# Magnetic Compton profiles of disordered Fe0.5Ni0.5 and ordered FeNi alloys

D. Beneaa, J. Minárb, H. Ebertc, L. Chionceld,e aFaculty of Physics, Babes-
Bolyai University, Kogalniceanustr 1, Ro-400084 Cluj-Napoca, Romania
bUniversity of West Bohemia, New Technologies - Research Center, Univerzitni
8, 306 14 Plzeň, Czech Republic  cDepartment of Chemistry, University of
Munich, Butenandstr. 5-13, D-81377 München, Germany  d Augsburg Center for
Innovative Technologies, University of Augsburg, D-86135 Augsburg, Germany  e
Theoretical Physics III, Center for Electronic Correlations and Magnetism,
Institute of Physics, University of Augsburg, D-86135 Augsburg, Germany

###### Abstract

We study the magnetic Compton profile (MCP) of the disordered Fe0.5Ni0.5 and
of the ordered FeNi alloys and discuss the interplay between structural
disorder and electronic correlations. The Coherent Potential Approximation is
employed to model the substitutional disorder within the single-site
approximation, while local electronic correlations are captured with the
Dynamical Mean Field Theory. Comparison with the experimental data reveals the
limitation of local spin-density approximation in low momentum region, where
we show that including local but dynamic correlations the experimental spectra
is excellently described. We further show that using local spin-density
approximation no significant difference is seen between the MCP spectra of the
disordered Fe0.5Ni0.5 and a hypothetical, ordered FeNi alloy with a simple
cubic unit cell. Only by including the electronic correlations, the spectra
significantly separate, from the second Brillouin zone boundary down to zero
momenta. The difference between the MCP spectra of ordered and disordered
alloys is discussed also in terms of the atomic-type decompositions. Finally
based on the presented calculations we predict the shape of the MCP profile
for the ordered FeNi alloy along the [111] direction.

###### pacs:

Valid PACS appear here

##  I Introduction

The dominant contribution, according to the non-relativistic limit of the
X-ray scattering comes from the interaction of the photon with the electron’s
charge Cooper (1985). The relativistic Compton amplitude does depend, however,
on the spin of the electron and on the polarization of the X-rays Platzman and
Tzoar (1970). To obtain the scattering cross section the Compton amplitude is
squared and summed over all final states consistent with the energy
conservation. Within the impulse approximation Chew and Wick (1952) the
magnetic scattering cross section measures the spin moments Cooper  _et al._
(1992); Carra  _et al._ (1996); Lovesey (1996) through the integrated
difference, $\int[n^{\uparrow}(\vec{p})-n^{\uparrow}(\vec{p})]\ d^{3}\vec{p}$,
in the momentum distribution $n^{\uparrow(\downarrow)}(\vec{p})$ of spin up
(down) electrons. The magnetic Compton scattering experiments combined with
theoretical calculations of the profile may provide also valuable information
about the exchange and correlation effects in materials.

The computed Magnetic Compton Profile (MCP) spectra for Ni Rennert  _et al._
(1983); Dixon  _et al._ (1998); Kubo and Asano (1990); Timms  _et al._ (1990);
Baruah  _et al._ (2000); Benea  _et al._ (2012); Chioncel  _et al._ (2014a,
b); Minár  _et al._ (2017) and Fe Collins  _et al._ (1989); Kubo and Asano
(1990); Tanaka  _et al._ (1993); Benea  _et al._ (2012) have been previously
reported in the literature. The analysis of spectra covers the aspects of
multiple scattering, core contribution, relativistic effects and electronic
correlations. The comparison with experimental measurements concerning the
shape of the MCP spectra and the values of spin moments were also discussed.
Along the [111] direction prominent features of the MCP for Fe and Ni are (i)
the negative polarization of $s-$ and $p-$bands at low momentum, (ii) dips in
the MCP profiles near $p_{z}=0\ a.u.$, and the (iii) periodic features due to
the Umklapp processes at momenta $\vec{p}=\vec{k}\pm n\vec{G}$, where
$\vec{G}$ is the reciprocal-lattice vector and $n\in Z$. Generally, the theory
overestimates the MCP spectra near $p_{z}=0$ irrespective of the band-
structure method used in the Density Functional Theory (DFT) calculations
Rennert  _et al._ (1983); Dixon  _et al._ (1998); Kubo and Asano (1990); Timms
_et al._ (1990); Baruah  _et al._ (2000); Collins  _et al._ (1989); Kubo and
Asano (1990); Tanaka  _et al._ (1993). This discrepancy have been attributed
to the inadequate treatment of the electron-electron correlations in the Local
Density Approximation (LDA) or its gradient corrected (GGA) type independent-
particle-models for the exchange correlations of DFT. It was shown during the
last decades that Dynamical Mean Field Theory (DMFT) Metzner and Vollhardt
(1989); Georges  _et al._ (1996); Kotliar and Vollhardt (2004), successfully
removes some of the observed inconsistencies Lichtenstein  _et al._ (2001);
Chioncel  _et al._ (2003); Minár  _et al._ (2005); Grechnev  _et al._ (2007);
Granas  _et al._ (2012) in the description of the ground state properties of
3d transition metal elements. DMFT based calculations for the MCP profiles
Benea  _et al._ (2012); Chioncel  _et al._ (2014a, b) of Fe and Ni, showed
indeed that the low momentum discrepancies in the MCP are reduced, however
high resolution measurements would be useful to investigate specific features
in the MCP profiles that are still not well described.

In this paper we report theoretical results on the MCP spectra for the
disordered Fe0.5Ni0.5 alloy using the exchange correlation potential of LDA
and the improved LDA+DMFT method Kotliar  _et al._ (2006). We show that the
discrepancies between the LDA and the experimental spectra at low momentum are
corrected including local dynamic correlations captured by DMFT. At the same
time we identify the correct magnitude of the Coulomb parameters on different
alloy components. To study the interplay of disorder and correlation in
momentum space, we compare MCP spectra of the Fe0.5Ni0.5 alloy with the
corresponding spectra of an ordered FeNi alloy with simple cubic symmetry that
has the same unit cell dimensions and chemical composition. We show that at
the LDA level the total MCP spectra for both ordered and disordered FeNi
alloys are similar, while only within LDA+DMFT the distinction between the two
becomes apparent. In particular at low momenta the MCP spectra obtained within
the CPA calculation is reduced, while for the ordered alloy the MCP spectra is
slightly enhanced. Based on the type decomposition of the MCP, we discuss
differences and similarities between these spectra. As the MCP spectra of
LDA+DMFT is found to be in agreement with the experiment in the disordered
case, and as we are unaware of any previous experimental measurements or
calculations of MCP spectra for the ordered FeNi alloy we thus, predict the
MCP shape of FeNi along the [111] direction.

In the following section (Sec. II) we provide an overview of the LDA+DMFT
computational procedure of the MCP profiles within the Korringa-Kohn-Rostoker
Green’s function formalism for the disordered, Sec.II.1, and ordered,
Sec.II.2, systems. The results are presented in section Sec. III: the total
MCP and type decompositions are discussed in Sec. III.1 and Sec. III.2 for the
disordered respectively ordered alloys. The results for the density of states
and magnetic moment calculations are presented in Sec. III.3, and finally the
paper is concluded in Sec. IV.

##  II Computational details

The most important ingredient in the analysis of the electronic momentum
density $n({\vec{p}})$ in disordered systems is the impurity configuration
averaged Green’s function. Such analysis was achieved for the first time by
Mijnarends and Bansil Mijnarends and Bansil (1976, 1979) within the muffin-tin
framework of the Coherent Potential Approximation, (CPA) Velický  _et al._
(1968); Yonezawa and Morigaki (1973); Elliott  _et al._ (1974); Ziman (1979),
formulated using the multiple scattering theory, the so-called KKR-CPA Györffy
(1972); Johnson  _et al._ (1986); Wang  _et al._ (1995); Vitos  _et al._
(2001) method. Here we present results using the spin-polarized relativistic
Korringa-Kohn-Rostoker (SPR-KKR) method Ebert (2000); Ebert  _et al._ (2011)
in the atomic sphere approximation (ASA). The exchange-correlation potentials
parameterized by Vosko, Wilk and Nusair Vosko  _et al._ (1980) were used for
the LSDA calculations. For integration over the Brillouin zone the special
points method has been used Monkhorst and Pack (1976). In addition to the LSDA
calculations, a charge and self-energy self-consistent LSDA+DMFT scheme for
correlated systems based on the KKR approach Minár  _et al._ (2005); Marco
_et al._ (2009); Minár (2011); Minár  _et al._ (2017) has been used. The many-
body effects are described by means of dynamical mean field theory (DMFT)
Metzner and Vollhardt (1989); Georges  _et al._ (1996); Kotliar and Vollhardt
(2004) and the relativistic version of the so-called Spin-Polarized T-Matrix
Fluctuation Exchange approximation Katsnelson and Lichtenstein (2002);
Pourovskii  _et al._ (2005) impurity solver was used. The realistic multi-
orbital interaction has been parameterized by the average screened Coulomb
interaction $U$ and the Hund exchange interaction $J$. Recent developments
allow to compute the dynamic electron-electron interaction matrix elements
exactly Aryasetiawan  _et al._ (2004). It was shown that the static limit of
the screened energy dependent Coulomb interaction leads to a U parameter in
the energy range of 1 and 3 eV for all 3d transition metals Aryasetiawan  _et
al._ (2004). As the $J$ parameter is hardly affected by screening it can be
calculated directly within the LSDA and is approximately the same for all 3d
elements, i.e J $\approx$ 0.9 eV. In our calculations we used values for the
Coulomb parameter in the range of U = 2.0 to 3.0 eV and the Hund exchange-
interaction J = 0.9 eV. The lattice parameter for both ordered and disordered
alloy was taken as $6.763\;a.u.$ and a BZ integration mesh of $62\times
62\times 62$ points was used.

The computation of Compton profiles within the SPR-KKR formalism Ebert (2000);
Ebert  _et al._ (2011) was worked out a decade ago Benea  _et al._ (2006);
Benea (2004). The Magnetic Compton Profile is given by the momentum
distribution of valence electrons projected along the scattering vector
$p_{z}$. The spin projected momentum density is expressed in terms of the
Green’s function in the momentum representation, constructed from the real-
space Green’s function, using the eigenfunctions of the momentum operator. The
electron momentum densities are usually calculated for the principal
directions $[001],[110],[111]$ using an rectangular grid of 200 points in each
direction. The maximum value of the momentum in each direction is $8\;a.u.$.
Here we present results only for the [111] direction which allows us to
compare with experimental data Kakutani  _et al._ (2003) for the disordered
alloy.

###  II.1 Magnetic Compton profiles for disordered alloys, type decompositions

Given the eigenfunctions of the momentum operator, real-space integration is
used to calculate $G_{m_{s}}(\vec{p},\vec{p},E)$. This integration is
performed over a unit cell, and summed over the cells, as described in Ref.
Benea  _et al._ , 2006. In the momentum representation the ensemble averaged
Green’s function is given by:

$\displaystyle G_{m_{s}}(\vec{p},\vec{p},E)$ $\displaystyle=$
$\displaystyle\frac{1}{\Omega}\,\sum_{q}\sum_{A}x_{q_{A}}\Big{[}-\sum_{\Lambda}\tilde{M}^{q_{A}}_{m_{s}\Lambda
m_{s}}+\,\sum_{\Lambda\Lambda^{\prime}}M^{q_{A}}_{m_{s}\Lambda}\Big{(}D^{q_{A}}\tau^{0q,0q}_{CPA}(E)\Big{)}_{\Lambda\Lambda^{\prime}}M^{q_{A}*}_{m_{s}\Lambda^{\prime}}\Big{]}$
$\displaystyle+$
$\displaystyle\,\frac{1}{\Omega}\,\sum_{q}{\sum_{q^{\prime}}}e^{-i\vec{p}(\vec{R}_{q}-\vec{R}_{q^{\prime}})}\sum_{A\neq
B}x_{q_{A}}x_{q^{\prime}_{B}}\sum_{\Lambda\Lambda^{\prime}}M^{q_{A}}_{m_{s}\Lambda}\Big{(}D^{q_{A}}\tau^{nq,n^{\prime}q^{\prime}}_{CPA}(\vec{p},E)\tilde{D}^{q^{\prime}_{B}}\Big{)}_{\Lambda\Lambda^{\prime}}M^{q^{\prime}_{B}*}_{m_{s}\Lambda^{\prime}}\;.$

We denote by $q(q^{\prime})$ the sites within the cells $n(n^{\prime})$. With
$\tau^{0q,0q}_{CPA}(E)$ we denote the site-diagonal and with
$\tau^{nq,n^{\prime}q^{\prime}}_{CPA}(\vec{p},E)$ the site-non-diagonal, parts
of the scattering path operator. In addition
$\tau^{0q,0q}_{CPA}(E)=\int_{BZ}\tau_{CPA}(\vec{k},E)d^{3}\vec{k}$. The type-
projected scattering path operators $D^{q_{A}}\tau^{0q,0q}_{CPA}(E)$ and
$D^{q_{A}}\tau^{nq,n^{\prime}q^{\prime}}_{CPA}(\vec{p},E)\tilde{D}^{q^{\prime}_{B}}$
appear as a consequence of the single-site approximation of the CPA, when
computing the configuration average
$\langle\tau^{nq,n^{\prime}q^{\prime}}_{\Lambda\Lambda^{\prime}}\rangle$
Faulkner (1982); Gonis (1992). Finally, $M^{q,A}_{m_{s}\Lambda}$ and
$\tilde{M}^{q,A}_{m_{s}\Lambda m_{s}}$ are the regular and irregular Compton
matrix elements for the alloy component A. The explicit form of these
expressions was presented in Refs. Benea  _et al._ , 2006 and Benea, 2004.

In order to proceed with the decomposition of the MCP we shall in the
following analyze Eq. (II.1). A specific site $q$ in the unit cell contains
the components A(B), with the concentrations $x_{q_{A(B)}}$. The site and
component diagonal Green function in momentum representation is:

$\displaystyle G_{m_{s}}^{{A},{A}}(\vec{p},\vec{p},E)$ $\displaystyle=$
$\displaystyle\frac{1}{\Omega}\,\sum_{q}x_{q_{A}}\Big{[}-\sum_{\Lambda}\tilde{M}^{q_{A}}_{m_{s}\Lambda
m_{s}}$ $\displaystyle+$
$\displaystyle\,\sum_{\Lambda\Lambda^{\prime}}M^{q_{A}}_{m_{s}\Lambda}\Big{(}D^{q_{A}}\tau^{0q,0q}_{CPA}(E)\Big{)}_{\Lambda\Lambda^{\prime}}M^{q_{A}*}_{m_{s}\Lambda^{\prime}}\Big{]}.$

The site-diagonal but component-non-diagonal ($A\neq B$) Green function is
obtained from the last term of Eq. (II.1):

$\displaystyle G_{m_{s}}^{A,{B\neq
A}}(\vec{p},\vec{p},E)=\frac{1}{\Omega}\,\sum_{q}x_{q_{A}}x_{q_{B}}\cdot$ (3)
$\displaystyle\sum_{\Lambda\Lambda^{\prime}}M^{q_{A}}_{m_{s}\Lambda}\Big{(}D^{q_{A}}\tau^{0q,0q}_{CPA}(\vec{p},E)\tilde{D}^{q_{B}}\Big{)}_{\Lambda\Lambda^{\prime}}M^{q_{B}*}_{m_{s}\Lambda^{\prime}}$

Accordingly, the spin resolved momentum densities in the disordered system are
obtained integrating the corresponding Green’s functions:

$n_{m_{s}}^{A,B;X}(\vec{p})={-\frac{1}{\pi}\Im\int_{-\infty}^{E_{F}}\left[G_{m_{s}}^{A,B;X}(\vec{p},\vec{p},E)\right]dE}\,$
(4)

With X we denote the functional form in the band structure calculation,
X=LSDA(+DMFT) and $m_{s}=\uparrow(\downarrow)$. Using the expressions for the
Green’s function the pure Eq. (II.1) and the mixed Eq. (3) contributions in
the momentum density can be obtained. The double integral of the spin momentum
density, projected onto the scattering direction ${\bf K}$, with
$\vec{p}_{z}||{\bf K}$, defines the magnetic Compton profile (MCP):

$J_{mag,\bf
K}^{A,B;X}(p_{z})=\int\int[n_{\uparrow}^{A,B;X}(\vec{p})-n_{\downarrow}^{A,B;X}(\vec{p})]dp_{x}dp_{y},$
(5)

###  II.2 Site decomposition of the Magnetic Compton Profile for ordered
alloys

For systems with more atoms in the unit cell, the MCP spectra is usually
decomposed into the site-projected contributions and the interference-like
terms similar to Ref. Benea  _et al._ , 2006. The unit cell sites
$q,q^{\prime}$ can be occupied by atoms of type A or B. The type- and site-
diagonal Green’s function has the form:

$\displaystyle G^{A,A}_{m_{s}}(\vec{p},\vec{p},E)$ $\displaystyle=$
$\displaystyle\frac{1}{\Omega}\sum_{q}\Big{[}\sum_{\Lambda\Lambda^{\prime}}M^{q}_{m_{s}\Lambda}\tau^{q,q}_{\Lambda\Lambda^{\prime}}(\vec{p},E)M^{q*}_{m_{s}\Lambda^{\prime}}$
(6) $\displaystyle-$ $\displaystyle\sum_{\Lambda}\tilde{M}^{q}_{m_{s}\Lambda
m_{s}}\Big{]}.$

The summation over the sites ($q$) in Eq. (6) is restricted to the sites
occupied by the same type of atoms. The site-off-diagonal Green’s functions
is:

$\displaystyle G^{A,B\neq A}_{m_{s}}(\vec{p},\vec{p},E)$ $\displaystyle=$
$\displaystyle\frac{1}{\Omega}\sum_{q}\sum_{q^{\prime}\neq
q}e^{-i\vec{p}\cdot(\vec{R}_{q}-\vec{R}_{q^{\prime}})}\cdot$ (7)
$\displaystyle\sum_{\Lambda\Lambda^{\prime}}M^{q}_{m_{s}\Lambda}\tau^{q,q^{\prime}}(\vec{p},E)_{\Lambda\Lambda^{\prime}}M^{q^{\prime}*}_{m_{s}\Lambda^{\prime}}$

Eq. (7) contains the product of Compton matrix elements
$M^{qA}_{m_{s}\Lambda}$, $M^{q^{\prime}B*}_{m_{s}\Lambda^{\prime}}$ with the
scattering path operator weighted by the phase factor
$\sum_{q,q^{\prime}}e^{-i\vec{p}(\vec{R}_{q}-\vec{R}_{q^{\prime}})}$. In
analogy with elementary formulas for X-ray diffraction by an assembly of
atoms, equations of type Eq. (7) can be interpreted as interference or
structure factor functions for the material. The momentum density and the
magnetic Compton profile are then computed using the formulas Eq. (4) and Eq.
(5). Accordingly, the magnetic Compton interference term is the MCP obtained
using the Green’s function of Eq. (7). Note that the Compton interference
function can be an alloy-type non-diagonal or diagonal depending on the
occupation of the $q(q^{\prime})$-sites. This interference term is an
incoherent scattering contribution and the corresponding MCP signal shall have
a weak amplitude for all directions along $p_{z}$.

##  III Results

In Fig. 1 we depict the unit cell for the ordered FeNi and the disordered
Fe0.5Ni0.5 alloys in the simple cubic geometry. The ordered sc-FeNi alloy
consists of alternating Fe and Ni layers with a unit cell containing two Fe
and two Ni atoms.

Figure 1: (Color on-line) Left: The simple cubic structure of the ordered
FeNi. Fe/Ni red/blue spehres. Right: the “CPA effective” atom (gray sphere) of
composition Fe0.5Ni0.5.

In-plane atoms have neighbors of the same type, while out-of-plane neighbors
are of different types. In this case, the calculation of the DMFT selfenergy
is performed for each type separately, which allows to use different
Coulomb/exchange parameters.

In the fcc-geometry, within the CPA the “effective” Fe0.5Ni0.5 atom has the
same neighbors in all directions. The DMFT (impurity) problem is still solved
for each component Fe/Ni, in addition the CPA equation is imposed for self-
consistency. The charge self-consistency involves the one “effective” atom
unit cell.

###  III.1 Total MCP and type decomposition spectra of Fe0.5Ni0.5 alloy

We have performed the LDA(+DMFT) calculations for the Fe0.5Ni0.5 alloy within
CPA. Table 1 summarizes the results for the spin and orbital magnetic moments.
The DMFT calculations were done for different values of the local Coulomb
interactions for Fe and Ni. As best values for the average Coulomb parameters
we identified $U_{Fe}=2eV$ and $U_{Ni}=3eV$. The average exchange parameter
was set to $J=0.9eV$.

The MCP spectra of Fe0.5Ni0.5 alloy are presented in Fig.2. With decorated
dashed blue/red lines we present raw LSDA/DMFT data. To allow the comparison
with the experimental spectra of Kakutani et al. Kakutani  _et al._ (2003) the
theoretical profiles have been broadened using a broadening parameter (full
width at half maximum FWHM) equal with the experimental momentum resolution
for recording the spectra, which was $\Delta p=0.42\;\;a.u.$. These are seen
in Fig.2a) with solid blue/red lines.

Figure 2: (Color on-line) a) Calculated total MCP of Fe0.5Ni0.5 alloy along
[111] direction. Blue solid line: LSDA(CPA); red solid line: LSDA(CPA)+DMFT.
The experimental spectra of Kakutani et al. Kakutani  _et al._ (2003) (black
circle). b) Type decomposition of MCP profiles of Fe0.5Ni0.5 alloy. LSDA/DMFT
results are represented by dashed/solid lines. The inset shows the mixed term.

In the [111] direction one clearly observe the significant discrepancy between
theory and experiment for $p_{z}<1.5\;a.u.$. The LDA+DMFT calculations capture
the correct behavior at low momenta, similarly to the situation in bulk Fe and
Ni Benea  _et al._ (2012); Chioncel  _et al._ (2014a, b). Many of the specific
features of the theoretical MCP can not be seen in the experimental profile
due to the relative limited resolution ($\Delta p\approx 0.42a.u.$). Within
the first zone, $p<p_{F}^{[111]}\approx 0.8\ a.u$, the theoretical spectra
predict a first peak marked with A and situated at $0.4\ a.u$. This peak is
absent in experiment but its Umklapp is observed experimentally (peak C).
Within the second zone, the theory predicts peaks marked with B and C and
outside the second zone the D and E peaks are visible. Further Umklapp
features can be observed for larger momenta as shoulders at $\sim 2.9a.u.$,
$\sim 3.5a.u.$ etc. Because of the relatively large broadening the
experimental spectra “melts” the peaks B, C and D, therefore the Umklapp of A
into the second Brillouin zone (C) is slightly overestimated. Furthermore the
higher momenta Umklapp shoulders in the experimental profiles are considerably
smeared out.

Based on Eq. (5), the total MCP along the [111] direction has been decomposed,
as seen in Fig. 2b), into the type-projected contributions. Both
$J_{mag}^{FeFe}$ and $J_{mag}^{NiNi}$ spectra show a pronounced dip at
$p_{z}=0$. At non-zero momenta we see that electronic correlations leads to
momentum density redistribution between different Brillouin zones. The
$J_{mag}^{FeFe}$ DMFT spectra is situated below the LSDA spectra for
$p_{z}<p_{F}^{[111]}\approx 0.8\;a.u.$ and within the further Brillouin zones
is above the LSDA. On the other hand the DMFT $J_{mag}^{NiNi}$ is situated
below the LSDA spectra for the entire range of momenta. The inset of Fig. 2b)
shows the mixed MCP term, $J_{mag}^{FeNi}$ obtained from the formula Eq. (4)
and the Green’s function Eq. (3). The mixed term shows no significant
correlation effects, its characteristic being the oscillatory structure.

The type-resolved spectra has been scaled according to the spin moment
obtained by self-consistent calculations, which is 1.57 $\mu_{B}$ in the LSDA
calculations and 1.56 $\mu_{B}$ in the LSDA+DMFT calculations with UFe = 2.0
eV and UNi = 3.0 eV, respectively. Iron gives the dominant contribution in
MCP, as a consequence of its large spin moment: 2.48 $\mu_{B}$ LSDA and 2.46
$\mu_{B}$ in LSDA+DMFT calculation, respectively. A significantly smaller Ni
spin moment is obtained 0.66 $\mu_{B}$ (LSDA) and 0.65 $\mu_{B}$ (LSDA+DMFT)
respectively.

###  III.2 Total MCP spectra and the site decompositions for the ordered FeNi
alloy

In Fig. 3a) we present the comparison between the total MCP [111] profile of
the ordered FeNi alloy obtained using the LSDA (black line) and the LSDA+DMFT
(red line) methods. The computed spectra (no applied broadening) are
normalized to the values of the magnetic moments obtained in LSDA/DMFT
calculations respectively.

Figure 3: (Color on-line) Calculated MCP of the ordered FeNi alloy along [111]
direction: panel a) the total MCP profiles computed with LSDA and DMFT for the
values of $U_{Fe/Ni}=2/3eV$ and $J=0.9eV$. Panels b) and c) the total MCP
profiles for the ordered and disordered FeNi alloys in LSDA and respectively
in DMFT.

Contrary to the disordered case, Fig. 2a) where correlation leads to a
depleted spectra around the zero momenta, we predict that correlation effects
enhance the MCP profile in the range up to $p_{z}\approx 1.5a.u.$. For larger
momenta $p_{z}>1.5a.u.$, similarly to the disordered case, the DMFT
corrections does not change much on the LSDA shape. We observe that at the
LSDA level the MCP spectra can hardly distinguish between the ordered and
disordered structures, Fig. 3b). Any broadening applied to the spectra to
account for the experimental resolution would make the spectra identical.
Including correlation effects Fig. 3c) the LSDA(CPA)+DMFT spectra separate
starting from the maximum value down to zero momenta and match the
experimental results. The MCP spectra the red line Fig. 3a) is our prediction
for the shape of the MCP of ordered FeNi along the [111] direction.

According to Eq. (6) the total spectra is further decomposed into the MCP Ni
Fig. 4a) and MCP Fe Fig. 4b) contributions. Note that Fe’s weight to the total
spectra is about four times larger than that of Ni. The reason why LSDA cannot
distinguish between the ordered and disordered structure Fig. 3b), become also
apparent: the Ni/Fe MCP-components for the ordered alloy under/over estimate
the corresponding spectra of the disordered alloy. The amount of under/over
estimation nearly compensate each other producing a similar total spectra.

Figure 4: (Color on-line) Type decomposition of MCP spectra for the ordered
FeNi alloy along [111] direction: panel a)/b) Ni respectively the Fe
components computed within LSDA (upper part) and DMFT (lower part).

The DMFT calculations for the disordered alloy produce slightly reduced
spectra for Fe (red solid line) in comparison with the LSDA (red dashed line)
as seen in Fig. 2b) . For Ni a slightly stronger reduction takes place. For
the ordered FeNi alloy, DMFT spectra of Fe/Ni are enhanced/diminished in the
low momentum region, however, the increase of the Fe’s MCP dominates the
decrease on the Ni side, and an overall enhanced spectra is obtained (Fig. 4a;
lower part). Fig. 4a, contains also the results of the Ni bulk MCP
calculations. We mark the first essential peaks as in Fig. 2a). Obvious
differences are seen below $p_{z}<2p_{F}~{}\approx 1.6a.u.$ (first two BZ).
For the bulk-Ni the first two peaks (A and B) are shifted and contained within
the first BZ: the first (A) is positioned at about $0.23a.u.$, the second (B)
is in the vicinity of $p_{F}$. The third, the fourth and the subsequent
Umklapp peaks are similar for all three spectra. The intensity of Umkalpps for
Fe0.5Ni0.5 are smeared out additionally because of disorder effects.
Electronic correlations enlarge the differences between the Ni-project spectra
of the disordered alloy with respect to the bulk and ordered FeNi.
Additionally the Ni contribution for the disordered alloy is further smeared
out.

###  III.3 Density of states and magnetic data analysis

In this section we discuss the ground state properties (DOS and magnetic
moments) for the FeNi alloys. In Fig. 5, we show the total and atom resolved
DOS for Fe0.5Ni0.5 (left column) and FeNi alloys (right column). The combined
effect of correlation and disorder is most remarkable for Fe’s DOS, as seen
also in the MCP spectra Fig. 2. There is a strong renormalization of the
spectral function towards the Fermi level on the majority spin channel (spin-
up). For the minority spin channel the weight of DOS is suppressed. In both
spin channels the spectra is broadened accordingly.

Figure 5: (Color on-line) The density of state results for the disordered
(left column) and ordered (right column) FeNi alloy. The DMFT results were
obtained the values U${}_{Fe}=2eV$, and U${}_{Ni}=3eV$ and J=0.9eV.

Our calculations show that the majority spin channel of Ni undergoes
relatively small changes upon addition of Fe. The minority states continue to
remain occupied in the Fe0.5Ni0.5 alloy, as in Ni pure. The spectral changes
are mainly limited to the weight reduction, in both spin channels, which is
more significant than the spectral weight transfer towards the Fermi level,
seen for the Ni majority spins.

Comparing the cubic ordered with the fcc-disordered alloy, one can easily
recognize that the LSDA(CPA)-DOS is more broadened because of the imaginary
part of the complex effective potential. In addition electronic correlations
have a less dramatic effect in the case of the ordered alloy, for the same
values of the local Coulomb and exchange parameters.

Concerning the magnetic moments, the alloys have a ferromagnetic ground state.
The Fe magnetic moment is in the range about $2.5-2.6\mu_{B}$ while a value of
about $0.6\mu_{B}$ is obtained for Ni, depending on the strength of the local
Coulomb parameters $U_{Fe/Ni}$. The Ni magnetic moment remains essentially at
its value in bulk-fcc. The fact that Fe in FeNi has a larger moment than in
fcc Fe at the same lattice constant can be explained by a smaller Fe-Ni
hybridization due to the more contracted $3d$-orbitals of Ni. A larger
hybridization tends to decrease the magnetic moment on an atom by filling the
minority spin 3d orbitals, which are more extended.

##  IV Conclusions

The self-consistent spin polarized electronic structure and the Magnetic
Compton profiles along the [111] direction have been computed for the
disordered Fe0.5Ni0.5 alloy. Disorder has been modeled using the CPA and the
electronic correlations were considered through a multi-orbital Hubbard model
solved with the DMFT. We showed that the discrepancy at low momenta due to the
inadequate treatment of electronic correlations in LSDA can be corrected using
DMFT. Note that DMFT has to be “active” on both alloy components. We have
checked that neglecting “electronic correlations” on one of components or
using improper values of the Coulomb interaction parameter does not provide a
good comparison with the experimental spectra of Kakutani Kakutani  _et al._
(2003). Most notably the LSDA(CPA)+DMFT with U${}_{Fe}=2eV$, U${}_{Ni}=3eV$
and J$=0.9eV$ resolves the discrepancy around $p_{z}=0$. Umklapp features of
the total-MCP spectra can be identified up to momenta $p_{z}<2p_{F}$. Subject
to electronic correlations, integrated spin-resolved momentum density show
significant changes while integrated spin resolved real-space densities
provide almost similar magnetic moments. Due to the limited momentum
resolution of the experimental spectra ($\Delta p=0.42a.u.$) no clear
comparison between theory and experiment can be performed for large $p_{z}$
momenta. High resolution measurements would be useful to identify specific
features of the computed MCP profile.

To study further the interplay between disorder and electronic correlations,
in momentum space, we have performed calculations for the MCP spectra of the
ferromagnetic ordered FeNi alloy. The calculation has been performed in the
supercell setup with a similar lattice parameter as the one of the disordered
alloy. Our results show that within the LSDA the MCP spectra for the ordered
and disordered alloys are similar. This additional shortcoming of the LSDA can
be explained by analyzing the type decomposition of MCP. Beyond the LSDA, the
MCP spectra of ordered and disordered alloys are different, thus we predict
within LSDA+DMFT the MCP shape of the FeNi alloy along the [111] direction.

An interesting conclusion maybe drawn from the results of the present work.
Namely, MCP appears to be more sensitive to changes in the strength of the
electronic correlations rather than in the different geometrical structures
(different disorder realizations). In other words, electronic correlations
affect the momentum distribution more significantly than the chemical bonding
induced by structural disorder. From an experimental point of view high
resolution measurements would be beneficial to resolve further the theoretical
features of the MCP profile at low momentum and the blurring of the Umklapp
features for high $p_{z}$.

$Fe_{0.5}Ni_{0.5}$ | DMFT $U_{Fe/Ni};J_{Fe/Ni}$ | LSDA
---|---|---
$2.0/0.0$ ; $0.9/0.0$ |  $2.0/2.3$ ; $0.9/0.9$ |  $0.0/2.0$ ; $0.0/0.9$ |  $2.0/3.0$ ; $0.9/0.9$
Fe | $m_{s}(\mu_{B})$ | 2.525 | 2.472 | 2.463 | 2.466 | 2.478
$m_{l}(\mu_{B})$ | 0.089 | 0.10 | 0.059 | 0.105 | 0.058
Ni | $m_{s}(\mu_{B})$ | 0.628 | 0.653 | 0.680 | 0.653 | 0.659
$m_{l}(\mu_{B})$ | 0.048 | 0.066 | 0.067 | 0.069 | 0.049
$FeNi$ | DMFT $U_{Fe/Ni};J_{Fe/Ni}$ | LSDA
$2.0/0.0$ ; $0.9/0.0$ |  $2.0/2.3$ ; $0.9/0.9$ |  $0.0/2.3$ ; $0.0/0.9$ |  $2.0/3.0$ ; $0.9/0.9$
Fe | $m_{s}(\mu_{B})$ | 2.594 | 2.596 | 2.572 | 2.599 | 2.573
$m_{l}(\mu_{B})$ | 0.109 | 0.108 | 0.064 | 0.106 | 0.064
Ni | $m_{s}(\mu_{B})$ | 0.578 | 0.584 | 0.60 | 0.584 | 0.598
$m_{l}(\mu_{B})$ | 0.037 | 0.040 | 0.04 | 0.045 | 0.051
Table 1:  Magnetic moments: spin and orbital components for the for the
disordered Fe0.5Ni0.5 and ordered FeNi alloy, computed with LSDA and for
different values of $U_{Fe/Ni};J_{Fe/Ni}$ using LSDA+DMFT.

##  V Acknowledgments

We are indebted to Prof. D. Vollhardt for very useful discussions and for
reading our manuscript. Financial support of the Deutsche
Forschungsgemeinschaft through FOR 1346, the CNCS - UEFISCDI (project number
PN-II-RU-TE-2014-4-0009) and of the COST Action MP1306 EUSpec are gratefully
acknowledged. JM would like to thank CEDAMNF project
(CZ.02.1.01/0.0/0.0/15_003/0000358) funded by the Ministry of Education, Youth
and Sports of Czech Republic.

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