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Multiphoton dissociation of diatomic molecule under the influence of intense laser beam
Volume 53A, number 5
PHYSICS LETTERS
14 July 1975
MULTIPHOTON DISSOCIATION OF DIATOMIC MOLECULE UNDER THE INFLUENCE OF INTkNSE LASER BEAM U. DEVI and M. MOHAN Department of Physics and Astrophysics, Universityof Delhi, Delhi-l 10007, India Received 29 January 1975 Transition probability of multiphoton dissociation of diatomic molecule in an intense laser beam has been calculated using a method due to Reiss. Numerical computations are done for the case of hydrogen molecular ion. Natural units have been used here.
Reiss’s method [l] used here for calculating transition probability is based upon unitary transformation which approximately removes the e.m. field from the problem. In the present paper the electron state of the molecule in the vibrational excitation process is considered as changing adiabatically with the internuclear distance r. Bunkin et a1..[2] have discussed the advantages of using Kratzer potential over that of Morse oscillator potential for the effective field of the electrons and nuclei. Initial state wave function used here is in the form of BornOppenheimer approximation while the continuum state wave function is a partial wave expansion which has the asymptotic form of a plane wave of unit amplitude plus a diverging spherical [2] wave. Reiss’s interaction operator in this case becomes (ie2,r.A) where (e2,) is the effective charge of the nuclei. The transition matrix element for the dissociation of diatomic molecule from the initial nuclear state [u(r) to the final continuum state &(r) in presence of an intense laser beam represented by a plane wave [3] of vector potential A and frequency o will be T=(l?f-- Ei) (~f(‘)lexp(ieZ,r.A)IEo(r)),
(1)
where N is the number of photons taking part in the process and for convenience A will be taken in z-direction. Wave functions .&Jr) and tf(r) are same as used by Bunkin et al. in their paper [2] and using selection rule k’= k + 1, M=O;weget M _Ef ‘E
_ 2 i
p=O
2ps1 t=O
2
b
c\/tT(eA4J2p+‘(-I
3p+~+Ir(u+2st2)c*‘Sg*‘~r(u-stn’-2f)
r=O j=O I 22P+I+sp! r(l tp t j/2)(1
s+3/2r(-,‘-s-~-ttj)r(r-2~tt-I)
where X = l/an - l/an’ and C= ik’+l( - p’(2a)2 exp (ink) 21i2, s(s + 1) = 2mA/fr2+ k(k + 1). Since the size of the molecule is very small compared to the wavelength of the e.m. field, the dipole approximation plane wave form for eA may be used [3] i.e. eA = ea cos of = iea {exp (iof) + exp (-iwf)},
421
Volume 53A, number 5
14 July 1975
PHYSICS LETTERS
N= 2
:c
/F@@+---=--
‘.r
<&----
27
8
9
10 11
Fig. 1.
[&4]
2P+l=
,l’E,),2 (p?7t11)
(+eu)2P+1 exp (- iNat)
.
(3)
Putting N = 2p t 1i-j’ and shifting the origin from j’= (N - 1)/2 to zero in the final expression we get coefficient exp (- iNot) known as transition amplitude r ik’+i(-)(3N-
1)I2+3p +1Zap 2 +NC *‘s 4 *‘“r(n’ts)r(2p
of
tNt2)r(-d-s)
P=O 1 ,s+3j222snS+2P((N-1)/2+p+1)f’((N+1)/2tp+3/2)~~ (‘p;N) j=O
P(j+l)P(2p+Nt2-j)l?(n’--s--.j)P(-n’-stj-2p-N-
* (3y,800)2p+N]2
.(4)
1)
Summation of this definite series is done using Euler transformation method [4]. Use’of computer has been made for numerical computation taking dissociation energy of Hz to be D = 2.78 eV and r. = 1.06& Za = 1.228. We have separated out the intensity dependent terms of eq. (3), named them “reduced transition probability” and plotted them against intensity parameter y = $ euuu. From eq. (3) as well as from fig. 1 it is clear that for initial values of intensity the dissociation probability varies linearly and for higher values of y,y 9 1, the curves of fig. 1 show nonlinear behaviour. This behaviour is due to’the fact that at high intensities power law dependence of probability on intensity, according to our calculation turns out to be a lnlr21/a In x = N - A where x = (+u)~, and A is intensity and N dependent term. This sort of behaviour has been observed experimentally also at high intensities which is quite different from the power law dependence of perturbation theory, a ln1r21/a In x = N.
References [ 1) H.B. Reiss, Phys. Rev. Al (1970) 803; Phys. Rev. Letters 25 (1970) 1149; Phys. Rev. D (1971). [2] F.V. Bunkin and 1.1. Tugov, Zh. Eksp. Teor. Fiz. 58 (1970) 1987; [Sovt. Phys. JETP 31 (1970) 31; Bunkin, Karapetyan and Prokhnov, JETP 47 (1964) 212; [Sovt. Phys. JETP 20 (1965) 1421. [3] M. Mohan and Thareja, Phys. Rev. A7 (1973) 34; J. Phys. B. Atom. Molec. PhyS.; U. Devi and S. Bhatnagar, J. Phys. B. Vol. 7, No. 10, 1974. [4] R.M. Morse and S. Feshback, Methods of Theoret. Phys. Vol. 1, (McGraw Hill Book Co., Inc.).
422
PHYSICS LETTERS
14 July 1975
MULTIPHOTON DISSOCIATION OF DIATOMIC MOLECULE UNDER THE INFLUENCE OF INTkNSE LASER BEAM U. DEVI and M. MOHAN Department of Physics and Astrophysics, Universityof Delhi, Delhi-l 10007, India Received 29 January 1975 Transition probability of multiphoton dissociation of diatomic molecule in an intense laser beam has been calculated using a method due to Reiss. Numerical computations are done for the case of hydrogen molecular ion. Natural units have been used here.
Reiss’s method [l] used here for calculating transition probability is based upon unitary transformation which approximately removes the e.m. field from the problem. In the present paper the electron state of the molecule in the vibrational excitation process is considered as changing adiabatically with the internuclear distance r. Bunkin et a1..[2] have discussed the advantages of using Kratzer potential over that of Morse oscillator potential for the effective field of the electrons and nuclei. Initial state wave function used here is in the form of BornOppenheimer approximation while the continuum state wave function is a partial wave expansion which has the asymptotic form of a plane wave of unit amplitude plus a diverging spherical [2] wave. Reiss’s interaction operator in this case becomes (ie2,r.A) where (e2,) is the effective charge of the nuclei. The transition matrix element for the dissociation of diatomic molecule from the initial nuclear state [u(r) to the final continuum state &(r) in presence of an intense laser beam represented by a plane wave [3] of vector potential A and frequency o will be T=(l?f-- Ei) (~f(‘)lexp(ieZ,r.A)IEo(r)),
(1)
where N is the number of photons taking part in the process and for convenience A will be taken in z-direction. Wave functions .&Jr) and tf(r) are same as used by Bunkin et al. in their paper [2] and using selection rule k’= k + 1, M=O;weget M _Ef ‘E
_ 2 i
p=O
2ps1 t=O
2
b
c\/tT(eA4J2p+‘(-I
3p+~+Ir(u+2st2)c*‘Sg*‘~r(u-stn’-2f)
r=O j=O I 22P+I+sp! r(l tp t j/2)(1
s+3/2r(-,‘-s-~-ttj)r(r-2~tt-I)
where X = l/an - l/an’ and C= ik’+l( - p’(2a)2 exp (ink) 21i2, s(s + 1) = 2mA/fr2+ k(k + 1). Since the size of the molecule is very small compared to the wavelength of the e.m. field, the dipole approximation plane wave form for eA may be used [3] i.e. eA = ea cos of = iea {exp (iof) + exp (-iwf)},
421
Volume 53A, number 5
14 July 1975
PHYSICS LETTERS
N= 2
:c
/F@@+---=--
‘.r
<&----
27
8
9
10 11
Fig. 1.
[&4]
2P+l=
,l’E,),2 (p?7t11)
(+eu)2P+1 exp (- iNat)
.
(3)
Putting N = 2p t 1i-j’ and shifting the origin from j’= (N - 1)/2 to zero in the final expression we get coefficient exp (- iNot) known as transition amplitude r ik’+i(-)(3N-
1)I2+3p +1Zap 2 +NC *‘s 4 *‘“r(n’ts)r(2p
of
tNt2)r(-d-s)
P=O 1 ,s+3j222snS+2P((N-1)/2+p+1)f’((N+1)/2tp+3/2)~~ (‘p;N) j=O
P(j+l)P(2p+Nt2-j)l?(n’--s--.j)P(-n’-stj-2p-N-
* (3y,800)2p+N]2
.(4)
1)
Summation of this definite series is done using Euler transformation method [4]. Use’of computer has been made for numerical computation taking dissociation energy of Hz to be D = 2.78 eV and r. = 1.06& Za = 1.228. We have separated out the intensity dependent terms of eq. (3), named them “reduced transition probability” and plotted them against intensity parameter y = $ euuu. From eq. (3) as well as from fig. 1 it is clear that for initial values of intensity the dissociation probability varies linearly and for higher values of y,y 9 1, the curves of fig. 1 show nonlinear behaviour. This behaviour is due to’the fact that at high intensities power law dependence of probability on intensity, according to our calculation turns out to be a lnlr21/a In x = N - A where x = (+u)~, and A is intensity and N dependent term. This sort of behaviour has been observed experimentally also at high intensities which is quite different from the power law dependence of perturbation theory, a ln1r21/a In x = N.
References [ 1) H.B. Reiss, Phys. Rev. Al (1970) 803; Phys. Rev. Letters 25 (1970) 1149; Phys. Rev. D (1971). [2] F.V. Bunkin and 1.1. Tugov, Zh. Eksp. Teor. Fiz. 58 (1970) 1987; [Sovt. Phys. JETP 31 (1970) 31; Bunkin, Karapetyan and Prokhnov, JETP 47 (1964) 212; [Sovt. Phys. JETP 20 (1965) 1421. [3] M. Mohan and Thareja, Phys. Rev. A7 (1973) 34; J. Phys. B. Atom. Molec. PhyS.; U. Devi and S. Bhatnagar, J. Phys. B. Vol. 7, No. 10, 1974. [4] R.M. Morse and S. Feshback, Methods of Theoret. Phys. Vol. 1, (McGraw Hill Book Co., Inc.).
422