Mispointing Calibration
The mispointing can be determined in a number of different ways. The most straightforward of these is to simply have the Alignment Group perform target and optical surveys that determine the exact position of all of the elements involved in the experiment. The survey reports then include the horizontal and vertical offsets of the spectrometers, which can be entered into one of the database files to correct the mispointing.
Unfortunately, this process is rather time-consuming, so it can only be used for experiments with few configuration changes. For experiments where many configuration changes are necessary, such as this one, another option must be used, such as using special mispointing runs. These runs can be taken after each configuration change using one of the fixed solid targets, such as the carbon foils or the beryllium oxide target. The necessary mispointing offsets can then be calculated from the data taken during the run.
Normally the carbon foils would be used for this process, in which the seven carbon foils are used as a target, and the center foil is focused upon during analysis. But, due to another one of the experiments in the SRC family of experiments, thirteen foils were used instead of the regular seven, which increased the difficulty in locating the center foil in the relevant histograms, so it was decided to use the single foil BeO target instead.
For each kinematic, then, a special optics run was taken using the BeO target. These runs, with additional information, are listed below. Note that the kinematics were not run in chronological order, so the run numbers below are not in an order one might expect.
| Kinematic | Run | Left Arm Angle | Right Arm Angle | # Events |
| 1a | 2077 | 22.5° | 12.5° | 75965 |
| 1b | 2281 | 22.5° | 12.5° | 77920 |
| 2 | 2152 | 12.5° | 12.5° | 64036 |
| 3 | 2210 | 38.5° | 12.5° | 188867 |
| 5 | 2402 | 30.0° | 18.0° | 61328 |
| 6 | 2359 | 13.5° | 18.0° | 18286 |
| 7 | 2371 | 46.0° | 18.0° | 59587 |
| 8 | 2479 | 31.5° | 22.0° | 51177 |
| 9 | 2506 | 21.0° | 22.0° | 13199 |
| 10 | 2532 | 41.5° | 22.0° | 44228 |
| 11 | 2423 | 14.5° | 22.0° | 17705 |
| 12 | 2448 | 48.0° | 22.0° | 46873 |
| 13 | 2549 | 37.5° | 20.5° | 37776 |
| 14 | 2562 | 34.5° | 21.0° | 53776 |
Kinematic 1 has two entries in this list because data was collected during two different time periods with movement of the spectrometers in between, requiring that the mispointing for that kinematic be calculated twice.
Target and Optical Survey
While it wasn't feasible to do a target and optical survey between each kinematic, there was one set of surveys performed prior to the experimental run. The survey reports can be downloaded from the Alignment Group's website or via the links below.
Optical Survey | Target Survey
From the optical survey, whose measurements are not necessarily tied to any specific reference frame, we find that prior to any initial spectrometer movement, the left arm spectrometer axis missed the ideal center of the target chamber by 2.31 mm upstream and 0.54 mm vertically up, while the right arm spectrometer axis missed the ideal center by 2.83 mm downstream and 0.66 mm vertically down.
In the individual spectrometer-based transport reference frames, which will be explained in the next section, we could record these values as:
LHRS: hsurv = 2.31 mm ± 0.5 mm, vsurv = −0.54 mm ± 0.5 mm
RHRS: hsurv = 2.83 mm ± 0.5 mm, vsurv = 0.66 mm ± 0.5 mm
In the survey reports, the horizontal offsets are not referred to as positive or negative, but in the transport frames as will be described later, "upstream" will effectively be positive for the LHRS, while "downstream" will be positive for the RHRS. Also, in both of these transport frames, vertically down is positive, as opposed to vertically up being positive as it is in the survey reports.
Geometry
Before moving on, it is helpful to stop and discuss the geometry of the systems involved, to better understand how the variables below are connected.
The mispointing process for each spectrometer involves the interaction of two different spatial orientations, the lab frame and the transport frame.
The lab frame consists of the z-axis pointing in the direction of the ideal beam, with its origin at the center of the target chamber, known as the ideal center. +z points downstream and −z points upstream. +y points vertically up, with −y pointing vertically down. The x-axis is the result of the cross product between the y and z axes; from above (with +z pointing to the right as in the diagram to the right) +x points up and −x down.
With this configuration, the x-z plane is parallel to the floor.
The transport frame is both rotated and translated from the lab frame. The z-axis of the transport frame lies along the spectrometer axis, with +z pointing toward the spectrometer and −z pointing away. Unlike the lab frame, however, in the transport frame the x-axis is the vertical axis, with +x pointing down and −x pointing up. The y-axis is defined by the cross product of the z and x axes; from above (with +z pointing to the right), +y points up and −y down.
One important result of this configuration is that for the left arm, +ytransport points upstream, while for the right arm,
+ytransport points downstream, which explains the situation with the signs on the hsurv variable in the previous section.Also, much like the lab frame, the y-z plane of the transport frame is parallel to the floor. This means that the ztransport axis is not directly parallel to the spectrometer axis. The z-axis is instead the projection of the spectrometer axis relative to the transport frame's y-z plane. That is, the z-axis of the transport frame is parallel to the floor, while the true spectrometer axis is not, as evidenced by its vertical offset.
One benefit of this particular arrangement is that, when viewed from above as in the diagram to the above-right, the vertical offsets have no effect, and the horizontal situation between the two frames can be handled as though it were two-dimensional.
With that geometry in mind, several variables can be defined.
Variable Definition
| Lab Frame | |
| xsurv | x-component of a vector from the ideal center to the center of the target, assumed to be 0 for the BeO target |
| ysurv | y-component of a vector from the ideal center to the center of the target, assumed to be 0 for the BeO target |
| zsurv | z-component of a vector from the ideal center to the center of the target, presumably given by the target survey |
| beamx | x-component of a vector from the ideal center to the true beamline (at z = 0), given by the analyzer variable rb.x |
| beamy | y-component of a vector from the ideal center to the true beamline (at z = 0), given by the analyzer variable rb.y |
| (no beamz) | |
| xoff | x-component of a vector from the ideal center to the origin of the transport frame, ultimate goal of the mispointing calibration |
| yoff | y-component of a vector from the ideal center to the origin of the transport frame, ultimate goal of the mispointing calibration |
| zoff | z-component of a vector from the ideal center to the origin of the transport frame, ultimate goal of the mispointing calibration |
| reactx | x-component of a vector from the ideal center to the interaction point in the target, given by the analyzer variables rpl.x and rpr.x |
| reacty | y-component of a vector from the ideal center to the interaction point in the target, given by the analyzer variables rpl.y and rpr.y |
| reactz | z-component of a vector from the ideal center to the interaction point in the target, given by the analzyer variables rpl.z and rpr.z, analogous to zsurv after offsets are applied |
| Transport Frame | |
| xtgt | x-component of a vector from the origin of the transport frame to the interaction point in the target, given by the analyzer variables exL.x and exR.x |
| ytgt | y-component of a vector from the origin of the transport frame to the interaction point in the target, given by the analyzer variables exL.y and exR.y |
| ztgt | not used |
| hoff | y-component of a vector from the origin of the transport frame to the ideal center (at z = 0), penultimate goal of the mispointing calibration, analogous to hsurv |
| voff | x-component of a vector from the origin of the transport frame to the ideal center (at z = 0), penultimate goal of the mispointing calibration, analogous to vsurv |
| Angles | |
| θs | angle between the lab frame and the transport frame, as seen from above |
| θ0 | angle between the ideal beamline and a line connecting the ideal center with the venier caliper, essentially the nominal angle of the spectrometer |
| L | distance between the ideal center and the venier calipers, assumed to be 9.9 m |
Offset Calculation
As mentioned above, the ultimate goal here is to calculate the three lab frame offsets (xoff, yoff, zoff), which themselves are calculated from the transport frame offsets (hoff, voff). The goal, then, is to calculate these offsets from the variables given.
In the diagram below, beamx, zsurv, and ytgt are all positive based on their respective frames. If counter-clockwise rotation is positive, then θs and θ0 are also positive.
Further, as hoff lies along the ytransport axis, it should also be positive in the diagram below.
Using the trigonometric identities, it can be shown that hoff can be calculated as
hoff = ytgt + zsurv sin θs − beamx cos θs
One problem with this equation is that it relies on θs, which isn't yet known. As a first-order approximation, θ0 can be used. Then, once a value for hoff has been found, it can be used to find a better approximation for θs, with the equation
θs = θ0 + sin−1(hoff / L)
One method, then, for calculating the values for hoff and θs is to use an iterative process, continually making substitutions until the values converge to their best values.
Another option is to directly substitute the equation for θs into the hoff equation, which results in a quadratic equation for hoff. Or, likewise, an equation for hoff can be substituted into the equation for θs, resulting in a quadration equation for θs.
No matter the method, once hoff and θs are known, they can be used to find the two horizontal lab frame offsets.
xoff = −hoff cos θs
zoff = hoff sin θs
zoff = hoff sin θs
The signs on the above equations come from the direction of the lab frame vectors associated with hoff. That is,
in the diagram above, if hoff is positive, then the origin of the transport frame is downstream of the ideal center,
and hoff points upstream. But xoff and zoff point from the ideal center to the
origin of the transport frame, so they point in the opposite direction. That makes xoff point in the negative direction
and zoff point in the positive direction (downstream), as defined by the lab frame, hence the signs.Like the horizontal offset, the vertical offset, voff can be calculated from the variables above, though the derivation is much simpler.
voff = beamy + xtgt
Since the horizontal planes of the two frames are parallel, the vertical offset is just a simple addition. If both variables are positive, beamy points up from the x-z plane of the lab frame, xtgt points down from the y-z plane of the transport frame, and voff is just the sum of the two.
Further, yoff will be exactly equal to voff, even though they point in opposite directions, as they're based on different reference frames, which allows them to both be positive in the directions that they point.
Spectrometer Angles
Before we go any further, it's important to bring up the spectrometer angles.
During the experiment, four cameras were set up to view the floor markings as the spectrometers were moved around the hall, two on each arm. The cameras closest to the pivot, the front cameras, had venier calipers attached to get a more exact floor reading.
The cameras furthest from the pivot, the back cameras, were there to help measure the mispointing of the arm relative to the pivot. That is, by noting what the front and back cameras were reading individually, a degree of mispointing could be determined.
Unfortunately, several problems plagued the camera process. First, the markings for the back cameras was not as detailed as the markings for the front, so the images captured by the back cameras are essentially worthless. Second, the camera capture software would often freeze up, and the images stored in the data stream for two of the kinematics are incorrect.
Nevertheless, for the kinematics in which the images are correct, the cameras are still useful in showing how exact the positioning of the spectrometers were.
Also, during the N-Delta experiment, three Doug-recommended procedures were followed to keep the spectrometer angle as simple as possible.
First, though the calipers had motors that could be used to adjust them, the calipers themselves were never moved relative to the cameras.
Second, rather than aiming for the center of the calipers, the "zero" point, a specific position on the calipers was used. For the left arm, that point was +29, and for the right arm, it was at -34.
Third, the spectrometer angles were kept to half-degrees. That is, rather than aiming for 18.2°, we used 18.0°. There was some concern that this could cause us to miss the reactions we were looking for, but we were assured that the angular acceptance of the spectrometers was large enough to accomodate this change.
By following these three procedures, we were told that our spectrometer angles would be almost exactly what we wanted without having to worry about fiddling with the calipers.
Assuming these procedures were indeed correct, the θ0 values for this calibration should be exactly as they were described, to a tenth of a degree.
 Kinematic 1a Right: 12.5°, Left: 25.5° |
 Kinematic 1b (incorrect) Right: 12.5°, Left: 25.5° |
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 Kinematic 2 (incorrect) Right: 12.5°, Left: 12.5° |
 Kinematic 3 Right: 12.5°, Left: 38.5° |
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 Kinematic 5 Right: 18.0°, Left: 30.0° |
 Kinematic 6 Right: 18.0°, Left: 13.5° |
 Kinematic 7 Right: 18.0°, Left: 46.0° |
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 Kinematic 8 Right: 22.0°, Left: 31.5° |
 Kinematic 9 Right: 22.0°, Left: 21.0° |
 Kinematic 10 Right: 22.0°, Left: 41.5° |
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 Kinematic 11 Right: 22.0°, Left: 14.5° |
 Kinematic 12 Right: 22.0°, Left: 48.0° |
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 Kinematic 13 Right: 20.5°, Left: 37.5° |
 Kinematic 14 Right: 21.0°, Left: 34.5° |
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Target Location
In order to properly calibrate the mispointing of the spectrometers using the BeO foil, it is necessary to first know where that foil actually is relative to the ideal center of the target chamber. According to the target survey, the BeO target, as well as the rest of the solid targets, is 15.32 mm upstream of the ideal center.
One way to test this positioning is to use the θs, hoff, and voff from the optical survey to calculate xoff, yoff, and zoff for a BeO run taken after the surveys but before the spectrometers were moved, such as with run 1201, a spot++ run.
A θs of 16.489°, an hoff of 2.31 mm, and a voff of -0.54 mm gives an xoff of -2.215 mm, a yoff of -0.54 mm, and a zoff of 0.656 mm. When those were used, the reconstructed target appeared at a reactz of -15.6 mm, approximately where the target survey said, indicating that the target survey and the optical survey are in agreement.
However, this does present a bit of a problem, as the target survey suggests that the center of the 4 cm LH2 target cell is a mere 0.06 mm upstream of the ideal center, while the 4cm LH2 reconstructed position clearly shows it to be approximately 6 mm upstream.
Further analysis indicates that all of the solid targets appear to be correctly centered at -15.32 mm, while the LH2 and LD2 targets are all approximately 6 mm upstream, not the fraction of a millimeter as written in the target surveys.
Offset Calculation
To calculate the offsets, code was written to go through each event in the calibration runs, assign the various beam and target variables, calculate the offsets, and store them in histograms. The histograms could then be fitted to a Gaussian curve, yielding the results below. All units are in millimeters or degrees.
| Kinematic | Run | L | zsurv | beamx | beamy | xtgt | ytgt | θ0 | θs | hoff | voff | xoff | yoff | zoff |
| LHRS | ||||||||||||||
| 1a | 2077 | 9900 | -15.32 | -3.81227 | 2.60288 | -2.64924 | 4.6731 | 25.5 | 25.5106 | 1.52442 | 0.0393538 | -1.37589 | 0.0393538 | 0.655582 |
| 1b | 2281 | 9900 | -15.32 | -3.84285 | 2.69468 | -2.70432 | 5.7368 | 25.5 | 25.5113 | 2.61641 | 0.0423368 | -2.35098 | 0.0423368 | 1.12017 |
| 2 | 2152 | 9900 | -15.32 | -3.84501 | 2.69833 | -2.67296 | 4.43006 | 12.5 | 12.3657 | 4.87021 | 0.0976965 | -4.75552 | 0.0976965 | 1.05207 |
| 3 | 2210 | 9900 | -15.32 | -3.83559 | 2.69392 | -2.70883 | 5.80877 | 38.5 | 38.4606 | -0.714683 | 0.0461911 | 0.557292 | 0.0461911 | -0.445293 |
| 5 | 2402 | 9900 | -15.32 | -3.83952 | 2.69194 | -2.69746 | 5.58369 | 30 | 30.0055 | 1.24891 | 0.0551383 | -1.09049 | 0.0551383 | 0.618673 |
| 6 | 2359 | 9900 | -15.32 | -3.84073 | 2.70194 | -2.68164 | 4.18905 | 13.5 | 13.7209 | 4.34018 | 0.0896211 | -4.22575 | 0.0896211 | 1.01945 |
| 7 | 2371 | 9900 | -15.32 | -3.84026 | 2.68627 | -2.69744 | 5.11492 | 46 | 16.1444 | -3.22391 | 0.0520981 | 2.24879 | 0.0520981 | -2.32695 |
| 8 | 2479 | 9900 | -15.32 | -3.78835 | 2.5235 | -2.57057 | 6.05016 | 31.5 | 31.6162 | 1.28767 | 0.0468762 | -1.08576 | 0.0468762 | 0.669747 |
| 9 | 2506 | 9900 | -15.32 | -3.85198 | 2.70403 | -2.69592 | 4.37087 | 21 | 21.0193 | 2.46491 | 0.052177 | -2.30708 | 0.052177 | 0.888025 |
| 10 | 2532 | 9900 | -15.32 | -3.84052 | 2.69362 | -2.69772 | 4.58993 | 41.5 | 41.743 | -2.6629 | 0.0452745 | 1.99269 | 0.0452745 | -1.7655 |
| 11 | 2423 | 9900 | -15.32 | -3.84119 | 2.70055 | -2.66621 | 4.37826 | 14.5 | 14.663 | 4.24567 | 0.0901928 | -4.12141 | 0.0901928 | 1.05867 |
| 12 | 2448 | 9900 | -15.32 | -3.84074 | 2.68825 | -2.70589 | 5.29946 | 48 | 48.0055 | -3.50603 | 0.0557607 | 2.33094 | 0.0557607 | -2.58976 |
| 13 | 2549 | 9900 | -15.32 | -3.85223 | 2.71412 | -2.72457 | 6.33884 | 37.5 | 37.4762 | 0.0714577 | 0.05275 | -0.0659175 | 0.05275 | 0.0476348 |
| 14 | 2562 | 9900 | -15.32 | -3.84366 | 2.68648 | -2.68651 | 5.91887 | 34.5 | 34.5114 | 0.412268 | 0.0641884 | -0.339902 | 0.0641884 | 0.227923 |
| RHRS | ||||||||||||||
| 1a | 2077 | 9900 | -15.32 | -3.81227 | 2.60288 | -2.56718 | 2.44221 | -12.5 | -12.3648 | 9.47271 | 0.177298 | -9.25511 | 0.177298 | -2.05874 |
| 1b | 2281 | 9900 | -15.32 | -3.84285 | 2.69468 | -2.71659 | 1.46048 | -12.5 | -12.3721 | 8.52632 | 0.0897959 | -8.31906 | 0.0897959 | -1.78016 |
| 2 | 2152 | 9900 | -15.32 | -3.84501 | 2.69833 | -2.74881 | -0.493299 | -12.5 | -12.357 | 6.57013 | 0.00939732 | -6.42367 | 0.00939732 | -1.33904 |
| 3 | 2210 | 9900 | -15.32 | -3.83559 | 2.69392 | -2.73135 | 1.67645 | -12.5 | -12.3608 | 8.71645 | 0.0827623 | -8.51087 | 0.0827623 | -1.85547 |
| 5 | 2402 | 9900 | -15.32 | -3.83952 | 2.69194 | -2.69211 | 4.39965 | -18 | -17.7068 | 12.7776 | 0.114348 | -12.1696 | 0.114348 | -3.93079 |
| 6 | 2359 | 9900 | -15.32 | -3.84073 | 2.70194 | -2.72198 | 4.33373 | -18 | -17.7926 | 12.6914 | 0.0877268 | -12.0575 | 0.0877268 | -3.91015 |
| 7 | 2371 | 9900 | -15.32 | -3.84026 | 2.68627 | -2.66995 | 4.58042 | -18 | -17.7626 | 12.9413 | 0.132668 | -12.3155 | 0.132668 | -3.9912 |
| 8 | 2479 | 9900 | -15.32 | -3.78835 | 2.5235 | -2.58995 | 2.93927 | -22 | -21.9184 | 12.1769 | 0.0854796 | -11.3083 | 0.0854796 | -4.5275 |
| 9 | 2506 | 9900 | -15.32 | -3.85198 | 2.70403 | -2.75264 | 2.74804 | -22 | -21.9208 | 12.027 | 0.0873014 | -11.1488 | 0.0873014 | -4.48187 |
| 10 | 2532 | 9900 | -15.32 | -3.84052 | 2.69362 | -2.72121 | 3.02801 | -22 | -21.9388 | 12.3445 | 0.0888009 | -11.45 | 0.0888009 | -4.59105 |
| 11 | 2423 | 9900 | -15.32 | -3.84119 | 2.70055 | -2.75976 | 2.7705 | -22 | -21.9163 | 12.0714 | 0.0778733 | -11.1924 | 0.0778733 | -4.48812 |
| 12 | 2448 | 9900 | -15.32 | -3.84074 | 2.68825 | -2.70238 | 3.11093 | -22 | -21.9146 | 12.3792 | 0.103533 | -11.4791 | 0.103533 | -4.59382 |
| 13 | 2549 | 9900 | -15.32 | -3.85223 | 2.71412 | -2.73416 | 2.0491 | -20.5 | -20.3782 | 11.0168 | 0.0884737 | -10.3271 | 0.0884737 | -3.83977 |
| 14 | 2562 | 9900 | -15.32 | -3.84366 | 2.68648 | -2.72903 | 2.35468 | -21 | -21.0026 | 11.441 | 0.0770575 | -10.6805 | 0.0770575 | -4.08142 |
Once the various offsets have been calculated, they can be entered into the "db_run.dat" database file, and the various runs can be replayed.
Results
To confirm these results, the three react variables can be used, where they are compared to the beam and surv variables. If the calibration is correct, then reactz should be equivalent to zsurv, and reactx and reacty should correspond to the respective beam variables.
| Kinematic | Run | beamx | beamy | zsurv | reactx | reacty | reactz |
| LHRS | |||||||
| 1a | 2077 | -3.81227 | 2.60288 | -15.32 | -3.88533 | 2.65963 | -15.2628 |
| 1b | 2281 | -3.84285 | 2.69468 | -15.32 | -3.90638 | 2.74677 | -15.3246 |
| 2 | 2152 | -3.84501 | 2.69833 | -15.32 | -3.90504 | 2.75054 | -14.9581 |
| 3 | 2210 | -3.83559 | 2.69392 | -15.32 | -3.91116 | 2.74694 | -15.2817 |
| 5 | 2402 | -3.83952 | 2.69194 | -15.32 | -3.90907 | 2.74132 | -15.2898 |
| 6 | 2359 | -3.84073 | 2.70194 | -15.32 | -3.90558 | 2.73705 | -15.1097 |
| 7 | 2371 | -3.84026 | 2.68627 | -15.32 | -3.90946 | 2.7451 | -15.3198 |
| 8 | 2479 | -3.78835 | 2.5235 | -15.32 | -3.87318 | 2.57638 | -15.2903 |
| 9 | 2506 | -3.85198 | 2.70403 | -15.32 | -3.92286 | 2.74678 | -15.1355 |
| 10 | 2532 | -3.84052 | 2.69362 | -15.32 | -3.9026 | 2.74862 | -15.3004 |
| 11 | 2423 | -3.84119 | 2.70055 | -15.32 | -3.89266 | 2.74198 | -15.1546 |
| 12 | 2448 | -3.84074 | 2.68825 | -15.32 | -3.91492 | 2.73473 | -15.2891 |
| 13 | 2549 | -3.85223 | 2.71412 | -15.32 | -3.91633 | 2.74143 | -15.2759 |
| 14 | 2562 | -3.84366 | 2.68648 | -15.32 | -3.90611 | 2.74523 | -15.3183 |
| RHRS | |||||||
| 1a | 2077 | -3.81227 | 2.60288 | -15.32 | -3.87644 | 2.66585 | -15.106 |
| 1b | 2281 | -3.84285 | 2.69468 | -15.32 | -3.89764 | 2.74406 | -15.1279 |
| 2 | 2152 | -3.84501 | 2.69833 | -15.32 | -3.90343 | 2.7552 | -15.1643 |
| 3 | 2210 | -3.83559 | 2.69392 | -15.32 | -3.88961 | 2.74698 | -15.167 |
| 5 | 2402 | -3.83952 | 2.69194 | -15.32 | -3.89027 | 2.74428 | -15.1914 |
| 6 | 2359 | -3.84073 | 2.70194 | -15.32 | -3.89366 | 2.76412 | -15.1623 |
| 7 | 2371 | -3.84026 | 2.68627 | -15.32 | -3.89297 | 2.73558 | -15.1849 |
| 8 | 2479 | -3.78835 | 2.5235 | -15.32 | -3.85506 | 2.57585 | -15.2002 |
| 9 | 2506 | -3.85198 | 2.70403 | -15.32 | -3.90851 | 2.74741 | -15.1469 |
| 10 | 2532 | -3.84052 | 2.69362 | -15.32 | -3.91155 | 2.7464 | -15.1691 |
| 11 | 2423 | -3.84119 | 2.70055 | -15.32 | -3.90493 | 2.75824 | -15.1878 |
| 12 | 2448 | -3.84074 | 2.68825 | -15.32 | -3.89118 | 2.73165 | -15.1674 |
| 13 | 2549 | -3.85223 | 2.71412 | -15.32 | -3.90947 | 2.76299 | -15.2622 |
| 14 | 2562 | -3.84366 | 2.68648 | -15.32 | -3.90042 | 2.73519 | -15.2547 |
As can be seen, the results are extremely close, indicating that the mispointing calibration has been done successfully.