Principles of Lasers
This new Fifth Edition of Principles of Lasers incorporates corrections to the previous edition. The text’s essential mission remains the same: to provide a wide-ranging yet unified description of laser behavior, physics, technology, and current applicati
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3.1. MOLECULES We will first consider the energy levels, and the radiative and nonradiative transitions in molecules. The considerations will be limited to a qualitative description of those features which are particularly relevant for a correct understanding of laser action in active media such as molecular gases or organic dyes. For a more extensive treatment of the wider subject of molecular physics the reader is referred to specialized texts..1/
3.1.1. Energy Levels The total energy of a molecule consists generally of the sum of four contributions: (1) electronic energy, Ee due to the motion of electrons about the nuclei; (2) vibrational energy Ev , due to the vibrational motion of the nuclei; (3) rotational energy Er , due to the rotational motion of the molecule; and (4) translational energy. We will not consider the translational energy any further since it is not usually quantized. The other types of energy, however, are quantized and it is instructive to derive, from simple arguments, the order of magnitude of the
O. Svelto, Principles of Lasers, c Springer Science+Business Media LLC 2010 DOI: 10.1007/978-1-4419-1302-9 3,
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Energy Levels, Radiative and Nonradiative Transitions
FIG. 3.1. Potential energy curves and vibrational levels of a diatomic molecule.
energy difference between electronic levels .Ee /, vibrational levels .Ev /, and rotational levels .Er /. The order of magnitude of Ee is given by Ee Š
„2 ma2
(3.1.1)
where „ D h = 2 , m is the mass of the electron, and a is the size of molecule. In fact, if we consider an outer electron of the molecule, the uncertainty in its position is of the order of a, then the uncertainty in momentum, via the uncertainty principle, is „ = a, and the minimum kinetic energy is therefore „2 = 2ma2 . For a diatomic molecule consisting of masses M1 and M2 we assume that the corresponding potential energy, Up , vs internuclear distance R, around the equilibrium distance R0 , can be approximated by the parabolic expression Up D k0 .RR0 /2 =2 (see Fig. 3.1). Then, the energy difference Ev between two consecutive vibrational levels is given by the well known harmonic oscillator expression 1=2 k0 (3.1.2) Ev D h0 D „ where D M1 M2 = .M1 C M2 / is the reduced mass. For a homonuclear molecule made of two atoms of mass M, the energy difference between two vibrational levels is then 2k0 1=2 (3.1.3) Ev D „ M We also expect that a displacement of the two atoms from equilibrium by an amount equal to the size of the molecule would produce an energy change of about Ee since this separation would result in a considerable distortion of the electronic wavefunctions. We can thus write Ee D k0 a2 = 2
(3.1.4)
3.1
Molecules
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From Eqs. (3.1.1), (3.1.3) and (3.1.4) one can eliminate a2 and k0 to obtain Ev D 2.m = M/1=2 Ee
(3.1.5)
For a homonuclear diatomic molecule, the rotational energy is then given by Er D „2 J.J C 1/ = Ma2 , where J is the rotational quantum number. Therefore, the difference Er in rotational energy between e.g. th
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