Difference between revisions of "DV MollerTrackRecon"

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===Different p<sub>2</sub><sup>1</sup> Values===
 
===Different p<sub>2</sub><sup>1</sup> Values===
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{| class="wikitable" align="center" border=1
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  |+ '''Differential Cross Section Scale for Different p<sub>2</sub><sup>1</sup> Values'''
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|-
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  ! <math>p_{2}'</math>
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  ! <math>\frac{d\omega}{d\Omega_{2}^'}</math>
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|-
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  | 10000 MeV
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  |
 +
|-
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  | 5000 MeV
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  |
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|-
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  | 1000 MeV
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  |
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|-
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  | 500 MeV
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  |
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|}
  
 
===Substituting for Moller range and energies===
 
===Substituting for Moller range and energies===

Revision as of 17:34, 4 January 2016

Moller Lund

LUND file with Moller events (with origin of coordinates occurring at each event)

2       1       1       1       1       0       0.000563654     3.53715 0       6.2002
1 -1 1 11 0 0 0.69 -2.4999 10993.7998 10993.80 0.000511 0 0 0
2 -1 1 11 0 0 -0.69 2.4999 6.5852 7.08 0.000511 0 0 0


From a GEMC run WITH the Solenoid ced is used to obtain the information from the eg12_rec.ev file.

      Event29.png


We take the phi angle from the Generated Event momentum as the initial phi angle. The obtain the final phi angle, we can look at the final position of the electron with in the drift chambers.

     Detector position.png

Examining the position from Timer Based Tracking, we can see that after rotations about first the y-axis, then the z-axis transforms from the detector frame of reference to the lab frame of reference.

Euler Angles

We can use the Euler angles to perform the rotations.

For the rotation about the y axis.

Euler1.png

And the rotation about the z axis.

Euler2.png

Transformation Matrix

The Euler angles can be applied using a transformation matrix

(cos(θ)0sin(θ)010sin(θ)0cos(θ)).(xyz)


=(xcos(θ)zsin(θ)yzcos(θ)+xsin(θ))



For event #29, in sector 3, the location of the first interaction is given by

Conversions.png


Converting -25 degrees to radians, θ=0.436332 which is the rotation the detectors are rotated from the y axis.

(cos(θ)0sin(θ)010sin(θ)0cos(θ)).(15.760237.43)

=(86.05880.221.845)

Finding ϕ=120 2π360; since "sector -1" =3-1=2*60=120 degrees

(cos(ϕ)sin(ϕ)0sin(ϕ)cos(ϕ)0001).(86.05880.221.845)

(43.029474.5291221.845)

This shows how the coordinates are transformed and explains the validity of using the TBTracking information to obtain a phi angle in the lab frame.


Phi shifts

Composite Fields.png


GeV graph.png


MeV graph.png


Total graph.png

Cross-section Area

Calculations of 4-momentum components

DV_Calculations_of_4-momentum_components

Summary of 4-momentum calculations

For 0ϕπ2 Radians
x=POSITIVE
y=NEGATIVE
For 0ϕπ2 Radians
x=POSITIVE
y=POSITIVE
For π2ϕπ Radians
x=NEGATIVE
y=NEGATIVE
For π2ϕπ Radians
x=NEGATIVE
y=POSITIVE


4 momentum calculations for different frames of reference
Electron Initial Lab Frame Moller electron Initial Lab Frame Moller electron Final Lab Frame Moller electron Center of Mass Frame Electron Center of Mass Frame Electron Final Lab Frame
p111000MeV p20 p2INPUT p2=E22m2 p1=E22m2 p1=E 21m2
θ10 θ20 θ2INPUT θ2=arcsin(p2p2 sin(θ2)) θ1=θ2+π θ1=arcsin(p1p1 sin(θ1))
E1=p21+m2 E2m E2=p 22+m2 E2=m(m+E1)2 E1=m(m+E1)2 E1EE2
p1(x)0 p2(x)0 p2(x)=p 22p 22(z)cos(ϕ2) p2(x)p2(x) p1(x)p2(x) p1(x)p1(x)
p1(y)0 p2(y)0 p2(y)=p 22p 22(x)p 22(z) p2(y)p2(y) p1(y)p2(y) p1(y)p2(y)
p1(z)p1 p2(z)0 p2(z)p2 cos(θ2) p2(z)=p 22p 22(x)p 22(y) p1(z)p2(z) p1(z)=p 21p 2(1(x)p 21(y)

Differential Cross Section

Moller Differential Cross Section

Using the equation from [1]

dσdΩ1=e48E2{1+cos4(θ2)sin4(θ2)+1+sin4(θ2)cos4(θ2)+2sin2(θ2)cos2(θ2)}


where α=e2cwith=c=1 and θ=θ1=θ2


This can be simplified to the form


dσdΩ1=α24E2(3+cos2θ)2sin4θ

Plugging in the values expected for 2 scattering electrons:



α2=5.3279×105


E106.031MeV


Using unit analysis on the term outside the parantheses, we find that the differential cross section for an electron at this momentum should be around

5.3279×1054×1.124×1016eV2=1.18×1021eV2=1.18×10211eV2×1×10181×1018=.0012GeV2

Using the conversion of


11GeV2=.3894mb


.00121GeV2=.0012111GeV2=.0012×.3894mb=.467×103mb



We find that the differential cross section scale is dσdΩ.5×103mb=.5μb


CM to Lab Frame

We can substitute in for θ


dσdΩ1=α24E2(3+cos2θ)2sin4θ


dσdΩ1=α24E2(3+cos2θ)2sin(θ)sin(θ)sin(θ)sin(θ)


Using,

sin(θ)=sin(θ2)=p2p2 sin(θ2)


dσdΩ1=α24E2(3+cos2θ)2p2p2 sin(θ2)p2p2 sin(θ2)p2p2 sin(θ2)p2p2 sin(θ2)



dσdΩ1=α2p424E2p42(3+cos2θ)2sin4(θ2)


Now, using the trigometric identity,

sin2t+cos2t=1cos2(θ)=1sin2(θ)


dσdΩ1=α2p424E2p42(3+1sin2(θ))2sin4(θ2)


dσdΩ1=α2p424E2p42(4sin(θ)sin(θ))2sin4(θ2)


dσdΩ1=α2p424E2p42(4p2p2 sin(θ2)p2p2 sin(θ2))2sin4(θ2)


dσdΩ1=α2p424E2p42(4p22p22 sin2(θ2))2sin4(θ2)


dσdΩ1=α2p424E2p42(168p22p22 sin2(θ2)+p42p42 sin4(θ2))sin4(θ2)


Substituting,

p2=E22m2



dσdΩ1=α2(E22m2)44E2p42(168p22p22 sin2(θ2)+p42p42 sin4(θ2))sin4(θ2)


dσdΩ1=α2(E22m2)24E2p42(168p22p22 sin2(θ2)+p42p42 sin4(θ2))sin4(θ2)


Substituting in for m, E2*,and E* α2=5.3279×105

dσdΩ1=(5.3279×105(((53.015MeV)2(.511MeV)2)24×(106.031MeV)2p42(168p22p22 sin2(θ2)+p42p42 sin4(θ2))sin4(θ2)


dσdΩ1=9.357×109p42(168p22p22 sin2(θ2)+p42p42 sin4(θ2))sin4(θ2)

Different p21 Values

Differential Cross Section Scale for Different p21 Values
p2 \frac{d\omega}{d\Omega_{2}^'}
10000 MeV
5000 MeV
1000 MeV
500 MeV

Substituting for Moller range and energies

Converting the number of electrons to barns,

L=iscatteredσiscattered×ρtarget×ltarget


where ρtarget is the density of the target material, ltarget is the length of the target, and iscattered is the number of incident particles scattered.


L=70.85kg1m3×1mole2.02g×1000g1kg×6×1023atoms1mole×1cm100cm×1m×1023m2barn=2.10×102barns


1L×4×107=1.19×106barns

FnlThetaCM.pngTheory new.png


Combining these plots, and rescaling the Final Theta in the Center of Mass for micro-barns, we find


XSect new zoom.png

DV_RunGroupC_Moller#Moller_Track_Reconstruction