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Rotational—vibrational energy transfer in collisions of Li+ with N2 and co in the 1–10 eV range
Volume
CHEMICAL
61, number 2
ROTATIONAL-VIBRATIONAL IN THE I-10
eV RAGE
ENERGY
THYSICS
TRANSFER
1 April
LETTERS
IN COLLISIONS
1979
OF Li+ WITH N2 AND CO
*
David A. MICHA. E. VILALLONGA Quanrum Tizeo~ Pro~ecr. Depurnnenrs of
CJtemistr_v and PJt> sirs. Unil ersiry of Ho&a.
GainesviUe. FJorida 335 2 I. VS.4
and J-P_ TOENNIES
Received 19 Oczober 1978; in finrttform IS December 1978
CAxdations of time-of-flight spectra for Li+ + h‘z and Li+ + CO hate been cxried out v.ith a model based on paircorrehtion functions and ztsirqle ph)sical parameter. Dependences on relative ener$es and detection amglesagree with periments for both sytems.
I_ Mroduction The purpose of this letter is to apply a recently developed theory of moleculal collisions [ I.21 to the ulculation of vibrational-rotational cross sections of scattered ions, measured with time-of-flight techniques [3]. The theoretical results are expressed in terms of pair-correlation functions of the molecular targets, following assumptions appropriate to the experimental conditions of quasi-elastic scattering in the eV range of coIlision energies_ The systems Lit + N2 and Lit -I-CO [4] have been chosen here as tests of the dynamic31 model, and in particular of the significance of correlation functions. These functions c3n be very readily cakulated and contain a wealth of information on collisional ener,T and momentum transfer_ They are also easily obtained for polyatomic targets, ~1hich h3ve so far been only sparingly studied in theory, because of the complexity of present models_ Previously, cakulations with an impuIse model have been carried out for Li+-CO [5]_ Extensive classical trajectory studies with realistic potentiak have also * PmtJysupported by NFS Grant 7710510. 238
ex-
been published for Li+-CO and Lii -N, [7], and shonv varying degree of agreement with experiment_ Our theoretical approach is also based on an impulsive
model, deduced from a rigorous theory of three-atom collisions and used here in 3 simplified version_ Satisfactory agreement is obtained within the present model with adjustment of a single parameter for severa! energies and angles of scattering. This agreement results from properfy incorporating the statistic31 distributions of initial states and describing the slow rotational motions within the pair-correlation functions_
2. Quasi-elastic collisions in the eV range In the mentioned experiments, Lii ions of sever31 eV loose energies e of the order of a few vibrational quanta of N2 or CO, so that E4
E = fi3k3/2hf,
K =
fk - k’l’
where
= ze(l
(1)
- cos 0) ,
(2)
is the initial relative momentum,M the reduced mass of the JY-BC system, and tl K the momentum tmnsfer for scattering angle 0. tzk
Volume
CHEMICAL
62. number 2
PHYSICS
At the collision energies and angles of interest the intermolecular forces at play are repulsive and steep, justifying two basic assumptions: (1) single collisions of the projectile Lit with each atom in the target, and (2) impulsive transfer in which internal potential energy changes rue negligible. With these assumptions the cross sections per unit solid angle and unit energy
are given by [ 1,2] d%r/dfide
=Co,(k',k)S(nn)(K,w)/~ , a
(3)
in which the sum extends over the target atoms a, and on is the cross section for scattering of hi+ by atom CI as seen in their own center-of-mass (c_m_) system Also, S(m) is the atom-pair self-correlation function PI
3_ The correlation
Fe(O)] exp[ir*
f&)]Ia)
,
(4)
in which P&) represents the position of atom a with respect to the target cm. at time t, w = $I and an average is indicated over a thermal distribution w-(T) of internal states LY. Because the scattering is quasi-elastic, and derection angles are larger than rninbow angles, uc can be calculated from atom-atom radial potentials for elastic scattering with relative velocity fik/Mand scattering angle 0. In eq_ (3) we have further neglected correlations between different atoms in the target because of random-phase averaging [?I. This however is not necessary but only convenient in applications of the model. The interaction potential I’was taken of the form V(R,r)
u&)
=Cu,(IR
a
= A, exp(--b,r)
-r-J),
(5a)
,
Gb)
where R is the relative position of Li+ and r represents target atom positions, all with respect to the target cm. The detailed form of u, is not essential, as long as its values are correct around the classical turning points.
functions
Calculation of the Scnn) functions for the present applicationspresent new problems, because E is much larger than rotational energy spacing_ so that sums over rotational states are impractical. Also, momentum transfers are much larger than in other applications [SJO], which means, indicating with (I the equilibrium diatomic distance, that KCI> 1 (i.e. large angular momentum transfer)_ We shall only outline our procedure for the mtermedkte Fourier transform b*“)(K_ t) of S(a), and refer elsewhere [9] for more details_ We consider a diatomic of reduced mass 111,moment of inertia I, and vibrational quanta ho,_ Firstly. for harmonic vibmtional motion one cm exactly calculate the correlation of displacements at each mo1ecuIar orientation [ lo], which gives F(K,
X (ffIexp[-iw.
1 April 1979
LETTERS
f)
=
fi
1,=--o
eXp(-irlwvr)
exp(---izcr,,)f,,(K,
t) , (6)
with II the chsnge in vibrational qumtum numbers, and (Y, = frw, /2k,T_Since collision times t,, are much shorter than rotational transfer times (At),, 22 ii/fzR, we use for f,,(r) 3 short-time expansion (see for example ref_ [ 111) ln
f;r(r) l
it’{
= ln Jr (0) f r&1(’‘(0)/f,j”‘(O) r,,‘“‘(0)/_&(O) -
v~“6%%,m
‘I J
(7)
with upper indices indicating the order of timederivatnes. These can be conleniently approximated in the Heisenberg picture using that hd 3 1, to obtain
f:)(O)= (iwl)r
i 0
d;r ekp [-(chg)~~$]
x (1 - ‘u?-)l,(xc)$)
)
(9
where f~wt = (mifi~/m,)~~~I is a rotational energy transfer, x0 = (mfih -/ t rrc)‘-/ZnrTrw,(shcrX) and i,, is a modified Bessel function_ From the inverse Fourier transform, the final result is S(@(K, sI,(K,
X
W) =cS,r(K, rr w)
=
a),
(2/rWb,)exp[-(w
expWq
)f;, (0)
,
(9a) from, - a,,)Q,;]
(9b) 239
2 that is a superposition of ganssian distributions centered at an - nov_
4.Tiie-of-flight
1 April 1979
CHEMICAL PHYSICS LETTERS
Volume 62. number2
I._?+ CO Ei = TO70 eV
S
6 = 432 deg [CM)
2 n = Q Q r;: z B %
spectra for Na and CO
Usingmassesmn = 14_0,mC =lZO,mo =16-O amu, diatomic parameters fi*/Z - Z-49 X 10-l eV, Jiw, = O-Z90 eV for N,,and 239 X 1O-4 eV, 0266 eV respectively for CO, as well as the interatomic potential constants A, = 307937 eV, b, = 5258 I A-1, A, = 190851 eV, 6, = 5.9491 ii-l, A, = 5237.46 eV, and b, = 4_8578 -4-l (obtained by fitting potential surfaces [6,17-l) calculations were carried out for experimental energies and detection angIes, and a vibrational temperature T = 300 K. It was noticed that the cakulated rotationaI peaks for rz = 0 were too high_ Arm&sing the model, it was concluded that the negiected Iong-range anisotropy might lead to an apparent increase of moments of inertia, so that we repIaced wl + 01/3. With this singIe adjustment, kept the same for both IS? and CO, good agreement was achieved for al1 energies and angIe.s.This may be seen in tabie I and figs. I and 2, where experimental results are compared with theoretical calculations. The experimental rotational energy transfers WR) in table 1 were obtained from the most probable fmal AER
62
65
66
66
70
72
Er (evr Fis I_ CakuIated time-uf-flightspectrafor li* + CO at relztive Ei= 7.070 eV and c-m. sngie 0 = 43.2”: --; experimental resuhs: ---_ Changesin vibrationalquantumnumber are indicatedwith )r = nf - Ri_
Et = 2070 eV Q = 43 2 deg [CM)
0
10
20
60
30
60
Fig. 2. Calculated dktributionlbf rotational quantum numbers if, corresponding to fig_ 1, for the icdiated vibrational quan-
tum numberchangesn = nf - “ix --; _--_
Zb!e I
experimentalresults:
Average rotational energy transfer C&R) for no vibrational excitation. and total average energy transfers (LIB for initial relative energies Eiand c-m. deiection angIe-~@cm__ Experimental results are from ref. [41. and theoretical ones as indicated in the test, with vibrationat temperature T= 300 I;
Li++co
Li++Nz CAL3/Ei
(=I@ (ev) esp.
d
t&R)
th.
esp_ b)
th_
cup. 3
th.
exp_ b)
th..
O-080 0_108
0.019 0.022
0.0262 0.0349
4.23 4.23 423
33.1 43-2 492
0.087 0.090 O-095
0.076 0.103 O-132
0.030 0.041 0.051
0.0253 0.0336 0_0428
0.085 0.092 0.130
7.07 7-07
372
0_142
0.131
0.036
0_0254
0.202
7.07 - ___
43-2 49.2
0.138 0.164
tAB/Ei
(ev)
0.177 0.229
o_oso 0.061
0.0338 0.0429
o-237 0.237
0.139
0.137 a1 86 0.242
O-032
0.027 0.041 0.046
3) G?&R>c&a&ted from the most probabie tinal PER xisuminggaussizn distributions. for& = 0. nf - ni = 0, in the cm. b)Taken from ref_ [41_
240
OXI
o-0264 O-0351 0.0447 SYSkm.
Volume 62, number 2
CHEMICAL PHYSICS LEl-l-ERS
and assumed gaussian distributions_ Rotational and total energy transfers (the first moments of spectral distriiutions) are on the average within about 10%. In fig. 1, the decrease in heights of vibrational peaks reflect dependences on varying momentum transfer K(Ei. Ef. O)_ Finally, the shoulder for largejf in fig. 2 reflects the contribution of C scattering, which is shifted from 0 scattering by mass effects.
I April 1979
AcknowIedgement The authors thank the Scientific Affairs Division of NATO for the award of a Research Grant_ CaIculations were partly supported by the Northeast Florida Regional Data Center of the University of Florida_
References 5. Discussion The agreemem between theoretical and experimental results is very encouraging, and suggests that correlation functions play a central role in energy-transfer collisions_ The model avoids the Born approximation, which is inapplicable for the present potentials. The assumptions of single collisions and of impulsive processes have been independently checked and appear justified [ IS]. Furthermore, the present model opens many possibilities for studying scattering by polyatomic molecules, in terms of their correlation functions_
[l] [ 31 [3] [4] [5]
[6] [7] [8] [9] [lo] [ 111 [I?] [ 131
D.A. Micha, Chem_ Phqs. Letters46 (1977) 188_ D.A. Michn. JI Chem. Phys_, to be pubhshedJ.P_ Toennies. Ann. Rev. Phys. Chem. 27 (1976) 225. R. Bijttner, IJ. Ross and J.P. Toennies, J. Chem. Phys. 65 (1976) 733. V. Philipp, H.J. Korsch and P. Eckeit, J. Phys. B9 (1976) 345. L.D. Thomas, W.P. Kramer and G.H.F_ Diercksen, Chem. Phys. 30 (1978) 33_ D. Poppe and R. Biittner. Chem. Phys_ 30 (1978) 375. L. ran Hove, Phys. Rev. 95 (19.54) 249. D.A. Micha. University of Florida. QTP Report 523 (May 1978)_ A-C. Zemach and R-J. Glauber,Phys_ Rev. 101 (1956) 118.1~9. N.H. March and M.P. Tosi, Atomic dynamics in liquids (Wle>. New York, 1976) ch. 3. V_ Staemmler. Chem_ Phys. 7 (1975) 17. L. Beard and D-k blicha, to bc published.
241
CHEMICAL
61, number 2
ROTATIONAL-VIBRATIONAL IN THE I-10
eV RAGE
ENERGY
THYSICS
TRANSFER
1 April
LETTERS
IN COLLISIONS
1979
OF Li+ WITH N2 AND CO
*
David A. MICHA. E. VILALLONGA Quanrum Tizeo~ Pro~ecr. Depurnnenrs of
CJtemistr_v and PJt> sirs. Unil ersiry of Ho&a.
GainesviUe. FJorida 335 2 I. VS.4
and J-P_ TOENNIES
Received 19 Oczober 1978; in finrttform IS December 1978
CAxdations of time-of-flight spectra for Li+ + h‘z and Li+ + CO hate been cxried out v.ith a model based on paircorrehtion functions and ztsirqle ph)sical parameter. Dependences on relative ener$es and detection amglesagree with periments for both sytems.
I_ Mroduction The purpose of this letter is to apply a recently developed theory of moleculal collisions [ I.21 to the ulculation of vibrational-rotational cross sections of scattered ions, measured with time-of-flight techniques [3]. The theoretical results are expressed in terms of pair-correlation functions of the molecular targets, following assumptions appropriate to the experimental conditions of quasi-elastic scattering in the eV range of coIlision energies_ The systems Lit + N2 and Lit -I-CO [4] have been chosen here as tests of the dynamic31 model, and in particular of the significance of correlation functions. These functions c3n be very readily cakulated and contain a wealth of information on collisional ener,T and momentum transfer_ They are also easily obtained for polyatomic targets, ~1hich h3ve so far been only sparingly studied in theory, because of the complexity of present models_ Previously, cakulations with an impuIse model have been carried out for Li+-CO [5]_ Extensive classical trajectory studies with realistic potentiak have also * PmtJysupported by NFS Grant 7710510. 238
ex-
been published for Li+-CO and Lii -N, [7], and shonv varying degree of agreement with experiment_ Our theoretical approach is also based on an impulsive
model, deduced from a rigorous theory of three-atom collisions and used here in 3 simplified version_ Satisfactory agreement is obtained within the present model with adjustment of a single parameter for severa! energies and angles of scattering. This agreement results from properfy incorporating the statistic31 distributions of initial states and describing the slow rotational motions within the pair-correlation functions_
2. Quasi-elastic collisions in the eV range In the mentioned experiments, Lii ions of sever31 eV loose energies e of the order of a few vibrational quanta of N2 or CO, so that E4
E = fi3k3/2hf,
K =
fk - k’l’
where
= ze(l
(1)
- cos 0) ,
(2)
is the initial relative momentum,M the reduced mass of the JY-BC system, and tl K the momentum tmnsfer for scattering angle 0. tzk
Volume
CHEMICAL
62. number 2
PHYSICS
At the collision energies and angles of interest the intermolecular forces at play are repulsive and steep, justifying two basic assumptions: (1) single collisions of the projectile Lit with each atom in the target, and (2) impulsive transfer in which internal potential energy changes rue negligible. With these assumptions the cross sections per unit solid angle and unit energy
are given by [ 1,2] d%r/dfide
=Co,(k',k)S(nn)(K,w)/~ , a
(3)
in which the sum extends over the target atoms a, and on is the cross section for scattering of hi+ by atom CI as seen in their own center-of-mass (c_m_) system Also, S(m) is the atom-pair self-correlation function PI
3_ The correlation
Fe(O)] exp[ir*
f&)]Ia)
,
(4)
in which P&) represents the position of atom a with respect to the target cm. at time t, w = $I and an average is indicated over a thermal distribution w-(T) of internal states LY. Because the scattering is quasi-elastic, and derection angles are larger than rninbow angles, uc can be calculated from atom-atom radial potentials for elastic scattering with relative velocity fik/Mand scattering angle 0. In eq_ (3) we have further neglected correlations between different atoms in the target because of random-phase averaging [?I. This however is not necessary but only convenient in applications of the model. The interaction potential I’was taken of the form V(R,r)
u&)
=Cu,(IR
a
= A, exp(--b,r)
-r-J),
(5a)
,
Gb)
where R is the relative position of Li+ and r represents target atom positions, all with respect to the target cm. The detailed form of u, is not essential, as long as its values are correct around the classical turning points.
functions
Calculation of the Scnn) functions for the present applicationspresent new problems, because E is much larger than rotational energy spacing_ so that sums over rotational states are impractical. Also, momentum transfers are much larger than in other applications [SJO], which means, indicating with (I the equilibrium diatomic distance, that KCI> 1 (i.e. large angular momentum transfer)_ We shall only outline our procedure for the mtermedkte Fourier transform b*“)(K_ t) of S(a), and refer elsewhere [9] for more details_ We consider a diatomic of reduced mass 111,moment of inertia I, and vibrational quanta ho,_ Firstly. for harmonic vibmtional motion one cm exactly calculate the correlation of displacements at each mo1ecuIar orientation [ lo], which gives F(K,
X (ffIexp[-iw.
1 April 1979
LETTERS
f)
=
fi
1,=--o
eXp(-irlwvr)
exp(---izcr,,)f,,(K,
t) , (6)
with II the chsnge in vibrational qumtum numbers, and (Y, = frw, /2k,T_Since collision times t,, are much shorter than rotational transfer times (At),, 22 ii/fzR, we use for f,,(r) 3 short-time expansion (see for example ref_ [ 111) ln
f;r(r) l
it’{
= ln Jr (0) f r&1(’‘(0)/f,j”‘(O) r,,‘“‘(0)/_&(O) -
v~“6%%,m
‘I J
(7)
with upper indices indicating the order of timederivatnes. These can be conleniently approximated in the Heisenberg picture using that hd 3 1, to obtain
f:)(O)= (iwl)r
i 0
d;r ekp [-(chg)~~$]
x (1 - ‘u?-)l,(xc)$)
)
(9
where f~wt = (mifi~/m,)~~~I is a rotational energy transfer, x0 = (mfih -/ t rrc)‘-/ZnrTrw,(shcrX) and i,, is a modified Bessel function_ From the inverse Fourier transform, the final result is S(@(K, sI,(K,
X
W) =cS,r(K, rr w)
=
a),
(2/rWb,)exp[-(w
expWq
)f;, (0)
,
(9a) from, - a,,)Q,;]
(9b) 239
2 that is a superposition of ganssian distributions centered at an - nov_
4.Tiie-of-flight
1 April 1979
CHEMICAL PHYSICS LETTERS
Volume 62. number2
I._?+ CO Ei = TO70 eV
S
6 = 432 deg [CM)
2 n = Q Q r;: z B %
spectra for Na and CO
Usingmassesmn = 14_0,mC =lZO,mo =16-O amu, diatomic parameters fi*/Z - Z-49 X 10-l eV, Jiw, = O-Z90 eV for N,,and 239 X 1O-4 eV, 0266 eV respectively for CO, as well as the interatomic potential constants A, = 307937 eV, b, = 5258 I A-1, A, = 190851 eV, 6, = 5.9491 ii-l, A, = 5237.46 eV, and b, = 4_8578 -4-l (obtained by fitting potential surfaces [6,17-l) calculations were carried out for experimental energies and detection angIes, and a vibrational temperature T = 300 K. It was noticed that the cakulated rotationaI peaks for rz = 0 were too high_ Arm&sing the model, it was concluded that the negiected Iong-range anisotropy might lead to an apparent increase of moments of inertia, so that we repIaced wl + 01/3. With this singIe adjustment, kept the same for both IS? and CO, good agreement was achieved for al1 energies and angIe.s.This may be seen in tabie I and figs. I and 2, where experimental results are compared with theoretical calculations. The experimental rotational energy transfers WR) in table 1 were obtained from the most probable fmal AER
62
65
66
66
70
72
Er (evr Fis I_ CakuIated time-uf-flightspectrafor li* + CO at relztive Ei= 7.070 eV and c-m. sngie 0 = 43.2”: --; experimental resuhs: ---_ Changesin vibrationalquantumnumber are indicatedwith )r = nf - Ri_
Et = 2070 eV Q = 43 2 deg [CM)
0
10
20
60
30
60
Fig. 2. Calculated dktributionlbf rotational quantum numbers if, corresponding to fig_ 1, for the icdiated vibrational quan-
tum numberchangesn = nf - “ix --; _--_
Zb!e I
experimentalresults:
Average rotational energy transfer C&R) for no vibrational excitation. and total average energy transfers (LIB for initial relative energies Eiand c-m. deiection angIe-~@cm__ Experimental results are from ref. [41. and theoretical ones as indicated in the test, with vibrationat temperature T= 300 I;
Li++co
Li++Nz CAL3/Ei
(=I@ (ev) esp.
d
t&R)
th.
esp_ b)
th_
cup. 3
th.
exp_ b)
th..
O-080 0_108
0.019 0.022
0.0262 0.0349
4.23 4.23 423
33.1 43-2 492
0.087 0.090 O-095
0.076 0.103 O-132
0.030 0.041 0.051
0.0253 0.0336 0_0428
0.085 0.092 0.130
7.07 7-07
372
0_142
0.131
0.036
0_0254
0.202
7.07 - ___
43-2 49.2
0.138 0.164
tAB/Ei
(ev)
0.177 0.229
o_oso 0.061
0.0338 0.0429
o-237 0.237
0.139
0.137 a1 86 0.242
O-032
0.027 0.041 0.046
3) G?&R>c&a&ted from the most probabie tinal PER xisuminggaussizn distributions. for& = 0. nf - ni = 0, in the cm. b)Taken from ref_ [41_
240
OXI
o-0264 O-0351 0.0447 SYSkm.
Volume 62, number 2
CHEMICAL PHYSICS LEl-l-ERS
and assumed gaussian distributions_ Rotational and total energy transfers (the first moments of spectral distriiutions) are on the average within about 10%. In fig. 1, the decrease in heights of vibrational peaks reflect dependences on varying momentum transfer K(Ei. Ef. O)_ Finally, the shoulder for largejf in fig. 2 reflects the contribution of C scattering, which is shifted from 0 scattering by mass effects.
I April 1979
AcknowIedgement The authors thank the Scientific Affairs Division of NATO for the award of a Research Grant_ CaIculations were partly supported by the Northeast Florida Regional Data Center of the University of Florida_
References 5. Discussion The agreemem between theoretical and experimental results is very encouraging, and suggests that correlation functions play a central role in energy-transfer collisions_ The model avoids the Born approximation, which is inapplicable for the present potentials. The assumptions of single collisions and of impulsive processes have been independently checked and appear justified [ IS]. Furthermore, the present model opens many possibilities for studying scattering by polyatomic molecules, in terms of their correlation functions_
[l] [ 31 [3] [4] [5]
[6] [7] [8] [9] [lo] [ 111 [I?] [ 131
D.A. Micha, Chem_ Phqs. Letters46 (1977) 188_ D.A. Michn. JI Chem. Phys_, to be pubhshedJ.P_ Toennies. Ann. Rev. Phys. Chem. 27 (1976) 225. R. Bijttner, IJ. Ross and J.P. Toennies, J. Chem. Phys. 65 (1976) 733. V. Philipp, H.J. Korsch and P. Eckeit, J. Phys. B9 (1976) 345. L.D. Thomas, W.P. Kramer and G.H.F_ Diercksen, Chem. Phys. 30 (1978) 33_ D. Poppe and R. Biittner. Chem. Phys_ 30 (1978) 375. L. ran Hove, Phys. Rev. 95 (19.54) 249. D.A. Micha. University of Florida. QTP Report 523 (May 1978)_ A-C. Zemach and R-J. Glauber,Phys_ Rev. 101 (1956) 118.1~9. N.H. March and M.P. Tosi, Atomic dynamics in liquids (Wle>. New York, 1976) ch. 3. V_ Staemmler. Chem_ Phys. 7 (1975) 17. L. Beard and D-k blicha, to bc published.
241