December 9, 1968
TO: Art Robinson and Ian Keaveny
FROM: Linus Pauling
SUBJECT: Structure of Fluorine
Here is an analysis of the problem of the structure of fluorine.
He assume that the molecule, with internuclear distance 1.44 Å, can be treated as two interpenetrating spheres, each with van der Waals radius 1.6 Å, and that van der Waals attraction will bring each atom into contact with as many atoms of neighboring molecules as possible.
We start with a hexagonal packing of molecules in the basal plane, with their axes perpendicular to this plane. By tipping all of the molecules in the same direction the atoms of the upper layer can be brought into contact with atoms in the lower layer, while retaining contact with the atoms in the same layer. A tilted hexagonal layer of molecules is thus obtained (rather, a layer of tilted molecules). There are two ways of tilting: either toward one of the nearest atoms, or at 30° from this direction, between the atoms. I think that calculation of the contact distances will show that the first is the better.
The amount of displacement of the top layer relative to the bottom is 0.5 Å, corresponding to a tilt of about 20.3o . The tilt is so small that it is convenient to refer to the double layer in the usual way for hexagonal layers, by use of the letters A, B, and C.
The tilt can be expected to lead to a decreased symmetry of the double layer, so that axes B and A will no longer be in the ratio 1.732. I think that a prediction of the amount of change in axial ratio could be made by allowing this ratio to change, as well as the tilt, and making a least-squares treatment of the van der Waals contact distances. A change 3 percent, as in alpha fluorine, is not unreasonable.
The structure of the crystal would result from packing these double layers over one another, so that the bottom atoms of the upper double layer fit into the triangular positions of the top atoms of the lower double layer. If the lower layer is described by A, the upper layer can be described by B; the same structure would be obtained by putting the upper layer in C.
The deformed layers can be fitted together only if the tilt for the second layer is in the same direction as for the first layer, or in the opposite direction. Tilts at 30° or 60° to the rectangular axes of a layer would not occur, unless the interaction energy between layers were great enough to deform the layers in the way required to achieve the fit.
If the second layer is tilted in the same direction as the first layer, the third in the same direction as the second, and so on, a monoclinic structure is achieved with the B axis 5.5 Å long. This unit of structure does not agree with that observed for fluorine or that observed for alpha-oxygen.
If the second layer is tipped in the opposite direction (along the 3.28 Å axis) from the first layer, the third opposite to the second, and so on there are two possible structures. If the sequence of layers is ABAB...an orthorhombic structure is obtained. If the sequence of layers is ABCABC.a monoclinic structure is obtained, with monoclinic angle about 103°, A = 3.28, B = 5.50, and C = 7.34 Å. This structure seems to be the probable one for alpha fluorine.
The coordinates are predicted to have the values x = 0.190, y = 0.049, z = 0.095, relative to an origin at the center of a fluorine molecule. I think that the parameter values need to be changed to agree with the choice of origin in the International Tables, probably by adding 0.250 to two of the parameters (I do not have the Tables at hand).
I think that we could carry out an argument involving isotropy of polarizability to make it reasonable that layers with alternating positive and negative tips would interact more strongly than layers tipped in the same direction. I do not see any simple way to account for the greater stability of the monoclinic structure of this sort than of the orthorhombic one.
The unit cell of oxygen does not correspond to the foregoing argument. Instead, it involves a tip in the direction at 90° or 30° from that described above for fluorine. A tip of this sort leads to long strings (along the B axis) of oxygen molecules in parallel rectangular orientation. I think that the magnetic interactions of the triplet molecules might well be responsible for the assumption of this structure, which is, I judge, antiferromagnetic, probably with antiparallel spins alternating along these rectangular rods. The ratio A/B differs from 1.732 by 10 percent, three times as much as for fluorine. Near the end of their fluorine paper the authors say that a tilt of about 3° is consistent with the data that they give (Barret, Meyer, and Wasserman) in J. Cham. Phys. 47, 592 (1967). I think that a larger tilt would be expected for this sort of tilting than for fluorine, perhaps 16°, and it might be worth while to look at this oxygen paper, to see how good the reported intensities seem to be. Perhaps we should check up on the rhombohedral oxygen phase and the high-temperature phases of both oxygen and fluorine.
Linus Pauling
LP:jj
December
9, 1968
