Crystals from balls

44

LABORATORY “QUANTA”

CRYSTALS FROM BALLS

G. I. KOSOUROV

The first step to understanding, explaining and predicting the properties of a crystal is to determine its structure. Knowing the configuration of atoms in a crystal lattice and understanding the symmetry of their arrangement, it is possible, for example, to say in advance whether the crystal will be piezoelective, that is, whether an electrical voltage will appear on its facets during mechanical rolling; will the crystal have a ferroelective transition, which will occur when the temperature is determined and will result in an electric field arising inside the crystal; will a light wave of a double particle appear in a substance if laser light is shined through it, etc. The structure of a crystal is a passport that can tell a lot about its owner.

You can get acquainted with the different “packing” of atoms in a crystal

45

with the help of simple aids. Using metal balls from bearings, we can build models of crystals using the same principles that nature uses to build crystals. It is precisely this practical acquaintance with some forms of crystal lattices that this article is devoted to. Not a single crystallographer neglects the construction of spatial models in which all the features of the arrangement of atoms in complex structures are clearly visible. But before we start hoofing, it is necessary to make a few theoretical notes.

The existence of the crystal lattice is due to the fusion of interactions between atoms. At short distances, repulsive forces predominate, which quickly increase when trying to lick atoms. At large distances, attractive forces that decrease relatively slowly with distance predominate. When atoms merge under the influence of attractive forces, the potential energy of interaction kills in the same way as the potential energy of a stone falling to the ground kills. At a distance at which the forces of attraction and repulsion become equal, the potential energy has a minimum, after which it increases due to the work of attraction. The dependence of potential energy on distance looks like the example shown in figure 1. In a state of equilibrium, atoms will occupy the place corresponding to the minimum potential energy. When there are many atoms, this will ultimately lead to the periodic repetition of some configuration that is most favorable in terms of energy for a small group of atoms, forming the so-called unit cell.

There are substances with very complex structures, such as some slicates, the unit cell of which contains more than two hundred atoms. Other substances, for example, many metals, form a crystal lattice according to a very simple law. Naturally, we will start with the simplest formations. In our experiments, the role of atoms will be played by metal balls, repulsion will be controlled by the forces arising when the balls come into contact, and the force of attraction will be replaced by gravity.

We stretch it onto the corresponding round can, box or piece of pipe

46

thin rubber film (for example, from a surgical glove) and secure it with a rubber ring. Let’s put two balls on the film. They will slightly bend the film and “attract” to each other. Their potential energy, depending on the distance, changes approximately as shown in figure 3, which is very similar to graph figure 1. By placing about thirty balls on the film and shaking the box slightly, we will see that the balls will be arranged in regular rows. The centers of the balls will lie at the vertices of equilateral triangles on a side equal to the diameter of the ball, and the balls themselves will fill the entire plane and form a network, which is called hexagonal *). Each ball is surrounded by six Sharimi touching it and each other. Their centers form regular hexagons. (Fig. 2).

If you turn your entire body around an axis passing through the center of any ball by one sixth of a revolution, then one ball will take the place of the others, and the general location of the system in space will remain unchanged. The network of balls will turn into itself. After six such turns, each ball will return to its original place. The crystallographer will say in this case that the symmetry axis of the sixth order, perpendicular to the plane of the balls, passes through the center of each ball. It is because of these axes that the network is called hexagonal. In addition to sixth-order symmetries, there are also third-order axes passing through the centers of the holes between the Sharimi. (The axis of symmetry of the third order is such a straight line, when turning around - each time at an angle of 120° - we return to the original picture. The body is incorrectly formed by the axis of symmetry of the first order, that is, it turns into itself only after a full rotation. For example, through the center of a circle perpendicular to its plane passes

*) From the Greek “hexa” - six and “gonia” - angle.

47

In one of these systems of holes the balls of the second layer will fall, forming a hexagonal network of collapsing balls, similar to the first. However, the attractive forces due to the elasticity of the film may not be enough to hold the balls of the second, much less the third layer. Therefore, knowing how the balls are placed in the bottom layer, we will make a tray from plywood in the shape of a regular triangle (Fig. 5), such that an integer number of balls are tightly placed along each side (in our model there are seven), and fill it with the balls of the first layer.

It makes no difference which system of holes the balls of the second layer are placed into, but for the third layer the systems of holes turn out to be nonequivalent: the centers of one system are located above the centers of the balls of the first layer, and the second system is located above the empty holes of the first layer. Let’s start by placing the balls in the holes lying above the balls of the first layer. In this case, the third layer will exactly repeat the arrangement of the balls of the first, the fourth - the second, etc. The elephant will repeat after one. We will get a not very stable paramid (Fig. 6), which, however, is only due to the fact that in our model the force of “attraction” acts only downward and the balls lying in the outer holes are easily squeezed out by the balls in the upper row.

This arrangement of balls is called the closest hexagonal packing. This is how beryllium, magnesium, cadmium, and helium are crystallized at low temperatures and pressures of more than five atmospheres. It has only one system of parallel densely packed words. Perpendicular to this system, the axis of a symmetrical triangle passes through the center of any ball. The decrease in symmetry is due to the fact that if it passes through the centers of the balls, for example, of four words, being an axis of the sixth order for them, then for odd words it will pass through the centers of the holes, and the order of their symmetrical relative to this axis will be only the third. However, the packing is called hexagonal because it can be considered as two hexagonal lattices of separate four and odd words, pushed into one another with drainage. Please also note that the empty holes in all layers are located one above the other and that channels pass through the entire hexagonal structure into which to insert rods with a diameter of 0.155 times the diameter of the ball. The centers of these channels are also the symmetry axes of the triangle. In figure 6 the hexagonal structure model is shown with inserted rods.

Now we will place the balls of the triangular layer into the holes lying above the free holes of the first layer. Depending on which system of holes we put the balls of the second layer in, we can get two paramids (Fig. 7 and 8). The first paramid is limited by regular triangles with hexagonal arrangement of balls in the planes of the faces, which are no different from the first layer of our arrangement, which lies at the base of the paramid. In other words, our paramid is one of the fifth regular polyhedra - a tetrahedron. The resulting packing contains four tightly packed words, the normals to which coincide with the symmetry axes of the tetrahedron triangle passing through its vertices. In this installation, the rows are repeated every two to three.

48

The lateral faces of the second paramid are equilateral right triangles, and the paramid itself is a part of the cube, cut off by a plane passing through the diagonals of the faces having a common vertex (Fig. 9). The laying of balls on the side faces of the paramid will occur on a square grid with rows parallel to the diagonals of the cube face.

It is easy to discover that we did not receive two different packages, but one, only with a different orientation. It is enough to find and remove the balls located on the edges of the triangle, and the faces of the cube will be exposed, just as by removing the balls lying along the edges of the cube, we gradually turn the cube into a triangle. This packing is called the densest cubic packing. So the crystallization of neon, argon, copper, gold, platinum, lead. The densest cubic packing has all the symmetrical elements of a cube. In particular, the third-order symmetry axes of the tetrahedron coincide with the spatial diagonals of the cube, which are also third-order symmetry axes for it. The construction of this package is based on an elementary cube of fourteen balls. Eight of them are located at the tops of the cube and six at the centers of its faces. If you carefully examine the second paramid (Fig. 8), this cube can be found at the top of the paramid. The densest cubic packing can also be considered as a set of four simple cubic lattices, shifted one into the other. With this consideration, the equality of all the balls in the pack becomes especially clear. By the very method of obtaining hexagonal and cubic dense packings by superimposing hexagonal layers, it is very clear that both packs, despite different symmetries, have the same density, or, as they say, the same filling factor.

If we make a square tray and place the balls on a square grid, we will also get a tight package. Although the balls in each layer are not tightly packed, the wells between the balls will be deeper, and therefore the layers will be more closely packed than in a hexagonal structure. Having finished laying, we get a tetrahedral paramid (Fig. 10), the side faces of which are

49

are equilateral triangles with hexagonal arrangement of balls. If we mentally add the same paramide, but with the vertex oriented downwards, we will get the third regular polyhedron after the terrahedra and the cube - an octahedron with eight faces. It is not difficult to prove that we have again obtained the densest cubic packing, only now the faces of the cube are parallel to the plane of the base. Remove the balls and the souls along the edges, and you will find on the upper section of the paramid five balls, forming the face of an elementary cube.

A number of physical experiments can be performed with the constructed models.

By rubbing a rubber film against each other, you can simulate the thermal movement of atoms. (You see how the correct placement of the balls is destroyed with the “increase in temperature.”)

Since one hexagonal layer fits into the relatively small holes of another, the layer turns out to be weakly connected, and slipping can easily occur in them. Try moving one hexagonal layer over another and you will see that there are three easy sliding directions in which the layer moves as a whole. The same thing happens in crystals. Sliding in these three directions is explained by the peculiarity of plastic deformation of crystals.

Models can be built from any balls. If it is not possible to remove the balls from the bearings, then you can use large beads or, in extreme cases, fish or small apples. Figures 11 and 12 show the unit cells of cubic and hexagonal packings glued together from real tennis balls. We recommend this comparative and convenient material for making a model, especially for a school physics classroom.

Arrays of collapsible spheres, both dense and otherwise, play a very important role in crystallography, and we will have occasion to talk about them later. In the meantime, stock up on marbles and build models!