An improved parametrization of the optical potential for pionic atoms

Nuclear Physics North-Holland

A519 (1990) 773-804

AN IMPROVED

PARAMETRIZATION FOR PIONIC J. KONIJN,

NXHEF-K,

C.T.A.M.

DE

OF THE OPTICAL ATOMS LAAT’

POTENTIAL

and A. TAAL’

P.O. BOX 4395, 1009 AJ Amsterdam,

The Netherlands

J.H. KOCH MKHEF-K,

P.O. BOX 4395,RX?9A3 Amsterdam, The Netherlands and Institute for Theoretical Physics,University of Amsterdam, The Netherlands Received

2 July 1990

Abstract: We try to find an improved

phenomenological optical potential for pionic atoms in two steps. First, we perform a new fit of the parameters in the standard optical potentials by using a large data base of 140 pionic atom data. A x2 is obtained that is substantially lower than with any previous parameter set. In a second step, we extend the number of parameters by using a more general form for the terms quadratic in the nuclear density. This extended version yields a reduction of the x2 by about a factor of two. The new fit includes a good description of shifts and widths of the deeply bound 3d orbits in heavy nuclei such as “‘Ta, na’Re, “%, ‘97Au, *“Pb and ‘*‘Bi. The value of the additional parameters in the extended version are difficult to interpret in the context of the chosen form. of the optical potential and indicate that higher-order effects play a role.

1. Introduction There are two main sources of info~ation on the pion-nucleus interaction at low energies: pion-nucleus scattering and pionic atoms. In pion scattering, only a few angular momentum states of the n-nucleus system contribute to the scattering wave function. For light targets and at the lowest energies one presently can measure (T,

=5 20

MeV),

usually

only the central

s-wave

and the more

peripheral

p-wave

contribute significantly. To change the relative importance of these partial waves or to bring in higher I-states one can increase the scattering energy. The situation is different in pionic atoms. After its capture, the pion finally cascades through circular orbits, each having a well defined angular momentum state f = n - I. As the orbits come close enough to the nucleus, the pion feels the strong interaction. This interaction shifts the energy of the pionic level with respect to the electromagnetic Coulomb value and - due to the possibility of pion absorption - leads to an increased width of the level. By measuring the position and width of pionic atom states, we can, therefore, get info~ation on the rr-nucleus interaction for a fixed, discrete angular momentum state at essentially constant energy (on the scale relevant for the r-nucleus interaction). In practice, the absorption process becomes so strong ’ Present address: University of Utrecht, Buys Ballot Laboratorium, 0375-9474/90/$03.50

@ 1990 - Elsevier

Science

Publishers

Princetonplein

B.V. (Noah-Holland)

5,3584 CC Utrecht.

114

J. Konijn et al. / An improved parametrization

as the pion cascades

down that the lowest possible

pion. Only the more peripheral targets

levels are hardly

levels - for example

reached

by the

the 5g and 4f levels in heavier

- are observable.

In a first generation of measurements, see e.g. the review by Backenstoss ‘), the shifts and widths of these more peripheral states for a variety of nuclei across the periodic system were measured. For the heaviest targets, such as *08Pb, no state lower than the 4f state could

be observed.

All lower-lying

states were too broad

to

be extracted from the emitted -y-ray spectrum. These data could be described quite satisfactorily by an optical potential with a few fitted parameters. It was also possible to relate this semi-phenomenological potential to the one needed to describe lowenergy pion-nucleus scattering. This situation changed when it became possible to extract the shift and width of the previously unobserved deeper bound and much broadened levels for many targets, e.g. the 3d level in *‘*Pb [refs. ‘-‘)I. It turned out that these states could not be described by the optical potentials obtained earlier. Particularly disturbing was the fact that the width of these states was smaller than expected. One possible explanation for these ‘anomalously narrow’ widths was that an additional nuclear s-wave repulsion, which is also indicated by scattering data, may expel the pion from the nuclear interior, thereby reducing the r-absorption 6-1o). However, the data base was too small to test this and other hypotheses. Recent

technical

advances, e.g. the use of high-resolution, Compton-suppressed 2-5,‘1-‘3) and the availability of intense rr- beams at pion solid state y-ray detectors factories, have yielded much more accurate data on pionic atoms than previously achieved, both for the peripheral and the low lying states. Given this extensive and accurate data base, we have performed a new fit of the optical potential parameters for pionic atoms. First, we keep the traditional parametrization, but perform a fit where we allow all parameters to vary to determine their optimal values for our data base. Since the nuclei range from light targets with N = 2 to nuclei with a large neutron excess, we then repeat our fits, but use a more general N- and Z-dependence for the terms quadratic in the density. In the following chapter, we first outline the previously used forms of the optical potential for pionic atoms and then describe our extended parametrization. The data set and previous optical potential fits are discussed in sect. 3. We then discuss in sect. 4 the results of new fits to a total of 140 pionic atom data points and present our new sets of parameters for both the traditional and extended forms of the phenomenological optical potentials. Sect. 5 briefly summarizes our results and places them into perspective.

2. The optical

potential

The optical potential used to describe pionic atoms and low-energy pion-nucleus scattering has two components: one part is derived from multiple scattering theory,

115

J. Konijn et al. / An improved parametrization

the other is purely

phenomenological

r-absorption.

By assuming

is important,

one

nucleons,

yielding

can

sum

the repeated

a potential K(r)=

Here /1 is the reduced

and accounts

for higher order effects such as

that only the elementary

s- and p-wave

scattering

of the pion

TN scattering from

the target

of the form 14)

-F

q(r)-V. [

pion-nucleus

ff(r) 1 +$&a(r)

1

v .

mass and

( +z>

q(r) = 1 a(r)=

l+z (

where

we have defined

{h+(r)+ h+(r)1 ,

>

-‘{e,p(r)+c,dp(r)],

(2b)

the densities Mr)

p(r) =p,(r)+p,(r),

= p,(r) -P,(r).

(3)

P,, and p,, are the neutron and proton densities, normalized to N and 2, respectively. The term involving q(r) in eq. (1) is due to the elementary r-nucleon s-wave interaction and b, and 6, represent the in-medium isoscalar and isovector scattering lengths, respectively. Analogously, a(r) contains the elementary p-wave scattering volumes, c,, and c,, respectively. However, this term enters non-linearly into the potential as shown by Ericson and Ericson 14). The parameter .$ depends crucially on the assumed range of the TN interaction, r,,, and of the nucleon-nucleon correlation length of the nuclear ground state, r,. Assuming both to be of zero range, as was done in the original derivation of this non-linear term, yields [ = 1. However, choosing r, = rxN = 0.5 fm yields 5-0.05. Nevertheless, in most fits the value [= 1 has been

used.

the numerator,

It was pointed the denominator

out several

years go 15) that for the term with c1 in

in eq. (1) should

have a different

form, namely

Most optical potential fits we are aware of have not taken this into account. To make a comparison with these earlier fits possible, we have also neglected this and used eqs. (1) and (2) when performing fits with 5 # 0.

2.1. PION

ABSORPTION

The optical potential, eq. (l), discussed so far is real and an important dynamical aspect of the pion-nucleus interaction is still absent: ?r-absorption. To include pion absorption into the optical potential, phenomenological terms are added in both the s-wave and p-wave term in eq. (1). One expects that absorption on a single

776

J. Konijn et al. / An improved parametrization

nucleon is highly suppressed due to energy and momentum conservation. If a stopped pion is to be absorbed on a single nucleon, the nucleon must have an initial momentum of roughly 500 MeV/ c. Therefore, this process is very unlikely and pion absorption occurs mainly on at least two nucleons. One therefore includes a phenomenological term proportional to the square of the nuclear density to account for possible absorption processes such as

a-+(pp)+pn,

93-lf(pn)+nn.

Of course, a n-- cannot be absorbed on two neutrons. 2.1.1. The conventional optical potential. An extension of the low-energy optical potential to include pion absorption is usually of the form shown in eqs. (4a) and (4b):

(4b)

This parametrization assumes that the absorption mechanism has zero range and the two nucleons involved are uncorrelated. When using these additional terms in fits to the data, the complex parameters one obtains will of course not only describe absorption, but also simulate other higher order effects in the pion nucleus interaction. Only the imaginary part of these “absorption parameters” is definitely associated with pion absorption, since this is the only open channel for a pion at this low an energy. Even though not derived rigorously, it has been customary to include the p-wave absorption term into the denominator of the p-wave term. We follow this tradition in our fits with &’# 0 to make comparison with the older fits possible. In some earlier fits, factors ZN/A2 were included, representing an estimate for the number of np pairs on which absorption can occur. In this paper, we will refer to the optical potential defined by eqs. (l), and (4a) and (4b) as the “standard form”. In sect. 4.2 below, we will fit the four real and two complex parameters in the optical potential shown in eqs. (4a) and (4b) to the experimental shifts and widths for 54 nuclei (in total 140 data points). 2.1.2. Extended parametrization of the optical potential. In all fits to pionic atom data up to now, only two overall strength parameters, B, and C,, were used for the absorption. Since our data set contains a sufficient number of nuclei with both 2 = N and 2 # N, we extend the phenomenological parametrization of the absorption terms in eq. (4). (The term “absorption” will be used from now on for simplicity, even though the assignment is unique only for the imaginary part). We will base our fits on a form that allows a more general quadratic dependence on the nucleon

J. Konijn et al. / An improved parametrization

777

densities: q(r)=

l+z (

{bop(r)+b,Sp(r)}+ >

(

1+m, 2mN >

x {&p2(r) + 4 p(r)&(r) + B2&(r)*l,

( +z>

(Y(r)= 1

-l{cop(r)

+c16p(r))+

(54 --1

( +z > 1

N

(5b) where Sp has been defined in eq. (3). The &, and C, parameters are now responsible

for the absorption

that 6p = 0. These strength

the absorption

parameters

thus describe

for the case

on a target with

N = 2, assuming that the shape of the densities is the same. The remaining parameters play a role, e.g., for nuclei with N > 2. Another way in which the four additional complex parameters will contribute, even if N = 2, is when the shapes of the neutron and proton distributions are different. If 6 absorption was indeed a zero-range process involving only two uncorrelated nucleons, these new parameters would allow us to identify the absorption strength for np and pp pairs from the coefficients of the corresponding densities. Furthermore, the fitted parameters would in this case be correlated such that the term quadratic in the neutron densities remains real, since there is no absorption on an nn pair. Our fits shown below will clearly indicate that such a simple interpretation is not possible. There have already been earlier, more limited attempts along these lines which tested the dependence of the absorption term on neutron and proton distributions. Carr

et al. 16) used an extended

parametrization

of the absorption

term to fit low

energy pion-nucleus scattering for several targets, extracting an energy-dependent C, parameter. Similarly, De Laat 4*5)showed that an improved fit of the “anomalous” 3d shifts and widths in heavy mesic atoms could be achieved by using B, and C, absorption terms. In this latter work, it was also shown that taking the neutron density different from the proton density 17) was important. Experimentally, very little information is available that allows one to significantly improve the phenomenological description of pion absorption, such as the dependence on the nuclear density. The ratio R of absorption on np pairs versus pp pairs is not well determined. For example, for “C Nordberg et al. 18) obtained R = 2.5 f 1.0, whereas Ozaki et al. 19) found a value of R = 5.0* 1.5 and Lee et al. *‘) quote R = 8.8f 1.3. The fact that this ratio is larger than the statistical expectation of np versus pp pairs indicates preferred absorption on np pairs. The quasi-deuteron absorption (QDA) model in fact assumes that absorption occurs in the nucleus on a correlated np pair in a relative s-wave, coupled to isospin T =O. Of course, the

778

J. Konijn et al. / An improved parametrization

absorption mechanism can also involve clusters of more than two nucleons, leading to a more complicated dependence on the nuclear matter distribution. The question of how many nucleons are involved in pion absorption is the subject of many investigations with low energy pion beams. A study of ten absorption channels with stopped pions on 6Li by coincidence detection of two outgoing nucleons was performed by DSrr et al. *‘) and the quasi-deuteron process was found to be responsible for 72.5% of the absorption strength. In another absorption experiment on 6Li with stopped pions, Isaak et al. 22) found that 51% of the absorption strength was due to quasideuteron absorption. Three other channels with outgoing np, nd and nt particles, each having a strength of about 10% were measured. Channels involving the coincidence of two charged particles leaving the nucleus after pionabsorption are weak (~1%). For heavy nuclei, the QDA may be the dominant process again. This is indicated, for instance, by Isaak et al. *“) and Shinohara et al. 24) who could explain their stopped pion absorption data for Ni isotopes and *09Bito a considerable extent by assuming quasi-deuteron absorption in the nuclear surface region. Clearly, the parametrization in terms of single nucleon densities can at best incorporate such higher-order processes in an average way through the fitted parameters. 2.2. CALCULATION

OF THE

LEVEL

SHIFTS

AND

WIDTHS

To calculate the energy levels of pionic atoms, the optical potential (1) and (5a, b) is used in a modified Klein-Gordon equation: {A + [(E - V’(r))‘-

p21Mr) = 2cLVdr)lcr(r),

I’s(r), eqs.

(6)

where E is the pion energy, including its rest mass m,. The reduced pion mass is P = m,[ 1 + (m,/ M)]-’ with M the nuclear mass. V”(r) is the Coulomb potential, given by v(r)=--a

I

fidr’,

with a the fine-structure constant and p,(r) the proton density. Since the optical potential V,(r) contains complex parameters, the eigenvalues of the Klein-Gordon equation (6) for a given pionic orbit (n, 1) are also complex & = Re (El) + i Im (En,) , (8) where the imaginary part of the eigenvalue yields the strong absorption width, r,, = -2 Im I?,,. The theoretical strong interaction energy level shift, E,~, is defined in this paper as (9) E,,~= Ez,l- Re (E,,) . The energy levels, Ez,, for a point charge with positive charge 2 are given by [see e.g. Scheck “)I Ez=rfi

{

I+

(

za

n-~_5+((~+f)2_(ZLy)2)1/2

2

-l/2

)l

*

(10)

779

J. Konijn et a.l / An improved parametrization

Using contains

for E,,!, a negative

this convention both strong

interaction

What is experimentally the relevant

orbits

observed

that

to a repulsive

difference

interaction,

shift. It

effects.

these

between

two orbits. For

are two circular

orbits,

of the X-ray, E=P nt,,

compared

corresponds

is the energy

feel the strong

(n + 1, I+ 1) and (n, 1). The energy

is therefore

value

as well as the finite-size

to the theoretical

Re (E,+,, ,+r)+ EZ,

/+1-n,

I,

(11)

prediction 1+1- (Re (E,,,) + EZ8

.

(12)

The contributions Ezr are due to effects such as vacuum polarization, orbital electron screening, Lamb shift and nuclear polarization as discussed in refs. 4*1’).These small corrections are calculated in perturbation theory. In practice, the strong interaction effects and finite size effects contained in E,,,,, ,+1 are down by about two orders of magnitude compared to the lower one and therefore the experimental shift of the lower level is defined exp &“,I

by

Ee,X,,,,+~,,,,,

-

W:+,,,+wn,,+

J%,,,+,

-

JCf,?

+

C’$‘I,

/+I

.

(13)

3. Data set and previous fits 3.1. SELECTION

OF EXPERIMENTAL

DATA

The experimental data base on the strong interaction shifts and widths used in our fits have been selected among the available data from literature for nuclei with 2 2 5. This data set contains a total of 140 data points. It includes recent pionic atom experiments on the pionic 1s and 2p levels 12) in 24Mg, *‘Al, 28Si, 93Nb, natR~,

and on the pionic

3d and 4f levels ‘) in ‘*ITa, “atRe, natPt, r9’Au, “*Pb,

209Bi, and

237Np. The selected pionic data are listed in appendix A, tables A.l-A.8. All experimental shifts are given with respect to the point nuclear Coulomb value, i.e. the shifts include the finite size effect. In the case of more than one result for a given level, the average experimental value is shown. If necessary, the errors in the strong interaction shifts in a few cases were adjusted to be 25% of the corresponding experimental widths. The reason for adjusting the errors in the shifts is that in previous analyses of the experimental pionic X-ray spectra systematic errors were ignored or underestimated. As a consequence, one obtains a more balanced weight in the fits among the experimental shifts and widths. The used errors are never smaller than the ones given in the corresponding original publications.

3.2. PREVIOUS

OPTICAL

POTENTIAL

FITS

For all the levels in our data base, we have calculated the shifts and widths predicted by previous optical potentials. The results are shown in tables A.l-A.8

780

J. Konijn et al. / An improved parametrization

of the appendix

for five often used parametrizations:

and Seki and Masutani parametrizations, defined

It is evident

26), Batty et al. 27*29)

quality

we show the total x2 for the shifts and the widths i.e. the quadratic

as x2 = (yexp - yca’c)2/&p,

experimental

by Tauscher

28). To give an idea of the general

differences

of these earlier of each level, divided

by the

error. Also shown is the sum of all x2 for the entire data set, x*(total). that the parameter

set by Batty et al. 27) provides

the lowest x2 value.

One should notice that the experimental data of tables A.l-A.8, used as input for our fits, differ from the data quoted originally by these authors, who often give the shift with respect to the complete electromagnetic nuclear energy of the state and not to the energy of the point nucleus. For the theoretical results shown in the tables, we have used different proton and neutron densities whenever this information was available “). The density parameters are given in table A.9 of the appendix. The proton and neutron density function are assumed to be represented by a two-parameter Fermi distribution pp(r)=N0(1+exp[41n3(r-c)/t])-‘, where No is the normalization constant chosen such that the volume integral equals the total number of protons or neutrons; c represents the half-density radius of the distribution and t the skin thickness. The Fermi parameters for the proton distribution 30) have been checked to reproduce the experimental muonic X-ray transition energies. For the calculation of the shifts and widths, a modified version of the computer code MESON 31) was used, based on the optical potential described in sect. 2. This

code

uses the Numerov

method

for integrating

the Klein-Gordon

equation with the optical and Coulomb potential. The code calculates the strong interaction shift with respect to the point Coulomb value and includes finite-size and deformation effects. The higher-order corrections (see sect. 2) and deformation effects 25) are calculated in perturbation theory. We will briefly review the situation for each level (n, I) separately. 3.2.1. Pionic Is leuel. The Is-level shifts, table A.l, are all negative, corresponding to a net repulsion. All level shifts of the higher orbits are positive, except for the recently measured shifts of deeply bound pionic 2p orbits 12) in 93Nb and “atRu. For the 1s levels, the predictions

of Batty et al. *‘) is the best describing

both shifts

and widths equally well. Common to all potentials are the poor values given for the Is-level shift in 180, and in the odd-A isotopes “B and 13C, see table A.l. The calculated shifts deviate, however, at most 10% from the experimental values. The shifts increase steadily with the charge 2 and isotopic effects seem hardly to be present. In contrast, the values for the width, table A.2, are more sensitive to isotopic lower than those for the effects. The widths for 13C, I80 and 22Ne are significantly This trend is correctly reproduced isotopes ‘*C, I60 and “Ne, respectively. potentials shown here. For the Is-level width in 160 and “Ne, the calculated are too small, up to about 25% below the experimental values.

by all values

J. Konijn et al. / An improved parametrization

781

The predictions for the Is-level shifts in “0, “B and i3C claim together 35% or more of the x2( &is), whereas the predictions for the Is-level widths in ‘60 and 20Ne contribute for over 45% to the x*(T,J for the 1s widths. 3.2.2 Pionic 2p level. For this level, the most data are available. shifts, shown in table A.3, are all attractive by Krell and Ericson

with two exceptions.

32), one expects the attractive

by the s-wave repulsion

for deeply

bound

pionic

The pionic

As already

p-wave interaction

2p

predicted

to be overcome

2p orbits in the region

of 2 = 36.

This is confirmed by the recently measured 2p shifts 12) in z:Nb and “i:Ru, &2p= -11 f 3 keV and -48 f 7 keV, respectively. This crossing over from net attraction to repulsion is correctly predicted by the potentials of Tauscher 26) and Batty 29). The data set contains several isotope chains. The level shift in the light isotopes does not change significantly, while for the heavier nuclei it decreases as N increases. The widths show no clear uniform

trend:

some increase,

some decrease

with the neutron

number. For Ca, the larget number of isotopes is available, showing an essentially constant width as N grows from 20 to 28 neutrons. This feature is only reproduced by the potential of Batty and Tauscher (5 = l), which however predicts the magnitude too small by 20%. For the x2 values of all 2p cases one finds that it varies considerably for the different parameter sets, ranging from about 300 to 1750 for the 58 2p data points. A detailed comparison “) of the calculated shifts and widths with the experimental ones shows that the majority of calculated values underestimates the experiment between 10 to 50%. The 2p-level shift in e.g. “0 is predicted too small for all sets, contributing 20% or more of the x2( sZp) for the 2p shift of each set, the same holds for the widths in isO, 24Mg and 26Mg with respect to x~(T~~), contributing about 40%. 3.2.3. Pionic 3d level. As can be seen in table 1, the 5 = 0 parameter set by Tauscher 26) yields predictions for the 3d-shifts and widths with the largest x2. The potential of Seki and Masutani 28), on the other hand, provides a poor description of the 3d shifts, while it does exceptionally well for the 3d widths. It underestimates the 3d shifts of the heavier elements 18’Ta, “atRe, ““‘Pt, 19’Au, ‘08Pb and ‘09Bi - the often discussed deeply bound states - by more than 60%. For this set 28) about 80% of the x2 value for the 3d level results from the shifts. The 3d widths of these heavier elements, which includes the “anomalously narrow widths”, are clearly overestimated by at least 35% for the sets in refs. 26,27,29),where about 80% of the x2 value is due to the widths. 3.2.4. Pionic 4flevel. All previously published parameter sets with 5 = 1 predict the 4f shifts worse than the 4f widths. Therefore, only about 25% or less of the x2 value shown in table 1 is due to the widths. Seki and Masutani 2”) predict all 4f shifts 15% to 40% smaller than experiment. The 4f-level shift in 237Np is only predicted well by Tauscher 2”). The calculated value ~~r(*~‘Np) claims over 40% of the x2(s4r) for the other sets with [= 1, while 40% or more of the x2(r4r) for the 4f widths of each set with t= 1 is ‘97Au.

782

J. Konijn

et al. / An improved parametrization

4. Parameter fits Given proceed

the set of 140 shifts

to fit the four real parameters,

parameters

B,, and C, for the standard

the extended complex

and

parametrization,

parameters,

Since a general we did not fit all Furthermore, for comparison with with 5 = 1, since

4.1. SELECTING

widths

for pionic

atoms

with A> 9, we now

b,,, b, , co and c, , and the complex form of the optical

which we then perform,

potential.

absorption

For the fit with

we have an additional

four

B1,* and C,,*.

search in such a large parameter space is rather time consuming, parameters at once, but according to the strategy outlined below. our main fits we did not allow 5 to vary, but kept 5 = 0. To allow the optical potentials used in the past, we also performed a fit this has been the most common choice.

A STARTING

POINT

We started the fit to the experimental values with only a limited set of data. To fit the s-wave parameters, b, and BO, we used the shifts and widths of the 1s levels of the seven nuclei with N =Z. For this first step, the less important p-wave parameters were kept fixed at the values from Batty et al. *‘). We then added the twelve 2p data for nuclei N = Z and fitted the parameters c0 and C,. The parameters obtained in this way were then kept fixed in a fit of b, and c, to the remaining data for s- and p-states. For the fits with the extended parametrization, starting points for the fit of the additional absorption parameters were obtained in two steps by first only using the additional data for nuclei with AS 104 to obtain starting points for B, and C, , the remainder of the data for finding starting values for Bz and C,.

4.2.

NEW

PARAMETER

SET

FOR

THE

STANDARD

FORM

The parameter sets obtained as explained above were used as the starting point for a fit where all strength parameters in eqs. (4a) and (4b), i.e. bo, b, , &, co, cl, and Co, were allowed to vary. The values resulting from this general search are shown in table 1. The errors assigned to the parameters have been calculated from the covariance matrix. From the total x2 we see that a substantial overall improvement is achieved. Even compared to the previous best fit by the potential of Batty et al. *‘), the total x2 is still reduced by about a factor of two. When we look at the fits to the shifts and widths of the different states, we see that the x2 per data is rather uniform, with the exception of the 3d level. Comparing again to the description by the potential of Batty, we see that the major part of the improvement is due to the better fit of the shifts. The

widths are also better fitted by the new parameters, but the improvement large. To allow a more direct comparison with the previous potentials,

is not as we also

11.16

X&f)

X&d)

X&P)

2 Xytal pw degreeof freedomXtotalper*atapoint 19.56

-0.0293 (5) -0.078 (7) 0.227 (8) 0.18 (3) 0. 0.0428 (15) 0. 0.076 (13)

Tauscher 26) f=l 119711

4.50 6.51 19.53 13.60 3.00 20.95 2.69 1.12

e r E r 8 r c r

-0.0296 (5) -0.077 (7) 0.172 (8) 0.22 (3) 0. 0.0436 (15) 0. 0.036 (13)

Tauscher 26) 5’=0 [1971]

5.21 5.61 37.23 22.40 12.35 35.28 1.71 3.13

B, B, Co Cc

X:tr>

b, co c1 Re Im Re Im

bo

Parameter

TABLE 1

5.79

9.92 2.50 6.53 3.92 6.19 9.28 5.70 0.92

-0.017 -0.13 (2) 0.255 (3) 0.17 -0.0475 0.0475 0. 0.090 (5)

Batty 27) 5=1 [ 19781

12.24

11.25 6.58 12.23 7.08 32.51 3.22 22.06 5.90

0.003 (8) -0.143 (6) 0.21 0.18 -0.15 (4) 0.046 (3) 0.11 (1) 0.09 (1)

Seki 28) 5=1 [ 19831

-

8.29

3.35 3.20 8.44 11.47 1.71 22.16 6.60 1.17

-0.023 -0.085 0.21 0.089 -0.021 0.049 0.118 0.058 (25) (25) (2) (8) (8)

(5) (5)

Batty 29) ‘$=l [1983]

3.30 3.11

2.62 2.10 1.88 2.84 3.24 8.03 3.88 0.97

0.024 (5) -0.090 (5) 0.272 (16) 0.107 (15) -0.261 (28) 0.0552 (13) -0.26 (5) 0.0640 (25)

Present fit [=O standard potential

Parameter sets for the standard form of the optical potential and resulting x2 per data point

(5) (5) (20) (21)

2.99 2.82

6.26 3.64 1.09

3.19

2.08 1.76 2.35

2.59

0.0546 (12) -0.14 (7) 0.105 (3)

-0.265 (26)

0.025 -0.094 0.273 0.184

Present fit &=l standard potential

--f m* --I mw -3 mrr -3 m, -4 m, rni4 m,--6 --6 mw

Units

!I R 9 a. 2 =: P

a B IL b

2

P

a 0 w ’

h k b J La:

784

J.

performed

Konijn et al. f An improved parametrization

a fit with 5 = 1 in eq. (1). This results

with our 5 = 0 parameter set. When we compare the old and new parameters changes.

The isoscalar

previous within

fits [except the range

significantly

s-wave parameter

in magnitude.

values,

better

‘“)I; the isovector

but both

The absorption

(10%) fit than

in table 1, we see several important

b, now has the opposite

for Seki and Masutani

of previous

in a slightly

Re B,, and

strength,

sign compared parameter

Im B, have

to

b, falls increased

Im I$,, is about 20% larger than

previously. If perturbation theory applied, a larger Im B,, would lead to an increase in the 1s widths for light nuclei by a corresponding amount. This is not the case, showing that higher order effects are important. We also see that the set of s-wave parameters, b,, b, , and BO, is essentially independent of the form chosen for the p-wave part of the potential, i.e. is the same for 5 = 0 and 6 = 1. As far as the p-wave parameters are concerned, the isoscalar p-wave scattering volume, cO, is now much larger and almost the same for both values of 5. The remaining two p-wave parameters, c, and Co, on the other hand are very different for the fits with 5 = 0 and 6 = 1. As was already mentioned, the largest x2 in table 1 occurs for the new fits for the widths of the 3d level. The data set of this state contains the anomalously narrow levels in “‘Ta, natRe, “atPt, 19’Au, *“Pb and *09Bi. As we can see in table A.6, the improved fit still cannot reproduce these widths. The x2 we obtain is approximately a factor of two larger than the parameter Masutani

28), which does remarkably

for the 3d widths set of Seki and

well for the 3d widths but yields a poor overall

description of the data. Therefore, even though our new parameters yield a substantial improvement for the description of the entire data set, these widths of the deeply bound 3d states remain “anomalous” within the framework of the standard optical potential.

4.3. FITS WITH

THE

EXTENDED

PARAMETRIZATION

We now describe the results of a fit to the data set with additional absorption parameters B,,* and C,,* given in eqs. (5a, b). The starting point for this search in a very large parameter space is described in sect. 4.1. Having more parameters resulted,

of course,

in an improved

fit. As can be seen in table

2, the total

x2

decreases by a factor of about two compared to the best fit we would achieve with the standard form for the optical potential. (Again, we also performed the fit with 4 = 1, the most common choice in previous parametrizations. The x2 for 5 = 1 is slightly lower than for 5 = 0). As the contributions to the total x2 from the different states in table 2 show, the fit is now very uniform, improving somewhat as one goes to higher (n, I). When searching in the large parameter space with .$=O for the smallest x2, we found two sets that differed only little as far as the total x2 is concerned. They are labelled “fit A” and “fit B” in table 2. In both fits, five data points are responsible for a very large fraction - about a quarter - of the total x2. They are T(ls) in 14N,

C,, C, C, C, C, C,

Re Im Re Im Re Im

B,

B,

B,

B,

B,

B,,

Cl Re Im Re Im Re Im

b, co

bo

Parameter

(2.00) 1.56 (1.80) 1.40

(2.09) 1.54 (1.84) 1.36

1.32 1.32 1.36 1.83 1.36 1.10 0.67 1.29

(1.32) (2.39) (1.36) (2.37) (1.36) (2.11) (2.57) (1.29)

(1.61) (2.46) (1.31) (2.41) (1.35) (2.11) (2.23) (1.28)

1.61 1.39 1.31 1.86 1.35 1.17 0.66 1.28

-0.25 (8) 0.062 (5) -1.2 (5) 0.18 (13) -0.55 (8) 0.950 (5)

-0.002 (6) -0.071(20) 0.280 (24) 0.33 (9) -0.13 (3) 0.0566 (17) -0.18 (11) -0.06 (7) 0.95 (3) -1.0741(17)

-0.002 (6) -0.070 (19) 0.270 (20) 0.34 (8) -0.13 (12) 0.0552 (15) 0. -0.07 (6) -0.158 (3) -0.8460 (15)

-0.209 (8) 0.063 (4) -1.7 (4) 0.12 (11) 1.9 (6) 0.949 (4)

fit B

fit A

t=o fits

1.49 1.37 1.22 1.78 1.24 1.12 0.61 1.35 (1.90) 1.48 (1.73) 1.33

(1.49) (2.45) (1.22) (2.31) (1.24) (2.19) (2.05) (1.35)

-0.10 (29) 0.105 (5) -2.3 (4) 0.06 (17) 2.41 (29) 1.786 (5)

-0.000 (5) -0.073 (5) 0.284 (26) 0.43 (8) -0.142 (26) 0.0553 (15) 0. -0.05 (6) 0.142 (26) -0.9300 (15)

fit A

[=l

fits

1.24 1.28 1.23 1.79 1.23 1.09 0.65 1.35 (2.02) 1.47 (1.79) 1.30

(1.24) (2.37) (1.23) (2.30) (1.23) (2.22) (2.37) (1.35)

-0.14(11) 0.100 (6) -1.6 (7) 0.31 (20) 0.08 (11) 0.695 (6)

-0.001 (6) -0.062 (20) 0.29 (3) 0.38 (11) -0.14 (3) 0.0570 (18) -0.14 (23) -0.08 (7) 0.58 (3) -0.9463 (18)

fit B

1.38 1.07 1.19 1.24 1.04 0.93 0.61 1.44 (1.88) 1.30 (1.65) 1.14

(1.38) (2.01) (1.19) (1.87) (1.04) (2.58) (1.88) (1.44)

0.8 (5) 0.63 (25) -5.1 (3.3) 1.2 (9) -1.3 (5) -1.46 (25)

-0.007 (5) -0.061 (16) 0.34 (6) 0.66 (25) -0.102 (26) 0.0561 (18) -0.19 (18) -0.09 (6) 0.641 (26) -0.7580 (18)

5=3.4(4) Units

Fitted parameter sets. The values given for ,y* between parentheses at the bottom of the table are calculated for the data set with 140 data points (see text)

TABLET

786

J. Konijn et al. / An improved parametrization

r(2p) in 48Ca, r(2p) in 75As, r(3d) in 93Nb and c(4f) in 237Np. The main difference in the x&, between fits A and B is that in fit B the x2 for these five points is somewhat higher, while the remaining 135 data points are fitted correspondingly better. We show in table 2 the total x2 per data-point and, between parentheses, the x2 including these five points. It is not clear why these points are particularly problematic. None of these levels belong to the puzzling “deeply bound” states discussed earlier. In looking at the values obtained for the individual shifts and widths in tables A.l-A.8 in the appendix - only the results for the fit B are shown - we see that most deeply bound states are now quite well described. For example, the shifts and “anomalously” narrow widths of the 3d states in target such as ‘*‘Ta and heavier (see tables A.5 and A.6) are now fit quite well. The same is also true for fit (A). We have repeated the fits with the p-wave parameter 5 = 1. No significantly different x2 was found. Again the same five data points caused about 25% of the total x2. Also for 5 = 1 a second parameter set (B in table 2) yielded a worse description of these problematic points, but the remaining 135 points were fit somewhat better. Again the deeply bound states are described quite well. What these fits A and B demonstrate is that the minimum in the extended parameter space is not unique and well defined. One can find x2-equivalent sets, that differ quite strongly in detail. For example, for 5 = 0 the value of ReC, changes from 1.9 in set A to -0.55 in set B. As the covariance matrix shows, this change is strongly correlated with the value of ReB,, which has a value of -0.16 in set A and 0.95 in B. Below, we will discuss the value of the parameters in the extended fit in more detail. In looking at the parameters in table 2, we notice that the s-wave scattering length, bO, is now close to zero as is the case for the free pion-nucleon isoscalar length. As far as the real parameters bo,, and co,, is concerned, the difference between the fits with 5 = 0 and 5 = 1 is very small. As one expects, it is mainly the parameters nonlinear in the nucleon density for the p-wave part that are most sensitive to the value for 5. As in previous fits, we find again a repulsive s-wave term Re &,, corresponding to a central strength of about 30 MeV. This is less than with the fit based on the standard form of the optical potential. The imaginary parts Im B,, and Im C, are very similar to our fits in table 1. The most interesting aspect about the extended parameter set are of course the values of the additional parameters B,,, and C,,,. If the absorption indeed was a process of zero range, involving only two uncorrelated nucleons, the extended parametrization would allow us to identify the np and pp absorption strength: If one rewrites the quadratic terms of the s-wave part of the potential as

(Bo-B,+B2)P~(r)+2(Bo-B2)pp(r)Pn(r)+(Bo+B1+B2)Pfi(r)

(15)

and analogously for the p-wave part, one could identify the absorption strength for a pp and np pair by looking at the coefficients of p”,(r) and pP( r) pn( r), respectively.

J. Konijn et al. / An improved parametrization

Inserting possible

the fitted parameters interpretation.

with absorption,

181

from table 2, one sees immediately

The sign of some of the imaginary

e.g. is negative

that this is not a

parts is not compatible

for the s-wave part. Furthermore,

the term propor-

tional to p;(r) is not real as one would expect if the above assumptions since it is not possible to absorb a rTT-on a neutron pair. Our fit parameters, can

only

important

be taken

as an indication

role: e.g. nucleon-nucleon

that

higher

correlations,

order multistep

processes

were true, therefore,

are playing

processes

an

or absorption

on larger nucleon clusters. The presence of a higher order effect would lead one to a different form for the optical potential, in particular for its density dependence. Of course, one can to some extent simulate these effects for the limited range of nuclear densities probed with pionic atoms by a certain choice of parameters in the optical potential. Without a model it is not possible to sort out which effect is responsible in our case for e.g. the sign and magnitude of the parameters B1,2 and C1,2 in our extended potential. Finally, to see if the parameter 5, which has been kept fixed at ,$ = 0 and 5 = 1, can be useful in simulating higher order effects, we also performed a fit where 6 was allowed to vary. As shown in table 2, a slightly improved x2 was obtained with 5=3.4 a value that is very large compared to the value of 0.05 < 5 < 1 one expects if it was due to the Lorentz-Lorenz effect it was meant to describe originally 33). The results in tables A.l-A.8 show that the extended form of the optical potential with its N- and Z-dependence, does not lead to an improvement of the description of the shifts and widths in a set of isotopes. For example, an essentially constant value for r(2p) in the various calcium isotopes, as experimentally observed, was already obtained with the standard form of the optical potential. With the extended form, the value for 44Ca begins to get smaller and is far too low for 48Ca. What the extended N- and Z-dependence does help to achieve is a better global fit to a large range of nuclei, from light nuclei with N = Z to heavy nuclei with N > Z. One trend is clearly indicated by the extended parametrization of the “absorption” terms,

the imaginary

for the proton

parts of the parameters

and neutron

B

densities,

+(N-Z)

0 7

B and C. If we assume

we can rewrite

B,+{y)2B2]p2(r)

the quadratic

equal shapes

terms as

(16)

for the s-wave part and analogously for the p-wave part. The fitted parameters in table 2 show that for a fixed A, the s-wave imaginary part decreases as we start e.g. from N =Z and then convert protons into neutrons. In contrast, the p-wave imaginary part increases as the neutron excess grows. As table 2 shows, the values for the parameters B1,2 and C1,2 are very large when compared to B, and Co, but through the powers of (N - Z)/A their influence is somewhat reduced. Still, if one considers a neutron rich target with (N - Z)/A of about 0.25, the imaginary part

788

J. Konijn et al. / An improved parametrization

of the s-wave optical potential decreases by about 30%, while the p-wave part increases by ca 50%! The changes in the real parts can be even larger. We are clearly not dealing

with a situation

small correction Rather

where the additional

terms relative

than admitting

we have also performed

parameters

B,,2 and C,,* represent

to the N = Z case.

the parameters several

to vary freely in order to minimize

fits with restrictions.

the x2,

In one case, the imaginary

parts of the p”(r) coefficients were constrained such that there was no ‘absorption on neutron pairs’, i.e. the coefficient of the pi(r) was kept real. This fit yielded a 2 xper data-pointof 2.2, about halfway between the best fits with the standard and extended forms. Other fits, where an nn term is absent altogether yield a &, data_pointslightly lower, but with an Im B,, and Im C,, that is negative. The trend of decreasing s-wave absorption gets of course amplified if we apply the ‘extended version’ of the optical potential to I= 0 states in targets with a large neutron excess. This is done in table 3 for states such as the 1s orbit in 48Ca and *“Pb. Of course, for the 1s state in *‘*Pb the Bohr radius is smaller than the nuclear radius and it is unlikely that any of the potentials, which were fitted to mesic states that mainly probe the nuclear periphery, will apply to states with such a large overlap between pion wavefunction and nuclear density. The examples shown in table 3 are therefore only meant to demonstrate the great sensitivity of these very deeply bound states to the optical potentials and to see how interesting it would be to obtain information about these states. For the strong interaction shifts there is a distinct difference between the extended potential and the standard one for 2”8Pb but not for the light isotopes 40Ca and 48Ca. For 48Ca, the 1s width predicted by the extended parametrization is now almost a factor two smaller than for the standard form. For 2osPb, of course, the situation is even more extreme: the extended parametrization predicts for the 1s state a width of only about 70 keV, far below the value obtained from any of the standard potentials. Another example for the increased sensitivity of these states to the form of the optical potential is the variation one sees among the predicted values for the T(ls), r(2s) and r(3s) widths for the extended potentials. It is practically impossible to reach these low lying states through the cascade of the pion after its capture into a high Bohr orbit of a nucleus. However, methods of directly producing a 6 into a bound state, thereby bypassing the cascade, have been proposed long ago 34-39). The first suggestion by Tzara 34) was to use pion photoproduction, while the (p, rr-) and more recently the (n, pm-) reaction have been discussed by Emery 37) and Toki et al. 39), respectively. A more effective way is probably to produce the pion recoillessly in orbit, only possible under special kinematic conditions 38) e.g. by the (p, 2pY) reaction. Clearly, these direct production methods also make it possible to study non-circular orbits, e.g. the 3s state. Such states would be difficult to study in the atomic cascade after capture, which involves mainly circular orbits. In table 3 we also give some strong interaction shifts and widths for non-circular pionic states, again showing a large model

dependence.

J. Konijn et al. / An improved parametrization

789

TABLE 3

Strong-interaction The * indicates

Type of potential

shifts, widths and that the Klein-Gordon

5

Binding ^^n..l.. ellelgjy pionic level r,._,r, LKC”J

binding energies calculated with the new parameter sets. equation with the point Coulomb potential does not provide solutions for s-states Shift with respect to ,ll:ArL **1UL11 pointnuclear [keV] value

0 1 0 1 3.4 1

B,(ls)

0 1 0 1 3.4

“‘Pb standard standard extended extended extended Toki et al. 39)

standard standard extended extended extended Toki et al. 39)

-1093.2 -1093.9 -1097.4 -1097.0 -1097.9 -1108

-1015.0 -1014.2 -1026.5 -1026.5 -1026.7 Uls)

0 1 0 1 3.4 1

0 1 0 1 3.4 1

0 1 0 1 3.4 1

Uls)

58.2 57.2 32.0 32.3 34.9

B,,(2s)

r(2s)

-1602.6 - 1602.6 -1633.3 -1627.1 -1636.3 -1642

-433.8 -433.1 -429.6 -430.1 -429.2 -416

137.9 133.6 12.5 18.8 37.2 183

B,(~P)

-512.3 -513.1 -500.7 -501.8 -500.6

-372.5 -372.6 -372.6 -372.6 -372.6

* * * * * *

* * * * * *

r(3s) 58.7 56.9 5.4 7.5 17.8 78

-372.7 -372.0 -373.0 -373.0 -373.0 -373

E(IS)

r(ls) 474.5 457.5 63.4 88.9 118.0 632

U3s) standard standard extended extended extended Toki et al. 39)

81.9 82.0 93.8 93.4 95.3 105

-6784.8 -6776.7 -6923.9 -6896.7 -6932.3 -6959

-2891.3 -2889.0 -2954.1 -2941.6 -2960.0 -2962

D=Vl

* * * * * *

Shift with respect to pointnuclear value

b+‘l B&P)

“Ca

standard standard extended extended extended

D-N

Width

D=Vl

40Ca standard standard extended extended extended Toki et al. 39)

Binding pionic level

WP)

1.51 1.53 1.65 1.65 1.60 1.7 WP)

1.51 1.50 1.29 1.29 1.26

&VP)

1.75 1.72 1.74 1.74 1.73 1 42P)

1.06 1.07 1.14 1.15 1.08

BA2p)

r(2p)

&(2P)

-5082.2 -5079.3 -5138.0 -5132.6 -5143.3 -5162

321.8 311.7 153.9 156.3 158.6 410

-1537.2 -1540.1 -1481.6 -1486.9 -1476.1 -1441

B,(~P)

r(3p)

E(3P)

-2383.2 -2382.2 -2407.6 -2405.5 -2411.6 -2418

108.2 104.8 51.7 52.0 54.2 410

-548.4 -549.4 -524.0 -526.2 -520.0 -509

B,(~P)

r(4p)

E(4P)

-1381.6 -1381.1 -1393.7 -1392.7 -1395.9 -1408

49.2 47.7 23.7 23.7 25.0 71

-248.0 -248.4 -235.8 -236.9 -233.6 -226

J. Konijn et al. / An improved parametrization

790

5. Summary and conclusions We have in this paper tried to improve the optical

potential

for pionic

atoms.

the phenomenological

For this purpose,

parametrization

of

we used 140 experimental

shifts and widths to find the set of optical potential parameters that yielded the best fit. Our data basis is larger than used in any of the previous fits and included states ranging standard

from the 1s state in l”B to the 4f state in 237Np. We started form for the optical

potential

by using

the

and were able to find a set of parameters

that yielded a significant improvement in the overall description by at least a factor of two over any older fits. A x& data-pointof about 3 was obtained. As in most earlier fits, we find that the part nonlinear in the nuclear density introduces a strong s-wave repulsion. The deeply bound states, such as the 3d state in 18’Ta, are not well described by this new parametrization. In particular, the widths still are predicted too broad. In the context of this standard form of the optical potential these states, therefore, remain a puzzle with their “anomalously narrow” widths. Nevertheless we think that our new fits using the standard a substantial improvement in the description

form of the optical potential represent of pionic atoms for a large variety of

states. Since our data base in addition to many targets with iV = 2 also contains a large amount of data for nuclei a considerable neutron excess, we proceeded with a more exploratory and speculative fit. We repeated the fit with an extended parametrization of the terms quadratic in the density. The new parameters play a role when N > 2 (or if the shapes

of the neutron

and proton

densities

are different).

A fit with this

larger number of parameters yields another improvement in the fit to a x’,~~data-point of about 1.4-1.1, depending on the value of 5. Part of the improvement is that now also the deeply bound states are described quite well. However, the value of some of the “absorption” parameters one obtains is such that any interpretation in terms of absorption on (uncorrelated) np or pp pairs is precluded. Furthermore, the unexpected magnitude of the parameters for the nonlinear terms leads to rather large changes in the potential for nuclei with the same A, but differing in their number of protons and neutrons. What the extended parametrization does achieve is a global improvement of the fit: one gets a good description for states in light nuclei, mostly N = 2, as well as for levels in heavy targets with a considerable neutron excess. What the potential with its extended N- and Z-dependence does not achieve is a better “local” description, e.g. the isotopes for a given element, such as for 40Ca-48Ca. Our fits have shown that a substantial improvement over the previously available phenomenological description within the context of the standard form of the optical potential is possible by using a large set of data and performing a search in the large parameter space. This can get one to a x2 per data-point of about 3. Further improvement is possible by extending the N and 2 dependence of the potential, but the value of some of the parameters cannot be interpreted in the context in which one derived the form of the phenomenological potential. Higher order effects,

J. Konijn et al. / An improved parametrization

791

e.g. nucleon correlation, preferred absorption by pairs in specific angular momentum states or by clusters with more than two nucleons etc., must be included in the potential if one wants to describe such a large variety of data with greater accuracy. These higher order effects will lead to a different density dependence of the optical potential. One can simulate these effects to some extent with the parameters of the present optical potentials, which contains terms linear and quadratic in p(r) for the limited nuclear densities probe by pionic atoms. Even though a fit with smaller x2 can be obtained with the extended form of the optical potential, the value of the best-fit parameters tells us that one should use a different basic form if one intends to improve beyond the standard parametrization in a meaningful fashion. Such a different form has to be motivated by theoretical models of e.g. the absorption mechanism. Clearly the direct production of negatively charged pions into very deeply bound orbits, which cannot be reached through the standard atomic cascade, will be very important for further studies of the low energy pion-nucleus interaction. They will allow us to test the presently available optical models under quite different conditions and to lead to a better understanding of the density dependence of the interaction, This work is part of the research programme of the NIKHEFK-K made possible by financial support from the Foundation for Fundamental Research on Matter (FOM) and the Netherlands Organization for the Advancement of Pure Research (NWO). References 1) G. Backenstoss, Ann. Rev. Nucl. Sci. 20 (1970) 467 2) C.T.A.M. de Laat, A. Taal, W. Duinker, A.H. Wapstra, J. Konijn, J.F.M. d’Achard van Enschut, P. David, J. Hartfiel, H. Janszen, R. von Mutius, C. Gugler, L.A. Schaller, L. Schellenberg, T. Krogulski, C. Petitjean and H.W. Reist, Phys. Lett. 8162 (1985) 81 3) A. Olin, J.W. Forsman, J.A. Macdonald, G.M. Marshall, T. Numao, P.R. Poffenberger, P. van Esbroek, G.A. Beer, D.I. B&ton, G.R. Mason, A.R. Kunselman and B.H. Olaniyi, Nucl. Phys. A439 (1985) 589 4) C.T.A.M. de Laat, Ph.D. Thesis, University of Technology Delft, March 1988 5) C.T.A.M. de Laat, P. David, H. Hanscheid, W. Lourens, C. Petitjean, F. Risse, Ch.F.G. RoseI, A. van der Schaaf, W. Schrieder, A. Taal and J. Konijn, Nucl. Phys. A, to be published 6) J.G.J. Olivier, M. Thies and J.H. Koch, Nucl. Phys. A429 (1984) 477 7) K. Masutani and R. Seki, Phys. Rev. C38 (1988) 867 8) D.H. Wright, M. Blecher, K. Masutani, R. Seki, R.L. Boudrie, R.L. Burman, M.J. Leitch, M. Alsolami, G. Blanpied, J.A. Escalante, C.S. Mishra, G. Pignault, B.M. Preedom, C.S. Whisnant and B.G. Ritchie, Phys. Rev. C35 (1987) 2258 9) D.H. Wright, M. Blecher, R.L. Boudrie, R.L. Burman, M.J. Leitch, B.G. Ritchie, D. Rothenberger, Z. Weinfeld, M. Alsolami, G. Blanpied, J.A. Escalante, C.S. Mishra, G. Pignault, B.M. Preedom and C.S. Whisnant, Phys. Rev. C36 (1987) 2139 IO) D.H. Wright, M. Blecher, B.G. Ritchie, D. Rothenberger, R.L. Burman, Z. Weinfeld, J.A. Escalante, C.S. Mishra and C.S. Whisnant, Phys. Rev. C37 (1987) 1155 11) A. Taal, Ph.D. Thesis, University of Technology Delft, March 1989 12) A. Taal, P. David, H. Hlnscheid, J.H. Koch, C.T.A.M. de Laat, W. Lourens, F. Risse, Ch.F.G. Rijsel, A. van der Schaaf, W. Schrieder and J. Konijn, Nucl. Phys. A511 (1990) 573

192

.I. Konijn et al. / An improved parametrization

13) J.F.M. d’Achard van Enschut, J.B.R. Berkhout, W. Duinker, C.W.E. van Eijk, W.H.A. Hesselink, T. Johansson, T.J. Ketel, J.H. Koch, J. Konijn, C.T.A.M. de Laat, W. Lourens, G. Van Middelkoop 14) 15) 16) 17) 18) 19) 20) 21) 22) 23) 24)

25) 26) 27) 28) 29) 30) 31) 32) 33) 34) 35) 36) 37) 38)

39)

and W. Poeser, Phys. Lett. B136 (1984) 24 M. Ericson and T.E.O. Ericson, Ann. of Phys. 36 (1966) 323 J. Hiifner, Phys. Lett. C21 (1975) 1 J.A. Carr, H. McManus and K. Stricker-Bauer, Phys. Rev. C25 (1982) 952 I. Angeli, M. Beiner, R.J. Lombard and D. Mas, J. of Phys. G6 (1980) 303 M.E. Nordberg, K.F. Kinsey and R.L. Burman, Phys. Rev. 165 (1968) 1096 S. Ozaki, R. Weinstein, G. Glass, E. Loh, L. Neimala and A. Wattenberg, Phys. Rev. Lett. 4 (1960) 533 D.M. Lee, R.C. Minehart, SE. Sobottka and K.O.H. Ziock, Nucl. Phys. A182 (1972) 20 M. Diirr, W. Fetscher, D. Cotta, J. Reich, H. Ullrich, G. Backenstoss, W. Kowald and H.-J. Weyer, Nucl. Phys. A445 (1985) 557 H.P. Isaak, P. Heusi, H.S. Pruys, R. Engfer, E.A. Hermes, T. Kozlowski, U. Sennhauser, and H.K. Walter, Helv. Phys. Acta 55 (1982) 477 H.P. Isaak, H.S. Pruys, R. Engfer, E.A. Hermes, F.W. Sclepiitz, A. Zglinski, T. Kozlowski, U. Sennhauser and H.K. Walter, Nucl. Phys. A392 (1983) 385 A. Shinohara, A. Yokoyama, S. Moriyasu, T. Saito and H. Baba, Nucl. Phys. A456 (1986) 701; G.W. Hoffmann, L. Ray, M. Barlett, J. McGill, G.S. Adams, G.J. Igo, F. Irom, A.T.M. Wang, C.A. Whitten Jr., R.L. Boudrie, J.F. Amann, C. Glashausser, N.M. Hintz, G.S. Kyle and G.S. Blanpied, Phys. Rev. C21 (1980) 1488 F. Scheck, Leptons, hadrons and nuclei (North-Holland, Amsterdam, 1983); J.H. Koch and F. Scheck, Nucl. Phys. A340 (1980) 221 L. Tauscher, Proc. Int. Seminar on T-Meson nucleus interaction, Strasbourg 1971, CNRS-Strasbourg p. 45 C.J. Batty, SF. Biagi, E. Friedman, S.D. Hoath, J.D. Davies, G.J. Pyle and G.T.A. Squier, Phys. Rev. Lett. 40 (1978) 931 R. Seki and K. Masutani, Phys. Rev. C27 (1983) 2799 C.J. Batty, E. Friedman and A. Gal, Nucl. Phys. A402 (1983) 411 H. de Vries, C.W. de Jager and C. de Vries, At. Data Nucl. Data Tables 36 (1987) 495 J.H. Koch and M.M. Sternheim, Los Alamos Scientific Lab. Report LA-5110, 1973 M. Krell and T.E.O. Ericson, Nucl. Phys. Bll (1969) 521 J. Hiifner, E.J. Moniz and J.M. Eisenberg, Phys. Lett. B47 (1973) 381 C. Tzara, Nucl. Phys. Bl8 (1970) 246 V.F. Dmitriev, Sov. Journ. Nucl. Phys. 17 (1973) 417 J.H. Koch, Phys. Lett. B59 (1975) 45 G.T. Emery, Phys. Lett. B60 (1976) 351 K. Kilian, CERN-EP/85-17; Proc. IUCF Workshop on nuclear physics new possibilities with recoilless kinematics using high quality proton beams, Bloomington, Indiana, October 15-17, 1984; Proc. of the Int. Symp. on dynamics on collective phenomena in nuclear and subnuclear long range interactions in nuclei, Bad Honnef. Fed. Rep. Germany May 4-7, 1987 H. Toki, S. Hirenzaki, T. Yamazaki and R.S. Hayano, Nucl. Phys. A501 (1989) 653

-3.25 L 0.08

-41.17zto.09 -4.79+0.15 -7.76~1~0.13 -11.721tO.22

-18.7zt0.4 -23.3 i 0.3

-31.6+0.5 -41.4+0X

-49.7 f 0.5 -63.9;t 0.9

-80.5* 1.2 -115.5* 1.4 -131.6+2.0

‘OB

‘lB ‘2c ‘% t4N

‘60 ‘80

19F “Ne

“Ne 23Na

“Mg 2’AI ‘sSi

rt%.fperdatapoint

Experiment

e(1s)

Experimental

‘f ‘)

5.21

4.50

_

9.92

-64.0 -76.1 -111.3 -128.4

_

11.25

-52.1 -64.6 -76.6 -113.8 -130.4

-64.0 -79.2 -110.8 -132.1

-51.6

-50.2

-50.4

-64.4 -79.7 -111.3 -132.8

-32.5 -39.7

-32.9 -40.3

-18.2 -25.3

-3.18 -4.68 -6.37 -8.47 -11.60 -18.2 -25.4

.$=l [19X3]

Seki ‘*)

-33.0 -42.6

-19.2 -24.3

-3.26 -4.59 -6.36 -8.35 -11.39

.$=l [I9781

Batty 27)

type potential

according

-33.2 -42.9

-19.3 -24.4

-3.53 -4.41 -6.74 -8.11 -11.91

-3.56

‘) ‘) ‘)

(=I [1971]

26)

Standard

shifts and predictions

.$=o [I9711

-4.43 -6.78 -8.14 -11.98

;;

Is-level

Tauscher

“) b) ‘) *)

Ref.

values of pionic

TABLE A.1

Appendix

3.35

-41.2 -49.5 -62.9 -77.2 -109.4 -129.5

-3.34 -4.31 -6.44 -7.91 -11.48 -18.5 -23.9 -32.1

6=1 [1983-j

Batty 29)

2.62

-39.8 -49.3 -63.2 -78.8 -114.5 -136.0

-3.16 -4.36 -6.66 -8.08 -12.27 -18.9 -23.2 -30.4

5=0

(present

1.24

1.32 2.59

-39.7 -49.4 -63.3 -78.6 -114.5 -135.6

-3.31 -4.24 -6.70 -7.94 -12.10 -19.1 -23.1 -31.0 -41.2 -49.4 -43.4 -79.3 -112.8 -134.9

5=1

1.38

-3.33 -4.21 -6.73 -7.95 -12.09 -19.1 -23.1 -31.2 -41.4 -49.1 -63.2 -79.3 -112.2 -134.8

5=3.4(4)

potential fit)

-3.31 -4.25 -6.71 -7.97 -12.12 -19.1 -23.1 -31.1 -41.1 -49.2 -63.3 -79.2 -112.8 -134.8

[=O

(present

optical

-3.15 -4.37 -6.60 -8.07 -12.18 -18.8 -23.3 -30.4

[=l

fit)

Extended

I and 2. All values are in keV

type potential

sets of tables

Standard

to the parameter

st? a

$h)’

i P

2_. j g

R. ‘c,

z

TABLE

A.2

Experiment

1.58i0.16 1.77kO.16 2.99 f 0.14 2.59 f 0.14 4.34 f 0.24 7.9hO.3 6.3 f 0.4 10.1 zto.7 15.4*0.4 10.8*0.7 17.1 k2.3 24.3 f 1.6 28.8 f 1.2 41*4

Uls)

‘OB “B ‘2C ‘3C ‘4N I60 ‘80 ‘9F “Ne “Ne 23Na “+Mg 2’AI “Si

Ref. [=l [ 19781 1.47 1.43 2.96 2.61 5.05 7.4 5.7 9.1 13.9 11.5 17.3 24.8 29.3 39.3 2.50

[=l [1971] 1.30 1.35 2.67 2.54 4.60 6.7 5.6 8.4 12.4 11.6 16.5 22.8 28.6 36.8 6.51

1.32 1.38 2.73 2.57 4.68 6.8 5.7 8.5 12.6 11.7 16.8 23.1 29.0 37.2

5.61

Batty 27)

[1971]

r=o

Tauscher 26)

6.58

27.3 35.2

31.9 40.7

2.10

12.4 17.3 23.6

13.1 18.8 25.8

3.20

6.4 9.9 14.3

6.4 9.8 14.4

29.7 39.3

27.1 35.3

1.32

11.2 18.0 25.7

12.3 17.2 23.6

2.08

5.2 9.7 15.0

6.3 9.8 14.2

(5.31) 7.8

4.90 7.3

4.86 7.3

7.6 5.22

[=O

1.67 1.41 3.14 2.61

(=l

1.28

29.5 39.2

11.1 18.0 25.6

9.8 15.0

5.2

(5.32) 7.8

1.67 1.41 3.16 2.61

Z=l

(present fit)

5.4

1.07

30.3 39.4

11.4 18.4 25.9

f 9 2 g. 3

a w b e

(5.25) 7.7

10.0 15.1

_. & 3

E

2

s 9:

1.66 1.44 3.07 2.61

[=3.4(4)

Extended optical potential

1.59 1.55 2.99 2.75

1.60 1.57 2.94 2.74

1.55 1.56 3.06 2.85

1.39 1.31 2.91 2.45 4.74 6.9 5.1 8.3 12.8 10.3 15.2 22.1 25.0 34.6

t=o

(present fit)

(=l [ 19831

Batty 29)

Standard type potential

5=1 [ 19831

Seki 28)

Standard type potential

Experimental values of pionic Is-level widths and predictions according to the parameter sets of tables 1 and 2. All values are in keV

0.00306 * 0.00016 0.0148 + 0.0003 0.0151*0.0004 0.030 f 0.003 0.125*0.004 0.122 f 0.004 0.190 i 0.007 0.275 + 0.010 0.271*0.010 0.590* 0.022 1.77kO.08 1.53 rtO.08 1.41*0.08 1.18*0.08 2.27*0.14 2.07*0.13 1.72kO.11 2.41 *to.16 3.17*0.21 2.74*0.19 2.43kO.19 3.04* 0.25 3.6*0.4 5.3 50.6

‘ZC

‘60 ‘80 19F “Mg 26Mg *‘Al *%i 3oSi ‘2s @Ca 42Ca “Ca 48Ca &Ti ‘8Ti “Ti 5’V “Cr “Cr 5% %n 56Fe *‘CU

Experiment

E(2P)

“)

4) 4) ‘1

9

PI PI PI PI “1 ‘1 *I 4) 9)

9 O)

J

‘1

:;

9

1;

Ref.

0.00193 0.0097 0.0097 0.017 0.088 0.092 0.130 0.208 0.202 0.412 1.22 1.11 1.03 0.95 1.75 1.62 1.57 2.01 2.57 2.44 2.03 2.55 3.4 3.2

[=O [ 19711 0.00217 0.0111 0.0108 0.020 0.102 0.100 0.148 0.237 0.222 0.472 1.41 1.26 1.12 0.89 1.97 1.74 1.57 2.05 2.88 2.58 2.10 2.65 3.6 3.6

&=l [1971]

Tauscher 26)

0.00265 0.0138 0.0121 0.024 0.132 0.117 0.180 0.304 0.263 0.617 1.91 1.58 1.28 0.76 2.54 2.03 1.62 2.26 3.73 3.03 2.20 3.00 4.2 4.4

[=l [ 19781

Batty *‘I

0.00258 0.0134 0.0116 0.023 0.129 0.111 0.170 0.294 0.249 0.594 1.82 1.47 1.14 o.s9 2.34 1.79 1.35 1.92 3.38 2.63 1.77 2.42 3.6 3.1

I$=1 [ 19831

Seki 28)

Standard type potential

0.00244 0.012s 0.0113 0.022 0.118 0.110 0.169 0.278 0.250 0.562 1.73 1.49 1.28 0.94 2.39 2.05 1.79 2.40 3.56 3.11 2.46 3.21 4.4 4.8

.$=l [ 19831

Batty 29)

0.588 1.75 1.60 1.43 1.06 2.45 2.10 1.19 2.25 3.36 2.85 2.48 2.78 3.5 4.0

0.00142 0.0147 0.0288 0.136 0.127 0.184 0.294 0.280

0.00269

@=O

0.00265 0.0138 0.0152 0.0293 0.132 0.125 0.190 0.291 0.277 0.581 1.74 1.58 1.42 1.14 2.36 2.07 1.78 2.20 3.32 2.79 2.52 2.72 3.6 4.3

.

g=o

0.00269 0.00140 0.0147 0.0283 0.134 0.127 0.183 0.291 0.279 0.581 1.72 1.58 1.41 1.07 2.41 2.09 1.81 2.27 3.33 2.86 2.48 2.78 3.6 4.0

. . . -. .._ __.

0.00272 0.0140 0.0149 0.0291 0.132 0.124 0.188 0.292 0.275 0.581 1.73 1.56 1.39 1,08 2.36 2.05 1.77 2.24 3.34 2.84 2.47 2.83 3.7 4.3

6 = 3.4 (4)

. ..-- -- --.---.-.--- --.-~ --.- -- - --

1.56 1.41 1.1s 2.35 2.07 1.80 2.24 3.33 2.83 2.52 2.76 3.7 4.3

0.00269 0.0139 0.0151 0.0289 0.131 0.125 0.189 0.292 0.277 0.582 1.74

&=l

(present fit)

Extended optical potential

[=l

(present fit)

Standard type potential

Experimental values of pionic Zp-level shifts and predictions according to the parameter sets of tables 1 and 2. All values are in keV

TABLE A.3

X2&&mdata-point

“)

4.9 f 0.8 2.7kO.9 1.2+0.7 -11*3 -48*7

natZn “a’Ge ‘SAS =Nb natRU

1; ‘) V

Ref.

Experiment

s(2p)

37.23

4.9 0.6 0.8 -12.5 -32.4

.$=O [1971]

19.53

5.4 0.5 0.7 -13.7 -34.9

.$=l [1971]

Tauscher 26)

6.53

7.2 -0.8 -0.1 -16.8 -43.0

[=l [1978]

Batty 27)

12.23

5.9 -2.8 -2.6 -23.4 -51.8

C=l [1983]

Seki *‘)

Standard type potential

8.44

7.2 1.4 2.0 -10.4 -32.1

.$=I [1983]

Batty 29)

TABLE A.3-continued

1.88

6.0 1.2 0.2 -19.1 -42.7

(=O

1.76

5.9 1.0 0.1 -19.1 -42.8

5=1

(present fit)

Standard type potential

1.36

5.6 2.0 0.7 -18.1 -38.8

<=o

1.23

5.6 1.8 0.6 -18.0 -38.9

5=1

1.19

5.8 1.5 0.8 -16.9 -38.2

f = 3.4 (4)

(present fit)

Extended optical potential

Experiment

‘60

1%

0.00136~0.00022 0.0068 * 0.0004 ‘80 0.0075 * 0.0004 19F 0.0112;tO.O019 24Mg 0.0725 *to.0018 26Mg 0.0811*00.0019 27A1 0.124*0.008 +3i 0.196+0.005 %i 0.196 f 0.008 32s 0.430*0.021 ‘%a 1.64kO.11 42Ca 1.65ztO.15 ‘Wa 1.64kO.12 %a 1.64kO.11 &Ti 2.39+0.15 48Ti 2.62*0.15 “Ti 2.15*0.27 slV 3.13zto.13 “Cr 4.3 * 0.4 52Cr 3.85 f 0.21 54Cr 3.84+0.29 55Mn 5.0*0.4 56Fe 6.75 * 0.21 natCU 11.5kO.8

WP)

Ref.

0.00079 0.0043 0.0041 0.0073 0.0511 0.0572 0.090 0.145 0.148 0.327 1.25 1.23 1.26 1.37 2.22 2.28 2.39 3.13 3.80 3.94 3.78 4.97 6.46 10.6

[I9711

(=O

0.00084 0.0048 0.0047 0.0088 0.0573 0.0618 0.099 0.158 0.159 0.353 1.35 1.33 1.33 1.38 2.35 2.37 2.42 3.17 3.99 4.03 3.88 5.05 6.52 11.0

[=l [1971]

Tauscher 26)

0.00102 0.0058 0.0055 0.0104 0.0695 0.0711 0.117 0.190 0.183 0.427 1.63 1.54 1.48 1.41 2.73 2.62 2.55 3.40 4.59 4.42 4.11 5.45 7.08 11.9

C=l [1978]

Batty 27)

0.00097 0.0055 0.0052 0.0101 0.0655 0.0664 0.108 0.176 0.169 0.392 1.47 1.39 1.33 1.26 2.44 2.33 2.27 3.00 4.04 3.88 3.62 4.74 6.14 10.2

[=l [ 19831

Seki ‘s)

Standard type potential

0.00084 0.0048 0.0047 0.0087 0.0590 0.0632 0.104 0.165 0.165 0.375 1.46 1.42 1.41 1.43 2.53 2.52 2.54 3.35 4.32 4.3 1 4.11 5.39 7.00 12.0

(=l [ 19831

Batty 29)

0.00099 0.0058 0.0060 0.0114 0.0700 0.0738 0.117 0.182 0.186 0.406 1.51 1.51 1.50 1.51 2.61 2.59 2.58 3.31 4.23 4.19 4.13 5.23 6.43 11.2

.$=o

0.00102 0.0060 0.0061 0.0117 0.0713 0.0746 0.119 0.185 0.188 0.412 1.53 1.52 1.51 1.50 2.62 2.59 2.57 3.31 4.26 4.20 4.12 5.21 6.46 11.2

5=1

(present fit)

Standard type potential

0.00105 0.0061 0.0060 0.0116 0.0741 0.0769 0.126 0.196 0.195 0.439 1.65 1.60 1.52 (1.29) 2.79 2.66 2.50 3.35 4.59 4.38 4.03 5.35 6.82 11.7

[=O

0.00105 0.0062 0.0061 0.0117 0.0741 0.0771 0.125 0.196 0.195 0.438 1.65 1.60 1.52 (1.29) 2.78 2.65 2.50 3.35 4.58 4.37 4.02 5.32 6.82 11.6

5=1

0.00103 0.0062 0.0069 0.0138 0.0758 0.0791 0.126 0.190 0.197 0.426 1.60 1.60 1.53 (1.26) 2.75 2.62 2.44 3.23 4.45 4.22 3.97 5.18 6.45 11.4

5= 3.4 (4)

(present fit)

Extended optical potential

Experimental values of pionic Zp-level widths and predictions according to the parameter sets of tables 1 and 2. All values are in keV.

TABLE A.4

14.0* 1.4 18.5 f 2.0 14.5* 1.7 64*8 77*24

natZn “We “AS 93Nb “a’RU

XZ(~&rdata-point

Experiment

U2P)

V *)

“) 1;

Ref.

13.6 17.1 20.8 59.2 76.5 13.60

22.40

[=l [1971]

13.2 16.9 20.6 60.0 77.8

t=o [1971]

Tauscher 26)

3.92

15.0 17.0 21.0 56.9 70.0

.!J=l [1978]

Batty *7)

7.08

12.9 14.5 17.8 46.8 57.1

.$=l [1983]

Seki ‘s)

11.47

14.8 18.3 22.3 63.0 80.7

5=1 [1983]

Batty *9)

A.4-continued

Standard type potential

TABLE

2.84

13.7 17.4 20.6 52.5 66.4

t=o

2.35

13.7 17.3 20.4 52.0 65.6

.$=l

(present fit)

Standard type potential

1.83

14.5 15.6 (19.6) 50.5 59.5

t=o

14.2 15.6 (19.3) 49.0 58.0 1.24

1.78

5 = 3.4 (4)

14.5 15.5 (19.3) 50.1 58.9

5=1

(present fit)

Extended optical potential

=Nb na’Ru -Ag “Wd 13*Ba “We ‘Ye “@Nd “‘Nd ‘66Er 16*Er ‘*ITa na’Re *% 197AU “‘Pb 209Bi

E(3d)

TABLE A.5

0.74 f 0.02 1.39 * 0.08 1.94*0.07 2.14*0.09 5.1*0.3 6.6 f 0.3 6.8 * 0.3 7.2 f 0.4 7.1*0.4 15.6ztl.O 14.6* 1.0 16.0* 1.3 16.5k1.3 22.8* 1.9 20.6 + 1.9 22.7 + 2.4 20.1*3.0

Experiment

1.12 1.75 2.08 6.1

7.4 7.5

8.7 8.9 16.5 17.8

19.6

30.0 22.1 33.7 27.9

36.9

:; ‘) “)

:;

‘; :;

“)

:; :;

“)

12.35

0.66

#$=o [1971]

3.00

21.4

23.9 18.2 22.5 24.7

16.4

7.8 8.0 15.5 14.8

6.8 6.9

1.77 1.12 2.03 5.5

0.67

5=1 [1971]

6.19

16.7

11.0 16.4 13.2

10.4

6.9 7.3 13.3 12.3

6.3 6.7

1.91 1.16 2.04 5.2

0.73

.$=l [ 19781

Batty *‘)

32.51

6.2

4.4 9.0 3.4 5.0

4.5

6.0 5.5 10.1 9.1

5.3 5.6

1.71 1.04 1.79 4.3

0.66

.$=l [1983]

Seki **)

Standard type potential Tauscher 26)

7

Ref.

_

1.71

0.70 1.14 1.85 2.04 5.4 7.0 6.7 7.9 7.6 14.4 15.2 15.3 16.8 22.5 20.6 22.3 25.7

l=l [ 19831

Batty *9)

0.76 1.24 1.97 2.26 5.7 7.0 7.0 8.4 8.3 14.5 14.7 11.3 13.5 20.2 18.1 18.3 20.0 3.19

3.24

[=l

0.75 1.23 1.97 2.25 5.6 7.0 6.9 8.3 8.3 14.4 14.6 11.1 13.3 20.1 17.9 18.0 19.8

[=O

(present fit)

Standard type potential

1.36

0.74 1.23 1.93 2.26 5.3 6.6 6.6 8.1 8.3 15.7 14.1 14.9 17.3 20.6 22.5 19.9 19.2

[=O

1.23

0.74 1.23 1.93 2.25 5.3 6.6 6.6 8.1 8.2 15.5 14.2 14.9 17.2 20.7 22.1 20.1 19.6

6=1

1.04

0.75 1.23 1.94 2.22 5.4 6.7 6.7 8.0 8.0 15.0 14.4 15.2 17.1 20.9 21.3 20.4 21.2

.$= 3.4 (4)

(present fit)

Extended optical potential

Experimental values of pionic 3d-level shifts and predictions according to the parameter sets of tables 1 and 2. All values are in keV

0.402 * 0.016 0.75 * 0.08

1.44*0.05 1.65 * 0.07

4.3 * 0.9

5.6* 6.5 f 8.8* 9.2*

“Nb -“Ru

-A, -%zd

t3sBa

‘?Ze ‘42Ce 14’Nd lsoNd

X2tr3d)pm

data-point

37*5 34*4 47*4 52*4

“A’pt ‘97A~ 20sPb

ZOSgi

19.7 zto.9 19.4* 1.0 24.6zt 1.5 29.7 f 2.7

‘=Er lasEr reiTa ““Re

1.0 0.9 1.2 1.1

Experiment

:;

“1 “1

z;

“) ‘;

:;

k) kI

k, h)

Ref.

Tauscher

3d-level

57.6 61.8 78.1 85.6 20.95

35.28

22.6 23.2 37.5 44.4

7.2 7.1 9.1 9.0

5.5

1.32 1.54

0.428 0.75

?$=l [1971]

26)

9.28

49.6 52.8 64.9 72.0

20.9 21.4 33.4 38.9

7.1 7.0 8.8 8.6

5.4

1.45 I.61

0.476 0.81

.$=l [1978]

Batty 27)

A.6

3.22

39.6 41.3 51.3 56.9

16.9 17.5 26.2 30.6

6.0 5.9 7.3 7.2

4.5

1.26 1.39

0.421 0.71

5=1 [1983]

Seki aa)

according

TABLE

type potential

and predictions

Standard

widths

64.2 67.6 88.0 97.0

24.0 25.6 40.5 48.2

7.7 7.7 9.7 9.6

6.0

1.36 1.54

0.441 0.75

E=O [1971-j

values of pionic

Wd)

&p&mental

22.16

57.9 62.6 77.8 85.5

23.2 23.5 38.3 45.2

7.3 7.2 9.2 9.1

5.5

1.32 1.57

0.425 0.77

8=1 [ 19831

Batty 29)

8.03

49.4 52.0 64.6 70.2

20.7 21.1 32.2 38.2

7.0 7.0 9.0 9.0

5.4

1.35 1.63

0.447 0.80

.$=o

(present

6.26

48.3 50.9 63.0 68.5

20.4 20.8 31.4 37.3

6.9 6.9 a.8 8.9

5.3

1.35 1.62

0.448 0.80

4=1

fit)

1.10

40.4 41.6 50.6 57.2

18.1 18.7 26.2 30.5

6.6 6.5 8.1 7.8

5.0

1.41 1.60

(0.470) 0.82

l=O

optical

1.09

40.4 41.8 50.5 57.0

18.1 18.6 26.4 30.7

6.6 6.5 8.1 7.8

5.0

1.40 1.60

(0.474) 0.82

5=1

0.93

39.8 41.4 49.2 55.8

18.1 18.3 26.0 30.5

6.6 6.4 8.1 7.9

5.0

1.45 1.65

(0.488) 0.84

5=3.4(4)

fit)

potential

are in keV

(present

Extended

1 and 2. All values

type potential

sets of tables

Standard

to the parameter

5 B B a B z z. S. s

2

; a

2 K

; 0 2a

X%&,r

“‘Ho ‘&Er ‘%r ‘7SLu ‘*‘Ta natRe ““% lq7Au ‘*‘Pb 209Bi 237Np

440

data-point

0.30 * 0.08 0.30* 0.03 0.30 * 0.03 0.60 * 0.07 0.56+0.04 0.76ztO.04 1.09 * 0.04 1.25 * 0.07 1.68 + 0.04 1.78 i 0.06 5.26zt0.19

Experiment

Experimental

Tauscher

4f-level

1.10 1.21 1.94 1.76

5.11

@) @) :;

s,

1.71

0.29 0.43 0.75 0.57

1; :;

0.25 0.29

[1971]

t=o

of pionic

Ref.

values

2.69

4.58

1.03 1.14 1.80 1.62

0.28 0.42 0.7 0.541

0.24 0.28

(=l [1971]

26)

Standard

5.70

4.09

1‘03 1.12 1.78 1.58

0.41 0.29 0.69 0.53

0.24 0.28

.$=l [1978]

Batty “f

22.06

3.32

0.91 0.98 1.57 1.39

0.26 0.36 0.61 0.47

0.21 0.25

5=1 [1983]

Seki a8)

according

type potential

shifts and predictions

6.60

4.26

0.98 1.08 1.73 1.54

0.27 0.39 0.67 0.51

0.23 0.27

[=i [1983]

Batty 29)

3.88

0.25 0.31 0.31 0.43 0.57 0.75 1.11 1.22 1.70 1.89 4.18

f=O

(present

3.64

0.25 0.31 0.31 0.44 0.58 0.76 1.12 1.24 1.73 1.91 4.26

[=l

fit)

0.67

0.24 0.32 0.30 0.44 0.58 0.77 1.07 1.24 X.65 1.78 (4.38)

c=o

optical

0.65

0.25 0.31 0.30 0.45 0.59 0.77 1.07 1.23 1.65 1.79 (4.42)

‘$=l

(present

Extended

0.61

0.26 0.31 0.30 0.46 0.59 0.77 1.08 1.22 1.66 1.82 (4.54)

[=3.4(4)

fit)

potential

1 and 2. All values are in keV.

type potential

sets of tables

Standard

to the parameter

TABLE A.?

xv-&r

data-point

0.19*0.04 0.20 f 0.07 0.3 1 f 0.05 0.41 f 0.05 0.59 * 0.05 0.77 f 0.04 0.98 f 0.05 1.24 f 0.09 3.X8*0.26

16sHo “5LU ‘s’Ta natRe “atpt

‘9’AU 2osPb 2WBi 237Np

Experiment

r(4f)

1.18 3.74 0.92

1.04 1.18 3.92 1.12

1.18

1.35 4.43

3.13

1.03

0.62 0.69

^)

0.61 0.68

0.67 0.73

0.13 0.24 0.31 0.41

5=1 [ 197x1

s) a)

0.12 0.22 0.30 0.40

[=l [1971]

Batty a’)

1;

[=O [I9711

Tauscher “)

0.13 0.23 0.42 0.31

Ref.

5.90

0.99 3.01

0.86

0.52 0.58

0.11 0.20 0.35 0.27

I$=1 [ 19x31

Seki as)

Standard type potential

1.17

1.16 3.87

1.02

0.60 0.67

0.12 0.22 0.39 0.30

C=l [19X3]

Batty r9)

0.97

1.16 3.65

1.03

0.62 0.70

0.13 0.24 0.42 0.32

.$=o

1.09

1.15 3.56

1.02

0.61 0.69

0.13 0.24 0.31 0.42

[=l

(present fit)

Standard type potential

1.29

1.18 3.36

1.04

0.64 0.71

0.14 0.25 0.43 0.33

t=o

1.35

1.19 3.34

1.05

0.64 0.71

0.14 0.25 0.43 0.33

[=l

1.44

1.20 3.28

1.05

0.66 0.72

0.14 0.25 0.33 0.44

5=3.4(4)

(present fit)

Extended optical potential

Experimental values of pionic 4f-level widths and predictions according to the parameter sets of tables 1 and 2. All values are in keV

TABLE A.8

%. P

B g b 2.

a bg

2

&

: C

h *. n 3 9:

Fermi distribution

18.998

24

24

24

“Cr

Wr

20

Ya

S°Cr

20

23

20

Ya

‘Wa

22

20

4oCr

5’V

16

328

50Ti

3.938

45.952

14

zosi

22

14

22

47.956

13

27A1

‘88i

4hTi

41.959

12

=Mg

4XTi

3.745

39.963

12

53.946

51.941

49.946

49.945 50.944

47.948

43.955

31.972

4.084 4.243

4.053

4.03 1

4.059

4.075

4.036

3.943

4.078

3.965

3.991

4.019

3.923

3.882

3.916

3.804

3.851

3.674

3.748

3.795

3.333

3.351

3.135

3.197

3.170

3.026

3.105

3.070

2.912

3.059

2.920

2.437

2.187

2.264

2.020

3.398

3.240

3.191

29.974

3.083

27.977

3.051

3.080

26.982

25.983

23.985

2.873

2.965

=Mg

22.990

21.991

10

I1

2.963

2.826

2.613

“Na

19.992

2.210

2.482

*‘Ne

9

10

19F

*‘Ne

17.999

8

14.003

15.995

2.014

-0.04

0.33

0

0.4

3.5

2.5

1.5

1

1.5

2.051

0.85

12

13.003

‘SO

6

7

6

12C

13C

0.78

3

2.227

1.937

2.027

2.046

10.013

11.009

8

5

“B

14N

5

‘OB

I”

A

I60

Z

Nucleus

54.938

5.127 5.342 5.438 5.636 5.912 5.936 6.005

4.986 5.129 5.299 5.328 5.400 5.721 5.774

92.906 101.904 106.905 108.905 113.903 137.905 139.905

44 47 47 48 56 58

“*Ru “‘Ag

19’AU

=‘NP

“sPb ZOSBi

6.870 7.300

6.687 7.001

208.980 237.095

83

6.554 195.970 78

93

6.443

186.970 75 ‘S’Re 196pt

6.850

6.768

6.347 180.948 73

“‘Ta

6.892

6.740

6.274

174.941 71 “sLu

6.652

6.650

6.174 167.932 68

lhXEr

6.554

6.570

6.161 165.932 68

‘66Er

196.967

6.372

6.125

164.930 67

207.977

6.454

6.001 149.921

60

“‘Nd 16SFfo

82

6.233 6.329

5.944

79

6.155

5.814 141.909 147.917

58

“We 60

5.468

‘@Nd

“Ye

13SBa

“‘kd

‘09Ag

“Nb

41

4.868

4.665

74.922

33

4.819

4.628

73.921

32

74Ge

4.447

4.440

63.929

=AS

4.493

4.362

62.930

4.139

4.045

30

4.184

4.088

29

55.935

parameter

(p)

and nuclear spin (I).

0.282

~0.025

2.5

4.5

1.5

2.5

0.22 0.105

3.5

3.5

1.5

2.5

0.265

0.31

0.32

-0.08

0.225

-0.23

0.24

t, = t, = 2.3 were used in all cases

64Zn

26

25

Z

deformation parameters

63cu

56Fe

55Mn

Nucleus

densities,

The surface

and neutron

in the calculations.

in eq. (14) for the proton

c, = cp was assumed

(c, t) parameters

For cases where no value for c, was available,

Proton and neutron

TABLE A.9

5

804

J. Konijn et al. / An improved parametrization

References to the appendix a) A. Olin et al., Nucl. Phys. A360 (1981) 426; L. Tauscher and S. Wychech, Phys. Lett. B62 (1976) 413; R.J. Harris et al., Phys. Rev. Lett. 20 (1968) SOS; G. Backenstoss et al., Phys. Lett. B25 (1967) 365; D.A. Jenkins et al., Phys. Rev. Lett. 17 (1966) 1; 1148 b) A. Olin et al., Nucl. Phys. A360 (1981) 426; L. Tauscher and S. Wycech; Phys. Lett. B62 (1976) 413; R.J. Harris et al., Phys. Rev. Lett. 20 (1968) 505; D.A. Jenkins et al, Phys. Rev. Lett. 17 (1966) 1; 1148 c) C. Fry et al., Nucl. Phys. A375 (1982) 325; L. Tauscher and S. Wycech; Phys. Lett. 862 (1976) 413; G. Backenstoss et al., Phys. Lett. B2J (1967) 365 d) C. Fry ef al., Nucl. Phys. A375 (1982) 325 e) L. Tauscher and S. Wycech, Phys. Lett. B62 (1976) 413; G. Backenstoss et al., Phys. Lett. B25 (1967) 365; D.A. Jenkins et al., Phys. Rev. Lett. 17 (1966) 1; 1148 f) I. Schwanner et al., Nucl. Phys. A412 (1984) 253; I. Schwanner et al., Phys. Lett. B96 (1980) 268 g) A. Olin et al., Nucl. Phys. A312 (1978) 361 h) B.H. Olaniyi et al., ICOHEPANS Versailles 1981, p. 3.52; M. Eckhause et al., Nucl. Phys. B44 (1972) 83 i) B.H. Olaniyi et al., ICOHEPANS Versailles 1981, p. 352 j) D.I. Britton et al., Nucl. Phys. A461 (1987) 571 k) A. Taal et al., Nucl. Phys. A511 (1990) 573 1) G. de Chambrier et al., Nucl. Phys. A442 (1985) 637 m) H. Koch et al., Phys. Lett. B29 (1969) 140 n) C.J. Batty et al., Nucl. Phys. A322 (1979) 445 o) C.J. Batty e? al., Nucl. Phys. A322 (1979) 445; K.E. Kir’yanov ef al., Sov. Journ. Nucl. Phys. 26 (1977) 685 p) R.J. Powers et al., Nucl. Phys. A336 (1980) 475 q) R. Kunselman et al., Nucl. Phys. A405 (1983) 627 r) R. Kunselman et al., Nucl. Phys. A405 (1983) 627; C.J. Batty et al., Nucl. Phys. A322 (1979) 445 s) R. Abela et al., Zeit. fiir Physik A282 (1977) 93; C.J. Batty et al., Nucl. Phys. A322 (1979) 445 t) R. Abela et al., Zeit. fiir Physik A282 (1977) 93 u) D.A. Jenkins, R.J. Powers and R.A. Kunselman; Phys. Rev. C2 (1970) 429 v) A.R. Kunselman et al., Phys. Rev. Cl5 (1977) 1801 w) Y. Tanaka et al., Phys. Lett. B143 (1984) 347 x) C.T.A.M. de Laat et al., Nucl. Phys. A, to be publish ed y) J.F.M. d’Achard van Enschut z) P. Ebersold et al., Phys. Lett. P. Ebersold et al., Nucl. Phys. (u) P. Ebersold et al., Nucl. Phys. p) J.F.M. d’Achard van Enschut 6) C.T.A.M. de Laat, Phys. Lett.

et al., Phys. Lett. B136 (1984) 24 B53 (1974) 48; A296 (1978) 493 A296 (1978) 493 ef al., Phys. Lett. B136 (1984) 24 B189 (1987) 7