Absolute polarization measurement by elastic scattering

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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.

LS.PNG

 

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) .

 

 

 

Annual Report for my department

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This report only covered some results as the limitation on page.

the report covered from April, 2011 to March, 2011. After the earth quake in March, 2011, i start the experiment on June and getting the system running on Sept. The data were collected from Sept to Dec. After that, i worked on C-13 polarization and moving the Lab. Then preparing a scattering experiment.

report2011_1

this is not the final version for publishing in the official report.

estimate the mean reaction rate given that no event in time interval

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The count follows Poisson Distribution:

P(n|\lambda) = \frac{(\lambda T)^n}{n!} Exp(-\lambda T)

this means, the probability of number of count, n, happened in time interval T given that the mean rate is \lambda .

now, we measured no count in time interval T, what is the mean rate for given confident interval?

if the count is zero, then the probability is:

P(0|\lambda) = Exp(-\lambda T )

and we can treat this as a new probability density function that, the probability of count rate is \lambda in time interval T.

normalized this pdf.

P(\lambda) = T Exp(-\lambda T )

we can see, for \lambda = 0 , the probability is 1. of course, if the count rate is 0, then no count is 100 %. therefore, we want to estimate the maximum count rate it will be to give “zero count “.

Now we have to introduce the Confident Interval (CL). this is the chance that the “interest” is true. in here, our interest is the “count rate is smaller then some value “. Thus, the total probability is:

1- CL = \int_0^(\lambda_0) P(\lambda) d\lambda

\lambda = \frac{1}{T} ln(\frac{1}{CL})

Lets give an example. suppose we count nothing in 10 second. what is the count rate for 95% Confident Interval?

plug in the equation and give \lambda = 0.005 .

Now, we can see how the Confident Interval means. 95% means, in 100 measurement of 10 second interval, there is 95% that no count. thus, in 1000 second time interval, there is at most 5 count. which is same to say  \lambda = 0.005 .

or, if we put the  \lambda = 0.005 into the Poisson distribution.

P(0|0.005) = Exp(-0.05) = 0.951

this is same that it has 95% give 0 count.

[Pol. p Target] resume experiment

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after a long summer break due to lab maintain.

we check the system to see weather it gives same results before.

the Hall probe reading is weird, so we calibrate it with water NMR signal.

and we redo the 30% laser duty, found that it is small, much smaller then expected.

===============================

after discussion with my professor on my PhD topic.

i like to study the spin by nuclear scattering experiment. the polarized spin target is a good spin detector.

one possibility is on the EPR paradox and the Bell’s inequality. my professor gave me a PhD thesis on proton-neutron spin experiment on EPR.

another possibility is on the localized special relativity and quantum entanglement. since these two are strongly related by spin. my professor gave me a book on spin statistic about that.

another unclear way is through the study of spin group, Lorentz group and Mobius group. by some transformation, a 3D rotation can transform into a 2X2 matrix and then reveal that spin can have classical picture with the help of complex number. that is a suggestion that L, the orbital momentum, and S, the spin, may be the same thing. moreover, the mathematical structure of L and S are the same for s=1. can we find a counterpart of l=1/2???

[Pol. p target]Fourier Spectrum of p-terphenyl

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the FID signal is:

the Fourier real and imaginary part, ( the central frequency is 12.6 MHz ):

The Amplitude of the Fourier spectrum, the time interval of fourier transform is from 30us to 197.96us.

if we use later time, we have:

If we subtracted the Background.

[Pol. p target] Principle of magnetic field optimization

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the Hartmann-Hahn Condition is:

\sqrt{ (\gamma_e H - \omega_{\mu w})^2 + P_{\mu w} \gamma_e^2 } = \gamma_p H

since we fixed the microwave power and frequency, the only parameter to change is the magnetic field. the solution of the magnetic field.

since, changing the magnetic field will also change the Larmor frequency of the proton and affect the pulse frequency of the NMR system. Thus, when changing the magnetic field, we have to find out the corresponding Larmor frequency to determine the NMR pulse frequency.

we first, measure the proton in water, since the proton can be regarded as free proton, and the Larmor frequency can be measured in high precision. the method we are used,

  1. set the NMR frequency in 12.2MHz, 12.4 MHz, etc.
  2. change the magnetic field such that the NMR signal is pure decay without any oscillation.

after acquire the data, we set the magnetic field and NMR frequency on crystal sample polarization.   there is only 1 magnetic field satisfy the Hartmann-Hahn condition and get a maximum polarization and NMR signal.

Note, the crystal field will broaden the peak of Larmor frequency, but the broadening is not shifting the center.

 

[ Pol. p target ] a short review

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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.

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