4.8: Questions

Quick questions

You should be able to answer these questions without too much difficulty after studying this TLP. If not, then you should go through it again!

1. Most single crystals contain:

a	No defects, since they must be perfect crystals.
b	Exactly one defect, hence the term 'single crystal'.
c	Many defects.
d	More defects than atoms, since every atom must generate at least one defect.

Answer

c

2. If quartz had optical properties such that the refractive indices for all vibration directions were equal, the crossed polariser experiment would show:

a	No light transmitted for any orientation.
b	Varying intensity as the orientation changes.
c	Uniform non-zero intensity of transmitted light regardless of orientation.
d	Circular dark patches to represent the symmetry of the optical properties.

Answer

a

3. Bubbles in a box behave in a similar way to grains in a crystal in several ways, but not in all. Which of the following statements is TRUE?

a	The geometry of the places where bubbles meet one another is different from the geometry of the junctions between grains in a real polycrystal.
b	The shape of the bubbles is different from the typical shape of a grain.
c	The way a bubble deforms when a load is applied is different from the way a grain deforms.
d	The three dimensional structure of bubbles in a box is unlike the three dimensional structure of a polycrystal.

Answer

c

4. Which of the following is false?

a	Quartz crystals have optically anisotropic properties.
b	Glass has no regular repeating crystalline structure.
c	Certain crystals may cleave easily along certain planes, defined by the crystal structure.
d	Crystal defects are not found in single crystals.

Answer

d

5. What does the 'shot model' fail to show?

a	Polycrystallinity
b	Crystalline defects
c	The third dimension of the structure
d	The difference between a vapour and a solid

Answer

c

Deeper questions

The following questions require some thought and reaching the answer may require you to think beyond the contents of this TLP.

6. Why is window glass transparent?

hint: Think carefully about the materials discussed in this package. Do the transparent ones have a common structure type (single crystal, polycrystal, amorphous)? How does the density of an amorphous material compare to a similar crystalline material?

6. a	Because it has a single crystal structure and each sheet is cut with the optic axis normal to the plane of the window.
   b	Because it has an amorphous structure with large interatomic spacing. Light waves can pass between widely spaced atoms without any interaction with the solid structure.
   c	Because sheets of glass are cut thin enough for light to pass through without any significant absorption.
   d	Because of the electronic nature of the bonds between the atoms in the glass.

7. A quantity of pure liquid aluminium is cooled slowly through its melting point. The solid is then left at room temperature for 100 years. What is the resulting structure?

a	A polycrystal with grains of identical chemical composition but different crystallographic orientation.
b	A polycrystal consisting of finely spaced lamellae with alternating composition.
c	A single crystal.
d	An amorphous solid with good mechanical strength.

Answer

A is incorrect. Window glass is not a single crystal.

B is incorrect. Think carefully about the materials discussed in this package. Do the transparent ones have a common structure type (single crystal, polycrystal, amorphous)? How does the density of an amorphous material compare to a similar crystalline material?

C is incorrect. Glass remains transparent even when it is very thick (although impurities such as water tend to decrease the transparency). This is why fibre-optic cables, made from glass, can carry signals for kilometres.

D is correct. The optical properties of materials largely depend on their electronic nature. Metals, which have lots of free electrons, interact strongly with light and are opaque and reflective. Window glass and quartz crystals have similar compositions and bonding, with few free electrons, and both are transparent. When glass crystallises, it may lose its transparency because the nature of the bonding at the grain boundaries changes, leading to optically-absorbing electronic states.

8. Self-diffusion is the diffusion of a species within a body of material made from the same species. In general, self-diffusion in a polycrystalline solid can occur through the bulk of the grains (lattice diffusion) or along the grain boundaries (grain boundary diffusion). Which of the following statements gives the best description of the relative contribution of each process to the overall diffusion rate?

hint: Re-read the sections in this tutorial on polycrystals and defects. Are grain boundaries likely to be easy paths for diffusion? Consider what happens when the temperature is increased - what is the overall effect on the rate of diffusion? Look at the micrographs of grains again. Is most of the material 'grain'or 'grain boundary'?

a	The contributions should be about the same in both cases.
b	The contribution from lattice diffusion will always be greater than the contribution from grain boundary diffusion.
c	The contribution from grain boundary diffusion will always be greater than the contribution from lattice diffusion.
d	The relative contributions of the two processes depend upon the temperature of the material.

Answer

A is incorrect. The structure of a grain boundary is very different to that of the bulk lattice, so in general one would expect the rate of bulk diffusion to be different in some way to the rate of lattice diffusion. In addition, the material contains different volume fractions of 'grain' and 'grain boundary'. Only in special cases do the two processes contribute equally to the overall diffusion rate.

B is incorrect. The contribution from lattice diffusion is sometimes greater than the contribution from grain boundary diffusion.

C is incorrect. The contribution from grain boundary diffusion is sometimes greater than the contribution from lattice diffusion.

D is correct. At low temperatures, diffusion is much easier along the grain boundaries, which are more disordered and less dense than the bulk lattice. Therefore grain boundary diffusion dominates. As the temperature is increased, the rate of diffusion along both pathways increases - diffusion is thermally activated. There is normally much more 'grain' than 'grain boundary' in a material, so at a high enough temperature the contribution to the overall rate of diffusion from lattice diffusion becomes greater than the contribution from grain boundary diffusion. Note that the definition of 'low' and 'high' temperature depends upon the material in question (its grain size, crystal structure etc.)

9. Imagine a polycrystalline solid with cubic grains of edge length D. When D = 10 μm, what percentage of the volume of solid lies within a grain boundary, if the grain boundary width d is 1 nm? What must the grain size D be if 10% of the volume lies within a grain boundary? Comment on your answers.

Answer

The following diagram shows two cubic grains side-by-side, with the dimensions D and d indicated.

The volume of a single grain is D³. The volume of the grain boundary on one face of the grain is 1/2D²d, where the factor of 1/2 accounts for each grain boundary being shared by two grains. The total volume of grain boundary per grain is therefore 6 x 1/2D²d. The percentage of material (by volume) occupying the grain boundaries is then

Percentage = 100 x 3D²d / D³ = 300d / D

So for D = 10 μm, the percentage of material on grain boundaries is

300 x 1 nm / 10 μm = 0.03%

Only a small proportion of the atoms of a polycrystal with a reasonably fine grain size lie within grain boundaries. One might therefore expect the properties of a polycrystal to be virtually identical to those of a single crystal, since the vast majority of the material is well ordered and crystalline within the grains. However, experimentally we observe that this is rarely the case. The grain boundaries in a material are of critical importance in many phenomena - see, for example, question 8 on diffusion.

If the percentage of material on grain boundaries is 10 %, then

D = 300d / percentage = 300 x 1 nm / 10 = 30 nm

A significant fraction of the material is in a 'grain boundary' environment only for the very finest microstructures. A material with a grain size of 30 nm might be classed as a 'nanomaterial'. Often, the properties of nanomaterials are very different from similar materials with conventional grain sizes. An interesting example is in the superplastic forming of ceramics with ultrafine grain sizes. The grain size of the ceramic in this article is around 200nm.

Which of the following material properties could show anisotropy? (answer yes or no for each)

a	Density (see )
b	Young's modulus (see )
c	Electrical conductivity (see )
d	Refractive index (see )

Answer

No. When defined as mass / volume, density is isotropic because its value is independent of the direction in which it is measured (provided that the volume sampled has dimensions greater than the dimensions of a unit cell of the material).

Yes. The Young's modulus of many materials is anisotropic. Graphite, for example, has a Young's modulus of about 1000 GPa in directions parallel to the planes of atoms, compared to 36 GPa in the 'c' direction (perpendicular to the planes). Examine the structure of graphite here. In polycrystals, the individual grains can have anisotropic properties but the bulk material may be isotropic if the grain orientations are truly random.

Yes. Electrical conductivity depends on the electronic nature of the bonding within the structure. If the bonding is different in different directions, as can be seen clearly in the crystal structures shown in this module, then the conductivity is also likely to vary. Polycrystalline materials are less likely to be electrically anisotropic than single crystals. The refractive index depends on the electronic nature of the bonding in the crystal, as seen in section Crystals 3. Materials can be optically isotropic (one unique refractive index), or uniaxially or biaxially anisotropic (two and three unique refractive indices respectively). In polycrystalline samples, each individual grain can exhibit optical anisotropy. Some single crystal materials, such as metallic single crystals, are optically isotropic and have only one unique refractive index.

Yes.

Open-ended questions

The following questions are not provided with answers, but intended to provide food for thought and points for further discussion with other students and teachers.

11. Think about some of the possible applications of materials showing optical anisotropy, like the quartz crystal.
12. How might you control the grain size of a material produced from a melt? How might the cooling rate and the chemical composition affect the results? Can you think of ways to change the grain structure after the material has solidified?
13. In this TLP, we have discussed pure materials. Real materials almost always contain some impurities. How might these impurities be incorporated into the crystal structure of a material? Consider the relative size of the impurity atoms and the host atoms. Are impurities always undesirable?
14. Graphite is sometimes used as a lubricant, and diamond can be used on the tips of cutting tools. In terms of the crystal structure, why might this be?
15. Why do the individual grains in a polycrystalline material, such as those in the photo of galvanised steel (on the Polycrystals page) appear to be different colours or shades, when the composition of every grain is approximately the same?