Solids Exam 2011 – Part A
a) State the one dimensional form of Fick’s first and second laws for diffusion. Describe how the
interface between two materials A and B would evolve as a function of time when they are brought
into contact if the materials have similar electronegavtivty and atomic radii.
{7}
b) The binary phase diagram of the magnesium lead system is shown below.
There are three single solid phases; -magnesium, -lead and the inter-metallic Mg2Pb. Lead is a
face centred cubic (Fm3m) structure (a=4.951 Å) whilst magnesium adopts a hexagonal structure
(a=3.21 Å, c=5.21 Å). The inter-metallic magnesium plumbide adopts a cubic (Fm3m) fluorite
structure with a lattice parameter a=6.81Å.
(i) In addition to the three solid phases shown in the Mg-Pb phase diagram above there are mixed
regions labelled 1-6. Identify the phases present in each of these numbered regions.
{3}
(ii) Use the phase diagram to identify any eutectic transformations that occur and clearly note their
temperature and composition. Write out the reversible material changes that occur at any such
transformation.
{4}
(iii) Consider cooling a 50 wt.% Pd solid solution from 700°C. What is the temperature at which the
first solid will form and what is its composition? After cooling to 500°C what are the composition
and relative weight fractions of the two components?
{4}
c) An x-ray powder diffraction experiment is performed on a 90 wt.% Pb material as it is cooled from
700°C to 100°C. Discuss the expected x-ray scattering pattern in terms of the material’s structure.
Compare the case when the sample is cooled slowly with the case when it is quenched, or cooled
rapidly. Clearly describe the position and intensities of the diffraction peaks in region 3 and 6 for
each case.
{7}
2.
a) Compare and contrast using suitable figures the Czochralski and Bridgman methods for producing
single crystals. Discuss the impurity profile that results in the final crystal using the two methods
and the effect of growth speed.
{7}
b) Potassium Niobate (KNbO3) belongs to the group of perovskite-type ferroelectrics of the barium
titanate family. At high temperatures it is cubic with a lattice parameter a=4.03Å. At 435°C KNbO3
undergoes a paraelectric to ferroelectric phase transition with an accompanying tetragonal
distortion of the structure with a c/a ratio of 1.0165. Below 225°C further structural changes occur
with a transition to a monoclinc structure.
(i) Use suitable diagrams to show how this structural distortion induces a polarisation within the
material and along which direction the polarisation vector points.
{3}
(ii) Discuss with a quick sketch how a diffraction pattern recorded using an energy of   0.1nm in
the vicinity of the (002) cubic peak would change as the sample goes through the phase transition at
435°C. Calculate and comment on the magnitude of any splitting observed. Would the experiment
be better performed using x-rays or neutrons? Justify your answer.
{7}
c) KNbO3 can be grown epitaxially by laser ablation onto MgO substrates at high temperatures. At
650°C the lattice parameter of MgO is 4.24 Å.
(i) Calculate the lattice mismatch at this deposition temperature.
(ii) Discuss using suitable figures how the system incorporates this strain as a function of the film
thickness.
(iii) As the film cools it will pass through the paraelectric – ferroelectric transition. Discuss how this
transition may be modified by the additional interaction of the KNbO3 film with the MgO substrate.
Consider the effects in the limits of a very thin and very thick KNbO3 film.
{8}
Answers – Question 1
State the one dimensional form of Fick’s first and second laws in for diffusion.
Fick’s laws relate to diffusion and the evolution of the concentration gradient:
1st Law: Flux is proportional to the concentration gradient. The proportionality constant is the
diffusion coefficient which is temperature dependent:
J  D(T )
dC
dx
{2}
2nd Law: The change in concentration as a function of time is balanced by the flux gradient:
C J

0
t x
or
C
 2C
 D T  2
t
x
{2}
State the one dimensional form of Fick’s first and second laws in for diffusion. Describe how the
interface between two materials A and B would evolve as a function of time when they are brought
into contact if the materials have similar electronegavtivty and atomic radii.
From Hume-Rothery rules we expect that the interface will for a solid solution whose structure will
evolve as a function of time and start to resemble an error function:
{3}
b(i) In addition to the 3 solid phase shown in the MgPb phase diagram there are 6 mixed regions
labelled 1-6. Identify the phases present in each of these numbered regions.
[unseen]
Region 1:
Region 2:
Region 3:
Region 4:
Region 5:
Region 6:
Liquid + alpha
Liquid + Mg2Pb
Liquid + Mg2Pb
Liquid + beta
alpha + Mg2Pb
beta + Mg2Pb
{½ mark each – total 3}
(ii) Identify any eutectic transformations that occur in the MgPb system. Clearly note their
temperature and composition and write out the reversible material changes that occur.
Eutectic occur at 460° and approx 67wt. % Pb and 240° and approx. 96 wt. % Pb
Reversible changes:
{2}

 alpha  Mg2Pb
Liquid 

and

 beta  Mg2Pb
Liquid 

{2}
(iii) Consider cooling a 50 wt.% Pd solid solution from 700°C. What is the temperature at which the
first solid will form and what is its composition? After cooling to 500°C what are the composition and
relative weight fractions of the two components?
[unseen]
First solid to form will at a temperature of approx. 560°C
At 500°C the solid will have a composition of approx. 34 wt.% Pb and the liquid 63 wt.% Pb
The relative weight fractions:
{1}
{2}
W 
{1}
CL  C 0 63  50
C C
50  34

 44.8% , WL  0  
 55.2% , or simply 100-W.
CL  C 63  34
CL  C 63  34
c) An x-ray powder diffraction experiment is performed on a 90 wt.% Pb material as it is cooled from
700°C to 100°C. Discuss the expected x-ray scattering pattern in terms of the material’s structure.
Compare the case when the sample is cooled slowly with the case when it is quenched, or cooled
rapidly. Clearly describe the position and intensities of the diffraction peaks in region 3 and 6 for each
case.
[unseen/interpretation]
Both phases have the space group –Fm3m which is the fcc cubic structure. Thus, HKL is either all
odd or all even. The Mg2Pb having a larger lattice parameter will have its diffraction peaks at angles
which are systematically lower than the beta-Pb.
{1}
In region 3, a mixture of the intermetallic and liquid phases exist. The liquid has no long range order
so will only contribute to the amorphous background scattering. As the temperature is lowered in
region3, a steady increase in the intensity of the intermetallic peaks is expected as the weight
fraction increases. A reduction in the Mg2Pb peak widths will be seen as the thermal displacement
term in the scattering factor reduces with reducing temperature.
{2}
At 240°C, the material crosses into region 6. The remaining liquid will form a eutectic structure
which is a series of narrow bands of inter-metallic and Pb. There will therefore be two contributions
to the intermetallic diffraction peaks – the original Mg2Pb, and the eutectic Mg2Pb. The later will be
from smaller polycrystalline grains so will be broad and we would expect to observe a two
component line shape, but with an overall increase in intensity. The slower the cooling rate the
larger these secondary Mg2Pb crystallites will be.
{2}
The Pb will crystallise with diffraction peaks appearing at higher scattering angles than the Mg2Pb the particle size will depend on the rate of cooling. Also expect to see some strain due to the
surrounding Mg2Pb.
{1}
From the relative weight fractions, we expect that the Pb peaks will correspond to 25% of the
diffracted signal with 75% associated with the inter-metallic.
{1}
Question 2:
Compare and contrast using suitable figures the Czochralski and Bridgman methods for producing
single crystals. Discuss the impurity profile that results in the final crystal using the two methods and
the effect of growth speed.
Impurities and grain growth is a diffusion process, thus in both cases the slower the temperature is
changed / sample withdrawn the more chance that the crystallites will be big and the impurities
have a chance to diffuse.
{1}
Czochralski:
Seed crystal pulled from the melt. As it is
withdrawn the sample is rotated.
Major advantage is that it is crucible free.
Impurities tend to be left in the melt on diffuse
to the edges of the crystals.
{1 for figure, 2 for discussion}
Bridgman:
Seed crystal at the end and the sample
withdrawn from a high temperature region.
Crucible can introduce impurities. All impurities
present in the solid remain in the crystal. Low k
impurities stay in liquid so impurities
concentrated at the end of crystal which is last
to freeze. Additional contamination from
crucible a problem at edges of crystal.
{1 for figure, 2 for discussion }
Potassium Niobate (KNbO3) belongs to the group of perovskite-type ferroelectrics of the barium
titanate family. At high temperatures it is cubic with a lattice parameter a=4.03Å. At 435°C KNbO 3
undergoes a paraelectric to ferroelectric phase transition with an accompanying tetragonal
distortion of the structure with a c/a ratio of 1.0165. Below 225°C further structural changes occur
with a transition to a monoclinc structure.
[unseen]
(i) Use suitable diagrams to show how this structural distortion induces a polarisation within the
material and along which direction the polarisation vector points.
The tetragonal distortion is associated with a shift of the Nb atom in the centre of the oxygen
octahedral which thereby breaks the inversion symmetry allowing a net electric charge.
{1}
polarisation along
the distorted
direction due to
the displaced
central atom. In
this case it is the
Nb in the oxygen
octahedral.
{1 Mark for the polarisation and 1 marks for the structural sketches – with clear comments on
which atoms are which.}
(ii) Discuss with a quick sketch how a diffraction pattern recorded using an energy of   0.1nm in
the vicinity of the (002) cubic peak would change as the sample goes through the phase transition.
Calculate and comment on the magnitude of any splitting observed. Would the experiment be better
performed using x-rays or neutrons, justify your answer.
The diffraction peak will show a splitting due to
the c and a directions no-longer being equal. A
peak associated with the (002) peak will appear
at a lower Bragg angle than the (200) and (020)
peaks which remain the same.
{2}
Use Bragg’s law to find the diffraction conditions
assume a=4.03:
Cubic phase: d(200)= 2.0150, 2 =28.735
Ferroelectric with c/a as given:
d(200)= 2.0150, 2=28.735°
d(002)= 2.0482, 2 =28.259°
Peak splitting is therefore 2=0.476°.
{2}
[unseen calculation and interpretation]
{1 Mark for sketch}
Better to use x-rays as material not magnetic and
the flux is higher. Better resolution means less
convolution and over-lap of peaks.
{2}
c
(i) Calculate the lattice mismatch at this deposition temperature.
Lattice miss-match:
4.24  4.03
4.03
 5.2% or
4.24  4.03
4.24
 4.95%
{1}
(ii) Discuss using suitable figures how the system incorporates this strain as a function of the film
thickness.
[unseen/interpretation]
For thin films the over layer is forced to lattice match to the substrate. For thicker over-layers the
strain is released through defects and missing bonds. The over-layer is said to be relaxed, this
happens above a critical thickness that is material and temperature dependent.
{3}
(iii) As the film cools it will pass through the paraelectric – ferroelectric transition. Discuss how this
transition may be modified by the additional interaction of the KNbO3 film with the MgO substrate.
Consider the effects in the limits of a very thin and very thick KNbO3 film. [unseen/interpretation]
A 5% lattice mismatch is quite high – the critical thickness will be very thin. The different thermal
expansion co-efficient of the two layers will have a larger effect in thin films, and may cause the
over-layer to be detached if the cooling rate is too high.
{1}
If the thickness of the niobate film is less than the critical thickness – the thin limit, the in-plane
lattice parameter of KNbO3 will be expanded causing a compression along the c-axis (through
Poisson’s ratio). This additional strain close to the paraelectric-ferroelectric transition is likely to
both modify the ordering temperature and also the direction of the polarisation axis.
{2}
In thicker films the KNbO3 film will be fully relaxed and the substrate will have a more limited effect.
The phase-transition should not be effected.
{1}