Gyromagnetic Ratio Calculator

The gyromagnetic ratio (γ) is the ratio of a particle's or system's magnetic moment to its angular momentum. It is a fundamental property used in nuclear magnetic resonance (NMR), magnetic resonance imaging (MRI), and quantum mechanics. The gyromagnetic ratio determines how rapidly a particle precesses in a magnetic field.

The gyromagnetic ratio is calculated using the formula γ = μ / J, where μ is the orbital magnetic moment (typically in J/T or A·m²) and J is the orbital angular momentum (typically in J·s or kg·m²/s). The result is expressed in units of rad/(s·T), GHz/T, or MHz/T depending on the context.

To calculate the gyromagnetic ratio, divide the magnetic moment by the angular momentum: γ = μ / J. For example, if the magnetic moment is 9.274 × 10⁻²⁴ J/T and the angular momentum is 1.0546 × 10⁻³⁴ J·s, the gyromagnetic ratio is approximately 8.794 × 10¹⁰ rad/(s·T). Make sure both inputs use consistent SI units.

The proton gyromagnetic ratio is approximately 2.675 × 10⁸ rad/(s·T), or about 42.577 MHz/T. This constant is crucial in NMR spectroscopy and MRI, as it determines the Larmor precession frequency of protons in a given magnetic field strength.

The gyromagnetic ratio is commonly expressed in rad/(s·T) in SI units, which is equivalent to C/kg. In NMR and MRI applications, it is often quoted in MHz/T or GHz/T. Converting between units: 1 rad/(s·T) = 1/(2π) Hz/T.

The g-factor is a dimensionless quantity that accounts for the deviation of a particle's magnetic moment from the classical value, while the gyromagnetic ratio is the actual ratio of magnetic moment to angular momentum with physical units. They are related by γ = g × (e / 2m), where e is charge and m is mass of the particle.

In MRI, the gyromagnetic ratio determines the Larmor frequency at which protons resonate in a magnetic field, given by ω = γ × B₀. Since hydrogen protons have a known gyromagnetic ratio, MRI scanners can tune their radiofrequency pulses precisely to excite and detect nuclear spin signals, forming the basis of magnetic resonance imaging.

Yes. By rearranging γ = μ / J, you can solve for any missing variable: μ = γ × J to find magnetic moment, or J = μ / γ to find angular momentum. Select the quantity you want to solve for in the 'Solve For' dropdown, enter the two known values, and the calculator will compute the third.