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A new look at nuclear excitation in an electron transition
Physics Letters B 272 ( 1991 ) 1-4 North-Holland
PHYSICS LETTERS B
A n e w look at nuclear excitation in an electron transition A. Ljubi~i6, D. Kekez Ruder Bogkovi{ Institute, P.O. Box 1016, YU-41001 Zagreb, Croatia, Yugoslavia
and B.A. Logan Ottawa-Carleton Institute of Physics, University of Ottawa, Ottawa, Canada K1N 6N5
Received 5 July 1991; revised manuscript received 16 September 1991
The mechanism for nuclear excitation in an electron transition (NEET) has been reanalyzed. In contrast to other calculations our analysis shows that the NEET process probability does not depend on the details of the nuclear transition but rather on the properties of the electron states. Our predictions are in very good agreement with the available experimental data.
Morita [ 1 ] was the first to consider and calculate the probability o f the excitation o f a nucleus in an electron transition ( N E E T ) ; there has been another recent theoretical analysis [ 2 ]. In these analyses the N E E T probabilities d e p e n d on the details o f both the nuclear and a t o m i c parameters. In this work we show that the N E E T probabilities d e p e n d on the details o f the atomic p a r a m e t e r s only. Consider the scattering o f a p h o t o n by two oscillators each with identical resonant energies mo which involve the same m u l t i p o l a r i t y in the excitation; the linewidths are different. W h e n the separation between the two oscillators R is much larger than the resonant wavelength 20 photon scattering is described by the sum o f two separate cross sections. The situation is both m o r e interesting and complex when R <20; in that case, in our scattering e x p e r i m e n t we cannot distinguish between the two oscillators any more and the system behaves as one oscillator, with one linewidth F. Simple q u a n t u m mechanical arguments reveal that the total linewidth f has to be the sum o f the linewidths Fj and/"2 o f the two oscillators. In that case the scattering cross section is given with the expression ngl22 2
a= ~
F2
( m _ m o ) : + ~ F 2,
(1)
where F = F ~ +F2, g~2 is a statistical factor and m is the p h o t o n energy. The result ( 1 ) means that we excite both oscillators in the scattering process. I f one o f the linewidths is much larger than the other the corresponding oscillator will d o m i n a t e the decay. The process can be visualized as a p h o t o n being absorbed by the oscillator with larger radiative width, being re-emitted and a b s o r b e d by the second oscillator, which than decays via the stronger oscillator. In essence the faster oscillator presents an additional channel for the slower oscillator, in analogy with the example o f the decay o f a nuclear level by internal conversion. This process can be presented with the diagram shown in fig. 1. This linearized picture can be used to study the cross section for p h o t o n scattering on two oscillators which have slightly different resonant energies COl and m2. We assume that each oscillator has several decay channels and that the radiative and total widths o f the first oscillator are much larger than the corresponding widths o f the second oscillator, i.e. F~ >> F2 and F~w >> F2T. The total cross section allows for scattering on each oscillator separately [3 ]. Bearing in m i n d that for R < 2 o in both scatterings the linewidths o f the stronger oscillator d o m i n a t e we can express the total scattering cross section as
0370-2693/91/$ 03.50 © 1991 Elsevier Science Publishers B.V. All rights reserved.
1
Volume 272, n u m b e r 1,2
PHYSICS LETTERS B
cross section for the resonant scattering of the photon on the electron system which can decay by several channels and we obtain
[a~Jo
p~ g~ FR(n'l'm'-,nlm)FT(nlm) 27t [ ( B,,,l.m,-Bnlm) --CON]2+ [ ½FT(nlm ) ]2' (4)
JOo oo caao
~,
%
07,
Fig. 1. Diagram representing resonant scattering of a photon on two closely spaced q u a n t u m oscillators ~1 and 7*2.
~rg]22 2
FIFIT
g2
28 N o v e m b e r 1991
FIFIT
p=F-2A
(co_co~)2+la/"]T 2 2zr (gO-- 092 ) 2+ a1/ ' l 2x (2)
where gi and g2 are the statistical factors of the two oscillators. The factor ½~zis a normalization factor. This general result can be applied to the NEET process if we regard the atom as consisting of two coupled oscillators corresponding to the electron and nuclear transitions, with the oscillator associated with the electron transition being the stronger of the two. When an electron is excited the nucleus can be excited in its de-excitation. The physical separation can be taken to be the Bohr radius which is <2o for the cases of interest. If the energies of the transitions are approximately equal, and of the same multipolarity, they satisfy the same condition as the coupled indistinguishable oscillators we have considered and we can apply eq. (2). We can regard the mechanism as proceeding as follows: a photon with an energy ¢ol equal to the energy difference between the (nTm') and (nlm) electron states given as the energy difference (Bn,l, m, --Bnfm) between the corresponding binding energies B is absorbed by the electron, it reemits the photon which excites the nuclear state with energy 092= CON. The NEET probability P is defined as Number of excited nuclear states P = Number of created electron vacancies "
where the nuclear statistical factor gN = (2•*+ 1 )/ (210+ 1 ) where I* and Io are respectively the total spins of the excited and ground states, and FT(nlm) and F• (n Tm'--, nlm) are the total width of the initial electron state and the radiative width for the transition between the two electron states, respectively. We can compare our result with the most recent calculation [2]. In this calculation P, the NEET probability per K-shell hole, is given by
(3)
In our case we obtain P if we divide eq. (2) by the
-21¢2/')2 zC~iK
,
(5)
where F is the electron level width and A is the nuclear-electron energy difference. Fc is the Coulomb energy of the two initial electron states and is defined as
Fc =ce
f drl dr2 ]gK(rl)[2] ~i(r2)I 2 Ir, --/'2 I '
(6)
where ~'K is the K-shell wave function and ~ui is the initial orbital electron wave function. The photon propagator QiK is given by
Q2K=
(gN/27~)CONFo
IB (e,m) [ 2 ,
(7)
where gN is the statistical weight and Fo is the ground state transition width of the excited nuclear level. Expressions for bB (.... )12 for different multipolarities are given in ref. [ 2 ]. Our result for P is completely different from the expressions obtained in earlier analyses [ 1,2]. The most important difference is that our result does not depend on the details of the nuclear transition. In contrast the result of Pisk et al. [2] gives P, being directly proportional to Fo; in Morita's analysis P was also proportional to nuclear parameters. Another difference is that our analysis gives a finite value for P even when the nuclear transition energy is exactly equal to the electron transition energy, in the previous analyses the transition probability diverges under this condition. The earlier theoretical and experimental papers
Volume 272, number 1,2
PHYSICS LETTERS B
have estimated E', the Coulomb interaction energy between the nucleus and the orbital electron. In our formalism E' is approximately given by (gNFRFX/ 27~) 1/2
The ultimate test of the accuracy of our analysis is the comparison of our predictions with the available experimental results for 237"NT~93,,1 a [4], 197A'79z~,U [ 5 ] , 1789Os [ 6-8]. All of the experiments involved transitions to the K-shell and the calculations of Leisi et al. [9] were used to obtain FT(K) values of 42.62 eV for 1869Os, 49.63 eV for ~97Au and 94.23 eV for 2337Np. The relevant radiative transitions are M4--,K and M5--,K for 189c~ 76~J~, M1--,K for 19779z~kUand L 3 - , K for 293~Np. The work of Schofield [10] was used as the basis for the estimation of these values. It was necessary to use extrapolations and interpolations and these were based on the well-known dependence of the probabilities on Z 4, and on the dependence on n-3 for different shells. We obtain a FR(M4--,K) of 0.053 eV and a F R ( M 5 - , K ) of 0.066 eV for 189¢~_ 76~.J~i, a f ' R ( M I ~ K ) of 0.0162 eV for 197Au, and a f'R(L3--,K) 0f45.81 eV for 293~Np. Our results for P, along with some predictions of previous analyses, are compared with the experimental values in table 1. In the case of 237Np our result is a dramatic improvement on the earlier prediction. The experimental uncertainty is very high for ~997Au and it is difficult to arrive at a very detailed conclusion. However, our prediction is compatible with the experimental result. There are three experimental results available for 189¢~^ 76,-,~. The earlier theoretical predictions [ 1,2 ] and
28 November 1991
ours all agree with the first experimental result [6]. Two more recent experimental results [7,8] are not in agreement with each other and both are lower than the theoretical prediction. It should be emphasized that all of these experiments are difficult and involve significant systematic errors. In the investigation of Otozai et al. [6] there are uncertainties in the value of the electron current and the GM counter efficiency. In the experiment of Saito et al. [7] there could be problems with the calculated bremsstrahlung spectrum. The most recent work used synchrotron radiation to excite the nuclear level; there was some uncertainty in determining the efficiency of the detector used to monitor the NEET production [ 8 ]. In all the experiments corrections have to be made for other possible nuclear excitation mechanisms. Although the most recent work involved a synchrotron source the photon source could not be tuned at the energy involved and indirect estimates had to be made of excitation via resonance fluorescence and by Compton excitation of nuclei [ 1 1,12 ]. This is unfortunate as a tunable photon source would allow a direct experimental estimate of secondary excitation contributions and this would allow more confidence to be placed in the experimental results. A superficial analysis of our mechanism may suggest that the nuclear state would decay with the halflife of the electronic state. This is prevented by the Pauli exclusion principle; when an electron jumps from a higher shell n2 into a lower shell nl it emits a photon which is absorbed by the nucleus. However, very quickly an electron from shell n3 jumps to shell
Table 1 Experimental and theoretical values for the NEET probability. Nucleus
Type of transition
Theory Pisk et al. [2]
our result
Experiment
Reference
23~Np
El
1.5)< 10 -7
2 . 6 X 10 - 4
(2.1 -+0.6) × 10 -4
[4]
197 79Au
M1
0.35)< 10 -4
0 . 2 2 × 10 -4
(2.2_+1.8)>(10 -4
[5]
189 76Os
MI E2
2.3)<10 -7 0.18)<10 7
1.06>( 10 7 1.25)< 10 -7
2.48)< 10 -7
2.31>(10 -7
(1.7-+0.2)>(10 -7 4.3>(10 -8 (5.7+1.7)>(10 -9
[6] [7] [8]
total
Volume 272, number 1,2
PHYSICS LETTERS B
/12 a n d t h i s p r e v e n t s t h e n u c l e u s f r o m e m i t t i n g a p h o t o n t o b e a b s o r b e d b y t h e i n i t i a l e l e c t r o n . T h e isol a t e d n u c l e u s d e c a y s w i t h its o w n half-life.
References [ 1 ] M. Morita, Prog. Theor. Phys. 49 (1973) 1574. [2] K. Pisk, Z. Kaliman and B.A. Logan, Nucl. Phys. A 504 (1989) 103. [3] A. Messiah, Quantum mechanics (North-Holland, Amsterdam, 1967) p. 856. [4] T. Saito, A. Shinohara and K. Otozai, Phys. Lett. B 92 (1980) 293.
28 November 1991
[5] H. Fujioka, K. Ura, A. Shinohara, T. Saito and K. Otozai, Z. Phys. A 315 (1984) 121. [6] K. Otozai, R. Arakawa and T. Saito, Nucl. Phys. A 297 (1978) 97. [7] T. Saito, A. Shinohara, T. Miura and K. Otozai, J. Inorg. Nucl. Chem. 43 (1981) 1963. [ 8 ] A. Shinohara, T. Saito, M. Shoi, A. Yokoyama, H. Baba, M. Ando and K. Taniguchi, Nucl. Phys. A 472 ( 1987 ) 151. [9] H.I. Leisi, I.H. Brunner, C.F. Perdrisat and D. Sherrer, Helv. Phys. Acta 34 (1961) 161. [ 10] J.H. Scofield, Phys. Rev. 179 (1969) 9. [ 11 ] I.S. Batkin, Sov. J. Nucl. Phys. 29 (1979) 464. [ 12 ] I.S. Batkin and M.I. Berkman, Sov. J. Nucl. Phys. 32 (1980) 502.
PHYSICS LETTERS B
A n e w look at nuclear excitation in an electron transition A. Ljubi~i6, D. Kekez Ruder Bogkovi{ Institute, P.O. Box 1016, YU-41001 Zagreb, Croatia, Yugoslavia
and B.A. Logan Ottawa-Carleton Institute of Physics, University of Ottawa, Ottawa, Canada K1N 6N5
Received 5 July 1991; revised manuscript received 16 September 1991
The mechanism for nuclear excitation in an electron transition (NEET) has been reanalyzed. In contrast to other calculations our analysis shows that the NEET process probability does not depend on the details of the nuclear transition but rather on the properties of the electron states. Our predictions are in very good agreement with the available experimental data.
Morita [ 1 ] was the first to consider and calculate the probability o f the excitation o f a nucleus in an electron transition ( N E E T ) ; there has been another recent theoretical analysis [ 2 ]. In these analyses the N E E T probabilities d e p e n d on the details o f both the nuclear and a t o m i c parameters. In this work we show that the N E E T probabilities d e p e n d on the details o f the atomic p a r a m e t e r s only. Consider the scattering o f a p h o t o n by two oscillators each with identical resonant energies mo which involve the same m u l t i p o l a r i t y in the excitation; the linewidths are different. W h e n the separation between the two oscillators R is much larger than the resonant wavelength 20 photon scattering is described by the sum o f two separate cross sections. The situation is both m o r e interesting and complex when R <20; in that case, in our scattering e x p e r i m e n t we cannot distinguish between the two oscillators any more and the system behaves as one oscillator, with one linewidth F. Simple q u a n t u m mechanical arguments reveal that the total linewidth f has to be the sum o f the linewidths Fj and/"2 o f the two oscillators. In that case the scattering cross section is given with the expression ngl22 2
a= ~
F2
( m _ m o ) : + ~ F 2,
(1)
where F = F ~ +F2, g~2 is a statistical factor and m is the p h o t o n energy. The result ( 1 ) means that we excite both oscillators in the scattering process. I f one o f the linewidths is much larger than the other the corresponding oscillator will d o m i n a t e the decay. The process can be visualized as a p h o t o n being absorbed by the oscillator with larger radiative width, being re-emitted and a b s o r b e d by the second oscillator, which than decays via the stronger oscillator. In essence the faster oscillator presents an additional channel for the slower oscillator, in analogy with the example o f the decay o f a nuclear level by internal conversion. This process can be presented with the diagram shown in fig. 1. This linearized picture can be used to study the cross section for p h o t o n scattering on two oscillators which have slightly different resonant energies COl and m2. We assume that each oscillator has several decay channels and that the radiative and total widths o f the first oscillator are much larger than the corresponding widths o f the second oscillator, i.e. F~ >> F2 and F~w >> F2T. The total cross section allows for scattering on each oscillator separately [3 ]. Bearing in m i n d that for R < 2 o in both scatterings the linewidths o f the stronger oscillator d o m i n a t e we can express the total scattering cross section as
0370-2693/91/$ 03.50 © 1991 Elsevier Science Publishers B.V. All rights reserved.
1
Volume 272, n u m b e r 1,2
PHYSICS LETTERS B
cross section for the resonant scattering of the photon on the electron system which can decay by several channels and we obtain
[a~Jo
p~ g~ FR(n'l'm'-,nlm)FT(nlm) 27t [ ( B,,,l.m,-Bnlm) --CON]2+ [ ½FT(nlm ) ]2' (4)
JOo oo caao
~,
%
07,
Fig. 1. Diagram representing resonant scattering of a photon on two closely spaced q u a n t u m oscillators ~1 and 7*2.
~rg]22 2
FIFIT
g2
28 N o v e m b e r 1991
FIFIT
p=F-2A
(co_co~)2+la/"]T 2 2zr (gO-- 092 ) 2+ a1/ ' l 2x (2)
where gi and g2 are the statistical factors of the two oscillators. The factor ½~zis a normalization factor. This general result can be applied to the NEET process if we regard the atom as consisting of two coupled oscillators corresponding to the electron and nuclear transitions, with the oscillator associated with the electron transition being the stronger of the two. When an electron is excited the nucleus can be excited in its de-excitation. The physical separation can be taken to be the Bohr radius which is <2o for the cases of interest. If the energies of the transitions are approximately equal, and of the same multipolarity, they satisfy the same condition as the coupled indistinguishable oscillators we have considered and we can apply eq. (2). We can regard the mechanism as proceeding as follows: a photon with an energy ¢ol equal to the energy difference between the (nTm') and (nlm) electron states given as the energy difference (Bn,l, m, --Bnfm) between the corresponding binding energies B is absorbed by the electron, it reemits the photon which excites the nuclear state with energy 092= CON. The NEET probability P is defined as Number of excited nuclear states P = Number of created electron vacancies "
where the nuclear statistical factor gN = (2•*+ 1 )/ (210+ 1 ) where I* and Io are respectively the total spins of the excited and ground states, and FT(nlm) and F• (n Tm'--, nlm) are the total width of the initial electron state and the radiative width for the transition between the two electron states, respectively. We can compare our result with the most recent calculation [2]. In this calculation P, the NEET probability per K-shell hole, is given by
(3)
In our case we obtain P if we divide eq. (2) by the
-21¢2/')2 zC~iK
,
(5)
where F is the electron level width and A is the nuclear-electron energy difference. Fc is the Coulomb energy of the two initial electron states and is defined as
Fc =ce
f drl dr2 ]gK(rl)[2] ~i(r2)I 2 Ir, --/'2 I '
(6)
where ~'K is the K-shell wave function and ~ui is the initial orbital electron wave function. The photon propagator QiK is given by
Q2K=
(gN/27~)CONFo
IB (e,m) [ 2 ,
(7)
where gN is the statistical weight and Fo is the ground state transition width of the excited nuclear level. Expressions for bB (.... )12 for different multipolarities are given in ref. [ 2 ]. Our result for P is completely different from the expressions obtained in earlier analyses [ 1,2]. The most important difference is that our result does not depend on the details of the nuclear transition. In contrast the result of Pisk et al. [2] gives P, being directly proportional to Fo; in Morita's analysis P was also proportional to nuclear parameters. Another difference is that our analysis gives a finite value for P even when the nuclear transition energy is exactly equal to the electron transition energy, in the previous analyses the transition probability diverges under this condition. The earlier theoretical and experimental papers
Volume 272, number 1,2
PHYSICS LETTERS B
have estimated E', the Coulomb interaction energy between the nucleus and the orbital electron. In our formalism E' is approximately given by (gNFRFX/ 27~) 1/2
The ultimate test of the accuracy of our analysis is the comparison of our predictions with the available experimental results for 237"NT~93,,1 a [4], 197A'79z~,U [ 5 ] , 1789Os [ 6-8]. All of the experiments involved transitions to the K-shell and the calculations of Leisi et al. [9] were used to obtain FT(K) values of 42.62 eV for 1869Os, 49.63 eV for ~97Au and 94.23 eV for 2337Np. The relevant radiative transitions are M4--,K and M5--,K for 189c~ 76~J~, M1--,K for 19779z~kUand L 3 - , K for 293~Np. The work of Schofield [10] was used as the basis for the estimation of these values. It was necessary to use extrapolations and interpolations and these were based on the well-known dependence of the probabilities on Z 4, and on the dependence on n-3 for different shells. We obtain a FR(M4--,K) of 0.053 eV and a F R ( M 5 - , K ) of 0.066 eV for 189¢~_ 76~.J~i, a f ' R ( M I ~ K ) of 0.0162 eV for 197Au, and a f'R(L3--,K) 0f45.81 eV for 293~Np. Our results for P, along with some predictions of previous analyses, are compared with the experimental values in table 1. In the case of 237Np our result is a dramatic improvement on the earlier prediction. The experimental uncertainty is very high for ~997Au and it is difficult to arrive at a very detailed conclusion. However, our prediction is compatible with the experimental result. There are three experimental results available for 189¢~^ 76,-,~. The earlier theoretical predictions [ 1,2 ] and
28 November 1991
ours all agree with the first experimental result [6]. Two more recent experimental results [7,8] are not in agreement with each other and both are lower than the theoretical prediction. It should be emphasized that all of these experiments are difficult and involve significant systematic errors. In the investigation of Otozai et al. [6] there are uncertainties in the value of the electron current and the GM counter efficiency. In the experiment of Saito et al. [7] there could be problems with the calculated bremsstrahlung spectrum. The most recent work used synchrotron radiation to excite the nuclear level; there was some uncertainty in determining the efficiency of the detector used to monitor the NEET production [ 8 ]. In all the experiments corrections have to be made for other possible nuclear excitation mechanisms. Although the most recent work involved a synchrotron source the photon source could not be tuned at the energy involved and indirect estimates had to be made of excitation via resonance fluorescence and by Compton excitation of nuclei [ 1 1,12 ]. This is unfortunate as a tunable photon source would allow a direct experimental estimate of secondary excitation contributions and this would allow more confidence to be placed in the experimental results. A superficial analysis of our mechanism may suggest that the nuclear state would decay with the halflife of the electronic state. This is prevented by the Pauli exclusion principle; when an electron jumps from a higher shell n2 into a lower shell nl it emits a photon which is absorbed by the nucleus. However, very quickly an electron from shell n3 jumps to shell
Table 1 Experimental and theoretical values for the NEET probability. Nucleus
Type of transition
Theory Pisk et al. [2]
our result
Experiment
Reference
23~Np
El
1.5)< 10 -7
2 . 6 X 10 - 4
(2.1 -+0.6) × 10 -4
[4]
197 79Au
M1
0.35)< 10 -4
0 . 2 2 × 10 -4
(2.2_+1.8)>(10 -4
[5]
189 76Os
MI E2
2.3)<10 -7 0.18)<10 7
1.06>( 10 7 1.25)< 10 -7
2.48)< 10 -7
2.31>(10 -7
(1.7-+0.2)>(10 -7 4.3>(10 -8 (5.7+1.7)>(10 -9
[6] [7] [8]
total
Volume 272, number 1,2
PHYSICS LETTERS B
/12 a n d t h i s p r e v e n t s t h e n u c l e u s f r o m e m i t t i n g a p h o t o n t o b e a b s o r b e d b y t h e i n i t i a l e l e c t r o n . T h e isol a t e d n u c l e u s d e c a y s w i t h its o w n half-life.
References [ 1 ] M. Morita, Prog. Theor. Phys. 49 (1973) 1574. [2] K. Pisk, Z. Kaliman and B.A. Logan, Nucl. Phys. A 504 (1989) 103. [3] A. Messiah, Quantum mechanics (North-Holland, Amsterdam, 1967) p. 856. [4] T. Saito, A. Shinohara and K. Otozai, Phys. Lett. B 92 (1980) 293.
28 November 1991
[5] H. Fujioka, K. Ura, A. Shinohara, T. Saito and K. Otozai, Z. Phys. A 315 (1984) 121. [6] K. Otozai, R. Arakawa and T. Saito, Nucl. Phys. A 297 (1978) 97. [7] T. Saito, A. Shinohara, T. Miura and K. Otozai, J. Inorg. Nucl. Chem. 43 (1981) 1963. [ 8 ] A. Shinohara, T. Saito, M. Shoi, A. Yokoyama, H. Baba, M. Ando and K. Taniguchi, Nucl. Phys. A 472 ( 1987 ) 151. [9] H.I. Leisi, I.H. Brunner, C.F. Perdrisat and D. Sherrer, Helv. Phys. Acta 34 (1961) 161. [ 10] J.H. Scofield, Phys. Rev. 179 (1969) 9. [ 11 ] I.S. Batkin, Sov. J. Nucl. Phys. 29 (1979) 464. [ 12 ] I.S. Batkin and M.I. Berkman, Sov. J. Nucl. Phys. 32 (1980) 502.