MATHEMATICAL
PROBLEMS
IN
THE
DEFINITION
OF
STANDARDS
BASED
ON
SCALES:
THE
CASE
OF
TEMPERATURE F.PAVESE
CNR,
Istituto
di
Metrologia
"G.Colonnetti",
strada
delle
Cacce
73,
Torino,
Italy.
[email protected]
The
paper
introduces
first
the
concept
of
“scale”
that
must
be
adopted
to
define
the
standards
of
some
intensive
physical
quantities,
such
as
temperature.
After
a
general
discussion
about
absolute,
semi‐empirical
and
empirical
scales,
the
focus
is
put
on
empirical
scales,
such
as
the
International
Temperature
Scale
of
1990
(ITS‐90),
and
on
the
mathematical
and
statistical
problems
which
arise
in
constructing
these
types
of
scales.
1
Standards
based
on
scales:
temperature
Some
physical
quantities,
one
being
temperature,
cannot
be
measured
in
the
same
way
as
other
quantities,
e.g.,
length.1
Once
the
unit
size
has
been
defined,
it
cannot
subsequently
be
labeled
“unit
interval”
and
used
to
measure
the
quantity
in
the
same
way
as
the
metre
in
length
measurements.
That
is,
an
additive
procedure
cannot
be
used
for
these
quantities,
by
which
its
value
is
determined
from
the
number
of
the
“unit
intervals”
contained
in
it.
Values
of
these
quantities,
instead,
can
only
be
determined
by
comparing
two
of
them,
one
of
which
is
taken
as
the
reference,
and
by
observing
whether
they
are
equal
–or
which
one
is
higher.
With
temperature,
this
can
be
done
by
observing,
in
accordance
with
the
zeroth
law
of
thermodynamics,
whether
there
is
–or
not–
a
heat
flow,
and
by
noting
its
direction.
To
assign
a
numerical
value
to
each
realisation
of
the
quantity,
one
has
first
to
“order”
the
measured
realisations,
that
is
to
establish
a
scale
in
which
the
relevant
parameter
–the
heat
flow
in
the
case
of
temperature–
1 Most of the general discussion is rearranged from Ref.1. PAVESE_AMCTM99.RTF
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always
changes
in
the
same
direction,
and
then
assign
a
sign
to
that
direction.
Accordingly
it
can
be
said
that
{...
T1>T2>T3
...}:
temperature
is
a
simply
ordered
manifold
2.
By
this
procedure,
one
cannot
yet
assign
a
numerical
value
to
the
different
temperatures,
but
only
give
them
a
serial
number
which
will
be
altered
by
the
arbitrary
addition
of
any
new
measurement.
Nor
can
one
say
yet
that
the
value
of
any
T2
is
closer
to
that
of
T1
than
to
that
of
T3
(in
fact,
one
cannot
assign
a
distance),
even
if
he
should
decide
to
take
the
interval,
say
{T1,T2},
as
the
unit
interval,
because
neither
intervals
nor
ratios
can
be
compared
by
means
of
heat
flow
measurements.
In
other
words,
the
metric
of
the
quantity
space
{T}
must
be
supplied
by
another
physical
quantity
P
whose
analytical
relationship
to
the
former
can
be
established
and
whose
metric
is
known.
The
metric
of
{T}
will
be
equal,
by
definition,
to
that
of
{P}
only
if
this
relationship
is
linear:
P
∝
T.
In
case
of
temperature,
actually,
things
are
more
complicated,
since
only
an
empirical
temperature
θ
can
be
measured,
P
∝
θ:
It
is
then
demonstrated
that
θ
≡
T.
1.1
Absolute
scales
“Absolute”
temperatures
are
defined
only
as
ratios
or
differences
of
values
and
normalization
constants
are
required
to
make
the
relationship
of
temperature
with
another
physical
quantity
determined.
These
definitions
may
be
said
to
provide
a
“blank
measuring
tape”,
which
is
necessary
for
temperature
measurement,
but
leaves
free
the
choice
as
regards
both
the
size
of
the
graduation
in
unit
intervals
and
the
position
of
the
“zero”
value
on
the
tape.
These
are
the
two
degrees
of
freedom
characteristic
of
any
linear
scale,
that
is,
as
already
pointed
out,
of
a
scale
having
the
same
metric
of
the
space
of
the
one
that
is
taken
to
represent
the
measuring
tool.
In
the
case,
for
instance,
of
the
unit
of
length,
the
metre,
it
is
appropriate
to
define
it
in
such
a
PAVESE_AMCTM99.RTF
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way
as
to
maintain
the
additive
property
in
the
classical
isotropic
space
(though
not
near
a
relativistic
curvature
of
the
space).
In
this
respect,
in
the
absolute
temperature
definition
of
Eq.
1
(that
is
Kelvin's
second
definition)
Q2/Q1
=
θ2/θ1
(1)
where Q is the quantity of heat, the metric of the space of heat - i.e., of energy - is used. Whereas in his first definition (Eq. 2)
W
=
(θ*,2
–θ*,1
)
Q
(2)
where W is the mechanical work, Kelvin used the metric of the space of mechanical work equivalent to heat. Incidentally, the transformation between the two definitions is not linear, but logarithmic (θ* = J logθ + const), reflecting the fact that the first defines a temperature ratio, the second a temperature difference (it is because temperature is defined in the heat (energy) space that the absolute lower end of the temperature scale is a zero –only asymptotically approached– and not a (negative) infinity). The definition in the space of heat is the modern one, preferred because it closely approximates the numerical values obtained using gaseous substances and, more specifically, because it matches the physical properties of a model-state of what is called “the ideal gas”, more or less approximated by real gaseous low-density substances. Starting from the modern generalization of the kinematic model, the particles of the ideal gas satisfy the equation of state
PAVESE_AMCTM99.RTF
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P
V
=
g
T
(3)
where
g
is
a
constant.
In
the
case
of
the
classical
ideal
gas,
g
is
defined
by
Rn
(n
being
the
amount
of
substance
and
R
the
gas
constant)
or
by
kN
(N
being
the
number
of
molecules
and
k
the
Boltzmann
constant).
Equation
3
is
sufficient
to
define
the
ideal
gas
model
for
absolute
temperature.
Let
us
remark
again
that
only
an
empirical
temperature
θ
can
be
measured.
None
of
the
definitions
of
temperature
given
defines
a
unique
scale.
Each
only
put
constraints
on
the
scale
form.
In
principle,
any
thermodynamic
law
can
be
used
to
assign
a
metric
to
the
thermodynamic
temperature
in
order
to
obtain
a
thermodynamic
scale,
provided
the
law
can
be
reduced
from
the
implicit
form
T
=
f(a1,...,
am;
x1,...,
xn)
(4)
where
xi
(i=1...n)
are
independent
variables
and
aj
(j=1...m)
are
constants,
to
a
form
depending
linearly
only
on
one
variable,
all
the
others
becoming
perturbations
to
be
measured
or
computed
(the
so‐called
“corrections”)
T
=
g(b1,...,
bk,…,
bp)
y
(5)
PAVESE_AMCTM99.RTF
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in
which
bk
become
known
constants
or
variables
b(T)
measured
or
independently
inferred,
and
y
is
a
single
independent
variable
(e.g.
pressure,
for
a
gas
thermometer).
In
this
form,
a
thermometer
is
fully
defined
by
its
analytical
equation
and
it
becomes
a
primary
thermometer
in
the
restricted
sense.
At
present,
such
a
primary
definition
can
only
be
realized
to
a
modest
accuracy,
except
at
very
low
temperatures.
Temperature
is
actually
defined
at
one
fixed
point
T0,
generally
the
triple
point
of
water.
Therefore,
one
free
parameter
is
left,
whose
value
for
each
specific
implementation
of
the
thermometer
is
determined
by
measurement,
y0
=
y(T0),
at
the
fixed
point.
Any
temperature‐dependent
physical
quantity
can,
in
principle,
be
used
to
measure
temperature.
Whether
a
physical
quantity
is
suitable
or
not
as
a
thermometer
is
a
matter
of
convenience;
a
choice
may
depend
as
well
on
the
required
precision
of
the
resulting
temperature
scale.
1.2
Semiempirical
scales
The
determination
of
thermodynamic
temperature
from
first
principles
is
a
difficult
experiment.
Every
measurement
is,
in
fact,
only
an
approximation,
because
of
imperfections
in
the
model
used
for
describing
the
basic
(“zeroth‐level”)
thermodynamic
law,
of
insufficient
control
of
the
secondary
experimental
parameters
which
are
included
in
the
model
(the
so‐called
“corrections”),
or
because
of
experimental
random
errors.
For
this
reason,
other
types
of
scales
can
be
preferred.
The
main
differences
between
them
are
listed
in
table
1.
Semi‐empirical
scales
differ
from
a
primary
thermodynamic
scale
in
that
the
number
of
stipulations
for
their
definition
is
increased.
Equation
5
may
be
the
(to‐date)
exact
representation
of
a
thermodynamic
law,
but
either
the
value
of
some
of
the
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parameters
bk
cannot
be
calculated
with
sufficient
accuracy,
or
accurate
calculations
are
not
estimated
conveniently.
Table
1.
Comparison
of
types
of
scales
Absolute
Scale
Physical
Law
“zeroth
level”
(PV
=
nRT
⇒
PV
=
nRθ)
+higher‐order
effects
Semiempirical
Scale
Empirical
Scale
•
Transducer
φ(θ)
Mathematical
Model
≡
Physical
Law
•
Fixed
Points
(physical
states)
•
Mathematical
Model
(“corrections”)
All
physical
parameters
measured
independently
•
Some
parameters
measured
“Calibration”
at
all
•
“Calibration”
at
(measurement
at
the
some
fixed
points
Physical
quantity
(T)
computed
Physical
quantity
(T)
computed
“Best
practice”
“Best
practice”
for
performing
for
performing
fixed
points
physical
states)
Scale
definition
for
the
physical
quantity
(T)
applied
•
Transducer
specs.
PAVESE_AMCTM99.RTF
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experiment
experiment
(interpol.
instrument)
•
Rules
for
Scale
•
“Best
practice”
for
performing
fixed
points
For
instance,
Curie's
law
for
paramagnetic
substances
is
χ = χ0 + Errore.
(6)
This
equation,
which
also
is
non‐linear
in
T,
can
be
exploited
to
determine
temperature
by
measuring
the
magnetic
susceptibility
χ,
but
the
three
constants
are
experimentally
determined
by
measuring
the
susceptibility
values
(χ1,...,
χ3)
at
three
reference
temperatures
(Tr,1,...,
Tr,3),
whose
values
are
defined
or
obtained
independently.
The
procedure
can
then
be
considered
a
calibration
of
the
experimental
setup.
In
semi‐empirical
scales
the
reference
points
are
two
or
more.
Since
they
are
taken
from
well‐defined
thermodynamic
states,
it
is
assumed
that,
when
the
state
is
exactly
reproduced,
the
thermodynamic
temperature
is
too,
independently
of
the
experimental
setup
or
of
the
experimentalist.
A
scale
so
defined
assumes
no
approximations
in
the
defining
thermodynamic
equation
and
no
simplifications
in
the
experiment,
as
in
the
case
of
a
primary
thermodynamic
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measurement.
Consequently,
the
scale
definition
need
not
specify
any
characteristics
of
the
experimental
equipment
or
any
measurement
procedure.
Measurements
have
to
be
performed
according
to
state‐of‐the‐art
techniques
required
by
the
accuracy
to
be
achieved.
With
respect
to
primary
measurements,
the
measurement
procedure
is
simplified
in
that
calibration
at
the
fixed
points
avoids
measuring
several
secondary
parameters
and
performing
cumbersome
calculations
of
the
corrections.
Two
defining
points
at
least
are
used,
as
said,
which,
being
fixed
by
the
scale
definition,
are
called
“fixed
points”
of
that
scale.
None
need
to
be
the
defining
point
of
the
kelvin
scale.
The
scale
definition
assigns
them
conventional
temperature
values.
Though
these
values
are
assumed
to
be
exact
thermodynamic
temperature
values,
they
are
actually
the
best
up‐to‐date
experimental
approximations.
The
empirical
temperature
θ
defined
by
these
scales
is
considered
to
be
satisfactory
enough
as
an
approximation
of
the
thermodynamic
temperature
T.
A
similar
scale
is,
since
1990,
the
part
of
the
International
Temperature
Scale
of
1990
(ITS‐90)3
defined
in
the
range
(3‐25
K),
realised
with
the
interpolating
gas
thermometer,
that
makes
use
of
three
fixed
points. 1.3
Empirical
scales
and
their
mathematical
problems
Quite
frequently,
the
thermodynamic
relationship
between
the
measured
physical
quantity
and
thermodynamic
temperature
is
not
known
to
sufficient
accuracy,
or
this
relationship
is
not
convenient
for
direct
use;
it
might
be
as
well
that
measurements
are
executed
with
instruments
which
cannot
or
need
not
be
fully
characterized.
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The
strategy
for
an
empirical
scale
is
fully
different
from
that
of
an
absolute
scale,
beginning
from
the
experimental
point
of
view,
as
shown
in
tables
1
and
2.
Table
2.
Difference
in
the
experimental
basis
Absolute
and
Semiempirical
Scales
(e.g.,
gas
law)
Empirical
Scales
(e.g.,
platinum
resistance
thermometry)
1)
Measurements
taken
at
discrete
values
{Pi,
θi}:
experimental
design
constrained
only
to
optimisation;
1)
Physical
device
used
as
transducer
(thermometer)
2)
Empirical
law
used
for
the
transducer
response;
2)
Physical
Law
between
two
quantities,
e.g.
T
=
f(P)
is
used
for
computing
all
θ
values.
3)
A
discrete
number
of
physical
states
is
considered
(θ0i);
4)
The
values
θ0i
are
stipulated
by
the
Scale
(θ0i
≈
T0i);
5)
Measurements
of
the
transducer
response
are
taken
only
at
these
points;
6)
Scale
definition
used
to
compute
all
other
θ
values
PAVESE_AMCTM99.RTF
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The
link
with
the
physical
world
is
obtained
through
the
experimental
data,
weakened
by
the
experimental
errors.
The
relative
merits
of
the
many
possible
empirical
definitions
are
compared
only
from
a
purely
mathematical
and
statistical
viewpoint.
In
all
instances,
the
relationship
between
an
empirical
scale
and
the
absolute
scale
must
be
known.
This
can
be
achieved
in
only
two
ways.2
a)
absolute
measurements
θ(P)
are
taken
at
several
independently‐defined
physical
states,
so
that
a
value
θ0i
≈
Ti
can
be
associated
to
each
of
them.
Then,
by
experimentally
reproducing
these
physical
states,
the
value
of
the
response
R
of
a
transducer
placed
in
the
experiment
can
be
measured
at
any
time
at
each
of
these
defined
temperatures,
R(θi).
The
selected
physical
states
can
be
unique
(e.g.,
triple
point
of
a
substance,
defined
by
thermodynamics)
or
individual
points
of
a
physical
law
(e.g.,
vapour
pressure):
in
the
latter
case
the
realisation
of
the
states
depends
on
measurements
of
another
physical
quantity
(P
in
the
example).
b)
the
measured
values
of
the
physical
quantity
used
for
the
absolute
scale,
e.g.,
P,
are
directly
related
to
the
physical
device
used
by
the
empirical
Scale.
This
is
achieved
by
placing
the
transducer
at
the
same
temperature
θ
with
the
absolute
2 In the following temperature scales are taken as an example. PAVESE_AMCTM99.RTF
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thermometer
and
by
measuring
its
response
R.
This
can
be
done
only
while
performing
absolute‐scale
experiments.
In
both
cases,
the
process
is:
P
⇒
T
≡
θ
⇐
R
(the
measured
quantities
are
in
bold)
For
an
empirical
Scale
aiming
at
approximating
the
absolute
Scale
within
a
given
uncertainty
ε,
its
functional
relationship
θ(R)
must
be
compatible
with
the
physical
law,
that
is,
for
any
θ
in
the
definition
interval,
it
must
be
Δθ = θ – Τ ≤ ε.
The
values
assigned
to
the
physical
quantity
at
selected
points
θ0i
=
Ti
must
be
known
within
the
above
uncertainty.
Summarising,
three
are
the
constitutive
elements
of
an
empirical
scale:
a)
a
physical
device,
the
interpolating
instrument,
giving
a
response
R
to
a
stimulus
T
b)
a
set
of
physical
states,
the
reference
or
fixed
points;
c)
a
mathematical
definition
to
relate
R
to
T.
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a)
Physical
device
–
Also
an
absolute
scale
is
obviously
modeling
the
physical
world:
equation
3
is
valid
for
a
particular
substance,
the
ideal
gas,
assumed
existing
and
unique,
at
least
in
the
limit
case
of
real
gases.
The
purpose
of
additional
terms
(corrections),
is
to
take
into
account
in
the
model
either
the
non‐idealities
of
the
real
substance
(eg,
purity,
compressibility,…)
or
the
technical
imperfections
of
the
experimental
apparatus.
An
absolute
scales
is
unique
by
definition.
On
the
contrary,
the
physical
device
used
in
an
empirical
scale
is
intrinsically
nonunique:
it
is
a
kind
a
reference
material,
whose
finite
tolerances
of
the
relevant
characteristics
are
specified.
Consequently,
the
response
Rj
to
the
same
stimulus
T
is
inherently
depending
on
the
specific
j‐th
device.
There
is
not
a
unique
model
θ
=
f(R).
Consequently,
each
device
must
be
calibrated,
i.e.,
the
specific
model
f(Rj)
found,
in
order
to
keep
the
Scale
unique,
not
exactly
but
at
least
within
a
given
uncertainty.
This
point
can
be
made
clearer
by
the
example
of
platinum
resistance
thermometry
(see
2.).
The
model
is
applicable
exclusively
to
a
specific
substance,
platinum,
as
the
present
theory
does
not
allow
to
extend
its
validity
to
other
metals
with
the
aimed
accuracy.
Other
restrictions
apply
as
well.
When
some
of
the
characteristics
of
the
interpolating
instrument
can
be
specifically
quantified,
as
for
platinum
in
the
ITS‐90,
the
range
of
the
accepted
values
for
these
parameters
must
be
defined
as
well
in
the
scale,
to
enable
one
following
this
prescription
to
reproduce
the
results,
viz
the
scale,
within
the
stated
uncertainty.
Should
this
not
be
possible,
as,
for
example,
with
diode
thermometers
or
thermistors,
results,
and
that
scale,
would
be
valid
only
for
the
specific
production‐lot
of
devices
from
which
results
were
obtained.
PAVESE_AMCTM99.RTF
submitted
to
World Scientific
:
25‐09‐ 2015
Therefore,
the
mathematical
interpolating
procedure
defined
by
the
scale
is
not
able
to
adequately
represent
the
measurements
performed
with
each
individual
unit
of
the
thermometer,
but
can
only
approximate
the
physical
behaviour
of
these
units.
In
other
words,
if
the
scale
definition
is
applied
to
two
specific
units
of
the
thermometer
placed
at
the
same
temperature,
there
will
be
a
measurable
difference
between
the
temperature
values
computed
using
their
calibration
tables
for
the
same
measured
temperature.
b)
Scale
uniqueness:
the
role
of
the
“fixed
points”
–
Should
the
non‐uniqueness
of
the
specified
transducer
be
irrelevant,
the
Scale
would
not
be
required
to
define
the
procedure
for
obtaining
the
experimental
data
necessary
to
determine
the
numerical
values
of
the
parameters
of
the
model
θ
=
f(R).
In
fact,
such
a
model
would
have
the
status
of
a
physical
law
and
the
scale
itself
could
be
classified
instead
as
semi‐empirical
(see
1.2
above).
On
the
contrary,
because
non‐uniqueness
inherently
exists,
the
Scale
must
specify
the
physical
states
to
be
used
for
the
calibration
of
each
transducer
and
stipulate
the
conventional
values
{θi}.
As
said
before,
it
must
be
possible
to
identify
these
physical
states
in
an
independent
way.
For
the
ITS‐90
below
0
°C,
the
fixed
points
for
resistance
thermometry
are
triple
points
of
ideally
pure
gases.
All
the
properties
associated
to
the
“physical
substance”
of
a
semi‐empirical
Scale
apply
to
these
“physical
states”
and
their
“physical
property”
T
must
be
unique.
The
thermometer
used
in
empirical
scales
is
said
to
be
an
“interpolating
instrument”
because
it
is
not
itself
required
to
reproduce
a
thermodynamic
property,
but
only
the
selected
function
θ
=
f(R)
which
is
used
to
interpolate
temperature
values
between
the
values
assigned
to
the
fixed
points.
PAVESE_AMCTM99.RTF
submitted
to
World Scientific
:
25‐09‐ 2015
c)
Mathematical
tools
–
Should
the
non‐uniqueness
of
the
specified
transducer
be
irrelevant,
the
procedure
for
defining
an
empirical
scale
would
consist
of
the
following
steps:
1.
select
a
transducer
for
use
as
the
physical
device
for
the
scale
realisation
(e.g.,
a
platinum
resistance
thermometer);
2.
consider
all
the
experiments
where
absolute
determinations
of
thermodynamic
temperature
have
provided
the
corresponding
values
of
the
response
(e.g.,
T(R))
of
specific
units
of
that
physical
device;
3.
select
a
suitable
mathematical
model
for
representing
the
functional
relationship
between
the
response
of
the
transducer
and
the
stimulus
(eg,
between
electrical
resistance
of
platinum
and
thermodynamic
temperature);
4.
best
fit
the
model
to
the
available
experimental
data.
The
resulting
set
of
numerical
values
of
the
parameters
of
the
relationship
(in
the
example
T(R),
or
θ(R))
will
be
valid
for
any
other
unit
of
the
transducer,
within
the
uncertainty
resulting
from
the
experimental
determinations.
One
important
point
is
the
smoothness
of
the
selected
mathematical
model
with
respect
to
the
underlying
physical
relationship
between
the
measured
quantity
and
thermodynamic
temperature.
In
the
case
of
temperature,
some
thermophysical
properties
are
making
use
of
the
first
or
second
derivative
of
temperature,
so
that
artifacts
in
their
relationship
to
temperature
can
arise
from
the
empirical
temperature
scale
itself.
In
order
to
model
accurately
the
relationship,
high‐order
polynomials
have
PAVESE_AMCTM99.RTF
submitted
to
World Scientific
:
25‐09‐ 2015
often
been
used,
which
tend
to
oscillate
between
the
constraints:
a
striking
evidence
of
this
are
the
differences
between
the
new
International
Temperature
Scale
of
1990
(ITS‐90),
where
the
smoothness
with
respect
to
the
thermodynamic
scale
has
been
controlled
more
carefully,
and
the
former
version,
the
International
Practical
Temperature
Scale
of
1968
(IPTS‐68).
Additionally,
data
are
generally
coming
from
several
independent
determinations
done
by
different
authors
in
different
times:
there
is
a
data
fusion
problem.
The
unavoidable
nonuniqueness
of
the
physical
device
puts
limits
to
the
accuracy
in
transferring
the
results
obtained
from
one
unit
of
the
physical
device
to
another.
The
empirical
Scale
must
set
the
rules
for
obtaining
the
relationship
for
any
single
unit,
without
resorting
to
a
direct
comparison
of
different
units
of
the
transducer.
The
following
steps
are
therefore
to
be
added:
5.
assume
the
differences
between
the
functional
relationship
established
in
4.
and
those
of
all
other
units
satisfying
certain
criteria
be
small;
6.
for
each
unit,
obtain
an
experimental
value
of
these
differences
at
a
limited
number
of
fixed
points
θi,
in
the
sense
of
point
b)
above.
These
are
actually
the
only
experimental
values
required
by
the
calibration
of
the
specific
unit.
Conventional
consistent
values
{Θi}
of
θi
are
assigned
by
the
Scale;
7.
adopt
a
mathematical
procedure
for
“correcting”
the
“reference
relationship”
4.
for
all
R
values
in
a
certain
range,
based
on
the
Δθ
(Ri)
values
measured
at
the
fixed
points.
This
PAVESE_AMCTM99.RTF
submitted
to
World Scientific
:
25‐09‐ 2015
procedure
can
be
said
to
interpolate
the
transducer
characteristics
between
the
fixed
points.
Step
7.
is
critical,
as
the
choice
of
the
procedure
aims
at
minimising
the
residual
effect
of
non‐uniqueness,
but
is
often
constrained
to
the
limited
choice
available,
for
physical
as
well
as
for
practical
reasons,
of
fixed
points.
For
the
mathematical
definition
of
an
empirical
scale,
an
equation
like
Eq.
5
becomes
a
purely
computational
model,
used
to
obtain
an
approximation
of
the
thermodynamic
temperature
T.
The
value
Rk
of
the
physical
quantity
is
obtained
from
the
equation
at
a
number
of
reference
temperatures
(the
temperatures
of
the
physical
states
mentioned
in
a))
equal
to
the
number
p
of
the
free
parameters
of
the
model
(7)
θk
=
g(a1,...,ap,Rk)
(k=1...p)
where
a1,...,ap
are
constant
parameters.
The
model
is
determined
empirically
and
must
prove
to
fit
θ
over
the
whole
defined
range
between
the
fixed
points
within
the
stated
uncertainty,
on
the
basis
of
experimental
θ
(R).
The
number
p
of
parameters
necessary
to
fit
Eq.
7
to
the
required
accuracy,
determines
the
number
of
the
fixed
points
necessary
to
calibrate
the
thermometer.
The
optimum
spacing
between
these
fixed
points
on
the
temperature
scale
is
determined
by
the
characteristics
of
the
model,
i.e.
by
the
mathematical
function
used,
but,
at
the
same
PAVESE_AMCTM99.RTF
submitted
to
World Scientific
:
25‐09‐ 2015
time,
it
is
obviously
constrained
by
the
availability
of
suitable
thermodynamic
states
in
nature.
The
mathematical
definition
should
be
the
simplest
function
(or
set
of
functions)
representing,
within
the
stated
accuracy,
the
relationship
between
the
measured
quantity
and
the
thermodynamic
temperature;
it
has
a
number
of
free
parameters
(this
is
characteristic
of
an
empirical
scale),
whose
numerical
values
must
be
obtained
from
a
"calibration"
of
the
interpolation
instrument
at
an
equal
number
of
reference
points.
The
number
of
these
definition
fixed
points
must
be
equal
to
the
number
of
free
parameters
and
their
position
must
be
such
as
to
permit
the
best
compromise
between
mathematical
requirements
and
what
is
available
in
nature.
The
temperature
value
of
the
reference
point
is
defined,
i.e.
fixed
by
the
scale
definition.
The
fact
that
the
model
θ
=
g(R)
is
purely
empirical
also
comes
from
the
fact
that
it
is
not
supposed
to
apply
to
a
specific
“substance”:
it
is
expected
to
be
(slightly)
different
for
each
practical
device
fabricated
with
a
portion
of
that
substance.
From
a
mathematical
point
of
view,
the
nonuniqueness
means:
a
single
set
of
parameters
{a}
of
any
chosen
model
θ =
g(R)
is
unable
to
represent
correctly
the
behaviour
of
the
physical
quantity
used
for
the
definition
of
the
empirical
scale
with
temperature
as
materialised
in
different
thermometers.
Non‐uniqueness
implies
that
the
any
relationship
experimentally
obtained
between
absolute
and
empirical
Scales
is
valid
only
for
the
thermometers
physically
mounted
in
that
experiment.
PAVESE_AMCTM99.RTF
submitted
to
World Scientific
:
25‐09‐ 2015
So,
which
is
the
meaning
of
measuring
the
thermometer
response
R =
f(θ)
?
It
is
necessary
to
“calibrate”
every
single
j‐th
thermometer,
i.e.
to
determine
its
specific
relationship
θ
=
f(Rj):
this
can
only
be
obtained
experimentally,
by
best
fitting
of
experimental
data
{Rij, θi}.
2
A
case
study:
the
ITS90
between
24.6
K
and
273.16
K
For
the
physical
quantity
temperature,
different
types
of
scale
definitions
have
been
preferred
depending
on
the
range
considered
and
on
the
uncertainty
required.
An
example
of
absolute
scale
is
Plank’s
law
used
for
the
ITS‐90
definition
of
temperature
higher
than
1235
K.
An
example
of
semi‐empirical
scale
is
vapour‐pressure
thermometry
between
0.65
K
and
5.0
K
or
gas
thermometry
between
3
K
and
24.6
K.
An
example
of
empirical
scale
if
the
temperature
scale
using
the
electrical
resistance
of
platinum
between
13.8
K
and
962
°C.
Only
the
part
below
273.16
K
will
be
considered
in
the
following.
2.1
The
ITS90
definition
In
the
temperature
range
considered,
an
electrical‐resistance
versus
thermodynamic‐temperature
function
is
used,
that
is
not
aimed
to
model
resistivity
of
ideally
pure
platinum
(ie,
of
platinum
as
a
physical
substance),
but
instead
the
resistance
of
real
samples
of
platinum
of
specified
quality.
The
specifications
consist
of
the
requirement
for
platinum
to
be
“pure”
and
“strain‐ free”
and
of
a
minimum
value
for
a
certain
electrical‐resistance
ratio,
everything
else
(including
stability
of
the
characteristics
with
time)
being
left
to
“best
practice”.
PAVESE_AMCTM99.RTF
submitted
to
World Scientific
:
25‐09‐ 2015
Also
explicit
from
the
construction
of
the
Scale
definition
is
that
each
sample
of
real
platinum
is
supposed
to
have
a
measurable
and
non‐reducible
different
characteristics
R(T)
—or
W(T)
=
(R(T)/R0)—
such
that
it
is
not
possible
to
find
a
model
suitable
to
represent,
within
the
aimed
uncertainty,
every
sample
of
platinum,
even
fulfilling
the
given
specifications.
This
effect
is
the
already
discussed
“non‐uniqueness”
of
the
interpolating
instruments:
each
unit
have
a
R(T)
characteristics
slightly
different
from
the
others.
Because
of
this,
each
single
platinum
sample
used
as
a
thermometer
must
be
calibrated
individually
according
to
the
scale.
To
make
the
calibration
more
convenient
than
the
use
of
a
thermodynamic
definition
of
the
scale
it
must
be
less
time
consuming
—i.e.,
in
a
sense,
more
“practical”.
To
be
simple,
or
practical,
the
calibration
should
require
the
measurement
of
each
thermometer
at
the
least
possible
number
of
temperatures
necessary
to
compute
the
values
of
the
empirical
model
parameters.
As
already
pointed
out,
it
is
therefore
intrinsic
in
the
procedure
the
fact
that
the
non‐uniqueness
is
never
eliminated,
but
only
limited
with
the
calibration
to
an
extent
considered
to
fit
the
purpose.
With
regard
to
the
calibration
points,
one
must
ensure
that
the
Scale
definition
is
not
“circular”,
i.e.,
the
measurements
at
the
calibration
points
can
be
done
without
resorting
to
other
temperature
standards,
and
are
independent
on
the
place
or
time
where
they
are
carried
out.
The
ITS‐90
in
the
above
range
therefore
indicates
a
number
of
physical
states
of
ideally
pure
substances,
that
can
be
reproduced
independently:
being
referred
to
the
real
physical
states,
the
Scale
does
not
make
any
prescription
about
their
realisation,
but
instead
leaves
it
to
“best
practice”.
The
temperature
value
of
these
states
cannot
be
obtained
from
the
ITS‐90
itself:
the
values
come
from
two
different
—and
PAVESE_AMCTM99.RTF
submitted
to
World Scientific
:
25‐09‐ 2015
conceptually
totally
distinct—
frames.
For
the
triple
point
of
water
the
value
is
defined
and
has
the
status
of
a
fundamental
constant:
it
is
the
basis
of
the
definition
of
the
unit
of
the
physical
quantity
thermodynamic
temperature.
All
the
others
are
best
estimates
of
experimental
data
obtained
with
thermodynamic
thermometers
measuring
the
same
physical
states:
this
not
only
means
that
there
is
an
uncertainty
associated
with
each
of
them,
but
also
that
they
could
in
the
future
result
not
fully
compatible
with
each
other.
With
regard
to
either
the
number
and
the
position
of
the
calibration
points
on
the
Scale,
this
depends
first
on
what
Nature
makes
available,
secondly
on
the
state‐of‐the‐art
of
the
realisation
of
the
different
candidates,
thirdly
on
the
mathematical
definition
of
the
ITS‐90:
for
this
reason
there
is
no
flexibility
in
their
choice,
i.e.,
they
are
“fixed”
points.
In
summary,
the
ITS‐90
basic
assumption
is
that
is
possible
to
find
a
model
essentially
valid
for
all
samples
of
the
substance,
platinum,
when
defined
by
suitable
specifications.
Sample‐to‐ sample
differences
are
small.
The
definition
is
made
in
two
steps:
1st
step:
The
method
reported
in
1.3
is
applied
for
defining
electrical
resistance
R0
=
f(T)
(actually
R0
=
f(θ))
of
a
few
“typical”
platinum
thermometers.
This
is
taken
as
the
“reference
function”
g0(θ)
for
the
platinum
matching
the
scale
specifications.
2nd
step:
Empirical
correction
functions
cj(θ)
are
applied
for
each
specific
platinum
thermometer:
PAVESE_AMCTM99.RTF
submitted
to
World Scientific
:
25‐09‐ 2015
Rj
=
g0(θ)
+
cj(θ)
(8)
The
number
of
parameters
of
these
functions
are
kept
to
a
minimum,
since
their
value
can
only
be
found
experimentally
by
“calibration”
of
each
thermometer
at
an
equal
number
of
reference
temperatures.
Since
one
must
not
need
to
refer
directly
to
the
thermometers
used
for
step
1,
the
Scale
must
define
a
set
of
the
physical
states
that
can
be
obtained
….
where
each
thermometer
must
individually
be
calibrated.
The
two‐step
method
does
not
eliminate
the
nonuniqueness
of
the
thermometers,
only
limits
its
maximum
extent.
This
means
that
residual
unaccounted
deviations
(type
B
errors)
from
the
true
temperature
will
occur
between
the
calibration
points.
As
a
consequence,
the
choice
of
the
number
and
position
of
the
calibration
points
cannot
be
left
to
user’s
choice,
since
the
mathematical
procedure
set
for
the
corrections
is
affected
by
that
choice.
For
this
reason
the
calibration
points
are
“fixed
points”
of
the
Scale.
2.2
The
ITS90
as
an
example
of
empirical
scale
and
the
concept
of
degree
of
equivalence
of
realisations
in
different
locations
or
times
Let
us
examine
separately:
A)
the
degree
of
equivalence
of
the
Scale
at
the
fixed
points
and
B)
the
equivalence
of
the
Scale
in
between
these
points.
The
ITS‐90
makes
a
difference
between
these
two
situations.
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2.2.1
Degree
of
equivalence
of
the
ITS‐90
at
the
fixed
points
The
values
of
the
electrical
resistance
R
of
each
individual
real
sample
of
platinum
used
as
a
thermometer
are
experimentally
determined
at
the
temperatures
of
a
number
of
physical
states
whose
realisation
does
not
depend
on
the
definition
of
a
temperature
value.
They
are
purely
thermodynamic
experiments
(phase
transitions),
their
practical
implementation
being
done
according
to
the
up‐to‐date
best
practice.
There
are
internal
Fixed point (physical quantity Q)
Q(Τ) Θlab1 = θ lab 1 R1
ITS-90:
Θab2 = θ lab 2 R2
• • •
Θ
= θ≈ Τ
Θlabk = θ
Θlabj = θ lab j Rj
d ef
• • •
lab k Rk
checks
for
the
correctness
of
the
procedures,
but,
eventually,
one
can
obtain
the
evidence
to
have
measured
R
at
the
correct
temperature
of
the
phase
transition
of
each
pure
substance
only
by
comparing
different
realisations
of
the
physical
state
and
different
practical
implementations.
Fig.1.
Laboratory
independent
realisations
of
the
physical
quantity
Q
(fixed
point).
Each
Laboratory
assigns
to
its
realisation
the
same
temperature
value,
the
one
of
the
ITS‐90
definition.
As
a
consequence,
all
practical
devices
for
each
fixed
point
should
be
considered,
at
least
at
start,
as
a
sample
of
the
same
statistical
distribution
containing
all
the
realisations
of
that
physical
state.
PAVESE_AMCTM99.RTF
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For
a
Laboratory
possessing
several
fixed
point
devices,
a
statistical
analysis
can
be
done
locally.
lab 1
lab 2
T
θrc = f(θι) θ0 ≡ Θ
lab j
lab 0
θk = θ0 + Δθ0k lab k
a)
b)
Fig.2.
Comparison
of
independent
Laboratory
realisations
at
a
fixed
point.
a)
Response
differences
Δ
R
jk
are
measured,
that
may
be
transformed
in
Δθ
jk
values.
b)
With
the
arbitrary
choice
of
one
Laboratory
as
a
“reference”
(e.g.,
Lab
0),
one
can
assign
the
ITS‐90
temperature
value
Θ
to
its
realisation
(θ0
=
Θ),
and
obtain
a
θ
k
value
for
each
k‐th
Laboratory.
From
the
{θ
k}
set
one
may
define
a
single
value
θ
rc
=
f(θ
k)
representing
the
comparison
“reference
value”.
In
general,
θrc
≠
Θ
≠
T.
In
principle,
it
is
θrc
that
should
be
assigned
the
value
Θ:
this
would
avoid
the
arbitrary
choice
of
a
reference
Laboratory.
In
a
comparison,
each
Laboratory
sample
becomes
part
of
the
larger
sample
tested
in
the
comparison.
Therefore
the
statistical
analysis
can
be
done
on
the
larger
comparison
sample.
The
analysis
will
tell:
if
the
sample
is
homogeneous;
which
elements
should
be
considered
as
outliers,
and
the
way
to
deal
with
them;
which
is
the
uncertainty
associated
to
the
sample.
The
analysis
should
also
state
a
confidence
interval:
all
practical
implementations
of
the
physical
state
that
fall
within
this
interval
are
not
statistically
different
from
each
other;
hence,
these
implementations
should
be
considered
strictly
equivalent.
The
temperature
value
assigned
to
these
implementations
cannot
PAVESE_AMCTM99.RTF
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be
other
than
the
one
defined
in
the
text
of
the
ITS‐90.
The
associated
uncertainty
at
each
fixed
point
considered
consists
mainly
of
two
parts:
i)
the
uncertainty
stemming
from
the
above
statistical
analysis,
due
to
the
comparison;
ii)
the
uncertainty
in
the
temperature
value
itself
assigned
by
the
ITS‐90
Supplementary
Information
for
all
fixed
points
but
the
triple
point
of
water.
Some
realisations
may
fall
outside
the
confidence
interval.
Still,
this
does
not
necessarily
imply
that
the
degree
of
equivalence
of
such
Scale
realisations
at
the
fixed
points
is
lower
than
the
full
(zero
difference).
In
fact,
at
least
for
a
category
of
low‐temperature
fixed
points
devices,
the
sealed
cells,
for
triple
point
of
gases
not
only
the
characteristics
of
the
solvent
are
stable
with
time,
but
also
those
of
the
solutes
(the
impurities).
As
a
consequence,
it
is
possible
to
make
a
fixed
correction
for
the
temperature
value
of
each
physical
device,
which
may
bring
the
realisation
within
the
confidence
interval.
For
the
purpose
of
defining
the
degree
of
equivalence
of
ITS‐90
realisations
of
different
Laboratories
in
the
range
considered,
should
a
reference
value
be
required
for
each
comparison
excercise,
it
should
be
set
equal
to
the
value
assigned
to
the
temperature
of
the
triple
point
of
water
by
the
definition
of
the
SI
unit
of
thermodynamic
temperature
and
to
every
other
fixed
point
by
the
ITS‐90.
And,
the
temperature
differences
(actually
ΔR
converted
in
ΔT)
of
each
laboratory
realisation
whose
values
are
falling
within
the
limits
of
non‐significance
—after
application,
if
necessary,
of
a
fixed
correction
for
the
practical
realisations
that
are
permanent—
should
be
set
to
zero.
For
the
equivalence
at
the
fixed
points
there
is
no
contribution
to
non‐equivalence
stemming
from
the
scale
definition
itself.
PAVESE_AMCTM99.RTF
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It
is
also
worth
noting
that
the
above
procedure
—and
the
equivalence
concept
itself—
does
not
depend
on
any
of
the
characteristics
of
the
interpolating
instrument.
On
the
other
hand,
no
further
degree
of
freedom
is
left
to
any
Laboratory
by
the
ITS‐90
definition
after
calibration
at
the
fixed
points
of
an
interpolating
instrument
passing
the
quality
tests
required
by
the
Scale
definition.
Therefore,
no
responsibility
(intrinsic
to
the
concept
of
equivalence)
can
be
charged
to
any
Laboratory,
for
example,
for
Scale
or
thermometer
non‐ uniqueness.
Finally,
the
degree
of
equivalence
of
two
realisations
assesses
that
each
of
the
two
Laboratories
has
the
expertise
and
the
methods
for
realising
and
maintaining
the
Scale
for
the
purpose
of
realising
the
Scale
of
the
physical
quantity
temperature
in
its
own
site
within
the
stated
degree.
Therefore,
no
sources
of
uncertainty
–such
as
thermometer
stability
on
transportation
or
over
periods
of
time
much
longer
than
necessary
to
the
local
use
(e.g.,
dissemination)–
should
be
included
in
the
error
budget
of
the
comparisons
since
they
are
irrelevant
to
the
use
of
the
Scale.
2.2.2
Degree
of
equivalence
of
the
ITS‐90
between
fixed
A
result
from
the
above
discussion
concerning
the
fixed
points
is
that
the
degree
of
equivalence
of
the
ITS‐90
between
fixed
points
can
be
computed
from
the
one
stated
at
the
fixed
points,
except
for
the
contribution
of
the
non‐uniqueness
of
the
ITS‐90,
which
comes
from
the
interpolating
instrument
and
can
only
be
assessed
experimentally.
PAVESE_AMCTM99.RTF
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However,
it
is
not
an
assessment
relative
to
any
particular
realisation
in
any
particular
Laboratory,
but
to
the
Scale
definition
itself.
Some
computations
have
been
done
before
1990
during
the
iter
of
approval
of
the
ITS‐90,
based
on
existing
data
for
the
previous
Scale,
the
IPTS‐68:
they
may
be
insufficient.
From
this
point
of
view,
further
comparisons
of
thermometers
at
temperatures
other
than
the
fixed
points
are
useful,
not
to
state
the
equivalence
in
itself,
but
to
obtain
a
better
knowledge
of
the
actual
non‐uniqueness
—either
of
the
ITS‐90
definition
or
of
the
thermometers—
for
the
evaluation
of
the
uncertainty
of
the
degree
of
equivalence
between
fixed
points.
This
because
it
is
clearly
impossible
to
establish
a
degree
of
equivalence
between
fixed
points
better
than
the
Scale
definition
itself.
Consequently,
because
no
responsibility
can
be
charged
to
any
Laboratory
in
interpolating
between
the
fixed
point
temperatures,
no
separate
assessment
of
the
degree
of
equivalence
seems
possible
for
these
temperatures.
Concerning
the
numerical
data
which
should
be
the
output
of
a
comparison
concerning
temperatures
between
fixed
points,
to
be
used
for
the
statement
of
degree
of
equivalence,
there
are
conceptual
difficulties
in
finding
a
definition
for
the
“reference
values”
for
the
temperature
scale
between
fixed
points.
On
the
contrary,
the
definition
of
an
uncertainty
band
is
possible,
based
on
the
uncertainty
values
associated
to
each
fixed
point,
propagated
at
all
the
intermediate
temperatures
and
augmented
by
further
components,
like
non‐uniqueness
of
the
thermometers
and,
in
the
relevant
ranges
where
multiple
definitions
are
allowed,
of
the
Scale.
PAVESE_AMCTM99.RTF
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3 Conclusions
After
introduction
of
the
different
types
of
scales,
with
particular
reference
to
temperature
scales,
the
problem
of
the
definition
of
reference
values
has
been
discussed
for
the
definition
of
the
degree
of
equivalence
of
realisations
in
different
locations
or
times.
For
the
ITS‐90
at
the
fixed
points,
it
has
been
shown
that
they
must
be
coincident
with
the
values
defined
by
the
Scale
definition.
For
any
other
temperature
between
fixed
points,
it
does
not
seem
possible
to
use
the
concept
of
“reference
value”.
References 1. Pavese
F.
and
Molinar
G.,
Modern
GasBased
Temperature
and
Pressure
Measurements
(1992),
Plenum
Press,
New
York.
2. Fodsick
R.L.
and
Rajagopal
K.R.,
Arch.
For
Rational
Mechanics
and
Analysis
81
(1983),
317‐20.
3. Preston‐Thomas
H.,
Metrologia
27
(1990),
3‐10.
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