Correlated radiative electron capture in ion-atom collisions

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Correlated radiative ele tron apture

in ion-atom ollisions

Ph.D. Dissertation

prepared underthesupervisionof

Prof. AndrzejWar zak

MarianSmolu howski Instituteof Physi s

Jagiellonian University

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Abstra t

Radiativedoubleele tron apture(RDEC)isaone-steppro esswhere twofree(or

quasi-free) target ele trons are aptured into a bound state of the proje tile, e.g. into an empty

K-shell, and the energy ex ess is released as a single photon. This pro ess an be treated

as a time inverse of a double photoionization. However, unlike in ase of photoionization

experiments, bareions areused duringRDEC observations. Thus,RDEC an be onsidered

asthe simplest, lean toolfor investigationof ele tron-ele tron intera tioninthepresen eof

ele tromagneti elds generated during ion-atom ollisions.

Withinthisdissertation,the

38

MeVO

8+

+Cexperiment, ondu tedatWesternMi higan

UniversityusingthetandemVandeGraaa elerator, isdis ussedandtherstexperimental

eviden e of the RDEC pro ess is presented. The ross se tion obtained experimentally is

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Abstrakt

Skorelowanyradia yjnywy hwytdwó helektronów(RDEC)jestpro esem,pod zas

które-godwaswobodne(albokwaziswobodne)elektronytar zywy hwytywanes¡dostanu

zwi¡zane-gopo isku(np. nieobsadzonejpowªokiK),aró»ni aenergiipomidzyko« owyma

po z¡tko-wym stanem eletronów emitowana jest w posta i pojedyn zego fotonu. Pro es ten mo»na

traktowa¢ jakoodwró eniew zasiepodwójnej fotojoniza ji. Jednak»e,wprze iwie«stwie do

eksperymentów dedykowany h fotojoniza ji, do obserwa jiRDEC stosujesi jony aªkowi ie

pozbawione elektronów, o pozwala na wyeliminowanie tªa po hodz¡ ego od elektronów nie

bior¡ y h bezpo±rednioudziaªu wbadanym pro esie. RDEC mo»eby¢wi traktowanyjako

najprostsze narzdzie do badania oddziaªywania elektron-elektron w obe no± ipola

elektro-magnety znego generowanegopod zaszderzenia.

Rozprawa ta po±wi ona jest pro esom atomowym za hodz¡ ym wzderzenia h O

8+

+C

przyenergii

38

MeVpod zaseksperymentuprzeprowadzonegoprzyu»y iual eleratoraVande GraaawWesternMi higanUniversity. Przedstawionezostaªowniejpierwszedo±wiad zalne

potwierdzeniepro esuRDEC.Uzyskanyeksperymentalnieprzekrój zynnyzostaªporównany

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A knowledgements

First,Iwouldliketogivemysin erethankstomysupervisor,ProfessorAndrzejWar zak.

He oered meadvi e, patiently supervisedmeandalwaysguidedmetothe orre tdire tion.

I have learned mu h fromhim, without hishelp Iwouldhave never nished mydissertation

su essfully.

Spe ial thanks are also given to Professor John A. Tanis. He is theone who invited me

to Western Mi higan University for my resear h during the 2008-2009 a ademi year. His

help anden ouragement made mefeel ondentenough to fulllmydesiresandto over ome

thedi ulties Ien ountered. Hisunderstanding, en ouragement and personalguidan ehave

provided a good basisfor mythesis. It is not su ient to express my gratitude withsu h a

few words.

IamverygratefultoProfessorBogusªawKamys,theHeadoftheNu learPhysi sDivision

at theInstitute ofPhysi s,Jagiellonian Universityand ProfessorPaulPan ella, theChair of

Physi s Department, Western Mi higan University, for the nan ial support of my stay at

WMU.

Additionally,Iowemymostsin eregratitudetoDr. AsgharKayanifor hispatien ewhile

tea hingmehowto operatethe WMUtandem Vande Graaa elerator and hiswillingness

to immediately solve any beam relatedproblems during my experiment. My sin ere thanks

should also go to Ri k Wel h and Allan Kern for their help with the maintenan e of the

experimental setup.

I would also like to thank JanuszKop zy«ski, Adam Malarz and Adam Mu ha for their

are and supportduringmywork at Jagiellonian University.

I amalso very gratefulto Prof. Thomas Stöhlker, theHead ofAtomi Physi s Group at

GSI, Darmstadt, who frequently invited me to join his group's experiments during whi h I

had a han e to learnthe se rets of theexperimental work ofatomi physi ists. I'm grateful

to Dr. AngelaBräuning-DemianandDr. ChristophorKozhuharovforinspiring onversations

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I am very grateful to my friends, David Cassidy, Buddhika Dassanayake, Maªgorzata

Maku h,DagmaraRozpdzikand AndrzejPezarski for always beingthere for me.

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Table of Contents

List of Tables viii

List of Figures x

List of Symbolsand Abbreviations xiv

1 Introdu tion 1

2 Atomi pro esses during ion-atom ollisions at low energy 4

2.1 Nonradiative ele tron apture (NREC) . . . 4

2.2 Radiative ele tron apture (REC) . . . 5

2.3 Bremsstrahlung . . . 7

2.3.1 Ele tron bremsstrahlung . . . 9

2.3.2 Nu leus-nu leus bremsstrahlung (NB) . . . 12

2.4 Multiele tron apture pro esses, non orrelated double radiative ele tron ap-ture (DREC) . . . 13

2.5 Proje tileionization ele tronloss . . . 15

3 Radiative double ele tron apture (RDEC) 17 3.1 Initial experiments dedi atedto RDEC . . . 17

3.2 Re enttheoreti al approa h to RDEC . . . 20

4 Experimental setup at Western Mi higan University 25 4.1 Van deGraa a elerator . . . 25

4.2 Beam line setupat Western Mi higanUniversity . . . 27

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5 Data analysis 34

5.1 PIXE analysisof thetarget material . . . 37

5.2 Proje tileK- andL-shell ele tron loss . . . 38

5.3 Ba kground pro esses . . . 40

5.4 Pile-upofsingleRECphotonsand its ontributionto theRDEC energyrange

of the spe trum . . . 42

5.5 Single spe tra analysis . . . 45

5.6 Coin iden e spe traanalysis . . . 47

6 The RDEC ross se tion 51

6.1 Experimentalvalueof theRDEC rossse tion. . . 51

6.2 Estimation ofthe

σ

RDEC

/σ

REC

ratio inthenonrelativisti approa h . . . 52 6.3 RDEC rossse tionbased ontheYakhontov approa h . . . 53

7 Monte Carlo simulations of the x-ray spe tra 55

8 Con lusions 60

A Statisti al analysis of the observed signal 62

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3.1 Comparisonofexperimentallyobtained RDEC rossse tions [War 95,Bed03℄

and the al ulated valuesgiven in[Yak97 ℄ and[Mik04a℄. . . 23

3.2 The REC (

σ

(1)

), RDEC (

σ

(2,γ)

) and DREC (

σ

(2,2γ)

) ross se tions and their

ratios asgiven in[Dru 07℄. . . 24

5.1 Results ofa

χ

2

testof theRDEC rangeoftheproton indu edspe trum. . . . 38

5.2 Ele tronloss rossse tionsfor oxygenionsat

38

MeVestimatedfromthedata presentedin[Bom89, Tan 91,Hip87℄. . . 39

5.3 Total rossse tionsfor theba kground pro essesthatweretaken intoa ount

during dataanalysis. . . 42

5.4 Probabilitiesand ountratesofthepro essesthatmight ontributetothex-ray

spe trumintheRDEC range. For moreinformation seetext. . . 43

5.5 Cal ulatedpositionsoftheRDECandRECpeaksinthex-rayspe trum

orre-spondingtodierent ombinationsoftheinitialandnalstatesofthe aptured

ele trons. Allvalues aregiven inkeV. . . 45

5.6 Results ofa

χ

2

testof theRDEC rangeofthe oin iden e spe tra. . . 47

5.7 Areas (

A

) ofthe shapesof theRDEC ontributions ttedto the sumof

q − 1

and

q − 2

spe tra. FWHMofall lineswassetto

0.3

keV whi histhewidthof the arbon Compton prole. . . 50

6.1 ComparisonoftheexperimentalvaluesoftheRDEC rossse tionandthe

R =

σ

1s

_RDEC

2 /σ

REC

ratio withtheonesobtained fromvarious theoreti alapproa hes . 54

7.1 Ratiosofthenumbersof ountsintheRDECandRECregionsobtainedduring

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8.1 Summary of the results of the theoreti al al ulations and the experiments

dedi ated tothe RDECpro ess.. . . 61

A.1 Quantilesof

χ

2

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List of Figures

2.1 Radiativeele tron apture(REC).Target ele tronis apturedinto the

proje -tile bound stateand the energy ex ess isemitted asasinglephoton. . . 6

2.2 Example of the x-ray spe trum registered in oin iden e with single ele tron

apture during U

92+

+N

2

ollisions at

309.7

MeV/u[Swi 00℄. . . 7 2.3 Compton proleofele trons in arbonatom. It an be noti ed thatthe

stru -ture ofthe

1s

line is mu h broader than thatfor

n = 2

states[Big75℄. . . 8 2.4 Example of bremsstrahlung pro ess an ele tron inthe ele tromagneti eld

of anion. . . 8

2.5 Radiative ele tron apture to ontinuum (RECC).Target ele tronis aptured

into a ontinuumstate of theproje tile anda photon isemitted. . . 9

2.6 Spe tra observed during the experiment [Bed 98 ℄ for: (a) Be-target, (b)

C-target. For ea h energy-target ombination two spe traaredisplayed. On the

r

transformed to the laboratory frame. Insetin (a)represents theexperimental

setup. . . 11

2.7 Example ontribution of various bremsstrahlung pro esses to the ontinuous

x-ray spe trum during Al + C ollisions at

1

and

4

MeV [Ish 06℄. It an be noti ed that SEBbe omesa dominating pro essat higher beamenergy. . . . 12

2.8 Bremsstrahlung pro esses observed during p + C ollisions at

2

MeV. Plot based onFig.3 in[Ish06℄. . . 13

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2.10 Single ele tron loss ross se tion as given in [Hip 87℄. Solid line PWBA

al ulationsfor He

2+

impa t,dot-dashedlinein ludes ontribution fromfree

ele tron impa t in CBE approximation. Symbols:

O

6+

,

△

Si

8+

,

▽

Si

13+

,

◦

and

×

O

7+

. . . 15

3.1 Typi alx-rayspe trumobtained during argonexperiment [War 95℄. . . 18

3.2 Typi alexperimental x-rayspe trumobtained for uraniumions [Bed 03℄. The Gaussian solidlineshows theexpe tedRDECpeak,whi hshould be observed a ording toYakhontov etal. [Yak96 ,Yak97 ℄. . . 19

3.3

Q

,universalfun tion ofthe dimensionless variable

ξ

[Mik04a℄. . . 21

3.4 Universalquantity

Q/H

al ulatedasa fun tion ofthedimensionlessvariable

ξ

[Mik04a℄. . . 22

3.5 The ratio oftheRDEC rossse tions to theex ited(

σ

(2)

2

1 S

) andground (

σ

(2)

1

1 S

) proje tilestatesasa fun tion ofadiaba ityparameter

ξ

[Nef05 ℄. . . 23

4.1 S hemati view of a lassi al Van de Graa a elerator: (1) lower roller, (2) upper roller, (3) harging ele trode, (4) ele trode olle ting positive harge, (5) voltage generator, (6) spheri al ele trode (high voltage terminal), (7) ion sour e, (8)extra ted ionbeam. . . 26

4.2 S hemati view of a tandem Van de Graaa elerator: Negative ion entering the a elerator (A

−

) is a elerated by the high terminal voltage. Some of its ele trons are removed while the ion passes through the stripping foil. The positive ion (A

+q

) is repelled by the high voltage terminal, thus additionally a elerated. . . 27

4.3 S hemati viewof theWMUvande Graaa elerator fa ility[Kay ℄. . . 28

4.4 The experimental target hamberin1:1 s ale. . . 29

4.5 Dete tion e ien y of ORTEC Si(Li)dete tors [ORTa ℄. . . 30

4.6 Experimentalsetup. . . 30

4.7 Exampleofatimespe trumregisteredduringtheexperiment. Thearrow indi- atesthewidthofatimewindowfortrue oin iden es( alibration

2

ns/ hannel). 32 4.8 S heme ofthedataa quisition system. . . 33

MeV O

7+

, (d)dot-dashed

line

2.375

MeV protons. . . 35 5.2 XraysregisteredforO

8+

+C ollisionsin oin iden ewithionswhi h aptured

(a)twoele tronsand(b)oneele tron,solidlinethesumoftheRECCompton

prole and the Gaussian shapeof theoxygenK-

α

linetted to thespe trum. 36 5.3 Proton indu ed x-ray spe trum. Solid line the RDEC range, dashed line

region onsidered duringba kground estimation. . . 37

5.4 Ba kground stru ture in the single x-ray spe trum. The bremsstrahlung

on-tribution in ludes all the relevant pro esses (SEB + AB + NB) dis ussed in

the text. Thespe trumis ompletely dominated bytheRECstru ture. . . . 41

5.5

N

RDEC

/N

REC

ratio in the

q − 1

oin iden e spe tra as a fun tion of beam intensity.. . . 44

5.6 O

8+

spe trumtaken without the arbon foil(red line)normalized to thedata

taken withthe foil(bla kline). . . 46

5.7 Double stru ture oftheREC lineresolved aftersubtra tion of theAlK-

α

line. 46 5.8 Possible RDEC transitions (a) and the stru ture of the produ ed x-ray

spe -trum (b) when equal ross se tions for all the partial pro esses are assumed.

Bla k line the sum of all ontributions. Additionally, orresponding RDEC

spe tra obtained experimentally insingle ( ) and double (d) harge ex hange

hannels arepresented. . . 48

5.9 The sum of spe tra registered in single and double harge ex hange hannels

withat ofallpossible ombinationsoftheRDECtransitions. Fitting

param-eters aregiven inTable 5.7. . . 50

7.1 Geometry of the experimental setup implemented in the Monte Carlo

simu-lation,

b

w

the beam diameter,

d

t

target thi kness in mm. The

x

-axis is perpendi ularto thepi tureplane. . . 56

7.2 Monte Carlo simulated x-ray spe tra: (a) no RDEC in luded, (b) the RDEC

ross se tion as given by Neodov, ( )

σ

1s

2 RDEC

= 3

b and

σ

1s

1 2s

1 RDEC

= 2.1

b ross se tions values for whi h MC simulation gives the results losest to the

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7.3 The RDEC range of the x-ray spe tra. Results of simulations: (a) no RDEC

in luded, (b) theRDEC rossse tion asgiven by Neodov, ( )

σ

1s

2 RDEC

= 3

b and

σ

1s

1 2s

1 RDEC

= 2.1

b rossse tions values for whi h MCsimulation gives the results losest to the experimental data. (d) Experimentally obtained single

spe trum . . . 58

A.1 Example of an experimentally obtained spe trumwith a stru ture within the

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η

Ele tron momentumwithin thetarget bound state

~ω

Photon energy inthe laboratoryframe

ℑ(p

z

)

Compton prole

ν

Sommerfeld parameterfor K-shellele tron

Ω

Dete tor solidangle

σ

Cross se tion

θ

Observationangle inthelaboratoryframe

ξ

Dimensionless parameter des ribing ollisionvelo ity (adiaba ity parameter)

A

Proje tilemass

A

t

Targetmass

b

Numberof ba kground ounts

b

w

Beam diameter [mm℄

d

Targetthi kness[parti les/ m

2

℄

d

t

Targetthi kness[mm℄

E

Proje tilekineti energy [MeV/u℄

E

B

Binding energy ofan ele troninthe bound stateof theproje tile

E

Bt

Binding energy ofan ele troninthe bound stateof thetarget

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m

e

Ele tron rest mass

m

p

Proton rest mass

N

Numberof ounts

p

Momentum

P

RDEC

ProbabilitythatthephotonisregisteredintheRDECrangeofthex-rayspe trum

P

REC

Probability thatthe photonisregistered intheRECrangeof thex-rayspe trum

q

Proje tile hargestate

T

Statisti al variableof the

χ

2

test

T

m

Maximumenergy transferduring ion-atom ollision (inthelaboratory frame)

T

r

Kineti energy of the freetarget ele tron al ulatedintheproje tileframe

v

Proje tilevelo ity

v

e

Ele tron velo ity

Z

Proje tileatomi number

Z

t

Targetatomi number

AB

Atomi bremsstrahlung

CBE

Coulomb Bornex hange

DREC

Double radiative ele tron apture

NB

Nu leusbremsstrahlung

NREC

Nonradiative ele tron apture

PWBA

Plain wave Bornapproximation

QFEB Quasifree ele tron bremsstrahlung

RDEC

Radiative doubleele tron apture

RECC

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REC

Radiative ele tron apture

RI

Radiative ionization

RR

Radiative re ombination

SEB

Se ondary ele tronbremsstrahlung

α = 1/137

Fine stru ture onstant

~

_{= 6.582 · 10}

−

₁₆

eV

·

s Plan k onstant

a

0 = 5.29 · 10

−

11

m Bohr radius

c = 2.997 · 10

8

m/s Speed oflight

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Introdu tion

Sin etherstobservationofthephotoele tri ee tbyHertz[Her 87 ℄anditsexplanation

by Einstein [Ein05 ℄ the intera tion between ele trons and light has been of onsiderable

attention. The fundamental pro ess o urring due to this intera tion is photoionization,

where absorption of aphotonof energy

~ω

results intheemissionofan ele tron:

A + ~ω → A

+

+ e

−

.

(1.1)

Simple photoionization experimentsusually are restri ted to neutral atoms, where the

inu-en e of the ele trons, whi h do not parti ipate in the pro ess dire tly, annot be negle ted.

This ompli ates omparison of theexperimentalresults withtheoreti alpredi tions.

However, basedon the prin iple of detailed balan e[Lan 79,Bey 03 ℄thephotoionization

anbestudiedviathetimereversedpro esses,i.e. radiativere ombination(RR)andradiative

ele tron apture(REC)[I h94 ,I h 96,Ei 95a ℄. Duringthesepro essesafree(RR)orloosely

bound (REC) ele tron is aptured to the bound state of the proje tile and a photon with

energy equal to thedieren e between thenaland initial ele tron statesisemitted. Unlike

singlephotoionizationofmultiele tron systems,REChasbeeninvestigated forbareion-atom

intera tions[Sto 92,Sto 94 ℄andoers lean onditionsforexplorationofphotoionizationwith

only one ele tron, allowing for observation of pure photon-ele tronintera tions.

During the last thirty years double photoionization has been of onsiderable interest

[Dal 94, and referen es therein℄. As a photon typi ally intera ts only with one ele tron,

double photoionization is aused by the ele tron-ele tron intera tion [Smi89℄. However,

double photoionization studies have been performed mainly for low

Z

atoms, su h as He [Ber 93,Tiw 82,Car 81℄,Ne[Sai 92,S h 93 ,Car 77 ℄,and Ar[Lab87,Car77℄. Thisisdueto

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ele tron orrelation ee ts di ult to observe. Fortunately, similarly to single

photoioniza-tion,doublephotoionization anbestudiedbymeansofthetimeinversedpro essRDEC,for

whi hthisba kgroundisabsent. Radiativedoubleele tron apture(RDEC)involvestransfer

of two target ele trons into a bound state of the proje tile with simultaneous emission of

a single photon [War95 , Bed03 ℄. Sin e bare ions are used during the experiment, RDEC

an be onsidered asthesimplest, lean toolfor investigationofele tron-ele tron intera tion

[War 95 ℄ inthepresen eofele tromagneti elds generated during ion-atom ollision. Thus,

investigation of the RDEC pro ess an provide ru ial information ne essary for a proper

des riptionofele tron orrelationswithinatomi systemsandprovidedatarequiredtodene

the wave fun tion of two orrelated ele trons intheproje tile ontinuum.

During the last twenty years the RDEC pro ess was addressed not only experimentally

[War 95 ,Bed03℄,butalsotheoreti ally[Mir 89,Yak 96 ,Yak 97℄. The al ulationswerefound

to be indisagreement withtheexperimentaldata[Bed 03℄andveri ation oftheRDEC

pro- ess wasnot possible. Themorere ent al ulationsnotonlyexplained previousexperimental

results, but also suggested the hoi e of low energy mid-

Z

(

Z ≤ 35

) ollision systems for observation of RDEC [Mik04a, Mik 04b , Dru07℄. It is also noted that for these systems

apture to an ex ited

1s

1 _2s

1

state might signi antly enhan e the pro ess and ontribute

to theobserved x-rayspe tra [Nef 05 ℄. These al ulations provided the main motivation for

yetanother experiment dedi ated to the RDEC pro ess. To fullytake advantage ofthe new

al ulations, two ollisionsystemsat two dierent a elerator omplexes were hosen:

•

Xe

54+

+Cat

20

MeV/u, to be performedat GSI, Darmstadt inGermany,

•

O

8+

+Cat

2.375

MeV/u,realizedbymeansoftheVan deGraaa elerator atWMU, Kalamazoo,MI, USA.

So far the WMUexperiment was arried out. During sixmonths oftheexperiment

prepara-tions and data taking,

43

days of beam time were used. At the moment when this thesis is beingwritten the GSIbeamtime is still pending.

Withinthisdissertation the O

8+

+Cat

38

MeV experiment isdis ussedand therst ex-perimentaleviden eoftheRDECpro essispresented. The rossse tionobtained

experimen-tally is ompared withthe latesttheoreti al al ulations [Mik04a ,Mik04b ,Nef 05 ,Dru07℄.

Thisthesisbeginswithanintrodu tiontothemostimportantatomi pro essesthato ur

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add tothe ba kgroundfor thex-rayspe trumregisteredduringtheexperiment andformulae

allowingforestimationsof ontributionsofthesepro essesaresuggested. Chapter3addresses

the RDEC pro ess in detail. The history of the experimental approa h and the theoreti al

al ulationsoftheRDEC rossse tionarepresented. Additionally,this hapterfo usesonthe

re ent theoreti al al ulations whi h were themain motivation for theexperiment dis ussed

inthis dissertation. Thegoalof theexperiment wastheobservation ofxraysemitted during

ollisions of bare oxygen ions with arbon atoms. The x-rays spe tra were registered in

oin iden e with ongoing parti les whi h underwent single or double harge ex hange. The

experimental setup whi h allowed for a hieving this goal is presented in Chapter 4. The

operation prin iple of a Van de Graa a elerator is explained and the onstru tion of the

target hamber, parti le spe trometer and x-raydete tor are des ribed in detail. Chapter5

isdedi atedto dataanalysis,withaparti ular fo uson pro esses thatmay ontributeto the

x-rayspe trumwithintheRDECregion. Variousapproa hestoestimationoftheba kground

and al ulationsof the rossse tionaredis ussed. InChapter6theexperimentallyobtained

RDEC rossse tion is ompared with thetheoreti al value and the possible reasons for the

obtained dis repan y are given. In Chapter 7 results of theMonteCarlo simulations of the

x-rayspe trumgenerated duringtheO

8+

+C ollisionsare ompared withtheexperimental

results. Finally, inChapter8 suggestionsfor further investigations of theRDEC pro ess are

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Atomi pro esses during ion-atom ollisions at low

energy

Intera tionbetweenanin omingionandatargetatommayleadtomanydierentatomi

pro esses,su has:

•

ionization, mainly ofthe target atom, asthe ele trons areusually lessbound to alight target than to a partiallyionizedproje tile,

•

ele tron transferfrom the targetto theproje tile,

•

ex itation of both target and proje tile states su hstates deex ite after the ollision emitting hara teristi xrays.

Withinthe followingse tionsthe mostimportant pro essesthatwere onsidered

ompet-itive to RDECfor thepresentedexperimentare dis ussed.

2.1 Nonradiative ele tron apture (NREC)

The Coulomb intera tion between the proje tile and the target ele trons an lead to a

pro ess alled Coulomb apture or nonradiative ele tron apture (NREC). Here, the energy

dieren e between the initial and nal state of the ele tron is onverted into the kineti

energyofthe ollisionpartners. Themost onvenientandfrequentlyuseds alingformulathat

estimates the ross se tion for nonradiative ele tron apture is the one given by S hla hter

[S h83℄. Itisasemiempiri alformulawhi hallowsfor al ulationofthe

σ

N REC

asafun tion of theproje tile energy for variousproje tiles withana ura y better than

30%

.

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The NRECpro ess o urs mainly at the velo itymat hing ondition

2.2 Radiative ele tron apture (REC)

Radiative ele tron apture (REC) isone of the best knownatomi pro esses observed in

heavy ion-atom ollisions. It wasrst observed inearlyseventies ofthelast entury [S h 72 ,

Kie 73,S h 74 ℄andsin ethattimehasbeenintensivelystudiedbothexperimentally[Kan 95 ,

Mok 95, Spi79, Sto 95b, Sto 92, Sto 94, Sto 95a, Sto 97b , Sto 97a , Sto 98 , Tan 81, Tan 87 ℄

and theoreti ally [Ei 95a , Ei 95b, Hin87, I h 94 , I h96, Soh 76 ℄. During this pro ess a

apture of a singletarget ele tron isfollowed bya photon emission (Fig.2.1). Energy

E

REC

of theemitted photonfulllsenergy onservation rule forthis pro ess. Thus, itis given by:

E

REC

= T

r

+ E

B

− E

Bt

+ −

→

v −

→

p ,

(2.2)

where

E

B

and

E

Bt

are the binding energies of the proje tile and target, respe tively,

−

→

v

proje tilevelo ityand

−

→

p

momentumoftheele tronintheboundstate ofthetarget.

T

r

=

(m

e

/m

p

)E

isthe kineti energy of thequasifree target ele tron al ulated intheproje tile's frame of referen e.

The REC line observed during experiments is mu h broader than the hara teristi

x-ray lines, as an be observed in Fig. 2.2, whi h is due to the velo ity distribution of target

ele trons. This distribution is des ribed by Compton prole

ℑ(p

z

)

[Big75 ℄, whi h gives the probabilityof ndinganele tronwithagivenmomentumproje tion

p

z

,where(forion-atom ollisions)

z

-axis is dened by the proje tile velo ity. The Compton prole depends on the target atomi number

Z

t

and its width in reases with in reasing

Z

t

. Moreover, the width depends on thebinding energy and is smallerfor looselybound ele trons, than for atightly

bound

1s

ele tronasshowninFig.2.3.

When the binding energy of the target ele tron is mu h smaller than

T

r

, the aptured ele tron an be treated as quasifree. This means that REC an be des ribed as apture of

a freeele tron (radiative re ombination - RR), whi h isthe timeinverse of photoionization.

(22)

Figure 2.1: Radiativeele tron apture(REC). Target ele tronis aptured into theproje tilebound

stateandtheenergyex essisemittedasasinglephoton.

given byStobbe [Sto 30 ℄, one an useit to al ulate theREC rossse tion via theprin iple

of detailed balan e.

Prin iple of detailed balan e des ribes the relation between the ross se tions for dire t

(

σ

i→f

) and time inverse (

σ

f →i

)pro esses [Lan79,Bey 03℄:

g

i

p

2 i

σ

i→f

(p

i

) = g

f

p

2 f

σ

f →i

(p

f

),

(2.3)

where

g

the numberof possible states given byangular momentumand spin ombinations and

p

the momentum of the parti le in the enter of mass system des ribe the size of a phase spa ea essible for theinitial (

i

) and nal(

f

) states.

Based on Eq. 2.3and theStobbe formulafor the photoionization ross se tion, the ross

se tionfor RECto theproje tile K-shellduring ollisionofabareionwithahydrogentarget

an be expressedintheform:

σ

REC

= 9.16

ν

3 1 + ν

2

2 exp(−4ν cot

−

1 (1/ν))

1 − exp(−2πν)

· 10

−

₂₁

[cm

2 ],

(2.4) where

ν = Z

t

e

2 _/~v

is the Sommerfeld parameter of the target K-shell ele tron and

v

the proje tilevelo ity. Thus,for fast ollisions, the REC rossse tions ales withenergy as:

σ

REC

∼

Z

t

Z

5

(23)

Figure 2.2: Example of the x-ray spe trum registered in oin iden e with single ele tron apture

during U

92+

+N

2

ollisions at

309.7

MeV/u[Swi00℄.

When this result is ompared with Eq. 2.1, one should noti e that the radiative ele tron

apture dominates for highenergy ollisions withlight targets.

The angular distribution of the REC photons, is given by the angular dierential REC

rossse tion al ulated withindipole approximation[S h 72,Kie 73 ℄:

dσ

REC

dΩ

=

3 8π

σ

REC

sin

2 _ϑ.

(2.6)

Finally,the doubledierential rossse tion

d

2 _σ

REC

/dΩdE

γ

an be expressedas:

d

2 σ

REC

dΩdE

γ

=

1 v

dσ

REC

dΩ

p=p

0 +p

z

ℑ(p

z

),

(2.7)

where

ℑ(p

z

)

istheComptonprole ofthe targetele trons. Thisformulades ribestheshape of theRECline registeredwithin thex-rayspe trumat agiven observationangle.

2.3 Bremsstrahlung

Whena hargedparti lepenetrates agaseous orsolidtargeta ontinuousx-rayspe trum

(24)

C

o

m

p

to

n

p

ro

fi

le

10 -3

10 -2

10 -1

10

0 p

z

-10

-5

0

5

10

Figure 2.3: Comptonproleofele tronsin arbonatom. It anbenoti ed thatthestru ture ofthe

1s

lineismu hbroaderthanthatfor

n = 2

states[Big75℄.

Figure2.4: Exampleofbremsstrahlungpro essanele tronintheele tromagneti eldofanion.

material, when a harged parti le is a elerated (or de elerated) in the Coulomb eld of

the target omponents. A s hemati explanation of this pro ess is presented in Fig. 2.4.

Bremsstrahlung was rst observed by Röntgen in1895 [Roe96 , Roe 98℄and sin e that time

hasbeen intensivelystudied [Ish 87,Ish 06 ,Mir89 ,Chu81 ,Jak 06 ,Lud 98℄.

Duringion-atom ollisionsboththeionandeje tedele tronsmayundergobremsstrahlung

pro esses. However, thetotalpowerradiated via bremsstrahlung isproportionalto

γ

4

(25)

Figure 2.5: Radiative ele tron apture to ontinuum (RECC). Target ele tron is aptured into a

ontinuumstateoftheproje tileandaphotonisemitted.

d−

→

_{v /dt ⊥ −}

→

v

) or

γ

6

(

d−

→

_{v /dt k −}

→

_v

) [Gri 01 ℄. Sin e

E = γmc

2

,where

m

is therest massof the movingparti le,the total radiated powerisproportional to

1/m

4

or

1/m

6

,respe tively. The

above means that ele trons lose energy via bremsstrahlung pro ess mu h more rapidly than

heavier hargedparti les. Thisiswhyele tronbremsstrahlungdominatesovertheion-related

pro esses.

Quasifree ele tron bremsstrahlung (QFEB), se ondary ele tron bremsstrahlung (SEB),

atomi bremsstrahlung(AB)andnu leus-nu leusbremsstrahlung(NB)aredominatingamong

various bremsstrahlung pro essesthat an o ur during ion-atom ollision. These pro esses

were taken into a ount during data analysis and are more thoroughly dis ussed in the

fol-lowing se tions.

2.3.1 Ele tron bremsstrahlung

Radiative ele tron apture to ontinuum RECC, sometimes referred to as quasi-free

ele tron bremsstrahlung (QFEB) is a pro ess where the target ele tron is aptured to the

proje tile ontinuum, whi h means it be omes a free ele tron. Energy onservation in this

=

1

2 m

e

v

2 ₌

m

e

m

p

E,

(2.8)

where

v

is the velo ity of the in oming ion in the laboratory frame (equal to the velo ity of the aptured ele tron in the proje tile referen e frame).

T

r

is the maximum energy (in the proje tile frame of referen e) of the photon emitted during the RECC pro ess. As the

maximum energy of theemitted photons is well dened, the spe trumof the emitted x-rays

will have a sharp edge at this value. This edgewasobserved, for example, during ollisions

of arbon ionswithC- and Be-targets[Bed 98 ℄, asshowninFig.2.6.

Eje tedtarget ele tronsmays atterinthe Coulombeldofothertargetnu lei,produ ing

additional bremsstrahlung. This pro ess isreferredto asse ondaryele tron bremsstrah-lung

(SEB). In this ase maximum energy (

T

m

) of the emitted photonsis equal to themaximum transfer ofthe kineti energy during ion-ele tron ollision, given by:

T

m

= 4T

r

= 4

m

e

m

p

E.

(2.9)

Thus,similarlytoRECC,SEBspe trumhasasharpedgeatthephotonenergyof

T

m

[Ish87℄. During the atomi bremsstrahlung pro ess (AB) proje tile ex ites a target ele tron to a

target ontinuumstate. Thisele tron an be re apturedbya targetatom withsimultaneous

emission of xrays. The ele tron an also lose only part of its energy but remain free. This

pro ess is referredto asradiativeionization (RI).

It was shown in [Ish06℄ that the relative ontribution of the above pro esses strongly

varies with proje tile energy. Theoreti al des ription of bremsstrahlung rossse tions given

in [Ish06℄ is in agreement with experimental data. Comparison of experimental data for

p+Al ollisionsat

1

and

4

MeVwiththeoreti al al ulationsarepresentedinFig.2.7. Simple s alingformulaedes ribingdoubledierential bremsstrahlung rossse tionswereproposedin

[Ish 06℄:

(~ω)

2 Z

2 d

2 σ

RECC

d(~ω)dΩ

= Z

t

f (

m

e

m

p

E

~ω

),

(2.10)

(~ω)

3 Z

2 d

2 σ

AB

d(~ω)dΩ

= f (

a

0 ω

v

p

),

(2.11)

(~ω)

2 Z

2 d

2 _σ

SEB

d(~ω)dΩ

= Z

2 t

f (

m

e

m

p

E

~ω

).

(2.12)

(27)

Figure 2.6: Spe tra observed during the experiment [Bed98℄ for: (a) Be-target, (b) C-target. For

ea h energy-target ombinationtwospe traare displayed. On thetop rawspe trumand lower

r

transformedto thelaboratoryframe. Inset in (a)representstheexperimentalsetup.

where

~ω

denotesphotonenergy,

a

0

istheBohrradiusand

f

isanuniversalfun tiondis ussed extensively in [Ish 06℄. The bremsstrahlung pro esses for protons intera ting with various

targets at wide range of energies were thoroughly studied for example by Folkmann [Fol84 ,

Fol75 ℄. Bymeansoftheaboveformulae,thebremsstrahlung ontributiontotheexperimental

(28)

Figure 2.7: Example ontributionof variousbremsstrahlungpro essesto the ontinuous x-ray

spe -trum during Al + C ollisions at

1

and

4

MeV [Ish06℄. It an be noti ed that SEB be omes a dominatingpro ess athigherbeamenergy.

2.3.2 Nu leus-nu leus bremsstrahlung (NB)

Nu leus-nu leusbremsstrahlungisa onsequen eoftheproje tiles atteringintheCoulomb

eld of the target nu lei. Emitted x-ray spe trum extends up to the proje tile energy. The

dierential rossse tion for NBpro ess an be al ulated fromthe formulagiven byMokler

[Mok 78 ℄:

dσ

N B

d(~ω)

= C

Z

2 _Z

2 t

(~ω)E

A

Z

A

−

Z

t

A

t

2 ,

(2.13)

C = ln

1 +

√

1 − x

2 x

!

· 4.3 · 10

−

₂₈

[cm

2 ],

(2.14)

x =

A + A

t

A

t

(~ω)

E

,

(2.15)

where

A

,

A

t

areproje tile andtarget massinatomi units,respe tively.

Fig. 2.8 shows ontribution of RECC, SEB, AB and NB to the x-ray spe tra obtained

(29)

Figure2.8: Bremsstrahlungpro essesobservedduringp+C ollisionsat

2

MeV.PlotbasedonFig.3 in [Ish06℄.

spe tra an be observed at

T

r

and

T

m

, respe tively. The NB ross se tion is signi antly lower than those of the ele tron bremsstrahlung pro esses. Thus, NB plays signi ant role

only within the x-rayspe trumrangeabove

T

m

.

2.4 Multiele tron apture pro esses, non orrelated double

ra-diative ele tron apture (DREC)

Duringasingleion-atom ollision aptureofmorethanonetargetele trontotheproje tile

bound state is possible. The simplest example of a non orrelated apture of two ele trons

is a double radiative ele tron apture (DREC) for whi h the apture of two ele trons is

a ompaniedbytheemissionoftwoindependentRECphotons(Fig.2.9). Duringthispro ess

the aptured ele trons do not intera t with ea h other and the apture of ea h of them an

be treated asaseparate pro ess.

Radiativenon orrelateddouble apturewastheoreti allyaddressedbyMeyerhof[Mey85℄.

Inthispapertheauthor al ulatedREC rossse tionasanintegralofthepro essprobability

(30)

dzρ(R) = Z

t

,

(2.18)

the REC rossse tionfor multiele tron target issimplygiven by

Z

t

σ

REC

(Z

t

= 1)

.

The same method wasapplied to non orrelated double radiative apture. If

P

0 (b)

is the probability of a single ele tron apture into the bare ion, and

P

1 (b)

is the orresponding probability ofele tron apture into theH-likeion, the rossse tionfor non orrelated double

ele tron apture an be expressedas[Mey85 ℄:

σ

DREC

=

Z

∞

0

(31)

Figure2.10: Singleele tronloss rossse tionasgivenin[Hip87℄. SolidlinePWBA al ulationsfor

He

2+

impa t,dot-dashedlinein ludes ontributionfromfreeele tronimpa tinCBEapproximation.

Symbols:

O

6+

,

△

Si

8+

,

▽

Si

13+

,

◦

and

×

O

7+

.

For doubleele tron apture(DREC) onenally obtains the rossse tion[Mey 85℄:

σ

DREC

= 0.13Z

t

1/2

σ

2 REC

(Z

t

)a

−

₀

2 .

(2.20)

The above formula wasveried experimentallybyBednarz [Bed 03℄.

When the aptured ele trons intera t with ea h other during the ollision, the pro ess

is referredto as orrelated apture. Radiative double ele tron apture (RDEC) is thebasi

exampleofa orrelatedpro essand anbetreatedasatimeinverseofdoublephotoionization.

Thus, due to ele tron-ele tron intera tion of the two aptured ele trons only one photon is

emitted andits energyistwo timesgreater thanthatof asingleRECphoton(Fig.2.9). The

RDEC pro ess isdis ussed indetailinChapter3.

2.5 Proje tile ionization ele tron loss

There is a variety of terminology used in the literature, when reering to ele tron

de-ta hment, whi h leads to onfusion. Here, anomen lature from [Bom89 ,Tan91, Hip87 ℄ is

(32)

whiletheremovalofthe ele tronfromtheproje tileboundstateisreeredtoasele tronloss.

Consequently, ele tron loss is the pro ess where an ele tron is removed from the proje tile

and remains freeafterwards:

A

q+

+ A

t

→ A

(q+x)+

+ A

t

+ xe

−

,

(2.21)

where

x

isthe numberofele trons lost bythe proje tile (

q + x ≤ Z

). Ele tronlosspro esses have been extensively studied during the late seventies an eighties for various elements and

harge stateswithinenergy range upto

10

MeV/u [Gra84,Gra 85,Ols78℄.

Bomanetal. [Bom89 ℄developedasimples alingformulaeforele tronloss rossse tion.

2 ],

(2.24)

where

q

denotestheinitial hargestateoftheion. Ithasbeenalso he kedbytheauthorsthat in ase of Si

8+

+ He ollisions at

1.0

MeV/u the ratio of single to double ele tron loss ross se tions

σ

8

1 /σ

8

2 ≈ 40

. Thus, it an be assumed that the double ele tron loss pro ess an be negle tedforthe aseofthemoretightlyboundK-shellele trons inO

6+

. As anbeobserved

inFig 2.10the singleele tron loss rossse tiondoesnot hange signi antly whenthebeam

energy isin reasedfrom

1

to

2

MeV/u. Thus,s alingformulaegiven byEqs2.22-2.24 an be used to estimatethe rossse tionwithin thisenergy range.

(33)

Chapter 3

Radiative double ele tron apture (RDEC)

Radiative double ele tron apture (RDEC) is a one-step pro ess for whi h two free (or

quasifree)targetele tronsare apturedintoboundstatesoftheproje tile, e.g. intoanempty

K-shell, and the energy ex ess is released as a single photon (Fig. 2.9). This pro ess hasto

be ompared with a two-step double radiative ele tron apture (DREC) during whi h two

ele trons are apturedindependently andtwo singleRECphotons areemitted.

Whilefor the DRECpro ess bothele trons an be treated separately(seeSe tion 2.3.2),

in ase of the RDEC one has to go beyond the independent ele tron model. Here, due

to ele tron-ele tron intera tion, transitions of two target ele trons into theproje tile bound

stateso urwithanemissionofonephoton withtheenergytwo timesgreater thanthatofa

single DRECphoton.

Ingeneral, aptured ele trons mayoriginate fromtwo dierent orbitals inthetarget and

arrivenally atdierent nalstatesintheproje tile. Thus, theenergy oftheRDEC photon

an beexpressedas:

E

RDEC

≈ 2T

r

+ E

_B

(1)

+ E

_B

(2)

− E

_Bt

(1)

− E

_Bt

(2)

+ −

→

v −

→

p

(1)

+ −

→

v −

→

p

(2)

,

(3.1)

where indi es

(1)

and

(2)

orrespond to ea h of the aptured ele trons. The width of the peakisabouttwi easlarge asthatofthe RECline. Roughly,itisdetermined bythesum of

Compton prolesof thetwoa tiveele trons [Mir 89℄.

3.1 Initial experiments dedi ated to RDEC

Therstexperiment dedi atedto RDECwasperformedatGSIin1994 with

11.4

MeV/u Ar

18+

(34)

exper-Figure3.1: Typi alx-rayspe trumobtainedduringargonexperiment[War95℄.

iment isgiven in [War 95℄. A typi al spe trumobtained duringthatexperiment ispresented

inFig.3.1. Asshowninthisgure,nosigni antline stru turerelatedtotheRDECpro ess

was observed. However, the number of ounts olle ted in theexpe ted RDEC energy

win-dowprovidedan upperlimitoftheRDEC rossse tionofabout

5.2

mb. Arough theoreti al estimate of the ross se tions ratio

σ

RDEC

/σ

REC

was also suggested, based on theprin iple ofdetailed balan eand onsideringRECasatimereversalofphotoionization. TheRECand

RDEC rossse tions an be written as:

σ

REC

(~ω) = Z

t

~ω

γβmc

2

2 σ

P I

(~ω),

(3.2)

σ

RDEC

(~ω

′

) = F Z

t

(Z

t

− 1)

~ω

′

2γβmc

2

2 σ

DP I

(~ω

′

),

(3.3)

where

σ

P I

and

σ

DP I

arethe rossse tionsforsingleanddoublephotoionization,respe tively. The fa tor

F

(

F ≤ 1

)des ribes thephase spa efra tionof doublephotoionization a essible for the RDEC pro ess. Thus,

σ

RDEC

/σ

REC

ratio an be expressed in terms of single and double photoionization rossse tionsas[War 95℄:

R =

σ

RDEC

σ

REC

= F (Z

t

− 1)

ω

′

2ω

σ

DP I

(~ω

′

)

σ

P I

(~ω)

,

(3.4)

or, asin aseof RDEC

~ω

′

≈ 2~ω

:

R =

σ

RDEC

σ

REC

= F (Z

t

− 1)

σ

P I

(2~ω)

σ

P I

(~ω)

σ

DP I

(2~ω)

σ

P I

(2~ω)

.

(3.5)

(35)

Figure 3.2: Typi alexperimental x-rayspe trumobtainedforuraniumions [Bed03℄. TheGaussian

solid line showstheexpe tedRDEC peak, whi h should be observeda ordingto Yakhontov et al.

[Yak96,Yak97℄.

The experiment [War 95℄provided an upper limitfor

R

of

3.1 · 10

−

₆

.

Thisexperiment stimulated theoreti altreatment oftheRDECpro ess [Yak96 ,Yak97℄.

Inthesepaperstheauthorspresentednonrelativisti al ulationsoftheRDECpro essadapted

to thekinemati s andthe energy rangeoftheAr

18+

+Cexperiment. The al ulations gave,

for this parti ular ollision system (Ar

18+

+ C at

11.4

MeV/u), the RDEC to REC ross se tions ratio

R

of

3.6 · 10

−

6

,whi his lose totheexperimental upper limit.

Moreover, these al ulations predi tedastrong enhan ement ofRDEC duringheavy

ion-atom ollisions at relativisti energies [Yak97℄. These al ulations were tested during the

se ond experiment dedi atedto RDEC.Here, bareuranium ions at an energy of

297

MeV/u were olliding with an Ar target at the ESR storage ring of the GSI fa ility [Bed 03 ℄. This

experiment showed thatfor the ollisionsystemunder onsiderationtheRDEC rossse tion

is ertainlyatleastthreeordersofmagnitudesmallerthanthetheoreti alpredi tion[Yak96 ,

Yak97 ℄. Fig. 3.2 shows a spe trum obtained during the experiment. Again, no signi ant

line stru turewhi h ouldbeassignedtotheRDECpro esswasobserved. TheGaussianline

inFig.3.2shownwithintheRDECregionofthespe trumrepresentstheshapeoftheRDEC

(36)

werereliable. Thisexperiment alsoprovided only anupper limitfor theRDEC rossse tion

value ofabout

10

mb.

3.2 Re ent theoreti al approa h to RDEC

In order to explain the disagreement between the uranium experiment [Bed 03℄ and the

theoreti altreatmentofRDEC[Yak96 ,Yak97℄,anewtheoreti alapproa hforthe orrelated

doubleele tron aptureintotheK-shellofbareionswasproposed[Mik04a ,Mik04b,Nef 05℄.

Here,abriefdes riptionofthisRDECtreatmentisgivenwiththenotationusedintheoriginal

papers. Indi es

(1)

and

(2)

orrespond to REC and RDEC, respe tively, and natural units

(~ = c = 1)

are usedthroughout the text.

Allthe ele tronsinvolvedinthepro ess were onsideredasnonrelativisti andtheenergy

ω

oftheemitted photonwaslimitedby

I

2K

≤ ω ≪ m

,where

I

2K

isthethresholdenergy for double photoionization oftheproje tile K-shelland

m

istheele tronmass. Insu h asethe Coulomb parameter (

αZ

,

α

denotes the ne stru ture onstant) is small (

αZ ≪ 1

) and the perturbation theorywithrespe tto theele tron-ele tron intera tion an beused.

Inthe referen e frameof thein ident iontheprobability

dW

for double ele tron apture into the K-shell of bare ion with the emission of a single photon per unit time is given by

[Mik04a ℄:

dW =

2π

V

2 |A|

2 d

−

→

k

(2π)

3 δ(2E

P

− ω − I

2K

),

(3.6)

where

E

P

is the one-ele tron energy within the initial ontinuum state,

ω = |

−

→

k | = k

is the energy of the emitted photon and

I

2K

= 2I

, with

I = η

2 _/2m

, the Coulomb potential for

singleionization and

η = mαZ

the hara teristi momentumoftheK-shellele tron, and

V

anormalization fa tor. Summation overall polarizations oftheemitted photonis assumed

inEq.3.6anddeltafun tionensurestheenergy onservation. Theamplitude

A

wasobtained from that for the double K-shell photoionization. Detailed des ription of this approa h is

given in[Mik04a℄.

Dividing Eq. 3.6 by the urrent ux of the in ident target ele trons

j = v/V

, where

v = p/m

is the absolute value of the initial velo ity of the in ident ele trons before the ollision withion, one obtains theee tivedierential rossse tion:

dσ

(2)

= 2π

ω

2 vV

|A|

2 dΩ

k

(37)

Figure3.3:

Q

,universalfun tionofthedimensionlessvariable

ξ

[Mik04a℄.

whi h denes the angular distribution of the RDEC photons emitted into an element of a

solid angle

dΩ

k

.

For theenergy regimeassumedinthese al ulations, itwaspossible to al ulatethetotal

rossse tionswithin theele tri dipoleapproximation. For ollisions ofheavy ionswithlight

target atoms the total ross se tion for radiative double ele tron apture (RDEC) into the

K-shell ofthe ionis given by:

σ

(2)

=

2

19 _Z

3 t

3πZ

5 Q(ξ),

(3.8)

where

ξ = η/p

is a dimensionless parameter,

σ

0 = α

3 _a

2

0

and

a

0

denotes the Bohr radius.

Q

is theuniversal fun tion of

ξ

,whi h an be obtained by numeri al integration (Fig. 3.3).

ξ ∼ 1

orresponds to the near-threshold domain, where the K-shell photoee t rea hes its maximum. For slow ollisions

(ξ ≫ 1)

the RDEC rossse tionin reases, whilein ase offast ollisions itde reasesrapidly. Moreover,ithastobepointedoutthattheRDEC rossse tion

rapidly dropswith the proje tile atomi number (

∼ Z

−

₅

) and in reases signi antly for low

energy ollisions.

Another value al ulated in [Mik04a℄ is the ross se tions ratio (

R = σ

(2)

_/σ

(1)

). The

(38)

Figure3.4: Universalquantity

Q/H

al ulatedasafun tionofthedimensionlessvariable

ξ

[Mik04a℄.

theprin iple ofdetailed balan e. As

σ

P I

isknownanalyti ally (Stobbeformula[Sto30℄),for the radiative ele tron apture into theK-shelloftheproje tileone obtains:

σ

(1)

=

2

10

3 π

2 _σ

0 Z

t

H(ξ),

(3.9)

H(ξ) =

ξ

2 ε

2 γ

exp(−4ξ cot

−

₁

ξ)

1 − exp(−2πξ)

,

(3.10)

where

ε

γ

= ω/I

isthedimensionless photon energy. Thentheratio

R

is given by:

R =

2

9 _Z

2 t

Q(ξ)

π

3 _Z

5 _H(ξ)

.

(3.11)

The fun tion

Q(ξ)/H(ξ)

is presentedinFig.3.4.

These al ulations are in disagreement with the previous relativisti approa h [Yak97℄,

whi hwasnotabletoexplainthe existingexperimentaldata [Bed03℄. Asshownin[Mik04a ℄

the enhan ement of the wave fun tion for therelativisti systems was al ulated in orre tly

byYakhontov [Yak97℄ andeven the orre ted value, whi h is

3

orders ofmagnitude smaller [Mik04a ℄, wasnot onrmedbytheexperiment[Bed 03 ℄. Therefore,theenhan ement ofthe

RDEC ross se tion for relativisti systems [Yak97 ℄ seems to be absent. This is similar to

thebehaviorofthe rossse tionfortheRECpro ess,where the rossse tionde reases when

(39)

Table 3.1: Comparison of experimentally obtained RDEC ross se tions [War95, Bed03℄ and the

al ulatedvaluesgivenin[Yak97℄ and[Mik04a℄.

Z

E

[MeV/u℄

ξ

Z

t

σ

(2)

[mb℄

Ref. [Mik04a℄ Ref. [Yak97℄ experiment

18 11.4 0.840 6 3.2 1.85

≤

5.2[War95℄

92 297 0.841 18 2.5

·

10

−2

5000

≤

10[Bed03℄

Figure 3.5: Theratioof theRDEC rossse tions totheex ited (

σ

(2)

2

1 _S

)and ground(

σ

(2)

1

1 _S

)proje tile statesasafun tionofadiaba ityparameter

ξ

[Nef 05℄.

However, ithasto be emphasizedthat the urrent estimate[Mik04a ℄of

σ

(2)

givesvalues

losertotheexperimentallyobtainedupperlimitsforboththenonrelativisti ase(Ar

18+

+C,

[War 95 ℄) andtherelativisti one (U

92+

+Ar,[Bed 03℄) (seeTable 3.1), whi hsuggests that

[Mik04a ℄ issofar themost reliable theoreti aldes ription ofRDEC.

In ontradi tiontopredi tionsgivenin[Yak97 ℄,thenew al ulationsshowthattheRDEC

)andDREC(

[mb℄

σ

(2,γ)

[mb℄

σ

(2,γ)

_σ

(1)

_σ

1.9

·10

−11

2.6

·

10

−2

ele tronsismu hsmallerthanthatoftheproje tile,they anbe onsideredasquasifreeinthe

proje tile's frame of referen e. In this referen e frame these ele trons appear asan ele tron

beam with velo ity

v

and on entration

n

e

= κρ

t

N

A

/M

t

, where

κ

is thenumber of valen e ele trons,

N

A

istheAvogadro'snumberand

ρ

t

and

M

t

arethedensityandmolarmassofthe target, respe tively. Hen e, withsubstituting

V = n

−

1 e

,Eq. 3.8 anbe rewrittenas:

σ

(2)

= (n

e

a

3

0 )

2

19 σ

0 3Z

5 Q(ξ).

(3.12)

Inaddition, the orrelated doubleele tron aptureinto the

1s2s

state in reases the ross se tion for the RDEC pro ess [Nef05℄. As shown in Fig. 3.5, the ratio of the ross se tion

for RDEC to the

1s2s

state,

σ

(2)

2

1 _S

, to the ross se tion for RDEC to the

1s

2

ground state,

σ

(2)

₁

1 S

,is stronglydependent onthe

ξ

value. As anbeseen fromFig.3.5,for

ξ ≫ 1

(i.e. slow ollisions) the rossse tion for ele tron apture to the

1s2s

state an greatly ex eed theone for the

1s

2

state apture.

Re ently,the al ulationsofNeodovandMikhailovwere ontinuedbyDrukarev[Dru 07℄,

who again addressed the high energy nonrelativisti limit (

ξ ≪ 1

) of the RDEC pro ess. As previously, a strong energy dependen e of the ross se tion was shown and the RDEC

probability was ompared with the one of non orrelated apture. Obtained values of the

REC, RDEC and DREC rossse tions for Ar

18+

+C andMg

12+

+ Cfor various proje tile

energies aregiven inTable3.2.

Thistheory[Mik04a,Mik 04b ,Nef 05 ,Dru07 ℄suggests thatthebest systemsfor

obser-vation of the RDEC pro ess are low energy ollisions of mid-

Z

ions with light solid targets. This theory wasa motivation for thenext RDEC experiment and a reason for the hoi e of

(41)

Experimental setup at Western Mi higan University

4.1 Van de Graa a elerator

Van de Graa a elerator is an ele trostati generator whi h uses a moving belt to

a - umulate very high, ele trostati ally stable voltage on a hollow metal sphere [Gra47℄. This

type of generators was developed by Robert J. Van de Graa at Prin eton University. The

rstmodelwasdemonstrated inO tober1929andin1931 aversionabletoprodu epotential

dieren e of 1MV wasdes ribed [Gra31℄.

A simple Van de Graagenerator ispresentedin Fig.4.1. A belt ofa diele tri material

runs over two rollers, one of whi h is surrounded by a hollow metal sphere high voltage

terminal. Two ele trodes an upperand a lower one are pla ed next to ea h roller. The

upper ele trode is onne ted to the sphere, while a high DC potential (with respe t to the

ground potential) isapplied to thelowerone apositive potential intheexample.

Duetothestrongele tri eldtheairaroundthelowerele trodeisionizedandthepositive

ions arerepelled from the ele trode and a umulated on thebelt. Thethey are transported

towards the upper ele trode whi h olle ts the harges from the belt and transports them

onto the spheri al olle ting ele trode. Thepotential of theHV ele trode in reases until the

speed ofits hargingequalstothespeedofdis harging. Themaximumpotential obtainedon

the HV ele trode dependson the radius of thesphere and insulating properties of thegases

surrounding it. SF

6

or a mixture of N

2

and CO

2

under a pressure even up to

20

bar are usually used[Hin 97℄. Thevalueofterminal voltage inVande Graaa eleratorsmayrea h

up to

15

-

20

MV [Edw 93 ,Bey 03℄.

If a sour e of positive ions is pla ed lose to the high voltage terminal, as in Fig. 4.1,

(42)

Figure4.1: S hemati viewofa lassi alVandeGraaa elerator: (1)lowerroller,(2)upperroller,

(3) harging ele trode, (4) ele trode olle ting positive harge, (5) voltage generator, (6) spheri al

ele trode(highvoltageterminal),(7)ionsour e,(8)extra tedionbeam.

towards the groundpotential. Final kineti energy of theions dependson their harge state

q

andis proportionalto theterminal voltage

V

terminal

:

E = qV

terminal

.

(4.1)

In a modern type of ion a elerators with a Van de Graa generator, the ele trodes at

entry and exit of theva uum tube aregrounded and thehigh-voltage terminal is lo ated at

the middle of the tube, asshown in Fig. 4.2 [Hin 97, Wed 99 ℄. A sour e of negative ions is

pla ed at the entran e ofthetubeand produ ed ions,usually singly harged, area elerated

within the tube towardsthehigh-voltage terminal, where two or moreele trons areremoved

from ea h ionasit passesthrough astripping foil. The hargestate of theion hanges from

negative to positive and the ion is repelled from the terminal and a elerated towards the

grounded exitof the tube. Compared to Van de Graaa elerators oftheordinary type,by

meansofthetandemVandeGraaa eleratorshigherparti leenergies anbeobtainedsin e

(43)

Figure 4.2: S hemati viewof atandemVande Graa a elerator: Negativeion entering the

a el-erator (A

−

)isa eleratedbythe highterminalvoltage. Someofitsele tronsareremovedwhilethe

ionpassesthroughthestrippingfoil. Thepositiveion(A

+q

)isrepelledbythehighvoltageterminal,

thusadditionallya elerated.

energy anbe al ulatedas:

E = (q + 1)V

terminal

,

(4.2)

where

q

isthe ion hargestate after passingthroughthestripping foil.

Experiment des ribed inthis dissertationwas performedat Western Mi higan University

using

6

MV tandem Van de Graa a elerator. The WMU a elerator was built by the High Voltage Engineering Corporation, the ompany founded byRobert Van de Graa.The

onstru tion of the a elerator allows for obtaining stablebeams of bareions of allelements

up to

9

Fwiththetotal kineti energy upto

40

MeV.

4.2 Beam line setup at Western Mi higan University

As hemati view oftheWMUa elerator beamlineispresentedinFig.4.3. A elerated

beam passes through a

90 ◦

analyzing magnet whi h allows for hoosing theappropriate ion

harge state. At this point, the nal energy of the beam is dened a ordingly to Eq. 4.2.

Then the beampassesthrough a poststripper followed bya swit hingmagnet whi hdire ts

desired harge state towardsthe experimental area. Forthe presentedexperiment a beamof

O

6+

wasextra ted fromthe a elerator operating at theterminal voltage of

5.43

MV,whi h produ ed the beam of energy equal

38

MeV.Then thebeam traversed through a

20 µ

g/ m

2

(44)

Figure4.3: S hemati viewoftheWMUvandeGraaa eleratorfa ility[Kay℄.

magneti eld of the swit hing magnet one ould hoose the ne essary harge state. When

the protonbeam wasa elerated, thestripperwasremoved fromthebeamline.

The experimental beam line farthest left, when looking along beam dire tion, was used

duringthis experiment. There,anex lusivelydesigned hamberfor asolidtargetwaspla ed,

whi h not only allowed for mounting up to four lms but also provided asimple me hanism

for target rotation. This was ne essary for optimization of the target position during the

experiment. Duringdataa quisitionthetargetlmwaspositionat

45 ◦

tothebeamdire tion,

fa ingthex-raydete torasshowninFig.4.4. Thissetupensuredadire tdete tionofemitted

photons, astheydidnottraversethroughthefoil,sotheunne essaryenergylosswasavoided.

It alsoallowed forusageofthewholea tive areaofthex-raydete tor, whi h wasnot overed

bythealuminumframeofthetargetholder. Thetargetfoilsusedduringtheexperimentwere

a few

µ

g/ m

2

thi kwhi h orrespondstothearealdensityoftheorderof

10

17

parti les/ m

2

.

Thetarget hamberwasdesigned ina waythatminimizes thedistan e between dete tor

windowandtarget enter. Thetotal rystal-targetdistan ea hievedwasabout

25

mm,whi h givesthe dete tion solidangle of

∆Ω = 0.044(1)

sr.

(45)

Figure4.4: Theexperimentaltarget hamberin 1:1s ale.

Emittedx-rayswereregisteredbyanORTECsingle rystalSi(Li)dete torpla ed

perpen-di ular to the beam dire tion. The rystal of

6

mm diameter and

3

mm thi kness, together with

7.5 µ

mBe-window,gavethe dete tione ien yintheenergyrange

2

-

4

keVbetterthan

90%

(Fig. 4.5). The dete torwasenergy alibrated witha standard

55

Fe alibration sour e.

Calibration pro edurewasfrequentlyrepeatedthroughout theexperimentinorderto ontrol

(46)

Figure4.5: Dete tione ien yofORTECSi(Li) dete tors[ORTa℄.

Figure4.6: Experimentalsetup.

Along the beam dire tion, a set of two ollimators was pla ed in front of the target

hamber. The distan e between ollimatorswas about

2

m. Collimators'apertures of

2

and

3

mm were to ensure a good beam ollimation. Additional ollimator between the target and magnet prevented s attered ions from entering the spe trometer and generating false

oin iden es (seeFig. 4.6).

Thetarget hamberwasfollowedbyamagneti spe trometer. Magneti eldofthedipole

magnetseparatednal hargestatesoftheionsanddire tedthemtowardsfoursurfa ebarrier