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O rder N um ber 9402673

D ielectronic recom bination on and electron impact excitation o f He-like ions and m ulti-electron processes in slow collisions o f highly charged ions w ith atom s Ali, Rami Moustafa, Ph.D. Kansas State University, 1993

UMI

300 N. Zeeb Rd. Ann Arbor, MI 48106



D IE L E C T R O N IC R E C O M B IN A T IO N O N A N D E L E C T R O N IM P A C T E X C IT A T IO N O F H E -L IK E IONS AND M U L T I-E L E C T R O N P R O C E SSE S IN SLO W CO LLISIO N S O F H IG H L Y C H A R G E D IO N S W IT H ATOM S by R A M I M O U STA FA ALI B.S., Yarmouk University, Jordan, 1986

A D ISSE R T A T IO N submitted in partial fulfillment of the requirements for the degree D O C T O R O F P H IL O S O P H Y Department of Physics College of Arts and Sciences K A N SA S STA TE U N IV E R S IT Y Manhattan, Kansas 1993 Approved by:

o .J. f a i L Major Professor


To my parents, Amneh and Moustafa Saffarini


TABLE OF CONTENTS L IS T O F F IG U R E S ............................................................................................... iii L IS T O F TA B LES ................................................................................................. viii A C K N O W L E D G E M E N T S ................................................................................... ix 1. IN T R O D U C T IO N ............................................................................................ 1 1.1 In tro d u ctio n ...................................................................................................... 1 1.2 Electron-Ion Collisions.................................................................................... 2

1.3 Slow Collisions of Highly Charged Ions with Atoms ..................................9 References .......................................................................................................20 2. E L E C T R O N -IO N C O L L ISIO N S ............................................................... 22 2.1 Paper I ............................................................................................................22 Dielectronic Recombination on Heliumlike Argon 2.2 Paper I I ........................................................................................................... 27 Dielectronic Recombination on and Electron-Impact Excitation of Heliumlike Argon 2.3 Paper I I I ..........................................................................................................37 Electron-Ion Recombination Experiments on the KSU EBIS 3. SLO W C O L L IS IO N S O F H IG H L Y C H A R G E D IO N S .......................48 W IT H A TO M S


3.1 Paper IV .........................................................................................................48 Multi-Electron Processes in 10 keV/u Ar,+ (5 < q < 17) on Ar Collisions 3.2 Paper V .......................................................................................................... 87 Q-value Measurements in Charge-Transfer Collisions of Highly Charged Ions with Atoms by Recoil Longitudinal Momentum Spectroscopy 3.3 Paper VI .........................................................................................................92 Angular Distributions in Charge-Transfer Collisions of Ar15+ with Ar 3.4 Paper VII ..................................................................................................... 112 On the Radiative Stabilization in Slow Double-Electron Capture Collisions of Highly Charged Ions with Neutral Atoms 4. S U M M A R Y A N D C O N C L U D IN G R E M A R K S .............................. 121 A P P E N D IX A .................................................................................................123 A P P E N D IX B .................................................................................................125 A P P E N D IX C .................................................................................................127

ii

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LIST OF FIGURES C hapter 1. Introduction 1 A schematic of the Kansas State University cryogenic electron-beam ion source

(CRYEBIS) facility .................................................................................................3 2 A schematic illustration of the dielectronic recombination process on heliumlike a r g o n ......................................................................................................................... 4 3 Electron energy dependence of the yield of Ar16+, Ar15+, ArH+ and their sum ........................................................................................................................... 6 4 Cross sections for dielectronic recombination on heliumlike argon ..................7 5 Differential cross sections at 0/a6 = 0° for K x-ray production due to dielectronic recombination on and electron-impact excitation of heliumlike argon ............ 8 6 A schematic illustration of electron capture in the molecular classical

over barrier model .................................................................................................. 11 7 A schematic of the general experimental setup used in the investigations of slow ion-atom collisions ........................................................................................ 13 8 Experimented and MCBM recoil production cross sections ............................ 15

9 Experimental and molecular classical overbarrier model average scattering angles ...................................................................................................................... 17 10 A comparison of the behavior of the experimental Prad and the model fraction FEl for Ar?+ (q = 9 - 17) on Ar ........................................................................ 18

C hapter 2. Electron-Ion Collisions 2.1 Paper I iii

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1 Plots vs electron energy of (a) the yields of Ar16+(ri|6) and ArIo+(n i5) and (b) the ratio n is/n ie ...................................................................................................24 2 Cross sections for dielectronic recombinationon heliumlike Ar ...................... 25 2.2 Paper II 1 Schematic of the EBIS and detection system ................................................. 29

2 Electron energy dependence of (a) the yields of Ar16+, Ar15+, Ar14+. and their sum, and (b) the ratio n 15/ n l6 ........................................................................... 29 3 Cross sections for dielectronic recombination on heliumlikeAr ......................30 4 Time-evolution of the Ar16+ fraction for on andoff resonance electron energies................................................................................................................... 32 5 A two-dimensional spectrum of Ar K x-ray energy vs electron energy ........ 33 6 Differential cross sections at 8(ab = 0° for K x-ray production due to dielectronic

recombination on and electron-impact excitation of A r16+ ............................33 7 Partial differential cross sections at 8iab = 0° for An = 1 dielectronic recombination on A r16+ ...................................................................................... 34 2.3 Paper III 1 Schematic of the EBIS and detection system ................................................... 39 2 Electron energy dependence of the yields of Ar16+, Arlo+. A r14+, and their sum ..........................................................................................................................40 3 Cross sections for dielectronic recombinationon heliumlike Ar ...................... 41 4 (a) Electron energy dependence of the yield of Ne8+. Ne7+ and their sum. (b) Cross sections for dielectronic recombination for heliumlike Ne .................... 42 5 Partial differential cross sections at 8tab = 0° for An = 1 dielectronic recombination on heliumlike Ar .......................................................................... 43 iv


6 Differential cross sections at diab — 0° for K x-ray production due to dielectronic

recombination on and electron-impact excitation of heliumlike a r g o n

44

7 Time evolution of the heliumlike fraction for on- and off-resonance electron energies .................................................................................................................. 44 8 D ata from DeWitt

et al.

Experimental and theoretical dielectronic

recombination cross sections on hydrogenlike Ar ............................................. 46 C hapter 3. Slow Collisions o f Highly Charged Ions with Atom s 3.1 Paper IV 1 Schematic of the experimental setup ................................................................. 53 2 A density plot of the projectile x-position vs recoil time-of-flight for the collision system Ar13+ on A r ...............................................................................................55 3 Projectile charge-change cross sections (a ,,,-* ) and total charge transfer cross section (aq =

a q,q-k) for 10 keV/u Ar?+ on Ar .........................................57

4 Experimental total transfer cross sections and MCBM geometrical, no charge transfer, and total charge transfer cross sections ............................................ 61 5 Experimental and MCBM recoil production cross sections c ql . The MCBM cross sections were obtained by suppressing possible target autoionization . 63 6 Experimental and MCBM recoil production cross sections crj. The MCBM

cross sections were obtained by allowing for target autoionization ............... 64 7 Energies of doubly and triply-excited ionic states relative to the lowest continuum limit shown in each example ............................................................ 68 8 Three examples of the stabilization of multiply-excited ionic states for the core

charge q = 16 .......................................................................................................... 71 9 Experimental and MCBM-Stabilization phenomenological cross sections

—i/> + y 0 R. l*>e resonant electron energy is £, = £ ,-£ ,, where £, and £, are the energies of the initial state U) and the resonant state |r ). The resonance cross section has the typical Lorentzian shape and is given by .r*1 S , r.U —i)f,lr—/ )

aDR' ' 2m , E ,

2 g,

(£,-£,)2+rJr(s)/4 '

In Eq. (5), g, and g, are the statistical weights of I s ) and If), Tr(s) is the total width of Is), r,(s —i) and f,(s —/) are the contributions to Tr(s) from the Auger

• Experiment Theory

KLL

B a o•4 at a b

KLN

Excitation Threshold

0.0

Electron Energy (IreV) FIG. 3. Cross sections for dielectric recombination on heliumlike Ar. The points are the extracted cross sections as discussed in the text. The solid line is the iheoretical calculation folded into the experimental resolution function.

30


ALL BHALLA, COCKE SCHULZ, AND STOCKLI

226

transition U) —1/> and the radiative transition |s> —1/>. If the total width is much smaller than the width of the experimental energy resolution function, the Lorentzian factor approaches a 6 function and Eq. (5) be comes

r r(s)

l m , E , g,

£ ( £ ,- £ , I . (6)

44

TABLE I. Integrated experimental and theoretical DR cross sections (in 10“'*cm1eV) and their ratios. Resonance •9,,,, / S H u g , SiMO, 2 ,,,, 6.588 0.90 5.942 K LL 7.233 0.85 6.160 KLM 1237 0.93 2.663 KLN' 'S calculated up lo £, = 2.96 keV corresponding to the lowest point on the nght-hand side of the K L N resonance.

When folded into the electron energy experimental reso lution function /(£,). = 2 > D R ,< £ .>

•

(I0>

f

The corresponding Auger energies (£,) and all the rates for the doubly excited Li-like argon were calculated ex plicitly, using the Hartree-Fock atomic model [24], for the following cases: U 2 p n p (n = 2 - 8 ), \ s l p n s (n = 2 - 8 ), U 2pnd In =3-6), \ s 2 p n f In =4,5), and ls 2 s n p (n =2-8). The n “* scaling law was used to obtain the resonance strengths for the above configurations but higher n , up to n =20. The scaled strengths are given by [25] ] £;(/>)= — Cl 1) n

where n m is the maximum n for which explicit calcula tions were done for each case mentioned above. The cor responding Auger energies for these high-n states were obtained from £ =3140-

13.6Z1

( 12)

where £„ is in units of eV, 3140 eV is the average direct excitation threshold (Is1— Is21) of Ar16*, and Z = 16.

The theoretical results so obtained are shown as a solid line in Fig. 3 and are in good agreement with experiment. To provide a comparison of theory and experiment that is independent of our choice of the width of the experi mental resolution function, we present in Table I the ex perimental and theoretical integrated cross sections (5) and their ratios. The discrepancies we had in our earlier results were traced to a channel-plate detector saturation effect. The signal strength, representing the ion yield, varies linearly with the instantaneous current in the detector for currents below a certain threshold, and nonlinearly for higher ones. The earlier discrepancies resulted from hav ing the yield of Ar16'* lying in the nonlinear zone over the whole electron energy range, while the yield of Ar15* was partially in this zone for the K L L and K L M resonances and in the linear zone for higher resonances. A similar saturation effect was found for the channeltron used here, but all data presented here were taken with sufficiently low voltages on the channeltron to ensure linearity. For the purpose of the experiment, the relative detec tor efficiency for the different charge states must be known. For charge states between 12* and 16+, the channeltron signal was measured to be approximately proportional to the incident charge, and this propor tionality was used to assign relative efficiencies for 14* -16* ions. The constant total ion yield in the sum curve of Fig. 2(a) confirms the validity of this efficiency assignment. Exclusive of any error in a „ we estimate an uncertainty of 9 % on the experimental cross section due to background subtraction, reproducibility, and relative detector efficiency. B. X-ray experiment The purpose of this experiment was to further investi gate DR on heliumlike argon by measuring differential and partial differential cross sections at 6 llb= 0 * and to measure the electron-impact-excitation differential cross section of heliumlike argon at the same angle. Since DR processes onto Arl!+ occur at electron energies that overlap with those of DR onto Ar16* and give rise to x rays with overlapping energies, it is important to have a pure Ar“ + target for such an experiment. In Fig. 4 we show the time evolution of the fraction of ArI6+ which was initially maximized by setting the electron-beam en ergy to a nonresonant value for 2 sec and switched onto the K L L resonance at t =0. By switching the electron en-

31


DIELECTRONIC RECOMBINATION ON AND ELECTRON-IMPACT . . .

U

227

off

a

e

0.9

a o c

£ ♦

«

D R S U to n a n c M

MO 0.7 0

50

100

150 200 Time (maec)

•00

too

300

2S0

FIG. 4. Time evolution of the Ar"* friction for on- and olf-reionance electron energies as discussed in the text. Inset: scanning time dependence of the Ar"* fraction. A plot of the K x-ray yield is also shown in the inset to allow the association of each time with the corresponding electron energy.

ergy to • resonant value, the Ar1** fraction dropped down by 13% in about 40 msec. A recovery time of 300 msec at a nonresonant £, was then needed for the frac tion to assume its initial value, demonstrating that the time constant for charge-state equilibrium on resonance is appreciably smaller than for ofT-resonance values. This short time constant on resonance energies required a modification to the experimental technique of part A in order to minimize the fraction of impurities. The -basic difference in the experimental technique from that of part A was that the argon-ion inventory was first prepared in the previously mentioned abundances by a 10-mA 2.4-keV electron beam for 2.0 sec before scan ning the electron-beam energy. After 2.0 sec of cooking time, the electron-beam energy was scanned between 2--1 and 3.8 keV up and down ten times in 6.0 sec before the ions were ejected and a new cycle started. This guaranteed that the electron energy did not spend enough time on any resonance to strongly afTect the charge-state equilibrium. The inset of Fig. 4 shows the scanning time dependence of the fraction of Ar1** for an up-down scan of 900 msec. Depletions of the fraction are seen at such times where the electron energy is resonant and recovery is observed otherwise. The data shows that the margin of error due to the target impurity cannot exceed 13%. This error should be even smaller, since lost DR process es onto Ar1** are partially substituted for by DR onto Ar,J*. The x rays were detected during the scanning time as a function of the electron-beam energy by a Si(Li) detector placed at 0* relative to the electron-beam direction. The Si(Li) was 2.3 m from the EBIS center and viewed the EBIS interior through the collector aperture and a 23-^m

Be window. The solid angle subtended by the detector was 4irX 10 "* sr. Shown in Fig. 3 is a two-dimensional spectrum of the data collected in the event mode in this experiment. The horizontal and vertical axes represent the incident elec tron energy and the stabilizing x-ray energy, respectively. A physical process occurring at electron energy E , and leading to x-ray production of energy E , is represented by a dot whose coordinates are (£,,£,). The area of each dot reflects the number of counts in that channel. A number of isolated peaks are seen at various locations; these represent DR resonances. The continua extending from £, = 2.93 keV to the end are due to both DR and EIE. By constructing two-dimensional windows, various parts of the spectrum could be projected onto the elec tron energy axis. A normalization factor was obtained by projecting the total two-dimensional spectrum onto the £, axis and subtracting a smooth background and then normalizing the area under the experimental K L L reso nance to the theoretical integrated K L L differential cross section. The differential x-ray cross section for a transition Is >—»|/> is given, when £, is less than the n — 1—n = 2 excitation threshold, by [26] do,(Oi,J

|

(13) where B , is the asymmetry parameter, J is the total angu lar momentum, and F2(0) is the Legendre polynomial of order 2. For the case discussed here

32


44

ALL BHALLA. COCKE, SCHULZ. AND STOCKLI

228

4.2

3

JC >V

4.0 38

KLL

I

Excitation

36 i w - -

iiiiiiiimiiMiiiiiiiiiiiiiimiiiiiMiiiiMiimiiiiiiM • < ! ■•

.

Mill•I I I-

•Hi.

•Ml' • m >-

2.8 2.6 24 20

2.5

•mu

3.0

3.5

3.8

Electron Energy (keV) FIG. 5. A two-dimensional spectrum of Ar K x-ray energy vs electron energy. The peaks represent dielectric recombination reso nances and the continua represent contributions from both dielectric recombination and clectron-impact excitation.

d i 7(0Ub=for K x-ray production due to dielectric recombination on and electron-impact exci tation of Ar14*. The points are the measured cross sections obtained through normalization as discussed in the text. The dotted lines are the individual contributions of dielectric recombination and electron-impact excitation folded into the experimental resolution function. The solid line is the sum of all contributions.

33


u

229

DIELECTRONIC RECOMBINATION ON AND ELECTRON-IMPACT. . .

TABLE II. Integrated experimental and theoretical differential DR cross sections (in I0~'° cm!eV/sr) and their ra tios. __ __ Resonance

dS„

dS„ dn

dn

d (l

■/

dn

3.586 3.586 1.00 3.556 3.961 0.90 1.639 1.652 0.99 ‘d S / d O calculated up to £, = 2.96 keV corresponding to the lowest point on the right-hand side of the K L N resonance. KLL KLN KLN*

TABLE III. K a , KB, and K y contributions to the integrated experimental and theoretical differential DR cross sections (in 10" 10 cm1eV/sr) and their ratios. d S „ „ / d£lhM r d S ,m Resonance dn / da da da 3.586 i.CG 3.586 K LL Ka 3.266 0.90 2.930 K LM 1.04 1.199 1.241 KLN* K LM

KB

0.656

0.94

0.696

0.94 0.446 0.418 KLN* calculated up to E , '■=2.96 keV corresponding to the lowest point on the right-hand side of the K L N resonance. Ky

‘d S / d fl

periment where differences are within the estimated ex perimental error of 15%. Figure 7 reveals many interesting features. A first ob servation is that the decay transitions are dominated by those giving rise to K a x rays, though K B and K y transi tions are also possible for the K L M and K L N resonances, for example. Another feature is the near absence of K B

8

7

• Experim ent Theory

KLU

KLL

Ka

6

5

Excitation

KLN

3 B

o

2

b|G 0 •u Itj 2 K/3 1 0

2 1

0

2.0

2.5

3.0

3.5

4.0

Electron Energy (keV)

FIG. 7. Partial differential cross sections at 0i4»“ Or f°r An “ 1dielectronic recombination on Ar14*. The points are the measured values and the solid lines are the theoretical calcula tions folded into the experimental resolution function.

transitions for the K L N resonance, which indicates that cascade transitions are quite weak. Although contribu tions from resonances higher than K L N to K y and higher K are observed, they need not necessarily be due to cascades but rather due to K 5 and higher K transitions that we could not resolve from K y . We also note that for the K L M resonance, the centroid of the K B group is shifted by about 10 eV to the left of that of the K a group, indicating that Li-like doubly excited stales giving rise to K B x rays are in general populated at lower E , than those giving rise to K a . This shift is also predicted by the theory. An =2 DR results, both experimental and theoretical, are shown in Fig. 6 riding on the direct excitation continua. Theoretical rates were calculated in the same manner as for An —1 but for the states with configurations of the form lr]/3/'and lr3/4f'. The dom inant channel for the decay of these states is by Auger emission leading to 1x2/ states, which decay radiatively to lx2 XS . The differential x-ray cross sections were calculat ed for 0Ub=O*. The experimental K M M and K M N areas were roughly estimated, and their ratios to theory were 0.98 and 0.93, respectively. An interesting feature ob served is how pronounced the An =2 DR K M M and K M N resonances are compared to the charge-state exper iment results. This occurs because these resonant states decay dominantly through L M M and L M N Auger decay followed by K a production, and thus do not achieve recombination. Now we discuss the contribution of direct excitation, which leads to the continua. Zhang and Sampson [27] have performed a distorted-wave calculation of total and partial (different M j ) EIE cross sections (crEIE) at impact energies of 3.16, 3.2, 3.4, and 3.7 keV for the following cases: lx2 lS0 -»lx2x)S 1,lx2p1 P1, and 1x2/7 !Po.i.j> whose thresholds are at impact energies of 3.107, 3.143, 3.125. 3.126, and 3.129 keV, respectively. Differential cross sections were obtained from these calculations us ing dam- 3 suggesting that these states must have been derived from multiply-excited projectile states through multiple autoionization processes. Theoretical ivestigations of multiply excited-states are essentially absent except for the calculations of radiative and non-radiative decay rates in triply-excited (31,31' , n l" ) N4+ by Vaeck and Hansen [32].

In order to account for the phenomenological cross sections a reasonable 66


stabilization scheme is needed. In the absence of even average aoutoionization and radiative rates and branching ratios for the enormous number of multiple-excited congfigurations of the projectile that are predicted by the MCBM, we propose a stabilization scheme that is based on the cumulative knowledge avilable from studies of doubly-excited states and the few experimental observations concerning multiplyexcited states. Before outlining the stabilization scheme, we first consider the energy levels of multiply-excited states. We begin with the doubly-excited configurations for which good knowledge of the evolution with q of the energy positions of the (4,4) and (5,5) manifolds relative to their nearest continuum levels turned out to be essential.

We have

used the General Purpose Relativistic Atomic Structure Program (GRASP2) [33] to calculate these energy levels for all incident projectile charge states.

In Fig.

7(a,b,c) we show three examples which illustrate this evolution. We observe that all states belonging to the (4,4) manifold in Ar6+

(core charge q = 8 ) are energetically

allowed to autoionize to the (3. oc) continuum limits, and the same is correct for a large fraction of such states in Ar12+

(core charge q = 14). However, none of

these states are allowed to autoionize to the (3, oo) continuum limits in A r14+" (core charge q = 16) and therfore the only available limits are (2,oo). A similar trend is observed for the (5,5) manifolds relative to the (4 ,oc) limits. This evolution has serious consequences for the stabilization of electrons on the projectile when autoionizing cascades are considered. The behavior is strongly modified for triplyexcited states as shown in Fig. 7(d,e,f). For example, for the same core charge q = 14 all the (4,4,4) in Ar11+

states are now energetically allowed to autoionize

to the (3,4, oc) limits. Similar evolution is observed for the core charge q = 16 where a large number of the (4,4,4) and (3,4,4) states can autoionize to the (3,4, oc) and (3,3, oc) continuum limits. We have investigated this evolution for configurations 67


J

3 .4 ,«

4 ,4 ,4

3,4,oo 3 .4 .4

F ig u re 7.

Energies of doubly and triply-excited ionic states relative to the

lowest continuum limit shown in each example. The horizontal borders of each box represent the lowest and highest energy levels in the corresponding manifold. The evolution with q from energetically allowed to not allowed autoionization of (4,4) states to the (3,oo) continuum limits (a,b,c) and the effect of introducing an additional electron (d,e,f) illustrate the need for proper knowledge of the characteristics of the multiply-excited states.

68


representing electrons in the same and in different shells of doubly, triply, quadruply, and qunituply-excited states which were frequently predicted by the MCBM. With the proper knowledge of the relevant energy levels we propose the following scheme:

(i) The multiply-excited states will dominantly stabilize through multiple Auger processes. Radiative stabilization is allowed if appropriate conditions are realized and will be discussed later.

(ii) Minimum electron rearrangement is dominant in the sense that only rwoelectron Auger transitions are allowed.

(iii) An Auger transition will proceed to the nearest energetically allowed continuum limit with unit probability; transitions to other continuum limits are assumed to be negligible, as is possible population of nonautoionizing multiply-excited states.

(vi) When many Auger transitions are possible, they proceed according to the following rules, (a) Transitions involving electrons in the same shell proceed first since the generally have higher Auger rates than electrons in different shells, (b) If two or more transitions involving electrons in the same shell are possible the one involving the more tightly bound electrons will proceed first (e.g. (4,4) will autoionize before (5,5)). (c) If two or more transitions involving electrons in different shells are possible the two electrons that spend more time in the vicinity of each other will autoionize first, otherwise the minimum ejected electron energy criterion is imposed and the two electrons giving rise to the lowest energy continuum electron will aotuionize first.

(v) In determining the energy levels and nearest continuum limits associated with a particular Auger transition in a certain configuration, only the electrons participating in the transition in addition to those in equal or lower lying n ’s are considered. Higher lying electrons are neglected. For example, in the configuration 69


(3.4,4,5,6) the two electrons in (4,4) will undergo the first transition and the relevant energy levels and continuum limits are those of (3,4,4), (3 ,3 ,oc), and (2,3,oc). (vi) Following each Auger transition a new configuration is realized and all the aforementioned rules are applied once more. If a transition results in the filling of a vacancy in the lowest empty (partially empty) shell, the new energy levels and continuum limits will be those of the initial core charge reduced by one unit. (vii) If the cascading process results in a final highly asymmetric doubly excited state such as (3, n2 > 6 ) or (4, ri2 > 8 ), for which the inner electron may stabilize via one or two photon emission processes, radiative stabilization of both electrons is assumed to take place. This situation is encountered when a highly excited electron is denied the opportunity to participate in an Auger process according to the previous rules. At the collision velocity of 0.632 a.u. this electron most likely posseses high angular momentum which reduces the Auger rate in favor of radiative stabilization [34]. Before discussing the results of the phenomenological cross sections some explanatory remarks are made. As mentioned ealier, the collision velocity of 0.632 a.u. results in a most likely population of high singular momentum states. We have also shown that the evolution of the energy levels relative to the continuum limits proceeds such that in some cases only a part of the states belonging to a certain manifold may autoionize with the ejection of low energy electrons. These states are usually the higher states in that manifold which correspond to moderately to high angular momentum states. In such cases, careful assessment of the possibility of an Auger transition with the ejection of a low energy electron is essential and has been allowed when reasonably probable. Otherwise the transition is taken to proceed to the next group of continuum limits. We show in Fig. 8 three illustrations of stabilization cascades for a core charge q = 16. We have applied the scheme 70

R ep ro d u ced with p erm ission of th e copyright ow ner. Further reproduction prohibited w ithout p erm ission .

Ar

11+

k= 1

Ar

10 +

k =2

Ar

12+

3— /iy

k=2

F ig u re 8 . Three examples of the stabilization of multiply-excited ionic states for the core charge q = 16. The cascades proceed from left to right, (a) The stabilization of one electron through multiple Auger transitions starting with a quintuply-excited Ar11+. (b) The stabilization of two electrons through multiple Auger transitions starting with sixtuply-excited Ar10+. (c) The stabilization of two electrons through mutiple Auger transitions and radiative stabilization of the final doubly-excited state starting with quadruply-excited Ar12+.

71


10

100

-i= 1

.i=5

8 6 4

40

2

0 4 6 8 1012141618 40 CNJ

4 6 8 1012141618

8

50

-i=2

i=6

6

30

20

CO

2

10

0 4 6 8 1012141618

4 6 81012141618

4 3

cr cr

i=7

2

1 0 4 6 8 1012141618

4 6 81012141618

1= 410

4 6 8 1012141618

4 6 8 1012141618

Projectile Charge State (q)

F ig u re 9.

Experimental (o) and MCBM-Stabilization (■•) phenomenological

cross sections ^ I?_i •

72


15

20

i=2

i=5

15

10

10

5

0

5

C

4 6 8 10 12 14 16 18

4 6 8 10 1 2 1 4 1 6 1 8

20

20 15

i=3

i=6

15

CM

6I O

•

0■ -to •

i t I

k«l

10 •

E le c tro sta tic D eflector

0•

(a)

# I p -1 0

o

10

-to

Z (m m )

Beam (b ) F IG . I. S c h e m a tic o f (a) th e e x p e rim e n ta l s e tu p , an d (b ) the c o o rd in a te system used in th e an a ly sis.

i f .

1 o

o

2 -D P ro jectile D e te c to r (B ackgam m on A node)

o

- to

2 -D Recoil D etector (Resistive Anode)

I

-w *0 •

F ie ld -F re e Drift Region —

Beam

“Ji®

k-2 i-5

\

o

to

Z (m m )

F IG . 2. T w o -d im en sio n al recoil p osition sp ectra for the d iffe re n t c o m b in atio n s o f p ro je ctile c h a rg e c h a n g e k and recoil c h a rg e s la te t. an d th e ir p ro je ctio n s o n to th e Z axis. T h e cen tro id s o f th e p ro je ctio n s a re in d ic a te d b y th e v ertical bars.

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volume

69, N umber 17

PHYSICAL

REVI EW

d iffe re n t e x tra c tio n fields (b e tw e en 5 a n d 15 V /c m ) th a t re s u lte d in d ifferen t tim e s o f flig h t an d d iffe re n t Z p o si tio n s for e a c h recoil c h a rg e s ta te . W ith th e lo n g itu d in a l m o m e n tu m tra n s fe r b e in g in d e p e n d e n t o f th e e x tra c tio n field. Z a co u ld be d e te rm in e d u sin g th e sim p le k in e m a tic e q u a tio n Z , " Z o + i',i(i .

(3 )

w h e re r , i a n d I, a re (he lo n g itu d in a l v elo city an d tim e o f flig h t o f th e i-lim e s io n ized re c o il, re sp ec tiv e ly . F o r th is p u rp o s e w e ch o se recoil c h a rg e s ta t e < “ 2 a n d p e rfo rm e d a lin e a r fit a c c o rd in g to Eq. (3 ). T h e u n c e r ta in ty in Z o is d o m in a te d by th e u n c e rta in ty in a lig n m e n t a n d th e u n c e r ta in ty in th e in te rc e p t o f th e lin e a r fit. A ll o th e r u n c e r ta in tie s w e re fo u n d to be neg lig ib le. T h e c e n tro id s o f th e Z p o sitio n s a re in d ic a te d by th e v e rtic a l b a rs in Fig. 2. W e o b se rv e t h a t th e se c e n tro id s h a v e n e g ativ e v a lu e s c o n siste n t w ith w h a t one w ould e x p e ct in e x o e rg ic co llisio n s w h ere th e reco ils a r e th ro w n b a c k w a rd s . H a v in g o b ta in e d th e n e ce ssa ry positio n a n d tim e in fo rm a tio n , th e lo n g itu d in a l m o m e n tu m tra n s fe r w as o b ta in e d from

(4)

P ,tm M iZ J t, .

T h e c o rre sp o n d in g Qg v alu es w e re th e n o b ta in e d u sin g Eq. (2 ) a n d a re listed in T a b le I. O n th e a v e ra g e th e Q v a lu e was a b o u t 25 eV fo r e a c h c a p tu r e d e le c tro n . W h ile re c o il c h a rg e s ta te s h ig h e r th a n i **5 h a v e b een o b se rv e d , th e y had s u b sta n tia lly lo w er s ta tis tic s to be c o n sid e re d . N e g le c tin g Q' in d u c e s sm all s h ifts to w a rd s s m a lle r Q v a lu e s re m in is c e n t o f th e k in e m a tic s h ift o fte n e n c o u n te re d in e n e rg y g a in sp e c tro s c o p y [1 5 ). T h e re a r e tw o w a y s to o b ta in an d th e re fo re th e m a g n itu d e s o f th e s h ifts . O n e w ay is to re c o n s tru c t th e recoil m o m e n tu m v e c to r fro m th e p o sitio n a n d tim e in fo r m a tio n , a te c h n iq u e u sed by F ro h n e e l at. [22] in th e ir reco il m o m e n tu m sp ec tro sc o p y s tu d ie s a t h ig h c o llision e n e rg ie s, a n d h e n c e o b ta in P A. A n o th e r, w h ic h we e m p lo y e d , is to use th e p ro je c tile a n g u la r d is trib u tio n s to d e te rm in e th e a v e r a g e s c a tte rin g a n g le s ( 0 ) . a n d by c o n se rv a tio n o f th e tra n s v e r s e c o m p o n e n t o f m o m e n tu m /» 1 * = F O0 .

(5 )

w h e re Pg is th e in c id e n t p ro je c tile m o m e n tu m . T h e a v er-

T A B L E I. E x p e rim en tal Qo. 8, Q '. Q. a n d p red icted M C B M _______________ Q values.

k

i

Co (eV )

1

1 2 3 4 5

28.7 5 t.5 76.5 106.4 136.2

t 1 1 2

9

C

Q‘

(m ru d )

(eV )

(eV )

0.96 i.JO 2.41 4.11

0.1

6.00

3.8

0.3

0.6 1.8

28.8 51 .8 77.1 108.2 140.0

± 4 .3 ± 6 .0 ± 7 .4 ± 8 .6 ± 9 .6

M CBM (eV ) 25.3 51.5 78.4 107.3 133.7

‘Q u o te d e rro rs a re I s ta n d a rd d ev iatio n .

LETTERS

26 O ctober 1992

jg e s c a tte rin g a ngles w ere o b ta in e d fro m th e a n g u la r d is trib u tio n s a n d th e shifts Q' w e re th e n d e te rm in e d . B oth 8 an d Q' a r e also listed in T a b le I. It is obvio u s (h a t the sh ifts a r e r a th e r sm all a n d n e g le c tin g th e m does not lead to s e rio u s de v ia tio n s from th e tr u e v a lu e s. It sh o u ld be n o te d t h a t w hile this is the first tim e recoil m o m e n tu m s p ec tro sc o p y is used to o b ta in Q v a lu e s for c a p tu re , recoil ion e n e rg y a n aly sis has been u s e d in th e p a st to d e te rm in e th e in e la stic e n e rg y loss o f in n e r-s h e ll e x c ita tio n processes for s m a ll im p a c t p a ra m e te r c o llisio n s w ith s in g ly c h arg ed p ro je c tile s [24]. In th e a b se n c e of a rigorous q u a n tu m m e c h a n ic a l t r e a t m e n t o f m u ltip le-ele c tro n c a p tu r e collisio n s, th e classical o v e rb a rrie r m o d el (25) was e x te n d e d by B a ra n y e l at. [26] a n d N ie h a u s (271 to in c lu d e s u c h collisions in an e ffort to g a in a b e tte r u n d e rs ta n d in g o f th e p h y sics in volved. T h e differen t versions o f th e m o le c u la r classical o v e rb a rrie r m odel (M C B M ) h a v e re a so n a b ly acc o u n te d for e x p e rim e n ta l m e a su re m e n ts o f c ro s s sec tio n s, peak w id th s a n d p ositions o f e n e rg y g a in s p e c tra , a n d a n g u la r d is trib u tio n s . W e have used th e m o d e l p ro p o se d by N ie h a u s to c a lc u la te Q values fo r th e d iffe re n t ( k . i ) c o m b in a tio n s th a t we m easured. In th e se c a lc u la tio n s w e as su m ed th e n u m b e r of e le c tro n s th a t b e c a m e m o le c u la r to be e q u a l to th e recoil c h arg e s ta t e i. T h e p re d ic te d values to g e th e r w ith th e shift c o rre c te d Q v a lu e s a re also listed in T a b le I. T h e a g re em en t b e tw e e n th e m e a s u re d Q values a n d th e predictions o f th e M C B M is su rp risin g ly very g o o d c o n sid e rin g th e c o m p le x ity o f th e p h y sics in volved a n d th e sim plicity o f th e M C B M a ssu m p tio n s. A lth o u g h th e ta rg e t th e rm a l s p re a d lim ited th e re so lu tion s u c h th a t a single Z -p o s itio n s p e c tru m w ith well resolved p e ak s could not be o b ta in e d , su ch s tr u c tu r e has p rim a rily been seen p reviously o n ly in th e s in g le -c a p tu re c h a n n e l. F o r m u ltip le c a p tu re , (h e h ig h d e n s ity o f final sta le s a n d th e loss o f p ro je c tile e n e rg y re so lu tio n d u e to th e e m issio n in flight o f a u to io n iz a tio n e le c tro n s has gen e ra lly re s u lte d in th e o b s e rv a tio n o f s tru c tu re le s s peaks [1 4 ,2 8 -3 0 1 . F o r such m u liip le - c a p tu re pro c e sse s, w e thus see no a d v a n ta g e o f c o n v e n tio n a l e n e r g y g ain sp ec tro sc o py over th e p re sen t te c h n iq u e , ev en fo r low p ro je c tile en erg ies. In d e e d , o u r m eth o d h a s th e a d v a n ta g e th a t the final c h a n n e ls a re easily id e n tifie d th ro u g h c o in c id en t c h a r g e - s ta te d e te rm in a tio n , a llo w in g u s to u n a m b ig u o u sly d e te rm in e th e n u m b e r o f e le c tr o n s c a p tu re d in itia lly , as given b y th e recoil c h a rg e s ta t e , a n d th e n u m b e r re m a in in g on th e p ro je c tile a fte r a u to io n iz a tio n , as given b y the p ro je c tile c h a r g e state. H o w e v e r, we a r e in th e process o f b u ild in g a c o ld gas je t to re d u c e th e ta rg e t lo n g itu d in a l m o m e n tu m s p re a d to a b o u t I a .u . for A r g a s. T h is c o rre sp o n d s to a re solution in Q o f a b o u t I a .u . (2 7 eV ) for a I-a .u . b e a m velocity and to 0 .2 a .u . (5 .5 e V ) for a 0.2a .u . b e a m velocity. W hile s u c h a re s o lu tio n co u ld be m u lc h e d b y p ro jectile e n e rg y g a in s p e c tro s c o p y a t rela tively low b e a m velocities, it is fa r fro m b e in g m a tc h ed fo r v e lo c itie s above 0.5 a.u. In c o n c lu sio n , we have u s e d reco il lo n g itu d in a l m om en2493

90


voLUMt 69, Number

i7

PHYSICAL

REVIEW

tu rn sp ec tro sc o p y to p ro v id e in fo rm a tio n on Q valu es in tw o -b o d y re a c tio n s. W e h ave a p p lie d it to a SO-keV A r ' 1* on A r collision s y ste m , a n d o b ta in e d Q v a lu e s fo r u p to q u in tu p le e le c tro n c a p tu r e co llisio n s. T h e m e a su re d v alu es c o m p a re w ell w ith th e p re d ic tio n s o f th e M C B M . W ith a cold ta rg e t gas j e t th e te c h n iq u e o ffers a d v a n ta g e s o v e r p ro je c tile tra n s la tio n a l e n erg y s p e c tro s c o p y a n d c an be e x te n d e d to in clu d e h ig h e r co llisio n v e lo c itie s a n d o th e r p ro je c tile and ta rg e t sp ecies. W e th a n k J. G ie se fo r e n lig h te n in g d iscu ssio n s. T h is w o rk w as s u p p o rte d b y th e D ivision o f C h e m ic a l S c i en ces. O ffice of B asic E n e rg y S c ie n c es, O ffice o f B asic E n e rg y R ese arch . U .S. D e p a r tm e n t o f E nergy.

'^ ’P resen t address: P h y sics D ivision, A rg o n n e N a tio n a l L a b o ra to ry . A rg o n n e, IL 60439-4843. I l l R . M a n n . C. L . C ocke. A . S. S c h la c h lc r. M. P rio r, a n d R. M a rru s . Phys. Rev. L e tt. 49. 1329 (1 9 8 2 ). 121 S . O h ta n i. Y . K aneko. M . K im u ra . N . K o b u y u sh i. T. Iw ai. A . M alsum oto. K. O k u n o . S . T a k a g i. H . T a w a r a . a n d S. T su ru b u ch i. J. P hys. B IS. L33S (1 9 8 2 ). 1)1 C . S ch m e issn e r. C. L. C o ck e. R. M a n n , a n d W . M ey e rh o f. Phys. R ev. A 3 0 . 1661 (1 9 8 4 ). [4] B. A. H u b er. H -J . K a h lc ri. a n d K. W ie sc m a n n . J. P h y s. B 17. 2 8 8 3 (1 9 8 4 ). 15) E. H . N ielsen. L. H . A n d e rs e n . A. B a ra n y . H . C e d e rq u is t. J . H ein e m eie r. P. H v elp lu n d . H. K nud scn . K. B. M acA d a m . and J. S o re n se n . J . Phys. B 18. 1789 (1 9 8 5 ). I6l H . C e d e rq u is t. L H . A n d e rs e n . A. B aran y . P . H v elp lu n d . H . K nudsen. E . H. N ielse n . J. 0 . K. P ed ersen , a n d J . S o re n s e n . J. P h y s. B 18. 3951 (1 9 8 5 ). 171 J . P. G iese. C . L C o c k e . W . W a g g o n er. L N . T u n n c li. a n d S . L. V arghese. P h y s. Rev. A 34. 377 0 (1 9 8 6 ). (81 R . W . M cC u llo u g h . S . M . W ilson, an d H . B. G ilb o d y . J . P hys. B 20. 2031 (1 9 8 7 ). 19) E. Y. K a m b e r. C. L. C o ck e . J. P. G iese. J. O . K. P e d e r sen . a n d W. W a g g o n e r. P hys. Rev. A 3 6 . 5 5 7 5 ( I 9 8 7 L (101 S . M . W ilson. R. W . M c C u llo u g h , a n d H. B. G ilb o d y . J . P hys. B 21. 1027 (1 9 8 8 ). (I ll E. Y. K a m b e r. N ucl. In s tru m . M eth o d s P hys. R es.. S ect. B 4 0 /4 1 .1 3 (1 9 8 9 ). [12] H . C e d e rq u is t. L. L ilje b y . C . B ie d erm an n . J. C . L evin. H .

(1)1

114] [151 [16! (17)

[181

LETTERS

26 O ctober 1992

R o lh a r d . K. O. G roeneveld. C. R . V ane, a n d I. A. S ellin. P h y s. Rev. A 3 9 .4 3 0 8 (1989). C . B ied erm an n . H C ed e rq u ist. L. R A n d e rse n . J. C. L ev in . R . T. S h o rt. S. B. E lston. J. P G ib b o n s. H. A n d e r s o n . L. Liljeby. and I. A. S ellin . P hys. R ev. A 41. 5889 (1 9 9 0 ). P . R o n cm . M N G ab o riau d . an d M . B a ra t. E urophys. L e tt. 16. 551 (19 9 1 ). C . L. C ocke and R. E. O lson. P hys. R ep. 2 0 5 . 153 (1 9 9 1 ). H . L ebius and B. A. H u b er. Z. P hys. D 23. 61 (1 9 9 2 ). R . S e h u e h . H. Schone. P. D. M ille r. H. F. K rau se. P. F. D in n e r . S . D atz. an d R. E. O lso n . Phys. R ev. L e tt. 60. 9 2 5 (1 9 8 8 ). R . D orn er. J U llrich. 0 . J a g u tz k i. S . L e n cin as. A. G e n s m a n le l. and H. S ch m id l-B d ck in g , in E lectro n ic a n d A t o m i c C o llisio n i. ed ited by W . R . M acG illiv ray . I. E. M c C a r th y , and M . C . S la n d a g e (A d a m H ilg er. B ristol.

1 9 9 2 ). p. 351. [191 W . W u ct al. (to be p u blished). [201 R . A li c t al. (to be published). [211 H . C ed e rq u ist. H. A ndersson. E. Beebe. C . B ied erm an n . L . B rostrom . A. E ngstrom . H . G a o . R. H u tto n . J. C. L ev in . L. Liljeby. M P ajek. T. Q u in te ro s. N S elb e rg . and P. S ig ray . in E lectro n ic a n d A to m ic C o llisio n s (R e f 1181). p. 391. [22] V. F ro h n e el al. (to be p u b lish e d ). [231 M a r tin P. S tockli. R. M. A ll. C . L C o ck e. M . L. A. R ap h a c lia n . P. R ic h ard , and T. N . T ip p in g . R ev. Sci. In s tru m . 6 3 . 2822 (1992). [241 G . H. M o rg an and E. E v e rh a rt. P hys. Rev. 128. 667 (1 9 6 2 ) : for a review, see 8. F a s tr u p . in M e th o d s o f E s p e r im e tita l Physics: A to m ic P h y sics, e d ite d by P R ic h a rd (A cad em ic. New Y o rk . 1 9 8 0 ). C h a p . 4. |2 5 | H . R y u fu k u , K. S asak i, and T . W a ta n a b e . Phys. Rev. A 2 1 . 745 (19 8 0 ). [26] A . B aran y . G. A stn er. H. C e d e rq u is t. H . D a n a re d . S. H u ld l. P. H velplund. A. J o h n so n . H . K n u d se n . L. L iljeb y . a n d K .-G . R eiu felt. N ucl. In s tru m . M eth o d s Phys. R es.. S e c t. B 9 . 397 (1 9 8 5 ). (271 A . N ie h a u s. J. Phys. B 19. 2925 (19 8 6 ). (28) P . H v elp lu n d . A. B arany. H. C e d e rq u is t. a n d J . O K. P e d e rs o n . J. Phys. B 20. 2515 ( 1 9 8 7 ). [291 H . C ed e rq u ist. H. A ndersson. G . A stn e r. P . H v elp lu n d . a n d J. O . K. Pedersen. P hys. R ev. L e tt. 6 2 . 1465 (1 9 8 9 ). (301 H . A n d erso n . H. C ed e rq u ist.. G . A s tn e r. P. H v elp lu n d . a n d J. O . K. Pedersen. P hys. S c r. 4 2 . 150 (1 9 9 0 ).

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3.3 P a p e r V I “Angular Distributions in Charge-Transfer Collisions of Ar15+ with Ar” R. Ali, C.L. Cocke, M.L.A. Raphaelian, and M. Stockli Phys. Rev. A, to be submitted


A n g u la r D istrib u tio n s in C harge-T ransfer Collisions of 50 keV A rIS+ w ith Ar

R. Ali, C.L. Cocke, M.L.A. Raphaelian, and M. Stockli J.R. Macdonald Laboratory Department of Physics Kansas State University, Manhattan, Kansas 66506-2604

A BSTRA CT We report on angular distribution measurements in 50 keV Ar15+ on Ar charge-transfer collisions. Distributions corresponding to the production of up to quintuply-charged recoils were obtained, and compared to the predictions of the molecular classical overbarrier model (MCBM) [1,2] under different assumptions. We found no evidence that the contributions of the tightly bound target electrons are overestimated by the MCBM, in contrast with the observations of Danared et al. [3] and Guillemot et al. [4], Previously reported [5] Q -values for the same processes were further compared to the model predictions under similar assumptions and the trends were found to be consistent with those of the angular distributions. PACS numbers: 34.70.+e, 34.50.Fa

93


I. IN T R O D U C T IO N Slow collisions of highly charged ions with multi-electron target atoms (u < la.u.) tire dominated by charge-transfer processes. Many electrons become active if highly charged ions are involved, and the transfer of even eight electrons is possible. Several experimental investigations of low energy collisions involving more them two electrons were reported. A number of cross sections were measured for projectile charge-change and phenomenological (final recoil and projectile charge states) processes [3,6-11], while some measurements involved the determination of recoil charge state fractions [12,13]. Few energy gain (Q-value) [5,14-17], Augerelectron [18-20], and photon [21-24] spectroscopic studies of such processes were carried out. In addition, some differential cross sections and angular dependences were obtained for large [3,4,25] and small [26,27] impact parameter collisions. Quantum mechanical treatment of such collisions is practically impossible due to the large number of channels involved. Instead, classical models have been widely used in an effort to reach some understanding of the physics of these processes. The classical overbarrier model, used by Ryufuku et al. [28] to account for electron capture cross sections in collisions of bare ions with one electron targets, was extended by Barany et al. [29] and Niehaus [1,2] to include multi-electron transfer processes. Both models fairly accounted for many of the experimental cross sections, energy gain spectra, and angular distributions. The molecular classical overbarrier model (MCBM) by Niehaus is more sophisticated and allows for more predictions to be made, one of which is target excitation. The MCBM has been subjected to a wide variety of tests and its predictive power regarding many of the general experimental features has been verified. It was however criticised [3,4] for overestimating the contributions of the tightly bound target electrons, and therefore target excitation, based on differential cross section measurements. On the other hand, Guillemot et 94


al. [4] found that taking the tightly bound electrons into account results in a better

agreement between the predicted and experimental average ^-values, in contrast with the earlier conclusion. The previous differential cross section measurements involved projectile charge states q < 11. Recently, we have carried out extensive cross section measurements [30] in Ar?+ on Ar collisions over a wide range of projectile charge states (5 < q < 17). In these measurements, all final recoil and projectile charge states were detected, and therefore proper and unambiguous comparison with the MCBM predictions was possible.

The measurements were shown to support the predictions on

target excitation.

In this paper, we present angular distribution and Q-value

measurements, for the production of up to quintuply-charged recoils, in the collision system 50 keV Ar15'*' on Ar.

The measurements are compared to the MCBM

predictions in different ways to further investigate the role of target excitation.

II. E X P E R IM E N T The experimental setup is shown in Fig.(la). The 50 keV Ar15+ beam was extracted from the Kansas State University cryogenic electron-beam ion source (KSU CRYEBIS) [31], and directed to the collision chamber.

Collimation was

provided by a four-jaw slit (0.8 mm width) and the collision chamber entrance aperture (0.4 mm diameter) which were separated by about 3.5 m, thus limiting the beam divergence to less than 0.01°. The collision chamber exit (3.2 mm diameter) allowed for scattering angles up to 33 mrad. The Ar gas target was furnished by a multi-channel array molecular jet, and the gas flow was adjusted to minimize double collisions

*ch that only about 2% of the projectiles that changed their

charge state and detected in coincidence with singly charged recoils were observed to have undergone double encounters. After the collisions, the recoil ions were extracted transverse to the beam direction by a uniform electric field (as 10V / c m ) 95


R ecoil D e te c to r

T im e -to -A n a lo g C onverter l Stop | |S ta r t

F ield —Free Drift R egion — P rojectile D etecto r

F ie ld -R e g io n - =_ ’J t Beam Gas Jet

t E le c tr o sta tic D eflector

(a) 2 -D P r o jectile D etecto r (B ack gam m on Anode) 2 - D R ecoil D e te c to r (R esistiv e A node)

Gas Jet

B eam

(b) F ig u re 1. Schematic of (a) the experimented setup, and (b) the coordinate system used in the analysis.

96


and detected by a two-dimensional positionsensitive channel-plate detector which had a 40 mm active diameter and a resolution of 0.11 mm. A coincident timeof-flight technique was used to determine the recoil charge states.

A parallel-

plate electrostatic deflector separated the final projectile charge states which were then detected by another two-dimensional position-sensitive channel-plate detector located 1.2 m downstream, which also had a 40 mm active diameter but a resolution of 0.5 mm. The beam divergence, the flight path to the detector, and the detector resolution gave an estimated angular resolution of about 1.1 mrad FWHM. Figure 1(b) shows the coordinate system adopted in the analysis.

The raw data are shown as a density plot in Fig. 2. Such a plot allows for an unambiguous high recoil charge state separation, taking advantage of the collision kinematics as manifested by the tilts in the density distributions. Each density distribution represents a combination (k , i ) where k is the projectile charge-change and i is the recoil charge state. By placing windows on each density distribution, the corresponding recoil and projectile position spectra could be obtained.

III. RESULTS AND D ISCUSSION The average Q-values were obtained using the recoil longitudinal momentum spectroscopy technique which was described in detail in a previous paper [5] and will not be discussed here. We will only use the reported values in the discussion. The projectile two-dimensional position spectra for severed (k, i) combinations are shown in Fig. 3 together with the correponding emgulax distributions. We notice that for the same projectile charge-cheinge k = 1, the angular distributions veiry considerably for the different recoil charge states. The peeik position emd width increase with increasing i. This is consistent with the observations made by Daneired et al. [3] and Guillemot et al. [4]. On the other hand, for the same i but different k the angular distributions are essentially the same as shown in Fig. 4, which is also 97


=3 40

4

5

R ecoil Tim e—o f - f lig h t (yus)

F ig u re 2 . A density plot of the projectile x-position vs recoil time-of-flight. Recoil charge state t = 1 is not shown to allow for proper display of the higher charge states.

98


I........t

k=l

10 ■

i=L

0-10

k=i i=2

10 0

m

-10

c k=l i=3

10

s i,

°-

H

** -1 0 ■ 10

k=l i=4-

b

T3 0 -10

k =2 i=5

1 o

o

10

-1 0

0

0

10

X (mm )

6

s 10 15 (mrad)

F ig u re 3. Two-dimensional projectile position spectra for different combinations of projectile charge-change(fc) and recoil charge state (*'), and their corresponding angular distributions.

99


i=2 k=l k=2 w c 3

i=3 k= 1 k=2

(0

u

-Q U «3

i=4 k=l k=2

b T3

0

5

9

10

15

(mrad)

F ig u re 4. Comparison of experimental angular distributions for the same recoil charge state (i) but different projectile charge-change (k).

100


consistent with the aforementioned observations. These observations suggest that the collision process may be best indicated by the recoil charge state i. and not the final projectile charge state q — k. which may be realized through post collisional relaxation. Before comparing the measurements to the MCBM predictions we first give a brief description of the model.

As the projectile A q+ approaches the target

B , the Coulomb barrier separating the target electrons from regions around A

start decreasing, and ceases to be effective for the target electrons in order of their increasing binding energies at well defined internuclear separations R \, where t is the index of the electron. These electrons are said to become molecular and are shared by the collision partners. On the way out, the barrier starts increasing and each molecular electron has a finite probability ( W t) to be captured by the projectile or recaptured by the target (1 - W t) at internuclear separations R°. These probabilities are determined from the phase space available to the electron on both A and B in the hydrogenic approximation. The MCBM predicts the recoil charge state i to result from the transfer of r

electrons to the projectile, in addition to the loss of

t — r

electrons through

autoionization if the target is left in a multiply excited state.

Such target

autoionization is expected if two or more tightly bound electrons are transferred to the projectile. Many reaction paths may lead to the same recoil charge state, each of which is described by a string ( j) whose elements Eire either 1 or 0 indicating capture by A or recapture by B . The positions of the elements indicate the indices of the electrons in order of increasing binding energies. For example, the string (j) = ( 01010100 ) describes a collision in which eight electrons became molecular out of which the second, the fourth, and the sixth most bound electrons were captured by A whereas the remaining electrons were recaptured by B . For this 101


particular example, the first and third electrons are expected to be left in excited states on the target, which may then autoionize and further increase the recoil charge state. An absolute cross section for the collision process described by the string ( j ) is obtained from products of probabilities Pf(j) = {"[, w [ ^ and geometrical cross sections A t = 7r[(f?J)2 —(f2}+1 )2] and is given by ( 6).

(3)

The differential cross section is then obtained by distributing the contributions A t P [ ^ evenly over the angular segment (&[J\

) corresponding to the impact

parameter range (6 = R t ,b = R t+1). The result is a step-like differential cross section. To compare with experimented differential cross sections the distributions 102


from all strings leading to the production of a particular recoil charge state should be added together and folded into the experimental resolution function. In Fig.

5, the experimental angular distributions are compared to the

MCBM predictions in two different ways. We first suppressed any possible target autoionization by taking the recoil charge state i to be equal to the number of transferred electrons r.

The corresponding distributions are shown in the first

column. We then allowed target autoionization to proceed whenever energetically allowed in the sense that states forbidden to autoionize tire assumed to be statistically negligible. The results so obtained tire shown in the second column. The MCBM step cross sections were folded with a Gaussian resolution function with FWHM= 1.23 mrad, which was chosen such that the experimental and MCBM distributions for the combination (k , i ) = (1,2) had similar total FWHM and was then kept fixed for all other combinations. The experimental and MCBM areas were normalized to each other.

We notice that the MCBM predictions (under

both assumptions) reproduce rather nicely the peak locations, widths, and relative heights considering the crudeness of the model and the complexity of the collision processes. The observed shoulder in the MCBM distributions for i = 4 may suggest that the model overestimates the contributions from the inner electrons. No such behavior however is observed for the higher recoil charge state i = 5 for which the inner electrons are predicted to play a more important role. In fact, the MCBM slightly underestimates their role. The average scattering angles were also obtained and are shown in Fig. 6 . The MCBM average scattering an gles seem to oscillate about the experimental ones, with no systematic evidence that it does over- or underestimate the contributions of the inner electrons. As for the two different assumptions used here, “target autoionization” and “transfer only”, we see no evidence that either one is preferred 103


T ra n s f e r o nly

k= 1 i= 1

T a rg e t a u to io n iz a tio n

k= 1 i=2

k= 1 i=3 CD

k= 1 i=4

k=2 i=5 o

5

10

15

6

F ig u re 5.

0

5

10

15

(mrad)

Comparison of experimental (o) and MCBM (solid line) angular

distributions under two different assumptions; suppressed target autoionization (transfer only) and allowed target autoionization.

104


7

Expt. Transfer only Target a u to io n iz a tio n L*

6

5 ctf

4

U

£

3

ICD

2 1 0

0

1

2

3

4

5

6

Recoil Charge State (i) F ig u re 6 . Comparison of experimental and MCBM average scattering angles under two different assumptions; suppressed target autoionization (transfer only) and allowed target autoionization.

105


over the other based on the angular distributions. However, we mentioned earlier that target autoionization was shown to play tin important role based on a large number of cross section measurements [30]. As for the difference in conclusions reached by us and by Danared et al. [3] and Guillemot et al. [4], it is possible that the answer lies in the projectile charge states used. To the best of our knowledge, the projectile charge state used here (q = 15) is the highest for which angular distributions have been reported in slow ion-atom collisions. It is also expected that the MCBM works better with increasing projectile charge states where the hydrogenic approximation used to calculate the capture probabilities becomes more justified. The previous assumptions can also be tested based on the average Qvalues. Table I compares the experimental and MCBM Q-values for the different combinations ( k , i ).

Within the experimental uncertainty, only the “transfer

only” assumption seem to overestimate the Q-values for the higher recoil charge states.

Although the outermost string gives as good agreement as the “target

autoionization” assumption, we believe that the collisions are best described by considering eight molecular electrons based on the cross section [30] and angular distribution measurements. Neglecting the outermost string, allowing for target autoionization gives better agreement than suppressing it. No contradiction similar to that reported by Guillemot et al. [4] regarding the role played by the inner electrons is found in this set of data when considering the angular distributions and the Q-values. Again, the reason might be the difference in the projectile charge states, where the highest charge state used by them was g = 11 .

IV. CON CLU SIO N We have presented angular distribution and Q-value measurements in chargetransfer collisions of 50 keV ArI5+ with Ar. The measurements have been 106


T able I.

Experimental and MCBM average Q-values (eV).

k

i

Expt.a

1

1

28.8±4.3

25.3

26.0

26.0

1

2

51.8±6.0

51.5

54.3

53.9

1

3

77.1±7.4

78.4

84.5

81.7

1

4

108.2±8.6

107.3

124.6

116.9

2

5

140.0±9.6

133.7

153.4

144.4

MCBM 6 MCBMC MCBM*

a Taken from ref. [5]. Quoted errors are one standard deviation. 6 Considering the outermost string only (ref. [5]).

c Considering eight molecular electrons and suppressing target autoionization (transfer only). d Considering eight molecular electrons and allowing target autoionization.

107


satisfactorily accounted for by the molecular classical overbarrier model.

No

evidence of systematic over- or underestimation of target excitation has been observed, in contrast with observations reported by other investigators [3,4]. Further investigations involving a wider range of projectile charge states, target species, and collision energies are needed to fairly judge the MCBM and determine its regions of applicability.

A C K N O W LED G EM EN T This work was supported by the Division of Chemical Sciences, Office of Basic Energy Sciences, Office of Basic Energy Research, U.S. Department of Energy.

108


References [1] A. Xiehaus, J. Phys. B 19, 2925 (1986). [2] A. Xiehaus, Xucl. Instrum. Methods Phys. Res. B31, 359 (1988). [3j H. Danared, H. Andersson, G. Astner, P. Defrance, and S. Rachafi. Physica Scripta 36, 756 (1987). [4] L. Guillemot, P. Roncin, M.N. Gaboriaud, H. Laurent, and M. Barat, J. Phys. B: At. Mol. Opt. Phys. 23,4293 (1990). [5] R. Ali, V. Frohne, C.L. Cocke, M. Stockli, S. Cheng, and M.L.A. Raphaelian, Phys. Rev. Lett. 69, 2491 (1992). [6 ] H. Klinger, A. Muller, and E. Salzbom, J. Phys. B: At. Mol. Phys. 8 , 230 (1975). [7] C.L. Cocke, R. DuBois, T.J. Gray, E. Justiniano, and C. Can, Phys. Rev. Lett. 46, 1671 (1981). [8 ] Edson Justiniano, C.L. Cocke, Tom J. Gray, R.D. DuBois, and C. Can, Phys. Rev. A 24, 2953(1981). [9] E. Justiniano, C.L. Cocke, T.J. Gray, R. DuBois, C. Can, W. Waggoner, R. Schuch, H. Schmidt-Bocking, and H. Ingwersen, Phys. Rev. A 29, 1088 (1984). [10] G. Astner, A. Barany, H. Cederquist, H. Danared, S. Huldt, P. Hvelplund, A. Johnson, H. Knudsen, L. Liljeby and K.-G. Rensfelt, J. Phys. B: At. Mol. Phys. 17, L877 (1984). [11] L. Liljeby, G. Astner, A. Barany, H. Cederquist,H. Danared, S. Huldt, P. Hvelplund, A. Johnson, H. Knudsen, and K.-G. Rensfelt, Physica Scripta 33, 310 (1986). [12] W. Groh, A. Muller, A.S. Schlachter, and E. Salzborn, J. Phys. B: At. Mol. Phys. 16, 1997 (1983). [13] J. Vancura, V. Marchetti, and V.O. Kostroun, Proc. 6th Int. Conf. on the 109


Physics o f Highly Charged Ions (Manhattan, KS, 1992) (New York: AIP) to

be published (1993). [14] P. Hvelplund, A. Barany, H. Cederquist, and J.O.K. Pedersen, J. Phys. B: At. Viol. Phys. 20, 2515 (1987). [15] H. Anderson, H. Cederquist, G. Astner, P. Hvelplund, and J.O.K. Pedersen, Physica Scripta 42, 150 ( 1990). [16] P. Roncin, M.N. Gaboriaud, VI. Barat, and H. Laurent, Europhys. Lett. 3, 53 (1987). [17] VI. Sakurai, H. Tawara, I. Yamada, M. Kimura, N. Nakamura, S. Ohtani, A. Danjo, M. Yoshino, and A. Matsumoto,

P to c.

6th Int. Conf. on the Physics of

Highly Charged Ions (Manhattan, K S, 1992) (New York: AIP) to be published

(1993). [18] P. Benoit-Cattin, A. Bordenave-Montesquieu, VI. Boudjema, A. Gleizes, S. Dousson, and D. Hitz, J. Phys. B: At. Mol. Opt. Phys. 21, 3387 (1988). [19] J. H. Posthumus

and R. Morgenstem, Phys. Rev. Lett. 6 8 , 1315 (1992).

[20] J. Vancura and V.O Kostroun, Proc. 6th Int. Conf. on the Physics of Highly Charged Ions ( Manhattan, KS, 1992) (New York: AIP) to be published (1993).

[21] S. Martin, A. Denis, J. Desesquelles, and Y. Ouerdane, Phys. Rev. A 42, 6564 (1990). [22] S. Martin, A. Denis, Y. Ouerdane, A. Salmoun, A. El Vlotassadeq, J. Desesquelles, M. Druetta, D. Church, and T. Lamy, Phys. Rev. Lett. 64, 2633 (1990). [23] S. Martin, Y. Ouerdane, A. Denis, and M. Carre, Z. Phys. D 21, s277 (1991). [24] S. Martin, A. Denis, A. Delon, J. Desesquelles, and Y. Ouerdane, Proc. 6th Int. Conf. on the Physics o f Highly Charged Ions (Manhattan, KS, 1992) (New

York: AIP) to be published (1993). 110


[25] H. Danared, H. Andersson, G. Astner, A. Barany. P. Defrance, £ind S. Rachafi. J. Phys. B: At. Mol. Phys. 20, L165 (1987). [26] H.

Schmidt-Bocking,M.H. Prior, R. Dorner, H.

Berg,J.O.K.Pedersen. C.L.

Cocke, M. Stockli, and A.S. Schlachter, Phys. Rev. A 37, 4640 (1988). [27] R. Hermann, M.H. Prior, R. Dorner, H. Schmidt-Bocking, C.M. Lyneis. and U. Wille, Phys. Rev. A 46, 5631 (1992). [28] H.

Ryufuku, K. Sasaki, and T. Watanabe, Phys.

Rev.A21, 745 (1980).

[29] A. Barany, G. Astner, H. Cederquist, S. Huldt, P. Hvelplund, A. Johnson, H. Knudsen, L. Liljeby, and K.-G. Rensfelt, Nucl. Instrum. Methods Phys. Res. B 9 , 397 (1985).

[30] R. Ali, C.L. Cocke, M.L.A. Raphaelian, and M. Stockli, Phys. Rev A, to be submitted. [31] Martin. P. Stockli, R.M. Ali, C.L. Cocke, M.L.A. Raphaelian, P. Richard, and T.N. Tipping, Rev. Sci. Instrum. 63, 2822 (1992).


3.4 P a p e r VTI “On the Radiative Stabilization in Slow Double-Electron Capture Collisions of Highly Charged Ions with Neutral Atoms” R. Ali, C.L. Cocke, M.L.A. Raphaelian, and M. Stockli J. Phys. B: At. Mol. Opt. Phys. 26, L177 (1993)

112


J . P h y s. B: Al. M ol. O p t. Phys. 26 (1993) L 177-L 1S 4. P rin te d in th e UK.

LETTER TO THE EDITOR

O n the radiative stabilization in slow double-electron capture collisions o f highly charged ions with neutral atoms R Ali, C L Cocke, M L A Raphaelian and M Stockli J R Macdonald Laboratory, Department of Physics, Kansas State University, Manhattan, KS 66506-2604, USA Received 24 November 1992 Abstract. We have measured the radiative stabilization probability in double-electron capture collisions for the systems 10 keV u~' Kr’* (ij “ 13-34) and Ar** (q = 6-17) on Ar, and examined its velocity dependence for Ar** on Ar. The measured probability exhibits pronounced structures as a function of q. A model incorporating the initial population, the collision velocity, and the core structure of the incoming projectile is proposed, and the gross features are fairly accounted for. The results support tne importance of populating asymmetric Rydberg states and reveal the crucial role played by the projectile core structure. The velocity dependence measurements favour a post-collision interaction mechanism for the population of the asymmetric Rydberg states, such as that proposed by Bachau et al. Electron capture processes are the dominant reaction channels in low energy highly charged ion-atom collisions. In particular, double-electron capture has been the subject of increasing interest in the past few years. It is established that such a process populates high lying projectile doubly excited states (n, n'). In the scope of the independent electron extended classical overbarrier models (ecbm) (Niehaus 1986, Barany et al 1985), symmetric or quasi-symmetric (n =* rt') population is predicted. Such states are widely assumed to be dominantly autoionizing, and little experimental evidence to the contrary has been reported for moderately charged projectiles. However, with the widespread use of the electron cyclotron resonance (e c r) ion sources and the electronbeam ion sources (ebis), a wealth of experimental data-with highly charged ( 7 . Luc-Koenig and Bauche (1990) reached similar conclusions using a configuration-average method. On the other hand, Chen and Lin (1993) studied the decay of the high lying doubly excited states of Ar16*. In their calculations, configuration interaction (ct) of the manifolds (n, n') within a certain series (n = constant, n') was considered while mixing with manifolds belonging to different series was ignored. They have shown that for a certain class of quasi-symmetric states the average fluorescence yields were quite large, and argued that the observed high P ,Jd may be an indication that such states are populated. This issue has also been considered by Vaeck et al (1992 and references therein) where the radiative stabilization of symmetric (4/, 4/') and asymmetric (31,14/') states in 0 6* has been examined. They argued against the pci effect and demonstrated that mixing between symmetric and asymmetric states is no guarantee for enhanced P r, d in the case of O6*. They attributed the radiative stabilization to the natural diversity of the decay rates of the symmetric and quasi-symmetric configurations. These calculations do not necessarily contradict the conclusions of Nikitin and Ostrovsky and Poirier since they only considered / ' s 6 . From the above discussion, it is obvious that, while our knowledge of double electron capture processes has expanded considerably, the experimental observations and the various theoretical investigations of both the formation and the subsequent decay of doubly excited states are yet to be reconciled. In this letter, we present experimental measurements of P rid for Kt’ * ( q = 13-34) and Ar"* (q = 6-17) capturing two electrons from Ar at a collision energy of 10 keV u _l ( u = 0.632 au). We also present the velocity dependence of Prad for the collision system Ar’* on Ar in the velocity range 0.1-0.75 au. To interpret our measurements, a model incorporating the initial population, the collision velocity, and the incoming projectile core structure is pro posed. The relevance of the pci effect to P „ d is also discussed. The experimental set-up has been described in detail elsewhere (Ali et al 1992). Briefly, the ion beams were extracted from the Kansas State University cryogenic ebis (Stockli et al 1992). The Ar gas target was furnished by a capillary array molecular jet. Following the collision, a parallel-plate electrostatic deflector separated the final projectile charge states which were then detected by a two-dimensional positionsensitive channel-plate detector. The recoil ions were extracted transverse to the beam direction by a uniform electricfield (=“ 10 V cm-1) and detected by another channel-plate detector. A coincident time-of-flight technique was used to determine the recoil charge states. The double capture channel was signalled by the detection of an Ar2* recoil ion, and Pnd was found as the ratio of the coincidence yield of projectiles which have kept both captured electrons to the yield of all projectiles in coincidence with Ar2* recoils.

114


L e t t e r to th e E d ito r

1179

In figure 1(a) and (b ) we show Prad as a function of the projectile charge state (q) for the collision systems Ki"" and Ar"" on Ar at u = 0.632 au. At a glance, one observes a general trend toward increasing Prad with increasing q, modulated by pronounced structures. This increase is expected since the radiative rates depend strongly on q (oc q* for An * 0 electric-dipolc transitions) while the autoionization rates are practically independent of q. In the Kr case, we observe a sharp increase in ^nd starting at q = 28. This is similar to the observation by Cederquist et al in the collision systems Xe9'" on Xe and He about q = 28, although structures could not easily be identified in their results. These structures, we believe, are strongly related to the core structures of the incoming projectiles and constitute a direct evidence that the initial population of the doubly excited states is only one of several aspects in realizing radiative stabilization. In order to explain the sharp increase in Prad for Xe’* on Xc and He and the unexpected similarities for both targets, Cederquist et al suggested an explanation based on the availability of 3d vacancies in the projectile electronic configuration. They argued that the collision velocities (0.1-0.2 au) allow a fair assumption that the angular momenta of the captured electrons are such that (I, l')> (3, 3). If the doubly excited states survive autoionization until one electron cascades down the yrast chain (the radiative decay chain down the energy levels characterized by I = n — 1 , see figure 2) to 4f, radiative stabilization will depend on the availability of 3d vacancies since An = 1, A/= -1 electric-dipole transitions (El) to 3d can compete with autoionization.

20 10 k e V /u Kr

10 k e V /u Ar o o Ar

o a Ar 15

10 5

o [ ---.-— --- --- --- 1 Z$

4

6

8

10 12 14 16 18

P ro je c tile C harge S ta te (q) F igure 1. T h e ra d ia tiv e sta b iliz a tio n p ro b a b ility P ,.4 as a fu n c tio n o f (he in c o m in g p ro je c tile c h a rg e s ta te q fo r th e c o llis io n sy stem s ( a ) K r** o n A r an d lb) A s ** on A r. T h e e r r o r b a rs c o m b in e s ta tistic a l a n d d o u b le -c o llis io n c o rre c tio n u n c erta in tie s. I o n iz a tio n L im it

Io n iz a tio n L im it

3d 3p,

t R y dberg \ E le c tro n Y r a s t- C h a in

F ig u re 2. S c h e m a tic d ia g ra m s illu s tra tin g th e y ra s t ch ain c a s c a d e o f the in n e r e le c tro n fo r th e in c o m in g p ro je c tile s (d ) K r’1* a n d ( 6 ) K r11* . T h e full circles re p re s e n t e le c tr o n s w hile th e o p e n circles re p re s e n t v a ca n c ie s. A c a s c a d e lead in g to in n e r e le c tro n e x c ita tio n by An = 0 is s h o w n in ( a ) w hile a c a s c a d e le a d in g to e x citatio n by An = I is s h o w n in (b ) .

115


LI 80

L e t t e r t o th e E d i to r

On the other hand, autoionization dominates if only An = 0, A/ = —1 El transitions are possible. Therefore, is expected to increase with increasing number of 3d vacancies. Although such assumptions may account for the sharp increase observed in the present measurements for the Kr case for q ^ 28, where 2p vacancies are available. is. We therefore propose s mode! thst extends

their assumptions to account for the detailed structure of the projectile core and incorporates the initial population and the collision velocity. The model assumptions are: (i) a fraction ( / ) of the double-electron capture flux feeds highly asymmetric Rydberg states (e.g. 61, 20/'). It is these states which are assumed to give rise to radiative stabilization. For simplicity, this fraction is assumed to have a constant value within each of the projectile charge state ranges that will be defined later, (ii) The double-electron capture populates high angular momentum states (/, /'). This is a fair assumption at the collision velocity of 0.632 au. These asymmetric Rydberg states with high angular momentum are metastable against autoionization. Indeed, such high Rydberg electron angular momentum states have been observed (Meyer et al 1988) and high /'-selectivity with increasing veiocityhas been confirmed, (iii) The inner electron undergoes El transitions favouring the highest energy gaps such that it ends up cascading down the yrast chain (An = 1, A/ = -1 El transitions). This yrast chain cascade proceeds until a singly excited core is formed (see figure 2) such that the inner electron is excited by An = 1 (e.g. l s :2sJ3d) or An =0 (e.g. Is 22s2p) above the ground state configuration, (iv) Meanwhile, the Rydberg electron also undergoes El transitions. However, these are slow transitions and it is assumed to remain a Rydberg electron by the time the inner electron realizes excitation by An =0 or 1. (v) In the L - S coupling scheme, the singly excited core is described by the terms 2S ~ ' L j . These terms are assumed to be populated statistically where the statistical weight of the ith term is given by W, = 27, + 1. (vi) Radiative stabilization is realized only for those terms that are not metastable against radiative decay, while the metastable ones lead to autoionization. The probability for realizing radiative stabilization is proportional to the fraction of states (F ei) that can decay via El transitions (2)

where W f' is the statistical weight of the term / which decays via El. We also assume that spin-changing El transitions lead to radiative stabilization in accordance with assumption (ii), since spin-orbit interaction is expected to cause substantial mixing of different spin states (e.g. singlet 'and triplet) of the same total angular momentum and parity. For the Kr1’* on Ar case, we recognize three regions. Region I (q = 13-18). This region is dominated by the cascade down the yrast chain, terminating with a 4f-*3d transition. Figure 3(a) shows that F E, varies between 0.9 and 1 .0 indicating that the projectile core structure plays little role in this range, i.e. the last transition proceeds promptly. In such a situation one might expect Pna to be determined by the competition between radiation and autoionization while the cascade down the yrast chain is proceeding. In the hydrogenic approximation, the radiative rate is proportional to q*, and thus for nearly constant autoionization rate, Prad should scale (for small P r3d) as q*. In figure 3(6) we show the product of F E, with q A, scaled to fit Prild (13). Qualitatively, the general trend seems to follow q‘ scaling.

116


L e t t e r to t h e E d ito r

■

L181

10 I

> 0.9

o.a

\o a i,

Kr

on Ar

• ;t

(b)-

y

E rp t. q -S c a lin g 1 20

18

12

P ro je c tile C harge S ta te (q)

Figure 3. (a) The model fraction FE, , and (6) the experimental Pttd and the q4-scaiing relative to P„d( 13) taking into account the fraction shown in (a) for Kr** (