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A simple chemical interpretation of the correlation between the Bloch-Simon structure index and the crystal structure of metals
Volume
60A, number
A
4
PHYSICS
LETTERS
I March 1977
SIMPLE CHEMICAL INTERPRETATION OF THE CORRELATION BETWEEN THE BLOCH-SIMON STRUCTURE INDEX AND THE CRYSTAL STRUCTURE OF METALS ES. MACHLIN Henry Krumb School of Mines, Columbia University, New York, New York 10027,
Received
27 December
In two recent publications [ 1,2] correlations were found between a structural index derived from application of Paul&force model potential eigenvalues to atomic spectral data and the stable crystal structure of s, p bonded elements as well as the c/a ratio of hcp s, p bonded div?lent etements. This structural index was first given a qualitative s = (‘I(0 - ro(0)/‘1(0 geometric interpretation [I]. In the later publication [2] this interpretation was changed by reinterpreting S in terms of the reciprocals of the rz, which in tern were interpreted to be a measu!epf the “orbital elecfronegativity”. The values rl = I (I + 1)/Z, where 1 - I is the quantum defect in the hydrogenic form of the radial SchrGdinger Fquation having eigenvalues given by E = -Z2/2(n t 1 - 1)2. Thus, rl is the radial maximum of the (unscreened) lowest valence eigenfunctions of the Paul&force model potential. The reason given for the change in interpretation was that the radii rl, at least for ions of high Z, are much too small to be compared directly with physically realistic bonding radii. I have discovered certain numerical correlations that suggest elementary chemical-geometric explanations for the c/a ratio of divalent hcp metals and the relative stability of hcp and fee s, p bonded metals. The first correlation is that c/a = r1 (f)/ro(ij.
1976
the? bon!ing
orbital radius approximates
closely to
rl(Ur&); (b) pz bonds preponderate between atoms in the [OOl] direction that are separated by the c lattice parameter; (c) s bonds preponderate between nearest-neighbor oriented atoms, in particular, along the [ 1001 direction. It appears that one valence electron is primarily associated with mainly s type bonding and the other electron with mainly p type bonding, the mainly s bonding being between nearest-neighbors and the mainly p bonding between the further neighbors lying in the c direction. Perhaps this is the basic significance behind the correlation, already previously noted by
Theoretical
(1)
This correlation is shown in fig. 1. The c/a values are experimental, except for Hg, which is obtained by Heine and Weaire [3] by linear’extrapolation of the c/a ratio for Cd-Hg hcp alloys. The r,(i) and ro(l> are calculated using the values of I evaluated by Simons and Bloch [4]. I suggest that this correlation may be interpreted as follows: (a) the ratio of the p bonding orbital radius to that of
USA
1.5
1.6
1.7 (I-sr’=
Fig. 1. c/a versus r,(Zj/ro(f) Hg obtained from ref. [ 31.
(Eq.
I)
I
I
I
I
1.8
1.9
2.0
2.1
r,dI/rot?,
for divalent
hcp metals.
Value for
333
Volume 60A, number 4
PHYSICS LETTERS
Engel and Brewer [5] and also found by St. John and Bloch [2], that the hcp structure tends to be stable at about 50% s character. The simple geometric-chemical bond concept suggested above implies the possibility that other structures may be stabilized at other values of r1 (i)/ro(i) satisfying the geometry of these structures. I have found that the fee structure can thus be interpreted as involving mainly s bonding between nearest-neigbor atoms and mainly p bonding between second neighbor atoms. This concept_ suggests that for s, p bonded fee metals the ratio rl(l)/ro(Z) should equal 42. The actual values for such metals range between 42 and 1.56. Above the latter value, the hcp structure is stable relative to the fee structure [ 1,2]. I believe that this result can be simply interpreted as follows: For ~1(I)/ro(l) values near 1/2 the bonds in the fee structure of s, p bonded metals are essentially unstrained, whereas those in the hcp structure are strained, because the corresponding c/a ratio deviates from the ideal ratio (1.633) and thus the nearest-neighbor s bonds in the latter structure not lying in the basal plane deviate in lengthAfromAthose lying in the basal plane. However, as rl (l)/rO(Z) values approach the value 1.633, the s and p bonds in the fee structure must be distorted whereas those in the hcp structure become unstrained. Given that the energies of the two structures are nearly the same then the changes in distortional energy described above can control the fcc/hcp relative stability. Thus, this interpretation is
Table 1 Stable structure
Element
r I (hlrdi)
fee
Ca Sr Al Pb
1.53 1.54 1.451 1.511
Be Mg Zn Cd Tl Li Na
1.587 1.659 1.895 1.851 1.681 2.142 2.318
hcp
334
7 March 1977
consistent with the fact that fee structures are stable relative to hcp structures below rI (l)/ro(i) Q 1.56 and vice versa for rl(@ro(l) > 1.56, for s, p bonded metals. See table 1. The bee s, p bonded metals do not seem to have their structural stability determined by such nonhybridized bonding concepts. Indeed, the fact that nearly all the known bee s, p bonded metals, with the exception of Ba, are singly valent suggests that only s or at most s, p hybridized “non-directional” bonds * operate in these bee structures, i.e. according to these concepts the bee s, p bonded metals should best obey the nearly free electron model. The correlations between structural parameters and the Simons’ orbital radii reported in this paper, the 100% successful separation of structures of elements [2], of AN B8-N compounds [2] and of more metallic compounds [6] using coordinates based on linear combinations of these radii leave little doubt that the cohesive energies of these phases depend solely on these radii. A first principle understanding of these results should prove fruitful. I should like to thank Dr. J.C. Phillips for having brought the work of Bloch and his coworkers to my attention. * The meaning of “nondirectional”
in the present context is that the symmetry of an sp bond (50% s character) is satisfied along any bond direction in the cubic structure because each atom is at a center of symmetry. Such “resonating” sp bonds then are “nondirectional”. It should be noted that pX, pp pz orbitals by symmetry considerations must lie along the cubic axes in the fee structure and similarly the pz orbital along the c axis of the hcp structure.
References [1] A.N. Bloch and G. Simons, J.A.C.S. 94 (1972) 8611. [2] J. St.John and A.N. Bloch, Phys. Rev. Lett. 33 (1974) 1095. [3] V. Heine and D. Weaire, Solid State Phys. 24 (1970) 392. [4] G. Simons and A.N. Bloch, Phys. Rev. B7 (1973) 2754. [S] N. Engel and L. Brewer, see: L. Brewer, in: Electronic structure and alloy chemistry of the transition elements, ed. P. Beck (Interscience, New York, 1963) p. 233. [6] ES. Machlin, T. Paul Chow and J.C. Phillips, submitted to Phys. Rev. Lett.
60A, number
A
4
PHYSICS
LETTERS
I March 1977
SIMPLE CHEMICAL INTERPRETATION OF THE CORRELATION BETWEEN THE BLOCH-SIMON STRUCTURE INDEX AND THE CRYSTAL STRUCTURE OF METALS ES. MACHLIN Henry Krumb School of Mines, Columbia University, New York, New York 10027,
Received
27 December
In two recent publications [ 1,2] correlations were found between a structural index derived from application of Paul&force model potential eigenvalues to atomic spectral data and the stable crystal structure of s, p bonded elements as well as the c/a ratio of hcp s, p bonded div?lent etements. This structural index was first given a qualitative s = (‘I(0 - ro(0)/‘1(0 geometric interpretation [I]. In the later publication [2] this interpretation was changed by reinterpreting S in terms of the reciprocals of the rz, which in tern were interpreted to be a measu!epf the “orbital elecfronegativity”. The values rl = I (I + 1)/Z, where 1 - I is the quantum defect in the hydrogenic form of the radial SchrGdinger Fquation having eigenvalues given by E = -Z2/2(n t 1 - 1)2. Thus, rl is the radial maximum of the (unscreened) lowest valence eigenfunctions of the Paul&force model potential. The reason given for the change in interpretation was that the radii rl, at least for ions of high Z, are much too small to be compared directly with physically realistic bonding radii. I have discovered certain numerical correlations that suggest elementary chemical-geometric explanations for the c/a ratio of divalent hcp metals and the relative stability of hcp and fee s, p bonded metals. The first correlation is that c/a = r1 (f)/ro(ij.
1976
the? bon!ing
orbital radius approximates
closely to
rl(Ur&); (b) pz bonds preponderate between atoms in the [OOl] direction that are separated by the c lattice parameter; (c) s bonds preponderate between nearest-neighbor oriented atoms, in particular, along the [ 1001 direction. It appears that one valence electron is primarily associated with mainly s type bonding and the other electron with mainly p type bonding, the mainly s bonding being between nearest-neighbors and the mainly p bonding between the further neighbors lying in the c direction. Perhaps this is the basic significance behind the correlation, already previously noted by
Theoretical
(1)
This correlation is shown in fig. 1. The c/a values are experimental, except for Hg, which is obtained by Heine and Weaire [3] by linear’extrapolation of the c/a ratio for Cd-Hg hcp alloys. The r,(i) and ro(l> are calculated using the values of I evaluated by Simons and Bloch [4]. I suggest that this correlation may be interpreted as follows: (a) the ratio of the p bonding orbital radius to that of
USA
1.5
1.6
1.7 (I-sr’=
Fig. 1. c/a versus r,(Zj/ro(f) Hg obtained from ref. [ 31.
(Eq.
I)
I
I
I
I
1.8
1.9
2.0
2.1
r,dI/rot?,
for divalent
hcp metals.
Value for
333
Volume 60A, number 4
PHYSICS LETTERS
Engel and Brewer [5] and also found by St. John and Bloch [2], that the hcp structure tends to be stable at about 50% s character. The simple geometric-chemical bond concept suggested above implies the possibility that other structures may be stabilized at other values of r1 (i)/ro(i) satisfying the geometry of these structures. I have found that the fee structure can thus be interpreted as involving mainly s bonding between nearest-neigbor atoms and mainly p bonding between second neighbor atoms. This concept_ suggests that for s, p bonded fee metals the ratio rl(l)/ro(Z) should equal 42. The actual values for such metals range between 42 and 1.56. Above the latter value, the hcp structure is stable relative to the fee structure [ 1,2]. I believe that this result can be simply interpreted as follows: For ~1(I)/ro(l) values near 1/2 the bonds in the fee structure of s, p bonded metals are essentially unstrained, whereas those in the hcp structure are strained, because the corresponding c/a ratio deviates from the ideal ratio (1.633) and thus the nearest-neighbor s bonds in the latter structure not lying in the basal plane deviate in lengthAfromAthose lying in the basal plane. However, as rl (l)/rO(Z) values approach the value 1.633, the s and p bonds in the fee structure must be distorted whereas those in the hcp structure become unstrained. Given that the energies of the two structures are nearly the same then the changes in distortional energy described above can control the fcc/hcp relative stability. Thus, this interpretation is
Table 1 Stable structure
Element
r I (hlrdi)
fee
Ca Sr Al Pb
1.53 1.54 1.451 1.511
Be Mg Zn Cd Tl Li Na
1.587 1.659 1.895 1.851 1.681 2.142 2.318
hcp
334
7 March 1977
consistent with the fact that fee structures are stable relative to hcp structures below rI (l)/ro(i) Q 1.56 and vice versa for rl(@ro(l) > 1.56, for s, p bonded metals. See table 1. The bee s, p bonded metals do not seem to have their structural stability determined by such nonhybridized bonding concepts. Indeed, the fact that nearly all the known bee s, p bonded metals, with the exception of Ba, are singly valent suggests that only s or at most s, p hybridized “non-directional” bonds * operate in these bee structures, i.e. according to these concepts the bee s, p bonded metals should best obey the nearly free electron model. The correlations between structural parameters and the Simons’ orbital radii reported in this paper, the 100% successful separation of structures of elements [2], of AN B8-N compounds [2] and of more metallic compounds [6] using coordinates based on linear combinations of these radii leave little doubt that the cohesive energies of these phases depend solely on these radii. A first principle understanding of these results should prove fruitful. I should like to thank Dr. J.C. Phillips for having brought the work of Bloch and his coworkers to my attention. * The meaning of “nondirectional”
in the present context is that the symmetry of an sp bond (50% s character) is satisfied along any bond direction in the cubic structure because each atom is at a center of symmetry. Such “resonating” sp bonds then are “nondirectional”. It should be noted that pX, pp pz orbitals by symmetry considerations must lie along the cubic axes in the fee structure and similarly the pz orbital along the c axis of the hcp structure.
References [1] A.N. Bloch and G. Simons, J.A.C.S. 94 (1972) 8611. [2] J. St.John and A.N. Bloch, Phys. Rev. Lett. 33 (1974) 1095. [3] V. Heine and D. Weaire, Solid State Phys. 24 (1970) 392. [4] G. Simons and A.N. Bloch, Phys. Rev. B7 (1973) 2754. [S] N. Engel and L. Brewer, see: L. Brewer, in: Electronic structure and alloy chemistry of the transition elements, ed. P. Beck (Interscience, New York, 1963) p. 233. [6] ES. Machlin, T. Paul Chow and J.C. Phillips, submitted to Phys. Rev. Lett.