## [ Pol. p target ] a short review

THe system is fairly acceptable now. the signal fluctuation is about ±30 unit. compare with the absolute value of 600 to 1200. it is fair enough.

we have a Hall probe now, but the measured magnetic field is quite different from what we expected before. we expect it should be around 0.300xxx but the measured value, is 0.33xxx that is mean, something is missing in our understanding.

after finishing the optimization, the system is ready for further development.

1. absolute polarization
2. spin echo
3. laser polarization dependency
4. Fourier analysis
5. T1 and T2 measurement
6. cross polarization between H1 and C13
in order to do the absolute polarization measurement, we have to lower the noise level. or, we can increase the magnetic field and reduce it back when measuring it. this requires to measure the T1 relaxation time. another way is spin echo method. since it can avoid the influence of the coil relaxation signal, which cover up the very beginning signal.
For the Fourier analysis, we have to use an external reference frequency for NMR system. currently, we use the same frequency for the pulse and for the reference frequency. Since the pulse frequency must be matching with the Larmor frequency ( more or less), which is the signal frequency. in principle, our signal must be a simple decay curve when exactly matching was archived. in that case, the Fast Fourier Transform will give is same peak at the edge of the spectrum, which is hardly identified. however, if we use an external reference frequency, problem can be solved, and we are able to obtain some peak at the middle of the frequency spectrum. By this, we can understand more about the crystal and the internal field and processes.
and also, when we cross polarize H1 and C13, we can use Fourier analysis to understand the effect much better.

## 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….

## Impedance matching and Smith Chart

the circles with center on x-axis of the Smith chart represent the real part of impedance.

the circles with center on the x = 1 are the imaginary part of the impedance.

and a circle with radius 1 and centered at (0,0) represent a coaxial line with 50Ω, the (-1,0) is zero length and (1,0) is ¼ wavelength from the load.

thus, we can construct the input impedance by Smith Chart very easily.

AND i always think, the normalized input impedance should be 1 in order to give minimum reflection. However, when the source impedance is not 50Ω, it is not the case. the impedance matching for normalized source impedance is

$z_{in} = z_s^*$

this is come form the Maximum Power Theory.

anyway, to construct the input impedance, we just need to  remember when circuit element is in series, we use normal Smith chart, if the circuit element is in parallel, we use a 180 degree rotated Smith chart, since adding by reactance is more easy.

there is an iphone app for this: rfCalc

it is very easy to use!

the question is how do i know the source impedance!???

## [ Be-12(p,n)B-12 ] the detector circuit

i joined a scattering experiment for $^{12}Be(p,n)^12B$. today the Beam-Line-Detector team was finished the BLD, and SHARAQ group is trimming the beam to the SHARAQ detector. i was worked on the neutron detector array, but i was absented for all preparation job due to a polarized proton target.

here is the detector circuit:

when the neutron hits on the plastics scintillator, it create s proton and further release photons. these photons is result of photoelectric effect and Compton scattering. the photons have different time on reaching the 2 ends of the scintillator due to the position. Thus, by analysis the timing, we can identify the position.

The Photo-Multipler Tube convert photon into electric signal. This signal go to a Splitter, the splitter simply copy the signal. one output is go to a Discriminator, which create a digitized signal. the PMT at the 2 ends will give different timing, and the Discriminator signals will be combined after the AND gate. If the neutron hit the scintrillator near either ends, the AND gate make sure that the later signal will be the trigger signal.

the trigger signal will enable the Charge-Digital-Converter (QDC) start integrate the delayed signal for 300ns. Since from the raw signal to the AND gate takes time, thus a delay will ensure a correct detection of the total energy for each signal. However, the signal will have exponential decay when traveling alone the PMT. the signal decay is:

$S = S_0 e^{- k x}$

For signal created at position r from the mid-point of the scintrillator , the 2 signals are:

$S_1 = S_0 e^{- k (L-r)}$

$S_2 =S_0 e^{-k ( L+r)}$

where L is the half length from the mid-point of the scintrillator. thus, if we use a geometric mean, we can get the original signal strength.

$S_0 = \sqrt{S_1 S_2} e^{k L}$

the trigger signal also combined with the discriminator signal for determination of the time. the trigger signal will define the t=0. by this, we can calculate back the location of the neutron. If we define the mid-point of the scintrillator be x=0. a neutron hits on position r will take different path and the time different for the scintrillator is:

$\Delta t_s = 2 r / c$

where c is the speed of light. we also have to add the time different due to the cable, thus, the total time different is:

$\Delta t = t_1 - t_2 = \frac{2r}{c} + t_0$

by fitting the statistic, the average of the position should be zero, thus the mean of Δt is the offset by the cable.

However, as we mention before, the signal will decay. since we are using Threshold Discriminator, different signal strength will give different time even for same start time. This is called Walk effect. For lower signal strength, the effect is stronger. and give larger Walk time. since the Walk time is always additional, the statistics of the QDC-TDC graph will has a tail at one side only.

In fact , there is a technique the tackle the Walk effect, which is by a Constant Fraction Discriminator. the principle is that, if we copy the signal and apply a negative fraction on 1 of them and summing up. the point of zero is always the same no matter the amplitude or the signal strength.

the Walk effect can be calculated by assuming the falling of the signal is just like a line with slope -m. with threshold -θ, and a decay ratio k. the time are:

$t_1 = \frac{\theta}{m}$

$t_2 = \frac{\theta}{k m} = \frac{1}{k} t_1$

$\Delta t = t_1-t_2 = \frac{k - 1}{k} \frac{\theta} {m}$

if we plot the signal strength versus time different, we have a curve in the form

$y = \frac{k-1}{k} \theta \frac{1}{x}$

From the equation, a smaller decay ratio, which corresponding to the location closer to the mid-point of the  scintrillator, the bigger the $1- \frac{1}{k}$ and give a border curve.

## Smith Chart ( quick guide )

for the coaxial cable is ¼ wavelength, everything in transmission line theory will be simplified.

the input impedance will be

$Z_{in} = Z_0 \frac{1-\Gamma}{1+\Gamma}$

where Γ is the characteristic impedance:

$\Gamma = \frac{ Z_L - Z_0 }{Z_L + Z_0}$

and the reflection wave and input wave ratio is :

$\rho = \left| \frac{V_-}{V_+} \right| = |\Gamma|$

these are 3 important equations on smith chart.

we can use normalized impedance, which is defined as

$z_{in} = Z_{in}/Z_0$

$z_L = Z_L / Z_0$

than  2 of the  3 equations will be normalized to :

$z_{in}= \frac{1-\Gamma}{1+\Gamma}$

$latex \Gamma = \frac{ z_L – 1 }{z_L + 1}$

by setting $z_{in} = r + i x$, we have :

$\Gamma = \frac{ 1- r^2 - x^2 }{(1+r)^2+x^2} - i \frac{2x}{(1+r)^2+x^2}$

by some algebra, we have :

$z_{in} = 1/ z_L$

which is the result for ¼ wavelength cable. by this,we have:

$z_L = \frac{r}{r^2+x^2}- i \frac{x}{r^2+x^2}$

after many equations, for impedance matching, we have following equivalent statements.

• impedance matching
• $z_L = 1$
• $\Gamma = 0$
• $z_{in} = 1$
• $\rho = 0$ , no reflected wave

OK. the math is over. The Smith Chart is the Cartesian coordinate for Γ, real axis on horizontal, imagine axis on vertical. and the circles, are the transformed coordinate of the input impedance. the transformation is called Möbiüs  transform.

a good impedance matching can be found at the origin of the chart, where $\Gamma = 0 + 0 i$, and if we read the input impedance coordinate, it is $z_{in} = 1 + 0 i$, which mean, impedance matched.

However, there is never so ideal in real world. the input impedance always has some imaginary part, or real part not equal to 1. so, what if, both of them are different by 0.2 or 10 Ohm?

## [Pol. p target] fix the NMR coil position

my colleague was improve the system by reducing the reflected wave. and also found the maximum condition for the timing. the microwave trigger should be delayed by 167 us and have a phase 180.

then, i get the polarization signal in FID area be 340 unit or up! compare to BG, which is just 25 unit. But in order to measure the polarization dependence or any dependence, i hope to have a small fluctuation on measurements.

when i try to fix the NMR coil by using optics mounting, i found that the input impedance shown on the network analyzer is very sensitive to position, only a very tiny different can change the impedance alot. That is why the NMR signal is not so reproducible, or with a large fluctuation.

when the experiment going well, suddenly, the signal gone, it only give signal like background. i checked the coil, put it out and found it is quite distorted. fix it in a good shape and do again. nothing.

then i thought, may be the network analyzer. then i restart. but don’t know how to operate…..

finally, it is the tuner circuit broken.

anyway, today found a way to stabilized the NMR coil. as long as moving it slowly, it can reproduce similar input-impedance with 5 ohm different. that result is done with a coil-before-fixed. Now the coil is in nice shape. it goes and leaves the sample more smoothly.

Todo:

1. setup the network analyzer and use it to get back the NMR signal
2. testing the stability of the signal
3. fin the crystal orientation.