Photon Factory Activity Report 2017 #35 (2018)

BL-6C/2016G674,2014G691

Temperature dependent local structures in an Inver alloy
Shinya HOSOKAWA,1,* Jens Rüdiger STELLHORN,1 Tomohiro MATSUSHITA,2Naohisa HAPPO,3 Koji KIMURA,4 Kouichi HAYASHI,4 and Kunio YUBUTA5
1Department of Physics, Kumamoto University, Kumamoto 860-8555, Japan
2Japan Synchrotron Radiation Research Institute (JASRI), Sayo 679-5198, Japan
3Graduate School of Information Sciences, Hiroshima City University, Hiroshima 731-3194, Japan
4Department of Physical Science and Engineering, Nagoya Institute of Technology, Nagoya 4668555, Japan5
Institute for Materials Research, Tohoku University, Sendai 980-8577, JapanFe and Ni Ka x-ray fluorescence holography (XFH) measurements were carried out on a single crystal of the traditional
Fe65Ni35 Invar alloy at 100 and 300 K to investigate the temperature dependence of the three-dimensional local structures
around the Fe and Ni atoms, respectively. Local structural information was obtained by detailed analyses using a L1regularized linear regression for the experimental data. At 100 K, the local atomic arrangements around both the elements
show fcc structures. At 300 K, however, only the image around Fe shows a bcc-like neighboring arrangement. From theseXFH data, we propose a model that with increasing temperature, an Fe atom with the low-spin state enters at the central
position of the fcc lattice and is stabilized. And then, large angular positional fluctuations are induced for the atoms at theface-centered positions around the central Fe atom only.
1 Introduction
In 1897, Guillaume [1] discovered an Invar alloy ofFe65Ni35, which has anomalously small thermal expansion
over a wide temperature range. It has been recognizedthat the Invar effect originates from the magnetism, i.e.,there is two types of electronic states in Fe, typicallyhigh-spin (HS) and low-spin (LS) states [2]. In this twostate model, the equilibrium potential energy is lower inthe HS state than in the LS one, while the atomic radius islarger in the former. This results in the compensation
between the usual thermal expansion and the increase of
number density of the smaller atomic size of LS statewith increasing temperature.
X-ray fluorescence holography (XFH) [3] is a newly
developed technique that enables one to draw threedimensional (3D) atomic images around a specificelement emitting fluorescent x-rays. Owing to theinterference between the direct incident x-rays and those
scattered by the surrounding atoms, the fluorescent x-ray
intensity from the emitter slightly modulates with theincident x-ray angles by about some 0.1% (called ahologram), from which 3D images can be obtained
without any special models.
We have measured Fe and Ni Ka XFH on the single
crystal Fe65Ni35 Invar alloy at 100 and 300 K to study thetemperature dependence of the local structures around theFe and Ni atoms independently [4].
2 Experiment
A single crystal of the Fe65Ni35 Invar alloy was grownin the Cooperative Research and Development Center for
Advanced Materials, Institute for Materials Research,
Tohoku University, using an optical floating zone method.
The crystallinity of the sample was examined by taking a
Laue photograph, and the concentration and homogeneity

over the sample were confirmed within the experimental
errors by an electron-probe micro-analysis.
Fe (6.403 keV) and Ni (7.477 keV) Kα XFH
measurements were carried out on the Fe65Ni35 single
crystal at 100 and 300 K using a cryostream apparatus at
BL6C in PF-KEK and at BL39XU and BL12B2 of theSPring-8. The sample was placed on a two-axes table of agoniometer. The measurements were performed in inverse
mode by changing two axes, the incident angle of 0◦ ≤ θ ≤75◦ in steps of 1◦ and the azimuthal angle of 0◦ ≤ f ≤ 360◦in steps of about 0.35◦, of the sample. The fluorescent xrays were collected using an avalanche photodiodedetector, capable of more than one million cps, via atoroidal graphite crystal energy analyzer. Details of theexperimental setup are given elsewhere [3].
Holographic oscillation data (hologram) were obtainedby subtracting the background from the fluorescent x-ray
intensities and normalizing them to the incident x-ray
intensities measured with an ion chamber. Usual
extensions of the holographic data to the 4π sphere weremade using the crystal symmetry and the measured x-ray
standing wave (XSW) lines. The holograms wererecorded at eight incident x-ray energies of 7.5-11.0 (8.512.0) keV for the Fe (Ni) Kα XFH in steps of 0.5 keV.
From the obtained holograms, 3D atomic configuration
images were reconstructed using a L1-regularized linear
regression, which is an inverse problem analysis based ona sparse modeling approach to the experimental
holographic data. Details of this analytical method aregiven elsewhere [5], and successfully adopted in recent
papers [6,7].
3 Results and Discussion
Figure 1 shows examples of the (a) Fe and (b) Ni Kαholograms of Fe65Ni35 single crystal measured at thePhoton Factory Activity Report 2017 #35 (2018)

incident x-ray energies of 8.5 keV, which are drawn under
an orthographic projection. The radial and angular
directions indicate θ and f, respectively, and themagnitudes are given as the color bars beside theholograms. A fourfold rotational symmetry including
XSW signals was observed in the holographic patterns,indicating a good quality of the sample crystal. Theholograms at 100 K look similar to those at 300 K exceptthat the magnitude of the oscillations are slightly larger in
100 K than in 300 K.

nearest-neighboring atoms disappear as indicated by
upward arrows, although the second-neighboring atoms
are clearly observed as a spread disk shape. Of special
interest is that there appears a relatively strong atomic
image at exactly the body- centered position as indicatedby the downward arrow.

(a)

(c)

(b)

(d)

(a) Fe
0.10%
0.05
0.00
-0.05
-0.10%(b) Ni
0.10%
0.05
0.00
-0.05
-0.10%

Fig. 1: Examples of the (a) Fe and (b) Ni Kα hologramsof Fe65Ni35 single crystal measured at 300 K at theincident x-ray energies of 8.5 keV. Taken from Ref. [4].
From these holograms, we can point out two
importances on the local structures of Fe65Ni35 Invar alloy
crystal. The first is that the magnitudes of the holographic
oscillations of both the Fe and Ni Kα are about 0.1%,
much weaker than those of the usual crystal samples ofsome 0.1%. The second is that the spectral features
around the Fe and Ni atoms are quite different from eachother, although based on the XRD data, Fe65Ni35 isbelieved to have the same fcc structures around the Feand Ni atoms.
Figure 2 shows 3D atomic images around the (a) Feand (b) Ni atoms measured at 100 K. The dashed lines
indicate the cubic unit cells, where the lattice constant, a,
is 0.3569 nm [8], and the central Fe/Ni atoms are locatedat the left-bottom corners. Every images in Fig. 2 aredrown as the 12% contour of the maximum of the Ni
images at 100 K. As seen in the figure, fcc lattices areclearly observed around both the Fe and Ni central atoms.
The atomic arrangements around the Fe and Ni atoms arevery similar to each other.
Figures 2(c) and (d) show the same 3D atomic images
measured at 300 K around Fe and Ni, respectively. The
atomic images around the Fe atoms in (c) are very
different from those at 100 K in (a). The fcc images of theFig. 2: 3D atomic images around the (a) Fe and (b) Ni
atoms at 100 K and (c)(d) those at 300 K. Taken from Ref.
[4].
Since the bcc structure is much sparse in spacecompared with fcc, we cannot believe that a fcc-bcc
transformation happens with increasing temperature.
Such a simple structural change should produce a muchlarger thermal expansion unlike the Invar effect.
The atomic image around the Ni atoms in (d) looks
mostly similar to that at 100 K in (b) indicating the fcc
structure. However, an additional image with a similar
size of that around the Fe atom is observed at the bcc
position of the cubic lattice as indicated by downward
arrow. This image is disk-like, where the positional
fluctuation in the angular direction is slightly larger thanthat in the radial direction.
From the present XFH results, we propose a new
structural model for the Invar effect of the Fe65Ni35 alloy.
At 100 K, the structure of this alloy is a fcc structure asshown in Fig. 3(a). With increasing temperature, thefraction of the Fe LS atoms becomes larger andcorrespondingly free volume increases owing to the small
diameter of the LS Fe compared with the HS Fe. The
largest vacancy in the fcc lattice is located at the bodycentered position of the cubic, and a LS Fe atom
preferring the bcc structure can enter there and form aquasi-simple cubic lattice as shown in Fig. 3(b).
Since the interatomic length of the quasi-simple cubic
is only 0.1792 nm, which is much smaller than the fcc
value of 0.2534 nm [8], the additional Fe atom must displace the neighboring atoms at the face-centered positions
even though the cubic center atom is a LS atom with thesmaller atomic size. Thus, the body-centered atom with
the LS state may make the positions of the face-centeredPhoton Factory Activity Report 2017 #35 (2018)

atoms unstabilized in the direction perpendicular to thecubic faces as shown in Fig. 3(c), because the central Fe
atom prefers to form a bcc lattice around it. As a result,
the XFH image around Fe at 300 K looks bcc-like asshown in Fig. 2(c), but the stabilized body-centered atoms
and unstabilized face-centered atoms co-exist.

Fig. 3: A structural model for the Invar effect at (a) 100 K,
and 300 K (b) without and (c) with the positional
fluctuations. Taken from Ref. [4].
On the other hand, the fcc lattice around the central Ni
atom highly preserved even entering an atom in the bodycentered position as shown in Fig. 2(d). This result may
originate from an intrinsic nature of Ni atom that prefers afcc structure at room temperature. As a result, the XFH
image around Ni at 300 K looks primitive cubic like inFig. 2(b), where the stabilized body-centered atoms andface-centered atoms coexist.
Owing to this additional LS Fe atom, the density of thisInvar alloy increases, and compensate for the normal
density decrease with increasing temperature. The HS-LS
spin transition does not solely happen with increasing
temperature, but is accompanied by the structural change
from fcc to quasi-primitive cubic structures.
In summary, Fe and Ni Ka XFH measurements werecarried out on a single crystal of the traditional Fe65Ni35
Invar alloy at 100 and 300 K to investigate thetemperature dependence of the 3D local structures aroundthe Fe and Ni atoms, respectively. Local structural
information was obtained by detailed analyses using a L1regularized linear regression for the experimental data. At
100 K, the local atomic arrangements around both theelements show fcc structures. At 300 K, however, only theimage around Fe shows a bcc-like neighboringarrangement. From these XFH data, we propose a modelthat with increasing temperature, an Fe atom with the LS
state enters at the central position of the fcc lattice and isstabilized. And then, large angular positional fluctuations
are induced for the atoms at the face-centered positions
around the central Fe atom only.
Acknowledgement
This work was performed under the inter-university
cooperative research program of the Cooperative
Research and Development Center for Advanced
Materials, Institute for Materials Research, Tohoku
University (Nos. 14G0014, 15G0004, and 16G0003).
This work was supported by JSPS Grant-in-Aid for
Scientific Research (B) (No. 17H02814), and those on
Innovative Areas “3D Active-Site Science” (Nos.
26105006 and 26105013) and “Sparse Modeling” (No.
16H01553). JRS gratefully acknowledges a financial support as Overseas Researcher under a JSPS fellowship (No.
P16796).
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* shhosokawa@kumamoto-u.ac.jp