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Dissociation of molecules in intense infra-red fields
Volume 62A, number 1
PHYSICS LETTERS
DISSOCIATION OF MOLECULES
11 July 1977
IN INTENSE INFRA-RED FIELDS
R.K. THAREJA a Department of Pure and Applied Physics. The Queen's University of Belfast, Belfast BT7 1NN, Northern Ireland Received 1 November 1976 Revised manuscript received 11 May 1977 A generalised analytic expression for the T-matrix for dissociation of diatomic hetro-nuclear molecules for arbitrary values of the vibrational (o) and rotational (K) quantum number of the initial state of the same electron terms, in an intense electromagnet field is evaluated.
In the last couple of years a considerable amount of work has been done in the field of dissociation, excitation etc. of diatomic molecules [ 1 - 3 ]. The transition probability for the dissociation of the molecule in an intense field may be calculated using the Nth order time dependent perturbation theory, which reduces essentially to calculating ( N - 1) sums over vibrational structure of intermediate transitions; in a manner similar to that used for calculating the multiphoton ionization o f atoms [4,5]. However, at high intensities not only the evaluation o f sums becomes complicated but the validity o f the theory itself is doubtful also [6]. For such high intensities the momentum translation approximation is very useful. Essentially it consists of applying a unitary transformation which approximately removes the incident e.m. field from the problem. The method has successfully been applied to multiphoton atomic transitions [7,81. In the present note, we apply the momentum translation approximation to evaluate the transition probability of diatomic hetro-nuclear molecule for arbitrary values of the vibrational (o) and rotational (K) quantum number of the initial state. It is assumed that N-quantum transition takes place without change in the electronic state of the molecule. In other words, the transition occurs from a vibrational levet-n to a vibrational level-n' of the continuum spectra of the same electron term (s = s'). As the effective field of the electrons and nuclei, we used the potential V(r) =
1 A Leverhulme Visiting Fellow.
A i r 2 - B/r. The values of the parameters A and B are determined by the bond dissociation energy, D = B2/4A, and the halfwidth of the potential well A, A = D~2/8, B = DA/Vc2. The T-matrix for a transition from an initial nuclear state ¢i(r) to a final nuclear state 0f(r) by the absorption of N-photons of the intense field of vector potential A is given by
where A, w, ~ are the amplitude, frequency and polarization vector of the incident wave and Jw(z) is the Bessel-function of order N. The wave function for the bound state is ¢i(r) = RvK(r) YKM(O, c))
(2)
where YKM(O, ¢) are the ordinary normalised spherical harmonics and 2 [['(2S + 2 + V)] 1/2 . . . . s -r/an RuK(r) - a3~2n-~P(2-~+ ~ l ~ t zrlan ) e
× ¢ ( - v , 2s + 2, 2r/an),
(3)
where s is the positive root of the equation s(s+ 1) = 2mA/h 2 + K ( K + 1), qb is the confluent hypergeometric function and n = s + 1 + v, v = 0, 1,2 .... is the vibration quantum number, a = Ii2/BM = aome2/mz2 is the coulomb unit of length. The energy of the state is given by
EoK = - D + hw0(o + 1) _ hWoXo(V" + 1)2 + Be(K +1)2 where w 0 = 2D/2x/~--ma is the vibrational frequency at o = O, WoX 0 = 3 (hD/mA) is the vibrational anharmonicity 19
Volume 62A, number 1
PHYSICS LETTERS
parameter, B e ='h2/2J is the rotational constant. Thus for the ground state (v = K = M = 0) the energy is approximately given by E 0 = - D + hWo/2. As for the final dissociation state 0f(r), we use qb'"
(2rr)3/2
fir) = ~ - -
K'M' ~ iK'e-i6K'RqK'(r) (4)
x YK,~r(O,4))YK,~r(OK,%) RqK,(r ) is the regular solution of the radial part of
11 July 1977
of degree v. Thus for v = 0, a(O) = 1 ; v -= 1, a(o1) = 2s + 2;a~ 1) etc.; and X = eAa/hc is a dimensionless intensity parameter. The eq. (6) gives the final expression for the Tmatrix for an arbitrary vibrational (v) and rotational (K) quantum number as a function of the incident intense field. The probability of multiphoton dissociation per unit time is given by ref. [9]
dW = -~- [Tfi(N)12 dp
(7)
the nuclear wave function given by where dp is the density of final states and is
Rqk,(r ) =
e 7r/2aq (2qr)KlF(s + 1 - i/qa)[ e -iqr
× qb(1 + s + i/qa, 2s + 2, 2iqr)
d p - rnpdg2 (5)
n' = --i]aq = --i [(B2m/ti2)/2Ef] 1/2 is the principal quantum number of the final state of the continuum spectra, q is the wave number of the dissociation product, with energy Ef = t~2q2 /2m = EvK + Nhw > 0, 6K' is the phase angle. OK, OK are the polar and azimuthal angles defining the direction of flight of the particles. Substituting eqs. (2) and (4) in eq. (1), and assuming X ~ 1 which corresponds to perturbation theoretical limit of momentum translation approximation, we get the T-matrix as N
,
7~iN) = iN(Nhw)(X) a(K )CM (N)
(6)
(2Ah) 3 where m and p are the reduced mass and momentum respectively, of the dissociating particles and d~2 is an element of solid angle. The method is applicable to multiquantum transitions involving arbitrary intensities and frequencies of the incident radiation. The salient features of the present calculations is the rapid convergence of perturbation series at high intensities. Thus we conclude that under the condition of non-relativistic dipole approximation, (wc/a) ~ 1, a relatively simple result, which is much easier to compute than the one obtained by the usual perturbation approach, is obtained. In conclusion, the author is thankful to Dr. M.H. Key for his interest in this investigation and to the referee for his suggestions.
where
a(K') = ~ 2 K' ( 2 K ' + I ) I ' ( N + 1)F{(N+K')[2+ 1} K'M' P{(N-I()/Z+I}F(N+K'+2) e ~r/2aq IC(s + 1 + n')l (v!) 1/2 c = v~ r(2s + 2) [aF(2s + 2 + u)] 1/2n 1)
M(N) = ~ a}O) 1 2(/+2s+2)a(nn,)(U+l+s+2) l=0 N.I nt × F(N+2s+l+3) (n + n') (N+2s+l+3) ×2F1
+2s+l+3,1-n'+s,
2s+2, n~-~n ]
where a} v) are the coefficient of Laguerre polynomials 20
References [1] F.V. Bunkin and I.I. Tugov, Zh. Eksp. Teor. Fiz. 58 (1970) 1987. [Sovt. Phys. JETP 31 (1970) 1971]; Phys. Rev. A8 (1973) 601,612, 620. [2] N.R. Isenor and M.C. Richardson, Appl. Phys. Lett. 18 (1971)6. [3] G.H. Dunn, Phys. Rev. 172 (1968) 1. [4] A. Gold and H.B. Bebb, Phys. Rev. 143 (1966) 1. [5] Y. Gontier and M. Trahin, Phys. Rev. A4 (1971) 1896. [6] H.R. Reiss, Phys. Rev. D4 (1971) 3533. [7] R.K. Thareja, Phys. Lett. 49A (1974) 54; R.K. Thareja and M. Mphan, ibid 57A (1976) 135. [8] N.K. Rahman and H.R. Riess, Phys. Rev. A6 (1972) 1252 [9] P. Roman, Advanced quantum mechanics (AddisonWesley, Mass. 1965).
PHYSICS LETTERS
DISSOCIATION OF MOLECULES
11 July 1977
IN INTENSE INFRA-RED FIELDS
R.K. THAREJA a Department of Pure and Applied Physics. The Queen's University of Belfast, Belfast BT7 1NN, Northern Ireland Received 1 November 1976 Revised manuscript received 11 May 1977 A generalised analytic expression for the T-matrix for dissociation of diatomic hetro-nuclear molecules for arbitrary values of the vibrational (o) and rotational (K) quantum number of the initial state of the same electron terms, in an intense electromagnet field is evaluated.
In the last couple of years a considerable amount of work has been done in the field of dissociation, excitation etc. of diatomic molecules [ 1 - 3 ]. The transition probability for the dissociation of the molecule in an intense field may be calculated using the Nth order time dependent perturbation theory, which reduces essentially to calculating ( N - 1) sums over vibrational structure of intermediate transitions; in a manner similar to that used for calculating the multiphoton ionization o f atoms [4,5]. However, at high intensities not only the evaluation o f sums becomes complicated but the validity o f the theory itself is doubtful also [6]. For such high intensities the momentum translation approximation is very useful. Essentially it consists of applying a unitary transformation which approximately removes the incident e.m. field from the problem. The method has successfully been applied to multiphoton atomic transitions [7,81. In the present note, we apply the momentum translation approximation to evaluate the transition probability of diatomic hetro-nuclear molecule for arbitrary values of the vibrational (o) and rotational (K) quantum number of the initial state. It is assumed that N-quantum transition takes place without change in the electronic state of the molecule. In other words, the transition occurs from a vibrational levet-n to a vibrational level-n' of the continuum spectra of the same electron term (s = s'). As the effective field of the electrons and nuclei, we used the potential V(r) =
1 A Leverhulme Visiting Fellow.
A i r 2 - B/r. The values of the parameters A and B are determined by the bond dissociation energy, D = B2/4A, and the halfwidth of the potential well A, A = D~2/8, B = DA/Vc2. The T-matrix for a transition from an initial nuclear state ¢i(r) to a final nuclear state 0f(r) by the absorption of N-photons of the intense field of vector potential A is given by
where A, w, ~ are the amplitude, frequency and polarization vector of the incident wave and Jw(z) is the Bessel-function of order N. The wave function for the bound state is ¢i(r) = RvK(r) YKM(O, c))
(2)
where YKM(O, ¢) are the ordinary normalised spherical harmonics and 2 [['(2S + 2 + V)] 1/2 . . . . s -r/an RuK(r) - a3~2n-~P(2-~+ ~ l ~ t zrlan ) e
× ¢ ( - v , 2s + 2, 2r/an),
(3)
where s is the positive root of the equation s(s+ 1) = 2mA/h 2 + K ( K + 1), qb is the confluent hypergeometric function and n = s + 1 + v, v = 0, 1,2 .... is the vibration quantum number, a = Ii2/BM = aome2/mz2 is the coulomb unit of length. The energy of the state is given by
EoK = - D + hw0(o + 1) _ hWoXo(V" + 1)2 + Be(K +1)2 where w 0 = 2D/2x/~--ma is the vibrational frequency at o = O, WoX 0 = 3 (hD/mA) is the vibrational anharmonicity 19
Volume 62A, number 1
PHYSICS LETTERS
parameter, B e ='h2/2J is the rotational constant. Thus for the ground state (v = K = M = 0) the energy is approximately given by E 0 = - D + hWo/2. As for the final dissociation state 0f(r), we use qb'"
(2rr)3/2
fir) = ~ - -
K'M' ~ iK'e-i6K'RqK'(r) (4)
x YK,~r(O,4))YK,~r(OK,%) RqK,(r ) is the regular solution of the radial part of
11 July 1977
of degree v. Thus for v = 0, a(O) = 1 ; v -= 1, a(o1) = 2s + 2;a~ 1) etc.; and X = eAa/hc is a dimensionless intensity parameter. The eq. (6) gives the final expression for the Tmatrix for an arbitrary vibrational (v) and rotational (K) quantum number as a function of the incident intense field. The probability of multiphoton dissociation per unit time is given by ref. [9]
dW = -~- [Tfi(N)12 dp
(7)
the nuclear wave function given by where dp is the density of final states and is
Rqk,(r ) =
e 7r/2aq (2qr)KlF(s + 1 - i/qa)[ e -iqr
× qb(1 + s + i/qa, 2s + 2, 2iqr)
d p - rnpdg2 (5)
n' = --i]aq = --i [(B2m/ti2)/2Ef] 1/2 is the principal quantum number of the final state of the continuum spectra, q is the wave number of the dissociation product, with energy Ef = t~2q2 /2m = EvK + Nhw > 0, 6K' is the phase angle. OK, OK are the polar and azimuthal angles defining the direction of flight of the particles. Substituting eqs. (2) and (4) in eq. (1), and assuming X ~ 1 which corresponds to perturbation theoretical limit of momentum translation approximation, we get the T-matrix as N
,
7~iN) = iN(Nhw)(X) a(K )CM (N)
(6)
(2Ah) 3 where m and p are the reduced mass and momentum respectively, of the dissociating particles and d~2 is an element of solid angle. The method is applicable to multiquantum transitions involving arbitrary intensities and frequencies of the incident radiation. The salient features of the present calculations is the rapid convergence of perturbation series at high intensities. Thus we conclude that under the condition of non-relativistic dipole approximation, (wc/a) ~ 1, a relatively simple result, which is much easier to compute than the one obtained by the usual perturbation approach, is obtained. In conclusion, the author is thankful to Dr. M.H. Key for his interest in this investigation and to the referee for his suggestions.
where
a(K') = ~ 2 K' ( 2 K ' + I ) I ' ( N + 1)F{(N+K')[2+ 1} K'M' P{(N-I()/Z+I}F(N+K'+2) e ~r/2aq IC(s + 1 + n')l (v!) 1/2 c = v~ r(2s + 2) [aF(2s + 2 + u)] 1/2n 1)
M(N) = ~ a}O) 1 2(/+2s+2)a(nn,)(U+l+s+2) l=0 N.I nt × F(N+2s+l+3) (n + n') (N+2s+l+3) ×2F1
+2s+l+3,1-n'+s,
2s+2, n~-~n ]
where a} v) are the coefficient of Laguerre polynomials 20
References [1] F.V. Bunkin and I.I. Tugov, Zh. Eksp. Teor. Fiz. 58 (1970) 1987. [Sovt. Phys. JETP 31 (1970) 1971]; Phys. Rev. A8 (1973) 601,612, 620. [2] N.R. Isenor and M.C. Richardson, Appl. Phys. Lett. 18 (1971)6. [3] G.H. Dunn, Phys. Rev. 172 (1968) 1. [4] A. Gold and H.B. Bebb, Phys. Rev. 143 (1966) 1. [5] Y. Gontier and M. Trahin, Phys. Rev. A4 (1971) 1896. [6] H.R. Reiss, Phys. Rev. D4 (1971) 3533. [7] R.K. Thareja, Phys. Lett. 49A (1974) 54; R.K. Thareja and M. Mphan, ibid 57A (1976) 135. [8] N.K. Rahman and H.R. Riess, Phys. Rev. A6 (1972) 1252 [9] P. Roman, Advanced quantum mechanics (AddisonWesley, Mass. 1965).