Lecture 9.1: Textbook Revisited, and
Calibration Examples from Several Experiments
Prof. Luke A. CorwinPHYS 733
South Dakota School of Mines & Technology
Oct. 25, 2013
L. Corwin, PHYS 733 (SDSM&T) Lec. 9: Calibration Examples Oct. 25, 2013 1 / 27
Outline
1 Textbook Example Redux
2 Calibration Example from NOvANOvA Experiment OverviewCosmic Rays for Attenuation Calibration
3 Papers for Remaining Experiments
4 References
L. Corwin, PHYS 733 (SDSM&T) Lec. 9: Calibration Examples Oct. 25, 2013 2 / 27
Textbook Example Redux
Plot from Lecture 7.2
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Textbook Example Redux
Lingering Questions
x E/keV141 725222 1131366 1845
We were all able to agree that the slope of the line, which is theMCA channel width, was about 5 keV. We disagreed on theintercept, getting values scattered around 22 keV; we alsodisagreed with the book [2], which claims
m = 5.00 keV b = 18 keV, (1)
where E(x) = mx+ b. This has bothered me ever since.
L. Corwin, PHYS 733 (SDSM&T) Lec. 9: Calibration Examples Oct. 25, 2013 4 / 27
Textbook Example Redux
Attempting to Resolve the Discrepancy
Lesson 1: Every physicist needs a little Captain (Ahab)
I neglected to give you one of the data points in the book(carbon).
I realized that the proper way to perform the calibration is tofit a line to the data.
Two fitting options: Exact analytical solution or ROOTfitting
Lesson 2: Always use all good data.
x E/keV98 505141 725222 1131366 1845
L. Corwin, PHYS 733 (SDSM&T) Lec. 9: Calibration Examples Oct. 25, 2013 5 / 27
Textbook Example Redux
Exact Analytical Solution
For the exact analytical solution, I use the method of leastsquares from Bevington and Robinson [1], a book I highlyrecommend.
We are not given the uncertainty on E, so I assumed theuncertainties were equal and used the solution given in thatcase.
∆′ = N∑x2i − (
∑xi)
2 (2)
b = 1∆′
(∑x2i
∑yi −
∑xi∑xiyi) (3)
m = 1∆′
(N∑xiyi −
∑xi∑yi) (4)
L. Corwin, PHYS 733 (SDSM&T) Lec. 9: Calibration Examples Oct. 25, 2013 6 / 27
Textbook Example Redux
Results of Least Squares Method
Some analytical method was probably used in the bookbecause it was published in 1978, before I was born.
The results from my spreadsheet calculations of the leastsquares method:
m = 4.99431637829324 b = 18.9250887878733 (5)
The slope agrees to two significant figures with the book, butnot the three claimed in the book.
The intercept still disagrees at two significant figures.
What will ROOT say?
L. Corwin, PHYS 733 (SDSM&T) Lec. 9: Calibration Examples Oct. 25, 2013 7 / 27
Textbook Example Redux
#include "TMath.h"
#include "TH1F.h"
#include "TF1.h"
#include "TStyle.h"
void CMN_CalibDemo(){
TH1F* h1 = new TH1F("h1","Calibration Points",400,-0.5,399.5);
Float_t common_error = 0.1; //keV
h1->Fill(98,505);
h1->SetBinError(h1->FindBin(98),common_error);
h1->Fill(141,725);
h1->SetBinError(h1->FindBin(141),common_error);
h1->Fill(222,1131);
h1->SetBinError(h1->FindBin(222),common_error);
h1->Fill(366,1845);
h1->SetBinError(h1->FindBin(366),common_error);
h1->SetMarkerStyle(20);
TF1* myline = new TF1("myline","[0]*x + [1]",0,400);
myline->SetParNames("slope","intercept");
gStyle->SetOptStat(0);
gStyle->SetOptFit(1);
h1->Fit("myline","R");
h1->Draw("PE1");
}L. Corwin, PHYS 733 (SDSM&T) Lec. 9: Calibration Examples Oct. 25, 2013 8 / 27
Textbook Example Redux
L. Corwin, PHYS 733 (SDSM&T) Lec. 9: Calibration Examples Oct. 25, 2013 9 / 27
Textbook Example Redux
Comparing Our Results
Full Results from ROOT using TF1
EXT PARAMETER
NO. NAME VALUE ERROR
1 slope 4.99432e+00 4.89452e-04
2 intercept 1.89251e+01 1.12873e-01
Source Slope, Channel Width m Intercept bBook [2] 5.00 keV 18 keV
Least Squares 4.994316 keV 18.92508879 keVROOT with TF1 4.99432 keV 18.9251 keV
Table : ROOT and the analytical method agree with each other butdisagree with the book. So, what do we do now?
L. Corwin, PHYS 733 (SDSM&T) Lec. 9: Calibration Examples Oct. 25, 2013 10 / 27
Textbook Example Redux
Our Options
1 Ideally, we would ask the authors of the book how they madetheir calculations; however, since the book is 35 years old,that may not be possible.
2 We could explore and test other fitting methods to see if wecan reproduce their results.
3 Assign the discrepancies as a systematic uncertainty
σ2E = σ2
mx2 + σ2
b [1] σm = 0.006 keV σb = 0.93 keV (6)
σE(x = 1100) =√
(0.006 keV)2 · 11002 + (0.93 keV)2 (7)
= 6.32 keV (8)
We then need to decide whether this is an acceptable systematicor use option 2. Which uncertainty (σm or σb) do we need tofocus on reducing?
L. Corwin, PHYS 733 (SDSM&T) Lec. 9: Calibration Examples Oct. 25, 2013 11 / 27
Calibration Example from NOvA
Calibration of NOvA Relative Energy Scale
Basic Information
Experiment: NOvA (a.k.a. NOνA) = NuMI Off-axis νeAppearance
NuMI = Neutrinos at the Main Injector
Instrument Calibrated: Avalanche Photo Diodes (APDs)
Standard: Cosmic Ray Energy Deposits.
Sources
NOvA Document Database http:
//nova-docdb.fnal.gov/cgi-bin/DocumentDatabase/
Public Documents 6554, 7534, 7546, 7556, 9245
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Calibration Example from NOvA NOvA Experiment Overview
NOvA Experiment Overview
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Calibration Example from NOvA NOvA Experiment Overview
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Calibration Example from NOvA NOvA Experiment Overview
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Calibration Example from NOvA NOvA Experiment Overview
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Calibration Example from NOvA NOvA Experiment Overview
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Calibration Example from NOvA NOvA Experiment Overview
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Calibration Example from NOvA Cosmic Rays for Attenuation Calibration
Motivation
Especially in the Far Detector, the fibers are very long
They attenuate light on the way from the interaction point tothe APD
We want to know how many photoelectrons (PEs) wereproduced based on track location and how many PEs weredetected by the APD (relative calibration)
The APDs output a quantity called Access Deficit Charge(ADC), which we want to convert into the number of PEsgenerated (PECorr)
The conversion from PECorr to actual energy deposited iscalled absolute calibration, and we are not covering thattoday.
L. Corwin, PHYS 733 (SDSM&T) Lec. 9: Calibration Examples Oct. 25, 2013 19 / 27
Calibration Example from NOvA Cosmic Rays for Attenuation Calibration
Selecting Cells
Figure : Use only cells where both neighbors are also illuminated
From the track through the detector, we know the cells thatare lit and the angle of the track through the detector.Since the tracks can traverse different lengths (L) in the cell,we actually measure ADC/cm.cy ≡ cos(θ) = Ly/L⇒ L = Ly/cy
L. Corwin, PHYS 733 (SDSM&T) Lec. 9: Calibration Examples Oct. 25, 2013 20 / 27
Calibration Example from NOvA Cosmic Rays for Attenuation Calibration
Attenuation Length Review
I = I0e−xλ (9)
I is the intensity after light has traveled a distance x
I0 is the initial intensity
λ is the attenuation length.
L. Corwin, PHYS 733 (SDSM&T) Lec. 9: Calibration Examples Oct. 25, 2013 21 / 27
Calibration Example from NOvA Cosmic Rays for Attenuation Calibration
Functional Form
To assess the affect of attenuation on our scintillationphotons, we want to fit the response A as a function ofdistance from the center of the cell W .
NOvA calls the attenuation length Xs
In Class Exercise (may work in teams)
Given that the light has two paths to reach the APD, W = 0 atthe center of the cell, and the attenuation length above, what isthe functional form of the response of the APD A(W )?
L. Corwin, PHYS 733 (SDSM&T) Lec. 9: Calibration Examples Oct. 25, 2013 22 / 27
Calibration Example from NOvA Cosmic Rays for Attenuation Calibration
A(W ) = A0
(eW−L/2Xs + e−
3L/2+WXs
)+ C
L. Corwin, PHYS 733 (SDSM&T) Lec. 9: Calibration Examples Oct. 25, 2013 23 / 27
Calibration Example from NOvA Cosmic Rays for Attenuation Calibration
Figure : Example fit result from NDOS.
L. Corwin, PHYS 733 (SDSM&T) Lec. 9: Calibration Examples Oct. 25, 2013 24 / 27
Calibration Example from NOvA Cosmic Rays for Attenuation Calibration
Figure : ADC distributions for different bins of W before (left) andafter(right) calibration.
L. Corwin, PHYS 733 (SDSM&T) Lec. 9: Calibration Examples Oct. 25, 2013 25 / 27
Papers for Remaining Experiments
Papers for Remainder of Class
ICARUS: http://arxiv.org/abs/hep-ex/0311040
Super-Kamiokande:http://arxiv.org/abs/hep-ex/0005014
BaBar: http://arxiv.org/abs/physics/0601138
IceCube: Guest Appearance from Prof. Bai
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References
References I
[1] P. R. Bevington and D. K. Robinson.Data Reduction and Error Analysis for the Physical Sciences.WCB/McGraw-Hill, Madison, WI, 2nd. edition, 1992.
[2] W.-K. Chu, J. W. Mayer, and M.-A. Nicolet.Backscattering Spectrometry.Academic Press, New York, NY, USA, 1978.
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