Hi,
-On the discussion of "average" vs "nominal" spin angle:
In Doug's routine, the spin profile parametrized as a 9th order
polynomial is evaluated relative to a "nominal" spin angle (which
happens to be theta(z=0)).
Doug's profile:
th(z) = th_nominal + \sum_i a_i z^i, i=1..9
= th_nominal + dth(z)
The yield-weighted average spin angle (over the allowed length of the
target) of Doug's profile is thus
<th>_yield = 1/40 * \int_{-20}^{20}[dz \rho(z) th(z)]
= th_nominal + <dth>_yield,
where \rho(z) is the target density distribution along z (normalized
to 1), and <dth>_yield is the yield-weighted average of the
polynomial terms.
While theta_nominal is a quantity of the detector that does not
depend on the reaction channel, the quantities \rho(z), <th>_yield
and <dth>_yield are dependent on the particular considered reaction,
or even on the sector.
The average angle <th>_yield can be determined from both ed elastic
and ep elastic asymmetry analysis; as such, this number is not yet
useful for any other reaction unless it is converted into the
"nominal" angle. In order to do this for the given reaction,
<dth>_yield = 1/40*\sum_i{\int_{-20}^{20}[dz \rho(z) a_i z^i]}, i=1..9
needs to be determined which is a simple number.
This said, comparing the spin angles from ed elastic and ep elastic
must only be done for the resulting theta_nominal, but not for the
yield-averaged numbers!
Chris has evaluated <th>_yield and <dth>_yield for ep elastic:
ep elastic(47): <th>_yield = 45.8 +- ??? degrees
<dth>_yield = -0.8
-> theta_nominal = 46.6
ed elastic(32): <th>_yield = 31.4
<dth>_yield = ???
There is deuterium data for both 47 (2005) and 32 (2004) degree
settings. Only the latter is available so far with the latest
recrunch, the former is being crunched right now.
For each of the 32 and 47 degree settings, the same theta_nominal
ought to be used by every analysis. The resulting <th>_yield for each
reaction channel is dependent on the specific yield distribution
\rho(z) which may be different in each channel. It may even depend on
the sector. In order to calculate the average angle in any reaction
channel from a given "universal" map, it needs to be averaged
over the specific yield (=evaluating <dth>_yield). Eugene has done
this for 32-deg=2004 d(e,e'n): <dth>_yield = <th>_yield - theta_nominal
en quasielastic(32): <dth>_yield (left sector) = -2.27 deg
<dth>_yield (right sector) = -2.42 deg
The resulting average <th>_yield is a number which can be quoted in a
paper for the considered reaction, but it is of no further use if the
extraction of observables relies on the same common spin angle map.
Nevertheless it is a good idea to compare the various <dth>_yield of
the various reaction channels (e,e'), d(e,e'p), (e,e'pi+), ...
Comparing ep elastic with en quasielastic, there seems to be some
significant difference, -0.8 deg vs -2.4 deg.
One should do the same exercise for MC-generated (cross-section
weighted) target distributions rho(z) for each channel, in order to
exclude that there are any surprises. Keep in mind that the
Q2bin-by-Q2bin target distributions in ep elastic from reconstructed
data looked rather funny, and they are prone to systematics in the
reconstruction. In MC, the tossed and accepted z distribution can be
used directly, without the reconstruction uncertainty.
Regards,
Michael
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