- 116 lated by Liu.

IX Figure

V-4. Elastic scattering minima calculated with DWPI for 118 164 MeV ir* using average bo.bi amplitudes in the Kisslinger potential.

V-5. Elastic scattering minima calculated with DWPI for 119 164 MeV TT° using the proper amplitudes for i& and it" in the Kisslinger potential.

V-6. Our 164 MeV elastic and inelastic data with curves 129 calculated with DWPI.

V-7. Our 230 MeV elastic and inelastic dita with curves 130 calculated with DWPI.

V-8. Our 164 MeV elastic and 2\ data with curves calcu- 136 lated by Arima et al.

V-9. Our 230 MeV elastic and 2\ data with curves calcu- 137 lated by Arima et al.

V-10. The 230 MeV elastic and 2\ data with curves caicu- 138 lated by Arima et al by varying the amplitudes bo,bj in the Kisslinger potential.

V-ll. Our 164 MeV elastic and 2\ data with curves calcu- 141 lated by Lee, Lawson, and Kurath using ZBM wave functions.

V-12. Our 164 MeV elastic and 2\ data with curves calculated 142 by Lee, Lawson, and Kurath using LSF wave functions.

V-13. Our 164 MeV elastic and 2\ data with curves calculated 143 by Lee, Lawson, and Kurath using s-d wave functions.

V-.14. Our 230 MeV elastic and 2\ data with curves calculated 149 by using Arima1s adjusted parameters in DWPI.

V-15. Our 230 MeV elastic and 2\ data with curves calculated 150 with DWPI using our best fit parameters.

B-l. Block diagram of the EPICS EVENT ANALYZER. 163

B-2. TEST FILE used during this experiment. 166

B-3. Block diagram of the main steps in calculating missing 172 mass for the energy loss histograms.

C-l. Typical drift chamber ion drift time and drift position 179 distributions.

C-2. Typical histograms of drift chamber event types. 181 x Figure

D-l. Measured relative spectrometer acceptances for 164 MeV 185 71"- with smooth curves of the chosen average acceptances.

D-2. Measured relative spectrometer acceptances for 230 MeV 186 ir* for jaws in both the open and partially closed positions, with smooth curves of the chosen average acceptances.

D-3. An example of a relatively hard to fit ir spectrum. 190

E-l. Charge density distribution for 180 measured at Bates 216 with curve obtained from our parameterization with a harmonic oscillator form ot the density.



Steven G. Tversen

ABSTRACT We have measured angular distributions for elastic and inelastic scattering of 164 and 230 MeV ir* from 18O. We present angular distri- butions for elastic scattering and for excitation of the 2J transition at 1.98 MeV, the 37 transition at 5.10 MeV, and an unresolved triplet of states consisting of the 4+ at 3.56 MeV, the o£ at 3.63 MeV, and the 7% at 3.92 MeV. The angular ranges covered are 17.5-80° for 1^=164 MeV and 17.5-70° for ^=230 MeV. In addition we have observed the 17 transition at 4.45 MeV and several transitions at excitation energies above 6 MeV, but detailed analysis of these transitions has not been undertaken. The elastic scattering angular distributions for 1-^=164 MeV have been analyzed with the optical model in an attempt to determine the difference in neutron and proton rms radii in 180. Our result is Arn_ =0.03± 0.03 fm. This is compared to theoretical predictions and other experimental results for 180. For the 2* transition we obtain an integrated cross section ratio Zo(iT~)/Za(-n+) of 1.86 ±0.16 for the 164 MeV data. We have analyzed this ratio in the collective model with the DWIA optical model code DWPI with suitable modifications to obtain separate neutron and proton deformation lengths (3R)n and (3R)-. Cur result for the 2\ transition 18 in 0 is (gR)n/(BR)p= 1.68± 0.20, indicating a large isovector deior- mation in this state. This is equivalent to a ratio of neutron to proton transition matrix elements Mn/iL, = (N 6n) / (Z 3 ) =1.25 x 1.68 = 2.10±0.25. These results are compares to inelastic transition rates from the literature and to determinations of Mjj/Mp using electromagnetic transition rates in . For the 37 transition we obtain an integrated cross section ratio of 0.89 ±0.06 for the 164 MeV data and 0.85 ±0.07 for the 230 MeV data. Applying the same techniques used for the 2i transition yields (3R)n/(BR)p = 0.69 ± 0.08 and 0.74 ±0.08 for 164 and 230 MeV respectively. That this ratio is smaller than unity is interpreted to be due to the partial blocking of core neutron excitations by the two valence in 180 outside thp. lo0 core in formation of the 37 state. The status of the analysis of the elastic scattering and zt tran- sition at 230 MeV is not satisfactory. The of the DWIA fits by us and others is poor, and the integrated cross section ratio for the 2t of 1.58±0.13 to (BR)n/(6R)p= 1.24± 0.14 which is inconsistent with the 164 MeV result. Only by the strictly phenomenological technique of varying the strength of the interaction potential to obtain reasonable fits to the elastic data is it possible to improve the fit to the 2t data. However, even after this is done we still obtain (BR)n/(gR)p= 1.26± 0.14 for the 2t transition at 230 MeV. We are led to the conclusion that this indicates a failure of the DWIA method. Our result is discussed in com- parison to the theoretical work which has been done using our data.


The study of is entering a new era. This

new age has been heralded by the availability of medium energy

probes: and which interact only electromagnetically

with the and nucleus, and nucleons, , and which

interact strongly. The present study relates the use of medium

energy pions as a probe of nuclear structure.

Although the pion was discovered in 1947 and has been used for experiments in ever since, it has only been in recent years with the advent of high intensity (10 -10 pions/ second), high resolution pion beams that tha use of pions for nuclear structure purposes has become widespread. One such "mespa factory" is LAMPF (Clinton P. Anderson Los Alamos Physics

Facility), where the experiments discussed here were performed.

Pions interact with nuclei via the . Thus, simply as a new probe of nuclear properties, pion experiments would bt valuable because they would allow comparison with results from experiments with other hadronic probes. However a more rewarding role for a new proba is its ability to illuminate properties which were previously difficult or impossible to study with older probes.

Certain special and properties of pions have long held the hope that the pion could be such a probe. In this dis- sertation we present evidence that this hope is being fulfilled.

Some of the interesting properties of pions are:

(1) The pion is the lightest known strongly interacting particle, having a mass of 139.56 MeV, or about 1/7 the nucleon mass. This means that recoil corrections in the analysis of pion-nucleon col-

lisions are small. It also means that the kinematics must be treated


(2) The pion exists in three charge states, IT" and ir°; in other words as an isospin triplet with T»l, T - +1,0,-1. The ir and ir z

have the same mass (139.56 MeV) and lifetime (26 ns); and the TT has a mass of 134.96 MeV and lifetime of 0.84 x 10 s.

(3) The pion has spin zero. It is thus a and can be created cr destroyed singly under the proper kinematic conditions.

An important consequence of che zero spin is the simplification of the theoretical treatment of TT-nucleus interactions.

(4) There are several resonances in the ir-nucleon scattering amplitudes. This is in contrast to nucleon-nucleon amplitudes which show no resonances.

(5) As a consequence of the strong resonances, at certain energies the pion is strongly absorbed by the nucleus. Such 'true absorption* deposits about 140 MeV in the nucleus, which is eq'jai to the for as many as a dozen nucleons.

Perhaps the two most important properties of the pion are its isospin and the of the ir-nucleon interaction. The existence of the pion in an isospin triplet enables single- and double-charge exchange reactions. Such reactions can be used to study exotic neutron or proton rich nuclei. Moreover, they may be sensitive tools for the study of TT-nucieus reaction mechanisms.

Of practical importance is also the fact that IT and IT induced reactions can be performed in an experimentally symmetric fashion, the change between them in many cases consisting of simply changing the polarities of magnetic elements in the incident parti- cle channel and the outgoing particle spectrometer. As mentioned above, the ir-nucleon interaction is characterized by several resonances. The properties of these are summarized in Table 1-1. Of these resonances the most important is the so-called (3,3) resonance (2T=3,2J«3) at T - 196 MeV with a total width of 122 MeV. For intermediate energy pion physics, that is pions with T < 400 MeV, it is this resonance which dominates the scene com- pletely. This is illustrated by the total ir-N cross sections shown in Fig. 1-1, and listed in Table 1-2.

If we denote

: -*• ir p Elastic p

°elastic •n~p •*• TT~p (1.1)

a~e x : 7T p + ir°n it can be shown that in terms of isospins T=l/2 and T=*3/2 and their associated scattering amplitudes A(l/2) and A(3/2)*


Elastic o?!2^1'2) +A(3/2)|2 (1.2)

a n —.1 A ("\ / 9 \ — A f ^ /0^ I ^ ex y ' Further, by the optical theorem

a + = a (3/2) a Im k T T °O/2) + (1.3) aT = 1/3 aT + 2/3 aT(l/2) . TABLE I-l. Characteristics of resonances in the pion-nucleon system.

T (MeV) Wave T(JT) Name Mass(MeV) Width(MeV)

196 + P33 3/2(3/2 ) A(1236) 1237 122 796 S31 3/2(1/2") A(1640) 1630 160 + 1387 F37 3/2(7/2 ) A(1950) 1940 210 517 l/2(l/2+) N(1470) 1460 260 604 D13 1/2(3/2") N(1520) 1515 115 620 1/2(1/2") N(1550) 1525 80 876 1/2(5/2") D15 N(1680) 1675 145 902 + F15 l/2(5/2 ) N(1690) 1690 125 949 1/2(1/2") N(1710) 1715 280 1937 G17 1/2(7/2") N(2190) 2190 300

aFrom Ref. 1. 300 350 TV(MeV)

Figure 1-1. TT p and ir~p total cross sections. The points are from the measurements of Ref. 2; and the curves are from a Breit-Wigner fit by the authors of Ref. 2, with typical values listed in Table 1-2. TABLE 1-2. Fitted values for the data measured In Ref. 2 for ir p and Tr~p total cross sections. The values in this table correspond to the curves in Fig. 1-1.

T^MeV) °tot(ir+P> atot(7r"p)

100 54.0 19.9 125 103.3 36.9 150 166.1 57.2 175 198.4 67.1 200 175.2 59.9 225 132.7 46.5 250 97.4 34.9 275 72.7 26.4 300 55.8 20.5 It follows that if the T-l/2 amplitude is zero, that is A(l/2) = 0,


elastic ' elastic * ex (1.4) a* : a " - 3:1.

At the peak of the (3,3) resonance the approximation A(l/2) = 0

is extremely good. This conclusion is verified by recalling that

the Breit-Wigner single-level formula gives cr (max) = 2TT X2(2J+1) =

for J-3/2. For the lab of 196 MeV, which cor-

responds to a CM momentum of 231.5 MeV and ft of 0.85 fm, the formula

gives cr (max) » 183 mb. This is to be compared to the value of about 200 mb from Table 1-2.

Another way of reaching the same conclusion is to note from

Fig. 1-1 that the ratio O /a is very nearly equal to 3. It may be noticed that this remains true over quite a wide region of energy. Thus over this large energy range pion-nucleus physics is considerably simplified because one can make the assumption, with- out introducing much error, that only T=3/2 amplitudes are involved.

There are some very interesting consequences of this fact. Not only are ir p(=ir n) elastic scattering cross sections a factor 9 larger than TT'n(=ir p) cross sections, but all other cross sections which are in any way proportional to the same amplitude (A(3/2)) behave similarly. Thus, for example, inelastic scattering ampli- tudes, which under certain model assumptions are directly related to the elastic scattering amplitudes, will show the same differences between TT and ir~ scattering. In the distorted wave theory of inelastic scattering, if an

inelastic trar.sif.ion id described within the collective model frame-

work, i.e. either as a rotation of a nonspherical shape or as a

vibration about a mean spherical shape, it is characterized by two

parameters: a deformation parameter 8 which is the measure of the

'eccentricity' of the ellipsoidal shape, and a radius R which is the

radius of the mean sphere. In this model the optical model potential

is a deformed potential whose spherical part gives rise to elastic


d •> d2 V(r) - V(r-R) - (8R)-§- V(r) + 1/2(3R)2 "fr V(r) + ... . (1.5) The first order transition potential is

V a (3 R) dV(r * * (1.6) a (B^R) • V(0) .

In the usual models for pion-nucleus interactions, e.g. the Kisslinger model, the potential V(r) is constructed in the impulse model appro- ximation from the pion-nuclson interaction and has components pro- portional to the elementary pion-proton amplitude (A ) and pion- neutron amplitude (A ). Thus in a transition in which N neutrons and Z take part

is [a(inel)j a [(3R)n • NA^ + (BR)p • ZA ] (1.7) or

h [a(if) ] = K"'[(3R)n • NA^ + (gR)p . ZA^] (1.8)

+h + + (3R)p • ZAu+p ] . (1.9) The constant K contains information on the reaction mechanism and is + _ approximately equal for ir and TT . Equating

A Vn * Vp " (1 1Q)

A A Vn " TT-P " "

and dividing through by A we get

h + lo(ir~)] = K«[(3R)n - N * (A /A~) + (BR)p • Z] (1.11)

+ h + [a(v )] = K-[(BR)n • N + (BR)p • Z • (A /A~)] . (1.12)

If we assume complete (3,3) dominance then A /A =3, and it follows


h [ff(TT~) ] = K- [3N- (8R)n + Z- (6R) ] (1.13)

]^ = K«[N«(pR)n + 3Z-(gR)p] . (1.14)

These two equations are the basis for our expectation that pion scattering can tell us more about the isovector nature of inelastic transitions [i.e., ($R) - (8R) ] than can the scattering of other .

For a transition with (SR) = (3R) = (3R), i.e., all isoscalar n p and no isovector component

P = La(Tr")/a(TT+)]3§ = (3N+Z)/(N+3Z) (1.15)

=1.0 for a self-conjugate nucleus

= 1.12 for 180, for example. If the average (gR) and (3R) are different, or equivalently if not n p

all the protons and neutrons participate in the transition, the

ratio p can be quite different from above. For example,

Z*0, pure neutron transition - p = 3

N=0, pure proton transition - p * 1/3 .

Thus the experimental measurement of cross sections for TT and TT

inelastic scattering for a given state can give us an insight into

whether all N neutrons and Z protons in the nucleus participate

(purely collective state), or whether just a few neutrons and/or

protons participate in the transition. In other words one can

investigate the isovector components of an inelastic transition.

This has been a long-standing hope in nuclear structure physics.

For example, Bohr and Mottelson in 1975 stated:1* •"The interaction and the presence of a neutron excess Imply that the deformation of the neutron and proton density distributions may be somewhat different and thus give rise to an isovector potential with a deformation different from that of the isoscalar poten- tial. The deformation of the isovector field could be obtained by comparing the cross sections for excita- tion of rotational states by inelastic scattering of neutrons and protons, of fr and TT", or by observing the isobaric analogs of the rotational excitations in the direct (p,n) process."

Many attempts have been made to unravel these features by

making the kind of comparisons suggested by Bohr and Mottleson.

Most such studies have compared experiments involving inelastic

scattering of different hadrons, and the results have been un-

rewarding. The primary reason for the inconclusive results is the

involved in comparing experiments, neutron and proton

scattering for example, for which both the experimental techniques

10 and the methods of data analysis are quite different. A large part

of the uncertainty can arise from absolute normalization differences

between two separate experiments, because such errors are generally

not quoted in the experimental results.

An early investigation into isovector components in transition

rates was made by Bernstein5 who compared electromagnetic transition

rates (obtained from lifetime measurements, Coulomb excitation, etc.)

to isoscalar transition rates from (a,a1) scattering experiments for

the 2 and 3. transitions in N>Z even-even nuclei.

In a measurement of electromagnetic transition rates one measures

only the proton transition matrix element M . In an experiment invol-

ving a hadronic probe, which interacts with both neutrons and protons,

the cross section is proportional to |M + M J2, where M is the neu-

tron transition matrix element. Comparing, for example, (a,a1) cross

sections to electromagnetic transition rates for the same transition

can to determination of the ratio M /M , which is equivalent to a

measurement of the isovector nature of the transition. This follows

from the definitions of isoscalar (IS) and isovector (IV) transition

rates IM n +P M 2 B(IS) I (1.16)

P B(IV) (1.17)

The isoscalar and electromagnetic transition rates will be equal when the ratio of neutron to proton contributions to the transition,

Mn/M , is equal to N/Z, i.e., a purely collective transition. This

is equivalent to having the ratio of deformations (which are pro-

11 portional to the transition matrix elements), e.g., 3 '/3 » equal act em to unity. Bernstein found the ratio 3 V3 to be within 30% of unity ota em

for all but a few cases for nuclei spanning the .

Subsequently Madsen and co-workers6 calculated differences in

transition rates for (p,pf) and (n,n') reactions and compared them

to electromagnetic rates. They were able to approximately reproduce

the experimental results which indicate that 3 t/3 differs from pp em

unity by about 15-25% for single-closed-shell nuclei (nuclei with

either valence neutrons or protons). From these results they pre-

dict isovector components (i.e. differences in $ , and 3 ,) for pp nn

these nuclei of up to 20%. Unfortunately the experimental uncertain-

ties, particularly for available (n,n') data, in total are larger than

the predicted effect.

More recently, Bernstein, Brown and Madsen7 have re-examined

the evidence for isovector transition rates. They observe that 3 '/3 > 1 for single-closed-shell (SCS) nuclei with valence ota em neutrons (i.e* M /M > N/Z). This is a result of the more probable n p participation of the valence neutrons in the transition compared

to participation from protons in the closed core. Similarly for

SCS proton valence nuclei 3 ,/3 < 1, or equivalently M /M < N/Z.

Departures of 3 ,i/3 from unity are predicted to be 10-40%. The

experimental results differ from unity by 10-60% for about two-

thirds of the cases considered (these results are different from

the 30% obtained by Bernstein in Ref. 5 because of the more ex-

tensive experimental results in the ensuing years). The results of

12 this study indicate that some sensitivity to the isovector com-

ponents is possible using conventional hadronic probes.

More recently Bainjm and co-workers have made similar compari-

sons8 of (p,pr) and (n,n') scattering using the results of proton

experiments with T ranging from 15-25 MeV and their own 11 MeV

neutron work. They find that the ratio B ,/& , differs from unity nn pp

by 10-20% for various N=50 and Z=50 nuclei. This magnitude of iso-

vector components agrees with the predictions of Madsen et al.

However, once again the sensitivity to isovector effects in such

analyses is small and the errors involved are large, because of the

completely different nature of the experiments being compared.

It is obvious that there is great interest in determining isovector components in inelastic transitions, and that more sensitive experiments are necessary. In this respect pions appear to be an excellent tool. The existence of the (3,3) resonance gives pions a valuable differential sensitivity to neutrons and protons.

Furthermore, experimental errors are minimized in a comparison of

IT and IT excitation of inelastic transitions because of the sym- metry of IT and TT scattering experiments and data analysis. It is true that the theory of pion-nucleus interactions is not well understood, but current theories attempt to construct pion-nucleus potentials from a knowledge of elementary pion-nucleon amplitudes.

This is a more fundamental approach than that used for proton- avd neutron-nucleus interactions, in which empirical optical model parameters are utilized in fitting the scattering data. This gives

13 us confidence that the ratio 6 /3 can be obtained reliably from n p

TT /TT~ inelastic scattering.

With the knowledge provided by the foregoing discussion we can

make an intelligent choice of the nucleus we wish to study. The

first major criterion is a nucleus for which the structure is well

understood so that our experimental results can be tested with the

known structure information. The light nuclei, those in the calcium

region and lighter, have been the most extensively studied. Next

we want a nucleus which exhibits collective activity so that some

inelastic states are strongly excited. However, the states should

not be of a purely collective nature, for which the ratio p2 = 1

and there would be no observable effect. Furthermore, the nucleus

should not be self-conjugate, for in such a case we would expect

the core neutrons and protons to participate equally in inelastic

transitions and there would be no observable enhancement of IT or

IT scattering. Thus the best choice would be a nucleus with a

closed core of either protons or neutrons, and one or two valence

neutrons or protons. Such nuclei are 13C, 180, and lf2Ca, for

example. Because of the less severe requirements on the necessary

energy resolution we chose 180. In retrospect 180 appears to be an

excellent choice - in a very recent paper Bernstein has once again

taken up the problem of the ratio M /M in single-closed-shell plus

valence nucleon nuclei, and he finds that this ratio is largest for

180. This result will be discussed further in the last chapter of

this dissertation.

14 Having discussed the reasons for doing pion scattering on nuclei and the choice of a target for such a study, we proceed with the description of the experiments, analysis of the data., and dis- cussion of the results in the following chapters.


The experiments reported in this dissertation were performed on

the Energetic Pion Channel and Spectrometer (EPICS) facility at the

Clinton P. Anderson Los Alamos Meson Physics Facility (LAMPF). The

LAMPF accelerator is an 800 MeV linear proton accelerator capable

of simultaneously accelerating H and H~ beams and supplying them

to several experimental areas. Experimental "Area A", which con-

tains several secondary beam lines, including EPICS, is shown in

Fig. II-l. The design intensity of the H beam into Area A is 1 mA,

but when our experiments were run (March 1978) the current was about

300 vA. A fairly detailed description of LAMPF and its facilities

is given in the LAMPF Users Handbook.10

A. The EPICS Facility

The complete EPICS facility consists of a high intensity pion

beam channel which produces pion beams from about 50-300 MeV and a

high resolution spectrometer. At present the EPICS system consists

of one spectrometer with a momentum analyzing capacity < 700 MeV/c,

and the channel which produces pions over a range of about

100-425 MeV/c.

EPICS was designed as a high intensity pion facility to make

possible a wide range of experiments, with the hope of measuring

differential cross sections down to a few nb/sr. To accomplish this

without sacrificing momentum resolution a dispersed beam design was

chosen.11 This means that, in one beam dimension (vertical at EPICS),

16 BEAM I KM * If



Figure II-I. Experimental areas at LAMPF. "Area A" is in the center and EPICS is in the upper left-hand quadrant of Area A.

17 the momentum of each particle is correlated with its verticle posi-

tion in the beam. The beam produced at the EPICS channel focal

plane, where the scattering target is placed, has a vertical extent

of 20 cm for a momentum bite of ± 1%.12

The scattered are analyzed by the spectrometer

which rotates in the horizontal plane. This requires a nearly

parallel horizontal beam. At the EPICS target the horizontal beam

dimension is about 7 cm. The outgoing particle momentum analysis

is done in the vertical direction in the spectrometer.

The EPICS facility is shown in Fig. II-2. The channel por-

tion consists of four dipole bsnding (BM01-4), a crossed

fields particle separator, four sets of movable jaws (FJ01-4), and

several focusing and steering magnets (FM's and SM's). The particle

separator removes protons, and to some extent electrons, from the

pion beam without disturbing the pions. The first set of jaws,

FJ01, helps to define the channel acceptance, the jaws between

BM03 and BM04, FJ04, can be used to adjust the momentum acceptance

of the channel. The focusing and steering magnets are intended

for fine tuning of the pion beam a~>.d are particularly effective at

low and high pion energies. They were not used during the course

of this experiment, mainly because the effect of these magnets on

the channel optics is not yet fully understood.

The vertical dispersion is created by the four dipole bends

in the vertical plane. This readily eliminates particles of the

opposite sign and different momentum from those desired. The






Figure II-2. A diagram of EPICS. The channel is to the left of the scattering chamber, and the spectrometer is to the right. characteristics of the channel portion only were tested in a tune-

up experiment which is reported in Ref. 12. The properties of the

beam at the target, measured in the tune-up experiment, are sum-

marized in Table II-1.

The spectrometer consists of three quadrupole and two dipole

magnets. A drawing of the spectrometer alone is shown in Fig. II-3,

and the spectrometer coordinate system in Fig. II-4. The quadrupole

triplet images the dispersion plane of the scattering target one-to-

one at its focus. Near this point are placed four multiwire pro-

portional chambers (MWPC). These are the front chambers (Fl-4),

and they record the positions (x,y) and angles (8,<|>) of each

detected particle. After bending through the dipoles the particles

are detected by four drift-type MWPC (Rl-4) placed near the spec-

trometer 'focal plane1. These rear chambers also record x,y,9,<(>

for each particle. The spectrometer can be rotated from about

-10° to +120°.

The EPICS system measures both the momenta of the incident

particle and those of the reaction products. For these purposes

the information required consists of eight chamber quantities -

positions and angles in the front (x^y^^,,^) and rear (Xf^yR*9^^

chambers - and the field settings of the channel and spectrometer.

These enable reconstruction of the trajectories of the outgoing

particles through the spectrometer dipoles, yielding their final

momenta. These same quantities also allow tracing of the trajec-

tories backward from the front chambers through the quadrupoles to

the scattering target. This specifies the interaction position in

20 TABLE II-l. Characteristics of the EPICS pion beam.

Beam Content Pion Flux (107 ir/s) T^MeV) y e P n TT

100 15(15) 3(50) 2(~35) 100 2.0 0.45 200 5(5) <1(8) 5(~4P"> 100 6.6 1.2 300 2(2) -0(2) 15(~6_0, 100 5.0 1.0

Relative numbers of various beam contami-ants, where pions = 100. Number outside (inside) parentheses are with (without) the separator operating. The pion flux is that at the time of this experiment, with a primary proton beam current of 300 yA-

TABLE II-2. Straggling AE FWHM for 230 MeV pions due to substances in the EPICS beam.

Material Thickness AE FWHM (inches) (KeV)

1. Scattering chamber window Mylar 0.010 13 2. bag He 12 2 3. Spectrometer entrance window Mylar 0.003 4 4. Front chamber box window Mylar 0.003 4 5. Front chamber box He 14 2 6. Front chambers 10 7. Front chamber box window Mylar 0.003 4 8. Spectrometer exit window Mylar 0.010 13

Total 52 KeV Targets

is 1. Thin H20 0.12 107 18 2. Thick H20 0.22 196







Figure II-3. A diagram of the EPICS spectrometer. The quadrupole focus, at the center of the front chamber box, is located at the position marked "detector". (o)

1 ••BEAM B



Figure II-4. (a) The spectrometer coordinate system is right-handed, with +z in the beam direction. 9 and <$> are the angles of the part- icle trajectory relative to the z axis (central trajectory) in the x-z and y-z planes respectively, (b) Coordinate system relative to the spectrometer elements.

23 the target in the dispersion plane, and thus the momentum of the

incident particle. (For details of spectrometer operation and

tuning see Appendix A).

Since the EPICS spectrometer is not an energy-loss spectrometer,

particles with a given energy loss do not arrive at any one point

on the focal plane. Only after software analysis of the incoming

and outgoing particle momenta, as described above, does one obtain

energy loss histograms of the type shown in Fig. II-5 for IT scatter-

ing on 180 at 230 MeV. The resolution (FWHM) for these runs is

about 500 keV, which was typical for the present experiments.

A significant contribution to the overall resolution is from

energy straggling due to various materials in the path of the beam.

Using the program STRAGL13 we have calculated the straggling due to

these sources and listed the results in Table II-2. Subsequent to

this experiment the spectrometer vacuum system has been extended and

the various windows reduced, and with a very careful tune-up a resolu-

tion of about 240 keV has been obtained. However there remain sources

other than those indicated in Table II-2 which must contribute about

200 keV to the measured resolution. The sources of these additional

contributions are not understood at present.

B. The 180 Targets

The 180 targets used in this experiment were in the form of

ice, enriched in 180. Their physical parameters are listed in

Table II-3.

24 l8O(r,f')l9O T, = 230 MeV

I U I I I 4 6 8 10 14 16 EXCITATION (MeV)

Figure II-5. An energy loss spectrum taken from a run during this experiment. The most prom- inent states observed are marked with vertical lines, and the states we have fit are labelled. TABLE II-3. 180 target content and target window impurities, as fractions of the total material.

Window Fraction of Target Material Impurity' Total

thin 0.884 (0.312 g/cm2) 17 0.018 0.040 foil (0.0013 cm) 27Al 0.006 Mylar (0.0064 cm) 0.009 12, 0.023 lH 0.019

thick a 0.914 2 o (0.553 g/cm ) L7 0.010 160 0 0.040 Mylar (0.0076 cm) 0.006 12, 0.016 0.013

Aluminum from Mylar superinsulation has been neglected.

26 A single target could not be used for the entire angular range.

For angles less than 40° a thinner target was needed in order to

resolve the 1.98 MeV transition from the elastic peak. This allowed

resolution of the 1.98 MeV transition in to angles as small as 17.5°.

For smaller angles the peak due to scattering from hydrogen in the

target obscured the 1.98 MeV transition, and no data was taken at angles smaller than 17.5°.

We had learned from a previous run of this experiment that light impurities caused great trouble in data analysis. The helium in the helium-filled scattering chamber contributed an unacceptable amount of background. It was therefore decided to use a large vacuum scattering chamber. Further, in the previous experiment a liquid HO18 target was used, and in order to confine it between 2 two thin Teflon windows (2 x .005 cm) it was found necessary to add

1.5% gelatin in order to increase the viscosity of the . The in the Teflon windows and the in the gelatin con- tributed significantly to the background. This made it necessary to take empty target (frame plus Teflon windows, but no H 018) runs at 2 each data point to enable proper background subtraction. Overall, the performance of this target left a lot to be desired.

Since a water target, even with the undesirable gelatin, could not be used in the vacuum scattering chamber without thick windows, it was decided to develop an ice target for the present experiment.

The ice target could be self-supporting and therefore not require gelatin or thick windows. Windows of minimal thickness, to prevent loss of water through sublimation into the vacuum of the scattering 27 chamber, could be used. It was determined that no losses took place

in vacuum with windows of .0025 cm Mylar. The design of the target

holder is shown in Fig. II-6.

The target was filled with liquid HO18, leaving adequate room 2

for expansion on freezing, and flat aluminum plates were clamped

outside and the water frozen in an ordinary freezer. After the

water had frozen the plates could be removed to yield a uniform ice

surface. The second copper frame, as in Fig. II-6, fitted with a

two-stage Freon pump,91* was then clamped to the first frame. The

pump was rated as being capable of attaining a low temperature of

-80° C, and operating at a capacity of 90 watts.

It was determined that the refrigerator unit, which had only

peripheral contact with the target, could not cope with the approxi-

mately 15 Watt radiant energy from the scattering chamber environ-

ment input into the target and frame. The problem was compounded

by the low thermal conductivity of ice. It was therefore found

necessary to surround the target by a shield. A sheet of

"superinsulation" (.0005 cm Mylar, aluminized on both surfaces) was

wrapped around the target. With this arrangement the target re-

mained stable for a period exceeding one week.

The thin target was made simply by pressing the water in the

frame to a smaller thickness, enabling use of the same size target

frame for both targets. The thin target had, in addition to the

aluminized Mylar shield, a sheet of .0013 cm aluminum foil on each

side for added insulation. The thickness of the various elements

in the windows and the wrapping are listed in Table II-3. The

28 to refrigerator


Figure II-6. Exploded diagram of the arrangement used to maintain the 18 frozen H20 . The refrigerator probe fits into a groove on the top of the frame which is attached to the other frame piece containing the ice.

29 background from these components was small, and could be calculated

and subtracted whenever necessary. We return to this point in the

next chapter.

C. The Spectrometer

The targets were housed in an evacuated scattering chamber 90 cm

in diameter. The outgoing particles exit through a .025 cm Mylar

window. There is an air space of approximately 23 cm between this

window and the entrance to the spectrometer quadrupoles. This space

was filled with helium in a thin plastic bag. The quadrupoles and

dipoles are under separate vacuum, with the box containing the front

chambers between them. The front chamber box contains Fl-4 and the

SI - which was not used in the present experiments, and

therefore removed. During these experiments the front chamber box

was filled with helium.

The rear detectors, Rl-4 and S2-3, were in air outside the

spectrometer exit window. In addition, a 2.5 cm-thick lucite

Cerenkov counter was mounted between S2 and S3, but during the pre-

sent experiments it was not used except as a proton absorber.

An ion chamber (IC2) was used as a pion beam monitor and was

placed on the outside wall of the scattering chamber at 0° to the

direct beam. There was a 2.5 cm lucite proton absorber mounted in

front of IC2. A schematic diagram of the spectrometer showing the

magnets and the particle detection system is presented in Fig. II-7.

A list of the active areas of the multiwire proportional chambers,

, and ion chamber is given in Table II-4. The detectors

will be discussed in more detail in the following sections.

30 RI.R2 3.R4



CHANNEL SPECTROMETER Figure II-7. Schematic diagram of EPICS showing the relative positions of the detectors and magnetic elements. The quadrupoles are depicted as lenses, the dipoles as boxes. A line at the right(left) side of a dipole represents an upward(downward) bend. The focussing properties are depicted by a solid line for the x, or vertical, dimension, and a dashed line for the y, or horizontal, dimension. OJ TABLE II-4. Characteristics of detectors used in these experiments.

Active Area Detector (X x Y x Z cm) Materials Gas Mixture Misc.

Front chambers Anode wires Fl - 4 28 x 20 x 1.9 = 0.002 cm gold-plated 73.6% Argon and tungsten 26.3% Isobutane 0.1% Freonb Rear chambers Cathode wires PI - 4 90 x 20 x 1.4 = 0.076 cm copper(10%)-clad aluminum Delay Lines ~ Q-clad 2-3-3a Scintillator EMI 9813 B S2 140 x 19 x 0.64 phototubes Scintillator EMI 9813 B S3 140 x 21 x 0.64 phototubes Ion chamber IC2 76 x 22 x 7.7 90% Argon 300 V battery 10% CO for plate voltage @ 780 mm absolute Monitor telescope scintillators EMI 9813 B MT1-3 20 x 12.5 x 0.64 phototubes

3M Company 'Bubbled through isopropyl alcohol at 20° C. D. Particle Detection System

1. Multiwire Proportional Chambers. A multiwire proportional

chamber (MWPC) is a high resolution, position-sensitive device for

detection of charged particles. A good general description of the

properties of MWPC is given in Ref. 14. Such chambers have the

disadvantage of obtaining maximum efficiency at rather low count

rates. For example, the front chambers in use on the EPICS spectrom-

eter at the time of these experiments were considered reliable at

instantaneous count rates less than 200 KHz. This limit could easily be reached at forward angles when extraneous particles from the direct beam illuminate the front chambers. To avoid this pro- blem at small angles thinner targets and smaller channel jaw openings need to be used. In general, however, these chambers presented no count-rate related problems in our present experiments. These chambers have the additional advantage that they present a low mass to the beam, so that their contributions to beam straggling is small.

The eight MWPC deployed at EPICS were all built at LAMPF.1S»16

They are two dimensional, containing an anode plane at positive high voltage and a parallel cathode plane, with its wires perpendicular to those of the anode plane. The avalanche occurs to the anode plane and the cathode carries a signal caused by positive ions formed during the avalanche. Both the cathode and anode in each chamber are operated as delay-line readout devices,i.e., the signal wires for the cathode and anode feed into two separate delay lines, each read out at both ends. The time difference between

33 signals received at each end of a delay line corresponds to a posi-

tion in that direction in the chamber. The time sum from both ends

of a delay line should equal the transit time of the delay line,

plus the drift time for the ions to reach the wire through the

chamber gas. Thus the time sum is uncertain by the ion drift time

for each event. If we take the time sum for the anode (AS) and sub-

tract the time sum for the cathode (CS) the ion drift time cancels.

Tnis quantity, AS-CS, is defined as the chamber check-sum, and is

a stringent test of a good event in each chamber.

2. Front Wire Chambers. The four chambers Fl-4 are packaged

in two pairs. Each chamber contains an anode with 4 mm wire spacing

and a cathode with wires spaced 1 mm. The wire planes in each pair

are less than 2 cm apart, and the pairs are spaced about 20 cm apart.

The use of the four chambers for calibration of each other is dis-

cussed in Appendix G.

It is assumed that an ionizing event in a chamber trigge-s a

signal on only one of the anode wires. If the difference between

the times of the two signals pa the anode is displayed in a histo-

gram each anode wire can be distinguished. The spatial resolution

can be no better than ±1/2 wire spacing, or ± 2 mm. The induced

cathode signal, on the other hand, is spread over several neighboring

wires (actually about 10 wires) near the anode avalanche and dis-

plays a continuous time spectrum. Because the time signal from the

cathode is characteristic of the centroid of this distribution the

resolution can be better than the wire spacing, and in fact is

about 0.5 mm for our chambers.

34 3. Rear Wire Chambers. As for the front chambers, the rear

chambers Rl-4 were contained in two sets of two each, Each chamber

contained an anode at positive high voltage with wires spaced 8 mm,

and a cathode with wires spaced 1 mm. The anode wires of all four

chambers were stretched in the y-direction, and span the momentum

dispersion (x) direction. There is an important difference in the

mode of operation of the rear chambers compared to the front

chambers: the rear chamber anodes were operated in the drift mode,

hence the larger wire spacing. To remove the ambiguities associated with the drift direction of the ions in the chamber, the anode planes of the two chambers in each pair were displaced relatively by 4 mm

so that the anode wires of one chamber were centered between those of its mate. The details of drift corrections may be found in

Appendix C.

The overall spatial resolution of the rear chamber anodes was about 0.25 mm. The cathode resolution was about 3 mm, appreciably larger than the wira spacing due to signal degradation on the long

signal wires.

4. Scintillators. At the time of these experiments provi- sions were made for only three scintillators: SI, S2, and S3.

However, SI was removed because of its effect on the resolution.

With SI removed a good scintillator event was defined as the logi- cal AND of S2 and S3 signals (denoted S2«S3). With a 7 m path length through the spectrometer between scintillators SI and S2

(supposing SI were being used), one could measure time of flight and separate pions and electrons over the entire range of EPICS

35 pions energies - T = 50-300 MeV. However, without Si this ability

is lost. It was felt that the background in our inelastic

scattering experiments would not be a serious problem. Further dis-

cussion of the problem of backgrounds will be discussed briefly in

the next chapter.

Although the particle separator was used during taking of all

the data for these experiments, there remains a. proton contamination

of about 3% at T = 164 MeV and - 8% at T = 230 MeV. To exclude

these protons from scintillator events a 2.5 cm-thick lucite absorber

was placed between S2 and S3.

5. Beam Monitors. In order to normalize the experimental

yields it is necessary to have a relative measurp of the pion beam

flux at the scattering target. This measurement must be reproducible

to as high a precision as possible over the duration of the experi-'

ment. Experiments at EPICS use several beam monitors - the re-

dundancy is valuable as a cross check on the stability of each. The

monitoring is done as either a direct measure of the pion beam flux

at the scattering target, or indirectly by measuring the primary

proton beam flux.

a. lonization Chamber //2. Originally two ion chambers

were placed inside the scattering chamber for direct monitoring of

the EPICS beam. Because they tended to leak into the chamber vacuum

they were replaced by a single ion chamber (IC2) outside the scat-

tering chamber at 0° to the beam. An absorber of 2.5 cm-thick

was positioned in front of the IC to remove protons from

36 the beam. At a spectrometer angle of about 25° IC2 interfered with the spectrometer and had to be removed for smaller angles. b. Monitor Telescope. For the present experiment a scattered beam monitor telescope (MT) consisting of three 20 x 12.5 cm (vert, x horlz.) by 0.64 cm-thick paddle scintlllators arranged colinearly, was used in triple coincidence, viewing the scattering target. It was placed at -30° to the pion beam, outside a 2.5 cm- thick lucite window in the scattering chamber. The window effective- ly removed protons from the scattered beam at the MT. This telescope was found to be an accurate motitor of beam- target interactions. Since IC2 and other incident pion beam monitors were also available, the telescope was primarily used to monitor the stability of the ice target. c. Primary Beam Current (1ACM02). The 1ACM02 monitor is an integrated signal from a toroid current-measuring device surrounding the primary proton beam. It is supplied by the accelerator opera- tions group to the experimental areas. The signal is gated by the accelerator "beam-on" gate so that it is only on when the beam is on, thus eliminating any "beam-off" background. If the proton beam remains on steadily and is properly steered so that all of it strikes the A-l production target, 1ACM02 is a good of the EPICS pion flux. d. Beam on Target (BOT). This monitor is an ion chamber placed in the A-l target which measures (primary proton) beam - (production) target interactions. This integrated signal is also supplied by the accelerator operations. It is a more reliable

37 measure of the pion flux in the EPICS channel than 1ACM02 because

it is sensitive to not only variations in the primary beam but also

to any variations in the A-l target (e.g., a hole in the target).

During the operation of EPICS, IC2 and BOT have been judged the

best monitors of the pion beam intensity. We found BOT to be ex-

tremely stable with respect to IC2 throughout the experiment. BOT

was therefore used for all beam normalizations. This was particular-

ly important at forward angles at which .IC2 could not be used.

6. Electronics. A block diagram of the electronics is given

in Fig. II-8. The raw detector signals are collected in standard

NIM devices, which are interfaced to a PDP-11/45 computer through

CAMAC modules. The pulse height and time information for each event

are written onto magnetic tape during data taking.

An evenu in the MWPC is not accepted unless a signal is received

from each end of both delay lines. The front chambers are used to

form the chamber event signal (Fl OR F2)•(F3 OR F4) = FRONT. Similar-

ly a scintillator event is defined, when signals are received from

both ends of the scintillators, by S2-S3. A good hardware event is

defined by EVENT = (S2-S3)-FRONT. When the computer is ready to

accept an event CAMAC provides a BUSY signal which together with

the EVENT forms the computer trigger, EVENT-BUSY. The trigger

allows the event into the computer software for further processing.

Both the EVENT and the EVENT-BUSY, along with real and accidental

signals from the various detectors, are input to sealers. The ratio

of the scaled EVENT-BUSY to EVENT is che computer live time - a

measure of the event-taking rate capabilities of the system.

38 Figure II-8. Diagram of the fast electronics used at EPICS for this exper- iment. The event circuit is at the top, and chamber circuits at the bottom. Raw signals enter from the left. Arrows represent the various sealer, ADC, TDC, and trigger inputs.

CO With the software used for this experiment we could process

20-25 events per second. This was sufficient to process all events

for ir scattering at spectrometer angles beyond about 40°. At

smaller angles the computer was run in "MAY PROCESS" mode. In this

mode all incoming events are written on tape, but an event is pro-

cessed on-line only if the computer is not busy. In contrast, if

data is taken in "MUST PROCESS" an incoming event must be processed

by the computer before another event is accepted, and only pro-

cessed events are written to tape. The advantage of running in

MAY PROCESS are obvious - the tapes can be replayed to retrieve

all events written, and no events are lost due to computer dead time.

A list of all scaled quantities, TDC, and ADC inputs is given

in Table I1-5.

7. Software. The event rejection and further processing are

done in the computer. A more detailed discussion of the software

details will be given in Appendix B. Briefly, the data processing

is done with a program called the EVENT ANALYZER in conjunction


lates all chamber and scintillator quantities from the raw times

and pulse heights, and projects the trajectory information to a

plane at the quadrupole focus and to the spectrometer 'focal plane'.

Also calculated is the target projection information x_,y,6_,„,

which enables calculation of the energy loss corresponding to the

event, and the storage of the event in the energy loss spectrum.

The TEST FILE performs a set of logical tests on each event.

These tests are of two types: "micro-tests" and "macro-tests".

40 TABLE II-5. Sealer, TDC, ADC Quantities8

Sealer TDC ADC

1. EVENT gate - RUN • 3EAM GATE S2P S2P 2. EVENT • BUSY S2N S2N 3. S2P singles S3P S3P 4. S2N singles S3N S3N 5. S2 - (S2P • S2N) RF 6. S3P singles Pb 7. S3N singles Nb 8. S3 - (S3P ' S3N) Rb 9. (S2 • S3) Lb 10. (S2 • S3) ace 11. Fl all 12. F2 all bfor 4 front 13. F3 all and 4 rear 14. F4 all chambers 15. Rl all 16. R2 all 17. R5 all 18. R6 all 19. Fl • (S2 • S3) 20. F2 • (S2 • S3) 21. F3 • (S2 ' S3) 22. F4 • (S2 • S3) 23. Rl • (S2 • S3) 24. R2 • (S2 • S3) 25. R5 ' (S2 • S3). 26. R6 • (S2 • S3) 27. IC1 gate = RUN 28. IC2 29. 1ACM02 30. BOT 31. BG • beam gates 32. S2P 33. S2N 34. S3P 35. S3N

t"(N) denote positive (negative) x direction; R(L) positive (negative) y direction; the label "all" with the chamber quantities means that a signal was received from both ends of both cathode and anode. These four signals are denoted P, N, R, L in the TDC column for which they were measured separately. The RF in the TDC column is the accelerator Rf signal. A dot product denotes the logical AND of two quantities.

41 Micro-tests are applied directly to raw or computed quantities. They

can be cuts (windows), or bit tests which check a particular compu-

ter bit for whether it is set or not. "Macro-tests" are logical

\ND or OR combinations of micro-tests, or previous macro-tests.

The DISPLAY PACKAGE enables construction and visual display (on

a Tektronix 4010 oscilloscope display) of histograms containing

either raw or computed quantities from the ANALYZER, and manipula-

tions on the data in the histograms. A histogram can be gated by

any test from the TEST FILE. At the end of each data run the data

can be saved; including the sealer and test outputs, and the histo-

grams. All on-line manipulations of the histograms can also be per-

formed on the saved histograms. By carefully choosing the display

histograms the user can avoid the need for much of the subsequent

replay of data.


In this chapter we discuss the analysis of the raw data to ob- tain differential cross sections. However, the majority of details will be presented in Appendix D. Of particular significance is the use of the hydrogen in the target for absolute cross section calcu- lations; the sets of hydrogen cross sections used are listed in

Appendix D. After discussing the method of analysis, we present the resultant data and compare it to the other available data on pion scattering from 180 at 163 MeV.17

The first step in data processing was to replay the data tapes imposing tighter restrictions on the criteria for good events. Care- ful cuts were made on the target x and y distributions in order to eliminate possible bad events in their tails. This removed about 5% of the total events. The energy loss histograms resulting from re- play under these conditions were used for extraction of final results.

A. Angular Binning

The spectrometer acceptance is about 6°. Of this approximately the central 2.5°, e±1.25°, subtends an approximately constant solid angle at the target. In our analysis of the data we have used the data either in the 2° bin around the center of this region from 6-1° to 6+1°, or in two 1° bins centered around 9-0.5° and 8+0.5°. Cal- culation of yields is identical for each bin. Only the pion survival fraction (SF) and solid angle, dfi, differ slightly from one bin to the next. Because these factors are known for the various bins, once we have the peak areas in each bin the yields can easily be calculated.

43 Details of the solid angle determination are presented in Appendix D.

B. Peak Fitting

An example of the energy loss spectra to be fitted has been shown

in Chapter II. The customary fitting method for the peaks in these

spectra has been via the interactive program PEKFIT"8 - capable of fit-

ting spectra on-line, or off-line using the saved histograms. PEKFIT

employs a skewed Gaussian shape to fit from one to three peaks simul-

taneously. For reasons discussed in Appendix D this routine was not

used for final data reduction, but was used on-line as a rough check

of data characteristics.

The final analysis was done with the fitting program AUTOFIT.19

This program applies a user-supplied standard line shape to perform

a least squares fit to the data. The peak areas are normalized to

the raw counts above the background contained within the limits of

the standard peak shape. (For this reason all known peaks must be

fit in order to get accurate areas when the peaks are spaced closely

enough to overlap). The exact level of the background is input by the

user and can take three forms: (1) linear, constant background; (2)

linear, sloping; or (3) non-linear background, in which case the given

background points are grouped in threes, a quadratic shape calculated

for each group, and the shapes smoothly joined.

From the areas calculated by AUTOFIT we first calculate a normal-

ized laboratory frame yield, and convert this to a CM frame cross sec-

tion. The errors in peak areas calculated by AUTOFIT are propagated

through the calculation of the cross section. These errors result from

44 the combined peak and background statistics plus an error due to the

estimated difficulty in fitting the peak. These statistical and fit-

ting errors are the only ones shown in the tabulations and curves of

the data.

C. Normalized Yield

The net peak area, A, is used to calculate the normalized lab fran"* yield. The yield is termed normalized because it is divided by one of the beam monitors, effectively normalizing to the (relative) incident pion flux. The monitor telescope (MT) was used to monitor the scattered beam. Our definition for normalized lab frame yield is:

Y. . = A/(CF-SF-dJ2-MT-N) (3.1)

lab where SF is the pion survival fraction over the spectrometer path length, d£2 is the relative spectrometer solid angle, CF is a correction term for computer live time and chamber efficiency, and N is the number 2 of target nucleons per cm for the unrotated target. Descriptions of these factors and how they are determined are presented in Appendix D.

D. Cross Sections

Because we know neither the absolute pion flux in the EPICS beam, nor the absolute spectrometer acceptance, we must use a known cross section to extract absolute differential cross sections for 180. This is easily done by using the hydrogen in the target to normalize the oxygen yields into cross sections. The advantages of this process are that the use of target hydrogen to calculate the oxygen cross sections cancels target thickness effects and corrects for possible target thick-

45 ness variations, as well as cancelling absolute solid angle and ab-

solute flux effects.

Due to our ignorance of the absolute pion flux and spectrometer

acceptance, we can only write a proportionality between lab frame

yield and CM frame differential cross sections for oxygen: f§ (3.2) where g is the Jacobian factor for the lab to CM transformation and

Y is the lab frame yield. A similar expression can be written for hydrogen:

§ (H)cM a g(H)Y(H)lab. (3.3)

Dividing (3.2) by (3.3) and rearranging gives (the constants of propor- tionality cancel and the. result is exact) :

18 18 f ( 0)cM - g( O).Y(^O)lab.[||(H)cM /g(H).Y(H)lab]. (3.4)

The last factor with a dependence on the hydrogen yield is the hydrogen normalization factor. The normalization factors calculated for use in this experiment are presented in Appendix D, in conjunction with the discussion of the irp cross sections used.

The hydrogen peak was within the focal plane acceptance for 8£40°.

For larger angles separate runs were made at magnet settings which would bring the hydrogen peak to the same position as the 180 elastic peak in the data runs. As the equation above shows, it is necessary to have a reliable set of up cross sections at the same energy in order to cal- culate the 180 cross sections. These up cross sections are generally not available and must be obtained by interpolation within a compiled data set of up cross sections. The procedure we have adopted is dis- cussed in detail in Appendix D.

E. Corrections

Two background corrections have been made in the data. These

are: (1) background from isotopic impurities in the H^O18, and

(2) background from the windows on tha ice target.

(1) From Table II-3 we see that about 6% of the oxygen in the target

is not 180. The elastic scattering from the 16»170 will appear as a

peak under the 180 elastic peak. Because 160 has no excited states below 6 MeV and the relative amount of 170 is so small, we have assumed

that these impurities do not affect the inelastic yields. The contam-

inant contribution in the elastic peak will vary due to yield differences in the 16»170 angular distributions relative to that of 180. We must,

therefore, have data on these nuclei to make a proper subtraction.

Data exist for pion elastic scattering on 160 at 162 and 240 MeV.20

These have been used to calculate the uumber of counts expected from the impurities, where we have treated all the target impurities as ^0.

The counts are subtracted from the total observed counts in the elastic peak.

Over most of the angular range of this experiment AUTOFIT cannot separate the isotopic impurities from the 180 elastic peak. In this case the single peak area is the area of 180 plus impurities. At more backward angles (>60°) the impurities are visible as a shoulder on the elastic peak and a doublet can be fit. In these cases the calculated impurity contribution is subtracted from the total doublet area.

47 As a result of this impurity subtraction we need to make an ad-

justment to the yields. Because the 180 yields result from only the

94% 180 in the target, but the hydrogen normalizations to absolute

cross sections result from the 100% hydrogen of the target, we must

correct for this difference. After the contributions due to oxygen

impurities have been subtracted we need to multiply elastic and in-

elastic yields by the factor 1/.94 = 1.064.

(2) The target was confined between two 1 mil (.00254 cm) Mylar

windows, and wrapped with several layers of aluminized Mylar super-

insulation. The amounts of impurities from these sources have been

listed in Table II-3. The amount of aluminum is small enough to

neglect. The oxygen has been included in the isotopic impurity sub-

traction described above. The hydrogen content has been neglected.

Angular distributions for elastic and inelastic scattering of pions

from 12C are available at 162 and 226 MeV.21 This enabled us to

calculate and subtract the proper amount of background due to the

carbon in the windows- However, the 12C groups move rapidly with

angle relative to 180 groups and this has to be kept in consideration.

The 12C elastic peak is unresolved from the 180 elastic peak at 6<35°

at 164 MeV and G<30° at 230 MeV, but it moves through the 180 2*

group in the region from ?5°-80° at 164 MeV and 57.5°-70° at 230 MeV.

Furthermore, the 12C 2 at 4.4 MeV moves through the 180 3. at angles

near 30°.

F. Beam Contaminants

Serious sources of background in the EPICS beam and spectrometer

48 are protons, muons, and electrons. In addition there is an appreciable

overall background due to neutrons produced around the accelerator, but

there is little chance of events from this source being mistaken for


Protons in the EPICS beam were largely removed by the separator, which was used during this entire experiment. Remaining protons were

excluded from the event trigger by the lucite absorber between S2 and

S3, and from the beam monitors IC2 and MT by absorbers in front of them.

Muon background at the spectrometer focal plane comes from two sources: (1) beam muons scattered into the spectrometer, and (2) muons from pion decays in the spectrometer. Because muons interact only elactromagnetically we expect contributions from source (1) to be small except at very small angles. At our smallest angle of 17.5° we do not expect the number of scattered beam muons to be large. Muons from source (2) reach the focal plane at a variety of positions with a variety of trajectories, and will be rejected by the software cuts on acceptable trajectories. The remaining few muons are expected to be randomly distributed across the focal plane and constitute a more or less flat background which is removed during peak fitting.

In many of the spectra at forward angles there is a noticeable shoulder on the high energy (left) side of the pion elastic peak.

This has been tentatively identified as muons resulting from late forward decays in the spectrometer, which cannot be rejected I/ any software cuts. Fortunately the contribution due to these is never more than a couple of percent of the elastic peak.

49 The electron contamination comes almost entirely from channel

electrons produced at the A-l target (experimental electron pro-

duction cross sections can be found in Ref. 22). At our energies

the proportion of electrons to pions is already small (: ae Refs. 12

and 22 ), and we expect the number scattered inti ' he spectrometer

to be much smaller for the same reasons as for beam muons.

G. Experimental Results

The primary emphasis in this experiment is on the investigation

of the characteristics of the 2 and the 3~ transitions in 180. How-

ever, since the spectrometer has a relatively flat acceptance over a

momentum range of 10% (approximately -5% to +5%) an excitation energy

range of about 20 MeV in 180 was, in principle, accessible. However,

because of the presence of hydrogen in the target it was possible to

observe weak inelastic groups at excitation energies greater than 5

MeV for 8^40° only when the very strong r^p elastic peak did not ob-

scure them. In order to look into possible weak excitation of levels

we have examined spectra summed over several adjoining angles in order

to improve statistics. Several of these spectra are reproduced in

Figs. 111-1,2. The following empirical conclusions can be drawn from

these spectra:

(1) There is little evidence for the excitation of any 1 states

(these are the states at 4.46, 6.20, 7.62 and (3.04) MeV).

(2) Evidence for excitation of almost all known low-lying 2 and

3 states is quite clear. The clearly excited states are the 2. at

1.98 MeV, 2* (plus some 4*) at 3.92 MeV, 3~ (plus some 2*) at 5.10

MeV, 3~ at 6.40 MeV, 2* (plus some 3~) at 8.21 MeV.

50 3500 I I I I I III-

3000 = 164 MeV


2000 - r 1(175-22.5°) 2(175-22.5°) 2(175-30°) 2(25-30°) DATA X 10 DATA X 10 O 15001- DATA X 10 o



4 6 8 10 12 14 16 EXCITATION (MeV)

Figure III-l. Composite summed spectrum for 164 MeV TT~ scattering from 180. Tne -2 to +3 MeV region results from summing runs at 17,5,20,22.5°; the 3-7 MeV region from summing runs at 25,27.5,30°; the 7-13 MeV region from summing runs at 17.5,20,22.5°; and the 13-18 MeV region from sum- ming all runs from 17.5 to 30° (the spectrum is broken up in this way so that the hydrogen peak does not appear). All states with definite or tentative J^ assignments are marked with vertical bars: the 2+.3~, 4+,6+ states are marked with bars 2,3,4,6 units long respectively. All other states are marked with bars one unit long. In addition to the levels indicated tliere are thirteen unidentified levels between 8.4 and 9.9 MeV, five others between 10.5 and 11.2 MeV, and nine above 12.6 MeV, including the T=2 states which are marked at 16.40 and 17.02 MeV.

51 5000 1 i i 1 (a) r^)'8O 4000 - TT= 164 MeV 60* S o-(fl): 40* CO 3000 z i s o o o 2000 - r

2 [, 32 2 4, 3 1000 r

i 1 i 8 10 12 14 !6 EXCITATION (MeV)


2000 -

V) 1500 -

3 O o 1000-

10 12 14 16 EXCITATION (MeV)

Figure III-2. High excitation region in IT scattering from 0 for (a) T = 164 MeV, the sum of runs from 40-60°, and (b) T = 230 MeV, the sum of runs from 40-55°. Prominent states are labelled with their probable Ju. Using the areas in the 37 peaks the ir~ spectra have been scaled so that the relative peak areas are equal to the relative T&fr~ cross sections in the region of interest.

52 (3) The 4* state at 3-56 MeV is observed only at large angles when

the excitation of the 2» state at 3.92 MeV has become weaker. The

4* state at 7.12 MeV is also clearly seen at these angles.

(4) It is difficult to identify states of higher angular momenta,

or states at energies much above 9 MeV. Certain highly suggestive

'bumps' are however observed at =9.5, 10.6 and 11.7 MeV. Not much

is known about the J71" of the dozens of states in 180 near these energies.

However, it is tempting to identify the bump at 11.7 MeV with the known

6 state at 11.69 MeV which has been suggested as a possible member of

the rotational band consisting of the 0 ground state, the 2^ at 1.98

MeV, and the 4* at 7.12 MeV.58

(A) In the forward angle summed spectra of 17.5° - 22.5° an =3

MeV wide enhancement is seen between 9 and 12 MeV excitation. It

obviously contains the 9.5, 10.6 and 11.7 MeV bumps within it.

(5) We find no evidence, beyond statistical fluctuation, for the

population of the T = 2 states at 16.4 and 17.0 MeV where the break-

up continuum is considerable.

Since there is insufficient statistics in any but the few states

below 6 MeV, and it is not possible to obtain detailed angular distri-

butions for any but these states, we will confine our discussion to

the following states: the olastic, 2 at 1.98 MeV, the unresolved

triplet of states - 4* at 3.56 MeV, 0* at 3.63 MeV, 2* at 3.92 MeV - which we will call the 'multiplet', and the 3~ at 5.10 MeV.

The final angular distributions obtained for elastic and in-

elastic scattering are shown in Figs. III-3 through 111-16 and tabu-

lated in Appendix F. We show angular distributions for data binned

53 in both 2° and 1° bins; although the multiplet has only been binned

in 2° bins because of its poor statistics. Fits to the data will be

presented in Chapter V. The differential cross sections are summed for

2° bins over the measured angular range by converting the integral

into a sum with an angular step size of 2.5° 8 max a, = 2TT r a(9)sin6d9 ->- a = 2TT V* a(9)'sin6'A6 (3.5) int Jo sum L^t 9min

where AQ = 2.5" = TT/72 radians. Therefore 2 x a = asu m = 3~6 y a(e)-sin9. (3.6) sum 36 xremifn 2 The sums and ratios of sums, always expressed in terms of p =

Ea(ir )/ECJ(TT ), are given in Table III-l.

Because data exist at 163 MeV from the spectrometer group at

SIN17, it is of interest to compare their data to our 164 MeV data.

The SIN data for elastic, 2 and 3, states are shown in Figs. III-

18, 19, and 20. Their reported ratios are included in Table III-l.

1. Elastic Scattering at 164 MeV. Referring to Fig. III-7 and

111-18 it can be seen that out to 40° the two sets of data are very

similar - the SIN data is very slightly higher than ours. The minima

occur at about the same angles in each set. However, our minima for

TT (TT ) are deeper by a factor 2(3). From 50° to 70° our data, for

both TT and IT , are about 20-25% higher than those of SIN.

2. Inelastic Scattering at 164 MeV: 2* State at 1.98 MeV. The

differences between our data and the SIN data are quite striking for

scattering to this state (see Figs. III-8 and 111-19). As listed in

Appendix E, the ratio of IT to IT differential cross sections for our

54 TABLE III-l. Summary of integrated cross section results for states in 808 .

o in mb p2 = Eo(Tr~)/Ea(Tr""<"<")) b TTT(MeV) Elastic 2i Multiplet Elastic Multiplet 3i Elastic 2j+ 3i+ (SIN) (SIN) (SIN)

164 ir 146.2(1.5) 4.57(22) 1.17(10) 4.03(11) 1 1-^(19) 0.89(6) 1.04(2) 1.29(7) 0.83(3) ir" 155.2(1.5) 8.57(32) 1.70(14) 3.69(11) -

+ 230 TT 107.5(1.1) 5.50(16) 1.48(10) 5.30(16) , , . n . ftsr7. IT" 108.7(8) 8.67(20) 2.26(10) 4.52(14) 1<01(7) 1'58<13> 1-52(17) 0.85(7)

Sums from 17.8 - 81° in all cases except when data is not available for some of the forward angles.

See Appendix D for a discussion of total errors in these ratios. data at 164 MeV is consistently about 1.8, except for a small region

in the vicinity of the minima at about 45°, while this ratio for the

SIN data is approximately 1.25 everywhere. In our inelastic data we

observed an approximately 2.5° shift between the TT and IT minima,

as we do in the c?.se of the elastic scattering* This shift is not

discernable in the SIN data because of large scatter in the data

points. Another serious difference occurs in the forward angle re-

gion in which the SIN data rises steadily, which is unlikely behavior

for a pure L = 2 transition. Our angular distributions, while not

turning over as much as may perhaps be expected (see Chapter V for

DWBA calculations), have definitely leveled off.

We cannot comment on the possible sources of error in the SIN

data. In part, at least, they are related to the proximity of the

strong elastic peak.

3. The 3" State at 5.10 MeV. From Figs. III-9 and 111-20 it

can be seen that the SIN-EPICS data are quite close. This is under-

standable because, unlike the 2 above, there are no strong inter-

fering states near the 3...

4. The Multiplet at 3.5 - 3.9 MeV. This group of states has

been analyzed because it is the only other prominent group in the

excitation energy region below 5 MeV, and looked as if it would be

interesting to examine. From the angular distribution in Fig. III-5

and the information in Table III-2 we can draw some general conclu-

sions. At angles less than 40° the centroid of the fitted peak is

about 3.9 MeV from the elastic, which suggests that over this region

the ?, is the dominant level. At 50°, near the 2 minimum, the

56 TABLE III-2. Experimental energies for all states observed in the present experiment.

a Fitted Centroids (MeV) Approximate Positions

Multiplet + Trr(MeV) *•* "42 "

164 IT+ 1.99(4) 3.84(14) 5.17(3) 6.35 7.15 8.25 11.70 TT~ 1.99(4) 3.65(15) 5.12(7) 6.35 7.10 8.20 11.SO

230 TT+ 1.96(3) 3.78(13) 5.10(3) 6.40 7.10 8.20 11.70 TT~ 1.97(2) 3.67(14) 5.10(3) 6.40 7.15 8.20 11.65

3Results of AUTOFIT. Result of hand fit in high excitation region. Errors are about ±150 keV.

TABLE III-3. Experimental energies for the 4,0,2 unresolved triplet of states at 3.55, 3.63, 3.92 MeV.

Approximate Centroid TTr(MeV) Range of 6 of Multiplet (MeV)

164 TT 25 - 50° 3.90(10) 52.5 - 62.5 3.65(6) 65 - 80 3.80(5) 25 - 35 3.90(6) 37.5 - 47.5 3.70(5) 50 - 80 3.60(8)

230 TT 20 - 37.5 3.85(5) 40 - 70 3.65(6) 20 - 32.5 3.85(7) 35 - 70 3.60(7)

57 centroid has shifted to about 3.65 MeV suggesting that the 3.56 MeV

b. level is contributing. At larger angles the mixture seems to be

relatively equal, the centroid occurring at an average of about

3.7 MeV.

5. Data at T - 230 MeV. As a consequence of the higher energy

the minima in elastic and inelastic scattering have shifted to smaller

angles. The minima are also shallower. We notice from Table III-l 2

that the ratios p for all states except for the multiplet have de-

creased. The decrease is substantial in the case of the 2.; but the

164 MeV and 230 MeV results are still within one

of each other.

The behavior of the angular distributions for the nultiplet is

quite similar to that at 164 MeV. At 9<35° the centroid averages

about 3.85 MeV, indicating mostly the excitation of the 2 state.

As the 2j minimum is approached the centroid shifts to about 3.6 MeV,

reflecting the dominance of the 4^T state at 3.56 MeV. From 60-70°

the centroid rises to about 3.7 MeV indicating an almost equal ad-

mixture of 22 and 4...

H» Overall Errors

Fitting errors only are given for the differential cross sections

in the tables and plots of angular distributions. These errors are

purely statistical. In addition there are errors of a systematic

nature arising from in the calculation of hydrogen

cross sections and in the determination of hydrogen normalization

factors for this experiment. These errors are estimated to be ±4%

58 for ir and ±4.5% for ir~ at 164 MeV, and ±42 for IT and ±6% for TT at

230 MeV. These errors affect the uncertainties in the ratios calculated with the summed cross sections and therefore have been ircluded in all quoted values. A detailed presentation of the error analysis is given in Appendix D.

59 10"

0 ( TT, IT ) 0





b •a • •


0.01 10 20 30 40 50 60 70 80 90 cm.

Figure III-3. Angular distributions for elastic scattering of 164 MeV 7T- from 180, for 2° binning of the spectrometer acceptance.

60 10

0 0 0(ir,ir ) 0

TT = 164 MeV 2*, 1.98 MeV -2° BINS

• 7T O IT


O.I • •

0.01 10 20 30 40 50 60 70 80 90 0c.ni.

Figure III-4. Angular distributions for excitation of the 2X state in 18C by 164 MeV TT±, for 2° bins.

61 10 I I 1

Odr.ir') 0



0.1 5 b •D


0.001 10 20 30 40 50 60 70 80 90 ft cm.

Figure III-5. Angular distributions for excitation of the unresolved triplet of states 180 - the 4+ at 3.56 MeV, 0| at 3.63 MeV and 2+ at 3.92 MeV - by 164 MeV TT1, for 2° bins.

62 10 i r

30 (ir -n ) l80

TT = 164 MeV 3", 5.10 MeV -2° BINS $ *e ° in

5 -o - b 0.1

0.0! j i i i I I 10 20 30 40 50 60 70 80 90 'cm.

Figure III-6. Angular distributions for excitation of the 3 state in 180 by 164 MeV IT1, for 2° bins.


— 10-


0.1 r


Figure III-7. Angular distributions for elastic scattering of 164 MeV TT* from 180, for 1° binning of spectrometer acceptance.

64 10 T r i i

8 . , ,18. 0(ir,irt r ) 0 TV 164 MeV % *V °0 2}, 1.98 MeV-1°BINS


b * "O 0.1 hi

I I 0 10 20 30 40 50 60 70 80 90

Figure III-8. Angular distributions for excitation of the 2t state in 180 by 164 MeV TT±, for 1° bins.

65 10

3 _ , i Jo. O(ir,ir ) 0

T1P« 164 MeV }h 3J.5.I0 MeV-l°BINS $5 0 IT' CO JE *• %

b #*> 0.1 ^??Mf

0 10 20 30 40 50 60 70 80 90 7c.m.

Figure III-9. Angular distributions for excitation of the 3~ state in 130 by 164 MeV TT±, for 1° bins.


E 10 r 1

cm. Figure 111-10. Angular distributions for elastic scattering of 230 MeV TT1 from 180, for 2° bins.

67 10 6 A I I o it*. o • o 'o (*•,*•') o TT = 230 MeV 2*. 1.98 MeV -2° BINS • IT O 7T '


b •o 0.1 -

0.01 I 10 20 30 40 50 60 70 80 cm.

Figure III-ll. Angular distributions for excitation of the 2* state in 180 by 230 MeV TT±, for 2° bins.

68 10 "I 1 1

l8 l8 0(irt1r') 0 TV = 230 MeV MULTIPLET -2° BINS

o IT

tfi .Q

b •!i T3 : ;


10 20 30 40 50 60 70 80

Figure 111-12. Angular distribution for excitation of the unresolved triplet in 180 - the 4+ at 3.56 MeV, 0^ at 3.63 MeV and 2+ at 3.92 MeV by 230 MeV n1, for 2° bins.

69 10 i i i

l8o (.,,-• )l8o

Tr = 230 MeV 37,5.10 MeV -2° BINS

O 17-

b 0.

0.01 I 10 20 30 40 50 60 70 80 cm.

Figure 111-13. Angular distributions for excitation of the 3" state in 180 by 230 MeV IT*, for 2° bins. l

70 : ' l \ ! ! 1 1 ;


% T r = 230 MeV Q V ELASTIC - I°BINS •C + s

I i • - o • • o • 10 o - °•# oo°2s b i °8 9f o

1 - •


<>• 0

1 1 I 1 h 10 20 30 40 50 60 80 6cm.

Figure 111-14. Angular distributions for elastic scattering of 230 MeV TT1 from 180, for 1° bins.

71 1 1 T

l8O(Tr,Tr')18 TV = 230 MeV

2* ,!.98MeV-l° BINS 10

o ir~ • o <*>

*b - O %


I I I I 0 10 20 30 40 50 60 70 80 ^c.rn.

Figure HI-15. Angular distributions for excitation of the 2^ state in 180 by 230 MeV •n±, for 1° bins.

72 f 10 r

Tr=230MeV 37,5.10 MeV -1° BINS

•IT* O TT- — I

b -o 0.

0-01, 10 20 30 40 50 60 70 80


Figure 111-16. Angular distributions for excitation of the 3~ state in 180 by 230 MeV TT1, for 1° bins.

73 i r i —

100 - tr - 0 0.0 = 163 MeV 0* 80 5.09 1.98 1 Z 47 W ° 3' 2 O o - SIN 60 1 I •'•• ll MM - THIS EXPT. IX* o d Jh 40 - ft 1

NUMBE R 20 -l n fi Jhn Awt Ilk. i 450 500 550 600


Figure 111-17. A comparison of spectra for 7r scattering on 180f from SIN and EPICS (thick line). The SIN spectrum was taken at 8 K=47° and the EPICS spectrum at 6. ."47.5 lab 1 l8O(7r,7r')l8O TV =163 MeV JANSEN et al (SIN) ELASTIC 10' • ir" • ir~





iJ 0.1 "ft

I I I I I II I 10 20 30 40 50 60 70 80 90 100 110

Figure 111-18. Angular distributions taken at SIN for elastic scattering of 163 MeV if* from18 0.

75 : i 1 I I I i 1 1 1 0{ir,v ) 0 TV 163 MeV JANSEN et.al (SIN) 10 - %,* *;. .98 MeV -


1 \ -

T3 • b - \, ii


11 i 1

) 01 1 1 1 ! 1 1 # If 0 10 20 30 40 50 60 70 80 190 1100 I ! 10 cm.

Figure 111-19. Angular distributions taken at SIN for excitation of the 2"J" state in 180 by 163 MeV TT±.

76 10 i i r I I I I O ( IT, IT ) O T, = 163 MeV JANSEN et.al.(SIN) S^.S.IO MeV + • IT ^ "IT

tn * * * 0.



j L 0 10 20 30 40 50 60 70 80 90 100 110 8 cm.

Figure 111-20. Angular distributions taken at SIN for excitation of the 3^ state in 180 by 163 MeV it*.


In recent years there has been a great deal of experimental and

theoretical work done on 180. In this chapter we will review the

present status of experimental and theoretical knowledge of 180. The

experimental quantities most commonly used to test wave functions will

he presented, and we will review the successes and failures of the

current theories in reproducing experimental results.

A. Experimental Information

The experimental information about 180 to be discussed in this

section are: (a) the spectrum of energy levels, (b) transfer reactions

leading to and from 180, (c) electromagnetic transition rates, and

(d) inelastic scattering. Much of the work done on 180 has been sum-

marized in the recent review by Ajzenberg-Selove.23

1. Energy Spectrum. In Fig. IV-1 the energy levels and decay

branches of 180 for levels up to about 7 MeV are shown. Table IV-1

lists the properties of all observed states at the time of publication

of Ref. 23. Ths values of mean life-time in the table are means of

selected best values, and the magnetic g-factors are the results of

recent measurements.21*»25

2. Transfer Reactions. Spectroscopic factors extracted from

stripping and pick-up reactions leading to states in 180 give infor-

mation about the single particle or single bole character of those

states. The * O(d,p)* 0 reaction has been performed by Moreh and

Daniels26 and by Li et al.,27 and spectroscopic factors have been

extracted for states in 180 up to 7.12 MeV excitation. The proton

78 E(MeV) 7.117 6.882 6.40* 6.351 6.201 5.53k 5.378^ 5.336-^

4.456 3.92K 3.635* 3.555-


+ 0 >fT* I 180

Figure IV-1. Energy levels and decay branches for low-lying states in 1188 0. Uncertain J1111 assignments are in parenthesis. Estimated weak branches are not shown.

79 TABLE IV-1. Energy levels of 1B180.', -

T (ps) or T (ps) or E(MeV±keV) E(MeV±keV) j";T m J ;T F (keV) cm T (keV) cm

0 0 ;1 8.283±3 3" 8±1 1.982 T =2,9±0.1 m thirteen g=-0.287±0.015 un- + assigned levels 3.555 4 xm=24.8±1.2 4- g=-0.62±0.10 1O.119±1C> 3" 16+4 3.635 0+ T =1.38±0.16 10.29±20 4+ m 2+ 10.38±20 3" 3.921 0.024±0.010 4.456 1" 0.065+0.015 five un- 5.099 3" 0.062±0.025 assigned levels 5.260 2+ 0.012±0.03 11.39+20 5.336 0+ 0.20±0.04 11.41+20 5.378 3+ <0.03 C4+) 11.62±20 5.531 2" <0.025 11.69±20 6+ 6.201 1 "" (3.2±0.6) j_ 11.82±20 12.04±20 (2+) 6.351 1,2 <0.035 12.25120 6.404 3" 0.03±0.015 (I") 12.33+20 6.882 <0.025 12.50±20 7.117 0.025 4+ V 12.53±20 6+ 7.620 1" r<2.5 7.75±20 1+4 five un- 7.84±14 assi^ned levels 7.96±20 (:3+, 4") (16.40±30) 8.040 <2.5 T=2 r<50 1- (17.02±30) 8.122±10 (19.0) 8.214±4 2+ 1.0±0.8 (23.4)

From Ref. 23.

Errors 2 keV or less are not given.

Uncertain assignments in parentheses.

80 pick-up reaction 19F(d,3He)18O was studied by Kaschl et al.30 The

spectroscopic factors deduced from these reactions are summarized in

Table IV-2a.

The ground state of 180 has been studied via the

ID IT 31,32 , p , 7 35-38 neutron pick-up reactions 180(p,d)170 and 1BO(d,tr'O.

The spectroscopic factors for transitions to the Id . ground state and

the 2s , first excited state in 170 provide measures of the d , and 1/2 5/2 s , occupancy in the ground state of 180. These spectroscopic factors 1 /2

are summarized in Table IV-2b.

The two nucleon transfer reaction 16O(t,p)18O, studied by Middleton and Pullen39 and analyzed by Fortune and Headley,36 gives information about the two particle properties of I80 levels with respect to the target nucleus 160. Results from various (t,p) reactions are summarized in Table IV-3.

The (d,p) reaction of Ref. 26 and (t,p) reaction of Ref.42 have been used to deduce wave functions of states in 180. These are listed i.n

Table IV-4, taken from the tabulation in Ref. 43. The label core exc

(standing for core excitation) refers to all components in the wave function not included in the other configurations listed, and will be discussed in the theoretical section of this chapter.

3. Electromagnetic Transitions. A large part of the knowledge about nuclear structure is obtained from the study of electromagnetic transition rates. They are, for example, the main source of information about the spin assignments of nuclear states. Before reporting the experimental knowledge of transition rates, we will outline their meaning.

81 TABLE IV-2. (a) Spectroscopic factors for the single nucleon transfer reactions 17O(d,p)l8O and 19F(d,3He)l80.a

17O(d,p)18OC 19F(d,3He)18O e E(MeV)b S, E =18 MeV C2S, E,=51.7 MeV d d

0 or 1.22 1.00 1.98 2+ 0.21+0.83(£=0+2) 1.39 + 3.56 4 1.57 £ 3.63 0+ 0.28 3.92 2+ 0.35+0.66(£=0+2) 4.46 1" 0.03 1.31 5.10 37 0.03 5.26 2+ 0.35 -vl. 2r 5.34 0+ 0.16 5.38 3+ 1.01 6.20 1" 0.03 0.70r 6.35 £3" 0.03 - 0.04

6.88 i 1.03 7.12 4+ 0.13d 7.67 0.4211 11.13 0.65 11.75 0.72f 12.25 0.89

aFrom Ref. 23, J from (d,p) reactions Refs. 26-29.

Nominal energy from Table IV-1.

CRef. 27.

From Ref. 26.

"Ref. 30, normalized to 1.0 for ground state transition.

Unresolved states.

82 TABLE IV-2. (b) Spectroscopic factors for the neutron pick-up reac- tions 18O(p,d)i70 and 18O(d,t)170,a

Ref. Reaction S(5/2+) Sum

31 (P,d) 1.60 0.22 1.82 0.14 32 1.10 0.11 1.21 0.10 33 (P,d) 1.64 0.24 1.88 0.15 34 1.31 0.07 1.38 0.05 35

aFrom the tabulation in Ref. 38. bUsing the (l80, 170*) reaction.

83 TABLE IV-3. States in 180 from UO(t,p)I8O.a

Ratios of E(MeV±keV) J1* Cross-SectionsC

0 0+ 1.980±5 2+ ot/ot = 0.0324±0.01d 3.549±5 3" or 4+ ot/0? = 0.433±0.12d 3.627±5 0+ 2t/2t = 0.298±0.09e 3.915±5 2+ itI it = 0.702±0.21e 4.449±5 5.090±5 3" 5.247±7 2+ 5.329±7 0+ 5.368±10 (2+)f 5.521+10 6.189±10 6.341±10 6.391±10

aFrom Ref. 23. From Ref. 41 CTabuIated in Ref. 67 , from Ref. 39. dAt 6. =5°. lab At peak of cross-section. Correctly identified in experxraents summarized in previous table.

84 TABLE IV-4. Experimentally determined percentages of components of wave functions for states of 1818O.aa

Reaction State Configuration (d,p) (t,p)

X 81% 56% 65.5% g.s. 15 18 3.5 core exc. 4 26 25.0

ot <18 42 3.63 MeV 41 core exc. 17

2! 56 48 1.98 MeV 23 26 (d5/2Sl/2) core exc. 21 26 2t 44 3.92 MeV 55 (d5/2Sl/2) core exc. 1

2t <8 10 5.25 MeV 32 20 (d5/2Sl/2} core exc. >60 70

1 80 3.56 MeV D'z .

The (d,p) and (t,p) columns are from a tabulation in Ref. 43, and the (P>p') column is from Ref. 34.

85 The probability for a transition between nuclear states to occur

is proportional to the matrix element of the transition operator (a one

body operator) between initial and final nuclear states

z P(ELM; J^ + JfMf) a | < ^ M |OE(LM) \^JM > \ (4.1)

where the transition operator 0w for electric transitions can be calcu-

lated from knowledge of the charge distribution. The transition rate

is defined as the sum of transition probabilities over all projections

of the angular momenta J , Jf

X T(EL; J± - Jf) = 2J + 1 ^ P(ELM; J^ - JfMf) (4.2) 1 M V 1 >Kf

which, applying the Signer-Eckart theorem, reduces to

T(EL; J± * Jf) a B(EL; J± -»• Jf)


B(EL J J < 0 (L) ' i * f> " irVl I *jfll E H*J1 > I' (4.4)

is the reduced transition probability, and the "double bar" matrix

elements are the reduced (via the Wigner-Eckart theorem) matrix ele-

ments. The reduced transition probabilities B(EL) are the quantities

most often quoted in the literature. These transition probabilities are

usually expressed in the sense of a decay from an excited state

to a state lower in energy, i.e., B(EL)+. However, inelastic scattering

experiments measure the rate of population of an excited state from the

ground state, i.e., B(EL)+.

86 For inelastic excitation of the 2 state, for example, we must compare our B(E2)t values to literature values by multiplying

B(E2H • (2Jf + 1) = 5 • B(E2)+.

It is common practice to define electromagnetic transition strengths in terms of Weisskopf units 'Wu). The purpose of this is to remove the strong dependence on the transition energy, giving a measure of the intrinsic transition probability which is determined solely by the wave functions of the states involved. The mean life- time of the state can be related to its width r in eV by the relation1*1*

T - 6.582 x 10~16 T"1 (eV) x BR (4.5) where T is the mean lifetime, and BR is the branching ratio for the decay. This width may in turn be expressed in Wu. The mean lifetime can also be used to calculate the reduced B(EL) using

1/T - 4.43 x 1019(^) K2L- 1)1112 ' < °E(L> > '* (4.6) x E /(2L + 1) sec " to calculate the reduced matrix element < 0 (L) > , where E is the E Y photon energy and k is the photon wave number = E /197.3 fm . The reduced matrix element is then related to B(EL) by I < 0 (L) > |2 B(EL) = (2L + 1) |j e2 fm2L . (4.7)

Equations (4.6) and (4.7) can be combined to yield a direct relation between the mean lifetime and the transition probability. In the general case for transition of multipolarity L

87 2L 2 2 (197.3) 2L(2L + 1) [(2L - I)!!] 1 1/x (5.57 x 1020)

e2 fm2L. (4.8)

Because we are most interested in the E2 and E3 transitions, we give

the expressions for B(E2) and B(E3) depending only on the photon

energy E and mean lifetime x

B(E2) = 8.16 x 10"10 1/E5 l/x e2fm4 (4.9)


B(E3) = 1.75 x 10~3 1/E7 l/x e2fm6. (4.10)

The transition probability can also be related to the collective

deformation 6 (discussed in the next section) of a state excited via

transfer of angular momentum L by .2

B(EL) .^ ^ (4.11)

where Z is the nuclear charge, and R is the RMS radius of the transi-

tion density. This is often taken to be the half density radius of

the deformed optical model potential.

Another experimental observable of importance in testing wave

functions is the quadrupole moment. This is defined in terms of the

L = 2 reduced matrix element between the same initial and final

states (J. = J = J) r j(n -i} V2 [(J + 1)(2J + 1)(2J + 3)J

x < J||OE(L = 2)||j > (4.12)

1/2 where the factor [16ir/5] arises from normalization of the spherical

harmonics. Experimental results for electromagnetic transitions and

quadrupole moments are given in Tab^e IV-5. Part (a) of Table IV-5

88 gives the deduced transition strengths, expressed as deformation

length 3 R, from analyses of t'.ie excitation of the 2 and 3" states

in 180 via inelastic scattering of various particles. Part (b) lists

several of the results obtained for B(E2) as a result of measurements

of the lifetimes for electromagnetic transitions.

4. Inelastic Scattering. Inelastic scattering analyzed macro-

scopically in the collective model provides information about the defor- mation of the states involved. In the collective model one attributes

the states to rotations of a statically deformed shape, or to oscillations of a shape about a spherical mean (with deformation characterized by the parameter Q ). The nuclear radius, R, is expanded about an equilibrium value R in terms of the deformation o

j>) = RQ(1 + $LYm(B,$)). (4.13)

This leads tc a first order deformed potential

VCR) - V(R > + (R - Ro) • ||| = V(RQ) + BLYm • ||| . (4.14) o o

The first term is the zeroth order, spherical term, and leads to elastic scattering. The higher order terms contribute to the scattering to excited states with angular momentum transfer L. In a standard collec- tive model analysis3 the inelastic cross section to first order has the

2 proportionality dainel a (8LR0V0) , where RQ is the equilibrium radius of the undeformed potential, and VQ is its depth. Thus the quantity characterizing inelastic scattering is not the deformation 6, but the defor- mation length, g R. It is this quantity which should be compared between different experiments.

Experimental results for deformations of the 2j and 3~ states in 18C have been listed in Table IV-5(a). Tor the cases in which the radius was

89 TABLE IV-5. (a) Transition strengths for the 2i 3 7 states in 180 by inelastic scattering.

Author Type

6 Groh" (e,e') 0.98 fm 1.40 fm 3.577 fm Lutz55 (a.a1) 0.S3 1.33 4.3 Harvey1*5 (a.a1) 1.08 0.89 3.145 Resmini56 (P,PT) 1.11 2.99 Escudie **3 (P.P1) 1.26 1.14 (DWBA) 2.93 1.08 1.08 (CCBA) Vandenboschs 7 (18O,18O*) 1.08 0.86 3.145

90 TABLE IV-5. (b) Transition strengths for states in 180 by lifetime measurements.

2 Transition (J± •* Jf), B(E2)+ in e

Author 1.98+0 3.56+1.98 3.63+1.98 3.92+0

Allen and Lawsona 8.2±0.8 12.1 ?7.1±4.0 0.97±0.6

Olness*8 9.25^'gJ 47.5+4.8 4.78+2.0

Berant1*9 7.49±0.38 3.34±0.18 McDonald50 8.00±0.48 3.38+0.46 2.80+1.2 Others 9.25+0.32b 3.34±0.16b 26.1±4.6C

tabulated in Ref. 47. bFrom "best" value of Ref. 23. Quoted in Ref. 67, average of other experimental values.

TABLE IV-5. (c) Measurements of the quadrupole moment of the 2! state of 180.

Author Q(2t) in e-fm2 Deduced B(E2)+ in e tm

Fewell51f -10.0±3.0 44.0±4.0a Void53 -3.1+?'? 40.2±1.4 Flaum52 -7.3±2.7 45.3±2.6 Dehnhard51 -7.6±2.0 38

Based on previous measurements.

91 not given in the reference it has been calculated as a weighted average

of the real and imaginary potentials.

Collective rotations of a statically deformed shape lead to forma-

tion of rotational bands with J77 = 0+, 2,4 The energy

spectrum of such a band is expected to follow the relation E a J(J + 1),

which predicts energies for the 2 * 4 , 6 levels to be in the ratio

1 : 3.3 : 7, Interestingly enough, we can find corresponding levels in

180 at 1.98 MeV (2+), 7.12 MeV (4+), and the 6+ at 12.55 MeV, which fall

in the proportionality 1 : 3.6 : 6.3.

This possibility has been investigated, particularly in the

inelastic proton scattering experiment of Escudie et al.1*3 From the

macroscopic coupled channel fits (CCBA) in this reference it is concluded

that the 2[ at 1.98 MeV and 4* at 7.12 MeV are members of the ground

state rotational band. This result had been previously proposed in

Ref. 56. It has also been proposed58 that the 0* at 3.63 MeV and 2*

at 5.25 MeV are members of a second rotational band. The results of

Ref. 43 do not support this hypothesis, chiefly because they find evi-

dence of very little collectivity in the 02 state. However, other

investigations, notably that of Ref. 36 , have found evidence for very

strong collectivity in the 0,.

In summary of the results from Ref. 43 , large collective compo-

nents were found in the 2 states at 1.98 and 5.25 MeV and the 4

states at 3.5 5 and 7.12 MeV. Also investigated were the first two

18 negative states of 0 and it was concluded that the 3X at

5.10 MeV is strongly collective, and the 1~ at 4.45 MeV contains little


92 B. Theoretical Description

In this section we will briefly discuss a small part of the large

body of theoretical work on 180. The examples presented in this

section have been chosen because they are among the more important works on 180, and also because most have been used in calculations done

for our 180 data, which are presented .n Chapter V.

The first task of a theory is to reproduce the level scheme reasonably well. From the wave functions predicted other observables such as one-nucleon transfer spectroscopic factors, two-nucleon transfer strengths, and electromagnetic transition rates can be calculated and compared to experimental values. The experimental data for 0 have imposed important constraints on the theoretical models: the wave functions must reflect the strong collectivity of some of the states, and the collectively enhanced transition rates must be correctly reproduced.

Electromagnetic transition rates predicted by the theoretical models are given in Table IV-6. The wave functions are summarized in

Table IV-7.

1. Two-Particle Excitations. In the simplest, shell model description, 1B0 consists of two correlated neutrons moving outside the closed i60 core, which is considered spherical and inert. We will restrict the valence neutrons ro move in only s-d shell orbits. From two neutrons in the d , orbit we can produce J' = 0 , 2 , 4 states. 5/2 If we enlarge the basis to include the s , orbit we can produce the .! /2 configurations (s , )2, j" = 0 , and (d , s • ). J =2,3. This 1/2 5/2 1/2

93 TABLE IV-6. (a) Theoretical predictions for electromagnetic transition rates for states of I80.

Transition (J ,->J-) ,B(E2)+ in e2f m«

2 Effective Author t-*oT 1.98-H) 3.56+1.98 3.63-KL.98 3.92-K) Charge Used

Cohen59 2.7 0.04 Inoue60 3.69 2.91 1.45 0.5 e Ellis, Engeland62 4.86 3.60 9.04 0.483a 0.5 e Lawson6 7 7.05 3.42 23.0 3.82b c Erikson, Brown68 9.32 2.70 39.94 1.08 0.5 e Morrison69 6.4 23.0 2.7 0.3 e

Using branching ratio of 4%.

Using branching ratio of 13%.

Used eeff=0.55e and 0.61e for the two-particle ds/2 and matrix elements respectively. Used ee££=0.753e for 4p-2h matrix elements.

TABLE IV-6. (b) Theoretical predictions for the quadrupole moment of 18 the 2: state in 0.

Author Q(2i) in e-fm2

Lawson6 7 -5.0 Erikson and Brown68 -5.51 Morrison69 -5.06 Engeland and Ellis71 -3.4

94 18 a TABLE IV-7. Theoretical predictions for the wave functions of the Oi and 2t states in O.

State Configuration Cohen59 Inoue60 Zuker61 Ellis62 Lawson67 Erikson68 Morrison69


2 (d5) 0.924 0.702 0.63 -0.774 0.791 (d5SX) 0.383 -0.641 0.44 -0.485 0.580 (d5d3) 0.189 [0.96] -0.056 0.084 [0.82]

0.11/ (4p-2h) 0.64 -0.25 deformed 0.347 0.356 0.57

Quantities in brackets are total amplitudes of (2p-0h) configurations. The Zuker wave functions are taken from the tabulation in Ref. 43.

Subscripts represent (2J+1). In various different calculations the distinction is made between the (4p-2h) states of a shell model calculation and the intrinsic deformed state from a coexistence type calculation.

V0 Or simple space reproduces the spins of all positive parity states below

18 or 5 MeV excitation in 0, as well as the extra 3 . The cL,2 bit can

be excluded from a study of low-lying states, because such configura-

tions are expected to lie above 6 MeV in excitation.

In this simple basis space calculations have been done by Cohan

et al.59 The two-body matrix elements of the residual interaction

were determined by a least squares fit to the known energy levels of

18,19,20Q< ye have calculated the wave functions from these matrix

elements by simple configuration mixing.

Inoue and co-workers included the d, ,_ orbit in their model

space, and energy levels and wave functions were calculated by intro-

ducing a central form for the residual interaction, and varying its

strength in order to get reasonable agreement with the experimental

level scheme.

However, such simple models, containing only two-particle states,

cannot predict the existence of the 2 level at 5.25 MeV or the 0- at

5.33 MeV. They also ft.il to reproduce the electromagnetic transition

rates and the observed single nucleon transfer spectroscopic factors.

It is not possible to form a negative parity state in this simple space,

because such a state would require participation from either the lp or

lf-2p shell. Experimentally the 1~ (at 4.45 MeV), 3~ (ut 5.10 MeV)

and 2~ (at 5.53 MeV) exist below 6 MeV excitation in 180.

2. Particle and Particle-Hole Excitations. Since the above

simple two-particle models are not very successful in accounting for

the observed properties of 180, it is clear that the configuration space

must be enlarged. The enlarged space contains the lp orbits, such that

core particles are excited from the lp into the s-d orbits. For sim-

96 piicity it is customarv to consider only 4-particle, 2-hole (4p~2h)

configurations, formed by exciting two core nucleons from the lp

orbits into the s-d orbits.

This approach has been taken by, among others, Zuker and Ellis

and Sngeland.62 Both of these works have used a "weak coupling" idea

first proposed by Arima et al.63 The basic tenet of the weak coupling model is that the structure of a nucleus is dominated by correlations between particles in the same major shell, and that the residual particle-hole interaction may be treated as a perturbation. For ex- ample, promoting two protons from the 160 core into the s-d shell makes

rhe structure in this shell equivalent to that of 20Ne, and the two holes in the p-shell make the structure of this shell equivalent to that of 11+C. Thus the core excited state may be written 20Ne(T = 0) x li+C(T = 1). Coupling these configurations weakly implies a "simple multiplication" of the amplitudes, as described in Ref. 61.

The model space used by Zuker is restricted to the d^-) si/2 and P-. /7 orbits. In this work the matrix elements of the residual interaction have been chosen from a search of those available in the literature for this model space. In an earlier work, Zuker, Buck and

McGrory6"* reported a method used to calculate the spectrum and wave functions for Op-Oh and 2p-2h configurations in 160. In Ref. 61 several of the matrix elements have been adjusted to improve the positions of the 180 ground state and the CL state of 160. Using the weak coupling scheme,61 the recalculated 160 wave functions have been coupled to the simple 2p states in 180 to obtain the weak coupling wave functions for

180. These wave functions are compared for some cases to an exact shell model calculation, and the correspondence is found to be quite good.

97 The work of Ellis and Engeland62 is significant in that it

employs the entire s-d shell basis for particles, and the entire lp

shell for holes. A central form is assumed for the particle-hole

interaction, and the model proceeds from first principles (i.e., no

fitting to the energy levels is attempted).

The negative parity states are studied carefully in the work of

Ellis and Engeland. The simplest way of making a state of negative

parity is to excite a particle from the p-shell into the s-d shell,

forming 3p-lh configurations. These are the only configurations con-

sidered in Ref. 62. However, because of the relatively high excitation

energy of the lowest negative parity state in 180, it is not unreasonable

to assume that it is possible to form a negative parity state by

exciting one of the valence particles from the s-d shell into the f-p

shell. As can be seen from the L = 3 spectroscopic factor to the 3

in Table IV-2(a), this has a small probability.

There are no transition rates quoted in the papers by Zuker.sl

But the Ellis and Engeland model is fairly successful in reproducing

these quantities. This model, as discussed in Ref. 27, also has been

successful in reproducing the spectroscopic factors for the (d,p)

reaction. So we can see that the choice of a large model space has led

to favorable results. It is interesting to note, however, that

microscopic fits to the (p,p') data of Ref. 43 using Zuker wave functions

were not particularly good.

3. Two-Particle and Collective Excitations. Experimental

evidence for collectivity of some states in 180 was presented in

Section A. 4. The core excited states discussed in the last section

98 are an attempt to describe this collectivity in a spherical shell model calculation. In principle, if enough configurations can be

included in the shell model basis, this approach should be able to give a good description of a collective nucleus. In practice a large basis calculation is very difficult to perform. A different approach is to introduce an intrinsic deformed state in addition to the shell model 2p-Oh states, and project states of good angular momentum from this state.

This approach was first suggested by Brown,65 and subsequently applied by various authors.66 Because of the collectivity associated with some levels in 180 it is an attempt to give a very physical inter- pretation of these states. The last three entries in Table IV-7 are variations of this approach. Each utilizes a single intrinsic deformed state obtained by promoting two protons from the K » -r-

Nilsson orbit into the K77 = -s- orbit.

The method used in shell model calculations, that of performing a least squares fit to the spectra of 180 and ueighboring nuclei and adjusting the matrix elements of the residual interaction to obtain the best fit, has also been used in calculations utilizing an intrinsic deformed state. The work of Lawson et al.67 is a variation of this method. In this work the wave functions have been varied to fit a variety of other experimental data - transition rates, static multipole moments, single nucleon transfer spectroscopic factors, and ratios of

(t,p) cross sections. The model space included one collective configur- ation each of J =0,2,4 plus all states formed by a closed 160 core and two neutrons in the d , and s , orbits, plus at most one 5/2 1/2 r

99 neutron in the d , orbit (nhey found the d , admixture to be less 3'2 3/2

than 5%). The 2h in the 4p-2h collective configuration were restricted

to the K71 » y Nilsson orbit.

Differing experimental results have been carefully considered

as constraints on the wave functions, and three possible sets of wave

functions are tabulated depending on the constraints used. The wave

functions listed in Table IV-7 are labelled Constrained II in Ref. 67

and are considered to give the best fit to the experimental data.

From these results the shell model matrix elements which reproduce

this set of wave functions are deduced.

Although the electromagnetic transition rates are used as experi-

mental constraints on the Jit, because so much additional data are

contained in the fit we have included the Lawson best fit values for

transitions in Table IV-6(a). Nearly all the data used in the fits are

well reproduced. The most notable exception is the ratio of spectro-

scopic factors for population of the first two states in 170 via the

reaction 180(d,t)170. This problem is discussed in Ref. 38.

The work of Erikson and Brown68 is less detailed than that of

Lawson et al., but also uses less experimental constraints to obtain

the "best" wave functions. The model space used is essentially the

same in both works. Erikson and Brown utilize cwo-body matrix elements

from previous calculations, and introduce empirical constraints as

necessary. One such constraint is that the collective band in 180 is

parallel to that of 20Ne, so they choose a moment of inertia near that

of 20Ne. The position of the collective bandhead in 180 is treated as

a variable.

100 The basic procedure is to construct overlap factors representing

the overlap in going from the 2p-2h states in 160 to the 4p-2h states

in 180. The interaction between the 2p-0h and 4p-2h states, involving

the overlap factors, leads to deformations of each which are described

by conventional deformation parameters. The position of the deformed

band is then varied, and the overlap factors are varied to obtain the

best fit to the experimental spectrum.

The wave functions deduced in this work are very similar to those

of Lawson et al. The electromagnetic transition rates are well

reproduced. No other experimental results are predicted in this paper.

The final work is that of Morrison et al.69 This work used a

Hartree-Fock method, projecting out states of good angular momentum from intrinsic Hartree-Fock states. The s-d shell basis is the same as used by Lawson et al. and Erikson and Brown, but the entire p-shell is included in the specification of the deformed state. This is a significant enlargement of the basis space over the other two calcula- tions discussed above. As can be seen from Table IV-7, the deformed components in the wave functions are quite different from the other two works, and for the 0 (ground state) and the 2 agree well with wave functions deduced from (d,p) and (t,p) experiments as listed in Table

IV-4. This work also reproduces the electromagnetic transition rates well.

It also contains an analysis of the (p,pT) data from Ref. 43. The data is fit better than with the wave functions used in Ref. 43, but, neverthe- less, the overall quality of fits is poor.

4. Effective Charges Used to Explain Transition Rates. As we have seen in the discussion of shell model calculations, the size of

101 basis space used is integral to the success of the model. But even

the most careful calculations are limited in the size of their space.

This truncation of the basis prohibits the model from correctly

reproducing transition rates in strongly collective nuclei. To

remedy this shortcoming an "effective charge" is assigned to both

neutrons and protons involved in the transition. In other words, in

the expressions for transition matrix elements the neutron is

assigned a charge e ff and the proton a charge 1+e ff, in units of

the electron charge e. The magnitude of effective charge needed is

related to the degree of truncation of the model space.

One might think that the introduction of an intrinsic deformed

state has taken into account all the needed core excitation. Unfor-

tunately this is not the case. Such models are also limited in the

model space chosen in that, while including 2p-Oh and 4p-2h configu-

rations, lp-lh excitations across 2fiw, which are responsible for the

introduction of effective charges, are neglected. For a positive

parity state such a lp-lh configuration involves the excitation of the

Single core particle to orbitals two major shells higher. The com-

ponents of such configurations in the wave functions are small. But,

as explained in Ref. 70, the effect of such excitations on the B(E2)

strengths is large because the amplitudes add constructively. Thus

there is need for introducing effective charges into the calculation

of transition rates in deformed state models as well.

In the tabulation of transition rates, Table IV-6(a), the effective

charges used in each model are given. It is interesting to note that

the effective charge needed in the work of Morrison et al.69 is

smaller than that used in the other works. This is a reflection of the

102 larger configuration space used. • It is obvious that the simple core plus valence neutron picture is very poor in predicting transition rates. Also notable is the general agreement among theoretical models on the Q(2j), and the general agreement with recent experiments measuring this quantity. This is in contradiction to the results of earlier experiments,72 which are now known to be in error.


In this final chapter we discuss recent attempts on our part, and

also by several others, to interpret our data. As outlined in the intro-

duction, because of its isospin characteristics, it has been hoped that

the pion would prove to be an especially sensitive probe of certain nu-

clear structure properties. Our data will help us to evaluate this pos-

sibility. Because 180 has been so extensively studied and its structure

is presumably well known, we can test our understanding of pion-nucleus

interactions in attempting to fit our data.

In describing various characteristicc of the data two approaches

have been taken. One approach is to employ schematic arguments based on

Tr-nucleon amplitudes as discussed in the introduction. The second ap-

proach is to actually perform DWIA calculations to calculate detailed

angular distributions, and obtain various ratios for IT ,TT cross sections.

Each of these approaches can in turn be applied in two ways: (1)

microscopic, using detailed nuclear wave functions and microscopic form

factors, and (2) macroscopic, or collective, in which the nucleus is des-

cribed by its collective properties such as deformations and collective

form factors. We will discuss various methods as they have been applied

to our 180 data.

Because much of the analysis presented in this chapter depends on

the use of the optical model of u-nucleus interactions we will first in-

troduce the TT-nucleus optical model. The derivation of an optical po-

tential from a generalized multiple scattering theory is presented in

Appendix E.

104 In our own calculations we have used the optical model code DWPI

which calculates Tr-nucleus elastic and inelastic scattering angular

distributions in the collective model framework, using the distorted

wave impulse approximation (DWIA). In all calculations we have used the

first order coordinate space representation of the Kisslinger potential

2 V,, = A • [-bo-k p(r) + bi $ • p(r)v" ] (5.1) K

with amplitudes bo,bi calculated by averaging elementary ir-nucleon ampli-

tudes over the 8 protons and 10 neutrons in I80. The elementary ir-nuc-

leon amplitudes have been taken from the phase-shift analysis of Roper et

al.75 The potential amplitudes we have used are listed in Table V-l.

In the above equation k is the pion cm wave number. The following cor-

rections to the optical model potential, discussed in Appendix E, have

not been made in our calculations: (1) nuclear binding and Fermi aver-

aging corrections, (2) the Lorentz-Lorenz effect, and (3) "true" pion absorption.

We have obtained the 180 charge density from the electron scat-

tering experiments of Bertozzi et al.76 These model independent re-

sults are shown in Fig. E-l in Appendix E. Attempts to fit a Woods-

Saxon density to these results were unsuccessful. A harmonic oscil- lator, or modified Gaussian, form fits the data quite well. The form of the density is

2 (r/w)2 P(r) = po [ l + a(r/w) ]e " . (5.2)

When the finite size of the proton is subtracted from the parameters of the fit one obtains the parameters used in the DWIA analysis: a=1.40, w=1.793 fm. The half-density radius, R, is taken to be 2.55 fm.

105 TABLE V-l. Potential amplitudes used in DWPI for our analysis. Units of bo ara fm5 and for bi are fm3=


164 7r+ -0.636 + 0.3611 3.474 + 8.5581 TT" -0.875 + 0.3751 3.925 + 9.5631

230 ir+ -0.473 + 0.313i -2.196 + 4.6931 ir~ -0.628 + 0.3291 -2.473 + 5.235i

106 From the harmonic oscillator density given in Eq. (5.2) we can

calculate a formula for the root-mean-square radius of the distribution

h = [a/zcflig)]3*.,,. (5.3)

With our point proton density parameters we get a value of 2.65 fm

for the rms radius of 180. For all calculations of inelastic transi-

tion strengths (0R) we have used the half-density radius of 2.55 fm read from Fig. E-l.

In the following we will describe the analysis of elastic and inelastic scattering separately.

107 A. Elastic Scattering

The ability to fit elastic scattering data is a basic requirement

for any model of Tr-nucleus interactions. The elastic scattering angular

distributions can be used to investigate those aspects of nuclear struc-

ture which relate to the ground state. The most fundamental of these

are ground state density distributions for protons and neutrons.

The proton distributions in nuclei are generally determined from

experiments. Because the electron interacts only

electromagnetically it is very difficult to deduce properties of the

neutron distribution from such studies. Hadronic probes, which interact

with both the protons and neutrons in the nucleus, are potentially good

tools for probing neutron distributions as well. However it is only very

recently that theoretical methods for extracting such information have

been developed.77 The isospin characterstics of pions, i.e. the fact

that the ir n and TT p amplitudes are enhanced over the IT p and ir n ampli-

tudes at energies near the (3,3) resonance energy, have long held the

hope that one day pions could be a sensitive probe of both the proton

and neutron distributions in nuclei. Because the ir "sees" mostly neu-

trons and the 7T "sees" mostly protons, it is expected that a comparison

of IT and IT elastic scattering, or total cross section data could give

information on the ground state neutron distribution.

A natural starting point for this pion investigation is an N^Z nu-

cleus for which it is likely that the neutron and proton distributions

have different sizes and/or shapes. Let us define

Ar = 5s - Js. (5.4) np n p '

108 We can estimate the difference in aeutron and proton rms r&f?4i, Ar , np

for a simple harmonic oscillator (HO) potential wall. (The HO well does not represent a realistic physical situation, but is commonly used be-

cause of its computational simplicity). Let us consider 80, for example.

It consists of a closed core of 8 protons and 8 neutrons plus two valence neutrons. We assume that these valence neutrons are in the s-d shell orbits. In the HO well these orbits have a larger radius than those of the p-shell, so we will get automatically Ar >0. For the HO h=[ I **?*'& Z n,-(2n+i-h)-b2]h (5.5) p 'p orbits orbits

where N is the total number of particles in the well, and n0 is the num- ber of particles in the particular orbit of interest. The quantity

(2n+J£-Jg) is tho HO principle , where n is the number of intermediate nodes in the wave function, and 2- is the orbital quantum number. For example, the Is orbit has n=l and £=0, the 2p has n=2 and

1=1, etc. The oscillator constant b is given by (fi/mco) 2. For 180, using the value b=1.7 fm, the 8 protons in the Is and Ip orbits give 2 h =2.55 fm. If the valence neutrons go into either the 2s or Id orbits we get = 2.69 fm. The net result is Ar =0.14 fm. n np The most sophisticated nuclear structure calculations possible today are those which use a self-consistent Hartree-Fock (HF) method. In such calculations the nucleus is built up by placing nucleons in each shell model orbit, and the HF pofential constructed from elementary nucleon- nucleon interactions. A solution is found when the HF energy is mini- mized. From the HF method detailed structure information can be derived, including the sizes of the neutron and proton distributions and thus Ar . np

109 In actual practice HF estimates of Ar are obtained by first deter-

mining tha ground state deformation. This is obtained by determining the

total as a function of the deformation. For most nuclei

such a plot is found to have a definite minimum corresponding to a defi-

nite value of the deformation. Ar is then calculated for this deforma- np

tion. Unfortunately, in the case of 180 the binding energy curve as a

function of deformation is found to be very shallow, i.e. 180 is what is

known in the trade as a very "soft" nucleus. It has no unique value of

deformation, and therefore Ar , in these calculations.

Recently a detailed shell model calculation for both 160 and 180 has

been done by Brown, Massen, and Hodgson.78 Their result for ls0 is Ar =-0.03 fm, which is consistent with the value of -0.02 fm calculated np by Negele using the density dependent Hartree-Fock method.79 For 180

they used Woods-Saxon wave functions in the configurations calculated by

Lawson, Serduke, and Fortune.67 Their result is Ar =0.19 fm. With np

twice as much deformed amplitude in the ground state wave function they

get Ar =0.18 fm, which is the value they prefer. It must be noted that

this calculation was done in a spherical shell model basis, which limits

the deformation to be zero. Because it is known that :80 has a deformed

component in the ground state it is difficult to attach too much physical

meaning to these results.

The size of the neutron distribution in 180 has been investigated

experimentally in four different experiments including ours. We describe

these below. These experiments use the fact that the optical model po-

tential may be decomposed in the following manner (see Eqs. (E.27) and


110 2 VR = A | k [ a0 • (Pn + P ) + Co/2 (pn - Pp) ]

) (5*6) + V- [ai • (pn+ p ) + ci/2 (pQ- Pp)] v J .

Here the a

respectively, and the ai and ex are those for the p-wave. In order to

be sensitive to differences in neutron and proton densities one should

attempt to sample a region in which either the parameter Co or ci is

relatively large. The c0 amplitude is large at low energies, and cx

is large at energies near resonance.

1. Total cross section measurements of Cooper et al. TT and IT

total cross sections from 160 and 180 were measured by Cooper et al80

in the energy range T =40-240 MeV at LAMPF. In their analysis of the

data the authors assumed that

* = ^ = ^ . (5.7) P P n

They then analyzed the total cross sections in terms of optical model predictions obtained by varying the neutron rms radius of 180. They used

a Kisslinger potential with free amplitudes, and a modified Gaussian den-

sity distribution for both protons and neutrons. The results were that

Ar (TT+-18O)-Ar (TT"-18O) = 0.19±0.02 fm. (5.8) np np

2. Measurement of elastic scattering differential cross section

ratios at low energies. Johnson et al81 have measured the scattering of

29 MeV iT on 160 and 180 with special care taken to obtain ratios of

I80 to 160 cross sections at each angle very accurately. They find that

the ratio a(x8O)/a(16O) has a pronounced maximum (with a value VL.6) at

0-70 . They analyze the ratio with an optical potential which is a

generalized form of Eq. (5.6) developed by Thies,8~ containing several

111 correction terms. By assuming Eq. (5.7) they attempt to fit the observed

ratios by varying the 180 neutron density parameters. The resultant

Ar is 0.18±0.02 fm. np 3. Measurement of differential cross sections at 163 MeV (SIN).

Jansen et al.17 have measured differential cross sections for ir and IT

elastic scattering from 180. They found that the shift in the first mini-

mum of TT relative to the IT minimum, A9 , , was 2.3°. In an earlier min

experiment20 done on ls0 the shift was 0.9°. They have analyzed these

results in terms of the black nucleus model of Johnson and Bethe.83

In the "sharp edge" version of this model minima of elastic scattering

occur at the zeroes of the Bessel function J^(qR), where q is the momen-

tum transfer which equals 2k#sin(9/2), and R is the radius of the black

disk. Here k is the pion lab wave number.

Since one cannot take account of differential Coulomb effects in

such a model they assume that these are completely reflected in the

A6 . =0.9° found for 160. They thus obtain the "Coulomb corrected" min 3 A6 . =2.3-0.9=1.4±0.3°. They claim that only 65% of this remainder can be understood in black nucleus terms as being due to the fact that N-Z=2 for 180, and the neutron density at a given point is larger than the proton density. For the remaining difference they thus require Ar = 0.08 fm. (Their result has also been quoted elsewhere8"* as being con- sistent with Ar =0.17 fm). np

4. The present experiment. In our experiment at T =164 MeV very

sharp minima for both 1T and TT were observed: the 7r minimum was at

44.0±0.2° and that for TT+ at 46.3+0.2°. This can be seen in Fig. V-l. Thus our A8 . =2.3±0.3° which is the same as that found by Jansen et al. irin J 112 EXPERIMENT Trr*!64 MeV ELASTIC

• I* bins • 2* bins

o I* bins a 2° bins

37 39 47 49 51

Figure V-l. Experimental points for 164 MeV ir elastic scattering in the vicinity of the minima. The lines are smooth curves drawn through the data to aid in locating the minima.

113 Unlike Jansen et al, however, we have chosen to analyze our data in terms

of an optical model calculation. In such a calculation differential

Coulomb effects are properly taken into account, and the absorption of

the pions is not replaced by the extreme assumption of blackness. Fur-

ther, in such a calculation the differences in scattering amplitudes for

TT and IT caused by the isovector part in Eq. (5.6) are properly taken

into account. We have therefore done a number of optical model calcula-

tions using the Kisslinger potential, Laplaciau potential (however, be-

cause of the general similarity of results using different potentials,

we used only the Kisslinger potential for our final calculations), Fermi

averaging and no Fermi averaging, and find that while the detailed pre-

dictions vary from calculation to calculation, the predictions for A0 . nun

remain remarkably stable. In Fig. V-2 we show the results of calculations

using the Kisslinger potential with free 7r-nucleon amplitudes averaged

over the 8 protons and 10 neutrons in 180. A harmonic oscillator point

proton density distribution (Eq. (5.2)) with a=1.40 and w=1.793 fm from

electron scattering76 was used, and Ar =0. We find that the fits to np the second maxima are not good but that the location of the minima are well reproduced with A0 . (DWPI)=2.0°. mm In order to determine if the predicted A6 . are sensitive to the mm

various corrections to the optical potential described in Appendix E we

requested Professor Liu85 to perform a momentum space calculation in

which binding energy and Fermi effects are included, and "true"

pion absorption is also taken into account. Figure V-3 shows the pre-

dictions of this calculation, done with Ar =0. As may be noted the np fits to the second maxima are much better, but once again A9 . =2.0°. mm

114 i r"—r

7^ = 164 MeV ELASTIC

10 o

~ 10 en


O.I r

I I ! J I 10 20 30 40 50 60 70 80 90 wc.m.

Figure V-2. Our 164 MeV elastic scattering data with curves calculated with DWPI.

115 ~ (Or J

b •o

0.1 r

0 10 20 30 40 50 60 70 80 90

Figure V-3. Our 164 MeV elastic scattering data with curves calculated by Liu.

116 This leaves only 0.3±0.3° of experimental shift to explain in terms

of a non-zero Ar . In order to determine the approximate sensitivity of np

the shift in elastic minima to the neutron distribution parameters we have

used a modified version of the elastic ir-nucleus scattering code PIRK.

The modifications allow us to separately vary the proton and neutron den-

sity parameters. We find that, fixing the proton distribution at the

values given by electron scattering, varying the size parameter for the

neutron distribution, w , by 0.02 fm, which is equivalent to a change in

rms radius of 0.03 fm, produces the 0.03° shift in the ir~ minimum - the

position of the Tr minimum is unaffected. We thus conclude that our data lead to Ar =0.03±0.03 fm. np To get a feeling for how the A9 . =2.0° shift in all the optical mm

model calculations described above comes about we have made some auxil-

liary calculations:

(1) To investigate purely Coulomb effects we have set differences in

the hadronic amplitudes to zero; that is, we have calculated DWPI elastic

scattering using the same average values bo ,bi for both IT and IT . As

shown in Fig. V-4 the difference in the 7r ,ir minima A9 . (Coul)=1.2°. min

(2) To determine the effect of the N-Z extra neutrons on the shift in minima we ran DWPI with the proper amplitudes bo ,bi for IT ,TT but with

the charge set to zero for both. This removes all Coulomb effects. The result from Fig. V-5 is that A6min(N-Z)=0.8°. While it is perhaps not correct to add these two separate effects to obtain the total shift it

is evident that the 2.0° predicted shift does arise from these two effects.

5. Critique of the optical model analyses. In a recent paper ft *7 Sternheim and Yoo investigated the problems associated with determining


AV6. bo,b,


40 42 44 46 48 50 ©CM

Figure V-4. Elastic scattering minima calculated with DWPI for 164 MeV IT- using average bo,bj amplitudes in the Kisslinger potential.

118 DWP3

Tn * 164 MeV ELASTIC IT* • ir* b.,bi w" b,, bi

40 44 48 48 50

Figure V-5. Elastic scattering minima calculated with DWPI for 164 MeV 7r° using the proper amplitudes for 11+ and TT" in the Kiss- linger potential.

119 neutron radii from pion data. As an example they considered the deter- mination of Ar for l*°Ca and 't8Ca from total cross section and elastic np

scattering angular distribution data. They used a Kisslinger potential

with Lorentz-Lorenz and angle transform corrections for all their calcu-

lations, and studied the interaction of the value of scattering ampli-

tudes bo,bi used in these calculations with the Ar determined from fit-

ting the data.

Sternheim and Yoo first fit the 't0Ca elastic scattering data with

free parameters and found that the fit was poor, particularly in the

depth of the minima. They then varied bi and found that much better fits

could be obtained by changing the imaginary part of bi to a value about

30% different from its free value. These calculations were done with

Alternatively they could obtain equally good fits by setting

Ar ^0 and again varying bi. Thus they could obtain for equally good

fits Ar =0 or 0.13 fm with very small changes in the amplitude bi.

They investigated fits to the '*8Ca elastic scattering data and found that

many equally good fits could be obtained, each corresponding to a dif-

ferent Ar with compensating differences in the amplitude bi.


They also calculated total cross sections for the different combina-

tions of parameters determined from elastic scattering fits and found

that once again the experimental differences Acr~ = o"~(*8Ca)-a~('t0Ca) did

not indicate a unique value for Ar H np

Thus from both elastic scattering angular distributions and from

total cross section measurements they conclude that unique answers can-

not be obtained for Ar unless one has exact knowledge of what ampli-

tudes bo,bi to use. Since the understanding of pion-nucleus physics 120 is not yet at the stage when such definitive answers can be given for what

are the correct optical model amplitudes to use, they conclude that neither

elastic scattering nor total cross sections, nor perhaps even their com-

bined analysis, can be used at present to determine such sensitive quan-

tities as Ar np B. Inelastic Scattering

A theory of inelastic scattering consists of two main aspects -

reaction dynamics and nuclear structure. We therefore require:

(1) Proper formulation of the ir-nucleus interaction, presumably based

on Tr-nucleon scattering amplitudes so that the incoming and outgoing

waves can be properly distorted.

(2) A description of the dominant mode of reaction mechanism, that is

direct reaction vs. compound nucleus, or single step vs. many step.

(3) A formulation of the transition amplitudes. This would once again

require knowledge of the pion-nucleon interaction and in addition the wave functions of the initial and final nuclear states.

Let us first consider points (2) and (3).

It is generally assumed that inelastic excitation of most states is via a direct single step process. In other words, inelastic excitation

is due to a one-body operator which, of course, can only excite one particle-one hole components with respect to the ground state of the

target. It is also known that the most important and strongest transi-

tions arise with the more or less coherent contribution of a large num- ber of such lp-lh components spanning a very large region of the con-

figuration space, including those which are two major shells away (2fiu>

121 excitations). Excitations in such a large configuration space cannot be

handled microscopically. Even with today's large computers one is rarely

able to span a single complete shell. Thus approximations have to be in-

troduced .

The simplest approximation is to forget about the microscopic ap-

proach and look upon inelastic excitation as collective excitations in-

volving excitation modes of the nuclear size and shape, e.g. rotations

or vibrations. In such an approach the entire content of nuclear

structure goes into defining measures of the amplitude of nuclear dis-

tortion. As mentioned in the introduction, in this collective model

description the nuclear density or potential is expanded in terms of a

surface displacement 5

2 V(r) =• V(r-R) - 6-|r V(r) + %6 -£r2 V(r) + ... . (5.9)

In this model the transition form-factor has the form of the derivative

of the density or potential and its amplitude is given by the displace-

ment 6

This displacement may be due to static or dynamic deformation, vibra-

tion, or "breathing", but it is always described in terms of the defor-

mation parameter (3R) where 8 is the measure of eccentricity of the

ellipsoidal shape and R is the equillibrium radius of the mean sphere.

The second approach is to attempt to do part of the problem micro-

scopically .and part collectively. In a manageable configuration space

all particle-hole excitations are considered explicitly and the rest

122 of the contribution is lumped into a collective parameter, generally

the effective charge. If the explicitly considered configuration space

is large enough the effective charge largely simulates the effect of

2fiw excitations which are almost always left out from explicit consid-

eration. If the configuration space is more restricted, effective

charges may be called upon to include the left out lfiu) components as

well. (We will see examples of both types of microscopic calculations

described below).

1. Collective Model, Amplitude Analysis. If one neglects all

aspects of reaction mechanism including Coulomb and nuclear distortions

one obtains Eqs. (1.11) and (1.12) of the introduction

K-[(BR)n • N (gR)p • Z] (5.11)

+ K-[(3R)n * N + (BR)p ' Z • (A /A~)] (5.12) which lead to

(8R)n/(8R)p = Z/N (5.13) where p is the ratio of [a(TT~)]^ to [a(Tr )P .

If we assume that only T=3/2, J=3/2 amplitudes are important, an assumption which is best at the peak of the (3,3) resonance, then

A /A~ =3, and we can obtain (gR) /(3R) from the ratio of the inte- n p grated cross sections, p2, directly. In Tables V-2(a), 3(a), we list the results obtained under this assumption.

We must reiterate that in these results all aspects of reaction mechanism are neglected. Of these, the ones which are not symmetric

123 TABLE V-2. Deformation lengths calculated from the 164 MeV data. The half-density radius R=2.55 fm, and relative normalization of all DWIA ir+ and TT~ curves is that of the integrated experimental cross section.

(a) Deformation lengths using measured p2 directly, and using DWIA in conjunction with amplitude arguments of Eqs. (5.15) and (5.16)

MeV Using DWIAa

State (BR)i,/(3R) (BR)n (6R)p (BR)n/(BR)p

2$ 1.86(16) 1.51(16) 1.19(3) 0.94(3) 1.30(9) 0.80(6) 1.62(17) 37 0.89(6) 0.71(5) 1.06(3) 1.22(3) 0.99(5) 1.33(8) 0.74(6)

aWh ere 6± bUsing Eqs. (5.11) and (5.12) directly.

(b) Deformation lengths using DWIA shape fits.

State (6R)n (BR) (BR)n/(BR)p

164 l\ 1.86(16) 1.09(6) 0.86(5) 1.19(10) 0.73(5) 1.62(19) 37 0.89(6) 1.05(6) 1.21(7) 0.98(7) 1.32(10) 0.74(7)

(c) Deformation lengths calculated using DWIA shape fits with facility to vary 3 and B directly, n p

T^MeV) State (BR) (SR) (BR).

164 2t 1.86(16) 1.21(12) 0.72(7) 1.68(20) 0.89(6) 0.97(8) 1.31(12) 0.74(8)

124 TABLE V-3. Deformation lengths calculated from the 230 MeV data. The half-density radius R=2.55 fm, and relative normalization of all DWIA ir+ and TT~ curves is that of the integrated experimental cross sections.

(a) Deformation lengths using measured p2 directly, and using DWIA in conjunction with amplitude arguments of Eqs. (5.15) and (5.16),

^=230 MeV Using DWIA

2 a State p (gR)n/(BR)p (fJR)^ (BR)^ (BR)n (BR)p (BR)n/(BR)p

2i 1.58(13) 1.27(12) 1.08(3) 0.97(2) 1.12(8) 0.90(5) 1.24(11) 37 0.85(7) 0.68(5) 1.00(3) 1.21(3) 0.91(6) 1.33(11) 0.68(7) aUsing Eqs. (5.11) and (5.12) directly.

(b) Deformation lengths using DWIA shape fits.

2 State p (BR)^- (BR)^ (BR)n (BR)p (SR)n/(BR)

230 2$ 1.58(13) 0.93(6) 0.83(5) 0.97(8) 0.78(5) 1,24(10) 37 0.85(7) 0.97(6) 1.1.7(6) 0.88(8) 1.29(12) 0.68(8)

(c) Deformation lengths calculated using DWIA shape fits with facility to vary B and B directly.

T (MeV) State p2 (BR) (BR) (BR) TT n p n j

230 2t 1.58(13) 0.96(9) 0.77(8) 1.24(14) 37 0.85(7) 0.84(8) 1.22(11) 0.69(8)

125 with respect to pion charge, such as Coulomb effects or pion absorption,

and those which are due to departures from (3,3) dominance are likely to + - produce the greatest errors. For example, since A /A can only be less

that 3 due to the presence of other than (3,3) amplitudes the values of

(8R) /(BR) obtained as described above are likely to be low. n p

2. Collective Model, DWIA Analysis. Extensive worit in nuclear

physics with many different projectiles has shown that an acceptable

description of distortions in the entrance and exit channels is provided

by optical potentials which fit elastic scattering in these channels.

We can therefore safely use pion-nucleus optical potentials which suc-

cessfully describe elastic scattering to generate proper distorted waves.

Two approaches are possible in this regard. We can use an optical poten-

tial generated by one of the theoretical prescriptions described in Ap-

pendix E, for example the Kisslinger potential, and choose its parameters

on the basis of free 7r-nucleon amplitudes. Or we can empirically ad-

just the parameters of the chosen potential until we best fit elastic

scattering. We have chosen to follow the first approach in our DWBA

analyses and have consistently used free parameters at both 164 and

230 MeV. The fit to elastic scattering angular distributions with these

parameters is quite good at 164 MeV, but rather poor at 230 MeV. This

has led some authors to adjust the parameters at 230 MeV to first obtain

a good fit to elastic scattering and then use the same parameters in the

description of inelastic scattering.

As discussed earlier, in the collective model the inelastic form

factor has the form of the derivative of the spherical potential. From

time to time it has been suggested that the parameters of the transition

126 potential, particularly its radius, should not be the same as those used

for generating the distorted waves, i.e. as for elastic scattering.

However, since there is no generally accepted procedure for such changes,

in our analysis we have used the same parameters for the transition po-

tential as for the distorted waves.

In Appendix E we have described the distorted wave impulse approxi-

mation (DWIA) code DWPI71* which was used in the present analysis. The

inputs used have been reiterated in Section A of this chapter, and the potential amplitudes used given in Table V-l. The DWIA cross section

is calculated assuming 0=1 and the actual value of 8 is determined by

fitting the curve to the data using the relation

The method of fitting the calculated curves to the data is somewhat arbitrary. We have generally chosen to optimize the fits to the data for 6- 35°. By normalizing in this manner the curves generally pass

through a majority of the data points. However, the overall quality of

the fits would not change if the curves for u and ir both were normali zed up to 10% higher or lower. The effect of this correlated error has been included in the errors for

8 - is largely sensitive to neutron deformations and 8 + to proton deformations. However in order to derive 8 and 8 from the values of n p

6 - and 8 + a separate relation is needed. Such a relation can be written analytically only under the assumption of (3,3) dominance, in which case

(8R) = [Z(a -1)] • [a(N+aZ)•(3R) + - (aN+Z)*(0R) _] (5.15) P TT TT

127 (BR) = [Nfa2-!)]"1 • [a(aN+Z)•(8R)_ - (N+aZ)•(BR)_+] (5.16) n Tr TT

where a = A /A , which equals 3 under the assumption of (3,3) dominance.

The results obtained under this assumption are listed in Tables V-2(b),

3(b). It might appear that all the gains by having gone from an ampli-

tude analysis to a DWIA analysis are lost by making the assumption of

(3,3) dominance at this stage. It turns out, however, that the structure

of Eqn. (5.15) and (5.16) is such that the sensitivity to the ratio A /A

is rather waak.

The compromise involved in using Eqs. (5.15) and (5.16) can be com-

pletely avoided if the distorted wave code can be modified to input

separate deformations for neutrons and protons. Such a modification has

Q C been made by C. Morris. We have therefore reanalyzed our data with

this modified version of DWPI using the following procedure.

Starting with the values of the deformation parameters B or B as n p determined above, a series of DWPI calculations for both TT+ and TT~ were done keeping B fixed and varying 8 • The ratio a /a^ . for IT" p n exp DWITTTA divided by that for TT+ was calculated in each case. When this ratio is

equal to unity we have the proper relative ratio for (3 /8 . At this

point the absolute values of B and 8 can be adjusted to provide the

best fit to the data by the same criteria as were used in the earlier

analysis. These results are given in Tables V-2(c), 3(c). Our final

curves for 164 and 230 MeV are presented in Figs. V-6 and V-7.

A comparison of the results obtained by the two procedures shows that

the assumption of (3,3) dominance in Eqs. (5.15) and (5.16) has not

introduced much error.

3. Microscopic Model, Amplitude Analysis. From the final value of

128 TT = 164 MeV • it* a w~ 10 ELASTIC





0 10 20 30 40 50 60 70 80 90 9. _

Figure V-6. Our 164 MeV elastic and inelastic data with curves calcu- lated with DWPI. The inelastic curves are the final results after re- normalization to the data as described in the text.

129 '•«>•

io 20 JO •«) 50 so mt 8c m.


0 10 20 30 <*0 50 6C

0 p

C ,0 20 10 -»C iO

F?lgure V-7. Our 230 MeV elastic and inelastic data with final curves calculated with DWPI.

130 P2=l.86(16) for the 2j transition excited by 164 MeV pions it is ob-

vious that the valence neutrons play a large role in this transition.

As mentioned earlier, in the collective model in which participation of

all nucleons in the nucleus is implied, this ratio should be 1.25. At

the other extreme in the closed, inert core plus valence neutron picture

it is predicted that P2=9. Our experimental values lie somewhere be-

tween these extremes. This indicates the desirability of performing microscopic calculations for our data in which core and valence pro- perties are treated in detail by utilizing thr ground and excited state wave functions, and by utilizing effective charges sufficient to repro- duce the observed transition rates.

At the time this experiment was first run neither we nor anyone else had done such a calculation for pion scattering on 180. Because we were not equipped to perform a microscopic calculation, in an early paper we presented a very qualitative schematic core plus valence argu- ment in the spirit of a microscopic approach. The framework for this argument is the concept of core polarization. The presence of the valence neutrons outside the (presumably) closed ls0 core tends to polarize the core; that is, the particles in the core are pulled into a shape more closely resembling that of the valence neutron orbits.

This leads to a deformation of the self-conjugate closed core which is largely isoscalar in character. Schematically we write, in analogy with Eqs. (5.11) and (5.12)

(Z+3Z)-(8R)C + 3(N-Z)-(BR)n P2 - (5.17) (3Z+Z)'(BR)C + (N-Z)-(BR)n j where we consider the closed core of N=Z neutrons and protons to have

131 a deformation (8R)C and the N-7 valence neutrons to have average defor-

2 mation (8R)n. With the value p = 1.86(16) we get (8R)n/(SR)c= 3.56(60).

If the deformation of the 160 core was due entirely to the polarizing

8 9

effect of the two valence neutrons, according to Bohr and Mottelson

(6R)c/(3R)n would be of the order of the number of valence particles

divided by the number of core particles (i.e. each valence particle

polarizes the core so that the core has an induced quadrupole moment

equal to that of the particle). For 180 this number is 2/16, i.e.

(8R)n/(ftR)c - 8. The fact that we measure a smaller number seems to

indicate a greater core participation than was anticipated, or, to

put it another way, an intrinsic core deformation.

a. Analysis of Brown. The core and valence model has been carried

a step farther by Brown,90 who suggested a description in terms of core

polarization effective charges and coexistence wave functions. The

effective charges, as discussed in Chapter IV, are a device for simu-

lating 2fioj excitations which are not explicitly included in the wave

functions of the nuclear states. Schematically they can bt described

by the polarization of the core - a valence neutron induces a core

polarization equivalent of e. neutrons and e2 protons, and a valence

proton similarly induces a core polarization equivalent of e2 neutrons

and ex protons. The magnitudes of e, and e2 are not necessarily equal

because the n-p is stronger than either of the n-n or p-p .

In the region of oxygen the effective charges are about e =0.49e and

70 e2= 0.61e.

The Brown model considers 2p-0h configurations with deformation 3 ,

and 4p-2h configarations with deformation 82 for both the ground state

132 18 and 2* state in 0. Overlaps Ax between the 2p-Oh parts of the 0x

and 2* states, and A2 between the 4p-2h parts are then calculated

In analogy with our Eq. (5.17) the ratio of TT~ to IT cross sections is then written in terms of amplitudes, except that instead of two neutrons one now has 2+2ej neutrons and 2e2 protons from the 2p-0h configurations, and instead of just 2 neutrons and 2 protons from the 4p-2h configurations one has 2+2e +2e neutrons and 2+2ej+2e2 protons. Thus one has

2 _ pI 1PJ-HJ--A1-[3(2+2eu^-r-iej]); +T 22e2] [61-A1.[(2+2e1) + 3-2eJ 2 2 (5.18)

-A2-[(2+2eij+2e2) + 3(2+2ei+2e2)]j

We have set up this equation for purely illustrative purposes. In order to actually use this equation the overlaps A and A2 must be properly calculated for the many components each of the 2p-0h and

4p-2h types. The algebra is non-trivial and will not be attempted here.

b. Analysis of Brown and Wildenthal. Brown and Wildenthal91 use multi-shell shell model wave functions similar to those of Zuker,

Buck and McGrory6'4 to do an amplitude analysis. In this calculation the overlaps are properly calculated for the large number of shell model amplitudes arising from the (sd) (p) space. As we have assumed above, Brown and Wildenthal also assume (3,3) dominance.

Effective charges e =1.5e and e =0.5e are used and p2=1.83 is pre- dicted for the 2 transition and p2=0.76 for the 3~ transition.

These effective charges give B(E2)=10 e^m1* and B(E3)=117 e2fm6

133 which may be compared with the experimental values B(E2)=9.25 e^m1*

and B(E3)=160 e2fms. The agreement with the B(E3) value is poorer

because the (sd) (p) space is too truncated for excitation of the 31

state. This can be taken into account by changing the values of ef-

fective charges used to e =1.5e and e =1.0e which give B(E3)=158 e2fm6. P n

The ratio p2 is then 0.90 for the 3^ transition.

4. Microscopic Model, DWIA Analysis. In the above discussion

of microscopic calculations reaction mechanism aspects have always

been neglected. As mentioned earlier in relation to the collective

model, a proper analysis must take reaction mechanism into account.

In other words microscopic model form factors should be used in a

DWIA analysis. There are several such calculations presently under-

way, and we report below on the current status of two of these.

a. Analysis of Arima et al. The first calculation is due to

Arima et al. 92 In choosing their model space these authors observe

that most calculations of 180 wave functions predict less than 15%

4p-2h components in the 0i and 2X wave functions. Because of these

small amounts they decided to simplify their calculations by consid-

ering only 2p-0h configurations in the 2s-ld shell, and compensate

for the effect of this truncation through the use of somewhat larger

effective charges than they would otherwise have needed. Arima et al

use the s-d shell wave functions of Inoue et al.60 With these wave

functions, :Ln order to explain the observed B(E2) values, effective charges e =2.05 and e =0.60 are required. Arima et al find that our p n 164 MeV data is best fitted by e =1.32 and e =0.68. These values lead P n to isoscalar (IS) and isovector (IV) effective charges

134 e(IS) = e + e =2.00 p n (5.19) e(IV) = e - e = 0.64 p n

If these values are used in a schematic amplitude analysis one would obtain

2 2 _ ("2E(IS) + e(iv)1 = (5 , . p [2e(IS) - £(IV)J lm91' t5*Z0)

Arima et al perform DWIA calculations using the Inoue wave func- tions and the above effective charges. They have used a hannomic oscil- lator density with oscillator constant w=1.7 fin. The rms radius for this distribution is =2.56 fm. They have used the Kisslinger potential with no angle transform and no Lorentz-Lorenz corrections, and free TT-nucleon phase-shifts from Roper et al75 to calculate the potential amplitudes. We see that, as far as the basic tools are con- cerned, this calculation is virtually identical to our calculations using DWPI, and we expect the elastic scattering angular distributions to be very similar to ours.

The Arima calculations for our 164 MeV data are shown in Fig. V-8.

The fit is quite staisfactory for both elastic and the 2\. Similar calculations done for the 230 MeV data, shown in Fig. V-9, result in poorer fits to the data. (Note that the Arima calculations of elastic scattering angular distributions are nearly identical to our results for both 164 and 230 MeV).

The poor fit to the 230 MeV data prompted Arima et al to vary the bo,bi potential amplitudes to improve the elastic fit. They then used the adjusted parameters to recalculate the 2\ angular distributions.

The results are shown in Fig. V-10. We see that the fits using the

135 1 1 1 I ; 1 1 l8 0( ir,»')l8O = 164 MeV ir* — o \ 10 r ELASTIC


\ 10 -

\ c o° " °° or O _ C- C \

_ I rr - # • v'*

\ -

2*, 1.98 MeV

— • TT*

•» ~ 0 TT -

I r I (

0.1 - w • ;

1 1 I 1 1 1 1 10 20 30 40 50 60 70 80 90 9cm.

Figure V-8. Our 164 MeV elastic and 2* data with curves calculated by Arima et al.

136 10*



TT • 230 McV 2*,l 98MeV 10 **<* „*



J I I I 10 20 30 40 50 60 70 80 "cm

Figure V-9. Our 230 MeV elastic and 2X data with curves calcu- lated by Arima et al.

137 .0*

I 10 cs T3 b •o

10 20 30 40 50 60 80

1 1 i 1 "TT T

18 18 3(X,T') TV • 230 MeV - ,1.98 MeV 10 - ——O r"

• • £ I - cj

- - \

**• 0.1 r-

I | | | 10 20 30 40 50 60 70 80 °c.m.

Figure V-10. The 230 MeV elastic and 21 data with curves calculated by Arima et al by varying the amplitudes bo,bi in the Kisslinger potential.

138 adjusted parameters are quite good for both the elastic and 2\ angular

distributions. Arima et al have used the same values of effective

charges for these calculations as they have at 164 MeV.

b. Microscopic Calculations of Lee et al. Lee, Lawson and Kurath93

(LLK) have done microscopic model DWIA calculations with snore sophis-

ticated wave functions than those used by Arima et al. They have used

a momentum space optical model code in which the formulation of the

optical potential is done in the manner of Londergan, McVoy and Moniz95

(LMM). Lee et al used three different types of wave functions. These are:

(1) The highly successful coexistence model wave functions of Lawson,

Serduke and Fortune (LSF) which are the result of empirical adjustments made to fit a great number of properties of the states in 180. (See

Chapter IV for a discussion of these and the other wave functions used in the calculations discussed in this chapter). In tne LSF model the effective charges required to explain several quadrupole transitions are: for the d-shell e =1.547 and e =0.547; for the s-shell e -1.612 p TI p and e =0.612; and in addition they ascribe an effective charge to the core which is equal to e =1.753 and e =0.753. P n

(2) The second calculation is done using the well known multi-shell shell model wave functions of Zuker, Buck and McGrory (ZBM) . These wave functions were obtained by assuming a closed 12C core and consid- ering active particles in the p and s-d shells for nuclei of A=*16-20.

These wave functions have been successfully used to fit a great variety of experimental results for nuclei in this region. Effective charges e =1.5 and e =0.5 are required to explain quadrupole transitions.93

139 (3) In the third calculation LLK used the s-d shell wave functions

of Cohen et al on the basis of essentially the same arguments as Arima

et al. That is, as long as p-shell components are less than 15%, one

can perhaps absorb their contribution by choosing slightly larger

effective charges. To explain the B(E2) value for the 2\ transition

the effective charges are e =1.9 and e =0.9. P n

The predictions of LLK are shown in Figs. V-11,12,13 along with

our data. The fits are absolute in the sense that they have not been

adjusted in any way to fit the inelastic data.

The first thing to notice is that through the second maximum the

curves from LSF and s-d shell wave functions are essentially identical.

In other words, the increased effective charges are successfully able

to simulate the effects of the detailed 4p-2h components as Arima et al

had surmised. The second significant point is that while ZBM results

for IT are essentially identical to the other two, for TT they are

about 15% lower - even at the first maximum. This would imply differ-

ences in neutron components in the ZBM wave functions relative to the


C. Discussion of Inelastic Results

In Figures V-6 through V-13 we can identify certain common aspects

of the failure of the DWIA attempts to fit the inelastic angular dis-

tributions. The forward angle behavior of all DWIA curves disagrees

with the experimental data at both energies. As one goes to smaller

angles the data show a rise in cross sections and an eventual flat-

tening at about 20°, whereas all DWIA curves reach a maximum at about

30° and decrease by as much as a factor two at smaller angles. Since

this happens in all the DWIA calculations whether they are done with

140 I I I I 18-, , .18- 0(ir,ir ) 0




I -


*-* b •o


j I 0 10 20 30 40 50 60 70 80 90 8cm.

Figure V-ll. Our 164 MeV elastic and 2\ data with curves calculated by Lee, Lawson, and Kurath using ZBM wave functions.

141 10



I I 10 20 30 40 50 60 70 80 90 9 cm.

Figure V-12. Our 164 MeV elastic and 2\ data with curves calculated by Lee, Lawson, and Kurath using LSF wave functions.

142 10 -



b •o


0 10 20 30 40 50 60 70 80 90 9 cm.

Figure V-13. Our 164 MeV elastic and 2i data with curves calulated by Lee, Lawson, and Kurath using s-d wave functions.

143 coordinate space or momentum space configurations and with collective

or microscopic form factors, we must conclude that the problem is a

general one and relates most likely to the DWIA method. We recall

that similar problems have been reported from time to time in the

literature about forward angle inelastic scattering of other nuclear

projectiles. However, it does appear that the discrepency is larger

in the (TT,TT ') case than in any other.

As far as the DWIA fits to the 164 MeV data are concerned they are

generally quite satisfactory in all other aspects. The location of the

minima and the secondary maxima are well reproduced. The height of the

second maxima are well reproduced in both Arima's and our calculations.

The LLK calculations do have a problem in this respect. They predict

the second maxima almost a factor two higher than the data. Also their

minima are almost a factor two deeper than the data. In our experience

these types of problems are generally related to the density parameters

used. Arima and we have used the latest electron scattering parameters,

as discussed in the introduction to this chapter and in Appendix E,

with half-density radius of 2.55 fm, whereas LLK have used a Woods-

Saxon density whose parameters were adjusted to fit the present data.

The values c =c =2.20 fm and a =a =0.569 fm were used. The n p n p resultant half-density radius of 2.20 fm used by LLK is quite a bit

less than that used by us. We believe this is the reason for the

aforementioned discrepency.

The 230 MeV data present considerable problems. Using the same

prescriptions which were successfully used for the 164 MeV data, i.e.

144 a Kisslinger potential, free ir-nucleon parameters, and the Bates

electron scattering density, the fits which were obtained to elastic

and inelastic scattering (Figure V-7) are quite poor. The elastic

scattering curves show much shallower minima than the data, and the

inelastic fits are even poorer. Arima obtains similarly disappointing

results (Figure V-9).

But there are worse problems. Not only are our DWIA fits to the

inelastic data poor, the numerical results obtained at 230 MeV are quite different from those at 164 MeV. At 164 MeV we obtained

(BR)n/(6R) = 1.68 ±0.20. At 230 MeV we obtain ($R)n/(SR) - 1.24±0.14, a result which is significantly different. Arima reports a similar problem in his microscopic model calculation.

There are two possible explanations for our problem. The trivial explanation is of course that our 230 MeV data are wrong. We can' easily rule this possibility out. The 164 and 230 MeV data were taken with the same targets, during the same run, and normalized to hydrogen cross sections in the same manner. Further, the 230 MeV data are in excellent agreement with our own earlier published data88 at 230 MeV which were taken with a different target, normalized in a different manner, etc. The internal consistency of the data is also clearly verified by the fact that in the extreme forward region,

17-25°, in which theoretical calculations for elastic scattering are essentially insensitive to variation of optical model parameters and where o -(Q) and a +(0) are nearly equal, the experimental data agree very well with the theoretical curves.

The only other possible explanation relates the two problems - poor fits and inconsistent results - to each other. As mentioned

145 earlier the 230 MeV data show sharper minima both in elastic and in-

elastic scattering than predicted. Since the minima are known to be

sharpest when the nucleus is at its blackest, that is when the pion

energy is nearest to the (3,3) resonance in the nucleus, it appears

that the data are indicating that the effective pion energy is closer

to that of the (3,3) resonance, i.e., is lower than the nominal

230 MeV. In other words the resonance amplitudes bi need to be ad-

justed. (As to why the 164 MeV parameters should not and did not

need adjustment is completely unclear in this line of thinking).

Arima et al therefore freely adjusted the amplitude parameters bo,bi

to obtain good fits to elastic scattering (see Figure V-10). Their

revised parameters are listed in Table V-4 along with their original

free ir-nucleon paramsters. It is indeed noticed that the largest

change is caused in the real part of the resonant amplitude bx which

changes from about -2.4 to +0.5 for TT and from -2.6 to -0.6 for TT .

This is what we expected. However, other parameters also change. The

largest change is an increase in the imaginary part of bo from about

+0.28 to a difficult to understand +0.80 for both TT and ir~.

With these modified parameters Arima et al obtain much improved

fits to the elastic scattering data as shown in Figure V-10. Though

better than before the fits are far from perfect - the predicted

minimum for TT is in agreement with the data, but for IT is at 2-3°

too large an angle. Similarly, while the depth of the minimum and the

height of the maximum for TT are well reproduced, those for TT ara

predicted significantly lower than the data. In any case, using these

adjusted parameters Arima et al report much improved fits to the

146 TABLE V-4. Potential parameters bo.bi used in calculations for the 230 MeV data of this experiment.

- Parameter 7 TT Calculation Type bo 1 bo b 1

Arima et al free -0.42 + 0.28i -2.36 + 4.60i -0.57 + 0.291 -2.65H1-5.151 This work free -0.47 + 0.311 -2.20 + 4.691 -0.63 + 0.331 -2.47Hh5.24i

Arima et al adjusted -0.71 + 0.831 0.47 + 5.041 -0.29 + 0.771 -0.65H1-6.161 This work adjusted -2.27 + 0.711 1.15 + 4.771 -1.32 + 1.241 -0.52H1-5.571 inelastic data. These fits are also shown in Figure V-10. The improve-

ment in shape is obvious. The results are truly remarkable if they

were indeed obtained using the same effective charges as at 164 MeV.

We are not entirely certain about this point since the calculations

were done only very recently in Japan and only fragmentary results

could be obtained up to the time of writing this thesis.

In view of the fact that at 164 MeV the fits obtained by our

collective model DWIA calculations and Arima's microscopic model

DWIA calculations were nearly identical, we have tried to reproduce

Arima's results by doing collective model calculations with his

parameters. We have not succeeded in these attempts. Our elastic fits,

shown in Figure V-14, are clearly different from his, especially in

the minima. Our inelastic fits differ even more from his. We do not

understand the reason for these differences. As a next step in these

investigations we have done our own parameter search starting with

Arima's best fit values. The resulting parameters are listed in

Table V-4. The elastic scattering fits obtained with these parameters

are shown in Figure V-15. The improvement in the region of the minima

is quite clear. The fits are actually better than those obtained by

Arima et al, especially for TT and especially in the location of the

minima. The inelastic predictions using these parameters are also

shown in Figure V-15. The improvement in the shape of the inelastic

fits is less marked, and even more disappointing is the fact that the

ratio (0R) /(8R) determined in the manner described in Section V-B-2 n p is found to be 1,26± 0.14, i.e., essentially unchanged from that

obtained before.

148 id1!



_1 I 1 I 10 20 30 40 90 60 SO "cm.

: 1 I I 1 1 1 I :

18JU..-1)18 : TT «230M«V 2| ,L98MeV 10 —— • x* - — — o

• E 1 C5 •a \ b L m 0.1 \

1:11 0 10 20 30 40 50 60 70 80

Figure V-14. Our 230 MeV elastic and 2~\ data with curves calculated by using Arima's adjusted parameters in DWPI.

149 C!

b T3

l8 18 0(rrT') I, * 230 MeV

10 r


b •o

0.1 r

10 20 30 40 50 60 70 80

Figure V-15. Our 230 MeV elastic and 2j data with curves calculated with DWPI using our best fit parameters.

150 In view of the above results it is difficult to understand how

Arima et al could obtain the good fits shown in Figure V-10 using the same effective charges at 230 MeV as they did at 164 MeV. We suspect that the effective charges needed by them for the 230 MeV data are actually different, since they themselves point out that their effec- tive charges give p2 = 1.91 and the experimental value at 230 MeV is

1.58±0.13. We must therefore defer further comment on this discre- pancy until more complete information is available on the work of

Arima et al.

Tn summary, the problem with the 230 MeV results does not have a simple explanation. Parameters adjusted to give better fits to the elastic scattering data help somewhat in fitting the shapes of the inelastic data but fail to solve the problem of the inconsistent re- sults for (6R) /(BR) . The problem evidently requires a better n p understanding of reaction mechanism for its resolution.

If we assume that the problem lies with the 230 MeV results and that the 184 MeV results are indeed correct, we can proceed to compare them with other results for 180 in the literature.

In Table IV-5(a) and (b) we have summarized the experimental in- formation available for the 0i to 2i and 0i to 3i transitions in 180.

For the 3i transition there is a very large spread in (3R) values ranging from 0.86 and 0.89 fm from (180,180 ) and (a,a1) reactions to 1.33 and 1.40 fm from (a,a') and (e,er). The average is approxi- mately 1.10 ± 0.25 fm. Our results, (3R) = 0.97 ± 0.08 and (8R) = n p

1.31 ± 0.12 are both within this range. For the 2i transition the the latest (p,p') and (ie0,180 ) values for (3R) are larger than the

151 earlier results of Kefs. 46 and 55. Although no errors are indicated

it appears that the results are consistent with ($R) =1.1 ± 0.1 fm.

The latest electromagnetic transition rates in the last row of Table

IV-5(b) as well as the recent values from Table IV-5(c) lead to an

average B(E2Hof about 45 e2fm** which also corresponds to CBR) of

1 3 1.1 fm, assuming R = 1.2A = 3.145 fm. Our results are (SR)Q =

1.21 ± 0.12 which is right on the edge of the errors, and (0R) =

0.72 ± 0.07 which is clearly outside the errors.

The only other type of comparison which can be meaningfully made

is if our ratio (gR) /(3R) = 1.68 ± 0.20 can be compared to another n p

independent determination of the same quantity. As mentioned briefly

in the introduction, Bernstein et al9 have recently drawn attention

to the fact that the transition rates in mirror nuclei can be simply

related to the matrix elements for isovector and isoscalar components,

or equivalently to the neutron and proton matrix elements, so that 18 h Mnn ||"B(E2)( Ne: 2r>ot (5.21) H 18 + p |_B(E2)( 0: 2| + 0 )J Z6P

In Table V-5 we summarize the results obtained from this equation

from the measured lifetimes for these two transitions. The value for

the ratio M /M equals 2.12 ± 0.20. Our result from pion inelastic n p scattering at 164 MeV is M /M = N/Z • (6R) /(BR) = 1.68 • 1.25 = n p n p

2.10 ± 0.25, i.e., it is in perfect agreement. This good agreement

is perhaps fortuitous, nevertheless it is quite clear that in spite of

experimental uncertainties both in lifetime measurements and in pion

inelastic scattering measurements, and uncertainties in the interpre-

tation of both experiments, there are no indications of any disagreement. 152 TABLE V-5. Calculations of transition matrix elements 1^ and M_ for the 2"J" state of 180 using electromagnetic transition rates.

18 18 M /Ma Source T(2t, O) T(2t, Ne) Mn M n p

Bernstein et al9 3.58 ps 0.73 ps 4.31(21) 1.73(7) 2.26(16) From literature 2.90 0.67 4.48(40) 1.92(7) 2.12(20) This experiment 2.10(25)

First two values corrected 10% for binding energy differences.

Lifetimes from Ref. 23.

153 We conclude this dissertation with the highly satisfying know-

ledge that, while much still remains to be learned about pion inter-

actions with nuclei, the unique capabilities of pions in shedding

light on new aspects of nuclear structure have been convincingly

demonstrated in the present work.


The properties of the channel part of EPICS are well known, 1 2 and are summarized in a report of the tune-up experiment.

Design details for both channel and spectrometer are discussed in

Refs. 11 and 12. This appendix contains a general discussion of

EPICS, with particular emphasis placed on the tuning of the spectrometer for data taking.

The channel operating conditions reported in Ref. 12 have been modified in one significant respect - the jaw openings. The

FJ04 horizontal opening (H) has been reduced from 3 cm to 2.5 cm, reducing the channel horizontal spot size from 6.4 cm to about

5.5 era. The vertical jaw setting (FJ04V) was determined by the size of the verticle opening in the target frame, 20 cm. To avoid the possibility of scattering from the edges of the frame a 6.25 cm

FJ04V setting was used. The pion flux at the target scales approxi- mately linearly with the FJ04 openings.

At forward spectrometer angles, especially for TT , the beam flux is high enough to affect the operation of the front chambers, and also to cause large computer dead times. The forward angle IT" runs can be handled by running the computer in "may process" mode

(every event written to tape, but an event is analyzed only when the computer is not busy). All the data can thei. be recovered by replaying the tape. For forward angle IT (forward of 25°), in addition, the pion flux must be reduced. This is accomplished by decreasing the FJ04H and FJ03V and H openings (the FJ03 are, for historical reasons, the last jaws in the channel, and affect the

155 pion flux in roughly the same manner as FJ04). For forward angle

u data we used a 10X reduced flux of pions.

As has been mentioned in Chapter II, the EPICS system measures

absolute momenta. The momentum of a scattered particle is measured

by its trajectory reconstruction in the spectrometer, and the

momentum of the incident particle corresponding to it is determined

by calculating its position at the target plane. This information

specifies the reaction kinematics. With such a system there is only

a very rough correlation between spectrometer focal plane position

and final momentum.

This method can be contrasted with that of focusing type energy-

loss spectrometers, which also utilize dispersed beams. In the

energy-loss configuration, if the beam momentum spread is not too

large, the quantity measured is the difference between the incident

and the outgoing momenta. This is done by focusing all particles

with the same momentum loss to the same point.

Event reconstruction in the EPICS system is done in the soft-

ware. This permits trajectory tracing and specification of the

event origin at the scattering target. This feature is also used

to reject non-target related events, i.e. accidentals.

Because of the large spectrometer phase space, particle flight

paths through the spectrometer vary over a wide range. In other

words, a particle hitting the far edge of the spectrometer focal

plane travels farther than one along the central trajectory. A

correction must be made for this variation if any time-of-flight

(TOF) information in the spectrometer is to be useful. This

156 correction is made possible by complete knowledge of particle paths

through the spectrometer. However, as mentioned in Chapter II, TOF

information was not obtained in this experiment.

Spectrometer design was done in several stages, utilizing

complimentary methods. Computer codes used were TRANSPORT , a

second order matrix transport optical program for a given beam

packet; DECAY TURTLE 97, which is TRANSPORT with and

3 8

several other refinements included; and MOTER ' , a fourth order ray

tracing program which handles individual particle trajectories. The

MOTER code was used with spectrometer maps to predict

the operating parameters of the spectrometer.

The spectrometer optics are parameterized, both in MOTER and in

actual operation, with nine quantities: (x,6,y,)_ , (x,9,y,)D r K


9 and are the angles in the x and y planes, respectively, of the particle trajectory relative to the central ray. Subscripts F and R refer to quantities at the front and rear chambers. The quantity

6 = [P~P=r.Q/,,J/pCT,o .. > is the percent deviation of the particle momentum from the spectrometer central momentum, as calculated firom the reaction kinematics. These quantities are considered in all combinations through fourth order (i.e. xp-x -x^-x or x^-x^•8p'5, etc). From analysis of the magnetic maps" it was found that 70 terms, first through fourth order, were of interest - but that each parameterized spectrometer quantity could be convergently fit by selecting a set of at most 26 terms. A satisfactory final solution could be found using no more than 24 terms for any one calculated 157 quantity. The terms calculated by MOTER from field maps can be com-

pared to their empirical values determined from real particle tra-

jectories in the spectrometer during the tune-up. They agree well

through second order. In actual operation, however, the empirical

values are used.

There are nine spectrometer quantities - 6 , <5. , x^, 9 , y^,

<4 , 9 , , , , , , £ - calculated with the polynomials for each event.

The particle delta, 6 , is first parameterized and then used in the

optimization of other of the spectrometer quantities. 6.. is the

percent momentum loss of a particle in the target. It is not useful

except in the case of a very thick target. The target positions -

(x,6,y,) - are projected backward through the quadrupoles from the

positions in the front chambers. Thesp quantities are used in making

target cuts for good events. The "angle checks", 9 ,, and $ ,, , are

the difference between the 9,4> angles of the particle trajectory

measured directly by the rear chambers and the 0, angles calculated

at the 'focal plane' by the trajectory reconstruction through the

spectrometer. The angle checks are used as further requirements for

good events. Finally, £ is the computed path length of particles

through the spectrometer, which, due to the large focal plane, can

vary widely. If time of flight information is desired a knowledge

of £ is necessary to correct for differences in path length.

As mentioned above, these spectrometer quantities are constructed

from polynomials through fourth order in the front and rear quantities

plus & . The polynomial coefficients are optimized by taking real

particle trajectories in the spectrometer. For example, to tune the

158 spectrometer parameters x and 9 a set of thin horizontal rods is

used as a target. The rods are well localized in the x (vertical)

direction and their positions are known. Particles scattered from

the rods are focassed by the quadrupoles onto the front chambers.

By adjusting the quadrupole tuning the image of the rods at the

chambers is sharpened. When the quadrupoles are properly tuned,

the x and 0T coefficients can be optimized by making the corres-

pondence Detween the known rod positions at the target and the front

and rear positions and angles read-out from the chambers for the

scattered events. A similar procedure is followed, using vertical

rods, to tune y and ij> .

The sequence of events leading to a "proper" spectrometer tune

is roughly as follows: (1) calibrate front and rear MWPC, (2) tune

6 , then remaining spectrometer multinomials, (3) make a spectrometer acceptance scan of solid angle vs.

±10%, which is essentially the entire spectrometer acceptance), and

(4) repeat step (3) for various standard jaw openings.

The details of MWPC calibration are discussed separately in

Appendix C. The spectrometer acceptance scan is done by moving the elastic scattering peak of some heavy target (56Fe in our case) across the focal plane in a number of steps from, for example,

6 = -10% to +10%. The yield is calculated at each point and stored in a solid angle data file. The acceptance scan must be repeated for each jaw setting used (the jaw settings were found to be well reproducible).

159 The Sp tunes done at the two energies of this experiment, 164 and

230 MeV, led to slightly different resolution. In practice, for

energies this close to each other, nothing is lost by using a single

spectrometer tune for both energies.


To reiterate some facts from Chapter II: the major EPICS soft-

ware features are the EVENT ANALYZER, the TEST FILE, and the DISPLAY

PACKAGE. Every good event, defined by (S2-S3)-FRONT (where FRONT =•

(Fl OR F2)-(F3 OR F4) ), is processed by the ANALYZER. In fact,

this is the only hardware restriction on acceptable events - all

further event processing is done in the software.


The ANALYZER is comprised of two parts: (1) a MACRO (PDP machine language) file which defines the CAMAC units (ADC's, TDC's,

sealers) and assigns their various outputs (for pulse heights,

times, and scaled events) to software channels; and (2) the FORTRAN processor file which takes the raw CAMAC data and calculates the relevant final quantities. It is this processor file which is referred to by the name ANALYZER.

At the time this experiment was run the ANALYZER had space for

150 integer data words and 150 real data words. The integer words were used for storage of the digitized time and pulse height infor- mation, and also real quantities truncated to integers for histogram display. The real data words contained quantities calculated in the

ANALYZER, such as chamber positions and the trajectory reconstruction information. In addition there were 150 integer constants and 300 real constants available in the ANALYZER. These were used for any- thing from chamber calibration, kinematic constant storage, or spectro- meter constant storage to display constants such as the channel off-

161 sets for each histogram.

A block diagram of the ANALYZER is shown in Fig. B-l. This

sequence is passed through for each event triggered in the computer,

with the exception that the first step, the reading of the drift

chamber and spectrometer polynomial data files, is done only for

the first event after the ANALYZER is installed (i.e. after a com-

puter dead-start).

After computing average scintillator times using times from

both ends, and particle energy loss from scintillator pulse heights,

the particle AE is tested (see section on TESTS). If the AE does

not fall within windows set for minimum ionizing particles then all

other calculations are skipped. In this case the ANALYZER is re-

entered before the end and such events are available for dot plot

and histogram arrays.

For a good minimum ionizing event we proceed to calculate the

positions in all chambers, front and rear. This information is

used to calculate the particle trajectory at the quadrupole focus.

The rear chamber positions are drift corrected (see Appendix C) in

the dispersion plane, and then the particle trajectory at the spectro-

meter 'focal plane1 is calculated.

Armed with all position and angle information, we are now in

a position to calculate the spectrometer quantities discuss 3d in

Appendix A. This is done by the subroutine MULTIS, which calculates

these quantities from the stored polynomial coefficients. This pro-

vides, among other things, the particle trajectory at the scattering

target. Until now we have assumed that the scattering target is

















Figure B-l. Block diagram of the EPICS EVENT ANALYZER.

163 perpendicular to the spectrometer central trajectory. The next step

is to correct the target quantities for the rotation of the target

with respect to the spectrometer. Also at this point the TOF cor-

rections for path length are entered info the ANALYZER.

The final manipulation consists of calculation of the missing

mass of the recoil target nucleus. This is the quantity which goes

into the energy loss spectrum, Fig. II-5. There is also an option,

via routine FUNNY, to calculate any given polynomial - this is used

as a debugging aid.

Next the data is arranged in bins for histogramming. A dot

plot package is available which plots, in a two-dimensional scatter

plot, any chosen binned data word vs. another for each event. This

option is activated on line by the user and the data words and dis-

play limits can be changed at will. The binned data is stored in

a pre-selected collection of histograms, which can be used to store

either raw or calculated quantities.

The last routine called from the ANALYZER, F1TP0L, is used for

tuning of the multinomials. When activated, it writes on a disk file,

for a specified number of events, the values of the nine data words

used to calculate the multinomials (xF,8F,yF,F,xR,9R,yR,R,6 ). A

separate routine constructs polynomials from these data words and

uses the data from the FITPOL disk file to calculate the polynomial

coefficients by methods discussed in Appendix A. This is the source

of the MULTIS coefficients.

One other option contained in the ANALYZER involves the quanti-

ties QAD. These are generated by a calibration chamber placed at

164 the entrance to the first quadrupole, and are used to aid in opti-

mization of the spectrometer angular resolution. This part is skip-

ped during normal data taking.


As discussed in Chapter II, the TEST FILE contains two classes

of tests: micro-tests4 GATE and BIT; and macro-tests, AND and OR.

The GATE micro-test can be applied to any data word and consists of

a cut placed on the binned data (i.e. a cut placed on channels re-

sulting from integerizing the data). An event falling above the

lower limit and below or equal to the upper limit passes the test,

all others fail. The BIT test queries whether a particular bit

(e.g. from a coincidence register) is set or not. The macro-tests

are logical AND's and OR's applied to previous micro- and macro-

tests. The only restriction on constructing test files is that, in

a given loop, all micro-tests must preceed all macro-tests.

In order to implement the TEST FILE a user first examines raw data histograms and makes cuts, then examines the calculated data histograms and makes cuts. With this information a TEST FILE con-

taining these cuts can be constructed and macro-tssts built upon it.

For convenience the file is structured in loops which are called separately and in order from the ANALYZER.

A typical TEST FILE is given in Fig. B-2. It contains four loops which are arranged in a somewhat logical order. The first loop tests only scintillator pulse height data. The second loop contains the chamber tests. The first ten tests in the chamber

165 CAT. 3 CAT 2

SS:i?::!J: AMD.II.11. AND. 12,13. AND.14.IS.

AKD.27.20. WO. ID, 11.12,13. AHC.IB.11.19. MB, 12,13.14.19. 1CM.3B,31,32,


IOIB KT CHEEK SET OCtK LE CHECK (2 * IW Tt IMEMTIOM CUTS f (*flj . FOCflL Ft, l^Ct) . rOCOL Fl« lfC-^> . FOCAL Ft LO» 3 COtffTER

IOK IBS. 184. AND.39.39. M4D 29-28.It.81.09. HMD 10.109. OHO ».Z6.t0.0B.0fl. AHD I 1. I I t. WTO 29.2*.13,03.31. 'CH9 nQRfi.. F - T | - - -.CH9 EFF flHP GOOD EL5C ;CHlfl NORN. F'S. KM. B9 OK :CH10 EFF onp DK EISEWEKE

Figure B-2. TEST FILE used during this experiment.

166 loop are those for the check sums - one for each chamber. The check

sum is defined as the difference between the time sum for signals on

the cathode and anode (see Appendix C on chamber calibration). Very

tight cuts are placed on the resulting distributions via the 10 check

sum tests. An event would fail the test if, for example, two anode wires fired for a single event or if one or more signals on the delay lines failed to be detected at the ends. The final test of a good chamber event is TEST 35, a coincidence of all front and rear chambers

(in our case chamber 9 was not working and was left out of this test).

The third loop contains most of the tests on quantities calcu- lated in the ANALYZER. The final test of a good spectrometer tra- jectory is the aberration cut, applied in TEST 70. This is a com- bination of satisfactory trajectories (x,9,y, values) at the target and front and rear chamber locations, plus reasonable 9, angle chacks, plus a good chamber event (TEST 35). We assume that a good part of the background, due almost entirely to pion decays in the spectrometer, is removed by application of this test.

The fourth loop contains two important groups of tests. One group, TESTS 94-107, handles the partitioning of the spectrometer acceptance into angular bins. The other group, TESTS 109-116, is used to calculate the chamber efficiencies.

The angular binning option is essential in measuring detailed angular distributions without moving the spectrometer for each point.

At EPICS it is the distribution which corresponds to the spectro- meter angular acceptance. This distribution had a total width of about 6°, but the central, relatively flat, region is slightly more

167 than 2° wide. The central region is divided for binning in two ways:

in four 0.5° bins, TESTS 101-104; and in two 1° bins, TESTS 106-107.

In addition the full 6° is accepted in TEST 105. These bins are all

gated by the aberration cuts. At this time the 1° bins are thought

to be reliable, and are consistent with the angular resolution of

the spectrometer, about 0.9°. The 0.5° bins have not been analyzed

for this experiment. The ability to use only 2° at a time makes it

necessary that the spectrometer be moved in 2° steps in order to

get non-overlapping, evenly spaced data points.

Front chamber efficiencies are calculated quite simply. The

definition of overall front chamber efficiency is ALL FRONT CHAMBERS

(TEST 29) divided by LOOP 2 COUNTER (TEST 36), where the LOOP 2

COUNTER counts all events in the i .amber loop.

As an example of the calculation of rear chamber efficiency we

will examine TESTS 109-110 which define the efficiency of chamber 5

(the first rear chamber). The rear chamber efficiency is calculated

for only good events in all the front chambers, i.e. those events

satisfying TEST 29. To calculate the efficiency of chamber 5 we

must determine the proportion of events recorded by chamber 5 out

of the total recorded by the other three rear chambers (6,9,10),

i.e. R -R -R -. The logical AND of TEST 11 and TEST 26 gives the

number a? events recorded in chambers 6,9,10. In addition we must

exclude all events which miss chamber 5 completely but are * recorded

in the other chambers, because such events do not represent a true

inefficiency in chamber 5. This is done by setting x and y gates

on events in chamber 6, which is adjacent to chamber 5, so that these

168 gates are somewhat smaller than the active areas of the chambers.

We do this with TESTS 81,89.

So in TEST 109 we have all events which were recorded in

chambers 6,9,10 (TEST ll'TEST 26), and which must have passed

through the active area of chamber 5 (TEST 81'TEST 89). TEST 110

is the same group of events which were recorded by chamber 5 as well. The ratio of TEST 110 to TEST 109 is the chamber 5 efficiency.

The product of individual chamber efficiencies is the overall rear chamber efficiency.


Data display and manipulation are done with a histogram dis- play package named DSP. It permits the display of any data word from the ANALYZER - the user specifies the data word, upper and lower channel display limits, and the packing density (called the bin size). Each histogram may be gated by any one of the tests from the TEST FILE, where the default is no gate. For example, if we wished to construct a histogram of the energy loss spectrum (from missing mass calculation, word 200) containing only good events from the first 1° angular bin, we would gate a histogram of data word 200 by TEST 106.

For a closer examination of the histogram data there are a num- ber of quick commands in DSP which enable the user to change axis scales or display limits. There are numbered cursors available for setting test gates or marking peaks. A fitting routine, PEKFIT102, can be called from DSP, with the ability to fit up to three peaks

169 in the spectrum. This fitting is useful for keeping track of on-

line data, or for fitting well separated peaks in the absence of

significant background, but was not used for the final analysis in

this experiment (see Chapter III and Appendix D).

Normally a small set of the histograms is saved for later ana-

lysis. This can be done automatically at the end of each data run,

and the saved histograms are written on a disk and signatured by

the run number.

Space is available in the display package DSP for as many as

eighty histograms to be set up - forty spaces reserved for on-line

data, and forty for reading in previously saved data. ' :he saved

histograms read into these spaces can also be subjected to all

possible DSP manipulations so that data from previous runs can be

examined while taking new data. Or the saved data can be examined

at any future time at the experimenter's leisure. The histograms

can also be listed channel by channel on a line printer for a

closer look at the data.

The EPICS software has evolved into a versatile system for

data handling. If the spectrometer is tuned carefully, the tests

and histograms chosen properly, and data taking done in must pro-

cess, the saved histogram files can be the source of much of the

final data - obviating the need for most of the time consuming

data raplay.

IV. Run Analysis

A problem for most experiments on EPICS is the sheer bulk of

170 data which can be generated in a period as short as a week. To

keep track of the operating parameters for each run there is an end

of run record which may be activated after each run. This informa-

tion is also saved with each histogram file written to disk. This record not only provides a handy reference for each run, but also calculates various correction and normalization factors used to cal- culate the experimental yield.

Briefly the information written contains the run kinematics, the EPICS magnet: DVM readings for all power supplies, the test results and the final sealer readings. The end of run routine calculates the final normalization and correction factor, so that the experimental yield is obtained by multiplying the peak area by the correction factor. Calculation of the cross sections from this information is discussed in detail in Appendix D.

V. Calculation of Missing Mass

A block diagram outlining the main steps in the missing mass calculation is shown in Fig. B-3. The missing mass is defined as the difference between the calculated mass of the recoil residual nucleus based on the incoming and outgoing pion momentum, and the ground state mass of the residual nucleus looked up in a mass table.

Thus, if the residual nucleus is left in an excited state, the out- going particle momentum will be lower than for the ground state transition, and the calculated mass of the recoil nucleus will be higher than its ground state mass. The difference between the two gives a positive value of missing mass which is equal to the energy of the excitation in the residual nucleus.





Figure B-3. Block diagrams of the main steps in calculating missing mass for the energy loss histograms.


I. Front Chamber Calibration

We have four front MWPC. Each contains an anode at positive high voltage with wires spaced 4 ram apart, and a cathode at zero voltage with 1 mm wire spacing. The negative ion avalanche in the chamber occurs at the anode, and we assume that only one anode wire collects the avalanche for each ionizing event. Thus the anode time spectrum will distinguish every wire, and is therefore "discrete".

The cathode signal is caused by the positive ions drifting from the avalanche site and is distributed over several wires (typically 8-10 wires) near the event - the cathode time spectrum is therefore "con- tinuous", the individual wires being indistinguishable.

The presence of four chambers makes calibration possible by utilizing particle positions in two chambers to calibrate a third, in all possible combinations of three of four chambers. The chambers are packaged in two pairs of two - chambers 1 and 2 together, and 3 and 4 together. The second and fourth chambers are rotated 90° with respect to the first and third. This means that the anodes of cham- bers 1 and 3 can be projected onto the cathodes of chambers 2 and 4; and the anodes of chambers 2 and 4 onto the cathodes of chambers 1 and 3. The pairs are separated by about 20 cm which gives a distance long compared to the separation of wire planes in each pair over which to define particle trajectories. This minimizes trajectory projection errors.

The anodes are calibrated first, and that information used to

173 calibrate the cathodes. To discuss the anode calibration we make the

following definitions:

AD * A - A^ = time difference between ends of anode delay line

CD • C_ - C R •• time difference between ends of cathode delay line

AS • A^ + A_ = time sum between ends of anode (C.I) delay line

CS =• CT + CB = time sum between ends of cathode delay line

TC = CS - AS = time check sum

We construct a quantity representing position along the anode plane:

2 3 XA = aQ + ax • AD + a2 • AD + a3 • AD + a4 • AD • TC (C.2)

where the last term is a correction for dispersion (signal degradation)

along the anode delay line. The quantity X is divided by the anode

wire spacing and integerized so that it represents the exact anode

wire fired. Multiplying the integerized value by the wire spacing

yields the position of this wire in the anc•?. . direction - the

so-called truncated anode position T . A

After a number of events are collected in the chamber being

calibrated we minimize the difference

2 2 X = 2 (XA - TA) (C.3)


by varying the coefficients aQ-a,. These > ome the anode calibration

coefficients for calculating ancde position.

With all anodes calibrated it is possible to calibrate the cathodes.

174 Because a single anode wire fires in each chamber for each event, the positions projected to the cathode plane from the two anode planes used for calibration form a discrete array. To minimize calibration errors from scatter about these discrete positions a large collection of events must be used for the calibration of the cathode planes. As an example we will discuss calibration of the cathode in chamber 4 using anode projections from chambers 1 and 3:



-t»> Z (b«om axis)

A.. = anode position chamber 1

A« = anode position chamber 3 (C.4)

C, - cathode position chamber 4 .

The positions of the chambers are defined relative to some arbitrary point on the z axis. Now the cathode position in chamber 4 can be projected from the anodes in chambers 1 and 3:

175 C. - A1 A - A Jt L = -J 1. (C.5) Z4 ' Zl Z3 " Zl


z. - z.. ^ ^ (C6)


4 3 - Ax) + An = Z • A3 + (1-Z)A1. (C.7)

We can now parameterize the cathode position as we did above for the


X = c + c, • CD + c • CD2 + c, • CD3 + c. • CD • TC (C.8) Col I0 i 4

where CD and TC are defined in equations (C.I). For a collection of

events we minimize the difference

2 2 X = Z (C4 - Xc) (C.9)


by varying c -c,. These are then the cathode calibration coefficients.

Chamber 2 cathode is also calibrated with chambers 1 and 3 anodes; and

the anodes in chambers 2 and 4 used to calibrate cathodes in chambers

1 and 3.

II. Drift Corrections for Rear Chambers

Because specification of the particle momentum in the spectrometer

depends on x as well as x^,, position resolution in the rear chambers

is just as important as in the front. However, if the rear chambers

176 were operated in the same manner as the front, the x (anode) re-

solution would be far worse due to signal dispersion along the ex-

tremely long delay line (the anode length of the chamber is 90 cm).

In order to obtain the necessary position resolution in the rear

chambers they are constructed and operated as drift chambers.

As is the case for the front chambers each of the four rear

chambers contains an anode and a cathode, and the chambers are

arranged in two pairs. However, ail four rear chambers are aligned

with anode plane along the spectrometer x plane, and cathode along y.

The anode plane is the drift plane, and it is at positive high voltage

with wires spaced 8 mm. The cathode plane is at zero voltage with wires spaced 1 mm apart. To improve the ion drift to the anode wires

each anode contains a grid of wires at a potential of -200 V spaced between the anode wires. Assuming that one anode wire fires per event the position of the anode event is determined by the difference of times read out from each end of the delay line. The resulting

time is integerized to give the wire of closest approach.

The drift properties of the chambers are used to determine how far the event passed from the wire, and to which side of the wire.

The distance from the wire is extracted from the ion drift time dis- tribution as described below. To decide which side of the wire the event passed requires a special arrangement. This is the reason for having a second anode plane in the chamber package - the two planes are offset by one-half the anode wire spacing so that the anode wires in each chamber lie midway between those in its mate.

Extraction of the distance from the wire the event passed is

177 easily done. When we take the sum of the times read out from each

end of the anode delay line we will get a time equal to the total

length of the delay line plus the drift time of the ions to the

nearest wire - the transit time of the signal from the position of

the event along the anode wire to its end is negligible. A typical

resulting drift time distribution is shown in Fig. C-l(a), offset

by the constant total time length of the delay line. This distri-

bution has been produced by illuminating the chamber in the EPICS

beam as uniformly as possible. Thus it is reasonable to assume

that the number of particles per unit width of the gaps between

anode wires is constant. It is obvious from the long tail on the

right, hand side of the distribution in Fig. C-l(a) that the drift

time is becoming non-linear for ions far from the wire. But be-

cause of the uniform illumination this distribution can be divided

into a number of bins containing equal counts, so that each bin re-

presents a particular position (i.e. fixed drift time) relative to

the wire. This result is shown in Fig. C-l(b), and has the proper

linearity of counts vs. position. An incoming event has a drift

time corresponding to a position in Fig. C-l(b) and that determines

its relative drift position. Now remains the determination of which

side of the wire the event passed.

To do this we need information f-om the other chamber in the

pair. Define x ,x~,x_,x, as the wire positions for an event in

rear chambers 1,2,3,4. The-i let d-,d_,d_,d, be the drift positions

(in absolute value) of events in each chamber. Then for each pair

of chambers we can specify four classes of events:

178 (a)



DRIFT TIME (b) c/)

O o


Figure C-l. (a) Typical drift chamber ion drift time distribution, (b) Plot of distribution in (a) re-binned so that a roughly equal number of events is in each bin. Each channel in this distribution corresponds to a (drift) position between chamber anode wires.

179 CHAMBER 2 d2



We call events of types 0 and @ "normal" events, and those of types

@ and @ "extraordinary". Next we must construct all possible positions

of the event in all chambers: x + d , x.. - d , x2 + d?, x? - d0, etc.

Upon examining these we see, for example, that an event of type © is

described by x - d and x2 + d?. An event of type © is described by

x1 + d1 and x_ - d,,. By taking the difference [ (x, - d^) - (x^ + d_)]

we get a distribution like that shown in Fig. C-2(a). Taking the dif-

ference [(x + d ) - (x^ - d2)] we get that shown in Fig. C-2(b).

Similarly, to include events of types © and ©, by forming [(x.. - d ) -

x_ - d2)] or [(x + d1) - (x2 + d2>] we get a distribution like that

of Fig. C-2(c).

For an event in thf: chamber all four of these quantities are

checked. The one which is closest to zero defines the event type,

180 (a)




Figure C-2. The distributions (a) and (b) result from "normal" events as described in the text. The distribution (c) results from "extraordinary" events- The peaks in the center of each distribution are good events for that particular event type.

181 and thus the side of the wire the event passed. Furthermore, if a

histogram of the minimum values for all events is constructed, the

FWHM of the resulting distribution will be the chamber resolution.


In this appendix we discuss details of angular binning, peak

fitting, yield calculations and cross section calculations. Be-

cause a knowledge of the hydrogen cross sections with which we have

calculated the absolute 180 cross sections is essential, details of

the phase-shift analvsis of the up data as we.11 as tables of inter-

polated cross section values at 164 and 230 MeV are also presented.

I. Angular Binning

The relative spectrometer solid angle, dfi, was tabulated by measuring the normalized yield (see section III) of the elastic peak of a heavy target as a function of 6 across the focal plane. An

iron target was used in this experiment, and the range of 6 values was -5% to +5%. For this scan the spectrometer angle was set at a flat portion of the angular distribution of elastic scattering from this target, i.e. at the second maximum. Yields were calculated for each angular bin at each <5 point, and the relative solid angles tabulated.

During the experiment the 180 elastic peak was placed near

6 = +4%, and the inelastic peaks were spread over lower 6 values. P ' p

The inelastic peaks remained well within limits of -5% to +3% over the entire angular range with no changes in the spectrometer field.

The large range of <5 enabled us to extract yields for the hydrogen peak to about 9 = 40°, without resorting to a separate run with hydrogen kinematics.

183 There are certain considerations involved in the binning pro-

cedure that are worth elaborating. Because the solid angles in the

two 1° bins comprising the 2° bin add to the solid angle in the 2°

bin, the peak areas in the two 1° bins should also add to those of

the 2° bin. The 1° bins were fit with AUTOFIT independently of the

2° bins, except that the sum of the backgrounds in the 1° bins was

set equal to that of the 2° bin. In approximately 50% of the cases,

the sum of the 1° peak areas was within 1-2% of that in the 2° bin.

In these cases the 1° areas were used in the yield calculations.

For the remaining cases the difference was larger that 2% but less

than 4%. In these cases the sum of 1° areas was normalized to the

2Q area.

In calculating yields we would in principle use our measured

spectrometer acceptances for 7T separately from IT and for the 1°

bins separately from the 2° bins. But because the spectrometer is

symmetric there is no reason to expect that the TT and ir acceptances

should be different. Our measured relative acceptances for n and n ,

though not absolutely identical, did not differ in a statistically

significant manner. For this reason an average smooth shape was used

for both ir and IT . The results of the smoothing are shown in Figs.

D-l and D-2.

The second set of points for TT at 230 MeV is the result of an

acceptance scan with the channel jaws partially closed. Closing the

jaws is necessary because the flux of elastically scattered particles

in ir runs at angles less than 25° is high enough to affect the

operation of the front MWPC. At both 164 and 230 MeV the channel jaws

were closed enough to reduce the pion flux by approximately a factor 184 + 5 +4 +3 +2 +1 0 -I -2 -3 -4 -5 8D(%)

Figure D-l. The points are the measured relative spectrometer acceptances (solid angles) on the focal plane for 164 MeV TT-, and the curves are the chosen smooth shapes. The relative scales of the full, 2° bin, and -1° bin acceptances are preserved; the +1° bin acceptance has been reduced by half. The TT~ acceptances have been multiplied by a factor 5 to account for the relative i&/vr beam flux.

185 10 9 T77--230MeV S x 7T+ 7 - FULL O 7T~ 6 5 2* BIN

5 2 -I'BIN -~ o o V)

JAWS CLOSE!? • I 9 w FULL - O 8 7

•o 6 5 2° BIN




+ 5 +4 +3 +2 +1 0 -I -2 -3 -4 -5 Sp (%)

Figure D-2. The measured relative acceptances for 230 MeV ir , with the chosen smooth shapes. The lower four curves are acceptances measured with the channel jaws in the partially closed positions. The relative scales of the full, 2° bin, and +1° bin acceptances are preserved; the +1° bin acceptance has been reduced by one-third. The TH7IT~ normaliza- tion is abouc 7,

186 10 for e<25°. At 230 MeV a separate set of relative solid angles

was measured with the jaws in the partially closed position. The

ratio dQ(open)/d$Kclosed) was found to be 8.68(34). This was not

done at 164 MeV; so instead of normalizing the cross sections for

the jaws partially closed position via separate acceptance data,

the jaws partially closed factor was accounted for in the hydrogen

normalization. It is encouraging to n-"te that this procedure gave

the same results as using the 230 MeV ratio for dO(open)/dft (closed)

at 164 MeV.

Because we have used a single acceptance shape for both it and

•n in all the bins at a particular energy, assuming that the shapes

should be the same due to symmetry, we have masked any real unknown

differences in the relative acceptances. For example, referring to

the 164 MeV curves in Fig. D-l, the worst deviation from the curve

in the 2° bin is the IT point at S = +1%, and this difference is

7%. The rest of the points all are within 5% of the curve. For the

1° bins the scatter of points is more noticeable. The IT points in

the range of 6 from -1 to -3% average 9% from the -1° bin curve.

There are similar problems with the ir points in the 6 range +1 to

-2% in the +1° bin. Over this region the average deviation from the

curve is 6%. This happens to be the region into which most of the

inelastic data falls. If this effect is more than statistical fluct-

uation, then the use of a single average curve would affect the

smoothness of the data, and a careful examination of the angular dis-

tributions may expose systematic deviations from the average accept- ance curves.

187 Similar comparisons can be made for the 230 MeV curves. Of

particular significance is the region of 6 from +3 to +5% for the

1° bins in the jaws open runs. The ir deviations are about 10% in

each bin, and the differences are in opposite diiactions! Also, in

the iS region from 0 to -3%, a particularly important region for the

3 state, the TT acceptances are more scattered than the IT . In

correlation with these observations we have noticed in our angilar

distributions that generally the IT curves are less smooth than the

TT .

II. Peak Fitting

Some of the options available to the user of AUTOFIT have been

mentioned in Chapter III. Now we wish to elaborate the reasons why

AUTOFIT was used for the final data reduction, and also some of the

difficulties encountered in fitting with AUTOFIT.

The major considerations in choosing a fitting routine for our

fits were flexibility in setting the background in ea<.h spectrum and

the ability of the routine to accurately fit peaks in cases where the

peaks are poorly resolved.

The first consideration is satisfied by AUTOFIT because the

exact level of background desired may be input by the user. We

found generally that a constant level of background was satisfactory,

thus we used a constant background in all our fits. In choosing the

proper background level in each spectrum we were constrained by the

assumption that the background per unit pion flux at the target

should be a smoothly varying function of angle. The pion flux was

188 normalized by the same factor that multiplies the area in calculation

of the yield (see following section). We held to this convention

guiding our choice of background levels throughout the fitting pro-


The capability of che AUTOFIT program to fit several overlapping peaks is one of its strongest points. The most difficult cases were provided by the forward angles at wnich the 2 peak at 1.98 MeV is almost obscured by the tail of the much stronger (as much as a factor

100) elastic peak. An example of such a case is shown in Fig. D-3 on a highly expanded scale. In these cases the difficulty of fitting is reflected in the size of our quoted errors - they are far larger than those due to statistics alone.

Cases in which the peak to background ratio is low are also dif- ficult to fit. In particular this was a problem for the small angle

3, areas where the background area could be as large as twice that of the peak. The problem is compounded somewhat by the presence on the focal plane of the hydrogen peak (see Fig. B-3), which is com- parable in size to the elastic peak in the rr runs. In Fig, D-3 the hydrogen peak has just separated from the 3.. peak, and this is in fact the first angle at which extraction of the 3^ area is pos- sible. It was felt that more reliable areas could be obtained by fitting the small angle data by hand, and this was done for 164 MeV ir and 7r over the angular range from 25-37.5°. To do this we chose a region or" +C.7 MeV about the peak in which to sum the counts and su._ract the background. The hand fits resulted in smoother angular distributions than the AUTOFIT e -eas provided in this region.

189 o


O(ir>+) 0

>I5O (D o in co 100

o o 50

/• \ 0 4 6 8 10 E*(MeV)

+ Figure D-3. An example of a relatively hard to fit TT spectrum. The 2l peak at 1.98 MeV sits high on the tail of the elastic peak, and the hydrogen peak has just separated from the 3^ at 5.09 MeV. For TT~ the hydrogen peak is a factor 9 smaller and thus interferes less at small angles. A final advantage in using AUTOFIT to fit difficult-to-resolve

peaks is the ability to use a standard peak shape. From inspection

of the spectra it is obvious that the peal-, shape is not symmetric.

By using the standard shape we were able to simulate the exact peak shape over a region up to 4 MeV wide. We were careful to use a dif- ferent standard shape whenever warranted. Because two thicknesses of target were used during the experiment separate shapes were used to fit the spectra from each. Different shapes were used for ir and ir spectra, and different shapes for 164 and 230 MeV data. Charact- eristics of the standard shapes used are tabulated in Table D-l.

For a given target, pion charge and energy the standard peak shape was not varied as a function of angle.

III. Yield Calculation

In Chapter III the equation for calculating normalized yields was given

Y = A/(CF-SF-dfl-MT-N) (D.I) where A is the peak area, CF is a hardware and software correction factor, SF is the pion survival fraction in the spectrometer, dft is the relative spectrometer solid angle, MT is the scattered beam monitor used (and has been corrected by a target rotation factor, 2 cose ), and N is the number of target nucleons per cm for the unrotated target.

The correction factor is the product of a software and hardware factor: CF = 1/(LT-CEFF). The factor LT is the computer live time,

191 TABLE D-l. Resolution widthsa observed in the present experiments.

Trr(MeV) Target FWHM(KeV)iT+ FWHM(KeV)if

164 thin 425 400 thick 550 525 230 thin 475 450

Widths of best-fit standard peak shapes used in AUTOFIT.

192 and is calculated by dividing EVENTS-BUSY by total EVENTS (where BUSY

indicates computer not busy). This quotient becomes less than 1.0 if

the rate of incoming events is high enough that the computer annot process them all. The CEFF factor is the overall chamber efficiency, equal to the product of front and rear chamber efficiencies, which are calculated separately. The details of chamber efficiency calcu- lation may be found in Appendix B.

The pion survival fraction, SF, represents the fraction of pions left in the spectrometer after decay during the flight from the tar- get to the 'focal plane'. It is calculated by SF * exp(-LfBycx), 2 -i where 6=v/c=p/E,Y= (1~S ) = E /m , CT = pion decay length = TV Tf TT IT TT

780.4 cm, and L = pion flight length. L is corrected to second order for paths of different, length through the spectrometer: L = L. - (3.35 x 10~2)-(5 + (3.5 x 10"5)-62 where L » 12.31 m, the U p p 0n length of the spectrometer central trajectory, and 6 = (p - p pcMTdC.L& as Pepec t«.)/ *specP t,. always, The factor dfl is the relative solid angle of the spectrometer for a given acceptance (bin size) and position on the focal plane.

The measurement of the relative solid angle has been described in section I.

Finally, the target angle is always set to be half the spectro- meter angle, where 6t = 0° is perpendicular to the beam. Assuming that the average position of the pion interaction, is at the center of the target, the target rotation ensures that the energy lost by the incident pion in reaching the center of the target is roughly the same as that of the outgoing pion exiting from the center of the

193 target. The energy of the pion beam is increased by one-half the

unrotated target energy loss so that the pion energy at the center

of the target is that desired. The cos9 factor corrects the yields

for the increased thickness of the rotated target.

IV. Cross Section Calculation

The equation relating 180 and H yields was derived in Chapter

III. We will start from equation (3.4) from that chapter:

18 §( 0)cM - g(18O).Y(18o)lab.[i|(H)CM /g(H).Y(H)lab]. (D.2)

The last factor on the right hand side of the equation is the hydro-

gen normalization factor. The hydrogen yields used in this factor

were determined in two ways. The first way was by setting the spectro-

meter for hydrogen kinematics, i.e. by bringing the hydrogen peak to

the same place on the focal plane as the 180 elastic peak, and ex-

tracting the hydrogen area. This was done at angles of 20,40,60,80°.

The other method was to fit the hydrogen peak in the spectra in which

it fell on the -5 to +5% of the focal plane for which reliable mea-

sures of relative dfi exist. In this way a hydrogen yield was ex-

tracted for all spectra with 9540°.

The cross sections we have used for TT p and TT p have been cal-

culated by D. Dodder102 with an energy dependent phase-shift analysis

of the experimental data, using the R-matrix formalism of Wigner and

Eisenbud. The channels considered are TT p,7T p,TT°n, and various ab-

sorption channels. The data used in the code come from a review by

Giacomelli 101*, plus the data of Bussey and Bertin. The energy

194 range spanned by the data is approximately 20-320 MeV.

In Table D-2 the n p and ir p data used in the code are summarized, and some indication is given of the errors involved in the fit. The

X per degree cf freedom for this particular run is fairly poor. How- ever, this was run with no normalization of the data. If the code is allowed to run freely choosing its own normalizations, the renormali- zation factors in the last column result for the TT p data at energies near 164 and 230 MeV. The worst case is 236.3 MeV for which all the data points are moved up 2%. A run resulting in the renormalizations 2 shown in Table D-2 gives a x Per degree of freedom of about 1.1 for + 2 ir p. The x Per degree of freedom for the IT p data is even worse + 2 than that for the up. A large part of the x is contributed by the data below 100 MeV. A more recent run, freeing the normalizations . 2 and including more data in the ir p channel, results in a x /dof of

2.2 for the TT p data. The t^drogen cross sections resulting from the Dodder code without any renormalization, which were used for calculation of our 180 cross sections, are shown in Table D-3.

Assuming the hydrogen cross sections used are reliable, the hydrogen normalization factor in equation (D.2) {should be constant as a function of angle for a given target and pion charge at each energy. Thus for a given energy and target thickness one average value of the normalization factor has been calculated from the hy- drogen runs, and used at all angles for that particular target for

180 cross section calculation. A summary of the factors used under various conditions is given in Table D-4. In this table the differences in target thickness, measured

195 TABLE D-2. (a) irTp data fit by Dodder.

Number of Number of Maximum Data , 8 2 Channel Type T^fcfeV) Data Points T^MeV) Data Points % Deviation x /Pt Renorm

•n p total 71.6-282.8 14 differential 20.8 10 78 3 21.5 6 81.7 10 24.8 3 94.5 3 30 4 95.9 10 30.5 10 98 14 31.4 3 100 11 37 6 114.1 5 6,26 7.90 0.953 39.5 10 124.8 3 0.13 - 1.003 40 5 142.9 14 3.19 0.85 1.013 45 4 166 14 3.85 0.93 1.010 51.5 10 194.3 15 4.57 0.65 1.303 53 3 214.6 14 4.47 2.31 0.997 58 6 236.3 14 9.11 3.18 1.020 67.4 10 263.7 13 9.13 5.99 1.019 75 2 291.4 11 polarization 246 7 247.5 7 310 4

Largest deviation of data from 2\t value at that energy, shown for energies in the vicinity of 164 and 230 MeVo

Renormalization factor for data at that energy. TABLE D-2. (b) ir"p data fit by Dodder.

Number of Number of Maximum Data Channel Type T^MeV) Data Points « T^MeV) Data Points % Deviation 2 x /Pt Renorm

Tf~p total 16.7-290.1 14 differential 30 4 144.1 8 9.60 6,02 _ 31.4 5 161.9 8 5.21 1.53 _ 35 10 191.9 10 6.69 1.76 _ 39 10 219.6 10 5.23 2.66 - 41.5 5 237.7 10 7.13 4.26 - 65 5 263.7 10 11.21 12.00 - 88.5 3 291.5 10 — - — 98 13 119.5 5 polarization 229 12 300 6 310 4 318 30 TT~p total 20-291.7 24 ->*rron differential 61 3 65 3 95 2 96.6 3 98 3 150 8

-.1 + + TABLE D-3. Predicted H(ir~,iT )H cross sections according to analysis of Dodder.

164 MeV 230 MeV

J. . 9 da(?r+) da(TT~) 9 da(ir+) da(7r") lab cm cm cm cm cm cm

(mb/sr) (mb/sr)

17.5 22.5 25.49 3.89 23.4 25.58 2.34 20 25.7 24.80 3.64 26.7 24.34 2.32 22.5 28.9 23.98 3.42 30.0 23.13 2.27 25 32.0 23.08 3.23 33.2 21.94 2.20 27.5 35.1 22.12 3.04 36.5 20.71 2.12 30 38.2 21.09 2.87 39.7 19.49 2.03 32.5 41.3 20.03 2.69 42.9 18.26 1.94 35 44.4 18.95 2.52 46.1 17.02 1.84 37.5 47.5 17.85 2.36 49.2 15.83 1.73 40 50.5 16.79 2.20 52.3 14.66 1.63 42.5 53.5 15.76 2.04 45 56.5 14.75 1.89 60 73.7 10.13 1.18 76.0 7.15 0.90 80 95.2 8.83 0.82 97.5 4.56 0.63

198 TABLE D-4. Hydrogen normalization factors, N, for all cases.

T (MeV) Target N(TT+) N(TT) N(TT+)/N(TT~)

164 thin, jaws closed 3.51(18) thin 3.50(13) 3.79(26) 1.08(8) b thick 3.10(10) 3.93(11)

230 thinC 3.47(16) 3.70(27) 1.07(9) d average 3.49(9) 3.81(13) 1.09(5)

Corrected for the measured change in channel acceptance: (jaws open)/(jaws closed) = 8.68(34).

Corrected for the measured ratio of target thicknesses: thick/thin = 1.77(4). See text of Appendix D for a discussion of how the anomalous 7T*" value was treated.

Corrected for the measured change in pion flux in the channel at 230 MeV compared to that at 164 MeV, corresponding to a given primary beam. The change was measured as (MT/BOT)23o/(MT/BOT)igA and had the values 1.95(3) for TT+ and 1.89(2) for iT.

For IT the average of the three good values has been taken.

199 spectrometer acceptance, and the pion flux differences between 164

and 230 MeV have been removed. This yields factors which are quite

close for all cases except that of the 164 MeV ir for the thick tar-

get. Because this is the only anomaly in the table we have assumed

it is incorrect. The average of the other three ir values in the

table has been taken and cross sections calculated with the incorrect

factor (164 MeV ir from 50-80") have been corrected by the ratio


This correction raises the affected cross sections by about 12.5%. 2 - + The ratios p of IT h summed cross sections do not change appreciably

due to this correction. In particular the elastic, 2 and 3. ratios

drop by about 0.5%, 1% and 3% respectively.

V. Total Errors

Only statistical errors due to the peak fitting have been re-

ported for the differential cross sections. Errors in the hydrogen

normalization factors are systematic and are not included in cross

section errors; however, it is proper to include these errors in

calculating errors in ratios of summed cross sections. The results

of such an error analysis are given in Table D-5. The errors in

hydrogen normalization have been estimated from uncertainties in the

Dodder fits to the irp data and uncertainties in the calculated nor-

malization factors (see Table D-4).

In addition, in the calculation of the ratio g /6 discussed in n p

Chapter V, we have included a ±10% uncertainty in the normalization

of the DWBA curves to the data points.

200 TABLE D-5. Total percentage errors calculated for the ratios p of summed cross sections.

Errors Errors in Errors Errors in Errors Tir(MeV) State in Sum Normalization Total in Sum Normalization Total in p2

164 elastic 0,9 4.0 4.1 1.0 4.5 4.6 6.2 2x+ 4.9 4.0 6.3 3.7 4.5 5.7 8.5 multiplet 8.8 4.0 9.7 8.1 4.5 9.3 13.4 3i~ 2.7 4.0 4.8 3.0 4.5 5.4 7.2

230 elastic 0.8 4.0 4.1 0.7 6.0 6.0 7.3 + 2l 2.9 4.0 4.9 2.3 6.0 6.4 8.1 muxtiplet 6.8 4.0 7.9 4.6 6.0 7.6 11.0 3i~ 3.0 4.0 5.0 3.0 6.0 6.7 8.4


The derivation of an optical model potential begins with the

formalism of multiple scattering theory. This formalism can be applied

to the scattering of any projectile from the nucleus, but we will treat

pions in particular in this discussion. The general idea is to express

the unknown pion-nucleus scattering amplitudes in terms of the

(supposedly) known pion-nucleon amplitudes. In practice even this much

is not usually done, but the pion-nucleus amplitudes are related in

some way to the elementary pion-nucleon amplitudes.

Once the multiple scattering expansion for ir-nucleus scattering

has been derived, one of two methods of solution can be chosen. The

multiple scattering equations can be solved directly, to all orders.

Due to the computation time involved this method is presently limited

to rather light nuclei. Alternatively, an optical potential can be

deduced from the multiple scattering series, such that the solution to

the appropriate wave equation containing this potential is a reasonable

approximation to the exact multiple scattering series solution. Details

of the derivation of the multiple scattering series and optical poten-

tials from it may be found in, for example, Refs. 107-109. Here we

will only skstch the derivation of an optical potential and note some

of the methods used to simplify the solution of the wave equation.

I. Multiple Scattering Theory

The material in this section is largely drawn from Ref. 114.

Multiple scattering theory (MST) is formulated in non-relativistic terms

and may be generalized into relativistic terms later. We wish to solve

202 the Schrodinger equation

where H is the nuclear Hamiltoniari, k is the pion kinetic energy, V, n 7T l

is the interaction potential of the pion with the i nucleon, and

is the A+l (nucleus + pion) body wave function. The ir-nucleus t-matrix is defined as

^ V ^ dr = ^T^'.q) (E.2)

where q',q are the cm momenta of the incident and scattered waves,

respectively, and T. is the i ir-nucleon t-matrix. The formal Lipp- mann-Schwinger equation for T in operator notation is

Ti = Vi + ViGo jtl Tj (E'3> where G = 1/(E-k -H ). This t-matrix represents the ir-nucleon scattering amplitude from the i nucleon in the nucleus. To eliminate the iT-nucleon potentials V. from Eq. (E.3) we define T. matrices

Ti = Vi + ViGoTi

Vi TiGo)Vi "*• Vi " (1 + TiGorl V

Putting this back into Eq. (E.3) we get the following set of equations

Ti " Ti + TiGo jSl Tj • ^'5>

By iterating these equations we obtain the Watson multiple scattering

203 series

A A T = .Z. T. = ,Z, T. + .Z. T.G T. + .1. T.G x.G T, + ... . (E.6) 1=1 i i=l 1 1*2 i o 2 i*J i o j o k

Note that Eq. (E.5) is a representation for the ir-nucleon t-matrix, but

unfortunately the propagator 'i contains the nuclear Hamiltonian H .

This means that our hope of simply using ir-nucleon amplitudes to solve

the TT-nucleus scattering problem cannot be realized exactly. But we

may interpret the T. as the t-matrices for u-nucleon interactions in

the nuclear medium.

Equation (E.6) has a useful physical interpretation: the first

term is the single scattering term, the second represents rescattering

from another nucleon (which cannot be the first one again), the third

represents three scatterings where the third can be from the same

nucleon as the first, etc. The antisymmetrization of the nuclear wave

functions, which we have conveniently neglected, is discussed in detail

in Ref. 108.

Various approximations are used in determining the T.. One such

approximation is to drop the H in the propagator G . This results in

T.(E) = T (E), called the impulse approximation. Another method, in-

serting a complete set of states i.nto Eq. (E.4)

V±|n> Ti in E-k -E

is called closure and leads to x.(E) - T.(E-1~) where 1" is an "effective"

interaction energy. A prescription for determining E must be chosen.

A common approximation in MST is that of the "fixed scatterer",

or fixed nucleon. This assumes that during the course of the pion's

204 interaction with the nucleus the positions of all nucleons are fixed

relative to the reference coordinate system. Recently the const .uences

or using this approximation have been studied 11 °»lia and corrections

for it proposed. The approximation of fixed nucleons neglects the

motion of the slow-moving nucleons relative to the medium energy (thus

fast-moving) pions. Further consequences are the neglect of nucleon

recoil after being struck by the pion (reasonable because m /m=l/7),

and some effects of nuclear correlations, e.g. binding. Such corre-

lations are small only in the extreme low-density limit, where it is

assumed that the pion feels no nuclear potential between scatterings.

These approximations are also important for the optical model because

the fixed nucleon approximation is generally employed in the derivation

of an optical potential.

.".I. Optical Model Potential

Because the customary method of applying MST to the ir-nucleus scattering problem is through the derivation of an optical potential, we will sketch its derivation from the multiple scattering series

Eq. (E.5). The elastic scattering is projected out by taking the ground state expectation value of Eq. (E.5)

<01 T-L |n> <01T. 10> = <0|x.|0> + Z ; n-K). (E.8) x l • n a—K — Hi "fin 7T n

Now we make the coherence approximation which requires the nucleus to remain in the grcund state. The nucleus may be excited to states |n> during each scattering, but between scatterings it must return to the ground state. Then only the J0><0| term in the sum remains and E =0, n

205 so that

<0|T |0> = <0|x |0> + <0| T_j | 0> g ^S <0|T.|0> (E.9)

where g=l/(E-k ). A consequence of this approximation is that all

elastic flux lost during intermediate excitations never returns to the

elastic channel, so that the resulting potential is likely to be too

absorptive. Summing over the A nucleons

T° = AT° + AT°g(^)T° (E A

where TC=<0|T|0>. All <0|T |0> are identical because of the anti- A . , o

symmetry of the states, which implies .E-<0|T.|0>= AT . It is con-

ventional to redefine T=(——)T , in which case


T = (A-1)T° + (A-i.)T°gT. (E.ll)

By comparing with Eq. (E.4) we obtain the first order optical potential

For nuclear states |0> of the independent particle model type

= / •*£,?•> e"1^-' xfr.q^,?') «

where the p,p are nucleon momentum operators. The full momentum

T dependence of the x(q ,q;p.,pf) arises because the T-matrices represent

the TT-nucleon scattering amplitudes in the nuclear medium. If the

"Fermi" momenta p.,pf are neglected the result is the factorization


VQ = (A-l) T(q\q) S(q'-q) (E.14)

206 where

is the nuclear form factor. Neglecting the motion of the nucleons via this approximation gives rise to the infamous angle transformation.

The solution of the ir-nucleus scattering problem is now effected by choosing a suitable form of the potential (E.12) or (E.14). In the work of Liu and Shakin112 or Landau and Thomas,11 3 for instance, T- matrices of the form of Eq. (E.13), remaining in momentum space, are used. A previously more common method is to transform to a coordinate space representation and obtain the so called Kisslinger potential. 116

Kisslinger proposed a potential containing the strong p-wave nature of the ir-nucleon interaction

V = (A-l)[d0 + di q'-ql S(q'-q) (E.16) o which leads to a coordinate space potential

2 VR = (A-l)[d0 q p(r") - di $-p( where the Fourier transform of |$(r,p)|2 is identified as the density p(r). The complex constants do,di are the s-wave and p-wave amplitudes, respectively, and are derived from averaging elementary iT-nucleon amplitudes over the Z protons and N neutrons in the nucleus. The p- wave amplitude, as a result of the (3,3) resonance, has a marked energy dependence. Reference 117 gives a complete derivation of the most common optical model potentials, and of the origin of the do,di. Due to the V operator in Eq. (E.17) the Kisslinger potential is 'non-local', and it is sensitive to the nuclear surface through the surface-peaked

207 Vp(r) term. An equivalent potential on the energy shell, i.e. when

N'l = |q!> can be deduced by making the substitution q'-q =-^(q-q')2+q2

2 1 2 VQ = (A-l)[d0 + dj q + igdi (q-q ) ]- (E.18)

In coordinate space this becomes the local Laplacian potential

2 2 dJ q p(r) + 3g(A-l) dx (V p(r)) . (E. 19)

Because of the absence of velocity-dependent terms it is termed local.

The Laplacian operator v2p(r) also leads to surface sensitivity, although

at a different portion of the nuclear surface from the Kisslinger poten-

tial. As noted, on-shell where jq'| = [q| the two potentials behave in

the same manner; but off-shell their behavior is quite different.

Nevertheless they give similar results for medium energy ir-nucleus

scattering and both fit the data fairly well.

III. Corrections to the Optical Potential

The optical potential we have used in our calculations is the

Kisslinger form, Eq. (E.17). Because it is only a first order potential

derived from the multiple scattering series, and has been derived in

the factorization approximation, certain corrections may be desired.

Some of the common possibilities are mentioned below.

A. Bound Nucleon Correction. In transforming from the free 77-

nucleon amplitudes to the 7r-nucleus system there is a kinematic correc-

tion to account for the lack of recoil of the heavy nucleus. A

schematic method of correcting for this, due to Ericson and Ericson,118

is to multiply the free ir-nucleon amplitudes by [(M + m -f 2M E )/M2]^, n 77 n 77 n where E is the total pion energy.

208 B. Nuclear Fermi Motion. The Fermi motion of the nucleons In

the nucleus necessitates several corrections. One such correction is

the angle transformation, which arises because in transforming from

the ir-nucleon to the 7r-nucleus cm system the effective angle of the pion

incident of the nucleon is affected by the direction of motion of the

struck nucleon. We recall that the nucleon momenta were neglected in

making the factorization approximation, so in general it is desirable

to make the angle transform correction. However, the angle transform

often is neglected because its effect is estimated to be on the order

of 10% for near resonance pions. The aagle transform can be approx-

imated by adding a kinematic term to the s-wave amplitude d0 (in

essence mixing some of the p-wave strength into the s-wave) as is

suggested by the Ericsons in Ref. 118, or in more exact (and more complicated) ways.

It is pointed out by authors solving optical models in momentum space representations112'11"3 that coordinate space representations do not properly include nuclear Fermi motion. Thus corrections for the neglect of Fermi motion are only approximate- In a momentum space representation (see, for example, Eq. (E.13)) the nucleon momentum operators are explicitly included, and the result is exact.

The Fermi motion also affects the interaction strengths in another way: the effective energy of the ir-nucleon scattering event differs for each separate nucleon in the Fermi distribution. This is sometimes taken into account by "Fermi averaging" the do,dj amplitudes over a typical nuclear Fermi distribution. Again it is the do which is pri- marily affected, the di being relatively insensitive to this effect.

209 C. The Lorentz-Lorenz Effect. The Lorentz-Lorenz (L-L) effect

is a pionic analog to the optical effect of the density of a medium on

its index of refraction. For a pion in the nuclear medium the L-I,

effect corrects for the change of potential as the nuclear density

changes. The form and approximate magnitude of this effect were first

derived by the Ericsons.118 Subsequently it was shown119 that the L-L

effect is equivalent to expansion of the optical potential to all orders

to include short-range nucleon correlations. The first order Kisslinger

potential with the L-L term is

VL-L) - (A-l) j d0 p(?)+ V . [ ^3^^ V ] j • (E.20)

The Ericsons took the parameter £ to be 1.0.

D. Higher Order Terms. A type of correlation not included in the

L-L correction is that due to "true pion absorption". The name refers

to the kinematical condition that a pion cannot be absorbed by or pro-

duced from a single nucleon - there must be at least two present.

Although the nucleus as a whole can take part by acting as the second

massive body in true absorption, it is assumed that a more likely

occurance is absorption by just two correlated micleons. This process

is parameterized by adding a cp2 term to the coordinate space optical

potentials, which represents scattering from one and then the other of

the correlated nucleons. There is no ready prescription for calculating

the constant c from first principles, so usually it is left as an ad-

justable parameter. Without the p2 term in the optical potential the

effects of true absorption would show up in the strengths of the d's,

if they were allowed to vary in fitting the data, and the imaginary

210 parts would be most sensitive to this effect.

The purpose of presenting the preceeding sketch of the optical

model formalism was to outline the approaches taken in calculating

7r-nucleus scattering, and to define commonly used terms. In the

following section we describe the optical model code DWPI which we

have used in our calculations, and discuss the inputs we have used.

IV. The Program DWPI

The DWPI program calculates angular distributions for elastic

and inelastic ir-nucleus scattering using the distorted wave impulse

approximation (DWIA). Since the program write-up discusses the

method of calculation in some detail we will not repeat the details

here. Briefly, the differential cross sections for inelastic

scattering are set up in terms of the square of the scattering amp-

litude, or T-matrix. The pion distorted waves are generated by numerically solving a Klein-Gordon equation containing the Coulomb

and nuclear potentials in the Hamiltonian. (The options available

for the ir-nucleus potentials will be discussed shortly).

The solutions are matched at a matching radius to Coulomb wave

functions derived from the Schrodinger equation in the absence of the nuclear potential. The phase shifts at the matching radius are cal-

culated and these are used to calculate the elastic scattering cross

sections. (This is the same procedure used in the program PIRK,121 written by the same authors, which calculates elastic scattering only).

The distorted pion waves generated from the Klein-Gordon equation

are then used in the calculation of the T-matrix elements for excitation

211 of inelastic transitions. At this point the calculation could proceed

either macroscopically or microscopically. In a macroscopic approach

the nucleus is deformed by expanding the density about a spherical

distribution. In a standard DWIA treatment the lowest order (spherical)

term leads to elastic scattering, and higher order terms couple to the

excited states. On the other hand, in a microscopic approach the T-

matrix elements are constructed with nuclear wave functions obtained

from a detailed structure calculation.

The present version of DWPI utilizes the macroscopic approach.

For an arbitrary deformed nuclear density p(r,c,a) the nuclear radius

may be expanded in terms of quanta ap characteristic of the defor-


) -RoU + ^a^Y^)]. (E.21)

This expansion may be generali2ed to the nuclear size parameter

c, assumed to have a 8, dependence

P) Co 11 + ^ aj if. V.9,<5>) J. CE.22)

The density, expanded to first order becomes122

p(r,c-co,a) = p(r,co,a) + (c ^! _„ uC C~C()

-p(r,c,,a) - [4W8,«]-c|lc=Co. (E.23)

The form factor is defined as F(r)= co * 8p/3c| . This leads to a c=c o

deformed potential, and the amplitude for transition to excited states

is then proportional to the matrix element of the deformed component

of the potential between the ground and excited states. The excited

212 state deformation is defined with the conventional deformation parameter

2 &0 = where I is the angular momentum transferred in the

excitation of the state. In using DWPI the input parameter &„ is nor-

mally set equal to unity and the inelastic cross sections later normal-

ized to the data via the relation da = 32 da . Coulomb excitation

of the inelastic states is calculated as an option in DWPI, and we have

always included it in our calculations, although the effect on the

cross sections is very small.

Several potential options are available in DWPI: the standard

Kisslinger form, Eq. (E.17), the local Laplacian, Eq. (E.19), and two

forms containing V2p(r) terms added to the Kisslinger potential - the

"phenomenological" and "modified" idsslinger forms. The angle trans-

form and L-L corrections are not done in DWPI, and none of the poten-

tials contain a p2 true absorption term. We have experimented with

Fermi averaging of the potential amplitudes, but this has not pro- duced significant changes in the calculated angular distributions.

For all of the calculations presented in Chapter V we have used

the standard Kisslinger potential. When used in the Klein-Gordon wave equation this potential is written

2 2E V = A[-bok p{r)+ bi v" • (p(r) $ ) ] (E.24) ft JN.

123 where the bo,bi are related to the do,di of Eq. (E.17) by

2 d0 - -bok /2E^ (E.25)

2 di = -bik£m/ (k 2Eu). (E.26)

Here k is the pion lab momentum, k is the pion momentum in the

213 7r-nucleus system, and E is the total pion lab energy. The poten-

tial can be decomposed into separate neutron and proton densities using

b0 p(r) - a0 ( Pn(r) + p (r) ) ± 1/2 c0 ( Pn(r) - pp(r) ) (E.27)

hi p(r) = ax ( Pn(r) + p (r) ) ± 1/2 cx ( Pn(r) - pp(r) ) (E.28)

where the coefficients a and c are related to the bo,bi by

-bo = i^~ [ao± (jir)Co] (E-29)

cm k cm f? ~bl = kTk~ Cai± (f?)ci]- (E*30) In the above four equations the minus (plus) sign holds for IT (IT ) .

This option for separate neutron and proton densities is not available

to us in our version of DWPI.

There are two nuclear density options available in DWPI. These

are the harmonic oscillator (or modified Gaussian)

2 (r/w) p(r) = p0 [l+a(r/w) ] e" (E.31)

and the two-parameter Fermi (or Woods-Saxon)

p(r) = p0 [1+ exp(•£=£) ] . (E.32)

In these w and c are the nuclear size parameters, a is the oscillator

constant, and t is the thickness parameter for the Fermi distribution.

In all our calculations we have used the * 80 charge density obtained

from the electron scattering experiments of Bertozzi et al76. These

model independent results are shown in Fig. E-l. We were unsuccessful

in fitting a Fermi form to this distribution, however a harmonic oscil-

lator form fits the data quite satisfactorily. The values of the param-

214 eters which give the fit shown in the figure are Po=O.895, a-1.40 and w=1.869 fm. These values give an rms radius (see Eq. (5.3))

-2.76 fm. When the finite size of the proton is subtracted from this radius we can calculate the parameters to use in our DWIA calcul- ations: a=1.40 and w=1.793 with an rms radius of 2.65 fin. For all calculations of inelastic transition strengths (SR) we have used the

'half-density' radius, R, which was determined visually from Fig. E-l to be 2.55 fm.

215 l80 Density (Bertozzi et.al.)

p = 0.895 1.0 o a = 1.40(1) Point: 0.8 w= 1.8690) w=l.793

0.6 0.09 -0.04 0.4 0.00


1.0 2.0 3.0 4.0 5.0 6.0 r(fm)

Figure E-l. Charge density distribution for 180 measured at Bates with curve obtained from our parameterization with a harmonic oscillator form of the density.


217 164


LA3 M CBOSS ;$3.03 CM C3DSS £3309 4MGLi »NGL£ SECTION SECTION acTio Ctd/S-*) MB/SB) 17.5 17.3 435.4090 4.4600 ->42.3000 6.S000 .'..015 .025 20.3 20.4 339.4000 3.0039 333.1000 5.1000 .964 .029 22.5 22.* 353.4616 3.0039 240.3000 5.9000 .950 .033 25.3 25.4 180.3672 4.3992 167.4000 6.7000 .929 .044 27.5 2B.0 125.7720 5.4144 112.6000 5.3000 .995 .057 30.9 30.5 dl.1032 1.5792 67.5000 1.3000 .932 .023 32.5 33.1 49.8576 1.3536 32.1000 .7000 .644 .022 35.3 35.6 27.2976 .7896 14.4000 .5000 .674 .027 37.5 38.1 12.1S24 .4512 b.3900 .4000 .525 .039 40.7 4.7499 .2256 2.0900 .1110 .439 .031 42.5 43.2 1.2634 .0451 .3260 .0290 .259 .025 45.3 45.7 .2098 .0169 .7470 .0460 3.560 .361 47.5 44.3 .4546 .0226 1.9600 .0500 4.312 .241 50.3 50.3 1.2295 .0677 j.2300 .0600 2.627 .153 52.5 53.3 2.1319 .0902 4.3400 .2200 2.036 .134 S5.3 55.9 2.7974 .0902 4.7400 .1500 1.594 .077 57.5 58.4 3.0692 .0677 4.4600 .1100 1.454 .049 SO.3 60.9 3.0230 .OSS'. 4.0500 .1100 1.343 .044 62.5 63.4 2.7195 .0903 3.3500 .3900 1.232 .053 bS.3 65.9 2.2673 .056 V 2.5000 .0700 1.103 .041 A7.S 68.5 1.7710 .0451 1.3300 .3400 1.033 .035 7U.3 71.0 1.1957 .03J9 1.1400 .3400 .953 .043 72.5 73.5 .7793 «u£03 .5970 .3260 .767 .039 75.3 76.0 .4512 .0130 .3380 .0310 .749 .075 77.5 7S.5 .2132 .0135 .1560 .3190 .732 .101 80.3 41.0 .1207 .0113 .3700 .3170 .590 .151

SO ^ •• 601.3695 5.9790 5 i.3439 5.3972

218 2DEG


C"t CM MOSS £**()•» CM C3DSS NGi_£ ftNGLi ScCTIOM s?cr:oN ITI3 17.5 17.a \ ^.7337 1.1167 6.4400 .9300 1.725 .573 20.3 20.4 3.7562 1.0942 7.0400 .9800 1.974 .605 22.5 22.9 4.2197 .9460 b.l»OO 1.3300 1.467 .390 25.0 25.4 3.3352 .5856 7.1100 1• V600 1.954 .475 27.5 28.0 it.2864 1.0829 6.5700 1.3400 1.533 .499 30.3 30.5 3.4968 .3610 5.5400 .3900 1.613 .200 32.5 33.1 3.3389 .3722 5.5300 .2800 1.556 .203 35.3 35.6 2.7523 .2820 :».2S00 .2200 1.549 .179 37.5 33.1 2.0755 .20 30 3.7300 .2500 1.797 .213 40.3 40.7 1.6807 .1241 2.7900 .1400 1.660 .149 42.5 43.2 1.31*9 .1354 1.9200 .1700 1.455 .197 45.3 45.7 .3866 .1139 1.5500 .1500 [.760 »HZ 47.5 4>).3 .5798 .0869 .3600 .1140 1.493 .297 50.0 50. -J .3643 .0445 • 4 790 .9990 1.315 .301 e>2. 5 53.3 .1918 .0316 .2560 .0640 1.335 .400 55.3 55.* .1054 .0215 .1400 .0350 1.405 .430 57.5 51.4 .0637 .0124 .1230 .0260 L.791 .499 f.0.3 60.9 .0464 .0107 .1580 .0270 3.624 1.020 62.5 63.4 .0714 .0206 .2010 .0270 2.915 .897 65.3 65.9 .1034 .0166 .2590 .3260 2.504 .474 67.5 68.5 .1410 .0147 .2680 .3170 ] .901 .232 70.3 71.0 .1691 .0135 .3070 .0210 .827 .193 72.5 73.5 .1455 .0102 .2720 .0160 1t.369 .171 75.3 76.0 .1196 .0090 .2380 .3190 11.990 .219 77.5 79.5 • 1128 .0102 .1940 . 0140 ]1.631 .192 *O. 3 41.0 .077S .0090 .1430 .0150 ] .902 .293

SJ<« = 13.8111 .9162 31.2460 1.1479

219 ?I PLJS [MJS

L43 CM Z» CROSS CM c=nss £9^03 INGL £ &NGL£ SECTION SECTION S4TI3 £3iO3 M9/S3) 1 T.3 17<3 • _ 20.3 20.4 - 22.5 22.9 - £5.3 25.* 1.3987 .5199 1.3900 .7200 .997 .63£ 27.5 29.0 1.0692 .5695 1.3400 .6900 1.254 .922 30.3 30.5 1.4551 .2369 1.3500 . ?2fiO .929 • 214 32.5 33.1 1.1844 .2369 .9960 .1400 .757 .192 35.3 35.6 .7061 . 1624 .56dO .1050 .904 .237 37.5 33.1 .6102 .12 + 1 .+980 .1210 .916 .259 <*0. 3 40.7 .4230 .0722 .5 + 00 .3840 1.277 .295 42.5 43. £ • 4365 .0369 .5360 .3710 1.228 • 2-J3 45.3 45.; . 2<»fcg .0744 • 3460 .0960 1.577 .620 <*7.5 48.3 .1658 .0575 .3640 .9950 2.195 .919 50.3 50.3 .1241 .0327 .2940 .3750 2.369 .869 52.5 53.3 .0797 .0240 .3200 .0630 4.013 1.479 55.3 55.3 .0773 .0178 .3000 .3410 3.983 1.041 57.5 54.4 .0835 .0125 .2570 .3300 3.079 .595 63.3 60.9 • 0840 .0199 .2+30 .0290 2.992 .503 f.2.5 63.4 .0746 .0 2*0 • 2230 .3260 2.936 .759 65.3 65.-* .0543 .01 34 .l?90 .3220 3.+33 .*52 67.5 61.5 .0846 .0122 .1050 .3150 1.241 .252 70.3 71.0 .0340 .0095 .3980 .3163 2.986 .939 72.5 73.5 .0380 .0070 .0745 .0120 1.960 .479 75.3 76.0 .0293 .3096 .0 469 .3129 1.599 .642 77.5 TH.5 - .3289 .3101 HO.3 H1.0 .0092 .0069 • 3?09 .3113 8.250 2.073

4.0213 6.1924 .'•997

220 O'.JS 1MU5

LA-j C:•< Z* CROSS :M C«SS kNGLi it SECTION SiCTION (Hd/SP) 17.5 17• S - - 20.3 20.* - - 22.5 22.9 » - 25.3 25.* 1.5679 .333* 1.1300 .2500 .753 .229 27.5 28.0 1.5905 .333* 1.2100 .3500 .761 .226

30.3 30 ft ^ 1.7597 .333* 1.3700 .2500 .779 .206 32.5 33.1 2.3801 .333* 1.5100 .2500 .63* .138 35.3 35.6 2.1206 .2256 1.5000 .2000 .75* .12* 37.5 3d.1 2.0868 .2256 1.5000 .2000 .767 .127 *0.3 *0 .7 2.1770 .135* 1.9700 .1*00 .905 .083 *2.5 *3.2 2.1996 .1692 1.7200 .1100 .792 .079 *5.3 *5.7 1.9*02 .1579 1.7100 .1600 .381 .109 *7.5 *4.3 1.6532 .135* 1.&400 .1*00 .993 .112 50.3 50.3 1.3997 .0902 1.3500 .1500 .96? .12* 52.5 53.3 1.1731 .0677 1*3500 .1100 .995 .107 55.3 55.3 .911* .05*1 .7710 .0630 .9*6 .095 57.5 5rt .6*07 .0305 .5100 .3*00 .796 .073 fcO.3 60 .9 .*771 .0226 .3630 .0330 .761 .079 f.2.5 *>i.:» .3*18 .0327 .2680 .0230 ,79* .111 65.3 65 .9 • ?*32 .021* .1530 .0210 .657 .102 67.5 f>%,• 5 .1590 .01*7 .1030 .3120 .6*9 .096 70.3 71,.0 .1092 .0122 .3375 .01*0 .309 .153 72.5 73,.5 .0839 .0030 .06** .3107 .767 .1*7 75.3 76..0 .0999 .019* .3706 .01*5 .706 .163 77.5 78..5 .09*9 .0109 .3775 .3123 .317 .160 «0.3 31..0 .11%* .0113 .09*1 .0151 .79* .1*3

S, 16.5969 .'+513 13. .3935


•>! PLJS

LA3 CH CM C*0SS CM C*3SS ANGLE SECTION SECTION 9 A no E3W0' (18/SR) 17.3 17.3 486.5064 9.3136 464.1000 6.5000 .954 .023 13.9 id.3 397.3944 4.1216 423.4000 5.6000 1.065 .026 19.5 19.9 378.2184 7.3320 352.1900 7.3000 .931 .025 20.5 20.9 300.8376 7.2192 313.1000 6.5000 1.057 .033 22.3 22.4 277.375? 7.4448 255.3000 5.4000 .922 .031 23.3 23.4 231.5784 4.7376 22V.0300 5. 4000 .999 .031 24.5 24.9 197.6256 5.8656 174.7000 4.4000 .994 .034 ?5.5 25.9 164.3496 5.3169 156.5000 3.3000 .?54 .044 27.0 27.5 139.7592 7.4v*fl 121.3000 2.6000 .371 .050 28.3 23.5 113.7024 2.5944 103.3000 2.1000 .906 .029 29.5 30.0 91.1424 U.'i920 73.9000 1.4000 .311 .022 30.5 31.0 73.2072 1.^664 60.9000 1.3000 .932 .022 32.3 32.5 55.8360 1.2409 35.5000 .3000 .555 .020 33.3 33.6 43.6536 1.0152 29.4000 .3000 .551 .024 34.5 35.1 32.0352 .5640 21.4000 .5000 .66d .022 •55.5 3o. 1 23.8008 .5640 15.5000 .6000 .697 .030 37.3 37.6 15.0024 .4512 7.5400 .'•300 .509 .032 38.0 33.5 9.9490 .3046 5.3300 .2900 .541 .034 39.5 40.2 5.3619 .3271 2.3400 .1500 .445 .033 40.5 ••1.2 3.7450 .3046 1.4300 .1000 .395 .042 42.3 »2.7 1.714.6 .090? .4500 .3630 .253 .042 43.3 43.7 .9024 • 0537 .2090 .3400 .232 .047 44.5 45.2 .3102 .0305 .5460 .0530 1.760 .255 45 o 5 46.2 .1545 .0130 .9220 .3640 5.966 .024 47.0 47.3 .3597 .0293 1.5000 .0700 4.461 .414 4rt. 3 »9.3 .5663 .0350 2.2500 .3800 3.973 .293 49.5 50.3 .99)3 .0462 2.7500 .3300 2.777 .152 50.5 51.3 1.3874 .0564 3.5000 .3900 2.595 .120 52.3 52. 3 1.6499 .0677 4.1300 .1600 2.233 .119 53.3 53. 4 2.4026 .0790 4.5200 .1500 1.391 .091 54.5 55.3 2.6959 .1466 4.5200 .1200 1.714 .103 5^.5 56.4 2.9779 .1579 4.3400 .1100 1.625 .094 57.3 57.^ 3.1020 .0790 4.5500 .1000 1.467 .049 5d.3 53.9 3.1246 .0564 4.4000 .1100 1.409 .043 5". 5 60.4 3.0230 .1015 4.2400 .1000 1.403 .053 *0. 5 61.4 3.0456 .1015 3.9100 .1000 1.294 .054 62. 3 62.9 2.80B7 .0790 3.4900 .3900 1.243 .047 *>j. 3 63. 9 2.6621 .0902 3.2700 .3300 1.229 .051 ">4, 5 65.4 2.3350 .0677 t'.55O0 .3700 1.135 .045 '•ti. 5 S6.4 2.2334 .0554 2.MOO .3700 1.0 79 .042 f>7,3 68.0 1.3499 .0451 2.3)00 .3600 1.130 .043 M.3 69.0 1.7033 .0451 1.5300 .3400 .936 .035 69, 5 70.3 1.2521 .0333 1«2300 .3500 1.022 .049 70.5 71.5 1.1506 .0333 1.0400 .3400 .904 .044 72.3 73.0 .8437 .0237 .i470 .0300 .767 .U42 73.3 74.0 .7309 .0226 .5550 .3270 .759 ,0t»<* 74.5 75.5 .5256 .0214 .3940 .3350 .750 .075 7a.3 76.5 .3903 .0203 .2910 .3310 .746 .038 77.J 73.0 .2583 .01 5B .1350 . 3 2 •} n .716 .099 79.3 79.0 .1771 .01+7 .1330 .3240 .751 .149 79.5 *U.5 ,095d .0201 .1200 .3130 1.253 .296 A 0 * 5 dl .5 .0673 .0213 .3977 .3103 l.*5l .436

l?09.*536 3.3121 1135.3784 6.3312

222 Ul VI O O ul UlUUUlUtUUUIMO O Ml

2 00 |-

o — Ul X

X O O *"• f *™* ^* ^ ^— r* H* ha »-— *-* O O O O ' O QD VI ^"4 O ^

•o •- "- •— in u • - L '_ L '_'_'--'_'_!.-- 1 _ !. L t. !. L '_'.'.?.---- 1 L ' '•''''''''*•'"'*'_ ^ ' V * w- 'V


Kll X o l ,, 1 1 . • i n n ^^ 0000000 u u ••» OOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO C0000000003300000 — •-• m

0000000000000000000000000000000000 000000000000 f> <=oooo


N> 164*E\/ 3-.a. 39M?V nEG 3

31 PLJS =>l MI

LA3 c«t caoss C« COSS ANGLE ANGLE SECTION SECTION 9ATI0 » 3S03 (5.5 06.4 .2200 .0226 .1520 .3210 .736 .122 67.0 68.0 .1703 .0130 .1550 .3220 .910 .161 63.3 69.0 .1557 .0158 .3333 .3136 .535 .103 69.5 70.5 .1045 .0152 .0979 .3195 .941 .223 70.5 71.5 .1078 .0155 .1020 .3160 .9*5 .201 72o3 73.0 .0807 .0099 .0632 .31 17 .794 .195 73.3 74.0 .0865 .0105 • 3S53 .3111 .755 .153 7*. 5 75.5 .1094 .0134 .3517 .3154 .569 .159 75.5 75.5 .0929 .0119 .3779 .3176 .337 .217 77.3 73.0 .0791 .0116 .J719 .3139 .921 .224 73.3 79.0 • 10*15 .0121 .3321 .3149 .757 .161 79.5 ^0.5 .1050 .0179 .3 959 .0107 .913 .195 90.5 41.5 .1072 .0192 .1120 .3120 1.345 .210

SJ* = 33.1817 .553* 26 .7563 .5000

224 ELASTIC 23E3

"I PLJS =»I -4MU5 LA3 C* C* CROSS £*30? C"» C33SS 9309 «NGL£ ANGLE SECTION SECTION 94iTIO % ™?J^ ^^ t 1/-5 17.9 392.6000 3.2900 vl5*0000 5 .VOOO 1.057 .027 20.3 20.4 263,9000 7.4000 273.4000 4,.5000 1.036 .034 22.5 23.0 169,3000 3.2000 16 7. )000 2,.5000 .999 .024 25.3 25.? 99,0000 1.3000 94.9000 1,.2000 .959 .017 27.5 28. 1 53.9000 1.1000 47.7000 1,.0000 .985 .026 30.0 30.6 25*7000 .6000 21.6000 i.6000 .940 .030 32.5 33.2 11.0000 .6000 9.1300 I.4500 .739 .057 35.3 35.7 4.5400 .4100 3.1700 .3200 .699 .095 37.5 39.3 2.6400 .3100 2.9600 i.3700 1.121 .192 40.3 40.9 2.7800 • 1700 4.3900 i.2100 1.468 .117 42.5 43.3 3.5400 .4000 4.9900 i.4200 1.410 .199 45,3 45.9 4,0700 .2500 3.3900 t.2900 1.302 .107 47.5 4(1. * 4.2100 .2900 5.0700 i.2600 1.204 .103 50.3 51.0 3.3300 .1800 4.4600 *,2000 I d64 .076 52-5 53.5 3.41QC .1100 3.S200 4.1300 1.062 .051 55.3 56.0 2.7200 .1400 2.5600 «,1100 .941 .063 57.5 58.6 1.7900 * "700 1.6100 4,0000 .999 .057 Si). 3 61.1 1.2000 .0400 1.1300 « 0500 .942 .052 '2.5 63. b .7110 .0290 .5770 3300 .812 .054 65.3 66.1 .4153 .0190 .3460 • 0260 .934 .073 67.5 68.7 «22£u .1)130 .1550 • 3140 .743 .092 70-3 71.2 .0950 .0090 .1050 • 3110 1 .105 .172

SJ* = 391.9950 3.9572 396.5442 2.3856

225 = 1 P'.JS "I SOS

LA3 CM CM C»OSS CM C*3SS ZHOR kNGLl 4N6L£ SECTIOM SECTION U[0 1 <4A /C2\ 17.5 17.9 6.9600 .8900 10.7200 1.0900 1.540 .250 20.3 20.4 6.7100 .6100 10.4000 .9500 1.550 .200 22.5 23.0 6.2300 .6400 10.1900 .6700 1.636 .199 25.3 25.5 5.6700 .3200 3.6000 .3900 1.517 .109 27.5 23.1 4.75O0 .3300 3.1100 .4200 L.7C7 .148 30.3 30.6 3.8200 .2500 5.9900 .3200 .565 -132 32.5 13.2 3.0600 .3000 4.6400 .3300 .516 .184 35.3 35.7 2.1800 .2400 3.3700 .3300 : .546 .250 37.5 38.3 1.4300 .2100 2.2400 .2900 i .566 .J07 »0.0 40.3 .9540 .1000 1.1400 .1200 i .190 .176 42,5 -3.3 .5690 .1660 .5910 .1610 l .039 .415 45.0 «t5.9 .3090 .0750 .2750 .3740 .990 .332 47.5 4(5.4 .1810 .0700 .2790 .3700 l .541 .711 50.0 51.0 .1710 .0440 .2970 .3590 l .737 .565 52.5 S3.5 .1670 .0350 .3440 .3470 z.060 .515 55.3 56.0 .2070 .0450 .4V00 .3470 .126 .515 37.5 59.6 .2360 .0420 .4 760 .3420 i .017 .401 hO.D 61.1 .2100 .0170 .4370 .0290 2 .031 .219 62.5 63.6 .1790 .0130 .3910 .0?30 2.129 .201 65.0 66.1 .1440 .00-10 .3170 .0220 2 .142 .189 67.5 6S.7 .1310 .0090 .2390 .3190 1.924 .196 70.3 71.2 c09OO .0070 .1490 3130 2 .100 .219

SJ1 = 20.0695 .5921 31.6135 .7193

226 20EG

PLJS c Z"4caoss CM C30SS EP^O^ NGLI 4M6L£ SECTION SECTIOM *»TID E**O* 1-3/ 3^ 7 17.5 17.9 20.3 20.4 1.4700 .6700 2.7000 .5000 1.337 .904 22.5 2J.0 1.5300 .3600 2.6300 .4000 1.719 ."•92 25.3 25.5 1.5200 . moo 2.3400 .2100 1.342 .211 27.5 28.1 1.2400 .1800 1.3400 .2200 1.494 .279 30.3 30.6 1.0700 .1400 1.1400 .1300 1.065 .219 32.5 33.2 .9550 .1730 1.0100 .1900 1.053 .276 35.3 35.7 .7250 .1740 .3620 .1730 .499 .267 17.5 33.3 .4(»40 .1270 .4560 .1650 1.027 .474 40.3 "•0.3 .2260 .0640 .5410 .3950 2.394 .799 42.5 43.3 .2100 .1110 .5150 .1520 2.452 1.435 45. 3 45.* .2040 .0640 .4760 .3960 2.299 .842 '•7.5 41.4 .W70 .0700 .•590 .3440 3.330 .931 50. :i 51.0 .1740 .0430 .4510 .0700 2.592 .756 52. 5 53.5 • 16?0 .0320 .3110 .3430 1.920 .463 5b .3 5o.Q .1520 .Qi5Q .3380 .3420 2*224 • S32 5 7.i 58.5 .1090 .0200 .2640 .0350 2.422 .549 ft0.3 M.I .0964 .0144 .1770 .3220 1.336 .357 62. 5 63.6 .0555 .0045 .1050 .3150 1.992 .396 *>5.3 66.1 .0431 .0079 .0793 .3145 1.340 . *»76 ft 7. 5 68.7 .0249 .0062 .051* • Jl 19 1.795 .560 70.3 71.2 .0122 .0045 .0 220 .0077 1.303 .917

SJX = 5.399'* .3657 3.2291 .3797

227 PL Ji 'I Hi >tJS

L.A-3 C« CM CROSS 1**0* c« coss -:=j*o* ftNGLi SECTION SECTION 9ATI3 E-?ttO=» ( ^ 17.5 17.9 _ ?0.3 -?0.4 j .0200 .7700 1.9900 . 4300 .931 .414 22.5 23.0 2.7100' .4400 1.6900 .3400 .624 .161 25.0 2-5.5 3.0800 .2400 2.5300 .2300 • 421 .098 27.5 29.1 3.1300 .2700 2.5400 .2500 .912 .106 30.0 30.6 3.3800 .2400 2.9100 .2400 .961 .094 2dm 5 J3.2 3.6400 .3200 3.4500 .2900 .949 .115 35.3 35.7 3.4400 .3400 2.9600 .3100 .960 .124 V.5 39.3 3.0700 .3000 2.5600 .3100 .934 .130 *o.a 40.9 2.3600 .1500 2.2100 .1600 .936 .090 42.5 43.3 2.1400 .3000 2.9000 .2600 .935 .179 4b. 3 45.9 1.5400 .1500 1.3100 .1500 .951 .129 47.5 4H.4 1.2100 .1500 .9590 .1190 .793 .139 50.0 51.0 .8610 .0850 .7980 .3350 .922 .129 52.5 53.5 .5800 .0560 .4610 .0510 .795 .117 55.3 56.0 .4040 .0530 .3000 .3410 .743 .141 57.5 58.6 .2520 .0270 .2360 .0320 .917 .154 60.3 M.I .1680 .0190 .1140 .3200 .679 .139 62.5 *S3.6 .1070 .0130 .3976 .3147 .919 .170 *>6.1 .0824 .0098 .0710 .0146 .962 .205 <>7.5 ii<5. 7 .0701 .0090 .3674 .0131 .961 .217 70.3 71.2 .0522 .0063 .3615 .3096 1.179 .232

19 .3259 .5775 16.4686 . V946

228 230*£\/ tL»STIC 1JE3


L*9 C^ CM C»0SS CM C«SS 4NIJL1 4N6L:: SECTION StCTION 3ATI3 M6/5H) !/.3 17.4 422.1000 24.2000 4<;3.30O0 7.7000 1.075 .0*4 Id.3 19.4 371.1000 11.3000 393.9000 5.5000 1.061 .036 19.5 IV.9 203.5000 5.7030 304.0000 5.3000 1.072 .029 20.9 246.0000 9.9000 ?5J.50OO 4.3000 1.030 .045 ?2'.O 22.5 180.6COO 3.5000 195.0000 3.4000 1.090 .029 23.3 23.5 157.9000 3.2000 147.4000 4.6000 .934 .035 ?o . 1400 _}.9'+00 .1500 1.026 .054 53.3 5».O 3.2000 .1100 3.4400 .1200 1.075 .053 T<», 5 55.5 3.0500 .1400 2.7900 .1100 .915 .055 35, 5 56.5 2.5500 .1100 2.4400 .1000 .957 .057 57.3 53. 1 2.0000 .0700 1.7400 .3900 .370 .054 5d,D 59.1 1.6900 .0600 1.5200 .3400 .905 .059 5^» 5 60.6 1.3200 .0500 1.2300 .3500 .932 .052 60. 5 61.5 1.1300 .3400 1.0500 .3500 .929 .055 62.3 63.1 .8170 .0350 .6940 .3350 .937 .055 63.j h<*. 1 .6510 .0290 .5020 .3300 .771 .057 64. 5 *>5.6 • ••660 .0220 .^290 .3270 .921 .072 *>3. 5 66.6 .39 30 .0190 .2900 .3220 .733 .065 *i 7. 3 68.2 .2940 .0160 .2110 .3210 .714 .091 "in.3 o9.2 .1660 .0110 .1350 .3170 .913 .116 *•>. 5 70.7 .1250 .0110 .1113 .3140 .999 .137 70.5 71.7 .0740 .0090 .0990 .31?0 1.339 .217

. 5Jv1 792.90 71 9.3212 307.2220 4.bO01

229 1DEG

= 1 P'.JS

LA3 CM CHOSS £S=?fH C*» C^DSS £'33« 4NGL£ 4NGl_£ SECTION StcriON £=»«39 (*d/S3) (*8/5R) 17.3 17.* 7.3300 1.1500 11.1500 1.+000 1.511 .302 13.3 13.* 6.5800 1.0*30 10.+900 1.3300 1.59* .301 19.5 19.9 7.1200 .9700 10.3*00 1.3600 1.522 .£55 20.5 20.9 6.3300 .9*00 10.1300 .3800 1.600 .275 22.3 22.5 6.*300 .7300 10.1600 .3700 1.590 .225 ^3.^ 23.5 6.0500 .5700 10.2500 1.3*00 1.59* .290 2*.5 25.0 5.9800 .3800 3.7700 .5200 l.*67 .127 ?5.5 25.J 5.*700 .3300 3.7900 .*O0 0 1.507 .121 27.3 27.5 +.9800 .3600 3.5100 .5000 1.709 .159 2a.3 23.6 "•.6600 .3230 3.3500 .3900 1.727 .1*5 29.5 30.1 *.2600 .2700 6.+300 .+000 1.509 .13* 30.5 31.1 3.5530 .2*10 5.5900 .3*00 1.500 .1** 3*.3 32.7 3.*500 .2900 5.+*00 • 3700 1.577 .170 "U.3 33.7 2.3800 .2600 4.3*00 .3100 l.*03 .165 3*. 5 15.2 2.5100 .2300 3.*500 .2300 1.375 .163 35.5 3o.2 1.9600 .2100 3.3*00 .2900 l.ro* .235 37.3 37.3 1.6500 .1900 2.3500 .2300 l.*2* .215 38.0 38.3 1.3*00 .1330 2.1300 .2600 1.627 .292 19.5 *3. 3 1.0*00 .0900 1.2*00 .1200 1.192 .155 *0.5 *1.3 .91(10 .09*0 1.3500 .1100 1.15* .170 *2.3 •+2.3 .65*0 .1270 .5*30 .1370 .983 .233 *3.3 *3.9 .*950 • 1190 .5390 .1290 1.039 .369 **.5 *5.* .35*0 .0 9*0 .2970 .3720 .339 .302 *3.5 *o.* .2920 .0330 .2610 .0630 .39* .356 *7.3 *7.9 .13*0 .0610 .2530 .3910 l.*57 .691 *3.3 *a.9 .1320 .0600 .2930 .3920 1.510 .733 *9.5 50.5 .1630 .0**0 .2760 .3510 1.5*3 .527 SO. 5 51.5 .1770 .0330 .3130 .3590 1.797 .510 S2.3 53.0 .1730 .0390 .2990 .3500 1.530 .*63 S3.3 5 + .0 .1660 .0310 .3380 .3*70 2.337 .b20 5*.; 55.5 .1350 .0*20 .*390 .3500 2.373 .60 3 56.5 56.5 .1990 .0300 • *550 .3*60 2.235 • *15 57.3 5 c* • 1 .2150 .0250 .*690 .3*50 2.131 .i29 5H.3 59.1 .2510 .0250 .*950 .3*20 1.397 .2*3 S9.5 60.6 .2100 .0210 .+370 .3300 2.031 .252 «»0. 5 61.5 .2250 • 0150 • *330 .3270 1.92* .132 b2.3 63.1 .1900 .0160 .3910 .02*0 2.053 .21* M.3 6*.l .1770 .0130 .3370 .3230 2.136 .207 6* • 5 65.5 .1630 .0130 .3550 .3200 2.113 .iO2 65.5 06.6 .1360 .0110 .2920 .0130 2.1*7 .213 67.0 hi .2 . 1*60 .0100 .2310 .3200 1.925 .190 69.? .1170 .0030 .20*0 .3170 1.7** .133 69.5 70.7 .0970 .0090 .2?*0 .3120 2.309 .227 70.5 71.7 .0650 .0070 .1530 .3120 1.395 .203

SJ* = ••0.6620 . -J35* 63.95S6 i . 1 7«>9

230 1DEG


17.4 ™ 1-1.3 Id.4 m IV. 5 19.9 2.2100 .5900 1.5600 .3500 .706 .246 20.5 20.9 1.8600 .3700 2.1700 .3500 1.167 .299 22.3 22.5 2.7300 .5200 1.5900 . +600 .609 .201 23.3 23*5 2.6500 .4800 1.7100 .3500 .645 .175 24.5 25.0 3.0300 .2900 2.6300 .3200 .969 .134 25.5 26.0 3.1600 .2600 2.4900 .£400 .749 .100 27.3 27.5 3.2000 .3000 2.3400 .3000 .731 .116 2a.o 29.6 3.1400 .2700 2.7900 .2500 .385 .110 29.5 30.1 3.5300 .2600 3.0000 .2900 .850 .103 30.5 31.1 3.3400 .2400 2.3800 .2500 .962 .097 32.3 32.7 3.7400 .3000 3.5000 .3100 .963 .113 33.3 33.7 3.5900 .2900 3.+300 .2900 .955 .110 34.5 35.2 3.4600 .2700 3.3200 .2700 .973 .104 35.5 J6.2 3.3900 .2600 3.0000 .2900 .995 .107 37.3 37.d 3.1900 .2600 2.5500 .2500 .799 .102 3a. 3 39.4 3.0100 .2500 2.6200 .2800 .970 .113 39.5 40.3 2.5200 .1300 2.2300 .1500 .985 .079 40.5 41.3 2.3000 .1400 2.2500 .1500 .993 .093 42.0 »2.3 2.J400 .2200 2.3200 .2100 .963 .121 43.3 • 3.9 2.0400 .2200 2.0100 .2200 .985 .151 44.5 45.4 1.6000 .1630 1.4200 .1400 .997 .125 45.5 46.4 1.5400 .1600 1.2900 .1300 .339 .121 4?.O 47.9 1.2900 .1200 .9480 .1340 .735 .124 44.3 4«.9 1.1400 .1200 .9320 .1350 .861 .149 4^.5 50.5 .9530 .0850 .5590 .0700 .592 .096 50.5 51.5 .4240 .0750 .7580 .0800 .915 .127 52.3 53.0 .6250 .0570 .4930 .3550 .789 .114 53.0 54.0 .5500 .0470 .4430 .3460 .905 .108 5-.5 55.5 .4440 .35^0 .3270 ,1460 .736 .137 55.5 •56.5 .3810 .0430 .2930 .0410 .769 .133 5 7. J 58.1 .2720 .0300 .2220 .0 380 .316 .166 5a.o 59.1 .2420 .0270 .1960 .,0 330 .910 .164 S9.5 60.6 .1950 .0230 .1050 .3240 .568 .149 60.5 61.6 .1570 .0150 .1200 .3210 .764 .155 *>2.3 63.1 .1130 .0150 .1140 .3190 .966 .203 *• J.O 64.1 .1010 .0130 .3578 .0155 .671 .176 64. 5 65.S .0897 .0136 .3370 .3150 .970 .203 f-5.5 .0733 .0094 .3592 .3121 .756 .179 67.3 6".2 .073? .00 94 .3776 .0155 1.053 .250 si. 3 •H.2 .0663 .0076 .3525 .0136 .943 .232 69. 5 70. 7 .0531 .007-5 .0 553 .0095 1.041 .232 70.5 71.7 .0499 .0056 .3563 .3099 1.329 .264

39.0H30 • 70 7fl 33.3610 .5755


I would like to extend my sincere thanks to my advisor Prof.

Kamal K. Seth for his guidance and instruction throughout this project.

Thanks are also due to Dr. Hermann Nann for many helpful discussions.

I wish to thank the many collaborators in these experiments -

Profs. C. F. Moore and D. Dehnhard; Drs. A. Obst, N. Tanaka and

R. Boudrie; and K. Boyer and W. Cottingame - for their assistance.

Special thanks are due Dr. H. A. Thiessen who acted as ray on-site

advisor at LAMPF, and Dr. C. L. Morris for his contributions, partic-

ularly his work on the theoretical codes used to interpret our data.

I am also grateful to Associated Western Universities, Inc. and the

Los Alamos Scientific Laboratory for financial support.

And my deepest gratitude is extended to my wife, Jan, for her

support and invaluable assistance in typing the manuscript.


1. W. 0. Lock and D. F. Measday, Intermediate Energy , (Metheun and Co., London, 1970); the original development is due to H. L. Anderson, E. Fermi, R. Martin, and D. E. Nagle, Phys. Rev. 91 (1953)155.

2. E. Pedroni, K. Gabathuler, J. J. Domingo, W. Hirt, P. Schwaller, J. Arvieux, C. H. Q. Ingram, P. Gretillat, J. Piffaretti, N. W. Tanner, and C. Wilkin, Nucl. Phys. A300(1978)321.

3. R. K. Bassel, G. R. Satchler, R. M. Drisko, and E. Rost, Phys. Rev. 128(1962)2693.

4. A. Bohr and B. Mottelson, Nuclear Structure, (Benjamin, New York, 1975)vol. II, p. 137.

5. A. M. Bernstein, Phys. Lett. 29B(1969)335; and A. M. Bernstein, Adv. in Nucl. Phys., vol. 3(1969)325.

6. V. A. Madsen, V. R. Brown, and J. D. Anderson, Phys. Rev. C12 (1975)1205.

7. A. M. Bernstein, V. R. Brown, and V. A. Madsen, Phys. Lett. 7IB (1977)48.

8. D. E. Bainum, R. W. Finlay, J. Rapaport, J. D. Carlson, and J. R. Comfort, Phys. Rev. Lett. 39(1977)443.

9. A. M. Bernstein, V. R. Brown, and V. A. Madsen, Phys. Rev. Lett. 42(1979)425.

10. H. Howard, B. Storms, and S. P. Slatkin, "LAMPF Users Handbook." (1974).

11. Los Alamos Scientific Laboratory Report LA-4534-MS, compiled by H. A. Thiessen and S. Sobottka (1970).

12. H. A. Thiessen, J. C. Kallne, J. F. Amann, R. J. Peterson, S. J. Greene, S. L. Verbeck, G. R. Burleson, S. G. Iversen, A. W. Obst, K. K. Seth, C. F. Moore, J. E. Bolger, W. J. Braithwaite, D. C. Slater, and C. L. Morris, Los Alamos Scientific Laboratory Report LA-6663-MS (1977).

13. R. G. Clarkson and N. Jarmie, Computer Phys. Comm. 2(1971)433; and N. Jarmie, M. S. Pindzola, and H. Bischel, ibid. 13(1978)317.

14. G. Charpak, Ann. Rev. Nucl. Science 20(1970)195; a recent review is F. Sauli, CERN-77-09(1977).

15. D. M. Lee, S. E. Sobottka, and H. A. Thiessen, Nucl. Instr. and Methods 104(1972)179; ibid. 109(1973)421; ibid. 120(1974)153; ibid. 111(1973)67.

233 16. C. L. Morris and G. W. Hoffmann, Nucl. Instr. and Methods 153 (1978)599.

17. J. Jansen, J. Zichy, J. P. Albanese, J. Arvieux, J. Bolger, E. Boschitz,, C. H. 0. Ingram, and L. Pflug, Phys. Lett. 77B(1978)359.

18. M. A. Oothoudt, unpublished.

19. J. R. Comfort, Argonne National Laboratory, unpublished.

20. J. ?. Albanese, J. Arvieux, E. Boschitz, C. H. Q. Ingram, L. Pflug, C. Wiedner, and J. Zichy, Phys. Lett. 73B(1978)119; and private communication (1977).

21. J. Piffaretti, R. Corfu, J.-P. Egger, P. Gretillat, C. Lunke, E. Schwarz, C. Perrin, and B. M. Preedom, Phys. Lett. 71B(1977) 324; the EPICS results are from H. A. Thiessen et al, to be published.

22. D. R. F. Cochran, P. N. Dean, P. A. M. Gram, E. A. Knapp, E. R. Martin, D. E. Nagle, R. B. Perkins, W. J. Schlaer, H. A. Thiessen, and E. D. Theriot, Phys. Rev. D6(1972)3085.

23. F. Ajzenberg-Selove, Nucl. Phys. A300(1978)l; ibid. A190(1972)l.

24. J. Asher, M. A. Grace, P. D. Johnston, J. W. Koen, P. M. Rowe, and W. L. Randolph, J. Phys. G2(1976)477.

25. Z. Berant, C. Broude, S. Dima, G. Goldring, M. Hass, Z. Shkedi, D. F. H. Start, and Y. Wolfson, Nucl. Phys. A235(1974)410.

26. R. Moreh and T. Daniels, Nucl. Phys. 74(1965)403.

27. T. K. Li, D. Dehnhard, R. E. Brown, and P. J. Ellis, Phys. Rev. C13(1976)55.

28. J. L. Wiza, R. Middleton, and P. V. Hewka, Phys. Rev. 141(1966)975.

29. D. Drain, B. Chambon, J. L, Vidal, A. Dauchy, and H. Beaumevieille, Can. J. Phys. 53(1975)882.

30. G. Th. Kaschl, G. J. Wagner, G. Mairle, U. Schmidt-Rohr, and P. Turek, Nucl. Phys. A155(1970)417; and Phys. Lett. 29B(1969)167.

31. J. C. Legg, Phys. Rev. 129(1963)272.

32. H. F. Lutz, J. J. Wesolowski, S. F. Eccles, and L. F. Hansen, Nucl. Phys. A101(1967)241.

33. M. Igarashi, M. Kawai, and K. Yazaki, Prog. Theor. Phys. 49(1973) 825.

234 34. M. Pignanelli, J. Gosset, F. Resmini, B. Mayer, and J. L. Escudie, Phys. Rev. 08(1973)2120.

35. J. C. Armstrong and K. S. Quisenberry, Phys. Rev. 122(1961)150.

36. H. T. Fortune and S. C. Headley, Phys. Lett. 51B(1974)136.

37. G. Mairle, K. T. Knopfle, P. Doll, H. Breuer, and G. J. Wagner, Nucl. Phys. A280(i977)97.

38. H. T. Fortune, M. E. Cobern, and G. E. Moore, Phys. Rev. C17(1978) 888.

39. R. Middleton and D. J. Pullen, Nucl. Phys. 51(1964)63.

40. A. A. Jaffe, F. de S. Barros, P. D. Forsyth, J. Muto, I. J. Taylor, and S. Ramavataram, Proc. Phys. Soc. 76(1960)914.

41. S. Hinds, H. Marchant, and R. Middleton, Nucl. Phys. 38(1962)81.

42. K. Kolltveit, R. Muthukrishnan, and R. Trilling, Phys. Lett. 26B (1968)423.

43. J. L. Escudie, R. Lombard, M. Pignanelli, F. Resmini, and A. Tarrats, Phys. Rev. C10(1974)1645.

44. I. S. Towner and W. G. Davies, Oxford Univ. Nucl. Phys. Lab. Report 27/69(1969).

45. B. G. Harvey, J. R. Meriwether, J. Mahoney, A. Bussiere da Nercy, and D. J. Horen, Phys. Rev. 146(1966)712.

46. J. L. Groh, R. P. Singhal, H. S. Caplan, and B. S. Dolbilkin, Can. J. Phys. 49(1971)2743.

47. H. G. Benson and B. H. Flowers, Nucl. Phys. A126(1969)332.

48. J. W. Olness, E. K. Warburton, and J. A. Becker, Phys. Rev. C7 (1973)2239.

49. Z. Berant, C. Broude, G. Engler, and D. F. H. Start, Nucl. Phys. A225(1974)55.

50. A. B. McDonald, T. K. Alexander, 0. Hausser, G. J. Costa, J. S. Forster, and A. Olin, Can. J. Phys. 52(1974)1381.

51. D. Dehnhard, Proc. Conf. on Medium and Light Nuclei, Florence(1977).

52. C. Flaum, J. Barrette, M. J. Levine, and C. E. Thorn, Phys. Rev. Lett. 39(1977)446.

235 53. P. B. Void, D. Cline, P. Russo, J. K. Sprinkle, R. P. Scharenberg, and R. J. Mitchell, Phys. Rev. Lett. 39(1977)325.

54. M. P. Fewell, D. C. Kean, R. H. Spear, and A. M. Baxter, J. Phys. G3(1977)L27.

55. H. F. Lutz and S. F. Eccles, Nucl. Phys. 81(1966)423.

56. F. Resmini, R. M. Lombard, M. Pignanelli, J. L. Escudie, and A. Tarrats, Phys. Lett. 37B(1971)2/5.

57. R. Vandenbosch, W. N. Reisdorf, and P. H. Lau, Nucl. Phys. A230 (1974)59.

58. G. L. Morgan, D. R. Tilley, G. E. Mitchell, R. A. Hilko, and N. R. Roberson, Nucl. Phys. A148(1970)480; and Phys. Lett. .32B (1970)353.

59. S. Cohen, R. D. Lawson, M. H. McFarlane, and M. Soga, Phys. Lett. 9 (1964)180.

60. T. Inoue, T. Sebe, H. Hagiwara, and A. Arima, Nucl. Phys. 59(1964)1; recoupled in the j-j scheme in Ref. 33.

61. A. P. Zuker, Phys. Rev. Lett. 23(1969)983.

62. P. J. Ellis and T. Engeland, Nucl. Phys. A144(1970)161; ibid. A181(1972)368.

63. A. Arima, H. Horiuchi, and T. Sebe, Phys. Lett. 24B(1967)129.

64. A. P. Zuker, B. Buck, and J. B. McGrory, Phys. Rev. Lett. 21 (1968)39.

65. G. E. Brown, Proc. Int. Conf. on Nucl. Phys., Paris, vol. 1 (1964)129.

66. P. Federman and J. Talmi, Phys. Lett. 15(1965)165; ibid. 19 (1965)490; see also T. Engeland, Nucl. Phys. 72(1965)68; and G. E. Brown and A. M. Green, Nucl. Phys. 85(1966)87.

67. R. D. Lawson, F. J. D. Serduke, and H. T. Fortuna, Phys. Rev. C14 (1976)1245.

68. T. Erikson and G. E. Brown., Nucl. Phys. A277(1977)l.

69. I. Morrison, R. Smith, P. Nesci, and K. Amos, Phys. Rev. C17 (1978)1485.

70. B. A. Brown, A. Arima, and J. B. McGrory, Nucl. Phys. A277(1977)77.

71. T. Engeland and P. J. Ellis, Phys. Rev. Lett. 36(1976)994.

236 72. A. M. Kleinfeld, K. P. Lieb, D. Werdecker, and U. Smilansky, Phys. Rav. Lett. 35(1975)1329; and A. Christy and 0. Hausser, Nucl. Data Tables 11(1973)281. 73. J. F. Petersen, D. Dehnhard, and B. F. Bayman, Phys. Rev. C15 (1977)1719- 74. R. A. Eisenstein and G. A. Miller, Computer Phys. Conrm. 11(1976) 95. 75. L. D. Roper, R. M. Wright, and B. T. Feld, Phys. Rev. 138(1965) B190. 76. W. Bertozzi, private communication (1977). The given point- proton distribution was fit by us with a harmonic oscillator form of the density distribution. 77. L. Ray, W. R. Coker, and G. W. Hoffmann, Phys. Rev. 018(1978)2641. 78. B. A. Brown, S. E. Massen, and P. E. Hodgson, private communi- cation (1978). 79. J. W. Negele, Phys. Rev. Cl(1970)1260. 80. M. D. Cooper, Proc. Conf. on Meson-Nuclear Physics, Pittsburgh (AIP Conf. Proc. 33)(1976)237. 81. R. R. Johnson, T. Marks, T. G. Masterson, B. Bassalleck, K. L. Erdman, W. Gyles, D. Gill, and E. Rost, to be published. 82. M. Thies, Phys. Lett. 63B(1976)43. 83. M. B. Johnson and H. A. Bethe, Comments Nucl. Part. Phys. 8(1978) 75. 84. C. Lunke, R. Corfu, J.-P. Egger, P. Gretillat, J. Piffaretti, E. Schwarz, J. Jansen, C. Perrin, and B. M. Preedom, Phys. Lett. 78B(1978)201. 85. L. C. Liu, private communication (1978). 86. Modifications by C. L. Morris, private communication (1978). 87. M. M. Sternheim and Kwang-Bock Yoo, Phys. Rev.. Lett. 41(1978)1781. 88. S. Iversen, A. Obst, K. K. Seth, H. A. Thiessen, C. L. Morris, N. Tanaka, E. Smith, J. t'. Amann, R. Boudrie, G. Burleson, M. Devereux, L. W. Swenson, P. Varghese, K. Boyer, W. J. Braith- waite, W. Cottingame, and C. F. Moore, Phys. Rev. Lett. 40(1978) 17.

237 89. A. Bohr and B. Mottelson, Nuclear Structure (Benjamin, New York, 1975) vol. I, p. 334.

90. G. E. Brown, private communication (1978) .

91. B. A. Brown and B. H. Wildenthal, private communication (1978).

92. A. Arima, R. Seki, K. Yazaki, K. Kume, and H. Ohtsubo, private communication (1978).

93. T.-S. H. Lee, R. D. lawson, and D. Kurath, private communication (1978).

94. Model LC-80 two-stage, FTS Systems, Stone Ridge, New York. Refrigerator rated at maximum low of -80° C, with a capacity of 90 W (306 BTU/h) at -60° C.

95. J. T. Londergan, K. W. McVoy, and E, J. Moniz, Ann. Phys. 86 (1974)147.

96. K. L. Brown, R. Rothacker, D. C. Carey, and C. Iselin, S1AC-91, Rev. 1(1974).

97. K. L. Brown and C. Iselin, CERN-74-2(1974).

98. H. A. Thiessen and M. M. Klein, Proc. Fourth Int. Conf. on Magnet Technology, Brookhaven(1972)p. 3.

99. K. Boyer, private communication (1978) .

100. J. F. .Atnann, unpublished.

101. R. Ridge, unpublished.

102. M. A. Oothoudt, unpublished.

103. D. Dodde'-, private communication (1978).

104. G. Giaconielli, P. Pini, and S. Stagni, CERN/HERA 69-1(1969).

105. P. J. Bussey, J. R. Carter, D. R. Dance, D. V. Bugg, A. A. Carter, and A. M. Smith, Nucl. Phys. B58(1973)363; and J. R. Carter, D. V. Bugg, and A. A. Carter, Nucl. Phys. B58(1973)378.

3 06. P. Y. Bertin, B. Coupat, A. Hivemat, D. S. Isabelle, J. Duclos, A. Gerard, J. Miller, J. Morgenstern, J. Picard, P. Vernin, and R. Powers, Nucl. Phys. B106(1976)341.

107. K. M. Watson, Phys. Rev. 105(1957)1388; and M. L. Goldberger and K. M. Watson, Collision Theory (Wiley, New York, 1964).

238 108. A. K. Kerman, H. McManus, and R. JL Thaler, Ann. Phys. 8(1959)551.

109. A. L. Fetter and K. M. Watson, Adv. in Theor. Phys., vol. I (1965)115.

110. W. R. Gibbs, B. F. Gibson, A. T. Hess, and G. J. Stephenson, Jr., Phys. Rev. 013(1976)2433.

111. L. C. Liu and C. M. Shakin, Phys. Rev. 016(1977)1965.

112. L. C. Liu and C. M. Shakin, Phys. Rev. C16(1977)333; L. C. Liu, Phys. Rev. 017(1978)1787.

113. R. H. Landau and A. W. Thomas, Nucl. Phys. A302(1978)461.

114. W. R. Gibbs, NATO Advanced Study Institute on Theoretical Methods in Medium-Energy and Heavy-Ion Physics, June 1978 (Los Alamos Scientific Laboratory Report LA-UR-78-1665).

115. L. S. Kisslinger and F. Tabakin, Phys. Rev. 09(1974)188; E. Kujawski and G. A. Miller, Phys. Rev. 09(1974)1205; G. A. Miller, Phys. Rev. 010(1974)1242.

116. L. S. Kisslinger, Phys. Rev. 98(195^)761.

11.'. G. A. Miller and J. E. Spencer, Ann. Phys. 100(1976)562; see also E. Rost and G. W. Edwards, Phys. Lett. 37B(1971)247.

118. M. Ericson and T. E. 0. Ericson, Ann. Phys. 36(1966)323.

119. J. Eisenberg, J. Huffner, and E. J. Moniz, Phys. Lett. 47B (1973)381.

120. J. Huffner, Phys. Reports 210(1975)1.

121. R. A. Eisenstein and G. A. Miller, Computer Phys. Comm. 8 (1974)130.

122. G. W. Edwards and E. Rost, Phys. Rev. Lett. 26(1971)735.

123. E. H. Auerbach, D. M. Fleming, and M. M. Sternheim, Phys. Rev. 162(1967)1683.

•ft- U.S. Government Printing Office: 1979-677-013/92 239