Atomic Spectroscopy

 

Our motivation for the study of atomic spectroscopy is to learn the rules that will be applicable to the more complicated case of molecular spectroscopy. Here we will study:

 

The Hydrogen Atom Wavefunctions

Selection Rules for Transitions

Line Broadening Mechanisms in Gas Phase Transitions

 

There are no rotational or vibrational motions to contend with in electronic spectroscopy. All of the transitions are electronic. This is convenient for a starting point since we avoid the complication of molecular transitions that can involve electronic, vibrational and rotational states simultaneously.

 

The Hydrogen Atom Wavefunctions

The simplest case is the hydrogen atom. It has one electron and one proton in the nucleus. The solutions for the electron in the hydrogen atom have the form:

where Rnl(r) is the radial wavefunction, Ylml(q,f) is the angular wavefunction, and we also have a spin wavefunction indicated by a and b. The angular wavefunction is the set of spherical harmonics whose properties we have investigated elsewhere. Ignoring normalization the spherical harmonics can be further separated as follows:

The one electron energy levels are given by

where m is the reduced mass of the nucleus and the electron. Since

where me is the mass of an electron and mN is the mass of a proton. We can show that m » me as follows 

The solutions are valid for any nucleus of charge Z with a single electron. We simply multiply the energy En by Z2 to obtain values for nuclei other than hydrogen. A natural unit of length that emerges from consideration of these solutions is the Bohr radius a0. One Bohr radius is 0.529 Å.

The quantum number n is known as the principal quantum number. For each value of n there is a degeneracy of 2n2. The degeneracy arises because of different value of l, ml, and ms. The angular momentum quantum number l can have integer values l < n.

l = 0 , 1, 2, 3, 4,… n-1

There correspond to the letter designations s, p, d, f, g, etc.

We have already seen that each value of l has 2l + 1 values of ml.

ml = -l, -l + 1, … 0, … l - 1, 1

The spin quantum number ms can have values of +1/2 or -1/2. The a and b in the wavefunction above represent the spin wavefunctions. Spin has an angular momentum although it is not a quantity that is easy to see intuitively. The best way to illustrate how to think about spin angular momentum is compare it to orbital angular momentum. We have seen that:

The magnitude of the orbital angular momentum vector is

and its projection onto the z-axis is mh.

Our classical picture of an electron rotating about the nucleus helps to understand the quantum mechanical angular momentum that arises from the wavefunction solutions to the hydrogen atom.

The spin angular momentum can be pictured as arising due to the magnetic momentum of a spinning electron, however, this physical picture is not necessarily correct. It is useful to have physical pictures in order to help your thinking about the observed states and transitions in atoms.

In both orbital and spin angular momentum we can know only the total angular momentum L and S and the z-projection Lz and Sz. Because Lx and Ly cannot be determined at the same time we illustrate the angular momenta as a cone. We know the magnitude along z and we know that L points somewhere in the cone.

The Figure shows L = Ö2h and Lz = h. Because L is somewhere in the cone there is no information in the x or y directions.

 

Electronic Transitions in Hydrogen

Electronic transitions in hydrogen are observed as series of discrete emission lines. Spectroscopists measure the transition energy in wavenumbers (cm-1). The wavenumber can be related to frequency and the wavelength as follows:

The transitions of the hydrogen atom are expressed in units of cm-1 by use of the Rydberg constant RH = 109,737 cm-1. The Rydberg constant is obtained by dividing the set of constants in the energy expression by hc (h is Plank's constant).

The highest energy series in the ultraviolet is known as the Lyman series. This involves transitions from n = 2, 3, 4, etc. to n = 1. The Balmer series is in the visible. This involves transitions from n = 3, 4, 5, etc. to n = 2. The Paschen series in the near infrared involves transitions from n = 4, 5, 6, etc. to n = 3. It is apparent from experimental observation that transitions are allowed from any quantum number n to any other quantum number n'. In other words there is radial selection rule. This is not true for angular momentum where there are selection rules.

 Selection Rules for Transitions in Atomic Hydrogen

Consider polarized radiation impinging on a hydrogen atom. For electric dipole allowed transitions we consider the matrix elements of the electric dipole moment. This is given by:

The components in the dipole operator correspond to polarizations along x, y, and z (for i, j, and k, respectively) of incident radiation. For example, if we want to consider only z-polarized radiation then we will define the operator as m = -er cos q. In the derivations that follow we consider all three polarizations at the same time. The quantum numbers for the initial state are n, l, ml, and ms. After the transition the quantum numbers are n', l', m'l, and m's. We have seen that the transition dipole moment is the central quantity that determines the magnitude of the transition probability per unit time:

We can write this out as

Note that the braket notation is just a shorthand for the integral over all space. The differential volume element is dt = r2drsinqsqdf. The r in the first integral arises from the dipole operator. The r2dr is the differential radial element. The x, y, and z polarization of incident radiation is indicated in the angular integral by the three factors in brackets. The Kroenecker delta function at the end of the expression means that:

This is a spin selection rule. The transition dipole will be zero if the spin quantum number changes. We also write this as Dms = 0. The significance, once again, is that ms should not change for an allowed transition.

Because of the separation of variables we can consider the three integrals over r, q, and f separately. If any one of them is zero then the transition dipole is zero. The f-integral is:

For the z-polarized component corresponding to the 1 above we have:

 

For z-polarized radiation the selection rule on ml is Dml = 0. To find the selection rules for x- and y-polarized radiation we use the relations:

and the fact that for any integral of the type

the exponent must be zero in order for the integral to be non-zero.

For example,

can be written as two integrals

or

which are non-zero if m'l = ml + 1 or m'l = ml - 1. The combined selection rule is Dml = 0, ±1. To find the selection rule for l we must integrate over q. From above we have the remaining terms to consider:

Where the sinq term is for x,y polarized light and the cosq term is for z-polarized light. Since we have eliminated the f dependence we now have only the associated Legendre polynomials remaining. The substitution x = cosq is made here.

To consider the selection rule for z-polarization we use the recursion relation:

Thus, we can replace xPlml(x) by

as shown below.

The orthogonality relations demand that l' = 1 + 1 or l' = 1 - 1. A similar reasoning holds for x- and y-polarized radiation. The selection rule for all three polarizations is Dl = ±1.

There are no symmetry restrictions on the radial integral in agreement with the experimental observation that transitions are allowed from any principal quantum number to any other.

 Line Broadening Mechanisms

For isolated atoms in the gas phase, the observed frequency of absorbed or emitted radiation will be shifted by the velocity of the atoms. This is an example of the Doppler effect. When a source emitting electromagnetic radiation of frequency n recedes with a speed v, the observer detects radiation of frequency:

where c is the speed of light. If the source approaches the observer the observed frequency is

Solving for v we find

If we now plug this expression into the Maxwell-Boltzmann distribution:

we find that the intensity is

The center frequency of the line is n0. Notice that the variance of the line is:

and the full-width at half-maximum is

Doppler broadening is an excellent example of inhomogeneous broadening. The intrinsic energy of each of the individual atoms is just n0. However, the observer sees a distribution due to an environmental effect, their velocity which is given a Maxwell-Boltzmann distribution.

 Given that Doppler broadening will always be present, what can we do to observe the intrinsic or natural linewidth? Recall that the intrinsic linewidth will be governed by population relaxation (T1) and pure dephasing (T2*) contributions. In the gas phase we can use Lamb dip spectroscopy, named after its discoverer, W. Lamb. The idea is a follows:

 

We first describe a configuration for excitation at frequencies higher than the peak frequency. The blue arrow represents radiation that will be absorbed by molecules that approach the observer. Although the frequency of the radiation does not change, the red arrow signifies the fact that the radiation will be absorbed by molecules receding from the observer after being reflected off a mirror. Thus, twice as much radiation will be absorbed in this configuration for the select population near n0. If the excitation laser is tuned at the peak of the absorption then particles moving perpendicular to the laser beam will absorb.

On the return path many of those molecules will still be in the excited state so they will not absorb. Thus, at the peak there will be less absorbance. There will be a dip in the absorption band at the peak. The width of this dip will correspond to the natural linewidth of the atom or molecule in the gas phase.

The Figure shows a Gaussian in frequency (reddish dashed line) and a Gaussian with a Lamb dip absorption at the peak of the band.

Of course, in order to perform this experiment we need a narrow bandwidth source of radiation. This type of spectroscopy is a general example of a phenomenon known as holeburning. Any time we use a narrow bandwidth laser to interrogate a select (homogeneous) population of molecules we perform a holeburning experiment. The natural linewidth or homogeneous linewidth is related to the lifetime of the atom. The lifetime broadening (also called uncertainty broadening) is

where t is the lifetime of the excited state or T1 time. However, in addition to the lifetime collisions may also cause pure dephasing. Collisions can also lead to broadening by either decreasing the lifetime or otherwise causing dephasing.