175. The Nature o f the Metallic Bond. II

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1 No. 7] The Nature o f the Metallic Bond. II The Transformation in Metallic Crystals By San-ichiro MIZUSHIMA, M.J.A, and Isao ICHISHIMA Tokyo Research Institute, Yawata Iron and Steel Co., Ltd., Ida, Kawasaki-City (Comm. Sept. 12, 1966) I. Covalent and Metallic Bonds. In this note an explanation of the data summarized in Part I1) will be explained. We would like to call attention to the point that our explanation has a significance as a whole; therefore, if a part of it will be picked up without any consideration of the remaining part, it will make no sense. Alkali metals may be called typical metals and the treatment as hydrogen-like atoms enables us to explain their properties fairly well, but hydrogen itself under the known experimental conditions does not exist in metallic state. As explained in Part I, almost all the cohesive energy is concentrated in atomic pair of nitrogen which is typically nonmetallic as hydrogen. We would like to note that typically nonmetallic substances contain atomic orbitals of is and 2p, and most characteristic one is found with half shell. From the wave functions we can calculate radial dependence of the probability functions /rr2`i/r* or the density of electron cloud and find a narrow peak in radial dependence. The number of nodes in the radial probability function is equal to (n - t --1), where n and l are, respectively, principal and azimuthal quantum numbers. Therefore, as we proceed to 2s, 3s,., 3p, 4p,.., the orbitals tend to spread over the crystal. Since half shell forms a stable electron configuration, it is not easy for it to release an electron or to accept one. However, as it has the maximum number of free spin electrons, these will form covalent bonds in which valence electrons are localized and cannot spread over the crystal to form metallic bonds. This tendency is most pronounced in is and 2p states. As n becomes larger, the number of nodes of the wave function increases, so that probability functions of the electrons tend to spread and overlap each other. Furthermore, even in half shell the Coulomb energy necessary for adding one more electron is fairly low. The same is the case for detaching one electron from the half shell, since the energy is roughly proportional to 1/n2. Thus valence electrons of atoms with high principal quantum

2 790 S. MIZUSHIMA and I. ICHISHIMA [Vol. 42, number n more easily move from one atom to another than those of low n. This is the atomic orbital approach for understanding on the increasing metallic nature which is stabilized by resonating among many valence bond structures. Since 3d wave function also shows a narrow peak as 1s and 2p functions, we may explain why a Mn with 3d half shell shows partial nonmetallic property and atoms with larger n (4d and 5d) form crystals with more metallic nature. II. Transformation in Iron Crystal, we have described the crystal structure of transition elements which show a certain regular behavior in the periodic table, with some exceptions such as Cr, Mn, and Fe. Let us discuss more in detail on this matter, with the structure of iron as the central figure. As explained before, Mn with stable 3d half shell has partially covalent character and, therefore, it will naturally follow that in the neighboring two elements, Cr and Fe, 3d-electrons will form bonds of partially directional character. For this reason we have considered the b.c.c, structure of a Fe and Cr is different in nature from apparently the same structure of alkali metals and Fe. The d-orbital energy level is five-fold degenerate, and the corresponding splitting of energy levels in crystal fields is observed in coordination compounds. Goodenough2~ explained the b.c.c, structure of a Fe by the combination of triply-degenerate orbitals ds (extended body-diagonally) which overlaps the same of the neighboring atom. The remaining d-orbitals d7 (doubly degenerate and extended axially) faces toward the next nearest neighbor atom and the exchange of electrons makes all the spins in the magnetic domain parallel (ferromagnetism) as first shown by Heisenberg. Thus, so far as the partial covalent character is concerned, the b.c.c, structure of a Fe and Cr belongs to the same category as the complicated structure of a Mn and actinides. As the temperature is raised, the interatomic distances in the crystal increase and the overlapping of electron clouds become less pronounced. Therefore, there will be a temperature above which the close packed structure is stable, since the directional bonds keeping this form can no more resist the thermal motion. An apt example of what has been explained above is afforded by the transformation in iron crystal. Roughly speaking the crystal of a Fe is formed by nondirectional metallic s-bonds and directional d- bonds (details will be discussed in our paper to be published). At the transition point the thermal motion dissociates the d-bonds in the sea of conduction electrons and make them join nondirectional bonds to form the f.c.c, structure of 'y Fe. At still higher temperature

3 No. 7] Nature of Metallic Bond. II 791 yfe is transformed to Fe, the b.c.e. structure of which is the same one as many metallic crystals take at higher temperature. The measure of the stability of a state is the free energy F which is related to enthalpy H and entropy S as: F = H -- TS. We consider the stability of the high temperature b.c.c. structure is due to the fact that this has more space for atoms to move than the f.c.c, structure: in other words the randomness is more pronounced to increase the entropy. The increase of entropy term in the high temperature b.c.c, structure overcomes enthalpy, resulting in the decrease of free energy to stabilize Fe. Alkali metals have the b, c, c, structure even at room temperature. As above mentioned, this is considered to correspond to the high temperature form which many metals show. Then it should follow that the crystal structure of alkali metals is more easily affected by thermal motion as can be seen from high compressibility and large thermal expansion coefficient. In other words the entropy term is large enough to keep them in the b.c.c, structure even at lower temperature. However, if the temperature is lowered sufficiently, the entropy term will become small enough so that the structure will become more closely packed. The b.c.c, structure of Li, for instance, changes to the h.c.p, and then by cold-working to the f.c.c, structure as the temperature is lowered. However, such is not the case for heavier alkali metals with large quantum number n, as the compressibility is too large to give rise to such structure change with temperature. Now let us return to the problem on the transformation in iron crystal. Notwithstanding the discontinuity shown at the transition points, nearly all the curves of the properties of Fe as functions of temperature seem to be the extensions of those of a Fe. From this we may consider 3d-orbitals of partial directional character is restored to a small extent, contributing to enthalpy, because the b, c, c, form is favorable for de-orbitals. However, there is no doubt that the entropy term plays the most important part in Fe as mentioned before. In Cu, Ag, and Au with closed d-shells there is no directional bonds as in Fe and furthermore the entropy term is small because of the stronger interatomic force. This will be the reason why the noble metals mentioned above, keeps the f, c, c, structure up to the melting point. III. The Crystal Structure of Transition Metals. Let us next discuss the crystal structure of other transition elements, V, Nb, Ta, and Mo, W with high cohesive energy, high hardness and high melt-

4 792 S. MIzusHIMA and I. ICHISHIMA [Vol. 42, ing point. All of them are in the b, c, c, structure, but in this case the enthalpy H plays an important role in contradistinction to the b.c.c, structure of alkali metals in which the entropy term TS is much more important than H as explained before. If the enthalpy has sufficiently a large value in the b.c.c, structure of the transition metals V, etc, mentioned above, (in other words, if d-bonds are very strong), we will not be able to see the structure change b.c.c.-~f.c.c., as in the case of Fe. Actually the transition metals (Va, VIa except Cr) do not show transformation. We have also mentioned that d-orbitals with high principal quantum number n, spread and overlap so much that the binding is not much affected by thermal motion. Therefore, such bonds are not the same as those formed by 3d-orbitals in a Fe. Let us next discuss the h.c.p, structure which is one of the close packed structures with the axial ratio c/a = However, except for Li, Na (low temperature form), Mg, Ca, and Sr, the actually observed ratio is different from this normal value, so that we have to consider the structure is deformed to some extent. There is almost no difference in energy between the h.c.p, and f.c.c, structures, but generally speaking the f.c.c, structure is more stable as we see from the crystal structure of rare gas elements, In He we observe the f, c, c, structure at low temperature under high pressure. However as the pressure is lowered, we observe a structure change: f.c.c.-->h.c.p.--b.c.c. This is due to the fact that the helium atom is very small and interatomic force is weak so that the entropy term plays an important part. We have seen the same structure change in Li which has also very small core. Thus we can understand that in both of He and Li the interatomic force is surely nondirectional, although He is nonmetallic and Li metallic. The same structure change observed for Ca, and Sr can be explained similarly. It is to be noted that Li, Ca, and Sr mentioned above have the hexagonal close packed structure with normal value of c/a, showing the nondirectional intercore forces. As to the deviation from the normal value of 1.633, the higher one (c/a= 1.8) is observed for Zn and Cd. This shows that innerlayer force is stronger than inter-layer one. The values of the axial ratio lower than c/a =1.633 are observed for nearly all the remaining elements in this structure, and, therefore, inter-layer force is stronger than inner-layer force for these crystals. Thus we may consider that the h.c.p, structure is a closed packed structure with a small contribution of directional bonds. Elements in IIIa and IVa groups have, respectively, only one

5 No, 7] Nature of Metallic Bond. II 793 and two d-electrons as compared with elements in Va and VIa groups having sufficient number of d-electrons to form the b, c, c, structure. In the h,c,p, structure, two kinds of close packed atomic layers A and B make the sequence ABAB, each layer being a plane of symmetry. Let us consider this structure in relation to d-orbitals. From the angular dependence of 3d wave functions we see that the electron cloud formed by dz2 spreads more than those of others. This may contribute in a way to the binding of inter-layer atoms. This will be the reason why nearly one half of the transition metals are in the stable h.c.p, structure. Along this line of d-bonding, we can conclude that c/a is less than the normal value for the crystal consisting of atoms with one 3d-electron (IIIa) and approaches to the normal value for those with two 3d-electrons (IVa). This is in agreement with the experimental results. We also see why the lanthanide elements are in general in h, c, p, with c/a less than the normal value. Similarly we can explain the transformation h,c,p.--~f.c,c, in Sc (IIIa group), Due to the thermal motion at higher temperature, the contribution of d2-orbital will be reduced and, therefore, the f, c, c, structure will become more stable. For IVa elements with two d- electrons which has a tendency of forming the b,c,c, structure will change from h.c,p, to b.c.c. This structure is due to the effect of the entropy term as in Fe in which, however, a small contribution of directional d-orbital may be considered. In VIIa elements we have more d-electrons, but because of the higher nuclear charge, the d-orbitals will be contracted. Therefore, the main contribution to binding will be made by dz2. This tendency is more pronounced in VIII1 elements. This will explain why c/a is almost equal to the normal value in VIIa elements and it is less than this in VIII1 elements. For VIII2 and VIII3 elements in which d- orbitals are more contracted, we can understand why these elements are in the f, c, c, structure. In addition to the orbital contraction we have to take into account the decreasing free spin with increasing atomic number. As explained in Part P~ the cohesive energy shows its maximum value at about VIa-group except for the first long period and the crystal structure changes regularly as h.c.p.-~b.c.c.--~h.c.p,--~f.c.c. from IIIa to VIII3, Ib. The corresponding changes of cohesive energy and crystal structure can be explained by the contribution of d-bonds which changes regularly with atomic number. 3d-orbital has a narrow peak in radial dependence and, therefore, 3d-bonds show directional character to a considerable extent. For this reason the

6 794 S. MIzusHIMA and I. ICHISHJMA [Vol. 42, structure of a Fe belonging to VIII1 group becomes b.c.c, and that of Co belonging to VIII2 group is usually a mixture of f.c.c, and h.c.p. We will try to understand why except for the first long period, the cohesive energy regularly changes with the number of valence electrons irrespective of the nature of electrons. References [1] [2] S. Mizushima and J. B. Goodenough: (1963). I. Ichishima: Proc. Japan Acad., Magnetism and the Chemical Bond. 42, (1966). Interscience Publishers