Optical-model analysis of exotic atom data

Nuclear Physics A372 (1981) 43344 Q North-Holland Publishing Company

OPTICAL-MODEL ANALYSIS OF EXOTIC ATOM DATA (In. Antiprotonic and sigma atoms C.J. BATTY Rutherford and Appkton Laboratories, C7:ilton, Didcot, Oxon, UK

Received 26 June 1981 Abstract: Data for antiprotonic and sigma atoms are fitted using a simple optical model with a potential proportional to the nuclear density. The potential strength can be related to the free hadtonnucleon scattering length using a model due to Deloff . A good overall representation of the data is also obtained with a black-sphere model .

1. Introduction In a previous paper t) (to be referred to as I) we have discussed the analysis of strong interaction shift (E) and width (T) measurements .for exotic atoms in terms of a simple optical model. In I it was shown that an optical potential proportional to the nuclear density was capable of fitting the directly measured strong interaction shift and width values for kaonic atoms over a wide range of nuclei . When yield values, which give the strong interaction width of the upper level are included, then somewhat poorer fits were obtained . Detailed calculations in which the shape of the potentialwas varied seemed to indicate that the use of a potential proportional only to the nuclear density is not entirely satisfactory and that some modification to this simple form may be required to get more exact fits to the data. Nevertheless the model is remarkably successful . In this simple model the hadron-nucleon scattering lengths are taken to be "effective" values and the fitted values for kaons differ considerably from recent determinations of the free kaon-nucleon scattering length . In I it was shown that the work of Deloff Z) gives a simple analytic relationship between the free and "effective" values in terms of a single parameter bq . It was found possible to get a very good fit to the kaon data by adjusting the value of bq. Fits to the experimental kaon data were also made t) using a simple model developed by Kaufmann and Pilkuhn s'a) in which the nucleus is represented by a totally absorbing ("black") sphere . The strong interaction shift (e) and width (T) can then be simply calculated from the radius of the black sphere, R and a parameter S which describes the reflection of the incident hadron wave at the surface of the sphere. It was found that the model gives a good overall representation of the kaon data using R = roA 1~3 and a constant value for S. 433

434

C.J. Batty / Optical model (II)

Further details about the calculations are given in I where kaonic atom data is analysed . In the present paper we are concerned with the analysis of strong interaction data for antiprotonic and sigma atoms. 2. Antiprotonic atoms s-s) The data for antiprotonic atoms were taken from the published literature and represent all currently available measurements for Z > 2. They consist of 13 measurements of energy shifts, 13 direct measurements of level widths together with 17 measurements of relative yields which can be used to give width values for the "upper" levels . These measurements cover a total of 8 levels and 18 different nuclei . The quality of the data, compared to that available for kaons, is however relatively poor . For example several of the shift and width measurements are consistent with the value zero and only 5 shift and 7 width measurements differ by more than 3 standard deviations from zero . Nevertheless useful information can be obtained from the measurements . A useful summary and tabulation of the measured values is given in ref. 9). 2 .1 . OPTICAL-MODEL ANALYSIS

The data were first fitted using a simple optical potential proportional to the nuclear density of the form given by eq . (2) of I. The complex parameter d = aR+iai, which determines the strength of the potential, was adjusted to give a best least-squares fit to the data . The results are given in table 1. Following the approach Tnst .~ 1 Analysis of antiproton data Data e,T e,I ;Y e, I; Y

aR (fm) 1 .OOf0 .32 1 .53t0 .27 1 .95 f 0 .32')

aI (fm) 2 .21f0 .26 2 .SOt0 .25 2 .22 t 0 .27')

X

x

2 X /F

26.6 53.9 46.0

1 .11 1 .32 1 .18

used for kaonic atoms the shift and directly measured width values only were first z analysed and as can be seen from the table a very good fit was obtained with X /F, the value of Xz per degree of freedom, equal to 1 .11 . Including the yield values again caused the value of Xz to increase but in this case Xz /F=1 .32 which still represents a very good fit to the data (see figs . 1, 2 and 3) . Including the yield values causes the value of aR to increase significantly although the errors associated with the two sets of values still overlap. The changes in aI are relatively much smaller.

435

C.J. Batty / Optical rnodcl (II) 1000

$HIF T(eV) n=5 n=2 n=3

n=4

n=7

~

r

100

10

0

I

10

PEAR SHIFTS

lag .

I

20

I

30

I

40

I

50

I

60

I

70

I

80

I

90

I Z

1 . Measured shifts in antiprotonic atoms . The continuous lines and ' are the fit to the data with d =1 .53 +12 .50 fm .

The values obtained for d are in reasonable agreement with that obtained by Robeson et al.') who analysed a similar set of data but in some cases with different density distributions . Their value was à = 0.85 t 0.38 + i (2.66 t 0.28) fm. The analysis of Poth et al. e) used a rather more restricted set of data and obtained à=2.Ot0.4+i(2 .Ot0 .4)fm. The shift, width and yield values were also fitted with a potential of the form given by eq. (3) of I where d is the difference between effective hadrorC-neutron and hadron-proton scattering lengths. This analysis gave d = -12.5 t 5 .5 + i(7.1 t 3.4) fm and X Z /F decreased by a small, but significant, amount . As in the kaon case these various sets of values for d, the "effective" antiprotonnucleon scattering length, are considerably different from the free values . For example the static potential of Bryan and Phillips 1°) gives ka~e = -0.915 + 10.695 fm. We notice that, as in the kaon case, the real parts of the free and effective interactions have opposite signs and the magnitude of the effective interaction is about twice that of the free interaction. For kaons we have shown that the model due to Deloff Z), who considers the hadron-nucleus interaction in terms of a multiple scattering approach, gives a very

436

C.l. Batty / Optical mark! (II) jL.iDTH(eV)

1000 .00

~ n=1 n=2

100 .00

n=3

n=4

n=5

n=8

10 .00

1 .00

0 .10

0 .01 PEAR YIDTHS

Fig. 2. Measured widths in antiprotonic atoms. The continuous lines and * are the fit to the data with d =1 .53 + i2.50fm. good fit to the data using the free kaon-nucleon scattering amplitudes and a single parameter bq which is related to the shape and range of the kaon-nucleon interaction . We have tamed out similar calculations for antiprotons using scattering lengths derived from the static and non-static potentials of Bryan and Phillips'. The results are given in table 2 for fits to the shift and widths only and for fits to the shift, width and yield values . The results obtained using the two sets of potentials are seen to be rather similar. However, unlike the results obtained with the simple optical potential, it is found that the fit to the data is much worse when the yield values are included . The values of 8 = aR + ia I deduced from the fit using the Deloff model are seen to be rather different from those obtained directly. In particular the value of aR determined directly is significantly bigger, particularly so when the yield values are included in the data to be fitted . This difference may indicate some fundamental problem with the model of Deloff or could be a consequence of the particular choice of values for the free scattering lengths which are not particularly well determined experimentally . Nevertheless the fit to the shift and width values only seems very satisfactory .

437

C.1. Batty / Optical model (II) 100

YIEL p(X)

3-2 75 4-3 8-7 50

7-6 9-B 6 -5

5-4

25

10

I

20

I

30

I

40

I

50

I

60

I

I

70

80

PBAR YIELDS

I

90

I 2

Wig. 3. Measured yields in antiprotonic atoms. The continuous lines and' are the fit to the data with d =1 .53+ i2.50 fm. 2.2 . BLACK~PHERE MODEL

As we have already discussed in I the black-sphere mode1 ) provides simple analytic formulae relating the strong interaction shift (e) and width (T) in hadronic atoms to the radius (R ) of the nucleus, which is taken to be represented by a totally 3~4

2 Results for antiprotons using model of Deloff TAHLE

Potential

Static

Non-static

aR (fm) ap (fm)

-0.83+i0.638 -0 .999+i0.751

-0 .865+i0.803 -0 .889+i0.811

Data b9 (fm~ aR (fm) aI (fm) Z xz X /F

e, T 1.18f0.11 0.50 f 0.20 1.72t 0.14 33 .7 1.35

e, I; Y 1.16f0.17 0.54 f0.29 1.69 f0.23 158.E 3.77

s, I' 1 .29t0.14 0.45f 0.20 1 .63t 0.13 36.1 1.44

e, I; Y 1.25t0.22 0.51 t0.30 1.60t0.24 183.5 4.37

438

C7. Batty/ Optical model (II) TAHLE 3 Antiproton results for black-sphere model Data e, T e, I' c, T, Y c, l; Y Y Y

1V(R) (nucleons) 1 .13 f 0.16 0.87 t 0.07 0.80 t 0.09

ro (fm) 1.594 t0.025 1.665 t 0.017 1.691 t 0.027

ô (degrees)

X

206.0 t4.5 208.1 t4.2 202.0 t4.5 201.9 t4.9 202.0 `) 202.0 `)

39.3 36 .2 95 .8 132.8 46 .9 75 .3

z

XZ

/F

1 .64 1.51 2.34 3.24 2.93 4.71

`) â fixed at this value.

absorbing black sphere, and a parameter S related to the reflection of the hadron wave at the surface of the sphere . Following the treatment used in I, the experimental data for antiprotons were also analysed and the results are presented in table 3. Again the parameter S was taken to be constant for all nuclei whilst the radius R was either parameterised in tis the form R = roA or taken to be the radius for each nucleus such that the number of nucleons N(R) outside the radius R is the same for all nuclei . In contrast to the situation for kaonic atoms it is found that either parameterisation for R gives equally good fits to the shift and width data . When the yield data is included the overall fit to the data is less satisfactory and there is then some preference for a parameterisation of the black-sphere radius R in terms of N(R) rather than the simple form R = roA~ is . This feature is seen more clearly when the yield data alone is fitted, but as was found for kaons the value of Xz is then relatively large. However in view of its extreme simplicity, the model works surprisingly well. 3. Signa atoms The strong interaction data for sigma atoms is limited, relatively inaccurate and covers only a restricted range of nuclei . Nevertheless some useful results can be obtained . In an experiment carried out at CERN, Backenstoss et al. tt) have measured the relative intensities of X-ray lines from E_ atoms in four elements . In an experiment at the Rutherford Laboratory, Batty et al. iz) have measured strong interaction shifts, widths and yields for O and for nuclei between Mg and S. In total there are 5 shift measurements, 3 direct measurements of widths and 9 measurements of relative yields . 3.1 . OPTICAL-MODEL ANALYSIS

The data were fitted using the simple optical potential given by eq . (2) of I at first fitting the shift and width values only and then the complete set of shift, width

439

Cl. Barry / Optical model (77) Tnsz.e 4 Analysis of sigma atom data aA

Data a, T r, T, Y

sI

(fm)

(fm)

Xz

X2/F

0,330 f0.046 0, 363 t0.048

0.138 f 0.056 0.202 t 0.025

1.4 12 .2

0.23 0.81

and yield measurements . The results are given in table 4. In both cases very good fits to the data are obtained (see figs . 4, 5 and 6) . The real part of the effective scattering length is seen to be relatively well determined and the two sets of parameters differ mainly in the strength of the imaginary part of d which is in any case less well determined . The value of d differs considerably from values of the free E--nucleon scattering length as determined by Alexander et al. t3 ) . This is clearly seen in fig. 7 where the free and effective scattering amplitudes are plotted. Again calculations were made using the model of Deloff together with the three free E--nucleon scattering 1000

SHIF T(eV)

n=3

n=4

100

10

1 SIGMA SHIFTS

I

20

I

30

I

40

I

50

I

60 Z

Fig. 4. Measured shifts in sigma atoms. The continuous lines and ; are the fit to the data with d = 0.36+ 10 .20 fm .

440

C.J. Batty / Optical model (II)

10000 . D0 WIpTH(eV)

n=4 1000 .00

100 .00

n=5

10 .00

1 .00

0 .10

0 .01

u

10

I

SIGMA WIDTHS

20

I

30

I

40

I

50

60 Z

Fig. 5. Measured widths in sigma atoms. The continuous lines and * are the fit to the data with d = 0.36 + 10 .20 fm .

length solutions A, B and C obtained by Alexander. The results are presented in table 5. The three solutions are seen to give significantly different fits to the data, Z the best fit being obtained with solution C which gives X /F = 0 .93 . However some caution should be used in interpreting these results since the three solutions do have large, and overlapping errors, as can be seen in fig. 7. These errors have not been directly taken into account in this analysis. 3.2 . BLACK-SPHERE MODEL

Following the methods used for kaonic and antiprotonic atoms, the strong interaction measurements for sigma atoms were analysed using the black-sphere model 3.a) . The results obtained are presented in table 6. Both the parameterisation R = ro .9 ti3 and the parameterisation for R in terms of a fixed number of nucleons N(R) outside the radius R are seen to give good fits to the data. The value for S, which depends on the quantity E/T, is seen to be considerably different from that obtained for antiprotonic and kaonic atoms. In the latter case the energy shift e is negative whilst for sigma atoms it is positive.

C.J. Batty / Optical model (II) 100

L

441

YIEL p(X) 4-3

5-4

6-5

75

50

9-8

25

T

10

20

30

SIGMA YIELDS

40

50

60 Z

Fig. 6. Measured yields in sigma atoms. The continuous lines and w are the fit to the date with d=0.36+i0 .20fm .

4. Discussion We have seen that the simple optical model with a potential proportional to the nuclear density is capable of giving a good fit to the data for kaonic, antiprooic and sigma atoms. For kaonic atoms there are some indications, however, that the model is not entirely satisfactory. For example for silicon it is only possible to get a good fit to the shift, width and yield data if the diffusivity of the nuclear density distribution is made unrealistically small. Nevertheless the model is remarkably successful in reproducing the overall data . Parameters obtained with this model are summarised for convenience in table 7 although the reader is referred to the individual tables for further details . Perhaps not too surprisingly the strength of the imaginary potential for antiprotons is found to be large, refiecting the very large probability for annihilation. For sigma hyperons the imaginary part is found to be rather small and significantly less than that for kaons. This small imaginary potential is of considerable relevance to the existence ~. of ~-hypernuclei which has been discussed in a number of papers ta~ ts

C1. Batty/ Optical model (II)

442 3

0 .6 A

B 0 .4

C

0 .2

0 .0

- .2

I

- .6

I

- .4

I

I

I

- .2

I

0 .0

I

I

0 .2

I

0 .4

SCATTERING LENGTHS

SIGMA

I a

F

Fig. 7. Real (a~ and imaginary (a~ parts of the scattering length for E-nucleon interactions. Points A, B, C are the values obtained by Alexander et al,' 3) for the free system. The other point d is the effective value deduced from E- atom data . TABLE 5

Results for sigma atoms using model of Deloff Solution

A

B

C

a (fm) ap (fm) b9 (fm) aR (fm) at (fm) a Xz X /F

-0 .175 0.275+i0.983 1 .26t0.26 0.161 t 0.009 0.235 f 0.041 26.1 1 .63

-0.8 0.6+i1.0 4.37 f 0 .35 -0.151 f 0 .008 0.371 t 0.010 77 .4 4.84

0.525 -0.592+i0.867 0 .573 f 0.031 0.427 f 0.013 0.189 f 0.020 14.8 0.93

TAHLE 6

Results for sigma atoms using black-sphere model Data s, I; Y e, 1, Y

N(R)

(nucleons) 3 .11 f 0.19

(fm)

(degrees)

8

XZ

X~lF

1.388 t 0 .030

124.7112.3 124.4 f 4.0

10.9 6.7

0.73 0.45

r°

443

C.J. Batty l Opdeal model (II) Twst.$ 7 Typical parameters for exotic atoms Particle

an (fm)

ai (fm)

N(R) (nucleons)

ro (fm)

K

0 .34t0 .04 1 .53t0 .27 0 .36t0.05

0 .84t0 .03 2.SOt0.25 0 .20t0 .03

3 .10t0 .14 0 .87t0 .07 3 .11t0 .19

1 .40f0.01 1 .67t0.02 1 .39t0 .03

F

In the simple model it is found that in all cases the effective hadron-nucleon scattering lengths differ considerably from the free values . However it has been found that the model due to Deloff using a. multiple-scattering formalism gives a very simple and direct relationship between the free and effective values in terms of a single parameter bq which is related to the shape and range of the interaction potential. For kaons, using recent determinations of the free scattering length the model is particularly successful. For antiprotons the fits are less satisfactory when the yield values are included (see table 2) but this may be a consequence of the choice of values for the frée scattering lengths which are not well determined experimentally . For sigma hyperons there are three sets of scattering lengths available. Good fits are obtained using two of the sets whilst the third gives a rather poor fit (see table 5) . Again however the free scattering lengths are poorly determined. A black-sphere model is found to give quite good fits for the data . For antiprotons the radius of the sphere is found to be significantly larger .than for kaons or sigma hyperons . Again this is presumably due to the large probability for annihilation to occur which is reflected in the low value for N(R), the number of nucleons outsïde the radius R. More accurate data would enable the simple optical model to be tested further and would allow the effective hadron-neutron and hadron-proton scattering amplitudes to be separately determined . Improved values for the antiproton-nucleon and sigma-nucleon free scattering amplitudes would allow the multiple-scattering treatment of Deloff to be investigated in more detail . References 1) 2) 3) 4) 5)

C.J . Batty, Nucl. Phys . A372, (1981) 418 A . Deloff, Phys. Rev . C21 (1980) 1516 W.B . Kaufmann and H . Pilkuhn, Phys. Lett. 62B (1976) 165 ; 74B (1978) 432 W.B. Kaufmann, Am. J . Phys . 4S (1977) 735 G . Backenstoss, A . Bamberger, T . Bunaciu, J . Egger, H . Koch, U. Lynen, H .G . Ritter, H .A . Schnitt and A. Schwitter, Phys. Lett. 41B (1972) 552 6) P .D. Barnes, S. Dytman, R .A . Eisenstein, W .C . Lam, J. Miller, R.B . Sutton, D .A. Jenkins, R .J . Powers, M. Eckhause, J .R . Kane, B.L . Roberts, R .E . Welsh, A.R . Kunselman, R .P. Redwine and R .E. Segel, Phys . Rev . Lett . 29 (1972)-1132

444

C.1. Batty / Optical model (II)

7) P. Robertson, T. King, R. Kunselman, J. Miller, R.J. Powers, P.D . Barnes, R.A . Eisenstein, R.B . Sutton, W.C . Lam, C.R . Cox, M. Eckhause, J.R . Kane, A.M . Rushton, W.F. Vulcan and R.E. Welch, Phyr . Rev. C16 (1977) 1945 8) H. Poth, G. Backenstoss, I. Bergstrom, P. Blum, J. Egge~, W. Fetecher, R. Guigas, R. Hagelberg, N. Hasaler, C.J. Herrlander, M. Izycki, H. Koch, A. Nilsson, P. Pavlopoulos, H.P. Povel, K. Roslhi, I. Sick, L. Simone, A. Schwuler, J. Stzarkier and L. Tauschet, Nucl . Phys . A294 (1978) 435 9) C.J . Batty, Rutherford Laboratory Report RL-80-094, Soviet J. Particlesand Nuclei, to be published 10) R.A. Bryan and R.J .N . Phillips, Nucl . Phys . BS (1968) 201 11) G. Backenstoss, T. Bunaciu, T. Egget, H. Koch, A. Schwitter end L. Tauseher, Z. Phys . A273 (1975) 137 12) C.J . Batty, S.F . Biagi, M. Blechet, S.D . Hoeth, R.A .J . Riddle, B .L . Roberts, J.D . Davier, G.J. Pyle, G.T.A. Squier and D.M. Asbury, Phys. Lett. 74B (1978) 27 13) G. Alexander, Y. Gell and I. Stumer, Phyr. Rev. ID6 (1972) 2405 14) A. Cial and C.B. Dover, Phys . Rev. Lett. 44 (1980) 379 15) C.J . Batty, Phys. Lett . 37B (1979) 324