View source for Elong-13-05-01-Azz-Method-2

==''A<sub>zz</sub>'' Formalism - Method 2==
::{| class="wikitable" style="text-align:center; border-collapse:collapse;" border="1"
| style="width: 200px;" | Target Material = ND3
|- 
| <math>z_{\mathrm{tgt}} = 3\mathrm{ cm}</math>
|-
| <math>p_f = 0.65</math>
|-
| <math>P_{zz} = 20\%</math>
|-
| <math>N_{A} = 6.0221413\cdot 10^{23}</math>
|-
| <math>\rho_{\mathrm{He}} = 0.1412 \mathrm{g/cm}^3</math>
|-
| <math>M_{\mathrm{He}} = 4.0026 \mathrm{g/mole}</math>
|-
| <math>\rho_{\mathrm{ND}_3} = 1.007 \mathrm{g/cm}^3</math>
|-
| <math>M_{\mathrm{ND}_3} = 20 \mathrm{g/mole}</math>
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| <math>I_{\mathrm{beam}} = 0.115 \mathrm{\mu A}</math>
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| <math>\delta F_1^d = 5\%</math>
|-
|}



{| class="wikitable" style="text-align:center; width:800px;" border="0" 
|style="width: 50px; height: 65px;" |(1) || <math>R_{\mathrm{Total}} = \mathcal{A}\left[ \mathcal{L}_{\mathrm{He}}\left( \frac{d^2\sigma_{\mathrm{He}}^u}{d\Omega dE'}\right) + \mathcal{L}_{\mathrm{N}}\left( \frac{d^2\sigma_{\mathrm{N}}^u}{d\Omega dE'}\right) + \mathcal{L}_{\mathrm{D}}\left( \frac{d^2\sigma_{\mathrm{D}}}{d\Omega dE'}\right) \right]</math>
|-
|style="width: 50px; height: 65px;" |(2) || <math>R_{\mathrm{Total}}   = \mathcal{A}\left[ \mathcal{L}_{\mathrm{He}}\left( \frac{d^2\sigma_{\mathrm{He}}^u}{d\Omega dE'}\right) + \mathcal{L}_{\mathrm{N}}\left( \frac{d^2\sigma_{\mathrm{N}}^u}{d\Omega dE'}\right) + \mathcal{L}_{\mathrm{D}}\left( \frac{d^2\sigma_{\mathrm{D}}^u}{d\Omega dE'}\left[ 1 + \frac{1}{2}P_{zz}A_{zz}^d \right]\right) \right]  </math>
|-
|style="width: 50px; height: 65px;" |(2a) || <math>R_{\mathrm{Total}}   = \mathcal{A}\left[ \mathcal{L}_{\mathrm{He}} \sigma_{\mathrm{He}}^u  + \mathcal{L}_{\mathrm{N}} \sigma_{\mathrm{N}}^u + \mathcal{L}_{\mathrm{D}} \sigma_{\mathrm{D}}^u\left( 1 + \frac{1}{2}P_{zz}A_{zz}^d \right) \right]  </math>
|}
where <br>
{| class="wikitable" style="text-align:center; width:800px;" border="0" 
|style="width: 50px; height: 65px;" |(3) || <math>\mathcal{A} = \left( \Delta\Omega \Delta E' \right)</math>
|-
|style="width: 50px; height: 65px;" |(4) || <math>\mathcal{L}_{\mathrm{He}} = \left[ \mathcal{N}_A  \frac{\rho_{\mathrm{He}}}{M_{\mathrm{He}}}\left(1 - p_f\right) \right] \cdot \left( \frac{I_{\mathrm{beam}}}{e} \right) \cdot z_{\mathrm{tgt}}</math>
|-
|style="width: 50px; height: 65px;" |(5) || <math>\mathcal{L}_{\mathrm{N}} = \left[ \mathcal{N}_A  \frac{\rho_{\mathrm{ND}_3}}{M_{\mathrm{ND}_3}} p_f \right] \cdot \left( \frac{I_{\mathrm{beam}}}{e} \right) \cdot z_{\mathrm{tgt}}</math>
|-
|style="width: 50px; height: 65px;" |(6) || <math>\mathcal{L}_{\mathrm{D}} = 3\left[ \mathcal{N}_A  \frac{\rho_{\mathrm{ND}_3}}{M_{\mathrm{ND}_3}} p_f \right] \cdot \left( \frac{I_{\mathrm{beam}}}{e} \right) \cdot z_{\mathrm{tgt}}</math>
|-
|style="width: 50px; height: 85px;" |(7) || <math>\sigma^u_X = \frac{d^2\sigma^u_X}{d\Omega dE'} = A_X \left( \frac{d\sigma}{d\Omega} \right) _{\mathrm{Mott}_{\mathrm{p}}} \left[ \frac{2\cdot \left(\frac{F_1^{X}}{A_X} \right)}{m_{p}}\tan^2\left( \frac{\theta_{e'}}{2} \right) + \frac{\left( \frac{F_2^X}{A_X}\right) }{\nu} \right]</math>
|-
|style="width: 50px; height: 65px;" |(8) || <math>\left( \frac{d\sigma}{d\Omega} \right) _{\mathrm{Mott}_{\mathrm{p}}}=\frac{1^2 \alpha^2 \hbar^2 c^2}{4E^2\sin^4\left( \frac{\theta}{2} \right)}\cos^2\left( \frac{\theta}{2} \right)</math>
|-
|style="width: 50px; height: 65px;" |(9) || <math>\sigma_D = \frac{d^2\sigma_{\mathrm{D}}}{d\Omega dE'} = \frac{d^2\sigma_{\mathrm{D}}^u}{d\Omega dE'}\left[ 1 + \frac{1}{2}P_{zz}A_{zz}^d \right] </math> 
|-
|}

Then 

{| class="wikitable" style="text-align:center; width:800px;" border="0" 
|style="width: 50px; height: 65px;" |(10) || <math>N_{Pol} - N_{u} = R^{Pol}_{\mathrm{Total}}t^{Pol}_{\mathrm{Total}} - R^{u}_{\mathrm{Total}}t^{u}_{\mathrm{Total}} </math>
|-
|style="width: 50px; height: 65px;" |(11) || <math>N_{u} = R^{u}_{\mathrm{Total}}t^{u}_{\mathrm{Total}} </math>
|-
|}

If we assume that <math>t^{Pol}_{\mathrm{Total}} \approx t^{u}_{\mathrm{Total}} \approx t</math>, then

{| class="wikitable" style="text-align:center; width:800px;" border="0" 
|style="width: 50px; height: 65px;" |(12) || <math>N_{Pol} - N_{u} = \left( R^{Pol}_{\mathrm{Total}} - R^{u}_{\mathrm{Total}}\right) t </math>
|-
|style="width: 50px; height: 65px;" |(13) || <math>N_{Pol} - N_{u} = \left( \mathcal{A}\left[ \mathcal{L}_{\mathrm{He}} \sigma_{\mathrm{He}}^u  + \mathcal{L}_{\mathrm{N}} \sigma_{\mathrm{N}}^u + \mathcal{L}_{\mathrm{D}} \sigma_{\mathrm{D}}^u\left( 1 + \frac{1}{2}P_{zz} A_{zz}^d \right) \right] - \mathcal{A}\left[ \mathcal{L}_{\mathrm{He}} \sigma_{\mathrm{He}}^u  + \mathcal{L}_{\mathrm{N}} \sigma_{\mathrm{N}}^u + \mathcal{L}_{\mathrm{D}} \sigma_{\mathrm{D}}^u \right] \right) t   </math>
|-
|style="width: 50px; height: 65px;" |(13a) || <math>N_{Pol} - N_{u} = \left( \mathcal{A}\mathcal{L}_{\mathrm{D}}\sigma_{\mathrm{D}}^u \left( \frac{1}{2}P_{zz} A_{zz} \right) \right) t</math>
|-
|style="width: 50px; height: 65px;" |(14) || <math>N_{u} = R^{u}_{\mathrm{Total}}t </math>
|-
|style="width: 50px; height: 65px;" |(15) || <math>N_{u} = \mathcal{A}\left[ \mathcal{L}_{\mathrm{He}} \sigma_{\mathrm{He}}^u  + \mathcal{L}_{\mathrm{N}} \sigma_{\mathrm{N}}^u + \mathcal{L}_{\mathrm{D}} \sigma_{\mathrm{D}}^u \right]t </math>
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|style="width: 50px; height: 65px;" |(16) || <math>\frac{N_{Pol} - N_{u}}{N_{u}} = \left( \frac{\mathcal{L}_{\mathrm{D}}\sigma_{\mathrm{D}}^u}{ \mathcal{L}_{\mathrm{He}} \sigma_{\mathrm{He}}^u  + \mathcal{L}_{\mathrm{N}} \sigma_{\mathrm{N}}^u + \mathcal{L}_{\mathrm{D}}\sigma_{\mathrm{D}}^u } \right) \frac{1}{2} A_{zz}  P_{zz}</math>
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|style="width: 50px; height: 65px;" |(17) || <math>\frac{N_{Pol} - N_{u}}{N_{u}} = f \frac{1}{2} A_{zz}  P_{zz}</math>
|-
|style="width: 50px; height: 65px;" |(18) || <math>A_{\mathrm{meas}}^{(2)} = \frac{N_{Pol} - N_{u}}{N_{u}}</math>
|-
|style="width: 50px; height: 65px;" |(18a) || <math>A_{\mathrm{meas}}^{(2)} = f \frac{1}{2} A_{zz}  P_{zz}</math>
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|style="width: 50px; height: 65px;" |(19) || <math>A_{zz} = \frac{2  }{f \cdot P_{zz}}A_{\mathrm{meas}}^{(2)}</math>
|-
|}

In order to get the uncertainty, we'd use

{| class="wikitable" style="text-align:center; width:800px;" border="0" 
|style="width: 50px; height: 65px;" |(20) || <math>\delta A_{zz} = \sqrt{\left( \frac{\partial A_{zz}}{\partial A_{\mathrm{meas}}^{(2)}} \delta A_{\mathrm{meas}}^{(2)} \right)^2 + \left( \frac{\partial A_{zz}}{\partial f} \delta f \right)^2 + \left( \frac{\partial A_{zz}}{\partial P_{zz}} \delta P_{zz} \right)^2 }</math>
|-
|style="width: 50px; height: 65px;" |(20a) || <math>\delta A_{zz} = \sqrt{\left( \delta A_{zz}^{\mathrm{Stat}} \right) ^2 + \left( \delta A_{zz}^{\mathrm{Dil}} \right) ^2 + \left( \delta A_{zz}^{\mathrm{Pol}} \right) ^2   }</math>
|-
|style="width: 50px; height: 65px;" |(20b) || <math>\delta A_{zz} = \sqrt{\left( \delta A_{zz}^{\mathrm{Stat}} \right) ^2 + \left( \delta A_{zz}^{\mathrm{Sys}} \right) ^2 }</math>
|-
|}

Ignoring <math>\delta A_{zz}^{\mathrm{Sys}}</math> for now (and in all of the plots I'm showing), then

{| class="wikitable" style="text-align:center; width:800px;" border="0" 
|style="width: 50px; height: 65px;" |(21) || <math> \delta A_{zz}^{\mathrm{Stat}} = \frac{\partial A_{zz}}{\partial A_{\mathrm{meas}}^{(2)}} \delta A_{\mathrm{meas}}^{(2)} = \frac{2}{f\cdot P_{zz}} \delta A_{\mathrm{meas}}^{(2)} </math>
|-
|style="width: 50px; height: 65px;" |(22) || <math> \delta A_{\mathrm{meas}}^{(2)} = \sqrt{ \left( \frac{\partial A_{\mathrm{meas}}^{(2)}}{\partial N_{Pol}} \delta N_{Pol} \right)^2 + \left( \frac{\partial A_{\mathrm{meas}}^{(2)}}{\partial N_{u}} \delta N_{u} \right)^2 }</math>
|-
|style="width: 50px; height: 65px;" |(23) || <math> \delta A_{\mathrm{meas}}^{(2)} = \sqrt{ \left( \frac{1}{N_u} \sqrt{N_{Pol}} \right)^2 + \left( -\frac{N_{Pol}}{N_u^2} \sqrt{N_u} \right)^2 }</math>
|-
|style="width: 50px; height: 65px;" |(24) || <math> \delta A_{\mathrm{meas}}^{(2)} = \sqrt{  \frac{N_{Pol}}{N_u^2} + \frac{N_{Pol}^2}{N_u^3}  }</math>
|-
|}

If we assume that <math>N_{Pol} \approx N_u \approx \frac{N}{2}</math>, then

{| class="wikitable" style="text-align:center; width:800px;" border="0" 
|style="width: 50px; height: 65px;" |(25) || <math> \delta A_{\mathrm{meas}}^{(2)} = \sqrt{  \frac{N/2}{N^2/4} + \frac{N^2/4}{N^3/8}  }</math>
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|style="width: 50px; height: 65px;" |(25a) || <math> \delta A_{\mathrm{meas}}^{(2)} = \sqrt{  \frac{2}{N} + \frac{2}{N}  }</math>
|-
|style="width: 50px; height: 65px;" |(25b) || <math> \delta A_{\mathrm{meas}}^{(2)} = \frac{2}{\sqrt{N}}</math>
|-
|}

which would yield

{| class="wikitable" style="text-align:center; width:800px;" border="0" 
|style="width: 50px; height: 65px;" |(26) || <math> \delta A_{zz}^{\mathrm{Stat}} = \frac{2}{f\cdot P_{zz}} \delta A_{\mathrm{meas}}^{(2)} </math>
|-
|style="width: 50px; height: 65px;" |(26a) || <math> \delta A_{zz}^{\mathrm{Stat}} = \frac{2}{f\cdot P_{zz}} \left( \frac{2}{\sqrt{N}} \right) </math>
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|style="width: 50px; height: 65px;" |(27) || <math> \delta A_{zz}^{\mathrm{Stat}} = \frac{4}{f\cdot P_{zz}\sqrt{t\cdot R_{\mathrm{Total}}}} </math>
|-
|}

Using the same formalism that HERMES used (which defines <math>F_{1_{\mathrm{HERMES}}}^d = \frac{(1 + Q^2/\nu^2)F_2^d}{2x(1+R)}</math> with <math>F_2^d=\frac{F_2^p + F_2^n}{2}</math> as a ''per nucleon'' quantity, which corresponds to the Bosted that uses ''per nucleus'' by <math>F_{1_{\mathrm{HERMES}}}^d = \frac{F_{1_{\mathrm{Bosted}}}^d}{A_{\mathrm{D}}} = \frac{F_1^d}{2}</math> -- [[Elong-13-05-01#Matching_HERMES_F1d.2C_dAzzd.2C_and_db1d|as described previously]]), we can extract <math>b_1^d</math> and its uncertainty by
{| class="wikitable" style="text-align:center; width:800px;" border="0" 
|style="width: 50px; height: 65px;" |(18) || <math> b_1^d = - \frac{3}{2}A_{zz} \left( \frac{F_1^d}{A_{\mathrm{D}}} \right)= - \frac{3}{2}A_{zz} \left( \frac{F_1^d}{2} \right)</math>
|-
|style="width: 50px; height: 65px;" |(19) || <math> \delta b_1^d =\sqrt{ \left(\frac{\partial b_1^d}{\partial A_{zz}} \delta A_{zz} \right)^2 + \left(\frac{\partial b_1^d}{\partial F_1^d} \delta F_1^d \right)^2 }</math>
|-
|style="width: 50px; height: 65px;" |(19a) || <math> \delta b_1^d =\sqrt{ \left[ - \frac{3}{2} \left( \frac{F_1^d}{2} \right)\delta A_{zz} \right]^2 + \left[ - \frac{3}{2} A_{zz} \left( \frac{1}{2} \right)\delta F_1^d \right]^2 }</math>
|-
|}


==Plots with using ''A<sub>zz</sub>'' Method 2 Calculations==

[[Image: 2013-05-02-method2.png]]

==Total Rates==

Ignoring our W>2 physics cut, then our total rates come out to:

[[Image: 2013-05-02-total-rates.png]]

==F1 from PDFs==
The plot below uses F1 calculated per nucleon from the PDF code, not from the Bosted code.

[[Image: 2013-05-02-pdf-f1.png]]