- Email: [email protected]
Velocity dependence of the total cross section for helium-argon, neon-krypton and neon-xenon scattering
Volume 8, number 1
CHEMICAL PHYSICS LETTERS
VELOCITY FOR
DEPENDENCE
HELIUM-ARGON,
OF
THE
NE’ON-KRYPTON G. D. LEMPERT,
1 Jan&y
TOTAL AND
CROSS
1971
SECTiON-
NEON-XENON
SCATTERING
S. J. B. CORRIGAN
Department of Physics, GowevStreef,
University College Londotz, London If.JClcI.UK
and Department
J. F. WILSON of Chemistry, University College London. Gower Street, Londo~~ WCl, UK
_ Received 3 August 1970 Revised manuscript received 11 November 1970 Velocity dependences of the total collision cross sections for scattering of helium by argon, neon by krypton and neon by xenon were measured using a time of flight method of velocity selection. For neon scattering glory oscillations were observed from which values of EY, (the product of the welI depth and the internuclear separation at the rninimumj were determined for various Lennard-Jones pntontials. Comnarison is made with the results from diffusion experiments and differential scattering measurements.
The widespread and fundamental influence of intermolecular forces in chemistry [l] suggests that their role on the molecular scale is as important as the role of thermo-dynamics in the properties and behavlour of the bulk phase. Although considerable advances have been made in recent years in improving the accuracy of the calculation of these forces between atoms at large separation where exchange forces are negligible [2], only in a few cases, -such as heliumhelium interactions [3], have there been accurate calculations of fhe short range forces. Molecular beam scattering techniques have -been recognized for some time as the most sensitive means of measuring intermolecular forces [4,.5]. The high energy experiments by Amdur and‘co-workers [6] have yielded reliable measurements of the short range forces between rare &as atoms. -The- intermediate and long range forces have been obtained through m&surements. of the differential and total elastic C-TOSS s&ions over a wide range of relative Fielocities [4]. &I-’ though potential parametdrs have been extensively measq;red for interactions invdlving an ,alk& atqm,’ there qe only- a few resylts_ [?-‘lO] available foe scattering i.n+olving non-alkali speciks. In this letter we.cjescribe a new ‘aphidach to ..the.problem of obtai+ig fhy %eliiciq dependence.:. .’ ,‘_. : -_ .__ “,._ -_ .-: ,_i .I
.
.
.
._-
‘,
,,_
’
;_ ..
of total collision cross sections and give some results that haire been obtained for helium-argon, neon-krypton and neon-xenon scattering_ An atomic beam of circular cross section was produced by an effusive source, passed through a scattering gas chamber, and detected by an electron bombardment ionizer followed by a quadrupole mass-spectrometer, electron multiplier and pulse counti& system. The source, the scattering gas chamber and the detector were In separate differentially pumped regions. Another differen tial pumping stage was provided before the detector to reduce the effusive background signal. The angular resolution of the apparatus, as defined by the angle subtended at the scattering gas chamber by the detector, was 6’. The velocity depe.\de&e of total collision cross sections was obtained by observing changes in transmitted intensities at different velocities ln the beam as t$e pres: ure of the gas in the scattering chamber was vtied. Velocity selection of the -atomic beam was bbtalned by a time of flight ~method. F&r the-source the bean? was pulsed by -a-narrow slit (=50 jfsec in time)- in a rotating disc. On the same &aft as the pulsing slit a second disc with a wider Sj!it trans+ted most of the pulsk, &t‘,cut off the slo.w~&toms sd that, before the ne$ ptilse,’ there wars a time when tie beam ,._._ ,.’ ‘. ,_..’ ._ .‘_ __ -. -67 _’
.,
.__I
.
..
.-
: ,
* ._.
-
.__’
., .
.,
_
_
. _
_
-.-
._ -.
..-_
_
Volume 8. number 1
CHEMICALPHYslCS LETTERS
1.
1‘January 1971
atoms were arriving ai the detector. -Pulses from the. multiplier were fed into four scalers which
were individually switched oti and off with variable delay time and gate width by circuits syn..chronised with the beam chopper. This enabled different scalers tb measure the beam intensity for different weli.definecj velocities. The instantaneous beam intensity was measured at the four selected delay times during each beam pulse by summing the gated signals for many (=104) consecutive pulses. The delay of one of the scalers was set at the peak of the time of flight distributiqn; th@ channel served as a reference intensity. -Another was fixed at a long delay time to give a measure of the background intensity. Two other delay times were varied to scan the velocity distribution. With these four channels it was possible in this way to measure the background and the beam intensjties at three different velocities in each cycle. and thus effectively simultaneously. There are several advantages in this method of velocity selection.- Firstly, the velocity dependence of a cross section can be determined without any measurement of the scattering gas pressure because the same pressure attenuates the intensities at each of the velocities selected. Secondly, since the fluctuations in the beam intensity, background pressure, scattering gas pressure and detector sensitivity are simultaneously common to each velocity, the errors due to these fluctuations cancel. This is verified by the scatter in the experimental data which does not exceed that which must arise from the statistics of counting. Thirdly, for a given resolution compared with a mechanically realisable slotted disc selector Ill], measurements can be extended to much higher velocities because the effective length of the selector is the length between the first disc and the detector. Fourthly, .by varying an electrical parameter (the detector gate open time) the resolution and transmission of this selector can be altered. ,In the_experiments reported here the velocity resolution (L ~11width at half maximum), which can be shown to be equal to detector gate open time divided by the delay time, ranges from 5% to 18% for the helium beam and from 3% to 5% for the neon b&am. No measurements were made of the absolute values of the total cross section. Fig. & qhows .the .veiocity dependence of the total cross section for helium-argon scattering for helium yelocjties of 700 to 2400_m/sec. Each. point corresponds tb many m+surements at that velocity with different scattering gas pressures. Tiie solid curve was_ calculated from .@_,
(
.’
.-
1
zil.o(’ LOO
600
1500zooO
6wlwo
300
HELIUM ATOM VELOCITY Imkc)
Fig. 1. Velocity dependence of the total cross section for thz scattering of helium atoms by argon. QV2/S is plotted against V on a log-log scale. Q =$
lFo (21+1) sin%l
for a Lennard-Jones (12,6) potential with the parameters obtained by Helbing et al. [‘i’] (E, well depth = 0.361 x lo-14 erg, r,, internuclear separation at the minimum = 3.40A). In eq. (1) k is the incoming wave number and the values of vl were obtained by numerically integrating the SchrBdinger equation for the scattering. The good agreement confirms the result of Helbing et al. within the range of these experiments. Fig. 2 shows the velocity dependence of the total cross section for neon-krypton and neonxenon scattering, for the velocity range 450 to 1200 m/set. For the velocity range of the neon experiments the total cross section for a LennardJones (12,6) potential is given by the WKB approx-
imation as [12] 2Er6 Q = 8.083 c+)
2’5
F (I-g2sin(2qm
- 38/4), (2) m where Lo is. the value of the angular momentum, 1, for which the phase shift attain6 its maximum value,. Qm, and $n = (ds/dZ2),,, The first term is the Schiff-Landau-Lifshitz total cross section and the second term contributes the glory oscillations. -The velocities at which the maxima and plinimd of the function QV2i5 occur-are sensitive principally to the product •7~ and also to the form of the’ interaction potential. [12]. The amplitude of the glory osdl!ations is much. less sensitive-to -the velocity and to the..potentia!. parameter& :’ -1 1 -_ -. .:. . +
Volume 8. number
PIiYSICS LETTERS
CHEMICAL
1
16-
YI .= 5 t e
lo-
f
20-
,r > ci
18-
NelKr
161L 12 -
NEON
ATOM
VELOCITY
(mlsec
1
Fig. 2. Velocity dependence of the total cross section for the scattering of neon atoms by krypton and xenon. Q@2/5 is plotted against V on a semi-log scale.
The velocity dependence of QVzi5 was calculated for a Lennard-Jones (l&6) potential from eq. (2), averaged over the scattering gas motion and corrected for the angular resolution of the apparatus [?]. These angular resolution corrections amounted to less than 3%. For each pair of gases several curves could be calculated with a minimum at the appropriate velocity to m.atch the data, these corresponding to different values 0fN in vrn = Qv-3/8)n;
N = 3/2,5/2,7/2,.
.. . (3)
which gave the best fit are shown as solid lines in fig. 2, and the corresponding values of E?-, are given in the second c&mm in table 1. The uncertainty in cyrn is determined by how accurately the velocity of the minimum can be established In these experiments this error is about 7%. The experimental data could also be fitted well with a Lennard-Jones (~,6) potential where n varies from 8 to 20. The values of errn for ?z = 8, 12, 16 and 20 are’given in table 1. From these data alone it is not possible to decide which value of R is the best representation oE.the true potential. Values of F and yrn have been obtained from diffusion experiments and from angular distribution measurements. The diffusion experiments of van Heijningen et al. [13] were anaIysed for a (12,6) potential and the product
them should only agree if this assumed form of the potential is correct. Thus this better agreement supports the observation [lo] that the (20,6) potential is a better representation than the (12,6) for neon-krypton ing.
and neon-xenon scatter-
This work was supported by a grant from the
Science Research
Council,
and one of us (G.D.L.)
acknowledges the support of an S.R. C. Postgraduate Studentship.
However, all but one (N=3/?) of these curves had a maximum within the observed velocity range which did not fit the data. The curves
Values of
,
~7~
Table 1 (lo-22 erg cm) for a Lennard-Jones (n, 6) potential
AoguIar distributions
Diffusion
This work
experiments
Assumed potential
(8,s)
(1% 6)
neon-krypton
2.18 f 0.20
3.13 f 0.22
3.33 f 0.23
3.47 r0.23
3.48 -L0.21
3.64 t0.32
3.88 5 0.35
neon-xenon
3.24 f 0.23
3.61 f 0.25
3.84 f 0.21
4.00 + 0.28
3672 * 0.37
3.94 t
4.00 LO.12
(16.6)
W’. 6)
(12.6)
W. 61
a.12
(20.6)
69.
. ,Volume 8, nutiber 1
;.
.
CHEMICAL-THYSICS LETT@S
..
RE:FtiRErjCEs
.’
111.Discussions’Faradsy Sot., .Vol. 40, intermolecular, For&es (1965). [Zj A; Dalgarno. Advan. Che& Phys. 12 (1567) 143. f3j P.E.PhStlipson. Phys.Rev; 125 (1962) 1981;~ *[4J R. E;Berasteln and J. T. Mtickerman, Advan. Chem. Phys. 12 (1967) 389. i-51H. Pauly and J. P. Toennfes, : in: Advances in atomic and molecular physics. -Vol. 1, ed, DR. Bates , (Academic Press. New York, 1965) p. 195. . [Gf I. Amdur and J. E. Jordan, Advan. Chem. Phys. IQ (IQ%} 29. [‘7j R. Hetbing. W. Gaide and H. Pauly.~ 2. Physik 208 (I 966) 215.
.:.
1 January 1971
C&em. Phys. Letters 4 (1363) ill. [Q] D.H.Wfnicur, -A&.Moursund,’ W.R.Dsvereaux. L-R. Martin and A. l$uppermsnn, J. Chem. Phys. 52 (1970) 3299. [lOj J. M.Parson. T.P.Scba&r, F. P.TuXly, P. E.Sis!ca, Y. C. Wong and.Y. T. Lee. J. Chem. Phys. 53 (lQ70) 2223. [ll] A. U.Hcstettler and R. B.Berns&in, Rev. Sci. Ins&. . .31 (1960) 872. fl2] R. B. Bernstein and T.3. P. O’Brien, Discwssiods Faraday Sot. 40 (lQ65) 35. [13] R, 3. J. van Heijningen, J. P. Harpe and J. J. M. Been&k&, Physica 38 (1968) 1.
[ 81 R. &. Bickes and k; B. Bernstein,
CHEMICAL PHYSICS LETTERS
VELOCITY FOR
DEPENDENCE
HELIUM-ARGON,
OF
THE
NE’ON-KRYPTON G. D. LEMPERT,
1 Jan&y
TOTAL AND
CROSS
1971
SECTiON-
NEON-XENON
SCATTERING
S. J. B. CORRIGAN
Department of Physics, GowevStreef,
University College Londotz, London If.JClcI.UK
and Department
J. F. WILSON of Chemistry, University College London. Gower Street, Londo~~ WCl, UK
_ Received 3 August 1970 Revised manuscript received 11 November 1970 Velocity dependences of the total collision cross sections for scattering of helium by argon, neon by krypton and neon by xenon were measured using a time of flight method of velocity selection. For neon scattering glory oscillations were observed from which values of EY, (the product of the welI depth and the internuclear separation at the rninimumj were determined for various Lennard-Jones pntontials. Comnarison is made with the results from diffusion experiments and differential scattering measurements.
The widespread and fundamental influence of intermolecular forces in chemistry [l] suggests that their role on the molecular scale is as important as the role of thermo-dynamics in the properties and behavlour of the bulk phase. Although considerable advances have been made in recent years in improving the accuracy of the calculation of these forces between atoms at large separation where exchange forces are negligible [2], only in a few cases, -such as heliumhelium interactions [3], have there been accurate calculations of fhe short range forces. Molecular beam scattering techniques have -been recognized for some time as the most sensitive means of measuring intermolecular forces [4,.5]. The high energy experiments by Amdur and‘co-workers [6] have yielded reliable measurements of the short range forces between rare &as atoms. -The- intermediate and long range forces have been obtained through m&surements. of the differential and total elastic C-TOSS s&ions over a wide range of relative Fielocities [4]. &I-’ though potential parametdrs have been extensively measq;red for interactions invdlving an ,alk& atqm,’ there qe only- a few resylts_ [?-‘lO] available foe scattering i.n+olving non-alkali speciks. In this letter we.cjescribe a new ‘aphidach to ..the.problem of obtai+ig fhy %eliiciq dependence.:. .’ ,‘_. : -_ .__ “,._ -_ .-: ,_i .I
.
.
.
._-
‘,
,,_
’
;_ ..
of total collision cross sections and give some results that haire been obtained for helium-argon, neon-krypton and neon-xenon scattering_ An atomic beam of circular cross section was produced by an effusive source, passed through a scattering gas chamber, and detected by an electron bombardment ionizer followed by a quadrupole mass-spectrometer, electron multiplier and pulse counti& system. The source, the scattering gas chamber and the detector were In separate differentially pumped regions. Another differen tial pumping stage was provided before the detector to reduce the effusive background signal. The angular resolution of the apparatus, as defined by the angle subtended at the scattering gas chamber by the detector, was 6’. The velocity depe.\de&e of total collision cross sections was obtained by observing changes in transmitted intensities at different velocities ln the beam as t$e pres: ure of the gas in the scattering chamber was vtied. Velocity selection of the -atomic beam was bbtalned by a time of flight ~method. F&r the-source the bean? was pulsed by -a-narrow slit (=50 jfsec in time)- in a rotating disc. On the same &aft as the pulsing slit a second disc with a wider Sj!it trans+ted most of the pulsk, &t‘,cut off the slo.w~&toms sd that, before the ne$ ptilse,’ there wars a time when tie beam ,._._ ,.’ ‘. ,_..’ ._ .‘_ __ -. -67 _’
.,
.__I
.
..
.-
: ,
* ._.
-
.__’
., .
.,
_
_
. _
_
-.-
._ -.
..-_
_
Volume 8. number 1
CHEMICALPHYslCS LETTERS
1.
1‘January 1971
atoms were arriving ai the detector. -Pulses from the. multiplier were fed into four scalers which
were individually switched oti and off with variable delay time and gate width by circuits syn..chronised with the beam chopper. This enabled different scalers tb measure the beam intensity for different weli.definecj velocities. The instantaneous beam intensity was measured at the four selected delay times during each beam pulse by summing the gated signals for many (=104) consecutive pulses. The delay of one of the scalers was set at the peak of the time of flight distributiqn; th@ channel served as a reference intensity. -Another was fixed at a long delay time to give a measure of the background intensity. Two other delay times were varied to scan the velocity distribution. With these four channels it was possible in this way to measure the background and the beam intensjties at three different velocities in each cycle. and thus effectively simultaneously. There are several advantages in this method of velocity selection.- Firstly, the velocity dependence of a cross section can be determined without any measurement of the scattering gas pressure because the same pressure attenuates the intensities at each of the velocities selected. Secondly, since the fluctuations in the beam intensity, background pressure, scattering gas pressure and detector sensitivity are simultaneously common to each velocity, the errors due to these fluctuations cancel. This is verified by the scatter in the experimental data which does not exceed that which must arise from the statistics of counting. Thirdly, for a given resolution compared with a mechanically realisable slotted disc selector Ill], measurements can be extended to much higher velocities because the effective length of the selector is the length between the first disc and the detector. Fourthly, .by varying an electrical parameter (the detector gate open time) the resolution and transmission of this selector can be altered. ,In the_experiments reported here the velocity resolution (L ~11width at half maximum), which can be shown to be equal to detector gate open time divided by the delay time, ranges from 5% to 18% for the helium beam and from 3% to 5% for the neon b&am. No measurements were made of the absolute values of the total cross section. Fig. & qhows .the .veiocity dependence of the total cross section for helium-argon scattering for helium yelocjties of 700 to 2400_m/sec. Each. point corresponds tb many m+surements at that velocity with different scattering gas pressures. Tiie solid curve was_ calculated from .@_,
(
.’
.-
1
zil.o(’ LOO
600
1500zooO
6wlwo
300
HELIUM ATOM VELOCITY Imkc)
Fig. 1. Velocity dependence of the total cross section for thz scattering of helium atoms by argon. QV2/S is plotted against V on a log-log scale. Q =$
lFo (21+1) sin%l
for a Lennard-Jones (12,6) potential with the parameters obtained by Helbing et al. [‘i’] (E, well depth = 0.361 x lo-14 erg, r,, internuclear separation at the minimum = 3.40A). In eq. (1) k is the incoming wave number and the values of vl were obtained by numerically integrating the SchrBdinger equation for the scattering. The good agreement confirms the result of Helbing et al. within the range of these experiments. Fig. 2 shows the velocity dependence of the total cross section for neon-krypton and neonxenon scattering, for the velocity range 450 to 1200 m/set. For the velocity range of the neon experiments the total cross section for a LennardJones (12,6) potential is given by the WKB approx-
imation as [12] 2Er6 Q = 8.083 c+)
2’5
F (I-g2sin(2qm
- 38/4), (2) m where Lo is. the value of the angular momentum, 1, for which the phase shift attain6 its maximum value,. Qm, and $n = (ds/dZ2),,, The first term is the Schiff-Landau-Lifshitz total cross section and the second term contributes the glory oscillations. -The velocities at which the maxima and plinimd of the function QV2i5 occur-are sensitive principally to the product •7~ and also to the form of the’ interaction potential. [12]. The amplitude of the glory osdl!ations is much. less sensitive-to -the velocity and to the..potentia!. parameter& :’ -1 1 -_ -. .:. . +
Volume 8. number
PIiYSICS LETTERS
CHEMICAL
1
16-
YI .= 5 t e
lo-
f
20-
,r > ci
18-
NelKr
161L 12 -
NEON
ATOM
VELOCITY
(mlsec
1
Fig. 2. Velocity dependence of the total cross section for the scattering of neon atoms by krypton and xenon. Q@2/5 is plotted against V on a semi-log scale.
The velocity dependence of QVzi5 was calculated for a Lennard-Jones (l&6) potential from eq. (2), averaged over the scattering gas motion and corrected for the angular resolution of the apparatus [?]. These angular resolution corrections amounted to less than 3%. For each pair of gases several curves could be calculated with a minimum at the appropriate velocity to m.atch the data, these corresponding to different values 0fN in vrn = Qv-3/8)n;
N = 3/2,5/2,7/2,.
.. . (3)
which gave the best fit are shown as solid lines in fig. 2, and the corresponding values of E?-, are given in the second c&mm in table 1. The uncertainty in cyrn is determined by how accurately the velocity of the minimum can be established In these experiments this error is about 7%. The experimental data could also be fitted well with a Lennard-Jones (~,6) potential where n varies from 8 to 20. The values of errn for ?z = 8, 12, 16 and 20 are’given in table 1. From these data alone it is not possible to decide which value of R is the best representation oE.the true potential. Values of F and yrn have been obtained from diffusion experiments and from angular distribution measurements. The diffusion experiments of van Heijningen et al. [13] were anaIysed for a (12,6) potential and the product
them should only agree if this assumed form of the potential is correct. Thus this better agreement supports the observation [lo] that the (20,6) potential is a better representation than the (12,6) for neon-krypton ing.
and neon-xenon scatter-
This work was supported by a grant from the
Science Research
Council,
and one of us (G.D.L.)
acknowledges the support of an S.R. C. Postgraduate Studentship.
However, all but one (N=3/?) of these curves had a maximum within the observed velocity range which did not fit the data. The curves
Values of
,
~7~
Table 1 (lo-22 erg cm) for a Lennard-Jones (n, 6) potential
AoguIar distributions
Diffusion
This work
experiments
Assumed potential
(8,s)
(1% 6)
neon-krypton
2.18 f 0.20
3.13 f 0.22
3.33 f 0.23
3.47 r0.23
3.48 -L0.21
3.64 t0.32
3.88 5 0.35
neon-xenon
3.24 f 0.23
3.61 f 0.25
3.84 f 0.21
4.00 + 0.28
3672 * 0.37
3.94 t
4.00 LO.12
(16.6)
W’. 6)
(12.6)
W. 61
a.12
(20.6)
69.
. ,Volume 8, nutiber 1
;.
.
CHEMICAL-THYSICS LETT@S
..
RE:FtiRErjCEs
.’
111.Discussions’Faradsy Sot., .Vol. 40, intermolecular, For&es (1965). [Zj A; Dalgarno. Advan. Che& Phys. 12 (1567) 143. f3j P.E.PhStlipson. Phys.Rev; 125 (1962) 1981;~ *[4J R. E;Berasteln and J. T. Mtickerman, Advan. Chem. Phys. 12 (1967) 389. i-51H. Pauly and J. P. Toennfes, : in: Advances in atomic and molecular physics. -Vol. 1, ed, DR. Bates , (Academic Press. New York, 1965) p. 195. . [Gf I. Amdur and J. E. Jordan, Advan. Chem. Phys. IQ (IQ%} 29. [‘7j R. Hetbing. W. Gaide and H. Pauly.~ 2. Physik 208 (I 966) 215.
.:.
1 January 1971
C&em. Phys. Letters 4 (1363) ill. [Q] D.H.Wfnicur, -A&.Moursund,’ W.R.Dsvereaux. L-R. Martin and A. l$uppermsnn, J. Chem. Phys. 52 (1970) 3299. [lOj J. M.Parson. T.P.Scba&r, F. P.TuXly, P. E.Sis!ca, Y. C. Wong and.Y. T. Lee. J. Chem. Phys. 53 (lQ70) 2223. [ll] A. U.Hcstettler and R. B.Berns&in, Rev. Sci. Ins&. . .31 (1960) 872. fl2] R. B. Bernstein and T.3. P. O’Brien, Discwssiods Faraday Sot. 40 (lQ65) 35. [13] R, 3. J. van Heijningen, J. P. Harpe and J. J. M. Been&k&, Physica 38 (1968) 1.
[ 81 R. &. Bickes and k; B. Bernstein,