Emission of K -, L - and M - Auger Electrons from Cu Atoms. Abstract

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1 Emission of K -, L - nd M - uger Electrons from Cu toms Mohmed ssd bdel-rouf Physics Deprtment, Science College, UEU, l in 17551, United rb Emirtes ssd@ueu.c.e bstrct The emission of uger electrons from the K-, L- nd M-sheels of two configurtions of copper toms is investigted in detil within the frme work of the momentum verge technique. b initio clcultions for the bound stte wvefunctions of the ionized toms re performed using Gussin computer code. The continuum wvefunctions of the emitted electrons re determined by employing n effective potentil vritionl pproch. Slight differences hve been noticed between the results of both configurtions. The trnsition rtes, energies nd widths of ll possible uger electrons re listed in Tbles. Slight differences hve been noticed between the results of the two types of copper toms Key Words: uger Spectroscopy PCS: 82.80P, H 1

2 1. Introduction Inner shell ioniztions of toms nd molecules could be initited vi electron, positron or photon interctions. In ll cses holes re creted followed by rerrngement processes. The most interesting process is the one in which n electron from n outer shell flls to fill hole. The energy difference between the two sttes is bsorbed by n electron belonging to higher orbit. Immeditely fter bsorption the electron leves the interction region yielding doubly ionized tom. The trnsition rte, (0), of the flying electron, usully referred to (fter the discoverer of the phenomenon by uger 1923 nd 1925) s uger electron, leds to vluble informtion bout the electronic structure of the originl system. This provides us with plusible explntion for the vigorous interest shown by industril institutions in the development nd employment of uger scnners. (For recent review of the dt, see Sfronov et l 2001). The min gol of the present work is to determine the uger rtes, energies nd widths of ll possible trnsitions from the K-, L- nd M-shells of two configurtions of copper toms. For this purpose, the momentum verge technique is employed, whilst the wvefunctions of the bound electrons re obtined using n b initio Gussin computer code. The continuum wvefunctions of the emitted electrons re evluted by employing n effective potentil vritionl pproch suggested by bdel-rouf et l (1999, 2002) for solving the corresponding Schrödinger eqution. In section 2 the mthemticl formlism is shortly presented. Section 3 contins the results nd discussions of our dt. Concluding remrks re lso given in the sme section 2

3 2. Mthemticl Formlism Theoreticlly, the uger trnsition rte (0) is determined by (Wentzel 1927): (0) = π τ 2 <φ 3 (r 1 ) φ c(r 2 ) r1 1 r2 φ 1(r 1 ) φ 2(r 2 ) > 2, (1) where τ = h 0 /2πe 2 = h 2 /4 π 2 me 4 = 2.42x10-17 sec, is the time tomic unit, φ c is the wvefunction of the emitted uger electron, (lso referred to s the continuum wvefunction). φ 3 is the wvefunction describing the electron tht filled the hole. The wvefunctions φ 1 nd φ 2 represent the originl bound sttes of these two electrons. They re determined in the present work using Gussin computer code. The wvefunction of the uger electron φ c > is clculted by solving Schrödinger s eqution (H-E) φ c > = 0 >, (2) where H is given by 2 h 2 H = V ( r) r 2m +. (3) V(r) is the effective potentil seen by the continuum electron t distnce r from the infinitely hevy nucleus of the doubly ionized tom. Following McGuire (1972), in the ngulr momentum verge scheme, the uger rte for the trnsition n 1 l 1 n 2 l 2 n 3 l 3 n c l c is connected with the two electrons uger rte (0) by two different forms: Cse I: n 1 l 1 n 2 l 2 = [(h 3 +1) (4 l 1 +2-h 1 ) (4l h 2 )] / [(4l 1 +2) (4l 2 +2)] ( i f ), (4) (0) 3

4 where h 3, h 1, nd h 2 re the numbers of holes in the subshells n 3 l 3, n 1 l 1 nd n 2 l 2, respectively. lso, the indices i nd ƒ specify n initil nd finl stte, respectively. Cse II: n 1 l 1 = n 2 l 2 = [(h 3 +1) (4 l 1 +2-h 1 ) (4l h 1 )] / [(4l 1 +2) (4l 1 +2)] ( i f ), (5) (0) Where (0) ( ) n1l1n 2l 2 n3l3ncl = (1/2) (2l c c + 1) (2l 1 + 1) (2l 1 + 1) h, g hg [Im( h, g)] 2 (6) h nd g re the spin nd orbitl quntum numbers. The corresponding uger widths re determined by Γ ( i) ( i f ) = f 3. Results nd Discussions In this section we present the finl results obtined fter lengthy investigtions of the problem under considertion. These re the uger rtes, uger widths Γ, nd uger energies E chrcteristic to K -, L - nd M - inner shell ioniztions for two configurtions of the copper tom, (to be referred to in the Tbles s Cu I nd Cu II). The following three subsections contin the results nd discussions of our uger trnsitions clculted for the inner shell ioniztions mentioned bove for Cu I nd Cu II. 4

5 3.1. K-Shell uger Trnsitions In Tble 1 we list the uger trnsition rtes 's (mesured in sec-1) corresponding the initil stte 1s 1 2s 2 2p 6 3s 2 3p 6 3d 10 4s 1 of Cu I. The uger trnsitions corresponding to the initil stte 1s 1 2s 2 2p 6 3s 2 3p 6 3d 9 4s 2 of Cu II re given in Tble 2. (Remember tht the superscripts refer to the numbers of electrons in the occupied orbits). From the Tbles we conclude the following remrks: (1) The trnsitions (1s 2s2s), (1s 2s2p), (1s 2p2p), nd (1s 2p3p) represent the dominnt contributions to the uger widths of the initil stte. The third trnsition is the lrgest one. (2) For Cu I nd Cu II, the uger rtes in which the outer electron is involved (e.g. (1s 3d4s) nd (1s 3d3d)) provide smll contributions to the uger widths, whilst (1s 3d4s) hs the smllest contribution in cse of Cu I nd (1s 4s4s) yields the smllest contribution for Cu II. The rtio of the uger widths of Cu I nd Cu II is (3) The resultnt trnsition energies for Cu I show tht E c increses with n nd l for identicl or different effective electrons. For exmple E c = Ry for the trnsition (1s 2s2s), E c = Ry for the trnsition (1s 2p2p) nd E c = Ry for the trnsition (1s 3s3s). E c = Ry for the trnsition (1s 3p3p) nd E c = Ry for the trnsition (1s 3d3d). Similr behvior is concluded from Tble 2. 5

6 Tble 1: uger trnsition rtes in sec -1 for initil sttes of Cu (I), the energy of the emitted electron (in Ry) nd the ngulr momentum of this electron (c). The number between prentheses is the power of 10. Cu (I) Initil Stte : 1s 1 2s 2 2p 6 3s 2 3p 6 3d 10 4s 1 Finl Stte l c E c (Ry) 1s 2 2p 6 3s 2 3p 6 3d 10 4s (13) 1s 2 2s 2 2p 4 3s 2 3p 6 3d 10 4s (13) 1s 2 2s 2 2p 6 3s 2 3p 6 3d 10 4s (12) 1s 2 2s 2 2p 6 3s 2 3p 6 3d 10 4s (11) 1s 1 2s 2 2p 6 3s 2 3p 4 3d 8 4s (7) (8) (10) 1s 2 2s 1 2p 5 3s 2 3p 6 3d 10 4s (13) 1s 2 2s 1 2p 6 3s 1 3p 6 3d 10 4s (12) 1s 2 2s 1 2p 6 3s 2 3p 6 3d 10 4s (13) 1s 2 2s 1 2p 6 3s 2 3p 6 3d 9 4s (12) 1s 2 2s 1 2p 6 3s 2 3p 6 3d (11) 1s 2 2s 2 2p 5 3s 1 3p 6 3d 10 4s (13) 1s 2 2s 2 2p 5 3s 2 3p 5 3d 10 4s (13) 1s 2 2s 2 2p 5 3s 2 3p 6 3d 9 4s (11) (12) 1s 2 2s 2 2p 6 3s 2 3p 6 3d (11) 1s 2 s 2 2p 6 3s 2 3p 6 3d 10 4s (10) 1s 2 2s 2 2p 6 3s 1 3p 6 3d 9 4s (11) 1s 2 2s 2 2p 6 3s 1 3p 6 3d (10) 1s 2 2s 2 2p 6 3s 2 3p 5 3d 9 4s s 2 2s 2 2p 6 3s 2 3p 5 3d (10) 1s 2 2s 2 2p 6 3s 2 3p 6 3d (9) Г =7.445(14) 6

7 Tble 2: uger trnsition rtes in sec -1 for CU (II), the energy of the emitted electron (in Ry) nd the ngulr momentum of this electron (c). The number between prentheses is the power of 10. Cu (II) Initil Stte : 1s 1 2s 2 2p 6 3s 2 3p 6 3d 9 4s 2 Finl Stte l c E c (Ry) 1s 2 2p 6 3s 2 3p 6 3d 9 4s (13) 1s 2 2s 2 2p 4 3s 2 3p 6 3d 9 4s (13) (12) 1s 2 2s 2 2p 6 3p 6 3d 9 4s (12) 1s 2 2s 2 2p 6 3s 2 3p 4 3d 9 4s (11) (12) 1s 2 2s 2 2p 6 3s 2 3p 6 3d 7 4s (7) (7) (10) 1s 2 2s 2 2p 6 3s 2 3p 6 3d 9 4s (13) 1s 2 2s 1 2p 6 3s 1 3p 6 3d 9 4s (12) 1s 2 2s 1 2p 6 3s 2 3p 5 3d 9 4s (13) 1s 2 2s 1 2p 6 3s 2 3p 6 3d 8 4s (12) 1s 2 2s 1 2p 6 3s 2 3p 6 3d 8 4s (11) 1s 2 2s 2 2p 5 3s 1 3p 6 3d 9 4s (13) 1s 2 2s 2 2p 5 3s 2 3p 5 3d 9 4s (13) (14) 1s 2 2s 2 2p 5 3s 2 3p 6 3d 8 4s (11) (12) 1s 2 2s 2 2p 5 3s 2 3p 6 3d 9 4s (11) 1s 2 2s 2 2p 6 3s 1 3p 5 3d 9 4s (12) 1s 2 2s 2 2p 6 3s 2 3p 6 3d 9 4s (10) 1s 2 2s 2 2p 6 3s 1 3p 6 3d 8 4s (10) 1s 2 2s 2 2p 6 3s 2 3p 5 3d 8 4s (10) (11) 1s 2 2s 2 2p 6 3s 1 3p 5 3d 8 4s (10) 1s 2 2s 2 2p 6 3s 2 3p 6 3d 8 4s (9) 1s 2 2s 2 2p 6 3s 2 3p 6 3d (8) Г =7.0988(14) 7

8 3.2. L -Shell uger Trnsitions In this cse two holes re possible, nmely 2s hole or 2p hole, leding to two sets of uger trnsitions, (corresponding to two initil ionized sttes) in ech configurtion of the copper tom. Tbles (3), (4) contin ll possible trnsitions for Cu I ions nd Tbles (5), (6) demonstrte ll possible uger trnsitions in cse of Cu II ions. Tble 3: uger trnsition rtes in sec -1 resulting from 2s-hole in Cu (I), the energy of the emitted electron (in Ry) nd the ngulr momentum of this electron (c). The number between prentheses is the power of 10. Cu (I) Initil Stte : 1s 2 2s 1 2p 6 3s 2 3p 6 3d 10 4s 1 Finl Stte l c E c (Ry) 1s 2 2s 2 2p 5 3s 1 3p 6 3d 10 4s (11) 1s 2 2s 2 2p 5 3s 2 3p 5 3d 10 4s (12) (12) 1s 2 2s 2 2p 5 3s 2 3p 5 3d 9 4s (11) (12) 1s 2 2s 2 2p 5 3s 2 3p 5 3d (10) 1s 2 2s 2 2p 6 3p 6 3d 10 4s (14) 1s 2 2s 2 2p 6 3s 1 3p 5 3d 10 4s (14) 1s 2 2s 2 2p 6 3s 1 3p 6 3d 9 4s ( 14) 1s 2 2s 2 2p 6 3s 1 3p 6 3d (13) 1s 2 2s 2 2p 6 3s 2 3p 4 3d 10 4s (12) (12) 1s 2 2s 2 2p 6 3s 2 3p 5 3d 9 4s (13) (14) 1s 2 2s 2 2p 6 3s 2 3p 5 3d (12) 1s 2 s 2 2p 6 3s 2 3p 6 3d 8 4s (12) (12) (13) 1s 2 2s 2 2p 6 3s 2 3p 6 3d (12) Г = (14) 8

9 Tble 4: uger trnsition rtes in sec -1 resulting from 2p-hole in Cu (I), the energy of the emitted electron (in Ry) nd the ngulr momentum of this electron (c). The number between prentheses is the power of 10. Cu (I) Initil Stte : 1s 2 2s 2 2p 5 3s 2 3p 6 3d 10 4s 1 Finl Stte l c E c (Ry) 1s 2 2s 2 2p 6 3p 6 3d 10 4s (11) 1s 2 2s 2 2p 6 3s 1 3p 5 3d 10 4s (13) (13) 1s 2 2s 2 2p 6 3s 1 3p 6 3d 9 4s (13) (13) 1s 2 2s 2 2p 6 3s 1 3p 6 3d (10) 1s 2 2s 2 2p 6 3s23p 4 3d 10 4s (14) (13) 1s 2 2s 2 2p 6 3s 2 3p 5 3d 9 4s (13) (14) (13) 1s 2 2s 2 2p 6 3s 2 3p 5 3d (12) (9) 1s 2 2s 2 2p 6 3s 2 3p 6 3d 8 4s (12) (14) (12) 1s 2 2s 2 2p 6 3s 2 3p 6 3d (10) (10) Г = (15) 9

10 Tble 5: uger trnsition rtes in sec -1 resulting from 2s-hole in Cu (II), the energy of the emitted electron (in Ry) nd the ngulr momentum of this electron (c). The number between prentheses is the power of 10. Cu (II) Initil Stte : 1s 2 2s 1 2p 6 3s 2 3p 6 3d 9 4s 2 Finl Stte l c E c (Ry) 1s 2 2s 2 2p 5 3s 1 3p 6 3d 9 4s (11) 1s 2 2s 2 2p 5 3s 2 3p 5 3d 9 4s (12) (12) 1s 2 2s 2 2p 5 3s 2 3p 6 3d 8 4s (11) (12) 1s 2 2s 2 2p 6 3s 2 3p 5 3d 9 4s (10) 1s 2 2s 2 2p 6 3p 6 3d 9 4s (14) 1s 2 2s 2 2p 6 3s 1 3p 5 3d 9 4s (14) 1s 2 2s 2 2p 6 3s 1 3p 6 3d 8 4s (14) 1s 2 2s 2 2p 6 3s 1 3p 6 3d 9 4s (12) 1s 2 2s 2 2p 6 3s 2 3p 4 3d 9 4s (12) (12) 1s 2 2s 2 2p 6 3s 2 3p 5 3d 9 4s (13) (14) 1s 2 2s 2 2p 6 3s 2 3p 5 3d 9 4s (12) 1s 2 s 2 2p 6 3s 2 3p 6 3d 7 4s (12) (11) (13) 1s 2 s 2 2p 6 3s 2 3p 6 3d 8 4s (12) 1s 2 2s 2 2p 6 3s 2 3p 6 3d (10) Г = (14) 10

11 Tble 6: uger trnsition rtes in sec -1 resulting from 2p-hole in Cu (II), the energy of the emitted electron (in Ry) nd the ngulr momentum of this electron (c). The number between prentheses is the power of 10. Cu (II) Initil Stte : 1s 2 2s 2 2p 5 3s 2 3p 6 3d 9 4s 2 Finl Stte l c E c (Ry) 1s 2 2s 2 2p 6 3p 6 3d 9 4s (11) 1s 2 2s 2 2p 6 3s 1 3p 5 3d 9 4s (13) (13) 1s 2 2s 2 2p 6 3s 1 3p 6 3d 8 4s (13) (12) 1s 2 2s 2 2p 6 3s 1 3p 6 3d 9 4s (10) 1s 2 2s 2 2p 6 3s23p 4 3d 9 4s (14) (13) 1s 2 2s 2 2p 6 3s 2 3p 5 3d 8 4s (13) (14) (13) 1s 2 2s 2 2p 6 3s 2 3p 5 3d (12) (12) 1s 2 2s 2 2p 6 3s 2 3p 5 3d 7 4s (12) (14) (12) 1s 2 2s 2 2p 6 3s 2 3p 6 3d 8 4s (10) (10) 1s 2 2s 2 2p 6 3s 2 3p 6 3d (10) Г = (14) From Tbles 3 nd 5 we notice the following points: (1)The dominnt contributions to the uger widths of the 2s ionized forms of the Cu I nd Cu II re obtined from the trnsitions (2s 3s3s), (2s 3s3p), (2s 3s3d) nd (2s 3p3d). The second trnsition yields the mximum contributions. 11

12 (2) The rtio between Г 's of the configurtions Cu I nd Cu II is equl to similr vlue ws obtined for 1s ionized toms, where the difference between the two set of trnsitions is miniml. The following remrks cn be lso noticed from Tbles 4 nd 6: (1)The dominnt contributions to the vlues of the uger widths, Г 's, corresponding to the 2p innershell ioniztions of Cu I nd Cu II re delivered by the trnsition (2p 3s3p), (2p 3p3p), (2p 3p3d) nd (2p 3d3d). The uger trnsition rtes (2p 3p3p) nd 3p3d) re superior. (2p (2)The rtio between the Г 's, of Cu I nd Cu II corresponding to the 2ptrnsitions is , which lies in the sme order of mgnitude of the rtions determined for 1s hole nd 2s hole trnsitions M - Shell uger Trnsitions The emitted uger electrons in this cse cn be originted by ny M-subshell hole, prt from 3d one. This is ttributed to the fct tht no trnsitions corresponding 3d-hole re llowed. Thus, only 3s or 3p hole decy is possible. The 3s 3p3p is energeticlly forbidden. ll possible 3s - uger trnsition rtes nd the corresponding widths of the initil sttes 1s 2 2s 2 p 6 3s 1 3p 6 3d 10 4s 1 nd 1s 2 2s 2 p 6 3s 1 3p 6 3d 9 4s 2 of Cu I nd Cu II, respectively, re given in Tbles 7 nd 8 12

13 Tble7: uger trnsition rtes in sec -1 resulting from 3s-hole in Cu (I), the energy of the emitted electron (in Ry) nd the ngulr momentum of this electron (c). The number between prentheses is the power of 10. Cu (I) Initil Stte : 1s 2 2s 2 2p 6 3s 1 3p 6 3d 10 4s 1 Finl Stte l c E c (Ry) 1s 2 2s 2 2p 6 3s 2 3p 5 3d 9 4s (12) (12) 1s 2 2s 2 2p 6 3s 2 3p 5 3d (11) 1s 2 2s 2 2p 6 3s 2 3p 4 3d 8 4s (12) (13) (14) 1s 2 2s 2 2p 6 3s 2 3p 6 3d (14) Г = (14) Tble 8: uger trnsition rtes in sec -1 resulting from 3s-hole in Cu (II), the energy of the emitted electron (in Ry) nd the ngulr momentum of this electron (c). The number between prentheses is the power of 10. Cu (II) Initil Stte : 1s 2 2s 2 2p 6 3s 1 3p 6 3d 9 4s 2 Finl Stte l c E c (Ry) 1s 2 2s 2 2p 6 3s 2 3p 5 3d 8 4s (12) (12) 1s 2 2s 2 2p 6 3s 2 3p 5 3d 9 4s (11) 1s 2 2s 2 2p 6 3s 2 3p 6 3d 7 4s (12) (13) (14) 1s 2 2s 2 2p 6 3s 2 3p 6 3d 8 4s (14) 1s 2 2s 2 2p 6 3s 2 3p 6 3d (12) Г = (14) 13

14 Tbles 7 nd 8 led to the following remrks: (1) The two uger trnsitions (3s 3d3d) nd (3s 4s3d) yield the dominnt contributions to the uger widths of both Cu I nd Cu II. (2) The energy of the continuum electron increses s n or l increses for equivlent or nonequivlent effective electrons. (3) The mximum vlues of E c re Ry for Cu I nd Ry for Cu II. (4) In generl the continuum energies of the 3s trnsitions re smller thn the continuum energies of the previously clculted trnsitions. (5) The rtio of Г 's of the two copper configurtions is which is nerly equl to the rtios clculted for the 1s hole, 2s hole nd 2p hole trnsitions. The results of 3p hole decys in Cu I nd Cu II re ccumulted in Tbles 9 nd 10. From the Tbles we conclude the following points: (1) The number of possible 3p uger trnsitions is reduced to (3p 3d3d) nd (3p 3d4s). lso the trnsition (3p 4s4s) is llowed. The first of these three trnsitions possesses the lrges contribution to the widths in both Cu I nd Cu II. (2) The uger trnsition (3p 3d3d) is the lrges trnsition corresponding to continuum energy E c 65 ev, which represents the pek of the uger spectrum. 14

15 Tble 9: uger trnsition rtes in sec -1 resulting from 3s-hole in Cu (I), the energy of the emitted electron (in Ry) nd the ngulr momentum of this electron (c). The number between prentheses is the power of 10. Cu (I) Initil Stte : 1s 2 2s 2 2p 6 3s 2 3p 5 3d 10 4s 1 Finl Stte l c E c (Ry) 1s 2 2s 2 2p 6 3s 2 3p 6 3d 8 4s (14) Scientific Reserch nd Essys (15) (12) 1s 2 2s 2 2p 6 3s 2 3p 6 3d (13) (12) Г = (15) Tble10: uger trnsition rtes in sec -1 resulting from 3p-hole in Cu (II), the energy of the emitted electron (in Ry) nd the ngulr momentum of this electron (c). The number between prentheses is the power of 10. Cu (I) Initil Stte : 1s 2 2s 2 2p 6 3s 2 3p 5 3d 9 4s 2 Finl Stte l c E c (Ry) 1s 2 2s 2 2p 6 3s 2 3p 6 3d 7 4s (14) (15) (12) 1s 2 2s 2 2p 6 3s 2 3p 6 3d 8 4s (14) (13) 1s 2 2s 2 2p 6 3s 2 3p 6 3d (10) Г = (15) 15

16 References bdel-rouf M.., El-Bkry S. Y. nd El-Bkry M. Y (2002)., Mod. Phys. Lett., 14: 877. bdel-rouf M.., El-Bkry S. Y. nd El-Bkry M. Y Ind. J. Phys B. (1999), 73: 711. uger P., J. Compt. Rend (1923), 117: 169. uger P. (1925), J. Phys. Rdium 6: 205. McGuire E. J., uger nd Coster Kronig Trnsitions in Inner Shell tomic Processes, ed. B. Crsmnn, cdemic Press-(1972). Sfronov U. I., Johnson W. R. nd lbritton J. R. (2001), t. Dt Nucl. Dt Tbles, 77: 215. Wentzel G. (1927), Zeitsch. Phys., 43:

Name Solutions to Test 3 November 8, 2017

Name Solutions to Test 3 November 8, 2017 Nme Solutions to Test 3 November 8, 07 This test consists of three prts. Plese note tht in prts II nd III, you cn skip one question of those offered. Some possibly useful formuls cn be found below. Brrier