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Rotationally inelastic cross sections and the parameters of the potential
Volume 208, number 3,4
CHEMICAL PHYSICS LETTERS
II June 1993
Rotationally inelastic cross sections and the parameters of the potential P.M. Agrawal,V. Garg ’ and ILR. Patidar SchoolofStudies
in Physics, Vikram University, Qjain (M.P.) 456 010, India
Received 10 February 1992; in final form 26 February I993
The dependence of rotationally inelastic cross sections on the parameters of the interaction potential has been investigated for a model system consisting of a homonuclear diatomic molecule and an atom. The potential, V(r, O)=Cema’[ 1t C& a,P,(cos 0) ] has been employed and the computations have been performed by the modified infinitearder sudden approximation. It is found that u( 0-J) =Ac:, where c,=b,/( 1+X Ib/l), with b,=a,/m, and A is a constant that does not depend on a, and 1. The limitations of this result with the variation in !, a, and energy have also been studied. The results are discussed to explain the existence of scaling and fitting laws, and to show the importance of expressing the potential in terms of a set of orthonormal functions, Y,(cos 0) = m P,( cos 0), in place of the Lcgendre polynomials, P,(cos 19).
1. Intruduction In recent years, attempts have been made to extract the potential parameters from the known experimental cross sections for rotationally inelastic collisions in molecular systems [ 1,2]. Empirical scaling and fitting laws that express the rotationally inelastic cross sections in terms of a few fitting parameters are also becoming more and more popular [ 3-61. Likewise, it can also be useful to arrive at some semiempirical expressions that give the dependence of cross sections on the parameters of the potential [ 7- 101. One of the simplest examples may be to explore the cross sections for a homonuclear diatomic molecule-atom system described by the potential V(r, O)=Ce-ar
(
1+ c a,P,(cos e) ) I >
(1)
where r is the distance between the centre of mass of the molecule and the atom, and 13is the angle between the vector r and the axis of the molecule. In this Letter, we attempt to show that for such systems the cross sections, 8(0-J), can be calculated ’ Permanent address: Department College, Ajmer 305 001, India. 204
of Physics, Government
from the knowledge of one cross section and the parameters al. We shall also see that if the potential ( 1) is rewritten as follows in terms of the orthonormal functions Yr(cos f3), V(r, O)=CeTar
1 t 1 b, Y,(cos 0) , I >
(2)
where b, =a,/m
,
(3)
and Y,(cose)=~P,(cose),
(4)
then eq. (2) not only becomes conceptually more meaningful but also leads to a conceptually simpler expression for the cross sections in terms of the potential parameter b,.
2. Calculation of cross sections The cross sections have been computed by using the modified infinite-order sudden approximation (IOSAM) [ 111. IOSAM cross sections, aIo-, are related to those calculated by the well known infiEkevier Science Publishers B.V.
Volume 208, number 3,4
nite-order sudden approximation the following relation:
(IOSA) [ 121, by
where b and kit respectively, denote wavevectors corresponding to final and initial translational energies. The calculations have been performed over a model system XZt Y. The masses of atoms X and Y have been taken as 14 and 4 amu, respectively. The bond length for XZ is chosen to be 1.0 A. For a homonuclear molecule X2, the odd coefftcients a,, u3, as, ... in eq. ( 1) are zero. The summation in eq. ( 1) has been taken up to 1=40, i.e. a,=0 for L-40. All the calculations have been performed for C= 800 eV, and (YZ3.0 A-‘.
3. Results and discussion It is found that the cross sections depend on the coefficients a, according to the following relation: a(O+l) =Ac:,
(5)
where A is a constant that does not depend on 1 and al; and c, is defined as
Table 1 0(0+/)/c: values as a function of I at three different values of total energy E (energy in eV, cross sections in A’). The potential given by eq. (I) with a2=a4=...=a10=0.01 has been used I
2 4 6 8 10 12 14 16 18 20 22 24 26 28 30
C lb,,).
(6)
The validity of eq. (5) has been noted extensively for a large number of calculations. Typical results are given in tables 1 and 2. Table 1 gives the values of a(O+l)/c: at three different values of total energy (E=0.05, 0.1 and 0.2 eV). These results show that the variation in a/c: with / is less than 10% for a given energy. Table 2 gives the value of a( 0+40) /c& as a function of a40 when, apart from udorall other values of ai are taken to be zero. These results show that value of A ( =a/&) does not vary (within 5%) with the magnitude of ai provided a& Ulim( I). It is found that alim( I) increases with the increase in I, e.g., at E=O. 1 eV, values of q,, at 1=2, 20 and 40 are about 0.3, 0.6 and 0.9, respectively. With the increase in E, ulim is found to decrease, e.g., values of al;,( 1~40) at E=O.l and 0.2 are about 0.9 and 0.4, respectively. The importance of b, over uI in eqs. (5) and (6)
0(0+1)/c:
values when
Ez0.05
Ez0.1
E=O.Z
351 336 328 323 321 309 311 315 315 310 302 303 296 296 304
603
1010
558 550 545 532 516 510 513 538 541 537 524 510 493 496
949 894 859 850 844 856 858 871 891 849 844 844 844 833
Table 2 0(0-+4O)/c~ values as a function of c4o at three different values of total energy E. In the potential except for the coefftcient a,,, all other coefftcients (a,) have been taken to be zero (energy in eV and cross sections in A*)
a, c,=b,/(l+
1 I June 1993
CHEMICAL PHYSICS LETTERS
0.05 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
a(O-dO)/c&
values when
E=0.05
E10.1
E=0.2
352 341 341 349 351 354 354 353 351 347 341
573 584 586 588 582 574 561 545 526 504 482
1030 986 991 965 928 884 834 776 719 661 604
is trivial as b1 is the coefficient of the normalised function Y,(cos @, whereas al is the coefftcient of the unnormalised function P,( cos 0). The importance of c, over b, may be understood, qualitatively, by realising that c, gives the relative weight of the Y,(cos 19) term in the total potential. Quantitatively, the significance of the term 1 + Z 1b,l in the denominator 205
Volume 208, number 3,4
CHEMICAL PHYSICS LETTERS
of eq. (6) is not clear. Various other values of cl such as b,l(JZ+CIb,I), b,l(l+CIa,I ), b,l(2+ have also been tried. Cb:)“‘, br/(l+Cb:)“2 Among all these the choice given by eq. (6) is found to be the best. In recent years the empirical scaling and fitting laws such as the exponential-gap law and the power-gap law have been found to be useful in expressing the entire matrix of cross sections in terms of a few fitting parameters. The validity of such fitting laws gives rise to an important question about the simplicity of the nature that leads to an expression for the cross sections in terms of a few parameters. Eq. (5) attempts to answer, at least partially, such a question. Eq. (5) shows that if in the expansion of the potential function in terms of Y,(cos 6’) its coefficient bl varies in some regular way then that regularity would be reflected in the variation of cross sections and from that a simple fitting law would emerge. ,When the initial state is not the rotational ground state of the molecule then the cross sections may be estimated by using eq. (5) and the IOS scaling law [ 41 which is within the framework of the IOSA. By this procedure one may see that if in the potential given by eq. ( 1) a,=0 for I>,/” then a(/ +I’ *I) would be zero for I> I”. In addition to the requirement of the low values of a, for the validity of eq. (5 ) one may also see that this equation may not hold good when a potential other than that given by eq. ( 1) is used or when the system is such that the IOSAM method of computation of cross sections is not valid. Despite such
206
1 I June 1993
limitations, in many situations this equation can be used to make at least an estimate of various cross sections from the knowledge of coefficients of I’!( cos 0) in the expansion of the angular part of the potential and the constant A that can be determined from experimental or theoretical cross sections. The converse may also be useful, i.e. from the known cross sections one may attempt to calculate the potential parameters. For more refinement eq. (5 ) may be further modified by investigating other types of potentials and dynamical methods.
References [ l] P.L. Jones, U. Hefter, A. Mattheus, I. Witt, K. Bergmann, W. Miller, W. Meyer and R. Schinke, Phys. Rev. A 26 (1982) 1283. [ 2 ] A.J. McCtiery and Z.T. Alwahabi,Phys. Rev. A 43 ( 199 I ) 611. [ 31 J.C. Polanyi and K.B. Woodall, J. Chem. Phys. 56 (1972) 1563. [4] T.A. Bnmner, N. Smith, A.W. Ka.rp and D.E. Pritchard, J. Chem. Phys. 74 ( I98 1) 3324. [ 51P.M. Agrawal and N.C. Agrawal, Chem. Phys. Letters 117 (1985) 451. [6] P.M. Agrawal, N.C. Agrawal and V. Garg, Chem. Phys. Letters 118 (1985) 213. [ 71 P.M. Agrawal and V. Garg, Current Sci. (India) 61 ( 1991) 43. [ 81 H.R. Mayne and M. Keil, I. Phys. Chem. 88 ( 1984) 883.. [ 91 L. Eno and H. Rabitz, J. Chem. Phys. 72 (1980) 2314. [IO] W. Erlewein, M. von Seggem and J.P. Toennies, 2. Physik 211 (1968) 35. [I l] P.M. Agrawal and L.M. Raff, J. Chem. Phys. 74 (198 1) 3292. [ 121 G.A. Parker and R.T Pack, 3. Chem. Phys. 68 (1978) 1585.
CHEMICAL PHYSICS LETTERS
II June 1993
Rotationally inelastic cross sections and the parameters of the potential P.M. Agrawal,V. Garg ’ and ILR. Patidar SchoolofStudies
in Physics, Vikram University, Qjain (M.P.) 456 010, India
Received 10 February 1992; in final form 26 February I993
The dependence of rotationally inelastic cross sections on the parameters of the interaction potential has been investigated for a model system consisting of a homonuclear diatomic molecule and an atom. The potential, V(r, O)=Cema’[ 1t C& a,P,(cos 0) ] has been employed and the computations have been performed by the modified infinitearder sudden approximation. It is found that u( 0-J) =Ac:, where c,=b,/( 1+X Ib/l), with b,=a,/m, and A is a constant that does not depend on a, and 1. The limitations of this result with the variation in !, a, and energy have also been studied. The results are discussed to explain the existence of scaling and fitting laws, and to show the importance of expressing the potential in terms of a set of orthonormal functions, Y,(cos 0) = m P,( cos 0), in place of the Lcgendre polynomials, P,(cos 19).
1. Intruduction In recent years, attempts have been made to extract the potential parameters from the known experimental cross sections for rotationally inelastic collisions in molecular systems [ 1,2]. Empirical scaling and fitting laws that express the rotationally inelastic cross sections in terms of a few fitting parameters are also becoming more and more popular [ 3-61. Likewise, it can also be useful to arrive at some semiempirical expressions that give the dependence of cross sections on the parameters of the potential [ 7- 101. One of the simplest examples may be to explore the cross sections for a homonuclear diatomic molecule-atom system described by the potential V(r, O)=Ce-ar
(
1+ c a,P,(cos e) ) I >
(1)
where r is the distance between the centre of mass of the molecule and the atom, and 13is the angle between the vector r and the axis of the molecule. In this Letter, we attempt to show that for such systems the cross sections, 8(0-J), can be calculated ’ Permanent address: Department College, Ajmer 305 001, India. 204
of Physics, Government
from the knowledge of one cross section and the parameters al. We shall also see that if the potential ( 1) is rewritten as follows in terms of the orthonormal functions Yr(cos f3), V(r, O)=CeTar
1 t 1 b, Y,(cos 0) , I >
(2)
where b, =a,/m
,
(3)
and Y,(cose)=~P,(cose),
(4)
then eq. (2) not only becomes conceptually more meaningful but also leads to a conceptually simpler expression for the cross sections in terms of the potential parameter b,.
2. Calculation of cross sections The cross sections have been computed by using the modified infinite-order sudden approximation (IOSAM) [ 111. IOSAM cross sections, aIo-, are related to those calculated by the well known infiEkevier Science Publishers B.V.
Volume 208, number 3,4
nite-order sudden approximation the following relation:
(IOSA) [ 121, by
where b and kit respectively, denote wavevectors corresponding to final and initial translational energies. The calculations have been performed over a model system XZt Y. The masses of atoms X and Y have been taken as 14 and 4 amu, respectively. The bond length for XZ is chosen to be 1.0 A. For a homonuclear molecule X2, the odd coefftcients a,, u3, as, ... in eq. ( 1) are zero. The summation in eq. ( 1) has been taken up to 1=40, i.e. a,=0 for L-40. All the calculations have been performed for C= 800 eV, and (YZ3.0 A-‘.
3. Results and discussion It is found that the cross sections depend on the coefficients a, according to the following relation: a(O+l) =Ac:,
(5)
where A is a constant that does not depend on 1 and al; and c, is defined as
Table 1 0(0+/)/c: values as a function of I at three different values of total energy E (energy in eV, cross sections in A’). The potential given by eq. (I) with a2=a4=...=a10=0.01 has been used I
2 4 6 8 10 12 14 16 18 20 22 24 26 28 30
C lb,,).
(6)
The validity of eq. (5) has been noted extensively for a large number of calculations. Typical results are given in tables 1 and 2. Table 1 gives the values of a(O+l)/c: at three different values of total energy (E=0.05, 0.1 and 0.2 eV). These results show that the variation in a/c: with / is less than 10% for a given energy. Table 2 gives the value of a( 0+40) /c& as a function of a40 when, apart from udorall other values of ai are taken to be zero. These results show that value of A ( =a/&) does not vary (within 5%) with the magnitude of ai provided a& Ulim( I). It is found that alim( I) increases with the increase in I, e.g., at E=O. 1 eV, values of q,, at 1=2, 20 and 40 are about 0.3, 0.6 and 0.9, respectively. With the increase in E, ulim is found to decrease, e.g., values of al;,( 1~40) at E=O.l and 0.2 are about 0.9 and 0.4, respectively. The importance of b, over uI in eqs. (5) and (6)
0(0+1)/c:
values when
Ez0.05
Ez0.1
E=O.Z
351 336 328 323 321 309 311 315 315 310 302 303 296 296 304
603
1010
558 550 545 532 516 510 513 538 541 537 524 510 493 496
949 894 859 850 844 856 858 871 891 849 844 844 844 833
Table 2 0(0-+4O)/c~ values as a function of c4o at three different values of total energy E. In the potential except for the coefftcient a,,, all other coefftcients (a,) have been taken to be zero (energy in eV and cross sections in A*)
a, c,=b,/(l+
1 I June 1993
CHEMICAL PHYSICS LETTERS
0.05 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
a(O-dO)/c&
values when
E=0.05
E10.1
E=0.2
352 341 341 349 351 354 354 353 351 347 341
573 584 586 588 582 574 561 545 526 504 482
1030 986 991 965 928 884 834 776 719 661 604
is trivial as b1 is the coefficient of the normalised function Y,(cos @, whereas al is the coefftcient of the unnormalised function P,( cos 0). The importance of c, over b, may be understood, qualitatively, by realising that c, gives the relative weight of the Y,(cos 19) term in the total potential. Quantitatively, the significance of the term 1 + Z 1b,l in the denominator 205
Volume 208, number 3,4
CHEMICAL PHYSICS LETTERS
of eq. (6) is not clear. Various other values of cl such as b,l(JZ+CIb,I), b,l(l+CIa,I ), b,l(2+ have also been tried. Cb:)“‘, br/(l+Cb:)“2 Among all these the choice given by eq. (6) is found to be the best. In recent years the empirical scaling and fitting laws such as the exponential-gap law and the power-gap law have been found to be useful in expressing the entire matrix of cross sections in terms of a few fitting parameters. The validity of such fitting laws gives rise to an important question about the simplicity of the nature that leads to an expression for the cross sections in terms of a few parameters. Eq. (5) attempts to answer, at least partially, such a question. Eq. (5) shows that if in the expansion of the potential function in terms of Y,(cos 6’) its coefficient bl varies in some regular way then that regularity would be reflected in the variation of cross sections and from that a simple fitting law would emerge. ,When the initial state is not the rotational ground state of the molecule then the cross sections may be estimated by using eq. (5) and the IOS scaling law [ 41 which is within the framework of the IOSA. By this procedure one may see that if in the potential given by eq. ( 1) a,=0 for I>,/” then a(/ +I’ *I) would be zero for I> I”. In addition to the requirement of the low values of a, for the validity of eq. (5 ) one may also see that this equation may not hold good when a potential other than that given by eq. ( 1) is used or when the system is such that the IOSAM method of computation of cross sections is not valid. Despite such
206
1 I June 1993
limitations, in many situations this equation can be used to make at least an estimate of various cross sections from the knowledge of coefficients of I’!( cos 0) in the expansion of the angular part of the potential and the constant A that can be determined from experimental or theoretical cross sections. The converse may also be useful, i.e. from the known cross sections one may attempt to calculate the potential parameters. For more refinement eq. (5 ) may be further modified by investigating other types of potentials and dynamical methods.
References [ l] P.L. Jones, U. Hefter, A. Mattheus, I. Witt, K. Bergmann, W. Miller, W. Meyer and R. Schinke, Phys. Rev. A 26 (1982) 1283. [ 2 ] A.J. McCtiery and Z.T. Alwahabi,Phys. Rev. A 43 ( 199 I ) 611. [ 31 J.C. Polanyi and K.B. Woodall, J. Chem. Phys. 56 (1972) 1563. [4] T.A. Bnmner, N. Smith, A.W. Ka.rp and D.E. Pritchard, J. Chem. Phys. 74 ( I98 1) 3324. [ 51P.M. Agrawal and N.C. Agrawal, Chem. Phys. Letters 117 (1985) 451. [6] P.M. Agrawal, N.C. Agrawal and V. Garg, Chem. Phys. Letters 118 (1985) 213. [ 71 P.M. Agrawal and V. Garg, Current Sci. (India) 61 ( 1991) 43. [ 81 H.R. Mayne and M. Keil, I. Phys. Chem. 88 ( 1984) 883.. [ 91 L. Eno and H. Rabitz, J. Chem. Phys. 72 (1980) 2314. [IO] W. Erlewein, M. von Seggem and J.P. Toennies, 2. Physik 211 (1968) 35. [I l] P.M. Agrawal and L.M. Raff, J. Chem. Phys. 74 (198 1) 3292. [ 121 G.A. Parker and R.T Pack, 3. Chem. Phys. 68 (1978) 1585.