The Pressure Induced Invar Effect in Fe
55
Ni
45
: An Experimental
Study with Nuclear Resonant Scattering - Supplementary Material
P. Guzman,
∗
S. H. Lohaus, C. M. Bernal-Choban, and B. Fultz
California Institute of Technology, Pasadena, California 91125, USA
J. Y. Zhao, G. Shen, M. Y. Hu, and E. E. Alp
Advanced Photon Source, Argonne National Laboratory, Argonne, Illinois 60439, USA
B. Lavina
Center for Advanced Radiation Sources, The University of Chicago, Chicago, IL 60637, USA and
Advanced Photon Source, Argonne National Laboratory, Argonne, Illinois 60439, USA
(Dated: September 24, 2024)
S1. ELINVAR EFFECT
S1.a. Measurements
Charles-
́
Edouard Guillaume discovered that the elastic
modulus had little variation with temperature for fcc al-
loys of Fe
55
Ni
45
[1–3]. These alloys, named “Elinvar,” are
used in many applications requiring elastic stability over
a broad range of temperatures, such as springs and tuning
forks [4]. For most materials, elastic constants generally
decrease with temperature owing to vibrational anhar-
monicity, and such a contribution is expected in Elinvar,
too. The elastic stability of Elinvar with temperature
may be caused by an opposing effect from the change of
magnetism with temperature, but there is no universal
agreement on the fundamental mechanism responsible for
the Elinvar effect.
FIG. S1. Pressure dependence of the isothermal bulk modulus
of Fe-45%Ni at RT (blue) and 392 K (red). The shading
marks the pressure region where the Elinvar effect is observed.
(Inset) Difference between the bulk modulus at RT and 392.
∗
pgguzman@caltech.edu
FIG. S2. Volume and pressure dependence of the isother-
mal bulk modulus of Fe-45%Ni at RT obtained from NRIXS
and NFS measurements. The shading marks the pressure re-
gion where the Elinvar effect is observed. The
∂
B
/∂
T
mag
and
∂
B
/∂
T
sum
were determined with
S
∆
T
mag
from [6]. The
∂
B
/∂
T
mag
*
and
∂
B
/∂
T
sum
*
was determined with
S
∆
T
mag
from
[7]. Refer to manuscript for description of
S
∆
T
mag
.
Between 0
°
C and 400
°
C, nearly constant values of
Young’s modulus and shear modulus of Fe
55.57
Ni
44.43
were reported in previous experimental studies [5]. The
bulk modulus can be expressed in terms of Young’s mod-
ulus and shear modulus using the following equation.
B
=
EG
3(3
G
−
E
)
(1)
From the results in [5], the bulk modulus is therefore
expected to remain nearly constant with temperature at
these temperatures below the Curie transition.
The isothermal bulk modulus is
B
T
=
−
V
∂P
∂V
T
(2)
The bulk modulus was obtained from our pressure-
dependent XRD measurements on Fe
55
Ni
45
by analyz-
ing the curvature of the unit cell volume as a function of
2
pressure at different temperatures. Results for the pres-
sure dependence of the bulk modulus in Fe
55
Ni
45
at room
temperature and 392K are shown in Fig. S1. The inset
of Fig. S1 shows the difference in bulk modulus at room
temperature and 392K for pressures below 5 GPa. The
Elinvar effect is observed for pressures below 3 GPa in
Fe
55
Ni
45
.
S1.b. Thermodynamics
The pressure is the response of the free energy to a
change in volume
P
=
∂F
∂V
T
(3)
Equation 3 gives a thermodynamic relation for the bulk
modulus of Eq. 2 by using the Helmholtz free energy
F
=
U
−
TS
B
T
=
V
∂
2
F
∂V
2
T
(4)
B
T
=
V
∂
2
U
∂V
2
T
−
V T
∂
2
S
∂V
2
T
(5)
where
U
is the internal energy,
T
is temperature, and
S
is entropy. The
B
T
should be the same for either positive
or negative changes in
V
, if the changes are small.
The temperature derivative of Eq. 5 gives the temper-
ature dependence of the bulk modulus. The first term on
the right-hand-side does not depend on temperature, so
∂B
T
∂T
=
−
V
∂
2
S
∂V
2
T
(6)
Determining a second derivative with the Fe
55
Ni
45
en-
tropy data obtained from NFS and NRIXS measurements
was attempted and is shown in Fig. S2. At pressures
below 3 GPa, the temperature dependence of the bulk
modulus is near zero. Interestingly, the magnetic contri-
bution undergoes a large change in sign near 10 GPa.
S2. TEMPERATURE DEPENDENCE OF THE
NORMALIZED MAGNETIZATION
The specific magnetization of an Fe
55
Ni
45
sample
having a mass of 46 grams is shown in Fig. S3. Mea-
surements were performed with a vibrating sample
magnetometer in a Physical Property Measurement
System (PPMS by Quantum Design) with an applied
magnetic field of 0.1 T from a temperature of 10 K to
400 K. Under ambient conditions, the sample is found
to be in a magnetic state equivalent to 90% of the state
of full magnetization.When considering a simple Ising
model to calculate the entropy associated with magnetic
disordering, the 10% of magnetic disorder seen at RT
corresponds to 25% of the magnetic entropy that would
be measured in a sample that begins in an aligned state
and becomes completely demagnetized.
FIG. S3. Fe
55
Ni
45
temperature dependence of the specific
magnetization from 10K to 400 K. The normalized specific
magnetization is shown in the inset.
S3. EFFECTS OF PRESSURE ON PHONON DOS
Figure S4 show
57
Fe phonon density of states (DOS)
curves for Fe
55
Ni
45
, plotted in pairs of sequential pressure
increments. Monotonic behavior of the phonons is ob-
served throughout most of the pressure range. However,
between the pressures of 7.2 and 12.8 GPa, the position of
the peak from the longitudinal modes remains fixed in en-
ergy. The lower part of the phonon DOS, dominated by
transverse modes, stiffens with pressure monotonically,
including through the Curie pressure.
S4. HYPERFINE MAGNETIC FIELD
DISTRIBUTIONS
The interference beats observed in nuclear forward
scattering (NFS) arise from
γ
-ray emissions of different
57
Fe nuclei transitioning from their excited state back to
their ground state. A distribution, P(HMF), of hyperfine
magnetic fields (HMF) can be fit to the beat patterns.
The NFS spectra were fitted with two asymmetrized
Gaussians and HMF distributions were obtained, shown
in Fig. S5.
3
FIG. S4.
57
Fe phonon DOS in Fe
55
Ni
45
shown in pairs with
pressure differences of approximately 2 GPa. Vertical line is
fixed at the average of the peak position of the longitudinal
modes from the 9.2 GPa and 11.1 GPa DOS curves.
REFERENCES
[1] C.-
́
E. Guillaume,
Recherches sur les aciers au nickel
, Di-
latations aux temperatures elevees; resistance electrique.
C. R. Acad. Sci.
125
, 235–238 (1897).
[2] C.-
́
E. Guillaume, The anomaly of the nickel-steels, Pro-
ceedings of the Physical Society of London,
32
, 374,
(1919).
[3] C.-
́
E.
Guillaume,
Nobel
Lecture
in
Physics,
https://www.nobelprize.org/uploads/2018/06/
guillaume-lecture.pdf
, (1920).
[4] E. Wasserman, Invar: Moment-volume instabilities in
transition metals and alloys, Handbook of Ferromagnetic
Materials,
5
, 237–322, (1990).
[5] Y. Tanji, Y. Shirakawa, and H. Moriya, Youngs modulus,
shear modulus, and compressibility of Fe-Ni fcc alloys, J
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34
, 417–421, (1970).
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Ni(fcc) Alloys at High Temperatures, Science Rep. Re-
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24
, 205–217, (1973).
[7] S. H. Lohaus, M. Heine, P. Guzman, C. M. Bernal-
Choban, C. N. Saunders, G. Shen, O. Hellman, D. Broido,
and B. Fultz, A thermodynamic explanation of the Invar
effect, Nature Physics,
19
, 1642–1648, (2023).
4
FIG. S5. Fe
55
Ni
45
HMF distribution obtained from CONUSS
fits at various pressures. The mean HMF at each pressure is
shown by the vertical dashed line.