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Nuclear capture in the atomic cascade of kaonic, antiprotonic and sigma hyperonic exotic atoms
NUCLEAR PHYSICS A ELSEVIER
Nuclear Physics A 589 (1995) 601-608
Nuclear capture in the atomic cascade of kaonic, antiprotonic and sigma hyperonic exotic atoms C.J. Batty a, R.E. Welsh b a RutherfordAppleton Laboratory, Chilton, Didcot, Oxon 0)(11 OQX, UK b Physics Department, College of William and Mary, Williamsburg, VA 23187 USA Received 7 March 1995
Abstract
Ratios of nuclear capture rates are shown to increase with increasing principal quantum number n for a given orbital quantum number l, in the high-n, high-/ states most relevant to studies of the exotic hadronic atoms of heavy particles. This surprising behaviour, which is confirmed by optical-model calculations, is in contrast to similar capture ratios in pionic atoms at low atomic number. Plots of the relevant hydrogenic atom wave functions are shown to explain such effects.
I. Introduction
With the production of beams of muons and pions at synchrocyclotrons, studies of exotic atoms were begun in the 1950's with the pioneering work of Fitch and Rainwater [1] at Columbia. In one of the earliest reviews of such studies, West [2] described the experimental and theoretical progress in a summary of the field that was at that time referred to as "mesonic atoms". In the mid 1960's, development of solid-state Ge and Si radiation detectors of high resolution gave impetus to more precise measurements of the properties of exotic atoms. Such measurements permit the extraction of precise information about the orbiting particle [3], such as charge/mass, and magnetic moment, and the interaction of such particles with nuclei. In addition, properties of nuclei [4], such as nuclear size, nuclear polarization, and neutron halo effects in heavy nuclei [5], have been studied. Furthermore, the mechanism of atomic capture of such heavy charged particles and atomic effects such as Stark mixing [6], trapping [7] and transfer of heavy negative particles between atoms of different species [8] have been studied extensively. 0375-9474/95/$09.50 © 1995 Elsevier Science B.V. All rights reserved SSD! 0 3 7 5 - 9 4 7 4 ( 9 5 ) 0 0 1 2 0 - 4
602
C.J. Batty, R.E. Welsh/Nuclear Physics A 589 (1995) 601-608
2. Atomic calculations Some of the first cascade calculations for muonic atoms were carried out by Eisenberg and Kessler [9] and later extended to pionic and kaonic atoms [10]. In this latter work, the nuclear capture probability for the pion or kaon was calculated from the overlap of the atomic wave function with the density distribution of the nucleus. For reasons of simplicity hydrogenic wave functions were used. In the late 1960's, Hiifner [11] wrote one of the first general purpose computer programs to describe the atomic cascade of negative muons in muonic atoms. This computer program has been the basis of much subsequent work. In 1971, this program was modified by Sapp [12] to include the effects of nuclear capture in low-Z pionic atoms. Experimentally measured values of strong interaction widths were used for the circular (n, l = n - 1) atomic states whilst the formula set out by West [2], which expresses ratios of absorption widths for states of given l and increasing n, was used for the higher, non-circular states. From the early 1970's to the present, this program, with the corrections discussed by Fried and Martin [13] included, has been used extensively to interpret exotic atom experiments. The validity of the West formula has been discussed by Turner and Jackson [14] for pionic atoms for orbitals of 1 > 1. They have also derived an improved formula for pionic s and p atomic states. Cascade programs are especially valuable in the interpretation of hadronic atoms. It is often possible to measure directly the strong interaction shift and width of only the lowest state in such atoms. Information about nuclear capture in higher atomic levels is then inferred from measurements of the relative yields of the atomic x-ray transitions. Thus, a prediction of the ratio of nuclear overlap of the various (n, l) atomic states is important to an accurate interpretation of experimental results.
3. Experiments on sigma hyperons Most recently, in measurements carried out at Brookhaven National Laboratory principally to extract the magnetic moment of the sigma hyperon [3], determinations have been made of nuclear shifts and widths and relative yields of kaonic atoms and sigma hyperonic atoms for principal quantum numbers as high as n = 11. When the approximation taken from West [2] was applied to the relative capture rates in high-n, high-/atomic states for these atoms, an unexpected result was obtained. In particular, the program of Hiifner, as modified by Sapp, suggested that in atomic states of orbital angular momentum quantum number l = 9, the so-called circular state for n = 10 has less overlap with the nucleus than several of the states of higher n with the same 1. When this result was first seen, it was suspected that the approximation taken from West might not apply for high-Z, high-n states in exotic, hadronic atoms. On testing the ratios predicted by the West formula against the predictions of a program using realistic wave functions and an optical-model representation of the nucleus, we were surprised to discover that very similar capture ratios were obtained.
CJ. Batty, R.E. Welsh/Nuclear PhysicsA 589 (1995) 601-608
603
We thus took steps to determine whether such results were physically meaningful and to try to understand the origin of the apparently surprising behaviour.
4. Theoretical background The strong interaction width F, for the atomic state (n, l) is given by
-4~fl ffJnl 12 I m U ( r ) r 2 dr
½F(n, 1) =
(1)
where ~0,t is the atomic wave function for the (n, l) state and U(r) is the optical potential representing the strong hadron-nucleus interaction. West [2] uses hydrogenic wave functions
=
l
(4Z3(n-l-l)') _ _
1/2
e-o/Z, tLZl+lt p)
(2)
normalized so that cx~
fo 4"n'r2qj2t dr = 1, and p = 2Zr/na o with a 0 = Bohr radius of the bound particle for n = 1, Z = 1: a o = he/Txe :. Lkn(x) is the associated Laguerre polynomial. Since p is small, West then approximates the terms e -p/2 and -2t+ 1.,+ ~1,t p) by their values at p = 0: -{(n+t)t}' 1)!(2/+ 1)!'
~+~, (0) = ( n - l Ln+
(3)
and
~On' - ~
1( 11c4z3
,n+,,
na o ] ~ n4a3o(n - / - 1 ) ! ( ( 2 /
+ 1)!} 2
)lj2r
Using Eq. (1) and assuming a uniform optical potential of radius R = r o A l / 3 ~ Z 1/3, West then obtains Z 81/3+4
r ( n , l) cc ~ a~t+
(n-~- I)!
n21+4(n -- I-- 1)![(21 + 1)!]2(21 + 3)
(5)
from which
F(m, l) = ( n )2l+4 (m + l ) , ( n - - l - - 1 ) ,
r(n, l)
--
(m-l-1)!(n+
t)!
o
(6)
And for the particular case of m = n + 1 r(n+l,l) r ( n, ~)
(
n
]21+4n+1+1
~-g-4S j
n - t
(7)
604
C.I. Batty, R.E. Welsh/NuclearPhysicsA 589 (1995) 601-608
5. Results
Fig. 1 shows a plot of the ratio of nuclear overlap for hydrogenic atoms for a range of n- and /-values, beginning with the circular states for l = 3 to l = 8 and extending to higher values in n as given by the formula (Eq. (6)) of West. As can be seen, overlap with the nucleus in the higher-/atomic states relevant to antiprotonic, sigma hyperonic and kaonic atoms appears to increase for states of principal quantum number above that of the circular state, dropping off only gradually with higher n. As a check on the validity of the West formula (Eq. (7)), the ratio F ( n + 1, l ) / F ( n , l) was also calculated by numerical solution of the Klein-Gordon equation for the Coulomb plus nuclear system including an optical potential proportional to the nuclear density [15] to describe the hadron-nucleus interaction. The Coulomb potential used was that due to the finite charge distribution of the nucleus together with the first order ot(Zol) vacuum-polarization potential. Details of the methods used are given in Ref. [15]. Calculations were made for l = 8, 9 and 10 states for ,~-Pb atoms for n-values up to n = 21 and for l -- 7, 8, 9 states for ~-Pb atoms up to n = 20. Typical results are shown in Fig. 2 where the optical-model calculations are seen to be in qualitative agreement with the predictions of the West formula for l = 8 states in ,~-Pb atoms. In particular the values for the ratio F ( n + 1, l ) / F ( n , l) greater than 1 are confirmed. The predictions of Eq. (7) are too high by a factor 1.13 at n = 9, decreasing to a factor of 1.01 at n = 20. Results for the other cases considered are similar. The poorest agreement is for l = 7 states in ~-Pb atoms where the factor varies from 1.26 at n = 8 to 1.02 at n = 19, presumably due to the increased distortion of the atomic wave function in the region of the nucleus by the strong antiproton annihilation.
~
2
............
5
10
n
15
orbital angular m o m e n t u m l = 3 to l = 8, to the relevant "circular" (n = l + 1, 1) states for a range of principal quantum number up to n = 17. The ratio of nuclear capture rates decreases monotonicallyonly for states of l < 3. Fig. 1. Ratio o f nuclear capture rates for states o f
C.J. Batty, R.E. Welsh/Nuclear Physics A 589 (1995) 601-608
605
E-Pb atom 1=8 2.4
2.2 ,....a
2 f.
t
1.8 + 1.6
1.4
1.2
0.8 0.6
I
8
I t I , ] I I h ] L I 11
10
12
14
16
I I I I I 11
18
L i i
20
L I
22
rl Fig. 2. R a t i o o f w i d t h s F ( n + l , l ) / F ( n , l) f o r , Y - - P b a t o m s w i t h l = 8 as a f u n c t i o n o f the p r i n c i p a l q u a n t u m n u m b e r n. T h e o p t i c a l - m o d e l c a l c u l a t i o n s are i n d i c a t e d b y a solid line a n d t h o s e f r o m Eq. (7) b y a d a s h e d line.
Calculations were also made for ~-p atoms since the West formula is again used in descriptions of the atomic cascade [6] where in this case, Stark mixing effects must also be included. For l = 0 states, which are of particular interest, the factor comparing the predictions of Eq. (7) to the " e x a c t " calculations varied between 0.95 at n -- 1, 0.99 at n = 2, reaching 0.999 at n = 8. The agreement between the two methods for calculating the widths of upper levels is particularly striking in this case. In order to gain some qualitative physical understanding of these results, the relevant hydrogenic wave functions were calculated. Fig. 3 shows plots for l = 8 atomic states of ,~--Pb atoms for principal quantum numbers extending from n = 9 to 17. Although a quantitative conclusion cannot be drawn directly from these plots, the radial nodes of the higher-n states are seen to give a rapid rise in the wave functions near r = 0. In Fig. 4 is shown a simultaneous plot of the n = 9, l = 8 and n = 12, l = 8 wave functions from r = 0 to well beyond the classical radius of the orbiting negative particle. In this plot, it is somewhat more apparent that the higher principal quantum number state rises more rapidly near the origin than the circular state. It is this increased overlap with the nucleus which gives rise to the larger strong interaction widths in the high-n states.
C.J. Batty, R.E. Welsh/Nuclear Physics A 589 (1995) 601-608
606
~k~
0
a) 9,8
c) 11,8
b) 10,8
d) 12,8
e) 13,8
f) 14,8
g) 15,8
h) 16,8
i) 17,8
200 400 0 r (fro)
200 400 0 r (fro)
200
400
r (fro)
Fig. 3. Plots of atomic hydrogenie wave functions for atomic states (n, l) = (9, 8) to (17, 8).
I/
o
0
2
4
6
100
200 r (fm)
Fig. 4. Plots of the hydrogenie wave functions for states (n, l) = (9, 8) and (12, 8). The inset shows how the (12, 8) atomic state rises faster near the origin than the circular state.
c.J. Batty, R.E. Welsh/Nuclear Physics A 589 (1995) 601-608
607
This picture is confirmed by inspection of Eq. (4) which gives an approximate form for the hydrogenic wave function close to r = 0. For l = 3, the coefficient of r; decreases monotonically as n increases. For l = 4 the coefficient initially increases for n = 5 and then decreases monotonically for larger n as expected from Fig. 1. For larger /-values the coefficient of r l again behaves in the way expected from Fig. 1. The complex wave functions obtained from the optical-model calculations have been compared with those obtained using the Coulomb potential alone, which are of course purely real. The real parts of the two sets of wave functions are found to be remarkably similar whereas the imaginary part obtained in the presence of the complex nuclear optical potential is typically less than 10 -2 of the real part in the nuclear region. This indicates that the use of hydrogenic wave functions for the estimation of the strong interaction width is a reasonable approximation.
6. Conclusions We conclude that hadronic atom capture ratios in high-Z atoms can show unexpected capture ratios when compared to the more familiar low-Z pionic atoms. In the comparisons we have made, the approach given by West [2] appears to provide a reasonable representation of the capture ratios from these atomic states over a wide range of Z, n and I. It is especially interesting to find that high-n states for a given l can show far higher nuclear overlap and thus higher capture rates than do the circular states. Indeed, for states of l = 19 one finds that some higher-n states have an overlap with the nucleus that yields hundreds of times higher capture rates than the circular (n = 20, l = 19) state. Such anomalous effects are more pronounced in kaonic, antiprotonic and hyperonic atoms where the last observed atomic transition can occur at n = 8 or higher. Finally we should comment that a cascade calculation for ~-Pb atoms using optical-model predictions for the capture ratios in high-n states for 1 = 8, 9, 10 gave very similar results to one using Eq. (7). The overall effect was to change the predicted x-ray intensity for circular transitions by less than 1%, although changes in some of the higher non-circular transitions were as large as 10%. The absolute changes in the fraction of ~ - captured from the various angular momentum states were always less than 1%.
Acknowledgements We thank J. Hiifner for the use of his cascade code. We gratefully acknowledge useful discussions with members of the BNL-E723 and CERN-PS174 Collaborations. One of us (R.E.W.) thanks the Rutherford Appleton Laboratory and the Oxford University Department of Physics for their hospitality. Work supported in part by the USNSF, grant PHY9224939.
608
CJ. Batty, R.E. Welsh/Nuclear Physics A 589 (1995) 601-608
References [1] V.L. Fitch and J. Rainwater, Phys. Rev. 92 (1953) 789. [2] D. West, Rep. Prog. Phys. 21 (1958) 271, see in particular eq. A9. [3] See, e.g., D.W. Hertzog et al., Phys. Rev. D 37 (1988) 1142; K.P. Gall et al., Phys. Rev. Lett. 60 (1988) 186; R.J. Powers et al., Phys. Rev. C 47 (1993) 1263; C.J. Batty et al., Phys. Rev. C 40 (1989) 2154. [4] C.J. Batty, E. Friedman, H.J. Gils and H. Rebel, Adv. Nucl. Phys. 19 (1989) 1. [5] P. Lubinski et al., Phys. Rev. Lett. 73 (1994) 3199. [6] C.J. Batty, Rep. Prog. Phys. 52 (1989) 1165, and references therein. [7] N. Morita et al., Phys. Rev. Lett. 72 (1994) 1180; R.S Hayano et al., Phys. Rev. Lett. 73 (1994) 1485. [8] See, e.g., K. Ishida et al., Hyperfine Interactions 82 (1993) 111. [9] Y. Eisenberg and D. Kessler, Nuovo Cimento 19 (1961) 1195. [10] Y. Eisenberg and D. Kessler, Phys. Rev. 123 (1961) 1472; 130 (1963) 2352. [11] J. Hiifner, Z. Phys. 195 (1966) 365; private communication. [12] W.W. Sapp, Ph.D. Thesis, College of William and Mary, Williamsburg, VA (1971), unpublished; W.W. Sapp et al., Phys. Rev. C 5 (1972) 690. [13] Z. Fried and A.D. Martin, Nuovo Cimento 29 (1963) 574. [14] M.J. Turner and Daphne F. Jackson, Phys. Lett. B 130 (1983) 362. [15] C.J. Batty, Nucl. Phys. A 372 (1981) 418; 433; Sov. J. Part. Nucl. 13 (1982) 71.
Nuclear Physics A 589 (1995) 601-608
Nuclear capture in the atomic cascade of kaonic, antiprotonic and sigma hyperonic exotic atoms C.J. Batty a, R.E. Welsh b a RutherfordAppleton Laboratory, Chilton, Didcot, Oxon 0)(11 OQX, UK b Physics Department, College of William and Mary, Williamsburg, VA 23187 USA Received 7 March 1995
Abstract
Ratios of nuclear capture rates are shown to increase with increasing principal quantum number n for a given orbital quantum number l, in the high-n, high-/ states most relevant to studies of the exotic hadronic atoms of heavy particles. This surprising behaviour, which is confirmed by optical-model calculations, is in contrast to similar capture ratios in pionic atoms at low atomic number. Plots of the relevant hydrogenic atom wave functions are shown to explain such effects.
I. Introduction
With the production of beams of muons and pions at synchrocyclotrons, studies of exotic atoms were begun in the 1950's with the pioneering work of Fitch and Rainwater [1] at Columbia. In one of the earliest reviews of such studies, West [2] described the experimental and theoretical progress in a summary of the field that was at that time referred to as "mesonic atoms". In the mid 1960's, development of solid-state Ge and Si radiation detectors of high resolution gave impetus to more precise measurements of the properties of exotic atoms. Such measurements permit the extraction of precise information about the orbiting particle [3], such as charge/mass, and magnetic moment, and the interaction of such particles with nuclei. In addition, properties of nuclei [4], such as nuclear size, nuclear polarization, and neutron halo effects in heavy nuclei [5], have been studied. Furthermore, the mechanism of atomic capture of such heavy charged particles and atomic effects such as Stark mixing [6], trapping [7] and transfer of heavy negative particles between atoms of different species [8] have been studied extensively. 0375-9474/95/$09.50 © 1995 Elsevier Science B.V. All rights reserved SSD! 0 3 7 5 - 9 4 7 4 ( 9 5 ) 0 0 1 2 0 - 4
602
C.J. Batty, R.E. Welsh/Nuclear Physics A 589 (1995) 601-608
2. Atomic calculations Some of the first cascade calculations for muonic atoms were carried out by Eisenberg and Kessler [9] and later extended to pionic and kaonic atoms [10]. In this latter work, the nuclear capture probability for the pion or kaon was calculated from the overlap of the atomic wave function with the density distribution of the nucleus. For reasons of simplicity hydrogenic wave functions were used. In the late 1960's, Hiifner [11] wrote one of the first general purpose computer programs to describe the atomic cascade of negative muons in muonic atoms. This computer program has been the basis of much subsequent work. In 1971, this program was modified by Sapp [12] to include the effects of nuclear capture in low-Z pionic atoms. Experimentally measured values of strong interaction widths were used for the circular (n, l = n - 1) atomic states whilst the formula set out by West [2], which expresses ratios of absorption widths for states of given l and increasing n, was used for the higher, non-circular states. From the early 1970's to the present, this program, with the corrections discussed by Fried and Martin [13] included, has been used extensively to interpret exotic atom experiments. The validity of the West formula has been discussed by Turner and Jackson [14] for pionic atoms for orbitals of 1 > 1. They have also derived an improved formula for pionic s and p atomic states. Cascade programs are especially valuable in the interpretation of hadronic atoms. It is often possible to measure directly the strong interaction shift and width of only the lowest state in such atoms. Information about nuclear capture in higher atomic levels is then inferred from measurements of the relative yields of the atomic x-ray transitions. Thus, a prediction of the ratio of nuclear overlap of the various (n, l) atomic states is important to an accurate interpretation of experimental results.
3. Experiments on sigma hyperons Most recently, in measurements carried out at Brookhaven National Laboratory principally to extract the magnetic moment of the sigma hyperon [3], determinations have been made of nuclear shifts and widths and relative yields of kaonic atoms and sigma hyperonic atoms for principal quantum numbers as high as n = 11. When the approximation taken from West [2] was applied to the relative capture rates in high-n, high-/atomic states for these atoms, an unexpected result was obtained. In particular, the program of Hiifner, as modified by Sapp, suggested that in atomic states of orbital angular momentum quantum number l = 9, the so-called circular state for n = 10 has less overlap with the nucleus than several of the states of higher n with the same 1. When this result was first seen, it was suspected that the approximation taken from West might not apply for high-Z, high-n states in exotic, hadronic atoms. On testing the ratios predicted by the West formula against the predictions of a program using realistic wave functions and an optical-model representation of the nucleus, we were surprised to discover that very similar capture ratios were obtained.
CJ. Batty, R.E. Welsh/Nuclear PhysicsA 589 (1995) 601-608
603
We thus took steps to determine whether such results were physically meaningful and to try to understand the origin of the apparently surprising behaviour.
4. Theoretical background The strong interaction width F, for the atomic state (n, l) is given by
-4~fl ffJnl 12 I m U ( r ) r 2 dr
½F(n, 1) =
(1)
where ~0,t is the atomic wave function for the (n, l) state and U(r) is the optical potential representing the strong hadron-nucleus interaction. West [2] uses hydrogenic wave functions
=
l
(4Z3(n-l-l)') _ _
1/2
e-o/Z, tLZl+lt p)
(2)
normalized so that cx~
fo 4"n'r2qj2t dr = 1, and p = 2Zr/na o with a 0 = Bohr radius of the bound particle for n = 1, Z = 1: a o = he/Txe :. Lkn(x) is the associated Laguerre polynomial. Since p is small, West then approximates the terms e -p/2 and -2t+ 1.,+ ~1,t p) by their values at p = 0: -{(n+t)t}' 1)!(2/+ 1)!'
~+~, (0) = ( n - l Ln+
(3)
and
~On' - ~
1( 11c4z3
,n+,,
na o ] ~ n4a3o(n - / - 1 ) ! ( ( 2 /
+ 1)!} 2
)lj2r
Using Eq. (1) and assuming a uniform optical potential of radius R = r o A l / 3 ~ Z 1/3, West then obtains Z 81/3+4
r ( n , l) cc ~ a~t+
(n-~- I)!
n21+4(n -- I-- 1)![(21 + 1)!]2(21 + 3)
(5)
from which
F(m, l) = ( n )2l+4 (m + l ) , ( n - - l - - 1 ) ,
r(n, l)
--
(m-l-1)!(n+
t)!
o
(6)
And for the particular case of m = n + 1 r(n+l,l) r ( n, ~)
(
n
]21+4n+1+1
~-g-4S j
n - t
(7)
604
C.I. Batty, R.E. Welsh/NuclearPhysicsA 589 (1995) 601-608
5. Results
Fig. 1 shows a plot of the ratio of nuclear overlap for hydrogenic atoms for a range of n- and /-values, beginning with the circular states for l = 3 to l = 8 and extending to higher values in n as given by the formula (Eq. (6)) of West. As can be seen, overlap with the nucleus in the higher-/atomic states relevant to antiprotonic, sigma hyperonic and kaonic atoms appears to increase for states of principal quantum number above that of the circular state, dropping off only gradually with higher n. As a check on the validity of the West formula (Eq. (7)), the ratio F ( n + 1, l ) / F ( n , l) was also calculated by numerical solution of the Klein-Gordon equation for the Coulomb plus nuclear system including an optical potential proportional to the nuclear density [15] to describe the hadron-nucleus interaction. The Coulomb potential used was that due to the finite charge distribution of the nucleus together with the first order ot(Zol) vacuum-polarization potential. Details of the methods used are given in Ref. [15]. Calculations were made for l = 8, 9 and 10 states for ,~-Pb atoms for n-values up to n = 21 and for l -- 7, 8, 9 states for ~-Pb atoms up to n = 20. Typical results are shown in Fig. 2 where the optical-model calculations are seen to be in qualitative agreement with the predictions of the West formula for l = 8 states in ,~-Pb atoms. In particular the values for the ratio F ( n + 1, l ) / F ( n , l) greater than 1 are confirmed. The predictions of Eq. (7) are too high by a factor 1.13 at n = 9, decreasing to a factor of 1.01 at n = 20. Results for the other cases considered are similar. The poorest agreement is for l = 7 states in ~-Pb atoms where the factor varies from 1.26 at n = 8 to 1.02 at n = 19, presumably due to the increased distortion of the atomic wave function in the region of the nucleus by the strong antiproton annihilation.
~
2
............
5
10
n
15
orbital angular m o m e n t u m l = 3 to l = 8, to the relevant "circular" (n = l + 1, 1) states for a range of principal quantum number up to n = 17. The ratio of nuclear capture rates decreases monotonicallyonly for states of l < 3. Fig. 1. Ratio o f nuclear capture rates for states o f
C.J. Batty, R.E. Welsh/Nuclear Physics A 589 (1995) 601-608
605
E-Pb atom 1=8 2.4
2.2 ,....a
2 f.
t
1.8 + 1.6
1.4
1.2
0.8 0.6
I
8
I t I , ] I I h ] L I 11
10
12
14
16
I I I I I 11
18
L i i
20
L I
22
rl Fig. 2. R a t i o o f w i d t h s F ( n + l , l ) / F ( n , l) f o r , Y - - P b a t o m s w i t h l = 8 as a f u n c t i o n o f the p r i n c i p a l q u a n t u m n u m b e r n. T h e o p t i c a l - m o d e l c a l c u l a t i o n s are i n d i c a t e d b y a solid line a n d t h o s e f r o m Eq. (7) b y a d a s h e d line.
Calculations were also made for ~-p atoms since the West formula is again used in descriptions of the atomic cascade [6] where in this case, Stark mixing effects must also be included. For l = 0 states, which are of particular interest, the factor comparing the predictions of Eq. (7) to the " e x a c t " calculations varied between 0.95 at n -- 1, 0.99 at n = 2, reaching 0.999 at n = 8. The agreement between the two methods for calculating the widths of upper levels is particularly striking in this case. In order to gain some qualitative physical understanding of these results, the relevant hydrogenic wave functions were calculated. Fig. 3 shows plots for l = 8 atomic states of ,~--Pb atoms for principal quantum numbers extending from n = 9 to 17. Although a quantitative conclusion cannot be drawn directly from these plots, the radial nodes of the higher-n states are seen to give a rapid rise in the wave functions near r = 0. In Fig. 4 is shown a simultaneous plot of the n = 9, l = 8 and n = 12, l = 8 wave functions from r = 0 to well beyond the classical radius of the orbiting negative particle. In this plot, it is somewhat more apparent that the higher principal quantum number state rises more rapidly near the origin than the circular state. It is this increased overlap with the nucleus which gives rise to the larger strong interaction widths in the high-n states.
C.J. Batty, R.E. Welsh/Nuclear Physics A 589 (1995) 601-608
606
~k~
0
a) 9,8
c) 11,8
b) 10,8
d) 12,8
e) 13,8
f) 14,8
g) 15,8
h) 16,8
i) 17,8
200 400 0 r (fro)
200 400 0 r (fro)
200
400
r (fro)
Fig. 3. Plots of atomic hydrogenie wave functions for atomic states (n, l) = (9, 8) to (17, 8).
I/
o
0
2
4
6
100
200 r (fm)
Fig. 4. Plots of the hydrogenie wave functions for states (n, l) = (9, 8) and (12, 8). The inset shows how the (12, 8) atomic state rises faster near the origin than the circular state.
c.J. Batty, R.E. Welsh/Nuclear Physics A 589 (1995) 601-608
607
This picture is confirmed by inspection of Eq. (4) which gives an approximate form for the hydrogenic wave function close to r = 0. For l = 3, the coefficient of r; decreases monotonically as n increases. For l = 4 the coefficient initially increases for n = 5 and then decreases monotonically for larger n as expected from Fig. 1. For larger /-values the coefficient of r l again behaves in the way expected from Fig. 1. The complex wave functions obtained from the optical-model calculations have been compared with those obtained using the Coulomb potential alone, which are of course purely real. The real parts of the two sets of wave functions are found to be remarkably similar whereas the imaginary part obtained in the presence of the complex nuclear optical potential is typically less than 10 -2 of the real part in the nuclear region. This indicates that the use of hydrogenic wave functions for the estimation of the strong interaction width is a reasonable approximation.
6. Conclusions We conclude that hadronic atom capture ratios in high-Z atoms can show unexpected capture ratios when compared to the more familiar low-Z pionic atoms. In the comparisons we have made, the approach given by West [2] appears to provide a reasonable representation of the capture ratios from these atomic states over a wide range of Z, n and I. It is especially interesting to find that high-n states for a given l can show far higher nuclear overlap and thus higher capture rates than do the circular states. Indeed, for states of l = 19 one finds that some higher-n states have an overlap with the nucleus that yields hundreds of times higher capture rates than the circular (n = 20, l = 19) state. Such anomalous effects are more pronounced in kaonic, antiprotonic and hyperonic atoms where the last observed atomic transition can occur at n = 8 or higher. Finally we should comment that a cascade calculation for ~-Pb atoms using optical-model predictions for the capture ratios in high-n states for 1 = 8, 9, 10 gave very similar results to one using Eq. (7). The overall effect was to change the predicted x-ray intensity for circular transitions by less than 1%, although changes in some of the higher non-circular transitions were as large as 10%. The absolute changes in the fraction of ~ - captured from the various angular momentum states were always less than 1%.
Acknowledgements We thank J. Hiifner for the use of his cascade code. We gratefully acknowledge useful discussions with members of the BNL-E723 and CERN-PS174 Collaborations. One of us (R.E.W.) thanks the Rutherford Appleton Laboratory and the Oxford University Department of Physics for their hospitality. Work supported in part by the USNSF, grant PHY9224939.
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