TLDR

• Wavelength: 0.0861 nm (14.4 keV)
• Measures: Energy gaps between nuclear states
• Uses: Detection of ${}^{57}Fe$, applicable only for solids
• Selection Rules: $\mathrm{\Delta }{m}_I=0,\pm 1$
• Formula: ${\nu }_L=|\gamma |\frac{B}{2\pi }$, $\frac{{\nu }_s-{\nu }_{ref}}{{\nu }_{ref}}\ast {10}^6=\delta$

Book chapter

Light Source

• Source Details:

  • 
  • $\gamma$-photons are emitted from the nuclear decay of a radioactive mother isotope (${}^{57}Co$).
  • ${}^{57}Co$ is typically produced via 3He bombardment of ${}^{55}Mn$.
  • Decay of ${}^{57}Co$ results in a metastable excited state of ${}^{57}Fe$, leading to a cascade of $\gamma$ transitions, with one at $E_0=14.4$ keV, close to the absorbing energy for ${}^{57}Fe$.

• Energy Characteristics:

  • The $\frac{3}{2}$ excited state has a lifetime of 100 ns, leading to a very narrow energy width $\mathrm{\Delta }E=4.55\times {10}^{-9}$ eV.
  • This narrow linewidth allows for selective detection of ${}^{57}Fe$ in solids.

Recoil Energy

• Problem with Recoil Energy:

  • The narrow width of $\gamma$ rays makes recoil energy significant, as photons transfer momentum to the nucleus, altering energy by an amount much larger than the Mössbauer linewidth.

• Avoiding Recoil:

  • Solids: In solids, molecular movement is restricted, reducing recoil effects.
  • Zero-Phonon Processes: These processes have a higher probability in solids, leading to sharp lines. The probability factor is given by the [Lamb-Mössbauer factor (f)].

The Mössbauer Experiment

• Experiment Setup:

  • Components:

    • 

  • Doppler Effect Usage:

    • Energy modulation is achieved by moving the source relative to the absorber, causing Doppler shifts in photon energy:
    • $E_{\gamma }=E_0(1+\frac{v}{c})$ where $v$ is the source velocity.

  • Transmission Setup:

    • A transmission arrangement is used:
      • 

  • Experimental Process:

    • The source is moved at Doppler velocities, and absorption probability is measured at each velocity by the overlap of the Doppler-shifted emission line with the absorption line.
    • 

Hyperfine Interactions

• Overview:
  • 

Isomer Shift

• Description:

  • Isomer shift occurs due to changes in the nuclear environment, typically relative to a reference (metallic iron).

• Key Factors:

  • Higher oxidation states result in shorter bond lengths and lower isomer shifts.
  • High-spin states have higher isomer shifts than low-spin states.
  • Four-coordination leads to shorter bond lengths and lower isomer shifts compared to six-coordination.
  • Covalent bonds and complexes with soft ligands show lower isomer shifts.

Electric Quadrupole Interaction

• Description:

  • Quadrupolar States: These states split into two due to the electric quadrupole interaction, with the ground state remaining unsplit.
  • 

• Mechanism:

  • Caused by the interaction of non-spherical nuclei with an inhomogeneous electric field from an asymmetric charge distribution of surrounding electrons.

Magnetic Hyperfine Splitting

• Description:
  • Magnetic Field Influence: The application of a magnetic field lifts degeneracy due to the Zeeman Effect.
  • Field Strength: Changing the field strength alters the Zeeman Effect, affecting the positions of the six peaks.
  •