Download Characterization of the phase transition in coper-gold alloys by X

Institute for Materials physics
Characterization of the phase transition in
coper-gold alloys by X-ray diffraction
Stand: April 2015
1 Introduction
In the framework of thermodynamic the transition of one phase in another phase is called phase
transition and can be described by the theorem of Ehrenfest. Well-known are phase transitions between liquid-solid phase or liquid-gas. In the given coper-gold alloy (relation 75 : 25 atom percent)
is a solid-solid transition possible. Here the atoms are moving from a ordered lattice to a disordered
lattice. Although is a change of the lattice type from fcc to a tetragonal distorted lattice recognizable. This change is visible by X-ray diffraction analysis, a common way in the materials physics
to investigate lattice types and lattice constants. Additional the material will be investigated by
calorimetry methods. Calorimetry methods are used in the materials physics to investigate phase
transitions in general. A combination of both techniques leads to an investigation of the full kinetic
of the phase transition.
Figure 1.1: phasediagram of Coper-Gold [after IFW11].
Figure 1.2: statistical distribution of the atoms on the lattice places:
(a) left: without ordered structure.
(b) right: with ordered structure.
Figure 1.3: Relation between ordered structure and lattice planes [Got07].
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2 Theory (short)
The lattice planes of a crystal can be determined with the help of Braggs law
2d · sin ϑ = nλ
n ∈ N,
(2.1)
where d is the lattice plane distance, ϑ the reflection angle and λ the wavelength. A special lattice
plane is described by his Millier Indexes (hkl) and is called then (hkl)-plane. The lattice plane
distance depends on the symetry of the crystal and his lattice parameter. The relation between
lattice plane distance and the lattice parameter a for different crystal symmetries is given in the
following equations:
For cubic with lattice parameter a:
dhkl = √
a
h2 + k 2 + l2
(2.2)
For tetragonal distorted with lattice parameters a and c:
1
d2hkl
=
h2 + k 2
l2
+
a2
c2
(2.3)
For a rhombic system with lattice parameters a,b and c:
1
d2hkl
=
h2 k 2
l2
+
+
a2
b2
c2
(2.4)
As a crystal transforms in an ordered phase, the crystal forms new lattice planes, called overstructure
(compare Figure 1.3).
The order parameter can be determined by
O
Iexp
η = F ·
Iexp
2
(fAu + 3fCu )F
(fAu − fCu )O
2
·
pF LFp DTF
·
· O ,
pO L O
DT
p
(2.5)
where O indicates a peak out of the overstructure. All other constants are listed in the main instruc-
tion (german only).
The growth of a crystal can be described by the help of the Johnson-Mehl-Avrami-Kogomorrow
Model. In this model a grain growth out of a nucleus under isothermal boundary conditions. The
fraction of transformed volume is given as
1
⇒ X(t) = 1 − e− 3 πN g
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3 tn
n
= 1 − e−K·t
(2.6)
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3 Experimental procedure
You will find the experimental setup ready for use in the lab. The sample (CuAu) is placed in the
heating chamber of the x-ray diffraction device D5000 by Siemens. The sample is contacted by a
thermocouple to read out the temperature and control the heating chamber by a PID controller.
The heating chamber allows, to heat up the sample to a maximum of 950 ℃(please keep the melting
temperature in mind). The heating chamber is closed by a cover with x-ray transparent beryllium
windows. The cover allows to establish a vacuum atmosphere in the chamber.
The x-ray diffractometer works with monochrome x-rays with a wavelength of λ = 1.54 Å.
3.1 Tasks
1: Measurement of a complete diffractogram of a disordered CuAu-Crystals at different temperatures
and of a ordered crystal after slow cooling (ca. 10 ℃ per minute).Measure with between 22°and
90°with a step size of 2ϑ = 0, 04°and a duration of 1 second per step. The sequence is arbitrary,
e.g.:
a) at roomtemperature
b) at 400℃
c) at 500℃
d) at 700℃
2: Cool the crystal from a high temperature (ca. 400℃) down to ca. 375℃ and observe the growing
of an overstructure peak (001)
a) The evolution of the phase will be observed with the grow rate of the (001)-Peak in relation
to the (111)-Peak.
b) For this: measure the intensity of the (111)-Peak before and after the measurement series.
3: For the heatflow DSC prepare a sample of ca. 20 mg. After initial heating through the transition
range and cooling below the transition temperature, observe the sample by isothermal holding
of the temperature.
3.2 Analysis
3.2 Analysis
The given points must be part of the practice report.
Part 1: Calculate the lattice parameter for the CuAu-Crystal in dependance of the temperature.
Please discuss the lattice parameters for different temperatures in the framework of the Lindemann law. The order parameter can be realized by senseful peak pairs. For the Lindemann
law use Debye-Waller Faktors out of table XX.
Part 2: Analyze the evolution of the order in framework of the JMAK-Model. The Avrami coefficient
must be in point of view and sensfully used. The order parameter can be calculated from the
ratio between (001)-Peak and (111)-Peak.
Part 3: Analyze the transition with respect to JMAK-Model. Here you can treat heatflow versus
time as derivation of the transformed sample mass. So use following equation as fit-function to
determine the Avrami coefficient.
dX(t)
= H · K · n · exp (−K · (t)n ) · tn−1
dt
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(3.1)
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Bibliography
[Deb14] Peter Debye, Interferenz von Röntgenstrahlen und Wärmebewegung,
Annalen der Physik Nr. 348, S. 49-95, 1914.
[FT98] M. Fanfoni & M. Tomellini, The Johnson-Mehl-Avrami-Kolmogorov model: A brief review,
IL NUOVO CIMENTO, Vol. 20D, N.7-8, Luglio-Agosto, 1998.
[Got07] Günter Gottstein, Physikalische Grundlagen der Materialkunde,
3. Auflage, Heidelberg, 2007.
[GM11] Rudolf Gross & Achim Marx, Festkörperphysik Kapitel 6,
Vorlesungsskript zur Vorlesung im SS 2011, Walther-Meissner-Institut, Lehrstuhl für Technische Physik (E23), S. 219,
http://www.wmi.badw.de/teaching/Lecturenotes/FKP/FKP_Kapitel6.pdf;
Stand: November 2011.
[IFW11] Internetauftritt des Leipniz-Instituts für Festkörper- und Werkstoffforschung Dresden,
Kapitel 15: Ordnung-Unordnung,
http://www.ifw-dresden.de/institutes/imw/lectures/lectures/pwe; Stand: Oktober 2011.
[Jam62] R.W. JAMES, The Optical Principles of the Diffraction of X-Rays,
The Crystalline State Vol. II, London, 1962.
[Wal27] Ivar Waller, Die Einwirkung der Wärmebewegung der Kristallatome auf die Intensität,
Lage und Schärfe der Röntgenspektrallinien,
Annalen der Physik, Band 83, 4. Folge, 1927.
[WBS97] Michael C. Weinberg, Dunbar P. Birnie III & Vitaly A. Shneidman,
Crystallization kinetics and the JMAK equation,
Journal of Non-Crystalline Solids, 219, S. 89-99, 1997.
[WK95] D. Waasmaier & A. Kirfel,
New Analytical Scattering-Factor Functions for Free Atoms and Ions,
Acta Crystallographica, A51, S. 416-431, 1995.
Bibliography
[hoehne1996] Höhne, G. and Hemminger, W. and Flammerheim, H.-J.,
Differential Scanning Calorimetry: An Introduction for Practitioners,
Springer-Verlag, 1996
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