## Single electron – single proton continuous solid effect

I made this long time ago, did not posted or published anywhere.

I planned to made a more beautiful journal paper in near future, may be this year.

CW Solid effect

somethings need to be clean up and more detail is needed.

The calculation is based on Hartman-Hahn and Tim Wenckebach, the difference is, they use Fermi-Golden Rule to approximate the polarization transfer rate, but I solved it exactly with computer and also deduced the analytical solution for high-frequency truncated Hamiltonian. This reveal the validity on the frequency truncation. And I will add a comparison with experimental data in the planned-to-do paper.

## Hartmann-Hahn Matching

in dynamic nuclear polarization by microwave induced cross-polarization ( polarization transfer from one spin to another, for example, from $^1H$ to $^{13}C$ ), the condition is called Hartmann-Hahn matching.

The Hamiltonian of the 2 kind of spin in lab frame is:

$H_0 = \omega_s S_z + \omega_I I_z + (A+B) S_z I_z$

where A and B are scalar coupling when the 2 spin in contact and the dipolar interaction.

$A = \frac{4\pi}{3} \gamma_s \gamma_I \rho(x)$

$B = \frac{1}{r_{sI}^2} \gamma_s \gamma_I ( 1- 3 cos^2(\theta_{sI}))$

the angle in dipolar interaction is relative to the direction of the external B-field. when the angle is at:

$cos^2(\theta_{sI}) = 1/3$

the dipolar interaction disappear and the spectrum line get thinner, and higher resolution. this is called the Magic Angle.

when a transverse oscillating field applied, and only affect the S spin, then the total Hamiltonian is :

$H= H_0 + \omega_R U^{-1}S_x U$

$U= e^{-\frac{i}{\hbar} \omega t S_z}$

where U is the rotation operator. a standard method is switch to the rotating frame along with the transverse field. the Hamiltonian in the rotating frame is:

$H_R = \omega S_z+U H U^{-1}$

$H_R =(\omega + \omega_s) S_z + \omega_R S_x + \omega_I I_z + (A+B) S_z I_z$

we can see that, during the Spin-Lock, i.e. $\omega + \omega_s = 0$, the longitudinal component of  S spin gone. But in general, it is not the case, thus, we have the rotation axis of S spin is titled. we can simplify the Hamiltonian by transform it in a titled coordinate, by another unitary transform which rotate on Sy axis.

$\tilde{U} = e^{\frac{i}{\hbar} \phi S_y }$

$tan(\phi) = \frac{\omega_R}{\omega+\omega_s}$

in this tilted axis, the rotating axis is on the $\tilde{S_z}$ axis with magnitude:

$\omega_{eff} = \sqrt{ (\omega + \omega_s)^2 + \omega_R^2 }$

the tilted Hamiltonian is :

$\tilde{H} = \omega_{eff} \tilde{S_z} + \omega_I I_z + (A+\tilde{B}) I_z ( \tilde{S_z} cos(\phi) - \tilde{S_x} sin(\phi) )$

For the interaction terms are small, the energy level is just like ordinary 2 spin system. but when

$\omega_{eff} = \omega_I$

which is the Hartmann-Hahn matching, the flip-flop exchange of the spin no need any energy and then the spin transfer. on the other hand, if:

$\omega_{eff} = - \omega_I$

the flip-flip forbidden transition happen.

In the case of electron spin to proton spin, if we apply a ESR freqeuncy, which is GHz order, so it is microwave, the power of the microwave have to be matched to the proton Larmor frequency.

$(\omega_{\mu w} - \gamma_s H )^2 + k P_{\mu w} = \gamma_I^2 H^2$

here i used the microwave wave B-field strength is proportional to the  voltage applied, and power is proportional to the square of voltage.

## Dynamic Nuclear Polarization

The Dynamic Nuclear Polarization (DNP) means we has a pumping source to change the population of nuclear spin, then create a polarization. in contrast, Static Nuclear Polarization (SNP) means thermal equilibrium of nuclear state population.

the introduction of the paper gives 7 applications on polarized nuclear spin.i only list some below:

1. the angular distribution on radiations can serve as a test on the theory of nuclear interaction
2. Polarized target can be used in scatter experiment
3. obtain detail information on static and dynamic interaction between nuclear spin and its environment.
4. increase the sensitivity of NMR

this paper focus on a general system and represents them by graphs ( called chart in the paper ). the graphs are based on electron spin ½ and nuclear spin also ½.

on section II, it give out the Spin Hamiltonian and use it for the discussion on the population distribution. by that, the author used the rate equations to related the population in each state. Then, he defined the Enhancement of polarization, which is the ratio between the population with saturating radiation to the thermal thermal distribution.

on section III, it mention about the first 2 successful dynamic nuclear polarization experiments around 1953-4. one group polarized the 6Li nucleus in metallic lithium. the other group polarized the 1H in solid DPPH.

The paper gives conditions for DNP, which is coupling between nuclear spin and an unpaired electron spin. the paramagnetic environment can be archived by

1. the conduction electron in metals or metal ammonia solution
2. the donor or acceptor electrons in semi-conductor
3. paramagnetic ions in diamagnetic solid
4. paramagnetic ions in solution