definitions
methods of measurement
Cp, CV, Cp – CV
relaxation method on PPMS
basics
entropy
Legend:
different contributions
more formulas
self-heating effects
basic
advanced
phase transitions
critical phenomena
crystal lattice
Schottky
conduction electrons
special
Formula labels in red ( L3 )are pointing to the corresponding page.
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basic definition of specific heat
Here is the definition of specific heat:
the amount of heat necessary to change temperature of 1 g of a substance by 1 K
A1
CV , p 
dQ
dT
optionally put any other
amount unit: kg, mol,
atom, …
V,p
it can be realized under constant volume or
pressure, both quantities are generally
different:
A2
C p  CV 
9 2VmT

where
 ……. linear expansion
Vm … molar volume
 ……isothermal compressibility
this holds for solids, for an ideal gas
C p  CV  R
(universal gas constant = 8.314 JK-1mol-1)
The difference between Cp and CV is usually negligible at low temperatures (below ~ 50100 K), but becomes more important with increasing temperature. Here is an example
for Cu:
T (K) Cp (JK-1mol-1) CV (JK-1mol-1) Cp – CV (JK-1mol-1) (Cp – CV)/Cp (%)
129.48
124.58
116.24
0.0063
1000
300
100
4
126.25
123.91
116.16
0.0063
3.23
0.67
0.08
0.00
2.50
0.54
0.07
0.00
As we usually measure Cp, but most of the theoretical models are based on calculating CV,
we have to take it into account for quantitative evaluations above ~ 100 K (depends also on
the aim of evaluation). This correction is included e.g. in the formula for anharmonic effects
in lattice vibrations (equation L4 ).
If we know the specific heat in the whole temperature range from zero up to certain
temperature T1, the total entropy can be easily calculated by integration:
S (T1 ) 
T1

0
C
T
dT
A3
Physical Property Measurement System
uses the relaxation method to determine heat capacity.
vacuum
wires
heater
thermometer
sample puck
T base
sample grease
platform
(3x3 mm)
t0
Heat pulse P(t) is applied, the thermal
response of the system (puck +
sample) is analyzed.
The basic formula is following:
Temperature (K)
34.57
34.43
34.29
34.14
Ctotal
34.00
33.86
33.71
33.57
0
2
4
6
8
10
12
14
Time (sec)
P0
0
The heat pulse is chosen as following:
P0
P(t )  
0
(0  t  t 0 )
(t  t0 )
16
18
20
dT (t )
 P(t )  K w (T (t )  Tb ) R1
dt
Experimentally, we get total heat
capacity (sample+platform+grease)
Ctotal by programming heater power
P(t) and measuring T vs time t, and
minimizing the difference between
measured temperatures and the
model R4 . Here, Tb is the base
temperature (see picture on previous
page), Kw is the heat coupling
through the wires.
t0
Formula R1 holds for platform without
any sample, or sample 100% thermally
attached to the platform. For real
samples, we have to solve R2 + R3 ,
instead of R1 . Ts and Tp are the sample
and platform temperature, Kg is the
thermal coupling between sample and
platform (mediated usually by grease).
Temperature (K)
34.57
34.43
34.29
34.14
34.00
33.86
33.71
33.57
0
2
4
6
8
10
12
14
16
18
20
The T(t) then usually looks like the red
curve, instead of the green one.
Time (sec)
Cplatform
Csample
dTp (t )
 P(t )  K w Tp (t )  Tb   K g Ts (t )  Tp (t ) 
dt
dTs (t )
  K g Ts (t )  Tp (t ) 
dt
R2
R3
More details on solving equation
R1
dT (t )
Ctotal
 P(t )  K w (T (t )  Tb )
dt
R1
and fitting the data:
we choose
(0  t  t 0 )
P
P(t )   0
0
(t  t0 )
Ton (0)  Tb
clearly
Ton (t0 )  Toff (t0 ),
Then the time dependence of temperature T(t) is find as
R4



Ton (t )  P0 1  e t / / Ctotal  Tb
T (t )  
t 0 / 
 ( t t 0 ) / 
T
(
t
)

P

1

e
e
/ Ctotal  Tb
0
 off

(0  t  t 0 )
(t  t0 )
Here, we know P0 and t0, Ctotal, Kw and Tb are unknowns. We find all
unknowns by minimizing the difference between measured
temperatures Ti and those obtained from R4 model at the same time ti.
where
  Ctotal / K w
 T (t )  T 
2
i
i
i
model
measured
Solving R2 + R3 is numerically more complex (more unknowns) but the principle is
the same. Note that we fit only to the expression for Tp since the thermometer in the
system is on the platform, there is no thermometer directly on the sample.
Special remarks when having heat producing samples (e.g. transuranium elements).
 Estimation of the lowest achievable temperature
1) calculate the heating power of your sample
example:
2mg of AmCoGa5
the heating power for 243Am is 6.3 W/mg
the mass of Am in AmCoGa5 is about 0.37 of the total mass, so we
get for the heating power of the whole compound ~ 2.4 W/mg.
for 2 mg, we have the power of 4.8 W.
2) the heating power of the sample causes a difference between the puck temperature
and the platform (or sample) temperature. It holds:
T platform
Prad 
H1
 K w (T )dT
T puck
The wire thermal conductivity Kw is known for PPMS wires (see e.g.
P. Javorský, Low-temperature
specific heat measurements on Transuranium samples, Technical Report JRC-ITU-TN-2002/47, December 2002
).
For a given puck temperature Tpuck, you can then find corresponding platform temperature
Tplatform (that is almost equal to the sample temperature) that satisfies relation H1 .
 Corrections for the temperature gradients
During the measurement, the temperature is measured on the puck and on the platform, the
latter one is assumed to be the sample temperature as well as the temperature of all the
wires. In reality, if your sample produces a heat, the sample is at higher temperature, and
the wires at lower temperature. To correct for this, we can apply following procedure:
1) From the data file calculate the sample heat capacity as usual (i.e. subtract
addenda), Csample(Tplatform) in J/K take the Tplatform (shown as sample temperature),
Tpuck and sample coupling .
-1
-1
Cp (Jmg K )
2) The wire conductivity of the PPMS puck
(Au-35%Pd wires; standard puck has 8 wires
with d = 76m, L = 6mm, He-3 puck has 4
wires with d = 76m) are known as well as the
wire heat capacity; the mass 1/3 of wires in the
He-4 puck is 1.2mg (1/3 is considered
generally for this type of experiment), so you
have to multiply by a factor of 1.2. (for other
pucks, e.g. He-3, recalculate Cp according to
number, material and diameter of the wires).
Au-35%Pd wire
10
1
0.1
0.01
1
10
T (K)
3) Calculate K g  100 K w
for all temperatures.
4) Calculate the radiation power using
H1
for all Tplatform.
5) Calculate the corrected sample and wire temperatures:
P
Twire  T platform  4rad
K
w
Tsample  T platform 
Prad
Kg
H2
6) The error due to the fact that sample is more hot:
C1  Csample (Tsample )  Csample (T platform)
H3
If the Cp(T) increases with increasing temperature (usual case), then this correction means
that the heat capacity Csample at Tplatform (that one you get in your data file) is higher than it
should be. Applying this correction you diminish it.
7) The error due to the fact that the wires are at lower temperature
C2  Cwire (Twire )  Cwire (T platform)
H4
As the Cwire decreases with T, this correction makes your Csample data larger because
originally the PPMS subtracted too high contribution for the wires than it really was.
8) Add both corrections, get corrected Csample(Tplatform) or Csample(Tsample) in J/K and
recalculate to J/mol.K or other units.
Generally, one can observe phase transitions of
 first-order type; in an ideal case we observe here a discontinuity in entropy.
 second-order type; in an ideal case we observe here a discontinuity in heat capacity.
These phase transitions can be of different origin, e.g.
the critical point of a substance (such as water)
 the transition between crystal structures
 the magnetic phase transition
 the trasition from ordered to disordered in liquid crystals
 the transition from superconducting to normal
 the transition from superfluid to normal fluid
….

In large majority of cases, we do not observe the “ideal” phase transition. The
transition is broadened due to many reasons (inhomogeneities etc.).
In case of first-order type transitions, we do not observe an ideal -function, but the
anomaly remains usually very sharp in temperature or other external parameter (e.g.
magnetic field).
UO2
GdPO4
60
2
6
80
Cp/T (J/mol.K )
Cp /T (J/mol.K2)
8
4
2
0
20.0
40
20
25.0
30.0
T (K)
35.0
40.0
0
0.0
1.0
2.0
T (K)
3.0
4.0
In case of second-order type transitions, critical temperature can be determined e.g. by
idealization of the specific heat jump under the constraint of entropy conservation as
shown in this figure. Circles are real experimental points, red line is the idealization
drawn in the way that the yellow and green areas are equal to satisfy the entropy
conservation.
2.5
HoNiAl
Cp/T (J/mol.K2)
2.0
1.5
1.0
T2
0.5
T1
0.0
0
4
8
12
T (K)
16
20
In the vicinity of the second-order phase transitions that correspond to order  disorder
phase transitions, one can study the critical phenomena. There are many theories
describing behaviour near the critical temperature. Following relation represents basic and
most simple description:
C p,V  t

C1
where  is the critical exponent and
More complex relation is written in the form: C p , V 
A

t

t
T  Tc
Tc
1  D t  B  Et
x
C2
C3
Here, , A, B,… are coefficients for T > Tc, whereas the same relation can be written
for T < Tc with primed coefficients. The first term represents the leading contribution to
the singularity in C. Assuming x > 0 then the Dtx vanishes at Tc and represents
contribution of higher order.
In some cases also a logarithmic dependence of C on t is considered.
Generally, the quantitative evaluation of critical phenomena is rather complex and
requires large number of high quality data measured in closest vicinity of Tc.
for more references see e.g. PRB 47 (1993) 14247
(note that we describe the heat capacity in solids, not gas where the heat capacity behaves somewhat different)
The lattice vibrations (phonons) are present in a solid at all temperatures and their heat
capacity and represent a major contribution to the total value in a wide temperature
region. One needs to determine or estimate this contribution even if phonons are of no
interest but other contribution are to be studied quantitatively.
It is practically impossible to determine lattice heat capacity exactly for a real material.
Approximations and models has to be used. Basically, there are two kinds of vibrations
and two models that describe them.
acoustic phonons
optic phonons
Debye model
Einstein model
The phonon spectrum in a crystalline solid consists generally of 3 acoustic and 3(p -1)
optic branches (p is the number of atoms per primitive cell).
Einstein model
The Einstein model assumes the same frequency E for all linear harmonic
oscillators. The heat capacity then depends on single parameter, usually called
the Einstein temperature.
N
CV  k B N A
ω E
θE 
kB

i 1
  Ei

 T



 Ei / T
2
e
(eEi / T  1)2
L1
(holds for 1 phonon branch)
Debye model
In the Debye model, frequency depends on the wave vector,  = c |k|, and all frequencies
up to maximum D occur with defined distribution. The heat capacity depends again on a
single parameter, the Debye temperature.
ω D
θD 
kB
T 
CV  3k B N A  
D 
3  /T
D

0
x 4e x
dx
x
2
(e  1)
(holds for 1 phonon branch)
L2
In principle, one can take different characteristic parameter (E or D) for every phonon
branch. Nevertheless, one can assume more simplifications in most cases. For example,
the heat capacity of acoustic phonons is usually well described by a single D and the
optic phonons can be also usually well grouped.
For lattice with single atom in the unit cell, there are only 3 acoustic branches and the
lattice heat capacity is well described by L2 (with the numerical pre-factor 9 instead of
3).
Low temperatures
at T << D,
simplifies to
(usually below  10K as D ~ 100K)
CV   T 3  234 R θ D3T 3
L3
(holds for 1 atom in unit cell = 3 phonon branches)
As the contribution of optic vibrations to the heat capacity at these low temperatures is
much smaller than that of acoustic ones, L3 can be used for majority of materials at
low T.
N 234 R
θ D ( N atoms )  3
useful formula when relating  to D:
β
Some notes on lattice heat capacity
 Many people use just the Debye model to describe the whole lattice with more than
one atom in the unit cell. This can well work in limited temperature range, but be
aware that the D value obtained from fit to experimental data depends on how many
phonon branches it describes!
- blue line is the correct use of L3
- red line, which originates from the fit to T 2 between
200 and 500 K2, is clearly misleading
Cp/T (J/mol.K2)
 Be aware that L3 holds only at T << D!
At higher T, 1) L3 is not valid, 2) optic phonons contribution is not negligible
Taking some limited temperature range at higher T, you can get similar T3
dependence (linear Cp/T vs T2), but you can’t use L3 . You would get wrong values
from the fit. See this example.
0.4
UCoGa5
0.3
0.2
0.1
T 2(K2)
0.0
0
100 200 300 400 500
Anharmonicity effects
Equations
L1
to L3 are valid in the harmonic approximation only.
The anharmonicity becomes more important at higher temperatures (~ 200 K and
above), depending strongly also on the given lattice properties. To calculate the heat
capacity in such a case is rather complex, so one tries to use simple approximations.
Such simplification is given by term
1
1   iT
L4
that should multiply lattice CV for every branch. This correction involves also the Cp-CV
difference that also becomes important at higher T. The parameters i can be generally
different for different branches, but are often fixed to one common value for simplicity.
For theoretical background see e.g.:
C.A. Martin, J.Phys.: Condens. Matter 3 5967 (1991)
The term “Schottky specific heat ” is used to label specific heat due to thermal excitations
in an energy-level system. Typical (and probably most frequent) example is the energylevel system in rare-earth ions due to crystal-field splitting of the ground-state multiplet.
The energy splitting is of the order of ~1 to ~100 K, usually well separated from higher
levels. It makes these excitations visible for specific heat below the room temperature.
The specific heat is the temperature derivative of energy
C

 
U 
E / k T

 ni Ei e i B Z
T T
S1
with the partition function Z   ni e  Ei / k BT
The equation S1 is also equivalent to
The final result is:
CSchottky
2
C  k B 2 T ln Z 
T

S3
 Ei / k BT
  n ( E / k T ) 2 e  Ei / k BT
n
(
E
/
k
T
)
e

i
i
B
i
i
B
 k BNA 

 Ei / k BT

 Ei / k BT 2
n
e

i
n
e
 i


S2

 
2


S4
In the simplest case of two-level system, Cschottky shows as an anomaly with
maximum at ~ 0.4*E.
5
4
CSch (J/mol.K)
E
3
2
1
0
0
50
100
150
T (K)
200
250
300
For a more complex level system, the Schottky specific heat is smooth function
without any clear anomalies. Nevertheless, it can be often fit to S4 . The crucial
difficulty is the separation of the Schottky term from other contributions (e.g.
lattice specific heat).
In a short qualitative way, it can be derived in a following way:
By increasing the temperature from 0 K to T, electrons at states around the Fermi
level gain energy ~ kT. The number of these electrons ~ NT/TF, where N is the
total number of electrons and TF is the Fermi temperature. We can write:
T
U  N k BT
TF
Cel 
U
T
T
 Nk B
 Nk 2B
 g(E F )k 2BT
T
TF
EF
The formula above gives correct qualitative dependence Cel ~ g(EF) T, where g(EF)
is the density of states at the Fermi level EF.
The full formula is derived e.g. in [N. W. Ashcroft, N. D. Mermin: Solid State Physics, Saunders College,
Philadelphia 1976] or [C. Kittel: Úvod do fyziky pevných látek, Academia Praha 1985]:
Cel  13 π 2g(E F )k 2BT  
E1
, where  is usually labeled as Sommerfeld coeffiecient.
For a free electron gas: g(E F ) 
3N
2E F
, and
k T
Cel  12 π 2 Nk B B
EF
E2
Experimentally, the  coefficient can be often easily determined from the low-temperature
specific-heat data, where the lattice part reduces to ~T3 dependence L3 . If only lattice
and electronic part contribute, we can write
C p / T  γ  βT 2 E3
60
β
Cp/T (mJ/mol.K2)
50
γ
40
In real metals, the  value is often different
from that obtained by free electron model E2 .
LuNiAl
30
To account for this difference, one can
introduce an effective mass m*, since g(EF)
is proportional to the electronic mass me.
Au
20
10
Cu
0
0
20
40
60
80
100
T2 (K2)
 (mJmol-1K-2):
Li Na K Fe Mn Cu Zn Ag Au Al Ga
E2
0.8 1.1 1.7 0.6 0.6 0.5 0.8 0.6 0.6 0.9 1.0
experiment 1.6 1.4 2.1 4.6 15.2 0.7 0.6 0.6 0.7 1.3 0.6
m * γ exp

m e γ f.e.
E4