ISSN 0031-918X, Physics of Metals and Metallography, 2017, Vol. 118, No. 9, pp. 827–838. © Pleiades Publishing, Ltd., 2017. Original Russian Text © V.V. Pogosov, V.I. Reva, 2017, published in Fizika Metallov i Metallovedenie, 2017, Vol. 118, No. 9, pp. 871–882.
THEORY OF METALS
On the Calculation of the Energies of Dissociation, Cohesion, Vacancy Formation, Electron Attachment, and the Ionization Potential of Small Metallic Clusters Containing a Monovacancy V. V. Pogosov* and V. I. Reva Zaporozhye National Technical University, Zaporozhye, 69063 Ukraine *e-mail:
[email protected] Received November 1, 2016; in final form, December 28, 2016
Abstract—In terms of the model of stable jellium, self-consistent calculations of spatial distributions of electrons and potentials, as well as of energies of dissociation, cohesion, vacancy formation, electron attachment, and ionization potentials of solid clusters of MgN, LiN (with N ≤ 254 ) and of clusters containing a vacancy (N ≥ 12) have been performed. The contribution of a monovacancy to the energy of the cluster and size dependences of its characteristics and of asymptotics have been discussed. Calculations have been performed using a SKIT-3 cluster at Glushkov Institute of Cybernetics, National Academy of Sciences, Ukraine (Rpeak = 7.4 Tflops). Keywords: metallic clusters, vacancy, cohesion energy, vacancy-formation energy, ionization potential, model of stable jellium DOI: 10.1134/S0031918X17070080
INTRODUCTION Frenkel’s theory of melting [1] suggests a discontinuous increase in the concentration of holes (vacancies) at the triple point and a decrease in the energy of vacancy formation with an increase in their concentration, which in turn stimulates the increase in their concentration. The equilibrium concentration of vacancies is calculated based on thermodynamic considerations if the energy of vacancy formation is known, which can be extracted from the data on the annihilation of positrons localized at a vacancy [2]. At the melting point, the concentration of vacancies in metals reaches only a fraction of a percent. Despite such small concentrations, the vacancies exert a significant effect on the properties of solids. It was established that the melting temperature of clusters on a substrate and of free clusters decreases with a decrease in their size [3–6]. Modern massspectrometric and calorimetric methods make it possible to investigate the process of premelting and the melting of clusters that consist of a finite number of atoms in great detail [7–12]. It turned out that, in the process of melting, the diffusion of surface vacancies into the bulk is more advantageous for clusters with unfilled electron shells than for clusters with filled shells (with a magic number of atoms) [8]. The mass-spectrometric measurements of the energy of dissociation of cluster ions of metals were investigated repeatedly and were commented in detail
(see, e.g., [13–17]). Traditionally, these data are used to calculate the cohesion energy of neutral clusters. A popular point of view developed in [18–21] based on thermodynamic considerations is that near the melting temperature the energy of the formation of vacancies in a cluster is the smaller, the smaller the cluster, and that the concentration of vacancies is independent of the cluster size. However, despite detailed mass-spectrometric and calorimetric investigations of the phase transition in clusters [7–9], the problem of the size dependence of the energy of vacancy formation, their concentration, and relation to the process of melting remains open. In various models, the energy characteristics of solid (vacancy-free) clusters have been investigated repeatedly, beginning with [22] (see [23, 24] and references therein). Nevertheless, no self-consistent calculations of the energy of vacancy formation in clusters and of the effect of the quantization of the electron spectrum on this energy have been performed to date. Therefore, one of topical problems that can be formulated in connection with the melting of small-sized aggregates is the investigation of the size-related behavior of the energy of electron attachment to clusters and of the energy of ionization of clusters containing a vacancy. The purpose of this work was the calculation, using the Kohn–Sham method, of the energy characteristics of clusters, both solid (vacancy-free) and containing a monovacancy, in particular of
827
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POGOSOV, REVA
sodium and aluminum clusters, in terms1 of the model of stable jellium.
The wave functions ψ i,v and the eigen-energies ε i,v are found by solving the set of wave equations
− ∇ 2ψ i,v (r ) + v eff ,v (r )ψ i,v (r ) = ε i,v ψ i,v (r ) 2m 2
PRINCIPLE RELATIONSHIPS Let a metallic cluster be a sphere. We will compare the characteristics of spheres with an identical number of atoms N at zero temperature. The radii of the solid spheres RN and the radii of spheres with a monovacancy at the center RN ,v differ from one another as follows:
RN = N
13
r0; RN ,v = (N + 1) r0, 13
(1)
where r0 is the radius of the Wigner–Seitz cell per one atom. In the model of stable jellium, the monovacancy is usually represented in the form of a spherical neutral hole with a radius r0 in a uniform positively charged background produced by ions. The distribution of the ion charge in the cluster with a vacancy is written using step functions θ
ρv (r ) = n θ(r − r0 )θ(RN ,v − r ),
(2)
where n = 3Z is the concentration of the uniform electron gas, and Z is the valence of the metal. For a solid (defect-free) cluster, ρ(r ) = n θ(RN − r ). The total energy of a metallic sphere with a vacancy at its center is written as a functional of the electron concentration (where e is an elementary positive charge) as follows:
∫
∫
+ δv
∫d r θ(r − r )θ(R 3
WS
0
∫
N ,v
(3)
− r )nv (r ),
where Ne
T s,v =
∑ε
i,v
∫
− d 3rnv (r )v eff ,v (r )
i =1
(4)
is the kinetic energy of noninteracting electrons, the number of which is N e = ZN . The last two terms in (3) take into account the structure of the ionic subsystem and the electron–electron interaction in the form of an Ashcroft potential (for details, see [25]). In the Kohn–Sham version, the profile of the electron distribution nv (r ) in the cluster with a vacancy is expressed via the one-electron wave functions N
nv (r ) =
∑ψ i =1
i,v (r )
2
.
with the effective one-electron potential
v eff ,v (r ) = eφv (r ) + v xc,v (r ) + δ v WS θ(r − r0 )θ(RN ,v − r ),
(7)
which includes the electrostatic φv (r ) and exchangecorrelation potential v xc,v [nv (r )] in the local-density approximation (LDA) [26]. The energy is counted from the vacuum level, i.e., from the energy of an electron with a zero kinetic energy, which is located far from (r @ RN ,v ) the sample, where no foreign charges are present. The spatial distribution of the electrostatic potential φv (r ) is found by solving the Poisson’s equation
∇ 2φv (r ) = − 4πe[nv (r ) − ρv (r )]
(8)
under a fixed condition
(4π r03 )
E N ,v = T s,v + e d 3r φv (r )[nv (r ) − ρv (r )] 2 3 3 + d rnv (r )ε xc,v (r ) − Δ ε d r ρv (r )
(6)
(5)
∞
∫ dr 4πr [ρ (r ) − n (r )] = Q e , 2
v
(9)
v
0
where Q is the total charge of the cluster. The symmetry of the problem makes it possible to separate the variables in Eq. (6). In this case, the oneelectron wave functions and the energies are characterized by the radial nr and orbital l quantum numbers. The step of discretization in r was approximately 0.002 a0 (a0 is the Bohr radius). The numerical integration of the Schrödinger equation for searching for the wave functions was performed by the Noumerov method.
Figure 1 displays the profiles of the electron distribution and of the effective potential for a solid cluster and for a cluster with a vacancy at the center containing identical numbers of atoms N = 12. Their radii differ according to Eq. (1). In the figure, the radii of the vacancies and of the clusters are shown. The insets show the profiles of the distributions of the electrons and of the effective potential far from the clusters. Although the electron distributions decrease rapidly, the tails of the potentials extend far (the calculations were performed to r = RN + 900a0 ). For the charged clusters, the electrostatic potential far from the surface decreases as Q/r. For large clusters, the spatial profile becomes similar to that near the surface of the semiinfinite metal, which exhibits a large number of Friedel oscillations.
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n(r ) × 103 , a0–3
15
829
0.006 1
10
1 0.003
2
2 3
5
3 0 12.0
12.5
13.0
Mg
r0 0
5
10
15
20
25
r, a0
Energy, eV
0 Mg
veff(r)
0.5
1
–10
1 2
0 2
–20
3
–0.5 70
3
0
5
77
10
15
80
20
25
r, a0
n(r ) × 103 , a0–3
8
1
0.006
6
1 2
0.003
3
4
2 0 3 12.0
2
12.5
13.0
Li
r0 0
5
10
15
20
25
r, a0
Energy, eV
0 veff(r) –4
1
–8
2
Li 0.5 1 0 –0.5 70
3
0
2 3
5
77
10
15
80
20
25
r, a0 Fig. 1. Self-consistent profiles of distribution of electrons n(r) and of the effective potential v eff(r) for the charged and neutral solid cluster (solid lines) and for a cluster with a monovacancy (dashed lines) containing identical numbers of atoms N = 12: (1) Q = –e, (2) Q = 0, and (3) Q = +e.
IONIZATION POTENTIAL AND THE ENERGY OF ELECTRON ATTACHMENT Clusters of atoms exhibit a structural periodicity that has the property of “spherical periodicity” rather than translational periodicity. The spherical periodicPHYSICS OF METALS AND METALLOGRAPHY
ity is determined by the existence of spherical layers of atoms (atomic shells, or coordination shells). The numbers of atoms in clusters with “occupied” atomic layers are N* = 13, 55, 147, 309, …. Another feature, which is characteristic only of metallic clusters, is
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POGOSOV, REVA 1s
6
1p
1d
2s
1f
2p
1g
2d
1h 3s 2f
4s 1i 3p 1j 2g 3d 1k
Li
Energy, eV
4
2h
IP
W0
2 EA 0 1
2
3
4 N
5
6
1/3
Fig. 2. Ionization potential IP and the energy of electron attachment EA calculated directly through the formulas (10) for solid clusters (points) and clusters with a monovacancy (empty circles).
determined by the degree of the filling of electron shells with increase in the number of atoms. The magic clusters of atoms (with occupied electron shells) possess an enhanced stability as compared to the nonmagic clusters, whose shells are occupied only partially. For example, the magic numbers of atoms are N** = 8, 20, 40, 58, … and N** = 13, 22, 35, 66, … for the clusters of Na and Al, respectively [15]. In the limit of N → ∞ , the difference between the clusters vanishes. By analogy with the above-said, for clusters with vacancies, we can introduce the numbers N* – 1, which minimally correspond to the sphericity of the problem. For these clusters, the calculations start with N = 12. By definition, the ionization potential IP and the energy of electron attachment EA are determined by the differences of the total energies: N −1
N
IPN ,v = E N ,ev − E N ,ev , N
N +1
EA N ,v = E N ,ev − E N ,ev , N −1 E N ,ev
(10)
N +1 E N ,ev
where are the energies of the sphere of radius RN ,v with an excessive charge Q = +e/–e and N
E N ,ev is the energy of a neutral sphere (Q = 0). Figures 2 and 3 display the results of the calculating IP and EA. The letters s, p, d, f, g, h, i, j, k, l, m, and n correspond to the orbital numbers l = 0, …, 11. The total pattern for the entire range of N is shown only for Li in Fig. 2. In this figure, the difference between the solid and defect clusters can be noted. With increasing N, beginning with 12, this difference for Li and Mg can be 0.1–1 eV. The maximum difference is observed upon the transition from the entirely occupied to the
empty shell. With increasing N, the difference decreases. The results of calculations for Mg and Li are given −1 3 in Fig. 3 in the coordinates N for both solid clusters and for clusters containing a monovacancy. For the clusters with a monovacancy, cv = 1 N , therefore, the −1 3
13
relationship N = cv is fulfilled. In our case, cv → 0 at N → ∞. If there is more than one vacancy but their concentration is small (the vacancies do not interact with one another), based on our figures, we can qualitatively observe the dependence of the energy characteristics on the concentration of vacancies. Figure 3 shows the experimental values of IPN and EAN available in the literature. Lithium is an anomalous metal, i.e., the ionization potential of clusters LiN is poorly described within the jellium model; here, the ab initio approach is more successful [15]. Figure 3 also displays the asymptotics
IPN = −μ 0 + α e RN , 2
(11)
EA N = −μ 0 − β e RN , 2
obtained based on the expansion of the chemical potential of electrons over the powers of the reciprocal radius of the neutral cluster
μ(RN ) = μ 0 + μ1 RN + O(RN−2 ). Here, μ 0 = −W 0, and W 0 is the work function of the metal at RN → ∞. The coefficients α = 1 2 − μ1 e 2 and β = 1 2 + μ1 e 2 contain the parameter μ1 = 2σ 0 n , which is characteristic of a given material [27]; and σ 0 is the specific energy of a flat surface (in the limit of N → ∞ ).
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ON THE CALCULATION OF THE ENERGIES OF DISSOCIATION
e2 , IPN ,v = −ε HO N ,v + 2C N+ ,v
(12)
e2 , EA N ,v = −ε LU N ,v − − 2C N ,v
2j 4d 1m 3g 2i 4p 1l 3f 2h 1k 4s 3d 1j 2g 3p 1i 2f 3s 1h 2d 1g 2p 1f 2s 1d 1p 1s
IP
the
partially
occupied
W0
4
2 – [16]
LU ± where ε HO N ,v ε N ,v and C N ,v are the energies of the upper occupied/lower unoccupied electron orbitals of the cluster and the electrical capacitances, respectively. Figure 4 shows the spectra for the solid clusters of Mg and Li. The figure demonstrates the filling of the electron shells with an increase in the number of electrons. Thick dashes denote filled electron levels; thin dashes correspond to unfilled (virtual) levels. The energies of the orbitals ε HO N (upper occupied level) (point at the vertical line) and ε LU N (lower unfilled level) (circle at the vertical line) are also shown.
For
Mg
6
Energy, eV
The ionization potential and the energy of electron attachment demonstrate a strong oscillation behavior caused by the spherical shell structure. These parameters slowly tend asymptotically to W 0, which is caused by the strong orbital degeneration and large angular quantum numbers l. In the experiments, the oscillations are much weaker. The use of the local spin density approximation (LSDA) [28] instead of the LDA leads to a weakening of oscillations. Using the Koopmans theorem, the formulas (10) can be rewritten as follows:
831
shells,
EA 0 0.2
2h
6
1k 3d 3p 4s 2g 3s 1i 1h 1j 2f 2d 1g 2p 1f 2s
1d
0.8
1.0
–1/3
1p
1s
IP
Li 4 – [14]
W0
Energy, eV
HO ε LU N ,v ≈ μ(RN ,v ). The maximum values of ε N ,v correspond to the entirely occupied shells and the magic numbers of atoms N** for the spherical solid clusters and clusters with a vacancy coincide by no means always. For Li, the following values were obtained: N** = 2, 8, 18, 20, 34, 40, 58, 68, 90, 92, 106, 132, 138, 168, 186, 196, 198, (230), 232, (252), 254; for the polyvalent Mg, N** = 1, 4, 9, 10, 17, 20, 29, 34, 45, 46, 53, 66, 69, 78, 93, 98, (99), {115}, (116), {126}, 127, 134, 153, 156, 169, 178, 199, 204, 219. The values for defect clusters that do not coincide with the corresponding numbers for the solid clusters are given in parentheses; while the values that do coincide are given in curly brackets.
2
EA 0 0.2
0.4
0.6 N
LU With increasing RN ,v , the values of ε HO N ,v and ε N ,v oscillate and tend to μ(RN ,v ) at R → ∞. The amplitude
0.8
1.0
–1/3
Fig. 3. Ionization potential IP and the energy of electron attachment EA calculated directly through the formulas (10) for solid clusters (points) and clusters with a monovacancy (empty circles); (crosses) experimental data for Mg [16] and Li [14]; dot-and-dash lines correspond to the asymptotics (12).
RN−3,v .
PHYSICS OF METALS AND METALLOGRAPHY
0.6 N
ε HO N ,v =
of the oscillations decreases approximately as Return now to Figs. 2 and 3. Let us introduce the difference Δ(IPN ) = IPN − IPN ,v . At the first glance, the sign of the difference is unexpected: Δ(I PN ) < 0 , i.e., the circles are located higher than points in the case of the same values of N, except for the clusters with N such that the maximum contribution comes for
0.4
the levels with low l (s, p, and partially d orbitals). In Fig. 2, these narrow regions are located between the vertical dot-and-dash lines. It followed from an anal-
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POGOSOV, REVA
0 LU
Energy, eV
HO
1h 2d 1g 2p 1f 2s 1d 1p 1s
–4
–8 Mg
0
10
20 N
30
40
0
HO
LU
Energy, eV
–2
1g 2p 1f 2s
–4
1d –6
1p 1s
Li 0
10
20 N
30
40
Fig. 4. Examples of spectra for solid clusters of Mg and Li.
ysis of the asymptotic behavior of IPN and IPN ,v (up to the terms ~RN−2) that Δ(IPN ) > 0 [27]. The main vacancy-related dependence is contained in the work function W 0(cv ) < W 0(cv = 0). In the case of small clusters with a monovacancy, the significant contribution is due to vacancies the concentration of which is cv ~ RN−3. As follows from Fig. 1, the behavior of v eff ,v (r ) is such that the electrons are pulled out by the vacancy from the center of the cluster to its surface and are grouped mainly in a spherical layer r0 < r < RN . Upon integration of (3) in spherical coordinates, this region makes the main contribution to the energy. This is confirmed by the spectral values of the energies that correspond to points (circles), e.g., for Li in Figs. 2 and 3. For N = 12, we have ε nr =0,l =0 = –6.36 (–5.83) eV; ε nr =0,l =1 = –4.79 (–4.71) eV; ε HO,LU nr = 0,l = 2 = –2.94 (–3.08) eV. For N = 19, we have ε nr = 0,l =0 = –6.57 (–6.1) eV, ε nr =0,l =1 = –5.24 (–5.16) eV; ε nr = 0,l =2 = –3.69 (–3.77) eV; and ε nr =1,l =0 = –3.25 HO
HO,LU
(–2.31) eV. With increasing N, the contribution from
the volume of the cluster becomes increasingly significant and in the asymptotics N → ∞ the points and circles change places, i.e., the difference Δ(IPN ) becomes positive. We self-consistently calculated the ionization potential IP and the energy of attachment of electrons EA using the general formulas (10), as well as the entire spectrum of the electron energies, including ε HO and ε LU. This permitted us to calculate the capacitances using expressions (10): − e2 −e 2 C , = , , N v 2(IPN ,v + ε HO 2(EA N ,v + ε LU N ,v ) N ,v ) (13) 2 eff e C N ,v = . HO LU IPN ,v + ε N ,v − EA N ,v − ε N ,v Analogous formulas for CN correspond to defectfree clusters. In the classical electrostatics, the capacitances of conducting spheres are determined by their outer radii RN ,v . The roughness of the surface on an atomic scale (the atoms have a finite volume) does not make it possible to precisely determine the position of the boundary [29, 30]. In the jellium model, the boundary of the ionic core always corresponds to the coordinate r = RN ,v . However, the electron cloud increasingly spills out beyond the boundary of the core with a decrease in its radius RN ,v . Moreover, this spilling out depends on the sign of the excessive charge of the cluster (see Fig. 1). Therefore, the quantities C Neff,v , C N+ ,v ,
C N+ ,v =
and C N− ,v are only equal to each other in the limit of N → ∞. Figure 5 displays the results of calculation of the capacitances C N and C N ,v , normalized with respect to their radii RN or RN ,v , respectively. The maximum difference is observed for intervals of N in which the filling of the s and p electron shells takes place. The alternating difference C N ,v RN ,v − C N RN
is mainly determined by the relationship between ε HO and ε LU for various l in the solid and defect clusters, which can change depending on the principal quantum number. The capacitance of the defect clusters upon the filling of the shells with small l is greater than that of the solid clusters; for large clusters, the opposite relationship is observed. Using the experimental data for the Li atoms (IP1 = 5.39 eV, EA 1 = 0.62 eV [31], R1 = r0 ) and the equality ε1HO = ε1LU for the unfilled shells as an example, we obtain C1eff r0 = 1.74 > 1. This value is in qualitative agreement with the calculated values presented in Fig. 5 for the smallest clusters. For incom-
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ON THE CALCULATION OF THE ENERGIES OF DISSOCIATION
C+/R
pletely filled electron shells, the cluster cannot be spherically symmetric, but will have some lower symmetry, e.g., spheroidal.
C–/R
1.2 1.0
Ceff/R
coh = N ε coh N − (N − 1)ε N −1.
(14)
1.2 1.0 0.25
In the model of a stable jellium, the quantity E at is the total energy of a sphere with radius r0 of the Wigner–Seitz cell. 1.2
0.30
(15)
dis n .
coh
coh
2σ 0 , nat RN
Ceff/R
(16)
(17)
where the last term can be written as −Z μ1 RN . As long ago as in the works of Frenkel and Langmuir (see [1]), it was noted that for some substances at low temperatures the following universal relation is fulfilled:
4πr02σ
q ≈ 2 3, which is based on the observable parameters: the average spacing between atoms r0, the specific surface energy σ, and the heat of evaporation q = ε coh (r0 ) (see table in [30]). Using this relation, the asymptotics (17) can be rewritten in the following form suitable for estimates: ⎛ ⎞ ε coh ≈ ε coh (r0 ) ⎜1 − 41 3 ⎟ . N ⎝ 9N ⎠ PHYSICS OF METALS AND METALLOGRAPHY
0.45
1.2
0.25
0.30
0.35
Fig. 5. Results of calculating electric capacitances for solid clusters (points connected by lines) and clusters with a monovacancy (circles). Capacitances are normalized with respect to radii of clusters.
n=2
(r0 ) −
0.40
1.2
1.0
The asymptotics of the size dependence of the cohesion energy (15) represents the well-known classical result [1]
εN = ε
N
–1/3
0.45
Li
N
∑ε
N
0.40
1.0
At N → ∞, ε coh → ε ∞coh ≡ ε coh (r0 ). The depenN dences ε coh (r0 ) were calculated by the Kohn–Sham method in terms of the LDA and are given in Table 1. The equation for the binding energy has the following form: coh εN = 1 N
0.35 –1/3
1.0 C–/R
NE at − E N E = E at − N . N N
C+/R
By definition, the cohesion energy ε coh N is the binding energy of atoms in a cluster per atom. This energy is determined by the difference between the total energy of N free atoms and the energy of a cluster consisting of N atoms:
ε coh N =
Mg
1.0
ENERGIES OF DISSOCIATION, COHESION, AND VACANCY FORMATION The energy of the dissociation of a neutral metallic (Me) cluster according to the reaction MeN → Me N −1 + Me at is equal to the difference
ε dis N = [E N −1 + E at ] − E N
1.2
833
Then, using formula (17) and the definition (14) as follows, we find the coincidence of the asymptotics dis ε coh N and ε N :
εN = ε dis
coh
(r0 ) −
2σ 0 . nat RN
(18)
Figures 6 and 7 display the energies of dissociation and of cohesion for solid clusters and for clusters conTable 1. Experimental and calculated values of the parameters Experiment [33] Metal Mg Li
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eV
ε vac ( r0 ) , eV [34]
ε coh ( r0 ) , eV
1.51 1.63
0.72 0.38
1.166 1.495
r0, a0
ε , eV
ε
3.34 3.28
0.84 0.37
2017
vac
Calculation
coh
,
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POGOSOV, REVA
0.8 Mg
Mg Energy of cohesion, eV
Energy of dissociation, eV
2.0
1.0
0
0.4
0
–0.4
–1.0 –0.8 –2.0 0.2
0.4
0.6
0.8
1.0
0.2
0.4
N–1/3
0.6 N
0.8
1.0
–1/3
2.0 Li
Li
1.6
Energy of cohesion, eV
Energy of dissociation, eV
1.2
1.2
0.8
0.4
0.8
0.4 – [13]
0 0.2
0.4
0.6
0.8
1.0
0 0.2
0.4
N–1/3
0.6 N
0.8
1.0
–1/3
Fig. 6. Calculated values of the energy of dissociation of
Fig. 7. Calculated values of the energy of cohesion of solid
solid clusters ε dis N (points connected by lines) and defect
clusters ε coh N (points connected by lines), defect clusters
clusters ε dis N ,v (circles); dot-and-dash lines correspond to the asymptotics (18).
ε coh N ,v (circles), and experimental values [13] (crosses); dotand-dash lines correspond to asymptotics (17).
taining a monovacancy. The size dependence of the energy of dissociation represents quantum oscillations with respect to its asymptotics. At large l, the values of
increasing N, while that of defect clusters increases. The comparison of the data given in Figs. 6 and 7 confirms the accuracy of the formula (16) and explains the difference in the local maxima in Fig. 7.
ε dis N for a defect cluster are located higher than those for a solid cluster; at small l, these values behave vice versa. For the solid and defect clusters, apart from the change in the order of the filling of electron levels, a significant difference in the behavior of the energy of dissociation of clusters with identical numbers of atoms N should also be noted: for small l, the energy of the dissociation of solid clusters decreases with
Thus, the conclusion can be drawn that the most stable defect-free clusters are those in which the last occupied levels are levels with small l and, in the case of defect clusters, the opposite is true. In the experiments, the size oscillations of ε dis N are apparently suppressed by temperature effects (see Fig. 9 in [13]).
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ON THE CALCULATION OF THE ENERGIES OF DISSOCIATION
The ab initio calculations of the energy of the formation of vacancies have been performed in a large number of works (see, e.g., [32]). The authors of [33] investigated the energy of the cohesion of an atom and the energy of the formation of a vacancy using the Padé expansion and the liquid drop model for stable jellium. In the designations of the work [30], the results of [33] look as follows:
( (1 − δ
) ).
ε coh (r0 ) = 4π r02σ 0 1 + δ1 r0 + δ 2 r02 ;
(19)
ε vac(r0 ) = 4π r02σ 0
(20)
1
r0 + δ 2 r02
The experimental values of the related quantities given in [33] are given in Table 1. The values of εv (r0 ) were obtained in our previous work as a result of selfconsistent calculations of the parameters of scattering of electron waves on a vacancy potential [34]. Using ε coh (r0 ) and ε vac(r0 ) from the table and δ 2 r02 = 0.41 (Li) and −0.015 (Mg) from [33], we find δ1 r0 = 0.64 (Li), 0.54 (Mg). The magnitudes δ1 and δ 2 are necessary to construct asymptotics for the energy of the vacancy formation. In the case of clusters, no self-consistent calculations of ε Nvac,v were performed because of the difficulties in the detailed elaboration of the process of vacancy formation. Therefore, it is of interest to clear the advantages of the vacancy formation via two mechanisms. In the case of the Schottky mechanism, an atom is extracted from the surface of a solid sphere and in the final state the vacancy proves to be at the center of the sphere as follows:
ε N ,v
vac,Sh
= [E N −1,v + E at ] − E N
(21)
coh = N ε coh N − (N − 1)ε N −1,v ,
where E N −1,v is the energy of a sphere containing a vacancy at the center of a radius r0 (the spherical layer between r = r0 and r = RN −1,v contains N − 1 atoms). In the case of the second mechanism [30], the number of atoms in the sphere remains unaltered, but a hole (vacancy) of radius r0 is blown at the center of the sphere as follows:
(
)
coh ε Nvac,blow = E N ,v − E N = N ε coh N − ε N ,v . ,v
(22)
The comparison of expressions (21) and (22) demonstrates the advantage of the second mechanism:
ε Nvac,blow = ε Nvac,Sh − ε dis ,v ,v N ,v .
(23)
Let us investigate the asymptotic behavior of the energy of formation of a vacancy. Its size dependence is determined by the difference of the total energies of the spheres calculated according to the formulas (21) and (22) in the limit of N → ∞, and is reduced to the difference of the total surface energies. PHYSICS OF METALS AND METALLOGRAPHY
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In the case of the mechanism of the blowing-out of a vacancy, using (10), and assuming that RN ,v = RN 1 + ζ 3 3 , where ζ ≡ r0 RN ! 1, and keeping a necessary order of the expansion, we obtain
(
ε N ,v
vac,blow
+ε
vac
)
( (1 + 1 + δ
= 4π RN ,v σ 0 1 + δ1 RN ,v + δ 2 RN ,v 2
(r0 ) −
4π RN2 σ 0
1
2
RN +
δ 2 RN2
) )
(24)
⎛ ⎞ 2 ⎟. (r0 ) ⎜1 + ⎜ 3N 1 3 1 − δ1 r0 + δ 2 r02 ⎟ ⎝ ⎠ In the case of the Schottky mechanism, assuming that RN3 −1,v = RN3 − r03 + r03, and using (23) and (24), we obtain =ε
ε N ,v
vac,Sh
vac
(
=ε
)
⎛ ⎞ 2 ⎟ (r0 ) ⎜1 + 13 2 ⎜ 3N 1 − δ1 r0 + δ 2 r0 ⎟ ⎝ ⎠ (25) 2σ 0 coh + ε (r0 ) − . 13 nat r0N
vac
(
)
As was noted above, if ε dis N ,v > 0, the mechanism of the blowing-out of a vacancy is more advantageous energetically than the Schottky mechanism, according to (23). In this case, the asymptotic dependence (25) only weakly depends on N and the dependence (24) demonstrates a decrease in the energy of the vacancy formation with increasing N. Figure 8 displays the results of the calculations of the energy of vacancy formation via two mechanisms. These calculations confirm formula (23), namely, the advantage of the mechanism of blowing-out a vacancy. All of the dependences exhibit strong oscillations. For some N, especially in the case of Mg, the values of ε Nvac,blow become negative in narrow ranges ,v of N. These regions are noted in Fig. 2 and, in the main text, comments are made concerning the hierarchy of electron states in these clusters. The above calculations and the LDA correspond to the concentration of atoms at zero temperature. It cannot be excluded that, at a density of atoms that corresponds to experimental temperatures of the phase transition, a lowered symmetry of clusters and the use of the LSDA for the exchange-correlation energy, these artifacts will be eliminated. In any case, the size behavior of the results of direct calculations agree with the behavior of their asymptotics. At a finite temperature T, the advantage of the appearance of vacancies in a cluster is estimated by the fulfillment of the thermodynamic condition
Δ FNvac,blow = ε Nvac,blow − T Δ S Nvac,blow ≤0 ,v ,v ,v
(26)
for the change in the free energy of the system caused by the formation of a vacancy in the cluster.
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POGOSOV, REVA
Mg
Mg
3
Energy of vacancy formation, eV
Energy of vacancy formation, eV
4
2
1 0.15
0.25
0.35
0.45
2
1
0
0.15
N–1/3
0.25
0.35 N
2.4
Li
1.2 2.0
1.6
1.2 0.15
0.25
0.35 N
0.8
0.4
0
–0.4
0.45
–1/3
0.15
0.25
0.35 N
Fig. 8. Calculated values of the energy of vacancy formavac,Sh tion via the Schottky mechanism ε N (21) for Mg ,v and Li; dot-and-dash lines correspond to the asymptotics (25).
Since in the case of the mechanism of the blowingout of a vacancy the number of atoms in the cluster remains unaltered, the entropy contribution to (26) is only due to the degenerate electron gas. The corresponding expression is as follows:
T Δ S Nvac,blow ,v 53 kT = 2π2 3 ⎜⎛ B2 ⎟⎞ ⎝ e ⎠ 3
2∞
∫ drr 0
2
0.45
–1/3
Li Energy of vacancy formation, eV
Energy of vacancy formation, eV
3
0.45
–1/3
Fig. 9. Calculated values of the energy of vacancy formation via the mechanism of the blowing-out of a bubble vac,blow (22). Corresponding asymptotics (24) are shown ε N ,v vac as dot-and-dash lines; ε ∞ ≡ ε vac (r0 ) dependences are given as horizontal dashed lines.
For the calculations in (27), there will be necessary equilibrium profiles of electrons in the model of stable jellium at given N and T. At zero temperature and at N = 12, these profiles are shown in Fig. 1.
CONCLUSIONS
⎡n1N 3,v (r ) − n1N 3(r )⎤ . ⎣ ⎦
(27)
Self-consistent calculations of the profiles of radial distributions of electrons and potentials of solid clusters and clusters with a monovacancy at the center
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ON THE CALCULATION OF THE ENERGIES OF DISSOCIATION
have been performed using the Kohn–Sham method in the model of stable jellium. This permitted us for the first time to determine the total energy of a neutral and charged defect cluster and based on the obtained results, to carry out direct calculations of the energies of dissociation, cohesion, vacancy formation, electron attachment, ionization potential, and of the electrical capacitance. For the example, simple metals Mg and Li have been chosen. The results of the calculations have been compared with the asymptotics and with the results for the defect-free clusters. The computer code for the calculations has been developed by us independently. For small aggregates, the ionization potential of a cluster with a vacancy is higher by 0.1–1 eV than the ionization potential of a solid cluster. The maximum difference is observed upon the transition from the completely occupied shell to the empty shell. With increasing N, this difference vanishes. The magic numbers of atoms for the solid clusters and clusters with a vacancy differ, especially in the case of Mg. The electric capacitances normalized with respect to the cluster radius are always more than unity and exhibit quantum size oscillations. In this case, the defect clusters with partially filled electron shells have the capacitances greater than those of the solid clusters. Quantum size dependences of the energy of vacancy formation according to the Schottky mechanism and the mechanism of the blowing-out of a bubble (vacancy) have been calculated for the first time, and their asymptotics have been determined. The asymptotics obtained based on these two mechanisms cardinally differ from one another. The cluster-size dependences of the cohesion energy contain local maxima. The clusters corresponding to these maxima are more stable; i.e., they have greater energies of binding, dissociation, and vacancy formation than their neighbors. For small clusters, these maxima correspond to the completion of the filling of a given electron shell with increasing N. The positions of the maxima for the defect-containing and defect-free clusters are different, which is caused not only by the differences in their sizes but also by the character of the behavior of the wave functions of the electrons. With increasing N, the energy of dissociation in the regions between the maxima either increases or exhibits a local minimum, whereas the energy of vacancy formation decreases monotonically. REFERENCES 1. Ya. I. Frenkel’, Kinetic Theory of Liquids (AN SSSR, Leningrad, 1959; Clarendon, Oxfor, 1946; Dover Publ., 1955). 2. A. V. Babich, V. V. Pogosov, and V. I. Reva, “Calculations of the probability of positron trapping by a vacancy in a metal and the estimation of the vacancy contribution to the work function of electrons and positrons,” Phys. Met. Metallogr. 117, 205–213 (2016). PHYSICS OF METALS AND METALLOGRAPHY
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PHYSICS OF METALS AND METALLOGRAPHY
Translated by S. Gorin
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