appearance probability model of electron anti … reu presentation.pdf · n + p h + 2 γ the...
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APPEARANCE PROBABILITY
MODEL OF ELECTRON ANTI-NEUTRINOS
ACCOUNTING FOR DIFFERENT REACTOR
DISTANCES
Maria Veronica Prado
Advisors: Dr. Glenn Horton-Smith
Dr. Larry Weaver
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KamLAND Experiment
56 nuclear reactors and one detector
Detector is located on the island of Honshu, Japan
Each nuclear reactor contains Uranium 235 and 238 & Plutonium 239 and 241
Fission occurs:
57.1% from U 235
7.8% from U 238
29.5% from Pu 239
5.6% from Pu 241
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KamLAND Experiment
Source: http://kamland.lbl.gov/Pictures/kamland-ill.html
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KamLAND Experiment
n
moderator reactor with
U235
U238
Pu239
Pu 241
two new
elements
after
fission
beta
decay
n p + e + v e
n p + e + v e
The reactors:
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KamLAND Experiment
The detector:
Source: http://kamland.lbl.gov/Pictures/kamland-ill.html
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KamLAND Experiment
The Liquid Scintillator inside the detector contains C9
H12 (pseudocumene) and C12 H26 (dodecane)
Some of the anti-neutrinos coming from the reactors
collide with protons found in these molecules
Inverse beta decay
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KamLAND Experiment
The detector:
Inverse beta
decay
v + p n + e e +
Positron moves through the
LS losing KE as it ionizes
atoms
Ionization of
atoms
+ e
UV/Gamma
photon
unstable atom Process A
Gamma photons: 10 keV
Optical photons: 1-3 eV
γ
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KamLAND Experiment
H
H H
Fluorescence
The UV photon
hits one of the
molecules and is
absorbed
The detector:
γ γ
A visible photon is
emitted from the
molecule
Optical
photon
DETECTED
The visible photon is detected
by all the photomultiplier
tubes
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KamLAND Experiment
Meanwhile still in the detector…
e + e + _
The positron loses energy
and comes to a stop where
it annihilates with an
electron
γ
γ
Gamma
photons
Gamma photons
Compton scatter or go
through the
Photoelectric Effect
_
e
_
e
_
e
_
e _
e
_
e
_
e
_
e
_
e
_
e _
e
_
e
Go through
Process A
DETECTED
Even though there are 3 separate
signals in the PMTs, it is detected
as only one
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KamLAND Experiment
Simultaneously in the detector…
v + p n + e e +
n + p H + γ 2
The neutron bounces off from the
atoms in the LS and moves
slower & slower until it is
absorbed
Recoil
(Deuteron)
Go through
Process A
Or very
rarely n + p C + γ 13
The neutron interacts
with hydrogen (H)
from the LS The gamma photon compton
scatters or goes through the
photoelectric effect with the
atoms in the LS. It produces a
detected signal called the
delayed coincidence.
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KamLAND Experiment
Source: kamland.lbl.gov/Pictures/picgallery.html
The detector:
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Number of Counts
Our main equation:
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A simple derivation
Flux of the
anti-neutrino
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A simple derivation
where,
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Terms
The number of counts at each energy prompt
The flux of anti-neutrinos expected at the detector
The cross section of one proton that could interact with the anti-
neutrinos coming into the detector
Probability that an electron anti-neutrino will stay an
electron anti-neutrino by the time it reaches the detector
Probability of detecting a reaction from the
reactions that have occurred (due to experimental
error)
The number of reactions
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Events graph with what was observed in the detector
KamLAND events graph
http://www.awa.tohoku.ac.jp/KamLAND/4th_result_data_release/4th_result_data_release.html
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KamLAND appearance probability graph
http://arxiv.org/pdf/1009.4771v2.pdf
Appearance probability from an average reactor length
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Our theoretical research
Make a change of variables:
Q P
N
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Our theoretical research
For simplicity, we shall call this:
We need to minimize for N(Ep):
where,
N(Ep) has a Poisson distribution because of the rare amount of
interactions at the detector
what we want to find empirically
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Our theoretical research
By taking the derivative of and
setting it equal to zero, we get
where,
transformation of the Q matrix
Our observed values
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Small proof
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Why we want to do this
Prove neutrino oscillations and KamLAND’s conclusions
empirically
Gain knowledge about how neutrinos behave, which
could lead to a better understanding of dark matter
Gain knowledge about neutrinos to be able to control
nuclear reactors efficiently by monitoring neutrinos
that leave
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Forming the Q matrix
For ex:
• Test for as many l’s as possible, binning them
• If the lies between 1.8 MeV-10 MeV, then plug the values into
the Q equation
• If the lies outside of that range, it does not contribute to the
detector, so we input zero for that matrix element
• Obtain a different Q matrix for each reactor
• Superpose all the Q matrices
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Forming the Q matrix
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Our ‘no oscillations’ graph
our ‘no oscillations’ graph (without
taking into account certain small
factors)
KamLAND’s ‘no oscillations’ graph
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Setting up the test
where, appearance probability if there were no oscillations
C Y
where,
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Setting up the test
Where R is a matrix containing the
orthonormal eigenvectors for each
eigenvalue, and D is a diagonal
matrix containing all the eigenvalues
of C
Since,
Has the smallest eigenvalue
element equal to zero
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The Binning
The greater the counts per bin, the smaller
the relative error
As a result, approaches a Gaussian
Why bin the Ep’s?
Why bin the l’s?
More functions than unknowns
A higher sum in each l column will provide
for a smaller error
Why find the eigenvalues of C?
If product of eigenvalues is big, error is
small when inverting C
If difference is big, magnifies error
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Why test it this way
Accounting for bias by using N0 prime to
calculate V inverse instead of N1 prime:
- This method gives each element in N1
prime their corresponding importance
according to how many number of counts they
each contribute and, therefore, how much
data they contain
For example:
while
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Why omit the smallest eigenvalue?
Contains
background noise
Contains
inverse eigenvalues
The smaller the eigenvalue, the
more noise error it contributes
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Error
Background noise in the data N1 from the experiment
Approximation of l values due to the l binning in Q0
Biggest error contributors:
Create N’1true with a specific P(l)
Add randomized background noise to N’1true
Create 1000 different P(l)s, each using a different
randomized N’1observed
Find the average P(l) and its standard deviation to obtain
different error bars for each P(l) entry
Producing reasonable error bars for our test of specific P(l)s :
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Omitting the smallest eigenvalue
16 Ep bins,11 l bins
Smallest error bars so far, but not as small as were expected
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16 Ep bins,11 l bins
Smaller error bars
Omitting vs not omitting smallest eigenvalue
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Testing KamLAND’s N
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Comparing Chi Squares
Chi square of the N between our estimate and
the N observed:
6.68
Chi square of the P between our
estimate and the closest straight line
of 0.44 without taking into account covariance:
9.65
The above chi square with covariance:
67.95
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The Covariance Matrix
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Conclusions
Obtained appearance probabilities for
11 values of L/Eν without assuming an
average L
Appearance probability cannot be
constant
Predicted N matched KamLAND’s
observed N
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Acknowledgements
Dr. Horton-Smith & Dr. Weaver
Dr. Corwin
KSU HEP Department
Kansas State University
NSF