In contrast, the treatment of the hcp structure is much less thorough, even in advanced level texts.  In fact, none of the more than 20 general, inorganic and physical chemistry textbooks we surveyed contained a complete and accurate illustration or description of the hcp unit cell analogous to those provided for the cubic structures. It seems improbable to us that the details of the hcp unit cell have not been worked out at the same level of geometric rigor as those of the cubic structures, but we have been unable to find a source, either printed or online, where this is done.  If the hcp unit cell has been treated rigorously, the results certainly have not found their way into modern texts.  The purpose of this paper is to provide such a description, along with useful illustrations, animations, and templates for 3D paper and glue models of the type developed by Birk and Yezierski (13). [Top][Back]

What’s Missing in Textbook Treatments of the hcp Unit Cell?

In every treatment that we examined, the hcp structure is described for atomic or metallic solids in terms of the abab… stacking of close-packed layers of spheres and the coordination number of 12 is readily illustrated.  However, there is less agreement on the preferred unit cell for the structure with some sources (1-4,8) presenting a hexagonal prismatic structure and others a rhombic prism (5-7,14,16-19).  In these unit cell descriptions, the hexagonal and rhombic faces coincide with close-packed layers that are separated by one intervening layer (Figure 1).  If we adopt the common definition of the unit cell that calls for the smallest repeating geometrical unit (3), then the rhombic prism is the preferred choice.  For a given atomic radius, the dimensions of the unit cell, as well as its shape and volume, are fixed, but the position of the boundaries with respect to atomic centers is arbitrary (15).  However, it is customary to place unit cell vertices at atomic centers. The illustrations found in textbooks and other traditional and online sources are similar to those in Figure 1, with or without the unit cell boundaries.  Such illustrations are incomplete because they neglect the contributions of atoms with centers outside of the unit cell that extend into the unit cell. The illustration that comes closest to the mark is found in John Winn’s physical chemistry textbook (4), but it doesn’t distinguish between atoms projecting into versus those projecting out from the unit cell. Nor does the author determine the atomic fractions involved.

(a)	(b)

Figure 1.  The hcp unit cell as commonly portrayed in textbooks with either hexagonal prismatic (a), or rhombic prismatic (b) structure.

The number of atoms contained in the hcp rhombic unit cell is 2, though we have not found any textbook or reference that clearly defines the contributions that sum to this number.  The subtlety lies in the middle, intervening, layer between the two rhombic faces.  The 4 atoms, whose centers define the vertices of the bottom rhombic face, form two triangular holes.  Due to the close proximity of these two holes, only one can be occupied by an atom of the second (middle) layer.  However, this atom is not wholly contained in the unit cell, since it projects through two faces of the unit cell into two adjacent unit cells (Figure 2).  The fractions of the central middle layer atom that are sliced off by the unit cell boundaries are exactly matched by fractions of two other middle layer atoms with centers in two adjacent unit cells that project into the first unit cell in the direction of the unoccupied hole.  Thus, the middle layer contributes exactly one atom as the sum of three separate fragments, and this, along with the half-atom contributed by each of the top and bottom layers constitute the two atoms contained in the unit cell.

Figure 2.  Top down view of hcp unit cell showing projection of middle layer atomic spheres (dashed circles) through the four rectangular faces of the unit cell.  The solid circles representing the top and bottom layers coincide. [Top][Back]

Calculation of Middle Layer Fractions

The fractional volume of the central atom that is sliced off by the unit cell boundary can be determined using the solids of revolution technique.  When a region in a plane is revolved about a line in the plane, a solid of revolution is generated.  By revolving the region bounded by the x-axis and graph of a circle whose center is at the origin about the x-axis we obtain a solid sphere.  Briefly explained, we partition the region under the curve, , where r is the radius of the sphere, into rectangles as shown in Figure 3a and revolve the result about the x-axis.  Many small discs are formed (Figure 3b).  Consider a particular disc, call it the ith disc, and let its center be distance p_i from the origin.  The base radius is and the altitude, or “thickness”, is .  Such a disc has volume .   The sum of the volumes of all such discs over a given interval will produce the volume of a solid and can be expressed as a Riemann sum .  As we let the “norm of the partition” approach zero (i.e. take the limit as ) the volume of the solid of revolution can be stated as  over a given interval [a,b]. 

Figure 3.  (a) The area bounded by the graph of a circle with radius r and the x-axis can be partitioned into rectangles centered at a distance p_i from the origin and having width and length.  (b) Rotation of the rectangles about the x-axis produces disks of thickness and radius. [Top][Back]

Our interest is with the volume of the middle layer atomic sphere that extends out of the unit cell.  To determine the interval [a,b] over which to integrate, we will consider a two-dimensional slice of the unit cell parallel to its base and passing through the center of the middle layer sphere.  This cross section is a rhombus that can be cut in half forming two equilateral triangles centered over the holes in the bottom layer (Figure 4).  The triangle whose center coincides with the center of the middle layer sphere has sides of length, 2r.  The distance from the center of the sphere to the edge of the triangle, , can be determined by considering the right triangle ABC formed by two altitudes of the equilateral triangle.  This 30°–60°–90° right triangle then gives us the information we need to find .   Using right triangle trigonometry,   or   .  Since   ,  .

Figure 4.  A rhombic cross section of the unit cell taken through the center of the middle layer sphere (shaded) shows the geometric relationship needed to obtain the integration interval to determine the fraction of the sphere volume extending through each of two adjacent faces of the unit cell. Circles (dashed) representing the top and bottom layer spheres are also shown for reference.

With the center of the middle layer sphere placed at the origin, we can determine the volume of the portion of the sphere that extends out of the unit cell by integrating over the interval .

Finally we can look at the ratio of the volume extending out of the unit cell to the total volume of the sphere.

Thus 11.51% of the total volume of the middle layer sphere extends out of one side of the parallelepiped.  Another 11.51% of this sphere extends out of an adjacent face of the unit cell, leaving 76.98% enclosed within the unit cell boundary.  The excluded portions are precisely matched by portions of two other middle layer spheres extending from adjacent unit cells into the first unit cell toward the center of the equilateral triangle centered over the unoccupied hole in the bottom layer (see Figure 2).

We have constructed a ‘cutaway’ 3-dimensional computer model of the hcp rhombic parallelepiped unit cell analogous to the illustrations of the cubic structures commonly included in textbooks (Figure 5a).  It shows the appropriate fractions of the three atoms that contribute to the sum of one middle layer atom.  An animated rotation of this 3-D model is helpful in distinguishing the middle layer atom that projects out of the unit cell from the two that project into it.  In addition, we have developed a ‘cutaway’ model and animation for the hexagonal definition of the hcp unit cell (Figure 5b). [Top][Back]

a

b

Figure 5.  Animated computer-generated ‘cutaway’ models of the complete hcp unit cell according to rhombic prismatic (a) and hexagonal prismatic (b) descriptions.

It seems intuitive that the packing efficiency as defined by the fraction of the unit cell that is occupied by the atomic spheres is the same for the hcp and ccp structures, i.e. 0.7405.  Textbooks, even those for advanced courses, routinely calculate this quantity for the three common cubic structures, and use the intuition argument for the hcp case.  The calculation of the packing efficiency for the hcp structure is only marginally more challenging than for its cubic counterpart and can be found in an earlier edition of Atkins (6) and also the inorganic text by Rodgers (2).  It may also seem intuitive that the two closest packed structures should have the same energy since they have the same number of nearest neighbors at the same distance, 2r, and the same number of next-nearest neighbors at a distance of r.  However, when more distant neighbors are considered, the two structures begin to differ in both the number of neighbors and the distances.  If pair-wise interactions are summed for the hcp and ccp structures, the ideal ccp geometry is about 0.01% more stable than the hcp (4).  This observation leads to the deeper and more challenging question of how nature chooses the particular crystal structure adopted by each atomic solid. [Top][Back]

Conclusions

The hcp structure is one of the most common and important crystal structures adopted by metals and other atomic solids. Both the hcp and ccp structures were worked out on paper and proposed for metallic solids by William Barlow, then curator of the London Science Museum, in 1883, long before experimental methods existed to confirm his predictions (19). These structures are now routinely described in chemistry textbooks; however, the hcp structure has not been treated at the same level of geometric rigor as the common cubic structures. We have shown how the fractions of atomic spheres projecting out from and into the rhombic prism unit cell may be calculated using the solids of revolution technique, and have prepared animated 3D ‘cutaway’ computer models and templates for paper and glue models to more accurately illustrate the hcp unit cell. We encourage textbook authors to consider updating their treatment of the hcp structure by accurately describing the complete unit cell. [Top][Back]

Supplementary Material

Computer animations of rotating 3D ‘cutaway’ models of the hcp unit cell according to both common (hexagonal and rhombic) descriptions are available. Paper templates for the construction of paper and glue models are also available.

Literature Cited

1. Shriver, D. F.; Atkins, P.W. Inorganic Chemistry, 3^rd Ed.; Freeman: New York, 1999.
2. Rodgers, G. E. Introduction to Coordination, Solid State, and Descriptive Inorganic Chemistry; McGraw-Hill: New York, 1994.
3. Housecroft, C. E.; Sharpe, A. G. Inorganic Chemistry; Prentice Hall: Upper Saddle River, NJ,, 2001.
4. Winn, J. S. Physical Chemistry; Harper: New York, 1995.
5. Meissler, G. L.; Tarr, D. A. Inorganic Chemistry, 2^nd Ed.; Prentice Hall: Upper Saddle River, NJ, 1999.
6. Atkins, P. W. Physical Chemistry, 5^th Ed.; Freeman: New York, 1994.
7. Levine, I. N. Physical Chemistry, 4^th Ed.; McGraw-Hill: New York, 1995.
8. Ball, D. W. Physical Chemistry; Brooks/Cole: Pacific Grove, CA, 2003.
9. Moore, J. W.; Stanitski, C. L.; Jurs, P. C. Chemistry, The Molecular Science, 2^nd Ed., Brooks/Cole: Pacific Grove, CA, 2003.
10. Ebbing, D. D.; Gammon, S. D. General Chemistry, 7^th Ed.; Houghton Mifflin: New York, 2002.
11. Olmsted, J.; Williams, G. M. Chemistry, 3^rd Ed.; Willey: New York, 2002.
12. Brown, T. L.; LeMay, H. E.; Bursten, B. E.; Burdge, J. R. Chemistry, the Central Science, 9^th Ed.; Prentice Hall: Upper Saddle River, NJ, 2003.
13. Birk, J. P.; Yezierski, E. J. J. Chem. Educ. 2003, 80, 157-159.
14. Robinson, W. R.; Tejchma, J. F. JCE Software; 1998, Vol. 5D, No. 2.
15. Suchow, L. J. Chem. Educ. 1976, 53, 226-227 [PDF].
16. West, A.R. Solid State Chemistry and its Applications; Wiley: New York, 1984; Chapter 7.
17. Borchardt-Ott, W. Crystallography; Springer: Berlin, 1993; Chapter 11.
18. Evans, R. C. An Introduction to Crystal Chemistry; Cambridge: London, 1946: Chapter 4.
19. Moore, W. J. Seven Solid States, An Introduction to the Chemistry and Physics of Solids; Benjamin: New York, 1967: Chapter 2. [Top][Back]