## 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.

## Overhauser effect

the Overhauser effect/method was posposed by Albert W. Overhauser on 1953, that, it can polarizing nuclei. or later called Dynamic Nuclear polarization.

the spin Hamiltonian is

$H = \gamma S\cdot B + \gamma_n I \cdot B + I\cdot A \cdot S$

where S is the electron spin, I is nuclear spin, B is external field strength and A is hyperfine structure coupling tensor. $\gamma$ is a constant such that:

$\gamma = g \frac{e \hbar}{2 m_e}$

it related to the Larmor (angular ) frequency, which is in radius by :

$\omega_0 = \gamma/\hbar$

it is same for proton with a corresponding g-factor and mass, or other nuclei with g-factor and charge and mass.  for example, proton Larmor frequency is:

$\omega_0(p)/ (2 \pi) = 42.5775 MHz T^{-1}$

$\omega_0(e)/ (2 \pi) = - 28.025 GHzT^{-1}$

$\omega_0(n)/ (2 \pi) = 0$

the Hamiltonian commute for both spin eigenstate. thus, we can write the energy as:

$E(m,M) = \gamma B_0 m + \gamma_n B_0 M + A m M$

where m is the eigenvalue of  $S_z$ and M is the eigenvalue for $I_z$. For spin-half electron and nuclei, there will be  4 levels by $m \times M = 4$. the selection rule is:

$\Delta m = \pm1, \Delta M = 0$

this is called the Overhauser Effect.

The Jeffries-Abragam Effect occur when a pumping radiation induced a “forbidden” transition”:

$\Delta m = \pm 1 , \Delta M = \pm 1$

## Production of High, Long-Lasting, Dynamic Proton Polarization by Way of Photoexcited Triplet States

This paper show a 42% of nuclear spin polarization on phenanthrene $C_{14}D_{10}$  doped in fluorene $C_{13}H_{10}$ by Dynamic Nuclear Polarization (DNP), or more specific, the Integrated Solid Effect (ISE) or Integrated Cross-Polarization (ICP), or Microwave-Induced Optical Nuclear Polarization (MIONP). they use 75GHz microwave at 1.4K.

the paper pointed out that conventional guest molecule is paramagnetic in ground state. That provided a channel for nuclear spin-relaxation and  reduce the polarization. in contrast, this paper use a paramagnetic triplet state and diamagnetic ground state. Thus, when the excitation laser is turned off, the nuclear-spin relaxation can prevented.

## 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
6. color centers

the detection of DNP can be via:

1. NMR
2. shift of EPR frequency
3. the β asymmetry or γ anisotropy from an oriented radioisotope

## [Pol. p target] Looking for a signal

the NMR signal was calibrated by water, since it has contains proton. the pule frequency is 12.78 MHz. this frequency is also equal to the Larmor frequency under a static B field.

$2 \pi f = \mu_p B /2$

the time of the pule is 5 μs. this should be the π/2 pulse. thus

$\mu_p B_R \tau_{\pi/2} /2 = \pi / 2$

we did a parameter search for any possible NMR signal. the parameters are the microwave power and the delay time.

(the following is a rough understanding, many places are unclear. )

the dynamic nuclear polarization requires a magnetic field and an microwave. the microwave is function as a power source or photo source. when nuclear spin flips, it requires a photo to provide the necessary energy. since our microwave frequency is fixed by the microwave cavity, so, in order to matching the correct frequency of absorption, we use a Sweeping magnetic field, when the magnetic field is at right magnitude, than the energy gap between spin up and spin down will equal to the photo energy and then absorbed.

after the electrons are all polarized by optical pumping, which means the electron polarization is saturated. the applied sweeping B-field and microwave will able to transfer the polarization to the proton.

thus, when the laser stopped by a chopper, the sweeping field and microwave will be turned on. the time delay between the laser and the sweeping field is called delay time.

the upper line is the chopper signal ( channel 4) . the next one is the sweeping field ( channel 2 ). the 3rd one is the reflected microwave, we can see a very small change on it, the base line are off screen, since the signal is very small compare to the overall signal ( channel 1). the last one is the RF Amp trigger signal, which control the microwave on-off ( channel 3 ).

## First experiment of 6He with a polarized proton target

this paper reported a first spin polarized proton solid target under low magnetic field ( 0.08 T ) and hight temperature ( 100K )

the introduction overview the motivation of a solid target.

• a polarized gas target is ready on many nuclear experiment.
• on the radioactive beam ( IR beam ), the flux of a typical IR beam is small, since it is produced by 2nd scattering.
• a solid target has highest density of solid.
• most solid target can only be polarized on low temperature ( to avoid environmental interaction to reduced the polarization )
• increase the experimental difficult, since a low temperature should be applied by a cold buffer gas.
• high field ( the low gyromagnetic  ratio ).
• high magnetic field make low energy scattered proton cannot get out from the magnetic field and not able to detect.
• a solid target can be polarized at high temperature and low magnetic field is very useful

the material on use is a crystal of naphthalene doped with pentacene.

the procedure of polarizing the proton is :

1. use optical pumping the polarize the electron of pentacene
• the population of the energy states are independent of temperature and magnetic field.
2. by Dynamic Nuclear Polarization (DNP) method  to transfer  the polarization of the electron to the proton.
• if the polarization transfer is 100% and the relaxation time is very long. the expected polarization of proton will be 72.8%

The DNP method is archived under a constant microwave frequency with a sweeping magnetic field. when the magnetic field and  microwave frequency is coupled. the polarization transfer will take place.

the next paragraph talks about the apparatus’s size and dimension, in order to fit the scattering experiment requirements.

the polarization measurement is on a scattering experiment with 6He at 71 MeV per nucleons. By measuring the polarization asymmetry $\epsilon$, which is related to the yield. and it also equal to the polarization of the target $P_t$  times the analyzing power $A_y$.

$\epsilon = P_t \times A_y$

with a reasonable guess of the target polarization. the analyzing power of  6He was found.

the reason why the polarization-asymmetry is not equal to the analyzing power is that, the target is not 100% polarized, where the analyzing power is defined. when the polarization of the target is 100%, both are the same.

in the analysis part. it used optical model and Wood-Saxon central potential to simulate the result. And compare the result from 6He to 6Li at same energy. the root mean square of 6Li is larger then 6He. it suggest the d-α core of 6Li may responsible for that.

they cannot go further discussion due to the uncertainly on the polarization of the target.