DC Super Layer 1:Layer 1

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DC Super Layer 1:Layer 1

From the ced simulations

Table 1: Superlayer 1 Change of Coordinates From Wire 1 to Wire 2(cm)

Coordinates(cm)	Layer 1	Layer 2	Layer 3	Layer 4	Layer 5	Layer 6
[math]\Delta x[/math]	1.22	1.22	1.22	1.22	1.21	1.22
[math]\Delta y[/math]	0.00	0.00	0.00	0.00	0.00	0.00
[math]\Delta z[/math]	.55	.56	.57	.57	.57	.57

While the wire placement is off, the difference in x and z position is matched by the calculated values. Using the geometric construction for determining angle theta to wire 2:

[math]\tan{\theta_2}=\frac{x_1+\Delta x}{z_1+\Delta z}\Rightarrow \theta_2=\arctan{\frac{x_1+\Delta x}{z_1+\Delta z}}[/math]

[math]\arctan{\frac{22.2092+1.212}{242.5779-0.5652}}=\arctan{\frac{23.4212}{242.0127}}=\arctan{0.0967}=5.53^{\circ}=\tan{\theta_2}[/math]

This equation can be solved for a hypothetical wire 0, which will allow the wire number to be the multiplicative factor for the change from the starting position.

[math]\tan{\theta_n}=\frac{x-\Delta x+n\Delta x}{z-\Delta z+n\Delta z}\Rightarrow \theta_n=\arctan{\frac{x+n\Delta x}{z+n\Delta z}}[/math]

where

[math]x_0=22.2092-1.212\approx 20.9972cm[/math]

[math]z_0=242.5779+0.5652\approx 243.1431cm[/math]

[math]\theta_n=\arctan{\frac{20.9972+n1.212}{243.1431-n0.5652}}[/math]

[math]\tan{\theta_n}=\frac{20.9972+n1.212}{243.1431-n0.5652}\Rightarrow \tan{ \theta_n}=-2.14437 + \frac{959.637}{430.189 - n}\Rightarrow 430.189 - n = \frac{959.637}{\tan{ \theta_n}+2.14437}[/math]

[math]\Rightarrow n = \frac{-959.637}{\tan{ \theta_n}+2.14437}+430.189[/math]

Using Mathematica, a series expansion about n=0 can be found:

In[2]:= Series[ArcTan[(20.9972+1.212 n)/(243.1431-0.5652 n)],{n,0,4}]

Out[2]= 0.0861437+0.00514708 n+9.67686*10^-6 n^2-2.72596*10^-8 n^3-2.22159*10^-10 n^4+O[n]^5

[math]\theta\approx 0.0861437+0.00514708 n+9.67686 \times10^{-6} n^2-2.72596 \times 10^{-8} n^3[/math]

This expression will find the angle theta in radians given the wire number. To convert from radians to degrees, we can multiply by 180 and divide by Pi.

In[3]:= 180(0.08614365821719583`+0.005147076698801422` n+9.676859262994711`*^-6 n^2-2.7259639443492433`*^-8 n^3-2.2215881081233252`*^-10 n^4+O[n]^5)/3.14159265359

Out[3]= 4.93567+0.294906 n+0.000554443 n^2-1.56186*10^-6 n^3-1.27288*10^-8 n^4+O[n]^5

[math]\theta\approx 4.93567+0.294906 n+0.000554443 n^2-1.56186*10^{-6} n^3[/math]

This tells us that the expression for theta will follow a function that comes from a series expansion. Using Mathematica, a line can be fitted to the data collected on the wire number to angle theta correspondence.

Table 4: Calculated Superlayer 1 Wire-Angle Theta Correspondence in Degrees

Wire Number	Layer 1
1	5.23
2	5.53
77	29.80
78	30.13
79	30.46
110	40.45
111	40.76
112	41.07

Declaring the data set:

In[4]:= data1={{1,5.23},{2,5.53},{77,29.80},{78,30.13},{79,30.46},{110,40.45},{111,40.76},{112,41.07}}

Out[4]= {{1,5.23},{2,5.53},{77,29.8},{78,30.13},{79,30.46},{110,40.45},{111,40.76},{112,41.07}}

Testing for a linear fit:

In[5]:= line1=Fit[data1,{1,n},n]

Out[5]= 4.90443 +0.323148 n

Examining the range limits for the angle theta for layer 1:

Taking the difference of the upper and lower limits in theta,

[math]\theta=\arctan{\frac{20.9972+n1.212}{243.1431-n0.5652}}[/math]

[math]\theta_{n=1}=\arctan{\frac{20.9972+(1)1.212}{243.1431-(1)0.5652}}=5.23\ \ \ \ \ \ \theta_{n=112}=\arctan{\frac{20.9972+(112)1.212}{243.1431-(112)0.5652}}=41.07[/math]

[math]\Delta degrees=41.07^{\circ}-5.23^{\circ}=35.84^{\circ}[/math]

Dividing by the change in wire numbers (112-1=111), we find

[math]\frac{\Delta degrees}{\Delta wire\ number}=\frac{35.84^{\circ}}{111\ wires}\approx\ \frac{0.323^{\circ}}{wire\ number}[/math]

This would imply that if the wires were evenly placed, their change in angle theta would increase by the factor of .323 degrees for each increase in wire number, starting obviously with wire 1 at 5.23 degrees. In addition, this implies that the bin spacing for each wire would be around .323 degrees in width.

Checking this, we can find the difference between wires 1 and 2,

[math]\frac{\Delta degree}{\Delta wire\ number}=\frac{5.5277^{\circ}-5.2311^{\circ} }{wire\ number}=\frac{.2965^{\circ}}{wire\ number} [/math]

Similarly, finding the difference between wires 111 and 112,

[math]\frac{\Delta degree}{\Delta wire\ number}=\frac{41.0740^{\circ}-40.7649^{\circ}}{wire\ number}=\frac{.0.3091^{\circ}}{wire\ number}[/math]

These differing values show that the bin width is not uniform in length, therefore a first order, linear fit, will not suffice. The change in bin size, with respect to the angle theta covered, is due to the tilt of the wire plane.

Figure 5.1.1: For a tilted plane, that is offset from the origin, the change in angle between uniformly spaced points depends on the tilt with respect to the beam line. The uniform distance between points, A, can be used to show the angle, [math]\alpha[/math] each wire bin covers can be found using the law of sines. The interior angle [math]\beta[/math]

Using the law of sines,

[math]\frac{\sin{\alpha}}{A}=\frac{\sin{\beta}}{B}[/math]

[math]\Rightarrow \alpha=\arcsin{\frac{A \sin{\beta}}{B}}[/math]

The range of [math]\beta[/math] runs from near to 65 to almost 180 degrees. At [math]\theta=25^{\circ}[/math] with respect to the positive z direction, a right triangle is created. Since the sine function is periodic and reaches its maximum at 90 degrees, this implies that the opposite side to this angle reaches its minimum. On either side of the right angle, as the angle [math]\beta[/math] either increases or decreases, the function [math]\sin{\beta} will be smaller. This implies for movement in angle \lt math\gt \alpha[/math] in a counter-clockwise direction, the angle [math]\beta[/math] will continue to increase, while the length of B will decrease, then increase, reaching it's minimum at [math]\theta=25^{\circ}[/math]. This also implies that [math]\alpha[/math] will increase in size, then decrease, hitting its maximum at the same spot where [math]\beta[/math] is at a minimum.

Testing for a quadratic fit:

In[6]:= quad1=Fit[data1,{1,n,n^2},n]

Out[6]= 4.89326 +0.324556 n-0.0000129154 n^2

The quadratic fit does not work since it's first derivative

[math]\theta'\approx \Delta \theta=0.324556 -0.0000258\ n[/math]

does not give the same spacing between low and high values of n as seen in the calculated data. The coefficient near the zeroth order for the derivative will have to be smaller than .31 to find a correlation that would agree with calculated values.

Testing for a polynomial of degree 3 fit:

In[7]:= polynomial1=Fit[data1,{1,n,n^2,n^3},n]

Out[7]= 4.93253 +0.297371 n+0.000566298 n^2-3.04016*10^-6 n^3

This fit best matches the data found in the calculated data

[math]\theta\equiv 4.93253 +0.297371 n+0.000566298 n^2-3.04016 \times 10^{-6} n^3[/math]

As discussed earlier, taking the 1st derivative of this function will give us the spacing of the bins as a function of wire number.

[math]\theta '\equiv 0.297371 + 0.0011326 n - 9.12048 \times 10^{-6} n^2[/math]

The derivative of this function then will tell us where the bin spacing is at a minimum and a maximum.

[math]\theta ''\equiv 0.0011326 - 0.000018241 n[/math]

[math]0.0011326 - 0.000018241 n=0\Rightarrow Bin\ Spacing\ Maximum\ at\ n=62.0909[/math]

This is near the point of the plane where [math]\theta=25^{\circ}[/math] as seen in the figure above.

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