- Email: [email protected]
Collision induced spectrum decay constant for Lennard-Jones interaction potential
Volume 50A, number 6
PHYSICS LETTERS
13 January 1975
COLLISION INDUCED SPECTRUM [)ECAY CONSTANT FOR LENNARD-JONES INTERACTION POTENTIAL F. BAROCCHI*
Istituto di Ricerca sulle Onde Elettromagnetiehe del C.N.R., Italy and R. VALLAURI and M. ZOPPI
Laboratorio di Elettroniea Quantistica del C.N.R., Firenze, Italy Received 25 November 1974 An expression of the eoUisioninduced spectrum decay constant has been derived for spherical molecules interacting with a (6-12) Lenna~d-Jonespotential Theoretical values for Hard-Sphereand Lennard-Jonespotential are compared with the experimental values for gaseous At, CH4, CF 4. Measurements of the decay constant of the exponential shaped collision-induced spectrum in low density gases can be performed with very good precision, better than 1% [ 1 - 3 ] . This enables us to extract information on the form of the potential which molecules experience during binary-collisions. A density dependent empirical form for the decay constant of the C.I.S. spectrum was first obtained by Fleury and Coworkers [4] which reads
A = G(KTIMo2)I/2 [1 + (p/pO)2],
(1)
where G is a numerical constant whose value was experimentally found ~ 3. At present such a density dependence has not been theoretically derived. A calculation by Mahan [5] shows that in the case of binary coUision, i.e. low densities, for hard-spheres potential and D.I.D. polarizabflities one has for the decay constant
AHS = 3 (KT/Mb2) 1/2,
(2)
where b is the hard-sphere diameter which in the conventional interpretation is identified with o. For Lennard-Jones potential the calculation can also be carried out easily as it follows. The decay constant A0 for an exponential spectrum which is determined by binary-collision process is
* Present address: Laboratorio di Elettronica Quantistiea del C.N.R., Via Panciatichi, 56/30 - 50127 Fizenze, Italy.
A 0 = (F(2)/F(2)~l/2 2 ' 0 J '
(3)
where the moments of the two-body intensity/(2) (¢o) are defined as +0.
Fi(2) = /
dco ~ / ( 2 ) (~).
(4)
The first two moments can also be expressed as [6]
02)_y f
(5)
0
F(22)=4~2KT/M)/{[d
fl(r)]2+6[~]2}r2dr(6)
where g(r) = exp [-V(r)/KT], is the radial distribution function and fl(r) is the two-body anisotropic polarizability which in the D.I.D. model, has the form fl(r) = 6a2]r 3. If one assumes that the short-range polarizability effect gives a small contribution to the firsttwo moments, expression (5) and (6) can be calculated for a ( 6 - 1 2 ) Lennard-Jones potential. With this assumption a simple calculation gives for A~J (LennardJones potential ALJ = [15 K T H8(Y)I1/2 Mo-=--2"H---~J '
(7)
where Hn(Y) are tabulated (7) functions of the reduced temperature parameter y = 2(e]KT) 112. 451
Volume 50A, number 6
PHYSICS LETTERS Table 1
Ar CH4 CF4
o X 108 (K/T)(a) (cm)
AHS (cm-1)
ALj (cm-1)
Aex.p1 (cm-)
2.50 1.98 1.95
11.67 16.27 5.67
12.44 17.26 6.02
12.55 [1] 15.5 [2] 4.92 [3]
3.405 3.817 4.70
(a)parameters obtained from ref. [8] page 1110. Table 1 gives b o t h the H.S. and L.J. calculated values o f A and the few precise experimentally determined values, which were extracted from the two body spectrum. The comparison shows that for Argon there is an almost exact agreement for the Lennard-Jones potential, while for CH 4 and CF 4 the experimental values result lower than the theoretical hard-sphere one, which suggests that for those molecules the anisotropy is affected by other mechanisms besides the dipole-induced dipole one.
452
13 January 1975
We like to thank Dr. G.C. Tabisz for making available to us his data prior to publication.
References [1] J.P. McTague, W.D. Hellenson, L.H. Hall: Journal de Physique 33 (1972) 241. [2] F. Barocchi, J.P. McTague: to be published. [3] G.C. Tabisz, D.P. Shelton, M.S. Mathur: to be published. [4] P.A. Fleury, J.R. Worlock and H.L Carter: Phys. Rev. Lett. 30 (1973) 591. [5] G.D. Mahan: Physics Letters 44A (1973) 287. [6] H.B. Levine, G. Birnbaum: J. of Chem. Phys. 55 (1971) 2914. [7] A.D. Buckingham, J.A. Pople: Trans. Faraday Soc. 51 (1955) 1173. [8] J.O. Hirsehfelder, C.F. Curtiss, R.B. Bird: Molecular theory of gases and liquids (Wiley N.Y., 1964).
PHYSICS LETTERS
13 January 1975
COLLISION INDUCED SPECTRUM [)ECAY CONSTANT FOR LENNARD-JONES INTERACTION POTENTIAL F. BAROCCHI*
Istituto di Ricerca sulle Onde Elettromagnetiehe del C.N.R., Italy and R. VALLAURI and M. ZOPPI
Laboratorio di Elettroniea Quantistica del C.N.R., Firenze, Italy Received 25 November 1974 An expression of the eoUisioninduced spectrum decay constant has been derived for spherical molecules interacting with a (6-12) Lenna~d-Jonespotential Theoretical values for Hard-Sphereand Lennard-Jonespotential are compared with the experimental values for gaseous At, CH4, CF 4. Measurements of the decay constant of the exponential shaped collision-induced spectrum in low density gases can be performed with very good precision, better than 1% [ 1 - 3 ] . This enables us to extract information on the form of the potential which molecules experience during binary-collisions. A density dependent empirical form for the decay constant of the C.I.S. spectrum was first obtained by Fleury and Coworkers [4] which reads
A = G(KTIMo2)I/2 [1 + (p/pO)2],
(1)
where G is a numerical constant whose value was experimentally found ~ 3. At present such a density dependence has not been theoretically derived. A calculation by Mahan [5] shows that in the case of binary coUision, i.e. low densities, for hard-spheres potential and D.I.D. polarizabflities one has for the decay constant
AHS = 3 (KT/Mb2) 1/2,
(2)
where b is the hard-sphere diameter which in the conventional interpretation is identified with o. For Lennard-Jones potential the calculation can also be carried out easily as it follows. The decay constant A0 for an exponential spectrum which is determined by binary-collision process is
* Present address: Laboratorio di Elettronica Quantistiea del C.N.R., Via Panciatichi, 56/30 - 50127 Fizenze, Italy.
A 0 = (F(2)/F(2)~l/2 2 ' 0 J '
(3)
where the moments of the two-body intensity/(2) (¢o) are defined as +0.
Fi(2) = /
dco ~ / ( 2 ) (~).
(4)
The first two moments can also be expressed as [6]
02)_y f
(5)
0
F(22)=4~2KT/M)/{[d
fl(r)]2+6[~]2}r2dr(6)
where g(r) = exp [-V(r)/KT], is the radial distribution function and fl(r) is the two-body anisotropic polarizability which in the D.I.D. model, has the form fl(r) = 6a2]r 3. If one assumes that the short-range polarizability effect gives a small contribution to the firsttwo moments, expression (5) and (6) can be calculated for a ( 6 - 1 2 ) Lennard-Jones potential. With this assumption a simple calculation gives for A~J (LennardJones potential ALJ = [15 K T H8(Y)I1/2 Mo-=--2"H---~J '
(7)
where Hn(Y) are tabulated (7) functions of the reduced temperature parameter y = 2(e]KT) 112. 451
Volume 50A, number 6
PHYSICS LETTERS Table 1
Ar CH4 CF4
o X 108 (K/T)(a) (cm)
AHS (cm-1)
ALj (cm-1)
Aex.p1 (cm-)
2.50 1.98 1.95
11.67 16.27 5.67
12.44 17.26 6.02
12.55 [1] 15.5 [2] 4.92 [3]
3.405 3.817 4.70
(a)parameters obtained from ref. [8] page 1110. Table 1 gives b o t h the H.S. and L.J. calculated values o f A and the few precise experimentally determined values, which were extracted from the two body spectrum. The comparison shows that for Argon there is an almost exact agreement for the Lennard-Jones potential, while for CH 4 and CF 4 the experimental values result lower than the theoretical hard-sphere one, which suggests that for those molecules the anisotropy is affected by other mechanisms besides the dipole-induced dipole one.
452
13 January 1975
We like to thank Dr. G.C. Tabisz for making available to us his data prior to publication.
References [1] J.P. McTague, W.D. Hellenson, L.H. Hall: Journal de Physique 33 (1972) 241. [2] F. Barocchi, J.P. McTague: to be published. [3] G.C. Tabisz, D.P. Shelton, M.S. Mathur: to be published. [4] P.A. Fleury, J.R. Worlock and H.L Carter: Phys. Rev. Lett. 30 (1973) 591. [5] G.D. Mahan: Physics Letters 44A (1973) 287. [6] H.B. Levine, G. Birnbaum: J. of Chem. Phys. 55 (1971) 2914. [7] A.D. Buckingham, J.A. Pople: Trans. Faraday Soc. 51 (1955) 1173. [8] J.O. Hirsehfelder, C.F. Curtiss, R.B. Bird: Molecular theory of gases and liquids (Wiley N.Y., 1964).