Hot band

In molecular vibrational spectroscopy, a hot band is a band centred on a hot transition, which is a transition between two excited vibrational states, i.e. neither is the overall ground state. In infrared or Raman spectroscopy, hot bands refer to those transitions for a particular vibrational mode which arise from a state containing thermal population of another vibrational mode. For example, for a molecule with 3 normal modes, ν 1 {displaystyle u _{1}} , ν 2 {displaystyle u _{2}} and ν 3 {displaystyle u _{3}} , the transition 101 {displaystyle 101} ← 001 {displaystyle 001} , would be a hot band, since the initial state has one quantum of excitation in the ν 3 {displaystyle u _{3}} mode. Hot bands are distinct from combination bands, which involve simultaneous excitation of multiple normal modes with a single photon, and overtones, which are transitions that involve changing the vibrational quantum number for a normal mode by more than 1. In molecular vibrational spectroscopy, a hot band is a band centred on a hot transition, which is a transition between two excited vibrational states, i.e. neither is the overall ground state. In infrared or Raman spectroscopy, hot bands refer to those transitions for a particular vibrational mode which arise from a state containing thermal population of another vibrational mode. For example, for a molecule with 3 normal modes, ν 1 {displaystyle u _{1}} , ν 2 {displaystyle u _{2}} and ν 3 {displaystyle u _{3}} , the transition 101 {displaystyle 101} ← 001 {displaystyle 001} , would be a hot band, since the initial state has one quantum of excitation in the ν 3 {displaystyle u _{3}} mode. Hot bands are distinct from combination bands, which involve simultaneous excitation of multiple normal modes with a single photon, and overtones, which are transitions that involve changing the vibrational quantum number for a normal mode by more than 1. In the harmonic approximation, the normal modes of a molecule are not coupled, and all vibrational quantum levels are equally spaced, so hot bands would not be distinguishable from so-called 'fundamental' transitions arising from the overall vibrational ground state. However, vibrations of real molecules always have some anharmonicity, which causes coupling between different vibrational modes that in turn shifts the observed frequencies of hot bands in vibrational spectra. Because anharmonicity decreases the spacing between adjacent vibrational levels, hot bands exhibit red shifts (appear at lower frequencies than the corresponding fundamental transitions). The magnitude of the observed shift is correlated to the degree of anharmonicity in the corresponding normal modes. Both the lower and upper states involved in the transition are excited states. Therefore, the lower excited state must be populated for a hot band to be observed. The most common form of excitation is by thermal energy. The population of the lower excited state is then given by the Boltzmann distribution.In general the population can be expressed as where kB is the Boltzmann constant and E is the energy difference between the two states.In simplified form this can be expressed as where ν is the wavenumber of the hot band and T is the temperature . Thus, the intensity of a hot band, which is proportional to the population of the lower excited state, increases as the temperature increases. As mentioned above, combination bands involve changes in vibrational quantum numbers of more than one normal mode. These transitions are forbidden by harmonic oscillator selection rules, but are observed in vibrational spectra of real systems due to anharmonic couplings of normal modes. Combination bands typically have weak spectral intensities, but can become quite intense in cases where the anharmonicity of the vibrational potential is large. Broadly speaking, there are two types of combination bands. A difference transition, or difference band, occurs between excited states of two different vibrations. Using the 3 mode example from above, 010 {displaystyle 010} ← 100 {displaystyle 100} , is a difference transition. For difference bands involving transfer of a single quantum of excitation, as in the example, the frequency is approximately equal to the difference between the fundamental frequencies. The difference is not exact because there is anharmonicity in both vibrations. However the term 'difference band' also applies to cases where more than one quantum is transferred, such as 100 {displaystyle 100} ← 020 {displaystyle 020} .