arXiv:cond-mat/0606024v1 [cond-mat.mtrl-sci] 1 Jun 2006
Structure peculiarities of cementite and their influence on the
A.K. Arzhnikov∗and L.V. Dobysheva
Physical-Technical Institute, Ural Branch of Russian Academy of Sciences,
Kirov str. 132, Izhevsk 426001, Russia
The iron carbide Fe3C is studied by the first-principle density functional theory. It is shown
that the crystal structure with the carbon disposition in a prismatic environment has the lowest
total energy and the highest energy of magnetic anisotropy as compared to the structure with
carbon in an octahedron environment. This fact explains the behavior of the coercive force upon
annealing of the plastically deformed samples. The appearance of carbon atoms in the octahedron
environment can be revealed by Mossbauer experiment.
PACS numbers: 75.50.Bb, 75.60.Cn, 71.15.Ap
∗Electronic address: firstname.lastname@example.org
The study of the properties of cementite, that is of iron carbide Fe3C, has been con-
ducted for a long time. Originally, cementite was investigated as a component of steels that
essentially affects their mechanical properties. Some attention was paid to the magnetic
properties in connection with nondestructive methods of the steel control.
A recent splash of interest in the Fe3C properties may be explained, first, by new pos-
sibilities of obtaining metastable compounds with mechanical alloying, implantation and
so forth (see, for example, Refs. [1, 2, 3, 4]). Besides, the monophase Fe3C by itself has
attracted considerable interest due to the peculiarities in the bulk modulus behavior  and
the instability of the magnetic state under pressure [6, 7]. Finally, an additional impact has
been given by the studies dealing with the chemical composition of the Earth’s core .
The cementite structure is believed to be known well enough  as for the arrangement of
iron atoms that is well detected by X-ray diffraction. The carbon disposition in the lattice
is still not clear. According to Ref.  the carbon atoms can occupy 4 positions between
the iron sites. Two of them (prismatic and octahedron, Fig. 1 and 2) were repeatedly spec-
ified as possible places of the carbon atoms. The two other named by authors of Ref. 
distorted prismatic and octahedron positions were not discussed earlier. Carbon in the two
last positions is very close to iron atoms and these configurations look improbable. The
contemporary experiments cannot still give unequivocal verification of the carbon positions.
The present-day experimantal data on cementite testify only that the carbon positions ac-
tually depend on the mechanical and thermal treatment. The relevant structural changes
manifest themselves in both the mechanical and magnetic properties. Fig. 3 taken from
Ref.  shows the coercive force as a function of the annealing temperature. Its uncommon
behavior may be attributed only to the redistribution of the carbon atoms in the Fe3C sys-
tem: without such a redistribution, annealing should increase the average grain size (Fig. 3),
reduce the imperfections of the sample, and decrease thus the coercive force. One can find
other experimental evidence for the changes in the carbon atoms positions, for example, the
change in the number of atoms in the nearest environment obtained in EELFS , changes
in the Mossbauer spectra  and so forth.
This paper presents the first-principle calculations conducted in order to determine the
position of carbon and its effect on the physical properties. Earlier theoretical investigations
of cementite have been carried out in numerous papers (see, for example, [6, 14] and their
references), but only in Ref.  a comparison of some characteristics of the two structures
with prismatic and octahedron positions of carbon was done, the relaxation of the crystal
lattice being not taken into account.
Models and methods of calculation
The crystal structure of Fe3C is an orthorhombic lattice with the lattice parameters
a=0.4523 nm, b=0.5089 nm, c=0.6743 nm . The unit cell contains two nonequivalent
iron (4 atoms of Fe 1 type and 8 atoms of Fe 2 type) and one carbon (4 atoms) positions.
The closest to iron atoms are carbon atoms, in the second sphere of Fe 1 atom there are 5
iron atoms at slightly different distances, and in that of Fe2 atom there are 8 iron atoms.
FIG. 1: Two unit cells of cementite with prismatic environment of the carbon atoms. Large black
balls show iron atoms, small white are carbon.
The calculations presented in this paper have been conducted with a full-potential lin-
earised augmented plane wave (FLAPW) method in the WIEN2k package . In the
FLAPW method the wave functions, charge density and potential are expanded in spherical
FIG. 2: Two unit cells of cementite with octahedron environment of the carbon atoms.
300 400500 600
700 800 900 1000 1100
Coercive force, A/m
Average grain size, nm
FIG. 3: The coercive force and average grain size of cementite as a function of annealing temper-
harmonics within non-overlapping atomic spheres of radius RMT and in plane waves in the
remaining space of the unit cell (interstitial region).The basis set is split into the core and
valence parts. The core states are treated with the spherical part of the potential and are
assumed to have a spherically symmetric charge density totally confined inside the muffin-
tin spheres; they are treated in a fully relativistic way. The expansion of the valence wave
functions inside the atomic spheres was confined to lmax= 10 and they were treated within
a potential expanded into spherical harmonics up to l = 4. We used an APW+lo-basis .
The wave functions in the interstitial region were expanded in plane waves with a cutoff
Kmaxdetermined by the relation RMTKmax= 7. The charge density was Fourier expanded
up to Gmax = 20. First atomic relaxation have been conducted in the scalar-relativistic
approximation for the valence electron so that the atoms are in the positions of minimum
total energy, where the calculated forces acting on the nuclei equal zero. A mesh of 60
special k-points was taken in the irreducible wedge of the Brillouin zone. Using the unit cell
obtained after relaxation, we have conducted calculations of the magnetic characteristics
including the spin-orbit coupling through the second variational method  (the number
of k-points here was increased up to 432).
Results and discussion
We have calculated periodical lattices with different positions of the carbon atoms in
the unit cell (Fig. 1 and 2), the lattice parameters being taken equal to the experimental
values. In the unit cell the atoms are shifted to the positions of minimum energy, that is,
the relaxation of the lattice is made.
Table I shows that most essential displacements (7 % for one distance) occur in the
lattice with octahedron environment of the carbon atoms; the distances between the iron
atoms in the lattice with prismatic position of carbon change by less than 2 %. In spite
of the fact that the displacements mightily decrease the energy of the cell with octahedron
position of carbon, the energy of the cell with prismatic position turns out to be lower,
(Eoct− Epris) ≈ 0.0075Ry/at (Table II). This confirms that the lattice with prismatic
carbon position and with account of the atomic relaxation has a minimum energy, so, it is
the ground state. The difference in energy between the two configurations of the carbon
positions testifies that on the one hand the deformation energy under mechanical treatment
is sufficient to shift the carbon from the prismatic positions to the octahedron ones, on the
other hand the probability of these shifts through thermodynamic fluctuations is small at
T < 700K.
TABLE I: Distances (nm) between the iron atoms (marked as Fe1 and Fe2) obtained from X-Ray
data (nonrelax) and after the atomic relaxation made for two positions of carbon atoms
nonrelax prismatic octahedron
Fe2-Fe2 0.2495 0.24590.2314
Fe1-Fe1 0.2653 0.26630.2629
0.2673 0.2671 0.2686
Fe1-Fe2 0.2700 0.2691 0.2715
TABLE II: Total energy (Ry) of the unit cell for  and  directions of magnetisation and
its difference EMAfor prismatic and octahedron environments of carbon atoms
The energy with the spin-orbit coupling included depends on the direction on magneti-
sation and is given in Table II.
The magnetic anisotropy energy EMAattracts most attention. Note that for transition
metals it is generally difficult to calculate because of its small value being a result of the
difference of two large quantities and close to the calculational inaccuracy. In the case
under study, EMA in the system with prismatic environment of carbon is much higher
than in the system with octahedron carbon position and is calculated more reliably. The
calculations show that magnetizing in the  direction is more preferable than in the 
one. The magnetic anisotropy energy - the difference in total energy between the states with
magnetization along these two axes per volume - equals:
EMA= E− E= 7 × 10−5Ry/cell = 7.9 × 105J/m3.
The easy-magnetization axis and the magnetic anisotropy energy correspond to those
obtained in the experiment (EMA= 6.97 × 105J/m3) made for a Fe3C monocrystal at a
temperature of 20.4 K . Some distinctions may be referred to the calculational inaccuracy
or to a possible intermixture of carbon in the octahedron environment in experiment.
For the system with carbon in octahedron environment EMA< 105J/m3and is seemingly
close to the magnetic anisotropy energy of pure iron 6 × 104J/m3. Using a simple model of
interacting magnetic moments at the iron sites along the easy axis, one can find the domain
wall width (see, for example, ) δ = πl?Eexch/EMA. Here l = 0.26nm is the closest
distance between the iron atoms, Eexchis the exchange energy per volume. Let us estimate
Eexchfrom the temperature of the ferromagnet-paramagnet phase transition (TC = 480K
): Eexch≈ kBTCnFe/(ZVcell) = 5.49×107J/m3, where kBis the Boltzmann constant, nFe
is the number of iron atoms in the unit cell, Z = 11÷12 is the number of nearest neighbors.
So, we obtain δpris≈ 6.1nm. Such a domain wall should be effectively pinned to the defects
larger than 10 nm. In Ref.  the samples studied were in the nanocrystalline state with
average grain size of nanocrystals ranging from 10 to 60 nm (Fig. 3). It is natural to assume
that they are places of the pinning of the domain walls. In the system with octahedron
environment of carbon, the domain-wall width is 4÷5 times larger (δoct≈ 30÷40nm), and
the nanocrystals lesser than 30 nm are of no significance in the formation of the coercive force.
During annealing of the deformed cementite, the carbon atoms move from the octahedron
positions to the prismatic ones, the total energy decreasing and the coercive force increasing
The decrease of the coercive force in the samples after annealing at temperature higher
than 700 K is due to a common mechanism: the degree of homogeneity of the crystal state
becomes higher with annealing temperature (Fig. 3).
Note that in spite of the large difference in EMAbetween the lattices with prismatic and
octahedron carbon positions the magnitudes of spin or orbital magnetic moments are not
very different in these two lattices (see Table III). Taking into account the fact that the
samples always contain some amount of other phases of iron and carbon, the magnetisation
measurements do not give a possibility to distinguish the carbon positions. The difference
TABLE III: Spin Mspin and orbital Morb magnetic moments (µB), electric field gradient Uzz
(×1021V/m2) and asymmetry parameter η.
Fe1 Fe2 Fe1Fe2
2.01 1.94 1.881.69
0.05 0.040.05 0.03
3.12 1.342.94 -2.37
η 0.12 0.640.88 0.22
in EMA for systems with equal spin moments and equal orbital moments results from a
difference in solely the electron-density distribution. Table III shows that the electric field
gradients Uzz at iron sites in the second inequivalent position are of opposite sign in the
systems with prismatic and octahedron environment of carbon. This leads to a quadrupole
interaction of different sign, and there should exist a difference in the Mossbauer spectra.
The large quadrupole splitting ∆ = 0.5e2QUzz= 1.44 × 10−27J, (eQ = 0.18 × 10−28m2, see
Ref. ) and the large share of these atoms in the cell nFe2= 8 allow one to believe that
a difference in the shape of the Mossbauer spectrum may be experimentally observed for
the carbon atoms in an octahedron or a prismatic position. Difficulties in interpreting such
Mossbauer experiments arise from the simultaneity of the electric and magnetic interactions.
The combined hyperfine interaction, the angle between the hyperfine magnetic field and the
electric field gradient, and the spatial averaging may essentially complicate the Mossbauer
With the help of the first-principle calculations we show that the magnetic anisotropy
energy of cementite is much higher when the carbon atoms are in the prismatic pores in
contrast with the structure when the carbon atoms occupy the octahedron pores. The former
structure has a lowest total energy. These calculations explain the experimental behavior
of coercive force as a function of the annealing temperature for the plastically deformed
samples. The experimental data and the results of calculations confirm a possibility of
different carbon disposition between the iron sites in cementite and the movement of carbon
atoms during the mechanical or thermal treatment. Such structural changes can be directly
detected in the Mossbauer experiment by a change in quadrupole splitting.
The authors are grateful to Prof. E. P. Yelsukov and Prof. A. I. Ul’yanov for helpful
discussions and for experimental data. This work was partially supported by INTAS (grant
03-51-4778), and RFBR (grant 06-02-16179).
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