Excitation of 2 1S, 2 1P and 3 1P, 3 3P levels of helium in He+ on He collisions at a few hundreds eV

319-333 AVRIL 319J. 41 1980,Physique (1980)
Classification
AbstractsPhysics
34.50
of 2 of in He+ on HeExcitation 2 1P and 3 3 3P levels helium 1S, 1P,
collisions at a few eVhundreds
G. G. J. Rahmat, Vassilev and Baudon
de 93430 FranceLaboratoire des Université Avenue J.-B.-Clément, Villetaneuse, Physique Lasers, Paris-Nord,
le 22 octobre révisé le 20 décembre le 20 décembre 1979, 1979, accepté 1979)(Reçu
On étudie l’excitation de l’hélium sur les niveaux 2 2 3 1P et 3 Résumé. - 1S, 1P, 3P, expérimentalement par
une moléculaire semi-d’ions He+ d’une centaine d’eV. des résultats utilise description impact L’interprétation
de la collision elle-même. On a une tech-du et un traitement mesuré, système He+2, semi-classique par diabatique
la section efficace différentielle l’excitation 2 à 100 et 150 eV de coïncidence 1P, (Lab),nique ion-photon, pour
et Les résultatsainsi la corrélation à 100 eV et des de diffusion de angles 10,25 12,5°. que angulaire ion-photon pour
sur le niveau 2 1S et montrent les voies 2 1P et 2 1Sobtenus des mesures effectuées précédemment que complètent
un secondaire de rotationnel. L’effet d’un tel mécanisme sur les sections efficacessont mécanisme couplées par type
2 différentielle et totale est examiné dans le cas où interviennent états de et un état Lasymétrie général 03A3g 03A0g.
= 3P-2 3S 3 889 a de mesurer lade coïncidence aux 3 (03BB Å) permis technique ion-photon, appliquée photons
l’excitation 3 3P à 120 et 150 eV La des de coïnci-section efficace différentielle pour (Lab). polarisation photons
audence à 150 eV et un de diffusion de montre l’état moléculaire final conduisant 13,5° (Lab) pour angle que
dans l’excitation 3 3P se manifestentniveau 3 3P a la 03A3. le nombre d’états symétrie impliqués (et qui Malgré grand
effets efficace on montre modèle extrêmement à 2 étatsdes Rosenthal dans la section par totale), qu’un simplifié
sont déterminées surtoutmodo la section efficace dont les différentielle, caractéristiques générales reproduit grosso
le mécanisme d’excitation. La même aux 3 1P-2 1Sprimaire technique expérimentale, appliquée photons par
= 5 016 a de mesurer la section efficace différentielle l’excitation 3 1P à 150 eV (03BB Å) permis pour (Lab). L’appli-
cation à l’excitation 3 1P du modèle à 3 états conduit à attribuer la 03A0 à l’état moléculaire ce symétrie final, que
de coïncidence faites à 150 eV et aux confirment les mesures de des de diffusion de polarisation photons angles 12,
et 13,5 15, 5° (Lab).
2014 at low Abstract. The excitations of Helium on 2 1P and 3 3 3P levels in He+ on He collisions 2 1S, 1P, energy
in frame of a semi-diabatic of are The data are the description He+2 , usinginvestigated experimentally. interpreted
2 1P differential cross sections at 100 and 150 eV and a semi-classical treatment of the collision. (Lab) angular
12.5 correlation functions at 100 eV and of 10.25 and havedeg. ion-photon (Lab) laboratory scattering angles
measured an coincidence These with measurements onbeen data, using ion-photon technique. together previous
1P 2 1S are rotational mechanism. The effects2 1S show that channels 2 and a excitation, coupled by secondary
and total cross sections of both excited channels are examined in the of such mechanisms on the differential général
case a 3 state model two states and one state. 3 3P differential cross sections have beenusing including 03A3g 03A0g
in 3 3P-2 3S measured at 120 and 150 eV coincidence the scattered ion and the (Lab) by detecting photon
= measurements at 150 13.5 show that the of the3 889 Coincidence (03BB Å). polarization eV, deg. (Lab) symmetry
3Pfinal molecular state to 3 3P is 03A3. In of the number of mechanisms involved in the 3 large secondary leading spite
effects in the total cross it is an twoexcitation Rosenthal shown that (which produce section), extremely simplified
calculation is able to the features of 3 3P differential cross which are determinedstate section, give général mainly
= the mechanism. A similar coincidence on the 3 1P-2 1S line 5 016 hastechnique (03BB Å) by primary operating
measure the 3 1P differential cross section at 150 eV A calculation of the 3 1P excitation been used to (Lab). using
a 3 state model is consistent with the measurements at 150 13.5 and 15.5 whicheV, 12, (Lab) polarization deg.
is 03A0.show that the of the final molecular state symmetry
- 1. Introduction. The theoretical treatment of the the lowest the of interactionsstates, by presence
excitation of one level of in He+ between different excited molecular states. TheseHelium, specific
He collisions at treaton low mechanisms make to relatively (few secondary energies impossible
hundreds is made even each atomic level in the foreV), generally difficult, except, presentseparately
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:01980004104031900320
is the lowest one which well 3 3P level is via a Z molecular state. In case, ’S) reached (2 separated spite
levels which a from the and for two state of the fact that mechanisms areupper many secondary
a in the whichapproximation [1, 2] gives very satisfactory agree- present, particular Eg Ilg coupling
ment with the results. In the same the 31P and at time,experimental populates level, couplings largeIg-Ig
of 2 3 S level the relative isolation the also distance which Rosenthal effects in the totalsimplifies produce
its since the excited level can be cross section the oscillations ofexperimental study [9, 10], Stueckelberg
characterized an loss the 3 3P differential cross section are rather wellunambiguously by energy analy-
as sis of the scattered ions As soon the second in a model the[3, 4]. reproduced very simple involving
excited level is the excitation(2 considered, only coupling.’S) primary Lg-Lg
mechanisms become much more and it iscomplex
- no more to this 2. 2 1 P and 2 18 excitations. 2.1 EXPERIMENTALpossible study process independently
- from the 2 3,1S excitation RESULTS. in coincidence the He+ excitations. The diffe- ionsBy detecting 2p
rential cross sections have been measured at 150 eV scattered in a direction and the UV given photons
the means of a scattered(in laboratory frame) by
- ion recoil atom coincidence Sterntechnique [5].
et al. to these results in a[6] attempted interpret
calculation where all excited states toleading 19
= He+ He* were introduced. As + (n 2) expected,
the calculated 2 3S differential cross section agrees
well with the while some experiment discrepancy
remains for the 2 ’S excitation. theParticularly,
value the 2 ’S cross sectioncalculated of average
to the 2 3S one is found smaller a factorrelatively by
of about 4 than the value. In order toexperimental
of the 2 1 S morea better and get understanding
= of the n 2 the excitations, productiongenerally
of has been also anHe(2 ’P) investigated using
coincidence method. Previous measure-ion-photon
correlation at anments of the ion-photon angular
of 150 eV have impact energy (Lab) [7] already
shown that the molecular state toleading
low thehas the At this rather energy H-symmetry.
i.e. the excitation primary mechanisms, couplings
the and channels and thebetween incoming lu Lg
excited molecular are radial states, essentially Lg-Lg
excluded fromall u-states can be then couplings;
excitation On another hand it is neces-the processes.
to introduce a rotational sub-sary coupling 03A3Lg-llg
toto the and sequent primary coupling, leading
It be thisvia a 77-statue. that may expected He(21 P)
will the mechanism secondary strongly modify Eg
and the 21 S
- finally population.amplitude 1. reduced differential cross section (DCS)Fig. a) Experimental
In the a 3 state model, for 2 Ip excitation at 100 as a function of the Lab-present paper, involving eV-Lab, PeX
a and of the reduced a at short intemuclear distance and scattering angle (upper scale) scattering angle :coupling Eg-lg
= 1 Because of instrumental the constraints, angularEcrn 8ern. is studied. This rotational coupling simple03A3g-llg is restricted to 8-17 The reduced DCS is related to(Lab). range deg. correct of the 2 ’Smodel allows the interpretation = usual the DCS a a. In the by : p 03B8cm. sin 8ern. present experiment
and 2 1 P excitation differential cross sections as well is in relative to ratio of the coincidencethe pex simply equal, value,
as correlation results. A similar mecha- number to the total number of counted the the angular photons along experi-
mental run. The error bars to the standard deviationnism has been Lefébure in order to correspond proposed by [8]
of the statistical noise. The two lines indicate wherebroken angles isthe excitations. This of view explain 3p point measurements are made Calculatedpolarization (see Fig. 3). b)
confirmed our measurements of the coïncidentby 2 1 P reduced DCS for excitation for the exit pn (i.e. channel),Ilg
31 P-21 S which a II in the 3 state without and with photon polarization assign ° model, (broken theline) (full line)
nuclear effect. Calculated reduced DCS theto the final molecular state. Thus it symmetry c) pLz in symmetry may
2 state Landau-Zener model. Rotational factors d) coupling fi.be that a 3 state model similarreasonably expected
The two branches from the threshold corres-starting angular (rth) for to the one would be convenient theprevious to factors inpond fi (upper branch) and f2 (lower branch) given 1 P to the 3 1 P3 excitation treatment. text. At the side of B 2 Contrarily these factors have right point signsopposite
measurements show that the and oscillates out of with to pn phase PLZ.level, polarization respect [7] 321
= 584 it is to measure :(2 ’ P-1 IS, À Á) possible
the when thecorrelation, (i) ion-photon angular
detector is moved in the collision photon plane,
2 1 P excitation cross the differential (ii) section,
the are detected towhen photons perpendicularly
the collision A of the plane. description apparatus
and correlation have beenangular typical diagrams
elsewhere 2 Ip excitation diffe-already given [7, 11].
rential cross sections measured at 100 and eV 150 eV
in are shown la and 2a. corre-(Lab) figures Angular
curves obtained lation at for 100 eV, scattering
of 10.25 and 12.5 are in 3.angles degrees given figure
These correlation measurements show thatclearly
= 1 sub-levels only m ± practically magnetic (refered
final direction of the to the molecular are axis) popu-
lated : this means that the of molecularthe symmetry
state which leads to the 21 P levels is H. In thefinally
differential cross besides oscillationssection, rapid
due to the nuclear of the onesymmetry Het system,
can a the observe modulation, very slight frequency
of which recalls a of oscillations.Stueckelberg type
marked oscillationsOn the other hand, Stueckelberg
in the 2 ’S differential cross sectionare observed
As noted above the of these(see Fig. 4a). frequency
oscillations is in with a good agreement semi-quantal
- 3. ’P-2 correlation for an Fig. Ion-photon (2 1 S) angular impact
of 100 eV-Lab. Error bars the standard deviationrepresent energy
The sine curves are least fits of theof the statistics. counting square
data. 10.25 The incident beam is ata) Scattering angle deg.-Lab.
180 and the final molecular 159.5 The azimutaldeg. axis R at deg.
= of the scattered ions is 90 Same as b) (a),angular spread ôçoi deg.
at 12.5 circles : incident beam atscattering angle deg.-Lab. Open
= 0 at 25 60 Full atdeg., Êf deg., ôçoi deg.
= 180 at 155 90 deg., Pf deg., ôçoi deg.
calculation into account states whiletaking only Lg
the calculated 2 ratio is about 4 times smaller’S/2 ’S
than the one.experimental
2.2 MODEL. - At these low MOLECULAR energies
where all u-states can be the( 100-150 eV) ignored,
= states to the n 2 excitations are fouronly leading
states and two states The (Fig. 5). energiesIg IIg E,
have been obtained the electronicby diagonalizing
hamiltonian in the first seven excited state sub-Eg
The two semi-diabatic states space [6, 13]. [8],IIg
represented by :
are correlated at infinite R to the 21 P 2 ’P and atomic
levels The two cores as well as the exter-respectively.
- Same as at an of 150 eV-Lab. nal molecular orbitals different 2. 1, Fig. figure energy having symmetries322
- 5. Potential curves for states into He+ + He*Fig. dissociating
= states states (n 2) : (full lines), (broken lines). Only2 Ig 2 ng
states 2 are considered in the 3 state calculation. Thelabelled 0, 1,
= between the n 2 levels have been energy splittings enlarged.
two The cor-by spin-orbitals [13]. configurations
to states and differ from each other(0) (2) responding
2 as electronic the momen-by spin-orbitals ; angular
as for 21S excitation at 150 eV-Lab.1, Fig. 4. - Same figure tum is a monoelectronic the rotationaloperator, Ly
reduced DCS obtained relative a) Pex, (in value)Experimental between states and vanishes. On thecoupling (0) (2) in coincidence scattered ions andby detecting energy-analyzed
other in the 3 where the hand, au, metastable recoil atoms. 3 state model calculation channel region R poten-b) (exit
2 state Landau-Zener model calculation. Rotational tial are states and arec) d) close, (1) (2) Z,,). energies very
factors i.e. and coupling fl, fl (upper branch) fi (lower branch). rotationally coupled.
On the side of is out of with to right A, pr respect pLZ.point phase
- THREE STATE MODEL. More2.3 SEMI-CLASSICAL
let us consider a 3 state wheregenerally problem
at the between the two states aboutcrossing Ilg states and of the same channel) (1), (0) (incoming Eg2 units is At small inter-atomic (au) quasi-diabatic. a localized Landau-Zenerare by symmetry, coupled
nuclear distance the state to(2) leading 2IIg at States andinteraction their crossing point Rl. (1)
He+ is well described the + He*(2 ’P) by single are in the R (2) nearly degenerated region R3
and it is configuration energetically very(Isa g) 23dng where are Follow-6a). they rotationally coupled (Fig.
state toclose to the (1) leading adiabatically 2 Eg Ankudinov et al. the time ing [14], development
He+ and described in the same + He*(2 ’S) region into as of the collision is divided shownintervals,
the + by configuration (core (lsO"g)2 3dQg 1 Eg 3dO"g). 6b. As each otherin the approach figure particles
Therefore in order to calculate the 2 lS and 2 1 P
= state crosses state at 0time 11 ; t (t 0), (0) (1)
excitations of a 3 state model Helium, simple involving to the when the corresponds turning point Ro ; par-
the state and the two excited (0) (1)incoming Eg Eg at time ticles the is crossed R1 t2.separate, point again
states will be used. In and another (2) fact, IIg Eg is time whichThe distance reached at R3 t3, beyond
state is to state at the (3) coupled (1) crossing point the different states.no interaction occurs between
2.4 however this occurs at aau; R13 - coupling The where the 0-1 interaction takes range AR, place,
intemuclear distance than the 0-1-2 inter-larger can be estimated from [22] :
actions and its effect on the 2 S excitation is des-
a iscribable suitable ratio. State by branching (0)
to state at the radially coupled (1) crossing point
= is of1.34 au. At the this where R is the radial AF the R1 present energy, coupling velocity, slope
and s is a dimensionless is rather weak since the the difference para-single configu- energy IsO"g(2pO"u)2
meter to In the case 100 which is a of state at close ration, (0) unity. present (R > eV,good description
all 1.57 the value of AR is differs from the excited distances, 1 AF 1 approximatelyconfigurations au), 1 9 323
0.1 au. AR is crossed in a time suf-Then the region
short to consider that the rotational ficiently coupling
is 1-2 in this has no effect. It thenregion practically
to treat a two state the justified crossing R, by (0, 1)
Landau-Zener model.
At the wave function of the can betime t, system
as :expanded
are the wave functions of states1 0), 11 >, 2 )
for a nuclear describe0, 1, 2, R(t) ; they separation
i.e. are ofsingle configurations, they eigenfunctions
an electronic approximate hamiltonian :
where V contains electronic correlation terms.
From the equation,time-dependent Schrôdinger
defined as :one for the gets amplitudes ai
the set of coupled equations :
As state on t via R thevectors 1 i > depend only,
can be operator êlôt replaced by :
R is the radial nuclear is the velocity ; polar anglee(t)
of the intemuclear axis Z and 0 the angular velocity
in is of thethe collision the plane. component Ly
electronic momentum on the nuclear rotationangular
axis The radial nuclear motion as well as they.
electronic term V states of same only sym-couple
and in the Onmetry, namely (0) (1) vicinity of Rl.
the other hand the nuclear rotation only couples
= states and for which AA + 1,(1)-(2) (0)-(2),
- 6. used in the 3 state is on axis Zu) Stmphhcd cnergy where Il the the molecular Fig. polL’l1l1.d diagram component
for 2 1 P model 21S and excitations : full lines : brokenstates; 2Eg of the electronic momentum In the forth-angular (1).
line : state. Same as as a function of ab) (a), time, during 2 ng the two andcoming applications configurations (0) collision at and t = 0 given energy impact parameter ; corresponds two will differ orbitals and the = by to the closest Illustration of the (2) spin coupling(R differentapproach Ra). c)
in the model.to channel 0-2 will be corresponding partial Ab A2 (in ignored present paths amplitudes 1)
and channel are when the transitionB1, B2 (in 2). A2, B2 produced the small time interval Within (tl’ ti) including
0 -+ 1 takes at time the rotational place ti, thencoupling acting
to are from when the transition 0 --+ 1 occurstl t3. Ai, B, produced
at the rotational from to t2, The two coupling acting t2 t3. d) partial
electronic can be in channel 1 in the 2 state Landau- In the He+-He the consideredA1, Az produced (’) collision, spin amplitudes
Zener model. The difference as and the so-called isbetween and is decoupled phase A 1 Ai completely A-representation propor-
L is orbital electronic momentum.tional to the shaded area. valid ; here the 324
where the o-1 takes the In the same ti only coupling place, coupled way :
for and are :al equations ao
time intervals and the]ti, t2[ ]t2, t3[, only During
rotational between the coupling quasi-degenerated
states and is to be then :considered, (1) (2)
Then the Landau-Zener oneusing approximation,
and at timesa relation between bo bi gets amplitudes The two 2 are assumed to differ configurations 1, byt 1 and t+ :
their external molecular orbital only, respectively nlQg
and then :nhcg,
= the Landau-Zenern. v), v where g exp(- being
parameter :
a rotational at smallGenerally coupling occurring
distance can be considered as a toconstant, equal
united atom limit In all its value at the the[15].
we shall take :region R R3
dE,dE1 dE, 0
= is is the thé 0-1 0- 1 interaction interaction and and Mi’ = AF V01 Vo 1 - - dR ----dRldE0
The of the transformation thatunitarity implies
- = 1 In all cases considered1 S 12 g 2. practical
is a short distance where all states arefurther, R1 1
well described This meansby single configurations.
that the interaction is small and that the Vol crossing
behaves In suchR, nearly diabatically (v « 1).
are Then the equations readily integratedcoupled conditions the that the remainsprobability p system
two between times t’, t" :in a state after the is close to given crossing unity
and :
with
z = now into account thewhere + al ia2. Taking As remains constant within theb2 approximately factors with :phase exp(1, u;), interval b the matrix of (tl , ti ) amplitudes bo,l,2
transforms as :
where :
side of are related tothe at the right t, amplitudes
left side of those at the t2 by :
where :
with :325
In same the way :
is obtained from 1!) J(t2 , 1i) by replacing ui byJ(t3 ,
= and a by fl £[O(t3) - 0(t2)]-
As no occurs the final differ from factorsonly by phase coupling beyond t3, amplitudes co) &(+ b(t3)
where : exp(- i.Ki),
= then the In the case where £ one the usualIn fact no interaction occurs before 0, 11, special gets
= can be written as for the in a twoinitial condition : bi(- oo) boi expression probability amplitude
well : state model :bi(tî ) = boi. (0, 1) Landau-Zener-Stueckelberg
as mentioned before that the Assuming crossing
and szbehaves R1 nearly diabatically neglecting
one the final into excitedterms, gets amplitudes
the where partial amplitudes (2) :and channels (1) (2) :
where (2) :
are related to different to state paths leading (1)
with the same difference as before.(Fig. 6d), phase x
In all cases studied thefurther, where £ gg 3,
rotational are coupling factors fii smoothly varying
functions to cos of b.(compared x) impact parameter
As a function of the and to impact parameter, probability The Piamplitudes A 1,2 Bl,2 correspond partial
takes the form :from state and following (0) différent possible paths starting
Theto final states and (1) (2) (see Fig. 6c). leading
for is then :transition 0 --+j probability Pj
= = where p(b) pl(b) P2(b).
When it reduces to the usual Landau-Zener-0, £ =
= the Landau- Stueckelberg probability :where Pl,2),PI,2 being Sl,2 JPI,2(1 -
and Zener relative to p A 1 A2,probability paths
andor and B, B2,
The factor varies p( 1 - p) ;,-1 - p slowly with b,
while the modulus of the difference
, phase x rapidly
and exhibits increases, PLZ regular Stueckelberg
oscillations cos As the(Fig. 7a). governed by x(b) are the andA1 phases developed along paths (Q1,2
factors are similar oscil-smooth functions of f’ b, or and andB, B2 A2,
The factor common to all (2) phase exp(- iw1) amplitudes
been omitted.has 326
- 8. Classical deflection function in a two c onfiguFig. O(l) path
such as that in 6d. The broken line is the deter-ration, given figure
mination full line is the is the threshold.01(l), 02(l); Olh angular
Points 1 and 2 to the two which interfere incorrespond amplitudes
the excited at a 0. The dine-channel, given scattering angle phase
rence two is between the the shaded area. When the nucleargiven by
is taken into 2 account, symmetry supplementary amplitudes,
to l’and 2’ must be added to the formercorresponding points
ones.
- 7. in transition the Landau-ZenerFig. a) Typical probability
as a function of the b. Transitionmodel, b) impact parameter
in 1 3 channels and 2, calculated in the state model.probabilities
The modulation seen on the oscillations isstrong Stueckelberg
the rotational factors.governed by coupling
It is to the hatched area in 8.equal figure
The for excitation 0 --+ j mayscattering amplitude
writtenthen be are in their relative lations but ampli-expected Pj(b),
is now modulated the factor :tude by
where :
oscillates in orAs a either consequence, phase Pj
out of with to to the phase respect PLz, according sign bi(O) dbi.
_ class l lis h the classical differential different crossof and the oscillations are 1(0) L (0) (see Fig. 7b), dampedmj = sin 0 dO
whenever 0. In conclusion the oscillationsrapid mj - section calculated i.along path
are relatedin are of the and Stueckelberg they type Pj 0 --+ is :The differential cross section for / excitation
to excitation whereas thethe mechanism, primary
slow modulation is connected to the secondary
i.e. to the rotational betweenmechanism, coupling
the two excited states (l, 2).
- 2 . 3 .1 cross sections. Similar conclu-Differential = When £ reduces to the Landau-Zener-0, Q1
sions are derived for the differential crosseasily section :differential cross Stueckelberg
in the semi-classical sections calculated approxi-
mation. The classical value of the momentum 1angular
is related to the b N + impact parameter by : ko h(l l2),
cos arewhere is the initial wave number. The classical oscillations, governed by X12, ko Stueckelberg
in but are modulated thedeflection function for a two also observed they by 0(l) path configuration aj,
has two branches which behave as shown in factor :01,2(l)
8. Each than the angularfigure specific angle, larger
threshold is obtained for two values of Oth 1,11,2
each of them related to one of the two diffe-being
rent The difference between the JWKB paths. phases
the since On the other of the two is : 82(0). hand, slightSI (0) - corresponding amplitudes 327
dif’erence between and due to the diffe- The term cos oscillates as a81(0) S2(8), containing x rapidly
rence between and the function of b and its contribution to the total crosspi(0) P2(0), why pLzexplains
minima are not zero. In indiscer- section vanishes. The the rotational exactly addition, squared coupling
of the two nuclei in the factors the of transitionnability system 4Hei produces i)2 represent probability
a nuclear interference 1 due effect : the to the action of rotational symmetry --> j, scattering coupling along
at center-of-mass 0 and n - 0 must be also written :amplitudes angles path i ; they may
be added and the differential cross section writes :
-
!
It to a four term = interference. At smallcorresponds where the andA01 quantities 0(t3) - 0(t2)
= > and the shift bet-angle, ! are the rotation of the1 Fj(O) 1 Fj(n - 0) phase A02 angles 0(t3) - 0(t1)
ween and increases with 0 more action of internuclear axis the rotationalFj(O) Fj(n - 0) rapidly during
than then the nuclear interference 1 and 2. The X12(0); symmetry coupling along paths generalrespectively
in as an undulation of small and E behaviour of these functions of and b is shown inappears amplitude uj
rather on the modulatedhigh frequency, 9a. The superimposed (b bmaxfigure largest impact parameters
oscillations. to the total cross section.Stueckelberg contribute predominantly
The actual calculation of and For above the these scattering amplitudes threshold, energies largely impact
differential cross sections has been carried out can be considered as in as much asusing large parameters
the method which is more semi-quantal [16] precise
than the semi-classical in the one, particularly angular
threshold In this the region. approximation scattering
is as a sum of wave amplitude expressed partial ampli-
limited to even values of 1 because of the nucleartudes,
symmetry :
are channel wave and :where numbers 0, j ko, kj
are the and obtained from abovep(1) simply fi,2(l)
semi-classical + expressions by replacing b by (1 -’,)Iko;
is the JWKB shift for wave 1,ç;(1) phase partial
evaluated i. It is assumed that the along path partial
vanishes when lies in the wave Rl classicallyamplitude
i.e. when : forbidden region,
is mass of the the reduced collision partners.where y
- 2.3.2 Total cross sections. The total cross sec- - 9. of the molecular axis : Rotation Fig. a) angles AO1, (broken
between times and in betweention for excitation 0 ---> at an is line) t2 t3 figure 6b ; A02 (full line) j, impact energy E
times and The numbers indicate values of E,t1 t3. impact energy as :expressed
in unit order of of the unit is that of the(the magnitude arbitrary
The dotted line the location of the maximumenergy defect). gives
of The scale on the hand side to theA02. A00 right corresponds
where the rotational term has been takenC, OB, coupling product
total cross sections in 1 to channels equal b) Typical (full/10.
and 2 a rotational line) (broken line) by secondary coupled coupling,
are of 9a. indi-the characteristics of which those The arrows figure
cate the extrema of the total cross obtained for :sections,
2 A .7r The dotted line where (k L. !1°o integer). gives 1 Q,,, QLz
is the Landau-Zener cross section.328
to small deflection has from been to obtain the adiabaticthey correspond angles E,(R) applied
10 The is then a of Stern et al. trajectory (0 - deg.). practically potentials [18]
to the line. This corresponds followingstraight
forms :asymptotic
where :
The constant interaction between states and (0) (1)
has been evaluated from the betweenenergy splitting On the other when b tends to zero, 0 ;hand, A01,2 --- + the two adiabatic atcorresponding states,
’is a monotonic function of whereasAO, b, increasing
= 1.34 au R, [17] : b.has a maximum for a value of A02 A00(E), bo(E)
Ab of Then in a rather interval parameterlarge impact
a as well as thearound value, bo, A02 keeps stationary
total crossAs a the probabilities ( f2)2. consequence As said before the rotational has beencoupling
section will exhibit around the halfoscillations, 6j assumed to be a constant in the interval R R3,
Landau-Zener total cross i.e. :section, and to vanish for R > The value of the rotationalR3.
matrix element at the united atom limit is :coupling
This duebehaviour, (see Fig. 9b). oscillating entirely
The effective of the rotational hasto the rotational has beensecondary 1-2, range R3 coupling coupling
confirmed the calculation in several cases. been estimated at each relative(or by practical impact energy
Oscillations in when the in such a that the term bestationary way clearly appear velocity vo), coupling Qj
is to over states and method the to the between integration equal energy splitting (1) (2)phase applied
at the atom i.e. :limit, impact parameter : separated
= which the value 2.0 au at 100 eV gives R3 (Lab)
and 1.48 au at’ 150 eV (Lab).
Differential cross sections for 21 S excitation at
150 eV and for 2 ’P at(Lab) (Fig. 4b)
The extrema of are obtained for values of E suchQj 100 eV and 150 eV have been(Fig. 1 b) (Fig. 2b)
that : calculated the using semi-quantal approximation.
Since the to 4a,comparison experimentat values (Fig.
is at small amade la, 2a) only relatively angles,
line has been used in the calculationstraight trajectory cross sections are out of Asand the two phase.
of the rotational factors 4d, ld, coupling fii (Fig. 2d).the total number of oscillations is smaller n, A00
It is seen that the 2 state tocalculation, corresponding than C. Since and Ab tend to zero as E -+ it oo, bo may
= £ 0 (Pig: 4c) clearly disagrees with the measured 2 lSbe that these oscillations be andexpected damped,
is differential cross section. The greatlyagreement more and more toward spaced high energies.
state whichthe introduction of by (3) improved Ilg
rise an inversion of to oscillations,gives Stueckelberg
at A of 4 where the factor vanishes.
- point figure fl 2.4 2 1S 2 1P APPLICATION TO AND EXCITATIONS.
At the same time the calculated differential crossThe 3 state model has been to the applied special
is section for channel i.e. for 2 1 P excitation seen(2), case of 2 ’S 2 1 P and excitations for ausing, Eo,
as 2 cross to be as the 1 S differential section,large diabatic obtained out of thepotential by fitting,
the effect of the which shows very secondaryimportant an to form the adiabaticcrossing region, analytical
rotational moreover it is in coupling ; good agreementpotentials of Bardsley [17] :
with the 2 ’P cross sections at 100experimental
and 150 eV. It be also noticed that the 3in au. may present
state model with the necessarily agrees polarization
In the of interest the for state is coincidence measurement on 2 1 P sincethe excitation region potential (2)
taken identical to that of state excited t he 2 1 P level is via a II state (1) (first populated only (Fig. 3).Eg
The same as that used for In as much as rather state). procedure Eo(R) large scattering angles (or