# Fine dipole moment - numbers


## Fine dipole moment

$\alpha_{QED} = \frac{e^2}{\hbar c} = \frac{d_{QED}^2}{a_B^2 \hbar c}$

this is the definition of fine dipole moment depending on the length $a$ between charges $e$.
The lengtha can be taken either as $a_B$ https://en.wikipedia.org/wiki/Bohr_radius or https://en.wikipedia.org/wiki/Classical_electron_radius

The dimensionality of dipole squared is $d^2=erg*cm^3$

## imagine that there are no free charges, but only dipole interactions

- let us compare with fine coupling in low-T glasses

$\alpha_{QSNS} = P \frac{\gamma^2}{\rho c_t^2}$

where $P=\frac{1}{erg*cm^3}$, $\gamma=erg$, and $\rho c^2= erg/cm^3$

To write down explicit dipole moment, we need to introduce length $\xi$

$\frac{\hbar c}{\xi} = \frac{\gamma^2}{\rho c_t^2} \frac{1}{\xi^3}$, therefore $\xi=\gamma \sqrt{\frac{1}{\hbar \rho c_t^3} }$
$\xi \sim 50A \sim 10* a_B$

Therefore $d^2_{QSNS} = P \gamma^2 \xi^6$

## compare $d^2_{QSNS}$ and $d^2_{QED}$

- this ratio $d^2_{QSNS}/d^2_{QED}$ implicitly depends on the ratio $a_B/\xi$

- for QSNSs, for electric dipole and elastic dipole moments compare
⏎
$e^2 a_B^2 \rightarrow P \gamma^2 \xi^6= \alpha_{QSNS} \rho c_t^2 \xi^6$
⏎
$e^2 a_B^2 \sim (5*10^{-10}esu*0.5*10^{-8}cm)^2 \sim 6 *10^{-36} erg*cm^3$
versus
$\alpha_{QSNS} \rho c_t^2 \xi^6 \sim 4*10^{-4} (4*10^5cm/s)^2 (5*10^{-7}cm)^6 \sim 800*125* 10^{-36} erg*cm^3 \sim 10^5 *10^{-36} erg*cm^3$
⏎
- **smaller $d^2_{QSNS}$**!!!!!
⏎
$d^2_{QSNS}/d^2_{QED} \sim 10^{-5}$⏎
⏎
## analogies

$a_B^2 \hbar c \rightarrow \rho c_t^2 \xi^6$

speculatively on the dimensionality basis $\hbar \rightarrow \rho c_t \xi^4$

$\hbar=10^{-27} erg*s$
$\rho c_t \xi^4 \rightarrow 1(g/cm^3) *4*10^{5}(cm/s) *(5*10^{-7})^4 \sim 25*10^{-21} erg*s$

just compare dimensionally $\hbar \rightarrow \rho c a_B^4$


# Parents

* Fine structure constant