Correlated radiative electron capture in ion-atom collisions

(1)

Anna Simon

Correlated radiative electron capture in ion-atom collisions

P h .D . D issertation

p rep ared under the supervision of Prof. A ndrzej W ar czak

M arian Smoluchowski In stitu te of Physics Jagiellonian U niversity

Kraków, A pril 2010

(2)

Anna Simon Correlated radiative electron capture in ion-atom collisions

A b stract

R adiative double electron cap tu re (RD EC) is a one-step process where two free (or quasifree) ta rg e t electrons are ca p tu red into a b ou n d s ta te of th e projectile, e.g. into an em pty K-shell, an d th e energy excess is released as a single photon. T his process can be tre a te d as a tim e inverse of a double photoionization. However, unlike in case of photoionization experim ents, bare ions are used during R D E C observations. Thus, R D E C can be considered as th e sim plest, clean tool for investigation of electron-electron interactio n in th e presence of electrom agnetic fields generated during ion-atom collisions.

W ith in th is dissertation, th e 38 MeV 0 8+ + C experim ent, conducted at W estern M ichigan U niversity using th e tan d em Van de G raaff accelerator, is discussed and th e first experim ental evidence of th e R D E C process is presented. T he cross section obtain ed experim entally is com pared to th e late st theoretical calculations.

(3)

A nna Simon Correlated radiative electron capture in ion-atom collisions

A bstrak t

Skorelowany radiacyjny wychwyt dwóch elektronów (RD EC) jest procesem , podczas które

go dwa sw obodne (albo kwaziswobodne) elektrony tarczy wychwytywane są do stan u związane

go pocisku (np. nieobsadzonej powłoki K), a różnica energii pom iędzy końcowym a początko

wym stan em eletronów em itow ana jest w p ostaci pojedynczego fotonu. Proces te n m ożna traktow ać jako odwrócenie w czasie podw ójnej fotojonizacji. Jednakże, w przeciwieństwie do eksperym entów dedykowanych fotojonizacji, do obserwacji R D E C stosuje się jony całkowicie pozbawione elektronów, co pozw ala n a wyeliminowanie t ła pochodzącego od elektronów nie biorących bezpośrednio u działu w badanym procesie. R D E C może być więc traktow any jako najp rostsze narzędzie do b ad an ia oddziaływ ania elektron-elektron w obecności pola elektro

m agnetycznego generowanego podczas zderzenia.

Rozpraw a t a poświęcona jest procesom atom ow ym zachodzącym w zderzeniach 0 8+ + C przy energii 38 MeV podczas eksperym entu przeprowadzonego przy użyciu alceleratora Van de G raaffa w W estern M ichigan University. Przedstaw ione zostało w niej pierwsze doświadczalne potw ierdzenie procesu R D E C . Uzyskany eksperym entalnie przekrój czynny został porów nany z wynikami najnow szych przew idyw ań teoretycznych.

(4)

A cknow ledgem ents

F irst, I would like to give my sincere th an k s to my supervisor, Professor Andrzej W arczak.

He offered me advice, p atien tly supervised m e and always guided m e to th e correct direction.

I have learned m uch from him, w itho u t his help I would have never finished my d issertatio n successfully.

Special th an k s are also given to Professor Jo h n A. Tanis. He is th e one who invited me to W estern M ichigan U niversity for my research during th e 2008-2009 academ ic year. His help and encouragem ent m ade me feel confident enough to fulfill my desires and to overcome th e difficulties I encountered. His u n derstanding, encouragem ent and personal guidance have provided a good basis for my thesis. It is n o t sufficient to express my g ratitu d e w ith such a few words.

I am very grateful to Professor Bogusław Kam ys, th e H ead of th e N uclear Physics Division a t th e In stitu te of Physics, Jagiellonian U niversity and Professor P aul Pancella, th e C hair of Physics D ep artm en t, W estern M ichigan University, for th e financial su p p o rt of my stay at W M U.

Additionally, I owe my m ost sincere g ratitu d e to Dr. A sghar K ayani for his patience while teaching me how to o p erate th e W M U tan d em Van de G raaff accelerator and his willingness to im m ediately solve any beam related problem s during my experim ent. M y sincere th an k s should also go to Rick Welch and A llan K ern for th eir help w ith th e m aintenance of the experim ental setup.

I would also like to th a n k Janusz Kopczyński, A dam M alarz an d A dam M ucha for th eir care and su p p o rt during my work at Jagiellonian University.

I am also very grateful to Prof. T hom as Stöhlker, th e H ead of A tom ic Physics G roup at GSI, D a rm sta d t, who frequently invited m e to join his g roup’s experim ents during which I h ad a chance to learn th e secrets of th e experim ental work of atom ic physicists. I ’m grateful to Dr. Angela B räuning-D em ian an d Dr. C hristophor Kozhuharov for inspiring conversations th a t led to m any ideas im plem ented in th is thesis.

(5)

A nna Simon Correlated radiative electron capture in ion-atom collisions I am very grateful to my friends, D avid Cassidy, B uddhika D assanayake, M ałg orzata M akuch, D agm ara R ozpędzik an d A ndrzej Pezarski for always being th ere for me.

Last, b u t not least, I w anted to th a n k my paren ts for th eir su p p o rt and encouragem ent.

(6)

Table o f C ontents

L ist o f Tables viii

L ist o f F igu res x

L ist o f S y m b ols and A b b rev ia tio n s x iv

1 In tro d u ctio n 1

2 A to m ic p rocesses d u rin g io n -a to m co llisio n s at low en erg y 4

2.1 N onradiative electron cap tu re ( N R E C ) ... 4

2.2 R adiative electron cap tu re ( R E C ) ... 5

2.3 B r e m s s tra h lu n g ... 7

2.3.1 E lectron b r e m s s tr a h lu n g ... 9

2.3.2 N ucleus-nucleus b rem sstrah lun g (NB) 12 2.4 M ultielectron cap tu re processes, noncorrelated double radiative electron cap tu re ( D R E C ) ... 13

2.5 P rojectile ionization - electron l o s s ... 15

3 R a d ia tiv e d ou b le electro n ca p tu re (R D E C ) 17 3.1 Initial experim ents dedicated to R D E C ... 17

3.2 R ecent theoretical approach to R D E C ... 20

4 E x p erim en ta l se tu p at W estern M ich igan U n iv ersity 25 4.1 Van de G raaff accelerator 25 4.2 Beam line setu p a t W estern M ichigan U niversity 27 4.3 D a ta acquisition s y s t e m ... 31

(7)

5 D a ta an a lysis 34

5.1 P IX E analysis of th e ta rg e t m a t e r i a l ... 37

5.2 P rojectile K- and L-shell electron loss 38

5.3 B ackground processes 40

5.4 Pile-up of single R E C photons and its con tribution to th e R D E C energy range of th e s p e c t r u m ... 42 5.5 Single sp ectra a n a l y s i s ... 45 5.6 Coincidence sp ectra a n a ly s is ... 47

6 T h e R D E C cross sectio n 51

6.1 E xperim ental value of th e R D E C cross s e c tio n ... 51 6.2 E stim atio n of th e cfrdec/p r e c ratio in th e nonrelativistic a p p r o a c h ... 52 6.3 R D E C cross section based on th e Y akhontov a p p r o a c h ... 53

7 M o n te C arlo sim u la tio n s o f th e x-ray sp ectra 55

8 C on clu sion s 60

A S ta tistica l an alysis o f th e o b served sign al 62

L ist o f R eferen ces 66

(8)

List o f Tables

3.1 C om parison of experim entally obtain ed R D E C cross sections [War 95, Bed 03]

an d th e calculated values given in [Yak 97] and [Mik 04a]... 23 3.2 T h e R E C ( e r ^ ), R D E C (cr^2,7)) and D R E C (cr^2,27)) cross sections and their

ratios as given in [Dru 07]... 24 5.1 R esults of a x 2 test of th e R D E C range of th e pro to n induced spectrum . . . . 38 5.2 E lectron loss cross sections for oxygen ions a t 38 MeV estim ated from th e d a ta

presented in [Bom 89, T an 91, Hip 87]... 39 5.3 T otal cross sections for th e background processes th a t were tak en into account

during d a ta analysis... 42 5.4 Probabilities and count rates of th e processes th a t m ight co n tribute to th e x-ray

sp ectrum in th e R D E C range. For m ore inform ation see te x t ... 43 5.5 C alculated positions of th e R D E C an d R E C peaks in th e x-ray sp ectru m corre

sponding to different com binations of th e initial and final sta te s of th e c ap tu red

electrons. All values are given in keV. 45

5.6 R esults of a x 2 test of th e R D E C range of th e coincidence s p e c tra ... 47 5.7 Areas (^4) of th e shapes of th e R D E C contributions fitted to th e sum of q — 1

an d q — 2 spectra. FW H M of all lines was set to 0.3 keV which is th e w idth of th e carbon C om pton profile... 50 6.1 C om parison of th e experim ental values of th e R D E C cross section and th e R =

a R D E c / a R E C ratio w ith th e ones obtained from various theoretical approaches . 54 7.1 R atios of th e num bers of counts in th e R D E C and R E C regions obtained during

M onte Carlo sim ulations com pared w ith th e experim ental value... 59

viii

(9)

A nna Simon Correlated radiative electron capture in ion-atom collisions 8.1 Sum m ary of th e results of th e theoretical calculations and th e experim ents

d edicated to th e R D E C process... 61

A .l Q uantiles of x 2 d istrib u tio n for D o F = 1 [Kam]. 64

(10)

List o f Figures

2.1 R ad iativ e electron ca p tu re (R E C ). T arget electron is cap tu red into th e projec

tile b oun d s ta te and th e energy excess is em itted as a single photon. 6 2.2 E xam ple of th e x-ray sp ectru m registered in coincidence w ith single electron

ca p tu re during U 92+ + N2 collisions a t 309.7 M eV /u [Swi 00]... 7 2.3 C om pton profile of electrons in carbon atom . It can be noticed th a t th e stru c

tu re of th e I s line is m uch broader th a n th a t for n = 2 sta te s [Big 75]... 8 2.4 E xam ple of b rem sstrahlu n g process - an electron in th e electrom agnetic field

of an ion. 8

2.5 R adiative electron cap tu re to continuum (R E C C ). T arget electron is c ap tu red into a continuum s ta te of th e projectile and a ph oton is em itted. 9 2.6 Spectra observed during th e experim ent [Bed 98] for: (a) B e-target, (b) C-

targ e t. For each energy-target com bination two sp ectra are displayed. O n th e to p - raw spectru m an d lower - sp ectrum after background su b tractio n . For presen tatio n purposes sp e ctra were m ultiplied by factors; for B e-target: 1/20 - 75 M eV /u, 10 - 290 M eV /u; for C -target: 1 / 8 - 7 5 M eV /u, 10 - 150 M eV /u, 50 - 290 MeV/u . D ashed line: SEB contribution, solid line: R E C C (relativistic approxim ation) + K -R E C + SEB. Arrows show th e R EC C -edge energy T r transform ed to th e lab o rato ry fram e. Inset in (a) represents th e experim ental

setup. 11

2.7 E xam ple contrib u tio n of various b rem sstrahlung processes to th e continuous x-ray sp ectru m during Al + C collisions at 1 and 4 MeV [Ish 06]. It can be noticed th a t SEB becomes a dom inating process a t higher beam energy. . . . 12 2.8 B rem sstrahlung processes observed during p + C collisions at 2 MeV. P lo t

based on Fig. 3 in [Ish 06]... 13 2.9 C om parison of th e D R E C an d R D E C processes... 14

(11)

A nna Simon Correlated radiative electron capture in ion-atom collisions 2.10 Single electron loss cross section as given in [Hip 87]. Solid line - PW B A

calculations for H e2+ im pact, dot-d ash ed line - includes con tribu tion from free electron im pact in C B E approxim ation. Symbols: □ - 0 6+, A - Si8+, V - Si13+, o an d x - 0 7+... 15

3.1 Typical x-ray sp ectrum obtained during argon experim ent [War 95]. 18 3.2 Typical experim ental x-ray sp ectru m obtained for u ranium ions [Bed 03]. T he

G aussian solid line shows th e expected R D E C peak, which should be observed

according to Y akhontov et al. [Yak 96, Yak 97]. 19

3.3 Q, universal function of th e dim ensionless variable £ [Mik 04a]. 21 3.4 U niversal q u a n tity Q / H calculated as a function of th e dimensionless variable

£ [Mik 04a]... 22

(21 (21

3.5 T h e ratio of th e R D E C cross sections to th e excited (<7215) an d ground ( v \ i s ) projectile sta te s as a function of adiabacity param eter £ [Nef 05]... 23 4.1 Schem atic view of a classical Van de G raaff accelerator: (1) lower roller, (2)

u p p er roller, (3) charging electrode, (4) electrode collecting positive charge, (5) voltage generator, (6) spherical electrode (high voltage term inal), (7) ion source, (8) ex tra cte d ion b e a m ... 26 4.2 Schem atic view of a tan d em Van de G raaff accelerator: N egative ion entering

th e accelerator (A - ) is accelerated by th e high term inal voltage. Some of its electrons are rem oved while th e ion passes th ro u g h th e stripp ing foil. T he positive ion (A +9) is repelled by th e high voltage term inal, thu s additionally accelerated... 27 4.3 Schem atic view of th e W M U van de G raaff accelerator facility [Kay]... 28

4.4 T h e experim ental ta rg e t cham ber in 1:1 scale. 29

4.5 D etection efficiency of O R TEC Si(Li) detectors [ORTa]. 30

4.6 E xperim ental setup. 30

4.7 E xam ple of a tim e spectru m registered during th e experim ent. T h e arrow indi

cates th e w idth of a tim e window for tru e coincidences (calibration 2 n s/ch an n el). 32 4.8 Scheme of th e d a ta acquisition sy stem ... 33

(12)

A nna Simon Correlated radiative electron capture in ion-atom collisions 5.1 E xperim ental single x-ray spectra. In all spectra: solid line - 38 MeV 0 8+.

(a) dashed line - 0 8+ d a ta taken w ithout th e carbon foil, (b) 0 8+ d a ta after su b tra c tio n of th e A1 K - a line, (c) d o tte d line - 38 MeV 0 7+, (d) dot-dashed line - 2.375 MeV p ro to n s... 35 5.2 X rays registered for 0 8+ + C collisions in coincidence w ith ions which cap tu red

(a) two electrons and (b) one electron, solid line - th e sum of th e R E C C om pton profile and th e G aussian shape of th e oxygen K - a line fitted to th e spectrum . 36 5.3 P ro to n induced x-ray spectrum . Solid line - th e R D E C range, dashed line -

region considered during background estim ation. 37

5.4 B ackground stru c tu re in th e single x-ray spectrum . T he brem sstrahlun g con

trib u tio n includes all th e relevant processes (SEB + AB + NB) discussed in th e tex t. T he spectru m is com pletely d om inated by th e R E C stru c tu re . 41 5.5 Nr d e c/ Nr e c ratio in th e q — 1 coincidence sp e ctra as a function of beam

intensity... 44 5.6 0 8+ spectrum tak en w ith ou t th e carbon foil (red line) norm alized to th e d a ta

taken w ith th e foil (black lin e)... 46 5.7 D ouble s tru c tu re of th e R E C line resolved after su b tra c tio n of th e A1 K - a line. 46 5.8 Possible R D E C tran sitio n s (a) an d th e stru c tu re of th e produced x-ray spec

tru m (b) when equal cross sections for all th e p a rtia l processes are assum ed.

Black line - th e sum of all contributions. Additionally, corresponding R D E C sp ectra ob tained experim entally in single (c) and double (d) charge exchange channels are presen ted ... 48 5.9 T h e sum of sp e ctra registered in single and double charge exchange channels

w ith a fit of all possible com binations of th e R D E C tran sitio n s. F ittin g p aram eters are given in Table 5.7... 50

7.1 G eom etry of th e experim ental setup im plem ented in th e M onte Carlo sim u

lation, bw - th e beam diam eter, dt - ta rg e t thickness in m m. T h e x-axis is p erpendicular to th e p icture p lan e... 56 7.2 M onte Carlo sim ulated x-ray spectra: (a) no R D E C included, (b) th e R D EC

cross section as given by Nefiodov, (c) o l^ , EC = 3 b and = 2.1 b - cross sections values for which M C sim ulation gives th e results closest to th e experim ental d a ta , (d) E xperim entally obtained single sp e ctru m ... 57

(13)

A nna Simon Correlated radiative electron capture in ion-atom collisions 7.3 T h e R D E C range of th e x-ray spectra. R esults of sim ulations: (a) no R D EC

included, (b) th e R D E C cross section as given by Nefiodov, (c) <yl^ , EC = 3 b and (Tji deq = 2.1 b - cross sections values for which M C sim ulation gives th e results closest to th e experim ental d a ta , (d) E xperim entally ob tained single

sp ectrum 58

A .l E xam ple of an experim entally obtain ed spectrum w ith a s tru c tu re w ithin th e A B ran g e ... 63

(14)

List o f Sym bols and A b b reviation s

V E lectron m om entum w ithin th e targ e t bound sta te huj P h o to n energy in th e lab o ra to ry fram e

%(Pz) C om pton profile

Som merfeld param eter for K-shell electron

n D etector solid angle

a Cross section

e O bservation angle in th e lab o ra to ry fram e

Dim ensionless param eter describing collision velocity (adiabacity param eter)

A P rojectile m ass A t Target m ass

b N um ber of background counts bw B eam diam eter [mm]

d Target thickness [p articles/cm 2]

dt Target thickness [mm]

E P rojectile kinetic energy [MeV/u]

Eb Binding energy of an electron in the bound s ta te of the projectile E ßt Binding energy of an electron in the bo un d s ta te of the targ et I B eam intensity [ions/s]

(15)

A nna Simon Correlated radiative electron capture in ion-atom collisions m e E lectron rest m ass

trip P ro to n rest m ass

N N um ber of counts

P M om entum

Pr d e c P ro b ab ility th a t th e p ho to n is registered in th e R D E C range of th e x-ray spectrum Pr e c P ro b ab ility th a t th e p h oton is registered in th e R E C range of th e x-ray spectrum

q P ro jectile charge s ta te

T S tatistical variable of th e

x2

test

Tm M axim um energy transfer during ion-atom collision (in th e lab o ra to ry fram e) T r K inetic energy of th e free ta rg e t electron calculated in th e projectile fram e V P ro jectile velocity

Ve E lectron velocity

z

P ro jectile atom ic num ber

Zt

T arget atom ic num ber AB A tom ic brem sstrah lu ng C B E Coulom b B orn exchange

D R E C D ouble radiativ e electron cap tu re NB Nucleus brem sstrah lun g

N R EC N onradiative electron cap ture PW B A P lain wave B orn approxim ation Q FEB Quasifree electron brem sstrah lu n g

R D E C R adiative double electron cap tu re R E C C R adiative electron cap tu re to continuum

(16)

A nna Simon Correlated radiative electron capture in ion-atom collisions R E C R adiative electron cap tu re

R I R adiative ionization R R R adiative recom bination

SEB Secondary electron brem sstrah lu ng

a = 1/137 Fine s tru c tu re constant

h = 6.582 • 10 16 eV-s P lanck co nstant ao = 5.29 • 10“ 11 m B ohr radius c = 2.997 • 108 m /s Speed of light

xvi

(17)

C hapter 1

Introduction

Since th e first observation of th e photoelectric effect by H ertz [Her 87] and its explanation by E instein [Ein 05] th e in teractio n betw een electrons and light has been of considerable atten tio n . T he fundam ental process occurring due to this interaction is photoionization, where ab sorption of a p h o to n of energy hu> results in th e emission of an electron:

A + hw ^ A + + e ~ . (1.1)

Simple photoionization experim ents usually are restricted to n eu tral atom s, where th e influ

ence of th e electrons, which do n o t p a rticip a te in th e process directly, cannot be neglected.

T his com plicates com parison of th e experim ental results w ith theoretical predictions.

However, based on th e principle of detailed balance [Lan 79, Bey 03] th e photoionization can be stu died via th e tim e reversed processes, i.e. rad iative recom bination (R R ) and radiative electron cap tu re (R EC ) [Ich 94, Ich 96, Eic 95a]. D uring these processes a free (RR) or loosely b ou nd (REC ) electron is ca p tu red to th e b o un d s ta te of th e projectile an d a photon w ith energy equal to th e difference betw een th e final and initial electron sta te s is em itted. Unlike single photoionization of m ultielectron system s, R E C has been investigated for b are ion-atom interactions [Sto 92, Sto 94] and offers clean conditions for exploration of photoionization w ith only one electron, allowing for observation of pure photon-electron interactions.

D uring th e last th irty years double photoionization has been of considerable interest [Dal 94, and references therein]. As a ph oton typically in teracts only w ith one electron, double photoionization is caused by th e electron-electron interactio n [Smi 89]. However, double photoionization studies have been perform ed m ainly for low Z atom s, such as He [Ber 93, Tiw 82, C ar 81], Ne [Sai 92, Sch 93, C ar 77], and A r [Lab 87, C ar 77]. T his is due to th e background contributions from o th er electrons for high Z system s, which m ake th e subtle

(18)

Anna Simon Correlated radiative electron capture in ion-atom collisions electron correlation effects difficult to observe. Fortunately, sim ilarly to single photoioniza

tion, double photoionization can be stu died by m eans of th e tim e inversed process - R D E C , for which th is background is absent. R adiativ e double electron cap tu re (RD EC) involves transfer of two ta rg e t electrons into a b o un d s ta te of th e projectile w ith sim ultaneous emission of a single p h oton [War 95, Bed 03]. Since b are ions are used during th e experim ent, R D EC can be considered as th e sim plest, clean tool for investigation of electron-electron interaction [War 95] in th e presence of electrom agnetic fields generated during ion-atom collision. Thus, investigation of th e R D E C process can provide crucial inform ation necessary for a proper description of electron correlations w ithin atom ic system s and provide d a ta required to define th e wave function of two correlated electrons in th e projectile continuum .

D uring th e last tw enty years th e R D E C process was addressed not only experim entally [War 95, Bed 03], b u t also theoretically [Mir 89, Yak 96, Yak 97]. T he calculations were found to be in disagreem ent w ith th e experim ental d a ta [Bed 03] and verification of th e R D E C pro

cess was not possible. T he m ore recent calculations n ot only explained previous experim ental results, b u t also suggested th e choice of low energy m id-Z ( Z < 35) collision system s for observation of R D E C [Mik 04a, M ik 04b, D ru 07]. It is also noted th a t for these system s cap tu re to an excited I s 12 s 1 s ta te m ight significantly enhance th e process an d con tribu te to th e observed x-ray sp e ctra [Nef 05]. These calculations provided th e m ain m otivation for yet an o th er experim ent dedicated to th e R D E C process. To fully take advantage of th e new calculations, two collision system s at two different accelerator complexes were chosen:

• X e54+ + C at 20 M eV /u, to be perform ed a t GSI, D a rm sta d t in Germany,

• 0 8+ + C a t 2.375 M eV /u, realized by m eans of th e Van de G raaff accelerator a t W M U, K alam azoo, MI, USA.

So far th e W M U experim ent was carried out. D uring six m onths of th e experim ent p rep ara

tions and d a ta taking, 43 days of beam tim e were used. A t th e m om ent w hen this thesis is being w ritte n th e GSI beam tim e is still pending.

W ith in this dissertatio n th e 0 8+ + C a t 38 MeV experim ent is discussed and th e first ex

perim ental evidence of th e R D E C process is presented. T h e cross section obtained experim en

tally is com pared w ith th e latest theoretical calculations [Mik 04a, M ik 04b, Nef 05, D ru 07].

T his thesis begins w ith an in trod u ctio n to th e m ost im p o rta n t atom ic processes th a t occur during ion-atom collisions. In C h ap ter 2 special a tte n tio n is paid to th e processes which

2

(19)

A nna Simon Correlated radiative electron capture in ion-atom collisions add to th e background for th e x-ray sp ectrum registered during th e experim ent and form ulae allowing for estim ations of contributions of these processes are suggested. C h ap ter 3 addresses th e R D E C process in detail. T he history of th e experim ental approach and th e theoretical calculations of th e R D E C cross section are presented. Additionally, th is chapter focuses on th e recent theoretical calculations which were th e m ain m otivation for th e experim ent discussed in this d issertation. T h e goal of th e experim ent was th e observation of x rays em itted during collisions of bare oxygen ions w ith carbon atom s. T h e x-rays sp ectra were registered in coincidence w ith ongoing particles which underw ent single or double charge exchange. T he experim ental setup which allowed for achieving this goal is presented in C hap ter 4. T he operatio n principle of a Van de G raaff accelerator is explained and th e construction of the ta rg e t cham ber, p article spectrom eter and x-ray d etector are described in detail. C h ap ter 5 is d edicated to d a ta analysis, w ith a p articu lar focus on processes th a t m ay contrib u te to th e x-ray sp ectrum w ithin th e R D E C region. Various approaches to estim atio n of th e background and calculations of th e cross section are discussed. In C h ap ter 6 th e experim entally obtained R D E C cross section is com pared w ith th e theoretical value and th e possible reasons for th e ob tained discrepancy are given. In C h ap ter 7 results of th e M onte Carlo sim ulations of th e x-ray sp ectru m generated during th e 0 8+ + C collisions are com pared w ith th e experim ental results. Finally, in C h a p te r 8 suggestions for fu rth er investigations of th e R D E C process are given, w ith indication of necessary im provem ents of th e experim ental setup.

(20)

C hapter 2

A tom ic processes during ion -atom collisions at low energy

In teractio n betw een an incom ing ion and a ta rg e t atom m ay lead to m any different atom ic processes, such as:

• ionization, m ainly of th e ta rg e t atom , as th e electrons are usually less b o u n d to a light ta rg e t th a n to a p a rtially ionized projectile,

• electron tran sfer from th e ta rg e t to th e projectile,

• excitation of b o th ta rg e t an d projectile sta te s - such sta te s deexcite after th e collision em itting ch aracteristic x rays.

W ith in th e following sections th e m ost im p o rta n t processes th a t were considered com pet

itive to R D E C for th e presented experim ent are discussed.

2.1 N onradiative electron capture (NR EC )

T he Coulom b interaction betw een th e projectile and th e ta rg e t electrons can lead to a process called Coulom b cap tu re or n onradiative electron cap tu re (N R E C ). Here, th e energy difference betw een th e initial and final sta te of th e electron is converted into th e kinetic energy of th e collision p artn ers. T h e m ost convenient and frequently used scaling form ula th a t estim ates th e cross section for n onradiative electron cap tu re is th e one given by Schlachter [Sch 83]. It is a sem iem pirical form ula which allows for calculation of th e o n r e c as a function of th e projectile energy for various projectiles w ith an accuracy b e tte r th a n 30%.

4

(21)

Anna Simon Correlated radiative electron capture in ion-atom collisions T he N R E C process occurs m ainly a t th e velocity m atching condition v « v e, where v e is th e velocity of th e cap tu red electron, b o u nd in th e ta rg e t atom . For v v e in nonrelativistic approxim ation, as shown for exam ple in [Eic 07], th e N R E C cross section scales as:

Z 5Z 5

& N R E C ~ y i iTô • (2-1)

2.2 R adiative electron capture (REC)

R adiative electron ca p tu re (REC ) is one of th e best known atom ic processes observed in heavy ion-atom collisions. It was first observed in early seventies of th e last century [Sch 72, Kie 73, Sch 74] and since th a t tim e has been intensively stu died b o th experim entally [Kan 95, M ok 95, Spi 79, Sto 95b, Sto 92, Sto 94, Sto 95a, Sto 97b, Sto 97a, Sto 98, T an 81, T an 87]

and theoretically [Eic 95a, Eic 95b, Hin 87, Ich 94, Ich 96, Soh 76]. D uring th is process a cap tu re of a single ta rg e t electron is followed by a p h oto n emission (Fig. 2.1). Energy Er e c

of th e em itted p h oton fulfills energy conservation rule for this process. Thus, it is given by:

Er e c = Tr + E ß - E ß t + ~v~P , (2-2) where E ß and E ß t are th e binding energies of th e projectile and targ e t, respectively, ~v - projectile velocity an d ~p - m om entum of th e electron in th e b ound s ta te of th e targ e t. Tr = (m e/ m p) E is th e kinetic energy of th e quasifree ta rg e t electron calculated in th e p ro jectile’s fram e of reference.

T h e R E C line observed during experim ents is m uch broader th a n th e characteristic x- ray lines, as can be observed in Fig. 2.2, which is due to th e velocity distrib u tio n of targ e t electrons. T his d istrib u tio n is described by C om pton profile 9 ( p z) [Big 75], which gives th e probability of finding an electron w ith a given m om entum projection p z , where (for ion-atom collisions) 2-axis is defined by th e projectile velocity. T h e C om pton profile depends on th e ta rg e t atom ic num ber Z t an d its w id th increases w ith increasing Z t . Moreover, th e w idth depends on th e binding energy and is sm aller for loosely b ound electrons, th a n for a tig htly bo u nd Is electron as shown in Fig. 2.3.

W hen th e binding energy of th e ta rg e t electron is m uch sm aller th a n Tr , th e cap tu red electron can be tre a te d as quasifree. T his m eans th a t R E C can be described as ca p tu re of a free electron (radiative recom bination - R R ), which is th e tim e inverse of photoionization.

As th e cross section for single photoionization can be calculated from th e well known form ula

(22)

Figure 2.1: Radiative electron capture (REC). Target electron is captured into the projectile bound state and the energy excess is emitted as a single photon.

given by Stobbe [Sto 30], one can use it to calculate th e R E C cross section via th e principle of detailed balance.

Principle of detailed balance describes th e relation betw een th e cross sections for direct (<Xi_/) and tim e inverse (cr/_ i) processes [Lan 79, Bey 03]:

gi pf cn^f {p i ) = gf p 2f cTf^i(pf), (2.3)

where g - th e num ber of possible sta te s given by angular m om entum and spin com binations and p - th e m om entum of th e particle in th e center of m ass system describe th e size of a

phase space accessible for th e initial (i) an d final ( / ) states.

B ased on Eq. 2.3 and th e Stobbe form ula for th e photoionization cross section, th e cross section for R E C to th e projectile K-shell during collision of a b are ion w ith a hydrogen targ e t can be expressed in th e form:

( is3 \ 2 e x p (—4 i/c o t- 1 (l/V )) ni r 2i

o r e c = 9 1 6 ( r r ^ J i _ exp(-27ri/) ' 10 |c m l - (2-4) where v = Z te2/ h v is th e Som merfeld p aram eter of th e ta rg e t K-shell electron and v - th e projectile velocity. T hus, for fast collisions, th e R E C cross section scales w ith energy as:

v r e c ~ — ę—• yO (^-5)

6

(23)

E [keV]

Figure 2.2: Example of the x-ray spectrum registered in coincidence with single electron capture during U92+ + N2 collisions at 309.7 MeV/u [Swi 00].

W hen th is result is com pared w ith Eq. 2.1, one should notice th a t th e radiativ e electron ca p tu re dom inates for high energy collisions w ith light targ ets.

T he angular d istrib u tio n of th e R E C photons, is given by th e angular differential R E C cross section calculated w ithin dipole approxim ation [Sch 72, Kie 73]:

doREC 3 , 2

— -77— = — V R E CSin -d.

a il on

Finally, th e double differential cross section d2ar e c/ d ü d E ^ can be expressed as:

d2CFREc 1 da r e c

(2 .6)

(KldE₇ V d£l P = P O + P z

(2.7)

where 9 ( p z) is th e C om pton profile of th e ta rg e t electrons. T his form ula describes th e shape of th e R E C line registered w ithin th e x-ray spectrum a t a given observation angle.

2.3 Brem sstrahlung

W hen a charged particle p en etrates a gaseous or solid ta rg e t a continuous x-ray spectru m is em itted. T his spectru m is a result of brem sstrahlu ng processes occurring in th e targ e t

(24)

Pz

Figure 2.3: Compton profile of electrons in carbon atom. It can be noticed that the structure of the Is line is much broader than that for n = 2 states [Big 75].

Figure 2.4: Example of bremsstrahlung process - an electron in the electromagnetic field of an ion.

m aterial, when a charged particle is accelerated (or decelerated) in th e Coulom b field of th e ta rg e t com ponents. A schem atic ex planation of th is process is presented in Fig. 2.4.

B rem sstrahlung was first observed by R öntgen in 1895 [Roe 96, R oe 98] and since th a t tim e has been intensively stu d ied [Ish 87, Ish 06, M ir 89, Chu 81, J a k 06, Lud 98].

D uring ion-atom collisions b o th th e ion and ejected electrons m ay undergo brem sstrah lun g processes. However, th e to ta l power rad ia te d via brem sstrahlung is proportional to ⁷4 (when

(25)

Figure 2.5: Radiative electron capture to continuum (RECC). Target electron is captured into a continuum state of the projectile and a photon is emitted.

d~v / d t -L 1?) or 76 {d~v / d t || 1?) [Gri 01]. Since E = j m c 2, where m is th e rest m ass of th e moving particle, th e to ta l rad ia te d power is prop ortion al to 1 /m 4 or 1 /m 6, respectively. T he above m eans th a t electrons lose energy via brem sstrahlung process m uch m ore rapidly th a n heavier charged particles. T his is why electron brem sstrahlung dom inates over th e ion-related processes.

Quasifree electron brem sstrahlu n g (Q FE B ), secondary electron brem sstrah lung (SEB), atom ic brem sstrahlu ng (AB) and nucleus-nucleus brem sstrahlung (NB) are dom inating am ong various brem sstrah lu ng processes th a t can occur during ion-atom collision. T hese processes were taken into account during d a ta analysis and are m ore thoroughly discussed in th e fol

lowing sections.

2 .3 .1 E le c t r o n b r e m s s t r a h lu n g

R adiative electron ca p tu re to continuum - R E C C , som etim es referred to as quasi-free electron b rem sstrahlu ng (Q FE B ) is a process where th e ta rg e t electron is c ap tu red to th e projectile continuum , which m eans it becomes a free electron. Energy conservation in this process is fulfilled by a p hoton emission (Fig. 2.5).

(26)

A nna Simon Correlated radiative electron capture in ion-atom collisions M axim um kinetic energy (Tr ) of th e involved electron, calculated for th e projectile fram e assum ing Tr E ß t , is given by:

Tr = \ m ev 2 = ^ E , (2.8)

2 m p

where v is th e velocity of th e incom ing ion in th e lab o rato ry fram e (equal to th e velocity of th e c ap tu red electron in th e projectile reference fram e). Tr is th e m axim um energy (in th e projectile fram e of reference) of th e p h o to n em itted during th e R E C C process. As th e m axim um energy of th e em itted photons is well defined, th e sp ectru m of th e em itted x-rays will have a sharp edge a t th is value. T his edge was observed, for exam ple, during collisions of carbon ions w ith C- an d B e-targets [Bed 98], as shown in Fig. 2.6.

E jected ta rg e t electrons m ay sc a tte r in th e Coulom b field of o ther ta rg e t nuclei, producing additional brem sstrahlung. T his process is referred to as secondary electron b rem sstrah-lung (SEB). In th is case m axim um energy (Tm ) of th e em itted photons is equal to th e m axim um tran sfer of th e kinetic energy during ion-electron collision, given by:

T rn = 4Tr = i — E . (2.9)

m p

T hus, sim ilarly to R E C C , SEB sp ectru m has a sharp edge at th e p hoton energy of T m [Ish 87].

D uring th e atom ic b rem sstrahlu n g process (AB) projectile excites a ta rg e t electron to a ta rg e t continuum sta te . T his electron can be recap tu red by a ta rg e t atom w ith sim ultaneous emission of x rays. T h e electron can also lose only p a rt of its energy b u t rem ain free. This process is referred to as rad iative ionization (RI).

It was shown in [Ish 06] th a t th e relative contribu tion of th e above processes strongly varies w ith projectile energy. T heoretical description of brem sstrahlu n g cross sections given in [Ish 06] is in agreem ent w ith experim ental d a ta . C om parison of experim ental d a ta for p + A1 collisions a t 1 and 4 MeV w ith theoretical calculations are presented in Fig. 2.7. Simple scaling form ulae describing double differential brem sstrah lun g cross sections were proposed in [Ish 06]:

( h u> ) 2 d ? aR E c c =

10

(27)

Figure 2.6: Spectra observed during the experiment [Bed 98] for: (a) Be-target, (b) C-target. For each energy-target combination two spectra are displayed. On the top - raw spectrum and lower - spectrum after background subtraction. For presentation purposes spectra were multiplied by factors;

for Be-target: 1/20 - 75 MeV/u, 10 - 290 MeV/u; for C-target: 1/8 - 75 MeV/u, 10 - 150 MeV/u, 50 - 290 MeV/u. Dashed line: SEB contribution, solid line: RECC (relativistic approximation) + K-REC 4" SEB. Arrows show the R.ECC edge energy Tj- transformed to the laboratory frame. Inset in (a) represents the experimental setup.

where hoj denotes p ho to n energy, ao is th e B ohr radius and / is an universal function discussed extensively in [Ish 06]. T h e brem sstrahlu n g processes for protons in teracting w ith various targ e ts a t wide range of energies were thoroughly stu died for exam ple by Folkm ann [Fol 84, Fol 75]. By m eans of th e above form ulae, th e brem sstrahlu ng contribu tion to th e experim ental d a ta can be estim ated from th e p ro to n d a ta .

(28)

A i

Figure 2.7: Example contribution of various bremsstrahlung processes to the continuous x-ray spec

trum during Al + C collisions at 1 and 4 MeV [Ish 06]. It can be noticed that SEB becomes a dominating process at higher beam energy.

2 .3 .2 N u c le u s - n u c le u s b r e m s s t r a h lu n g ( N B )

Nucleus-nucleus brem sstrah lun g is a consequence of th e projectile scatterin g in th e Coulomb field of th e ta rg e t nuclei. E m itte d x-ray spectrum extends up to th e projectile energy. T he differential cross section for NB process can be calculated from th e form ula given by Mokler [Mok 78]:

(2.13)

(2.14)

(2.15) A t E ’

where A, A t are projectile and ta rg e t m ass in atom ic units, respectively.

Fig. 2.8 shows con tributio n of R E C C , SEB, AB and NB to th e x-ray sp e ctra obtained during collisions of 2.0 MeV protons w ith carbon. U pper lim its of R E C C and SEB x-ray

12

(29)

Figure 2.8: Bremsstrahlung processes observed during p + C collisions at 2 MeV. Plot based on Fig. 3 in [Ish 06].

sp ectra can be observed a t T r an d Tm , respectively. T he NB cross section is significantly lower th a n those of th e electron b rem sstrah lu ng processes. T hus, NB plays significant role only w ithin th e x-ray spectrum range above T m .

2.4 M ultielectron capture processes, noncorrelated double ra

diative electron capture (DR EC )

D uring a single ion-atom collision cap tu re of m ore th a n one ta rg e t electron to th e projectile b o und s ta te is possible. T h e sim plest exam ple of a noncorrelated ca p tu re of two electrons is a double rad iativ e electron ca p tu re (D R E C ) for which th e ca p tu re of two electrons is accom panied by th e emission of two independent R E C photons (Fig. 2.9). D uring this process th e c ap tu red electrons do n o t in teract w ith each other and th e cap tu re of each of them can be tre a te d as a sep arate process.

R ad iativ e noncorrelated double ca p tu re was theoretically addressed by M eyerhof [Mey 85].

In this p ap er th e a u th o r calculated R E C cross section as an integral of th e process probability given as a function of im pact p aram eter. T h e single electron cap tu re cross section can be

(30)

Figure 2.9: Comparison of the DREC and RDEC processes.

calculated as an integral of th e probability P(b) of an electron cap tu re given as a function of th e im pact p aram eter b:

ro c

<?r e c= / db2irbP(b). (2.16)

Jo In case of R E C , P(b) is given by:

/ OO

dz p( R) , (2.17)

-OO

R being th e projectile to ta rg e t atom distance and p - th e electron density. As th e electron density is norm alized:

ro c ro c

/ db2irb / dzp{R) = Z u (2.18)

^ 0 J—00

th e R E C cross section for m ultielectron ta rg e t is sim ply given by Z t<jREc{Zt = 1)-

T he sam e m etho d was applied to noncorrelated double radiativ e capture. If Po(b) is th e probability of a single electron cap tu re into th e bare ion, and P\{b) is th e corresponding probability of electron cap tu re into th e H-like ion, th e cross section for noncorrelated double electron cap tu re can be expressed as [Mey 85]:

roc O 'D REC =

ro c

= / db2nbP0{b)Pl {b). (2.19)

Jo

14

(31)

E / M ( M e V / u )

Figure 2.10: Single electron loss cross section as given in [Hip 87]. Solid line - PWBA calculations for He2+ impact, dot-dashed line - includes contribution from free electron impact in CBE approximation.

Symbols: □ - 0 6+, A - Si8+, V - Si13+, o and x - 0 7+.

For double electron cap tu re (D R E C ) one finally obtains th e cross section [Mey 85]:

^d r e c = 0.13Zt ^ a \ EC( Zt )a 0 2. (2.20)

T h e above form ula was verified experim entally by B ednarz [Bed 03].

W hen th e c ap tu red electrons in teract w ith each other during th e collision, th e process is referred to as correlated cap ture. R adiative double electron ca p tu re (R D E C ) is th e basic exam ple of a correlated process an d can be tre a te d as a tim e inverse of double photoionization.

T hus, due to electron-electron in teraction of th e two c ap tu red electrons only one ph oton is em itted an d its energy is two tim es greater th a n th a t of a single R E C p h o to n (Fig. 2.9). T he R D E C process is discussed in detail in C h ap ter 3.

2.5 P rojectile ionization - electron loss

T here is a variety of term inology used in th e literatu re, when reffering to electron de

tachm ent, which leads to confusion. Here, a nom enclature from [Bom 89, T an 91, Hip 87] is

(32)

A nna Simon Correlated radiative electron capture in ion-atom collisions while th e removal of th e electron from th e projectile b ound s ta te is reffered to as electron loss.

Consequently, electron loss is th e process where an electron is removed from th e projectile and rem ains free afterw ards:

A q+ + A t -► A ^ +x^+ + A t + xe , (2.21)

where x is th e num ber of electrons lost by th e projectile (q + x < Z) . E lectron loss processes have been extensively stud ied during th e late seventies an eighties for various elem ents and charge sta te s w ithin energy range up to 10 MeV/ u [Gra 84, G ra 85, Ols 78].

B om an et al. [Bom 89] developed a sim ple scaling form ulae for electron loss cross section.

T h e single electron loss cross section for oxygen ions a t 1 MeV/ u can be estim ated as:

• for q = 5 :

<Ą = (3.27 • 1 0 - 18)Z?-98[cm2], (2.22)

• for q = 6 :

<Ą = (8.83 • 10“ 19)Z i°-78[cm2], (2.23)

• for q = 7 :

<r\ = (2.22 • 10“ 19)Z i°-33[cm2], (2.24)

where q denotes th e initial charge sta te of th e ion. It has been also checked by th e au thors th a t in case of Si8+ + He collisions at 1.0 M eV /u th e ratio of single to double electron loss cross sections cr8/cr8 ~ 40. T hus, it can be assum ed th a t th e double electron loss process can be neglected for th e case of th e m ore tig h tly bo u n d K-shell electrons in 0 6+. As can be observed in Fig 2.10 th e single electron loss cross section does not change significantly when th e beam energy is increased from 1 to 2 MeV/u . Thus, scaling form ulae given by Eqs 2.22-2.24 can be used to estim ate th e cross section w ithin th is energy range.

16

(33)

C hapter 3

R ad iative double electron capture (R D E C )

R adiative double electron ca p tu re (R D E C ) is a one-step process for which two free (or quasifree) ta rg e t electrons are ca p tu red into b o un d sta te s of th e projectile, e.g. into an em pty K-shell, and th e energy excess is released as a single photon (Fig. 2.9). T his process has to be com pared w ith a tw o-step double radiative electron ca p tu re (D REC) during which two electrons are ca p tu red independently an d two single R E C photons are em itted.

W hile for th e D R E C process b o th electrons can be tre a te d sep arately (see Section 2.3.2), in case of th e R D E C one has to go beyond th e independent electron m odel. Here, due to electron-electron interaction, tran sitio n s of two ta rg e t electrons into th e projectile bound states occur w ith an emission of one p h oton w ith th e energy two tim es greater th a n th a t of a single D R E C photon.

In general, ca p tu red electrons m ay originate from two different orbitals in th e ta rg e t and arrive finally at different final sta te s in th e projectile. T hus, th e energy of th e R D E C photon can be expressed as:

Er d e c « 2Tr + E g + E {V - E ^ l - E {£ + , (3.1)

where indices (1) and (2) correspond to each of th e c ap tu red electrons. T h e w id th of th e peak is ab o u t twice as large as th a t of th e R E C line. Roughly, it is determ ined by th e sum of C om pton profiles of th e two active electrons [Mir 89].

3.1 Initial experim ents dedicated to RDEC

T he first experim ent dedicated to R D E C was perform ed at GSI in 1994 w ith 11.4 M eV /u A r18+ ions from UNILAC im pinging upo n a carbon foil. A detailed description of th is exper-

(34)

3O

X -R a y Energy (keV)

Figure 3.1: Typical x-ray spectrum obtained during argon experiment [War 95].

im ent is given in [War 95]. A typical sp ectrum obtained during th a t experim ent is presented in Fig. 3.1. As shown in this figure, no significant line s tru c tu re related to th e R D E C process was observed. However, th e num ber of counts collected in th e expected R D E C energy win

dow provided an up p er lim it of th e R D E C cross section of ab o u t 5.2 mb. A rough theoretical estim ate of th e cross sections ratio <Jr d e c/ &r e c was also suggested, based on th e principle of detailed balance an d considering R E C as a tim e reversal of photoionization. T he R E C and R D E C cross sections can be w ritten as:

/ \ 2

o SEc < M = Z, j a p , ( M

&RDEc i ^ 0 = F Z t { Z t - 1) (

(3.2)

(3.3) v 2 7ß m c2 /

where a p j and a d p i are th e cross sections for single and double photoionization, respectively.

T h e factor F (F < 1) describes th e phase space fraction of double photoionization accessible for th e R D E C process. Thus, <Jr d e c/&r e c ratio can be expressed in term s of single and double photoionization cross sections as [War 95]:

ctrec

or, as in case of R D E C hw' « 2h u

D o-r d e c T ,o -p i(2 h w ) a Dp i ( 2 h w ) n = --- = £ — 1J-

ctrec api(hu>) api(2hu>)

(3.4)

(3.5)

18

(35)

photon energy [keV]

Figure 3.2: Typical experimental x-ray spectrum obtained for uranium ions [Bed 03]. The Gaussian solid line shows the expected RDEC peak, which should be observed according to Yakhontov et al.

[Yak 96, Yak 97].

T h e experim ent [War 95] provided an up p er lim it for R of 3.1 • 1CT6.

T his experim ent stim u lated theoretical tre a tm e n t of th e R D E C process [Yak 96, Yak 97].

In these papers th e auth o rs presented nonrelativistic calculations of th e R D E C process ad ap ted to th e kinem atics an d th e energy range of th e A r18+ + C experim ent. T h e calculations gave, for this p articu lar collision system (A r18+ + C a t 11.4 M eV /u), th e R D E C to R E C cross sections ratio R of 3.6 • 1CT6, which is close to th e experim ental u p p er lim it.

M oreover, these calculations predicted a strong enhancem ent of R D E C during heavy ion- atom collisions at relativistic energies [Yak 97]. T hese calculations were teste d during th e second experim ent dedicated to R D E C . Here, bare uranium ions a t an energy of 297 M eV /u were colliding w ith an A r ta rg e t at th e E S R storage ring of th e GSI facility [Bed 03]. This experim ent showed th a t for th e collision system under consideration th e R D E C cross section is certainly at least th ree orders of m ag n itud e sm aller th a n th e theoretical prediction [Yak 96, Yak 97]. Fig. 3.2 shows a spectru m obtain ed during th e experim ent. Again, no significant line stru c tu re which could be assigned to th e R D E C process was observed. T h e G aussian line in Fig. 3.2 shown w ithin th e R D E C region of th e spectrum represents th e shape of th e R D EC line which should be observed in th e spectrum , if th e theoretical calculations [Yak 96, Yak 97]

(36)

A nna Simon Correlated radiative electron capture in ion-atom collisions were reliable. T his experim ent also provided only an upper lim it for th e R D E C cross section value of ab o u t 10 mb.

3.2 R ecent theoretical approach to RDEC

In order to explain th e disagreem ent betw een th e uranium experim ent [Bed 03] and the theoretical tre a tm e n t of R D E C [Yak 96, Yak 97], a new theoretical approach for th e correlated double electron ca p tu re into th e K-shell of b are ions was proposed [Mik 04a, M ik 04b, Nef 05].

Here, a brief description of th is R D E C tre a tm e n t is given w ith th e n o tatio n used in th e original papers. Indices (1) and (2) correspond to R E C and R D E C , respectively, and n a tu ra l units (h = c = 1) are used th ro u g h o u t th e tex t.

All th e electrons involved in th e process were considered as nonrelativistic and th e energy to of th e em itted p ho ton was lim ited by I-²K < w <C m , where I-²i< is th e threshold energy for double photoionization of th e projectile K-shell an d m is th e electron m ass. In such case th e Coulom b p aram eter ( a Z , a denotes th e fine s tru c tu re constant) is small ( o Z <c 1) and th e p e rtu rb a tio n th eory w ith respect to th e electron-electron in teractio n can be used.

In th e reference fram e of th e incident ion th e probability d W for double electron cap ture into th e K-shell of b are ion w ith th e emission of a single p hoton p er u n it tim e is given by [Mik 04a]:

9-7r f] If

d W = v i lAl2^ 0{2Ep “ w “ / 2 x ) ’ (3-6)

where E p is th e one-electron energy w ithin th e initial continuum sta te , u> = \ k \ = k is th e energy of th e em itted ph oton an d h x = 2 /, w ith I = rj2/2rn. th e Coulom b p o ten tial for single ionization and r] = m a Z - th e ch aracteristic m om entum of th e K-shell electron, and V - a norm alization factor. Sum m ation over all polarizations of th e em itted p h o to n is assum ed in Eq. 3.6 an d d e lta function ensures th e energy conservation. T h e am p litude A was obtained from th a t for th e double K-shell photoionization. D etailed description of th is approach is given in [Mik 04a].

Dividing Eq. 3.6 by th e current flux of th e incident ta rg e t electrons j = v / V , where

V = p / m is th e absolute value of th e initial velocity of th e incident electrons before th e collision w ith ion, one obtains th e effective differential cross section:

( 3 ' 7 )

2 0

(37)

£

Figure 3.3: Q, universal function of the dimensionless variable Ç [Mik 04a].

which defines th e angular d istrib u tio n of th e R D E C photons em itted into an elem ent of a solid angle (ifi/,..

For th e energy regim e assum ed in these calculations, it was possible to calculate th e to ta l cross sections w ithin th e electric dipole approxim ation. For collisions of heavy ions w ith light ta rg e t atom s th e to ta l cross section for rad iative double electron cap tu re (RD EC) into th e K-shell of th e ion is given by:

919y 3

where £ = rj/p is a dimensionless p aram eter, ctq = a 3ag an d ao denotes th e B ohr radius.

Q is th e universal function of £, which can be obtained by num erical integration (Fig. 3.3).

£ ~ 1 corresponds to th e n ear-threshold dom ain, where th e K-shell photoeffect reaches its m axim um . For slow collisions (£ 1) th e R D E C cross section increases, while in case of fast collisions it decreases rapidly. M oreover, it has to be poin ted out th a t th e R D E C cross section rapidly drops w ith th e projectile atom ic num ber ( ~ Z ~ 5) and increases significantly for low energy collisions.

A nother value calculated in [Mik 04a] is th e cross sections ratio (R = c r ^ / c r ^ ) . T he R E C cross section can be expressed in term s of th e photoionization cross section ( a pi ) via

(38)

Figure 3.4: Universal quantity Q / H calculated as a function of the dimensionless variable Ç [Mik 04a].

th e principle of detailed balance. As a p i is known analytically (Stobbe form ula [Sto 30]), for th e rad iativ e electron ca p tu re into th e K-shell of th e projectile one obtains:

c (1) = ^ ^ c r o Z t H i Ç ) ,2 10

H ( 0 = £ exp(—4£cot £) e2 1 - exp(-27r£)

where e7 = iv/1 is th e dimensionless p h oton energy. T h en th e ratio R is given by:

R = 29Z t2Q ( 0

(3.9)

(3.10)

(3.11) 7t3Z 5H ( £ ) '

T h e function Q(Ç)/H(Ç) is presented in Fig. 3.4.

These calculations are in disagreem ent w ith th e previous relativistic approach [Yak 97], which was n o t able to explain th e existing experim ental d a ta [Bed 03]. As shown in [Mik 04a]

th e enhancem ent of th e wave function for th e relativistic system s was calculated incorrectly by Y akhontov [Yak 97] and even th e corrected value, which is 3 orders of m agn itud e sm aller [Mik 04a], was not confirm ed by th e experim ent [Bed 03]. Therefore, th e enhancem ent of th e R D E C cross section for relativistic system s [Yak 97] seems to be absent. T his is sim ilar to th e behavior of th e cross section for th e R E C process, where th e cross section decreases when th e projectile energy increases.

2 2

(39)

Table 3.1: Comparison of experimentally obtained RDEC cross sections [War 95, Bed 03] and the calculated values given in [Yak 97] and [Mik 04a].

z E

[MeV/u]

z t

(j(2) [mb]

Ref. [Mik 04a] Ref. [Yak 97] experiment

18 11.4 0.840 6 3.2 1.85 <5.2 [War 95]

92 297 0.841 18 2 .5 1 0 -2 5000 <10 [Bed 03]

Figure 3.5: The ratio of the RDEC cross sections to the excited an(i ground (0^ 5 ) projectile states as a function of adiabacity parameter Ç [Nef 05].

However, it has to be em phasized th a t th e current estim ate [Mik 04a] of a 1'2"1 gives values closer to th e experim entally obtain ed up p er lim its for b o th th e nonrelativistic case (A r18+ + C, [War 95]) and th e relativistic one (U92+ + Ar, [Bed 03]) (see Table 3.1), which suggests th a t [Mik 04a] is so far th e m ost reliable theoretical description of R D EC .

In contradiction to predictions given in [Yak 97], th e new calculations show th a t th e R D EC cross section strongly depends on th e ta rg e t atom ic num ber and electron density. One can expect m uch larger values of a 1'2"1 in case of slow collisions of m ulticharged ions w ith a solid s ta te ta rg e t w ith low atom ic num ber Z t [Mik 04a]. As th e orbital velocity of th e ta rg e t valence

Correlated radiative electron capture in ion-atom collisions

Anna Simon