Wave Motion in Time and Space

A wave is a disturbance that travels from one location to another, and is described by a wave function that is a function of both space and time. If the wave function was sine function then the wave would be exressed by

where A is the amplitude of the wave, w is the angular frequency of the wave and k is the wave number. The negative sign is used for a wave traveling in the positive x direction and the positive sign is used for a wave traveling in the negative x direction.

The tricky part of understanding wave motion is recognizing the differences between the time and space behavior. Hopefully the animation at left can help. The top part of the animation shows a sine wave pulse traveling from left to right. A red dot has been identified to highlight the time behavior of a point at a specific location.

The graph at lower left shows the time history of the displacement of this red dot as the wave passes by. This graph represents the wave as a function of time for a specific location. Nothing happens for the first few seconds until the leading edge of the wave reaches the location of the red dot. Once the wave reaches that location, the position oscillates up and down with time as the wave passes through. After the wave pulse has passed through, the displacement returns to zero.

The graph at lower right represents a snap shot (a "photograph") of the wave at t=27 seconds. The graph is blank for the first 26 seconds, then the graph appears all at once, and does not change. This graph represents the wave as a function of position at a specific time.

What is a Wave?

Definition of a Wave

Webster's dictionary defines a wave as "a disturbance or variation that transfers energy progressively from point to point in a medium and that may take the form of an elastic deformation or of a variation of pressure, electric or magnetic intensity, electric potential, or temperature."


The most important part of this definition is that a wave is a disturbance or variation which travels through a medium. The medium through which the wave travels may experience some local oscillations as the wave passes, but the particles in the medium do not travel with the wave. The disturbance may take any of a number of shapes, from a finite width pulse to an infinitely long sine wave.

Examples which illustrate the definition

Have you ever "done the wave" as part of a large crowd at a football or baseball game? A group of people jumps up and sits back down, some nearby people see them and they jump up, some people further away follow suit and pretty soon you have a wave travelling around the stadium. The wave is the disturbance (people jumping up and sitting back down), and it travels around the stadium. However, none of the individual people the stadium are carried around with the wave as it travels - they all remain at their seats.

Longitudinal sound waves in air behave in much the same way. As the wave passes through, the particles in the air oscillate back and forth about their equilibrium positions but it is the disturbance which travels, not the individual particles in the medium.

Transverse waves on a string are another example. The string is displaced up and down as the wave travels from left to right, but the string itself does not experience any net motion.

Other Single Crystal Surfaces

Although some of the more common metallic surface structures have been discussed in previous sections (1.2-1.4), there are many other types of single crystal surface which may be prepared and studied. These include

• high index surfaces of metals
• single crystal surfaces of compounds

These will not be covered in any depth, but a few illustrative examples are given below to give you a flavour of the additional complexity involved when considering such surfaces.

High Index Surfaces of Metals
High index surfaces are those for which one or more of the Miller Indices are relatively large numbers. The most commonly studied surfaces of this type are vicinal surfaces which are cut at a relatively small angle to one of the low index surfaces. The ideal surfaces can then be considered to consist of terraces which have an atomic arrangement identical with the corresponding low index surface, separated by monatomic steps (steps which are a single atom high).

As seen above, the ideal fcc(775) surface has a regular array of such steps and these steps are both straight and parallel to one another.

By contrast a surface for which all the Miller indices differ must not only exhibit steps but must also contain kinks in the steps. An example of such a surface is the fcc(10.8.7) surface - the ideal geometry of which is shown below.

Real vicinal surfaces do not, of course, exhibit the completely regular array of steps and kinks illustrated for the ideal surface structures, but they do exhibit this type of step and terrace morphology. The special adsorption sites available adjacent to the steps are widely recognised to have markedly different characteristics to those available on the terraces and may thus have an important role in certain types of catalytic reaction.

For further information on the structure of metal surfaces you should take a look at :

Single Crystal Surfaces of Compounds
The ideal surface structures of the low index planes of compound materials can be easily deduced from the bulk structures in exactly the same way as can be done for the basic metal structures. For example, the NaCl(100) surface that would be expected from the bulk structure is shown below :

In addition to the relaxation and reconstruction exhibited by elemental surfaces, the surfaces of compounds may also show deviations from the bulk stoichiometry due to surface localised reactions (e.g. surface reduction) and/or surface segregation of one or more components of the material.

For further information on the surface structure of compound materials you should take a look at :

Particulate Metals

As mentioned in the Introduction, macroscopic single crystals of metals are not generally employed in technological applications.

Massive metallic structures (electrodes etc.) are polycrystalline in nature - the size of individual crystallites being determined by the mechanical treatment and thermal history of the metal concerned. Nevertheless, the nature and properties of the exposed polycrystalline, metal surface is still principally determined by the characteristics of the individual crystal surfaces present. Furthermore, the proportions in which the different crystal surfaces occur is controlled by their relative thermodynamic stabilities. Thus, a macroscopic piece of an fcc metal will generally expose predominantly (111)-type surface planes.

A more interesting case for consideration is that of metals in a highly dispersed system - the classic example of which is a supported metal catalyst (such as those employed in the petrochemical industries and automotive catalytic converters). In such catalysts the average metal particle size is invariably sub-micron and may be as small as 1 nm . These metal particles are often tiny single crystals or simple twinned crystals.

The shape of these small crystals is principally determined by the surface free energy contribution to the total energy. There are two ways in which the surface energy can be reduced for a crystal of fixed mass / volume :

1. By minimizing the surface area of the crystallite
2. By ensuring that only surfaces of low surface free energy are exposed.

If matter is regarded as continuous then the optimum shape for minimizing the surface free energy is a sphere (since this has the lowest surface area/volume ratio of any 3D object) - this is why liquid droplets in free space are basically spherical.

Unfortunately, we cannot ignore the discrete, atomic nature of matter and the detailed atomic structure of surfaces when considering particles of the size found in catalysts. If, for example, we consider an fcc metal (eg. Pt) and ensure that only the most stable (111)-type surfaces are exposed, then we end up with a crystal which is an octahedron.

( Note : there are 8 different, but crystallographically-equivalent, surface planes which have the (111) surface structure - the {111} faces. They are related by the symmetry elements of the cubic fcc system).

A compromise between exposing only the lowest energy surface planes and minimizing the surface area is obtained by truncating the vertices of the octahedron - this generates a cubo-octahedral particle as shown below, with 8 (111)-type surfaces and 6 smaller, (100)-type surfaces and gives a lower (surface area / volume) ratio.

Crystals of this general form are often used as conceptual models for the metal particles in supported catalysts.

The atoms in the middle of the {111} faces show the expected CN=9 characteristic of the (111) surface. Similarly, those atoms in the centre of the {100} surfaces have the characteristic CN=8 of the (100) surface.

However, there are also many atoms at the corners and intersection of surface planes on the particle which show lower coordination numbers.

This model for the structure of catalytic metal crystallites is not always appropriate : it is only reasonable to use it when there is a relatively weak interaction between the metal and the support phase (e.g. many silica supported catalysts).

A stronger metal-support interaction is likely to lead to increased "wetting" of the support by the metal, giving rise to :

• a greater metal-support contact area
• a significantly different metal particle morphology

For example
In the case of a strong metal-support interaction the metal/oxide interfacial free energy is low and it is inappropriate to consider the surface free energy of the metal crystallite in isolation from that of the support.

Our knowledge of the structure of very small particles comes largely from high resolution electron microscopy (HREM) studies - with the best modern microscopes it is possible to directly observe such particles and resolve atomic structure.

Relaxation & Reconstruction of Surfaces

The phenomena of relaxation and reconstruction involve rearrangements of surface ( and near surface ) atoms, this process being driven by the energetics of the system i.e. the desire to reduce the surface free energy. As with all processes, there may be kinetic limitations which prevent or hinder these rearrangements at low temperatures.

Both processes may occur with clean surfaces in ultrahigh vacuum, but it must be remembered that adsorption of species onto the surface may enhance, alter or even reverse the process !

I. Relaxation
Relaxation is a small and subtle rearrangement of the surface layers which may nevertheless be significant energetically, and seems to be commonplace for metal surfaces. It involves adjustments in the layer spacings perpendicular to the surface , there is no change either in the periodicity parallel to the surface or to the symmetry of the surface.

We can consider what might be the driving force for this process at the atomic level ....

If we use a localised model for the bonding in the solid then it is clear that an atom in the bulk is acted upon by a balanced, symmetrical set of forces.

On the other hand, an atom at the unrelaxed surface suffers from an imbalance of forces and the surface layer of atoms may therefore be pulled in towards the second layer.

(Whether this is a reasonable model for bonding in a metal is open to question !)

The magnitude of the contraction in the first layer spacing is generally small ( < 10 % )- compensating adjustments to other layer spacings may extend several layers into the solid.

II. Reconstruction
The reconstruction of surfaces is a much more readily observable effect, involving larger (yet still atomic scale) displacements of the surface atoms. It occurs with many of the less stable metal surfaces (e.g. it is frequently observed on fcc(110) surfaces), but is much more prevalent on semiconductor surfaces.

Unlike relaxation, the phenomenon of reconstruction involves a change in the periodicity of the surface structure - the diagram below shows a surface, viewed from the side, which corresponds to an unreconstructed termination of the bulk structure.

This may be contrasted with the following picture which shows a schematic of a reconstructed surface - this particular example is similar to the "missing row model" proposed for the structure of a number of reconstructed (110) fcc metal surfaces.

Since reconstruction involves a change in the periodicity of the surface and in some cases also a change in surface symmetry, it is readily detected using surface diffraction techniques (e.g. LEED & RHEED ).

The overall driving force for reconstruction is once again the minimization of the surface free energy - at the atomic level, however, it is not always clear why the reconstruction should reduce the surface free energy. For some metallic surfaces, it may be that the change in periodicity of the surface induces a splitting in surface-localized bands of energy levels and that this can lead to a lowering of the total electronic energy when the band is initially only partly full.

In the case of many semiconductors, the simple reconstructions can often be explained in terms of a "surface healing" process in which the co-ordinative unsaturation of the surface atoms is reduced by bond formation between adjacent atoms. For example, the formation of a Si(100) surface requires that the bonds between the Si atoms that form the new surface layer and those that were in the layer immediately above in the solid are broken - this leaves two "dangling bonds" per surface Si atom.

A relatively small co-ordinated movement of the atoms in the topmost layer can reduce this unsatisfied co-ordination - pairs of Si atoms come together to form surface "Si dimers", leaving only one dangling bond per Si atom. This process leads to a change in the surface periodicity : the period of the surface structure is doubled in one direction giving rise to the so-called (2x1) reconstruction observed on all clean Si(100) surfaces [ Si(100)-(2x1) ].

Summary
The minimisation of surface energy means that even single crystal surfaces will not exhibit the ideal geometry of atoms to be expected by truncating the bulk structure of the solid parallel to a particular plane. The differences between the real structure of the clean surface and the ideal structure may be imperceptibly small (e.g. a very slight surface relaxation ) or much more marked and involving a change in the surface periodicity in one or more of the main symmetry directions ( surface reconstruction ).

Energetics of Solid Surfaces

All surfaces are energetically unfavourable in that they have a positive free energy of formation. A simple rationalisation for why this must be the case comes from considering the formation of new surfaces by cleavage of a solid and recognizing that bonds have to be broken between atoms on either side of the cleavage plane in order to split the solid and create the surfaces. Breaking bonds requires work to be done on the system, so the surface free energy (surface tension) contribution to the total free energy of a system must therefore be positive.

The unfavourable contribution to the total free energy may, however, be minimised in several ways :

1. By reducing the amount of surface area exposed
2. By predominantly exposing surface planes which have a low surface free energy
3. By altering the local surface atomic geometry in a way which reduces the surface free energy

Of course, systems already possessing a high surface energy (as a result of the preparation method) will not always readily interconvert to a lower energy state at low temperatures due to the kinetic barriers associated with the restructuring - such systems (e.g. highly dispersed materials such as those in colloidal suspensions or supported metal catalysts) are thus "metastable".

It should also be noted that there is a direct correspondence between the concepts of "surface stability" and "surface free energy" i.e. surfaces of low surface free energy will be more stable and vice versa.

One rule of thumb, is that the most stable solid surfaces are those with :

1. a high surface atom density
2. surface atoms of high coordination number

(Note - the two factors are obviously not independent, but are inevitably strongly correlated).

Consequently, for example, if we consider the individual surface planes of an fcc metal, then we would expect the stability to decrease in the order

fcc (111) > fcc (100) > fcc (110)

Warning - the above comments strictly only apply when the surfaces are in vacuum. The presence of a fluid above the surface ( gas or liquid ) can drastically affect the surface free energies as a result of the possibility of molecular adsorption onto the surface. Preferential adsorption onto one or more of the surface planes can significantly alter the relative stabilities of different planes - the influence of such effects under reactive conditions (e.g. the high pressure/high temperature conditions pertaining in heterogeneous catalysis ) is poorly understood.

Surface Structure of bcc Metals

A number of important metals ( e.g. Fe, W, Mo ) have the bcc structure. As a result of the low packing density of the bulk structure, the surfaces also tend to be of a rather open nature with surface atoms often exhibiting rather low coordination numbers.

I. The bcc (100) surface
The (100) surface is obtained by cutting the metal parallel to the front surface of the bcc cubic unit cell - this exposes a relatively open surface with an atomic arrangement of 4-fold symmetry.

 The diagram below shows a plan view of this (100) surface - the atoms of the second layer (shown on left) are clearly visible, although probably inaccessible to any gas phase molecules.

II. The bcc (110) surface
The (110) surface is obtained by cutting the metal in a manner that intersects the x and y axes but creates a surface parallel to the z-axis - this exposes a surface which has a higher atom density than the (100) surface.

The following diagram shows a plan view of the (110) surface - the atoms in the surface layer strictly form an array of rectangular symmetry, but the surface layer coordination of an individual atom is quite close to hexagonal.

III. The bcc (111) surface
The (111) surface of bcc metals is similar to the (111) face of fcc metals only in that it exhibits a surface atomic arrangement exhibiting 3-fold symmetry - in other respects it is very different.

In particular it is a very much more open surface with atoms in both the second and third layers clearly visible when the surface is viewed from above. This open structure is also clearly evident when the surface is viewed in cross-section as shown in the diagram below in which atoms of the various layers have been annotated.