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https://doi.org/10.1038/s41567-023-02142-z
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A thermodynamic explanation of the Invar
effect
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Supplementary Information: A thermodynamic explanation of the Invar effect
S. H. Lohaus,
1
M. Heine,
2
P. Guzman,
1
C. M. Bernal-Choban,
1
C.
N. Saunders,
1
G. Shen,
3
O. Hellman,
4
D. Broido,
2
and B. Fultz
1
1
Dept. of Applied Physics and Materials Science,
California Institute of Technology, Pasadena, CA 91125, USA
2
Dept. of Physics, Boston College, Chestnut Hill, MA 02467, USA
3
Advanced Photon Source, Argonne National Laboratory, Argonne, IL 60439, USA
4
Dept. of Molecular Chemistry and Materials Science,
Weizmann Institute of Science, Rehovoth 76100, Israel
(Dated: May 31, 2023)
This Supplementary Information presents data and analyses of (a) lattice parameter and thermal
expansion from synchrotron X-ray diffraction, (b) time spectra from nuclear forward scattering and
hyperfine magnetic field distributions obtained from them, (c) heat capacity measurements and an
assessment of the magnetic entropy, (d) phonon partial DOS curves of Fe and Ni, obtained from
computation and from the joint analysis of neutron and NRIXS spectra.
A. Lattice Parameter and Thermal Expansion
The lattice parameter of Fe
65
Ni
35
was measured as
a function of pressure at two temperatures (295 K and
390 K) as described in the Methods. Fig. S1 shows that
below 3 GPa the lattice parameter of Invar is the same
for both temperatures, so there is no detectable thermal
expansion between room temperature and 390 K (see in-
sert). Above the Curie transition the volume increases
with temperature, and the thermal expansion takes a
more typical value of
β
= 3
.
4
·
10
−
5
K
−
1
. To our knowl-
edge, these are the first direct measurements of the ther-
mal expansion of Fe
65
Ni
35
as a function of pressure.
0
2
4
6
8
10
12
P
(GPa)
3.50
3.52
3.54
3.56
3.58
3.60
Lattice parameter (
Å
)
Curie transition
295 K
390 K
0
2
4
6
8
P
(GPa)
0
1
2
3
4
(
K
1
)
1e
5
FIG. S1. Lattice parameter and thermal expansion of Invar as
a function of pressure. Lattice parameter vs. pressure were
measured by synchrotron X-ray diffraction at two tempera-
tures, with samples inside diamond anvil cells. The insert
shows the volumetric thermal expansion coefficient
β
from
these data. Solid lines are polynomial fits.
B. NFS spectra
Nuclear forward scattering (NFS) measures temporal
interference patterns of photons emitted as
57
Fe nuclei
make transitions back to their ground states after being
excited by a synchrotron flash. Magnetism causes a nu-
clear Zeeman splitting of the hyperfine levels, and tran-
sitions from different hyperfine energy levels give rise to
interference beats in the NFS spectra. The measured
magnetic beat patterns (intensity modulations) can be
fitted to a distribution
P
(
B
hf
) of hyperfine magnetic
fields (HMF). Figure S2 shows the evolution of the NFS
spectra and their fits as the magnetic beat patterns are
changed under pressure. The spectra were fitted to ex-
tract the HMF distributions
P
(
B
hf
) shown in Fig. S3.
The magnetization
M
(
P
) presented in Fig. 3 of the main
manuscript is the normalized mean of these HMF distri-
butions.
At low pressures, the pronounced magnetic beats of
Fig. S2 correspond to a near-ferromagnetic state. The
beats broaden with increasing pressure, and the spec-
trum at 4.4 GPa shows an almost exponential decay, con-
sistent with a near-paramagnetic state. The decrease in
magnetization within this region is shown in Fig. S3 as
the HMF distributions move toward lower fields with in-
creasing pressure.
The broad beats in the NFS spectra above 4.6 GPa
(Fig. S2) are nearly independent of pressure. They orig-
inate from the sample thickness, where photons emitted
at different depths interfere with each other. We there-
fore fitted the spectra above the Curie transition with a
superposition of paramagnetic patterns for two sample
thicknesses that are close to the nominal thickness of our
samples. Nevertheless, since the time window of these
measurement (about 110 ns) is constrained by the bunch
spacing in the APS storage ring, the fits cannot reliably
distinguish this thickness effect from beats from a small
magnetic field. Fortunately this does not affect the re-
sults for the magnetic entropy below the Curie transition
that is presented in the main manuscript. A previous
2
40
60
80
100
120
Time (ns)
10
3
10
5
10
7
10
9
10
11
10
13
10
15
Intensity
0.0
0.1
0.9
1.3
2.8
3.8
4.4
5.6
6.3
8.1
12.8
21.3
(GPa)
FIG. S2.
NFS spectra of Invar at labeled pressures. Solid
lines are the fits used to extract the HMF distributions.
conventional M ̈ossbauer spectrometry measurements on
Invar alloys containing
57
Co radioisotope [1], showed that
the magnetization and HMF go to zero at a Curie tran-
sition in pressure.
For calculating the magnetic entropy of Fig. 4a of the
main manuscript, we use the magnetization normalized
at ambient conditions up to the Curie transition from the
NFS experiments. The total entropy at the Curie tran-
sition is calibrated by the heat capacity measurements
(see Eq. 6 of the main manuscript). In reality
M <
1
at ambient conditions, however our computed entropy is
nearly unaffected by this. If, instead, the magnetization
is normalized with data taken at 20 K (not shown here),
only the curvature of the entropy has a minor increase,
but its total change up to the Curie transition is exactly
same because it is calibrated by heat capacity. See [2] for
this alternative analysis.
0
10
20
30
40
B
hf
(T)
0.0
0.5
1.0
1.5
2.0
P (
B
hf
)
0.0
0.1
0.9
1.3
2.8
3.8
4.4
5.6
6.3
8.1
12.8
21.3
P
(GPa)
FIG. S3. Hyperfine magnetic field (HMF) distributions of In-
var at different pressures. The distribution
P
(
B
hf
) was found
by fitting the NFS data of Fig. S2. The fitted HMF distribu-
tions consist of two asymmetrized Gaussians. Vertical lines
mark the mean of each distribution.
C. Heat Capacity
The heat capacity of Invar was measured by differen-
tial scanning calorimetry (DSC) from room temperature
to 773 K, and is displayed in Fig. S4a. These measure-
ments are in good agreement with [3, 4]. The phonon
contribution to the heat capacity was calculated using
the measured phonon DOS,
g
(
ε
), as [6]:
C
P,
ph
(
T
) = 3
N k
B
Z
∞
0
g
(
ε
)
ε
k
B
T
2
×
e
ε/k
B
T
e
ε/k
B
T
−
1
2
d
ε
(1)
where
ε
is the phonon energy and
n
ε,T
= (exp(
ε/k
B
T
)
−
1)
−
1
the Planck distribution of phonon occupancy. The
remaining heat capacity comes from electrons and spins.
3
300
400
500
600
700
T
(K)
3.0
3.2
3.4
3.6
3.8
4.0
C
P
(
k
B
/atom)
T
C
= 515 K
(a)
C
P
total (DSC)
C
P
ph+el
Pepperhoff (1978)
Moriya (1973)
300
400
500
600
700
T
(K)
0.00
0.05
0.10
0.15
S
mag
(
k
B
/atom)
(b)
FIG. S4. Heat capacity contributions and magnetic entropy of
Invar. (a) Heat capacity of Invar measured by DSC, compared
to labeled references. Dashed curve shows the phonon con-
tribution (measured by NRIXS) together with the electronic
heat capacity (fitted at high temperatures). Gray errors show
the standard deviation of three sequential DSC measurements
on two different samples (see Methods). (b) Change in mag-
netic entropy above 300 K calculated with Eq. 2.
The broad magnetic transition can be clearly identified
in the measured heat capacity, with a Curie transition
at 515 K. Some magnetic short-range order is apparent
above the Curie temperature, but above 650-700 K the
heat capacity is dominated by phonons and electrons.
The magnetic entropy shown in Fig.S4b was obtained as:
∆
S
mag
(
T >
300) =
Z
T
300K
C
P,
mag
(
T
′
)
T
′
dT
′
(2)
The change in magnetic entropy between RT and the
Curie transition is ∆
S
mag
(
T
C
) = 0
.
11
±
0
.
005
k
B
/atom,
where the uncertainty was propagated from errors in the
calorimetry measurements.
D. Partial Phonon DOS
NRIXS is a resonant technique that measures the par-
tial phonon density of states (DOS) of
57
Fe in the mate-
rial. The partial phonon DOS from Fe and Ni atoms in
fcc alloys were previously shown to be very similar [5],
which is not surprising due to their similar masses and
electron configurations. Calculations are able to compute
individual DOS contributions from Fe and Ni to the total
phonon DOS of Fe
65
Ni
35
Invar. Figure S5a shows those
calculations at ambient pressure and temperature, con-
firming the similar contributions from Fe and Ni to the
phonon DOS spectrum.
In addition to NRIXS, we also measured the phonon
spectrum of Invar with inelastic neutron scattering (INS)
at ambient conditions, for which Ni has about a 50%
larger scattering cross-section than Fe.
Combining
NRIXS with INS, and accounting for the different weights
and cross-sections, allows us to extract the partial DOS of
Fe, Ni, and the total DOS shown in Fig. S5b. The exper-
imental entropy computed from the (normalized) partial
Fe DOS and from the total DOS are extremely similar
(Fig. S5c), they differ by approximately 0.14%. There
are minor differences around 30 meV, but when comput-
ing the total entropy by integrating the entire DOS up
to 40 meV, both entropies are almost indistinguishable.
The partial Fe DOS is sufficient to compute the total
entropy of Invar.
[1] Rhiger, D. M ̈ossbauer Experiments in Iron-Nickel Alloys
at High Pressures. Ph.D. dissertation in physics, Univ.
Washington, Seattle (1972).
[2] Lohaus, S. H. Experimental study on the thermodynamic
interactions of phonons and magnetism in Fe systems.
Ph.D. dissertation in Materials Science, California Insti-
tute of Technology, Pasadena (2023).
[3] Tanji, Y., Asano, H. and Moriya, H. Specific Heats of Fe-
Ni (fcc) Alloys at High Temperature.
Sci. Reports Res.
Institutes, Tohoku Univ.
24
, 205 (1973).
[4] Bendick, W., Ettwig, H. H. and Pepperhoff, W. Anomalies
in specific heat and thermal expansion of FCC iron alloys.
J. Phys. F: Met. Phys.
8
, 2525 (1978).
[5] Lucas, M. S., Mauger, L., Mu ̃noz, J. A., Halevy, I., Hor-
wath, J., Semiatin, S. L., Leontsev, S . O., Stone, M. B.,
Abernathy, D. L., Xiao, Y., Chow, P and Fultz, B. Phonon
densities of states of face-centered-cubic Ni-Fe alloys.
J.
Appl. Phys.
113
, 17A308 (2013).
[6] Fultz, B.
Phase Transitions in Materials, 2
nd
ed
. Cam-
bridge University Press, Cambridge (2020).
4
0
10
20
30
40
Energy (meV)
0.00
0.02
0.04
0.06
DOS (1/ meV)
(a)
partial Fe
partial Ni
total
0
10
20
30
40
Energy (meV)
0.00
0.01
0.02
0.03
0.04
0.05
DOS (1/ meV)
(b)
partial Fe
partial Ni
total
0
10
20
30
40
Energy (meV)
0
1
2
3
Entropy (
k
B
/ atom)
(c)
partial Fe
total
FIG. S5. Partial densities of states of Fe and Ni, their contri-
butions to the total DOS, and effects on the phonon entropy.
(a) Calculated partial phonon DOS of iron and nickel at am-
bient pressure and temperature. They are weighted by their
atomic composition in Fe
65
Ni
35
and add up to the total DOS
of Invar. (b) Experimental partial phonon DOS of iron (from
NRIXS measurements) and nickel (computed from the INS
and NRIXS measurements), compared to the total DOS from
INS corrected using the NRIXS measurements. (c) Cumu-
lative experimental entropy computed from the partial iron
DOS and total DOS of panel b.