## Absolute polarization measurement by elastic scattering

The magnitude of proton polarization can be measured by NMR technique with a reference. Because the NMR gives the free-induction decay signal, which is a voltage or current. For Boltzmann polarization using strong magnetic field and low temperature, the polarization can be calculated. However, when a reference point is not available, the absolute magnitude of proton polarization can be measured using proton-proton elastic scattering. The principle is the nuclear spin-orbital coupling. That creates left-right asymmetry on the scattering cross section.

Because of spin-orbital interaction:

$V_{ls}(r) = f(r) \vec{l} \cdot \vec{s} ,$

where $f(r)$ is the distance function, $\vec{l}$ is the relative angular momentum, $\vec{s}$ is the spin of the incident proton. In the following picture, the spin of the incident proton can be either out of the plane ($\uparrow$ ) or into the plan ($\downarrow$). When the proton coming above, the angular momentum is into the plane ($\downarrow$). The 4 possible sign of the spin-orbital interaction is shown. We can see, when the spin is up, the spin-orbital force repulses the proton above and attracts the proton below. That creates an asymmetry in the scattering cross section.

The cross section is distorted and characterized using analysing power $A_y$. Analyzing power is proportional to the difference between left-right cross-section. By symmetry (parity, time-reversal) consideration, $A_y = 1 + P sin(2\theta)$ (why?), in center of mass frame. In past post, the transformation between difference Lorentz frame. The angle in the $A_y$ has to be expressed in lab angle. The cross section and $A_y$ can be obtained from http://gwdac.phys.gwu.edu/ .

In scattering experiment, the number of proton (yield) is counted in left and right detectors. The yield should be difference when either proton is polarized. The yield is

$Y(\theta, \phi) = L \epsilon \sigma_0 (1 + cos(\phi)A_y(\theta) P) ,$

where $L$ is the luminosity, $\epsilon$ is the detector efficiency, $\sigma_0$ is the integrated cross-section of un-polarized beam and target of the detector, $P$ is the polarization of either the target or beam. When both target and the beam are polarized, the cross section is

$\sigma = \sigma_0 (1 + (P + P_T)A_y + P P_T C_yy),$

where $C_yy$ is spin-spin correlation due to spin-spin interaction of nuclear force.

Using the left-right yield difference, the absolute polarization of the target or the beam can be found using,

$\displaystyle A_y P = \frac{Y_L - Y_R}{Y_L + Y_R} ,$

where $Y_L = Y(\phi =0)$ and $Y_R = Y(\phi=\pi)$.

## Magnetic Dipole Moment & Gyromagnetic Ratio

I always confuses on the definition, and wiki did not have any summary. so,

The Original definition is the Hamiltonian of a magnetic dipole under external magnetic field $\vec{B}$,

$H = -\vec{\mu}\cdot \vec{B}$,

where $\vec{\mu}$ is magnetic dipole moment (MDM). It is

$\vec{\mu} = g \frac{q}{2 m} \vec{J} = g \frac{\mu}{\hbar} \vec{J} = \gamma \vec{J}$.

Here, the $g$ is the g-factor, $\mu$ is magneton, and $\vec{J}$ is the total spin, which has a intrinsic factor $m\hbar / 2$ inside. $\gamma$ is gyromegnetic ratio.

We can see, the g-factor depends on the motion or geometry of the MDM. For a point particle, the g-factor is exactly equal to 2. For a charged particle orbiting, the g-factor is 1.

Put everything into the Hamiltonian,

$H = -\gamma \vec{J}\cdot \vec{B} = -\gamma J_z B = -\gamma \hbar \frac{m}{2} B [J]$,

Because energy is also equal $E = \hbar f$, thus, we can see the $\gamma$ has unit of frequency over Tesla.

———————————–

Take electron as an example, the MDM is Bohr magneton $\mu_{e} = e\hbar/(2m_e)$. The MDM is,

$\vec{\mu_e} = g_e \frac{e}{2 m_e} \vec{S} = g_e \frac{\mu_e}{\hbar}\vec{S} = \gamma_e \vec{S}$.

The magnitude of MDM is,

$|\vec{\mu_e}|= g_e \frac{e}{2 m_e} \frac{\hbar}{2} = \gamma_e \frac{\hbar}{2} [JT^{-1}]$,

The gyromagnetic ratio is,

$\gamma_e = g_e \frac{\mu_e}{\hbar} [rad s^{-1} T^{-1}]$.

Since using $rad s^{-1}$ is not convenient for experiment. The gyromagnetic ratio usually divided by $2\pi$,

$\gamma_e = g_e \frac{\mu_e}{2\pi\hbar} [Hz T^{-1}]$.

————————————-

To evaluate the magnitude of  MDM of  single particle state, which has orbital angular momentum and spin, the total spin $\vec{J} = \vec{L} + \vec{S}$. However, the g-factor for $\vec{L}$ is difference from that for $\vec{S}$. Thus, the MDM is not parallel to total spin. We have to use Landé Formula,

$\left< JM|\vec{V}|JM'\right> = \frac{1}{J(J+1)} \left< JM|(\vec{J}\cdot\vec{V})|JM\right> \left$

or see wiki, sorry for my laziness.

The result is

$g=g_L\frac{J(J+1)+L(L+1)-S(S+1)}{2J(J+1)}+g_S\frac{J(J+1)-L(L+1)+S(S+1)}{2J(J+1)}$

For $J = L \pm 1/2$,

$g = J(g_L \pm \frac{g_S-g_L}{2L+1})$

## Q-factor

Q-factor is Not Q-value. in Q-factor, the Q is for Quality.

Q-factor is a dimensionless factor for showing the degree of reasonace of a coil or an Oscillator.

Higer value means a better coil. The definition is the resonance frequency over the Full Width Half Maximum (FWHM).

$Q = f/\Delta f$

High value means:

1. high sensitivity
2. noise reduction due to narrow band of absorption.
3. over damped with long decay time.
4. higher energy stored
5. lower energy loss

For complex electric circuit, the Q factor is:

Q = | im Z/ re Z|

It seem that there is a conflict between impedance matching.

## Finding a 90 degree pulse of the NMR system

Tuning of NMR System check the pdf… so tired to rewrite again….

## T1 and T2 measurement

Measuring  T1 in NMR, we apply follow  pulse sequence:

$\pi_x \longrightarrow \tau \longrightarrow (\pi/2)_x$

according to previous post on density matrix in operator form, we can evaluate the polarization. suppose the magnetization is pointing with the external B-field.

$\rho_R = \rho = k\sigma_z$

$\downarrow \pi_x$

$\rho_R = - k \sigma_z$

$\downarrow \tau$

$\rho_R = -k \left( 1- 2 e^{-\tau/T_1} \right) \sigma_z$

$\downarrow (\pi/2)_x$

$\rho_R = k \left( 1- 2 e^{-\tau/T_1} \right) \sigma_y$

in lab frame:

$\rho = k \left( 1-2 e^{-\tau/T_1} \right) (\sigma_y cos(\omega_0 t) - \sigma_x sin(\omega_0 t) )$

if the NMR coil is placed alone with x-axis. the magnetization is proportional to :

$\left<\sigma_x\right> = 2 k\left( 1- 2 e^{-\tau/T_1} \right) sin(\omega_0 t)$

the amplitude of the magnetization is only a function of $\tau$. bu measuring the amplitude with different $\tau$, we can determine the $T_1$.

To measure the T2, we use follow pulse sequence:

$(\pi/2)_x \longrightarrow \tau \longrightarrow (\pi)_y \longrightarrow \tau$

again we use the same initial state, but this time, we are going watch it as lab frame.

$\rho = k\sigma_z$

$\downarrow (\pi/2)_x$

$\rho = - k \sigma_y$

$\downarrow \tau$

$\rho = -k (\sigma_y cos( \omega_0 \tau ) - \sigma_x sin(\omega_0 \tau) ) e^{- \tau/T_2}$

$\downarrow \pi_y$

$\rho = -k (\sigma_y cos( \omega_0 \tau ) + \sigma_x sin(\omega_0 \tau) ) e^{- \tau/T_2}$

$\downarrow \tau$

$\rho = - k \sigma_ye^{- 2 \tau/T_2}$

at  the last step, the free-induction decay is not decay but a revert process of decay. we see that the state back to its beginning state! thus, this method also called the spin echo. this can be see in pictorial  way. after the $(\pi/2)_x$ pulse, the spin go to – y axis. now, due to the incoherence of each spin, some spin are faster and some are slower. after a time $\tau$, the $\pi_y$ pulse flips all the spin by 180 degree. now, the “slower” spin become ahead of the “faster” spin. After the same time interval, the “faster” spin will catch up the “slower” spin and all the spin becomes coherence at that moment again! thus, the amplitude of the signal will become large again. However, since the spins are not at same Larmor frequency, some will flip more than 180 degrees while some flip less, thus, the decay are still there.

Note:

the FID ( free induction decay) rate is not exactly equal to T2, since there are many other way to make the spin polarization decay. the FID decay rate $T_2^*$ should be :

$\frac{1}{T_2 ^*} = \frac{1}{T_2} + \frac{1}{T_{others}}$

and the spin echo method can eliminate the others.

## [Pol. p target] Matching Impedance of NMR coil

i had played with the impedance of coax cable. connect the signal generator at 50Ω output and a 4meter long coax to a CRO with 50Ω input with a tee. although the web said, the coax impedance is 52, there is no observed different. the peak-to peak signal does not depend on the frequency.

then connect a 12.8MHz coax to the tee and open at the other end. in theory, it should be zero input-impedance and he CRO reading should be

$V_{in}= 1 - cos( 2\pi L/\lambda)$

where the CRO reading is the input voltage. when the frequency adjusted to 12.8MHz, which is quata-wavelength, the input voltage is equal to 1.

when connect the end of the coax with a 50Ω resistor, the input voltage does not depends on frequency, as expected.

i haven’t try to short the end of the coax.

after that. i going to matching the impedance of the tuner and coil. we use a short coax to connect the input to the tuner before, i replaced it with a 12.8MHz coax. the reason is, the little bit mismatching of the impedance can be saved by the length of the cable, such that:

$Z_{in} = Z_0^2/Z_L \approx Z_0$

and i opened the tuner to see the circuit inside:

The above is the circuit diagram. there is a fixed capacitor with 2.2pF in parallel. i cannot identify the type of the 2 variable capacitor.  the coax cable can be put in port 1 or port 2 and the coil put in the other port. different configuration has different behavior.

i found that, the input is in port 2 and the coil is in port 1, and the box doesn’t ground but just wrapped with metal sticker. i took it out and grounded to NMR system.

For the input at port 2, the total impedance of the coil and tuner is:

$Z_L= - i /(\omega (C_0+C_p + C_s/(1-\omega^2 L C_s)))$

The first things to notice is the impedance depends on the frequency. which mean, the impedance matching can only on particular frequency. when the Cs adjusted to matching the frequency. the impedance solely depends on Cp. when the driving voltage gone, the LC circuit will oscillate at natural frequency:

$\omega_n = 1/\sqrt{ L ( C_s + C_0 C_p / (C_0+C_p) ) }$

Thus, i tired to measure to inductance of the coil by a parallel resistance. but i cannot find any suitable wire to convert BNC cable to wires. after a long time finding, i gave up and wait unit work with my partner.

the input in port 2 is not a common config, so, i changed it to port 1. and the impedance is :

$Z_L=i\omega (C_s +L/(1-\omega^2L(C_0+ C_p)))$

The impedance also depends on frequency. and the natural frequency when the driving voltage gone is:

$\omega_n = 1/\sqrt{L(C_0+C_p)}$

there is one way to tune the Cp to match the input frequency and make load impedance solely depends on Cs. by using a pulse signal. and measure the natural frequency of the LC loop, such that the natural frequency is same as input frequency. However, the Low Pass Filter only let frequency less then 1MHz pass and out frequency is 12.8MHz. can i use other pulse source? may be, if i have a mixer.

so, i matching the impedance by very naive way. i use the method on testing the impedance matching. i use a continuous signal source and fixed the frequency at 12.8MHz, then connect it with a coax cable to 50Ω CRO input by a tee, then connect th tee with a 12.8MHz coax cable. the other end of the cable connect to a 50Ω resistor. this setup should be matched impedance. so, i record the input voltage on the CRO. and replace the resistor with the tuner, which port 2 connected to the coil. then adjust the capacitors (both) so that the CRO voltage is same as 50Ω resistor.

by solving the load impedance formula of port 1 configuration, there are mulitple solution for Cp and Cs to give 50Ω. and i think, as long as the impedance is 50Ω at 12.8MHz, any configuration can do the job.

later, i try to find the water NMR signal. although i cannot find any. but the noise level reduced to ±5mV. more or less equal to the background.

i played with the NMR program. the record data is counted by point, so the CRO horizontal setting should set to 5000 points over the screen.

and just before i leave, i don’t know what wrong, the program doesn’t read the CRO signal…

## 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