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Momentum transfer and total differential cross sections in the impulsive Na+-N2 and Na+-CO collisions
Chemical Physics 125 ( 1988) 4 15-424 North-Holland, Amsterdam
MOMENTUM TRANSFER AND TOTAL DIFFERENTIAL IN THE IMPULSIVE Na+-N, AND Na+-CO COLLISIONS
CROSS SECTIONS
S. KITA, H. TANUMA and M. IZAWA ’ Research Institute for Scientific Measurements, Tohoku University, I-I Katahira-Nichome, Sendai 980, Japan
Received 20 January 1988; in final form 17 June 1988
Total differential cross sections (DCS) in Na+-N2 and Na+-CO collisions were measured over a wide range of laboratory angles 5 6 & 90” at laboratory energies 100
1. Introduction
Large-angle scatterings of atomic ions (or atoms) from N2 and CO molecules have been studied extensively over a wide range of center-of-mass (c.m. ) collision energies, 1 SE5 500 eV [ l-51. These studies are mostly concerned with the investigation of the energy transfer mechanism in collisions. The N2 and CO molecules are isoelectronic closed-shell systems with the same molecular mass. The measured energy loss spectra for N2 and CO targets, however, exhibit distinctly different structures. The difference is ascribed to the difference in rotational excitation, which is due to the fact that N2 is a homonuclear molecule, while CO is heteronuclear. The total differential cross sections (DCS) in the laboratory (lab) frame for K+-N2 and K+-CO collisions both display sudden decrease at certain large angles (cut-off angles), and show remarkably different features around the cut-off angles [ 11. The difference in the cross sections for the two systems is due to the fact that one cut-off angle appears for K+-N2 collisions while two cut-off angles exist for K+-CO. An attempt was made to explain these observations ’ Present address: Photon Factory, National Laboratory for High Energy Physics, 1-I Oho, Tsukuba 305, Japan.
with the impulsive approximation, but a full understanding was not attained. We also found that the total DCS for K+ ions produced in moderate energy K-N2 and K-O, collisions showed the similar cut-off structure at large angles [ 6 1. Therefore the cut-off phenomenon in the total DCS seems to be common to heavy particle collisions. In this article we report the study of differential large-angle scattering in moderate energy ion-molecule collisions by measuring the total DCS and energy loss spectra for the Na+-N, and Na+-CO systems. The classical trajectory (CT) calculation was done to complement the measurements. In high-energy Na+-N, and Na+-CO collisions, the energy transferred into the internal degrees of freedom of the molecules is so large that the neutral atoms are produced by dissociation [ 7 1. According to the time-of-flight (TOF) analyses of the fragment atoms [ 71 and the scattered Na+ ions [ 5,8], electronic excitation can be neglected for these collision systems at the energies employed in this study. These collision systems, therefore, are suitable to study the large angle scattering of projectile ions which are accompanied only by the momentum transfer. The total DCS for the Na+-N2 and Na+-CO collisions both show the distinct cut-off structures at large angles, which are the same with the earlier observa-
0301-0104/88/$03.50 0 Elsevier Science Publishers B.V. ( North-Holland Physics Publishing Division )
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S. Kita et al. /Total DCS in Na+-N, and Na +-CO collisions
tions for K+-N2 and K+-CO collisions [ I]. The cutoff of the DCS is attributed to the big momentum transfer in the collisions. The cut-off structures in the total DCS for the Ar-O2 collisions, which were ascribed to individual rotational state-to-state excitation, have recently been observed at the low energies of E= 79.7 and 97.0 meV [ 91. The experimental results in the present study are somewhat similar to this low-energy experiment. In the impulsive molecular collisions, the cut-off position is determined by the angular momentum transfer as is in the low-energy collisions, whereas the sharp falling-off of the DCS is due to the effect of vibrational excitation.
2. Apparatus The crossed-beam apparatus which was used for the measurements is similar to that of Lee et al. [ lo] and is described in detail elsewhere [ 111. The two beams cross each other perpendicularly in the main chamber, and the particles scattered in-plane are detected by a rotatable multiplier. The primary Na+ ions are formed by surface ionisation of Na20*A120,.2Si02 on a heated platinum wire. The ions are accelerated to the desired energies of 100 < ElabG 450 eV, and are collimated by two slits into an angular spread of approximately 0.5” full width at half-maximum (fwhm). For the TOF measurements the Na+ ion beam is pulsed with a pair of condenser plates in front of the collimating slits [ 12 1. The flight path length from the scattering center to the detector is approximately 50 cm. The overall angular resolution for the scattered particles is approximately 0.8 ’ fwhm. The N2 and CO beams are injected into the collision volume as a pulsed supersonic beam, which is obtained by adiabatic expansion of the source gases through a nozzle of 0.1 mm diameter followed by a skimmer [ 13 1. The stagnation pressure p. in the nozzle beam source used is around 1300 Torr. The mean velocity of the nozzle beams is approximately 7.9 x lo4 cm s- ‘. Target density ratio of the CO beam to Nz beam, n( CO)/n(N*), at an identical pressure p. is estimated to be approximately 0.95. Fig. 1 exhibits the Newton diagram for the Na+NZ and Na+-CO collisions with various reduced energy transfers AE/E in the c.m. system. The quantities v. and v, in the figure are the velocity vectors of
Fig. 1. Newton diagram for the Na+-N2 and Na+-CO collisions. 0, is the cut-off angle for a certain energy transfer AE/E.
the projectile ions and target molecules, respectively. In this study, the intensity of the scattered particles was measured as a function of the angle 8 with respect to the velocity vector v,. In the elastic scattering (AE/E=O) of the Na+ ions from the N2 and CO molecules, the recoiled ions are observed over the full angular range of 0 < 8~ 180”, which can be seen in fig. 1. In the inelastic collisions with energy transfer AElE~0.33, on the other hand, the range of kinematically accessible lab angles for scattered Na+ ions extends only to a cut-off angle 0, ( c 90 o ). The angle 0, is given by &=sin-’
(
1/z
!f l-G+%
[ m
sWW0) 0
1 ,
>
(1)
where m and Mare the masses of the projectile Na+ ions and the target molecules of Nz and CO, respectively, and 6,, is the value of 0, for V. = 0. The cut-off angle 0, is, thus, related with the energy transfer AE/ E. One can obtain, therefore, information on the energy transfer by measuring the cut-off structure in the total DCS.
3. Experimental results 3.1. Total dlflerential cross sections Angular dependence of the total DCS in the laboratory (lab) frame measured at Elab= 100 eV is shown in fig. 2. The open and full circles represent the DCS for the Na+-N, and Na+-CO collisions, respectively. The open triangles in fig. 2 show the DCS of Na+-Ar collisions for comparison. The integral cross sections for Na+-N, and Na+-CO collisions are
S. Kita et al. /Total DCS in Na +-N, and Na+-CO collisions
417
%(degl Fig. 2. Total DCS in the lab frame measured at Elab= 100 eV. (0) DCS for Na+-N2 and (0 ) DCS for Na+-CO collisions. (A ) Elastic DCS for Na+-Ar collisions.
Fig. 3. Intensities of the scattered particles at Efab=350 eV as a function of the lab angle 8. (0 ) Na+ ions scattered from N2 and ( l ) from CO molecules. (A ) Neutrals for Na+-N, and (A ) for Na+-CO collisions.
nearly equal to those for Na+-Ar at the same collision energies [ 14,15 1. The measurements of Na+ scattering from Ar atoms were made under the nozzle source condition same as that for N2 and CO beams, p. = 1300 Torr. The number density ratio of Ar target to NZ, n(Ar)/n(Nz), was approximately 1X5. The experimental values normalized by the target density are shown in the figure. Since only the relative DCS could be measured in this study, the experimental DCS of Na+-N2 at 85 10” are fitted to the values calculated with the spherical potential determined experimentally [ 141. The DCS for the two systems Na+-N, and Na+CO in fig. 2 are the same within the experimental uncertainty over the whole angular range measured in this work. According to the TOF analysis at ,?&,G 350 eV, the Na” ions are scattered only elastically from Ar atoms. The total DCS measured for Na+-N2 and Na+-CO collisions show the angular dependence similar to the elastic DCS of Na+-Ar in the angular range of 9s49”, which is indicated by an arrow in the figure, but fall off rapidly at larger angles. Fig. 3 exhibits angular dependences of the intensity of the scattered ions and neutrals at Et,,=350
eV. The open and full circles indicate the intensity of the Naf ions scattered from N, and CO molecules, respectively. The open and full triangles indicate the intensity of the neutrals for the Na+-N, and Na+CO collisions, respectively. Intensities of the projectile ion beam and the target beams were the same for both collision systems. The neutrals were detected directly with a secondary electron multiplier by rejecting the scattered ions with a positive high voltage V,. When the high voltage V, is not supplied in our apparatus the ions and neutrals are detected simultaneously. The scattered ion intensity was therefore deduced by taking the difference between the two measurements of the total signals (ion+ neutral) and the neutrals. Although the ion intensity I( 19)shown in fig. 3 represents the relative total DCS a( 19))the angular dependence of the neutrals is apparent. This is because the energy of the neutrals depends on the scattering angle and the detection efficiency of the multiplier depends strongly on the impinging energy of the particles. The TOF experiments showed that the detected neutrals were the fragment C, N and 0 atoms which resulted from the collisional dissociation [7].
S. Kita et al. /Total DCS in Na+-N, and Na +-CO collisions
418
The scattered ion intensities for the two collision systems in fig. 3 have almost the same magnitude and show the same angular dependence at lab angles 19536’. A distinct difference between the ion intensities of these two systems, however, can be seen at 8s 36”. The ion intensity in Na+-N, collisions decreases rapidly at the angles larger than 8= 44”, while the angular dependence of the ion intensity for Na+CO changes its slope at 8x 36 and 5 1’. The angles of 36, 44 and 5 lo are indicated in the figure by arrows 1, 2 and 3, respectively. The difference in the scattered ion intensities between the Na+-N2 and Na+CO collisions was observed clearly at the collision energies of Elab 2 200 eV. The angles indicated with the arrows in figs. 2 and 3 are considered to be the cut-off angles in the molecular collisions, as discussed in the earlier work for K+-N2 and K+-CO collisions [ I]. 3.2. Energy loss spectra Fig. 4a shows the experimental energy loss spectrum of Na+ ions scattered from Nz molecules at I
I
I
Elab= 350 eV and the cut-off angle 8, = 44 a (arrOW 2 in fig. 3 ). In the present study, the energy loss spectra o(AE/E, 19) were deduced from the measured TOF spectra a( t, 0) with the relation of a(AE/E, 0) = a( t, 6J) dt/dAE. Around the cut-off angle, the ion signal distributes over a wide range of the flight time, 15 5 t 5 40 ps, with a broad and weak tail. To get enough intensity at the weak tail, with which our discussion will be concerned, the TOF measurements were made at the sacrifice of the time resolution, At x 0.5 us. The measured spectrum in fig. 4a shows a sharp cut-off at the largest energy loss, (AEIE), = 0.69. The cut-off position in the energy transfer spectra is given kinematically by ($)~=l-(~~sin20+
N&CO
(2)
A&
0(e)=
(d)
.
Since the experimental angular and energy resolutions are not taken into account in the data analysis, the energy loss spectra diverge at the cut-off energy. The divergence at (AE/E), originates from the Jacobian factor dt/dAE. The total differential cross section g( 0) is given by
I-
I
$$sin(XJ)
I
a(~,
e) dm.
(3)
0
0.4
0
ue
u.4
”
AE/E
Fig. 4. Energy loss spectra for Na+-N2 and Na+-CO collisions at I&,= 350 eV and cut-off angles. (a) Spectra measured for Na+Nz at 0=44” and (b) for Na+-CO at 6’=36”. (c) Spectra computed for Na+-N2 at 0~43” and (d) for Na+-CO at 0~36”.
At 8< e,, the majority of the ion signal distributes at the energy loss AE/E< (AEIE),, and the cut-off condition of eq. ( 2 ) gives no noticeable effect on the total DCS. At 8> & on the other hand, the energy loss distribution is restricted strongly by the cut-off condition, and then the total DCS fall off steeply as shown in fig. 3. Fig. 4b exhibits the experimental energy loss spectrum for the Na+-CO collisions at Elab= 3 50 eV and the first cut-off angle 0, = 36’ (arrow 1 in fig. 3 ). In this case the Jacobian peak is located at AEIEc0.78. In contrast to the spectrum of fig. 4a for the Na+-N2 collisions, a prominent peak A, which is located at AEIEx0.28, is seen in the spectrum of fig. 4b. As will be discussed below, peak A originates from the interaction on the O-atom side, while the broader peak B is due to the interaction on the C-atom side. Increasing the scattering angle, signal B vanishes rapidly, which causes the falling-off of the DCS as can be seen in fig. 3, and peak A broadens and shifts to larger
S. Kita et al. /Total DCS in Na +-N, and Na +-CO collisions
energy loss. If the angle is further increased, signal B disappears completely and signal A begins to vanish around 8= 5 1O. Therefore, the total DCS decrease steeply at 19>5 lo, which also can be seen in fig. 3. Thus the energy loss distribution in the spectra is reflected on the structure of the falling-off of the total DCS. For Na+-N, collisions at Elab= 100 eV and I% 49 o (the arrow in fig. 2), the energy loss spectrum also shows the cut-off structure around AE/E=0.64. Contrary to the spectra for high-energy collisions, a prominent peak is located at lower energy loss, AE/ ExO.1. This energy dependence of the energy loss spectra brings about the energy dependence of the cutoff structure in the total DCS in figs. 2 and 3, which will be discussed below. 3.3. Cut-off angles
angles and squares are for Na+-CO. At Elab= 100 eV, no noticeable structures in the DCS for Na+-CO could be found. The arrows labelled C, N and 0 in fig. 5 represent the cut-off angles for the binary Na+C, Na+-N and Na+-0 interactions, where the target velocity V, is ignored. The measured cut-off angles, ( 1) , (2 ) and (3 ), slowly approach the values given by the binary encounter model as the collision energy increases. This tendency has also been found by Inouye et al. [ 1 ] in K+-N2 and K+-CO collisions.
4. Classical trajectory calculation and discussion
The total DCS as well as the energy loss spectra were computed with the CT method. Interaction potentials for the collision systems were approximated by the additive interaction, Vh2,
Fig. 5 shows the energy dependences of the cut-off angles determined from the DCS measurements for the Na+-N2 and Na+-CO collisions. Open circles are the experimental values for Na+-N2, and open triI
I
1
I
I
50-
419
rz3,
r3,)=
Ur12)+
Ur23)+
Wr3,)
,
(4)
where the subscript 1 means Na + ion, 2 and 3 are constituent atoms of the N2 and CO molecules, r12, r23 and r3, are the interatomic distances. The ionatom interactions V( r, 2) and V( r31) were calculated with the statistical electron gas model [ 16 1. A Morse function was taken as the intramolecular potentials V(r23)forN2andC0 [17]. The collision geometry employed in the calculation is shown in fig. 6. The initial velocity vector v. lies in the yz plane and is parallel to the z axis, and b represents the impact parameter. The center of mass of the molecule is on the origin. a and /3are the polar and azimuthal angles, respectively, defining the moZ
t
I
600
0
E
Lob
(eV)
Fig. 5. Energy dependences of the cut-off angles 0,. ( 0 ) Experimental results for Na+-N2 collisions; (A ) and (o), experiments for Na+-CO. (0 ) Theoretical results obtained with the CT calculation of the DCS for Na+-N2 collisions; (A ) and ( n ) theoretical values evaluated from the DCS for Na+-CO. (-) CT calculations for the molecular orientation (Y= Ifi1 =90”; (---) CT calculation for the orientation ar= 120” and /3=90”. (---) Computations with the hard-shell model. The arrows C, N and 0 are the cut-off angles determined with the binary encounter model.
Fig. 6. Collision
geometry
for the classical trajectory
calculation.
S. Kita et al. /Total DCS in Na+-N, and Na +-CO collisions
420
lecular orientation in the space-fixed coordinate system. The CT calculation was carried out for the molecular orientation of -l
Here, N,, is the number of points chosen for the molecular orientation and vibrational initial phase. Its I
I
I
I
I
\
I
I
I
No+-N2,C0
‘\ :
I
ELab =lOOeV
‘\
I
20
I
I
40
I
I
60
I
I
80
0 (deg)
Fig. 7. Total DCS in the lab frame calculated with the CT method for the Na+-N2 and Na+-CO collisions at I&,= 100 eV. (p) DCS for Na+-N2 collisions; (---) DCS for Na+-CO. (---) DCS evaluated with the spherical potential of eq. (6).
value was 990 for the Na+-N, collisions and 1980 for Na+-CO. N(B) is the number of trajectories which fall into the angular window of A& 1o centered at each scattering angle 8. The interval Ab of impact parameter was taken 0.02 A. The total number of the trajectories was approximately 60000 for Na+-N, and 120000 for Na+-CO. The DCS a( 0) in the lab frame for the Na+-N, and Na+-CO collisions in fig. 7 show almost the same angular dependence, which is in good agreement with the experiments in fig. 2. For the Na+-N, collisions, a small hump in the computed DCS a( 0) can be seen around 8= 47’) which is indicated by an arrow in the figure. The spherical potentials averaged over the molecular orientation for Na+-N2 and Na+-CO, which were determined from the Na+-N, Na+-C and Na+0 interactions, can be approximated by V(R) =4010 exp( -3.82R)
eV
(6)
at the intermolecular distance of 0.8
S. Kita et al. /Total DCS in Na+-NJ and Na+-CO collisions I
,
I
I
I
I
No+-
20
40 0
60
(deg)
Fig. 8. Total DCS in the lab frame computed with the CT method for the Na+-N, and Na+-CO collisions at _I&= 350 eV. (-) DCS for Na+-N2; (---) DCS for Na+-CO.
loo% @ b
0 (deg) Fig. 9. Total DCS in the c.m. frame calculated with the CT method for the Na*-N2 and Na+-CO collisions at I&,= 350 eV. (-) DCS for Na+-N,; (---) DCS for Na+-CO.
4.2. Cut-off angles The cut-off angles evaluated with the CT calculation of the total DCS are given in fig. 5. The full circles represent the results computed for the Na+-N,
421
collisions. The full triangle is the result for the interaction on the C-atom side and the full square is that on the O-atom side in the Na+-CO collisions. The calculations are in good agreement with the experithe values calculated at ments. For Na+-N,, Elab= 100 and 200 eV are also shown. The value for Elab= 100 eV is the position of the small hump (the arrow in fig. 7 ) . 4.2.1. Energy loss spectra The energy loss spectra were computed with the CT method for the collisions at Elab= 350 eV to get information on the cut-off phenomenon. The trajectories used in the computation fell into the angular window A&2” centered at each cut-off angle in the computed DCS. The calculated cut-off angles are 43’ for Na+-N,, and 36 and 50” for Na+-CO. Fig. 4c exhibits the energy loss spectrum computed for the Na+-N, collisions at cut-off angle e,= 43 O.The spectrum shows a prominent peak at the cut-off energy (AE/E),=0.70, and reproduces fairly well the experiment in fig. 4a. The computed spectrum for the Na+-CO collisions at the first cut-off angle e,= 36’) which is shown in fig. 4d, also agrees qualitatively well with the experiment of fig. 4b. The cut-off angle f3, is related with the cut-off energy (AE/E), by eq. (2). The cut-off energy (AE/ E), given by eq. (2) is 0.70 for Na+-N, collisions at &=43’, and 0.78 and 0.62 for Na+-CO at &=36 and 50’) respectively. The trajectories, which fall into the window A(AE/E) ~0.02 at each cut-off energy, were analysed in order to clarify the cause of the sharp cut-off of the total DCS. According to the analysis, the peak at the cut-off energy is composed predominantly, 80°h, of the trajectories of the initial molecular orientations 60ia< 120” and 70 < lfil~90”. It was also found from the analysis that these trajectories are accompanied by the large angular momentum transfer. The collisions in the perpendicular configuration, cr= 1/?I = 90”, are typical ones to interpret the sharp cut-off in the DCS, which will be discussed in section 4.2.2. 4.2.2. Collisions in the perpendicular configuration Fig. 1Oa and the full curve in fig. 1Ob show the scattering lab angle and the reduced energy transfer AE/ E, respectively, as a function of the impact parameter b for the Na+-N, collisions at Elab= 350 eV. The col-
S. Kita et al. /Total DCS in Na+-N2 and Na+-CO collisions
422,
Y, tdeg 1
b (8)
Fig. 10. (a) The scattering lab angle 6’(b), (b) the reduced excitation energy AE(b)/E, and (c) the angular momentum J(b) calculated as a function of the impact parameter b for the perpendicular Na+-N, collisions at Elab= 350 eV. The collision geometry is inserted in (b). The full curve in (b) denotes the total excitation energy AE/E, and the broken curve gives the rotational excitation energy AL?,/E. The upper scale represents the molecular orientation yCat the turning point.
lision geometry is inserted in fig. lob, where the initial velocity vector u. and the molecular axis are inplane and at right angles, i.e. cr = p= 90”. The c.m. angle 8 is a monotonic function of the impact parameter b, i.e. @= 180, 137 and 40” for b= 0,0.6 and 1.2 A, respectively. The energy transfer is so big at small impact parameter b (large c.m. angle) that the deflection function B(b) has a maximum around b=0.65-0.7 A (19= 11%127”), where the energy AE/E is approximately 0.7. The maximum angle 0,,,, in the deflection function, which is 42.5”, is equivalent to the maximum lab angle in the Newton diagram. For the Na+-CO collisions in the perpendicular configuration, there are two different deflection functions corresponding to the collisions on the Catom and O-atom sides. The deflection functions for
E iat,= 350 eV give the angles &!,,,,= 36’ (C-atom side) and 50” (O-atom side). The solid curves in fig. 5 represent the energy dependences of the maximum angle &,,, in the deflection function 6(b) calculated for the collisions in the perpendicular configuration. These curves agree well with the experimental values (open symbols) and three-dimensional CT calculations (full symbols) at Elab3 200 eV. The curve calculated for Na+-N2 collisions at Elab < 200 eV, however, rises rapidly with decreasing the energy, and deviates largely from the experiment (open circle) and three-dimensional CT calculation (full circle) at Elab= 100 eV. The small hump around 8=47” in fig. 7 for Na+Nz collisions at Elab= 100 eV arises from the maximum in the function /3(b) for the molecular orientations only around (r=120” and /3=90”, which is different from the result for higher energies. The energy dependence of &,, for the Na+-N2 collisions at the molecular orientation (Y= 120” and p=90” is shown by the dashed curve in fig. 5. This curve represents well the energy dependence of the experiments and three-dimensional CT calculations over the whole energy range. 4.2.3. Cut-offstructure In order to get further information on the cut-off angle and the falling-off structure in the total DCS, additional CT calculation with a rigid-rotor model [ 19 ] was carried out for the Na+-N, collisions at &b = 350 eV. In the high-energy collisions, the rotational energy transfer AEJE exceeds the dissociation energy at large scattering angles [ 71. However, the intramolecular distance rz3 remains almost at the equilibrium value r, of the isolated molecule during the collisions, because rotational excitation takes place suddenly. We can therefore safely use the rigidrotor model to estimate the angular momentum transfer. The rigid-rotor calculation shows that the cut-off angle appears at 6, z 45’) which is nearly equal to the result obtained with the above vibrotor calculation in fig. 8, 0, = 43 ‘. The DCS by the rigid-rotor calculation at large angles of 6> 45 ‘, however, fall with a slope smaller than that by the vibrotor calculation in fig. 8. These results suggest that the cut-off position is determined by the angular momentum transfer, while steepness of the falling-off of the DCS is given predominantly by vibrational excitation. One
S. Kita et al. /Total DCS in Na+-NJ and Na +-CO
can, therefore, conclude that the strong energy dependence of the cut-off structure, i.e. sharper fallingoff of the DCS at high energy, is due to the effect of vibrational excitation. 4.2.4. Rotational rainbow eflect Fig. 1Oc shows the angular momentum transferred to the Nz molecule as a function of the impact parameter b, where the collisional condition is taken the same as that of figs. 10a and lob. The angular momentum J takes the maximum value J,,,x720 around bs0.65 A. The dashed curve in fig. lob represents the rotational energy transfer A&/E, which was evaluated with the intramolecular distance rz3( = re) at the turning point in the collision [ 71. The angular momentum J(b) and the rotational energy transfer A&(b) /E by the vibrotor calculation in fig. 10 are both nearly equal to those obtained by the rigidrotor model. As can be seen in fig. lob, the energy transfer in the collisions around the maximum angle e max,which corresponds to the cut-off angle & is exclusively due to rotational excitation. The cut-off position is, therefore, determined by the angular momentum transfer. The upper scale in fig. 10 gives the.molecular orientation yc at the closest approach, y being the angle between the molecular axis and the intermolecular distance vector R.The maximum in the deflection function appears at the collisions of yc = 50”) which is the rotational rainbow angle as discussed previously [ $81. Therefore, the cut-off positions and their energy dependence can be qualitatively explained by the rotational rainbow effect [ 3,201. The hard-shell model is very useful to evaluate the rotational energy in the impulsive collisions [ 581. We computed the rotational rainbow positions as a function of c.m. angle 8 with the two-dimensional hard-shell model proposed by Bosanac [ 2 11, and determined the Newton diagram. Function R*(y) defining the hard-shell size is assumed to be given by the equipotential surface, following Hasegawa et al. [ 5 1. Increasing the scattering angle @, the rotational rainbow energy AERIE increases rapidly. As a result, the Newton diagram shows a cut-off angle 19,. The dotted curves in fig. 5 exhibit the energy dependences of the cut-off angles obtained with the hardshell model. As shown, the computations reproduce qualitatively well the experiments. One rotational rainbow appears in the energy loss
collisions
423
spectra for the scattering ofNa+ from the homonuclear molecule N2 and two rotational rainbows for Na+-CO collisions, which correspond to one cut-off angle for Na+-N, and two cut-off angles for Na+CO in the total DCS. As reported previously in ref. [ 5 1, the rotational rainbow position Al&/E at a fixed angle weakly depends on the collision energy and hence the cut-off position should also behave the same. The rotational rainbow originates from the collisions at the molecular orientation yRx 50”. For ion-molecule collisions around the rotational rainbow angle yR, the repulsive potential of the COlliSiOn system is expected to be determined by the ion-atom interaction of the closest pair. This means that the ion-molecule collisions around the rainbow angle can be regarded approximately as the ion-atom interactions. Therefore the cut-off angles ( 1 ), (2) and (3 ) are close to 6, of the Na+-C, Na+-N and Na+-0 interactions, respectively.
5. Conclusion The angular and energy depencences of the total DCS measured for the Na+-N2 and Na+-CO collisions could be well reproduced with the classical trajectory calculation using the simple additive potentials based on the Na+-N, Na+-C and Na+-0 interactions. The characteristic difference in the DCS for the high-energy Na+-N, and Na+-CO collisions is due to the difference in the momentum transfer. The cut-off positions in the DCS, 0,, were interpreted mostly by assuming the collisions to be in the perpendicular configuration, and were found to be related to the rotational rainbow effect.
Acknowledgement We are grateful for valuable discussion with Professors H. Inouye (Bunri University Tokushima, Tokushima, Japan), I. Kusunoki, Y. Sato (Tohoku University, Sendai, Japan) and Dr. M. Nakamura (Institute for Molecular Science, Okazaki, Japan). We are also indebted to Mr. T. Hasegawa for his contribution to this study. This work was financially supported in part by a Grant-in-Aid for Scientific
424
S. Kita et al. /Total DCS in Na+-NJ and Na+-CO collisions
Research from the Ministry of Education, Science and Culture of Japan.
References [ 1] H. Inouye, K. Niurao and S. Kita, Chem. Phys. Letters 33 (1975) 241. [2] R. Btittner, U. Ross and J.P. Toennies, J. Chem. Phys. 65 (1976) 733. [ 31 W. Schepper, U. Ross and D. Beck, Z. Physik A 290 ( 1979) 131. [ 41 U. Gierz, J.P. Toennies and M. Wilde, Chem. Phys. Letters llO(1984) 115. [ 51 T. Hasegawa, S. Kita, M. Izawa and H. Inouye, J. Phys. B 18 (1985) 3775. [ 61 S. Kita, T. Hasegawa and H. Inouye, unpublished results. [ 71 S. Kita, T. Hasegawa, A. Kohlhase and H. Inouye, J. Phys. B20 (1987) 305. [ 81 M. Nakamura, S. Kita and T. Hasegawa, J. Phys. Sot. Japan 56 (1987) 3161.
[ 9) M. Faubel and G. Kraft, J. Chem. Phys. 85 ( 1986) 2671. P.R. LeBreton and D.R. Herschbach, Rev. Sci. Instr. 40 (1969) 1402. [ 111 H. Inouye, M. Izawa, S. Kita, K. Takahashi and Y. Yamato, Bull. Res. Inst. Sci. Meas. Tohoku Univ. 32 (1984) 41. [ 121 S. Kita and H. Inouye, J. Phys. B 12 (1979) 2339. [ 131 M. Izawa, S. Kita and H. Inouye, J. Appl. Phys. 53 (1982) 4688. [ 141 S. Kita, K. Noda and H. Inouye, Chem. Phys. 7 ( 1975) 156. [ 15 ] S. Kita, K. Noda and H. Inouye, J. Chem. Phys. 63 ( 1975 ) 4930. [ 16 ] T. Ishikawa, S. Kita and H. Inouye, Bull. Res. Inst. Sci. Meas. Tohoku Univ. 24 (1976) 101. [ 171 K.P. Huber and G. Herzberg, Molecular spectra and molecular structure, Vol. 4. Constants of diatomic molecules (Van Nostrand-Reinhold, New York, 1979). [ 181 M. Karplus, R.N. Porter and R.D. Sharma, J. Chem. Phys. 43 (1965) 3259. [ 191 R.A. LaBudde and R.B. Bernstein, J. Chem. Phys. 55 ( 1971) 5499. [20] R. Schinke and J.M. Bowman, in: Molecular collision dynamics, ed. J.M. Bowman (Springer, Berlin, 1983) p. 6 1. [ 211 S. Bosanac, Phys. Rev. A 22 ( 1980) 26 17.
[ 10lY.T. Lee, J.D. McDonald,
MOMENTUM TRANSFER AND TOTAL DIFFERENTIAL IN THE IMPULSIVE Na+-N, AND Na+-CO COLLISIONS
CROSS SECTIONS
S. KITA, H. TANUMA and M. IZAWA ’ Research Institute for Scientific Measurements, Tohoku University, I-I Katahira-Nichome, Sendai 980, Japan
Received 20 January 1988; in final form 17 June 1988
Total differential cross sections (DCS) in Na+-N2 and Na+-CO collisions were measured over a wide range of laboratory angles 5 6 & 90” at laboratory energies 100
1. Introduction
Large-angle scatterings of atomic ions (or atoms) from N2 and CO molecules have been studied extensively over a wide range of center-of-mass (c.m. ) collision energies, 1 SE5 500 eV [ l-51. These studies are mostly concerned with the investigation of the energy transfer mechanism in collisions. The N2 and CO molecules are isoelectronic closed-shell systems with the same molecular mass. The measured energy loss spectra for N2 and CO targets, however, exhibit distinctly different structures. The difference is ascribed to the difference in rotational excitation, which is due to the fact that N2 is a homonuclear molecule, while CO is heteronuclear. The total differential cross sections (DCS) in the laboratory (lab) frame for K+-N2 and K+-CO collisions both display sudden decrease at certain large angles (cut-off angles), and show remarkably different features around the cut-off angles [ 11. The difference in the cross sections for the two systems is due to the fact that one cut-off angle appears for K+-N2 collisions while two cut-off angles exist for K+-CO. An attempt was made to explain these observations ’ Present address: Photon Factory, National Laboratory for High Energy Physics, 1-I Oho, Tsukuba 305, Japan.
with the impulsive approximation, but a full understanding was not attained. We also found that the total DCS for K+ ions produced in moderate energy K-N2 and K-O, collisions showed the similar cut-off structure at large angles [ 6 1. Therefore the cut-off phenomenon in the total DCS seems to be common to heavy particle collisions. In this article we report the study of differential large-angle scattering in moderate energy ion-molecule collisions by measuring the total DCS and energy loss spectra for the Na+-N, and Na+-CO systems. The classical trajectory (CT) calculation was done to complement the measurements. In high-energy Na+-N, and Na+-CO collisions, the energy transferred into the internal degrees of freedom of the molecules is so large that the neutral atoms are produced by dissociation [ 7 1. According to the time-of-flight (TOF) analyses of the fragment atoms [ 71 and the scattered Na+ ions [ 5,8], electronic excitation can be neglected for these collision systems at the energies employed in this study. These collision systems, therefore, are suitable to study the large angle scattering of projectile ions which are accompanied only by the momentum transfer. The total DCS for the Na+-N2 and Na+-CO collisions both show the distinct cut-off structures at large angles, which are the same with the earlier observa-
0301-0104/88/$03.50 0 Elsevier Science Publishers B.V. ( North-Holland Physics Publishing Division )
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S. Kita et al. /Total DCS in Na+-N, and Na +-CO collisions
tions for K+-N2 and K+-CO collisions [ I]. The cutoff of the DCS is attributed to the big momentum transfer in the collisions. The cut-off structures in the total DCS for the Ar-O2 collisions, which were ascribed to individual rotational state-to-state excitation, have recently been observed at the low energies of E= 79.7 and 97.0 meV [ 91. The experimental results in the present study are somewhat similar to this low-energy experiment. In the impulsive molecular collisions, the cut-off position is determined by the angular momentum transfer as is in the low-energy collisions, whereas the sharp falling-off of the DCS is due to the effect of vibrational excitation.
2. Apparatus The crossed-beam apparatus which was used for the measurements is similar to that of Lee et al. [ lo] and is described in detail elsewhere [ 111. The two beams cross each other perpendicularly in the main chamber, and the particles scattered in-plane are detected by a rotatable multiplier. The primary Na+ ions are formed by surface ionisation of Na20*A120,.2Si02 on a heated platinum wire. The ions are accelerated to the desired energies of 100 < ElabG 450 eV, and are collimated by two slits into an angular spread of approximately 0.5” full width at half-maximum (fwhm). For the TOF measurements the Na+ ion beam is pulsed with a pair of condenser plates in front of the collimating slits [ 12 1. The flight path length from the scattering center to the detector is approximately 50 cm. The overall angular resolution for the scattered particles is approximately 0.8 ’ fwhm. The N2 and CO beams are injected into the collision volume as a pulsed supersonic beam, which is obtained by adiabatic expansion of the source gases through a nozzle of 0.1 mm diameter followed by a skimmer [ 13 1. The stagnation pressure p. in the nozzle beam source used is around 1300 Torr. The mean velocity of the nozzle beams is approximately 7.9 x lo4 cm s- ‘. Target density ratio of the CO beam to Nz beam, n( CO)/n(N*), at an identical pressure p. is estimated to be approximately 0.95. Fig. 1 exhibits the Newton diagram for the Na+NZ and Na+-CO collisions with various reduced energy transfers AE/E in the c.m. system. The quantities v. and v, in the figure are the velocity vectors of
Fig. 1. Newton diagram for the Na+-N2 and Na+-CO collisions. 0, is the cut-off angle for a certain energy transfer AE/E.
the projectile ions and target molecules, respectively. In this study, the intensity of the scattered particles was measured as a function of the angle 8 with respect to the velocity vector v,. In the elastic scattering (AE/E=O) of the Na+ ions from the N2 and CO molecules, the recoiled ions are observed over the full angular range of 0 < 8~ 180”, which can be seen in fig. 1. In the inelastic collisions with energy transfer AElE~0.33, on the other hand, the range of kinematically accessible lab angles for scattered Na+ ions extends only to a cut-off angle 0, ( c 90 o ). The angle 0, is given by &=sin-’
(
1/z
!f l-G+%
[ m
sWW0) 0
1 ,
>
(1)
where m and Mare the masses of the projectile Na+ ions and the target molecules of Nz and CO, respectively, and 6,, is the value of 0, for V. = 0. The cut-off angle 0, is, thus, related with the energy transfer AE/ E. One can obtain, therefore, information on the energy transfer by measuring the cut-off structure in the total DCS.
3. Experimental results 3.1. Total dlflerential cross sections Angular dependence of the total DCS in the laboratory (lab) frame measured at Elab= 100 eV is shown in fig. 2. The open and full circles represent the DCS for the Na+-N, and Na+-CO collisions, respectively. The open triangles in fig. 2 show the DCS of Na+-Ar collisions for comparison. The integral cross sections for Na+-N, and Na+-CO collisions are
S. Kita et al. /Total DCS in Na +-N, and Na+-CO collisions
417
%(degl Fig. 2. Total DCS in the lab frame measured at Elab= 100 eV. (0) DCS for Na+-N2 and (0 ) DCS for Na+-CO collisions. (A ) Elastic DCS for Na+-Ar collisions.
Fig. 3. Intensities of the scattered particles at Efab=350 eV as a function of the lab angle 8. (0 ) Na+ ions scattered from N2 and ( l ) from CO molecules. (A ) Neutrals for Na+-N, and (A ) for Na+-CO collisions.
nearly equal to those for Na+-Ar at the same collision energies [ 14,15 1. The measurements of Na+ scattering from Ar atoms were made under the nozzle source condition same as that for N2 and CO beams, p. = 1300 Torr. The number density ratio of Ar target to NZ, n(Ar)/n(Nz), was approximately 1X5. The experimental values normalized by the target density are shown in the figure. Since only the relative DCS could be measured in this study, the experimental DCS of Na+-N2 at 85 10” are fitted to the values calculated with the spherical potential determined experimentally [ 141. The DCS for the two systems Na+-N, and Na+CO in fig. 2 are the same within the experimental uncertainty over the whole angular range measured in this work. According to the TOF analysis at ,?&,G 350 eV, the Na” ions are scattered only elastically from Ar atoms. The total DCS measured for Na+-N2 and Na+-CO collisions show the angular dependence similar to the elastic DCS of Na+-Ar in the angular range of 9s49”, which is indicated by an arrow in the figure, but fall off rapidly at larger angles. Fig. 3 exhibits angular dependences of the intensity of the scattered ions and neutrals at Et,,=350
eV. The open and full circles indicate the intensity of the Naf ions scattered from N, and CO molecules, respectively. The open and full triangles indicate the intensity of the neutrals for the Na+-N, and Na+CO collisions, respectively. Intensities of the projectile ion beam and the target beams were the same for both collision systems. The neutrals were detected directly with a secondary electron multiplier by rejecting the scattered ions with a positive high voltage V,. When the high voltage V, is not supplied in our apparatus the ions and neutrals are detected simultaneously. The scattered ion intensity was therefore deduced by taking the difference between the two measurements of the total signals (ion+ neutral) and the neutrals. Although the ion intensity I( 19)shown in fig. 3 represents the relative total DCS a( 19))the angular dependence of the neutrals is apparent. This is because the energy of the neutrals depends on the scattering angle and the detection efficiency of the multiplier depends strongly on the impinging energy of the particles. The TOF experiments showed that the detected neutrals were the fragment C, N and 0 atoms which resulted from the collisional dissociation [7].
S. Kita et al. /Total DCS in Na+-N, and Na +-CO collisions
418
The scattered ion intensities for the two collision systems in fig. 3 have almost the same magnitude and show the same angular dependence at lab angles 19536’. A distinct difference between the ion intensities of these two systems, however, can be seen at 8s 36”. The ion intensity in Na+-N, collisions decreases rapidly at the angles larger than 8= 44”, while the angular dependence of the ion intensity for Na+CO changes its slope at 8x 36 and 5 1’. The angles of 36, 44 and 5 lo are indicated in the figure by arrows 1, 2 and 3, respectively. The difference in the scattered ion intensities between the Na+-N2 and Na+CO collisions was observed clearly at the collision energies of Elab 2 200 eV. The angles indicated with the arrows in figs. 2 and 3 are considered to be the cut-off angles in the molecular collisions, as discussed in the earlier work for K+-N2 and K+-CO collisions [ I]. 3.2. Energy loss spectra Fig. 4a shows the experimental energy loss spectrum of Na+ ions scattered from Nz molecules at I
I
I
Elab= 350 eV and the cut-off angle 8, = 44 a (arrOW 2 in fig. 3 ). In the present study, the energy loss spectra o(AE/E, 19) were deduced from the measured TOF spectra a( t, 0) with the relation of a(AE/E, 0) = a( t, 6J) dt/dAE. Around the cut-off angle, the ion signal distributes over a wide range of the flight time, 15 5 t 5 40 ps, with a broad and weak tail. To get enough intensity at the weak tail, with which our discussion will be concerned, the TOF measurements were made at the sacrifice of the time resolution, At x 0.5 us. The measured spectrum in fig. 4a shows a sharp cut-off at the largest energy loss, (AEIE), = 0.69. The cut-off position in the energy transfer spectra is given kinematically by ($)~=l-(~~sin20+
N&CO
(2)
A&
0(e)=
(d)
.
Since the experimental angular and energy resolutions are not taken into account in the data analysis, the energy loss spectra diverge at the cut-off energy. The divergence at (AE/E), originates from the Jacobian factor dt/dAE. The total differential cross section g( 0) is given by
I-
I
$$sin(XJ)
I
a(~,
e) dm.
(3)
0
0.4
0
ue
u.4
”
AE/E
Fig. 4. Energy loss spectra for Na+-N2 and Na+-CO collisions at I&,= 350 eV and cut-off angles. (a) Spectra measured for Na+Nz at 0=44” and (b) for Na+-CO at 6’=36”. (c) Spectra computed for Na+-N2 at 0~43” and (d) for Na+-CO at 0~36”.
At 8< e,, the majority of the ion signal distributes at the energy loss AE/E< (AEIE),, and the cut-off condition of eq. ( 2 ) gives no noticeable effect on the total DCS. At 8> & on the other hand, the energy loss distribution is restricted strongly by the cut-off condition, and then the total DCS fall off steeply as shown in fig. 3. Fig. 4b exhibits the experimental energy loss spectrum for the Na+-CO collisions at Elab= 3 50 eV and the first cut-off angle 0, = 36’ (arrow 1 in fig. 3 ). In this case the Jacobian peak is located at AEIEc0.78. In contrast to the spectrum of fig. 4a for the Na+-N2 collisions, a prominent peak A, which is located at AEIEx0.28, is seen in the spectrum of fig. 4b. As will be discussed below, peak A originates from the interaction on the O-atom side, while the broader peak B is due to the interaction on the C-atom side. Increasing the scattering angle, signal B vanishes rapidly, which causes the falling-off of the DCS as can be seen in fig. 3, and peak A broadens and shifts to larger
S. Kita et al. /Total DCS in Na +-N, and Na +-CO collisions
energy loss. If the angle is further increased, signal B disappears completely and signal A begins to vanish around 8= 5 1O. Therefore, the total DCS decrease steeply at 19>5 lo, which also can be seen in fig. 3. Thus the energy loss distribution in the spectra is reflected on the structure of the falling-off of the total DCS. For Na+-N, collisions at Elab= 100 eV and I% 49 o (the arrow in fig. 2), the energy loss spectrum also shows the cut-off structure around AE/E=0.64. Contrary to the spectra for high-energy collisions, a prominent peak is located at lower energy loss, AE/ ExO.1. This energy dependence of the energy loss spectra brings about the energy dependence of the cutoff structure in the total DCS in figs. 2 and 3, which will be discussed below. 3.3. Cut-off angles
angles and squares are for Na+-CO. At Elab= 100 eV, no noticeable structures in the DCS for Na+-CO could be found. The arrows labelled C, N and 0 in fig. 5 represent the cut-off angles for the binary Na+C, Na+-N and Na+-0 interactions, where the target velocity V, is ignored. The measured cut-off angles, ( 1) , (2 ) and (3 ), slowly approach the values given by the binary encounter model as the collision energy increases. This tendency has also been found by Inouye et al. [ 1 ] in K+-N2 and K+-CO collisions.
4. Classical trajectory calculation and discussion
The total DCS as well as the energy loss spectra were computed with the CT method. Interaction potentials for the collision systems were approximated by the additive interaction, Vh2,
Fig. 5 shows the energy dependences of the cut-off angles determined from the DCS measurements for the Na+-N2 and Na+-CO collisions. Open circles are the experimental values for Na+-N2, and open triI
I
1
I
I
50-
419
rz3,
r3,)=
Ur12)+
Ur23)+
Wr3,)
,
(4)
where the subscript 1 means Na + ion, 2 and 3 are constituent atoms of the N2 and CO molecules, r12, r23 and r3, are the interatomic distances. The ionatom interactions V( r, 2) and V( r31) were calculated with the statistical electron gas model [ 16 1. A Morse function was taken as the intramolecular potentials V(r23)forN2andC0 [17]. The collision geometry employed in the calculation is shown in fig. 6. The initial velocity vector v. lies in the yz plane and is parallel to the z axis, and b represents the impact parameter. The center of mass of the molecule is on the origin. a and /3are the polar and azimuthal angles, respectively, defining the moZ
t
I
600
0
E
Lob
(eV)
Fig. 5. Energy dependences of the cut-off angles 0,. ( 0 ) Experimental results for Na+-N2 collisions; (A ) and (o), experiments for Na+-CO. (0 ) Theoretical results obtained with the CT calculation of the DCS for Na+-N2 collisions; (A ) and ( n ) theoretical values evaluated from the DCS for Na+-CO. (-) CT calculations for the molecular orientation (Y= Ifi1 =90”; (---) CT calculation for the orientation ar= 120” and /3=90”. (---) Computations with the hard-shell model. The arrows C, N and 0 are the cut-off angles determined with the binary encounter model.
Fig. 6. Collision
geometry
for the classical trajectory
calculation.
S. Kita et al. /Total DCS in Na+-N, and Na +-CO collisions
420
lecular orientation in the space-fixed coordinate system. The CT calculation was carried out for the molecular orientation of -l
Here, N,, is the number of points chosen for the molecular orientation and vibrational initial phase. Its I
I
I
I
I
\
I
I
I
No+-N2,C0
‘\ :
I
ELab =lOOeV
‘\
I
20
I
I
40
I
I
60
I
I
80
0 (deg)
Fig. 7. Total DCS in the lab frame calculated with the CT method for the Na+-N2 and Na+-CO collisions at I&,= 100 eV. (p) DCS for Na+-N2 collisions; (---) DCS for Na+-CO. (---) DCS evaluated with the spherical potential of eq. (6).
value was 990 for the Na+-N, collisions and 1980 for Na+-CO. N(B) is the number of trajectories which fall into the angular window of A& 1o centered at each scattering angle 8. The interval Ab of impact parameter was taken 0.02 A. The total number of the trajectories was approximately 60000 for Na+-N, and 120000 for Na+-CO. The DCS a( 0) in the lab frame for the Na+-N, and Na+-CO collisions in fig. 7 show almost the same angular dependence, which is in good agreement with the experiments in fig. 2. For the Na+-N, collisions, a small hump in the computed DCS a( 0) can be seen around 8= 47’) which is indicated by an arrow in the figure. The spherical potentials averaged over the molecular orientation for Na+-N2 and Na+-CO, which were determined from the Na+-N, Na+-C and Na+0 interactions, can be approximated by V(R) =4010 exp( -3.82R)
eV
(6)
at the intermolecular distance of 0.8
S. Kita et al. /Total DCS in Na+-NJ and Na+-CO collisions I
,
I
I
I
I
No+-
20
40 0
60
(deg)
Fig. 8. Total DCS in the lab frame computed with the CT method for the Na+-N, and Na+-CO collisions at _I&= 350 eV. (-) DCS for Na+-N2; (---) DCS for Na+-CO.
loo% @ b
0 (deg) Fig. 9. Total DCS in the c.m. frame calculated with the CT method for the Na*-N2 and Na+-CO collisions at I&,= 350 eV. (-) DCS for Na+-N,; (---) DCS for Na+-CO.
4.2. Cut-off angles The cut-off angles evaluated with the CT calculation of the total DCS are given in fig. 5. The full circles represent the results computed for the Na+-N,
421
collisions. The full triangle is the result for the interaction on the C-atom side and the full square is that on the O-atom side in the Na+-CO collisions. The calculations are in good agreement with the experithe values calculated at ments. For Na+-N,, Elab= 100 and 200 eV are also shown. The value for Elab= 100 eV is the position of the small hump (the arrow in fig. 7 ) . 4.2.1. Energy loss spectra The energy loss spectra were computed with the CT method for the collisions at Elab= 350 eV to get information on the cut-off phenomenon. The trajectories used in the computation fell into the angular window A&2” centered at each cut-off angle in the computed DCS. The calculated cut-off angles are 43’ for Na+-N,, and 36 and 50” for Na+-CO. Fig. 4c exhibits the energy loss spectrum computed for the Na+-N, collisions at cut-off angle e,= 43 O.The spectrum shows a prominent peak at the cut-off energy (AE/E),=0.70, and reproduces fairly well the experiment in fig. 4a. The computed spectrum for the Na+-CO collisions at the first cut-off angle e,= 36’) which is shown in fig. 4d, also agrees qualitatively well with the experiment of fig. 4b. The cut-off angle f3, is related with the cut-off energy (AE/E), by eq. (2). The cut-off energy (AE/ E), given by eq. (2) is 0.70 for Na+-N, collisions at &=43’, and 0.78 and 0.62 for Na+-CO at &=36 and 50’) respectively. The trajectories, which fall into the window A(AE/E) ~0.02 at each cut-off energy, were analysed in order to clarify the cause of the sharp cut-off of the total DCS. According to the analysis, the peak at the cut-off energy is composed predominantly, 80°h, of the trajectories of the initial molecular orientations 60ia< 120” and 70 < lfil~90”. It was also found from the analysis that these trajectories are accompanied by the large angular momentum transfer. The collisions in the perpendicular configuration, cr= 1/?I = 90”, are typical ones to interpret the sharp cut-off in the DCS, which will be discussed in section 4.2.2. 4.2.2. Collisions in the perpendicular configuration Fig. 1Oa and the full curve in fig. 1Ob show the scattering lab angle and the reduced energy transfer AE/ E, respectively, as a function of the impact parameter b for the Na+-N, collisions at Elab= 350 eV. The col-
S. Kita et al. /Total DCS in Na+-N2 and Na+-CO collisions
422,
Y, tdeg 1
b (8)
Fig. 10. (a) The scattering lab angle 6’(b), (b) the reduced excitation energy AE(b)/E, and (c) the angular momentum J(b) calculated as a function of the impact parameter b for the perpendicular Na+-N, collisions at Elab= 350 eV. The collision geometry is inserted in (b). The full curve in (b) denotes the total excitation energy AE/E, and the broken curve gives the rotational excitation energy AL?,/E. The upper scale represents the molecular orientation yCat the turning point.
lision geometry is inserted in fig. lob, where the initial velocity vector u. and the molecular axis are inplane and at right angles, i.e. cr = p= 90”. The c.m. angle 8 is a monotonic function of the impact parameter b, i.e. @= 180, 137 and 40” for b= 0,0.6 and 1.2 A, respectively. The energy transfer is so big at small impact parameter b (large c.m. angle) that the deflection function B(b) has a maximum around b=0.65-0.7 A (19= 11%127”), where the energy AE/E is approximately 0.7. The maximum angle 0,,,, in the deflection function, which is 42.5”, is equivalent to the maximum lab angle in the Newton diagram. For the Na+-CO collisions in the perpendicular configuration, there are two different deflection functions corresponding to the collisions on the Catom and O-atom sides. The deflection functions for
E iat,= 350 eV give the angles &!,,,,= 36’ (C-atom side) and 50” (O-atom side). The solid curves in fig. 5 represent the energy dependences of the maximum angle &,,, in the deflection function 6(b) calculated for the collisions in the perpendicular configuration. These curves agree well with the experimental values (open symbols) and three-dimensional CT calculations (full symbols) at Elab3 200 eV. The curve calculated for Na+-N2 collisions at Elab < 200 eV, however, rises rapidly with decreasing the energy, and deviates largely from the experiment (open circle) and three-dimensional CT calculation (full circle) at Elab= 100 eV. The small hump around 8=47” in fig. 7 for Na+Nz collisions at Elab= 100 eV arises from the maximum in the function /3(b) for the molecular orientations only around (r=120” and /3=90”, which is different from the result for higher energies. The energy dependence of &,, for the Na+-N2 collisions at the molecular orientation (Y= 120” and p=90” is shown by the dashed curve in fig. 5. This curve represents well the energy dependence of the experiments and three-dimensional CT calculations over the whole energy range. 4.2.3. Cut-offstructure In order to get further information on the cut-off angle and the falling-off structure in the total DCS, additional CT calculation with a rigid-rotor model [ 19 ] was carried out for the Na+-N, collisions at &b = 350 eV. In the high-energy collisions, the rotational energy transfer AEJE exceeds the dissociation energy at large scattering angles [ 71. However, the intramolecular distance rz3 remains almost at the equilibrium value r, of the isolated molecule during the collisions, because rotational excitation takes place suddenly. We can therefore safely use the rigidrotor model to estimate the angular momentum transfer. The rigid-rotor calculation shows that the cut-off angle appears at 6, z 45’) which is nearly equal to the result obtained with the above vibrotor calculation in fig. 8, 0, = 43 ‘. The DCS by the rigid-rotor calculation at large angles of 6> 45 ‘, however, fall with a slope smaller than that by the vibrotor calculation in fig. 8. These results suggest that the cut-off position is determined by the angular momentum transfer, while steepness of the falling-off of the DCS is given predominantly by vibrational excitation. One
S. Kita et al. /Total DCS in Na+-NJ and Na +-CO
can, therefore, conclude that the strong energy dependence of the cut-off structure, i.e. sharper fallingoff of the DCS at high energy, is due to the effect of vibrational excitation. 4.2.4. Rotational rainbow eflect Fig. 1Oc shows the angular momentum transferred to the Nz molecule as a function of the impact parameter b, where the collisional condition is taken the same as that of figs. 10a and lob. The angular momentum J takes the maximum value J,,,x720 around bs0.65 A. The dashed curve in fig. lob represents the rotational energy transfer A&/E, which was evaluated with the intramolecular distance rz3( = re) at the turning point in the collision [ 71. The angular momentum J(b) and the rotational energy transfer A&(b) /E by the vibrotor calculation in fig. 10 are both nearly equal to those obtained by the rigidrotor model. As can be seen in fig. lob, the energy transfer in the collisions around the maximum angle e max,which corresponds to the cut-off angle & is exclusively due to rotational excitation. The cut-off position is, therefore, determined by the angular momentum transfer. The upper scale in fig. 10 gives the.molecular orientation yc at the closest approach, y being the angle between the molecular axis and the intermolecular distance vector R.The maximum in the deflection function appears at the collisions of yc = 50”) which is the rotational rainbow angle as discussed previously [ $81. Therefore, the cut-off positions and their energy dependence can be qualitatively explained by the rotational rainbow effect [ 3,201. The hard-shell model is very useful to evaluate the rotational energy in the impulsive collisions [ 581. We computed the rotational rainbow positions as a function of c.m. angle 8 with the two-dimensional hard-shell model proposed by Bosanac [ 2 11, and determined the Newton diagram. Function R*(y) defining the hard-shell size is assumed to be given by the equipotential surface, following Hasegawa et al. [ 5 1. Increasing the scattering angle @, the rotational rainbow energy AERIE increases rapidly. As a result, the Newton diagram shows a cut-off angle 19,. The dotted curves in fig. 5 exhibit the energy dependences of the cut-off angles obtained with the hardshell model. As shown, the computations reproduce qualitatively well the experiments. One rotational rainbow appears in the energy loss
collisions
423
spectra for the scattering ofNa+ from the homonuclear molecule N2 and two rotational rainbows for Na+-CO collisions, which correspond to one cut-off angle for Na+-N, and two cut-off angles for Na+CO in the total DCS. As reported previously in ref. [ 5 1, the rotational rainbow position Al&/E at a fixed angle weakly depends on the collision energy and hence the cut-off position should also behave the same. The rotational rainbow originates from the collisions at the molecular orientation yRx 50”. For ion-molecule collisions around the rotational rainbow angle yR, the repulsive potential of the COlliSiOn system is expected to be determined by the ion-atom interaction of the closest pair. This means that the ion-molecule collisions around the rainbow angle can be regarded approximately as the ion-atom interactions. Therefore the cut-off angles ( 1 ), (2) and (3 ) are close to 6, of the Na+-C, Na+-N and Na+-0 interactions, respectively.
5. Conclusion The angular and energy depencences of the total DCS measured for the Na+-N2 and Na+-CO collisions could be well reproduced with the classical trajectory calculation using the simple additive potentials based on the Na+-N, Na+-C and Na+-0 interactions. The characteristic difference in the DCS for the high-energy Na+-N, and Na+-CO collisions is due to the difference in the momentum transfer. The cut-off positions in the DCS, 0,, were interpreted mostly by assuming the collisions to be in the perpendicular configuration, and were found to be related to the rotational rainbow effect.
Acknowledgement We are grateful for valuable discussion with Professors H. Inouye (Bunri University Tokushima, Tokushima, Japan), I. Kusunoki, Y. Sato (Tohoku University, Sendai, Japan) and Dr. M. Nakamura (Institute for Molecular Science, Okazaki, Japan). We are also indebted to Mr. T. Hasegawa for his contribution to this study. This work was financially supported in part by a Grant-in-Aid for Scientific
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S. Kita et al. /Total DCS in Na+-NJ and Na+-CO collisions
Research from the Ministry of Education, Science and Culture of Japan.
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