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Cascade calculation for K− mesic atoms
1
~ 8.8
Nuclear Physics 43 (I963) 363--366 @ North-Holland Publishing Co., Amsterdam [
Not to be reproduced by photoprint or microfilm without writtell permission from the publisher
CASCADE CALCULATION FOR K - MESIC A T O M S J. R. ROOK
Nuclear Physics Department, Oxford Received 20 August 1962 Abstract: The competition between X-ray and Auger emission and nuclear absorption in the decay
of K - mesic atoms is calculated. It is suggested that nuclear capture occurs from states with high angular momentum but probably not from circular states.
1. Introduction
It has been suggested by Wilkinson 1) that the number of fast hyperons observed in the decay of K - mesic atoms in nuclear emulsion might indicate the presence of particle clusters in the nuclear surface. The reader is referred to ref. 1) for details of the argument; it is sufficient for our purposes to note that an essential assumption is that the K - meson is captured from states of high angular momentum. There are at least three competing processes, nuclear absorption, X-ray emission and electron emission via the Auger effect. Each mesic atom state is described by quantum numbers n and l, where l specifies the orbital angular momentum of the state and n its energy. The orbital angular momentum number can be any positive integer, or zero, while n is any integer larger than 1. Thus for each energy specified by n there are n + 1 degenerate states with different angular momenta. The probability for dipole X-ray decay from a state (nl, 11) to a state (n2, 12) is given by
Px oc (Enl-En2) 3D,,,t,,,2z2, 2
I/1-s
l =
1,
(1)
where E,, and E.2 are the energies of the states and D is the dipole matrix element 2, a). The quantity Px depends on the nuclear charge Z as Z 4. The probability for Auger decay between two states is given approximately by 3) PA 0C Z 3
1 2 n212 ~ Dnlll, #Enl--En2
1/1--/21
1
(2)
and is independent of Z. To obtain this formula we assume that the electron is emitted in a plane wave state and is initially in the ls atomic state. Since our cascade calculation depends only on the order of magnitude of the respective transition probabilities these formulae may be sufficient. It should be noted that Auger transitions with l 1 = lz are allowed but we found that these transitions were not important. The dipole matrix element favours transitions for which l 2 ~-- l 1 - 1 and n 2 = n 1 - - l , but in eq. 363
364
J . R . ROOK
(1) the first factor favours large changes in n. For this reason Px tends to favour large changes in n and the most favoured transition will be that to the circular state n2 = /2 + 1. This is the argument of Wightman 4) indicating that the cascade proceeds through the circular orbits and hence that nuclear capture occurs from these states. Unfortunately in eq. (2) the Auger effect is large for small changes in n and is larger than the probability for X-ray decay for the high principal quantum numbers. The probability for nuclear absorption from a state described by quantum numbers nl and ll is given for a nucleus with mass number A by
Pn oc 4).lh(r)W(r)~).a,(r)
(3)
f
where ~b,lh(r ) is the unperturbed atomic wave function given in ref. 2) and we assume W
W(r) =
l÷ex (r¢)
(4)
with W = 30 MeV, R = 1.07 A ~ fm and a = 0.62 fm. The probability PN depends only weakly on nl but strongly on l 1. We started our cascade calculation with nl = 15 since for higher n~ the calculation becomes more difficult and the assumptions involved in the derivation of eq. (2) become less valid. For historic reasons we considered the "emulsion nucleus" with Z = 41 and A = 93. 2. Results and Discussion
We give in table 1 the atomic state from which nuclear absorption takes place for each value of angular momentum at the n = 15 level. A unique set of results for each entry in table 1 has been obtained by assuming at each step of the calculation that the next meson state is that for which the value of the transition probability is largest. TABLE 1 Atomic state f r o m which absorption takes place for each value o f l at the n = 15 level l=
0
1
2
3
4
5
6
7
8
9
10
11
12
13
Capture
n=
15,
/ n=
12
12
12
12
11
11
10
10
11
10
9
8
7
6
14 5
state
t l =
0
0
0
0
0
1
1
2
3
4
4
4
4
4
4
In more detail it would be necessary to take account of the respective branching ratios but usually one decay mode from a given state is strongly preferred. Further a more detailed calculation would not be justified since the calculation of the Auger rate is rather crude. It has previously been assumed s-7) that decay takes place from circular orbits but we see from table 1 that this can only be the case if the meson is in a circular orbit at the n = 15 level.
CASCADE CALCULATION FOR K- MESIC ATOMS
365
We have no information at present concerning the population of states at the n = 15 level but it is of interest to see the consequences o f some plausible assumptions. I n the first case we assume a statistical, 2 l + 1, distribution at the n = 15 level. In this case the ratio of nuclear capture f r o m meson states with 1 = 0, 1, 2, 3, 4 is 25: 24 : 15 : 17 : 140, respectively. In fig. 1 we show the radial dependence of the absorption probability corresponding to this assumption. For comparison we show also the radial dependence obtained by assuming that the cascade proceeds by circular orbits so that the meson is captured in the n = 5, l = 4 state. We see that to a certain extent the surface nature of the interaction is lost. The probability of two nucleon events, assuming circular orbits and no correlations in the nucleus, has been estimated
: 5
113 rCf rn}
Fig. 1. R a d i a l d ependence o f the a b s o r p t i o n p r o b a b i l i t y c o r r e s p o n d i n g to different a s s ume d p o p u l a tions o f the n = 15 levels.
to be about 2 ~ for emulsion nuclei 7). A recalculation assuming a statistical distribution at the n = 15 level gives roughly 4~o which is well short of the experimental s) value of 20 ~o. Thus the inclusion of the Auger effect does not invalidate Wilkinson's argument if the statistical assumption is roughly correct. We may alternatively assume that the low angular m o m e n t u m states are strongly populated at the n = 15 level. This for example would be the case if the selection rules on Auger emission were valid up to very high principal quantum numbers. We see f r o m table 1 that this assumption leads to capture f r o m s states and Wilkinson's hypothesis would be invalid. In fig. 1 we show the radial dependence of the absorption probability assuming s state capture. It is not possible at present to carry out a reliable theoretical estimate of the relative population of states at the n = 15 level but some information can probably be obtained from measurements on the K - mesic X-rays. The X-ray decay predominates from the n = 8 level down and hence only those mesons which were in the l = 11, 12, 13 and 14 states at the n = 15 level can yield X-rays. In consequence we see that the X-ray yield provides a measure of the population of these states and that the
366:
~. R. ROOK
relative yields below n = 8 provides a measure of the relative populations of the high angular momentum states at the n = 15 level. In conclusion we see that the observation of a reasonable yield of X-rays indicates nuclear capture from states of high angular momentum, but not necessarily circular states, and consequently would support Wilkinson's hypothesis. These conclusions should not depend in qualitative terms on Z since only in a small region of n do the X-ray and Auger transitions compete. In consequence even an order of magnitude change in the X-ray yield will have little effect on the cascade although it will effect the value of n for which nuclear capture takes place. I would like to thank Professor D. H. Wilkinson for suggesting this work and Dr. P. B. Jones for helpful discussions. I also thank the Director of the Oxford University Computing Laboratory for the use of Mercury. References 1) D. H. Wilkinson, in Prec. Int. Conf. on Nuclear Structure (North-Holland Publ. Co., Amsterdam
1,960) 2) 3) 4) 5) 6) 7) 8)
L. I. Schiff, Quantum mechanics (McGraw-Hill, New York, 1955) Y. Eisenberg and D. Kessler, Nuovo Cim. 19 (1961) 1195 H. A. Bethe and F. de Hoffmann, Mesons and fields (Row Peterson, Evanston, Ill., 1955) P. B. Jones, Phil. Mag. 3 (1958) 33 D. H. Wilkinson, in Prec. Rutherford Jubilee Int. Conf. (Heywood, London, 1961) J. R. Rook, to be published K - European collaboration, Nuovo Cim. 14 (1959) 315
~ 8.8
Nuclear Physics 43 (I963) 363--366 @ North-Holland Publishing Co., Amsterdam [
Not to be reproduced by photoprint or microfilm without writtell permission from the publisher
CASCADE CALCULATION FOR K - MESIC A T O M S J. R. ROOK
Nuclear Physics Department, Oxford Received 20 August 1962 Abstract: The competition between X-ray and Auger emission and nuclear absorption in the decay
of K - mesic atoms is calculated. It is suggested that nuclear capture occurs from states with high angular momentum but probably not from circular states.
1. Introduction
It has been suggested by Wilkinson 1) that the number of fast hyperons observed in the decay of K - mesic atoms in nuclear emulsion might indicate the presence of particle clusters in the nuclear surface. The reader is referred to ref. 1) for details of the argument; it is sufficient for our purposes to note that an essential assumption is that the K - meson is captured from states of high angular momentum. There are at least three competing processes, nuclear absorption, X-ray emission and electron emission via the Auger effect. Each mesic atom state is described by quantum numbers n and l, where l specifies the orbital angular momentum of the state and n its energy. The orbital angular momentum number can be any positive integer, or zero, while n is any integer larger than 1. Thus for each energy specified by n there are n + 1 degenerate states with different angular momenta. The probability for dipole X-ray decay from a state (nl, 11) to a state (n2, 12) is given by
Px oc (Enl-En2) 3D,,,t,,,2z2, 2
I/1-s
l =
1,
(1)
where E,, and E.2 are the energies of the states and D is the dipole matrix element 2, a). The quantity Px depends on the nuclear charge Z as Z 4. The probability for Auger decay between two states is given approximately by 3) PA 0C Z 3
1 2 n212 ~ Dnlll, #Enl--En2
1/1--/21
1
(2)
and is independent of Z. To obtain this formula we assume that the electron is emitted in a plane wave state and is initially in the ls atomic state. Since our cascade calculation depends only on the order of magnitude of the respective transition probabilities these formulae may be sufficient. It should be noted that Auger transitions with l 1 = lz are allowed but we found that these transitions were not important. The dipole matrix element favours transitions for which l 2 ~-- l 1 - 1 and n 2 = n 1 - - l , but in eq. 363
364
J . R . ROOK
(1) the first factor favours large changes in n. For this reason Px tends to favour large changes in n and the most favoured transition will be that to the circular state n2 = /2 + 1. This is the argument of Wightman 4) indicating that the cascade proceeds through the circular orbits and hence that nuclear capture occurs from these states. Unfortunately in eq. (2) the Auger effect is large for small changes in n and is larger than the probability for X-ray decay for the high principal quantum numbers. The probability for nuclear absorption from a state described by quantum numbers nl and ll is given for a nucleus with mass number A by
Pn oc 4).lh(r)W(r)~).a,(r)
(3)
f
where ~b,lh(r ) is the unperturbed atomic wave function given in ref. 2) and we assume W
W(r) =
l÷ex (r¢)
(4)
with W = 30 MeV, R = 1.07 A ~ fm and a = 0.62 fm. The probability PN depends only weakly on nl but strongly on l 1. We started our cascade calculation with nl = 15 since for higher n~ the calculation becomes more difficult and the assumptions involved in the derivation of eq. (2) become less valid. For historic reasons we considered the "emulsion nucleus" with Z = 41 and A = 93. 2. Results and Discussion
We give in table 1 the atomic state from which nuclear absorption takes place for each value of angular momentum at the n = 15 level. A unique set of results for each entry in table 1 has been obtained by assuming at each step of the calculation that the next meson state is that for which the value of the transition probability is largest. TABLE 1 Atomic state f r o m which absorption takes place for each value o f l at the n = 15 level l=
0
1
2
3
4
5
6
7
8
9
10
11
12
13
Capture
n=
15,
/ n=
12
12
12
12
11
11
10
10
11
10
9
8
7
6
14 5
state
t l =
0
0
0
0
0
1
1
2
3
4
4
4
4
4
4
In more detail it would be necessary to take account of the respective branching ratios but usually one decay mode from a given state is strongly preferred. Further a more detailed calculation would not be justified since the calculation of the Auger rate is rather crude. It has previously been assumed s-7) that decay takes place from circular orbits but we see from table 1 that this can only be the case if the meson is in a circular orbit at the n = 15 level.
CASCADE CALCULATION FOR K- MESIC ATOMS
365
We have no information at present concerning the population of states at the n = 15 level but it is of interest to see the consequences o f some plausible assumptions. I n the first case we assume a statistical, 2 l + 1, distribution at the n = 15 level. In this case the ratio of nuclear capture f r o m meson states with 1 = 0, 1, 2, 3, 4 is 25: 24 : 15 : 17 : 140, respectively. In fig. 1 we show the radial dependence of the absorption probability corresponding to this assumption. For comparison we show also the radial dependence obtained by assuming that the cascade proceeds by circular orbits so that the meson is captured in the n = 5, l = 4 state. We see that to a certain extent the surface nature of the interaction is lost. The probability of two nucleon events, assuming circular orbits and no correlations in the nucleus, has been estimated
: 5
113 rCf rn}
Fig. 1. R a d i a l d ependence o f the a b s o r p t i o n p r o b a b i l i t y c o r r e s p o n d i n g to different a s s ume d p o p u l a tions o f the n = 15 levels.
to be about 2 ~ for emulsion nuclei 7). A recalculation assuming a statistical distribution at the n = 15 level gives roughly 4~o which is well short of the experimental s) value of 20 ~o. Thus the inclusion of the Auger effect does not invalidate Wilkinson's argument if the statistical assumption is roughly correct. We may alternatively assume that the low angular m o m e n t u m states are strongly populated at the n = 15 level. This for example would be the case if the selection rules on Auger emission were valid up to very high principal quantum numbers. We see f r o m table 1 that this assumption leads to capture f r o m s states and Wilkinson's hypothesis would be invalid. In fig. 1 we show the radial dependence of the absorption probability assuming s state capture. It is not possible at present to carry out a reliable theoretical estimate of the relative population of states at the n = 15 level but some information can probably be obtained from measurements on the K - mesic X-rays. The X-ray decay predominates from the n = 8 level down and hence only those mesons which were in the l = 11, 12, 13 and 14 states at the n = 15 level can yield X-rays. In consequence we see that the X-ray yield provides a measure of the population of these states and that the
366:
~. R. ROOK
relative yields below n = 8 provides a measure of the relative populations of the high angular momentum states at the n = 15 level. In conclusion we see that the observation of a reasonable yield of X-rays indicates nuclear capture from states of high angular momentum, but not necessarily circular states, and consequently would support Wilkinson's hypothesis. These conclusions should not depend in qualitative terms on Z since only in a small region of n do the X-ray and Auger transitions compete. In consequence even an order of magnitude change in the X-ray yield will have little effect on the cascade although it will effect the value of n for which nuclear capture takes place. I would like to thank Professor D. H. Wilkinson for suggesting this work and Dr. P. B. Jones for helpful discussions. I also thank the Director of the Oxford University Computing Laboratory for the use of Mercury. References 1) D. H. Wilkinson, in Prec. Int. Conf. on Nuclear Structure (North-Holland Publ. Co., Amsterdam
1,960) 2) 3) 4) 5) 6) 7) 8)
L. I. Schiff, Quantum mechanics (McGraw-Hill, New York, 1955) Y. Eisenberg and D. Kessler, Nuovo Cim. 19 (1961) 1195 H. A. Bethe and F. de Hoffmann, Mesons and fields (Row Peterson, Evanston, Ill., 1955) P. B. Jones, Phil. Mag. 3 (1958) 33 D. H. Wilkinson, in Prec. Rutherford Jubilee Int. Conf. (Heywood, London, 1961) J. R. Rook, to be published K - European collaboration, Nuovo Cim. 14 (1959) 315