Eur. Phys. J. C (2018) 78:212 Page 3 of 14 212
2.2 Time-of-flight spectroscopy
The idea of using time-of-flight (TOF) spectroscopy for
a measurement of the light neutrino mass is explained in
detail in Ref. [34]. In the following, we will recapitulate the
approach briefly and explain the motivations for investigat-
ing this technique for a keV-scale sterile neutrino search as
well.
In contrast to the standard mode of operation, as described
in the last section, TOF spectroscopy allows to measure not
only the count-rate, but a full TOF spectrum at a given retard-
ing potential qU . The TOF as a function of the energy is
given by integrating the reciprocal velocity over the center
of motion, which we will assume for simplicity to be on the
z-axis,
T (E,ϑ) =
dz
1
v
=
z
stop
z
start
dz
E + m
e
c
2
− qΔU(z)
p
(z) · c
2
, (4)
where E and ϑ are the initial kinetic energy and polar angle
of the electron, respectively. z
start
and z
stop
are the positions
on the beam axis between which TOF is measured, ΔU(z) is
the potential difference as a function of position z and p
(z)
the parallel momentum. By assuming adiabatic conservation
of the magnetic moment, p
(z) can be expressed analytically
as a function of the potential ΔU (z) and magnetic field B(z)
(derivation see Ref. [34]). If these are known, the integral in
Eq. (4) can be solved numerically.
Since the TOF is a function of the energy, the β-spectrum
can be transformed into a TOF spectrum dN/dτ ,giventhe
initial angular distribution of the β-decay electrons. A fea-
ture in the β-spectrum such as a sterile neutrino contribution
would then also have a corresponding effect on the TOF spec-
trum if the retarding energy qU is sufficiently low. Like the
β-spectrum (2), the TOF spectrum can as well be expressed
as a superposition of a component with a heavy neutrino mass
m
h
and a light neutrino mass m
l
:
dN
dτ
(m
ν
e
) = sin
2
θ
dN
dτ
(m
h
) + cos
2
θ
dN
dτ
(m
l
). (5)
For each of these two components, the TOF spectrum can
then formally be obtained from the β-spectrum with neu-
trino mass m
l
and m
h
, respectively, using the transformation
theorem for densities [56]:
dN
dτ
=
ϑ
max
0
E
0
qU
dϑ dEg(ϑ)
dN
dE
(E,ϑ)δ
(
τ − T (E,ϑ)
)
, (6)
where g(ϑ) denotes the angular distribution and dN /dE(E,ϑ)
the response corrected energy spectrum, which itself is a
function of the β-spectrum (1) for a given neutrino mass. If
angular changes from inelastic scattering processes in the tri-
tium source are neglected, the angular distribution is approx-
imately independent from the energy spectrum and given by
isotropic emission
g(ϑ) =
1
2
sin(ϑ) (7)
within the angular acceptance interval given by the default
KATRIN field settings with ϑ
max
=
√
B
S
/B
max
= 50.77
◦
.
The response corrected energy spectrum dN /dE(E,ϑ) in
Eq. (6) is given in good approximation by the β-spectrum
(3), convolved with the inelastic energy loss function in the
tritium source,
dN
dE
(E|ϑ) =
dΓ
dE
⊗ f
loss
(E,ϑ)
= p
0
(ϑ) ·
dΓ
dE
+
∞
n=1
p
n
(ϑ) ·
dΓ
dE
⊗ f
n
(E) (8)
where the f
n
is the energy loss spectrum of scattering order
n which can be approximately defined via recursive convo-
lution through the single scattering energy loss spectrum f
1
.
This can be written as
f
n
= f
n−1
⊗ f
1
(n > 1). (9)
The probability p
n
that an electron is scattered n times
depends on the emission angle ϑ and is given by a Poisson
law
p
n
(θ) =
λ
n
(ϑ)
n!
e
−λ(ϑ)
. (10)
The average number of scattering processes λ is given in
terms of the column density ρd of the tritium source, the
mean free column density ρd
free
and the scattering cross sec-
tion σ
scat
as
λ(ϑ) =
1
0
dx
ρd · x
ρd
free
· cos ϑ
=
1
0
dx
ρd · x · σ
scat
cos ϑ
. (11)
Since the probability of n-fold scattering is a function of
the emission angle (10), the response corrected energy spec-
trum (8) itself becomes dependent on the angle. Note that
the scattering model is simplified, since angular changes in
collisions are neglected and the scattering probabilities are
averaged over a hypothetical uniform density profile in the
source. We would like to clarify that in our actual imple-
mentation the n-fold energy loss spectra are not generated
via convolution but via Monte Carlo, which yields, however,
equivalent results. Furthermore, using Eq. (4), the radial start-
ing position is always assumed to be r = 0, which is not the
case in KATRIN, but we do not expect significant changes in
the spectral shape for outer radii. For analysis of real exper-
imental data a fully realistic treatment would be necessary,
yet for a principle sensitivity study these approximations are
reasonable.
The benefits of a TOF measurement can be understood
from Fig. 2, where the TOF (4)asafunctionofE for dif-
ferent angles is shown. It can be seen that energy differences
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