3d-electrons contribution to cohesive energy of 3d-metals
In this paper a model for 3d-subsystem of transition 3d-metals has been proposed and used for calculation of the cohesive energy dependent on 3d-band filling of particular metal, its bandwidth and effective intra-atomic interaction value. It has been shown that the model enables one to explain the...
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irk-123456789-1570412019-06-20T01:29:41Z 3d-electrons contribution to cohesive energy of 3d-metals Didukh, L. In this paper a model for 3d-subsystem of transition 3d-metals has been proposed and used for calculation of the cohesive energy dependent on 3d-band filling of particular metal, its bandwidth and effective intra-atomic interaction value. It has been shown that the model enables one to explain the observed peculiarities of cohesive energy effect on the atomic number. The nature of two parabolic dependencies of cohesive energy on 3d-band filling has been clarified. The calculated values of cohesive energy are close to those experimentally obtained for Sc-Ti-V-Cr-Mn-Fe series. В роботi запропоновано модель пiдсистеми 3d-електронiв перехiдних металiв та застосовано її для розрахунку енергiї зв’язку, залежної вiд заповнення 3d-зони конкретного металу, ширини цiєї зони та величини ефективної внутрiшньоатомної взаємодiї. Показано, що модель дозволяє пояснити спостережуванi особливостi залежностi енергiї зв’язку вiд атомного номера. Пояснено природу двох параболiчних залежностей енергiї зв’язку вiд заповнення 3d-зони. Обчисленi значення енергiї зв’язку є близькими до отриманих експериментально для ряду Sc-Ti-V-Cr-Mn-Fe. 2018 Article 3d-electrons contribution to cohesive energy of 3d-metals / L. Didukh // Condensed Matter Physics. — 2018. — Т. 21, № 1. — С. 13701: 1–7. — Бібліогр.: 26 назв. — англ. 1607-324X PACS: 71.10.Fd, 71.15.Nc DOI:10.5488/CMP.21.13701 arXiv:1706.06979 http://dspace.nbuv.gov.ua/handle/123456789/157041 en Condensed Matter Physics Інститут фізики конденсованих систем НАН України |
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In this paper a model for 3d-subsystem of transition 3d-metals has been proposed and used for calculation of
the cohesive energy dependent on 3d-band filling of particular metal, its bandwidth and effective intra-atomic
interaction value. It has been shown that the model enables one to explain the observed peculiarities of cohesive
energy effect on the atomic number. The nature of two parabolic dependencies of cohesive energy on 3d-band
filling has been clarified. The calculated values of cohesive energy are close to those experimentally obtained
for Sc-Ti-V-Cr-Mn-Fe series. |
| format |
Article |
| author |
Didukh, L. |
| spellingShingle |
Didukh, L. 3d-electrons contribution to cohesive energy of 3d-metals Condensed Matter Physics |
| author_facet |
Didukh, L. |
| author_sort |
Didukh, L. |
| title |
3d-electrons contribution to cohesive energy of 3d-metals |
| title_short |
3d-electrons contribution to cohesive energy of 3d-metals |
| title_full |
3d-electrons contribution to cohesive energy of 3d-metals |
| title_fullStr |
3d-electrons contribution to cohesive energy of 3d-metals |
| title_full_unstemmed |
3d-electrons contribution to cohesive energy of 3d-metals |
| title_sort |
3d-electrons contribution to cohesive energy of 3d-metals |
| publisher |
Інститут фізики конденсованих систем НАН України |
| publishDate |
2018 |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/157041 |
| citation_txt |
3d-electrons contribution to cohesive energy of 3d-metals / L. Didukh // Condensed Matter Physics. — 2018. — Т. 21, № 1. — С. 13701: 1–7. — Бібліогр.: 26 назв. — англ. |
| series |
Condensed Matter Physics |
| work_keys_str_mv |
AT didukhl 3delectronscontributiontocohesiveenergyof3dmetals |
| first_indexed |
2025-07-14T09:22:53Z |
| last_indexed |
2025-07-14T09:22:53Z |
| _version_ |
1837613684347633664 |
| fulltext |
Condensed Matter Physics, 2018, Vol. 21, No 1, 13701: 1–7
DOI: 10.5488/CMP.21.13701
http://www.icmp.lviv.ua/journal
3d-electrons contribution to cohesive energy of
3d-metals
L. Didukh
Ternopil Ivan Puluj National Technical University, Ternopil, Ukraine
Received July 18, 2017, in final form October 19, 2017
In this paper a model for 3d-subsystem of transition 3d-metals has been proposed and used for calculation of
the cohesive energy dependent on 3d-band filling of particular metal, its bandwidth and effective intra-atomic
interaction value. It has been shown that themodel enables one to explain the observed peculiarities of cohesive
energy effect on the atomic number. The nature of two parabolic dependencies of cohesive energy on 3d-band
filling has been clarified. The calculated values of cohesive energy are close to those experimentally obtained
for Sc-Ti-V-Cr-Mn-Fe series.
Key words: cohesive energy, 3d-metals, electron correlations, energy spectrum, orbital degeneracy
PACS: 71.10.Fd, 71.15.Nc
1. Introduction
In figure 1, the experimental findings are presented for cohesive energy of 3d-metals depending
on the atomic number (see [1], figure 3.8). Similar dependence on the atomic number is observed
for melting temperatures and boiling points of 3d-metals [1, 2]. According to Friedel’s theory, the
cohesion energy of d-metals is defined as a sum of energies for the occupied single-electron states of
a valence band [3]. If the simplest rectangular density of states is used and equivalence of d-bands is
assumed, then a parabolic dependence of the cohesive energy on the number of d-electrons follows. The
parabolic dependence of cohesive energy on the atomic number is in satisfactory agreement with the
experimentally found dependencies for transition 4d- and 5d-metals. However, for 3d-metals, as one can
see from figure 1, there are substantial qualitative discrepancies between the experimental and theoretical
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Figure 1. (Colour online) Dependence of cohesion energy on the atomic number for transition 3d-metals.
This work is licensed under a Creative Commons Attribution 4.0 International License . Further distribution
of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI.
13701-1
https://doi.org/10.5488/CMP.21.13701
http://www.icmp.lviv.ua/journal
http://creativecommons.org/licenses/by/4.0/
L. Didukh
results. Clarification is necessary regarding this “two-hump” dependence and the anomaly associated
with Mn. It has been shown in papers [4–6] that in non-degenerate (Hubbard) model, the cohesive energy
Ecoh =
n
2
(2 − n)w − νU,
where w is the half-width of s-band, U represents the magnitude of intra-atomic Coulomb repulsion
of two electrons with opposite spins on the same lattice site, n stands for the electron concentration, ν
is either the doubly occupied (with two electrons on the same site) states concentration for n < 1 or
empty sites (holes) concentration for n > 1. As a consequence, with interaction taken into account in the
Hartree-Fock approximation, two symmetrical parabolic dependencies Ecoh(n) (for n < 1 and n > 1) as
well as the peculiarity for Mn are explained.
An explanation of Ecoh(n) dependence asymmetry within the s-band model was proposed in pa-
pers [7, 8] by taking into account the electron transfer caused by electron-electron interaction, namely the
correlated hopping of electrons. This way we obtain two asymmetric parabolic dependencies for Ecoh(n)
with respect to n = 1. It is worth noting that the role of correlated hopping has been recently studied in
papers [9–11] within the context of electron interactions in strongly correlated electron systems.
In the present work, developing the models [1–12], the observed peculiarities of cohesive energy
effect on the atomic number in 3d-metals are interpreted within the framework of the model with five-
fold orbital degeneracy of the band with taking into account the dependence of intraatomic interactions
and 3d-band widths in 3d-metals on their filling.
2. The model
1. We represent the Hamiltonian of the model generalized for five-fold orbitally degeneracy (gener-
alized Hubbard model) in the following form
H =
∑
i jmσ
′
tim, jm(n)a
†
imσajmσ + Hint. (1)
Here, the first term describes the delocalization energy of 3d-electrons, which provides the metallic
bonding of atoms in crystals, tim, jm is the delocalization integral of electrons, a†imσ , ajmσ are the
operators of creation and annihilation of electrons on the lattice site in state m (m = 1, 2, 3, 4, 5) with
spin σ (σ =↑, ↓). In the considered model, the transfer integral tim, jm(n) depends on the 3d-electrons
concentration nd in transition 3d-metal. This important peculiarity of the model allows us to calculate
the 3d-electrons contribution in the cohesive energy of particular 3d-metals. This is the distinction of
the present model from the one proposed in paper [12] where the energy band width was assumed
independent of the band filling and from “non-Hubbard-type” approach in papers [13–15].
Hint describes the intra-atomic interactions in the considered model and is a generalization of the
Coulomb interaction term
U
∑
i
ni↑ni↓ (2)
of the orbitally non-degenerate Hubbard model.
Let us assume that
Hint = H1 + H2 + H3 , (3)
where
H1 = U
∑
im
nim↑nim↓ , (4)
H2 = U ′
∑
imm1σ,
m,m1
nimσnim1σ̄ , (5)
H3 = U ′′
∑
imm1σ,
m,m1
nimσnim1σ . (6)
13701-2
3d-electrons contribution to cohesive energy
Here, nimσ is an operator of electron number in the state m with spin σ on site i. The expression (4)
describes the Coulomb repulsion of two electrons with opposite spins in the same orbital m, the expres-
sion (5) represents the Coulomb interaction of electrons in different orbital states (σ̄ denotes the spin
projection opposite to σ). The expression (6) describes the Coulomb interaction of electrons on different
orbitals with taking into account the intra-atomic exchange interaction, U > U ′ > U ′′ (for example,
see [16]).
2. The energy spectrum of electrons, with Hint taken into account in the Hartree-Fock approximation
in the absence of magnetic ordering, is given by the expression
Em(k) = −µ + tm(k), (7)
where tm(k) is a Fourier component of the transfer integral tim, jm. The chemical potential µ is renormal-
ized by taking into account the Coulomb and exchange interactions in the Hartree-Fock approximation
and is to be found from the equation for d-electron concentration
nd =
µ∫
−w
ρ(ε)dε, (8)
where ρ(ε) is the electron density of states with taking into account the orbital degeneracy.
3. In the orbitally non-degenerate model, the cohesive energy is defined by the following expression
Ecoh = −
1
N
∑
i jσ
t(n)〈a†iσajσ〉 − νU, (9)
where the first term is the electron delocalization energy and the second one takes into account the
delocalization energy lowering by polar states. In the Hartree-Fock approximation, we have
ν =
n2
4
for n > 1, (10)
ν = 1 − n +
n2
4
for n < 1. (11)
Using the model unperturbed rectangular density of electron states, we obtain
Ecoh =
1
2w(n)
(w2 − µ2) − νU, (12)
where
µ = w(n)(n − 1). (13)
In the framework of orbitally non-degenerate model, the formula (12) reflects characteristic features of
the experimentally found dependency of cohesion energy on the atomic number in 3d-metals (1).
4. Let us extend the above considerations onto the model with orbital five-fold degeneracy. We define
the cohesive energy as
Ecoh =
µ∫
−w
ε ρ(ε)dε − νUeff(nd), (14)
where Ueff(nd) is an effective intra-atomic interaction which includes both Coulomb and intra-atomic
(Hund’s rule) exchange terms, ν generalizes the expressions (10), (11) of the non-degenerate model.
Following Friedel, we take
ρ(ε) =
5
w(nd)
, (15)
where w(nd) is 3d-band half-width which is dependent on the band filling. Then,
µ =
w(nd)nd
5
− w(nd), (16)
13701-3
L. Didukh
thus,
Ecoh = 10w(nd)
[
nd
10
−
(nd
10
)2
]
− νUeff(nd). (17)
When correlation effects can be neglected, then the cohesive energy maximum corresponds to the band
center, as was obtained in the Friedel’s theory [3].
3. Application for 3d-metals
1. Let us interpret a general character of the Ecoh dependency by formula (17) on the energy parameters
and the band filling in 3d-metals in terms of configurational (atom-band) model of transition metal [17,
18]. Accordingly, in the first approximation by nd we mean the atomic values of the corresponding
element, then moving on to estimations for transition metals, this statement will be corrected by taking
into account s−d-transitions, which cause deviations of 3d-band filling from atomic values of nd.
Given the peculiarities of intra-atomic interactions, represented by the expressions (4)–(6), for the
case of nd < 5, the Hund’s polar states 3dn+1 will be relevant, for the case of nd > 5, the non-Hund polar
states (atomic configurations of electrons with opposite spins on different orbitals) play the role. The case
of nd = 5 should be considered separately, because in this case, the intra-atomic Coulomb interaction is
present at the same orbital, which is considerably greater than interactions at different orbitals. For these
distinct regions of nd, the characteristic effective magnitudes of intra-atomic interactions and averaged
values of bandwidth will be selected.
Let us take for all nd < 5 the same values of w(nd) and Ueff(nd) (w1 and U1, respectively). Besides,
generalizing the expression (10) for non-degenerate model, we obtain
ν1Ueff = 5
(nd
10
)2
U1. (18)
Here, the intra-atomic interaction is taken into account in the Hartree-Fock approximation, ν1 is the polar
3dn+1-states concentration. Therefore, for nd < 5
Ecoh
10w1
=
[
nd
10
−
(nd
10
)2
]
−
(nd
10
)2 U1
2w1
. (19)
In the case of nd > 5, we put values w2 and U2 of formula (17) in correspondence with quantities w(nd)
and Ueff(nd), and for formula (11), the corresponding expression is
ν2Ueff =
[
5 − nd + 5
(nd
10
)2
]
U2. (20)
In accordance with electron-hole symmetry, here ν2 is the “hole” 3dn−1-states concentration. For the
cohesive energy in this case we obtain
Ecoh
10w2
=
nd
10
(
1 +
U2
w2
)
−
(nd
10
)2
(
1 +
U2
10w2
)
−
U2
2w2
. (21)
From formulae (19) and (21) we have that Ecoh(nd) reaches its maximum values at
nd =
10w1
2w1 +U1
(22)
in the case of nd < 5 and for the case of nd > 5, the corresponding value is
nd =
10(w2 +U2)
2w2 +U2
. (23)
One can see that taking into account the intra-atomic interaction shifts the maximum of Ecoh(nd) to the
left or to the right from the band center.
13701-4
3d-electrons contribution to cohesive energy
The reasonable estimate of the intra-atomic interaction magnitude is the halfbandwidth [19]. So, we
take U1 = w1 in formula (22), which leads to nd ≈ 3. This corresponds to the atomic value nd = 3 for
vanadium. In analogous way one can interpret the existence of themaximum value of Ecoh by formula (23)
for nd ≈ 7 (atomic value of nd for cobalt). Hence, by formulae (19) and (21), two parabolic dependencies
can be obtained with maxima specified by formulae (22) and (23). This can be put in correspondence
with the experimentally found dependence of cohesive energy on atomic number, shown in figure 1.
2. For an estimation of the cohesive energy magnitudes for particular 3d-metals, we use the values
for 3d-band widths and 3d-band fillings given in [1] (table 2.1), where W is the bandwidth given by [20]
(table 20.4) and Wexp are experimental findings [21].
One can see that there is a satisfactory agreement between experimental values [20] and calculated
values [1, 2] of the band widths for metals of Sc-Ti-V-Cr series.
In table 1, the results of cohesion energy calculation by formula (17) with data from table 2 are
summarized. Here, E1 are cohesion energies for the atomic values nd (s−d-transitions neglected) and
Wexp used. E2 is cohesive energy from data for nd and Wexp of table 2, E3 are values of cohesive energy
from data for nd and W of table 2. Eexp are the experimentally found values for cohesive energy. In
these calculations, an effective intra-atomic interaction has been taken equal to half-width of conduction
band [19].
Note that for Sc, Ti, V, Cr, the values of W and Wexp are close and reliable, while for Mn, Fe, Co, Ni,
these values are contradictory. For this reason, in table 2 the values for E1 and E2 for metals of the second
group are omitted. The calculated values are close to those experimentally obtained for Sc-Ti-V-Cr-Mn-Fe
sequence. Values E3 for Co and Ni are substantially lower than the experimental findings Ecoh (not listed
in table 1), this can signal of either the inadequacy of the data listed in table 2 or the necessity to go
beyond the framework used for obtaining the expressions (19) and (21), in particular, taking into account
the inter-orbital transitions of electrons and go beyond the Hartree-Fock approximation. In figure 2, the
values of the cohesive energy are marked red. For 3d-electrons in metals, positioned to the right of Mn,
the description in terms of “configurational” (atomic) model is more appropriate, as noted in chapter 3.5
of monograph [22]. Summarizing, one can state that the proposed model not only elucidates the nature
of the two-hump dependence of cohesive energy on the atomic number which is observed in transition
3d-metals and the peculiarity of Ecoh for Mn but also leads to the values of cohesive energy close to those
experimentally found for metals of Sc-Ti-V-Cr-Mn-Fe series.
Table 1. The obtained results for cohesion energy. E1 are cohesion energies for atomic values nd and
Wexp used. For E2 data for nd and Wexp of table 2 were used, for E3 data for nd and W of table 2 were
used. Eexp are experimental data [1, 2].
Metal E1, eV E2, eV E2, eV Ecoh, eV
Sc 2.9 4.0 3.3 3.9
Ti 4.6 5.4 5.0 4.85
V 5.6 5.4 5.4 5.31
Cr 4.1 4.1 4.1 4.10
Mn 2.6 2.92
Fe 4.0 4.28
Table 2. Values for 3d-band fillings given in [1, 2], 3d-band widths W given by [20] and experimental
findings Wexp by [21].
Metal Sc Ti V Cr Mn Fe Co Ni
nd 1.76 2.90 3.98 4.96 5.98 6.94 7.86 8.97
W (2w), eV 5.13 6.08 6.77 6.56 5.60 4.82 4.35 3.78
Wexp, eV 6.2 6.6 6.8 6.5 8.5 8.5 6.9 5.4
13701-5
L. Didukh
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Figure 2. (Colour online) Values of cohesion energy E3 are marked red, experimental findings are marked
blue.
3. The results obtained from the calculation of Ecoh(nd) allow one to interpret the dependencies of
melting temperatures Tm for 3d-metals on the atomic number (analogous to the ones shown in figure 1),
as the melting temperature
Tm =
0.04Ecoh
kB
, (24)
where kB is Boltzmann constant [1, 2].
4. Conclusions
In this paper, a model for 3d-subsystem of transition 3d-metals has been proposed and used for
calculation of the cohesive energy dependent on 3d-band filling of a particular metal, its bandwidth and
the effective intra-atomic interaction value.
It has been shown that the model allows one to explain the observed peculiarities of cohesive energy
on atomic number. The nature of two asymmetric “parabolic” dependencies of cohesive energy on 3d-
band filling has been clarified. The calculated values of cohesive energy are close to the experimentally
obtained for Sc-Ti-V-Cr-Mn-Fe series. The obtained results can be extended for explaining the peculiarities
of melting temperatures of transition 3d-metals on their atomic number and other systems [23–26] for
which strong Coulomb correlations determine the peculiarities of the energy spectrum.
Acknowledgements
Fruitful discussions with Prof. I. Stasyuk are gratefully acknowledged by the author.
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Тернопiльський нацiональний технiчний унiверситет iменi Iвана Пулюя, Тернопiль, Україна
В роботi запропоновано модель пiдсистеми 3d-електронiв перехiдних металiв та застосовано її для роз-
рахунку енергiї зв’язку, залежної вiд заповнення 3d-зони конкретного металу, ширини цiєї зони та вели-
чини ефективної внутрiшньоатомної взаємодiї. Показано,що модель дозволяє пояснити спостережуванi
особливостi залежностi енергiї зв’язку вiд атомного номера. Пояснено природу двох параболiчних за-
лежностей енергiї зв’язку вiд заповнення 3d-зони. Обчисленi значення енергiї зв’язку є близькими до
отриманих експериментально для ряду Sc-Ti-V-Cr-Mn-Fe.
Ключовi слова: енергiя зв’язку, 3d-метали, мiжелектроннi взаємодiї, енергетичний спектр, орбiтальне
виродження
13701-7
https://doi.org/10.5488/CMP.17.43704
https://doi.org/10.1002/pssb.201350313
https://doi.org/10.1103/PhysRevB.95.125133
https://doi.org/10.1051/jphys:01977003806069700
https://doi.org/10.1007/s11106-008-0006-3
https://doi.org/10.1103/PhysRevB.54.5326
https://doi.org/10.1063/1.4916823
https://doi.org/10.1088/0022-3719/8/13/019
https://doi.org/10.1007/s11106-008-0005-4
https://doi.org/10.1016/0304-8853(91)90829-Y
https://doi.org/10.12693/APhysPolA.122.532
https://doi.org/10.5488/CMP.9.1.161
https://doi.org/10.1080/15421406.2016.1254507
Introduction
The model
Application for 3d-metals
Conclusions
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