Are Landscapes for Constrained Optimization

Physica Scripta. Vol. 57, 321È323, 1998

Are Landscapes for Constrained Optimization Problems Statistically Isotropic ? Garrison W. Greenwood1¤ and Xiaobo (Sharon) Hu2 1 Computer Science Department, Western Michigan University, Kalamazoo, MI 49008, U.S.A. 2 Computer Science and Engineering Department, University of Notre Dame, Notre Dame, IN 46556, U.S.A. Received June 23, 1997 ; accepted August 23, 1997

PACS Ref : 0250Fz

Abstract Correlation in the Ðtness landscape corresponding to an optimization problem can tell much about the landscapeÏs topology. It has previously been suggested that a long random walk is sufficient to gather this statistical information. In this paper we will show that Ðtness landscapes for constrained optimization problems are statistically anisotropic. This means a single random walk over this type of landscape is insufficient for determining correlation.

1. Introduction There has been considerable recent interest in using algorithms based on population genetics to search for optimal solutions to difficult combinatorial optimization problems. Perhaps the best known of these are the genetic algorithms. Implicit is the idea that the set of all possible solutions can be thought of as a collection of genotypes in an abstract sequence space where each distinct genotype represents a unique solution. Genotypes are considered neighbors in this sequence space if they di†er by a single mutation (i.e., only a single problem parameter has been altered). Associated with each genotype is a real number which is a measure of Ðtness where a high Ðtness value signiÐes a high quality solution. The sequence space and Ðtness values taken together form a Ðtness landscape. Searching then becomes an evolutionary process which starts at one genotype and moves by some procedure, including mutation and selection, toward higher Ðt genotypes. DeÐning an e†ective search operator for an evolutionary algorithm requires some knowledge of the Ðtness landscape topology [1]. Unfortunately, the sequence space of most optimization problems is extremely large which prohibits complete enumeration. Weinberger [2] suggested using a random walk to gather statistical information about the Ðtness landscape. Starting at some random genotype g , the 0 walk next visits a randomly chosen neighbor. Repeating this process produces a sequence of Ðtness values f , f , . . . . 0 1 Weinberger assumed that since there is some underlying distribution of Ðtness values, a random walk in any direction is sufficient to gather statistics. The degree of correlation between two genotypes s steps apart in this random walk is ÈÈÈ ¤ E-mail : garry.greenwood=wmich.edu

given by the correlation function S f f T [ S f T2 t t`s t p2 f where S É T means the expected value over all pairs s steps apart. If a high degree of correlation exists, then the landscape is ““smoothÏÏ in the sense that neighboring points di†er in Ðtness by only a small amount. Conversely, a low correlation means the landscape is ““ruggedÏÏ since neighboring points di†er markedly in Ðtness. Highly uncorrelated landscapes have a large number of local optima and any adaptive walk (i.e., a walk restricted to Ðtter neighbors) is likely to stop very quickly [1]. Most of this previous work presumes that the landscape is statistically isotropic. In other words, independent of where the random walk begins, the statistical information is invariant and only depends on genotype distance ; a sufficiently long walk in any direction will determine the correlation present in the landscape. This is analogous to gathering statistics from a single run of a stationary random process. Unfortunately, statistical isotropy is not valid for constrained combinatorial problems. In these problems the goal is to optimize an objective function subject to a set of parametric constraints. Genotypes are considered feasible if and only if they do not violate any of the constraints. R(s) \

2. Discussion Let S denote the Ðnite sequence space of a combinatorial optimization problem and let F U S deÐne the feasible region. (For unconstrained optimization problem F \ S.) An objective function f is deÐned on S such that f : g ] R where g ½ F. The general combinatorial optimization problem is to Ðnd the genotype g* ½ F such that f (g*) P f (g) for all g ½ F but g D g*. (We are assuming a maximization problem in order to maintain a consistency with the concept of Ðtness.) One of the most popular methods of handling infeasibility in an evolutionary search process is through the use of an external penalty function [3]. This penalty function artiÐcially decreases a genotypeÏs Ðtness if constraints are violated. That is, Ðtness(g) \

G

f (g) f (g) ] penalty(g)

if g ½ F, otherwise. Physica Scripta 57

322

G. W . Greenwood and X. Hu

Fig. 2. An example graph for the maximum independent set problem.

Fig. 1. Two random walks in sequence space. The darkened areas represent low Ðtness regions associated with infeasible genotypes.

The penalty for infeasibility is often quite severe so that there is a low probability of survival. The key point here is that an otherwise smooth Ðtness landscape can now have numerous ““sinkholesÏÏ of extremely low Ðtness corresponding to infeasible genotypes. It is important to emphasize that these sinkholes are not isolated points. Indeed, sinkholes are typically entire subregions of the Ðtness landscape. These sinkholes destroy any notion of an underlying distribution which could be used to assign Ðtness values since the true Ðtness is artiÐcially decreased for infeasible genotypes. Consequently, the correlation present in the landscape can di†er dramatically depending on where the initial starting point of a random walk is located. This is shown in Fig. 1 where two random walks will discern a di†erent degree of correlation. Even repeating random walks from the same starting point can produce di†erent results depending on whether or not an infeasible region is traversed. Both adaptive and random walks on landscapes with infeasible regions are analogous to runs from a nonstationary random process ; statistics gathered from a single run will not be indicative of the true underlying correlation. Hence, landscapes with sinkholes are statistically anisotropic. 3. An example problem To Ðx ideas consider a case of the maximum independent set problem which is known to be NP-complete. The problem is deÐned as follows. Let G \ (V , E) be an undirected graph where V is the set of N vertices and E is the set of edges. The objective is to Ðnd a set VŒ U V such that #i, j ½ VŒ , the edge Si, jT ¾ E and o VŒ o is maximum. For example, Fig. 2 shows a graph where VŒ \ M2, 4, 7, 9N is an independent set (though not maximal with respect to G) while 2, 8 ½ VŒ is infeasible because S2, 8T ½ E. An encoding for this problem is a binary string (x , . . . , 1 x ) where x \ 1 indicates the ith vertex is in VŒ . Each bit in N i the string can be interpreted as a gene and the entire bit string as a genotype. A neighboring genotype has the identical pattern of 1Ïs and 0Ïs except one bit position is Ñipped. Note that a particular genotype may be infeasible since it violates some constraint. For instance, the genotype could have x \ 1 and x \ 1 but Si, jT ½ E. j i Physica Scripta 57

The Ðtness of a genotype can be computed using a function suggested by Back and Khuri [4]. An obvious Ðtness measure is given by the number of vertices in the independent set ; genotypes with a large number of 1Ïs are considered highly Ðt. Observe that neighboring genotypes di†er in Ðtness by ^1. Hence (in the absence of sinkholes) a smooth correlated Ðtness landscape exists. Let N \ o V o and deÐne e ½ M0, 1N such that ij 0 if Si, jT ¾ E, e \ ij 1 if Si, jT ½ E.

G

Then the Ðtness of a genotype is given by

A

B

N N ; x [N Æ x Æ ; x e . i i j ij j/1 i/1 The Ðrst term is the obvious Ðtness measure while the second term decreases the Ðtness by a penalty for every vertex j ½ VŒ such that Si, jT ½ E. This penalty term is what creates sinkholes in the Ðtness landscape of the maximum independent set problem. For this particular problem these sinkholes can be quite deep. Indeed, while the Ðtness of a feasible genotype is strictly positive, the Ðtness of an infeasible genotype can be considerably less than zero. As a test case two independent random walks of several thousand steps were conducted on the Ðtness landscape of a maximum independent set problem with N \ 24. The correlation R(s) was computed and the results are plotted in Fig. 3. Observe that there is a marked di†erence in the correlation of the landscape beyond 10 steps despite the fact these two random walks began at the same initial point in the sequence space. This shows that any presumption of statistical isotropy is unfounded. 4. Final remarks The purpose of reducing the Ðtness is to insure infeasible genotypes have a low probability of surviving in an evolu-

Fig. 3. Correlation determined by two independent random walks starting at the same point. Note that the plot is semi-logarithmic.

Are L andscapes for Constrained Optimization Problems Statistically Isotropic ? tionary process. In this paper we have only considered static penalties where the depth of a sinkhole is solely a function of the constraint violations (i.e., the greater the violation, the greater the depth). There are, however, a number of other penalty methods that have been proposed [3]. SpeciÐcally, 1. Death penalty : infeasible genotypes are simply rejected. 2. Adaptive penalty : penalties are increased or decreased depending on the ratio of feasible to infeasible genotypes previously encountered in the search process. 3. Dynamic penalty : penalties are increased as the search time increases. The death penalty method is just a variation of the static penalty method. It is implemented by simply assigning a Ðxed, extremely low Ðtness value to all infeasible genotypes regardless of the severity of the constraint violation. That is, Ðtness(g) \

G

f (g) if g ½ F, [b otherwise.

b should be of sufficient magnitude so that the genotype has a zero probability of surviving regardless of the survival criteria used in the evolutionary process. Moreover, all sinkholes will be of identical depth as opposed to the static penalty method where the depths can vary. The adaptive and dynamic penalty methods alter the penalty based upon feedback from the search process itself which makes the depth of a sinkhole time variant. Consequently, any underlying distribution of Ðtness values is from a non-stationary process which again makes the Ðtness landscape anisotropic. It is clear that statistically anisotropic landscapes are found in a broad range of constrained optimization problems. We should expect the Ðtness landscape for constrained optimization problems to be moderately uncorrelated as the

323

sinkholes increase the number of local optima. As a consequence, adaptive walks in anisotropic landscapes will tend to be rather short and provide little useful information. Even long random walks can give inconsistent results depending on how frequently they encounter and traverse sinkholes. One possible means of estimating the correlation would be to conduct a large number of random walks so that correlation is determined from an ensemble. Alternatively, a series of random walks starting at distinct points could be conducted. By restricting each walk to within a small neighborhood of its starting point, correlation could be at least locally determined. The overall ruggedness of the landscape could then be formed from a composite of these local correlations. Finally, we have assumed that the optimization problem was a maximization problem. These same concepts apply equally as well to minimization problems since min [ f (g)] is equivalent to max [[f (g)] for g ½ F. Acknowledgements This investigation was supported in part by an External Research Program Grant from HewlettÈPackard Laboratories, Bristol, England, and by DARPA/Army under contract number DABT63-97-C-0048. Dr. HuÏs work was also partially supported by NSF under grant number MIP-9701416.

References 1.

Kau†man, S., ““The Origins of OrderÏÏ (Oxford University Press, New York 1993). 2. Weinberger, E., J. Biol. Cybern. 63, 325 (1990). 3. Michalewicz, Z. and Schoenauer, M., Evolutionary Computation 4, 1 (1996). 4. Back, T. and Khuri, S., Proc. 1st IEEE Conf. on Evolutionary Computation, 531 (1994).

Physica Scripta 57


Bose–Einstein Condensation in Harmonic Oscillator Potentials Xue-Xi Yi,1,2 Hai-Jun Wang2 and Chang-Pu Sun3 1 Institute of Theoretical Physics, Northeast Normal University, Changchun 130024, China 2 Institute of Theoretical Chemistry, Jilin University, Changchun 130023, China 3 Institute of Theoretical Physics, Academia of Sinica, Peking, 100081, China Received April 11, 1997 ; accepted August 13, 1997

PACS Ref : 03.50.Fi, 05.30.Jp, 64.60.-i, 32.80.Pj

Abstract BoseÈEinstein Condensation (BEC) of both the ideal and the weakly interacting cold alkali gases conÐned by anisotropic harmonic oscillator potentials is investigated in this paper. It is shown that the transition point is shifted towards lower temperatures because of the trapped potentials, and the speciÐc heat below the transition point is no longer proportional to T 3@2. Expressions of modiÐed condensation temperature, internal energy and the speciÐc heat are derived for the ideal trapped gases explicitly. Moreover, the e†ect of interactions among the atoms on the transition temperature is also given and discussed.

The recent observation of BoseÈEinstein Condensation (BEC) in ultracold trapped alkali gases [1È3] have created a wave of renewed interest in this phenomenon. BEC is a purely quantum statistical phase transition, which is characterized by a macroscopic population of the ground state below the transition point T . Experiments on BEC demonc strated that below T (about a few microkelvin), several c thousand atoms are found in the ground state. Because the gases are dilute, in most textbooks of statistical mechanics, e.g. [4], the theory of BEC is formulated for noninteracting bosons in a three-dimensional box. This treatment has been extended to power-law potentials by de Groot [5], Bagnato et al. [6] who found that the transition temperatures and the speciÐc heat depend on the shape of the potentials. BEC in one- and two-dimensional systems is only possible for sufficiently conÐning potentials [7È9]. All these investigations are based on the use of the thermodynamic limit and the assumption that the ground state energy was negligible. However, recent BEC experiments on alkali gases were performed with Ðnite numbers of particles [1È3]. For these relatively low numbers (about 109), the e†ects caused by the above approximation are nonvanishing. In this paper, we study BEC in anisotropic harmonic oscillator potentials, because this kind of potentials is a good approximation to the recent BEC experiments. We will derive analytical expressions of the density of states for a system of a Ðnite number of particles in an anisotropic harmonic oscillator potential, a direct application of this result to study BEC of atoms trapped in anisotropic harmonic oscillator potentials is given, meanwhile the e†ect of the nonvanishing ground state energy on BEC of atomic gases in trap potentials is evaluated. We indeed Ðnd marked differences from the usual treatments : the transition point is shifted toward lower temperatures by the trap potentials, the speciÐc heat below the onset of condensation is no longer proportional to T 3@2, the e†ect of the ground energy is nonvanishing, and the interactions between the atoms Physica Scripta 57

increase or decrease the transition temperature, according to whether the scattering length is negative or positive. In order to express the density of states for a system of ultracold atoms trapped in a potential, we consider a range of parameters describing the BEC of atoms [3]. The potentials for the centre-of-mass motion of a single atom in the ground electronic state can be approximated as a threedimensional anisotropic harmonic oscillator potential with frequencies u \ 235, u \ 410, u \ 745 Hz in the y, z and y z x x directions, respectively. Because the trap potential forms a Ðnite barrier, there are several thousand energy levels within the trap. In these several thousands of energy levels about 109 sodium atoms are distributed, therefore, it is too low a number to use the thermodynamic limit. Nevertheless, the Ðnite number e†ects on BEC can be discussed in many ways [10, 11]. In this paper, we discuss the Ðnite number e†ects using the density-of-state approach [12]. The energy eigenvalues for an atom trapped in an anisotropic oscillator potential read ; +u (n ] 1 ). (1) i i 2 i/x, y, z Under real experimental conditions the temperature is high on the scale of the trap level spacing, namely, k T ? +u B i (i \ x, y, z). Therefore, within the canonical ensemble, the partition function E \ E(n , n , n ) \ N x y z

= e~b(nxux`nyuy`nzuz) ; Q(b) \ ; e~bEN \ nx, ny, nz/0 N 1 (2) \ < 1 [ e~bui i/x, y, z of such a system without interactions can be expanded as follows : Q(b) ^ a b~3 ] a b~2 ] a b~1 ] a ] O(b) 2 1 0 ~1 with

(3)

1 a \ , 2 u u u x y z a \ 1 a (u ] u ] u ), (4) 1 2 2 x y z a \ 1 a (u2 ] u2 ] u2 ] 3u u ] 3u u ] 3u u ), 0 12 2 x y z x y y z z z a \ 1 ] 1 (u /u ] u /u ] u /u ~1 8 24 x y x z y z ] u /u ] u /u ] u /u ), (5) y x z x z y where, b \ k T , k is the Boltzmann constant, and the conB B tribution of the ground state was singled away for special

Bose Einstein Condensation in Harmonic Oscillator Potentials treatment. On the other hand, by making use of

P

= e~bEo(E) dE (6) Q(b) \ ; e~bEN \ 0 N the partition function can be calculated equally, where o(E) is the density of states. Comparing with eq. (3), it is proved that the density of states o(E) takes the form :

325

ground state population n can be determined by the E 0 min independent part of the r.h.s. of eq. (13), because n remains 0 unchanged when the parameter E is varied. Thus, we min have : a n2 n \ N [ 1 a b~3C(3)f(3) ] a b~1 ln (be ) [ 1 . 0 2 2 0 0 6b2

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o(E) \ b ] b E ] b E2 ] É É É (7) 0 1 2 This expansion cannot contain the terms Er with r being negative or non-integer, since they become zero after comparison with the direct calculation given by eq. (3). Substituting eq. (7) into eq. (6), one can easily Ðnd :

The Ðrst two terms in eq. (14) are just the results in the thermodynamic limit [6], whereas the last two terms are direct corrections of the Ðnite number e†ects. The corrections depend on the ground state energy e and the shape of 0 the trap potentials. If one introduces a temperature

Q(b) \ b b~1 ] b b~2 ] 2b b~3 ] É É É (8) 0 1 2 Comparison of eq. (8) with the direct calculation (3) shows that

1 2N 1@3 T0\ , (15) c k a f(3)C(3) B 2 which denotes the critical temperature of an inÐnite number atoms trapped in anisotropic oscillator potentials [6], then the critical temperature T for a Ðnite number of atoms c takes the form

b \ a , b \ a , b \ 0.5a . (9) 0 0 1 1 2 2 There is an additional constant term b in the density of 0 states in comparison with the result of Grossmann [12]. As we see below, this additional term will result in a shift of the transition point. Now, let us consider a system of N noninteracting bosons such that the population N(E ) of a state with energy E is i i given by the BoseÈEinstein distribution 1 . N(E ) \ i eb(Ei~k) [ 1

(10)

Here, we set the statistical weights corresponding to the state E , g \ 1. k stands for the chemical potential, which is i i determined by the constraint that the total number of particles in the system is N : = N \ ; N(E ). (11) i i Around the transition point, k ^ e is a good approx0 imation with error 1/n (e and n denote the ground state 0 0 0 energy and ground state population, respectively). Using the density-of-state approximation [10], eq. (10) can be rewritten as follows :

P

=

o(E) dE ]n . (12) min exp [b(E [ e )] [ 1 Emin 0 Here, we single out the ground state population n , because 0 we are interested in BEC where the ground state plays a key role. n denotes the population of all the other states with min energies below E , for the case of particles trapped in a min cavity with volume N \ L3, E takes a value which is at min least 400h2/8ML2 [10] to avoid the error of converting from sum to integral. Using eq. (7), we can obtain that

N\n ] 0

C

N \ n ] n ] 1 a b~3C(3)f(3) 0 min 2 2

A

B

C

1 b n2 k T 0 1 B c T \T 0 1[ c c 3 b 3 C(3)f(3) 2 2 b ln (k T 0 e ) 0 B c 0 T 02k2 . (16) ] c B 3b f(3)C(3) 2 The last term in square brackets is negative, since (e /k T 0) > 1. Hence, for a Ðnite number of atoms trapped 0 B c in anisotropic harmonic potentials, the temperature at which the BEC occurs is lower than T 0 . To measure this c e†ects is in reach of current experiments [1È3]. Equation (16) demonstrates that the tighter the conÐnement (larger u , u , u ), the lower the critical temperature. x y z Generally speaking, the speciÐc heat is more interesting from the experimental point of view, since the lowtemperature behavior of the speciÐc heat c is generally V treated as the hallmark of onset of BEC. It is well known that the speciÐc heat can be derived from the internal energy, which is expressed by

D

bU \ n be ] bU 0 0 min = Eo(E) dE. (17) ]b exp [b(E [ e )] [ 1 Emin 0 The Ðrst term on the right-hand side is the ground state energy, the second term stands for the energy of the other state below E , the third term denotes the energy of the min states above E . Substituting eq. (7) into eq. (17), one min easily Ðnds

P

C

e2 j3 a 0 bU \ n be ] 2 b~3C(4)f(4) ] 0 0 3b(j [ 1) 2

D

] e3(0.5j2 ] j ] ln (j [ 1)) 0

BD

j2 E2 ] j ] ln (j [ 1) [ min 2 b

a a n2 [ 0 ln [be (j [ 1)] ] 1 0 b b2 6

A

(13)

where j \ E /e , C(n) and f(n) denote the Gamma funcmin 0 tion and the RiemannÏs zeta function, respectively. The

] a [b~2C(3)f(3) [ e2(0.5j2 ] j ] ln (j [ 1))] 1 0 a n2 ] 0 [ bE [ be ln [be (j [ 1)] . min 0 0 b 6

C

D

(18)

The j-dependent terms can be dropped, because j appears only in the higher order terms [10]. In T 0 term, the speciÐc c Physica Scripta 57

326

X. X. Y i, H. J. W ang and C. P. Sun

heat c \ (1/N)(LU/LT ) below T is given by V c a 2 T 3 c \ k C(4)f(4) 2 C(3)f(3) V B N T0 c a 2@3 1 5@3 T 2 a ] 3k [C(3)f(3)]5@3 2 1 N B 2 T0 c a 1@3 1 4@3 T n3 a . (19) ] k [C(3)f(3)]1@3 2 0 N 2 T0 3 B c The last two terms on the right-hand side are caused by the Ðnite number of atoms e†ect. It is thus absent in the thermodynamic limit [6]. At the onset of BEC, the relative importance of the three distributions is about 1 : N~4@3a~4@3a : N~4@3a~5@3a , in all current experiments 2 1 2 0 on BEC of atomic alkali gases, the number of atoms trapped in a potential is at least 104, the frequency of the oscillator potential is taken to be about 0.5 ] 103 Hz, therefore, the last two terms in eq. (19) can be dropped and the T 3-law can be treated as a hallmark of the atomic BEC in harmonic oscillator potentials. It is worth noting that there are not only e†ects due to the thermodynamic limit but also e†ects caused by the interactions between the atoms. In the end of this paper, we will discuss the e†ect of the interactions between the atoms on the transition temperature in details. The atoms contained in anisotropic oscillator potentials interact with one another through binary collisions, which are characterized by the s-wave scattering length a. Using the mean-Ðeld approximation, the interaction energy between the atoms is cn(r) proportional to the local density n(r) [6]. c denotes the interaction constant which depends only on the s-wave scattering length a at low temperature, c \ 2n+2a/M, where M is the mass of an atom. Under the local density approximation (LDA) [13], the density of the gas is given by

AB

A B AB AB A B AB AB

n(r) \ exp [[bV (r)]/j3

B A B

B

c c ]a 1] E o(E) \ a 1 ] 0.5 1 0 jk T jk T B B c E2. (21) ] 0.5a 1 ] 1.5 2 jk T B In Ðrst order approximation, the e†ects of the interactions between the atoms are only to modify the parameter in the expression of o(E). The modiÐed results are equal to increasing or decreasing of the frequency for the light Ðeld used to trap the atoms, according to whether the scattering length is negative or positive. Following the above procedure, the transition temperature is given by

A

c 1 T \ T c/0 [ T c/0, c c j k T0 c 0 B c + j \ 0 2nMk T 0 B c Physica Scripta 57

Acknowledgements One of us (X. X. Yi) would like to thank Prof. J. C. Su for his helpful discussions. This work is supported by a Foundation endowed by the Institute of Theoretical Physics, the Academia of Sinica of China.

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where j \ +/J2nMk T is the thermal de Broglie waveB length. V (r) stands for the trapped potentials. The LDA is a good approximation when k T ? +u (i \ x, y, z) [14]. B i Because the gas is weakly interacting, n(r) can be expanded in powers of c. Retaining the Ðrst term, the density of states takes the form

A

where T 0 was represented by eq. (15), T c/0 stands for the c c transition temperature for c \ 0. Equation (22) shows that if the scattering length is negative, T is larger than T c/0. As c c expected, the correction is proportional to the ratio c/j . In 0 most textbooks of statistical mechanics, the problem of BEC of imperfect Bose gases was discussed with the assumption that the scattering length a is positive. For alkali atoms, however, the problem is much more complex, since the molecular potential curves which can typically support many bound states are not known precisely. Some of the atoms (e.g. cesium) are believed to have a positive a [15], and others (e.g. lithium) to have a negative a [16]. In this paper we show that binary collisions with negative a increase the critical temperature. With the same procedure, we can derive the speciÐc heat below the critical temperature, the correction of the interactions between the atoms to the speciÐc heat is also proportional to c/j . 0 In conclusion, we have discussed the BEC of atomic gases trapped in anisotropic oscillator potentials, it was shown that corrections due to the e†ect of Ðnite atoms and the ground state energy are small, but observable. The transition point was shifted toward lower temperature in comparison with the case in the thermodynamic limit. The T 3-law for the low-temperature behavior of c may still be V used to detect the onset of BEC.

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References 1. 2. 3. 4.

5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16.

Anderson, M. H., Ensher, J. R., Matthews, M. R., Wieman, C. E. and Cornell, E. A., Science 269, 198 (1995). Bradley, C. C., Sackett, C. A., Tollett, J. J. and Hulet, R. G., Phys. Rev. Lett. 75, 1687 (1995). Davis, K. B. et al., Phys. Rev. Lett. 75, 3969 (1995). London, F., ““SuperÑuids IIÏÏ (Wiley, New York 1954) ; Huang, K., ““Statistical MechanicsÏÏ (Wiley, New York 1987) ; Landau, L. D. and Lifshitz, E. M., ““Course of Theoretical PhysicsÏÏ Vol. 5 : ““Statistical PhysicsÏÏ (Pergamon, London 1959). de Groot, S. R., Hooyman, G. J. and ten Seldam, C. A., Proc. R. Soc. London Ser. A 203, 266 (1956). Bagnato, V., Pritchard, D. E. and Kleppner, D., Phys. Rev. A35, 4354 (1987). Bagnato, V. and Kleppner, D., Phys. Rev. A44, 7439 (1991). Rehr, J. J. and Mermin, N. D., Phys. Rev. B1, 3166 (1970). Masut, R. and Mullin, W. J., Am. J. Phys. 47, 493 (1979). Grossmann, S. and Holthaus, M., Z. Phys. B97, 319 (1995). Ketterle, W. and van Druten, N. J., Phys. Rev. A54, 656 (1996). Grossmann, S. and Holthaus, M., Z. Naturforsch. 50a, 921 (1995). Oliva, J., Phys. Rev. B39, 4197 (1989), and references therein. Chou, T. T., Chen Ning Yang and Yu, L. H., Phys. Rev. A53, 4257 (1996). Gribakin, G. F. and Flambaum, V. V., Phys. Rev. A48, 546 (1995). Cote, R. et al., Phys. Rev. A50, 399 (1994) ; Moerdijk, A. J. et al., Phys. Rev. Lett. 72, 40 (1994) ; ibid 73, 518 (1994).


Theory of Flute and Current-Convective Instability in a Thin Magnetically Confined Plasma Layer E. Abu-Assali, A. A. Rukhadze and B. Shokri Theoretical Department, General Physics Institute, Vavilova 38, 117942, Moscow, Russia Received July 15, 1997 ; accepted July 30, 1997

PACS Ref : 2160jz

Abstract The Ñute and current-convective instabilities of a magnetized plasma layer are investigated when the inhomogeneity size of the density near the layer surface is much smaller than the Ðeld inhomogeneity (a sharp boundary model) and as a result the geometrical optic approximation is not applicable. It is studied both cases of thick (in comparison with a Ðeld inhomogeneity) and thin plasma layers in the limits of collisionless and collisional plasmas. The obtained growth rates of instabilities are compared with those found in the geometrical optic frame.

1. Introduction It is well known the general analytical method for the stability problem of a magnetically conÐned plasma with arbitrary density when the Ðeld oscillations may be assumed potential [1]. However we are interested only in instability of short wavelength oscillations described by a geometrical optic approximation. Long wavelength oscillations, with wavelengths larger than the inhomogeneity size of a magnetically conÐned plasma, were studied in [2, 3] where instabilities of interest cannot be developed. In these papers only slow instabilities, stipulated by charged particles Larmor drift, were treated. At the same time it is well known that in the presence of either real drift of charged carriers, stipulated by magnetic Ðeld line curvature (gravitational drift), or electron longitudinal drift (current in plasma), it is possible the development of fast instabilities with a phase velocity larger than the particle thermal velocity. They are known as Ñute (or gravitational) and current-convective instabilities where the spectrum for the collisionless plasma case in the geometrical optic approximation is given by [1]

where v2 \ T /M, v2 \ T /M. The quantity of g is often Ti i s e eff called gravitational acceleration, stipulated by the magnetic Ðeld line curvature R. In this case L ln N k2 M X2 (3) ] z Lx k2 m i y It is obvious that oscillations can be unstable if (L ln N)/ Lx [ 0 (curvature and inhomogeneity are opposite), where longitudinal wave numbers stabilize the instability. In the limit k \ 0 this instability is known as Ñute. z Current-convective instability has very similar nature and is caused by longitudinal current drift of electrons i.e. u \ u p OZ, u \ 0. In this case from (1) we obtain [1] e i k u L ln N k2 M u2 \ z X ] z X2 (4) i Lx k k2 m i y y It is obvious that instability may take place with any sign of (L ln N)/Lx if k D 0, but with decreasing k the stabilization z z of instability takes place. u2 \ g

eff

2. Basic equation and mathematical treatment Below, we will generalize the above results for the plasma layer case with thickness a and will be restricted to longwavelength oscillations with wavelengths larger than the inhomogeniety of the plasma boundary and in the presence of real drift of charged particles. We begin from PoissonÏs equation for arbitrary inhomogeniety of plasma media conÐned by an external magnetic Ðeld [1]

A GC

B

If charged carriers drift is stipulated by the curvature of the magnetic Ðeld lines, then [1]

il u2 a La +2U \ ; 1 ] u [ k Æ u v2 Ta a u ] il [ k Æ u a ] 1[J ` k v z Ta k v2 y Ta ] (u [ k Æ u ] il )X a a v2 L0 u ] il [ k Æ u a ] Ta ] J ` X2 Lx k v a z Ta L2 u ] il [ k Æ u a k2 [ ]J y Lx2 ` k v z Ta LU v2 L0 u ] il [ k Æ u Ta a ]U[ J ` Lx X2 Lx k v a z Ta where the operator

v2 ] v2 g s 4 eff u p OY , u \ Ti RX X i i

LT L L0 L ln N a] a \ Lx Lx Lx LT a

k (k Æ u) L ln N k2 M u2 \ y ) ] z )2 i Lx k2 k2 m i

(1)

Here u \ u [ u is the relative velocity of electron and ion e i volume drift, k and k are longitudinal and azimuthal wave z y numbers a with frequency u. Moreover, it is assumed that the magnetic Ðeld B is along the OZ axis and the plasma 0 inhomogeneity is along OX and X \ eB /Mc is the ion i 0 Larmor frequency. In obtaining (1) it is assumed that k ? y k , L ln N/Lx and therefore k2 \ k2 ] k2 ] (L ln N/Lx)2 ^ z y z k2 and the plasma is assumed sufficiently dense, y 4ne2N ? )2. u2 \ i Li M

(2)

A

A

A

B

BA A

B

BD

BH

(5)

Physica Scripta 57

E. Abu Assali, A. A. Rukhadze and B. Shokri

328

where

acts on all quantities placed on its right side, and

P

J (x) \ x e~x2@2 `

x

(6) et2@2 dt i= is the well-known function connected with the error function. Equation (5) was obtained in the limit k2 v2 M Ta > 1 (7) X2 a Here u \ J(4ne 2 n )/m and X \ (e B )/(m c) are the a a a a a 0 a La Langmuir and Larmor frequencies of a (a \ e, i) type particles respectively, v is the thermal velocity, l is the colliTa a sion frequencies of a type particles, k , k are longitudinal z y and transverse wave numbers with frequency u. Equation (5) is valid in all regions of the coordinate X, inside a plasma layer 0 O x O a as well as outside it (x P a, x O 0). Therefore it allows us to investigate the problem of Ñute and current-convective instabilities in a thin magnetically conÐned plasma layer. In the cold plasma limit and relatively high frequency range in which the following inequalities hold X2 ? k2v2 , u [ k Æ u ] il ? k v (8) a z Ta a z Ta for simplicity we assume the magnetic Ðeld to be very strong and both electrons and ions are magnetized i.e. X ? u, l . e e It allows us to e†ectively simplify eq. (5). In bulk of plasma it may be reduced to the form

A G

B

il u2 a La +2U \ ; 1 ] u [ k Æ u v2 Ta a u ] il [ k Æ u v2 a ] Ta ] 1]J ` k v X2 z Ta a L2 u ] il [ k Æ u a k2 [ U. (9) ]J y Lx2 ` k v z Ta Equation (9) is completed by boundary conditions on the surfaces x \ 0, x \ a derived by integrating it over an inÐnitely thin (in comparison with the wavelength) boundary layer near these surfaces. These boundary conditions take the form [4, 5]

A

A

BA

B

BH

M'N \ 0, x/0, a il LU u2 a La ]; 1] u [ k Æ u X2 Lx a a LU X k U a y \ 0. [ ] Lx u [ k Æ u ] il a x/0, a Assuming the solution of (9) to have the form

G

A

A

B

BH

(10)

7

c exJk2 ] k2 , x O 0, 1 y z (11) U \ c eix ] c e~ix, 0 O x O ] , 2 3 x P a. c e~xJk2 ] k2, 4 y z Substituting these solutions into the boundary conditions, we Ðnd the following dispersion relation [(1 ] b)2i2 ] k 2 ] k 2 [ c2 ] tanh (ia) z y ] 2i(1 ] b)Jk 2 ] k 2 \ 0 z y Physica Scripta 57

(12)

G A

A

B

il a i2 \ k2 ] (1 ] b)~1 k2 ] ; 1 ] y z u[k Æ u a/i, e a u2 u ] il [ k Æ u a a ] La 1 [ J ` v2 k v Ta z Ta u2 u ] il [ k Æ u il La J a a a b\ ; 1] k v u [ k Æ u X2 ` a z Ta a/i, e a u2 k u ] il [ k Æ u La y a a . c\ ; J (13) (u [ k Æ u )X ` k v a/i, e a a z Ta In thick layer limit, when o ia o ? 1, eq. (12) goes over to the dispersion relation for oscillations of semi-bounded plasma conÐned by a magnetic Ðeld corresponding to both plasma surfaces

A

C

B

A

A

BDH

B

B

(1 ] b)i ] Jk2 ] k2 ^ c \ 0 (14) y z In this case it is obvious that i2 [ 0. Another thing to pay attention to is the oddness of eq. (14) with respect to k , y which naturally causes the appearance of the odd power of u(k ). Hence on the surface of the semi-bounded plasma y there exists one-directional surface waves for which u(k ) D y u([k ). It is characteristic only for plasma conÐned by y magnetic Ðelds. In the model of mirror reÑection of carriers from an inÐnitely thin surface such an e†ect was not observed. In dense plasma, when u2 ? X , from eq. (14) we Li i2 Ðnd the following spectrum for surface waves of a thick layer, a o k o ? 1, under the condition that their phase velocy ity is larger than both drift and thermal velocities of carriers [4, 5] u2 \ ^

k k2 M X2 y (k Æ u)X ] z i i 2k2 m ok o il y y 1] e u

(15)

When the carrier volume drift is determined by the curvature of the magnetic Ðeld lines, from (2) and (14) we Ðnd k2 M X2 i u2 \ ^ o k o g ] z y eff 2k2 m il y 1] e u

(16)

For collisionless (l > u) plasma comparing (15) with (3) e we notice that instead of the characteristic scale of plasma inhomogeneity 1/L D o (L ln N)/Lx o in (13) appears k . 0 y Obviously only the surface waves, for which the curvature is positive are unstable. Moreover the Ðnite value of the longitudinal wave number k , as in case (3), plays a stabilizing z role. Since in the geometrical optics approximation it is assumed that k L ? 1, we conclude that the increment of y 0 Ñute instability (16) (when k ] 0) is larger than the short z wavelength ones. In collisional plasma (l ? u) the second term will be e important if u \ 0 and it introduces Maxwell relaxation of the density of charged carriers in thick a layer of a dense plasma. An analogical situation takes place for plasma with current i.e. when u \ u p OZ, u \ 0. In this case from (14) e i we Ðnd [4, 5] k k u k2 M X2 i u2 \ ^ y z e X ] z i 2k2 m ok o il y y 1] e u

(17)

Flute and Current Convective Instability in Magnetically ConÐned Plasma L ayer Here also 1/L D o (L ln N)/Lx o is changed to k . For col0 y lisionless (l > u) plasma the oscillations will be stabilized e by increasing k , only spiral perturbations elongated along z the magnetic Ðeld are unstable. Moreover since k L ? 1 we y 0 again conclude that long-wavelength modes have larger increments than short wavelength ones. In collisional plasma with (l ? u) the stabilizing e†ect is dissipative. e With decreasing layer thickness a and in the limit a o i o > 1 the one-directedness of waves completely disappear. In this limit eq. (12) takes the form [4, 5] (1 ] b)2i2 ] k2 ] k2 [ c2 ] 2 y z

1]b Jk2 ] k2 \ 0 y z a

(18)

This equation, in contrast to (16), is even with respect to k . y This means that in the considered limit two one-direction waves on the opposite layer surfaces combine and as a result one-directedness disappears. In the second order with respect to a o i o > 1 it weakly appears. Supposing k2 ? k2 y z and u2 ? X2, eq. (18) may as a result be reduced to Li i 2 X2 k2 MX2 u2 i i u4 [ z [ (k Æ u)2X2 \ 0. (19) 1] i il o k o a u2 k2 m y Li y 1] e u

A

B

One of roots of this equation corresponds to unstable oscillations and is strongly simpliÐed for Ñute (k ] 0) oscilz lation modes and will have the following form

S

k g y eff (20) 2 X2 i 1] o k o a u2 y Li Comparing (20) with (17), we notice that in thin plasma layers when k \ 0 there exist surface waves with growth z rate depending on the thickness a if o k o a \ 2)2/u2 > 1. y i Li Moving to current convective instability in the collisionless limit (l > u) we get the simple expression [4] e q[u0 kz u t m k2 X2 t y u2 [ u2 4 i when , 0 1 ] 2/(o k o a)(X2/u2 u2 \ r M k2 z y i Li t m m k2 t[ y u2 \ u2 . k2 u2 when 0 s M y M k2 z (21) u2 \ [

Here we see the decrease of the growth rate of currentconvective instabilities in thin plasma layers in comparison with the growth rate of such modes in thick layers. If (l ? u), i.e. in a collisional plasma layer, from eq. (19) e we get the spectrum [4, 5] u2 \ [

X ok ou i z 2 X2 i 1] o k o a u2 y Li

Here upper case corresponds to the Ðrst spectrum of (23), that shows the unchangeable character of current convective instabilities strongly elongated along the Ðeld when we move from collisional plasma to the collisionalless one. Moreover (23) shows that instability in this limit is dissipative and it is positive for longitudinal conductivity 3. Discussion At the end we brieÑy discuss the possibility of manifestation of current-convective instability, as considered above, in experiments [6]. For explaining these experiments, investigations [2, 3] were realized. We must remind the reader that in experiment [6] a thin plasma layer with thickness a O 1 mm was created by a low energy electron beam (with energy O1 keV and density n ^ 1011 cm~3) in a heavy gas b with M ^ 200 É 10~24 g (Xe) under the pressure P ^ 0 10~3 torr. The average radius of the beam and consequently the plasma layer is r ^ 1 cm, i.e. r ? a and the layer is p p assumed to be plane. The gas was completely ionized at ten microseconds (n \ ^1013 cm~3). After that the beam was max turned o† and the plasma was disintegrated at ten microseconds. The whole system is placed under a strong longitudinal magnetic Ðeld with a strength of 20 kGauss and the wavelength is 25 cm. In [2, 3] the azimuthal inhomogeneity of plasma layers, observed in experiments [6], is explained by growing drift instability in the stage of disintegrating without current plasma. In spite of quite convincing accordance of theoretical estimates with observed facts in experiments it has not been clear whether current-convective instability could be developed in the plasma production stage. To answer to this question we remind the reader that current-convective instability develops when ) ? u ? k u, k v , l (24) i z z Te e In experiments [6] k D n/L D 10~1 cm~1 , z min A u ^ 109 cm/s, X ^ 2 É 106 s~1 . Therefore, obviously k u ^ i z 108 s~1 and the inequality X ? k u trivially does not hold. i z Consequently, the development of current-convective instability should not be expected in the stage of plasma creation and it remains the explanation, given in [2, 3], only. On the other hand, the current-convective instability with increment (23) can be developed in the mentioned experiments if l B k u. The temperature and density of the e z plasma must be measured with high accuracy to see if this instability can be developed [5]. In addition we note that in stronger magnetic Ðeld (up to 50 kGauss) and longer systems (up to L P 100 cm), to which modern relativistic A plasma microwave electronics leads, the danger of development of current-convective instability is real.

(22) References 1.

if m k2 X2 y u2 [ u2 4 i , 0 M k2 2 X2 z i 1] o k o a u2 y Li m 1@3 m k2 y u2 \ u2 . k2 u2l if u\i e 0 M y M k2 z

A

B

329

2. 3. 4. 5.

(23)

6.

Alexandrov, A. F., Bogdankevich, L. C., Rukhadze, A. A., ““Principle of Plasma ElectrodynamicsÏÏ (Heidelberg : Springer 1984). Krinetski,V. B., Rukhadze, A. A. and Shokri,B., Short Commun. Phys. N11È12, 38È42 (1995) (P. N. Lebedev Institute, Moscow). Rukhadze, A. A. and Shokri B., Phys. Lett. A 232, 115 (1997). Abu-Assaly, E., Rukhadze, A. A. and Shokri, B., Short Commun. Phys (P. N. Lebedev Institute, Moscow) N9È10, 25 (1997). Abu-Assaly, E. and Rukhadze, A. A., Short Commun. Phys. (P. N. Lebedev Institute, Moscow) N9È10, 30 (1997). Loza, O. T. et al., Soviet Plasma Phys. 23, 228 (1997). Physica Scripta 57


The X-ray Characteristic L-Spectrum of Sn, Emitted by Metallic Vapour P. S. Antsiferov Institute for Spectroscopy, Troitsk, Moscow Region 142 092, Russia Received June 9, 1997 ; accepted June 16, 1997

PACS Ref : 32.30.R

Abstract The X-ray L-spectrum of free Sn atoms is detected and compared with that emitted by solid state. Metallic vapour is obtained and excited by means of an electron beam. The splittings and shifts of X-ray lines of free atoms are found. The measured values of shifts are in the range *j/j + 10~4. The accuracy of the resulting wavelengths of the X-ray lines for free atoms is about dj/j + 10~5

an X-ray tube. Now this technique is applied to the study of the spectrum of another type È L-spectrum, emitted by transitions of inner-shell electrons with changes of the main quantum number of 2È3 and 2È4. The Sn atom was chosen as an element under investigation.

2. The experimental procedure 1. Introduction The energy of the inner-shell transition of an electron in a single-ionized atom depends on the state of outer-shell electrons. That is the reason for the so-called chemical shifts of X-ray characteristic spectral lines, and up to now a lot of experiments dealing with the changes in X-ray spectra of particular elements in di†erent chemical bonds have been carried out [1]. For the same reason we should expect, that the X-ray spectra of free atoms will di†er from those emitted by solid state. The experimental study of X-ray spectra of free atoms is much less developed. Only few special publications, dealing with this problem can be found (see, e.g., [2]). Special interest in the X-ray spectra of free atoms arises in connection to use them for reference purposes. The main part of the information on wavelengths of characteristic X-ray lines is available only for the solid state of the emitting element [3]. Nevertheless, sometimes it is necessary to know the wavelengths of characteristic lines of free atoms for energy calibration of X-ray spectra of multiply charged ions [4]. More general interest to study X-ray spectra of vapours may be connected with the problems of X-ray wavelength standards. In the range of accuracy dj/ j + 10~6 [ 10~7, which is available now through X-ray interferometry [5], the emitted wavelength can be sensitive to small admixtures in the particular piece of metal used for the anode of the X-ray tube, while the radiation of free atoms is totally free from any type of chemical shift. The present work continues the investigations [6] where a simple method of detection of X-ray spectra of metallic vapours was proposed and the K X-ray spectrum of Fe was studied. The idea was to use an intense electron beam for the evaporation of metals and at the same time for the excitation of the metallic vapour. The vapour spectra are compared with the conventional solid state spectra, emitted by Physica Scripta 57

The scheme of the experiment is shown in Fig. 1. There are two main parts of the set up : the spectral source unit and spectroscopic unit. The source unit contains an electron gun and target, made from the metal under investigation. The Pierce-type electron gun can operate in two regimes, which allow to have two modes of spectral source È (a) free atom spectra mode and (b) solid state spectra mode. In source mode (a) the electron gun produces a convergent beam (energy 9 keV, current 15 mA), focused on the surface of the target (focus diameter about 0.5 mm). The power density of the beam is high enough for evaporation of the target, and itÏs energy is sufficient to produce L-shell vacancies in Sn atoms. Pulsed mode with pulse durations of 0.15 s, one pulse per 20 s, ensures moderate rate of erosion of the Sn target. Another regime of gun operation is used for the mode (b) of the spectral source. Here the gun is used as a cathode of the X-ray tube with constant current 0.2 mA, the target can not melt and serves as an anode. This is achieved by the lowering of the gun heating current. The target itself can be

Fig. 1. The general scheme of the experiment.

T he X ray Characteristic L Spectrum of Sn, Emitted by Metallic V apour

331

Fig. 2. The X-ray spectra of Sn L : left axis-metal, right axis-vapour a1,2 (circles).

Fig. 5. The X-ray spectra of Sn L : left axis-metal, right axis-vapour b4 (circles).



moved in the vertical direction a distance of about 1 mm (see Fig. 1), itÏs position 2 is used in (a) source mode, and position 1 È in (b) source mode. The spectral unit is constructed as an X-ray monochromator with curved crystal and an entrance slit (0.1 mm width, 0.5 mm height) on the Rowland circle [7]. An X-ray gas Ðlled proportional photon counter is used for the detection of the radiation (see Fig. 1). A quartz crystal with a 2d spacing of 6.68 Ó and radius of curvature R \ 580 [ 640 mm for di†erent lines was used as dispersion element. The principle of the monochromator is the following : if the central ray (slit È midpoint of the crystal) satisÐes the Bragg condition j \ 2d sin r (r is the sliding angle in the midpoint of the crystal), practically the whole surface of the crystal takes part in the reÑection of radiation with given wavelength j. The spectral scanning is produced by the motion of the unit with curved crystal and photon counter in such a way, that the entrance slit e†ectively moves along the Rowland circle (see Fig. 1). The advantage of this scheme is the possibility of the usage of the detector without spatial resolution, which just detects all the quanta reÑected by the surface of the crystal. We need a relatively narrow spectral range in the vicinity of the line under investigation (about 10 mÓ), and it allows to get high efficiency of the spectrometer unit [7]. The ratio of distances slit-crystal and slit-beam is 60, it makes the transverse size of the electron beam large enough (0.5 mm) to produce X-rays going from the entrance slit in such a solid angle, that they cover the whole surface of the crystal. The total experimental procedure includes three steps. The Ðrst step is obtaining reference lines from the solid target. The current of the electron gun is 0.2 mA, the target can not be melted by the beam (mode (b) of the spectral source). The vertical position of the target corresponds to the anode spot just in front of the entrance slit (position 1 in Physica Scripta 57

332

P. S. Antsiferov

Fig. 1). The reference spectrum of X-ray lines from solid state is scanned. At the second step È mode (a) of the spectral source, the beam melts the target and produces vapour. The target is shifted down a distance 1È1.5 mm (position 2 in Fig. 1) and now a cloud of exited vapour is in front of the entrance slit. The X-rays emitted by the surface of the target are screened and can not reach the crystal and the photon counter. X-ray radiation excited by the electron beam in the vapour reaches the surface of the crystal and thus we can scan the spectrum of free atoms. Third step È control scanning of the reference lines from the solid target, as it was done in the Ðrst step. 3. The results of measurements The spectra of L , L , L and L lines, emitted by free a1, 2 b1 b3 b4 atoms of Sn together with reference solid state lines are shown in Fig. 2ÈFig. 5. One can see, that free atom lines are shifted to the short wavelength side. The change of the position of the maximum of each line was measured. The position of the maximum is deÐned by means of Ðtting of the upper part of the line proÐle (several points j with levels of i intensity higher than half maximum) by a quadratic parabola I \ [p (p [ j )2 ] p . The total accuracy of meai 1 2 i 3 surement was adopted as a square sum of the statistical accuracy and half of the shift of maximum of reference lines, obtained on the Ðrst and on the third steps of the experimental procedure. The results of the shift measurements are given in Table I. Most likely, the main part of the line shifts is connected with the shift of the upper level of the particular transition. Then it is possible to see, that deeper M-levels have a tendency to be shifted less (Fig. 6). The spectra of L and L lines are given in Fig. 7 and b15 c1 Fig. 8. These spectra show the short wavelength structure in the case of free atoms. Corresponding solid state spectral


lines have unresolved short wavelength asymmetric wings. The upper level for both lines is the same È N , and the IV main components of the free atom structure are similar for these lines. This fact conÐrms the assumption, that the changes of lines are connected mostly with upper states. 4. Discussion of the results One of the most dangerous sources of systematic errors is the possible presence of ions in the vapour. The removing of outer-shell electrons cause shifts of the characteristic X-ray lines [8]. It was mentioned [6], that the visible spectrum of

Table I. T he shifts of the maximusm of the X-ray characteristic L -lines of Sn emitted by metallic vapour (v) with respect to lines emitted by solid state (s) Line

j [ j , mÓ s v

(j [ j )/j s v s

L a1 L b1 L b3 L b4

0.77 ^ 0.06 0.61 ^ 0.04 0.61 ^ 0.09 0.29 ^ 0.13

(21.5 ^ 1.6) ] 10~5 (18.1 ^ 1.1) ] 10~5 (18.5 ^ 2.7) ] 10~5 (8.6 ^ 4) ] 10~5

Fig. 6. The energy shifts E [ E of M levels of Sn. v s Physica Scripta 57

Fig. 8. The X-ray spectra of Sn L : left axis-metal, right axis-vapour c1 (circles).

T he X ray Characteristic L Spectrum of Sn, Emitted by Metallic V apour Fe vapour (it was also excited by the same electron beam) contains lines belonging mostly to neutral atoms, and on distances from the target of 1 mm and more one can see only transitions from the lowest excited conÐgurations of neutral Fe. It is also possible to make simple estimations, showing that the admixture of ions is negligible in our case. The velocity of Sn atoms in the vapour can be estimated as v \ 2 Æ 104 cm/s (T about 600 ¡K). Until reaching the detection volume of the vapour cloud, the evaporated atom spends a certain time, t \ L /v, in the volume of the electron beam (where L is the distance from the surface of target to the midpoint of the entrance slit, it is about 2 mm) and t \ 10~5 s. The ionization cross section of an outer shell electron by 9 keV electrons of the beam can reach 10~18 cm2. Taking into account the electron beam density (B5 Æ 1019 1/(s Æ cm2)) we can get the mean ionization time t as 2 Æ 10~2 s. The ratio t/t \ 0.5 Æ 10~3 gives an estiion ion mate of possible ion admixture. Due to possible ion trapping by the space charge of the beam the ion number in the volume of the beam can not be higher, than it is necessary for neutralization of the beam space charge. Such ion density in our case is 5 Æ 109 cm~3. The density of the emitting vapour atoms, estimated from the value of the absolute X-ray Ñux, is 1013È1014 cm~3, so this e†ect can not inÑuence the present results either. 5. Conclusion The main result of the present work are the measured shifts of wavelengths of characteristic X-ray L-lines of free atoms

333

of Sn with respect to those emitted by solid state (Table I). The measurements allow to use free atom spectral lines as reference lines with the accuracy dj/j + 10~5. The structure of L and L lines emitted by free atoms, (which correb15 c1 sponds to short wavelength asymmetrical wings of these lines in solid state spectra), can be of interest for theoretical interpretation. The measured values of M-level shifts (Fig. 6) can also be of interest for the calculations of electron band structure of solid state for strong coupling case. The technique, used in the present work, can be useful for the problems of X-ray wavelength standards.

Acknowledgements The work is supported by Russian Foundation of Basic Research, grant No 95-02-05810a.

References 1. 2. 3. 4. 5. 6. 7. 8.

Agarwal, B. K., ““X-ray SpectroscopyÏÏ (Springer Verlag, Berlin Heidelberg New York 1991). Britov, I. A., Mstibovskaja, L. E., and Rabinovich, L. G., Sov. Izvestija Academii Nauk USSR (in Russian) 40, 303 (1976). Bearden, J. A., Rev. Mod. Phys. 39, 78 (1967). Antsiferov, P. S. and Movshev, V. G., Z. Phys. D21, 317 (1991). Kessler, Jr. E. G. et al., Phys. Rev. A26, 2696 (1982). Antsiferov, P. S., Opt. Spectrosc, 81, 645 (1996). Antsiferov, P. S., Sov. Zhur. Tehnicheskoi Fiziki (in Russian), 65, 168 (1995). House, L. L., Astrophys. J., Suppl. Ser. No.155, 18, 21 (1969).

Physica Scripta 57


Theoretical analysis of the Doubly Excited 3lnl ¾ States of Sodiumlike Copper R. Bruch,1 U. I. Safronova,*1 A. S. Shlyaptseva,1 J. Nilsen2 and D. Schneider2 1 Department of Physics, University of Nevada, Reno, Reno, NV 89557, U.S.A. 2 Lawrence Livermore National Laboratory, P.O. Box 808, Livermore, CA 94551, U.S.A. Received June 18, 1997 ; accepted August 29, 1997

PACS Ref : 32.80D, 31.20.G, 31.50

Abstract Energy levels, radiative transition probabilities and autoionization rates for CuXIX including the 1s22s22p53lnl@ (n \ 3 [ 8, l@ O n [ 1) and 1s22s2p63lnl@ (n \ 3 [ 4, l@ O n [ 1) doubly excited states were calculated using the multi-conÐgurational HartreeÈFock method (Cowan code). Contributions of relativistic e†ects were discussed in comparison with data obtained by the multi-conÐgurational DiracÈFock method (YODA code). Application of the theoretical data for interpretation of experimental spectra obtained by double-electron capture in slow ion-atom collisions is discussed.

1. Introduction This paper reports new results for the doubly excited states of Na-like copper associated with di†erent transitions including both radiative and non-radiative processes from conÐgurations 1s22s2p53lnl@ (n \ 3 [ 8, l@ O n [ 1) and 1s22s2p63lnl@ (n \ 3 [ 4, l@ O n [ 1). Studies of the 1s22s2p53lnl@ conÐgurations in the Na-like sequence are of continuous interest from both the experimental and theoretical points of view. Experimentally, they are studied by photon emission and Auger spectroscopy. To our knowledge, the Ðrst measurements of 3sÈ2p transition (2p63s 2S È2p53s2 2P ) in Na-like Cu was done by 1@2 3@2 Feldman and Cohen in 1967 [1] using photon emission spectroscopy. By using a low-inductance vacuum-spark between electrodes which were made from the elements under investigation, the lines of highly ionized elements (TiÈCu) have been recorded in the 10 ÓÈ30 Ó range. The 1s22s2p53s3p 4D7@2È1s22s22p53s3d 4G transition in Cu 9@2 XIX was observed by the beam-foil technique [2]. The identiÐcation of this transition was based on the Na-like isoelectronic study (SVIÈCuXIX) of the di†erence in wavenumbers between experimental values and theoretical predictions by Cowan code [3]. Recently, X-ray spectra of Na-like Cu were presented in papers [4È7]. Spectra of multiply charged copper ions produced in X-pinch plasma were observed by Mingaleev et al. [4]. A fast-optics Johann spectrograph and detection with spatial resolution were used to measure these spectra both in hot, dense plasma and in relatively cold recombining plasma expanding outward from a ““hot spotÏÏ. A 100 ns eximer laser-produced copper plasma was used in the work [5] to investigate the Na-like Cu n \ 4-2 satellite ÈÈÈ * Permanent address : Institute of Spectroscopy, Russian Academy of Sciences, 142092, Russia. To whom all correspondence should be addressed at : Department of Physics, University of Notre Dame, IN 46556, U.S.A. Physica Scripta 57

emission in a higher density and lower temperature regime than in paper [4]. X-ray spectra of Na-like Cu were also obtained by using a pulsed-laser plasma source with various duration and pulse shapes [6, 7]. IdentiÐcation of spectra were made by using the DiracÈFock method (YODA code [8]). Auger electron emission from Na-like Cu ions excited in collisions of 170 keV Cu XXI on He was reported in the paper by Hutton et al. [9]. The dominant spectral structures were due to Auger decay of states with 1s22s22p53l3l@ conÐgurations in Cu XIX. Special attention was paid to the metastable level 1s22s22p53s3p 4D which was studied in 7@2 detail and the absolute energy of this level was measured. A detailed theoretical study of the doubly excited states of Na-like ions was done only for ions associated with the 1s22s22p53l3l@ conÐgurations [6, 8, 9]. Energies, collision strengths and oscillator strengths for the Ðne structure transitions involving 1s22s22p63lÈ1s22s22p53l@3lA and 1s22s22p63lÈ1s22s2p63l@3lA in Na-like ions with nuclear charge number 22 O Z O 62 were calculated in paper [10]. The values of the energy and mixing coefficients were obtained by the Cowan code using a scalar factor of (Z-3.9)/Z. In the paper by Chen [11], Auger and radiative transition energies and rates were calculated for 18 ions with atomic numbers in the range 18 O Z O 92 using the multiconÐguration DiracÈFock method. It should be noted that the Tables in [11] include the numerical results only for levels with the 1s22s22p53s2 and 1s22s22p53s3p conÐgurations and ions with Z \ 18, 22 and 26. The wave-lengths, radiative transition rates, Auger rates, satellite intensity factor, and Ñuorescence yields were presented in paper [8] for dielectronic satellites of 14 neon-like ions (Ar8`ÈW64`) including the 1s22s22p63lÈ1s22s22p53l@3lA and 1s22s22p63lÈ 1s22s2p63l@3lA transitions. The calculations of these data were based on multiconÐguration relativistic bound states and distorted-wave Dirac continuum for the electron (YODA code). It should be noted that the results for Na-like Cu were presented only in paper [8]. The present paper is devoted to theoretical calculation of energy levels, radiative transition probabilities and autoionization rates for CuXIX including 1s22s22p53lnl@ (n \ 3 [ 8, l@ O n [ 1) and 1s22s2p63lnl@ (n \ 3 [ 4, l@ O n [ 1) doubly excited states. This study was performed using the multiconÐgurational HartreeÈFock method (Cowan code). The contribution of relativistic e†ects was discussed in comparison with the data obtained by the multi-conÐgurational DiracÈFock method (YODA code). The application of these theoretical data to experimental spectra obtained by

T heoretical Analysis of the Doubly Excited 3lnl States of Sodiumlike Copper double-electron capture in slow ion-atom collisions is discussed. 2. Energy levels, radiative transition probabilities and autoionization rates We carried out detailed calculations of radiative transition probabilities and autoionization rates for the doubly excited states including the 1s22s22p53lnl@ and 1s22s2p63lnl@ (n \ 3, 4) conÐgurations. Table I lists the conÐgurations for the 20 even and 19 odd complexes. These complexes, deÐned by whole momentum J, include the following number of elements in each of the matrixes : even parity J\1/2

J\3/2

J\5/2

J\7/2

J\9/2

J\11/2

J\13/2

J\15/2

89

135

131

90

48

18

5

1

J\1/2

J\3/2

J\5/2

J\7/2

J\9/2

J\11/2

J\13/2

J\15/2

87

134

126

88

47

17

2

0

odd parity

The atomic energy levels, radiative transition probabilities and autoionization rates were calculated by the Cowan code [3]. It was found (see, for example, Pindzola et al. [12]) that using this code, one could obtain good agreement with experimental energies by scaling the electrostatic Slater parameters using a di†erent factor (0.80 in paper [12] and 0.85 in our case) to make a correction due to correlation e†ects. It should be noted that all the levels including the 1s22s22p53lnl@ and 1s22s2p63lnl@ conÐgurations are located above only one threshold : 1s22s22p6. As a result we calculated autoionizing rates by using the following processes : 1s22s22p53lnl@ F 1s22s22p6 ] elA, 1s22s2p63lnl@ F 1s22s22p6 ] elA, where the orbital quantum number lA \ s, d, g, i for even parity and lA \ p, f, h, k for odd parity states. The value of Table I. L abeling of conÐgurations for even and odd complexes N

Even parity complex

N

Odd parity complex

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

2s22p63s 2s22p63d 2s22p53s3p 2s22p53p3d 2s2p63s2 2s2p63s3d 2s2p63p2 2s2p63d2 2s22p53s4p 2s22p53s4f 2s22p53p4s 2s22p53p4d 2s22p53d4p 2s22p53d4f 2s2p63s4s 2s2p63s4d 2s2p63p4p 2s2p63p4f 2s2p63d4s 2s2p63d4d

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19

2s22p63p 2s22p53s2 2s22p53s3d 2s22p53p2 2s22p53d2 2s2p63s3p 2s2p63p3d 2s22p53s4s 2s22p53s4d 2s22p53p4p 2s22p53p4f 2s22p53d4s 2s22p53d4d 2s2p63s4p 2s2p63s4f 2s2p63p4s 2s2p63p4d 2s2p63d4p 2s2p63d4f

335

an energy was chosen as an average value among the 1s22s22p53lnl@ doubly excited states and was equal to 27.6 Ry. The theoretical results of our calculations are given in Tables IIÈIV. In all these Tables column (1) is relevant to the name of the conÐguration, column (2) to the intermediate coupling momentum term deÐned by L S , column 0 0 (3) to the L S and column (4) to the J quantum numbers of the level. All others columns show theoretical results. Table II lists the energies relative to the ground state and weighted radiative rates gA for odd parity levels of the 1s22s22p53lnl@ r and 1s22s2p63lnl@ conÐgurations. This Table is organized by increasing energy values within each conÐguration. ConÐgurations are also ordered according to the energy increase. We have compared our data calculated by Cowan code including di†erent number of conÐgurations and di†erent scaling parameters. Data in columns ““aÏÏ and ““bÏÏ were obtained by including only the 1s22s22p53lnl@ and 1s22s2p63lnl@ conÐgurations, but the data in ““aÏÏ are ab initio data (scaling parameter equal to 1) and data in ““bÏÏ are done with a scaling parameter equal to 0.85. We can see that the di†erence between two values for the energy is ^10 000È 25 000 cm~1 for levels of the conÐguration 2s22p53p2. It should be noted that the scaling parameter is an input parameter in the multiconÐguration HartreeÈFock method (MCHF) implemented in Cowan code to take into account correlation e†ects since the ab initio MCHF method includes correlation e†ects only partially (see [3]). This difference in results given in columns ““aÏÏ and ““bÏÏ shows that correlation e†ects are particularly important for Na-like doubly excited states. Sometimes the order of energies is different in the two columns (see an asterisk for such levels). Column ““dÏÏ represents data obtained by YODA code. We can see that energy values given in this column di†er for some levels by 10 000È30 000 cm~1 when compared to data in column ““cÏÏ. This di†erence can be explained by the inÑuence of correlation e†ects, since the YODA code is one version of ab initio MDHF methods. Data given in column ““cÏÏ were obtained using Cowan code with a scaling parameter of 0.85 and with an inclusion of the 1s22s22p53l4l@ and 1s22s2p63l4l@ conÐgurations. Let us note that the inÑuence of upper conÐgurations on the data for lower conÐgurations is not very strong. This conclusion can be conÐrmed by results given in Table II. Data in column ““cÏÏ were obtained by including 44 and 45 conÐgurations of odd and even parity, respectively, 1s22s22p63lnl@ (n \ 3 [ 8, l@ O n [ 1), 1s22s22p53lnl@ (n \ 3 [ 8, l@ O n [ 1) and 1s22s2p63lnl@ (n \ 3 [ 4, l@ O n [ 1). From comparison of values given in columns ““bÏÏ and ““cÏÏ of Table II it is seen, that the di†erence in energy values is 1000 cm~1. Also the values of gA in r columns ““bÏÏ and ““cÏÏ agree much better. Table III lists Auger energies E , weighted sums of radiA ative transition probabilities gA , autoionization rates A r a for levels of the 1s22s22p53l3l@ and 1s22s2p63l3l@ conÐgurations calculated by the Cowan (a) and YODA (b) codes. Table III is organized by increasing energy within each conÐguration. ConÐgurations are also ordered by energy. We already mentioned above that energy values given in columns ““aÏÏ and ““bÏÏ di†er by 1È4 eV. The values of autoionization rates A agree very well for many levels, a especially for levels with the largest values of A . It should a be noted that the values of A given in Table III are in the a Physica Scripta 57

336

R. Bruch, U. I. Safronova, A. S. Shylaptseva, J. Nilsen and D. Schneider

Table II. Energies relative to the ground state (E in 103 (cm~1)), weighted sums of radiative decay (gA in r s~1), calculated by Cowan code : a È n \ 3 and without scaling, b È n \ 3 and with scaling, parameter 0.85, c È n \ 3, 4, 5, 5, 7, 8 and scaling parameter 0.85, d - Y ODA code for Na-like Cu E

gA r

A

Conf. 1

L S 0 0 2

LS 3

J 4

2s22p53s2 2s22p53s2 2s22p53p2 2s22p53p2 2s22p53p2 2s22p53p2 2s22p53p2 2s22p53p2 2s22p53p2 2s22p53p2 2s22p53p2 2s22p53p2 2s22p53p2 2s22p53p2 2s22p53p2 \2s22p53p2 \2s22p53p2 \2s22p53p2 2s22p53p2 2s22p53p2 2s22p53p2 \2s22p53p2 \2s22p53p2

(1S) (1S) (3P) (1D) (3P) (1D) (1D) (3P) (1D) (3P) (3P) (3P) (3P) (3P) (1S) (1D) (3P) (3P) (3P) (3P) (3P) (1S) (3P)

2P 2P 4P 2P 4P 2F 2P 2D 2D 4P 4D 2D 4D 4S 2P 2F 4D 2S 4S 2D 2P 2P 2P

1.5 0.5 1.5 0.5 2.5 3.5 1.5 1.5 2.5 0.5 3.5 2.5 0.5 1.5 1.5 2.5 1.5 0.5 1.5 2.5 0.5 0.5 1.5

7668 7836 8286 8294 8301 8310 8314 8335 8335 8345 8351 8352 8414 8420 8459 8485 8486 8487 8523 8529 8599 8629 8631

7674 7842 8296 8307 8311 8320 8323 8339 8341 8354 8358 8359 8415 8418 8453 8493 8491 8492 8528 8535 8577 8618 8608

7675 7842 8296 8307 8312 8321 8324 8340 8342 8354 8359 8360 8415 8419 8453 8494 8492 8493 8529 8536 8574 8618 8606

7652 7819 8267 8276 8283 8292 8296 8317 8317 8328 8334 8335 8397 8403 8442 8467 8465 8468 8506 8511 8586 8618 8611

2s22p53s3d 2s22p53s3d 2s22p53s3d 2s22p53s3d 2s22p53s3d 2s22p53s3d 2s22p53s3d 2s22p53s3d 2s22p53s3d 2s22p53s3d 2s22p53s3d 2s22p53s3d 2s22p53s3d 2s22p53s3d 2s22p53s3d 2s22p53s3d 2s22p53s3d 2s22p53s3d 2s22p53s3d 2s22p53s3d 2s22p53s3d 2s22p53s3d 2s22p53s3d

(3P) (3P) (3P) (3P) (3P) (3P) (3P) (3P) (1P) (1P) (1P) (3P) (3P) (3P) (3P) (3P) (3P) (1P) (1P) (1P) (3P) (3P) (3P)

4P 4P 4F 4P 4F 4F 4F 4D 2F 2P 2P 4D 2F 2P 2D 4F 4D 2D 2F 2P 2P 2F 2P

0.5 1.5 4.5 2.5 3.5 2.5 1.5 3.5 2.5 0.5 1.5 0.5 3.5 1.5 2.5 1.5 2.5 2.5 3.5 1.5 0.5 2.5 1.5

8404 8414 8432 8433 8439 8449 8463 8471 8473 8483 8503 8530 8552 8567 8574 8608 8614 8623 8629 8670 8702 8728 8759

8412 8424 8437 8439 8443 8452 8464 8471 8473 8481 8501 8525 8537 8558 8557 8613 8617 8624 8630 8660 8687 8712 8738

8412 8424 8437 8440 8443 8452 8464 8471 8473 8480 8500 8524 8536 8556 8555 8613 8617 8624 8630 8658 8684 8711 8734

2s22p53d2 2s22p53d2 2s22p53d2 2s22p53d2 2s22p53d2 2s22p53d2 2s22p53d2 2s22p53d2 2s22p53d2 2s22p53d2 2s22p53d2 2s22p53d2 2s22p53d2 2s22p53d2 2s22p53d2

(3F) (3F) (3F) (3F) (3F) (3F) (3F) (3F) (3P) (3F) (3F) (1D) (1D) (3P) (1D)

4D 4D 4D 4D 4G 4G 2F 4G 4P 4F 2G 2D 2D 4P 2P

0.5 1.5 2.5 3.5 5.5 4.5 2.5 3.5 2.5 4.5 3.5 1.5 2.5 1.5 0.5

9205 9208 9214 9224 9233 9234 9241 9241 9253 9256 9257 9259 9268 9270 9278

9215 9218 9223 9232 9240 9240 9245 9246 9257 9259 9260 9260 9269 9271 9278

9216 9219 9224 9233 9240 9240 9245 9246 9257 9259 9260 9260 9268 9271 9278

Physica Scripta 57

a 5

b 6

c 7

d 8

a 9

b 10

c 11

5.981 ] 12 3.397 ] 12 2.917 ] 11 7.137 ] 11 1.426 ] 11 9.816 ] 11 9.087 ] 11 1.096 ] 11 5.274 ] 11 4.641 ] 11 4.440 ] 11 3.540 ] 11 9.041 ] 11 1.489 ] 12 2.217 ] 12 1.592 ] 12 2.138 ] 12 4.533 ] 10 4.035 ] 12 5.749 ] 11 1.980 ] 12 1.487 ] 12 4.311 ] 12

5.703 ] 12 3.225 ] 12 2.099 ] 11 6.534 ] 11 1.403 ] 11 8.155 ] 11 9.116 ] 11 1.095 ] 11 4.140 ] 11 5.269 ] 11 4.576 ] 11 3.707 ] 11 7.503 ] 11 1.060 ] 12 1.227 ] 12 8.936 ] 11 6.113 ] 12 4.306 ] 10 8.575 ] 11 5.765 ] 11 5.467 ] 12 7.288 ] 11 5.549 ] 11

5.65 ] 12 3.19 ] 12 2.02 ] 11 6.21 ] 11 1.39 ] 11 7.87 ] 11 8.63 ] 11 1.08 ] 11 4.06 ] 11 4.98 ] 11 4.52 ] 11 3.60 ] 11 7.15 ] 11 1.01 ] 12 1.23 ] 12 8.61 ] 11 7.60 ] 12 4.29 ] 10 6.97 ] 11 5.77 ] 11 6.03 ] 12 7.09 ] 11 6.71 ] 11

8387 8397 8413 8416 8421 8432 8447 8453 8456 8465 8486 8514 8534 8550 8556 8591 8596 8605 8611 8652 8682 8709 8741

1.978 ] 11 9.729 ] 11 1.961 ] 11 9.877 ] 11 2.365 ] 12 5.415 ] 12 4.085 ] 12 5.671 ] 12 8.470 ] 12 7.593 ] 12 2.739 ] 13 1.702 ] 13 4.181 ] 12 3.268 ] 13 4.291 ] 12 3.738 ] 12 2.469 ] 12 1.206 ] 13 9.875 ] 12 1.040 ] 14 9.423 ] 13 5.249 ] 12 7.761 ] 13

3.893 ] 11 1.489 ] 12 1.956 ] 11 8.948 ] 11 2.345 ] 12 5.531 ] 12 4.801 ] 12 5.459 ] 12 8.626 ] 12 8.355 ] 12 2.697 ] 13 1.804 ] 13 4.260 ] 12 4.507 ] 13 4.630 ] 12 5.444 ] 12 2.314 ] 12 1.009 ] 13 9.149 ] 12 9.017 ] 13 8.701 ] 13 4.856 ] 12 7.420 ] 13

3.57 ] 11 1.47 ] 12 1.93 ] 11 8.91 ] 11 2.36 ] 12 5.55 ] 12 4.79 ] 12 5.49 ] 12 8.64 ] 12 8.16 ] 12 2.48 ] 13 1.74 ] 13 4.53 ] 12 4.48 ] 13 4.60 ] 12 5.21 ] 12 2.34 ] 12 1.00 ] 13 9.18 ] 12 8.45 ] 13 8.16 ] 13 4.94 ] 12 6.94 ] 13

9188 9191 9196 9206 9214 9216 9224 9224 9235 9237 9239 9242 9249 9252 9261

1.564 ] 11 3.738 ] 11 4.933 ] 11 9.289 ] 11 4.189 ] 11 3.723 ] 11 1.186 ] 12 2.268 ] 12 8.360 ] 11 3.505 ] 11 1.194 ] 12 4.203 ] 12 5.798 ] 12 5.600 ] 12 2.733 ] 11

1.614 ] 11 3.707 ] 11 4.628 ] 11 8.282 ] 11 4.245 ] 11 3.772 ] 11 1.252 ] 12 2.339 ] 12 8.492 ] 11 3.558 ] 11 1.522 ] 12 5.002 ] 12 4.917 ] 12 6.156 ] 12 2.317 ] 11

1.58 ] 11 3.63 ] 11 4.49 ] 11 7.96 ] 11 4.15 ] 11 3.69 ] 11 1.15 ] 12 2.24 ] 12 9.15 ] 11 3.49 ] 11 1.28 ] 12 4.86 ] 12 4.34 ] 12 6.05 ] 12 2.02 ] 11

T heoretical Analysis of the Doubly Excited 3lnl States of Sodiumlike Copper

337

Table II. Continued E A a 5

gA r

Conf. 1

L S 0 0 2

LS 3

J 4

b 6

c 7

d 8

a 9

b 10

c 11

2s22p53d2 2s22p53d2 2s22p53d2 2s22p53d2 2s22p53d2 2s22p53d2 2s22p53d2 2s22p53d2 2s22p53d2 2s22p53d2 2s22p53d2 2s22p53d2 2s22p53d2 \2s22p53d2 \2s22p53d2 \2s22p53d2 \2s22p53d2 2s22p53d2 2s22p53d2 2s22p53d2 2s22p53d2 2s22p53d2 2s22p53d2 2s22p53d2 2s22p53d2 2s22p53d2 2s22p53d2 2s22p53d2 2s22p53d2 2s22p53d2

(1G) (1D) (3P) (3P) (3F) (3P) (1G) (1G) (3P) (1G) (3P) (3F) (3P) (3F) (1S) (3F) (3F) (3F) (1D) (3P) (3P) (3P) (3P) (1G) (3P) (1G) (3F) (3F) (3P) (1S)

2H 2F 4D 4D 4F 4P 2G 2F 2D 4G 4S 4F 4D 4G 2P 2G 2G 2F 2F 4D 2S 4D 2D 2H 2P 2F 2D 2D 2P 2P

5.5 3.5 3.5 2.5 1.5 0.5 4.5 2.5 1.5 3.5 1.5 2.5 0.5 3.5 1.5 4.5 2.5 3.5 2.5 1.5 0.5 1.5 2.5 4.5 0.5 3.5 2.5 1.5 1.5 0.5

9283 9290 9292 9292 9294 9295 9317 9324 9326 9338 9356 9356 9374 9408 9409 9421 9426 9431 9440 9445 9452 9462 9468 9469 9495 9511 9514 9523 9537 9586

9282 9287 9289 9290 9292 9294 9311 9319 9320 9330 9346 9349 9361 9413 9390 9425 9423 9431 9437 9440 9448 9457 9464 9465 9483 9499 9499 9506 9519 9561

9280 9287 9289 9290 9292 9294 9309 9319 9319 9329 9346 9348 9360 9413 9390 9425 9423 9431 9437 9440 9449 9458 9464 9463 9483 9497 9497 9504 9517 9560

9264 9271 9274 9274 9277 9278 9299 9307 9309 9320 9339 9339 9357 9390 9392 9402 9408 9413 9422 9427 9434 9444 9449 9450 9477 9492 9495 9504 9518 9568

2.683 ] 11 1.911 ] 13 4.729 ] 12 1.128 ] 13 4.832 ] 12 9.566 ] 11 2.388 ] 11 7.412 ] 13 2.954 ] 13 6.002 ] 13 3.244 ] 13 9.752 ] 13 3.993 ] 13 3.236 ] 11 2.476 ] 13 3.496 ] 11 9.335 ] 13 3.076 ] 13 6.471 ] 13 1.159 ] 13 2.163 ] 11 4.700 ] 12 1.491 ] 12 2.350 ] 11 4.673 ] 13 1.942 ] 14 2.765 ] 14 1.975 ] 14 1.776 ] 14 2.455 ] 13

2.948 ] 11 2.188 ] 13 9.336 ] 11 9.640 ] 12 4.943 ] 12 8.771 ] 11 2.603 ] 11 3.647 ] 13 9.099 ] 13 7.038 ] 13 3.799 ] 13 1.308 ] 14 4.575 ] 13 3.364 ] 11 3.008 ] 13 3.546 ] 11 7.537 ] 13 2.646 ] 13 5.288 ] 13 1.215 ] 13 1.608 ] 11 2.975 ] 12 1.080 ] 12 2.554 ] 11 4.095 ] 13 2.505 ] 14 1.860 ] 14 1.848 ] 14 1.641 ] 14 2.248 ] 13

2.84 ] 11 2.26 ] 13 9.75 ] 11 1.16 ] 13 5.32 ] 12 8.39 ] 11 2.51 ] 11 9.14 ] 13 3.58 ] 13 7.15 ] 13 3.67 ] 13 1.30 ] 14 4.41 ] 13 3.35 ] 11 2.95 ] 13 3.47 ] 11 7.37 ] 13 2.47 ] 13 5.11 ] 13 1.22 ] 13 1.63 ] 11 2.51 ] 12 1.56 ] 12 2.46 ] 11 3.84 ] 13 1.86 ] 14 2.38 ] 14 1.77 ] 14 1.52 ] 14 2.04 ] 13

2s2p63s3p 2s2p63s3p 2s2p63s3p 2s2p63s3p 2s2p63s3p 2s2p63s3p 2s2p63s3p

(3S) (3S) (3S) (1S) (1S) (3S) (3S)

4P 4P 4P 2P 2P 2P 2P

0.5 1.5 2.5 0.5 1.5 0.5 1.5

9248 9258 9281 9322 9342 9427 9434

9254 9264 9287 9318 9338 9412 9418

9254 9264 9287 9318 9338 9412 9418

9230 9240 9264 9306 9326 9410 9417

5.088 ] 11 2.746 ] 12 1.263 ] 12 1.164 ] 13 2.778 ] 13 3.948 ] 12 3.696 ] 12

5.470 ] 11 1.736 ] 12 1.241 ] 12 1.205 ] 13 3.011 ] 13 4.288 ] 12 3.251 ] 12

5.62 ] 11 1.71 ] 12 1.29 ] 12 1.19 ] 13 2.98 ] 13 4.38 ] 12 3.19 ] 12

2s2p63p3d 2s2p63p3d 2s2p63p3d \2s2p63p3d \2s2p63p3d 2s2p63p3d 2s2p63p3d 2s2p63p3d 2s2p63p3d 2s2p63p3d 2s2p63p3d 2s2p63p3d 2s2p63p3d 2s2p63p3d 2s2p63p3d 2s2p63p3d 2s2p63p3d 2s2p63p3d 2s2p63p3d 2s2p63p3d 2s2p63p3d 2s2p63p3d 2s2p63p3d


4F 4F 4F 4F 2D 2D 2F 4D 4D 4D 2F 4P 4P 4D 4D 2D 2P 2D 2P 2F 2F 2P 2P

1.5 2.5 3.5 4.5 2.5 1.5 2.5 0.5 1.5 2.5 3.5 0.5 1.5 3.5 2.5 1.5 0.5 2.5 1.5 2.5 3.5 1.5 0.5

10034 10041 10055 10073 10076 10083 10098 10109 10110 10115 10118 10131 10132 10132 10133 10161 10177 10182 10184 10242 10244 1025 1026

10041 10048 10062 10081 10080 10086 10098 10105 10108 10114 10118 10128 10129 10130 10131 10150 10167 10172 10173 10222 10223 91023 31023

10040 10047 10061 10079 10079 10085 10097 10103 10106 10113 10117 10126 10127 10129 10129 10149 10166 10170 10172 10217 10218 41022 71023

10015 10022 10036 10055 10058 10065 10079 10091 10092 10097 10100 10114 10114 10114 10115 10142 10160 10164 10166 10224 10225 91024 21024

1.455 ] 12 3.517 ] 12 3.230 ] 12 1.873 ] 12 1.504 ] 13 9.654 ] 12 3.614 ] 13 8.408 ] 11 1.301 ] 12 5.549 ] 12 4.936 ] 13 4.133 ] 11 8.714 ] 11 9.361 ] 12 1.513 ] 12 1.882 ] 13 1.289 ] 13 2.408 ] 13 2.043 ] 13 8.316 ] 12 6.198 ] 12 15.445 ] 12 41.279 ] 12

1.726 ] 12 4.400 ] 12 3.896 ] 12 1.855 ] 12 1.491 ] 13 9.833 ] 12 3.539 ] 13 1.062 ] 12 1.511 ] 12 7.376 ] 12 4.776 ] 13 4.099 ] 11 8.740 ] 11 1.273 ] 13 1.630 ] 12 2.064 ] 13 1.386 ] 13 2.671 ] 13 2.218 ] 13 9.640 ] 12 6.793 ] 12 6.053 ] 12 1.215 ] 12

1.67 ] 12 4.30 ] 12 3.86 ] 12 1.75 ] 12 1.48 ] 13 9.79 ] 12 3.51 ] 13 1.06 ] 12 1.52 ] 12 7.90 ] 12 4.65 ] 13 3.77 ] 11 8.11 ] 11 1.46 ] 13 1.57 ] 12 2.08 ] 13 1.40 ] 13 2.71 ] 13 2.25 ] 13 1.03 ] 13 7.16 ] 12 5.96 ] 12 1.12 ] 12

Physica Scripta 57

338


Table III. Auger energies (E in eV ), autoionization rates (A in s~1), weighted sums of radiative decay ( gA A a r in s~1), and branching ratios for Na-like Cu, calculated by Cowan È a and Y ODA È b codes E

gA r

A

A a

Conf. 1

L S 0 0 2

LS 3

J 4

a 5

b 6

a 7

b 8

a 9

b 10

BR a 11

2s22p53s2 2s22p53s2

(1S) (1S)

2P 2P

1.5 0.5

280.2 301.0

278.3 298.9

5.66 ] 12 3.20 ] 12

5.58 ] 12 2.93 ] 12

2.56 ] 12 2.48 ] 12

3.20 ] 12 2.95 ] 12

0.644 0.608

2s22p53s3p 2s22p53s3p 2s22p53s3p 2s22p53s3p 2s22p53s3p 2s22p53s3p 2s22p53s3p 2s22p53s3p 2s22p53s3p 2s22p53s3p 2s22p53s3p 2s22p53s3p 2s22p53s3p 2s22p53s3p 2s22p53s3p 2s22p53s3p 2s22p53s3p 2s22p53s3p

(3P) (3P) (1P) (3P) (1P) (1P) (3P) (1P) (3P) (3P) (3P) (3P) (3P) (3P) (1P) (3P) (3P) (3P)

4S 4D 2D 4D 2P 2D 4P 2S 2D 4P 2P 4D 4D 4P 2D 2P 2D 2S

1.5 2.5 1.5 3.5 0.5 1.5 2.5 0.5 2.5 0.5 1.5 0.5 1.5 1.5 2.5 0.5 1.5 0.5

310.6 313.7 314.5 315.8 316.1 318.7 318.9 321.5 329.8 331.4 331.7 333.4 335.2 338.0 339.1 346.3 351.1 355.3

307.7 311.3 312.3 313.3 314.2 316.9 317.0 319.8 329.6 330.1 330.9 331.6 333.0 335.8 336.9 347.0 350.8 358.2

5.53 ] 10 9.71 ] 11 4.82 ] 12 0.00 ] 00 2.60 ] 12 5.05 ] 12 4.14 ] 12 3.05 ] 12 4.88 ] 12 1.41 ] 12 2.95 ] 12 8.09 ] 11 5.05 ] 12 2.77 ] 12 7.48 ] 12 3.49 ] 12 3.87 ] 12 1.99 ] 12

7.04 ] 10 9.68 ] 11 4.51 ] 12 0.00 ] 00 2.47 ] 12 5.02 ] 12 4.00 ] 12 2.86 ] 12 4.44 ] 12 1.43 ] 12 7.98 ] 11 1.33 ] 12 5.94 ] 12 3.28 ] 12 7.25 ] 12 3.31 ] 12 3.63 ] 12 2.44 ] 12

0.00 ] 00 2.90 ] 10 1.97 ] 11 1.00 ] 06 1.94 ] 11 1.35 ] 11 1.42 ] 11 1.79 ] 11 3.12 ] 11 4.37 ] 13 1.70 ] 10 9.00 ] 09 1.04 ] 11 3.00 ] 09 2.09 ] 11 7.08 ] 13 1.43 ] 11 1.92 ] 14

8.15 ] 08 4.36 ] 10 2.31 ] 11 1.17 ] 07 1.53 ] 11 1.76 ] 11 1.77 ] 11 2.28 ] 11 7.43 ] 11 4.77 ] 13 5.37 ] 10 2.92 ] 10 2.10 ] 11 1.42 ] 05 3.75 ] 11 7.32 ] 13 5.32 ] 11 4.01 ] 14

0.000 0.152 0.140 1.000 0.130 0.096 0.171 0.105 0.277 0.984 0.022 0.022 0.076 0.043 0.144 0.976 0.129 0.995

2s22p53p2 2s22p53p2 2s22p53p2 2s22p53p2 2s22p53p2 2s22p53p2 2s22p53p2 2s22p53p2 2s22p53p2 2s22p53p2 2s22p53p2 2s22p53p2 2s22p53p2 2s22p53p2 2s22p53p2 2s22p53p2 2s22p53p2 2s22p53p2 2s22p53p2 2s22p53p2 2s22p53p2

(3P) (1D) (3P) (1D) (1D) (3P) (1D) (3P) (3P) (1D) (3P) (3P) (1S) (3P) (3P) (1D) (3P) (3P) (3P) (3P) (1S)

4P 2P 4P 2F 2P 2D 2D 4P 4D 2D 4D 4S 2P 4D 2S 2F 4S 2D 2P 2P 2P

1.5 0.5 2.5 3.5 1.5 1.5 2.5 0.5 3.5 2.5 0.5 1.5 1.5 1.5 0.5 2.5 1.5 2.5 0.5 1.5 0.5

357.3 358.6 359.2 360.4 360.7 362.7 363.0 364.5 365.1 365.2 372.0 372.5 376.7 381.5 381.7 381.8 386.1 387.0 391.7 395.7 397.2

354.5 355.6 356.5 357.6 358.1 360.7 360.7 362.0 362.8 362.9 370.6 371.4 376.2 379.2 379.4 379.0 384.1 384.7 394.0 397.1 398.1

2.05 ] 11 6.42 ] 11 1.39 ] 11 8.07 ] 11 8.91 ] 11 1.09 ] 11 4.10 ] 11 5.19 ] 11 4.55 ] 11 3.64 ] 11 7.42 ] 11 1.04 ] 12 1.22 ] 12 7.31 ] 12 4.30 ] 10 8.71 ] 11 7.07 ] 11 5.81 ] 11 5.98 ] 12 6.83 ] 11 7.03 ] 11

2.11 ] 11 6.48 ] 11 2.84 ] 10 8.61 ] 11 8.26 ] 11 2.83 ] 10 4.23 ] 11 4.17 ] 11 2.74 ] 11 2.55 ] 11 7.33 ] 11 1.28 ] 12 1.92 ] 12 1.49 ] 12 1.60 ] 09 1.62 ] 12 3.85 ] 12 4.16 ] 11 1.61 ] 12 4.89 ] 12 1.45 ] 12

1.97 ] 13 8.62 ] 12 1.80 ] 10 8.91 ] 11 5.47 ] 12 5.77 ] 11 1.59 ] 11 1.26 ] 13 1.34 ] 11 1.21 ] 11 3.90 ] 13 1.59 ] 13 4.79 ] 13 6.30 ] 10 1.51 ] 11 6.83 ] 11 1.51 ] 12 1.61 ] 11 1.87 ] 14 1.96 ] 14 3.23 ] 13

3.44 ] 12 1.39 ] 13 2.42 ] 10 1.24 ] 12 9.14 ] 12 6.89 ] 11 2.09 ] 11 1.64 ] 13 9.66 ] 10 1.25 ] 11 6.66 ] 13 2.13 ] 13 5.25 ] 13 5.11 ] 11 2.65 ] 11 1.09 ] 12 2.02 ] 12 1.22 ] 11 3.46 ] 14 3.36 ] 14 8.93 ] 13

0.975 0.964 0.437 0.898 0.961 0.955 0.700 0.980 0.702 0.666 0.991 0.984 0.994 0.033 0.875 0.825 0.895 0.624 0.984 0.999 0.989

2s22p53s3d 2s22p53s3d 2s22p53s3d 2s22p53s3d 2s22p53s3d 2s22p53s3d 2s22p53s3d 2s22p53s3d 2s22p53s3d 2s22p53s3d 2s22p53s3d 2s22p53s3d 2s22p53s3d 2s22p53s3d 2s22p53s3d 2s22p53s3d 2s22p53s3d 2s22p53s3d 2s22p53s3d 2s22p53s3d 2s22p53s3d 2s22p53s3d 2s22p53s3d


4P 4P 4F 4P 4F 4F 4F 4D 2F 2P 2P 4D 2F 2D 2P 4F 4D 2D 4F 2P 2P 2F 2P

0.5 1.5 4.5 2.5 3.5 2.5 1.5 3.5 2.5 0.5 1.5 0.5 3.5 2.5 1.5 1.5 2.5 2.5 3.5 1.5 0.5 2.5 1.5

371.6 373.1 374.8 375.1 375.5 376.6 378.1 379.0 379.2 380.1 382.6 385.5 387.0 389.4 389.5 396.5 397.1 397.9 398.7 402.2 405.5 408.6 411.7

369.4 370.6 372.6 373.0 373.6 374.9 376.8 377.6 377.9 379.1 381.6 385.1 387.6 390.4 389.6 394.7 395.3 396.4 397.1 402.2 406.0 409.3 413.2

3.80 ] 11 1.50 ] 12 1.94 ] 11 8.89 ] 11 2.35 ] 12 5.52 ] 12 4.78 ] 12 5.45 ] 12 8.60 ] 12 8.15 ] 12 2.51 ] 13 1.77 ] 13 4.29 ] 12 4.57 ] 12 4.55 ] 13 5.17 ] 12 2.33 ] 12 9.98 ] 12 9.12 ] 12 8.61 ] 13 8.35 ] 13 4.82 ] 12 7.15 ] 13

1.53 ] 11 8.84 ] 11 0.00 ] 00 8.74 ] 11 2.12 ] 12 5.00 ] 12 3.92 ] 12 5.12 ] 12 7.54 ] 12 7.22 ] 12 2.73 ] 13 1.77 ] 13 3.55 ] 12 3.85 ] 12 3.27 ] 13 3.54 ] 12 2.03 ] 12 1.06 ] 13 8.41 ] 12 1.01 ] 14 9.02 ] 13 4.38 ] 12 7.11 ] 13

2.40 ] 13 3.76 ] 12 0.00 ] 00 5.00 ] 09 3.77 ] 11 6.11 ] 11 6.10 ] 10 6.41 ] 11 5.34 ] 11 1.18 ] 12 9.93 ] 11 7.24 ] 13 5.35 ] 12 1.68 ] 12 2.09 ] 13 4.00 ] 13 2.00 ] 11 3.00 ] 09 7.34 ] 11 2.77 ] 13 3.36 ] 12 4.69 ] 12 1.79 ] 13

5.39 ] 11 2.61 ] 12 1.76 ] 04 2.48 ] 09 4.96 ] 11 9.03 ] 11 2.12 ] 12 9.65 ] 11 7.56 ] 11 2.46 ] 10 4.72 ] 10 8.93 ] 13 7.81 ] 12 2.43 ] 12 4.27 ] 13 7.50 ] 12 4.46 ] 11 1.33 ] 10 1.38 ] 12 9.62 ] 13 4.46 ] 12 5.31 ] 12 4.27 ] 13

0.992 0.910 0.000 0.033 0.562 0.399 0.049 0.485 0.271 0.225 0.137 0.891 0.909 0.688 0.648 0.969 0.340 0.002 0.392 0.563 0.074 0.854 0.501

Physica Scripta 57


339

Table III. Continued E

gA r

A

A a

Conf. 1

L S 0 0 2

LS 3

J 4

a 5

b 6

a 7

b 8

a 9

b 10

BR a 11

2s22p53p3d 2s22p53p3d 2s22p53p3d 2s22p53p3d 2s22p53p3d 2s22p53p3d 2s22p53p3d 2s22p53p3d 2s22p53p3d 2s22p53p3d 2s22p53p3d 2s22p53p3d 2s22p53p3d 2s22p53p3d

(3S) (3S) (3S) (3D) (3S) (3D) (3S) (3S) (3D) (3D) (3D) (3D) (1P) (1D)

4D 4D 4D 4G 4D 4G 2D 2D 4G 4P 4G 2F 2P 2G

0.5 1.5 2.5 3.5 3.5 4.5 2.5 1.5 5.5 0.5 2.5 3.5 0.5 3.5

407.1 408.5 410.6 412.0 413.0 413.0 413.3 414.0 414.7 415.6 415.7 415.9 417.2 417.2

403.6 405.1 407.3 409.3 410.0 410.2 410.6 411.4 411.7 413.4 413.3 413.4 415.0 415.1

2.11 ] 10 2.57 ] 11 3.22 ] 11 5.32 ] 10 1.02 ] 11 6.34 ] 10 8.71 ] 11 1.76 ] 12 8.25 ] 10 1.06 ] 12 3.53 ] 11 1.30 ] 11 1.22 ] 12 6.89 ] 10

7.43 ] 09 2.45 ] 11 3.58 ] 11 0.00 ] 00 0.00 ] 00 0.00 ] 00 7.87 ] 11 1.46 ] 12 0.00 ] 00 9.30 ] 11 2.01 ] 11 0.00 ] 00 9.94 ] 11 0.00 ] 00

1.00 ] 09 5.10 ] 10 1.90 ] 10 2.90 ] 10 3.00 ] 09 2.75 ] 11 6.30 ] 11 5.50 ] 11 0.00 ] 00 1.01 ] 11 3.02 ] 11 1.00 ] 09 2.30 ] 11 5.00 ] 09

6.98 ] 08 5.23 ] 10 1.32 ] 10 4.76 ] 10 1.03 ] 04 3.72 ] 11 8.12 ] 11 6.29 ] 11 2.00 ] 05 1.65 ] 11 3.62 ] 11 1.10 ] 09 3.20 ] 11 1.29 ] 10

0.086 0.442 0.262 0.814 0.190 0.977 0.813 0.555 0.000 0.160 0.837 0.058 0.274 0.367

2s22p53p3d 2s22p53p3d 2s22p53p3d 2s22p53p3d 2s22p53p3d 2s22p53p3d 2s22p53p3d 2s22p53p3d 2s22p53p3d 2s22p53p3d 2s22p53p3d 2s22p53p3d 2s22p53p3d 2s22p53p3d 2s22p53p3d 2s22p53p3d 2s22p53p3d 2s22p53p3d 2s22p53p3d 2s22p53p3d 2s22p53p3d 2s22p53p3d 2s22p53p3d 2s22p53p3d 2s22p53p3d 2s22p53p3d 2s22p53p3d 2s22p53p3d 2s22p53p3d 2s22p53p3d 2s22p53p3d 2s22p53p3d 2s22p53p3d 2s22p53p3d 2s22p53p3d 2s22p53p3d 2s22p53p3d 2s22p53p3d 2s22p53p3d 2s22p53p3d 2s22p53p3d 2s22p53p3d 2s22p53p3d 2s22p53p3d 2s22p53p3d 2s22p53p3d 2s22p53p3d 2s22p53p3d 2s22p53p3d 2s22p53p3d 2s22p53p3d

(3D) (3D) (3D) (3P) (3D) (3P) (3D) (3D) (3P) (3D) (3P) (1D) (3D) (3S) (1D) (3D) (3S) (1D) (1D) (3D) (1D) (3D) (3D) (3P) (3D) (3P) (3D) (3D) (3P) (3D) (3D) (1P) (3P) (1D) (3D) (3P) (1P) (3P) (3D) (3P) (1D) (1D) (3D) (1D) (3D) (1S) (1S) (3D) (3D) (3P) (1S)

4P 4F 4S 4D 4P 4F 4F 4F 4F 4D 4F 2F 4D 4D 2D 2D 2D 2P 2S 2D 2P 2G 4G 2F 2D 4F 2S 2P 2F 4G 2F 2P 4D 2G 4D 2P 2D 4D 4F 4F 2F 2D 4D 2P 2P 2D 2D 2G 2S 2D 2D

1.5 2.5 1.5 3.5 2.5 4.5 1.5 4.5 1.5 2.5 3.5 2.5 3.5 0.5 1.5 2.5 1.5 0.5 0.5 2.5 1.5 4.5 2.5 3.5 2.5 1.5 0.5 1.5 2.5 3.5 2.5 1.5 3.5 4.5 2.5 1.5 1.5 0.5 1.5 2.5 3.5 2.5 0.5 1.5 0.5 2.5 1.5 3.5 0.5 1.5 2.5

417.9 418.6 418.8 419.4 419.5 419.6 420.1 421.3 421.5 421.5 422.5 422.7 423.3 423.5 424.5 424.8 425.5 426.4 427.4 428.4 429.9 430.4 432.0 433.5 433.7 434.3 435.0 435.1 435.3 435.3 436.6 437.5 437.9 438.9 441.1 441.5 442.6 443.0 443.4 443.5 443.5 445.7 445.7 449.0 451.4 451.7 452.3 452.7 455.9 456.1 457.3

415.9 417.1 417.2 417.6 417.5 417.6 418.7 419.6 420.0 420.4 421.1 421.4 422.1 422.4 423.6 424.0 424.7 426.1 427.3 427.4 429.2 430.9 429.3 432.8 433.4 433.9 435.8 434.5 434.2 434.6 437.0 438.0 435.6 436.5 439.6 440.2 441.8 441.8 443.3 442.3 442.1 445.3 445.1 449.3 452.4 454.1 454.6 453.4 457.9 459.3 460.3

1.52 ] 12 9.95 ] 11 5.22 ] 12 2.30 ] 11 7.84 ] 11 7.83 ] 10 6.93 ] 12 3.80 ] 11 1.21 ] 13 1.38 ] 12 2.99 ] 11 6.45 ] 12 2.74 ] 11 5.98 ] 12 6.18 ] 12 1.06 ] 13 7.86 ] 12 2.98 ] 13 7.39 ] 12 4.45 ] 13 4.40 ] 13 4.00 ] 11 1.34 ] 11 3.43 ] 11 7.88 ] 12 1.82 ] 13 2.57 ] 13 2.88 ] 13 8.31 ] 12 5.25 ] 10 1.56 ] 13 4.45 ] 11 6.58 ] 10 7.22 ] 10 1.13 ] 13 1.08 ] 13 9.38 ] 13 1.02 ] 11 3.47 ] 13 1.73 ] 13 3.10 ] 11 6.68 ] 13 4.80 ] 13 9.86 ] 13 6.74 ] 13 8.37 ] 13 5.74 ] 13 3.32 ] 11 3.09 ] 13 1.39 ] 13 4.72 ] 13

6.79 ] 11 1.09 ] 12 5.66 ] 12 0.00 ] 00 3.00 ] 11 0.00 ] 00 6.20 ] 12 0.00 ] 00 1.00 ] 13 1.49 ] 12 0.00 ] 00 5.82 ] 12 0.00 ] 00 4.79 ] 12 4.15 ] 12 1.06 ] 13 5.13 ] 12 2.78 ] 13 7.64 ] 12 2.86 ] 13 3.38 ] 13 0.00 ] 00 1.02 ] 12 0.00 ] 00 7.08 ] 12 1.81 ] 13 1.93 ] 13 1.83 ] 13 1.55 ] 13 0.00 ] 00 4.70 ] 12 1.66 ] 12 0.00 ] 00 0.00 ] 00 1.07 ] 13 8.78 ] 12 1.94 ] 13 4.24 ] 10 1.36 ] 14 1.28 ] 13 0.00 ] 00 8.56 ] 13 2.28 ] 13 1.24 ] 14 7.55 ] 13 1.22 ] 14 4.35 ] 13 0.00 ] 00 3.36 ] 13 1.42 ] 13 1.90 ] 13

2.20 ] 11 3.70 ] 10 6.21 ] 11 2.00 ] 09 2.40 ] 10 4.80 ] 10 7.71 ] 11 2.95 ] 11 1.77 ] 12 8.20 ] 10 5.00 ] 09 6.76 ] 11 3.80 ] 10 6.30 ] 11 1.52 ] 11 4.60 ] 11 1.69 ] 12 4.64 ] 12 3.74 ] 11 4.24 ] 12 7.49 ] 11 2.10 ] 13 2.91 ] 11 7.39 ] 12 4.92 ] 13 7.76 ] 13 4.29 ] 13 1.62 ] 13 2.34 ] 13 3.20 ] 10 7.52 ] 12 3.90 ] 10 8.00 ] 09 0.00 ] 00 1.00 ] 12 2.23 ] 12 5.88 ] 13 4.10 ] 10 3.12 ] 13 8.06 ] 11 5.36 ] 11 1.67 ] 13 3.40 ] 13 8.45 ] 11 2.88 ] 13 3.19 ] 13 9.51 ] 13 1.38 ] 13 3.91 ] 13 6.35 ] 13 2.14 ] 14

1.56 ] 11 2.44 ] 11 6.24 ] 11 5.07 ] 09 3.07 ] 09 4.41 ] 10 9.92 ] 11 3.30 ] 11 1.43 ] 12 1.22 ] 11 1.13 ] 09 6.36 ] 11 3.24 ] 10 6.82 ] 11 1.64 ] 10 4.70 ] 11 1.30 ] 12 6.18 ] 12 1.43 ] 12 2.17 ] 12 5.59 ] 11 2.94 ] 13 2.59 ] 10 1.57 ] 11 8.25 ] 12 3.45 ] 13 4.26 ] 13 5.32 ] 13 7.42 ] 13 9.97 ] 12 8.79 ] 12 1.28 ] 12 4.87 ] 11 4.58 ] 10 2.45 ] 12 1.39 ] 12 1.27 ] 13 1.09 ] 11 1.31 ] 14 1.25 ] 12 6.18 ] 11 3.55 ] 13 9.79 ] 13 5.74 ] 11 4.76 ] 13 9.56 ] 12 9.77 ] 13 1.72 ] 13 4.89 ] 13 1.78 ] 14 3.66 ] 14

0.368 0.182 0.323 0.065 0.155 0.860 0.308 0.886 0.368 0.263 0.118 0.386 0.526 0.174 0.090 0.206 0.463 0.237 0.092 0.364 0.064 0.998 0.929 0.994 0.974 0.945 0.769 0.692 0.944 0.830 0.743 0.260 0.493 0.000 0.348 0.453 0.715 0.445 0.782 0.219 0.933 0.601 0.586 0.033 0.461 0.696 0.869 0.997 0.717 0.948 0.965

Physica Scripta 57

340



gA r

A

A a

Conf. 1

L S 0 0 2

LS 3

J 4

a 5

b 6

a 7

b 8

a 9

b 10

BR a 11

2s2p63s2

(1S)

2S

0.5

446.7

446.0

6.28 ] 12

3.56 ] 13

3.15 ] 13

2.98 ] 12

0.909

2s22p53d2 2s22p53d2 2s22p53d2 2s22p53d2 2s22p53d2 2s22p53d2 2s22p53d2 2s22p53d2 2s22p53d2 2s22p53d2 2s22p53d2 2s22p53d2 2s22p53d2 2s22p53d2 2s22p53d2 2s22p53d2 2s22p53d2 2s22p53d2 2s22p53d2 2s22p53d2 2s22p53d2 2s22p53d2 2s22p53d2 2s22p53d2 2s22p53d2 2s22p53d2 2s22p53d2 2s22p53d2 2s22p53d2 2s22p53d2 2s22p53d2 2s22p53d2 2s22p53d2 2s22p53d2 2s22p53d2 2s22p53d2 2s22p53d2 2s22p53d2 2s22p53d2 2s22p53d2 2s22p53d2 2s22p53d2 2s22p53d2 2s22p53d2 2s22p53d2

(3F) (3F) (3F) (3F) (3F) (3F) (3F) (3F) (3P) (3F) (3F) (1D) (1D) (3P) (1D) (1G) (1D) (3P) (3P) (3F) (3P) (1G) (1G) (3P) (1G) (3P) (3F) (3P) (1S) (3F) (3F) (3F) (3F) (1D) (3F) (3P) (3P) (1G) (3P) (3P) (1G) (3F) (3F) (3P) (1S)

4D 4D 4D 4D 4G 4G 2F 4G 4P 4F 2G 2D 2D 4P 2P 2H 2F 4D 4D 4F 4P 2G 2F 2D 2G 4S 4F 2P 2P 4G 4G 2G 2F 2F 2D 2S 4D 2H 2D 2P 2F 2D 2D 2P 2P

0.5 1.5 2.5 3.5 5.5 4.5 2.5 3.5 2.5 4.5 3.5 1.5 2.5 1.5 0.5 5.5 3.5 3.5 2.5 1.5 0.5 4.5 2.5 1.5 3.5 1.5 2.5 0.5 1.5 3.5 2.5 4.5 3.5 2.5 1.5 0.5 1.5 4.5 2.5 0.5 3.5 2.5 1.5 1.5 0.5

471.3 471.6 472.3 473.4 474.3 474.3 474.9 475.0 476.4 476.7 476.7 476.8 477.8 478.1 479.0 479.2 480.0 480.4 480.5 480.8 481.0 482.8 484.0 484.1 485.3 487.4 487.7 489.1 492.8 495.8 496.9 497.2 498.0 498.7 499.1 500.2 501.3 501.9 502.0 504.4 506.1 506.2 507.1 508.7 513.9

468.7 469.1 469.7 470.9 471.8 472.1 473.1 473.1 474.5 474.7 475.0 475.4 476.3 476.7 477.7 478.1 479.0 479.3 479.4 479.7 479.8 482.4 483.4 483.6 485.1 487.4 487.4 489.6 493.9 493.7 495.9 495.2 496.5 497.6 498.3 499.2 500.4 501.2 501.1 504.5 506.3 506.7 507.9 509.6 515.8

1.58 ] 11 3.63 ] 11 4.49 ] 11 7.91 ] 11 4.15 ] 11 3.69 ] 11 1.15 ] 12 2.23 ] 12 9.14 ] 11 3.48 ] 11 1.29 ] 12 4.88 ] 12 4.33 ] 12 6.06 ] 12 2.02 ] 11 2.81 ] 11 2.18 ] 13 8.83 ] 11 1.13 ] 13 5.34 ] 12 8.62 ] 11 2.49 ] 11 8.88 ] 13 3.58 ] 13 6.93 ] 13 3.67 ] 13 1.30 ] 14 4.43 ] 13 2.97 ] 13 3.36 ] 11 7.24 ] 13 3.47 ] 11 2.43 ] 13 4.89 ] 13 1.22 ] 13 1.63 ] 11 2.52 ] 12 2.44 ] 11 1.57 ] 12 3.85 ] 13 1.79 ] 14 2.38 ] 14 1.77 ] 14 1.53 ] 14 2.05 ] 13

8.33 ] 10 2.38 ] 11 2.80 ] 11 6.90 ] 11 0.00 ] 00 0.00 ] 00 9.24 ] 11 2.09 ] 12 5.17 ] 11 0.00 ] 00 8.69 ] 11 4.51 ] 12 5.48 ] 12 5.89 ] 12 1.81 ] 11 0.00 ] 00 1.84 ] 13 4.67 ] 12 1.13 ] 13 4.81 ] 12 8.14 ] 11 0.00 ] 00 7.32 ] 13 2.94 ] 13 6.02 ] 13 3.28 ] 13 9.69 ] 13 4.01 ] 13 2.48 ] 13 2.12 ] 10 9.25 ] 13 0.00 ] 00 3.08 ] 13 6.35 ] 13 1.15 ] 13 1.20 ] 11 4.39 ] 12 0.00 ] 00 1.32 ] 12 4.54 ] 13 1.86 ] 14 2.68 ] 14 1.91 ] 14 1.72 ] 14 2.36 ] 13

1.00 ] 09 1.00 ] 09 2.90 ] 10 6.10 ] 10 2.20 ] 10 1.00 ] 09 1.18 ] 13 2.80 ] 10 6.27 ] 11 0.00 ] 00 8.58 ] 11 2.59 ] 12 9.41 ] 11 2.06 ] 11 2.83 ] 12 4.05 ] 12 1.05 ] 14 7.92 ] 12 4.43 ] 13 2.58 ] 12 5.42 ] 12 1.56 ] 12 2.39 ] 14 2.40 ] 11 1.27 ] 14 1.95 ] 12 6.12 ] 13 2.30 ] 13 2.48 ] 13 3.90 ] 10 8.88 ] 13 9.20 ] 10 1.73 ] 13 1.30 ] 14 4.50 ] 11 2.40 ] 10 2.28 ] 11 2.43 ] 12 1.67 ] 12 1.42 ] 13 3.28 ] 14 4.57 ] 12 1.72 ] 12 3.67 ] 13 2.90 ] 13

3.99 ] 08 1.45 ] 09 2.82 ] 10 1.25 ] 11 1.71 ] 10 1.25 ] 09 1.35 ] 13 4.99 ] 08 8.02 ] 11 3.75 ] 08 1.12 ] 12 1.58 ] 12 8.07 ] 11 3.58 ] 11 2.97 ] 12 6.33 ] 12 1.34 ] 14 1.55 ] 13 6.19 ] 13 3.39 ] 12 6.86 ] 12 2.54 ] 12 2.51 ] 14 4.97 ] 11 1.44 ] 14 2.87 ] 12 6.57 ] 13 2.11 ] 13 2.05 ] 13 5.29 ] 10 1.28 ] 14 1.12 ] 11 3.25 ] 13 2.19 ] 14 2.91 ] 10 3.75 ] 10 1.76 ] 10 3.58 ] 12 5.07 ] 11 1.68 ] 13 4.15 ] 14 6.37 ] 12 4.34 ] 12 3.90 ] 13 2.99 ] 13

0.012 0.011 0.279 0.382 0.389 0.026 0.984 0.091 0.805 0.000 0.841 0.680 0.566 0.120 0.966 0.994 0.975 0.986 0.959 0.660 0.926 0.984 0.942 0.026 0.936 0.175 0.739 0.509 0.769 0.482 0.880 0.726 0.851 0.941 0.129 0.227 0.266 0.990 0.864 0.425 0.936 0.103 0.037 0.490 0.739

2s2p63s3p 2s2p63s3p 2s2p63s3p 2s2p63s3p 2s2p63s3p 2s2p63s3p 2s2p63s3p

(3S) (3S) (3S) (1S) (3S) (3S) (3S)

4P 4P 4P 2P 2P 2P 2P

0.5 1.5 2.5 0.5 1.5 0.5 1.5

476.0 477.2 480.1 483.9 486.5 495.5 496.3

473.9 475.1 478.1 483.3 485.8 496.2 497.1

5.63 ] 11 1.72 ] 12 1.29 ] 12 1.21 ] 13 3.02 ] 13 4.35 ] 12 3.16 ] 12

1.22 ] 11 1.22 ] 12 0.00 ] 00 1.12 ] 13 2.61 ] 13 3.08 ] 12 2.59 ] 12

1.57 ] 11 1.04 ] 11 3.20 ] 10 7.79 ] 13 5.24 ] 13 4.29 ] 13 7.13 ] 13

2.58 ] 11 1.27 ] 12 1.25 ] 10 1.20 ] 14 8.20 ] 13 6.47 ] 13 1.06 ] 14

0.358 0.195 0.130 0.928 0.874 0.952 0.989

2s2p63p2 2s2p63p2 2s2p63p2 2s2p63p2 2s2p63p2 2s2p63p2 2s2p63p2 2s2p63p2

(3P) (3P) (1D) (1D) (3P) (3P) (3P) (1S)

4P 4P 2D 2D 2P 4P 2P 2S

0.5 1.5 2.5 1.5 0.5 2.5 1.5 0.5

522.7 524.8 524.9 525.7 527.7 529.0 531.6 540.0

521.1 523.0 522.8 523.9 526.5 527.2 530.4 540.6

1.93 ] 12 4.24 ] 12 1.53 ] 13 1.62 ] 13 2.07 ] 13 7.36 ] 12 4.08 ] 13 7.68 ] 12

1.17 ] 12 5.76 ] 12 1.44 ] 13 1.02 ] 13 1.80 ] 13 4.52 ] 12 3.53 ] 13 5.96 ] 12

7.40 ] 11 2.33 ] 12 7.13 ] 12 6.52 ] 12 1.55 ] 11 3.86 ] 12 1.80 ] 12 1.10 ] 13

7.14 ] 11 8.47 ] 12 1.41 ] 13 9.45 ] 12 2.67 ] 11 5.91 ] 12 2.37 ] 12 1.81 ] 13

0.435 0.687 0.736 0.616 0.015 0.759 0.150 0.742

Physica Scripta 57


341


gA

A

A a

r

Conf. 1

L S 0 0 2

LS 3

J 4

a 5

b 6

a 7

b 8

a 9

b 10

BR a 11

2s2p63s3d 2s2p63s3d 2s2p63s3d 2s2p63s3d 2s2p63s3d 2s2p63s3d 2s2p63s3d 2s2p63s3d

(3S) (3S) (3S) (3S) (1S) (1S) (3S) (3S)

4D 4D 4D 4D 2D 2D 2D 2D

0.5 1.5 2.5 3.5 1.5 2.5 1.5 2.5

537.8 538.0 538.3 538.7 546.8 547.4 552.5 552.6

536.6 536.7 537.0 537.3 547.2 547.7 554.4 554.4

3.95 ] 11 7.93 ] 11 1.20 ] 12 1.61 ] 12 7.00 ] 11 1.06 ] 12 2.77 ] 12 3.78 ] 12

0.00 ] 00 0.00 ] 00 0.00 ] 00 0.00 ] 00 0.00 ] 00 5.95 ] 09 1.45 ] 12 2.05 ] 12

0.00 ] 00 3.00 ] 09 4.00 ] 09 0.00 ] 00 3.43 ] 13 2.77 ] 13 5.73 ] 13 6.36 ] 13

3.20 ] 07 1.09 ] 10 2.27 ] 10 7.70 ] 06 4.83 ] 13 4.21 ] 13 7.22 ] 13 7.71 ] 13

0.000 0.015 0.020 0.000 0.995 0.994 0.988 0.990

2s2p63p3d 2s2p63p3d 2s2p63p3d 2s2p63p3d 2s2p63p3d 2s2p63p3d 2s2p63p3d 2s2p63p3d 2s2p63p3d 2s2p63p3d 2s2p63p3d 2s2p63p3d 2s2p63p3d 2s2p63p3d 2s2p63p3d 2s2p63p3d 2s2p63p3d 2s2p63p3d 2s2p63p3d 2s2p63p3d 2s2p63p3d 2s2p63p3d 2s2p63p3d


4F 4F 4F 2D 4F 2D 2F 4D 4D 4D 2F 4P 4P 4D 4D 2D 2P 2D 2P 2F 2F 2P 2P

1.5 2.5 3.5 2.5 4.5 1.5 2.5 0.5 1.5 2.5 3.5 0.5 1.5 2.5 3.5 1.5 0.5 2.5 1.5 2.5 3.5 1.5 0.5

573.4 574.3 576.0 578.2 578.3 579.0 580.5 581.3 581.6 582.4 583.0 584.1 584.3 584.4 584.4 587.0 589.0 589.6 589.8 595.4 595.5 596.8 597.3

571.2 572.1 573.8 576.5 576.1 577.4 579.2 580.6 580.8 581.4 581.8 583.5 583.6 583.6 583.5 587.0 589.2 589.7 589.9 597.1 597.2 599.2 599.7

1.67 ] 12 4.29 ] 12 3.84 ] 12 1.48 ] 13 1.75 ] 12 9.78 ] 12 3.51 ] 13 1.06 ] 12 1.52 ] 12 7.91 ] 12 4.64 ] 13 3.77 ] 11 8.10 ] 11 1.56 ] 12 1.46 ] 13 2.08 ] 13 1.40 ] 13 2.70 ] 13 2.25 ] 13 9.88 ] 12 6.77 ] 12 5.96 ] 12 1.10 ] 12

8.72 ] 11 2.89 ] 12 2.11 ] 12 1.33 ] 13 0.00 ] 00 8.25 ] 12 3.39 ] 13 5.00 ] 11 4.88 ] 11 4.19 ] 12 4.48 ] 13 0.00 ] 00 2.68 ] 10 1.86 ] 11 7.76 ] 12 1.77 ] 13 1.17 ] 13 2.14 ] 13 1.83 ] 13 6.35 ] 12 4.33 ] 12 4.15 ] 12 8.45 ] 11

2.00 ] 09 1.00 ] 10 6.40 ] 10 6.30 ] 10 0.00 ] 00 4.60 ] 10 3.46 ] 11 2.00 ] 09 1.20 ] 10 6.10 ] 10 1.40 ] 10 1.00 ] 09 8.00 ] 09 2.20 ] 10 1.28 ] 11 1.26 ] 12 1.31 ] 12 9.87 ] 11 9.73 ] 11 2.29 ] 13 2.28 ] 13 2.72 ] 12 3.79 ] 12

2.97 ] 09 1.51 ] 10 6.75 ] 10 7.98 ] 10 1.53 ] 07 4.84 ] 10 8.00 ] 11 7.45 ] 09 3.75 ] 09 8.64 ] 10 1.52 ] 11 4.33 ] 08 3.65 ] 09 1.47 ] 10 2.48 ] 11 1.42 ] 12 2.16 ] 12 2.78 ] 11 1.39 ] 12 1.24 ] 13 1.26 ] 13 1.04 ] 12 1.15 ] 12

0.005 0.014 0.118 0.025 0.000 0.185 0.056 0.004 0.031 0.044 0.002 0.005 0.038 0.078 0.066 0.195 0.157 0.180 0.148 0.933 0.964 0.646 0.873

2s2p63d2 2s2p63d2 2s2p63d2 2s2p63d2 2s2p63d2 2s2p63d2 2s2p63d2 2s2p63d2 2s2p63d2 2s2p63d2 2s2p63d2 2s2p63d2 2s2p63d2 2s2p63d2 2s2p63d2 2s2p63d2

(3F) (3F) (3F) (3F) (3P) (3P) (3P) (1D) (1D) (3F) (1G) (1G) (3F) (3P) (3P) (1S)

4F 4F 4F 4F 4P 4P 4P 2D 2D 2F 2G 2G 2F 2P 2P 2S

1.5 2.5 3.5 4.5 0.5 1.5 2.5 2.5 1.5 2.5 3.5 4.5 3.5 0.5 1.5 0.5

634.6 634.9 635.4 635.9 639.6 639.8 640.0 642.1 642.3 643.0 643.0 643.2 643.7 647.4 647.9 654.0

633.1 633.3 633.7 634.1 638.9 639.1 639.2 642.0 642.3 642.7 642.9 643.2 643.5 648.1 648.4 656.5

6.61 ] 11 1.00 ] 12 1.35 ] 12 1.70 ] 12 3.38 ] 11 6.83 ] 11 1.03 ] 12 1.11 ] 12 7.11 ] 11 1.20 ] 12 1.33 ] 12 1.72 ] 12 1.38 ] 12 4.00 ] 11 7.42 ] 11 4.15 ] 11

0.00 ] 00 0.00 ] 00 0.00 ] 00 0.00 ] 00 0.00 ] 00 0.00 ] 00 0.00 ] 00 2.95 ] 10 3.06 ] 10 2.08 ] 10 0.00 ] 00 0.00 ] 00 0.00 ] 00 2.41 ] 09 5.37 ] 09 9.32 ] 10

3.40 ] 10 1.80 ] 10 2.20 ] 10 1.18 ] 11 2.40 ] 10 4.00 ] 10 3.07 ] 11 3.46 ] 12 5.57 ] 12 1.91 ] 12 3.73 ] 13 5.25 ] 13 1.55 ] 13 2.58 ] 11 4.90 ] 10 1.67 ] 13

1.52 ] 10 8.41 ] 09 1.33 ] 10 7.98 ] 10 1.86 ] 10 1.04 ] 10 9.01 ] 10 1.93 ] 12 3.54 ] 12 1.63 ] 12 2.15 ] 13 6.88 ] 13 4.68 ] 13 4.76 ] 10 1.31 ] 10 1.41 ] 13

0.171 0.097 0.115 0.409 0.124 0.190 0.641 0.949 0.969 0.905 0.996 0.997 0.989 0.563 0.209 0.988

range 1011È1014 s~1 and a deviation of results obtained by the two di†erent methods is not larger than 2 times. It should be noted that data in column ““aÏÏ were obtained by including 44 and 45 conÐgurations of odd and even parity, respectively, 1s22s22p63lnl@ (n \ 3 [ 8, l@ O n [ 1), 1s22s22p53lnl@ (n \ 3 [ 8, l@ O n [ 1) and 1s22s2p63lnl@ (n \ 3 [ 4, l@ O n [ 1). That can explain the di†erence in results given in columns ““aÏÏ and ““bÏÏ. The same conclusion can be made analysing the data for A . Table III lists r branching ratios for level of 1s22s22p53l3l@ and 1s22s2p63l3l@ conÐgurations calculated by the Cowan code including the above mentioned conÐgurations. Tables IV lists the Auger energies E , the weighted sum A of radiative transition probabilities gA , autoionization r

rates A , and branching ratios for levels originating from a 1s22s22p53l4l@ and 1s22s2p63l4l@ conÐgurations. There is a total of 776 levels for 24 conÐgurations of even (12) and odd (12) parity. As we discussed above Auger spectra should include levels with largest values of non-radiative decays (A [ 1013 s~1). For such levels the branching ratios are a about 1 since the radiative rate is about 1012 s~1 (see Table IV). Let us note that the levels of the 1s22s22p53l4l@ and 1s22s2p63l4l@ conÐgurations are in the region of 600È920 eV and are partially overlapped with the corresponding levels from the 1s22s22p53l3l@ and 1s22s2p63l3l@ conÐgurations which are in the region 280È650 eV. The largest values of A ([1014 s~1) are observed for 3 levels from a the 1s22s22p53l4l@ conÐgurations : 2s22p53p4p(1S) 2P 1@2 Physica Scripta 57

342


Table IV. Auger energies (E in eV), autoionization rates A (A in s~1), weighted sums of radiative decay (gA in s~1), and a r branching ratios for Na-like Cu, calculated by Cowan code

Table IV. Continued Conf. 1

L S 0 0 2

LS 3

J 4

E A 5

gA r 6

A a 7

BR 8

Conf. 1

L S 0 0 2

LS 3

J 4

E A 5

gA r 6

A a 7

BR 8

2s2p63p4s 2s2p63p4s

(3P) (3P)

2P 2P

0.5 1.5

807.8 812.3

1.52 ] 12 3.07 ] 12

3.32 ] 13 3.45 ] 13

0.978 0.978

2s22p53s4p 2s22p53p4s 2s22p53p4s

(3P) (3P) (1S)

4P 4P 2S

0.5 0.5 0.5

625.3 657.2 679.1

7.33 ] 11 1.64 ] 12 1.99 ] 12

3.56 ] 13 2.05 ] 13 9.40 ] 13

0.990 0.961 0.990

2s22p53p4p 2s22p53p4p 2s22p53p4p 2s22p53p4p 2s22p53p4p 2s22p53p4p 2s22p53p4p 2s22p53p4p 2s22p53p4p 2s22p53p4p

(3S) (1P) (3D) (3P) (3P) (3D) (3P) (3P) (1S) (1S)

2P 2S 2P 4D 4D 4D 4S 2P 2P 2P

0.5 0.5 1.5 0.5 1.5 0.5 1.5 1.5 0.5 1.5

664.6 665.4 670.0 673.5 675.0 682.5 687.0 688.4 695.6 697.0

1.30 ] 12 1.21 ] 12 2.38 ] 12 1.36 ] 12 2.12 ] 12 1.14 ] 12 2.41 ] 12 3.13 ] 12 1.28 ] 12 2.72 ] 12

3.40 ] 13 1.08 ] 13 6.28 ] 13 6.31 ] 13 2.50 ] 13 5.26 ] 13 1.28 ] 13 3.59 ] 13 9.98 ] 13 1.32 ] 14

0.981 0.947 0.991 0.989 0.979 0.989 0.955 0.979 0.994 0.995

2s2p63p4f 2s2p63p4f 2s2p63p4f 2s2p63p4f

(3P) (1P) (1P) (1P)

2G 2G 2D 2D

4.5 3.5 1.5 2.5

862.7 863.2 863.8 864.4

2.06 ] 13 1.67 ] 13 2.26 ] 12 9.92 ] 11

1.05 ] 13 1.24 ] 13 1.95 ] 13 1.69 ] 13

0.836 0.856 0.972 0.990

2s2p63d4s 2s2p63d4s

(3D) (3D)

2D 2D

1.5 2.5

862.7 863.2

4.13 ] 12 1.89 ] 12

1.18 ] 13 1.32 ] 13

0.920 0.977

2s2p63d4p 2s2p63d4p 2s2p63d4p

(3D) (1D) (1D)

2F 2F 2F

3.5 2.5 3.5

880.7 884.1 884.7

3.17 ] 13 7.30 ] 12 9.46 ] 12

1.00 ] 13 1.36 ] 13 1.38 ] 13

0.717 0.918 0.921

2s2p63d4d 2s2p63d4d

(1D) (3D)

2D 2S

1.5 0.5

904.4 910.0

8.77 ] 11 7.16 ] 11

1.07 ] 13 1.87 ] 13

0.980 0.981

2s22p53p4d 2s22p53p4d 2s22p53p4d 2s22p53p4d 2s22p53p4d 2s22p53p4d 2s22p53p4d 2s22p53p4d 2s22p53d4p 2s22p53d4p 2s22p53d4p 2s22p53d4p 2s22p53d4p 2s22p53d4p 2s22p53d4p 2s22p53d4p 2s22p53d4p 2s22p53d4p 2s22p53d4p 2s22p53d4p 2s22p53d4d 2s22p53d4d 2s22p53d4d 2s22p53d4d 2s22p53d4d 2s22p53d4d 2s22p53d4d 2s22p53d4d 2s22p53d4d 2s22p53d4d 2s22p53d4d 2s22p53d4d 2s22p53d4d 2s22p53d4d 2s22p53d4d 2s22p53d4d 2s22p53d4d 2s22p53d4d

(3D) (3P) (3D) (3P) (3P) (3D) (1S) (1S) (1D) (3F) (1F) (3D) (3D) (3D) (3D) (3F) (1D) (1F) (1P) (1P) (3P) (1D) (3F) (3D) (3D) (3D) (3D) (3D) (1P) (3F) (3F) (1D) (3D) (3P) (1P) (1P) (1P) (1P)

4P 2D 2S 4F 2D 4P 2D 2D 2F 2D 2D 2P 4P 4F 4P 4F 2P 2D 2D 2S 2D 2D 2F 4P 2F 4G 2G 2F 2P 2P 4G 2D 4D 2F 2F 2F 2P 2P

0.5 2.5 0.5 1.5 2.5 0.5 1.5 2.5 2.5 2.5 1.5 1.5 2.5 1.5 0.5 1.5 1.5 2.5 2.5 0.5 2.5 2.5 3.5 2.5 3.5 2.5 3.5 2.5 0.5 1.5 2.5 2.5 3.5 3.5 2.5 3.5 1.5 0.5

679.7 689.5 690.1 695.0 708.5 709.8 717.4 717.4 717.3 721.6 721.8 722.7 723.5 725.2 727.7 737.2 738.7 741.8 747.0 747.4 739.9 740.4 741.1 741.3 743.8 746.6 747.2 748.5 748.9 749.6 759.4 760.2 760.7 761.8 766.1 768.0 768.9 770.2

2.60 ] 12 3.99 ] 13 2.46 ] 13 7.78 ] 12 5.94 ] 13 2.13 ] 13 1.61 ] 13 8.87 ] 12 5.34 ] 12 3.98 ] 12 2.28 ] 12 2.38 ] 12 3.66 ] 12 2.26 ] 12 1.30 ] 12 7.39 ] 11 1.84 ] 12 3.59 ] 12 1.28 ] 12 1.42 ] 12 1.90 ] 13 2.60 ] 13 3.78 ] 13 4.70 ] 13 5.42 ] 13 1.09 ] 13 8.17 ] 12 3.85 ] 13 1.03 ] 13 1.06 ] 13 2.94 ] 13 4.51 ] 13 3.58 ] 13 5.63 ] 13 2.88 ] 13 9.87 ] 12 1.45 ] 13 3.29 ] 12

1.85 ] 13 1.07 ] 13 2.37 ] 13 1.83 ] 13 1.50 ] 13 1.64 ] 13 3.57 ] 13 3.04 ] 13 1.66 ] 13 2.39 ] 13 1.03 ] 13 4.29 ] 13 5.23 ] 13 3.86 ] 13 4.37 ] 13 1.49 ] 13 1.98 ] 13 4.54 ] 13 1.94 ] 13 8.47 ] 13 1.16 ] 13 2.34 ] 13 9.17 ] 13 4.05 ] 13 1.68 ] 13 1.57 ] 14 7.64 ] 13 5.39 ] 13 3.94 ] 13 3.83 ] 13 3.29 ] 13 2.64 ] 13 1.53 ] 13 1.03 ] 13 6.40 ] 13 2.33 ] 14 5.45 ] 13 5.24 ] 13

0.934 0.617 0.657 0.904 0.602 0.607 0.899 0.954 0.949 0.973 0.948 0.986 0.988 0.986 0.985 0.988 0.977 0.987 0.989 0.992 0.786 0.844 0.951 0.838 0.713 0.989 0.987 0.894 0.885 0.935 0.870 0.779 0.774 0.595 0.930 0.995 0.938 0.970

2s2p63d4f 2s2p63d4f

(1D) (1D)

2H 2H

5.5 4.5

918.1 918.3

4.76 ] 13 3.92 ] 13

1.29 ] 13 1.30 ] 13

0.764 0.769

2s22p53d4f 2s22p53d4f 2s22p53d4f 2s22p53d4f 2s22p53d4f 2s22p53d4f

(3D) (1P) (3D) (3D) (1P) (1P)

2G 2F 2G 2H 2G 2G

4.5 3.5 3.5 4.5 3.5 4.5

754.2 758.2 759.1 759.2 776.6 780.7

2.58 ] 13 2.06 ] 13 2.26 ] 13 3.07 ] 13 1.26 ] 13 3.57 ] 13

1.11 ] 13 1.51 ] 13 3.77 ] 13 3.64 ] 13 9.67 ] 13 9.30 ] 13

0.812 0.855 0.930 0.922 0.984 0.963

2s2p63s4s 2s2p63s4s

(3S) (1S)

2S 2S

0.5 0.5

767.8 771.9

1.68 ] 12 2.19 ] 12

1.46 ] 13 1.61 ] 13

0.946 0.936

2s2p63s4p 2s2p63s4p

(1S) (1S)

2P 2P

0.5 1.5

788.0 789.7

3.54 ] 12 6.99 ] 12

4.98 ] 13 4.95 ] 13

0.966 0.966

2s2p63s4d 2s2p63s4d

(1S) (1S)

2D 2D

1.5 2.5

809.6 810.1

4.21 ] 12 5.30 ] 11

3.92 ] 13 3.94 ] 13

0.974 0.998

Physica Scripta 57

(697 eV), 2s22p53d4d(3D) 4G (747 eV), 2s22p53d4d(1P) 5@2 2F (769 eV). 7@2 3. IdentiÐcations of spectra Results of our new identiÐcation of the Auger spectrum of Na-like Cu [9] are given in Fig. 1 and Table V. Auger spectra of Na-like Si, Ar, Ti, Sc, Fe and Cu ions were pre-

Fig. 1. Auger spectra produced by F-like Fe17` (a) and Cu20` (b) colliding on He.


343

Table V. Comparison of experimental Auger energies with theoretical energies (eV) Theor. energy Peak

Rel. int.

Exp. energy

a

b

ConÐg.

L SJ 2P

1

31

277.5

280.2

278.3

2s22p53s2(1S)

3

225

312.8

313.7 315.8

311.3 313.3

2s22p53s3p(3P) 2s22p53s3p(3P)

4

30

324.0

321.5

319.8

2s22p53s3p(1P)

5

39

356.2

355.3

358.2

2s22p53s3p(3P)

6

32

360.4

362.7 363.0 364.5 365.1

360.7 360.7 362.0 362.8

2s22p53p2(3P) 2s22p53p2(1D) 2s22p53p2(3P) 2s22p53p2(3P)

7

39

368.8

371.6

369.4

2s22p53s3d(3P)

8

65

371.6

372.0 373.1

370.6 370.6

2s22p53p2(3P) 2s22p53s3d(3P)

9

69

375.8

374.8 375.1 375.5

372.6 373.0 373.6

2s22p53s3d(3P) 2s22p53s3d(3P) 2s22p53s3d(3P)

10

27

384.3

382.6 385.5 387.0

381.6 385.1 387.6

2s22p53s3d(1P) 2s22p53s3d(3P) 2s22p53s3d(3P)

A1

35

408.6

407.1 408.5 410.6

403.6 405.1 407.3

2s22p53p3d(3S) 2s22p53p3d(3S) 2s22p53p3d(3S)

A2

35

411.0

412.0 413.0 413.0

409.3 410.0 410.2

2s22p53p3d(3D) 2s22p53p3d(3S) 2s22p53p3d(3D)

A3

42

415.7

414.7 415.6 417.9

411.7 415.6 415.9

2s22p53p3d(3D) 2s22p53p3d(3D) 2s22p53p3d(3D)

A4

27

417.6

418.6 418.8 419.4

417.1 417.2 417.6

2s22p53p3d(3D) 2s22p53p3d(3D) 2s22p53p3d(3P)

A5

26

419.2

419.5 419.6 420.1

417.5 417.6 418.7

2s22p53p3d(3D) 2s22p53p3d(3P) 2s22p53p3d(3D)

C1

36

471.7

472.3 473.4

469.7 470.9

2s22p53d2(3F) 2s22p53d2(3F)

C2

40

475.9

474.3 474.9 475.0 476.4 476.7

472.1 473.1 473.1 474.5 474.7

2s22p53d2(3F) 2s22p53d2(3F) 2s22p53d2(3F) 2s22p53d2(3P) 2s22p53d2(3F)

sented in [9] to perform an isoelectronic study of doubleelectron capture processes in slow multicharged ion-atom collisions. The main idea of the paper [9] was to analyse some general trends in these spectra, however no detailed identiÐcation was given. The Auger spectrum of Na-like Fe was identiÐed and studied in detail elsewhere [13È15]. In this paper we have identiÐed the Auger spectrum of Na-like Cu following Cu20` ] He collisions. In Fig. 1 the Auger spectrum of Na-like Cu is shown together with the spectrum of Na-like Fe. The present identiÐcation of Na-like Cu was based on our theoretical results, obtained by Cowan and YODA codes. Moreover, due to the common trends in these two spectra a similar identiÐcation as for the Na-like Fe spectra (see [13È15]) has been used. Thus unambiguous

3@2 4D 5@2 4D 7@2 2S 1@2 2S 1@2 2D 3@2 2D 5@2 4P 1@2 4D 7@2 4P 1@2 4D 1@2 4P 3@2 4F 9@2 4P 5@2 4F 7@2 2P 3@2 4D 1@2 2F 7@2 4D 1@2 4D 3@2 4D 5@2 4G 7@2 4D 7@2 4D 9@2 4G 11@2 4P 1@2 4P 3@2 4F 5@2 4S 3@2 4D 7@2 4P 5@2 4F 9@2 4F 3@2 4D 5@2 4D 7@2 4G 9@2 2F 5@2 4G 7@2 4P 5@2 4F 9@2

identiÐcation of peak ““1ÏÏ arises from the 2s22p53s2 2P 3@2 level. This level is well separated from the other levels. It should be noted that gA and A are almost equal for this r a level. Also the identiÐcation of the most intense peak ““3ÏÏ, assigned to the metastable level 2s22p53s3p 4D , seems to 7@2 be reasonable since the transition from this level can be observed for a few other Na-like ions [9]. Moreover, peaks ““4ÏÏ and ““5ÏÏ have been assigned to the levels 2s22p53s3p 2S with two di†erent intermediate terms (1P 1@2 and 3P). In addition, four levels associated with the 2s22p53p2 conÐguration can be assigned to peak ““6ÏÏ, namely (3P)2D , (1D)2D , (3P)4P , (3P)4D , since the 3@2 5@2 1@2 7@2 branching ratios for these levels are almost equal to 1. To identify peaks ““7ÏÏ and ““8ÏÏ in the same way, we have chosen Physica Scripta 57

344


the transitions originating from the level 2s22p53s3d (3P)4P (for peak ““7ÏÏ) and from levels 2s22p53p2 1@2 (3P)4D , 2s22p53s3d (3P)4P (for peak ““8ÏÏ) with the 1@2 3@2 largest values of branching ratio. Furthermore, we have identiÐed peak ““9ÏÏ arising from the metastable level 2s22p53s3d 4F and the levels 2s22p53s3d 4P and 9@2 5@2 2s22p53s3d 4F , while there energies are very close to the 7@2 energy of this metastable level. Peak ““10ÏÏ has been identiÐed as transitions owing to the levels 2s22p53s3d 4D and 1@2 2s22p53s3d 2F . Moreover, the level 2s22p53s3d 2P is 7@2 3@2 expected to contribute to this peak too. We have divided peak ““AÏÏ (see [13]) into 5 parts : A (408.6 eV), A (411.0 eV), 1 2 A (415.7 eV), A (417.6 eV), and A (419.2 eV). We used the 3 4 5 levels belonging to the 2s22p53p3d conÐguration to describe this peak. Finally, some levels of the 2s22p53d2 conÐguration (see Table V) are assigned to the peaks ““C1ÏÏ and ““C2ÏÏ (see [13]). In conclusion, our comprehensive calculations will be useful for the interpretation and designing of new experiments and will shed more light on understanding of complex collision proceses, in particular double-electron transfer involving multicharged ions.

Physica Scripta 57

References 1. Feldman, U. and Cohen, L., J. Opt. Soc. Am. 57, 1128 (1967). 2. Jupen, C., Engstrom, L., Hutton, R. and Trabert, E., J. Phys. B 21, L347 (1988). 3. Cowan, R. D., ““The Theory of Atomic Structure and SpectraÏÏ (University of California Press. Berkeley, 1981). 4. Mingaleev, A. R. et al., Quantum Electron. 23, 397 (1993). 5. Bollanti, S. et al., Physica Scripta 51, 326 (1995). 6. Akulinichev, V. V., Pivinsky, E. G., Shlyaptseva, A. S., Kantsyrev, V. L. and Golovkin, I. E., Physica Scripta 51, 714 (1995). 7. Shlyaptseva, A. S. et al., Physica Scripta 52, 377 (1995). 8. Nilsen, J., At. Data Nucl. Data Tables 41, 131 (1989). 9. Hutton, R., Schneider, D. and Prior, M. H, Phys. Rev. A 44, 243 (1991). 10. Zhang, H. L., Sampson, D. H., Clark, R. E. H. and Mann, J. B., At. Data Nucl. Data Tables 57, 1 (1989). 11. Chen, M. H., Phys. Rev. A 40, 2365 (1989). 12. Pindzola, M. S. et al., Phys. Rev. A 49, 993 (1994). 13. Schneider, D., Chen, M. H., Chantrenne, S., Hutton, R. and Prior, M. H., Phys. Rev. A A40, 4313 (1989). 14. Bliman, S., Bruch, R., Altick, P. L., Schneider D. and Prior, M. H., Phys. Rev. A 53, 4176 (1996). 15. Bruch, R., Safronova, U. I., Shlyaptseva, A. S., Nilsen, J. and Schneider, D., J. Quant. Spectrosc. Radiat. Transfer (accepted for publication).