EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCH (CERN)

 CERN-LHCb-DP-2018-003 August 26, 2020

Monitoring radiation damage in the LHCb Tracker Turicensis

The LHCb Silicon Tracker group

C. Abellan Beteta1, M. Atzeni1, V. Battista2, A. Bursche9, B. Dey10, A. Dosil
Suarez4, C. Elsasser1, A. Fernandez Prieto4, J. Fu3, E. Graverini2, I.
Komarov2,b, E. Lemos Cid4, F. Lionetto1, A. Mauri7, A. Merli3, P.R. Pais6, E.
Perez Trigo4, M. del Pilar Peco Regales5,a, P. Stefko2, O. Steinkamp1, B.
Storaci1, M. Tobin8, A. Weiden1 .

1Physik-Institut, Universität Zürich, Zürich, Switzerland
2Institute of Physics, Ecole Polytechnique Fédérale de Lausanne (EPFL),
Lausanne, Switzerland
3Università degli Studi di Milano, Milano, Italy
4Instituto Galego de Física de Altas Enerxías (IGFAE), Universidade de
Santiago de Compostela, Santiago de Compostela, Spain
5Max-Planck-Institut für Kernphysik (MPIK), Heidelberg, Germany
6European Organization for Nuclear Research (CERN), Geneva, Switzerland
7Nikhef National Institute for Subatomic Physics, Amsterdam, Netherlands
8Chinese Academy of Sciences, Beijing, China
9Sezione INFN di Cagliari, Cagliari, Italy
10Institute of Particle Physics, Central China Normal University, Wuhan,
Hubei, China

anow at AGH University of Science and Technology, Krakow, Poland
bnow at Deutsches Elektronen-Synchrotron, Hamburg, Germany

This paper presents the techniques used to monitor radiation damage in the
LHCb Tracker Turicensis during the LHC Runs 1 and 2. Bulk leakage currents in
the silicon sensors were monitored continuously, while the full depletion
voltages of the sensors were estimated at regular intervals by performing
dedicated scans of the charge collection efficiency as a function of the
applied bias voltage. Predictions of the expected leakage currents and full
depletion voltages are extracted from the simulated radiation profile, the
luminosity delivered by the LHC, and the thermal history of the silicon
sensors. Good agreement between measurements and predictions is found.

Published in JINST 15, P08016

© CERN on behalf of the LHCb collaboration, license CC-BY-4.0.

##  1 Introduction

The LHCb experiment [1] at the LHC was designed to study decays of hadrons
containing beauty and charm quarks. Its main goal is to perform precision
measurements of $C\\!P$ violating observables, and to search for signatures of
potential physics beyond the Standard Model in rare decays of such hadrons.

The layout of the LHCb detector during the LHC Runs 1 and 2 is illustrated in
Fig. 1. The detector is a single-arm forward spectrometer that covers polar
angles from about 15 mrad to 250 mrad and consists of a vertex locator, a
tracking system comprising a single planar tracking station upstream of the
spectrometer dipole magnet and three tracking stations downstream of this
magnet, two Ring Imaging Cherenkov detectors, a calorimeter system comprising
electromagnetic and hadronic calorimeters, and a muon system. In the LHC Run
1, the Silicon Tracker (ST) comprised the tracking station upstream of the
dipole magnet, called Tracker Turicensis (TT), and the central part in each of
the three tracking stations downstream of the dipole magnet, called Inner
Tracker (IT); the outer part of the downstream tracking stations was formed by
a straw-tube detector. The tracking system has been dismantled in the context
of a substantial detector upgrade for the LHC Run 3. The TT will be replaced
by a new silicon strip detector, while the downstream tracker will consist of
scintillating fibre planes [2, *LHCb-TDR-015].

Figure 1: Layout of the LHCb detector: vertex locator (VELO), Ring-Image
Cherenkov counters (RICH1 and RICH2), tracking stations upstream (TT) and
downstream (T1-T3) of the spectrometer dipole magnet, scintillating pad
detector used for trigger purposes (SPD), pre-shower detector (PS) and
electromagnetic calorimeter (ECAL), hadronic calorimeter (HCAL), and muon
stations (M1-M5). LHCb uses a right-handed cartesian coordinate system with
its origin at the nominal interaction point, the $z$-axis pointing along the
LHC beam axis and the $y$ axis pointing upwards.

The TT employed conventional AC-coupled float-zone
$\rm{\it{p^{+}}}\mbox{-}on\mbox{-}{\it{n}}$ silicon micro-strip sensors that
are 500 ${\,\upmu\rm m}$ thick and have 512 strips with a pitch of 183
${\,\upmu\rm m}$ and a length of 96 $\rm\,mm$. The total silicon volume per
sensor is 4512 mm3. The initial full depletion voltages of the sensors were
determined from measurements of the bulk capacitance as a function of applied
bias voltage and were found to range between 135 and 275 V.

Figure 2: Layout of the TT, with the LHC beam pipe passing through an opening
in the centre of the detection layers. The four detection layers are labelled
TTaX, TTaU, TTbV and TTbX. The four different types of read-out sectors
employed in each of the detection layers are indicated by different shading:
read-out sectors close to the beam pipe consist of a single silicon sensor,
other read-out sectors consist of two, three or four silicon sensors that are
connected together in series.

The TT consisted of one detector box containing four detection layers arranged
in two pairs: the first two layers were centered around $z=232$ cm, and the
last two around $z=262$ cm. An ambient temperature of about 8∘C was maintained
inside the box. The box was flushed with dry gas ($N_{2}$) to avoid
condensation on cold surfaces. Silicon sensors within each detection layer
were electronically grouped into read-out sectors consisting of one, two,
three or four sensors, as illustrated in Fig. 2. All sensors within a read-out
sector were connected in series to a front-end read-out hybrid that carried
four 128-channel Beetle chips [4].

The Beetle front-end read-out chip amplifies and shapes the signals from the
read-out strips and samples the signal amplitude every 25 ns, corresponding to
the LHC bunch-crossing frequency of 40 MHz. The time offset between the LHC
bunch clock and the Beetle sampling time is an adjustable parameter. The
output data of the Beetle chips are multiplexed, digitised and transmitted
optically to the LHCb TELL1 read-out boards [5]. In the TELL1 boards,
algorithms for common-mode subtraction, cluster finding and zero-suppression
are executed during normal data taking. It is, however, also possible to store
the raw sampled amplitudes from each input channel of the Beetle chip for
offline analysis. This non-zero-suppressed read-out mode was employed for the
studies discussed in this paper.

Figure 3: Left: evolution of the integrated luminosity for $pp$ collisions in
each year of LHC operation. Right: expected 1-$\mathrm{\,Me\kern-1.00006ptV}$
neutron equivalent fluence per proton-proton collision, with the TTaU detector
layer overlaid. These results were obtained from a Fluka simulation of the
LHCb detector, assuming a proton-proton collision energy of 14 TeV.

LHCb collected an integrated luminosity of about 3 $\mbox{\,fb}^{-1}$ at
proton-proton collision energies of $\sqrt{s}=$7 and 8
$\mathrm{\,Te\kern-1.00006ptV}$ during Run 1 of the LHC (2010–2012), and about
6 $\mbox{\,fb}^{-1}$ at 13 $\mathrm{\,Te\kern-1.00006ptV}$ in Run 2
(2015–2018). The evolution of the integrated luminosity is shown in Fig. 3
(left) for each year of data taking. Expected particle fluences at collision
center-of-mass energies of $\sqrt{s}=7$, $8$ and $14$
$\mathrm{\,Te\kern-1.00006ptV}$ have been derived from simulation using Fluka
[6, *Battistoni:2007zzb], a general purpose tool for calculations of particle
transport and interactions with matter, and a detailed description of the
geometry of the LHCb experimental area [8, 9]. Maps of the particle fluence at
$\sqrt{s}=7$ and $8$ TeV are scored with a resolution of $[1,1]$ cm, while the
map at $\sqrt{s}=14$ $\mathrm{\,Te\kern-1.00006ptV}$ is scored with a
resolution of $[2.5,2.5]$ cm and subsequently interpolated to obtain a map
with a resolution of $[1,1]$ cm. All maps are produced at a $z$-coordinate
corresponding to the location of the TTaU layer. The statistical uncertainty
on the expected fluence is modeled as a power-law function of the radial
position from the beam pipe, $a_{0}\times(r/r_{0})^{\alpha}$, where
$a_{0}=0.7\times 10^{-2}$ cm-2 , $r_{0}=1$ cm, and $\alpha$ = 0.918. The
expected 1-$\mathrm{\,Me\kern-1.00006ptV}$ neutron equivalent fluence per
proton-proton collision at a center-of-mass energy of $\sqrt{s}=14$
$\mathrm{\,Te\kern-1.00006ptV}$ at the location of one of the detection layers
of the TT is shown in Figure 3 (right). The highest expected
1-$\mathrm{\,Me\kern-1.00006ptV}$ neutron equivalent fluence, in the innermost
region of the TT, corresponds to about $10^{13}$ cm-2 per
$1\,\mbox{\,fb}^{-1}$ of integrated luminosity collected at a proton-proton
collision energy of $\sqrt{s}=14$ $\mathrm{\,Te\kern-1.00006ptV}$. Since
particles are primarily produced in the forward direction, the expected
fluence falls off by almost three orders of magnitude from the innermost to
the outermost region of the TT detector.

Two methods were employed to monitor the radiation damage to the silicon
sensors and its impact on the detector performance. The first method,
discussed in Section 2, is based on the continuous measurement of detector
leakage currents. These measurements are compared to the expected radiation-
induced increase of the bulk leakage current as a function of received fluence
at an ambient temperature of $8^{\circ}$ C. The second method, discussed in
Section 3, is based on the analysis of collision data accumulated during
dedicated Charge Collection Efficiency (CCE) scans. These CCE scans were
performed at regular intervals and allow an estimation of the full depletion
voltage of the sensors. The results of these measurements are compared to
predictions derived from the Hamburg model [10], which parametrises the
evolution of the effective doping concentration of the silicon bulk as a
function of received fluence and temperature. Section 4 presents a summary of
the different studies and discusses the expected evolution of the TT
performance in terms of radiation damage.

##  2 Leakage current measurements

Bias voltage for the TT was supplied by a commercial high-voltage (HV)
system111CAEN SY1527 crates with A1511B modules, by CAEN S.p.A., Viareggio,
Italy. with 152 channels supplying up to 500 V at a maximum current of 10 mA
per HV channel. The innermost six silicon sensors in each detection layer,
which received the highest fluence and therefore experienced the largest
increase in leakage current, were connected to individual HV channels. In the
outer regions of the detection layers, groups of two, three, four, nine or
twelve silicon sensors were connected to a common HV channel. The current
drawn by each HV channel was monitored with a resolution of 1.0 $\mu$A, and a
maximum interval of 120 minutes allowed between consecutive readings. For the
purpose of this analysis, the leakage current, $I_{\text{leak}}$, for a given
HV channel is determined for each LHC fill as the maximum current observed in
that fill222A typical LHC fill is several hours in duration.. Measurements
from fills with special detector configurations (e.g. higher or lower bias
voltage) are discarded in the leakage current measurements. The leakage
current has a temperature dependent behaviour given by

$\frac{I_{\text{leak}}(T_{1})}{I_{\text{leak}}(T_{2})}=\Bigg{(}\frac{T_{1}}{T_{2}}\Bigg{)}^{2}\exp\Bigg{[}\frac{E_{eff}}{2k_{B}}\Bigg{(}\frac{1}{T_{2}}-\frac{1}{T_{1}}\Bigg{)}\Bigg{]},$
(1)

where $T_{1,2}$ are different temperatures, $k_{B}$ is the Boltzmann constant
and $E_{eff}=1.21$ eV is the effective energy for the temperature dependence
of the bulk current in irradiated silicon sensors, shown to be temperature-
independent in Ref. [11]. All measured currents are normalised to a nominal
temperature of 8∘C, using measurements of the ambient temperature inside the
detector box and the well-known temperature dependence of the bulk leakage
current in silicon [12].

The normalized leakage currents, scaled to 8∘C and per silicon volume, are
shown as a function of integrated delivered luminosity and time in Fig. 4 for
the innermost sensors in the second and third detection layers, indicated as U
and V layers, respectively.

Figure 4: Evolution of the leakage current for the different HV channels connected to one-sensor read-out sectors in TT as a function of (a) the delivered integrated luminosity measured and (b) time. The red curves show channels in the detector layer TTaU, the blue one in TTbV. The predicted evolution is shown in black while the grey band shows its uncertainty, computed from the uncertainty on the model parameters, on the Fluka simulation and on the temperature measurements. The uncertainty does not account for the range of fluence expected across the sectors shown. Table 1: Parameters for the leakage current predictions. Parameter | Value
---|---
$\alpha_{I,0}$ | $(6.67\pm 0.09)$ | $\times 10^{-17}$ | A/cm
$\alpha_{I,1}$ | $(7.23\pm 0.06)$ | $\times 10^{-17}$ | A/cm
$\alpha_{I,2}$ | $(3.08\pm 0.07)$ | $\times 10^{-18}$ | A/cm
$k_{0}$ | $(4.2\enspace\pm 0.5\enspace)$ | $\times 10^{13}$ | s-1
$E_{a}$ | $(1.11\pm 0.05)$ |  | eV

The change $\Delta I_{\text{leak}}$ in the leakage current is expected to be
linear as a function of the fluence, $\Phi$, that the sensor has been exposed
to, for a range between $10^{11}$ and $10^{15}$ 1-MeV n / cm2 [10]. So $\Delta
I_{\text{leak}}$ can be described as

$\Delta I_{\text{leak}}=\alpha\cdot\Phi\cdot V,$ (2)

where $\alpha$ is the current-related damage rate and $V$ the silicon volume.
The coefficient $\alpha$ shows a behaviour dependent on temperature, $T_{a}$,
and on time, $t$, as the irradiation occurs alternating with annealing, i.e.
periods without collisions [13]. Short term annealing (on the order of a few
days to a month) is described by the sum of a constant and exponential terms,
where the first term describes the effects of stable damage. An additional
logarithmic term is used to describe annealing effects for time periods longer
than a year [14],

$\alpha(t,T_{a})=\alpha_{I,0}+\alpha_{I,1}\exp\Big{(}-t/\tau_{I}(T_{a})\Big{)}+\alpha_{2}\log\Big{(}t/t_{0}\Big{)}.$
(3)

The temperature dependence of the time constant $\tau_{I}$ in Eq. (3) can be
described by the Arrhenius relation [15],

$\tau_{I}(T_{a})=\Bigg{[}k_{I,0}\exp\Bigg{(}-\frac{E_{a,I}}{k_{B}T_{a}}\Bigg{)}\Bigg{]}^{-1},$
(4)

where $k_{I,0}$ is the frequency factor and $E_{a,I}$ the activation energy of
the process. The values of these parameters are listed in Table 1.

Figure 4 also shows the predicted evolution of $I_{\text{leak}}$ based on this
formalism using the actual running conditions in LHCb (i.e. instantaneous
luminosity, duration of the fills, ambient temperature) and the
1-$\mathrm{\,Me\kern-1.00006ptV}$ neutron equivalent fluence computed for a
given sector by integrating the simulated Fluka radiation map over the sensor
area.

The predictions are in good agreement with the evolution of the leakage
currents in data. Significant decreases in the currents are observed, as
expected, in the data during periods with no irradiation, i.e. in the long
shutdown, LS1, of the LHC in 2013 and 2014, extended stops of the accelerator
during the winter season and short technical stops throughout the year (one
week about every three months).

##  3 Depletion voltage measurements

Non-ionizing energy loss (NIEL) damage in the silicon bulk leads to a change
in the effective doping concentration, $n_{\text{eff}}$, of the bulk and
therefore to a change in the full depletion voltage $V_{\text{depl}}$ since
the two quantities are proportional to each other according to the relation

$V_{\text{depl}}=\frac{q}{2\varepsilon\varepsilon_{0}}n_{\text{eff}}D^{2},$
(5)

where $q$ is the electron charge, $\varepsilon\varepsilon_{0}$ the effective
permittivity of silicon and $D$ the full thickness of the sensor.

As mentioned in Section 1, the initial full depletion voltage of all silicon
sensors was determined in $C-V$ scans at the foundry and during the
construction of the detector. For a subset of the installed detectors, the
evolution of $V_{\text{depl}}$ during operation has then been measured from
scans of the charge-collection efficiency as a function of the applied bias
voltage (CCE scans). Such CCE scans require particles from LHC collisions to
be recorded using special data taking settings. They are performed about two
to four times a year, in particular once at the beginning of LHC operation in
spring and once at the end of operation before Christmas. Table 2 summarises
the CCE scans performed between 2011 and 2018. Few CCE scans have been removed
from the analysis due to operational inconsistencies, and are not listed in
the table.

Table 2: Summary of CCE scans dates and corresponding integrated delivered luminosity. Date | Integrated Luminosity [fb-1]
---|---
| 7 TeV | 8 TeV | 13 TeV
2011-07-14 | 0.48 | – | –
2011-09-07 | 0.81 | – | –
2011-10-25 | 1.24 | – | –
2012-04-05 | 1.26 | 0.01 | –
2012-07-02 | 1.26 | 0.66 | –
2012-09-28 | 1.26 | 1.46 | –
2015-06-03 | 1.26 | 2.21 | 0.00
2015-10-20 | 1.26 | 2.21 | 0.23
2016-04-23 | 1.26 | 2.21 | 0.37
2016-08-04 | 1.26 | 2.21 | 1.34
2017-06-05 | 1.26 | 2.21 | 2.28
2017-07-12 | 1.26 | 2.21 | 2.51
2017-09-12 | 1.26 | 2.21 | 3.21
2017-10-27 | 1.26 | 2.21 | 3.81
2017-11-21 | 1.26 | 2.21 | 4.16
2018-04-17 | 1.26 | 2.21 | 4.26
2018-06-26 | 1.26 | 2.21 | 5.07
2018-09-23 | 1.26 | 2.21 | 6.26

While in the nominal data taking configuration the applied bias voltage
$V_{\text{bias}}$ was 300 V for all TT read-out sectors, during a CCE scan the
$V_{\text{bias}}$ of one detection layer (TTaU) is scanned in a 60 V to 400 V
range. The other detection layers are operated at the nominal bias voltage.
The data from these layers and from the other detectors of the LHCb tracking
system are used to reconstruct the trajectories of charged particles produced
in the proton-proton collisions.

Each CCE scan consists of 66 different data taking configurations. For each of
the 11 bias voltage steps, 6 sampling time steps are performed. As the drift
velocity of charge carriers and therefore the signal collection time depends
on the applied bias voltage, in fact, a scan of the signal sampling time of
the Beetle front-end ASIC is performed for each bias voltage setting in the
range $\delta t=-9.38,\,-4.69,\,0.00,\,4.69,\,9.38,\,14.06$ ns with respect to
the nominal sampling time. A minimum number of 500000 events are recorded for
each calibration step, which roughly corresponds to a total of 2–3 hours of
data taking depending on the beam configuration at the time of the CCE scan.
These scans are mainly taken during the beam commissioning phase, when only a
small number of proton bunches circulate in the accelerator, in order to
minimize the loss of useful data for physics analysis.

\begin{overpic}[width=455.24408pt,scale={.25},tics=20]{figs/TT_signal_LV}
\put(52.0,55.0){(a) $V_{\text{bias}}=100$~{}V} \end{overpic}

\begin{overpic}[width=455.24408pt,scale={.25},tics=20]{figs/TT_signal_HV}
\put(52.0,55.0){(b) $V_{\text{bias}}=400$~{}V} \end{overpic}

Figure 5: Signal height distribution for hits associated to tracks in the TT
for two different bias voltage values. The fitted distribution is described in
the text.

Tracks reconstructed by the rest of the LHCb tracking system, including the TT
layers operating in nominal conditions, are used to interpolate the hit
position in the scanned layer. Track quality requirements are applied to
remove tracks with poor track fit quality or originating from random
combinations of track segments from different parts of the tracking system.
The pedestal-subtracted ADC values of the three read-out strips closest to
this interpolated hit position are then summed to give an estimate of the
collected charge. As an example, Fig. 5 shows the signal height distribution
in a TT read-out sector for $V_{\text{bias}}=100$ V and $400$ V with $\delta
t=0.00$ ns. The incomplete depletion of the silicon bulk at
$V_{\text{bias}}=100$ V induces the clearly visible decrease in signal
amplitude.

The most probable value (MPV) of the collected charge is extracted from a fit
of this distribution, using a model consisting of the sum of two Gaussians to
describe the residual contribution of noise hits due to wrong track
extrapolations (blue line), convolved with the sum of two Landau distributions
to describe the contribution from signal hits (red line). The second Landau
distribution describes a signal component due to photon conversions in the
beam pipe and detector material upstream of the TT, creating $e^{+}e^{-}$
pairs that will cross the TT at the same position and deposit twice as much
charge as a single track. The MPV and the width of this second Landau
distribution are fixed to twice the corresponding values of the first Landau.

\begin{overpic}[width=455.24408pt,scale={.25},tics=20]{figs/TT_pulse_LV}
\put(43.0,15.0){(a) $V_{\text{bias}}=100$~{}V} \end{overpic}

\begin{overpic}[width=455.24408pt,scale={.25},tics=20]{figs/TT_pulse_HV}
\put(43.0,15.0){(b) $V_{\text{bias}}=400$~{}V} \end{overpic}

Figure 6: Pulse shapes recorded in the TT for two different bias voltage
values. The fitted function is shown in Eq. (6).

The MPVs extracted for different sampling times $\delta t$ at a given bias
voltage are then fit with the empirical function

$f(\delta t|\tau,t_{0},A)=A\cdot\exp\Bigg{(}-\frac{\delta
t-t_{0}}{\tau}\Bigg{)}\cdot\Bigg{[}\frac{1}{2}\Bigg{(}\frac{\delta
t-t_{0}}{\tau}\Bigg{)}^{2}-\frac{1}{6}\Bigg{(}\frac{\delta
t-t_{0}}{\tau}\Bigg{)}^{3}\Bigg{]}$ (6)

describing the semi-Gaussian signal shape output by the pulse shaping circuit
employed in the Beetle chip, which is composed of a differentiator (CR)
followed by an integrator (RC) [16, 17]. The shaping time $\tau$, time offset
$t_{0}$ and amplitude $A$ are free parameters of the fit. As an example,
Figure 6 shows the results of these fits for one of the innermost sensors and
for $V_{\text{bias}}=100$ V and $400$ V. The integral of the fitted function
between its two zeroes $t_{0}$ and $t_{0}+3\tau$ is taken as a measure of the
collected charge at the given bias voltage.

\begin{overpic}[width=455.24408pt,scale={.25},tics=20]{figs/TT_voltage_early}
\put(25.0,60.0){(a)} \end{overpic}

\begin{overpic}[width=455.24408pt,scale={.25},tics=20]{figs/TT_voltage_late}
\put(25.0,60.0){(b)} \end{overpic}

Figure 7: Integrated charge as a function of the applied bias voltage for a
central TT read-out sector measured in the CCE scans from (a) April 2012 and
(b) September 2017. The decrease of $V_{\text{depl}}$ is clearly visible.

\begin{overpic}[width=455.24408pt,scale={.25},tics=20]{figs/TT_ratio_hist}
\put(79.0,58.0){(a)} \end{overpic}

\begin{overpic}[width=455.24408pt,scale={.25},tics=20]{figs/TT_systematics_r}
\put(25.0,58.0){(b)} \end{overpic}

\begin{overpic}[width=455.24408pt,scale={.25},tics=20]{figs/TT_systematics_spline}
\put(79.0,58.0){(c)} \end{overpic}

Figure 8: Distributions for all analysed read-out sectors of (a) the ratio $r$
defined in Eq.7; (b) the relative difference between $V_{\text{depl}}$ and
$V_{\text{depl}}^{\text{2011-07-04}}$ as discussed in the main text; (c) the
differences in $V_{\text{depl}}$ obtained from a fifth-order spline and from a
linear interpolation. The solid black line in (a) highlights the value of the
average $\overline{r}$.

This quantity, referred to as “integrated charge”, is shown as a function of
$V_{\text{bias}}$ in Fig. 7. The three highest $V_{\text{bias}}$ data points
are fit with a constant to determine the plateau value. The data points are
interpolated with a fifth-order spline [18] $S$ and the depletion voltage is
determined as the bias voltage at which $S(V_{\text{bias}})$ reaches the
fraction $\overline{r}=94\%$ of its value in the plateau. The value
$\overline{r}=94\%$ was chosen based on an analysis of the earliest CCE scan,
taken on 2011-07-14. First, the ratio

$\displaystyle
r=S\left(V_{\text{depl}}^{\text{2011-07-14}}\right)\,/\,S\left(\text{plateau}\right)$
(7)

was calculated for each analysed read-out sector, where the true depletion
voltage at the time of the CCE scan, $V_{\text{depl}}^{\text{2011-07-04}}$,
was estimated from the full depletion voltage of the sensor as measured in
$C-V$ scans during the construction of the detector, corrected for the
expected change in full depletion voltage according to the Hamburg model and
the integrated fluence received at the time of the CCE scan. The distribution
of $r$ for all analysed read-out sectors is shown in Fig. 8(a). The values of
$r$ obtained for the different read-out sectors were then averaged to
determine $\overline{r}$.

The systematic uncertainty on $V_{\text{depl}}$ due to the choice of
$\overline{r}$ was estimated by applying the full method to the first CCE scan
and comparing the values of $V_{\text{depl}}$ extracted from the spline fits
with $\overline{r}=94\%$ to the values $V_{\text{depl}}^{\text{2011-07-04}}$
extracted from the initial $C-V$ scans and the Hamburg model. The residual
distribution of $V_{\text{depl}}$ and $V_{\text{depl}}^{\text{2011-07-04}}$ is
shown in Fig. 8(b); the spread of this distribution is taken as systematic
uncertainty.

A second source of systematic uncertainty is the choice of a fifth-order
spline to interpolate the measurements. To estimate this uncertainty, the
entire procedure was repeated using a simple linear interpolation between
consecutive measurements instead of the spline fit. The distribution of the
differences in $V_{\text{depl}}$ obtained using the two types of interpolation
are shown in Fig. 8(c) for all analysed read-out sectors; the spread of this
distribution is taken as systematic uncertainty. These two uncertainties are
summed in quadrature.

Table 3: Parameters for the depletion voltage simulation by the Hamburg model [10]. Parameter | Value
---|---
$n_{c,0}$ | $(3.28\pm 0.26)$ | $\times 10^{-10}$ | cm-3
$c$ | $2.29\color[rgb]{1,1,1})$ | $\times 10^{-13}$ | cm2
$g_{c}$ | $(1.60\pm 0.04)$ | $\times 10^{-2}$ | cm-1
$g_{a}$ | $(1.40\pm 0.14)$ | $\times 10^{-2}$ | cm-1
$g_{r}$ | $(5.70\pm 0.09)$ | $\times 10^{-2}$ | cm-1
$k_{a,0}$ | $(2.4\enspace\pm 1.0\enspace)$ | $\times 10^{15}$ | s-1
$k_{r,0}$ | $(1.5\enspace\pm 1.1\enspace)$ | $\times 10^{15}$ | s-1
$E_{aa}$ | $(1.09\pm 0.03)$ |  | eV
$E_{ar}$ | $(1.31\pm 0.03)$ |  | eV

The change $\Delta n_{\text{eff}}$ in the effective doping concentration is
described in the Hamburg model by three different mechanisms, described in
detail in Refs. [19, 20, 21, 10]:

  1. 1.

a contribution associated to the effective (incomplete) removal of donor atoms
and the addition of stable acceptor atoms due to radiation-induced changes in
the band structure of the silicon crystal. This contribution, referred to as
“stable damage”, is described by

$\Delta n_{c}(\Phi_{\text{1
MeV-n,eq}})=n_{c,0}\,\big{[}1-\exp(-c\,\Phi_{\text{1
MeV-n,eq}})\big{]}+g_{c}\,\Phi_{\text{1 MeV-n,eq}};$ (8)

  2. 2.

a temperature-dependent contribution due to the annealing of the induced
defects, which can be described by

$\Delta n_{a}(\Phi_{\text{1 MeV-n,eq}},\,t,\,T_{a})=\Phi_{\text{1
MeV-n,eq}}\,g_{a}\exp\bigg{(}-\frac{t}{\tau_{a}(T_{a})}\bigg{)};$ (9)

  3. 3.

a contribution due to the combination of individual defects in the silicon
lattice, leading to stable defects (“reverse annealing”), described by

$\Delta n_{r}(\Phi_{\text{1 MeV-n,eq}},\,t,\,T_{a})=\Phi_{\text{1
MeV-n,eq}}\,g_{r}\bigg{(}1-\frac{1}{1+t/\tau_{r}(T_{a})}\bigg{)}.$ (10)

The sum of the three contributions describes the total change of
$n_{\text{eff}}$:

$\Delta n_{\text{eff}}=\Delta n_{c}+\Delta n_{a}+\Delta n_{r}.$ (11)

The values of the model parameters are listed in Table 3. Their values were
measured in dedicated irradiation campaigns and can be found in Reference
[10]. The parameters $\tau_{a}$ and $\tau_{r}$ have a temperature dependence
described by the Arrhenius relation, cf. Eq. 4, with parameters $E_{aa}$ and
$k_{a,0}$ and $E_{ar}$ and $k_{r,0}$, respectively, and the ambient
temperature $T_{a}$ is taken from sensors placed inside the detector boxes.

Figure 9 shows the measured depletion voltages for a one-sensor and for a two-
sensors TT read-out sectors, highlighted in Figure 11(a), and their predicted
evolution based on the Hamburg model described above. This calculation uses
the fluence estimated from the Fluka simulations333Fluence is calculated by
integrating the radiation map over the sensor area., the actual running
conditions and the temperature measurements in the detector boxes.

\begin{overpic}[width=455.24408pt,scale={.25},tics=20]{figs/TT_time_evolv_inner}
\put(80.0,60.0){(a)} \end{overpic}

\begin{overpic}[width=455.24408pt,scale={.25},tics=20]{figs/TT_time_evolv_outer}
\put(80.0,60.0){(b)} \end{overpic}

Figure 9: Measured effective depletion voltage (red points) in the area of two
TT sensors: one just above the beam pipe (a) and one further away (b), whose
positions are indicated in Figure 11(a). The black point and error bars
correspond to the CCE scan used to calibrate the ratio $r$. The statistical
contribution to the total uncertainty is indicated by the solid error bars.
The predicted evolution of the depletion voltage, based on the initial
depletion voltage measured after sensor production, the running conditions and
the Hamburg model, is shown as a solid black line. The grey bands show the
uncertainty on the predicted evolution of $V_{\text{depl}}$, while the black
dashed lines account for the $\pm 2.5$ V uncertainty on the measurement of the
initial depletion voltage $V^{0}_{\text{depl}}$ in $C-V$ scans.

\begin{overpic}[width=455.24408pt,scale={.25},tics=20]{figs/TT_deplVphiN_sector}
\put(80.0,60.0){(a)} \end{overpic}

\begin{overpic}[width=455.24408pt,scale={.25},tics=20]{figs/TT_deplVphiN}
\put(80.0,60.0){(b)} \end{overpic}

Figure 10: Measured evolution of $V_{\text{depl}}$ as a function of the 1-MeV
neutron equivalent fluence obtained from the running conditions and Fluka for
a TT sector in the area close to the beam pipe (a) and for all of the read-out
sectors in the TT (b). The innermost sectors are subdivided in regions
separating tracks traversing the detector at a distance larger or smaller than
75 mm from the beam axis. The error bars of the data points display the sum of
the statistical and systematic uncertainty. The solid black curve shows the
predictions based on the stable damage part of the Hamburg model, the grey
shaded region its uncertainty due to the parameter uncertainty of the model.
The initial depletion voltage $V^{0}_{\text{depl}}$ for the Hamburg model
prediction in (b) is averaged among all sectors, and the dashed black lines
show the standard deviation of the distribution of the initial
$V^{0}_{\text{depl}}$ values.

\begin{overpic}[width=455.24408pt,scale={.25},tics=20]{figs/TT_sectors}
\put(42.0,-7.0){(a)} \end{overpic}

\begin{overpic}[width=478.00766pt,scale={.25},tics=20]{figs/tt_map_3478.pdf}
\put(40.0,0.0){(b)} \end{overpic}

\begin{overpic}[width=478.00766pt,scale={.25},tics=20]{figs/tt_map_7211.pdf}
\put(40.0,0.0){(c)} \end{overpic}

Figure 11: Sketch highlighting the TT sectors analysed during the various CCE
scans (a). The greyed out sectors were excluded due to poor statistics. The
two sectors highlighted with dashes, just above the beam pipe and below the
beam pipe on the left hand side, are those which evolution is shown in Figure
9(a) and Figure 9(b), respectively. Absolute change in $V_{\text{depl}}$ in
the innermost region of TT in January 2013 and September 2018 is shown in (b)
and (c) respectively. The white sector in (c) became inoperative in August
2017 due to an HV connector issue.

For the read-out sectors closest to the LHC beam pipe, dedicated measurements
only including tracks traversing the sensors within 75 mm from the beam axis
are performed. Fig. 10 shows the measured values of $V_{\text{depl}}$ of the
different TT read-out sectors in the different CCE scans as a function of the
1-$\mathrm{\,Me\kern-1.00006ptV}$-neutron equivalent fluence estimated from
the Fluka simulations and the integrated luminosity collected in LHCb. It also
shows the expected evolution of $V_{\text{depl}}$ based on the stable damage
contribution $n_{c}$.

Fig. 11(c) shows the absolute change of $V_{\text{depl}}$ in the innermost
part of the detector between July 2011 and September 2018.

Good agreement between the measurements and the predicted evolution of
$V_{\text{depl}}$ is observed, with no type inversion seen in the silicon
sensors of the TT at the end of the LHC Run 2.

##  4 Conclusion

The evolution of the radiation damage in the LHCb Tracker Turicensis has been
monitored using measurements of leakage currents and effective depletion
voltages. The latter were performed with data collected in dedicated charge
collection efficiency scans. The obtained results show good agreement with
predictions based on phenomenological models. At the end of 2018, the
innermost sensors, which experience the highest fluence, had not yet reached
the point of type inversion, and no modifications to the operation procedure
of the detector were performed in its last year of operation. The detector has
been dismantled and will be replaced as part of the LHCb upgrade [2, *LHCb-
TDR-015] during the long shutdown LS2 of the LHC.

## Acknowledgements

E. G. gratefully acknowledges support by the Forschungskredit of the
University of Zurich, grant no. FK-17-108. The work of M. A. is supported by
the Swiss National Science Foundation (SNF) under contract BSSGI0_155990.

## References

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space $\\{\gamma\in\Gamma_{m,D^{\prime}}:\lVert\gamma\rVert_{\infty}\leq M\\}$
corresponding to observations

$\displaystyle\mathcal{E}_{3}^{(r)}:\quad$
$\displaystyle\big{(}\pi_{JK}\tilde{\Lambda}_{\gamma}+\varepsilon\mathbb{W}\big{)}(U)_{U\in\mathbb{H}_{r}}=\big{(}\langle\pi_{JK}\tilde{\Lambda}_{\gamma},U\rangle_{\mathbb{H}_{r}}+\varepsilon\sum_{j,k}g_{jk}\langle
U\phi_{j}^{(r)},\phi_{k}^{(0)}\rangle_{L^{2}(\partial
D)}\big{)}_{U\in\mathbb{H}_{r}},$ $\displaystyle\mathcal{E}_{4}^{(r)}:\quad$
$\displaystyle\big{(}\pi_{JK}\tilde{\Lambda}_{\gamma}+\varepsilon\mathbb{W}\big{)}(\pi_{JK}U)_{U\in\mathbb{H}_{r}}=\big{(}\langle\pi_{JK}\tilde{\Lambda}_{\gamma},\pi_{JK}U\rangle_{\mathbb{H}_{r}}+\varepsilon\sum_{\begin{subarray}{c}j\leq
J,\\\ k\leq K\end{subarray}}g_{jk}\langle
U\phi_{j}^{(r)},\phi_{k}^{(0)}\rangle_{L^{2}(\partial
D)}\big{)}_{U\in\mathbb{H}_{r}},$

where we recall the projection $\pi_{JK}$ was defined in 24, and
$g_{jk}=\mathbb{W}(b_{jk}^{(r)})\overset{iid}{\sim}N(0,1)$. By the triangle
inequality, we decompose
$\Delta(\mathcal{E}_{1}^{(r)},\mathcal{E}_{2}^{(r)})\leq\Delta(\mathcal{E}_{2}^{(r)},\mathcal{E}_{3}^{(r)})+\Delta(\mathcal{E}_{3}^{(r)},\mathcal{E}_{4}^{(r)})+\Delta(\mathcal{E}_{4}^{(r)},\mathcal{E}_{1}^{(r)})$.
We control each of the terms on the right.

#####  $\Delta(\mathcal{E}_{2}^{(r)},\mathcal{E}_{3}^{(r)})$:

Lemmas 23a, 4, and the proof of Lemma 9 yield

$\Delta(\mathcal{E}_{2}^{(r)},\mathcal{E}_{3}^{(r)})\leq\tfrac{1}{2}\varepsilon^{-1}\times\sup_{\gamma\in\Gamma_{m,D^{\prime}}:\lVert\gamma\rVert_{\infty}\leq
M}\lVert\tilde{\Lambda}_{\gamma}-\pi_{JK}\tilde{\Lambda}_{\gamma}\rVert_{\mathbb{H}_{r}}\leq
C\varepsilon^{-1}\min(J,K)^{-\nu},$

for a constant $C=C(\nu,r,D,D^{\prime},M,m).$

#####  $\Delta(\mathcal{E}_{3}^{(r)},\mathcal{E}_{4}^{(r)})$:

We note that
$\big{(}\pi_{JK}\tilde{\Lambda}_{\gamma}+\varepsilon\mathbb{W}\big{)}(\pi_{JK}U)_{U\in\mathbb{H}_{r}}$
is a sufficient statistic for
$\big{(}\pi_{JK}\tilde{\Lambda}_{\gamma}+\varepsilon\mathbb{W}\big{)}(U)_{U\in\mathbb{H}_{r}}$
by independence of $(g_{jk})_{j\leq J,k\leq K}$ from $(g_{jk}:j>J\textnormal{
or }k>K)$, so that $\Delta(\mathcal{E}_{3}^{(r)},\mathcal{E}_{4}^{(r)})=0$ by
23c.

#####  $\Delta(\mathcal{E}_{4}^{(r)},\mathcal{E}_{1}^{(r)})$:

Using 23b as in the proof of Theorem 24, it is clear that the experiment
$\mathcal{E}_{1}^{(r)}$ is equivalent to observing

$\big{(}{\textstyle\sum_{j\leq J,k\leq
K}}\big{(}\langle\tilde{\Lambda}_{\gamma}\phi_{j}^{(r)},\phi_{k}^{(0)}\rangle_{L^{2}(\partial
D)}{u}_{jk}+\varepsilon g_{j,k}{u}_{jk}\big{)}\big{)}_{u\in\ell^{2}},$

where $\ell_{2}$ denotes the space of square-summable real sequences. Since
$(g_{jk}):=(\mathbb{W}(b_{jk}^{(r)}))\overset{d}{=}(g_{j,k})$ (the latter
being the noise in $\mathcal{E}_{1}^{(r)}$) and, for
$U=\sum_{j,k}u_{jk}b_{jk}^{(r)}$,

${\textstyle\sum_{j\leq J,k\leq
K}}\langle\tilde{\Lambda}_{\gamma}\phi_{j}^{(r)},\phi_{k}^{(0)}\rangle_{L^{2}(\partial
D)}{u}_{jk}=\langle\pi_{JK}\tilde{\Lambda}_{\gamma},\pi_{JK}U\rangle_{\mathbb{H}_{r}},$

we deduce $\Delta(\mathcal{E}_{4}^{(r)},\mathcal{E}_{1}^{(r)})=0$.∎
Complementing the two previous theorems, let us remark that in fact
$\delta(\mathcal{E}_{2}^{(r)},\mathcal{E}_{1}^{(r)})=0$ and similarly
$\delta(\mathcal{E}_{2}^{(0)},\mathcal{E}_{0})=0$, as is shown in the proof of
the lower bound in the following theorem.

In our final result we prove that the rates obtained in Theorems 1 and 2 hold
with $P_{\varepsilon}^{\gamma}$ replaced by the law
$P_{\varepsilon,P}^{\gamma}$ of the data $Y=(Y_{p,q})_{p,q\leq P}$ in the
electrode model 7, for an appropriate number of electrodes $P$ and for
appropriate sets $(I_{p})_{p\leq P}$. The same conclusion holds as well in the
‘discrete spectral’ model 8 – the proof of this fact is similar (in fact
simpler) and omitted.

######  Theorem 27.

Let $P_{\varepsilon,P}^{\gamma}$ denote the law of the data
$Y=(Y_{p,q})_{p,q\leq P}$ in the electrode model 7. Recall the parameter set
$\Gamma_{m_{0},D_{0}}^{\alpha}(M)$ defined in 3.

  1. A.

Suppose $m_{0},D_{0},\alpha,M$ satisfy the conditions of Theorem 1. Suppose
that the sets $(I_{p})_{p\leq P}$ satisfy the conditions of Theorem 24 and
that $P\geq\varepsilon^{-\mu(d-1)}$ for some $\mu>1$. Then there exists a
measurable function $\hat{\gamma}^{\prime}$ of the data $Y\sim
P_{\varepsilon,P}^{\gamma}$ such that, for $C,\delta$ the same constants as in
Theorem 1,

$\sup_{\gamma\in\Gamma^{\alpha}_{m_{0},D_{0}}(M)}P_{\varepsilon,P}^{\gamma}(\lVert\hat{\gamma}^{\prime}-\gamma\rVert_{\infty}>C\log(1/\varepsilon)^{-\delta})\to
0,~{}\text{as }\varepsilon\to 0.$ (66)

  2. B.

Suppose $D_{0},D,m_{0},\alpha,M,c,\delta^{\prime}$ are as in Theorem 2. Let
$P\in\mathbb{N}$ be arbitrary, and let $(I_{p})_{p\leq P}$ be arbitrary
disjoint measurable sets. Then for all $\varepsilon$ small enough,

$\inf_{\tilde{\gamma}^{\prime}}\sup_{\gamma\in\Gamma^{\alpha}_{m_{0},D_{0}}(M)}P_{\varepsilon,P}^{\gamma}(\lVert\tilde{\gamma}^{\prime}-\gamma\rVert_{\infty}>c\log(1/\varepsilon)^{-\delta^{\prime}})>1/4,$
(67)

where the infimum extends over all measurable functions
$\tilde{\gamma}^{\prime}=\tilde{\gamma}^{\prime}(Y)$ of the data $Y\sim
P_{\varepsilon,P}^{\gamma}$.

###### Proof.

We consider statistical experiments denoted $\mathcal{E}_{0}$,
$\mathcal{E}_{1}^{(r)}$, $\mathcal{E}_{2}^{(r)}$ defined as at the start of
Section D.2, but here for the smaller parameter space
$\Gamma^{\alpha}_{m_{0},D_{0}}(M)$. The definition of the Le Cam deficiency
ensures that it cannot increase upon considering a smaller parameter space, so
that Theorems 24 and 26 continue to hold for these experiments.

##### A:

Consider the loss function
$L(\gamma,\rho)=\mathbbm{1}\\{\lVert\gamma-\rho\rVert_{\infty}>C\log(1/\varepsilon)^{-\delta}\\}.$
The associated risk $R$ of the decision rule $\hat{\gamma}$ in the continuous
model 12 is
$P_{\varepsilon}^{\gamma}(\lVert\hat{\gamma}-\gamma\rVert_{\infty}>C\log(1/\varepsilon)^{-\delta})$,
so that $\sup_{\gamma}R\to 0$ as $\varepsilon\to 0$, for any $r\in\mathbb{R}$,
by Theorem 1.

In view of 60, there exists a measurable function $\hat{\gamma}^{\prime}$ of
the data in model 7 whose risk $R^{\prime}$ satisfies $R^{\prime}\leq
R+\delta(\mathcal{E}_{0},\mathcal{E}_{2}^{(0)}).$ [In fact, as the proof
shows, all Markov kernels involved in bounding
$\delta(\mathcal{E}_{0},\mathcal{E}_{2}^{(0)})$ arise from deterministic maps,
and hence $\hat{\gamma}^{\prime}$ can be taken to be non-randomised.] The
limit 66 (which is $\sup_{\gamma}R^{\prime}\to 0$) will follow from showing
that $\delta(\mathcal{E}_{0},\mathcal{E}_{2}^{(0)})\to 0$. The triangle
inequality gives that
$\delta(\mathcal{E}_{0},\mathcal{E}_{2}^{(0)})\leq\delta(\mathcal{E}_{0},\mathcal{E}_{1}^{(0)})+\delta(\mathcal{E}_{1}^{(0)},\mathcal{E}_{2}^{(0)})$,
where we may freely choose the parameters $J,K$ of the intermediate model. For
some positive constant
$\mu^{\prime}<\min\\{(2d-2)(\mu-1)/(3d),(2d-2)\mu/(5d-2)\\},$ take
$\nu>1/\mu^{\prime}$ and choose $J=K$ of order $\varepsilon^{-\mu^{\prime}}$.
Then, by Theorems 24 and 26 and the triangle inequality, we have for a
constant $C$

$\delta(\mathcal{E}_{0},\mathcal{E}_{2}^{(0)})\leq
C(\varepsilon^{\mu-1}J^{3d/(2d-2)}+J^{(5d-2)/(2d-2)}\varepsilon^{\mu}+\varepsilon^{-1}J^{-\nu})\to
0,$

as required.

##### B:

Consider the loss function
$\tilde{L}(\gamma,\rho)=\mathbbm{1}\\{\lVert\gamma-\rho\rVert_{\infty}>c\log(1/\varepsilon)^{-\delta^{\prime}}\\}.$
In view of Theorem 2, the associated minimax risk in the continuous model,
$\tilde{R}_{\textnormal{minimax}}=\inf_{\rho}\sup_{\gamma}P_{\varepsilon}^{\gamma}(\lVert\gamma-\rho\rVert_{\infty}>c\log(1/\varepsilon)^{-\delta^{\prime}})$
is greater than 1/4, at least for $\varepsilon$ small enough. Then 61 implies
that the minimax risk $\tilde{R}_{\textnormal{minimax}}^{\prime}$ in model 7
satisfies
$\tilde{R}_{\textnormal{minimax}}^{\prime}>1/4-\delta(\mathcal{E}_{2}^{(r)},\mathcal{E}_{0}).$
The lower bound 67 will follow from showing that
$\delta(\mathcal{E}_{2}^{(0)},\mathcal{E}_{0})=0$. By 23b, it suffices to show
that the data in model 7 has law matching that of some subset of the data
observed in model 12 with $r=0$. Such a subset, recalling the concrete
interpretation 11 of model 12, is given by $(Y(T_{pq}))_{p,q\leq P}$, where
$T_{pq}=\psi_{p}\otimes\psi_{q}=\langle\cdot,\psi_{p}\rangle_{L^{2}(\partial
D)}\psi_{q}$. To see this, observe that $\langle
T_{pq}(\phi_{j}^{(0)}),\phi_{k}^{(0)}\rangle_{L^{2}(\partial
D)}=\langle\phi_{j}^{(0)},\psi_{p}\rangle_{L^{2}(\partial
D)}\langle\psi_{q},\phi_{k}^{(0)}\rangle_{L^{2}(\partial D)}$, hence

$\displaystyle\langle\tilde{\Lambda}_{\gamma},T_{pq}\rangle_{\mathbb{H}_{0}}$
$\displaystyle=\sum_{j,k}\langle\tilde{\Lambda}_{\gamma}\phi_{j}^{(0)},\phi_{k}^{(0)}\rangle_{L^{2}(\partial
D)}\langle\phi_{j}^{(0)},\psi_{p}\rangle_{L^{2}(\partial
D)}\langle\psi_{q},\phi_{k}^{(0)}\rangle_{L^{2}(\partial
D)}=\langle\tilde{\Lambda}_{\gamma}\psi_{p},\psi_{q}\rangle_{L^{2}(\partial
D)},$ $\displaystyle\mathbb{W}(T_{pq})$
$\displaystyle=\sum_{j,k}g_{jk}\langle\phi_{j}^{(0)},\psi_{p}\rangle_{L^{2}(\partial
D)}\langle\psi_{q},\phi_{k}^{(0)}\rangle_{L^{2}(\partial D)}.$

The noise variables
$g^{\prime}_{p,q}:=\langle\mathbb{W},T_{pq}\rangle_{\mathbb{H}_{0}}$ are
jointly normally distributed with mean zero and covariances

$\begin{split}\operatorname{Cov}(g^{\prime}_{p,q},{g}_{l,m}^{\prime})&=\sum_{j,k}\langle\phi_{j}^{(0)},\psi_{p}\rangle_{L^{2}(\partial
D)}{\langle\phi_{j}^{(0)},\psi_{l}\rangle}_{L^{2}(\partial
D)}{\langle\psi_{q},\phi_{k}^{(0)}\rangle}_{L^{2}(\partial
D)}\langle\psi_{m},\phi_{k}^{(0)}\rangle_{L^{2}(\partial D)}\\\
&=\langle\psi_{p},\psi_{l}\rangle_{L^{2}(\partial
D)}{\langle\psi_{q},\psi_{m}\rangle}_{L^{2}(\partial
D)}=\delta_{pl}\delta_{qm},\end{split}$

so that indeed $(Y(T_{pq}))_{p,q\leq P}\overset{d}{=}(Y_{p,q})_{p,q\leq P}$ as
claimed.∎

Acknowledgements. The authors would like to thank Tapio Helin for helpful
discussions at the outset of this project. We are further grateful to five
anonymous referees and an associate editor for their valuable remarks and
suggestions. We would particularly like to acknowledge one referee for
pointing out the reference [39] which allowed us to generalise our results to
include $d=2$. RN was supported by ERC grant No. 647812. KA was supported by
the UK EPSRC grant EP/L016516/1.

##  Appendix E Notation index

$D\subseteq\mathbb{R}^{d}$, $d\geq 2$, a bounded domain, which is taken to
mean a connected open set with smooth boundary $\partial D$.

$\Gamma_{m,D^{\prime}}=\\{\gamma\in C(D):\inf_{x\in D}\gamma(x)\geq
m,~{}\gamma=1\text{ on }D\setminus D^{\prime}\\}$, some $m\in(0,1)$ and some
domain $D^{\prime}$ compactly contained in $D$ (i.e. the closure
$\bar{D}^{\prime}$ is a subset of $D$). $C(D)$ denotes the real-valued bounded
continuous functions from $D$ to $\mathbb{R}$. $C_{u}(D)$ denotes the real-
valued uniformly continuous functions on $D$.

$\Gamma^{\alpha}_{m,D^{\prime}}(M)=\\{\gamma\in\Gamma_{m,D^{\prime}}:~{}\lVert\gamma\rVert_{H^{\alpha}(D)}\leq
M\\}$.

$\gamma\in\Gamma_{m,D^{\prime}}$ a conductivity function, $\gamma_{0}$ its
‘true’ value for some statistical theorems.

$\theta_{0}=\Phi^{-1}\circ\gamma_{0}$ for $\Phi$ described in Section 3.1.
$\theta\in C_{u}(D)$ a function of the form $\Phi^{-1}\circ\gamma$ for
$\gamma\in\Gamma_{m,D^{\prime}}$ (usually denoting a generic draw from the
prior $\Pi$ of 15).

$m_{0},D_{0}$ a lower bound and support set for the ‘true’ $\gamma_{0}$;
$m_{1},D_{1}$ a lower bound and support set for any draw
$\gamma=\Phi\circ\theta$ from the prior $\Pi$ of Section 3.1.

$\lVert\cdot\rVert_{\infty}$ the usual supremum norm on $C(D)$ or
$C(\mathbb{R})$.

$\overline{v}$ the usual complex conjugate of a number or function
$(\overline{v}(x):=\overline{v(x)})$.

$u_{\gamma,f}$ the (weak) solution to the Dirichlet problem 1
($\operatorname{\operatorname{\nabla}\cdot}(\gamma\operatorname{\nabla}u)=0$
in $D$, $u=f$ on $\partial D$).

$H^{s}$ an $L^{2}$-Sobolev space of complex-valued functions (carefully
defined in Appendix A); $H^{1}_{0}(D)$ the traceless subset of $H^{1}(D)$.
Inner products here are linear in the first argument and conjugate-linear in
the second.

$\mathcal{H}_{s}=\big{(}H^{\min\\{1,s+3/2\\}}(D)\cap
H^{1}_{\textnormal{loc}}(D)\big{)}/\mathbb{C}$ ($H^{1}_{\textnormal{loc}}$ is
defined in Appendix A, and $/\mathbb{C}$ means we identify functions $f,f+c$
which are equal up to a scalar $c\in\mathbb{C}$).

$H^{s}_{\diamond}(\partial D)=\\{g\in H^{s}(\partial D):\langle
g,1\rangle_{L^{2}(\partial D)}=0\\}$, $L^{2}_{\diamond}(\partial
D)=H^{0}_{\diamond}(\partial D)$.

$(\phi_{k}^{(r)})_{k\in\mathbb{N}\cup\\{0\\}}$ an orthonormal basis of
$H^{r}(\partial D)$ consisting of real-valued eigenfunctions of the
Laplace–Beltrami operator on $\partial D$, with corresponding eigenvalues
(sorted so they increase with $k$) $\lambda_{k}\geq 0$. More details in
Appendix A.

$\psi_{p}=c_{p}\mathbbm{1}_{I_{p}}$ indicator functions of some disjoint
measureable subsets $(I_{p})_{p\leq P}$ of $\partial D$, scaled to be
orthonormal.

$\frac{\partial}{\partial\nu}$ the outward normal derivative at the boundary
of a domain (i.e. usually on $\partial D$) defined in a trace sense.
$\operatorname{tr}$ denotes the usual trace map, taking $V:D\to\mathbb{C}$ to
its ‘boundary values’ $\operatorname{tr}V=V|_{\partial D}:\partial
D\to\mathbb{C}$.

$\Lambda_{\gamma}:H^{s+1}(\partial D)/\mathbb{C}\to H^{s}_{\diamond}(\partial
D)$ the Dirichlet-to-Neumann map, taking $f$ to $\gamma\frac{\partial
u_{\gamma,f}}{\partial\nu}|_{\partial D}.$

$\tilde{\Lambda}_{\gamma}=\Lambda_{\gamma}-\Lambda_{1}$.

$\lVert\cdot\rVert_{*}=\lVert\cdot\rVert_{H^{1/2}(\partial D)/\mathbb{C}\to
H^{-1/2}(\partial D)}$, where $\lVert\cdot\rVert_{A\to B}$ denotes the
operator norm between Banach spaces $A$ and $B$.

$\mathcal{L}(A,B)=\\{T:A\to B\text{ linear s.t. }\lVert T\rVert_{A\to
B}<\infty\\}.$

$\mathcal{L}_{2}(A,B)=\\{T\in\mathcal{L}(A,B):\lVert
T\rVert_{\mathcal{L}_{2}(A,B)}^{2}:=\sum_{k}\lVert
Te_{k}^{(A)}\rVert_{B}^{2}<\infty\\}$ the space of Hilbert–Schmidt operators
from $A$ to $B$ for separable Hilbert spaces $A$ and $B$, with
$(e_{k}^{(A)})_{k\in\mathbb{N}}$ an orthonormal basis of $A$.

$b_{jk}^{(r)}(f)\equiv(\phi_{j}^{(r)})\otimes\phi_{k}^{(0)}(f)=\langle
f,\phi_{j}^{(r)}\rangle_{H^{r}(\partial D)}\phi_{k}^{(0)}$,
$j,k\in\mathbb{N}$. These form an orthonormal basis of $\mathbb{H}_{r}$
(considering real linear combinations) and of $\mathcal{L}_{2}(H^{r}(\partial
D)/\mathbb{C},L^{2}_{\diamond}(\partial D))$ (considering complex linear
combinations).

$\mathbb{H}_{r}=\big{\\{}T:H^{r}(\partial D)\to L^{2}(\partial
D),~{}T=\sum_{j,k=1}^{\infty}t_{jk}b_{jk}^{(r)}:t_{jk}\in\mathbb{R},\sum_{j,k=1}^{\infty}t_{jk}^{2}<\infty\big{\\}},$
a (real) Hilbert space for the inner product $\langle
S,T\rangle_{\mathbb{H}_{r}}=\sum_{j,k=1}^{\infty}s_{jk}t_{jk}=\sum_{j,k=1}^{\infty}\langle
S\phi_{j}^{(r)},\phi_{k}^{(0)}\rangle_{L^{2}(\partial D)}\langle
T\phi_{j}^{(r)},\phi_{k}^{(0)}\rangle_{L^{2}(\partial D)}.$ We can also view
$\mathbb{H}_{r}$ as the subset (closed under vector addition and _real_ scalar
multiplication) of $\mathcal{L}_{2}(H^{r}(\partial
D)/\mathbb{C},L^{2}_{\diamond}(\partial D))$ comprising those operators which
map real-valued functions to real-valued functions.

$Y=\tilde{\Lambda}_{\gamma}+\varepsilon\mathbb{W}$ the observed data in model
12, where $\mathbb{W}$ is a Gaussian white noise indexed by the Hilbert space
${\mathbb{H}_{r}}$, and $\varepsilon$ is a noise level which tends to zero for
our asymptotic results. Concrete interpretation of this model given in 11.

$P^{\gamma}_{\varepsilon}=P^{\gamma}_{\varepsilon,r}$ the law of $Y$,
$E^{\gamma}_{\varepsilon}$ the corresponding expectation operator,
$\operatorname{Var}_{\gamma}$ the corresponding variance operator.

$p^{\gamma}_{\varepsilon}(Y)=\exp\big{(}\frac{1}{\varepsilon^{2}}\langle
Y,\tilde{\Lambda}_{\gamma}\rangle_{\mathbb{H}_{r}}-\frac{1}{2\varepsilon^{2}}\lVert\tilde{\Lambda}_{\gamma}\rVert_{\mathbb{H}_{r}}^{2}\big{)}$
the probability density of the law of $Y$ w.r.t. the law of
$\varepsilon\mathbb{W}$.

$\ell(\gamma)=\log p^{\gamma}_{\varepsilon}$ the log-likelihood function.

$\xi_{\varepsilon,\delta}=(\log(\varepsilon^{-1}))^{-\delta}$ for
$\varepsilon,\delta>0$.

$\Pi$ a prior for $\theta\in C_{u}(D)$ described in Section 3.1. $\Pi$ also
denotes the induced prior on $\gamma=\Phi\circ\theta\in\Gamma_{m_{1},D_{1}}$
and the induced prior on $\Lambda_{\gamma}$.

$\Pi(\mathrel{\;\cdot\;}\mid Y)$ the corresponding posterior.
$E^{\Pi}[\mathrel{\;\cdot\;}\mid Y]$ the posterior expectation.

$(\mathcal{H},\lVert\cdot\rVert_{\mathcal{H}})\subseteq(H^{\alpha}(D),\lVert\cdot\rVert_{H^{\alpha}(D)})$
the RKHS of a base prior $\Pi^{\prime}$ from which $\Pi$ is constructed.

$\Phi$ a ‘regular link function’ used in the prior construction (see Section
3.1).

$\zeta:D\to[0,1]$ a smooth cutoff function used in the prior construction.

$\pi_{JK}$ the ${\mathbb{H}_{r}}$-orthogonal projection map onto
$\operatorname{span}\\{b_{jk}^{(r)}:j\leq J,k\leq K\\}$ (see 24).

$K(P,Q)=K(p,q)=E_{X\sim p}\log((p/q)(X))$ for distributions $P,Q$ with
densities $p,q$ (the _Kullback–Leibler_ , or just KL, divergence)

$B_{KL}^{\varepsilon}(\eta)=\\{\theta\in
C_{u}(D):K(p^{\theta_{0}}_{\varepsilon},p^{\Phi\circ\theta}_{\varepsilon})\leq(\eta/\varepsilon)^{2},\operatorname{Var}_{\gamma_{0}}(\log(p^{\theta_{0}}_{\varepsilon}/p^{\Phi\circ\theta}_{\varepsilon}))\leq(\eta/\varepsilon)^{2}\\}.$

$N(S,\rho,\delta)$ the covering numbers of the set $S$ for metric $\rho$, i.e.
the smallest number of $\rho$-balls of radius $\delta$ needed to cover $S$.

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Départment de Mathématiques d’Orsay
Université Paris-Sud
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Department of Pure Mathematics and Mathematical Statistics
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