On NMR noise reduction

February 4, 2011

GoLuckyRyan Basic, theory , , , Leave a comment

there are many ways to reduce the signal-to-noise ratio in NMR.

here, we will show the simplest one – by measuring it many times.

the reason behind this is the different of statistic nature between signal and noise.

for signal, the correlation between each measurement is strong. or to say, if you measured the signal is around 10 this time, it will probable measure it around 10 next time.

however, the noise is different, it is random in nature. so, in average, it will equal to zero.

in mathematics, we use the standard deviation \sigma and covariance cov  to show it.

\sigma(S_T)=\sigma(\sum S_i) = \sqrt{ \sum ( \sigma( S_i)^2 + 2cov(S_i,S_j) ) }

\sigma(N_T)=\sigma(\sum N_i) = \sqrt{ \sum ( \sigma( N_i)^2 + 2cov(N_i,N_j) ) }

since the signal should be correlated, cov(S_i , S_j) = \sigma(S_i) \sigma(S_j) thus,

\sigma(S_T) = n \sigma(S_i)

but the noise should have no correlation, cov(N_i, N_j) = 0

\sigma(N_T) = \sqrt{n} \sigma(N_i)

Thus, the signal-to-noise level will be

\sigma( S/N _T ) = \sqrt{n} \sigma (S/N)

For 100 times measurements, the signal-to-noise ratio will be 10 times stronger.

[Pol. p target] Changing to the small setup

February 4, 2011

GoLuckyRyan circuits, Lab Log, others , , , , , , , , , Leave a comment

Motivation

In order to convert the NMR single to the absolute magnetic field strength of the polarization. the thermal polarization ( polarization at thermal equilibrium ) should be measure and used to calibrate the NMR signal.

the thermal polarization is given by the Boltzmann statistic. The excited and the ground state population is:

\frac{ N_{\uparrow}} { N_{\downarrow} } = Exp \left( - 2 \frac { \mu_p B } {k_B T } \right)

the thermal polarization is the ratio of the different of spin-up and spin-down to the total spin.

P_{thermal} (B , T ) = tanh \left( \frac{ \mu_p B }{ k_B T } \right)

where \mu=p is the proton magnetic moment.

\mu_p = g_p \mu_N = g_p \frac{e \hbar }{2 m_p} = 1.410606662 \times 10^{-26} J T^{-1}

k_B is the Boltzmann constant.

k_B = 1.3806504 \times 10^{-23} J K^{-1}

The proton magnetic moment is small, the thermal polarization can be approximated as a linear function:

P_{thermal} ( B, T) = \frac{\mu_p B}{ k_B T}

since our polarization is on T = -5 ^oC and B = 0.05 T , thus, the thermal polarization is:

P_{thermal} (0.05, 268.15) = 1.90509 \times 10^{-7}

which is very small to be detected, or to say, the signal is smaller then the noise level.

the small system has a better sensitivity, down to 10^{-7}.

Set-Up

TO-do

  • Connect the Static field power and water cooling system
  • connect the Controler Unit
  • connect the magnetic field sweeping
  • Test Run

First experiment of 6He with a polarized proton target

February 4, 2011

GoLuckyRyan experimental, papers reading , , , , , , , , , , , , , , Leave a comment

DOI : 10.1140/epjad/i2005-06-110-5

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.