24 Hyperfine Structures and Broadening Mechanisms
Devendra Mohan
Contents
1. Hyperfine Structure of Spectral Lines
2. Magnetic Hyperfine Structure
3. Electrical Hyperfine Structure
4. Isotope Shift
5. Width of Spectrum Lines
6. Natural Broadening
7. Doppler Broadening
8. Collision Broadening
Summary
The students will be able to learn about Hyperfine Structure of Spectral Lines, Magnetic Hyperfine Structure, Electrical Hyperfine Structure, Isotope Shift, Width of Spectrum Lines, Natural Broadening, Doppler broadening and Collision Broadening
1. Hyperfine Structure of Spectral Lines
So far the nuclei have been considered as positive point charges of infinite mass. But the splitting of the electronic energy levels has been observed through high precision experiments that cannot be explained on the basis that nuclei are point charges of infinite mass. The splitting that is much smaller than the fine structure splitting is of the order of 10-3 – 1 cm-1 is called hyperfine splitting or hyperfine structure. This hyperfine structure can be explained in terms of properties of the nuclei. In 1924, Pauli suggested that nucleus has a total angular momentum I, called nuclear spin. The eigenvalue of the operator I2 is h2I (I+1) where I is nuclear spin quantum number. The largest measurable component of the angular momentum I is hI. I is different for different states of a nucleus and can have integral or half integral values. I is considered for the ground state of a given nucleus. If I> 0, the nucleus has a magnetic dipole moment and has a quadrupole moment for I ≥1 that can interact with electric field gradient produced by the atomic electrons.
There are nuclear effects that produce hyperfine structure:
(i) The interaction between an intrinsic magnetic dipole moment of the nucleus and a magnetic field
produced by the motion of the atomic electrons. The hyperfine splitting is smaller than fine structure
splitting by an order of 10-3 because nuclear magnetic dipole moment is smaller than the electronic
magnetic dipole moment .
(ii) The interaction between quadrupole moment of the nucleus, due to a non-spherical symmetrical nuclear
charge distribution, and electric field gradient produced by the atomic electrons.
(iii) The energy levels of isotopes show small displacement relative to each other as isotopes have
different masses. Thus transitions and magnetic dipole moment lie at slightly different frequencies.
2. Magnetic Hyperfine Structure
The nucleus has a spin angular momentum I that is related to nuclear magnetic moment µN by
where g1 is a dimensionless number called nuclear g factor or nuclear Lande’ factor.
βN = eh/2Mp is nuclear magneton and has value 5.0509 x 10-27 J T-1.
At the site of the nucleus, atomic electrons exhibit an effective magnetic field B directed along the atomic axis of rotation J,
B = CJ
where C is a quantity proportional to the internal magnetic field.
The interaction of the nuclear magnetic dipole moment in the magnetic field produced by the atomic electrons is given by
Hmhf = -µN.β
whereHmhf is the Hamiltonian for magnetic hyperfine structure (mhf). Using above equations
Hmhf= – C g1βNI.J = a I.J
where a = -C g1 βN is called magnetic dipole interaction constant and contains product of nuclear quantity g1 that is proportional to the moment and electronic quantity C which is proportional to the magnetic field produced by the atomic electrons. The sign of ‘a’ depends on the sign of g1 that varies for different nuclei.
The sign of C is also variable.
The figure depicts the coupling of angular momentum vectors where F is used as the quantum number for the total angular momentum F of an atom with quantized values h√F9F+1).
The vector model tells, with LS coupling between electrons, L and S precess rapidly around their resultant, J. The magnetic field produced by the electrons couples I and J and causes these vectors to precess slowly around their resultant F.
According to vector addition, F can take the following values
F = I + J, I + J – 1,………….,|I – J|
giving 2I+1 values if J>I and 2J+1 values if I>J.
The levels corresponding to F are 2F+1 fold degenerate. However, the degeneracy can be removed by the magnetic field. Using F = J + I, F2 is
F2 = (J + I) . (J + I) = J2 + I2 + 2J.I
J.I = (1/2)[F2 – J2 – I2] = (1/2)[F(F+1)- J (J+1) – I(I+1)]
Therefore
Emhf= (a/2)[F(F+1) – J(J+1) – I(I+1)]
The energy difference between two neighboring hyperfine levels F and F-1 is called hyperfine separation and is proportional to F because (∆E)mhf=aF.
The interaction constant ‘a’ depends upon the type of orbital of the interacting electrons as well as upon the values of the nuclear magnetic moments. For p, d, f ……orbitals (l≠0) that have zero probability density |Ψ(0)|2 at the nucleus.
The interaction is a dipole-dipole type between the s orbital magnetic moment and the orbital & spin magnetic moment of the electrons. The dipole-dipole interaction is averaged to zero for an electron because of spherical symmetry of the s orbital. But s orbital has non zero probability density |Ψ(0)|2at the nucleus that gives rise to Fermi contact term.
The coefficient ‘a’ can be positive or negative depending on the considered J level. The nuclear magnetic moment can be in the same direction or in the opposite direction to the vector I.
The field B can be oriented parallel to J or in the opposite direction.
Due to the negative charge of the electron, the field BL created at the nucleus by the orbital motion of an electron is in the opposite direction to the vector L and the field BS created at the nucleus by the spin magnetic moment is parallel to S.
The projection of the resultant of these field on J representing the mean value of B of the field at the nucleus can either be parallel to J or in the opposite direction for the assembly of atomic electrons depending on the respective values of L and S in L-S coupling or different J in jj coupling,
The figure shows the hyperfine structure of 2P1/2 – 2S1/2 transition of Na. The nuclear spin of Na is 3/2 and the values of F are 2 and 1 for both 2P1/2and 2S1/2 level.
According to the selection rules ∆F = 0, ±1.
The transition 2P1/2 – 2S1/2 splits into four and the width of 2P1/2level is about ten times smaller than that of 2S1/2term.
It is noteworthy that for nuclei, in which magnetic moment has the same sign as I, the levels of higher F have higher energy and hyperfine structure is normal.
3. Electrical Hyperfine Structure
The energy difference between the ground level and the excited levels of an atom is ~ 1014Hz. On the other hand the hyperfine splitting between levels of the same J is of the order of ~109Hz. Therefore, the effect of hyperfine structure requires very small energy correction. It is necessary to take into account the very small but finite size of the nucleus for this correction to be applied. As the distribution of charge within the nucleus is often not spherically symmetric, it leads to the existence of a quadrupole moment Q of the nucleus. The quadrupole moment Q is positive if the nuclear charge distribution is elongated along the direction of I (prolate) and is negative if the distribution is flattened (oblate). It may be noted that a nucleus whose charge distribution is spherically symmetric has no electric quadrupole moment.
The interaction of electric quadrupole moment with the electric field gradient produced by atomic electrons leads to first order energy shift
qJ is proportional to the electric field gradient.
For I = 0 or ½, nuclear quadrupole moment vanishes and therefore the energy shift also vanishes. Adding the electric quadrupole correction, the total hyperfine structure energy correction is given by
The dependence of Electric quadrupole interaction on the quantum number F is different from that of the magnetic hyperfine structure. The figure shows the magnetic hyperfine structure and electric hyperfine structure for J = 3/2
4. Isotope Shift
Hyperfine structure does not exist for Isotopes with I = 0. But in transitions between energy levels in a mixture of I = 0 of the same element, a structure may be observed that is called the isotope shift. There can be two effects that cause the isotope shift:
(i) the mass effect and
(ii) the volume effect
(i) Mass Effect
The mass effect is considered to be normal which is due to the motion of the nucleus, as it is not infinitely heavy. For hydrogenic atoms, the reduced mass of the electron
used into the Schrodinger equation depends on the mass M of the nucleus and varies with the isotope under consideration. Energy levels E (M) for an atom whose nucleus has finite mass M is raised above the fictitious level E (∞) for an atom whose nucleus is infinitely heavy.
The wavenumber shift for the observed spectral line is
The displacement due to the mass effect reduces the term value thereby raises the energy and as this displacement decreases with increasing M, the line due to heavier isotope has greater wavenumber. If the two isotopes of mass M and M+1are considered, then
However, the specific mass effect is due to the interactions between the different outer electrons. The specific mass shift falls off as 1/M2 with increasing mass like the normal shift, and is very small in heavy elements.
ii) Volume Effect
The observed isotopic shift is much greater than caused by the motion of the nucleus for the heavy elements and hence the volume effect becomes important. It is prominent when the electron configuration contains unpaired s electrons. Pauli and Peierls provided the first explanation on the basis of variation of volume of the nucleus and of charge distribution within the nucleus between one isotope and another. The nucleus has a charge density over a finite volume. The s electrons have a finite charge probability distribution at the nucleus. The s electrons are then no longer under the influence of pure Coulomb field. The largest relative change in the potential occurs near the origin and therefore the volume effects have great influence for s orbits.
5. Width of Spectrum Lines
A transition between two states is not seen as infinitely sharp line though there is a definite line shape and purely monochromatic spectral lines does not exist. However there is a broadening of spectral lines:
(i) Natural broadening
(ii) Doppler broadening and
(iii) Collision broadening
The spectral line emitted by an isolated atom (at rest) shows its natural shape. Let the atom gain some translational velocity after being given some thermal energy. There is a change in frequency of the emitted radiation according to Doppler Effect due to a velocity component in the direction of the observer. A certain width called Doppler width is anticipated at a given temperature because Maxwellian distribution of velocities at a given temperature leads to different velocities with various probabilities. Thus the Natural and Doppler broadening is present even in the absence of neighbors. In case, the atoms are not isolated, there will be an additional broadening effect due to collision among atoms.
Natural broadening and Collision broadening are homogeneous while the Doppler broadening is inhomogeneous broadening which refers to the mechanisms that affect the line shape of every atom or molecule of the sample in the same way. The line shape is, in general, Lorentzian. In inhomogeneous broadening atoms or molecules contribute to different parts of the line profile and is a statistical effect and therefore, broadened line is Gaussian in shape.
6. Natural Broadening
An accelerating charge radiates according to the laws of classical electrodynamics. This radiation carries off energy, which is due to particles kinetic energy. Charged particles accelerate less than a neutral particle of the same mass, under the influence of a given force. There is a recoil force that the radiation evidently exerts a force back on the charge. Thus a vibrating electric charge is continually damped by the radiation of energy and hence the energy E of such an oscillator decreases exponentially with time as
E = E0exp(-γt)
where E0 is initial energy at time t = 0. E is the energy at any later time t and ˠ =
Here ω0 is frequency of emitted radiation.
Classically, spectrum has a Lorentzian shape given by the light intensity I
Here 2γ is full width at half maximum, i.e., the interval between the two points where the intensity drops at half maximum value as is clear from the figure.
As it has already been discussed that even for an isolated atom at rest, a spectral line will be broadened slightly that further implies that there will be indefiniteness in energy, ∆E.
Let ∆t be the time available for measurement of energy and using Heisenberg’s uncertainty principle
For a transition from ground state (τ= ∞, ∆E = 0) to a state (τ)
7. Doppler Broadening
An excited atom drop back to the original state with the excess energy emitted in the form of electromagnetic radiation. The frequency v0 of the emitted radiation is related to the corresponding wavelength (For an atom at rest) by
Here c is the velocity of light.
Let us consider that an atom while radiating is moving with a velocity vx towards an observer in x-direction. The frequency of the emitted radiation, according to Doppler Effect, is observed to be
In case, the atom moves away from the observer, the frequency of emitted radiation is observed to be
Now consider an assembly of N atoms moving freely and behaving as the molecules of a gas. The vx can have all possible values from 0 to ∞.
Therefore, according to equations (a) and (b), all frequency about v0 is possible and hence spectral line will have an infinite width. But, the number of atoms having large velocities is very small, the intensity that is proportional to number of atoms, falls off rapidly about a central maximum.
In thermal equilibrium, the probability p (vx) that an atom of mass M moving in x direction has velocity between vx and vx + dvx is given by Maxwell velocity distribution function.
The number of atoms in x direction is therefore,
dN(v x) = Np(vx)dvx
where p(vx) is the Maxwell velocity distribution function for the x component of the velocity v and is
The intensity emitted in the frequency interval υ and υ + dυ is denoted by I (v) dv and is proportional to the number of radiating atoms in the velocity component between vx and vx+dvx irrespective of the value of velocity components along y axis (vy) and z axis (vz). This is due to the fact that the Doppler shift is based upon the component of velocity of atom moving towards or away from the observer.
where C and C’ are constants of proportionality. This equation is Gaussian and the frequency at which intensity drops to half its maximum value is
where T is temperature in K and MN is the mass number of the atom and
v−v0 can be found
Figure depicts a plot of the Gaussian line shape. ∆ω is FWHM
Multiplying the above Equation by 2 to obtain full width at half maximum
As ∆υ is proportional to υ0, therefore, the smaller the frequency the Doppler width is not significant. Also, the Doppler width is inversely proportional to square root of mass of the atom, therefore hydrogen lines are diffused whereas Cd and Hg lines are sharp. Further, as Doppler broadening is proportional to √T, therefore, cooling the source reduces broadening.
8. Collision Broadening
Atoms in gaseous forms are not isolated and at high temperatures, in a gas, atoms collide with other atoms, ions or walls of the container. As a result there is a sudden change in the atomic radiation. The collisions of the radiative particles interrupts the radiative process that leads to broadening of a spectral line. Hence, truncated wavetrain is obtained instead of the long wavetrains, which means, with every collision; the oscillations are momentarily interrupted. After the collision the atom resumes its motion with the same frequency with a completely random initial phase with a possible amplitude change.
The figure shows that at every collision the phase of the wave changes abruptly.
In the presence of collisions the linewidth of radiation is greater as compared to that of uninterrupted process. In other words, mean life time τ0 (average) between the collisions is larger than the collision time.
The frequency of collision depends on the gas pressure and therefore collision broadening is also called pressure broadening. However, the collisions between gas atoms limits their radiative lifetime, that is quite the same as life broadening (without the interruption of the collision in the initial state of the atom).
The relative intensity I is
where ω0 = 2πυ0 and υ0 is the frequency with which the atom radiates between the collisions. The intensity drops to half its value at
The linewidth at half its maximum intensity is
Summary
