Flux Oddities

Intresting development on the poalrization/diffusion front. Looked into the flux from a $G_E^n$ style cell. The results are in table 2. These results indicate that there may be an additional degree of difficulty in extracting useful information from the upper coil.

Table 1: Flux Calculations. These are rough calculations to show the effect of the different magnetic field and pickup coil configurations.

	Target	Pumping	Transfer
	Chamber	Chamber	Tube	Total
Set-Up	Flux	Flux	Flux	Flux	Comment
Old Configuration	52.721	-4.114	-0.122	48.485
Offset	52.844	-3.083	-0.630	49.131	Shifted, same size coils
Rotate Coils	-0.0519	-0.236	-0.078	-0.366	rotated coils only
Rotate Field	52.844	4.695	-0.648	56.892	$G_E^n$ config., old cell
Lengthen Transfer Tube	52.844	3.076	-0.507	55.413
``Inflate'' Pumping Chamber	52.844	9.219	-0.648	61.415
Long Tube, Big PC	52.844	6.197	-0.507	58.535	$G_E^n$ config., $G_E^n$ cell

The size and relative sign of the flux lead me to re-evaluate the diffusion calcualtion, as well as the calibration constant used.

$$S_{NMR} = \mu_{^3He} \cdot G \left[P_u n_u \Phi_u + P_l n_l \Phi_l \right]$$ (1)

$P_u$ is obtained from EPR, $n_u$ and $n_l$ are easily calculable - provided we understand our temperatures. $\Phi_u$ and $\Phi_l$ are above. What remains unknown is $G$ and $P_l$. What we really care about is $P_l$. We need to separate out this factor from the equation.

Another approach would be to

$$S^{cold}_{NMR} = \mu_{^3He} \cdot G \left[P_u^\prime n_u^\prime \Phi_u + P_l^\prime n_l^\prime \Phi_l \right]$$ (2)

Taking the ratio of the two

$$\frac{S_{NMR}}{S^{cold}_{NMR}} = \frac{P_u n_u \Phi_u + P_l n... ...l}{P_u^\prime n_u^\prime \Phi_u + P_l^\prime n_l^\prime \Phi_l}$$

At equilibrium, $P_u^\prime = P_l^\prime$ and $n_l^\prime$, $n_u^\prime$, $n_l$, $n_u$, $\Phi_u$, and $\Phi_l$ can be calculated. However, there is no good way of knowing $P^\prime_u$, since EPR cannot be done at room temperature. (n.B., this information has been gained in the past via water calibration)

Still another approach would be to look at the signal from the upper coil:

$$S^{upper}_{NMR} = \mu_{^3He} \cdot G^{up} \left[P_u n_u \Phi_u^{up} + P_l n_l \Phi_l^{up} \right]$$ (3)

And the ratio of upper to lower coil signals:

$$\frac{S^{upper}_{NMR}}{S_{NMR}} = \frac{G^{up}}{G} \frac{P_u... ...u^{up} + P_l n_l \Phi_l^{up}}{P_u n_u \Phi_u + P_l n_l \Phi_l}$$ (4)

This situations gives us a different $G$, as well as different fluxes. For these fluxes, the uncertainty is large and the overall value is small. So, calulating them directly does little to accurately determine $P_l$.

Now, if we assume that the temperature tests results are accurate, then we can calculate $n_u$ and $n_p$. From that we can calculate the ratio $P_u/P_l$ (which is what has been done already, and is included in my most recent polarization numbers.) What has not been done is to extract the correct $P_l$. There is an indication that relative sign of the flux ratio helps us (this manages to offset the effect of diffusion). However, we must proceed very carefully in determining whether or not the beam was on before the NMR measurement. The ratio of the upper and lower coil measurements may help this, but it is not yet clear.

The missing pieces, as I see them, are: $P_u^\prime$, $G^{up}$, $\Phi_u^up$ and $\Phi_l^{up}$