Bonding in crystals, mechanical and dielectric properties

Question 1

Introduce the main type of chemical bonds. Give an example of solid for each type of bonding

Answer 1

Covalent: sharing atoms
Ionic: loosing atoms or gaining
Metallic: boding between free moving electron (delocalised) and positive ions
Van der Waals: molecule are electrically neutral. In addition some molecules are also polar (have two oppositely charged poles). Thus when they interact there is hydrogen bond. There are also the non polar molecules which however temporarily can be polar for a brief period of time since the location of the electron around the atom are constantly changing. Thus the briefly polar molecule can attract the neighbouring molecule making it polar as well thus there is a network of polar molecules connected by weak van der waals forces.
Molecular: it is a dipole dipole interaction that happens between a very electronegative atom and a hydrogen atom. The highly electro negative atom causes the hydrogen to take on a slightly positive charge

Question 2

What s cohesive and lattice energy and how to determine them experimentally?

Answer 2

Cohesion is the ability of a molecule to be attracted to identical molecules. Whereas adhesion is the property of a molecule to attracted to a different molecule.

So, cohesive energy is the energy required to completely separate a substance into its individual atoms or molecule. Can be calculated through the change in enthalpy (joules)

Lattice Energy is required to separate an ionic solid into its positive and negative ions.

Both can be measured experimentally through calorimetry that is by measuring the heat of reaction

Question 3

What determines lattice energy of ionic crystals? Explain the meaning of the madelung constant

Answer 3

The lattice energy is the energy required to separate an ionic compound into its positive and negative ions. What determines the lattice energy is:
charge of the ions: the stronger the ionic bond the higher the lattice energy
size of the ions:
Distance between the ion: less energy is requires to sperate ions with greater distance
Electronegativity: the more the electronegativity the higher the lattice energy.

The madelung constant is a dimensionless factor that is specific to each compound. the madelung constant allows to calculate with more accuracy the electrostatic interaction between ions keeping in mind the geometrical arrangement of the crystal.

The calculation is done by calculating the electrostatic energy of each ion and then summing that up. this is then simplified using the macluarain expression

Question 4

Explain the meaning of an ionic radii. How do cation and anion radii determine the structure of ionic crystals? (First Pauling’s rule)

Answer 4

In crystal lattice an ionic radii is a measure if the size of atom’s ions and it is calculated by measuring the distance between the nucleii of two neighbouring atoms. the raddi are measure in pico meter. the ionic radii is really important to understand the structure of the crystal: infact according to pauling’s first rule cations and anions will arrange themselves in a way that the sum of their ionic radii is equal to the distance between them which means that they will try to arrange themselves as densely as possible in order to have a more stable structure. By calculating the ratio of the radii of cation by the radii of the anion that sorrounds it we can obtain a coordination number that allows su to determine the type of polyhydron a crystal structure is.

Question 5

Give at least two examples of closed packed structures of ionic crystals. What determines their stability.

Answer 5

Sodium chloride: is a face centred cubic structured also known as close packed structure. One Na+ is surrounded by six Cl- giving stability to the molecule
Cesium Fluoride: is.a hexagonal close packed structure with each Ca++ surrounded by 12 F-.

the stability is given by many factors but the most important is the reduction of repulsive forces thus having a more stable structure.

Question 6

What determines lattice energy of van der Waals crystals? Explain changes in the lattice energy of crystals of noble gases

Answer 6

Not sure

Question 7

How to elucidate the nature of chemical bonds? which properties should be measured and what information should be extracted?

Answer 7

A chemical bond is a strong attraction between atoms. One way to understand the nature of a chemical bond is to use calorimetry and the born harber cycle, These methods allow us to calculate the lattice energy of a compound which in turn allows us to understand the chemical bond type. for example there are ranges that the lattice energy corresponding to a bond exists. comparing to those allows us to determine the chemical bond. So the properties to look for a are the thermal ones and extract the energy and radii

Question 8

What is the bulk modulus? How to determine it experimentally?

Answer 8

It is the measure of the stiffness of a meterial.
The formula is given by - (delta P)x(delta V)/V
The unit is pascals.
Experimentally it is measured using a compression test

Question 9

Derive an equation of state for solids, p = p(V )

Question 10

Introduce stress and strain and explain their relation. What is the difference between compressive, tensile and shear strain?

Answer 10

Stress: the internal distribution of forces within a body to maintain the equilibrium when two opposite force are applied. Force/A (Pascals)
Normal stress is when the force applied is perpendicular to the cross section area:
Normal stress can be
tensile if the material is stretched and
compressive stress if the material is compressed. Tensile stress has a positive sign whereas the compressed material has a negative sign.

The deformation that occurs within a body is described through strain. Strain is calculated as the fraction of change of the over the original property.

The relationship between the two can be described using a stress and strain diagram. We can obtain the diagram by performing a tensile test .

there is a linear part described by hooke’s law sigma = E*epsilon
For larger deformations the relationship is non linear so we have permanent plastic deformation

Shear stress:
the external force is applied perpendicular to the material thus internal forces are directed parallel to the cross sectional are.

Question 11

37. What are elastic constants? How to determine them experimentally?

Answer 11

Young’s modulus: obtained through the linear part of the stress-strain curve. (uniaxial tensile testing)

Poisson’s ratio: dimensionless property that described the lateral deformation due to longitudinal load (compression testing)

Shear modulus: measures the rigidity of the material and it is gives as the ratio of shear stress over shear strain. (shear testing)

All these testing procedures are similar with the load force being applied differently.

Question 12

Explain the meaning of Poisson’s ratio. Give examples of materials with positive, zero, and negative Poisson’s ratio.

Answer 12

Poisson’s ratio describes the lateral deformation of a material under uniaxial load. Materials can be differentiated by the sing of the ratio.

Positive: expansion: (rubber, most metals)
Zero: isotropic or homogeneous solids
Negative: metamaterials

Question 13

Sketch the stress-strain curve of a solid. Indicate the regions of the elastic and plastic deformation. What is the difference between ductile and brittle materials?

Answer 13

In the stress-strain curve we can notice two areas: the elastic and plastic

The separating point between them is the yield strength.

Ductile material have similar compressive and tensile strength whereas brittle material have compressive strength larger than tensile strength.

Ductility: is a measure of the ability of a material to deform plastically before fracture. they fracture at very large strain.

Materials that fracture at no strain thus have no plastic deformation are called brittle material.

Question 14

Introduce permittivity and polarisability. what is complex permittivity?

Answer 14

Given an electric field polarisability is the ability of molecules is positive charges to align in the direction of the electric field and negative charges to align on the opposite side.
Permittivity is the ability if a material to store electrical energy in electric field. it is also defined the resistance of a matter to electric field.

The absolute permittivity is given by a real part and a complex part. the real part is related to the energy stored in the capacitor whereas the complex part is related to the energy stored in the resistor.

Question 15

41. Sketch real and imaginary parts of the permittivity obtained in the Debye model of relaxation. Give an example of a material that this model can be applied to.

Answer 15

When we talk about the debye relaxation time we are principally talking about the phenomena where an electric field is switched of and how the dipole moment goes to zero. The dipoles in order to randomize would require collision with other dipoles and that would happen over a relaxation time tao. The time taken will be roughly the time taken between collisions. However when apply an alternating electric field we can study the frequency dependance of polarization. The maxima for both real and imaginary part is reached hen wt = 1 thus frequency is tao^-1

Question 16

Explain operation principle of the microwave oven

Answer 16

A microwave contains three main components. A vacuum tube called a magneteron, a waveguide that directs the energy to the food and a chamber that contains the mcirowave radiation. The microwave heats the food by not only heating the outside but also hte inside. THat is is done by taking advantage of the presnce of water in our food. Water is positiviely charged at one end and negatively charged on the other end. To give these molecules more energy we expose it to EM waves with electric and magnetic field that changed rapidly. The water molecules will try to orientate in the direction of the elctric field hwoever the alterantic electric and magnetic field will rock the molecules back nad forth creating heat by disrupting the bond between the neighbourig water molecules.

Question 17

What is phonon? Explain the difference between acoustic and optical phonons.

Answer 17

a phonon is a quanta of lattice vibration and it is based upon k that goes from -pi/a to pi/a. The energy is given by hw.

Acoustic phonons propagate like sound waves and have low frequencies and have a linear dispersion relation

Optical phonons have a more complicate dispersion relation and high frequencies

Question 18

Sketch phonon dispersion relation for a monoatomic chain. Indicate sound velocity

Answer 18

In a monoatomic chain the dispersion is linear.
The velocity of wave it is a know fact is express thoruhg omega and K. w/k gives V_s. thus a linear replationship where k is inversly proportional to lambda however if lambda increases or k dicreseases the particles scatter more and more this measn the velocity of the wave decreses which is shown as bend on k as pi/a

Question 19

Compare compressive and shear waves. How do they propagate in solid and liquids?

Answer 19

Compressive or longitudinal waves are waves in which the particle of the medium propagates in the direcition of the wave. They travel at a speed that is dependant of the bulk moduus in solid

Whereas, transverse or shear waves propagate perpendicular to the medium. This is caused due to shear forces thus thevvelocity is also dependant on the shear modulus in solids.

In general compressive waves moves faster and can exist in any medium whereas shear waves can only exist in solids because in liquids and solids the bodns between molecules aren’t strong enough to not be be broken by the shear force.

Question 20

Sketch phonon dispersion for a diatomic chain comprising two atoms with different mass

Answer 20

The dispersion relation for a diatomic chain with two dofferent mass is give by an opticla and acoustic branch.

For k=0 for the acoustic branch the two masses will have the same branch.

for k= pi/a the small mass is stationary for the acoustic branch

For the optical branch when the omega is minimum the bigger mass is stationary

for k=0 the masses in the optical branch are moving in totally different directions so will move out of phase

Question 21

Explain the splitting between transverse and longitudinal optical phonons at q=0. WHat does it singal?

Answer 21

energy differnece between Lo and TO and the brillouin zone centre. Anharmonic interaction of the lattice vibration LO mode has lower frequency than TO. It shows the ionincity.

Question 22

48. Elucidate interaction of light with an optical phonon. What makes ionic crystals reflect light?

Question 23

How to study phonons experimentally? Compare and contrast different technique.

Answer 23

We can’t use light due to very small lambda thus momentum very small than what is required.
X-rays that have lambra of 1 amstrong. the frequency is much larger than that of phonon. So yes but very sophisticated stuff but electron carry charge which implied they interact with other electrons. So electorns are not accepted
We need a non interacting particle in the range of Mev so neutrons

Question 24

Explain the principle of inelastic X-ray/neutron scattering. What determines the q-range accessible in the experiment.

Answer 24

The principle of the experiment lied in the interaction of the particle with the lattice.
In the process of interaction the external particle: exchanges momentum with the phonons
and exchanges energy with the phonon.

The momentum and energy conservation law must be obeyes. Mean;
Kin = kout = q
Ein = Eout + hbarw(q)

Answer 25

Raman scattering features a monochromatic loight source and a sample. The light can scatter, transmit or be absorbed. when the light scatters inelastically it is Raman whereas elastic is Rayleigh.

Question 26

How to measure heat capacity?

Answer 26

heat capacity is the amount of heat requied to raise the temperature of the substance by a certain amount. It can be done and measure by using a calorimeter.

Question 27

Compare Debye and Einstein models of the heat capacity

Answer 27

Classicaly according to doulong petit law the specific heat at room temperature is constant and equal to 3R. However DP. could not explain the specific heat approaching 0 at low temperature. EInstein takes the oscillator to have discrete energies. The ain difference lies in the fact that Einstein takes the the oscillation as small independent systems wheread Debye takes them as a large number of independent oscillations. THis is where Eisntein fails. According to DP the specific heat is temperature independent, all solids have the same specific heat however according to the data the C-v decreases for low T. Einstie nuse the planck’s QM theory to say that at Hight T C-v is 3R and at low T the Cv drops exponentially however the neisntein theory did not fit perfectly the data for low T. Einstein considered atoms to to have indeppendant vibrational motion but Debye retook that and said that these are couples motions and each atom won’t have one single freq but we would have a range of freq until the cut off frequency the cut off frequency.

Question 28

Explain assumptions of the Debye model (of the heat capacity) and the meaning of the Debye temperature. Elucidate the low-temperature and high-temperature limits

Answer 28

According to DP the specific heat is temperature independent, all solids have the same specific heat however according to the data the C-v decreases for low T. Einstie nuse the planck’s QM theory to say that at Hight T C-v is 3R and at low T the Cv drops exponentially however the neisntein theory did not fit perfectly the data for low T. Einstein considered atoms to to have indeppendant vibrational motion but Debye retook that and said that these are couples motions. In Debey we are dealing with elastic standing waves that are longitudinal -. -. and transversal . -. i. -. i.

Elastic waves will have quantized energies. The total numbe rof vibrational modes is 3N.

There isa cut off frequency the Debye frequency. At that Frequecncy we have the debeye temperature. SO at T greater than the debeye temperature the law follows a T cubes proportionality that fits the experimental data.

Question 29

Introduce the Debye-Waller factor. How to determine it experimentally? What is its temperature dependence?

Question 30

Define thermal expansion. Why do crystals expand upon heating?

Answer 30

THe thermal exapnsion is due to anharmonocoty as if we consider a harmonic crystal the average distance between the atoms change will be zero but if we consider anharmonicity the potential iwll move towards one direction more so there will be a change in distance as we raise the temperature. WHich means thermal expansion.

Question 31

Introduce the Gru¨neisen parameter (Γ). What sign of Γ would you expect for longitudinal and transverse phonon modes?

Answer 31

THe Gruneisen parameter is used to relate the phonons to the volume.

Question 32

Sketch temperature dependence of the crystal volume and thermal expansion coefficient

Question 33

What is thermal conductivity and thermal diffusivity?

Answer 33

Thermal conductivity is the rate of heat transfer through a unit thickness

Answer 34

Thermal conductivity can be measured through the study of the temperature gradient in a metal bar.

Question 35

66. Sketch temperature dependence of thermal conductivity (lattice contribution). Explain the trends

Answer 35

On the left we have a T3 dependance because our mean free path is contant at low temperature as the temperature starts to go up and up our heat capacity becomes a constant but our mean free path has an inverse relationship to temperature becasue of Umklamp scattering. Umklapp scattering will be the dominant source of phonon phonon scattering at room atemperature and above with the mean free path being proporotional to the inverse of temperature.

Question 36

What are main assumptions of the Drude model? Discuss their applicability in the context of real crystals

Answer 36

electrons move in a straight line and thhey collie with ions and instantaneuosly they come in equilibirum afte that. There is no collision between electrons. The elctrons have a probability of collision of 1/tao.