Materials Science and Engineering/Doctoral review questions/Interesting Questions

List Of "Must Know" Questions

• What is the fundamental flaw in the regular solution model?
• What happens to the bandgap of a material under hydrostatic pressure?

What are the types of magnetism?

Ferromagnetism

Ferromagnetism is the "normal" form of magnetism with which most people are familiar, as exhibited in horseshoe magnets and refrigerator magnets. It is responsible for most of the magnetic behavior encountered in everyday life. The attraction between a magnet and ferromagnetic material is "the quality of magnetism first apparent to the ancient world, and to us today," according to a classic text on ferromagnetism.

Ferromagnetism is defined as the phenomenon by which materials, such as iron, in an external magnetic field become magnetized and remain magnetized for a period after the material is no longer in the field.

All permanent magnets are either ferromagnetic or ferrimagnetic, as are the metals that are noticeably attracted to them.

Historically, the term ferromagnet was used for any material that could exhibit spontaneous magnetization: a net magnetic moment in the absence of an external magnetic field. This general definition is still in common use. More recently, however, different classes of spontaneous magnetisation have been identified when there is more than one magnetic ion per primitive cell of the material, leading to a stricter definition of "ferromagnetism" that is often used to distinguish it from ferrimagnetism. In particular, a material is "ferromagnetic" in this narrower sense only if all of its magnetic ions add a positive contribution to the net magnetization. If some of the magnetic ions subtract from the net magnetization (if they are partially anti-aligned), then the material is "ferrimagnetic". If the ions anti-align completely so as to have zero net magnetization, despite the magnetic ordering, then it is an antiferromagnet. All of these alignment effects only occur at temperatures below a certain critical temperature, called the Curie temperature (for ferromagnets and ferrimagnets) or the Néel temperature (for antiferromagnets).

Antiferromagnetism

In materials that exhibit antiferromagnetism, the spins of electrons align in a regular pattern with neighboring spins pointing in opposite directions. This is a different manifestation of magnetism. Generally, antiferromagnetic materials exhibit antiferromagnetism at a low temperature, and become disordered above a certain temperature; the transition temperature is called the Néel temperature. Above the Néel temperature, the material is typically paramagnetic.

The antiferromagnetic behaviour at low temperature usually results in diamagnetic properties, but can sometimes display ferrimagnetic behaviour, which in many physically observable properties is more similar to ferromagnetic interactions.

The magnetic susceptibility of an antiferromagnetic material will appear to go through a maximum as the temperature is lowered; in contrast, that of a paramagnet will continually increase with decreasing temperature. However, more complicated behavior may result if the magnetic structure is more complicated.

Ferrimagnetism

In physics, a ferrimagnetic material is one in which the magnetic moment of the atoms on different sublattices are opposed, as in antiferromagnetism; however, in ferrimagnetic materials, the opposing moments are unequal and a spontaneous magnetization remains. This happens when the sublattices consist of different materials or ions (such as Fe2+ and Fe3+).

Ferrimagnetic materials are like ferromagnets in that they hold a spontaneous magnetization below the Curie temperature, and show no magnetic order (are paramagnetic) above this temperature. However, there is sometimes a temperature below the Curie temperature at which the two sublattices have equal moments, resulting in a net magnetic moment of zero; this is called the magnetization compensation point. This compensation point is observed easily in garnets and rare earth - transition metal alloys (RE-TM). Furthermore, ferrimagnets may also exhibit an angular momentum compensation point at which the angular momentum of the magnetic sublattices is compensated. This compensation point is a crucial point for achieving high speed magnetization reversal in magnetic memory devices.

Paramagnetism

Paramagnetism is a form of magnetism which occurs only in the presence of an externally applied magnetic field. Paramagnetic materials are attracted to magnetic fields, hence have a relative magnetic permeability greater than one (or, equivalently, a positive magnetic susceptibility). The force of attraction generated by the applied field is linear in the field strength and rather weak. It typically requires a sensitive analytical balance to detect the effect. Unlike ferromagnets paramagnets do not retain any magnetization in the absence of an externally applied magnetic field, because thermal motion causes the spins to become randomly oriented without it. Thus the total magnetization will drop to zero when the applied field is removed. Even in the presence of the field there is only a small induced magnetization because only a small fraction of the spins will be orientated by the field. This fraction is proportional to the field strength and this explains the linear dependency. The attraction experienced by ferromagnets is non-linear and much stronger, so that it is easily observed on the door of one's fridge.

Superparamagnetism

Superparamagnetism is a phenomenon by which magnetic materials may exhibit a behavior similar to paramagnetism even when at temperatures below the Curie or the Néel temperature. This is a small length-scale phenomenon, where the energy required to change the direction of the magnetic moment of a particle is comparable to the ambient thermal energy. At this point, the rate at which the particles will randomly reverse direction becomes significant.

Diamagnetism

Diamagnetism is a weak repulsion from a magnetic field. It is a form of magnetism that is only exhibited by a substance in the presence of an externally applied magnetic field. It results from changes in the orbital motion of electrons. Applying a magnetic field creates a magnetic force on a moving electron in the form of F = Qv × B. This force changes the centripetal force on the electron, causing it to either speed up or slow down in its orbital motion. This changed electron speed modifies the magnetic moment of the orbital in a direction opposing the external field.

Consider two electron orbitals; one rotating clockwise and the other counterclockwise. An external magnetic field into the page will make the centripetal force on an electron rotating clockwise increase, causing it to speed up. That same field would make the centripetal force on an electron rotating counterclockwise decrease, causing it to slow down. The orbiting electrons create magnetic fields themselves, and in both cases, the change in B due to the electron's change in velocity is in the opposite direction to the external B field. Since the material originally had no net magnetic field from its orbiting electrons (because their orbits were aligned in random directions), the result is that the induced B field opposes the applied B field, and these repel each other.

All materials, except helium, show a diamagnetic response in an applied magnetic field. In fact, diamagnetism is a very general phenomenon, because all paired electrons, including the core electrons of an atom will always make a weak diamagnetic contribution to the material's response. However, for materials which show some other form of magnetism (such as ferromagnetism or paramagnetism), the diamagnetism is completely overpowered. Substances which only, or mostly, display diamagnetic behaviour are termed diamagnetic materials, or diamagnets. Materials that are said to be diamagnetic are those which are usually considered by non-physicists as "non magnetic", and include water, DNA, most organic compounds such as petroleum and some plastics, and many metals, particularly the heavy ones with many core electrons, such as mercury, gold and bismuth.

Superdiamagnetism

Superdiamagnetism (or perfect diamagnetism) is a phenomenon occurring in certain materials at low temperatures, characterised by the complete absence of magnetic permeability (i.e. a magnetic susceptibility \ \chi_{v} = −1) and the exclusion of the interior magnetic field. Superdiamagnetism is a feature of superconductivity. It was identified in 1933, by Walter Meissner and Robert Ochsenfeld (the Meissner effect).

Superdiamagnetism established that the superconductivity of a material was a stage of phase transition. Superconducting magnetic levitation is due to superdiamagnetism, which repels a permanent magnet, and flux pinning, which prevents the magnet floating away.

Why does the electrical conductivity suddenly increase in metals at very low temperatures?

Superconductivity is a phenomenon occurring in certain materials at extremely low temperatures, characterized by exactly zero electrical resistance and the exclusion of the interior magnetic field (the Meissner effect).

The electrical resistivity of a metallic conductor decreases gradually as the temperature is lowered. However, in ordinary conductors such as copper and silver, impurities and other defects impose a lower limit. Even near absolute zero a real sample of copper shows a non-zero resistance. The resistance of a superconductor, on the other hand, drops abruptly to zero when the material is cooled below its "critical temperature". An electrical current flowing in a loop of superconducting wire can persist indefinitely with no power source. Like ferromagnetism and atomic spectral lines, superconductivity is a quantum mechanical phenomenon. It cannot be understood simply as the idealization of "perfect conductivity" in classical physics.

What happens to metals (such as Na) under extremely high pressures?

Metallic hydrogen results when hydrogen is sufficiently compressed and undergoes a phase change; it is an example of degenerate matter. It consists of a crystal lattice of atomic nuclei (namely, protons), with a spacing which is significantly smaller than a Bohr radius. Indeed, the spacing is more comparable with an electron wavelength (see De Broglie wavelength). The electrons are unbound and behave like the conduction electrons in a metal. As is the dihydrogen molecule H2, metallica hydrogen is an allotrope.

Why is the sky blue? Why are sunsets red?

Rayleigh scattering causing the blue hue of the sky and the reddening at sunset

Rayleigh scattering (named after Lord Rayleigh) is the scattering of light or other electromagnetic radiation by particles much smaller than the wavelength of the light. It can occur when light travels in transparent solids and liquids, but is most prominently seen in gases. Rayleigh scattering of sunlight in clear atmosphere is the main reason why the sky is blue. Rayleigh and cloud-mediated scattering contribute to diffuse light (direct light being sunrays).

Why doesn't an electron combine with a proton in an atom?

With regard to the uncertainty principle, this is an energetically unfavorable event. The uncertainty in the velocity of an electron becomes high as it approaches the proton. The kinetic energy can be high, and the electron needs to compromise in terms of position and be a particular distance from the proton.

Reference:

Solymer and Walsh. Electrical Properties of Materials, 7th Edition, p. 30

Write down Fick's first and second laws, as generally as possible. What is the origin of these equations?

Figure 1. Model of a single adatom diffusing across a square surface lattice. Note the frequency of vibration of the adatom is greater than the jump rate to nearby sites. Also, the model displays examples of both nearest-neighbor jumps (straight) and next-nearest-neighbor jumps (diagonal). Not to scale on a spatial or temporal basis.

First law

Fick's first law is used in steady-state diffusion, i.e., when the concentration within the diffusion volume does not change with respect to time $(\,J_{\mathrm {in} }=J_{\mathrm {out} })$. In one (spatial) dimension, this is

$J=-D{\frac {\partial \phi }{\partial x}}$

where

• $J$ is the diffusion flux in dimensions of [(amount of substance) length⁻² time^-1], example ${\bigg (}{\frac {\mathrm {mol} }{m^{2}\cdot s}}{\bigg )}$

• $\,D$ is the diffusion coefficient or diffusivity in dimensions of [length² time⁻¹], example ${\bigg (}{\frac {m^{2}}{s}}{\bigg )}$

• $\,\phi$ is the concentration in dimensions of [(amount of substance) length⁻³], example ${\bigg (}{\frac {\mathrm {mol} }{m^{3}}}{\bigg )}$

• $\,x$ is the position [length], example $\,m$

$\,D$ is proportional to the velocity of the diffusing particles, which depends on the temperature, viscosity of the fluid and the size of the particles according to the Stokes-Einstein relation. In dilute aqueous solutions the diffusion coefficients of most ions are similar and have values that at room temperature are in the range of 0.6x10^-9 to 2x10^-9 m²/s. For biological molecules the diffusion coefficients normally range from 10^-11 to 10^-10 m²/s.

In two or more dimensions we must use $\nabla$, the del or gradient operator, which generalises the first derivative, obtaining

$J=-D\nabla \phi$.

Second law

Fick's second law is used in non-steady or continually changing state diffusion, i.e., when the concentration within the diffusion volume changes with respect to time.

${\frac {\partial \phi }{\partial t}}=D\,{\frac {\partial ^{2}\phi }{\partial x^{2}}}\,\!$

Where

• $\,\phi$ is the concentration in dimensions of [(amount of substance) length^-3], [mol m^-3]
• $\,t$ is time [s]
• $\,D$ is the diffusion coefficient in dimensions of [length² time^-1], [m² s^-1]
• $\,x$ is the position [length], [m]

It can be derived from the Fick's First law and the mass balance:

${\frac {\partial \phi }{\partial t}}=-\,{\frac {\partial }{\partial x}}\,J={\frac {\partial }{\partial x}}{\bigg (}\,D\,{\frac {\partial }{\partial x}}\phi \,{\bigg )}\,\!$

Assuming the diffusion coefficient D to be a constant we can exchange the orders of the differentiating and multiplying by the constant:

${\frac {\partial }{\partial x}}{\bigg (}\,D\,{\frac {\partial }{\partial x}}\phi \,{\bigg )}=D\,{\frac {\partial }{\partial x}}{\frac {\partial }{\partial x}}\,\phi =D\,{\frac {\partial ^{2}\phi }{\partial x^{2}}}$

and, thus, receive the form of the Fick's equations as was stated above.

For the case of diffusion in two or more dimensions the Second Fick's Law is:

${\frac {\partial \phi }{\partial t}}=D\,\nabla ^{2}\,\phi \,\!$,

which is analogous to the heat equation.

If the diffusion coefficient is not a constant, but depends upon the coordinate and/or concentration, the Second Fick's Law becomes:

${\frac {\partial \phi }{\partial t}}=\nabla \cdot (\,D\,\nabla \,\phi \,)\,\!$

An important example is the case where $\phi$ is at a steady state, i.e. the concentration does not change by time, so that the left part of the above equation is identically zero. In one dimension with constant $\,D$, the solution for the concentration will be a linear change of concentrations along $\,x$. In two or more dimensions we obtain

$\nabla ^{2}\,\phi =0\!$

which is Laplace's equation, the solutions to which are called harmonic functions by mathematicians.

What is k in Bloch's Theorem?

Bloch wave in silicon

A Bloch wave or Bloch state, named after Felix Bloch, is the wavefunction of a particle (usually, an electron) placed in a Particle in a one-dimensional lattice (periodic potential)|periodic potential. It consists of the product of a plane wave envelope function and a periodic function (periodic Bloch function) $\,u_{nk}(r)$ which has the same periodicity as the potential:

$\psi _{n\mathbf {k} }(\mathbf {r} )=e^{i\mathbf {k} \cdot \mathbf {r} }u_{n\mathbf {k} }(\mathbf {r} ).$

The result that the eigenfunctions can be written in this form for a periodic system is called Bloch's theorem. The corresponding energy eigenvalue is Є_n(k)= Є_n(k + K), periodic with periodicity K of a reciprocal lattice vector. Because the energies associated with the index n vary continuously with wavevector k we speak of an energy band with band index n. Because the eigenvalues for given n are periodic in k, all distinct values of Є_n(k) occur for k-values within the first Brillouin zone of the reciprocal lattice.

More generally, a Bloch-wave description applies to any wave-like phenomenon in a periodic medium. For example, a periodic dielectric in electromagnetism leads to photonic crystals, and a periodic acoustic medium leads to phononic crystals. It is generally treated in the different forms of the dynamical theory of diffraction.

The plane wave wavevector (or Bloch wavevector) k (multiplied by the reduced Planck's constant, this is the particle's crystal momentum) is unique only up to a reciprocal lattice vector, so one only needs to consider the wavevectors inside the first Brillouin zone. For a given wavevector and potential, there are a number of solutions, indexed by n, to Schrödinger's equation for a Bloch electron. These solutions, called bands, are separated in energy by a finite spacing at each k; if there is a separation that extends over all wavevectors, it is called a (complete) band gap. The electronic band structure is the collection of energy eigenstates within the first Brillouin zone. All the properties of electrons in a periodic potential can be calculated from this band structure and the associated wavefunctions, at least within the independent electron approximation.

A corollary of this result is that the Bloch wavevector k is a conserved quantity in a crystalline system (modulo addition of reciprocal lattice vectors), and hence the group velocity of the wave is conserved. This means that electrons can propagate without scattering through a crystalline material, almost like free particles, and that electrical resistance in a crystalline conductor only results from things like imperfections that break the periodicity.

What is so unique about diamond?

The slightly misshapen octahedral shape of this rough diamond crystal in matrix is typical of the mineral. Its lustrous faces also indicate that this crystal is from a primary deposit.

A diamond is a transparent crystal of tetrahedrally bonded carbon atoms and crystallizes into the face centered cubic diamond lattice structure. Diamonds have been adapted for many uses because of the material's exceptional physical characteristics. Most notable are its extreme hardness, its high dispersion index, and extremely high thermal conductivity (900 – 2320 W/m K), with a melting point of 3820 K (3547 °C / 6420 °F) and a boiling point of 5100 K (4827 °C / 8720 °F). Naturally occurring diamonds have a density ranging from 3.15 to 3.53 g/cm³, with very pure diamond typically extremely close to 3.52 g/cm³.

Diamonds can be identified via their high thermal conductivity. Their high refractive index is also indicative, but other materials have similar refractivity. Diamonds do cut glass, but other materials above glass on Mohs scale such as quartz do also. Diamonds easily scratch other diamonds, but this damages both diamonds.

Why do we use effective mass, rather than just mass, for mobility of electrons in a metal?

In solid state physics, a particle's effective mass is the mass it seems to carry in the semiclassical model of transport in a crystal. This mass is usually stated in units of the ordinary mass of an electron me (9.11×10^-31 kg).

It can be shown that, under most conditions, electrons and holes in a crystal respond to electric and magnetic fields almost as if they were free particles in a vacuum, but with a different mass. When electron is moving inside the solid material, the force between other atoms will affect its movement and it will not be described by Newton's law. So we introduce the concept of effective mass to describe the movement of electron in Newton's law. The effective mass can be negative or different due to circumstances.

Effective mass is defined by analogy with Newton's second law ${\vec {F}}=m{\vec {a}}$. Using quantum mechanics it can be shown that for an electron in an external electric field E:

$a={{1} \over {\hbar ^{2}}}\cdot {{d^{2}\varepsilon } \over {dk^{2}}}qE$

where $a$ is acceleration, $\hbar =h/2\pi$ is reduced Planck's constant, $k$ is the wave number (often loosely called momentum since $k$ = $p/\hbar$ for free electrons), $\epsilon (k)$ is the energy as a function of $k$, or the dispersion relation as it is often called. From the external electric field alone, the electron would experience a force of ${\vec {F}}=q{\vec {E}}$, where q is the charge. Hence under the model that only the external electric field acts, effective mass $m^{*}$ becomes:

$m^{*}=\hbar ^{2}\cdot \left[{{d^{2}\varepsilon } \over {dk^{2}}}\right]^{-1}$

For a free particle, the dispersion relation is a quadratic, and so the effective mass would be constant (and equal to the real mass). In a crystal, the situation is far more complex. The dispersion relation is not even approximately quadratic, in the large scale. However, wherever a minimum occurs in the dispersion relation, the minimum can be approximated by a quadratic curve in the small region around that minimum. Hence, for electrons which have energy close to a minimum, effective mass is a useful concept.

In energy regions far away from a minimum, effective mass can be negative or even approach infinity. Effective mass, being generally dependent on direction (with respect to the crystal axes, is a tensor. However, for most calculations the various directions can be averaged out.

Effective mass should not be confused with reduced mass, which is a concept from Newtonian mechanics. Effective mass can only be understood with quantum mechanics.