Underlying principles

As was mentioned in Chapter 3, the spectra of molecules are much more complicated than those of atoms due to the influence of the vibration of the covalent bonds, and the angular momentum associated with molecular rotation. Due to the limitations of space, only an abbreviated summary of the relevant aspects is given here. During this overview, the underlying reason for the choice of molecular tellurium for the purpose of frequency-locking a blue laser will become evident.

The effects of molecular rotation will be considered first. A diatomic molecule such as Te2 can be considered to have two rotational degrees of freedom since the moment of inertia around the axis of the bond is very low; the moments of inertia (I) around the two axes perpendicular to the bond are virtually identical. In many cases, it is reasonable to approximate the diatomic molecule as a rigid rotator, although in reality the bond length increases for faster rotation due to centrifugal distortion. This leads to the following equation (Banwell 1994) for the potential (in cm-1_{) of the J’th} rotational sub-level above that of the rotational ground state ε0:

2

( 1) - ( 1)

j BJ J DJ J

ε = + + 2 _(6.1)

Here, J is the rotational quantum number (J=0,1,2,…), D is the centrifugal distortion constant, which compensates for the fact that the real molecule is not a rigid rotator, and B (cm-1_{) is the rotational constant, defined as:}

6. Laser locking to 130Te2 resonance line 2 8 h B Ic π = (6.2)

In Eqn. 6.2, h is Planck’s constant and I is given by:

2 0

I =µr (6.3)

Here, r0 represents the bond length (286 pm), and µ the reduced mass. Pure rotational transitions correspond to the absorption or emission of electromagnetic radiation in the microwave region of the spectrum; the selection rule governing such transitions is ∆J=±1. Interaction with electromagnetic radiation can, however, only occur for heteronuclear molecules because there is no dipole component change during the rotation of homonuclear molecules (Banwell 1994). As a consequence of this, the rotational structure of homonuclear molecules such as 130Te2 can only be investigated spectroscopically by probing transitions in which there is also a change in the electronic energy level.

Vibrations of the covalent bond are another source of internal energy in molecules. The vibration results from the elasticity of the bond, which allows the position of the bound molecules to oscillate around their mean locations. The potential energy of the system as a function of internuclear distance is described empirically by the Morse function (Banwell 1994), which leads to the following equation for the energies of the vibrational states:

1 1 2 2 e xe ν ε =ω ⎡_⎢ − ⎛_⎜ν + ⎞ ⎛_{⎟ ⎜}⎤_⎥ ν + ⎝ ⎠ ⎝ ⎠ ⎣ ⎦ 1⎞ ⎟ (6.4)

6. Laser locking to 130Te2 resonance line

Here, ωe is an oscillation frequency and xe is an ‘anharmonicity constant’,

which is always small and positive. The selection rule for vibrational transitions is ∆ν = ±1, ±2, ±3…, but the transition probability decreases rapidly with increasing ∆ν and transitions with ∆ν = ±3 typically have negligible intensity. It follows from Eqn. 6.4 that the energy change for an absorption from ν=0→ν=1 is: ∆ =ε ω_e

(

1 2− x_e

)

, while for ν=0→ν=2, the energy change is: ∆ =ε 2ω_e

(

1 3− x_e

)

. Since xe is small, the energy change in

the latter case is roughly double that of the former. Therefore, the ν=0→ν=1 transition is referred to as a fundamental absorption, whereas the ν=0→ν=2 transition is referred to as the first harmonic overtone. Vibrational transitions of molecules correspond to the absorption or emission of radiation in the infra-red spectral region. As was the case for molecular rotations, the molecule can only interact with light if the vibration involves a change in the dipole moment of the molecule. Therefore, homonuclear diatomic molecules can be probed neither by pure rotational spectroscopy nor by pure vibrational spectroscopy. To perform spectroscopy of such molecules, electronic transitions can instead be studied.

In general, transitions of molecules from one electronic state to another tend to correspond to the emission of light in the UV-visible spectral range. Electronic transitions can be probed spectroscopically in all molecules since changes in the electronic configuration of the molecule always lead to a change in the dipole moment. For molecules in general, the lowest electronic energy states is denoted by the letter X, and the next levels are called A, B, C, etc (Eckbreth 1996). The electronic states of molecules are categorised

6. Laser locking to 130Te2 resonance line according to the quantum number (Λ) describing the axial component of the total orbital angular momentum (Herzberg 1950). These types of state are denoted by the term symbols Σ, Π, ∆, Φ, etc., for Λ = 0, 1, 2, 3, etc.

Vibrational and rotational transitions may occur together with the electronic transitions; therefore, each electronic transition consists of a large number of individual lines. The total energy change can, by the Born-Oppenheimer approximation, be written as (Banwell 1994):

total electronic vibrational rotational

ε ε ε ε

∆ = ∆ + ∆ + ∆ (6.5)

The rotational transitions that may occur together with changes in the electronic energy state are governed by the general selection rule ∆J= -1,0,1. These three situations are denoted respectively as the P, Q, and R branches. If however, Λ = 0 in both electronic states of the transition, then ∆J = 0 is forbidden and there is no Q branch.

An electronic transition may be accompanied by transitions between any vibrational states, but absorption transitions originating in the lowest vibrational level are strongest since that level is most highly populated. The intensity of molecular absorption lines is also affected by the Franck-Condon principle, which states that, during the electronic transition, there is unlikely to be a substantial change in the internuclear separation or of the molecular velocity (Herzberg 1950). An example of this situation is given in Figure 6.1, in which two electronic energy levels are shown. The electronic potential is plotted for each level as a function of internuclear separation; the minimum of this function corresponds to the bond length. The wavefunctions (ψ) for

6. Laser locking to 130Te2 resonance line selected vibrational levels are also shown. The internuclear separation for a particular vibrational level is described by a probability density function which is equal to ψ2. The consequence of the Franck-Condon principle is that the transition corresponds to a vertical line on Figure 6.1.

Figure 6.1. Franck-Condon principle for electronic-vibration transitions. The potential curves for two electronic states and the wavefunctions for selected vibrational levels are shown. For the internuclear separation depicted here, the best overlap of the wavefunctions occurs for ν’ = 2, ν’’ = 0. This diagram has been taken from Herzberg (Herzberg 1950)

The pairs of levels for which the maximum of ψ2 occurs at the same internuclear separation are thus ‘well-overlapped’, and these transitions have highest probability. Having briefly addressed the general basis of molecular spectra, some further details will now be given concerning the spectra of molecular tellurium, which was studied during the present work.

6. Laser locking to 130Te2 resonance line

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