International Journal of Scientific Research Engineering & Technology (IJSRET), ISSN 2278 – 0882
Volume 3, Issue 5, August 2014
Surface Energy Calculation of Low Index Group 1 Alkali Metals of the
Periodic Table Using the Modified Analytical
Embedded Atom Method
1
Akpata Erhieyovwe, 2 Enaibe. A. E and 3Iyoha. A.
1
2
Department of Physics, University of Benin, Benin City, Edo State, Nigeria.
Department of Physics, Federal University of Petroleum Resources, Effurun, Delta State, Nigeria.
3
Department of Physics, Ambrose Alli University, Ekpoma, Edo State, Nigeria
Abstract
The surface energies (Es) of the low index (100), (110) and (111) Group 1 bcc alkali metals of the periodic table
(Li,Na,K,Rb,Cs) have been calculated using the Modified Analytical Embedded Atom Method (MAEAM).The result
shows that the surface energy of each (hkl) plane of Group 1 bcc alkali metals decreases down the Group. That is
(EsLi  EsNa  EsK  EsRb  EsCs). It also shows that the differences between the surface energies of metal of
subsequent group of period one (1) decreases down the group for each hkl plane. That is (EsLi-EsNa)  (EsNaEsK)  (EsK-EsRb)>(EsRb-EsCs). The results obtained also shows that Es(111) ‹ Es(110) ‹ Es(100) for each (hkl) plane. This
is not completely in agreement with the results obtained by other methods such as Analytical Embedded Atom
Method (AEAM),[11] Finnis Sinclair[4], Tight binding[11], Zhang et al [11] and experiment[9,5] which shows that
Es(110) ‹ Es(100) ‹ Es(111) .
Keywords: Embedded atom method, Alkali metals, Body Centered Cubic, Modified Analytical Embedded Atom
Method, Surface Energy
1.
INTRODUCTION
The elements in group 1 of the periodic table are collectively known as alkali metals. They are lithium, sodium,
potassium, rubidium, caesium and francium. Francium (Fr) is a short-lived radioactive element about which little is
known [1]. A detailed knowledge of the structure and energy of surfaces is important for understanding many surface
phenomena such as adsorption, oxidation, corrosion, catalysis, crystal growth etc. [12,3]. Surface energies for
materials with Face-Centered Cubic (fcc), Body-Centered Cubic (bcc) and diamond structures have been calculated
[8] by the original Embedded Atom Method (EAM) developed by Daw and Baskes[3] and Modified Embedded Atom
Method (MEAM) extended by Baskes [2].
In this paper, the surface energies of the bcc alkali metals (Li,Na,K,Rb,Cs) of Group 1 of the periodic table
have been calculated by using the Modified Analytical Embedded Atom Method (MAEAM) developed by Zhang et al
[15]. The Modified Analytical Embedded Atom Method (MAEAM) is an extension of the initial Embedded Atom
Method (EAM) and the Analytical Embedded Atom Method (AEAM) [5,6] by adding a modified term to describe the
energy change due to the non-spherical distribution of electrons and deviation from the linear superposition of atomic
electron density.
The physical properties of the alkali metals are shown in table 1.
Property
Table 1: Physical properties of the alkali metals. [1]
Li
Na
K
Sodium
Potassium
Lithium
Cs
Caesium
19
37
55
Atomic number
3
Electron configuration (outer)
1s22s1
2p22s1
3p64s1
4p65s1
5p66s1
Isotopes (in order of abundance)
7.6
23
39,41,40
85.87
133
Relative atomic mass
6.94
22.99
39.096
85.48
132.91
Metallic radius (A)
1.55
1.90
2.35
2.48
2.67
Ionic radius (Ǻ)
0.60
0.95
1.33
1.48
1.69
1.57
2.03
2.16
2.35
Atomic radius (Ǻ)
1.33
11
Rb
Rubidiu
m
www.ijsret.org
874
International Journal of Scientific Research Engineering & Technology (IJSRET), ISSN 2278 – 0882
Volume 3, Issue 5, August 2014
Density (gcm-3)
0.54
First ionization energy (kJmol-1)
0.97
0.86
1.53
1.90
518.32
493.24
418
401.28
376.2
Electronegativity (A/R)
1.15
1.0
0.9
0.9
0.85
Electrode potential (Eθ)
-3.02
-2.71
-2.92
2.99
-3.02
Enthalpy of sublimation (kJmol)
Melting point (K)
152.3
108.47
89.95
85.69
78.71
454
371
336
312
302
Boiling point (K)
1615
1156
1033
959
942
65
28.300
25.900
310
7
Crimson
Bright yellow
Lavender
lilac
Lilac
Bright
blue
Softest
95.5
55.8
71
1
Abundance
million(p.p.m))
Flame
(parts
per
Physical state
Hardest
Metallic bonding
Weak
M.
bonding
13.1
Atomic volume (cm3mol-1)
Decreases
down
the group
23.1
2. THEORETICAL CONSIDERATION AND CALCULATIONS
Zhang et al [16] modified Johnson’s Analytical EAM, in a similar way to Pasianot.[7]. They added a modified
analytical energy term M(P) to the total energy expression for the EAM to express the difference between the actual
total of a system of atoms and that calculated from the original EAM using a linear superposition of spherical atom
electron densities. While the analytical EAM energy modified term M(P) take into account the angularity term [7]
P   f 2 (rm )
2
2
2
rmx
 rmy
 rmz
m
rm2
.
(1)
The modified analytical EAM energy modified term [10] does not include angularity term.
P   f 2 (rij )
ji
(2)
where f(rij) is the separation distance of atom j from atom i. Again while the modified term M(P) in the analytical
EAM of Wangyu Hu et al [10] contain both exponential form and natural logarithm.

  P 2 
M Pi    1  exp  In   

  Pe   

(3)
the modified term in Zhang et al [11] only have exponential form

M ( Pi )    i  e
 e 
2
  P 2 
  1  
  Pe  


(4)
In the MAEAM the surface energy Es of a system is calculated using equation (5) below
Es  
E i  E c 
(5)
As
N
where Ei is the total energy of the system, Ec is the cohesive energy and As is the area of a periodic cell containing
one atom. In calculating the total energy of the system, the following input parameters were taken as constants. [14]
F0  E C  E1f
(6)
where E1f is mono-vacancy formation energy.
n
(C11  2C12 )(C11  C12 )
(216E1f C 44 )
www.ijsret.org
(7)
875
International Journal of Scientific Research Engineering & Technology (IJSRET), ISSN 2278 – 0882
Volume 3, Issue 5, August 2014
(C12  C 44 ) n 2 F0


5126 .4
8
 E  E1f 
fe   C

  
3
(8)
5
(9)
where C11, C12 and C44 are elastic constants of the bcc metal considered and   a
3
2
is the atomic volume for bcc
metals and a is the lattice constant.
 E1 f (51519C 44  57111C12  57111C11 )

7
75582360
(33327C 44  52563C12  52563C11 )
k1 
431189920
(147456C11  147456C12  59049C 44 )
k2 
302329440
k0 
k3 
1536(4C 44  C11  C12 )
66134565
(10)
(11)
(12)
(13)
According to the Modified Analytical Embedded Atom Method, the total energy of a system Ei is given as [15, 17]


1
E i   Fi    (rij )  M(Pi )
2 i j
i 

(14)
where F(i) is the energy to embed an atom in site i with electron density i . rij is the separation
distance between atoms i and j, ( rij) is the interaction potential between atoms i and j and M(Pi) is
the modified term. It describes the energy change due to the non-spherical distribution of electron Pi .
(15)
i 
f (rij )

i j
Pi   f 2 (rij )
(16)
i j


  i

F (  i )   F0 1  n ln  i
 

  e   e 

2
n
(17)
4
12
r

r

r

(18)
(rij )  k 0  k1  ij   k 2  ij   k 3  ij 
r
r
r
 ie 
 ie 
 ie 
where rie is the first nearest neighbour distance at equilibrium. The cut-off distance rc of interaction potential is taken
to lie between the second and the third nearest neighbour distance.

M ( Pi )    i  e
 e 
2
r 
f (rij )  f e  ij 
 rie 
3.
  P 2 
  1  
  Pe  


(19)
6
(20)
PRESENTATION OF RESULTS
Table 2: Input parameters for selected bcc metals [13]
Metals
Li
Na
K
Rb
Cs
a(nm)
0.35093
0.42096
0.5321
0.5703
0.6141
Ω (nm3)
0.021609
0.037299
0.075327
0.092743
0.115794
Ec (eV)
1.63
1.113
0.934
0.852
0.804
E1f (eV)
0.48
0.34
0.34
0.341
0.322
C11 (eVnm-3)
84
46
28
18
15
www.ijsret.org
C12 (eVnm-3)
71
39
23
15
9
C44 (eVnm-3)
55
26
16
10
13
876
International Journal of Scientific Research Engineering & Technology (IJSRET), ISSN 2278 – 0882
Volume 3, Issue 5, August 2014
Ω = atomic volume, Ec = cohesive energy, E1f = mono-vacancy formation energy, C11, C12 and C14, are elastic
constants, a = lattice constant. [13]
Metals
Li
Na
K
Rb
Cs
Table 3: Calculated model parameters for group 1 bcc alkali metals.
Fo
n
α
Fe
Ko
k1
k2
1.1500
0.1055
-0.0015 0.1720
-0.0692 0.0006
-0.0001
0.7730
0.1302
-0.0015 0.0977
-0.0490 0.0004
-0.0001
0.5940
0.1540
-0.0016 0.0547
-0.0491 0.0005
-0.0001
0.5110
0.1347
-0.0010 0.0441
-0.0491 0.0004
0.0000
0.4820
0.1592
-0.0015 0.0373
-0.0465 0.0003
0.0000
Table 4:
Metals
Li
Na
K
Rb
Cs
e
1.376
0.782
0.438
0.353
0.298
Pe
0.237
0.076
0.024
0.016
0.011
Computed Surface Energy for (100) Surfaces
rij
i
Pi
F
Mpi

0.979
0.139
-1.149
0.000
-0.273
0.556
0.054
-0.772
0.000
-0.193
0.311
0.017
-0.593
0.000
-0.193
0.251
0.011
-0.510
0.000
-0.194
0.212
0.008
-0.481
0.000
-0.183
k3
0.0001
0.0001
0.0001
0.0001
0.0001
Ei
-1.423
-0.966
-0.787
-0.705
-0.664
Es
269.7
133.1
83.5
72.6
59.4
Table 5:
M
s

e
Computed Surface Energy for (110) Surface.
(rij)
i Pi
Fr
M(pi)
2
Li
0.688176
0.83339
0.12893
1.14976
-1.2E05
0.20516
Na
0.390798
0.47322
0.04158
0.77276
-1.2E05
0.14526
0.91803
124.6613
K
R
b
0.218858
0.26503
0.01304
0.59374
-1.3E05
0.14516
0.73891
78.07222
0.51083
0.48177
-8.1E06
-1.2E05
0.65655
68.08885
0.149168
0.00841
0.00605
8
0.14572
Cs
0.21373
0.18063
3
-0.13734
-0.61912
55.54689
0.1765
Pe
0.1183
7
0.0381
1
0.0119
5
0.0077
8
0.0055
3
Table 6:
Metal
s
Li
Na
K
Rb
Cs
e
1.20430
9
0.68389
6
0.38300
1
0.30887
5
0.26104
4
Pe
0.20719
4
0.06681
6
0.02095
6
0.01362
9
0.00973
5
Ei
1.35493
Es
253.0751
Computed surface energy for (111) surfaces
i
0.90592
0.51444
9
0.28810
6
0.23234
6
0.19636
6
Pi
0.13420
1
0.04327
7
0.01357
3
0.00882
8
0.00630
5
Fr
1.14949
0.77248
0.59345
0.51063
Mpi
0.00021
0.00021
0.00022
0.00014
(rij)/2
Ei
Es
-0.23915
-1.38885
181.1221
-0.16931
-0.94201
89.25393
-0.16916
-0.76283
55.9216
-0.16985
-0.68063
48.73779
0.48152
0.00021
0.15999
0.64172
39.80333
4.
DISCUSSION
The results of our calculation of surface energies for the five alkali metals Li, Na, K, Rb, Cs are presented in this
session. The calculations have been done in Microsoft Excel – an electronic spread sheet and are limited to low index
surfaces. By default the calculations in Microsoft Excel are correct to the eighth decimal place and as much as
possible accumulation of floating point errors has been avoided.
www.ijsret.org
877
International Journal of Scientific Research Engineering & Technology (IJSRET), ISSN 2278 – 0882
Volume 3, Issue 5, August 2014
o
The input parameters for our calculation include lattice Constant (a(nm) and a( A )), cohesive energy (Ec(eV)),
mono-vacancy formation energy (E1f(ev)), and elastic constants C11(eVnm-3), C12(eVnm-3) and C44(eVnm-3) Their
values for the bcc alkali metals considered in his work are presented in table 2. Model parameters that are required for
the surface energy calculation are presented also in table 3. These parameters are calculated using equations 6 to 13.
Compiled values in the table are in good agreement with published results[13] expect for the values of  which
differs by about 1%.
As detailed out in session 2, the calculation for (100) surface is tabulated in table 4. For this case As = a 2 . In
a 3
the first plane, there are four nearest neighbours with r 
and four next nearest neighbours in the second plane
2
with r = a .
The result for (110) surface is presented in table 5. For this case, As a 2 2 . Only one plane was considered
with four nearest neighbours and two next nearest neighbours.Similarly, the result for (111) surface is shown in table
6. The results for (111) surface is strikingly lower by almost 60% than values obtained by Yan-Ni and Zhang[13] for
all the metals considered in this work.
5. CONCLUSION
The surface energies of the low index (100), (110) and (111) bcc alkali metals (Li,Na,K,Rb,Cs) in Group 1 of the
periodic table have been calculated using the Modified Analytical Embedded Atom Method (MAEAM).The result
shows that the surface energy (Es) of each (hkl) plane of Group 1 bcc alkali metals of the periodic table decreases
down the group. That is (EsLi  EsNa  EsK  EsRb  EsCs). It also shows that the differences between the surface
energies of metal of subsequent group of period one (1) decreases down the group for each hkl plane. That is (E sLiEsNa)  (EsNa-EsK)  (EsK-EsRb)>(EsRb-EsCs). The results obtained also shows that Es(111) ‹ Es(110) ‹ Es(100) for each
(hkl) plane. This is not completely in agreement with the order among the three low-index surface energies which
shows that Es(110) ‹ Es(100) ‹ Es(111) [13],[14],[9] and[5].
REFERENCES
1. Bajah S.T, Teibo S.O, Onwu G and Obikwere A (2002): Senior Secondary Chemistry 3.New edition Pp 61
2. Baskes M.I. (1987): Application of the Embedded Atom Methods to Covalent Materials: A semi empirical
potential for silicon, P. 2667.
3. Daw, M.S. and Baskes, M.I. (1983): Embedded-atom methods: Derivation and application to impurities,
surfaces and other defects in metals. Pp. 6443 – 6453.
4. Finis, M.W. and Sinclair, J.E. (1984): A Simple Empirical N-body potential for Transition Metals, Phil, Mag
A. Vol. 50, No. 1, Pp. 45 – 55.
5. Grenga H.E, R. Kumar (1976): Surf. Sci. 61, P 283.
6. Johnson, R.A. (1988): Alloy Models with the Embedded Atom Methods pp 1 – 3
7. Pasianot R, Farkas D and Savino E.J(1991): Empirical Many-Body Interatomic Potential For Bcc Transition
Metals. Phys. Rev. B 43 P6952.
8. Smith J.R, Banerjea. A. (1987): Phys. Rev. Lett. 59. P2451
9. Sundquist.B.E, (1964): Acta. Mtall.12. P67.
10. Wangyu Hu, Xiaolin Shu, B. Zhang. (2002): Point-defect properties in bcc transition metals with AEAM
interatomic potentials. Mat. Sci. vol. 23, issues 1-4, P. 175-189.
11. Yan, N.W. and Zhang, J.M. (2007): Surface Energy Calculation of the fcc Metals by using the modified
analytical embedded atom method.
12. Young F.W, Catheart J.V, Gwathmey A.T,(1956): Acta. metal. 4. P145.
13. Yan-Ni W, Jian-Min Zhang.(2008). Comp.mat. Sci. 42. P.282.
14. Zhang B.W, Hu W.F, Shu X.L (2003): Theory of EAM and its application to material science, Hunan
University press, Changsha .
15. Zhang B.W, Ouyan Y.F, Liao S.Z, Jin Z.P.(1999): Phys. Rev. B 262. P218
16. Zhang J.M, Wei X.M, and Xu K.W (2006): Co-Energy of Surface and grain boundry in Ag film. Appl. Surf.
Sci. 252. P. 4936.
17. Zhang B.W, Hu W.Y, Shu X.L, Huang B.Y. (1999): J. Alloy. Compd. 287. P159.
www.ijsret.org
878