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Molecular emission spectrum

Contents

Idea

A molecular emission spectrum for a specific molecule describes how that molecule interacts with electromagnetic radiation. It is important to understand why only some molecules interact with (far) infrared radiation while others do not, to understand the greenhouse effect.

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An important concept in this context is that of a dipole.

Details

Theory, Quantum Mechanics

Dipole transitions are by far the most dominant transitions for the interaction of molecules and (ultraviolett, visible and infrared) light, at least under the circumstances found in the Earth’s atmosphere today. In this paragraph we will take a look at the quantum mechanical description to get an explanation for this.

In a first approximation we are going to neglect the spins of all elementary particles involved. Taking these into account is usually called fine structure (e.g. spin-orbital interactions for electrons) and hyper fine structure (electron-nucleus interactions).

As a further simplification, we will consider a single particle in a potential, which can be applied to a single electron in a molecule, for example.

In the Schrödinger picture the impulse operator of a particle in an external electromagnetic field is

P=ieA P = -i \hbar \nabla - e \vec{A}

where ee is the electric charge and A\vec{A} is the vector potential of the field.

Writing

p=i \vec{p} = -i \hbar \nabla

and using the Coulomb gauge

A=0 \nabla \cdot \vec{A} = 0

we get for the Hamiltonian

H=p 22m+V(r)+eApm+e 22mA 2 H = \frac{\vec{p}^2}{2 m} + V(\vec{r}) + \frac{e \vec{A} \cdot \vec{p}}{m} + \frac{e^2}{2 m} \vec{A}^2

Remember that we are looking at a single particle of mass mm and electric charge ee. So the interaction term UU of electromagnetic field and particle in the Hamiltonian is proportional to Ap\vec{A} \cdot \vec{p}.

In order to describe a plane wave we use for the vector potential

A(r,t)=A 0ecos(krωt) \vec{A} (\vec{r}, t) = A_0 \vec{e} cos(\vec{k} \cdot \vec{r} - \omega t)

with some unit vector e\vec{e}.

If we define a transition operator

T=e ikrωmep T = \frac{e^{i \vec{k} \cdot \vec{r}}}{\omega m} \vec{e} \cdot \vec{p}

then we get for the interaction term

U(r,t)=T 0(Te iωt+T *e iωt) U(\vec{r}, t) = T_0 (T e^{-i \omega t} + T^{*} e^{i \omega t})

where T 0T_0 is an abbreviation for the constant amplitude:

T 0=eωA 0 T_0 = e \; \omega \; A_0

The wavelength of ultraviolett to infrared light is rather long compared to the molecules that are present in Earth’s atmosphere: In this case we have

kr1 \vec{k} \cdot \vec{r} \ll 1

and can therefore approximate the exponential function in the definition of TT by the first terms:

T1+ikr+...ωmep T \approx \frac{1 + i \vec{k} \cdot \vec{r} + ...}{\omega \; m} \vec{e} \cdot \vec{p}

References

  • Peter W. Atkins, Ronald S. Friedman (Author): “Molecular Quantum Mechanics”, Oxford University Press, USA; 5 edition (December 30, 2010)

  • Ingolf V. Hertel, C.P. Schulz: “Atome, Moleküle und optische Physik 1: Atomphysik und Grundlagen der Spektroskopie”, Springer, Berlin, 1st edition (Dezember 2007)

  • Ingolf V. Hertel, C.P. Schulz: “Atome, Moleküle und optische Physik 2: Moleküle und Photonen - Spektroskopie und Streuphysik”, Springer, Berlin, 1st Edition (May 2011)“