[Sane-analysis] Draft of technote for nitrogen correction

[Oscar Rondon]

Hi Hoyoung,

Thank you for sharing the technical note. For SANE's 14N target, the RSS
analysis of the polarized nitrogen correction applies with the only
changes being:

- The proton asymmetry of the unpaired proton in 15N, A_{15} = -1/3 A_1,
stays the same for 14N, A_{14} = -1/3. A_{14} effectively represents the
polarization of the unpaired proton in polarized nitrogen, either 14N or
15N.

As  discussed in the section following eq. (12) of the RSS technote,
ref. [1] of your draft, the proton spin is aligned opposite the
nitrogen's spin 1/3 of the time. This value applies to both 14N and 15N,
(Table 3 in http://dx.doi.org/10.1103/PhysRevC.60.035201.)

- the 14N polarization P_{14} is parallel to the polarization of the
free protons P_1, because the magnetic moment of 14N is positive, unlike
 the 15N moment, which is negative. Given the expected size of the
correction, approximating the P_{14}/P_1 ratio by EST may be OK.

A fancier option would be to fit world data on 14N vs H polarizations,
including PSI, E143 and E155x technical run data if available for 14N
which Don Crabb may have, like we did for 15N for RSS, see plots here:

https://userweb.jlab.org/~rondon/sane/target/15NVSP.EPS
https://userweb.jlab.org/~rondon/sane/target/FIT15NPC.GIF

Original Don's plot (doesn't have E155x data):
https://userweb.jlab.org/~rondon/sane/target/15NH3P.BMP

Other 14N vs p fits would work, too.

- Finally, the rate from the polarized proton in 14N and the free proton
are related by the respective dilution factors, eqs. (8) and (7) of the
RSS TN. Substituting A_{14} = -1/3 A_1 and solving eq. (13) of that TN
for A_1, the 14 N correction is

C_{14N} = 1 - 1/3 * f_{14}(W)/f_1(W) * P_14/P1

The denominators of f_{14} and f_1 cancel because they are the total
rates from all nuclei in the target, so the ratio f_{14N}/f_1 is just
the ratio of cross sections per nucleon of 14N and of 3*H (this factor
of 3 comes from the ammonia chemical formula). In DIS, this last ratio
of cross sections is well approximated by the EMC Bjorken x-dependent
factor g(x), so one gets eq. (15) of the RSS TN.

In the resonances, one needs to apply some parameterization, like
F1F209, to calculate the ratio as a function of W. For an accurate
result one would need to calculate a ratio of radiated rates. However,
since the correction is very small, Born model cross sections like
F1F209 can be used directly for most of the inelastic W range. A plot of
Born vs radiated rates for the RSS nitrogen correction is posted here:
https://userweb.jlab.org/~rondon/sane/target/15N-p_rss.pdf

(This result applies to 14N, too, since 15N and 14N cross sections agree
within about 7%).

The magnitude of the correction can be estimated with the approximation
~1 for the ratio of 14N(per nucleon)/H cross sections and P_{14}/P_1 ~
0.14 for the range of proton target polarizations measured during SANE:

C_{14N} = 1 - A_{14}/A_1 * f_{14}/f_1 * P_{14}/P_1
        ~ 1 - 1/3        * 1/3 * 1    * 0.14
        = 1 - 0.016 = 0.984

with a +/- 0.003 uncertainty (~20% of the second term in the last line).
As expected, it's very small and it's < 1 for 14N, unlike that for 15N
which is > 1 due to the minus sign in the P_{15}/P_1 ratio.

Given the size of the correction, I would just apply a 0.984 factor to
the physics asymmetries A1 and A2, and increase the normalization
systematic error by 0.3% relative or so.

For the region W < 1200 MeV, where C_N grows, our statistical errors
seem too large to apply a more accurate correction, but the systematic
error would be bigger. From the plot of Born vs radiated rates, the
correction seems to double in this region, so a 0.6% or so error might
be OK.

Cheers,

Oscar



Hoyoung Kang wrote:
>  Sorry for not attaching the file. Here it is.
> 
>  Hoyoung
> 
>>  Dear all,
>>
>>  I wrote a rough draft about nitrogen correction. The attached is my
>> technote.
>>
>>  But I've not yet decided how to approximate the ratio of cross sections
>> and asymmetries.
>>
>>  Any comment would be very appreciated.
>>
>>  regards,
>>  Hoyoung
>>
>>
>>
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