The recoil-free processes arising from the M¨ossbauer effect permit resonant absorption with an ex- tremely high energy resolution. Energy levels of a nucleus in a solid are modified by the local environment of the nucleus. Very small energy changes that result from the hyperfine interactions between the nucleus and its surrounding electrons can be detected by measuring the energy depen- dence of the resonant absorption of M¨ossbauerγ-rays by nuclei. The M¨ossbauer spectrum is thus characterized by features including the number, shape, position, and relative intensity of various absorption lines that result from the the nature of hyperfine interactions and their time dependence.
2.3.1 Isomer Shift
The isomer shift is proportional to the electron density at the nucleus, with effects from its neighbors.
The electronegativity, covalency, bond strength, and electronic screening of the neighboring atoms can influence the charge density at the iron nucleus. As a result, the isomer shift provides a direct probe of the valence state of the M¨ossbauer active isotope.
Isomer shift of absorption lines results from the Coulomb interaction between the nuclear charge distribution over the nuclear volume and the electronic charge density over this volume. Excitation from ground state to excited state causes an increase in the nuclear volume, and the difference between electronic densities at the nucleus is different depending on the chemical environment. If
counts
-8 -6 -4 -2 0 2 4 6 8
2
1
-
2 + velocity (mm/s)δ
Δ
-
Figure 2.1: The nuclear energy levels of 57Fe in a non-zero electric field gradient demonstrate the quadrupole splitting, ∆, that results from the splitting of the excited state into two substates, and the isomer shift,δ, that results from the difference in energy levels between the source and absorber.
the Coulomb interaction were the only hyperfine interaction affecting nuclear energy levels, the ground state and excited states would be unsplit, but the separation between the states would be different in the source and absorber. This causes a shift in the absorption line,δ, that is the isomer shift.
An example of the role of isomer shift is in distinguishing Fe2+ and Fe3+. The valence states of Fe2+, 1s22s22p63s23p63d6, and Fe3+, 1s22s22p63s23p63d5 differ only by a d-electron. The 3s- electron spends a fraction of time further from the nucleus than the 3d-electrons. The electrostatic potential experience by the d-electrons depends on the screening effects of the inner electrons. By adding ad-electron, the attractive Coulomb potential is reduced, allowing the wave function of the 3s electrons to expand, reducing its charge density at the nucleus. As a result, removal of the 6th 3d-electron in going from Fe2+ to Fe3+ increases the charge density at the nucleus and produces a sizable isomer shift.
2.3.2 Electric Quadrupole Splitting
The nuclear charge distribution was assumed spherical in the discussion of the Coulomb interaction and the resulting isomer shift. However, nuclei have non-spherical charge distributions when their nuclear angular momentum quantum number is I>12, causing a nuclear quadrupole moment. A quadrupole moment interacts with an electric field gradient caused by an asymmetric electronic charge distribution. When the nuclear quadrupole moment experiences the asymmetric electric field, an electric quadrupole interaction causes the nuclear energy levels to split. This split occurs in correspondence with the different alignments of the quadrupole moment with respect to the principal axis of the electric field gradient. In the simple case of57Fe in the presence of a non-zero electric field gradient, the excited state hasI=32, which splits into the two substatesmI=±12 andmI=±32. This results in a two-line absorption spectrum separated by the quadrupole splitting, ∆. Fig. 2.1 shows the effect of the isomer shift and quadrupole splitting on the M¨ossbauer absorption spectrum for57Fe in an electric field gradient.
2.3.3 Hyperfine Magnetic Field Splitting
Hyperfine magnetic field splitting occurs in ferromagnetic, ferrimagnetic, and antiferromagnetic materials. When a nucleus is placed in a magnetic field, there is an interaction between the spin of the nuclear states and the magnetic field. The spins can be oriented with different projections along the magnetic field, modifying the energies of nuclear transitions. The energy perturbations that result from the hyperfine magnetic splitting are referred to as the nuclear Zeeman effect.
For states with a nuclear angular momentum quantum number greater than I>0, this lifts the degeneracy and causes splitting into 2I+1 substates. For 57Fe, which is ferromagnetic at room temperature, the ground state has I=12, which splits into two substates, and the excited state has I=32, which splits into four substates. Given the selection rule, ∆mI=0, ±1, this gives six allowed transitions, and results in a six-line absorption spectrum. Fig. 2.2 shows the M¨ossbauer spectrum that results from the effect of magnetic splitting in bcc Fe at 300K. The overall splitting of the absorption lines is proportional to the total magnetic field at the nucleus, and the transition
-8 -6 -4 -2 0 2 4 6 8
1 - 2
1 + 2 1
2
velocity (mm/s)
counts
Figure 2.2: The effect of magnetic splitting on the nuclear energy levels of57Fe is demonstrated for bcc Fe at 300K.
probabilities between the nuclear substates determine the intensities of the lines. The M¨ossbauer spectrum can therefore give information on the relative orientation of the magnetic field at the nucleus with respect to the direction of the incidentγ-ray.
The biggest contribution to the hyperfine magnetic field at the nucleus results from the coupling of the nucleus and its electrons. This coupling is called the Fermi contact interaction. The Fermi contact interaction between a nucleus and ans-electron can be written as follows:
Hs=−16π 3 βDX
(s↑ −s↓)E
(2.2)
where thes↑ ands↓are the spin up and spin downs-electron densities at the nucleus andβ is the Bohr magneton. The charge densities of the spin up and spin downs-electrons may be different even in filled s-shells if the atoms contains another partially filled magnetic shell, such as the 3d-shell.
The exchange interaction between the spin ups-electron and spin up polarizedd-shell is attractive,
Doppler Drive Source Sample Detector
Data acquisition device (NI DAQmx) PC LabVIEW Pb
NIM USB
Pre-amp Amplifier
Single Channel Analyzer (SCA) Function
Generator
Figure 2.3: Schematic of the M¨ossbauer spectrometer in use at Caltech. The spectrometer is arranged in transmission geometry and shielded by lead bricks. The signal from the detector is outputted to a series of electronics for pulse shaping, amplification, and selection. The detector signal and the timing of the doppler drive interface with a National Instruments data acquisition card. This card converts the analog signal to TTL pulses that are read by Labview software.
Reproduced from [4].
but the interaction between the spin downs-electron andd-shell is repulsive. Thus, the spin density terms in Eq.2.2do no cancel, resulting in a Fermi contact interaction field.