Engineering & Science
Mössbauer Spectroscopy: Free-Nucleus Recoil Energy & Doppler Velocity
Calculate free-nucleus gamma-ray recoil energy E_R = E_gamma^2 / (2 * M * c^2) (meV) and Mössbauer spectrometer linear Doppler velocity v = (Delta_E / E_gamma) * c (mm/s) for 57Fe isomer shifts and hyperfine splitting.
Why recoilless crystal lattices let millimeter-per-second drives resolve nano-eV nuclear hyperfine levels!
When a free gas-phase 57Fe nucleus emits a 14,412.5 eV gamma ray, momentum conservation kicks the nucleus backward with recoil kinetic energy E_R = E_gamma^2 / (2 * M * c^2) = (14,412.5)^2 / (2 * 56.9354 * 931.4941e6 eV) = 1.9583 meV—over 400,000 times wider than the natural excited-state linewidth Gamma = 4.66 neV, completely destroying resonant nuclear absorption! Rudolf Mössbauer (Nobel Prize 1961) discovered that locking 57Fe into a solid crystal transfers recoil momentum to the macroscopic crystal mass (M -> infinity, E_R -> 0) for a recoilless Lamb-Mössbauer fraction f!
Because resonant absorption is now 4.66 neV sharp, oscillating the radioactive 57Co source on a tiny electromechanical speaker drive at millimeters per second tunes gamma-ray energy via first-order Doppler shift v = (Delta_E / E_gamma) * c! Shifting a 14,412.5 eV 57Fe gamma ray by a magnetic hyperfine splitting Delta_E = 4.80e-7 eV (480 neV) requires a linear source velocity of just v = (4.80e-7 / 14,412.5) * 299,792,458,000 mm/s = 9.984 mm/s!
| Mössbauer Isotope Transition | Gamma Energy E_gamma (eV) | Free-Atom Recoil E_R (meV) | Doppler Scale (v for 480 neV) |
|---|---|---|---|
| 57Fe (M = 56.9354 u, Mars Rover MIMOS II) | 14,412.5 eV | 1.9583 meV | 9.984 mm/s |
| 119Sn (M = 118.9033 u, Organotin Chemistry) | 23,879.5 eV | 2.5736 meV | 6.026 mm/s |
| Invalid Mass or Energy (M <= 0 or E <= 0) | Physical Guard | "Invalid Nuclear Inputs" | Requires E_gamma > 0, M > 0 |
Free-Nucleus Gamma Recoil Kinetic Energy E_R (meV)
Uses CODATA atomic mass energy equivalent 1 u * c^2 = 931,494,100 eV and converts E_R into millielectronvolts (* 1000).
=IF(OR(B2 <= 0, B3 <= 0), "Invalid Nuclear Inputs", ROUND(((B2 ^ 2) / (2 * B3 * 931494100)) * 1000, 4))Mössbauer Spectrometer Doppler Drive Velocity v (mm/s)
Converts target energy shift B4 (eV) into source velocity v = (Delta_E / E_gamma) * 299,792,458,000 mm/s.
=IF(OR(B2 <= 0, B3 <= 0), "Invalid Nuclear Inputs", ROUND((B4 / B2) * 299792458000, 3))Try the example
Use the original values or change them to test your understanding.