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Surface energy calculation of alkali metals with the empirical electron surface model
Materials Chemistry and Physics 123 (2010) 658–665
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Surface energy calculation of alkali metals with the empirical electron surface model Baoqin Fu, Wei Liu, Zhilin Li ∗ College of Materials Science and Engineering and College of Chemical Engineering, Beijing University of Chemical Technology, Beijing 100029, China
a r t i c l e
i n f o
Article history: Received 11 January 2010 Received in revised form 3 March 2010 Accepted 10 May 2010 Keywords: Metals Surfaces Surface properties Computer modelling Simulation
a b s t r a c t The empirical electron surface model (EESM) based on empirical electron theory (EET) and dangling bond analysis method has been used to establish a database of surface energy for low index surfaces of alkali metals such as Li, Na, K, Rb and Cs. Since the lattice electrons of alkali metals occupy large proportion of the total electron number for valence shell, the contribution of these lattice electrons cannot be neglected in surface energy calculation. Therefore, the necessary modification to EESM will be presented in this paper. The calculated results are in agreement with experimental and other theoretical values. And the calculation results show that the surface energy is anisotropic. As we predicted, the surface energy of the close-packed plane (1 1 0) is the lowest one of all index surfaces. It is also found that the dangling bond electron density and the spatial distributions of lattice electron have great influences on surface energy of various index surfaces. EESM will provide one good basis for the research of surface science phenomena, and since abundant information about valence electronic structure is generated from EET, the model may be extended to the surface energy estimation of more metals, alloys, ceramics and so on. © 2010 Elsevier B.V. All rights reserved.
1. Introduction Surface energy determines the stability of surfaces, and surface energy anisotropy determines the equilibrium shape of crystals. So the detailed knowledge of the structure and property of crystal surfaces is important for the understanding of many surface phenomena such as adsorption, oxidation, corrosion, catalysis and crystal growth [1–4]. However, the surface energies of metals are difficult to determine experimentally and just few data exist [5–8], and these data include uncertainties of unknown magnitude. Therefore, to determine the surface energy theoretically is very essential. During the last decade there have been several methods to calculate the surface energy of metals, such as first-principles calculations [9–17] and semi-empirical methods [18–26]. However, the first-principle calculation is a tough job, especially for a largescale structure, due to not only the computer time required but also the reliability of the calculated results, which depends strongly on the selection of basis-set, exchange-correlation functional and cutoff energy. And most semi-empirical methods are based on some existing experimental results. And the methods only focused on a
few systems in which the needed parameters and functions have already been known. The empirical electron surface model (EESM) has been found remarkably successful in the calculation of the surface energies of transition metals [3,4] and diamond cubic crystals, of which the crystal cohesive energy is mainly dependent on the number and the spatial distribution of the covalent electrons in the crystal structure. The starting point of EESM is the valence electron structure (VES) calculated with the empirical electron theory in solid and molecule (EET) established by Yu [27–29]. And a brief English introduction of EET can be found in Refs. [3,4]. The reliability of EET has been extensively examined in the fields of metals, alloys, metallic compounds and ceramics [3,4,30–39]. However, when that model is used in the calculation of the surface energies of alkali metals, large errors are produced when the calculation results are compared with the results of experiments and other calculations. So the EESM has to be modified in this case. In this paper, EESM is modified and the calculation results of the surface energies of alkali metals are in well accordance with results of experiment and other calculations. 2. Summary of EESM
∗ Corresponding author at: College of Materials Science and Engineering, Beijing University of Chemical Technology, Box 23, 15 of Beisanhuan East Road, Beijing 100029, China. Tel.: +86 10 64421306; fax: +86 10 64437587. E-mail addresses: [email protected], [email protected] (Z. Li). 0254-0584/$ – see front matter © 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.matchemphys.2010.05.034
Surface energy () is one of the most important static physical quantities charactering metal surface. And the surface energy is usually defined as the additional value of the free energy per unit
B. Fu et al. / Materials Chemistry and Physics 123 (2010) 658–665
area of increase on a particular crystal surface, and therefore, the surface energy of crystal surface is given by the expression: =
E S
(1)
where E represents the additional value of the free energy with the forming of new surface and S is the added area of the crystal surface. In EET, valence shell electrons are divided into four types [28,29] according to the spatial distribution and the action characteristics of these electrons in the formation of solids or molecules, such as dumb pair electrons, magnetic electrons, covalent electrons (bonding electrons) and lattice electrons. The dumb pair electrons represent either a bonding and an anti-bonding electron with their resultant bonding power mutually cancelled by each other and their spins opposite to each other, or a pair of non-valence electrons of opposite spins deeply sinking down to the atomic orbit. The magnetic electrons here represent the localized electrons from the single-occupied orbit. The covalent electron, also from the singleoccupied orbit, is paired with another electron of opposite spins from other atom as elements are alloyed, contributing to atomic bonding. The lattice electron is present in a space among 3 or 4 atoms or even more than 6 atoms. When the s energy band is wide, the electron might reach an energy level above the Fermi energy and is called lattice electron. The number of valence electrons of an atom in solids, including the covalent electrons and the lattice electrons, equals to the valency of the atom in solids. In EET, the crystal cohesive energy is mainly dependent on the number and the spatial distribution of the four type electrons in the crystal structure. The contribution of the four type electrons to the crystal cohesive energy and the calculation formulae can be found in Refs. [29,30]. And the hybridization states of alkali metals, the number and the distribution of these electrons, are listed in Appendix A. For these alkali metals, all valence shell electrons belong to one of the following two types: covalent bond electron and lattice electron. And the numbers of lattice electrons of alkali metals are large, so lattice electrons should be taken into account in the calculation of surface energies. Correspondingly, the crystal cohesive energy for the alkali metals equals to the sum of the energy caused by these covalent electrons and lattice electrons. Therefore, when two nearest crystal planes of the alkali metals form two new crystal surfaces, the energy requirement for the procedure, i.e., the additional free energy (E), is the very sum of the additional free energy caused by the covalent electrons and lattice electrons (Ec and El ): E = Ec + El
(2)
In EESM, the additional free energy caused by the covalent electrons (Ec ) is equal to the sum of the bond energies of all bonds between two nearest crystal planes [3,4]: Ec =
X˛ E˛
where the factor g represents the contribution of the couple effect between spin and orbit of d electrons to the bonding capability. For 4, 5 and 6 periods, g is equal to 1, 1.35 and 1.70, respectively. ˛ , ˇ and are contents of s, p and d electrons in covalent bonds, respectively: ˛ =
lCh + l Ct nT
ˇ =
mCh + m Ct nT
=
nCh + n Ct nT
b=
31.395 n − 0.36ı
(7)
The factors n and ı in the denominator reflect the total effect of the screen, coulomb, exchange and interrelated interaction of the inner electrons in solids. The values of n and ı of the alkali metal elements are all 1 [28,29]. And the calculation formula for the additional free energy caused by the lattice electrons (El ) is: El = b
Zp nl f D
(8)
where the coefficient ZP , is the ratio of the additional free energy caused by the lattice electrons and the crystal cohesive energy of one atom, which is related to atomicity of the related crystal plane with a finite area, the distribution of lattice electrons and the surface orientation, and will be calculated in Section 4; nl represents the number of lattice electrons of the alkali metal atom and f represents the bonding capability of lattice electron:
f =
2nl nT
(9)
In formula (8), D is equivalent bond distance and equals to the arithmetic mean of the length of the bonds which are distributed around the space of lattice electron:
˛
where n˛ represents the number of covalent electron pairs on ˛ bond, D˛ represents the bond length and f represents the bonding capability of covalent electron: √ (5) f = ˛ + 3ˇ + g 5
(6)
where l, m, n and l , m , n represent the sum of the numbers of s, p, d covalent electron and lattice electron of the h and t states, respectively. The terms and are parameters for the h and t states, respectively, and value 1 when the s electron is covalent electron or 0 when the s electron is lattice electron. The values of these parameters can be taken from Refs. [28,29], and some values of the alkali metal elements are given in Appendix A. nT , Ch and Ct represent the total numbers of covalent electron and lattice electron, the relative compositions for the h state and for the t state of hybrid level, respectively. In formula (4), b is screen factor upon the core electron:
(3)
where ˛ represents bond name, E˛ represents the bond energy of ˛ bond and X˛ , obtained by dangling bond analysis method (DBAM), is the equivalent dangling bond number of ˛ bond on one particular crystal plane. The analysis of X˛ is presented in Section 4 in detail. In EET, the calculation formula for bond energy (E˛ ) of one bond in which the two atoms are the same is: n˛ (4) E˛ = bf D˛
659
D =
I˛ D˛
˛
(10) I˛
˛
where I˛ represents the equivalent bond numbers of the bonds in the structure unit. Therefore, the key to this surface energy model is the calculation of VES and the analysis of the equivalent dangling bond number. 3. The VES and the bond energies of alkali metals These alkali metals, the crystal cell is shown in Fig. 1 and the lattice constants are shown in Table 1, belong to A2 type crystal structure, i.e., the body centered cubic (bcc) structure and the atom
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of the electron number on all the covalent bonds in the structure unit, i.e.:
nc =
nA =
I˛ n˛ = nA (IA + IB rB )
(15)
˛
nc
IA + IB rB
=
2nc IA + IB rB
(16)
following: DA = (0.5a)2 + (0.5a)2 + (0.5a)2 , DB = a. The equivalent bond numbers (I˛ ) can be calculated with formula [28,29]:
For certain hybrid level of an atom, the covalent electron number and single bond radius can be found in Table 2 [28,29]. By selecting a suitable ˇ value and substituting the number of covalent electrons and the single bond radii of the atoms of the corresponding hybrid levels, the number of covalent electron pair nA on the strongest covalent bond can be figured out. By substituting the obtained nA into the ratio equation, the numbers of covalent electron pairs on the B bonds can be calculated. The hybrid states of the atoms and the corresponding distributions which are in accordance with the actuality have to be determined. In EET, the bond lengths calculated with the calculated n˛ are called theoretical bond lengths. Substituting the obtained n˛ into the bond length formula (12) yields the theoretical bond lengths of all the bonds:
I˛ = iM iS iK
D˛ = 2R (1) − ˇ log n˛
Fig. 1. Crystal cell of alkali metals.
coordinates are (0 0 0) and (0.5 0.5 0.5), respectively. So the experimental bond lengths which cannot be neglected can be given as the
(11)
where ˛ (˛ = A or B) represents bond name, iM represents the reference atom number in the structure unit, iS represents the equivalent bond number for a reference atom to form ˛ bond and iK is a parameter, which equals 1 when the two atoms that form the bond are of the same kind or 2 when the two atoms are of different kinds. Therefore, the equivalent bond numbers of ˛ bond are: IA = 2 × 8 × 1 = 16, IB = 2 × 6 × 1 = 12. According to formula (10), the equivalent bond distance (D ) can be calculated. According to the Pauling bond length formula [28,29,40]: D˛ = 2R(1) − ˇ log na
(12)
where D˛ represents the bond length of ˛ bond; R(1) represents the single bond radius of the atom which forms the bond; n˛ represents the number of covalent electron pairs on the bond and ˇ is a parameter with units of length. Here a modification for Pauling’s original theory is made as the follows:
⎧ ⎪ ⎨ 0.0710 (nm)
0.0600 (nm) ˇ= ⎪ 0.0710 − 0.22ε (nm)
⎩
M when nM ˛ < 0.25 or n˛ > 0.75 ≤ 0.700 when 0.300 ≤ nM ˛ (13) M when nM ˛ = 0.250 + ε or n˛ = 0.750 − ε
where 0 ≤ ε < 0.050, nM ˛ represent the largest n˛ in the structure. Therefore, the following ratio equation can be obtained: rB =
nB = 10(DA −DB )/ˇ nA
(14)
A structure unit should be electron neutral, so the covalent electron of all atoms in the structure unit (there are 2 atoms in the unit cell of the alkali metals) should be distributed on all ˛ covalent bonds in it. In other words, the total covalent electron number ( nc ) of all atoms in a structure unit should equal the sum ( I˛ n˛ ) Table 1 The lattice constants of the alkali metals.
Bond length difference (BLD) D˛ , is the absolute value of the difference between the experimental bond length and the theoretical one, i.e.:
a = b = c (nm)
˛ = ˇ = (◦ )
Li Na K Rb Cs
0.35092 0.42906 0.53200 0.57000 0.61400
90 90 90 90 90
D˛ = D˛ − D˛
(18)
In EET, D˛ < 0.005 nm is the criterion to determine whether the given atom state and the corresponding valence electron distribution accord with reality. If D˛ < 0.005 nm, the applied hybrid levels of the atom are believed being in accordance with the actual state, therefore, the obtained corresponding parameters are the possible VES of the calculated crystal. Through programming the calculation according to the above equations and substituting the parameters of all hybrid levels of alkali metal atoms, all the possible existing hybrid levels of the atoms can be obtained. So the VES of these alkali metals can be calculated, and the calculation results with the smallest bond length difference D˛ are shown in Table 3. According to the VES of these alkali metals and formulae ((4)–(7)), the bond energy can be calculated. And the calculation results are tabulated in detail in the last column of Table 3. 4. Dangling bond analysis of the related crystal plane of alkali metals The DBAM is to investigate the number and the type of the dangling bonds on one particular crystal surface. And based on the calculated bond energy (see Table 3), Ec of one particular crystal plane can be calculated. Therefore, the DBAM is one of the important steps of the surface energy model. Formula for the equivalent dangling bond numbers (X˛ ) on one particular crystal plane can be deduced out [3,4]: X˛ = iP iK
Alkali metals
(17)
iD NP
(19)
NP
where iP represents the reference atom number on one particular plane with the area S and iK is a parameter, which equals 1 when the two atoms that form the bond are of the same kind or 2 when the two atoms are different kinds; iD represents the equivalent dangling bond number for a reference atom to form one type
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Table 2 Hybridization table of alkali metals. Metals
Li
1 2 3 4
Na
A type
B type
C type
nc
nl
R(1) (nm)
nc
nl
R(1) (nm)
nc
nl
R(1) (nm)
0 0.25309 0.65108 1
1 0.74691 0.34892 0
0.1326 0.12400 0.11046 0.0986
0 0.46143 0.53498 1
1 0.53857 0.46502 0
0.1326 0.11691 0.11441 0.0986
1 1 1 1
0 0 0 0
0.1326 0.12542 0.10579 0.0986
1 2 3 4
0 0.46143 0.53498 1
1 0.53857 0.46502 0
0.15733 0.14504 0.14308 0.1307
1 1 1 1
0 0 0 0
0.15733 0.14816 0.1374 0.1307
K
1 2 3 4
0 0.46143 0.53498 1
1 0.53857 0.46502 0
0.19628 0.18794 0.18661 0.1782
1 1 1 1
0 0 0 0
0.1964 0.18628 0.17440 0.167
Rb
1 2 3 4
0 0.46143 0.53498 1
1 0.53857 0.46502 0
0.2087 0.20270 0.20175 0.1957
1 1 1 1
0 0 0 0
0.2087 0.19633 0.18183 0.17279
Cs
1 2 3 4
0 0.46143 0.53498 1
1 0.53857 0.46502 0
0.2214 0.2226 0.22279 0.224
1 1 1 1
0 0 0 0
0.2214 0.20870 0.19381 0.18453
Table 3 The VES and bond energies of alkali metals (including the number of lattice electrons and equivalent bond distance). Alkali metal (hybrid level)
D (nm)
Bond name (˛)
D˛ (nm)
D˛ (nm)
n˛
Dn˛ (nm)
E˛ (kJ mol−1 )
Li (A3)
0.32405
A B
0.30391 0.35092
0.30294 0.34996
0.06996 0.01523
0.00096 0.00096
12.48715 2.35307
Na (B3)
0.39621
A B
0.37158 0.42906
0.37297 0.43045
0.05991 0.00929
0.00139 0.00139
6.18145 0.83026
K (B3)
0.49127
A B
0.46073 0.53200
0.45883 0.53011
0.06225 0.00617
0.00189 0.00189
5.22073 0.44787
Rb (B2)
0.52636
A B
0.49363 0.57000
0.49526 0.57162
0.05426 0.00456
0.00162 0.00162
3.91568 0.28507
Cs (B3)
0.56699
A B
0.53174 0.61400
0.53055 0.61281
0.06356 0.00441
0.00119 0.00119
4.61060 0.27705
bond with the atoms of the same neighbor crystal plane; NP (NP = 1, 2, . . .) represents the crystal plane number passed through by the bond. As is shown in Fig. 2, there is one atom on the (1 0 0) plane with the area of a2 nm2 in the unit cell of the alkali metals. Every atom with the atoms of the nearest neighbor crystal plane forms four A bonds, and with the atoms of the second neighbor crystal plane forms one B bond. When the (1 0 0) plane forms the outer surface, and these bonds will form dangling bonds in the outer surface. And
Fig. 2. The spatial distribution of dangling bond on alkali metals (1 0 0) crystal plane.
based on formula (16), the equivalent dangling bond numbers on the (1 0 0) plane are XA = 1 × 1 × 4 × 1 = 4 and XB = 1 × 1 × 1 × 2 = 2. As is shown in√ Fig. 3, there are two atoms on the (1 1 0) plane with the area of 2a2 nm2 in the unit cell of the alkali metals. Every atom with the atoms of the nearest neighbor crystal plane forms two A bonds and two B bonds. So the equivalent dangling bond numbers on the (1 1 0) plane are XA = 2 × 1 × 2 × 1 = 4 and XB = 2 × 1 × 2 × 1 = 4. As is shown in Fig. 4, √ there is one atom on the (1 1 1) crystal plane with the area of 3a2 nm2 in the alkali metal crystal cell. Every atom with the atoms of the nearest neighbor crystal plane forms three A bonds, with the atoms of the second neighbor crystal plane forms three B bonds, and with the atoms of the third neighbor crystal plane forms one A bond. Therefore, the equivalent dangling bond numbers on the (1 1 1) plane can be obtained. They are XA = 1 × 1 × (3 × 1 + 1 × 3) = 6 and XB = 1 × 1 × 3 × 2 = 6. As is shown in Fig. 5, √ there is one atom on the (2 1 0) crystal plane with the area of 5a2 nm2 in the alkali metal crystal cell. Every atom with the atoms of the nearest neighbor crystal plane forms two A bonds, with the atoms of the second neighbor crystal plane forms one B bond, with the atoms of the third neighbor crystal forms two A bonds, with the atoms of the fourth neighbor crystal plane forms one B bond. So the equivalent dangling bond numbers on the (2 1 0) plane are XA = 1 × 1 × (2 × 1 + 2 × 3) = 8 and XB = 1 × 1 × (1 × 2 + 1 × 4) = 6.
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B. Fu et al. / Materials Chemistry and Physics 123 (2010) 658–665
Fig. 5. The spatial distribution of dangling bond on alkali metals (2 1 0) crystal plane.
structure. And there are four tetrahedral interstices around one octahedral central site, which is the geometric center of the four tetrahedral interstices. To simplify the calculation of the coefficient Zp , the lattice electrons in the tetrahedral interstices are equivalently distributed in the octahedral central sites, and the process is interpreted in Fig. 6. It is assumed that the contribution of lattice electrons to the crystal cohesive energy is equal to the very sum of the energy of the charge effect between the octahedral central sites and the atoms around the octahedral central sites: El =
Fig. 3. The spatial distribution of dangling bond on alkali metals (1 1 0) crystal plane.
5. The coefficient Zp In EET, lattice electron is presented in a space among 3 or 4 atoms or even more than 6 atoms. So as is shown in Fig. 6, the lattice electrons of alkali metals are presented in both octahedral and tetrahedral interstices (o sites and t sites) of the crystal
Eoi
(20)
where El represents the crystal cohesive energy caused by the lattice electrons; Eoi is the energy of the charge effect between the octahedral central site and various adjacent atoms. Therefore, the additional free energy caused by the lattice electrons (El ) in the forming of new surface is related to the charge effect between the octahedral central sites and the adjacent atoms. As is shown in Fig. 7, there are six adjacent atoms around one octahedral central site. Because the distances of the octahedral central site to the adjacent √ atoms are different, i.e. d1 = d2 = a/2 = / d3 = d4 = d5 = d6 = a 2, the charge effect between the octahedral central sites and the atoms around the octahedral central sites are also different and are in inverse proportion to these distances, i.e., Eoi ˛1/di (i = 1, 2, . . ., 6). Therefore, the fraction (fi ) of Eoi in the crystal cohesive energy caused by the lattice electrons of one atom can be deduced out. fi =
Eoi = El
E
1/d
oi = Eoi
no
i
(21)
1/di
i
Fig. 4. The spatial distribution of dangling bond on alkali metals (1 1 1) crystal plane.
Fig. 6. The spatial distribution of the lattice electrons of alkali metals (the octahedral and tetrahedral interstices).
B. Fu et al. / Materials Chemistry and Physics 123 (2010) 658–665
663
According to the above analysis and formulae ((20)–(22)), a computer program written in Microsoft Visual Basic language has been compiled to analyze and calculate the coefficient Zp of various index surfaces. So for the alkali metals, the coefficients Zp of the (1 0 0), (1 1 0), (1 1 1) and (2 1 0) plane are Z(1 0 0) = 0.528595, Z(1 1 0) = 0.861929, Z(1 1 1) = 1 and Z(2 1 0) = 1.390524.
6. Calculation of surface energy of the corresponding surface
Fig. 7. The distances of the octahedral central site to the adjacent atoms in the crystal cell of alkali metals.
where no represents the number of octahedral central sites corresponding to one atom. As is shown in Fig. 6, there are two atoms and six octahedrons in the unit cell of alkali metals, that is, no = 3. Based on the above analysis and with the similar idea of DBAM, the calculation formula for the coefficient Zp on one particular crystal plane can be deducted out. Zp =
o
According to the calculated bond energy in Table 3 and the equivalent dangling bond number of the related crystal plane calculated in Section 4, the additional free energy caused by the covalent electrons (Ec ) can be calculated. And based on the values of the lattice electron number and equivalent bond distance in Tables 2 and 3, the coefficient Zp calculated in Section 5 and formulae ((7)–(10)), the additional free energy caused by the lattice electrons (El ) can also be calculated. Therefore, the calculation formula for the surface energy () of one particular clean and ideal surface can be derived from formula (1).
=
Ec + El = 2S
X˛ E˛ + b
Zp nl f D
˛
(23)
2S
(22)
fi Np iK
i
where o represents the reference octahedral central site on the particular plane (h l k) with the area of S, or in the middle part of the region and its neighbor plane, Np represents the crystal plane number between the reference octahedral central site (o) and the i adjacent atom of the neighbor crystal plane (not including the atoms of the (h l k) index plane of o site) and iK is a parameter, which equals 1 when the reference octahedral central site (o) is located at the cleavage plane or 0.5 when the reference octahedral central site (o) is located in the middle part of two neighbor cleavage plane.
The physical meaning of the number 2 in this formula is that when the interaction of electrons and atoms between two nearest crystal planes is broken, there will be two crystal surfaces in the region. So the total area is two times of one crystal plane in one unit crystal cell. Through programming the calculation according to these parameters and the surface energy formula (23), we can calculate the surface energy of the corresponding clean and ideal surface of the alkali metals. The calculation results are tabulated in detail in Table 4.
Table 4 The surface energies of alkali metals. Alkali metal Li
Na
K
Rb
Cs
a b c d e
Plane
Ec (eV)
El (eV)
S (nm2 )
(eV nm−2 )
Other theoretical results (eV nm−2 ) a
Experiment (eV nm−2 )
b
(1 0 0) (1 1 0) (1 1 1) (2 1 0)
0.56646 0.61523 0.92285 1.18169
0.24173 0.39416 0.45730 0.63589
0.12314 0.17415 0.21329 0.27536
3.28144 2.89801 3.23533 3.30036
3.263 , 3.6 3.475a , 3.163c , 3.1b 3.688a , 3.406c , 4.5b 3.894c
(1 0 0) (1 1 0) (1 11) (2 1 0)
0.27348 0.29069 0.43603 0.56416
0.30419 0.49601 0.57546 0.80019
0.18409 0.26035 0.31886 0.41164
1.56894 1.51086 1.58611 1.65720
1.65a , 2.1b 1.581a , 1.8b 1.794a , 2.6b
1.631d , 1.625e
(1 0 0) (1 1 0) (1 1 1) (2 1 0)
0.22572 0.23500 0.35251 0.46072
0.24533 0.40003 0.46411 0.64536
0.28302 0.40026 0.49021 0.63286
0.83217 0.79329 0.83292 0.87387
0.888a , 1.1b 0.844a , 1.0b 0.95a , 1.4b
0.906d , 0.813e
(1 0 0) (1 1 0) (1 1 1) (2 1 0)
0.16824 0.17415 0.26123 0.34239
0.28539 0.46536 0.53990 0.75075
0.32490 0.45948 0.56274 0.72650
0.69811 0.69591 0.71181 0.75233
0.7a , 0.94b 0.65a , 0.75b 0.738a , 1.1b
0.731d , 0.688e
(1 0 0) (1 1 0) (1 1 1) (2 1 0)
0.19689 0.20263 0.30394 0.39951
0.21256 0.34661 0.40213 0.55917
0.37700 0.53315 0.65298 0.84299
0.54304 0.51508 0.54066 0.56862
0.581a , 0.75b 0.512a , 0.62b 0.575a , 0.87b
0.592d , e
The full charge density(FCD) method in the generalized gradient approximation (GGA) [7]. The modified broken-bond rule [41]. The first-principles molecular dynamics method [18]. Experiment [5]. Experiment [6].
3.263d , 3.281e
664
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7. Results and discussion The (1 0 0), (1 1 0), (1 1 1) and (2 1 0) surface energies for 5 alkali metals such as Li, Na, K, Rb and Cs, are presented in Table 4 in eV nm−2 , respectively. For comparison we also present some theoretical results and experimentally derived values. These theoretical methods include the full charge density (FCD) method in the generalized gradient approximation (GGA) in Ref. [9], the modified broken-bond rule in Ref. [41], and the first-principles calculation in Ref. [14], and so on. The experimental values in Refs. [7,8] are determined from measurements of the surface tension of liquid metals extrapolating through the liquid–solid phase transition. It must be pointed out that experimental measurements of the surface energy are more commonly found for polycrystalline materials. But for alkali metals, there is little surface energy data in the literature. The relaxation and reconstruction of the atomic positions are neglected, which may lead to errors of up to a few percent. From Table 4, the surface energy values calculated in this paper are close to the values come from the other Refs. From Table 4, it can be clearly seen that, the calculated surface energy shows a strong anisotropy, the surface energy values of the (1 0 0), (1 1 0), (1 1 1) and (2 1 0) surface are different and the closepacked (1 1 0) surface energy is the lowest of all these index surfaces as predicted. And the order of the energies of various index surfaces of Li and Cs is (1 1 0) < (1 1 1) < (1 0 0) < (2 1 0) , but the order of Na, K and Rb is (1 1 0) < (1 0 0) < (1 1 1) < (2 1 0) . This is caused by the differences of their VES and lattice constants. And according to the analysis of dangling bond, the spatial distribution of dangling bond of the (1 0 0), (1 1 0), (1 1 1) and (2 1 0) crystal surface is also different. So the dangling bond electron density and the spatial distribution of covalent bonds have a great influence on surface energy of various index surfaces. EESM has been found remarkably successful in calculating surface energies of transition metals [3,4], where the number of the covalent electrons takes up major constituent of the total electron number for valence shell. However, in the case of alkali metals with more lattice electrons, the neglect of the contribution of lattice electrons will lead to the calculation results inconsistent with the other theoretical results and experimentally derived values. For instance, without consideration of the effect of lattice electrons, the (1 1 0) surface energies of Li, Na, K, Rb and Cs are 1.76638, 0.55827, 0.29356, 0.18951 and 0.19003 eV nm−2 , respectively, and the relative errors are over 50% (relative to the other theoretical results or experimentally derived values), but the relative errors of our results in this paper are less than 10%. Therefore, the contribution of lattice electrons to the surface energies of alkali metal cannot be neglected. Similarly, for alkaline earth metals with more lattice electrons, the lattice electrons should be considered as well, which will be reported in the future. The meaning of this paper is not only to extend the EESM to alkali metals with more lattice electrons, and to thereby fill out that gap in the literature, but also to provide comprehensive surface energy results for alkali metals, transition metals [3,4] and diamond cubic crystals. Such extensive results obtained with the same theoretical model, are useful to both theorist and experimentalists. The EESM can be provided for the research of surface energy of the more metals, alloys, ceramics, compound and so on.
8. Conclusions Since the lattice electrons of alkali metals occupy large proportion of the total electron number for valence shell, the contribution of these lattice electrons cannot be neglected in surface energy calculation. Therefore, the necessary modification to the empirical electron surface model (EESM) has been presented in this paper.
With EESM, we have built a database of the (1 0 0), (1 1 0), (1 1 1) and (2 1 0) surface energy for the alkali metals like Li, Na, K, Rb and Cs, and the spatial distribution of dangling bonds of various index surfaces and the distribution of lattice electrons has been analyzed with dangling bond analysis method (DBAM). The dangling bond electron density and the spatial distribution of lattice electrons and covalent bonds have a great influence on surface energy of various index surfaces. Under the first-order approximation, the results are close to the other theoretical values and the experimental values. These surface energy values may be used as a consistent starting point for study of surface science phenomena. The calculated surface energy shows a strong anisotropy. And the close-packed (1 1 0) surface energy is the lowest of these index surfaces as predicted. This research model can be further extended to the surface energy estimation of more crystals, which will be the next research objects. Acknowledgements This work was supported by the Beijing Natural Science Foundation (Grant No. 2072014) and the Research Fund for the Doctoral Program of Higher Education of China (Grant No. 200800100006). Appendix A. Hybridization states in EET of the elements In EET, dumb pair electrons, magnetic electron, covalent electron, and lattice electron are denoted by ||, ↑, 䊉 and , respectively. When the covalent electron and lattice electron are equivalent electrons, they are denoted by 䊉, or s , p , respectively. The hybridization states of these elements are as the follows. A.1. A type of H and Li
h state: s1 t state: s␣ p1−␣
s 䊉
䊉
l = 1, m = 0, n = 0, = 0; l = 0.94879 (˛), m = 0.05121, n = 0, = 1. H: R(1)h = 0.03708 nm, R(1)t = 0.0280 nm. Li: R(1)h = 0.1326 nm, R(1)t = 0.0986 nm.
A.2. B type of H, Li, Na, K, Rb and Cs
h state: s1 t state: s␣ p1−␣
s 䊉
䊉
s 䊉 䊉
䊉
l = 1, m = 0, n = 0, = 0; l = 0.9982 (˛), m = 0.0018, n = 0, = 1. H: R(1)h = 0.03708 nm, R(1)t = 0.0280 nm. Li: R(1)h = 0.1326 nm, R(1)t = 0.0986 nm. Na: R(1)h = 0.15733 nm, R(1)t = 0.1307 nm. K: R(1)h = 0.19628 nm, R(1)t = 0.1782 nm. Rb: R(1)h = 0.2087 nm, R(1)t = 0.1957 nm. Cs: R(1)h = 0.2214 nm, R(1)t = 0.224 nm.
A.3. C type of Li, Na, K, Rb and Cs
h state: s1 t state: s␣ p1−␣
l = 1, m = 0, n = 0, = 1; l = 0.5 (˛), m = 0.5, n = 0, = 1. Li: R(1)h = 0.1326 nm, R(1)t = 0.0986 nm. Na: R(1)h = 0.15733 nm, R(1)t = 0.1307 nm. K: R(1)h = 0.1964 nm, R(1)t = 0.1670 nm. Rb: R(1)h = 0.2087 nm, R(1)t = 0.17279 nm. Cs: R(1)h = 0.2214 nm, R(1)t = 0.18453 nm.
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References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20]
F.W. Young, J.V. Cathcart, A.T. Gwathmey, Acta Metall. 4 (1956) 145. C.A. Chang, J. Appl. Phys. 67 (1990) 566. B.Q. Fu, W. Liu, Z.L. Li, Appl. Surf. Sci. 255 (2009) 8511. B.Q. Fu, W. Liu, Z.L. Li, Appl. Surf. Sci. 255 (2009) 9348. I. Galanakis, N. Papanikolaou, P.H. Dederichs, Surf. Sci. 511 (2002) 1. V.K. Kumikov, Kh.B. Khokonov, J. Appl. Phys. 54 (1983) 1346. W.R. Tyson, W.A. Miller, Surf. Sci. 62 (1977) 267. F.R. de Boer, R. Boom, W.C.M. Mattens, A.R. Miedema, A.K. Niessen, Cohesion in Metals, North-Holland, Amsterdam, 1988. L. Vitos, A.V. Ruban, H.L. Skriver, J. Kollar, Surf. Sci. 411 (1998) 186. M. Methfessel, D. Hennig, M. Scheffler, Phys. Rev. B 46 (1992) 4816. H.L. Skriver, N.M. Rosengaard, Phys. Rev. B 46 (1992) 7157. P. Blonski, A. Kiejna, Vacuum 74 (2004) 179. M.J.S. Spencer, A. Hung, L.K. Snook, I. Yarovsky, Surf. Sci. 513 (2002) 389. K. Koko, P.T. Salo, R. Laihia, K. Mansikka, Surf. Sci. 348 (1996) 168. Z. Crljen, D. Sokcevic, R. Brako, P. Lazic, Vacuum 71 (2003) 101. Y. Mishin, D. Farkas, M.J. Mehl, D.A. Papaconstantopoulos, Phys. Rev. B 59 (1999) 3393. P.J. Feibelman, Phys. Rev. B 53 (1996) 13740. J.M. Zhang, D.D. Wang, K.W. Xu, Appl. Surf. Sci. 252 (2006) 8217. C. Barreteau, D. Spanjaard, M.C. Desjonqueres, Surf. Sci. 433–435 (1999) 751. M.I. Baskes, Phys. Rev. B 46 (1992) 2727.
[21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35] [36] [37] [38] [39] [40]
[41]
665
C. Bombis, A. Emundts, M. Nowicki, H.P. Bonzel, Surf. Sci. 511 (2002) 83. T.M. Trimble, R.C. Cammarata, Surf. Sci. 602 (2008) 2339. T.M. Trimble, R.C. Cammarata, K. Sieradzki, Surf. Sci. 531 (2003) 8. J.M. Zhang, F. Ma, K.W. Xu, Appl. Surf. Sci. 229 (2004) 34. J.M. Zhang, D.D. Wang, K.W. Xu, Surf. Sci. 253 (2006) 2018. E. Aghemenloh, J.O.A. Idiodi, S.O. Azi, Comput. Mater. Sci. 46 (2009) 524. R.H. Yu, Chin. Sci. Bull. 23 (1978) 217. Z.L. Liu, Z.L. Li, W.D. Liu, Electron Structure of Interface and their Properties, Science Publishing House, Beijing, 2002, pp. 7–13 (in Chinese). R.L. Zhang, The Empirical Electron Theory in Solids and Molecules, Jinlin Science and Technology Publishing House, Changchun, 1993, pp. 268–278 (in Chinese). W.D. Xu, R.L. Zhang, R.H. Yu, Sci. China Ser. A 32 (1989) 351. Z.L. Li, J. Xu, B.Q. Fu, Solid State Sci. 10 (2008) 1434. Z.L. Li, H.B. Xu, S.K. Gong, J. Phys. Chem. B 108 (2004) 15165. Z.L. Li, W. Liu, Y.Q. Wu, Mater. Chem. Phys. 105 (2007) 278. Z.L. Li, Q. Huang, Y.Q. Wu, Z.F. Li, Sci. China Ser. E 50 (2007) 462. Z.L. Li, Z.F. Li, Q. Huang, Sci. China Ser. E 50 (2007) 664. Z.L. Liu, Z.L. Li, Z.G. Sun, Metall. Mater. Trans. 30A (1999) 2757. Z.L. Liu, Z.L. Li, Z.G. Sun, Chinese Sci. Bull. 41 (1996) 367. Z.L. Li, H.B. Xu, S.K. Gong, Sci. China Ser. E 46 (2003) 234. Z.L. Liu, Z.G. Sun, Z.L. Li, Prog. Nat. Sci. 8 (1998) 134. L. Pauling, The Nature of the Chemical Bond and the Structure of Molecules and Crystals: An Introduction to Modern Structural Chemistry, Cornell University Press, New York, 1960, pp. 230–256. Q. Jiang, H.M. Lu, M. Zhao, J. Phys.: Condens. Matter 16 (2004) 521.
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Materials Chemistry and Physics journal homepage: www.elsevier.com/locate/matchemphys
Surface energy calculation of alkali metals with the empirical electron surface model Baoqin Fu, Wei Liu, Zhilin Li ∗ College of Materials Science and Engineering and College of Chemical Engineering, Beijing University of Chemical Technology, Beijing 100029, China
a r t i c l e
i n f o
Article history: Received 11 January 2010 Received in revised form 3 March 2010 Accepted 10 May 2010 Keywords: Metals Surfaces Surface properties Computer modelling Simulation
a b s t r a c t The empirical electron surface model (EESM) based on empirical electron theory (EET) and dangling bond analysis method has been used to establish a database of surface energy for low index surfaces of alkali metals such as Li, Na, K, Rb and Cs. Since the lattice electrons of alkali metals occupy large proportion of the total electron number for valence shell, the contribution of these lattice electrons cannot be neglected in surface energy calculation. Therefore, the necessary modification to EESM will be presented in this paper. The calculated results are in agreement with experimental and other theoretical values. And the calculation results show that the surface energy is anisotropic. As we predicted, the surface energy of the close-packed plane (1 1 0) is the lowest one of all index surfaces. It is also found that the dangling bond electron density and the spatial distributions of lattice electron have great influences on surface energy of various index surfaces. EESM will provide one good basis for the research of surface science phenomena, and since abundant information about valence electronic structure is generated from EET, the model may be extended to the surface energy estimation of more metals, alloys, ceramics and so on. © 2010 Elsevier B.V. All rights reserved.
1. Introduction Surface energy determines the stability of surfaces, and surface energy anisotropy determines the equilibrium shape of crystals. So the detailed knowledge of the structure and property of crystal surfaces is important for the understanding of many surface phenomena such as adsorption, oxidation, corrosion, catalysis and crystal growth [1–4]. However, the surface energies of metals are difficult to determine experimentally and just few data exist [5–8], and these data include uncertainties of unknown magnitude. Therefore, to determine the surface energy theoretically is very essential. During the last decade there have been several methods to calculate the surface energy of metals, such as first-principles calculations [9–17] and semi-empirical methods [18–26]. However, the first-principle calculation is a tough job, especially for a largescale structure, due to not only the computer time required but also the reliability of the calculated results, which depends strongly on the selection of basis-set, exchange-correlation functional and cutoff energy. And most semi-empirical methods are based on some existing experimental results. And the methods only focused on a
few systems in which the needed parameters and functions have already been known. The empirical electron surface model (EESM) has been found remarkably successful in the calculation of the surface energies of transition metals [3,4] and diamond cubic crystals, of which the crystal cohesive energy is mainly dependent on the number and the spatial distribution of the covalent electrons in the crystal structure. The starting point of EESM is the valence electron structure (VES) calculated with the empirical electron theory in solid and molecule (EET) established by Yu [27–29]. And a brief English introduction of EET can be found in Refs. [3,4]. The reliability of EET has been extensively examined in the fields of metals, alloys, metallic compounds and ceramics [3,4,30–39]. However, when that model is used in the calculation of the surface energies of alkali metals, large errors are produced when the calculation results are compared with the results of experiments and other calculations. So the EESM has to be modified in this case. In this paper, EESM is modified and the calculation results of the surface energies of alkali metals are in well accordance with results of experiment and other calculations. 2. Summary of EESM
∗ Corresponding author at: College of Materials Science and Engineering, Beijing University of Chemical Technology, Box 23, 15 of Beisanhuan East Road, Beijing 100029, China. Tel.: +86 10 64421306; fax: +86 10 64437587. E-mail addresses: [email protected], [email protected] (Z. Li). 0254-0584/$ – see front matter © 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.matchemphys.2010.05.034
Surface energy () is one of the most important static physical quantities charactering metal surface. And the surface energy is usually defined as the additional value of the free energy per unit
B. Fu et al. / Materials Chemistry and Physics 123 (2010) 658–665
area of increase on a particular crystal surface, and therefore, the surface energy of crystal surface is given by the expression: =
E S
(1)
where E represents the additional value of the free energy with the forming of new surface and S is the added area of the crystal surface. In EET, valence shell electrons are divided into four types [28,29] according to the spatial distribution and the action characteristics of these electrons in the formation of solids or molecules, such as dumb pair electrons, magnetic electrons, covalent electrons (bonding electrons) and lattice electrons. The dumb pair electrons represent either a bonding and an anti-bonding electron with their resultant bonding power mutually cancelled by each other and their spins opposite to each other, or a pair of non-valence electrons of opposite spins deeply sinking down to the atomic orbit. The magnetic electrons here represent the localized electrons from the single-occupied orbit. The covalent electron, also from the singleoccupied orbit, is paired with another electron of opposite spins from other atom as elements are alloyed, contributing to atomic bonding. The lattice electron is present in a space among 3 or 4 atoms or even more than 6 atoms. When the s energy band is wide, the electron might reach an energy level above the Fermi energy and is called lattice electron. The number of valence electrons of an atom in solids, including the covalent electrons and the lattice electrons, equals to the valency of the atom in solids. In EET, the crystal cohesive energy is mainly dependent on the number and the spatial distribution of the four type electrons in the crystal structure. The contribution of the four type electrons to the crystal cohesive energy and the calculation formulae can be found in Refs. [29,30]. And the hybridization states of alkali metals, the number and the distribution of these electrons, are listed in Appendix A. For these alkali metals, all valence shell electrons belong to one of the following two types: covalent bond electron and lattice electron. And the numbers of lattice electrons of alkali metals are large, so lattice electrons should be taken into account in the calculation of surface energies. Correspondingly, the crystal cohesive energy for the alkali metals equals to the sum of the energy caused by these covalent electrons and lattice electrons. Therefore, when two nearest crystal planes of the alkali metals form two new crystal surfaces, the energy requirement for the procedure, i.e., the additional free energy (E), is the very sum of the additional free energy caused by the covalent electrons and lattice electrons (Ec and El ): E = Ec + El
(2)
In EESM, the additional free energy caused by the covalent electrons (Ec ) is equal to the sum of the bond energies of all bonds between two nearest crystal planes [3,4]: Ec =
X˛ E˛
where the factor g represents the contribution of the couple effect between spin and orbit of d electrons to the bonding capability. For 4, 5 and 6 periods, g is equal to 1, 1.35 and 1.70, respectively. ˛ , ˇ and are contents of s, p and d electrons in covalent bonds, respectively: ˛ =
lCh + l Ct nT
ˇ =
mCh + m Ct nT
=
nCh + n Ct nT
b=
31.395 n − 0.36ı
(7)
The factors n and ı in the denominator reflect the total effect of the screen, coulomb, exchange and interrelated interaction of the inner electrons in solids. The values of n and ı of the alkali metal elements are all 1 [28,29]. And the calculation formula for the additional free energy caused by the lattice electrons (El ) is: El = b
Zp nl f D
(8)
where the coefficient ZP , is the ratio of the additional free energy caused by the lattice electrons and the crystal cohesive energy of one atom, which is related to atomicity of the related crystal plane with a finite area, the distribution of lattice electrons and the surface orientation, and will be calculated in Section 4; nl represents the number of lattice electrons of the alkali metal atom and f represents the bonding capability of lattice electron:
f =
2nl nT
(9)
In formula (8), D is equivalent bond distance and equals to the arithmetic mean of the length of the bonds which are distributed around the space of lattice electron:
˛
where n˛ represents the number of covalent electron pairs on ˛ bond, D˛ represents the bond length and f represents the bonding capability of covalent electron: √ (5) f = ˛ + 3ˇ + g 5
(6)
where l, m, n and l , m , n represent the sum of the numbers of s, p, d covalent electron and lattice electron of the h and t states, respectively. The terms and are parameters for the h and t states, respectively, and value 1 when the s electron is covalent electron or 0 when the s electron is lattice electron. The values of these parameters can be taken from Refs. [28,29], and some values of the alkali metal elements are given in Appendix A. nT , Ch and Ct represent the total numbers of covalent electron and lattice electron, the relative compositions for the h state and for the t state of hybrid level, respectively. In formula (4), b is screen factor upon the core electron:
(3)
where ˛ represents bond name, E˛ represents the bond energy of ˛ bond and X˛ , obtained by dangling bond analysis method (DBAM), is the equivalent dangling bond number of ˛ bond on one particular crystal plane. The analysis of X˛ is presented in Section 4 in detail. In EET, the calculation formula for bond energy (E˛ ) of one bond in which the two atoms are the same is: n˛ (4) E˛ = bf D˛
659
D =
I˛ D˛
˛
(10) I˛
˛
where I˛ represents the equivalent bond numbers of the bonds in the structure unit. Therefore, the key to this surface energy model is the calculation of VES and the analysis of the equivalent dangling bond number. 3. The VES and the bond energies of alkali metals These alkali metals, the crystal cell is shown in Fig. 1 and the lattice constants are shown in Table 1, belong to A2 type crystal structure, i.e., the body centered cubic (bcc) structure and the atom
660
B. Fu et al. / Materials Chemistry and Physics 123 (2010) 658–665
of the electron number on all the covalent bonds in the structure unit, i.e.:
nc =
nA =
I˛ n˛ = nA (IA + IB rB )
(15)
˛
nc
IA + IB rB
=
2nc IA + IB rB
(16)
following: DA = (0.5a)2 + (0.5a)2 + (0.5a)2 , DB = a. The equivalent bond numbers (I˛ ) can be calculated with formula [28,29]:
For certain hybrid level of an atom, the covalent electron number and single bond radius can be found in Table 2 [28,29]. By selecting a suitable ˇ value and substituting the number of covalent electrons and the single bond radii of the atoms of the corresponding hybrid levels, the number of covalent electron pair nA on the strongest covalent bond can be figured out. By substituting the obtained nA into the ratio equation, the numbers of covalent electron pairs on the B bonds can be calculated. The hybrid states of the atoms and the corresponding distributions which are in accordance with the actuality have to be determined. In EET, the bond lengths calculated with the calculated n˛ are called theoretical bond lengths. Substituting the obtained n˛ into the bond length formula (12) yields the theoretical bond lengths of all the bonds:
I˛ = iM iS iK
D˛ = 2R (1) − ˇ log n˛
Fig. 1. Crystal cell of alkali metals.
coordinates are (0 0 0) and (0.5 0.5 0.5), respectively. So the experimental bond lengths which cannot be neglected can be given as the
(11)
where ˛ (˛ = A or B) represents bond name, iM represents the reference atom number in the structure unit, iS represents the equivalent bond number for a reference atom to form ˛ bond and iK is a parameter, which equals 1 when the two atoms that form the bond are of the same kind or 2 when the two atoms are of different kinds. Therefore, the equivalent bond numbers of ˛ bond are: IA = 2 × 8 × 1 = 16, IB = 2 × 6 × 1 = 12. According to formula (10), the equivalent bond distance (D ) can be calculated. According to the Pauling bond length formula [28,29,40]: D˛ = 2R(1) − ˇ log na
(12)
where D˛ represents the bond length of ˛ bond; R(1) represents the single bond radius of the atom which forms the bond; n˛ represents the number of covalent electron pairs on the bond and ˇ is a parameter with units of length. Here a modification for Pauling’s original theory is made as the follows:
⎧ ⎪ ⎨ 0.0710 (nm)
0.0600 (nm) ˇ= ⎪ 0.0710 − 0.22ε (nm)
⎩
M when nM ˛ < 0.25 or n˛ > 0.75 ≤ 0.700 when 0.300 ≤ nM ˛ (13) M when nM ˛ = 0.250 + ε or n˛ = 0.750 − ε
where 0 ≤ ε < 0.050, nM ˛ represent the largest n˛ in the structure. Therefore, the following ratio equation can be obtained: rB =
nB = 10(DA −DB )/ˇ nA
(14)
A structure unit should be electron neutral, so the covalent electron of all atoms in the structure unit (there are 2 atoms in the unit cell of the alkali metals) should be distributed on all ˛ covalent bonds in it. In other words, the total covalent electron number ( nc ) of all atoms in a structure unit should equal the sum ( I˛ n˛ ) Table 1 The lattice constants of the alkali metals.
Bond length difference (BLD) D˛ , is the absolute value of the difference between the experimental bond length and the theoretical one, i.e.:
a = b = c (nm)
˛ = ˇ = (◦ )
Li Na K Rb Cs
0.35092 0.42906 0.53200 0.57000 0.61400
90 90 90 90 90
D˛ = D˛ − D˛
(18)
In EET, D˛ < 0.005 nm is the criterion to determine whether the given atom state and the corresponding valence electron distribution accord with reality. If D˛ < 0.005 nm, the applied hybrid levels of the atom are believed being in accordance with the actual state, therefore, the obtained corresponding parameters are the possible VES of the calculated crystal. Through programming the calculation according to the above equations and substituting the parameters of all hybrid levels of alkali metal atoms, all the possible existing hybrid levels of the atoms can be obtained. So the VES of these alkali metals can be calculated, and the calculation results with the smallest bond length difference D˛ are shown in Table 3. According to the VES of these alkali metals and formulae ((4)–(7)), the bond energy can be calculated. And the calculation results are tabulated in detail in the last column of Table 3. 4. Dangling bond analysis of the related crystal plane of alkali metals The DBAM is to investigate the number and the type of the dangling bonds on one particular crystal surface. And based on the calculated bond energy (see Table 3), Ec of one particular crystal plane can be calculated. Therefore, the DBAM is one of the important steps of the surface energy model. Formula for the equivalent dangling bond numbers (X˛ ) on one particular crystal plane can be deduced out [3,4]: X˛ = iP iK
Alkali metals
(17)
iD NP
(19)
NP
where iP represents the reference atom number on one particular plane with the area S and iK is a parameter, which equals 1 when the two atoms that form the bond are of the same kind or 2 when the two atoms are different kinds; iD represents the equivalent dangling bond number for a reference atom to form one type
B. Fu et al. / Materials Chemistry and Physics 123 (2010) 658–665
661
Table 2 Hybridization table of alkali metals. Metals
Li
1 2 3 4
Na
A type
B type
C type
nc
nl
R(1) (nm)
nc
nl
R(1) (nm)
nc
nl
R(1) (nm)
0 0.25309 0.65108 1
1 0.74691 0.34892 0
0.1326 0.12400 0.11046 0.0986
0 0.46143 0.53498 1
1 0.53857 0.46502 0
0.1326 0.11691 0.11441 0.0986
1 1 1 1
0 0 0 0
0.1326 0.12542 0.10579 0.0986
1 2 3 4
0 0.46143 0.53498 1
1 0.53857 0.46502 0
0.15733 0.14504 0.14308 0.1307
1 1 1 1
0 0 0 0
0.15733 0.14816 0.1374 0.1307
K
1 2 3 4
0 0.46143 0.53498 1
1 0.53857 0.46502 0
0.19628 0.18794 0.18661 0.1782
1 1 1 1
0 0 0 0
0.1964 0.18628 0.17440 0.167
Rb
1 2 3 4
0 0.46143 0.53498 1
1 0.53857 0.46502 0
0.2087 0.20270 0.20175 0.1957
1 1 1 1
0 0 0 0
0.2087 0.19633 0.18183 0.17279
Cs
1 2 3 4
0 0.46143 0.53498 1
1 0.53857 0.46502 0
0.2214 0.2226 0.22279 0.224
1 1 1 1
0 0 0 0
0.2214 0.20870 0.19381 0.18453
Table 3 The VES and bond energies of alkali metals (including the number of lattice electrons and equivalent bond distance). Alkali metal (hybrid level)
D (nm)
Bond name (˛)
D˛ (nm)
D˛ (nm)
n˛
Dn˛ (nm)
E˛ (kJ mol−1 )
Li (A3)
0.32405
A B
0.30391 0.35092
0.30294 0.34996
0.06996 0.01523
0.00096 0.00096
12.48715 2.35307
Na (B3)
0.39621
A B
0.37158 0.42906
0.37297 0.43045
0.05991 0.00929
0.00139 0.00139
6.18145 0.83026
K (B3)
0.49127
A B
0.46073 0.53200
0.45883 0.53011
0.06225 0.00617
0.00189 0.00189
5.22073 0.44787
Rb (B2)
0.52636
A B
0.49363 0.57000
0.49526 0.57162
0.05426 0.00456
0.00162 0.00162
3.91568 0.28507
Cs (B3)
0.56699
A B
0.53174 0.61400
0.53055 0.61281
0.06356 0.00441
0.00119 0.00119
4.61060 0.27705
bond with the atoms of the same neighbor crystal plane; NP (NP = 1, 2, . . .) represents the crystal plane number passed through by the bond. As is shown in Fig. 2, there is one atom on the (1 0 0) plane with the area of a2 nm2 in the unit cell of the alkali metals. Every atom with the atoms of the nearest neighbor crystal plane forms four A bonds, and with the atoms of the second neighbor crystal plane forms one B bond. When the (1 0 0) plane forms the outer surface, and these bonds will form dangling bonds in the outer surface. And
Fig. 2. The spatial distribution of dangling bond on alkali metals (1 0 0) crystal plane.
based on formula (16), the equivalent dangling bond numbers on the (1 0 0) plane are XA = 1 × 1 × 4 × 1 = 4 and XB = 1 × 1 × 1 × 2 = 2. As is shown in√ Fig. 3, there are two atoms on the (1 1 0) plane with the area of 2a2 nm2 in the unit cell of the alkali metals. Every atom with the atoms of the nearest neighbor crystal plane forms two A bonds and two B bonds. So the equivalent dangling bond numbers on the (1 1 0) plane are XA = 2 × 1 × 2 × 1 = 4 and XB = 2 × 1 × 2 × 1 = 4. As is shown in Fig. 4, √ there is one atom on the (1 1 1) crystal plane with the area of 3a2 nm2 in the alkali metal crystal cell. Every atom with the atoms of the nearest neighbor crystal plane forms three A bonds, with the atoms of the second neighbor crystal plane forms three B bonds, and with the atoms of the third neighbor crystal plane forms one A bond. Therefore, the equivalent dangling bond numbers on the (1 1 1) plane can be obtained. They are XA = 1 × 1 × (3 × 1 + 1 × 3) = 6 and XB = 1 × 1 × 3 × 2 = 6. As is shown in Fig. 5, √ there is one atom on the (2 1 0) crystal plane with the area of 5a2 nm2 in the alkali metal crystal cell. Every atom with the atoms of the nearest neighbor crystal plane forms two A bonds, with the atoms of the second neighbor crystal plane forms one B bond, with the atoms of the third neighbor crystal forms two A bonds, with the atoms of the fourth neighbor crystal plane forms one B bond. So the equivalent dangling bond numbers on the (2 1 0) plane are XA = 1 × 1 × (2 × 1 + 2 × 3) = 8 and XB = 1 × 1 × (1 × 2 + 1 × 4) = 6.
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B. Fu et al. / Materials Chemistry and Physics 123 (2010) 658–665
Fig. 5. The spatial distribution of dangling bond on alkali metals (2 1 0) crystal plane.
structure. And there are four tetrahedral interstices around one octahedral central site, which is the geometric center of the four tetrahedral interstices. To simplify the calculation of the coefficient Zp , the lattice electrons in the tetrahedral interstices are equivalently distributed in the octahedral central sites, and the process is interpreted in Fig. 6. It is assumed that the contribution of lattice electrons to the crystal cohesive energy is equal to the very sum of the energy of the charge effect between the octahedral central sites and the atoms around the octahedral central sites: El =
Fig. 3. The spatial distribution of dangling bond on alkali metals (1 1 0) crystal plane.
5. The coefficient Zp In EET, lattice electron is presented in a space among 3 or 4 atoms or even more than 6 atoms. So as is shown in Fig. 6, the lattice electrons of alkali metals are presented in both octahedral and tetrahedral interstices (o sites and t sites) of the crystal
Eoi
(20)
where El represents the crystal cohesive energy caused by the lattice electrons; Eoi is the energy of the charge effect between the octahedral central site and various adjacent atoms. Therefore, the additional free energy caused by the lattice electrons (El ) in the forming of new surface is related to the charge effect between the octahedral central sites and the adjacent atoms. As is shown in Fig. 7, there are six adjacent atoms around one octahedral central site. Because the distances of the octahedral central site to the adjacent √ atoms are different, i.e. d1 = d2 = a/2 = / d3 = d4 = d5 = d6 = a 2, the charge effect between the octahedral central sites and the atoms around the octahedral central sites are also different and are in inverse proportion to these distances, i.e., Eoi ˛1/di (i = 1, 2, . . ., 6). Therefore, the fraction (fi ) of Eoi in the crystal cohesive energy caused by the lattice electrons of one atom can be deduced out. fi =
Eoi = El
E
1/d
oi = Eoi
no
i
(21)
1/di
i
Fig. 4. The spatial distribution of dangling bond on alkali metals (1 1 1) crystal plane.
Fig. 6. The spatial distribution of the lattice electrons of alkali metals (the octahedral and tetrahedral interstices).
B. Fu et al. / Materials Chemistry and Physics 123 (2010) 658–665
663
According to the above analysis and formulae ((20)–(22)), a computer program written in Microsoft Visual Basic language has been compiled to analyze and calculate the coefficient Zp of various index surfaces. So for the alkali metals, the coefficients Zp of the (1 0 0), (1 1 0), (1 1 1) and (2 1 0) plane are Z(1 0 0) = 0.528595, Z(1 1 0) = 0.861929, Z(1 1 1) = 1 and Z(2 1 0) = 1.390524.
6. Calculation of surface energy of the corresponding surface
Fig. 7. The distances of the octahedral central site to the adjacent atoms in the crystal cell of alkali metals.
where no represents the number of octahedral central sites corresponding to one atom. As is shown in Fig. 6, there are two atoms and six octahedrons in the unit cell of alkali metals, that is, no = 3. Based on the above analysis and with the similar idea of DBAM, the calculation formula for the coefficient Zp on one particular crystal plane can be deducted out. Zp =
o
According to the calculated bond energy in Table 3 and the equivalent dangling bond number of the related crystal plane calculated in Section 4, the additional free energy caused by the covalent electrons (Ec ) can be calculated. And based on the values of the lattice electron number and equivalent bond distance in Tables 2 and 3, the coefficient Zp calculated in Section 5 and formulae ((7)–(10)), the additional free energy caused by the lattice electrons (El ) can also be calculated. Therefore, the calculation formula for the surface energy () of one particular clean and ideal surface can be derived from formula (1).
=
Ec + El = 2S
X˛ E˛ + b
Zp nl f D
˛
(23)
2S
(22)
fi Np iK
i
where o represents the reference octahedral central site on the particular plane (h l k) with the area of S, or in the middle part of the region and its neighbor plane, Np represents the crystal plane number between the reference octahedral central site (o) and the i adjacent atom of the neighbor crystal plane (not including the atoms of the (h l k) index plane of o site) and iK is a parameter, which equals 1 when the reference octahedral central site (o) is located at the cleavage plane or 0.5 when the reference octahedral central site (o) is located in the middle part of two neighbor cleavage plane.
The physical meaning of the number 2 in this formula is that when the interaction of electrons and atoms between two nearest crystal planes is broken, there will be two crystal surfaces in the region. So the total area is two times of one crystal plane in one unit crystal cell. Through programming the calculation according to these parameters and the surface energy formula (23), we can calculate the surface energy of the corresponding clean and ideal surface of the alkali metals. The calculation results are tabulated in detail in Table 4.
Table 4 The surface energies of alkali metals. Alkali metal Li
Na
K
Rb
Cs
a b c d e
Plane
Ec (eV)
El (eV)
S (nm2 )
(eV nm−2 )
Other theoretical results (eV nm−2 ) a
Experiment (eV nm−2 )
b
(1 0 0) (1 1 0) (1 1 1) (2 1 0)
0.56646 0.61523 0.92285 1.18169
0.24173 0.39416 0.45730 0.63589
0.12314 0.17415 0.21329 0.27536
3.28144 2.89801 3.23533 3.30036
3.263 , 3.6 3.475a , 3.163c , 3.1b 3.688a , 3.406c , 4.5b 3.894c
(1 0 0) (1 1 0) (1 11) (2 1 0)
0.27348 0.29069 0.43603 0.56416
0.30419 0.49601 0.57546 0.80019
0.18409 0.26035 0.31886 0.41164
1.56894 1.51086 1.58611 1.65720
1.65a , 2.1b 1.581a , 1.8b 1.794a , 2.6b
1.631d , 1.625e
(1 0 0) (1 1 0) (1 1 1) (2 1 0)
0.22572 0.23500 0.35251 0.46072
0.24533 0.40003 0.46411 0.64536
0.28302 0.40026 0.49021 0.63286
0.83217 0.79329 0.83292 0.87387
0.888a , 1.1b 0.844a , 1.0b 0.95a , 1.4b
0.906d , 0.813e
(1 0 0) (1 1 0) (1 1 1) (2 1 0)
0.16824 0.17415 0.26123 0.34239
0.28539 0.46536 0.53990 0.75075
0.32490 0.45948 0.56274 0.72650
0.69811 0.69591 0.71181 0.75233
0.7a , 0.94b 0.65a , 0.75b 0.738a , 1.1b
0.731d , 0.688e
(1 0 0) (1 1 0) (1 1 1) (2 1 0)
0.19689 0.20263 0.30394 0.39951
0.21256 0.34661 0.40213 0.55917
0.37700 0.53315 0.65298 0.84299
0.54304 0.51508 0.54066 0.56862
0.581a , 0.75b 0.512a , 0.62b 0.575a , 0.87b
0.592d , e
The full charge density(FCD) method in the generalized gradient approximation (GGA) [7]. The modified broken-bond rule [41]. The first-principles molecular dynamics method [18]. Experiment [5]. Experiment [6].
3.263d , 3.281e
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7. Results and discussion The (1 0 0), (1 1 0), (1 1 1) and (2 1 0) surface energies for 5 alkali metals such as Li, Na, K, Rb and Cs, are presented in Table 4 in eV nm−2 , respectively. For comparison we also present some theoretical results and experimentally derived values. These theoretical methods include the full charge density (FCD) method in the generalized gradient approximation (GGA) in Ref. [9], the modified broken-bond rule in Ref. [41], and the first-principles calculation in Ref. [14], and so on. The experimental values in Refs. [7,8] are determined from measurements of the surface tension of liquid metals extrapolating through the liquid–solid phase transition. It must be pointed out that experimental measurements of the surface energy are more commonly found for polycrystalline materials. But for alkali metals, there is little surface energy data in the literature. The relaxation and reconstruction of the atomic positions are neglected, which may lead to errors of up to a few percent. From Table 4, the surface energy values calculated in this paper are close to the values come from the other Refs. From Table 4, it can be clearly seen that, the calculated surface energy shows a strong anisotropy, the surface energy values of the (1 0 0), (1 1 0), (1 1 1) and (2 1 0) surface are different and the closepacked (1 1 0) surface energy is the lowest of all these index surfaces as predicted. And the order of the energies of various index surfaces of Li and Cs is (1 1 0) < (1 1 1) < (1 0 0) < (2 1 0) , but the order of Na, K and Rb is (1 1 0) < (1 0 0) < (1 1 1) < (2 1 0) . This is caused by the differences of their VES and lattice constants. And according to the analysis of dangling bond, the spatial distribution of dangling bond of the (1 0 0), (1 1 0), (1 1 1) and (2 1 0) crystal surface is also different. So the dangling bond electron density and the spatial distribution of covalent bonds have a great influence on surface energy of various index surfaces. EESM has been found remarkably successful in calculating surface energies of transition metals [3,4], where the number of the covalent electrons takes up major constituent of the total electron number for valence shell. However, in the case of alkali metals with more lattice electrons, the neglect of the contribution of lattice electrons will lead to the calculation results inconsistent with the other theoretical results and experimentally derived values. For instance, without consideration of the effect of lattice electrons, the (1 1 0) surface energies of Li, Na, K, Rb and Cs are 1.76638, 0.55827, 0.29356, 0.18951 and 0.19003 eV nm−2 , respectively, and the relative errors are over 50% (relative to the other theoretical results or experimentally derived values), but the relative errors of our results in this paper are less than 10%. Therefore, the contribution of lattice electrons to the surface energies of alkali metal cannot be neglected. Similarly, for alkaline earth metals with more lattice electrons, the lattice electrons should be considered as well, which will be reported in the future. The meaning of this paper is not only to extend the EESM to alkali metals with more lattice electrons, and to thereby fill out that gap in the literature, but also to provide comprehensive surface energy results for alkali metals, transition metals [3,4] and diamond cubic crystals. Such extensive results obtained with the same theoretical model, are useful to both theorist and experimentalists. The EESM can be provided for the research of surface energy of the more metals, alloys, ceramics, compound and so on.
8. Conclusions Since the lattice electrons of alkali metals occupy large proportion of the total electron number for valence shell, the contribution of these lattice electrons cannot be neglected in surface energy calculation. Therefore, the necessary modification to the empirical electron surface model (EESM) has been presented in this paper.
With EESM, we have built a database of the (1 0 0), (1 1 0), (1 1 1) and (2 1 0) surface energy for the alkali metals like Li, Na, K, Rb and Cs, and the spatial distribution of dangling bonds of various index surfaces and the distribution of lattice electrons has been analyzed with dangling bond analysis method (DBAM). The dangling bond electron density and the spatial distribution of lattice electrons and covalent bonds have a great influence on surface energy of various index surfaces. Under the first-order approximation, the results are close to the other theoretical values and the experimental values. These surface energy values may be used as a consistent starting point for study of surface science phenomena. The calculated surface energy shows a strong anisotropy. And the close-packed (1 1 0) surface energy is the lowest of these index surfaces as predicted. This research model can be further extended to the surface energy estimation of more crystals, which will be the next research objects. Acknowledgements This work was supported by the Beijing Natural Science Foundation (Grant No. 2072014) and the Research Fund for the Doctoral Program of Higher Education of China (Grant No. 200800100006). Appendix A. Hybridization states in EET of the elements In EET, dumb pair electrons, magnetic electron, covalent electron, and lattice electron are denoted by ||, ↑, 䊉 and , respectively. When the covalent electron and lattice electron are equivalent electrons, they are denoted by 䊉, or s , p , respectively. The hybridization states of these elements are as the follows. A.1. A type of H and Li
h state: s1 t state: s␣ p1−␣
s 䊉
䊉
l = 1, m = 0, n = 0, = 0; l = 0.94879 (˛), m = 0.05121, n = 0, = 1. H: R(1)h = 0.03708 nm, R(1)t = 0.0280 nm. Li: R(1)h = 0.1326 nm, R(1)t = 0.0986 nm.
A.2. B type of H, Li, Na, K, Rb and Cs
h state: s1 t state: s␣ p1−␣
s 䊉
䊉
s 䊉 䊉
䊉
l = 1, m = 0, n = 0, = 0; l = 0.9982 (˛), m = 0.0018, n = 0, = 1. H: R(1)h = 0.03708 nm, R(1)t = 0.0280 nm. Li: R(1)h = 0.1326 nm, R(1)t = 0.0986 nm. Na: R(1)h = 0.15733 nm, R(1)t = 0.1307 nm. K: R(1)h = 0.19628 nm, R(1)t = 0.1782 nm. Rb: R(1)h = 0.2087 nm, R(1)t = 0.1957 nm. Cs: R(1)h = 0.2214 nm, R(1)t = 0.224 nm.
A.3. C type of Li, Na, K, Rb and Cs
h state: s1 t state: s␣ p1−␣
l = 1, m = 0, n = 0, = 1; l = 0.5 (˛), m = 0.5, n = 0, = 1. Li: R(1)h = 0.1326 nm, R(1)t = 0.0986 nm. Na: R(1)h = 0.15733 nm, R(1)t = 0.1307 nm. K: R(1)h = 0.1964 nm, R(1)t = 0.1670 nm. Rb: R(1)h = 0.2087 nm, R(1)t = 0.17279 nm. Cs: R(1)h = 0.2214 nm, R(1)t = 0.18453 nm.
B. Fu et al. / Materials Chemistry and Physics 123 (2010) 658–665
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20]
F.W. Young, J.V. Cathcart, A.T. Gwathmey, Acta Metall. 4 (1956) 145. C.A. Chang, J. Appl. Phys. 67 (1990) 566. B.Q. Fu, W. Liu, Z.L. Li, Appl. Surf. Sci. 255 (2009) 8511. B.Q. Fu, W. Liu, Z.L. Li, Appl. Surf. Sci. 255 (2009) 9348. I. Galanakis, N. Papanikolaou, P.H. Dederichs, Surf. Sci. 511 (2002) 1. V.K. Kumikov, Kh.B. Khokonov, J. Appl. Phys. 54 (1983) 1346. W.R. Tyson, W.A. Miller, Surf. Sci. 62 (1977) 267. F.R. de Boer, R. Boom, W.C.M. Mattens, A.R. Miedema, A.K. Niessen, Cohesion in Metals, North-Holland, Amsterdam, 1988. L. Vitos, A.V. Ruban, H.L. Skriver, J. Kollar, Surf. Sci. 411 (1998) 186. M. Methfessel, D. Hennig, M. Scheffler, Phys. Rev. B 46 (1992) 4816. H.L. Skriver, N.M. Rosengaard, Phys. Rev. B 46 (1992) 7157. P. Blonski, A. Kiejna, Vacuum 74 (2004) 179. M.J.S. Spencer, A. Hung, L.K. Snook, I. Yarovsky, Surf. Sci. 513 (2002) 389. K. Koko, P.T. Salo, R. Laihia, K. Mansikka, Surf. Sci. 348 (1996) 168. Z. Crljen, D. Sokcevic, R. Brako, P. Lazic, Vacuum 71 (2003) 101. Y. Mishin, D. Farkas, M.J. Mehl, D.A. Papaconstantopoulos, Phys. Rev. B 59 (1999) 3393. P.J. Feibelman, Phys. Rev. B 53 (1996) 13740. J.M. Zhang, D.D. Wang, K.W. Xu, Appl. Surf. Sci. 252 (2006) 8217. C. Barreteau, D. Spanjaard, M.C. Desjonqueres, Surf. Sci. 433–435 (1999) 751. M.I. Baskes, Phys. Rev. B 46 (1992) 2727.
[21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35] [36] [37] [38] [39] [40]
[41]
665
C. Bombis, A. Emundts, M. Nowicki, H.P. Bonzel, Surf. Sci. 511 (2002) 83. T.M. Trimble, R.C. Cammarata, Surf. Sci. 602 (2008) 2339. T.M. Trimble, R.C. Cammarata, K. Sieradzki, Surf. Sci. 531 (2003) 8. J.M. Zhang, F. Ma, K.W. Xu, Appl. Surf. Sci. 229 (2004) 34. J.M. Zhang, D.D. Wang, K.W. Xu, Surf. Sci. 253 (2006) 2018. E. Aghemenloh, J.O.A. Idiodi, S.O. Azi, Comput. Mater. Sci. 46 (2009) 524. R.H. Yu, Chin. Sci. Bull. 23 (1978) 217. Z.L. Liu, Z.L. Li, W.D. Liu, Electron Structure of Interface and their Properties, Science Publishing House, Beijing, 2002, pp. 7–13 (in Chinese). R.L. Zhang, The Empirical Electron Theory in Solids and Molecules, Jinlin Science and Technology Publishing House, Changchun, 1993, pp. 268–278 (in Chinese). W.D. Xu, R.L. Zhang, R.H. Yu, Sci. China Ser. A 32 (1989) 351. Z.L. Li, J. Xu, B.Q. Fu, Solid State Sci. 10 (2008) 1434. Z.L. Li, H.B. Xu, S.K. Gong, J. Phys. Chem. B 108 (2004) 15165. Z.L. Li, W. Liu, Y.Q. Wu, Mater. Chem. Phys. 105 (2007) 278. Z.L. Li, Q. Huang, Y.Q. Wu, Z.F. Li, Sci. China Ser. E 50 (2007) 462. Z.L. Li, Z.F. Li, Q. Huang, Sci. China Ser. E 50 (2007) 664. Z.L. Liu, Z.L. Li, Z.G. Sun, Metall. Mater. Trans. 30A (1999) 2757. Z.L. Liu, Z.L. Li, Z.G. Sun, Chinese Sci. Bull. 41 (1996) 367. Z.L. Li, H.B. Xu, S.K. Gong, Sci. China Ser. E 46 (2003) 234. Z.L. Liu, Z.G. Sun, Z.L. Li, Prog. Nat. Sci. 8 (1998) 134. L. Pauling, The Nature of the Chemical Bond and the Structure of Molecules and Crystals: An Introduction to Modern Structural Chemistry, Cornell University Press, New York, 1960, pp. 230–256. Q. Jiang, H.M. Lu, M. Zhao, J. Phys.: Condens. Matter 16 (2004) 521.