Reduced Gyromagnetic Ratio of Proton

Theorem

$\ds \gamma_{\mathrm p} / 2 \pi$

$\approx$

$\ds 4 \cdotp 25774 \, 785 \times 10^7 \, \mathrm {Hz \, T^{-1} }$

$\ds $

$\approx$

$\ds 4 \cdotp 25774 \, 785 \times 10^3 \, \mathrm {Hz \, Gs^{-1} }$

By definition, the reduced gyromagnetic ratio of a proton is given as:

$\dfrac {\gamma_{\mathrm p} } {2 \pi}$

where $\gamma_{\mathrm p}$ denotes the gyromagnetic ratio of the proton.

Then we have:

$\ds \gamma_{\mathrm p}$

$\approx$

$\ds 2 \cdotp 67522 \, 18744 \, (11) \times 10^8 \, \mathrm {rad \, s^{-1} \, T^{-1} }$

$\ds \leadsto \ \ $

$\ds \dfrac {\gamma_{\mathrm p} } {2 \pi}$

$\approx$

$\ds \dfrac {2 \cdotp 67522 \, 18744 \, (11) \times 10^{-15} } {2 \pi} \, \mathrm {Hz \, T^{-1} }$

$\ds $

$\approx$

$\ds 4 \cdotp 25774 \, 785 \times 10^7 \, \mathrm {Hz \, T^{-1} }$

by calculation

$\ds $

$\approx$

$\ds 4 \cdotp 25774 \, 785 \times 10^3 \, \mathrm {Hz \, Gs^{-1} }$

by calculation

$\blacksquare$

1964: Milton Abramowitz and Irene A. Stegun: Handbook of Mathematical Functions ... (previous) ... (next): $2$. Physical Constants and Conversion Factors: Table $2.3$ Adjusted Values of Constants

which gives the mantissa as $4 \cdotp 257 \, 707$, with an uncertainty of $\pm 13$ corresponding to the $2$ least significant figures