Structures PART A – QUESTIONS 2010
1. (a) Briefly describe the production of Neutrons with specific regard to the different source types
and explain what is meant by the term thermal neutron. Compare the energy of a thermal neutron
with an x-ray of the same wavelength.
{5}
(ii) Describe how a neutron Powder diffraction profile from a solid fcc cubic structure would vary as
it undergoes the following magnetic phase transitions.
1
a paramagnetic to ferromagnetic transition
2
a ferromagnetic to anti-ferromagnetic transition
3
an anti-ferromagnetic to incommensurate transition
{6}
(b) The phase diagram of Fe-Ti is shown below for Ti compositions less than 30 wt. %. Ferrite is the
bcc form of Iron and Fe2Ti is an inter-metallic with a hexagonal structure.
A
B
C
Ferrite + Fe2Ti
(i)
What components are present in Regions A, B and C?
{3}
(ii)
Where does the eutectic transform occur and what is the reversible material change that
occurs at this point?
{2}
(iii)
With the aid of clear diagrams describe the micro-structural changes that occur as a 10 wt. %
alloy is cooled slowly.
{5}
(c) The magnetic structure of the Laves Phase inter-metallic Fe2Ti is rather uncertain. The magnetic
structure appears to be linked to compositional uniformity of the samples. Some authors report a
ferromagnetic structure whilst others suggest that for stoichiometric material the magnetic
structure is anti-ferromagnetic with a Néel temperature of 263 K.
(i) Describe how the neutron powder pattern of the 15 wt.% sample would vary as a function of
temperature. Justify why this is a good composition with which to study the magnetic structure of
Fe2Ti and how could you use a neutron data study to probe the magnetic structure of the sample?
{4}
2a
(i) By using suitable sketches show what is meant by the term rotational symmetry of an
object. Show using a geometrical argument (for a 2D lattice) that the only allowed rotational
symmetries in a crystal correspond to rotation angles of 2 n where n=1,2,3,4 or 6.
(ii) Consider the object shown below:
{6}
Identify the point symmetry operations present and thereby identify the point group to which it
belongs.
{4}
b) The two allotropes of Tin that are encountered at normal pressure and temperature are α-tin and
β-tin which are more commonly known as grey tin, and white tin respectively. Below 13.2°C a
structural phase transition occurs and the tetragonal structure of the white tin transforms into the
diamond structure of grey tin.
(i) Of the two allotropes, one is metallic and the other insulating. Identify which allotrope is metallic
justify your answer based on the bonding likely to be found in these structures.
{3}
(ii) Describe at least two parameters that you could use as the order parameter to follow this phase
change. Discuss how you would distinguish between a first and second order phase transition.
{4}
(c) (i) Barium Titanate undergoes a cubic (fcc) to tetragonal (bct) phase transition at 120°C. Sketch
the Perovskite structure above and below the phase transition clearly showing how an electric
dipole is induced in the material.
{4}
(ii) If this phase transition in a powder of BaTiO3 is measured by x-ray diffraction describe how
the shape of the diffraction will change in the vicinity of the (100) diffraction peak.
{4}
ANSWERS:
1. (a) Briefly describe the production of Neutrons with specific regard to the different source types
and explain what is meant by the term thermal neutron. Compare the energy of a thermal neutron
with an x-ray of the same wavelength.
Neutrons are produced in either a reactor (ILL) or through spallation (ISIS).
In the fission process, neutrons from a previous decay cause spontaneous decay in other heavy
nuclei (U235). In the Spallation process, proton beams cause neutrons to be ejected from a heavy
nucleus such as Ta.
[3 Marks]
A thermal neutron is one whose energy has been moderated to thermal energies (100meV). It has a
wavelength of 0.1nm. An x-ray of the same wavelength has an energy of 12keV.
[2 Marks]
(ii) Describe how a neutron Powder diffraction profile from a solid fcc cubic structure would vary as it
undergoes the following magnetic phase transitions.
(i)
a paramagnetic to ferromagnetic transition
(ii)
a ferromagnetic to anti-ferromagnetic transition
(iii)
an anti-ferromagnetic to incommensurate transition
In an fcc structure only those peaks associated with HKL being either all odd or all even will appear.
The nuclear peaks come from a point source so there is no sin/ fall-off.
Magnetic peaks come from the electron distribution so will show the sin/ fall-off.
[3 Marks]
The paramagnetic to ferromagnetic transition:
Magnetic peaks at the same q as the nuclear peaks
The ferro- to anti-ferromagnetic transition:
Magnetic peaks now appear at the half-order positions
The anti- to incommensurate transition: [UNSEEN]
The magnetic super-structure is now incommensurate with the nuclear long-range order.
The magnetic peaks reflect the periodicity of the magnetic ordering only and will move away from
the half-position
[3 Marks]
(i)
What components are present in Regions A, B and C?
Region A:
Liquid + Ferrite
Region B:
Liquid + Fe2Ti
Region C:
Austentite and Ferrite
[3 Marks]
(ii)
Where does the eutectic transform occur and what is the reversible material change that
occurs at this point?
[UNSEEN]
Eutectic occurs at 15 wt. % Ti and 1289°C.
Liquid
Ferrite andFe2Ti
[2 Marks]
(iii)
With the aid of clear diagrams describe the micro-structural changes that occur as a 10 wt. %
alloy is cooled slowly.
{5}
On cooling the liquid will initially enter region A which contains both ferrite and liquid. On cooling
below 1289°C the remaining liquid will be at the eutectic composition and will solidify in a lamellar
structure of alternating Fe and Fe2Ti layers. The ratio of the eutectic ferrite and Fe2Ti will be
approximately 50%.
[3 Marks + 2 Marks for a suitable sketch]
Describe how the neutron powder pattern of the 15 wt.% sample would vary as a function of
temperature. How could you use the neutron data to study the magnetic structure of Fe2Ti?
[UNSEEN]
The eutectic structure will have approximately equal quantities of the bcc ferrite (the sum of HKL is
even) and pure, stoichiometric hexagonal Fe2Ti - formed through the diffusion of the two
components. Expect to observe equal peak widths in the diffraction pattern.
There will be no diffraction peaks in the liquid due to the lack of long-range order.
On cooling through the eutectic transition the ferrite will exhibit ferromagnetism but these peaks
will be at the same location as the bcc peaks. The ordering temperature is likely to be very high. On
cooling to below 263 K any anti-ferromagnetism in the sample will be associated with the Fe2Ti. New
peaks will appear due to the doubling of the magnetic periodicity and will be directly related to the
hexagonal structure.
[4 Marks]
2a
(i) By using suitable sketches show what is meant by the term rotational symmetry of an
object. Show using a geometrical argument (for a 2D lattice) that the only allowed rotational
symmetries in a crystal correspond to rotation angles of 2 n where n=1,2,3,4 or 6.
An object has n-fold rotational symmetry if it is invariant under a rotation of 2 n
.
[2 Mark]
Allowed crystallographic orientations found by constructing:
[2 Marks]
In row A, points 1 and p separated by (m-1)a where m is an integer. In row B points 1’ and p’
separated by X=sa. S and M are positive integers which yields:
3   s  m  
cos    
. Thus  can only have values corresponding to angles given by n=1,2,3,4 or
2
6.
[2 Marks]
(ii) Consider the object shown below (UNSEEN):
Diads, tetrads and Mirror planes as shown. Space group 4mm
[4 Marks]
b) The two allotropes of Tin that are encountered at normal pressure and temperature are α-tin and
β-tin which are more commonly known as grey tin, and white tin respectively. Below 13.2°C a
structural phase transition occurs and the tetragonal structure of the white tin transforms into the
diamond structure of grey tin.
(i) Of the two allotropes, one is metallic and the other insulating. Identify which allotrope is metallic
justify your answer based on the bonding likely to be found in these structures. (UNSEEN)
The diamond structure is associated with COVALENT bonds and is found in many insulators. Metallic
bonding is associated with dense packed structures, therefore White Tin is metallic.
[3 Marks]
(ii) Describe at least two parameters that you could use as the order parameter to follow this phase
change. Discuss how you would distinguish between a first and second order phase transition.
Order parameters could include:
Lattice parameter – absolute values or c/a ratio
Volume changes
Resistivity
[2 Marks]
First phase transition has a discontinuity in the first derivative of the order parameter and has latent
heat. Second order phase transitions do no show latent heat and are associated with a discontinuity
in the second derivative of the order parameter.
[2 Marks]
(c) (i) In Barium Titanate undergoes a cubic (fcc) to tetragonal (bct) phase transition at 120°C. Sketch
the Perovskite structure above and below the phase transition clearly showing how an electric dipole
is induced in the material.
The tetragonal distortion is associated with a shift of the Ti atom in the centre of the oxygen
octahedral which thereby breaks the inversion symmetry allowing a net electric charge. [1 Mark]
[1 Mark for the polarisation and 2 marks for the structural sketches.]
(ii) If this phase transition in a powder of BaTiO3 is measured by x-ray diffraction describe how
the shape of the diffraction will change in the vicinity of the (100) diffraction peak.
The diffraction peak will show a splitting
due to the c and a directions no-longer
being equal. A peak associated with the
(001) peak will appear at a lower Bragg
angle than the (100) and (010) peaks which
remain the same.
[4 Marks]