11institutetext: The Australian National University
11email<EMAIL_ADDRESS>
http://www.michaeltimothybennett.com/

# Is Complexity an Illusion?

Michael Timothy Bennett
11 0000-0001-6895-8782

###### Abstract

Simplicity is held by many to be the key to general intelligence. Simpler
models tend to “generalise”, identifying the cause or generator of data with
greater sample efficiency. The implications of the correlation between
simplicity and generalisation extend far beyond computer science, addressing
questions of physics and even biology. Yet simplicity is a property of form,
while generalisation is of function. In interactive settings, any correlation
between the two depends on interpretation. In theory there could be no
correlation and yet in practice, there is. Previous theoretical work showed
generalisation to be a consequence of “weak” constraints on implied by
function, not form. Experiments demonstrated choosing weak constraints over
simple forms yielded a $110-500\%$ improvement in generalisation rate. Here we
show that all constraints can take equally simple forms, regardless of
weakness. However if forms are spatially extended, then function is
represented using a finite subset of forms. If function is represented using a
finite subset of forms, then we can force a correlation between simplicity and
generalisation by making weak constraints take simple forms. If function
determined by a goal directed process (e.g. natural selection), then
efficiency demands weak constraints take simple forms. Complexity has no
causal influence on generalisation, but appears to due to confounding.

##  1 Introduction

Complexity is a quality of systems we find difficult to understand. Formal
analogues include entropy [1], compression [2, 3] and even fractal dimension
[4]. Physicist Leonard Susskind believes complexity may be the key to a
unified theory of physics [5, 6]. Cyberneticist Francis Heylighen recently
argued that goals are attractors of dynamical systems that self organise in
complex reaction networks [7]. If complex reaction networks do self organise
as he argues [8], then it goes some way towards explaining the origins of goal
directed behaviour, and thus life. Finally, many hold that complexity is the
key to general intelligence. Language models like GPT-4 amount to compressed
representations of human language [9]. Simpler objects can be compressed to
greater extents, because they exhibit self similarity. As a result, some hold
that compression is general intelligence, meaning a general reinforcement
learning agent like AIXI [10] can use Solomonoff Induction [11] to maximise
expected reward across a wide range of environments. Yet for all that
complexity seems to be at the heart of every matter, it has profound flaws. As
a qualitative indicator of how subjectively difficult a system might be to
understand, it makes perfect sense. It makes far less as an indicator of
anything objective. Ockham’s Razor is the epistemic principle that simpler
statements are more likely to hold true. It can be understood as the claim
that our subjective perceptions of complexity reflect an objective property of
our environment. There is no obvious reason this should be the case, and yet
it is [12]. Simpler statements tend to be more accurate representations of
reality. The aforementioned Solomonoff Induction formalises Ockham’s Razor,
meaning AIXI is based on the premise that simpler models are more accurate
representations of the environment. AIXI is a superintelligence in the sense
that it maximises Legg-Hutter intelligence [13]. However, Jan Leike later
showed that any claim regarding AIXI’s performance is “entirely subjective”.
Legg-Hutter intelligence is measured with respect to a fixed Universal Turing
Machine (UTM), and AIXI is only optimal if it uses exactly the same UTM [14].
This calls into question the viability of complexity based induction systems
in interactive settings. Their performance is subjective, and from a pragmatic
standpoint it is only objective performance claims that matter. Leike’s result
suggests there could be no correlation between objective performance and
subjective complexity. This concurs with what seems intuitively obvious, that
complexity is an aspect of interpretation. Complexity is a measure of form,
not function. So why does the subjective perception of simplicity tend to
correlate with objective performance?

####  1.0.1 What exactly is complexity supposed to indicate?:

As it is used in AGI research [10], complexity is intended to help us infer
the program which generated or caused data. If one can identify that which
caused past data, then one can “generalise” to predict the outcomes in future
interactions, to maximise performance [15]. We are concerned with adaptation
or “the ability to generalise” [16], not any specific circumstance. This is
because any system can eventually identify cause given enough data and memory
(by simply rote learning every outcome until it has a complete behavioural
specification of the causal program). So assuming one can correctly infer
cause, then the amount of data one requires to do so is the sole measure of
performance. We refer to this as sample efficiency. The more sample
efficiently one can infer cause, the greater one’s ability to generalise and
adapt to any desired end. Thus, we take intelligence to be a measure of the
sample efficiency in generalisation.

####  1.0.2 Key questions:

We build upon previous work [17, 15], in which:

  1. 1.

Maximising simplicity of policies was proven unnecessary and insufficient to
maximise sample efficiency [17, prop. 3].

  2. 2.

Maximising policy “weakness” was proven necessary and sufficient to maximise
sample efficiency [17, prop. 1,2] and identify cause [15]. In experiments,
weak policies outperformed simple by $110-500\%$.

Our purpose here is to extend this work, and to establish:

  1. 1.

Is complexity just an artefact of abstraction?

  2. 2.

Why do sample efficiency and simplicity tend to be correlated?

Results:  We begin by presenting a formalism. Our results are only meaningful
if one accepts that our formalism is reflective of reality, so we provide an
argument to the effect that it is (lest we be accused of straw-manning
complexity). Second, we show that the complexity of all behaviours is equal in
the absence of an abstraction layer (a general formalisation of any
interpreter). In other words, complexity is a subjective “illusion”. We
further show that if the vocabulary is finite then weakness can confound
simplicity and sample efficiency. Third, we argue that abstraction is goal
directed. If the vocabulary is finite, and tasks uniformly distributed, then
weak statements take simple forms.

##  2 The Formalism

The following definitions are shared with [18, 19], which apply them to
biological and philosophical perspectives.

######  Definition 1 (environment)

.

  * •

We assume a set $\Phi$ whose elements we call states.

  * •

A declarative program is $f\subseteq\Phi$, and we write $P$ for the set of all
declarative programs (the powerset of $\Phi$).

  * •

By a truth or fact about a state $\phi$, we mean $f\in P$ such that $\phi\in
f$.

  * •

By an aspect of a state $\phi$ we mean a set $l$ of facts about $\phi$ s.t.
$\phi\in\bigcap l$. By an aspect of the environment we mean an aspect $l$ of
any state, s.t. $\bigcap l\neq\emptyset$. We say an aspect of the environment
is realised111Realised meaning it is made real, or brought into existence. by
state $\phi$ if it is an aspect of $\phi$.

######  Definition 2 (abstraction layer)

.

  * •

We single out a subset $\mathfrak{v}\subseteq P$ which we call the vocabulary
of an abstraction layer. The vocabulary is finite unless explicitly stated
otherwise. If $\mathfrak{v}=P$, then we say that there is no abstraction.

  * •

$L_{\mathfrak{v}}=\\{l\subseteq\mathfrak{v}:\bigcap l\neq\emptyset\\}$ is a
set of aspects in $\mathfrak{v}$. We call $L_{\mathfrak{v}}$ a formal
language, and $l\in L_{\mathfrak{v}}$ a statement.

  * •

We say a statement is true given a state iff it is an aspect realised by that
state.

  * •

A completion of a statement $x$ is a statement $y$ which is a superset of $x$.
If $y$ is true, then $x$ is true.

  * •

The extension of a statement $x\in{L_{\mathfrak{v}}}$ is
$E_{x}=\\{y\in{L_{\mathfrak{v}}}:x\subseteq y\\}$. $E_{x}$ is the set of all
completions of $x$.

  * •

The extension of a set of statements $X\subseteq{L_{\mathfrak{v}}}$ is
$E_{X}=\bigcup\limits_{x\in X}E_{x}$.

  * •

We say $x$ and $y$ are equivalent iff $E_{x}=E_{y}$.

(notation) $E$ with a subscript is the extension of the subscript222e.g.
$E_{l}$ is the extension of $l$..

(intuitive summary) $L_{\mathfrak{v}}$ is everything which can be realised in
this abstraction layer. The extension $E_{x}$ of a statement $x$ is the set of
all statements whose existence implies $x$, and so it is like a truth table.

######  Definition 3 ($\mathfrak{v}$-task)

For a chosen $\mathfrak{v}$, a task $\alpha$ is a pair
$\langle{I}_{\alpha},{O}_{\alpha}\rangle$ where:

  * •

${I}_{\alpha}\subset L_{\mathfrak{v}}$ is a set whose elements we call inputs
of $\alpha$.

  * •

${O_{\alpha}}\subset E_{I_{\alpha}}$ is a set whose elements we call correct
outputs of $\alpha$.

${I_{\alpha}}$ has the extension $E_{I_{\alpha}}$ we call outputs, and
${O_{\alpha}}$ are outputs deemed correct. $\Gamma_{\mathfrak{v}}$ is the set
of all tasks given $\mathfrak{v}$.

(generational hierarchy) A $\mathfrak{v}$-task $\alpha$ is a child of
$\mathfrak{v}$-task $\omega$ if ${I}_{\alpha}\subset{I}_{\omega}$ and
${O}_{\alpha}\subseteq{O}_{\omega}$. This is written as
$\alpha\sqsubset\omega$. If $\alpha\sqsubset\omega$ then $\omega$ is then a
parent of $\alpha$. $\sqsubset$ implies a “lattice” or generational hierarchy
of tasks. Formally, the level of a task $\alpha$ in this hierarchy is the
largest $k$ such there is a sequence
$\langle\alpha_{0},\alpha_{1},...\alpha_{k}\rangle$ of $k$ tasks such that
$\alpha_{0}=\alpha$ and $\alpha_{i}\sqsubset\alpha_{i+1}$ for all $i\in(0,k)$.
A child is always “lower level” than its parents.

(notation) If $\omega\in\Gamma_{\mathfrak{v}}$, then we will use subscript
$\omega$ to signify parts of $\omega$, meaning one should assume
$\omega=\langle{I}_{\omega},{O}_{\omega}\rangle$ even if that isn’t written.

(intuitive summary) To reiterate and summarise the above:

  * •

An input is a possibly incomplete description of a world.

  * •

An output is a completion of an input [def. 2].

  * •

A correct output is a correct completion of an input.

####  2.0.1 Learning and inference definitions:

Inference requires a policy and learning a policy requires a proxy, the
definitions of which follow.

######  Definition 4 (inference)

.

  * •

A $\mathfrak{v}$-task policy is a statement $\pi\in L_{\mathfrak{v}}$. It
constrains how we complete inputs.

  * •

$\pi$ is a correct policy iff the correct outputs $O_{\alpha}$ of $\alpha$ are
exactly the completions $\pi^{\prime}$ of $\pi$ such that $\pi^{\prime}$ is
also a completion of an input.

  * •

The set of all correct policies for a task $\alpha$ is denoted
$\Pi_{\alpha}$.333To repeat the above definition in set builder notation:
${\Pi_{\alpha}}=\\{\pi\in L_{\mathfrak{v}}:{E}_{I_{\alpha}}\cap
E_{\pi}={O_{\alpha}}\\}$

Assume $\mathfrak{v}$-task $\omega$ and a policy $\pi\in L_{\mathfrak{v}}$.
Inference proceeds as follows:

  1. 1.

we are presented with an input ${i}\in{I}_{\omega}$, and

  2. 2.

we must select an output $e\in E_{i}\cap E_{\pi}$.

  3. 3.

If $e\in{O}_{\omega}$, then $e$ is correct and the task “complete”.
$\pi\in{\Pi}_{\omega}$ implies $e\in{O}_{\omega}$, but $e\in{O}_{\omega}$
doesn’t imply $\pi\in{\Pi}_{\omega}$ (an incorrect policy can imply a correct
output).

(intuitive summary) To reiterate and summarise the above:

  * •

A policy constrains how we complete inputs.

  * •

A correct policy is one that constrains us to correct outputs.

In functionalist terms, a policy is a “causal intermediary”.

######  Definition 5 (learning)

.

  * •

A proxy $<$ is a binary relation on statements, and the set of all proxies is
$Q$.

  * •

$<_{w}$ is the weakness proxy. For statements $l_{1},l_{2}$ we have
$l_{1}<_{w}l_{2}$ iff $\lvert E_{l_{1}}\rvert<\lvert E_{l_{2}}\rvert$.

  * •

$<_{d}$ is the description length or simplicity proxy. We have
$l_{1}<_{d}l_{2}$ iff $\lvert l_{1}\rvert>\lvert l_{2}\rvert$.

By the weakness of a statement, we mean the cardinality of its extension. By
the weakness of an extension we mean its cardinality. Likewise, when we speak
of simplicity with regards to a statement, we mean its cardinality. The
complexity of an extension is the simplest statement of which it is an
extension.

(generalisation) A statement $l$ generalises to a $\mathfrak{v}$-task $\alpha$
iff $l\in\Pi_{\alpha}$. We speak of learning $\omega$ from $\alpha$ iff, given
a proxy $<$, $\pi\in{\Pi}_{\alpha}$ maximises $<$ relative to all other
policies in ${\Pi}_{\alpha}$, and $\pi\in{\Pi}_{\omega}$.

(probability of generalisation) We assume a uniform distribution over
$\Gamma_{\mathfrak{v}}$. If $l_{1}$ and $l_{2}$ are policies, we say it is
less probable that $l_{1}$ generalizes than that $l_{2}$ generalizes, written
$l_{1}<_{g}l_{2}$, iff, when a task $\alpha$ is chosen at random from
$\Gamma_{\mathfrak{v}}$ (using a uniform distribution) then the probability
that $l_{1}$ generalizes to $\alpha$ is less than the probability that $l_{2}$
generalizes to $\alpha$.

(sample efficiency) Suppose $\mathfrak{app}$ is the set of all pairs of
policies. Assume a proxy $<$ returns $1$ iff true, else $0$. Proxy $<_{a}$ is
more sample efficient than $<_{b}$ iff

$\left(\sum_{(l_{1},l_{2})\in\mathfrak{app}}\lvert(l_{1}<_{g}l_{2})-(l_{1}<_{a}l_{2})\rvert-\lvert(l_{1}<_{g}l_{2})-(l_{1}<_{b}l_{2})\rvert\right)<0$

(optimal proxy) There is no proxy more sample efficient than $<_{w}$, so we
call $<_{w}$ optimal. This formalises the idea that “explanations should be no
more specific than necessary” (see Bennett’s razor in [17]).

(intuitive summary) Learning is an activity undertaken by some manner of
intelligent agent, and a task has been “learned” by an agent that knows a
correct policy. Humans typically learn from “examples”. An example of a task
is a correct output and input. A collection of examples is a child task, so
“learning” is an attempt to generalise from a child, to one of its parents.
The lower level the child from which an agent generalises to parent, the
“faster” it learns, the more sample efficient the proxy. The most sample
efficient proxy is $<_{w}$.

##  3 Arguments and Results

The distinction between software and hardware is unsuitable for reasoning
about cause [15]. The performance of a software agent in an interactive
setting is subjective [14], as its behaviour depends on hardware which
interprets it. Hardware is an “abstraction layer” between software and the
surrounding environment. If we are to understand complexity, we must
understand what the concept entails in the absence of such abstraction layers.
We must ascertain what is objective rather than subjective, and so we must
begin at the level of the environment rather than the agent. To this end,
previous work [17, 15, 20] proposed a “de facto” pancomputational [21] model
of all conceivable environments and aspects thereof, which we refine and
extend. While the formalism used here is a departure from past work, it is
equivalent with respect to those previously published proofs we reference
[17]. The formalism is not computational in the sense of relying on symbols,
quantities, or any other high level abstraction interpreted by a human mind.
The assumptions we make are extremely weak.

Axiom 1:  When there are things, we call these things the environment.

Axiom 2:  The environment has at least one state444Or, if the reader prefers,
there are as many environments as there are states.. If there is more than one
state, then there is at least one “dimension” along which things can differ.

A dimension is a set of points, for example time. We do not make the
additional claim that such sets of points must be ordered. Each state is the
environment at a different point in one or more dimensions. States are
“extended” along dimensions, meaning no two states can occupy the same points
in all dimensions (in other words, states are like the environment perceived
from different positions in time or some other dimensions by an omniscient
observer). These are all the assumptions we need for our version of
pancomputationalism. We don’t even need to speculate about any internal
structure states might have, or what dimensions might be. Any “fact” about a
state’s internal structure can be defined by its relation to other states. The
existence of sets is implied by the existence of states (because there is more
than one “thing” that exists), and a fact is just the set of states in which
it holds (the truth conditions of a “declarative program”).

Universality Claim:  Axioms 1 and 2 hold for every conceivable environment.

As truth is defined in existential terms, there is nothing which is not a fact
of the environment, and no environment which we cannot describe as a set of
facts. In other words, this is a minimalist formalism of everything. An
environment could be like our own, or not. It could be deterministic, in which
case states follow a sequence. It could be non-deterministic, in which case
they don’t555In any case, the difference between deterministic and non-
deterministic seems meaningless when you consider that DFAs and NFAs are
equivalent [22].. It could be fantastical with magic and true names. It could
even be a world constructed through the subjective experience of its
inhabitants. All that matters is that an environment has states, and from the
relations between them we obtain the set of all facts. Facts as relations
between states let us avoid anything like a universal set, because states are
otherwise irreducible. An aspect of a state is just a set of facts about that
state. With aspects in hand, we can define abstraction. An aspect is akin to a
logical statement. It has a truth value given a state. We also define its
extension (def. 2), which is all other aspects of which it is a part. This
implies a heirarchy or “lattice” of aspects. Intuitively, an abstraction layer
is like a window through which one can view part of the environment. A laptop
computer could function as an abstraction layer, as could all or part of the
system in which an embodied and embedded organism enacts cognition [23]. In
precise terms an abstraction layer is implied by a vocabulary, which is a set
of declarative programs. A vocabulary implies a formal language whose rules
are determined by relations between states. An abstraction layer implies a set
of “$\mathfrak{v}$-tasks” (def. 3), each of which is behaviour that defines a
system. The formalisation of policies as causal intermediaries between inputs
and outputs [24] then develops this into a causal depiction of goal directed
behaviour (assuming a policy is implied by the inputs and outputs). For
example, an organism could be a policy for a $\mathfrak{v}$-task which is
behaviour that organism might enact. Again, we must emphasise this is a first
principles approach. We do not assume symbols and Turing machines. Inputs,
outputs and policies are all just sets of declarative programs. Whether
something is goal directed is determined by the relations between states. The
environment makes only one sort of value judgement (existence or non-
existence), and is otherwise impartial. Goal directed behaviour is a value
judgement, so we formalise this impartiality as a uniform distribution over
tasks. Finally, we must now define complexity. The claims we make in the rest
of this paper pertain to this notion of complexity.

Complexity of Extension:  If two statements have the same extension, then the
simpler is the smaller set (def. 5). The complexity of an extension is the
simplest statement of which it is the extension. This is like other formal
notions of complexity [2, 3], but facilitates comparison of abstraction
layers.

###  3.1 Implications for Complexity

######  Proposition 1 (subjectivity)

If there is no abstraction, complexity can always be minimized without
improving sample efficiency, regardless of the task.

###### Proof

In accord with definition 2, the absence of abstraction means the vocabulary
is the set of all declarative programs, meaning $\mathfrak{v}=P$. It follows
that for every $l\in L_{\mathfrak{v}}$ there exists $f\in\mathfrak{v}$ such
that $\bigcap l=f$. Statements $l$ and $\\{f\\}$ are equivalent iff
$E_{l}=E_{\\{f\\}}$, which is exactly the case here because $\bigcap l=f$.
[17, prop. 1,2] shows maximising weakness is necessary and sufficient to
maximise the probability of generalisation, which means weakness maximises
sample efficiency (is the optimal proxy). This means sample efficiency is
determined by the cardinality of extension. For every correct policy $l$ of
every task in $\Gamma_{\mathfrak{v}}$ there exists $f\in\mathfrak{v}$ s.t.
$E_{l}=E_{\\{f\\}}$. Policy complexity can be minimised regardless weakness,
because the simplest representation of every extension is simplicity $1$.
$\blacksquare$

In this sense, complexity is an illusion created by abstraction. In the
absence of any particular abstraction, all behaviours (extensions) are implied
by statements of the same complexity. To be clear, we are not repeating the
claim made by others [14] that if the interpreter used by a complexity based
induction system matches one used to to compute an objective value for
complexity, then that induction system will be optimal in the sense of
eventually learning the correct policy666To quote verbatim: “Legg-Hutter
intelligence is measured with respect to a fixed UTM. AIXI is the most
intelligent policy if it uses the same UTM.” [14]. We are claiming that if
interpretation is truly objective, then $\mathfrak{v}=P$ and complexity has
absolutely nothing to do with intelligence777Intelligence here meaning not
just eventual generalisation, but the efficiency thereof.. There is no
objective notion of complexity. However, when we take empirical measurements
it is inevitably through an abstraction layer, for which $\mathfrak{v}\neq P$.
In that context simpler forms have been observed to generalise more
efficiently. This raises the question; what additional assumptions can we make
that would explain the correlation?

####  3.1.1 Time, space and causal confounding:

We now make the additional assumption that vocabularies are finite. Every
aspect of the world in which we exist appears to be spatially extended,
meaning no two things occupy the same space at the same time. For the sake of
understanding complexity we assume this is true of all environments. We hold
that this justifies the assumption of a finite vocabulary, because in our
spatially extended environment the amount of information which can influence a
system is finite, within a finite area [25].

######  Proposition 2 (confounding)

If the vocabulary is finite, then policy weakness can confound888$A$ confounds
$B$ and $C$ when for example $A=\mbox{``{\tt badly injured}''}$ causes
$B=\mbox{``{\tt died}''}$ and $C=\mbox{``{\tt picked up by ambulance}''}$, and
it looks like $C$ causes $B$ because $p(B\mid C)>p(B\mid\lnot C)$, and yet it
may be that $p(B\mid C,A)<p(B\mid\lnot C,A)$.sample efficiency with policy
simplicity.

###### Proof

We already have that policy weakness causes sample efficiency, in that it is
necessary and sufficient to maximise it in order to maximise sample
efficiency. Continuing from proof 1, in a finite vocabulary, there may not
exist $f\in\mathfrak{v}$ s.t. $E_{l}=E_{\\{f\\}}$, which means the complexity
of all extensions will not be the same. If we choose any vocabulary in which
weaker aspects take simpler forms, then simplicity will be correlated with
weakness and so will also be correlated with sample efficiency. This means we
would choose $\mathfrak{v}$ s.t. for all $a,b\in L_{\mathfrak{v}}$, the
simpler statement has the larger extension, meaning $a<_{w}b\leftrightarrow
a<_{d}b$. For example, suppose $P=\\{a,b,c…\\},\ a=\\{1,2,4\\},\
b=\\{1,3,4\\},\ \mathfrak{v}=\\{a,b\\},\
L_{\mathfrak{v}}=\\{\\{a\\},\\{b\\},\\{a,b\\}\\}$, then it follows
$\\{a,b\\}<_{w}\\{a\\},\ \\{a,b\\}<_{w}\\{b\\},\ \\{a,b\\}<_{d}\\{a\\},\
\\{a,b\\}<_{d}\\{b\\}$.  $\blacksquare$

####  3.1.2 Why confounding tends to occur:

We now briefly argue that abstraction is goal directed. This means the tasks
an abstraction layer tends to represent are those it is best suited to
represent, which implies weak constraints take simple forms. There are several
reasons an abstraction layer is biased toward particular goals, depending upon
the context in which we consider complexity. In the case of a computer, a
human has specifically designed each abstraction layer to express that which
is needed for a purpose. What separates x86 from a higher level abstraction
layer like Numpy is that the former has a more general intended purpose,
expressing “weaker” constraints. We tend to construct abstraction layers to be
as versatile as possible whilst satisfying a particular need. More generally
natural selection favours adaptation, which means generalisation, which is
maximised by preferring weaker policies [17]. Biological cognition is not
limited to the brain [26], meaning the mind is not neatly confined within a
well defined neurological abstraction layer. Instead the multiscale competency
architectures observed in living organisms [27] amount to self organising
abstraction layers. Because natural selection favours adaptable organisms,
these abstraction layers will be selected to represent the weakest policies
which constitute fit behaviour (a weaker policy is more adaptable). More
generally, we speculate that phase transitions motivate the emergence of self
preserving goal directed behaviour, by destroying some physical structures and
preserving others. Such goal directed abstraction must minimise the size of
vocabularies at higher levels, whilst also maximising the weakness of the
policies they can express (two opposing pressures). This is because a larger a
vocabulary exponentially increases the space of outputs and policies [28],
which may conflict with with finite time and space constraints. A larger
vocabulary would make inference and learning less tractable (more “complex” in
the sense of being a more difficult search problem that takes up more time).
To maximise the weakness of policies in higher levels of abstraction, while
minimising the size of the vocabulary in which they’re expressed, weaker
policies must take simpler forms.

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# Construction of Precision sMDT Detector for the ATLAS Muon Spectrometer
Upgrade

D. Amidei  N. Anderson  A. Chen  E. Carpenter  L. Cooperrider  T. Dai  E.
Diehl  C. Ferretti  Y. Guo  J. Li  X. Meng  K. Nelson  V. Pillsbury  E. Salzer
T. Schwarz  L. Simpson  Z. Wang  C. Weaverdyck  C. Wei  Z. Yang  M. Yuan  B.
Zhou  J. Zhu

###### Abstract

This paper describes the small-diameter monitored drift-tube detector
construction at the University of Michigan as a contribution to the ATLAS Muon
Spectrometer upgrade for the high-luminosity Large Hadron Collider at CERN.
Measurements of the first 30 chambers built at Michigan show that the drift
tube wire position accuracy meets the specification of 20 $\mu$m. The
positions of the platforms for alignment and magnetic field sensors are all
installed well within the required precision. The cosmic ray test measurements
show single wire tracking resolution of $\rm 100\pm 7\;\mu m$ with an average
detection efficiency above $\rm 99$%. The infrastructure, tooling, techniques,
and procedures for chamber production are described in detail. The results
from the chamber quality control tests of the first 30 constructed chambers
are reported.

##  1 Introduction

The ATLAS Muon Spectrometer (MS) is the largest muon spectrometer ever
constructed [1]. Currently in the barrel (pseudo-rapidity $|\eta|<$ 1.05)
Resistive Plate Chambers (RPC) provide fast timing trigger-signals to identify
events containing muons and Monitored Drift-Tube (MDT) chambers provide
precision tracking. In the end-cap (1.05 $<|\eta|<$ 2.7) the muon trigger
comes from Thin Gap Chambers (TGC) and precision tracking from MDTs and in the
innermost region the new Small-Wheel [2] (consisting of MicroMegas and small-
gap TGC detectors). The barrel contains 20 layers of MDTs in three concentric
stations, each with a single tube resolution of 80 $\mu$m, plus 3 stations of
RPCs on the middle and outer MDT stations. The MDTs are arranged in 16 sectors
around the beam line, covering $2\pi$ in alternating "large" and "small"
chamber sizes, referred to as "Large" and "Small" sectors. The ATLAS MS will
be upgraded [3], scheduled in 2026-28, for the high-luminosity LHC (HL-LHC)
operations. One major MS upgrade project is to add an extra triplet-station of
small-gap RPC (sRPC) chambers to the barrel inner MS station to improve the
muon Level-1 trigger efficiency. To accommodate the new sRPCs, the MDTs of 8
Barrel-Inner-Small (BIS) sectors will be replaced with small-diameter (15 mm)
MDTs (sMDT’s) designed and developed by the Max Planck Institute (MPI) [4],
which are half of the thickness of the MDTs.

Figure 1: Drawing of a BIS2 sMDT chamber. Quoted dimensions are in mm.

The 8 Large sectors have space for inserting the new sRPCs, so the current
MDTs will remain. MPI and the University of Michigan (UM) are sharing equally
the production of a total of 100 sMDT chambers. An sMDT chamber consists of 2
tube multi-layers (ML), each containing 4 tube layers of 70 (BIS1) or 58
(BIS2-6) tubes. The geometry of a BIS2 is shown in figure 1, while figure 2
shows a complete BIS4 chamber built at UM.

For the US site, the tubes are constructed and tested at Michigan State
University (MSU), then shipped to UM, where they are fully retested [5] before
chamber construction. A spacer frame (parts made at IHEP111Institute for High
Energy Physics, Protvino, Russia), containing the RASNIK in-plane alignment
system [6] (from NIHKEF222National Institute for Subatomic Physics,Amsterdam,
Netherlands), is glued between the two ML’s. Platforms (from Saclay333Saclay
Nuclear Research Centre, Saclay, France) for mounting the magnetic field
sensors and global alignment devices [7] are glued on top of the chamber,
together with the chamber kinematic mounts on supporting structures (from
IHEP). Temperature sensors assembled with cables (from NIHKEF) are distributed
on both ML’s. The Faraday cages (from IHEP) cover the front-end electronics
(from LMU444Ludwig Maximilian University, Munich, Germany) with 2 HV
distribution boxes (from MPI) mounted on top of them.

(a)

(b)

Figure 2: Chamber view: a) top with platforms, support structures, temperature
cable routing, sensors readout panel, gas pipes, Faraday cages, and HV
splitter boxes. CCC and AP platforms are intended for the global alignment
sensors; (b) side with kinematic mounts on the support structures holding the
chamber to a overhead crane. The spacer bars are visible between the two ML’s.

This paper reports the sMDT chamber construction infrastructure, assembly
procedure, quality control, and performance test results at UM.

##  2 Infrastructure and Tooling for sMDT Construction and Testing

From 2017 to 2020 UM built the infrastructure for sMDT chamber construction as
well as two prototype precision sMDT chambers to certify the tooling and
chamber assembly procedures. The mass production of the sMDT chambers at UM
started in April 2021. The sMDT chamber construction infrastructure and
tooling are described in this section.

###  2.1 High bay and large granite table

The chamber construction makes use of 2 large ($20\times 20$ m2) high bay
research labs (see figure 3(a)) in the UM physics building. These are equipped
with multiple overhead cranes and a clean room portion outfitted with a
$3\times 7$ m2 granite table with 25 micron flatness. This clean room utilizes
a dedicated HVAC system to maintain ${}\pm 0.5^{\circ}C$ temperature stability
and ${}\pm 5\%$ humidity stability. The chamber parts (tubes, structural
elements and platforms) are assembled on the granite table using precision
jigging and glued together using an automated epoxy dispensing machine.

(a)

(b)

Figure 3: (a) Map of the sMDT construction labs in the high bay area. (b) The
glue-machine and a constructed sMDT chamber on the granite table in the
temperature and humidity controlled room.

Installation of the chamber’s gas system and front-end electronics is done
after the chamber is moved to mobile stations in the space outside the clean
room.

A sub-room in the corner (labeled as "ELX Testing" in the room map) was built
to keep the humidity below 40% for chamber cosmic ray testing. Another bay-
area is used for storing and shipping chambers as it has two overhead cranes
and a roll up outside door for loading chambers into a shipping container. A
small clean room in this bay area is used for clean parts storage and gas-bar
pre-assembly.

###  2.2 Automatic glue machine

A large gantry-mounted, computer driven glue dispensing machine was designed
and built at UM. It controls the 3D motion (marked X, Y, and Z on the machine)
of the dispensing tip as well as the drive piston for the epoxy cartridge (A
drive). The gantry is assembled from $\rm 7.6cm\times 15.2cm$ aluminum
extrusions which ride on carriages on rails mounted directly on the surface of
the granite table (see figure 4(a)).

(a)

(b)

Figure 4:  (a) Four motion actuators for 3D motion of the glue dispenser, all
computer-controlled. (b) Gluing the spacer frame on the bottom multi-layer of
a chamber.

The two rails on the granite are aligned to be parallel within 7 micro-
radians. The X drive engages a toothed rail on one side of the granite, which
moves the gantry along the long dimension of the table with a precision of 0.1
mm. The linear rail carriages and rail are kept clean and lubricated to
minimize frictional drag. The Y, Z, and A drives are linear
actuators555https://www.isel.com/en/linear-units-les4.html, and
https://www.isel.com/en/linear-units-les5.html. The gantry Y axis, with a
range of 2.5 m, travels along the length of the tubes (see figure 4(b)). The
actuator along the Z axis has a range of 0.5 m to allow the movement of the
gluing cartridge over the whole vertical dimension of a chamber, from the
first layer to the supporting structures at the top. The A drive expels epoxy
from the cartridge and is programmed to move in steps of 0.26 mm to dispense
0.5 cc epoxy at a time. The control software is written with LabView[8] which
interfaces with the motor drivers provided by the manufacturers. Specific
sequences of gantry motions and epoxy dispensed at each location on the gantry
path are designed and tested to determine the configurations used for gluing
each tube layer, spacer frame, and support structure.

###  2.3 Precision combs

The precision combs are the jigging used to precisely position tubes for
chamber construction and were designed by MPI and made in Germany for both
production sites. Combs are used on both the chamber readout (RO) and high
voltage (HV) ends. Each endplug has a precision brass surface of diameter
$5.00\,\genfrac{}{}{0.0pt}{1}{+0.00}{-0.01}$ mm which is concentric to the
tube-center (therefore to the wire) within 5 microns rms. The endplug
precision surface is captured between two half-holes in adjacent stacked
combs. A full comb set consists of 9 stacked plates with precision holes
(paired half-holes) of diameter $5.00\,\genfrac{}{}{0.0pt}{1}{+0.01}{-0.00}$
mm on the interfaces between adjacent plates (see fig 5(a)). Each plate is
fixed to the one underneath with 17 screws tightened to 7.5 Nm torque. The
base comb is 50 mm thick for rigidity and subsequent combs are 13.077 mm thick
except for a 45.600 mm thick comb which sets the gap for the spacer frame.

The combs locate the tubes on a 15.100 mm horizontal pitch, and a vertical
pitch (tube layer) of 13.077 mm. During assembly of the tubes into a chamber,
the precision cylinder of each tube’s endplugs are placed in the two end-combs
as shown in figure 5(b). The lower comb’s half-cylinders constrain the
horizontal locations of the tubes’ precision cylinders. The upper comb is
screwed to the lower comb to provide a vertical constraint on tube position.

(a)

(b)

Figure 5:  (a) Upper and lower combs constrain the location of the endplugs
with high precision. The smaller holes a the top of the picture allow
insertion of ground anchors which are screwed into the gaps between tubes
after the epoxy has cured; (b) Tubes placed in the chamber construction combs.
The heights of the endplug precision cylinders are measured using a height
gauge.

Besides four base middle combs fixed to the table shown in figure 6(b), there
is another set of four temporary combs used on top of the bottom ML when
gluing the fifth tube layer on the spacer frame bars. These temporary combs
replace the four middle combs on the granite which are not accessible when
gluing the top ML. In addition there is a set of four weights with 7.5 mm
radius semi-circles cut in the bottom surface which are placed on top of each
freshly glued tube layer to control the straightness and potential bowing of
the tubes.

####  2.3.1 Precision jigging survey and setup on granite table

The set up of the combs on the granite was an iterative process, alternating
between adjusting the jigging positions and making precision measurements of
the jigging. Positions were measured with both a FARO-
arm666https://www.faro.com/en/Resource-Library/Tech-Sheet/techsheet-8-axis-
edge-faroarm-scanarm for 3D-measurements with precision of 10 $\mu$m and a
height gauge777https://www.mitutoyo.com/webfoo/wp-
content/uploads/2104_LH-600E.pdf for measuring height relative to the granite
table surface with a precision of 5 $\mu$m. Figure 6(a) shows the two base
precision end-combs which are bolted at opposite ends of two crossbars. The RO
comb is in the foreground and the height gauge is visible measuring the HV
comb at the far side of the table. The middle base-combs, shown in figure
6(b), are connected to the two cross-bars and ensure tube straightness when
tubes are glued. The middle base-combs are aligned both along the tube
direction (yellow-dashed lines in figure 6(b)) and in height to better than 50
$\mu$m using the FARO-arm.

Each end comb base is clamped onto the granite table at two points using 100
microns precision shims under the combs. The combs are also bolted to heavy
side bars which set the length of the jigging. The base combs need to be set
parallel to each other and perpendicular to the wire direction. The wire
direction must be aligned with the glue machine Y axis and this alignment is
periodically verified and the Y gantry position tuned if necessary.

(a)

(b)

Figure 6: Set up of base and mid-combs: (a) the two precision base end-combs
are connected to two sidebars by bolts indicated by the white/red arrows. The
yellow/red arrows point to the hold-down clamps of the base jigging; (b) the
middle base-combs are connected to the two sidebars.

The squareness of the base-comb setup was checked using the Faro-arm to
measure the diagonals defined by precision half-circular cylinders placed on
comb notches 4 and 67 (as in figure 9(a)). These diagonals differ by 15
microns from the average and the angles between the comb assembly and the long
base-plates have a maximum deviation of 0.0018∘ = 0.03 mrad from 90∘, as shown
in figure 7.

Figure 7: Base comb layout geometry and measured distances and angles.

####  2.3.2 Measured precision of the combs

As mentioned previously, there are two sets of precision combs, one for the
chamber RO-end and the other for the HV-end. Figure 8 shows one set of the
combs stacked on the granite table. The comb geometry precision was measured
three ways: comb flatness, and the horizontal and vertical pitches of the tube
positions defined by the holes in the combs.

Figure 8: One set of the precision combs stacked on the granite table.

The flatness of the base-combs was measured with the height gauge using
precision rods placed in the combs precision half-circles (see figure 9(a)).

(a)

(b)

Figure 9: Precision comb geometry measurements: (a) flatness using the height
gauge; (b) wire pitch using the FARO-arm to determine the center positions of
the comb half-cylinders.

Both base combs, when first set up on the granite, had a bowed shape, where
the ends were about 80 microns higher than the middle. This same shape was
seen in subsequent layers as they were added to the base comb. Flattening of
the base combs was achieved by an iterative process of loosening the screws
holding the ends of the base to the side bars, applying downward pressure on
the outer ends and re-tightening the screws followed by height measurements,
and repeating the whole process until the combs were sufficiently flat. The
measured shapes after the flatness tuning of the readout (RO) and high voltage
(HV) side base combs are shown in figure 10 (green data-points). The measured
standard deviation (RMS) of the comb flatness is smaller than 10 $\mu$m for
both base-combs. The initial indexed tube locations above 60 show higher
deviations, at the level of 20 microns. After the construction of the first
twelve sMDT chambers, the RO and HV base comb shapes were re-flattened using
the method described above The blue data-points in figure 10. show and
improvement of the average flatness to 1 $\mu$m or better and of the RMS from
9 $\mu$m to 5 $\mu$m.

Figure 10: Base comb cylinder center height measurements relative to the
target wire height above the granite surface for HV and RO sides, showing the
initial tuning and the re-tuning after assembly of the 12th chamber.

The Faro-arm with a precision spherical probe tip was used to measure the
half-cylinder center positions on the combs to determine the wire pitch (see
figure 9(b)). Measurements of the deviations of the base comb cylinder center
pitch (wire pitch) from the nominal value (15.100 mm) are shown in figure 11.
The horizontal wire pitch were measured to be $15.1017\pm 0.0005$ mm
($15.1019\pm 0.0004$ mm) for the RO (HV) comb, with an RMS of about 4 microns.

Figure 11: Measurements of the base-comb wire pitch, for both HV and RO side.

The complete geometry of the vertical-pitch (tube layer pitch) on both RO and
HV sides was measured on the granite table with 5 microns precision using the
height gauge and shown in figure 12.

Figure 12:  Summary of the measured precision comb vertical tube-pitch, for
both RO and HV side after set up on granite table. Values are in mm.

The tube layer pitch was measured to be $13.077\pm 0.005$ mm ($13.076\pm
0.005$ mm) and the spacer frame height $45.604\pm 0.005$ mm ($45.599\pm 0.005$
mm) for the HV (RO) side.

###  2.4 Spacer frame assembly jigging and in-plane alignment system

The spacer frame consists of five aluminum extrusions with rectangular cross-
sections of dimensions $\rm 880\times 30\times 50\;mm^{3}$. The in-plane
optical devices are housed inside the spacer frame cross-bars as shown in
figure 13(a). The bars and two side support panels are assembled using the
jigging shown in figure 13(b). Precision end blocks are glued to the ends of
the cross-bars and bolted to two side-plates to form a rigid frame. The side
panels are needed to preserve the relative positions of the cross-bars before
the spacer frame cross-bars are glued onto a ML and are removed at the end
chamber assembly.

The spacer frame jigging consists of two rigid aluminum L-shaped bars firmly
bolted to the table at right angles. The angle between them was set using
FARO-arm measurements. The jigging bars are machined with cutouts to match the
side panel profile and several L-shaped brackets are mounted on them to
reinforce the clamping. One side-panel and one cross-bar are clamped to the
jigging during the assembly process to ensure the flatness and perfect
rectangular shape of the frame while the glue is curing.

(a)

(b)

Figure 13:  (a) Assembly of the in-plane system parts into the cross-bar; (b)
spacer frame assembling with a jigging setup on the granite table. The
locations of the in-plane devices, LEDs and CCDs, inside the cross bars are
indicated as well as the four RASNIK lines, (blue lines).

The in-plane alignment system consists of 4 RASNIK lines, 2 diagonal, and 2
straight, along the sides of the chamber. The RASNIK system is a three-point
system (coded mask, lens, CCD) that is sensitive to the relative movement of
any of the three elements. A schematic of a RASNIK system with readout system
(provided by NIKHEF and customized at UM as part of the chamber assembly
tooling) is shown in figure 14. The coded mask is back-illuminated by an LED
and imaged with a lens onto a CCD. Analysis of the image of the coded mask
reveals displacements transverse to the surface of the mask accurate to 1
micron.

The in-plane system has 2 LED-Masks and 2 CCDs. Each CCD views sequentially
images of each LED-Mask to provide one straight and one diagonal measurement.
Reference RASNIK measurements are taken after chamber assembly on the granite
table when the chamber is considered to be in perfect alignment while still
held by the precision jigging. RASNIK measurements taken after the chamber is
moved from the granite reveal chamber deformations when subtracted from the
granite reference RASNIK measurements.

Figure 14: RASNIK system showing LED illuminating a coded mask on left,
imaging lens in the center, and CCD on the right. The CCD image is read out
and analysed by a computer.

###  2.5 Survey of the jigging for sensor platform installation

Two different jigging sets (one for BIS1 and another for BIS2-6 chambers),
provided by MPI, are used to install the sensor platforms on top of each
chamber while it is still on the combs, so that the platform locations can be
set and measured with precision. The combs determine the geometric grid of
wires within a chamber. Knowledge of where one chamber’s wires are relative to
its neighbors in the same plane is provided by a set of Axial-Praxial (AP)
sensors. The exact locations of 3 planes of each AP platform relative to the
wire grid must be accurate within 0.20 mm and known to within 0.03 mm.
Furthermore each chamber has two Hall sensors (for measuring local magnetic
field direction and intensity) and some chambers have CCC (Chamber to Chamber
Connection) sensors for measuring a small (large) chamber’s location relative
to chambers in the adjacent large (small) sector. The AP platforms and CCC
platforms must be glued with X, Y and Z positions accuracy each better than
200 microns compared to nominal designed positions. The precision on the
measured 3D-positions are each required to be within 30 microns when
referenced to the granite surface (Y-axis), endplug surface (X-axis), and the
first wires on both sides of the chamber (Z-axis). The B-sensor platforms must
be glued with positions better than 500 microns compared to nominal designed
positions.

(a)

(b)

Figure 15: (a) Four AP plates on the top corners of a sMDT chamber, each held
by a screw to a L-shaped jigging; (b) a FARO-arm probe is used to check and
adjust locations of AP platforms before and after gluing. Final positions are
measured after glue is cured.

The AP jigging references the comb precision holes to set each AP platform
within 40 microns of a reference location. The AP jigging supporting
structures, two close to the end-combs and four mounted on the base cross
bars, are tuned and measured so that all the surfaces of the L-shaped jigging
are parallel to the surface of the granite table. Each L-shaped jigging (see
figure 15(a)) holds one platform and allows fine 3D adjustments to be made via
6 contact points, two referenced to the first wire, one relative to the tube
RO endplug surface and three to the surface of the granite table. Each AP-
plate’s position is verified by FARO-arm measurements as shown in figure 15(b)
before and after gluing.

(a)

(b)

Figure 16: (a) Assembly plate to mount the B-field and CCC platforms for
installation on chamber; (b) survey of the big plate with the FARO-arm.

The CCC-plates and B-sensor plates are mounted on a big precision plate where
their positions are fixed relative to four blocks matching the AP-plate
positions on the chamber (see figure 16(a)). Surveys on two assembly plates
(for BIS1 and BIS2-6 chambers) were done using the FARO-arm on the granite
table (see figure 16(b)). Mistakes were found on both of them. One was fixed
by removing one of the four blocks and re-gluing it on the correct position on
the plate. The other plate was rectified by gluing a 120 microns shim to the
reference block which sets the X position of the platforms on the chamber,
thus correcting the relative position of the platforms respect to the block.
All the fixes were further verified by dry runs of installation of the
platforms on a chamber.

###  2.6 Chamber test stations

Two chamber test stations were designed and built at UM. One is used for
installation of on-chamber services (temperature sensors, cables, on-chamber
gas system, temperature and RASNIK readout panel, HV and RO electronics
hedgehog cards888Hedgehog cards are front-end electronics cards which attach
directly to the tube endplugs., and Faraday cages) and for performing a gas
tightness test. The other test station is dedicated to electronics and cosmic
ray tests.

####  2.6.1 Chamber service installation and leak test station

The chamber services installation is done by mounting the chamber on a
rotation cart consisting of a chamber-frame (provided by Protvino) mounted on
rotation bearings on a movable base (figure 17 (a)). The chamber is attached
to the frame using the chamber kinematic mounts. On the cart the chamber can
be rotated through $360^{\circ}$ on the base allowing easy access of all parts
of the chamber, for example, when gluing temperature cables on top and bottom
of the chamber. Two removable small tables can be connected to the chamber-
frame for services installation tooling and parts. The rotation feature is
also used to do a test of the in-plane alignment system by taking RASNIK
measurements at different rotation angles and checking for a characteristic
set of deflection measurements. Chamber ground cables, front-end hedgehog
cards and Faraday cage installations on chambers are carried out in this
station. An end-view of a chamber mounted on the rotation cart is shown in
figure 17(b), where one gas-bar is installed on the bottom ML of the chamber,
and one gas-bar is on the small table ready to be installed.

(a)

(b)

Figure 17: (a) A rotation cart for chamber service installation and testing;
(b) Installation of the gas-bars on the rotation cart.

####  2.6.2 Cosmic ray test station

The chamber cosmic ray testing is performed inside a room where humidity is
kept below 40%. The station itself consists of a scintillator trigger system
mounted in a movable frame, a chamber cart with a gas control panel, HV/LV
power supplies, and a data acquisition (DAQ) system [9]. The schematics of the
cosmic ray test system, including an sMDT chamber, trigger detector, front-end
electronics, the DAQ modules and data flow, is shown in figure 18.

Figure 18: Block diagram of the sMDT readout electronics and the DAQ system.

Figure 19 (a) shows an sMDT chamber on the mobile cart with the scintillator
($1.2\times 0.6~{}m^{2}$) placed on top to provide fast trigger signal for the
cosmic ray test.

(a)

(b)

Figure 19: (a) Scintillation pad on a mobile support over a test chamber on a
mobile cart connected with gas and HV; (b) online DAQ and monitoring systems
for the cosmic ray test.

The DAQ system (see figure 19(b)) is located outside the humidity-controlled
chamber test room and consists of two computers (a Windows computer for DAQ
configuration and controls and a linux computer for data-taking), a VME crate
containing the DAQ interface and trigger modules, HV power supplies for
chamber and for the trigger scintillator, and a NIM crate containing the
trigger coincidence modules. The firmware of the DAQ and the software for
online data monitoring were developed at Michigan. During the cosmic ray test
the muon tracks, hit map, and the ADC and TDC spectra for each sMDT tube are
displayed online, as shown in figure 19(b).

##  3 sMDT Base Chamber Construction on Granite Table

This section describes the assembly of the base chamber on the granite table
using the jigging described in Section 2 as well as measurements of the
chamber mechanical precision done on the granite table after chamber assembly.

###  3.1 Spacer frame and in-plane alignment system assembly

The spacer frame parts are first cleaned with IPA and then are assembled with
the in-plane devices and tested before being glued to the chamber. The
mechanical shape of the frame is determined by the assembly jigging while its
flatness, referenced to the surface of the granite table, is checked with 25
micron shims.

In the spacer frame the optical devices used in the four RASNIK lines are
installed as shown in figure 13. The LED+mask and CCDs are mounted on the end
cross-bars of the spacer frame, and the lenses for each RASNIK line are
mounted on the center cross-bar. Images are taken after spacer frame is
assembled to certify that the hardware is fully functional. Final reference
RASNIK measurements are taken after the chamber is fully assembled and still
held by the precision combs.

###  3.2 Gluing an sMDT base chamber

The tubes that have passed the quality control tests (including straightness,
wire tension, gas leak, and dark-current measurements [5]) are cleaned with
IPA before chamber gluing. They are arranged in eight groups by layer and
their barcode ID is recorded in the database together with their final
positions on the chamber. In addition, the precision combs are also cleaned
with IPA and the tube ground screws inserted into the dedicated holes in the
combs. An sMDT chamber is then assembled by gluing eight tube layers and a
spacer frame with the precision jigging set up on the granite table. This
sequence takes 5 days: two days to glue the bottom multi-layer, one day to
glue the spacer frame and the first tube layer of the second multi-layer, and
the last two days to glue the remaining three tube layers on the top multi-
layer.

Tube layer gluing is carried out by two operators (as in figure 20) placing
the tubes on precision combs with all endplug plastic insulators adjusted to
touch the RO-side end comb, so that tubes are lined up to allow proper
electronics RO card installation ater on. The automatic glue-machine dispenses
Araldite 2011999https://krayden.com/technical-data-sheet/huntsman-
araldite-2011-technical-data-sheet/ glue dots at 11 equidistant positions in
each groove between the tubes.

(a)

(b)

Figure 20: (a) Two operators are placing a tube on the comb following the
automatic glue dispenser on the lower layer of the tubes. (b) Weights on tube
layers after gluing.

The pre-assembled spacer frame is glued between the two multi-layers (MLs) of
a chamber using
DP190101010https://multimedia.3m.com/mws/media/1235570O/dp190-scotch-weld-
technical-data-sheet.pdf epoxy. The glue is dispensed on the top of the bottom
ML tubes in a pattern following the spacer frame extrusion profile. Four
temporary combs are placed on top of the bottom ML to help constrain the tube
positions when the first tube layer of the top ML is placed on the two end
combs and glued on the spacer bars.

After a layer of tubes is glued with the lower assembly combs, the next level
of combs are screwed on the top of the former one on both sides with a torque
of 7.5 Nm for each screw.

After the tubes are glued on the combs each day, 4 weights, each with the
bottom side machined to reproduce a tube layer half-cylinder profile, are
placed on top of the tube layer while supported on precision riser blocks, as
shown in figure 20(b). The riser blocks set both the height and the lateral
location of the weights, and are also used to prevent the outermost tube from
bowing out. After the glue cures overnight, the weights are removed, and the
ground-screws pre-loaded on comb layers 2, 3, 4, 6, 7, and 8 (counting from
the bottom) are driven into the gap between the tubes.

###  3.3 Installation of the platforms and the chamber supporting structure

The ATLAS Muon spectrometer utilizes two types alignment sensors to determine
the relative position of each chamber. The Axial-Praxial (AP) alignment system
uses RASNIK sensors, and the CCC alignment system using SacCam sensors [3].
The AP system requires four platforms on each chamber, and the CCC system
requires 1 or 2 platforms on selected chambers. Each chamber also utilizes two
magnetic field sensors, which require two B-sensor platforms on each chamber.
The location of each platform must be known to better than 30 $\mu$m, and the
platforms must be glued on the chamber within 200 $\mu$m (500 $\mu$m in the
case of the B-sensor platforms) of their target positions. The installation
process of the plates and support structures takes two days: one day to glue
the AP plates, and another day to glue the platforms for the CCC alignment and
the B-field sensors, as well as the chamber support structures.

The AP-plates installation jigging was described in Section 2.5. The FARO-arm
is mounted on the base cross-bar of the chamber jigging and used to set up the
coordinate system for the two AP-plates’ installation on the same side. The
FARO-arm local coordinate system is defined by three perpendicular surfaces:
the granite table, the side surface of the base cross bar, and the outside
surface of the RO side end comb. The normal directions to these three planes
form the coordinate axes (X, Y, Z). The intersection point of three surfaces
determine the origin of the FARO-arm coordinate system. The cylinder centers
of the top layer end combs are measured using FARO-arm to define the line of
the first wire in this coordinate system. The FARO-arm measures then the 3D
distances of an AP plate relative to the tube RO endplug surface (X), to the
granite surface (Y), and to the first (or last) wire (Z), shown in figure 21.

Figure 21: AP and CCC plates on top of the chamber. The blue and yellow arrows
indicate the measurement points relative to the reference wires (in red) and
the inner RO comb surface.

Placing the AP-plates in correct positions is critical, since the assembly
plate jigging (described in Section 2.5) relies on the AP plates to set the
position of for B-field and CCC platforms. Multiple measurements with the
FARO-arm and adjustments of the set screws are made for each AP plate before
gluing to ensure the AP plate position is within the tolerances.
DP490111111https://multimedia.3m.com/mws/media/82790O/dp490-scotch-weld-tm-
adhesive.pdf epoxy is applied to the top layer of the tubes to bind each AP-
plate, then a new set of measurements is taken and further adjustment of the
AP plate position is performed if needed. The final position of the AP plates
is measured after the glue has cured.

The CCC and B-sensor plates are mounted on the assembly plate and glued on to
the chamber after the AP-plate glue is set. Figure 22(a) shows applying glue
on a B-sensor plate which is mounted on the assembly plate prior to gluing on
chamber. Figure 22(b) and (c) show lowering the assembly plate on top of the
chamber, and checking that the assembly plate blocks are correctly positioned
on the AP-plates.

(a)

(b)

(c)

Figure 22: (a) Glue being applied to B-field and CCC-plates on the assembly
plate; (b) assembly plate being placed on the AP-plates; (c) check of the
assembly plate alignment with the AP-plates on chamber.

The chamber support structures (one can be seen in figure 21 clearly) contain
the rail support bearings. Three kinematic mounting points, required for the
chamber installation in the ATLAS experiment, are located in these structures.
A 150 micron thick insulating tape is applied on the bottom side of the two
support structures after they have been cleaned using IPA. DP190 glue is
dispensed by the automatic glue machine on to the top of the tube layer
matching the position and profile of each support structure. Two gauge blocks
set the position of each support structure relative to the end-comb positions,
and one common gauge block is used to set the position of the ends of the
support structure relative to the first and last tubes.

###  3.4 Measurements of the base chamber mechanical precision

The chamber mechanical precision is measured by the 3D positions of the tubes
and platforms, using the height gauge and the FARO-arm on the granite table,
respectively. The platform positions are first measured while the chamber is
still held by the precision combs. Next, the precision combs are removed,
except the base combs and the tube positions are measured with the height
gauge. The in-plane system RASNIK lines are also read out to record the base-
line measurements of the chamber shape.

####  3.4.1 Platform position measurements

The platform positions are measured by the FARO-arm using the coordinates
described in Section 3.3. Several distances corresponding to reference points,
as indicated by the arrows shown in figure 21, are measured on the three
perpendicular surfaces of each AP and B-field platform to determine their 3D
positions relative to the reference positions. Each FARO-arm measured distance
is compared with its nominal value and the deviation from the target value is
calculated.

Figure 23 shows the measured AP-plate position offsets (in X, Y, Z) from the
targets for the first 30 sMDT chambers built at UM. All AP-plates are placed
well within the specified 200 microns maximum deviation from the nominal
position. Similar results are obtained for B-field sensor positions, where the
maximum tolerance is 500 microns.

Figure 23: RO and HV AP plate position offset in X, Y, and Z from target
values for the first 30 UM sMDT chambers.

In the case of the CCC-plates, an 8 mm precision sphere in a precision conical
socket is measured to determine the center position of the plate. A small
clamping arm, as shown in figure 24(a), was designed to hold the sphere
rigidly allowing the FARO-arm probe to sample the surface of the sphere to
perform this measurement.

(a)

(b)

Figure 24: (a) A small clamping arm was devised to hold the sphere rigidly
while the CCC plate is measured with the FARO-arm spherical probe; (b)
Measured X, Y, and Z offsets of the CCC plate positions relative to target
values for the first 30 UM sMDT chambers. The vertical black line indicates
when the assembly plate was changed.

Figure 24(b) plots the deviation of the CCC sphere center X, Y, Z from to the
target positions. Note that, by design, not all the chambers are instrumented
with CCC platforms and installation jigging was changed after building the
first 8 BIS1 chambers. All CCC plates were installed at the target positions
within the required accuracy of 200 microns, except for the Y-coordinates for
a few early chambers which were out of the tolerance by a maximum of 47
$\mu$m.

####  3.4.2 Tube position measurements

The tube height relative to the granite table surface is measured via the
precision endplug cylinder surfaces using the height gauge with a flat probe,
as shown in figure 25(a).

(a)

(b)

Figure 25: (a) Tube height measured on the endplug precision cylinders with a
height gauge. (b) Histograms of the wire height offset from the layer mean for
the first 30 UM sMDT chambers.

Before the measurements the granite surface used for the measurement is
cleaned. The flat probe is frequently checked and tuned with a precision block
to be parallel to the granite table. The zero of the height gauge is set
before starting the measurement at a specific location on the granite surface
and re-checked if an abnormal value is measured.

The histograms in figure 25(b) show the tube height deviations from the
average in each tube layer for both the HV and RO sides. The RMS of these
distributions is below 11 microns indicating the level of flatness of all the
tube layers. The average height of each tube layer is recorded for both RO and
HV sides of the chamber, from which the Y-pitch (vertical) of the tube layers
and the spacer frame are determined. Figure 26 shows all the tube layer
Y-pitch deviation from the nominal value for each built module.

Figure 26: Tube layer pitch offset respect to the nominal value (13.077 cm)
measured on both RO and HV sides for the first 30 sMDT chambers built at UM. A
black zero line and a pink band of ${}\pm 25$ microns are added to guide the
eye.

The specified distance between the two multi-layers is 45.600 mm. The
distribution of the spacer frame distances is shown in figure 27. The
separation between the two multi-layers was found to be systematically lower
than the target value for the first 20 chambers. Subsequent chambers were
given an additional 50 micron mylar tape on the spacer frame cross-bars to
increase the spacer frame distance to match the specification.

Figure 27: Spacer frame Y-pitch measurements for both RO and HV side for the
first 30 sMDTs built at UM. The pink line is at the nominal value of 45.600
mm.

####  3.4.3 Chamber shape deformation measurements

The reference RASNIK measurement is made at the end of the construction while
the chamber is still in the jigging on the granite table. The chamber is then
removed from the granite table and mounted on a rotation cart. On the rotation
cart RASNIK measurements are taken at $22.5^{\circ}$ intervals from which the
granite reference values are subtracted to show the deformation of the chamber
as it is rotated through $360^{\circ}$.

Figure 28: RASNIK measurements in chamber coordinates X (top) and Y (bottom)
for all four RASNIK lines as the chamber is rotated through $360^{\circ}$. The
data points value is the averages over the first 30 modules built at UM, while
the error is the standard deviation of these measurements.

Figure 28 shows the result of these measurements for the first 30 sMDT
chambers as deformation in the X and Y directions for all four LED-CCD RASNIK
lines. The X direction is transverse to the layer of tubes (local XZ-plane)
and the chamber is very stiff along this axis, hence there is very little
deflection (a few microns) over $360^{\circ}$. The Y direction is
perpendicular to the planes of tubes, and much more affected by gravity, hence
the deflection is larger with a typical peak-to-peak variation of roughly 0.2
mm. These results, besides confirming the functionality of the system after
the cabling to the the readout panel, show that the optical in-plane devices
cover a range of deformation of several hundredths of a millimeter.

##  4 Chamber Services Installation and Testing

The chamber services are installed when the chamber is on the rotation cart.
The services installed are: temperature sensors, ground foil, gas manifolds,
hedgehog electronics cards, Faraday cages, and ground straps. As soon as the
chamber is removed from the jigging all endplugs are covered with protective
caps to keep dust from entering the tubes.

Temperature sensors, together with their readout cables, are glued in twelve
positions on the chamber surface: six on the bottom surface and six on the top
tube layer. Thermal contact between temperature sensors and tubes is made with
thermal paste epoxy. All temperature sensor cables are routed to the top
surface of the chamber through a hole in the support structure at the RO side,
seen in figure 2(a). The RASNIK cables for in-plane system control and readout
are also routed along the top surface. A readout panel is built and installed
on the RO side support structure of the chamber for connection to both the
temperature sensor cables and RASNIK cables. A perforated foil is wrapped over
each end of the chamber, providing common ground connections from the tube
walls to each RO and HV hedgehog card. This foil becomes the internal wall of
the Faraday cage that contains the electronics. Ground pins are screwed to the
grounding anchors (installed during the gluing operations) and tightened up
against this foil. These pins are installed after installation of the gas
system, and before the hedgehog cards are installed.

###  4.1 Gas-bar assembly, cleaning, and installation on chamber

The sMDT gas distribution system uses molded plastic parts, shown in figure
29(a), which plug together in sets of 4 (to feed in series 4 tubes in a
vertical stack across the 4 tube layers in a ML). On both sides of a ML either
58 or 70 of such stacks are connected to two gas-bars for parallel gas flow in
and out of each ML. Gas input and output are at opposite corners to balance
the impedance for uniform gas flow through all tubes.

All gas fittings are first ultrasonically cleaned with IPA, which is the only
lubricant used during assembly. Each gas-bar is also ultrasonically cleaned in
a bath of de-ionized water. After drying, each bar is wiped down with IPA. The
inside length of the bar is then cleaned out using air pressure. Each bar is
again wiped down with IPA.

(a)

(b)

(c)

(d)

Figure 29: Gas-bar components and assembly: (a) Molded plastic gas fittings
with o-rings form stacks of 4 (two different kinds); (b) blue 3-D printed
jigging pre-aligns the 4 pieces of a stack; (c) stacks being mounted on a gas-
bar manifold using more 3-D printed jigging parts to aid in alignment; (d) a
gas-bar manifold with installed stacks, ready to be mounted on a chamber.

Each stack of 4 fittings is assembled using double o-rings between components.
A sandwich jig, shown in figure 29(b) is used to align components for
assembly. The sandwich jig and several other tools were made with a 3D printer
to simplify gas-bar assembly (see figure 29(c)). Gas stacks are attached with
M2.5X6 screws to the gas-bar with an o-ring in between each stack and the gas
manifold. Different gas-bar configurations are made for the two sides of the
chamber. Completed gas-bars are stored in lint-free bags prior to installation
on chamber.

The gas-bar cleaning process was modified partway through chamber production
because it was found that metal pieces left from the machining process could
block a gas hole. This problem was found during cosmic ray testing where it
was seen that some sets of 4 tubes connected to the same stack showed low
efficiencies. To investigate this problem, the electronics and gas-bar were
removed from the chamber. Further inspection revealed that small circular
metal chips left over from hole drilling were blocking the holes in the gas-
bar. The affected chambers had to have electronics and gas-bar removed and the
gas-bar carefully inspected for chips and re-cleaned. After reassembly, the
tube efficiency was restored to normal values. The gas-bar cleaning procedure
was subsequently modified to include the poking of each gas-bar hole to remove
any metal fragments. Since this procedure was adopted no further issues with
low efficiency stacks have been observed.

Figure 30 shows the gas-bar installation on a chamber. The ground foil is
first laid and held by ground screw nuts onto the chamber. Each gas-bar is
inserted into all the tube endplugs simultaneously using multiple special
brass tools and three specially designed jigging clamps (see figure 30(a)).

(a)

(b)

(c)

Figure 30: Gas-bar installation on a chamber: (a) fitting a pre-assembled gas-
bar on the top ML of a chamber. The brass tools are used to make sure that all
gas fittings are fully seated on their corresponding endplugs before beginning
to install o-rings. The same tools are used to force an o-ring over the
endplug precision surface where it is compressed by the gas fitting; (b) after
the back o-ring is in place the signal-cap, which holds the outer o-ring, is
screwed onto the endplug. The signal-cap has cross drilled holes which allow
the gas from the fitting to enter the endplug and the tube body; (c) Two
multi-layers with gas-bars and signal-caps installed.

O-rings, after ultrasonic cleaning with IPA, are placed onto the signal caps.
Each signal cap is inserted into each gas connector using a special driver
shown in figure 30(b). Complete gas-bar installation on both multi-layers on
one side of a chamber is shown in figure 30(c).

The gas-bars are connected to an on-chamber gas distribution system using
Swagelock fittings and valves connect the gas-bars as shown in figure 31. A
high precision gas pressure gauge is temporarily connected to the gas system
for chamber pressure measurements. The temperature of the chamber is monitored
with 12 temperature sensors distributed on both sides of the chamber and
readout through the patch panel shown in the picture.

Figure 31: Completing the on-chamber gas system are 6 mm stainless steel and
copper tubing pieces, Swagelok valves and fittings which connect the gas-bars
to external supply lines. A high precision gas pressure gauge is here
connected for the gas leak test, and the RO Faraday cage is already installed.

###  4.2 Front-end electronics installation

The schematic of the electrical connections to an MDT drift tube is shown in
figure 32. For an sMDT tube, the 383 $\Omega$ resistor in the HV side, shown
in the figure, is replaced with a 330 $\Omega$ resistor. The RO and HV
hedgehog (HH) cards consist mainly of passive components and mount directly on
the signal and grounds via sockets on the cards.

Figure 32: Electrical connections to an MDT drift tube [10]. Resistors values
are in ohms ($\Omega$). For an sMDT tube, the 383 $\Omega$ resistor is
replaced with a 330 $\Omega$ resistor.

There are two configurations of the HH and RO cards. One type has 24 channels
with configuration of 4$\times$6 to span the 4 layers of a ML, 6 tubes wide.
The other has 20 channels with configuration of 4$\times$5, which is installed
on the ends of each ML. A total of 20 (16) 4$\times$6 cards plus 4 4$\times$5
cards are installed in the BIS1 (BIS2-6) chambers. HV is distributed through
four pairs of the input HV and ground cables. The HV HH cards are chain
interconnected via PCB jumpers.

Faraday cages (FC) are installed around the HH cards to reduce noise. Two HV
distribution boxes are mounted on top of the FC and distribution cables are
routed through two holes of the FC. The FC on the RO side has the mezzanine
cable connection holes to allow mounting the mezzanine cards to read out the
chamber signals. The mezzanine cards hold the active components of the front-
end electronics, the amplifier and time-to-digital converter chips.

###  4.3 Chamber gas tightness measurement

The ATLAS upper limit of the leak rate is $\rm 1\times
10^{-5}[mbar\,liter/s]\times(2\,N_{tube}),$ where $\rm 2\,N_{tube}$ denotes
total number of endplugs included in the measurement. The tube gas volume is
$\rm V(tube)\approx 250cm^{3}$, therefore the volume of each ML is $\rm
V_{ML}=N_{tube}\times 250~{}cm^{3}.$ Using this information, the upper limit
of the chamber leak rate can be converted to a limit on the chamber pressure
drop per hour in the measurements, which is 0.288 mbar/hour for a ML of an
sMDT chamber.

Since the chamber pressure depends on gas temperature, the measured pressure
change ($\Delta P$) over time interval ($\Delta t$) must be corrected to be
compared to the specified leak rate limit. The chamber normalized leak rate is
determined by

$\left(\frac{\Delta P}{\Delta t}\right)_{norm}=\frac{\Delta P_{corr}}{\Delta
t}$

where $\Delta P_{corr}$ is the change in pressure corrected with the
temperature variation over a time interval $\Delta t$, calculated as:

$\Delta P_{corr}=P_{f}~{}\frac{T_{ref}}{T_{f}}-P_{i}~{}\frac{T_{ref}}{T_{i}}$

where $P_{i},~{}P_{f},~{}T_{i},~{}T_{f}$ are initial and final pressure and
temperature, and $T_{ref}=293.15^{\circ}$ K is used as reference temperature.

The chamber gas output valves are connected to a pressure gauge (see figure
31), which can monitor each ML gas pressure. The chamber is pressurized with
Helium gas, then a Helium sniffer is used to detect leaks from connection
points and/or individual tubes. Most leaks can be fixed quickly by replacement
of o-rings, gas stacks, or signal caps. A few tubes have developed leaks which
cannot be fixed. For these, the wire is removed, the end plugs sealed to
isolate the tube from the gas system, and the signal caps modified to isolate
the tube from both the HV and RO electronics.

The final leak rate measurement is performed with all HH cards and the Faraday
cages installed. The resistance of all 12 temperature sensors is measured in
Ohms, converted into degrees Celsius and averaged for each ML to correct the
initial and final chamber ML pressures readout during the leak test, which
check the pressure drop as a function of time. The measured ML leak rates of
the first 30 sMDT chambers built at UM are shown in figure 33. The pink
dashed-line in the plot indicates the upper limit, and blue (red) triangles
are the measured leak rates for ML1 (ML2) of produced chambers at UM, all well
below the allowed upper limit.

Figure 33: Multi-layer gas leak rate of the first 30 sMDT chambers built at
UM. The dashed line represents the ATLAS maximum acceptable gas leak rate.

###  4.4 Ground cable connections

The chamber grounding follows a "star" design that avoids ground-loops. The
schematic of the ground connectivity is shown in figure 34, where the pictures
show how the grounding cables (in green color) connect the gar-bars, chamber
side-panels, and the chamber support structures.

Figure 34: Grounding cable connections schematics. Pictures show detailed
connections to the gas-bar (a), side-panel (b), support structure (c), and
between the gar-bar and the support structure.

##  5 Electronics and Cosmic Ray Test

The final QA/QC tests include measurements of the noise level and efficiency
of each tube, as well as the chamber tracking efficiency and resolution. These
tests are carried out in the cosmic ray test station which has been described
in Section 2.6.2.

(a)

(b)

Figure 35: (a) A 24 channel RO HH and mezzanine card mounted on a sMDT
chamber; (b) 24 front-end electronics cards on a BIS1 chamber RO side.

Each chamber runs for a minimum of 24 hours at the operating HV (+2730 V) to
ensure that the current drawn is at a level below 2 nA times number of tubes.
Tubes which draw excessive current are individually treated with negative HV
(up to -3 kV) for about one hour to reduce the current below the 2 nA limit.
Finally, the readout mezzanine cards are installed on top of the RO HH cards.
One mezzanine card contains a board with three ASD
(Amplifier/Shaper/Discriminator) ASIC chips, each with 8 channels, and another
board with one 24 channels high-performance time-to-digital converter (HPTDC)
chip on top of the ASD board. Figure 35(a) shows a mezzanine card installed on
a chamber. Two Chamber Service Modules (CSMs) are used to multiplex the
readout from all the HPTDCs of a chamber (each CSM can multiplex up to 18
HPTDCs). The DAQ system with a scintillator trigger and the online monitoring
system are connected to the test chamber for measurements. An offline analysis
program (detailed description of the algorithms in [11]) was developed to
measure the performance of each tube and ultimately the efficiency and the
resolution.

###  5.1 Noise rate measurements

Each tube noise rate is measured in a series of data runs at different readout
thresholds with both HV-off and HV-on (at the nominal working voltage of +2730
V). A 10 KHz pulse-generator is used as trigger to read out noise hits. The
noise rate can be calculated based on the recorded number of hits and the
number of triggers during the data taking period (typically $\Delta t$ = 10
minutes) as:

$Noise\;rate=\frac{N_{hits}}{N_{trigger}}\times\;time_{window}\;,$

where ${time}_{window}(1.55\;\mu$s) is the HPTDC readout time window;
$N_{hit}$ is the number of hits in each channel recorded in the test time
$\Delta t$, and $N_{trigger}=10{\rm~{}KHz}\times\Delta t$.

(a)

(b)

Figure 36: (a) Noise rate vs. threshold for a representative sMDT chamber. (b)
Chamber average noise rate measured at 39 mV threshold for the first 30
modules.

The measured noise rate vs. ASD threshold for a representative sMDT chamber
built at UM is shown in figure 36(a). When the readout threshold is larger
than $\sim$30 mV, the chamber hit rate is dominated by the cosmic rays, as
seen from the red curve measured with HV-on. The specification of the maximum
tube noise rate is 500 Hz at the readout threshold of 39 mV. Figure 36(b)
shows the average noise rate measured at 39 mV threshold for the first 30 sMDT
built at UM, all well below 500 Hz, except Module-4 with noise rate close to
the specification in the HV-on measurement. This chamber will be closely
evaluated again during the final commissioning phase at CERN.

###  5.2 Tube spectra from cosmic ray data

Each tube’s response to charged particles is tested with cosmic rays. Figure
37 shows typical (a) time (TDC) and (b) charge (ADC) spectra for the hits of a
tube on a representative sMDT chamber together with their fits (described in
[11]). Both the leading and trailing edge of the TDC distribution are fit with
a Fermi-Dirac function in order to get the time offset ($T_{0}$) and the
maximum time $T_{Max}$. The ADC distribution is fit with a skew normal
function.

(a)

(b)

Figure 37: Distributions of cosmic ray hits recorded by a typical tube: (a)
time, (b) charge.

The measured hit drift time, calculated as the difference between the hit time
and the $T_{0}$, is then corrected with the time-slew correction using the
charge information. The time-slew effects are due to the jitter introduced by
different times of crossing the ASD threshold by different amplitude signals.

###  5.3 Tube and chamber efficiencies

Figure 38(a) shows an example of the chamber response uniformity based on the
number of hits per tube in the cosmic ray run.

The definitions of efficiency rely on having successfully reconstructed a
track. A muon track is built starting from the sMDT hits for each triggered
event by fitting the hits drift distances obtained from the (corrected) drift
times via an $R(t)$ function. This time-to-space relationship is obtained with
an auto-calibration program that converts the drift time to the drift distance
through a procedure of minimizing the tracking residuals of all the collected
muon tracks in a test run. An example of a comic ray muon track reconstructed
in an sMDT chamber built at UM is hown in figure 38(b).

(a)

(b)

Figure 38:  (a) Cosmic ray run hit map for the 8 tube layers as a function of
the tube number for a representative sMDT chamber; (b) example of a track
reconstructed in a sMDT chamber by a fit tangent to the drift circles (blue
circles). Grey (white) circles are drift tubes read out by a different (the
same) mezzanine card.

The tube efficiency is defined as the ratio between the number of hits
recorded on a track passing through that tube and the number of reconstructed
tracks passing through the gas-volume of that tube. Figure 39(a) shows this
single tube detection efficiency, which is on average $\rm 99.02\pm 0.01$% for
the first 30 chambers built at UM.

(a)

(b)

Figure 39: (a) Tube and (b) tracking efficiency for the first 30 sMDT chambers
built at UM.

The tube layer efficiency is defined as the number of hits in that layer
belonging to a track divided by the number of tracks passing through that
layer. This latter efficiency includes the tube efficiency as well as the
inefficiency of the tube walls and the space between tubes (dead regions).
Figure 39(b) shows the tube layer efficiency averaged over a chamber (called
tracking efficiency). The average tracking efficiency for each tube layer is
$94.52\pm 0.03$ %.

Out of the 14688 tubes of the first 30 UM sMDT chambers, 13 were lost due to
wire slippage (insufficient crimping) or irreparable gas leaks (cracked
endplug or improper endplug swaging).

###  5.4 Tracking resolution

The ATLAS muon spectrometer measures the momenta of muons by reconstructing
the curvatures of their trajectories in the magnetic field, and thus the
spatial tracking resolution will in turn determine the resolution of the
momentum measurement.

The single tube resolution is determined by fitting the reconstructed muon
tracks and evaluating the residual distributions. The tracking residual is the
difference $r_{\textrm{drift}}-r_{\textrm{fit}}$, where $r_{\textrm{drift}}$
is the measured drift radius and $r_{\textrm{fit}}$ is the radius obtained by
the track fit. Two types of residuals are introduced, biased and unbiased. The
biased one is the residual when the track is fit with all hits. The unbiased
one is the tracking residual of a hit from a track reconstructed by removing
that hit from the fit. Figure 40 shows the biased and unbiased residual
distributions.

(a)

(b)

Figure 40: (a) Biased and (b) unbiased residual distribution for a
representative sMDT chamber built at UM with a double Gaussian function fit.

A residual distributions is fitted with a double Gaussian function with common
mean (see [12] for details) and the single wire resolution is defined as [13]:

$\sigma_{wire}=\sqrt{\,\sigma_{B}\cdot\sigma_{U}\,}$ (5.1)

where $\sigma_{B}\;(\sigma_{U})$ is the biased (unbiased) residual width
calculated as amplitude weighted average of the two Gaussian components,
narrow ($n$) and wide ($w$), of the fit:

$\sigma_{\alpha}=\frac{A_{n}\cdot\sigma_{n}+A_{w}\cdot\sigma_{w}}{A_{n}+A_{w}},\hskip
28.45274pt(\alpha=B,\ U).$ (5.2)

(a)

(b)

Figure 41:  (a) Average resolution vs. radius of the 30 sMDT chamber built at
UM. The errors are the standard deviations of the 30 data-point at that drift
radius; (b) chamber resolution for the first 30 sMDT chamber built at UM.

The multiple Coulomb scattering effect on low energy cosmic ray muons are
corrected when computing the resolution [11]. The single wire resolution vs.
the tube radius for a representative sMDT chamber built at UM is shown in
figure 41(a). The measured tracking resolution (integrated over all radii) for
the first 30 chambers made at UM shown in figure 41(b), is $100\pm 7$ $\mu$m
to be compared with the expected value of 106 $\mu$m [14]. The resolution of
the constructed chambers is better than the specification because the expected
value was estimated using the first generation of the electronics (in
particular the ASD-1), while the sMDT built at UM are tested with mezzanine
cards instrumented with the final ASD-2 which has a higher gain.

All chambers constructed at UM so far meet the performance specifications both
for tracking resolution and efficiency.

##  6 Database

All the information about each chamber construction precision measurements,
gas tightness, and electronics tests as well as the overall performance are
assembled into a local database, ready to be checked and/or displayed. The
data from the various measurements and tests stations is usually saved in text
files, either directly or after an initial decoding/analysis on the output
data from a test station. This format of the local data storage allows
converting to different databases (MS Access, MySQL, Oracle, JSON, …) for the
task. The most natural choice is a relational database, since all information
to be saved is relative to many single units (as tubes, sensors, plates, tube-
layers, multi-layers, …), each one with a list of properties, test parameters
and results: all data is correlated.

With the goal to store, quickly retrieve, and eventually show the data, a ROOT
[15] based approach is used. C++ classes read each chamber (and corresponding
tubes) data files and organize them into a (multiple) set of measurements with
conditions and relative data into a TTree of the ROOT file. The main blocks of
this TTree are described below.

Construction:

Includes the IDs of the tubes, their locations on the chamber and the
measurements of the tube heights relative to the granite table. From this data
the average height of a tube layer is calculated and then each layer and
spacer frame Y-pitch are determined. Also, the positions of all the platforms
(AP, B-field, and CCC) glued on the chamber are recorded.

In-plane alignment:

Stores the RASNIK system data from the final measurement on the granite table
(the base-line measurements) and at different inclination angles on the
rotation cart.

Gas leak rate:

Saves the gas pressure for each chamber ML together with the temperature at
different times to determine the gas leak rate.

Noise level:

Retains the noise rate for each tube for HV-off and HV-on runs at 8 different
ASD thresholds.

Electronics performances:

Collects each tube efficiency together with its cosmic ray test TDC and ADC
spectra, as well as the tracking efficiency, the chamber biased and unbiased
hit residual distributions, the single wire resolution, and the resolution vs
drift radius.

In all these ROOT-based DB TTrees, the tube barcode ID is the primary key for
retrieving information from the measurements on a tube, either as a single
unit (wire tension, dark current, and so on, see [5]) or part of an assembled
chamber (tube heights over the granite table surface, efficiency, resolution,
and so on.) While filling the TTree, a set of standard plots for each test is
automatically saved, ready to show trends and/or discrepancies between a
chamber and all the others already tested or reference values. A dedicated set
of files with common formats, which are followed by both Michigan and the MPI
production sites, is prepared and uploaded to the CERN
website121212https://atlas-mdt-phase2-qc.web.cern.ch/ to share the results by
both sites on the sMDT chamber production quality control. This common
database on sMDT production will be used for future sMDT commissioning and
operations in the ATLAS experiment.

##  7 Conclusion

With the elaborate infrastructure built at UM, 30 sMDT chambers have been
constructed meeting the stringent precision and quality requirements. These
chambers are all thoroughly tested with cosmic rays and achieve an single tube
efficiency above 99%, and tracking resolution of $\rm 100\pm 7\;\mu m$.

There are three major steps in the chamber construction and certification
(base chamber gluing, services installation, and final cosmic ray testing),
each requiring 10 working days on average. The three steps are carried out in
parallel in order to meet the production schedule.

Of the 30 chambers constructed at UM, the first batch of 24 chambers has
already been delivered to CERN. 20 more chambers are still to be constructed
at UM and will be delivered to CERN by the end of 2023. The final
commissioning of all 50 Michigan chambers prior to installation on the ATLAS
cavern is planned to begin in 2024 at CERN.

## Acknowledgments

We would like to thank Hubert Kroha and Oliver Kortner (MPI), Rinat
Fakhrutdinov (IHEP, Protvino), Reinhard Schwienhorst (MSU), Tristan Du Pree
(NIKHEF), Pierre-François Giraud (Saclay), and their team members for their
close collaborations and valuable technical discussions. We also thank the UM
ATLAS faculty, Jianming Qian and Shawn McKee for their strong support, and our
undergraduate students, Tyler Coates, Kathryn Ream, Dylan Ponman, Saarthak
Johri, and Yuxin Wan, for their great technical assistance.

This work was supported in part by the U.S. Department of Energy grant DE-
SC0012704 and in part from the U.S. National Science Foundation PHY-1948993.

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