Deconvolution of the energy loss function of the KATRIN experiment

0.99 MB·

by  Volker Hannen

Deconvolution of the energy loss function of the
KATRIN experiment
V. Hannen, I. Heese, C. Weinheimer
7 Institut fu¨r Kernphysik, Westf¨alische Wilhelms-Universit¨at Mu¨nster,
1 Wilhelm-Klemm-Str. 9, 48149 Mu¨nster, Germany
0
2 A. Sejersen Riis

n
Department of Physics and Astronomy, University of Aarhus, Ny Munkegade, DK-8000
a
J Aarhus C, Denmark

7
K. Valerius
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Institut fu¨r Kernphysik, Karlsruher Institut fu¨r Technologie,

] Hermann-von-Helmholtz-Platz 1, 76344 Eggenstein-Leopoldshafen, Germany
t
e
d
-
s
n
i
.
s Abstract
c
i The KATRIN experiment aims at a direct and model independent determi-
s
y nation of the neutrino mass with 0.2 eV/c2 sensitivity (at 90% C.L.) via a
h measurement of the endpoint region of the tritium beta-decay spectrum. The
p main components of the experiment are a windowless gaseous tritium source
[
(WGTS), differential and cryogenic pumping sections and a tandem of a pre-

1 and a main-spectrometer, applying the concept of magnetic adiabatic collima-
v tion with an electrostatic retardation potential to analyze the energy of beta
6
decay electrons and to guide electrons passing the filter onto a segmented sili-
6
con PIN detector.
0
8 One of the important systematic uncertainties of such an experiment are due
0 to energy losses of β-decay electrons by elastic and inelastic scattering off tri-
. tiummoleculeswithinthesourcevolumewhichaltertheshapeofthemeasured
1
0 spectrum. To correct for these effects an independent measurement of the cor-
7 responding energy loss function is required. In this work we describe a decon-
1 volution method to extract the energy loss function from measurements of the
:
v response function of the experiment at different column densities of the WGTS
i using a monoenergetic electron source.
X
r Keywords: Neutrino mass, electron scattering, deconvolution
a
Email address: [email protected](V.Hannen)
Preprint submitted to Journal of Astroparticle Physics January 30, 2017
1. Introduction
TheKArlsruheTRItiumNeutrino(KATRIN)experimentaimsatdetermin-
ing the neutrino mass in a model independent way from the kinematics of tri-
tium β-decay. The observable in this case is an ”average electron anti-neutrino
mass”givenbytheincoherentsumofneutrinomasseigenstatesweightedbythe
squaredelementsofthemixingmatrix. TheexperimentcombinesaWindowless
Gaseous Tritium Source (WGTS) and a high resolution electrostatic retarding
spectrometer(MAC-Efilter)tomeasurethespectralshapeofβ-decayelectrons
close to the endpoint energy at 18.6 keV with an unprecedented precision. KA-
TRIN’s sensitivity to the neutrino mass will be 0.2 eV/c2 (at 90% C.L.) after 3
years worth of data taking [1]. An observed mass signal of 0.35 eV/c2 will have
a 5σ significance at the expected level of statistic and systematic uncertainties.
In order to reach the desired sensitivity, all systematic effects of the measure-
ment must be well under control with the major systematic uncertainties being
allowed to contribute no more than ∆m2 = 0.0075 eV2/c4 to the systematic
error budget.
An overview of the KATRIN experiment is shown in figure 1. The experi-
ment starts with the WGTS where molecular T gas is injected at the center of
2
(a) (b) (c) (d) (e) (f) (g)
Figure 1: Overview of the KATRIN experiment. The main components are: (a) calibration
and monitoring system, (b) windowless gaseous tritium source, (c) differential and (d) cryo-
genic pumping sections, (e) pre-spectrometer, (f) main spectrometer, (g) detector system.
Overalllengthca. 70m.
the source and removed at both ends by turbo-molecular pumps. The T gas
2
is kept at a constant temperature of 30 K within the source that is operated
at a column density of 5·1017cm−2. The operational parameters of the source
cryostataremonitoredbyacomplexsensornetworkandadedicatedcalibration
andmonitoringsectionattherearofthesourcesystem[2]. About1010 β-decay
electronsareemittedpersecondintotheacceptedforwardsolidanglewithpitch
angles less than θ =51◦ and are guided magnetically through the transport
max
section to the spectrometer tandem consisting of pre- and main-spectrometers.
Thetaskofthetransportsectionmadeupofadifferentialpumpingsectionand
acryo-pumpingsectionistosuppresstheflowofT moleculesintothedirection
2
ofthespectrometersbyatleastafactorof1014 inordertoreduceexperimental
background from tritium decays within the spectrometers. A first energy dis-
crimination is performed by the pre-spectrometer which rejects the low energy
part of the β spectrum (up to 300 eV below the endpoint) and thereby reduces
therateofelectronsgoingintothemainspectrometertoapproximately103s−1.
2
Like the pre-spectrometer the main spectrometer operates as a so-called MAC-
E filter [3] and has the task to perform a precise energy analysis of the decay
electrons.
In a MAC-E filter electrons are guided magnetically against an electrostatic
retardation potential that can only be surpassed by electrons with sufficiently
highlongitudinalenergywithrespecttotheelectricfield. Herethelongitudinal
energy is given by E = E · cos2θ with E being the kinetic energy of
(cid:107) kin kin
the electron and θ being the angle between electron momentum and magnetic
field direction. The transverse energy is accordingly given by E = E ·
⊥ kin
sin2θ. The spectrometer acts as a high pass filter with a transmission function
describingtheobservedelectronrateasafunctionoftheelectronsurplusenergy
(see section 3.1). To reduce the amount of transversal energy of the electrons
that is not analyzed by the spectrometer, the technique of magnetic adiabatic
collimationisused. Theideaisthatthemagneticguidingfielddropsbyseveral
orders of magnitude from the entrance of the spectrometer to the analyzing
plane, where the electric potential reaches its maximum. If the gradient of the
magnetic field is small enough, such that the field is approximately constant
alongonecyclotronloopoftheelectronmovement,themagneticmomentofthe
cyclotron motion µ = E /B (non-relativistic) is constant, and as B drops the
⊥
transversalenergyoftheelectronsisconvertedintolongitudinalenergyE that
(cid:107)
can be analyzed by the spectrometer. By varying the electric potential of the
spectrometeritisthenpossibletoscantherelevantregionaroundtheendpoint
energyoftritiumβ-decayandaccumulateaspectrum. Electronswithsufficient
energy to pass the spectrometer are finally detected by a 148 pixel silicon PIN
detector [4] at the end of the setup.
Among the main systematical uncertainties of the experiment are energy
losses from inelastic scattering of electrons in the source, fluctuations of the
source density, fluctuations of the spectrometer analyzing potential, uncertain-
tiesinthetransmissionfunctionanduncertaintiesinthefinalstatedistribution
of the daughter molecules left after the decay reaction. A sophisticated cal-
ibration and monitoring system is being set up to keep the aforementioned
systematiceffectsundercontrol. Whilethereissomeinformationontheenergy
loss of 18.6 keV electrons in gaseous tritium or quench condensed deuterium
from the former neutrino mass experiments in Troitsk and Mainz [5], precise
experimental information on energy losses of electrons with energies near the
endpoint of the tritium β spectrum are only available for molecular hydrogen
as target gas [6, 7].
Ameasurementoftheenergydifferentialscatteringcrosssectionof18.6keV
electronsoffmoleculartritiumisthereforehighlydesirable. Suchameasurement
can be performed using a monoenergetic source of electrons mounted upstream
of the WGTS to determine the response function of the overall experiment at
different column densities of the source. A deconvolution method suitable to
extracttheenergylossfunctionfromthedatawillbepresentedinthefollowing
sections.
Once the energy loss function for tritium is known with sufficient accuracy,
thesamemeasurementsetupcanbeusedforanindependentcheckofthecolumn
3
density of the WGTS during intervals between the regular measurement cycles
of the KATRIN experiment [1].
2. Energy loss function
The processes contributing to the energy loss of electrons traversing the
molecular tritium gas within the WGTS are excitation of rotational and vi-
brational states of the T molecules, excitation of electronic molecular states,
2
dissociation and ionization of the molecules.
Aseev et al. [5] report on measurements of energy losses of electrons in
gaseous tritium and in quench condensed deuterium films. Because of the
limited energy resolution of a few eV the shape of the energy loss spectrum
was not directly extracted from the data in their analysis, but approximated
by a Gaussian representing electronic excitations and dissociation and a one-
sided Lorentzian curve representing the continuum caused by ionization of the
molecules. The parameters of the two functions were then adapted to fit the
observedintegralenergyspectraobtainedwithan18.6keVmono-energeticelec-
tron source for gaseous tritium or from 17.8 keV mono-energetic conversion
electrons from a 83mKr film covered by various thicknesses of D absorbers.
2
In both cases, energy losses caused by rotational and vibrational excitations
of the molecules without electronic excitation could not be resolved and were
neglected.
More detailed information is available for the scattering of 25 keV electrons
from molecular hydrogen gas [6, 7] where direct measurements of the energy
loss function with resolutions down to 40 meV have been performed. This in-
formation about the scattering of electrons from molecular hydrogen has been
implemented into a computer code by F. Glu¨ck [8] that can be used in sim-
ulations to generate energy losses ∆E and scattering angles ∆ϕ in individual
scattering events. The spectral shape produced with this routine is shown in
figure 2. It is used in a toy Monte Carlo simulation of the WGTS to evaluate
the deconvolution methods described in the following sections.
The probability for an electron of kinetic energy E to lose a specific amount
of energy ∆E in a single scattering event is described by the differential energy
loss function dσ . For our purpose, we normalize the function by the total
d∆E
inelastic scattering cross section σ , obtaining1
tot
1 dσ (cid:90) E/2
f(∆E)= · with f(∆E)d∆E=1. (1)
σ d∆E
tot 0
The total inelastic scattering cross section for 18.6 keV electrons off gaseous
tritiumisgivenbyσ (T )=(3.40±0.07)·10−18 cm2 [5]. Theabovementioned
tot 2
code by F. Glu¨ck [8] for scattering of 18.6 keV electrons off hydrogen gives a
total inelastic cross section of σ (H )=3.7·10−18 cm2.
tot 2
1The integral over the energy losses runs up to E/2 since the incoming electron and the
secondary electron in an ionisation process (assuming E is larger than twice the ionisation
energy)areidenticalquantumparticles.
4
0.8 100
0.4
sum
0.7
section0.6 00..32 ution10-1 einlaeslatisct,i c∆, ϕ∆eϕl=in3el.=3◦0.5◦
alized cross 000...435 00..0111 13 15 17 gular distrib1100--32
m0.2 n
or a10-4
n
0.1
0.0 10-5
0 10 20 30 40 50 0 2 4 6 8 10
∆E [eV] scattering angle [deg]
Figure2: Left: Normalizedenergylossfunctionf(∆E)forscatteringoffhydrogenaccording
to [8] (bin width 0.1 eV). Right: corresponding angular distribution for elastic and inelastic
scattering,normalizedto1.
3. Deconvolution method
In the following sections we describe suitable mathematical methods to ex-
tract the energy loss function of 18.6 keV electrons in gaseous tritium from a
series of measurements of the overall response function of the experiment at
different column densities of the WGTS.
3.1. Response function
1.2
y
t
bili1.0
a
b
o0.8
r
p
n 0.6
o ρd=0
si
mis0.4 ρ1d
s ρ d
tran0.2 ρ23d
0.0
0 10 20 30 40 50
E qU [eV]
−
Figure3: SimulatedresponsefunctionsoftheKATRINexperimentfordifferentcolumnden-
sities.
Figure 3 displays simulated response functions of the KATRIN experiment
at different column densities ρd = 0 and ρ d < ρ d < ρ d assuming a mono-
1 2 3
energeticelectronsourcewithnarrowangularemissioncharacteristics,i.e. pitch
5
angles w.r.t. the magnetic field lines θ ≤ O(1◦). Shown is the transmission
probability as a function of the nominal surplus energy E = E −qU of the
s
electrons, i.e., the difference between the setpoint energy E of the electron
source and the retardation potential qU of the main spectrometer.
The response function at non-zero column density of the tritium source is
givenbyasummationovercontributionscorrespondington-fold(i.e. no,single,
double, etc.) scattering of the electrons within the tritium source, weighted by
the probabilities P for n-fold scattering
n
R(E )=P ·T (E )+P ·T (E )⊗f(∆E)+P ·T (E )⊗f(∆E)⊗f(∆E)+... .
s 0 e s 1 e s 2 e s
(2)
HereT (E )istheexperimentaltransmissionfunctionofthemainspectrometer
e s
for the given electron source and f(∆E) the sought after energy loss function
neglecting small scattering angles. Given that we only consider a small energy
intervalofupto50eVbelowtheendpointenergyof18.6keVoftheβspectrum,
the dependence of the energy loss function f(∆E) on the kinetic energy of
the electrons can be neglected. The scattering probabilities are normalized
such that (cid:80)∞ P = 1. The experimental transmission function T (E ) is
n=0 n e s
determined from a measurement of the response function R(E ) with an empty
s
tritium source and hence without scattering. It is then equal to the analytical
transmission function of the spectrometer T(E ) convolved with a function S
s e
describing the energy spread and angular distribution of the electron source
T (E )= R(E )| =T(E )⊗S . (3)
e s s ρd=0 s e
In the simulations, the energy spread of the source is described by a Gaussian
smearing of the energy setpoint with a width of σ = 0.2 eV. It is assumed
e
that the electron source has a small angular divergence with starting angles
θ ≤ 0.5◦ := θ relative to the magnetic field direction at its location at
e e,max
the rear end of the WGTS. Within this narrow cone, the emission angles of the
electrons are assumed to be isotropically distributed. Suitable electron sources
that emit single electrons at adjustable total energy and adjustable emission
anglehavebeendevelopedandtestedwithintheKATRINcollaboration[9,10].
The above mentioned numerical values for the energy and angular spread are
compatiblewiththecharacteristicsofthephoto-electronsourceusedduringthe
commissioning of the KATRIN main spectrometer [11].
The analytical transmission function of the spectrometer T(E ) for such a
s
source is given by the following relation [5]

0 for E <qU
 1−(cid:112)1−E−qU Be
T(Es)= 11−(cid:113)1−E⊥,AEE,maBxABBAe ffoorr qqUU +≤EE ≤qU +<EE⊥,,A,max (4)
⊥,A,max
where B is the magnetic field at the electron source, B the magnetic field
e A
at the analyzing plane of the spectrometer and E⊥,A,max =Esin2(θe,max)BBAe is
6
the maximum remaining transversal energy component in the analyzing plane.
Defining
(cid:15) (E ) = T (E )
0 s e s
(cid:15) (E ) = T (E )⊗f(∆E) (5)
1 s e s
(cid:15) (E ) = T (E )⊗f(∆E)⊗f(∆E)
2 s e s
...
as the n-fold scattering functions we can rewrite equation 2 to obtain
R(E )=P ·(cid:15) (E )+P ·(cid:15) (E )+P ·(cid:15) (E )+... . (6)
s 0 0 s 1 1 s 2 2 s
If we manage to determine the single scattering function (cid:15) (E ) from measured
1 s
response functions we can, with the knowledge of T (E ), extract the energy
e s
loss function f(∆E) using suitable deconvolution methods.
3.2. Scattering probabilities
ThemeanfreepathoftheelectronswithinthetritiumgasinsidetheWGTS
can be expressed in terms of a mean free column density (ρd) which the
free
electrons pass before an interaction and which is calculated taking the inverse
ofthetotalscatteringcrosssection(ρd) =1/σ . Theactualcolumndensity
free tot
seenbyanelectrontraversingtheWGTSatanangleθrelativetothesymmetry
axis, i.e. the magnetic field axis, is given by ρd/cosθ. Neglecting possible
scattering angles ∆ϕ in the scattering processes for the moment, the mean
number of expected scatterings is
ρd ρdσ µ
µ(θ)= = tot = 0 . (7)
(ρd) cosθ cosθ cosθ
free
The probability for an n-fold scattering is given by a Poissonian distribution:
µn(θ)
P (µ(θ))= exp(−µ(θ)) with n=0, 1, 2, ... . (8)
n n!
We have to take into account that electrons generated by the electron source
followanangulardistributionwhichis,forourpurpose,assumedtobeisotropic
within a narrow interval between 0◦ ≤ θ ≤ θ . If we further take into
e e,max
accountthatthemagneticfieldatthelocationoftheelectronsourcewillbelower
than the field within the WGTS, the starting angles have to be transformed
according to
(cid:32) (cid:114) (cid:33) (cid:114)
B B
θ =arcsin sinθ · WGTS ≈θ · WGTS , (9)
e B e B
e e
resultinginanglesθ ofelectronmomentarelativetothemagneticfielddirection
within the WGTS.
To obtain average scattering probabilities we weigh the values from equation 8
7
with g(θ) = sinθ, corresponding to the probability function of an isotropic
distribution, integrate over the given range of angles and normalize
(cid:90) θmax (cid:46)(cid:90) θmax
P (µ ) = g(θ)P (µ(θ))dθ g(θ)dθ (10)
n 0 n
0 0
(cid:90) θmax (µ /cosθ)n (cid:46)
= sinθ· 0 exp(−µ /cosθ)dθ (1−cosθ ).
n! 0 max
0
Equation 10 delivers approximate values for the scattering probabilities, as it
does not take into account changes in the direction of the electrons during
scatterings. Secondly this equation also assumes a homogeneous distribution
of tritium molecules in transverse direction within the WGTS. Compared to
scattering probabilities extracted from the simulations accounting for scatter-
inganglesinelasticandinelasticscatteringdescribedinsection4thedeviations
totheresultscalculatedwith10werefoundtobeonthe<10−3 level,however.
The small difference is due to the fact, that the scattering angles for inelastic
scattering in our energy loss range of interest (∆E < 50 eV) are strongly for-
ward peaked with a mean of ∆ϕ=0.5◦ and elastic scattering is a subdominant
process (see figure 2, right).
Toobtainmoreprecisevaluesforthescatteringprobabilities,adetailedsimula-
tion using a 3-dimensional description of the column density within the WGTS
is required, which is beyond the scope of this paper.
3.3. Extraction of the single scattering function
In order to determine the single scattering function (cid:15) (E ) we have to per-
1 s
formmeasurementsoftheresponsefunctionR(E )atdifferentcolumndensities.
s
Neglecting multiple scattering events with more than three interactions of the
electrons with the tritium gas inside the WGTS2, we can set up a system of
linear equations for measurements at three column densities labeled a, b and c:
Ra(E )−Pa·T (E ) = Pa·(cid:15) (E )+Pa·(cid:15) (E )+Pa·(cid:15) (E ),
s 0 e s 1 1 s 2 2 s 3 3 s
Rb(E )−Pb·T (E ) = Pb·(cid:15) (E )+Pb·(cid:15) (E )+Pb·(cid:15) (E ),
s 0 e s 1 1 s 2 2 s 3 3 s
Rc(E )−Pc·T (E ) = Pc·(cid:15) (E )+Pc·(cid:15) (E )+Pc·(cid:15) (E ), (11)
s 0 e s 1 1 s 2 2 s 3 3 s
which we can write as a matrix equation:
 Pa Pa Pa 
1 2 3
R(cid:126) −P(cid:126)0·Te(Es)=P·(cid:126)(cid:15) with P= P1b P2b P3b  . (12)
Pc Pc Pc
1 2 3
Taking higher scattering orders into account would require additional measure-
ments at further non-zero column densities and would increase the dimension
2The probability of a 4-fold scattering process with the maximum energy loss under con-
sideration∆E<50eVisbelow8%atthemaximumcolumndensityof5·1017 cm−2 usedin
thesimulations.
8
of the system of linear equations to be solved. Whether the inclusion of only
threescatteringordersprovidessufficientlyaccurateresultswillbeevaluatedin
section 4.3.
Multiplying with the inverse of P, which is calculated using the Gauss-
Jordan algorithm from the ROOT software package [12], we obtain
(cid:16) (cid:17)
(cid:126)(cid:15)=P−1· R(cid:126) −P(cid:126) ·T (E ) (13)
0 e s
from which we can calculate the single scattering function (cid:15) (E ).
1 s
3.4. Deconvolution of the energy loss function
As described in section 3.1, the single scattering function is the result of the
convolution of the experimental transmission function of the spectrometer with
the energy loss function. This convolution is calculated taking the integral
(cid:90) E/2
(cid:15) (E )=T (E )⊗f(∆E)= T (E −∆E)f(∆E)d∆E. (14)
1 s e s e s
0
In our case, where the values of the functions in question are only known at N
equallydistributeddiscretemeasurementpointsdefinedbytheappliedretarda-
tion voltage U , the integral is replaced by a sum
i
N−1
(cid:88)
(cid:15) (E−qU )= T (E−qU −∆E )f(∆E ). (15)
1 i e i j j
j=0
The latter equation can be rewritten in N ×N matrix form
(cid:15)(cid:126) =T ·f(cid:126), (16)
1 e
where the T matrix is constructed from the discrete transmission function
e
T (E =E−qU ) as3
e s,i i
 
T (E ) 0 ··· 0
e s,0
 Te(Es,1) Te(Es,0) 0 ··· 0 
 
T = Te(Es,2) Te(Es,1) Te(Es,0) 0 ··· 0  . (17)
e  . . . . . 
 . . . . . 
 . . . . . 
T (E ) T (E ) ··· T (E )
e s,N−1 e s,N−2 e s,0
One could now try to solve equation 16 by multiplying with the inverse of
the T matrix. The latter, however, is close to being singular and cannot
e
easily be inverted numerically. We therefore have to apply more sophisticated
methodstodeconvolvetheenergylossfunctionfromthematrixequation. Inthe
following two methods are applied to the problem: the so-called Singular Value
Decomposition (SVD) [13] and the iterative Stabilized Biconjugate Gradient
method [14].
3The zeroes in the right upper corner of Te are caused by the transmission condition
E−qUi−∆Ej ≥0.
9
3.4.1. Singular Value Decomposition
The Singular Value Decomposition (SVD) is a method to deal with systems
of linear equations given by a matrix equation A·(cid:126)x=(cid:126)b that are either singular
or numerically very close to singular and is able to provide useful, although not
necessarily unambiguous, solutions to the given problem. It is based on the
theoremthatanyM×N matrixAwithM ≥N canbewrittenasaproductof
an M ×N column-orthogonal matrix U, an N ×N diagonal matrix W whose
elementsaretheso-calledsingularvaluesw ≥0,andthetransposeofanN×N
i
orthogonal matrix V [13]:
 
w 0
1
A=U·W·VT =U· w2 ·VT . (18)
 ... 
0 w
N
As U and V are orthogonal the inverse of equation 18 can be written as
A−1 =V·W−1·UT =V·[diag(1/w )]·UT . (19)
i
Problems arise when some of the singular values w are either zero or so small
i
that their values are dominated by numerical rounding errors. Using the SVD
method it is still possible to construct an approximate solution vector (cid:126)x that
will minimize the residual r given by
r ≡|A·(cid:126)x−(cid:126)b| . (20)
ForthatpurposealldiagonalelementsofW−1 wherethesingularvaluesw are
i
below a chosen threshold value w are set to zero, thereby removing infinite
thr
or problematically large matrix elements. The matrix constructed using the
modified diagonal matrix W˜ −1 is the so-called pseudoinverse matrix A˜−1, and
the solution vector (cid:126)x is then given by
(cid:126)x≈A˜−1·(cid:126)b=V·W˜ −1·UT·(cid:126)b , (21)
which, translatedtoouroriginalproblemofdeconvolutingtheenergylossfunc-
tion from the measured single scattering function (see equation 16), becomes
f(cid:126)≈T˜−1·(cid:15)(cid:126) =V·W˜ −1·UT·(cid:15)(cid:126) . (22)
e 1 1
Whatremainstobesettledistheoptimalthresholdvaluew forsuppression
thres
oftheproblematicsingularvalues. Thiscanonlybedeterminedbyinvestigating
the influence of the deconvolved energy loss function on the extracted neutrino
mass values in simulated neutrino mass runs of the KATRIN experiment. Such
a study, applying a toy Monte Carlo simulation of the experiment, is presented
in section 4.
10