i came upon some insight into the physics behind t20 measurements. maybe
people are already aware of this, but i have never heard it mentionned
at any analysis meetings or whatnot. in addition, six months ago, i seem
to remember a then-unresolved discussion at an analysis meeting regarding
where, as a function of Q2, one should concentrate their t20 analysis in
order to be of most benefit to theorists.
for those who haven't already looked at it, check out the following:
J. L. Forest et al., Phys Rev C 56, 646 (1996)
specifically, check out pgs. 651-653.
you can read it for yourself, but here's what i got out of it. i might
have interpreted the paper incorrectly, so i encourage you to read it for
yourself.
1) because of the tensor component of the N-N interaction, deuterium
has a density that depends on m_z, the projection of the total spin
along the quantization axis.
2) the m_z = 0 density is toroidally shaped, like a donut lying in the
x-y plane. the donut has a certain thickness, t.
3) the m_z = +1 or -1 densities are dumbbell shaped, like two balls
centered at z = +d/2 and -d/2.
t20 analysis can help determine the values of t and d. here's how:
4) the charge form factors for deuterium, F^C_{m_z}(q), are related to
the fourier transform of the deuteron density, rho^{m_z}(\vec{r}):
F^C_{m_z}(q) = (1/2)*int{rho^{m_z}(\vec{r})*exp(i*q*z)*d^3\vec{r}}
5) figure 7 in the paper shows that F^C_{+1 or -1}(q) has minima at q =
3.6fm-1 and 12.6fm-1. one can argue that, at its minima,
F^C_{+1 or -1} is highly determined by the dumbbell-ness of the m_z
= +1 or -1 state. thus, in this way, F^C_{+1 or -1} is sensitive
to the value of d.
6) similarly, the minima of F^C_{0}(q) is sensitive to the value of t.
7) t20 also has minima and maxima. at its extrema, t20 has a simple
mathematical form:
[(F^C_{0}(q))^2 - (F^C_{1}(q))^2]
t20 = -sqrt(2) * ---------------------------------
[(F^C_{0}(q))^2 + (F^C_{1}(q))^2]
8) the minima of t20 occur at F^C_{1}(q) = minimum; the maxima of t20
occur at F^C_{0}(q) = maximum. thus, by determining the extrema of
t20, you are ultimately measuring the d and t values for deuterium.
specifically, the first minimum of t20 provides information on d. the
first maximum of t20 provides information on t. d, in turn, is directly
related to the diameter of the maximum-density torus (see figure 3). one
can see from the figure that there is not much model uncertainty at
maximum density. thus, it would seem to me that determining the actual
value of the first t20 minimum is not of too much theoretical
significance. (additionally, in a lecture that donnelly gave to us
graduate students, i seem to remember him saying that the minima isn't of
much interest, but maybe i'm wrong here; one of the other students at the
lecture can help here).
on the other hand, the first maximum of t20 has never been measured (at
least not that i am aware of). as it will give information on the value
of the thickness of the density for the m_z = 0 deuteron, there is
definite merit to aiming for it.
i realize that, in some sense, the data is what it is, and we cannot "get
higher Q2 values just because we want to". thus, i know that we will
not be able to get high-enough Q2 data to reach the maximum. but maybe
this will help focus the analysis some.
or maybe all of my thoughts are wrong. someone more knowledgable can
decide.
aaron
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