THE LATTICE PARAMETER OF GAMMA IRON AND
IRON-CHROMIUM ALLOYS
By
Zhiyao Feng
Submitted in partial fulfillment of the requirements
For the degree of Master of Science
Thesis Advisor: Dr. David. Matthiesen
Department of Materials Science and Engineering
CASE WESTERN RESERVE UNIVERSITY
May, 2015
CASE WESTERN RESERVE UNIVERSITY
SCHOOL OF GRADUATE STUDIES
We hereby approve the thesis/dissertation of
Zhiyao Feng
candidate for the degree of Master of Science in Materials Science and Engineering.
Committee Chair
Dr. David Matthiesen
Committee Member
Dr. Matthew Willard
Committee Member
Dr. Frank Ernst
Date of Defense
Mar. 26. 2015
*We also certify that written approval has been obtained for any proprietary material contained
therein.
II
Table of Contents
ACKNOWLEDGEMENTS ................................................................................................................ 11
ABSTRACT ......................................................................................................................................... 13
CHAPTER ONE.
INTRODUCTION...................................................................................... 14
CHAPTER TWO.
LITERATURE REVIEW ......................................................................... 15
2.1
Lattice Parameter and Lattice Parameter Expansion of Pure Iron ........................................ 15
2.2
Lattice Parameter of α-Fe (ferrite) with Binary Additions of Transition Metals .................. 19
2.3
Lattice Parameter of γ-Fe (Austenite) in Literature .............................................................. 26
CHAPTER THREE.
CHAPTER FOUR.
THESIS OBJECTIVES........................................................................ 29
EXPERIMENTAL PROCEDURE......................................................... 32
4.1
Experimental Equipment Introduction ................................................................................. 32
4.2
Determination of Temperature Offset ................................................................................... 33
4.3
High Temperature X-ray Experiment Procedure .................................................................. 34
CHAPTER FIVE.
RESULTS AND DISCUSSION ................................................................ 36
III
5.1
Starting Material ................................................................................................................... 37
5.2
Adjustment of Sample Height .............................................................................................. 40
5.3
Determination of Errors ........................................................................................................ 42
5.4
Lattice Parameter Determination of AHC 100.29 Powder at Room Temperature ................ 48
5.5
Temperature Offset Determination ....................................................................................... 52
5.6
Variation of Lattice Parameter with Temperature ................................................................. 55
5.6.1
Measurements of AHC 100.29 Iron Powder at High Temperatures ................................. 56
5.6.2
Measurement of AMES Fe – Cr Powder at High Temperatures....................................... 60
5.6.3
Measurement of CrA Fe – Cr Powder at High Temperatures ........................................... 62
CHAPTER SIX.
CHAPTER SEVEN.
CONCLUSIONS .......................................................................................... 70
SUGGESTIONS FOR FUTURE WORK............................................ 73
APPENDIX 1: AHC 100.29 POWDER LATTICE PARAMETER DATA..................................... 74
APPENDIX 2: AMES FE – 1CR POWDER LATTICE PARAMETER DATA ............................ 76
APPENDIX 3: ASTALOY CRA FE–CR POWDER LATTICE PARAMETER DATA ............... 78
IV
REFERENCES .................................................................................................................................... 80
V
List of Tables
Table 1. Resulting equations for the linear least squares regression analysis on Hume-Rothery’s lattice
expansion data of α and γ-phases of Fe as a function of temperature. .................................................. 19
Table 2. Lattice parameters and thermal expansion coefficients for Fe austenite in literature [7, 17-20].
............................................................................................................................................................... 27
Table 3. Data of X-ray diffraction peaks of α-phase pure iron with 0.04 second / degree scan rate and
0.02° step size scanned from 30° to 150° . .......................................................................................... 44
PU
PU
P
Table 4. Data of X-ray diffraction peaks of α-phase pure iron with 0.1 second / degree scan rate and
0.05° step size, scanned from 40° to 90° . ........................................................................................... 45
PU
PU
P
Table 5 Lattice parameter of α-phase iron calculated from the data in Table 3 and Table 4. ................ 47
Table 6. Comparison of the result for linear least squares regression analysis performed on data of
AHC 100.29 and Sutton and Hume-Rothery. ........................................................................................ 59
Table 7. Result for linear least squares regression analysis performed on the data of Ames Fe – Cr
sample. .................................................................................................................................................. 61
Table 8. Results of linear least squares regression analysis performed on the Astaloy CrA sample data.
............................................................................................................................................................... 64
VI
Table 9. Summary of lattice parameters of the γ-phase and the α-phase at 20℃ of AHC 100.29, Ames
PU
Fe – Cr, and Astaloy CrA samples......................................................................................................... 64
Table 10. Results of linear least squares regression analysis performed on the data of AHC pure iron
sample, Ames Fe – Cr sample, and Astaloy CrA Fe – Cr sample. ........................................................ 71
VII
List of Figures
Figure 2-1. (a) Face-centered cubic arrangement of iron atoms and (b) body-centered cubic
arrangement of iron atom. ..................................................................................................................... 16
Figure 2-2. (a) (100) plane in face-centered cubic lattice and (b) (110) plane in body-centered cubic
lattice ..................................................................................................................................................... 17
Figure 2-3. Variation of lattice parameter of pure iron with temperature [7]. ....................................... 19
Figure 2-4. Comparison between Sutton and Hume-Rothery’s experimental data [1] and Vegard’s law
about the change in lattice parameter of α iron with binary additions of chromium. ............................ 22
Figure 2-5 Change in lattice parameter of α iron with binary additions of chromium [2]. ................... 23
Figure 2-6. Change in lattice parameter of α iron with binary additions of chromium plotted with
measured data points [2]. ...................................................................................................................... 24
Figure 2-7. Change in lattice parameter of α iron with binary additions of chromium from Sutton and
Hume-Rothery’s [1] and Abrahamson and Lopata’s data [2]. ............................................................... 26
Figure 2-8. Literature data of lattice parameter of austenite as a function of temperature [7, 17-20]. .. 28
Figure 3-1. Equilibrium binary phase diagram of Fe-N system [21]. ................................................... 30
VIII
Figure 4-1. (a) The exterior appearance and (b) interior chamber of high temperature X-ray system. . 32
Figure 4-2. (a)The front and (b) back view of the thermocouple on the sample holder. ....................... 33
Figure 5-1. SEM image of Astaloy AHC 100.29 iron powder. ............................................................. 38
Figure 5-2. SEM image of Ames Fe-1Cr powder. ................................................................................. 39
Figure 5-3. SEM image of Astaloy CrA powder. .................................................................................. 40
Figure 5-4. Effect of sample displacement on the diffraction peak position and peak shape................ 42
Figure 5-5 Extrapolation of measured lattice parameters of AHC 100.29 powder against Nelson-Riley
function, calculated from the data in Table 3. ....................................................................................... 46
Figure 5-6 Extrapolation of measured lattice parameters of AHC 100.29 powder against Nelson-Riley
function, calculated from the data in Table 4. ....................................................................................... 47
Figure 5-7. Diffraction Pattern of AHC 100.29 pure iron powder at 20℃ by Scintag X-1 Cu source.
PU
............................................................................................................................................................... 49
Figure 5-8. The variation of 𝐬𝐬𝐬𝐬𝐬𝐬𝐬𝐬 with 𝛉𝛉. .......................................................................................... 50
Figure 5-9. Extrapolation of measured lattice parameters of AHC 100.29 powder .............................. 52
IX
Figure 5-10. Unit cell structure of Al 2 O 3 . ............................................................................................. 55
Figure 5-11. Comparison between literature value [23] of c lattice parameter of Al 2 O 3 and
experimental values. .............................................................................................................................. 55
Figure 5-12. Resulting comparison between the lattice expansion of iron with temperature in this study
and in Sutton and Hume-Rothery [1]. ................................................................................................... 58
Figure 5-13. Comparison of the variation of lattice parameter of the γ-phase with temperature between
AHC 100.29 pure iron powder and literature values. ............................................................................ 60
Figure 5-14. Variation of lattice parameter with temperature of AMES Fe – Cr sample. ..................... 61
Figure 5-15. Variation of lattice parameter with temperature of Astaloy CrA Fe – Cr powder. ............ 64
Figure 5-16. Comparison of the lattice parameter of the α-phase between Abrahamson’s data [2] and
the extrapolated values in Table 9. ........................................................................................................ 65
Figure 5-17. Variation of lattice parameter of γ-phase with different chromium content. .................... 66
Figure 5-18 Variation of temperature interval of the “A 3 point” with different chromium content. ..... 67
Figure 5-19. Binary phase diagram of Fe – Cr system [30]. ................................................................. 69
X
ACKNOWLEDGEMENTS
I wish to express my sincere gratitude to my advisor, Prof. David Matthiesen, for his
guidance, advice, and encouragement during my study.
I also wish to thank my committee members, Prof. Matthew Willard and Prof. Frank
Ernst, for their help, encouragement, and serving in my thesis committee.
The information, data, or work presented herein was funded in part by the Advanced
Research Projects Agency-Energy (ARPA-E), U.S. Department of Energy, under Award
Number DE-AR0000194.
The information, data, or work presented herein was funded in part by an agency of the
United States Government. Neither the United States Government nor any agency thereof,
nor any of their employees, makes any warranty, express or implied, or assumes any legal
liability or responsibility for the accuracy, completeness, or usefulness of any information,
apparatus, product, or process disclosed, or represents that its use would not infringe
privately owned rights. Reference herein to any specific commercial product, process, or
service by trade name, trademark, manufacturer, or otherwise does not necessarily
constitute or imply its endorsement, recommendation, or favoring by the United States
Government or any agency thereof. The views and opinions of authors expressed herein
11
do not necessarily state or reflect those of the United States Government or any agency
thereof.
I additionally thank all my friends and colleagues as well as the faculty and staff in
Department of Material Science for their help and cooperation during my study.
Finally, I would like to thank my parents for their invaluable support and encouragement.
12
The Lattice Parameter of Gamma Iron and Iron – Chromium Alloys
ABSTRACT
by
ZHIYAO FENG
The lattice parameters of the gamma iron as a function of chromium alloy content were
determined using high temperature X-ray diffraction measurements for three Fe-rich
alloys. The three concentrations were: (1) pure iron powder, (2) 1 at.% Cr, and (3) 1.8
at.% Cr, and the temperature range was between 800℃ to 1300℃. Linear relationships
between lattice parameter and temperature were observed and determined in all three
samples. The lattice parameters of the γ-phase of the three samples at room temperature
were determined by extrapolating the high temperature data to 20℃. The values are
(0.3572±0.0005) nm, (0.3604±0.0003) nm, and (0.3609±0.0003) nm for pure iron powder
and iron-chromium powders with 1 at.% Cr and 1.8 at.% Cr, respectively. A linear least
𝑛𝑛𝑛𝑛
squares regression analysis yielded: 𝑎𝑎 = 0.0021(𝑎𝑎𝑎𝑎.%) × 𝐶𝐶𝐶𝐶𝐶𝐶 + 0.3575(𝑛𝑛𝑛𝑛).
13
Chapter One. Introduction
The objective of this thesis is to determine the lattice parameter of the gamma iron, at
20℃, in solid solutions for iron-chromium alloy powders with 1.0 at.% and 1.8 at.% Cr,
and to compare the results with the lattice parameter of the γ-phase in pure iron powder.
The determination of the lattice parameter of α-phase in Fe – Cr alloys with Cr content up
to 5.37 at.% [1]and up to 3.19 at.% [2] were reported in literature, but in neither case was
the lattice parameter of the γ-phase reported. This thesis describes the experimental
procedure for the determination of the γ-phase lattice parameter of three powder samples.
Commercially available AHC 100.29 iron powders, Astaloy CrA iron-1.8 at.% Cr
powders were both manufactured by Hӧganӓs Inc. using a water atomized process. In
addition, a 1.0 at.% Cr alloy powder was custom manufactured for this study by Ames
Laboratory using a gas atomization process. The γ-phase lattice parameter of these
powder samples were determined at temperatures between 800℃-1300℃, from which a
linear least squares regression analysis gave the lattice parameter of the γ-phase at 20℃.
14
Chapter Two. Literature Review
Due to the pervasive usage of iron and iron-based alloys in industry, understanding the
phase relationships of pure iron and iron alloys is crucial. Thus, a great deal of research
has been reported on the lattice parameters of pure iron phases and solid solutions of
different transition metals in iron [1].
2.1 Lattice Parameter and Lattice Parameter Expansion of Pure Iron
Figure 2-1 shows the face-centered cubic (γ-phase, austenite) and body-centered cubic
(α-phase, ferrite and δ-phase) arrangements of iron atoms. The structures were created
using CrystalMaker version 9.0.3(619) for Windows [3]. The space groups of the α-Fe
and the δ-Fe are both Im3m, and the space group of the γ-Fe is Fm3m. For a perfect
crystal, there are 12 nearest neighbors for face-centered cubic and 8 nearest neighbors,
and 6 next nearest neighbors for body-centered cubic. The 6 next nearest neighbors for
body-centered cubic are only 15% larger than the nearest neighbors. Face-centered cubic
is a close packed structure, with four {111} close packed planes. Body-centered cubic is
less space filling than face-centered cubic structure, in which case there are no close
packed planes, only close packed directions <111>. The planes of highest atomic density
in face-centered cubic and body-centered cubic are shown in Figure 2-2. Assuming hard
spheres touch along the diagonal direction, and the radius of iron atom is r and the lattice
15
parameters of γ Fe and α Fe are 𝑎𝑎𝛾𝛾 and 𝑎𝑎𝛼𝛼 , respectively, the relation between atomic
radius and the lattice parameters are:
4𝑟𝑟𝐹𝐹𝐹𝐹−𝛾𝛾 = √2𝑎𝑎𝛾𝛾
1
4𝑟𝑟𝐹𝐹𝐹𝐹−𝛼𝛼 = √3𝑎𝑎𝛼𝛼
2
Using Eq. 1 and the corresponding slopes and intercepts in Table 1, the lattice parameters
of γ Fe and α Fe at 20℃ are (0.35696±0.00007) nm and (0.28658±0.00002) nm. The
P
lattice parameters of γ Fe, 𝑎𝑎𝛾𝛾 , is 1.246 times the value of 𝑎𝑎𝛼𝛼 , which is a little bit larger
than the result derived by Eq.1 divided by Eq. 2. Eq. 1and Eq. 2 were derived using a
hard sphere model, yet, the arrangements of iron atoms of both α and γ-phase may
deviate a little from the ideal situation, and thus result in a discrepancy between
calculated value and experimental value.
(a)
(b)
Figure 2-1. (a) Face-centered cubic arrangement of iron atoms and (b) body-centered cubic
arrangement of iron atom.
16
(a)
(b)
Figure 2-2. (a) (100) plane in face-centered cubic lattice and (b) (110) plane in body-centered
cubic lattice
The lattice parameter of α-phase pure iron has been determined many times, and various
values have been reported by different studies. Owen reported the value 0.28605(0) nm at
15℃ in 1937 [4]. Sutton and Hume-Rothey reported the value 0.28662(1) nm at 20℃ in
1955 [1]. This value in nanometer units was converted from the one reported in kX units,
by multiplying the kX values by 0.100202 [5]. Abrahamson and Lopata determined the
lattice parameter value of iron as (0.28662±0.00002) nm [2] in 1966. Both
Hume-Rothery’s and Abrahamson’s data are close to the data in Pearson’s Handbook [6]
0.28664 nm.
The lattice expansion experimental results, reported by Basinski, Sutton and
Hume-Rothey are plotted in Figure 2-3 [7], indicating the exact temperature of phase
transformation α ↔ γ and γ ↔ δ is 910℃ and 1390℃, respectively. It was pointed out in
experimental details that at 1388℃, the A 4 temperature, which is the phase transformation
17
temperature from γ-Fe to δ-Fe, the diffraction lines from both the γ-phase and the δ phase
were observed. However, such effect was not mentioned at the A 3 temperature, which is
the phase transformation temperature from α-Fe to γ-Fe. The experimental procedure
described in the literature [7] indicated the X-ray exposures were performed twice on the
iron sample, and the sample was cooled to room temperature, 20℃ , in between. The
P
α-phase diffraction peaks were observed in the first X-ray exposure, and the γ-phase
peaks were observed in the second experiment. The time length of each measurement was
not mentioned.
The lattice parameter of the γ-phase is larger than those of α-phase and δ phase due to its
face-centered cubic structure, which will be addressed in the next section. A linear least
squares regression analysis of lattice expansion as a function of temperature, according to
Eq. 3, was performed on this data, the resulting slopes and intercepts of α-Fe and γ-Fe are
listed in Table 1.
𝑎𝑎 = 𝑎𝑎0 + 𝑆𝑆 ∙ T
18
3
Figure 2-3. Variation of lattice parameter of pure iron with temperature [7].
Table 1. Resulting equations for the linear least squares regression analysis on Hume-Rothery’s
lattice expansion data of α and γ-phases of Fe as a function of temperature.
Phase
α
γ
Intercept 𝑎𝑎0 (nm)
Value
Standard Error
0.28649
2 × 10−5
0.35679
7 × 10−5
Slope S (nm/oC)
4.30 × 10−6
Slope S (nm/oC)
8.59 × 10−6
Intercept 𝑎𝑎0 (nm)
3 × 10−8
6 × 10−8
2.2 Lattice Parameter of α-Fe (ferrite) with Binary Additions of Transition Metals
Vegard [8] observed a linear relation between the crystal lattice constant and
19
concentration in some ionic salt alloys in an early application of X-ray analysis and
postulated Vegard’s law in 1921, which has been extensively used in material science,
mineralogy and metallurgy for nearly a century. According to Vegard’s law, the unit cell
lattice parameter of an alloy, at constant temperature, varies linearly with the
concentration of the substitutional elements [9]. For instance, consider an alloy A x B 1-x ,
𝑎𝑎(𝐴𝐴,𝐵𝐵) = 𝑥𝑥𝑎𝑎𝐴𝐴 + (1 − 𝑥𝑥)𝑎𝑎𝐵𝐵
4
The lattice constants of α-Fe and α-Cr at 20℃ are (0.28662±0.00002) nm and
P
(0.2884±0.0001) nm [1, 10]. Applying the lattice parameters of Fe and Cr into Eq. 4, the
lattice parameter of the binary alloy would be a function of the composition:
𝑎𝑎(𝐹𝐹𝐹𝐹,𝐶𝐶𝐶𝐶) = 𝑥𝑥𝐹𝐹𝐹𝐹 ∙ 0.28662 (𝑛𝑛𝑛𝑛) + (1 − 𝑥𝑥𝐹𝐹𝐹𝐹 ) ∙ 0.2884(𝑛𝑛𝑛𝑛)
5
Vegard’s law was postulated on empirical evidence. In the later extensions of the rule to
metallic alloys, most of the systems have been found not to obey Vegard’s law [11-13].
The same phenomenon happens in the α-Fe system as well, several positive and negative
deviations from this law have been reported in the study of Abrahamson and Lopata [2].
Raynor [14] discovered that the deviation from Vegard’s law were proportional to the
difference of the solute and solvent valencies and electron / atom ratio.
Sutton and Hume-Rothery [15] measured the lattice parameters of dilute solid solutions
20
of titanium, vanadium, chromium, manganese, cobalt and nickel in α-Fe, and in all cases
a lattice expansion was observed. For equal atomic percentages of the solute atoms, the
elements to the left of iron, in the periodic table Ti, V, Cr, and Mn, expanded the lattice
spacing of α-Fe because of their larger size compared with iron atoms. However, those
elements to the right of iron, Co and Ni, which are smaller than iron atoms, also showed
an expansion of the lattice due to the exchange repulsion between nearly filled d shells.
When the distance between two atoms decreases, the electron clouds approach each other
and their charge distributions gradually overlap. The electron density in this region would
decrease due to the Pauli exclusion principle [16]. The positively charged nuclei of the
atoms are then incompletely shielded from each other and therefore exert a repulsive
force on each other, and therefore the lattices of the alloys are expanded.
Figure 2-4 shows the lattice expansion of α iron by adding Cr. The lattice parameter in Fe
– Cr system increases linearly with composition, but greater than would be expected from
Vegard’s law. By adopting linear least squares regression analysis to the data,
𝑛𝑛𝑛𝑛
𝑎𝑎 = 5.4 × 10−5 �
� ∙ 𝐶𝐶𝐶𝐶𝐶𝐶 + 0.286620(𝑛𝑛𝑛𝑛)
𝑎𝑎𝑎𝑎. %
6
where 𝑎𝑎 is the lattice parameter of the α-phase Fe – Cr alloy and 𝐶𝐶𝐶𝐶𝐶𝐶 is the composition
of Cr in the alloy. The standard error of the intercept and slope is 4 × 10−6 nm and
1 × 10−6 nm / at.%, respectively. By using Eq. 6, the lattice parameters of the α-phase of
21
Fe – Cr alloy at 20℃ with 1.0 at.% Cr (Ames powder) and 1.8 at.% Cr (Astaloy CrA
powder) are 0.2866(7) nm and 0.2867(2) nm, respectively.
Figure 2-4 shows the data for Fe – Cr alloys from Sutton and Hume-Rothery [1] and the
linear least squares regression result Eq. 6, whilst the green line shows the lattice
parameters expected in accordance with Vegard’s law. Results reported by Sutton and
Hume-Rothery show a slightly greater expansion of the lattice parameter than would be
expected from Vegard’s law.
Figure 2-4. Comparison between Sutton and Hume-Rothery’s experimental data [1] and
Vegard’s law about the change in lattice parameter of α iron with binary additions of chromium.
In Abrahamson and Lopata’s work [2], the lattice parameter of α-phase iron alloyed with
22
different concentrations of Cr was measured. The reproducibility for repeat runs on any
sample was ±0.00001 nm. The plot is shown in Figure 2-5. The slope of the line was
determined by linear least squares regression analysis with the origin as 0.28662 nm,
which is the lattice parameter of pure iron in Abrahamson and Lopata’s work. The slope
of the line in Figure 2-5 is 9.21 × 10−5 𝑛𝑛𝑛𝑛/at. %.
𝑎𝑎 = 9.21 × 10−5 �
𝑛𝑛𝑛𝑛
� ∙ 𝐶𝐶𝐶𝐶𝐶𝐶 + 0.28662(𝑛𝑛𝑛𝑛)
𝑎𝑎𝑎𝑎. %
7
Figure 2-5 Change in lattice parameter of α iron with binary additions of chromium [2].
Though Abrahamson and Lopata did not present the original data which were plotted in
Figure 2-5, the data points in Figure 2-5 were measured and converted into nm unit in
order to reproduce the linear least squares regression analysis, which is shown in Figure
2-6. The resulting equation for a linear least squares regression analysis, with the origin
set as 0.28662 nm is:
23
𝑛𝑛𝑛𝑛
𝑎𝑎 = 9.4 × 10−5 �
� ∙ 𝐶𝐶𝐶𝐶𝐶𝐶 + 0.28662(𝑛𝑛𝑛𝑛)
𝑎𝑎𝑎𝑎. %
8
The slope is 9.4 × 10−5 𝑛𝑛𝑛𝑛/at. %, which is slightly greater than the value reported by
Abrahamson and Lopata. The standard error of the slope is 3 × 10−6 𝑛𝑛𝑛𝑛/𝑎𝑎𝑎𝑎. % .
According to Eq. 8, the lattice parameter of α iron alloyed with 1.0 at.% Cr is
(0.28671±0.000002) nm, and that of α iron alloyed with 1.8 at.% Cr is
(0.28679±0.000002) nm.
Figure 2-6. Change in lattice parameter of α iron with binary additions of chromium plotted
with measured data points [2].
Figure 2-7 shows Abrahamson and Lopata’s results compared with those found by Sutton
and Hume-Rothery. Abrahamson and Lopata’s results indicate a larger increase in the
24
lattice parameters of α iron with binary addition of Cr.
Abrahamson and Lopata measured the lattice parameters using an X-ray diffractometer
with molybdenum radiation, at a temperature maintained at 22° ±1℃. The 2θ values of the
P
diffraction peaks were simply determined by bisection of the peak. A computer program
(not described in detail [2]) was used to extrapolate the lattice parameter values
calculated by each diffraction peak to θ=90° where the error of lattice parameter was
minimized. Sutton and Hume-Rothery’s X-ray measurement was conducted with cobalt
radiation, and the experimental temperature was 20℃. It was mentioned in the literature
[1] that the powder photographs were taken in two 19 cm Unicam cameras, and the exact
lattice parameters were determined by standard extrapolation methods. Neither the
assumption nor the mathematical function of the standard extrapolation methods was
discussed. The analysis method of data as well as the experimental condition could result
in the discrepancy between the results measured by the two research groups.
25
Figure 2-7. Change in lattice parameter of α iron with binary additions of chromium from
Sutton and Hume-Rothery’s [1] and Abrahamson and Lopata’s data [2].
2.3 Lattice Parameter of γ-Fe (Austenite) in Literature
The lattice parameter of austenite at elevated temperatures has been measured by quite a
few research groups [7, 17-20]. The variation of the lattice parameter of austenite 𝑎𝑎𝛾𝛾
with temperature were summarized in Onink’s paper [20]. As the lattice parameter of
austenite 𝑎𝑎𝛾𝛾 obeys a linear relationship with temperature, a linear least squares analysis
was performed on their data, which are listed in Table 2. The linear fit lines of these data
are plotted in Figure 2-8. It is obvious that some scatter exists in these literature data. The
first four data, by Basinski (1955) [7], Goldschmidt (1962) [17], Kohlhaas (1967) [18],
and Gorton (1965) [19], were acquired by using X-ray diffraction, while the last data was
26
measured by high temperature neutron diffraction by Onink (1993) [20]. Basinski (1955)
measured the lattice parameter of γ-Fe almost in the entire γ-phase temperature region,
while Onink (1993) only did the measurement between 907℃-977℃.
In this study, the lattice parameter of the γ-phase pure iron (AHC 100.29 powder) was
measured by high temperature X-ray diffractometer. The reproducibility of the results in
Table 6 can then be confirmed if the resulting least squares linear regression fitting curve
of γ-phase pure iron (AHC 100.29 powder) lies in the range of the other lines in Figure
2-8.
Table 2. Lattice parameters and thermal expansion coefficients for Fe austenite in literature [7,
17-20].
Reference
𝑎𝑎𝛾𝛾 (𝑇𝑇)
Temperature
Range (℃)
Basinski [7]
916-1388
Goldschmidt [17]
912-1255
Kohlhaas [18]
950-1361
Gorton [19]
920-1070
Onink [20]
907-977
𝑛𝑛𝑛𝑛
)∙T
℃
𝑛𝑛𝑛𝑛
0.35753(nm) + 7.80902 × 10−6 ( ) ∙ T
℃
𝑛𝑛𝑛𝑛
0.35838(nm) + 7.38413 × 10−6 ( ) ∙ T
℃
𝑛𝑛𝑛𝑛
0.35826(nm) + 7.08650 × 10−6 ( ) ∙ T
℃
𝑛𝑛𝑛𝑛
0.35668(nm) + 8.97104 × 10−6 ( ) ∙ T
℃
0.35680(nm) + 8.56751 × 10−6 (
27
Figure 2-8. Literature data of lattice parameter of austenite as a function of temperature [7,
17-20].
28
Chapter Three.
Thesis Objectives
The ARPA-E project TEN Mare, which stands for Transformation Enabled Nitride
Magnets absent rare earth, is developing a highly magnetic iron nitride alloy, α”-Fe 16 N 2
phase, which can be used in magnets for electric vehicles and renewable power
generators. Nitrogen austenite is the precursor of the α”-Fe 16 N 2 phase. In order to obtain
100% α”-Fe 16 N 2 phase as the product, the nitrogen content in the nitrogen austenite has
to be exactly at 11.1 at.%.
The maximum nitrogen solubility in α-phase in pure iron is about 0.3 at.% [21], however,
the maximum solubility of nitrogen in γ-phase iron is notably extended to 10.3 at.%
according to the equilibrium binary phase diagram [21] in Figure 3-1. According to the
study of the late Professor Gary. Michael, in Department of Materials Science and
Engineering of Case Western Reserve University, Calphad calculations predict alloying
iron with chromium can enhance the solubility of nitrogen in austenite. Therefore, in
addition to the pure iron powder, two types of Fe – Cr alloy powders with different Cr
compositions are used in this project.
29
Figure 3-1. Equilibrium binary phase diagram of Fe-N system [21].
It is very critical to measure the nitrogen content in nitrogen austenite. The most
straightforward way of determining interstitial nitrogen content is by calculating the
lattice parameter of the nitrogen austenite from the X-ray diffraction peak positions. The
linear relation between the lattice parameter of nitrogen austenite and nitrogen content in
pure iron was determined by Jack in 1973 [22]. As described in Chapter Two, the change
in lattice parameter of α-Fe with binary additions of Cr less than 6 at.% was studied by
Abrahamson and Lopata [2], and Sutton and Hume-Rothery [1]. Various groups
investigated the linear relationship between the γ-phase parameter and temperature and
the thermal expansion coefficient for the γ-phase. Nonetheless, there has been no study
30
about the lattice parameter of the γ-phase with a binary addition of Cr, or the lattice
parameter of the γ-phase for Fe – Cr alloys as a function of nitrogen content.
In order to know the interstitial nitrogen content in the γ-phase, the lattice parameter of
the γ-phase with no nitrogen interstitial atoms at room temperature should be determined
first. High temperature X-ray diffraction measurements were conducted on AHC 100.29,
AMES Fe – Cr powder and Astaloy CrA powder in the high temperature region, and then
the γ-phase lattice parameter was extrapolated to room temperature.
31
Chapter Four. Experimental Procedure
4.1 Experimental Equipment Introduction
The lattice parameter of the α-phase and the γ-phase of all the samples was monitored by
a high temperature X-ray diffractometer (Scintag X-1 advanced X-ray diffractometer).
The setup of the system is shown in Figure 4-1 (a), particularly the hot stage chamber
(Edmund Bühler HDK 2.4) where the samples were mounted, with electrical and water
cooling connections attached. Ultra high purity 5.0 grade nitrogen gas was fed into the
chamber and flowed out through a bubbler. The interior chamber of the Scintag X-ray hot
stage is shown in Figure 4-1 (b). The X-rays go through the beryllium X-ray window as
marked in Figure 4-1 (b). Powder samples were placed onto the molybdenum heating
element.
(a)
(b)
Figure 4-1. (a) The exterior appearance and (b) interior chamber of high temperature X-ray system.
32
The front and back view of the sample holder is shown in Figure 4-2. The type S PtRh
(EL10) thermocouple is welded to the back of the molybdenum holder, so the
temperature of the sample surface is typically different from the thermocouple reading. A
similar high temperature lattice parameter experiment was performed on Al 2 O 3 powder
to calibrate the temperature, resulting in determination of a temperature offset, which will
be discussed in section 5.5.
(a) Front view of thermocouple
(b) Back view of thermocouple
Figure 4-2. (a)The front and (b) back view of the thermocouple on the sample holder.
4.2 Determination of Temperature Offset
The configuration of the sample carrier is shown in Figure 4-2. The thermocouple was
welded to the bottom surface of the sample carrier, while the samples are spread evenly
on the top surface of the sample carrier, thus it is plausible that the thermocouple reading
temperatures would be different from the sample temperature. Hence, the determination
33
of the temperature offset is of crucial importance.
The temperature offset was determined by comparing the measured lattice parameters of
a well known material, Al 2 O 3 , with the accepted values found in the [22. The high
temperature experiment on Al 2 O 3 was done twice in order to demonstrate that the
relationship between the thermocouple reading and the sample temperature was
reproducible.
Because the main purpose of this study was to obtain the lattice parameter of the γ-phase
of iron powder, or Fe – Cr powders, the temperature region of interest is above 900℃.
Hence, experiments on Al 2 O 3 were conducted from 890℃ to 1410℃.
4.3 High Temperature X-ray Experiment Procedure
After placing the powder evenly onto the molybdenum sample carrier and adjusting the
sample height, an X-ray scan (Scintag X-1 Cu source) was conducted on Astaloy AHC
100.29 iron powder to obtain a diffraction pattern from which the lattice parameter of
pure α iron at room temperature can be determined. The start and stop angles of the scan
were set to 30° and 150° , step size was 0.02° , and the scan rate was 0.04 second / degree.
P
P
P
Under nitrogen gas flow, AHC 100.29 powder was heated by manually increasing the
current value of the power supply (LT-800 manufactured by Lambda Electronics Inc.).
34
After maintaining at each of the temperatures for ten minutes to stabilize the temperature
of the system, X-ray diffraction scans were performed on the sample. The start and stop
angles of the scan were set to 40° and 90° , step size was 0.05° , and the scan rate was 0.1
P
P
P
second / degree in order to decrease the time length of each scan, thereby reducing the
risk of oxidation of the sample powder or the sample carrier. The same procedures were
performed on Ames Fe-1Cr powder and Astaloy CrA powder.
35
Chapter Five. Results and Discussion
In the case of pure iron, phase transformations take place during heating and cooling in
the solid state. Both the α and δ phases are body-centered cubic, while the γ lattice is
face-centered cubic. Four special points in both iron and steel, A 1 , A 2 , A 3 , A 4 have been
identified [24]:
723℃: A 1 point, eutectoid transformation temperature.
769℃: A 2 point, Currie temperature of α iron.
912℃: A 3 point, α↔γ phase transition temperature.
1392℃: A 4 point, γ↔δ phase transition temperature.
The sequence of phase changes of pure iron is unique. The face-centered cubic structure
of the γ-phase is stable at temperatures between the A 3 and A 4 . The body-centered cubic
structure of the α and δ phases is stable at temperatures lower than A 3 and higher than A 4 ,
respectively. The volume, and thus the lattice parameter, of each unit cell of iron
increases in the transformation from α→γ, i.e. during heating at the A 3 point, but the
volume of the unit cell decreases by 50% in the transition from γ→δ, i.e. during heating
at the A 4 point [7]. However, the volume of the iron sample, bulk or powder, shrinks in
36
the transformation from α→γ, and expands in the transformation from γ→δ, because the
γ-phase structure has 2 more Fe atoms per unit cell than the α and δ phase structures.
5.1 Starting Material
The water atomized AHC 100.29 pure iron powder (supplier Höganäs) of particle size
less than 20μm was used to determine the lattice parameter of pure iron. The morphology
of AHC 100.29 pure iron powder was evaluated by SEM observation (FEI Nova Nanolab
200), which is shown in Figure 5-1. The SEM image of AHC 100.29 pure iron powder
showed irregular shaped particles with rough surfaces were present. X-ray diffraction
measurement was performed on the AHC 100.29 iron powder at room temperature, as
well as elevated temperatures. The measurements produced diffractograms with
diffraction peaks related to the crystal structure, and the lattice parameters were
calculated based on the positions of the peaks. The procedure of calculating the precise
lattice parameter will be discussed in detail in section 5.4.
37
Figure 5-1. SEM image of Astaloy AHC 100.29 iron powder.
The SEM image of Ames Fe-1Cr powder with particle size less than 20μm is shown in
Figure 5-2. The SEM image of Ames pure iron powder showed spherical particles with
smooth surfaces were present due to the gas atomization production process.
38
Figure 5-2. SEM image of Ames Fe-1Cr powder.
The Astaloy CrA (supplier Astaloy Höganäs) is a pure prealloyed Fe – Cr powder with
1.8 at.% Cr. The SEM image of Astaloy CrA powder of particle size less than 20 μm is
presented in Figure 5-3. Water atomization results in the irregular shape of particles with
rough surfaces.
39
Figure 5-3. SEM image of Astaloy CrA powder.
5.2 Adjustment of Sample Height
Differences in sample height influence the accuracy of the peak position of the X-ray
diffraction pattern. When using a standard sample holder for powdered samples, the top
surface of the powdered sample is in the exact right position of the focal plane. Since the
hot stage is not a standard sample holder, it is necessary to adjust the height of the hot
stage before each set of experiments. One of the sets of diffraction pattern acquired
during calibration, of the ferrite (110) peak, is shown in Figure 5-4. The adjustment of the
sample height was realized by adjusting the micrometer screw above the hot stage
chamber. In Figure 5-4, the position and shape of the ferrite (110) peak was changed after
40
each adjustment, in order from 1 to 5, and finally reach the same two theta degree of the
ferrite (110) peak as the value obtained by using standard sample holder. The peak shifted
to a higher degree when the sample surface was higher than the focal plane, and shifted to
a lower degree when it was lower than the focal plane. The adjustment of sample height
should be done prior to each X-ray measurement using the hot stage.
Turning the micrometer screw clockwise one semi-circle would raise the height of the hot
stage for 0.27 mm, and result in about 0.125° increase in the 2𝜃𝜃 degree of the ferrite
(110) peak. The mean thermal expansion coefficient of α-Fe from 0℃ to 916℃ is
14.8 × 10−6 /℃, and that of γ-Fe from 916℃ to 1388℃ is 24.7 × 10−6 /℃ [7]. For a
sample with thickness less than 1 mm, the increase in thickness, in the entire temperature
range from room temperature to 1300℃, would be less than 0.023 mm. And therefore the
diffraction peak shift resulted from the expansion of sample thickness should be less than
0.011°. The step size was set to be 0.05° for high temperature measurements, so a peak
shift less than 0.011° should be negligible.
41
Figure 5-4. Effect of sample displacement on the diffraction peak position and peak shape.
5.3 Determination of Errors
Error is defined by Webster as “the difference between a calculated or observed value and
the true value” [25]. There are two types of errors which are considered in this study: (1)
systematic error, and (2) random error.
Systematic errors generally come from measuring instruments because of faulty
calibration or bias on the part of the observer. These errors can be avoided typically by
calibrating the instrument using standard samples and operating the instrument correctly.
On the other hand, random errors originate from unknown and unpredictable changes in
42
the experiment. Random errors usually show a Gaussian normal distribution. The mean
value, m, of a number of measurements of the same quantity is the best estimate of that
quantity, and the standard error of the estimate is expressed as:
Standard Error =
𝜎𝜎 2 =
𝜎𝜎
√𝑁𝑁
1
∙ �(𝑥𝑥𝑖𝑖 − 𝑥𝑥̅ )2
𝑁𝑁 − 1
1
𝑥𝑥̅ = ∙ � 𝑥𝑥𝑖𝑖
𝑁𝑁
9
10
11
where 𝜎𝜎 is the standard deviation of the measurements, 𝑁𝑁 is the number of
measurement, 𝑥𝑥𝑖𝑖 is a random variable and 𝑥𝑥̅ is the mean value. The denominator of Eq.
10 is the number of degrees of freedom left after determining 𝑥𝑥̅ from N observations
[26].
In this study, the systematic errors mainly come from the temperature difference between
the sample surface and the sample holder, where the thermocouple is connected. The
determination and correction of systematic error will be discussed in detail in the next
chapter. The random error introduced by procedures, such as the subtle differences of
sample height, conditions of the X-ray diffractometer, and lattice parameter calculations,
were determined as described below.
The AHC 100.29 α-phase pure iron powder was scanned by an X-ray diffractometer at
43
room temperature ten times. After each experiment, the sample holder was emptied and
then reloaded with fresh iron powder. In the first five iterations of the X-ray scans, the
starting angle was 30° and the stop angle was 150°. The (110), (200), (211), (220), (310),
and (222) diffraction peaks of α iron lie in this 2θ region. The scan rate was 0.04 second /
degree and the step size was 0.02° . In another five X-ray measurements, the starting and
P
stop angles were 40° and 90° , respectively. The scan rate was increased to 0.1 second /
P
P
degree and the step size was increased to 0.05° . The faster X-ray scan setting parameters
P
were the same as the high temperature experiments, and the slower ones were the same as
the room temperature experiments. The peak data was analyzed using software Origin
9.0.0 [27] using a Gaussian function. The 2θ values of the ten X-ray diffraction patterns
are listed in Table 3 and Table 4.
Table 3. Data of X-ray diffraction peaks of α-phase pure iron with 0.04 second / degree scan rate
and 0.02° step size scanned from 30° to 150° .
P
P
P
2θ (degree)
Exp. #
(110)
(200)
(211)
(220)
(310)
(222)
1
44.7361
65.065
82.4025
99.0477
116.545
137.3161
2
44.7392
65.0803
82.4132
99.0401
116.4805
137.4996
3
44.7208
65.1186
82.4209
99.0477
116.5566
137.5914
4
44.7269
65.0956
82.4132
99.0432
116.5183
137.4079
44
5
44.7331
65.0803
82.4209
99.0432
116.5183
137.3161
Table 4. Data of X-ray diffraction peaks of α-phase pure iron with 0.1 second / degree scan rate
and 0.05° step size, scanned from 40° to 90° .
P
P
P
2θ (degree)
Exp. #
(110)
(200)
(211)
1
44.7278
65.0988
82.4050
2
44.7266
65.0786
82.4231
3
44.7312
65.0928
82.4079
4
44.7195
65.0851
82.4133
5
44.7243
65.0954
82.4237
The lattice parameters of the α iron were calculated based on the 2θ values above using
Bragg’s law. The calculated lattice parameters were plotted against the Nelson-Riley
function
𝑐𝑐𝑐𝑐𝑐𝑐2 θ
𝑠𝑠𝑠𝑠𝑠𝑠θ
+
𝑐𝑐𝑐𝑐𝑐𝑐2 θ
θ
, as shown in Figure 5-5 and Figure 5-6. The Bragg’s law and
Nelson-Riley extrapolation method will be discussed in detail in section 5.4. After
applying linear least squares regression analysis on each set of data points in Figure 5-5
and Figure 5-6, each set of data points results in a precise lattice parameter of α iron,
which is the value of the intercept of the linear-fit curve. The precise lattice parameters of
45
α-phase pure iron of the ten experiments are listed in Table 5.
Figure 5-5 Extrapolation of measured lattice parameters of AHC 100.29 powder against
Nelson-Riley function, calculated from the data in Table 3.
46
Figure 5-6 Extrapolation of measured lattice parameters of AHC 100.29 powder against
Nelson-Riley function, calculated from the data in Table 4.
Table 5 Lattice parameter of α-phase iron calculated from the data in Table 3 and Table 4.
Lattice parameter of α-phase iron
Lattice parameter of α-phase iron
calculated from the data in Table 3
calculated from the data in Table 4
1
0.28670
0.28672
2
0.28667
0.28669
3
0.28666
0.28674
4
0.28667
0.28668
5
0.28669
0.28664
47
Using Eq. 9, Eq. 10, and Eq. 11, the standard error of the lattice parameter of α-phase
iron obtained from the slower X-ray scan (data in Table 3) is 0.00001 nm, and that
obtained from the faster X-ray scan (data in Table 4) is 0.00002 nm. The experiment
settings of the slower scans are the same as all the room temperature X-ray experiment
settings, and those of the faster scans are the same as all the high temperature X-ray
experiment settings, so the standard error of room temperature and high temperature
X-ray tests are determined to be 0.00001 nm and 0.00002 nm, respectively.
5.4 Lattice Parameter Determination of AHC 100.29 Powder at Room Temperature
As described in Chapter 3, the lattice parameter of AHC 100.29 pure iron powder was
measured from 30° to 150° at room temperature by an X-ray diffractometer (Scintag
P
X-1 Cu Source). The diffraction pattern of AHC 100.29 pure iron powder at 20℃ is
shown in Figure 5-7.The process of determining the lattice parameter from the diffraction
pattern is straightforward, and high precision can be obtained. The interplanar spacing of
a certain plane in a crystal can be calculated from Bragg’s law [28]:
nλ = 2dsinθ
12
where n is an integer, λ is the wavelength of the incident X-ray photon, d is the
interplanar spacing, and θ is the angle between the incident ray and the scattering planes.
48
A Cu X-ray source was used which has a wavelength λ of 0.15418 nm.
Figure 5-7. Diffraction Pattern of AHC 100.29 pure iron powder at 20℃ by Scintag X-1 Cu
P
source.
The lattice parameter 𝑎𝑎 of a cubic lattice is proportional to the interplanar spacing 𝑑𝑑ℎ𝑘𝑘𝑘𝑘
of any particular set of lattice planes:
𝑑𝑑ℎ𝑘𝑘𝑘𝑘 =
𝑎𝑎ℎ𝑘𝑘𝑘𝑘
√ℎ2 + 𝑘𝑘 2 + 𝑙𝑙 2
13
The three integers h, k, l are the Miller indices of a family of lattice planes [28]. After
measuring the Bragg angle 𝜃𝜃 for this set of planes, d can be determined using Eq. 12,
and thus 𝑎𝑎 can be calculated from Eq. 13. However, it is 𝑠𝑠𝑠𝑠𝑠𝑠𝜃𝜃, not 𝜃𝜃, that is included in
the Bragg’s law. Therefore, the precision of 𝑑𝑑 and 𝑎𝑎 lies in the precision of 𝑠𝑠𝑠𝑠𝑠𝑠𝜃𝜃
49
instead of 𝜃𝜃 [28]. The value of sinθ changes slowly with 𝜃𝜃 when 𝜃𝜃 approaches
𝜋𝜋
2
(=1.57 radians=90° ), as shown in Figure 5-8, and therefore, in this region, the error in
P
𝑠𝑠𝑠𝑠𝑠𝑠𝜃𝜃 caused by a given error in 𝜃𝜃 decreases as 𝜃𝜃 increases. Hence, a very accurate
value of sinθ can be obtained from a measurement of 𝜃𝜃 which itself is not perfectly
𝜋𝜋
precise, if 𝜃𝜃 is in the neighborhood of 2 .
Figure 5-8. The variation of 𝐬𝐬𝐬𝐬𝐬𝐬𝐬𝐬 with 𝛉𝛉.
Unfortunately, the X-ray scan cannot be performed at or close to an angle of
𝜋𝜋
2
because
of equipment limitation. The X-ray tube and the detector would collide at a high angle
𝜋𝜋
close to 2 , due to the geometry of the conventional Scintag X-ray diffractometer . Since
the estimated values of 𝑑𝑑, as well as 𝑎𝑎, approach the true value more closely as 𝜃𝜃
increases, the true value of 𝑑𝑑 and 𝑎𝑎 can be obtained by simply plotting the measured
50
values against 2𝜃𝜃 and extrapolating to 2𝜃𝜃 = 𝜋𝜋, though this curve is not linear. A more
precise method is to plot the measured value of 𝑎𝑎 against certain functions of 𝜃𝜃, rather
than against 𝜃𝜃 directly. The function
𝑐𝑐𝑐𝑐𝑐𝑐2 θ
𝑠𝑠𝑠𝑠𝑠𝑠θ
+
𝑐𝑐𝑐𝑐𝑐𝑐2 θ
θ
, which was first reported by Nelson
and Riley [29] in 1944 holds quite accurately down to very low values of 𝜃𝜃 and not just
at high angles. The calculated lattice parameter 𝑎𝑎 was plotted against the Nelson-Riley
function
𝑐𝑐𝑐𝑐𝑐𝑐2 θ
𝑠𝑠𝑠𝑠𝑠𝑠θ
+
𝑐𝑐𝑐𝑐𝑐𝑐2 θ
θ
in Figure 5-9. A linear least squares regression model was utilized
to fit the curve and extrapolate to 2𝜃𝜃 = 𝜋𝜋, i.e.
𝑐𝑐𝑐𝑐𝑐𝑐2 θ
𝑠𝑠𝑠𝑠𝑠𝑠θ
+
𝑐𝑐𝑐𝑐𝑐𝑐2 θ
θ
= 0. The precise lattice
parameter of AHC 100.29 powder obtained by this method is (0.28663±0.00001) nm.
This value is in good agreement with (1) the value of 0.28664 nm in Pearson’s handbook
[6], (2) the value of (0.28662±0.00002) nm by Abrahamson’s work [2], and (3) the value
of 0.28662(1) nm reported by Hume-Rothery [7].
51
Figure 5-9. Extrapolation of measured lattice parameters of AHC 100.29 powder against
Nelson-Riley function.
5.5 Temperature Offset Determination
The general method used was described in Section 4.2. The space group of Al 2 O 3 is 𝑅𝑅3� 𝑐𝑐,
most commonly referred to a hexagonal unit cell. The calculation of the interplanar
spacing in hexagonal unit cell is
1
4 ℎ2 + ℎ𝑘𝑘 + 𝑘𝑘 2 𝑙𝑙 2
=
×
+ 2
𝑑𝑑 2 3
𝑎𝑎2
𝑐𝑐
14
where h,k,l are the Miller indices of a certain plane, and a and c are the lattice parameters
of the a-axis and the c-axis in the hexagonal unit cell. Therefore, hk0 data are required for
the a-axis and 00l for the c-axis parameters. However, there were only one hk0 data and
52
one 00l data collected in the 2θ range from 40° to 80° . Therefore the lattice parameter of
P
P
Al 2 O 3 was determined using the software CrystalDiffract version 6.0.5 (200) for
Windows [3], instead of using Nelson Riley extrapolation method.
The unit cell structure of Al 2 O 3 , created by CrystalMaker version 9.0.3(619) for
Windows, is shown in Figure 5-10. The c parameter, OC in Figure 5-10, was used for
comparison. The data reported by Munro [23] and the data from the two previously
described experiments were plotted in Figure 5-11. The reproducibility of the temperature
of the samples is well demonstrated by the high agreement of the c lattice parameter
values of Al 2 O 3 samples in the two experiments. A linear least squares regression
analysis was performed on data points of experiment 1 and experiment 2 (11 data points
in total). The resulting equation is:
c = 1.34 × 10−6 (
𝑛𝑛𝑛𝑛
) ∙ 𝑇𝑇 + 0.12954 (𝑛𝑛𝑛𝑛)
℃
15
and the standard error of the intercept and the slope are, 4 × 10−5 𝑛𝑛𝑛𝑛 and 3 ×
10−8 𝑛𝑛𝑛𝑛/℃ respectively.
The resulting equation for linear least squares regression analysis performed on Munro’s
data is:
53
c = 1.342 × 10−6 �
𝑛𝑛𝑛𝑛
� ∙ 𝑇𝑇 + 0.129680(nm)
℃
16
with a standard error of the intercept and the slope of 8 × 10−6 𝑛𝑛𝑛𝑛 and
6 × 10−9 𝑛𝑛𝑛𝑛/℃, respectively.
Combing Eq. 15 and Eq. 16 to calculate the temperature offset,
𝑛𝑛𝑛𝑛
1.34 × 10−6 � ℃ � ∙ 𝑇𝑇𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 + 0.12954 (nm)
17
𝑛𝑛𝑛𝑛
= 1.342 × 10−6 � ℃ � ∙ 𝑇𝑇𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 + 0.129680(nm)
within the error of the intercepts and slopes in Eq. 15 and Eq. 16, yields the following
equation for the sample temperature:
𝑇𝑇𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 = 𝑇𝑇𝑚𝑚𝑒𝑒𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎 − 104.3 (℃)
18
The standard error of 𝑇𝑇𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 is ±0.4oC. This means there is a constant 104.3oC offset
between 𝑇𝑇𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 and 𝑇𝑇𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 .
54
Figure 5-10. Unit cell structure of Al 2 O 3 .
Figure 5-11. Comparison between literature value [23] of c lattice parameter of Al 2 O 3 and
experimental values.
5.6 Variation of Lattice Parameter with Temperature
55
X-ray diffraction patterns of the three powders were obtained during heating. The more
diffraction peak information that is obtained, the better the precision of the calculation of
lattice parameter will be at each temperature. However, on the other hand, thermal
agitation decreases the intensity of a diffracted beam because it has the effect of smearing
out the lattice planes, and thermal vibration causes a greater decrease in the reflected
intensity at high angles than at low angles [28]. The high angle diffraction peaks, the ones
at 2θ degree larger than 90° , of both α-phase and γ-phase, are of low intensity even at
P
room temperature. At the same time, although the experiment was conducted under
nitrogen flow, the possibility of the existence of slight amount of oxygen in the system
cannot be ruled out, so an overlong scan time would cause oxidation of the sample
powder as well as the sample carrier. Therefore, the start and stop angles of the scan were
set to 40° and 90° , step size was 0.05° , and the scan rate was 0.1 second / degree. Ferrite
P
P
P
(110), (200), and (211) peaks, and austenite (111), (200), (220), and (311) peaks, lie in
this angle range. Each X-ray scan took about 10 minutes.
5.6.1
Measurements of AHC 100.29 Iron Powder at High Temperatures
In this work, temperatures of the samples were increased by adjusting the current value of
the power supply. Twenty X-ray diffraction patterns were measured for the AHC 100.29
powder. Each intensity spectrum was fitted by employing a Gaussian function as the peak
56
profile, using the software package Origin 9.0.0. For each resolved peak the
corresponding lattice parameter was calculated using Bragg’s law Eq. 12, and then Eq. 13.
𝑐𝑐𝑐𝑐𝑐𝑐2 θ
The lattice parameters were then plotted against the function ( 𝑠𝑠𝑠𝑠𝑠𝑠θ +
𝑐𝑐𝑐𝑐𝑠𝑠2 θ
θ
), and
extrapolated to zero. The intercept thus derived is the precise lattice parameter of the
sample powder at the set temperature. Temperatures were corrected using Eq. 18, and the
“true” temperatures are listed in the Appendix.
The coexistence of diffraction peaks from both the α-phase and the γ-phase occurred at
two temperatures. According to Gibbs phase rule, the pure iron sample couldn’t be in a
two-phase region. Therefore, the sample did not reach equilibrium at these temperatures.
The AHC 100.29 iron powder underwent a phase transformation from the α-phase to the
γ-phase, starting from a temperature between 905.7℃ to 931.7℃ , and ending at a
P
P
temperature between 954.7℃ to 980.7℃ , thus the “A 3 point” lies in the temperature
P
P
interval between 905.7℃ to 931.7℃ .
P
P
The black data points in Figure 5-12 show the variation of lattice parameter of AHC
100.29 iron with temperature. The α-phase diffraction peaks didn’t disappear until the
temperature rose above 954.7℃ , however, only the strongest peaks, diffracted from (110)
P
planes in α-Fe, were clearly observable in diffraction patterns obtained at 931.7℃ and
P
954.7℃ . Since the lattice parameter values derived by the 2θ position of the strongest
P
57
peak are not necessarily very accurate, the two lattice parameter data points at 931.7℃
P
and 954.7℃ are not in a line with the data points of the α-phase at lower temperatures in
P
Figure 5-12. The calculation of the lattice parameter of the α-phase did not include the
data points at 931.7℃ and 954.7℃ due to the lack of accuracy. All the temperatures
P
P
shown in Figure 5-12 are the corrected temperatures.
Figure 5-12. Resulting comparison between the lattice expansion of iron with temperature in this
study and in Sutton and Hume-Rothery [1].
The data points of Sutton and Hume-Rothery [1] are also plotted in Figure 5-12. For each
phase of the data in this study and in Sutton and Hume-Rothery’s study, linear least
squares regression analysis of the data according to Eq.19 was performed.
58
𝑎𝑎 = 𝑆𝑆 ∙ 𝑇𝑇 + 𝑎𝑎0
19
The results are shown in Table 6. Both Figure 5-12 and Table 6 visually and qualitatively
indicate that the results of Hume-Rothery and Sutton were reproduced in this study. The
calculated lattice parameter of the γ-phase and the α-phase in AHC 100.29 pure iron
powder at 20℃ are (0.3572±0.0005) nm and (0.2866±0.0001) nm, respectively in this
P
study, within the error of Hume-Rothery and Sutton’s value (0.35694±0.00007) nm and
(0.28659±0.00002) nm.
Table 6. Comparison of the result for linear least squares regression analysis performed on data of
AHC 100.29 and Sutton and Hume-Rothery.
Phase
Standard
Error for S
(nm/℃ )
𝑎𝑎0 (nm)
4.7 × 10−6
2 × 10−7
0.2865
4.30 × 10−6
3 × 10−8
0.2865
S (nm/℃ )
P
P
AHC 100.29
α
γ
Sutton &
α
Hume-Rothery
γ
8.5 × 10−6
8.61 × 10−6
4 × 10−7
0.3570
6 × 10−8
0.3568
Standard
Error for
𝑎𝑎0 (nm)
1 × 10−4
5 × 10−4
2 × 10−5
7 × 10−5
The data points of AHC 100.29 pure iron in the γ-phase region were plotted in Figure
5-13, together with the literature values, previously mentioned in Table 2 in Section 2.3.
The variation of the lattice parameter of the γ-phase with temperature lies exactly in the
59
range of the literature values.
Figure 5-13. Comparison of the variation of lattice parameter of the γ-phase with temperature
between AHC 100.29 pure iron powder and literature values.
5.6.2
Measurement of AMES Fe – Cr Powder at High Temperatures
The precise lattice parameter of the α-phase of Ames Fe – Cr powder at 20℃ was
P
determined to be (0.28666±0.00001) nm. The data of the high temperature experiment on
Ames Fe – Cr powder is plotted in Figure 5-14. Nineteen diffraction patterns of Ames Fe
– Cr powder were collected at high temperatures. The γ peaks were first observed when
the temperature increased to 939.7℃ , and the α peaks didn’t disappear until 952.7℃ .
P
P
Therefore, the phase transformation from α↔γ started somewhere between 917.7℃ to
P
60
939.7℃ , and ended at a temperature between 952.7℃ to 980.7℃ . The “A 3 point” in
P
P
P
Ames Fe – Cr powder is thus determined to be between 917.7℃ and 939.7℃ . A linear
P
P
least squares regression analysis based on Eq. 19 was performed on the data, and the
results are shown in Table 7. According to Eq. 19, the lattice parameter of the γ-phase and
the α-phase of Ames Fe – Cr powder at 20℃ is (0.3604±0.0003) nm, and
P
(0.2868±0.0001) nm, respectively.
Figure 5-14. Variation of lattice parameter with temperature of AMES Fe – Cr sample.
Table 7. Result for linear least squares regression analysis performed on the data of Ames Fe –
Cr sample.
61
AMES FeCr
Phase
S (nm/℃)
Standard
Error for S
(nm/℃)
𝑎𝑎0 (nm)
α
4.2 × 10−6
2 × 10−7
0.2867
γ
5.6.3
5.2 × 10−6
2 × 10−7
0.3603
Standard
Error for
𝑎𝑎0 (nm)
1 × 10−4
3 × 10−4
Measurement of CrA Fe – Cr Powder at High Temperatures
The same experimental procedure was conducted on CrA iron powder. The precise lattice
parameter of the α-phase of CrA iron powder at 20℃ was determined to be
P
(0.28686±0.00001) nm. Twenty-one diffraction patterns of CrA powder were collected at
high temperatures. The data point at 979.7℃ is not in a line with the other data points of
P
the γ-phase due to the inaccurate calculation of lattice parameter from only one
diffraction peak. This data point was not included in the calculation of the lattice
parameter of the γ-phase of Astaloy CrA Fe – Cr powder at 20℃ .
P
At elevated temperatures, the γ-phase diffraction peaks were first observed at 979.7℃ ,
P
and the α-phase diffraction peaks did not disappear until temperature was raised above
1005.7℃ . Therefore, the phase transformation from α↔γ started somewhere between
P
941.7℃ to 979.7℃ , and ended at a temperature between 1005.7℃ to 1034.7℃ . So the
P
P
P
P
“A 3 point” in Astaloy CrA Fe – Cr powder should lie between 941.7℃ to 979.7℃ , which
P
62
P
is higher than in pure iron powder. The increase of the apparent “A 3 point” may due to
the slower kinetics of phase transformation from α-phase to γ-phase. Such effect can be
resulted from the decrease of interface mobility with the increase of Cr content. The
variation of the lattice parameter with the temperature of Astaloy CrA powder is plotted
in Figure 5-15. Linear least squares regression analysis of the fit to Eq. 19 was performed
on the data of Astaloy CrA powder as well. The results are listed in Table 8. According to
Eq. 19, the lattice parameters of the γ-phase and the α-phase of Astaloy CrA powder at
20oC are (0.3609±0.0003) nm and (0.2869±0.0002) nm, respectively.
The lattice parameters of the γ-phase, and the α-phase in AHC 100.29 pure iron powder,
Ames Fe – Cr (1.0 at.% Cr) powder, and Astaloy CrA Fe – Cr (1.8 at.% Cr) powder are
listed in Table 9.
63
Figure 5-15. Variation of lattice parameter with temperature of Astaloy CrA Fe – Cr powder.
Table 8. Results of linear least squares regression analysis performed on the Astaloy CrA sample
data.
Phase
S (nm/℃ )
P
Standard
Error for S
(nm/℃ )
𝑎𝑎0 (nm)
2 × 10−7
0.2868
Standard
Error for
𝑎𝑎0 (nm)
P
Astaloy CrA
α
γ
4.0 × 10−6
4.4 × 10−6
2 × 10−7
2 × 10−4
3 × 10−4
0.3608
Table 9. Summary of lattice parameters of the γ-phase and the α-phase at 20℃ of AHC 100.29,
P
Ames Fe – Cr, and Astaloy CrA samples.
Sample
Lattice Parameter of
γ-Phase at 20℃ (nm)
Lattice Parameter of
α-Phase at 20℃ (nm)
0.3572±0.0005
0.2866±0.0001
P
AHC 100.29
64
P
AMES Fe-Cr (1.0 at. pct. Cr)
0.3604±0.0003
0.2868±0.0001
Astaloy CrA Fe-Cr (1.8 at. pct. Cr)
0.3609±0.0003
0.2869±0.0002
The extrapolated lattice parameters of the α-phase in AHC 100.29 pure iron powder,
Ames Fe – Cr (1.0 at.% Cr) powder, and Astaloy CrA Fe – Cr (1.8 at.% Cr) powder at
20℃ in Table 9 are plotted together with Abrahamson’s data [2] in Figure 5-16. The
P
extrapolated lattice parameters agree with Abrahamson’s data within errors.
Figure 5-16. Comparison of the lattice parameter of the α-phase between Abrahamson’s data [2]
and the extrapolated values in Table 9.
The data of the lattice parameters of the γ-phase at 20℃ in Table 9 are plotted in Figure
P
5-17.
65
Figure 5-17. Variation of lattice parameter of γ-phase with different chromium content.
A linear least squares regression analysis was performed on the γ-iron lattice parameter
data in Table 9. The variation of the γ-iron lattice parameter with binary addition of Cr is:
𝑎𝑎 = 0.0021 �
𝑛𝑛𝑛𝑛
� × 𝐶𝐶𝐶𝐶𝐶𝐶 + 0.3575 (𝑛𝑛𝑛𝑛)
𝑎𝑎𝑎𝑎. %
20
The standard error of the slope and the intercept are 7 × 10−4 𝑛𝑛𝑛𝑛/𝑎𝑎𝑎𝑎. % and 9 ×
10−4 𝑛𝑛𝑛𝑛, respectively.
66
Figure 5-18 Variation of temperature interval of the “A 3 point” with different chromium
content.
According to the equilibrium phase diagram of Fe – Cr binary system [30], Figure 5-19,
in the low Cr region, the “A 3 temperature” decreases with the increase of Cr content. Yet,
Figure 5-18 shows the binary addition of chromium into iron increases the apparent A 3
temperature. On the other hand, the coexistences of the diffraction peaks of the α-phase
and the γ-phase in two or three sets of data in all three samples indicate the high
temperature X-ray measurements were conducted under conditions that deviated from
equilibrium. Wits [31] investigated experimentally the effect of the composition of binary
alloying elements on the ferrite to austenite transformation kinetics in iron alloy. The
interface mobility, 𝑀𝑀, is given by
67
M = 𝑀𝑀0 ∙ exp(−
𝑄𝑄
)
𝑅𝑅𝑅𝑅
𝑀𝑀0 = 𝛿𝛿𝑓𝑓 ∗ /𝑅𝑅𝑅𝑅
𝑓𝑓 ∗ = 𝑘𝑘𝑘𝑘/ℎ
21
22
23
where 𝑀𝑀0 is a pre-exponential factor, 𝛿𝛿 is the atomic diameter, 𝑄𝑄 and 𝑓𝑓 ∗ are the
activation energy and frequency for the atoms crossing the interface. In pure iron, with
𝛿𝛿 ≈ 0.3𝑛𝑛𝑛𝑛, it follows that 𝑀𝑀0 ≈ 0.8 𝑚𝑚𝑚𝑚𝑚𝑚 ∙ 𝑚𝑚 ∙ 𝐽𝐽−1 ∙ 𝑠𝑠 −1. The interface mobility for Fe –
Cr alloy with 2 at.% Cr was determined to be (0.37 ± 0.07) 𝑚𝑚𝑚𝑚𝑚𝑚 ∙ 𝑚𝑚 ∙ 𝐽𝐽−1 ∙ 𝑠𝑠 −1 ,
indicating a slower ferrite to austenite transformation kinetics in Fe – 2Cr alloy than in
pure iron. This phenomenon can be caused by complicated effects such as the interatomic
interactions at the interface, the strain energy caused by density differences between the
two phases, or the segregation of solute atoms at the interface, etc. Therefore, the
transformations from ferrite to austenite show “higher” transformation temperatures and
longer time-lag in Ames Fe – Cr (1.0 at.% Cr) powder and Astaloy CrA Fe – Cr (1.8 at.%
Cr) powder than in AHC 100.29 pure iron powder.
68
Figure 5-19. Binary phase diagram of Fe – Cr system [30].
69
Chapter Six.
Conclusions
High temperature X-ray experiments were conducted on AHC 100.29 pure iron powder,
AMES Fe – Cr powder (1.0 at.% Cr), and Astaloy CrA powder (1.8 at.% Cr).
The systematic error in this study originated from the temperature difference between the
sample surface and the thermocouple reading temperature. The systematic error was
overcame by comparing the lattice parameter of Al 2 O 3 at elevated temperatures with
literature values, and the temperature difference between the real temperature of the
sample and the reading temperature was thus determined to be 104.3℃. The random error
mainly came from the subtle differences of experimental operation, the conditions of the
equipment, and the lattice parameter calculations. The random error was determined by
performing the entire procedure, including loading sample, performing X-ray scan, and
doing Nelson-Riley extrapolation, on pure iron powder for five times, and then the
standard errors of the calculated lattice parameters were determined to be 0.00001 nm
and 0.00002 nm for room temperature tests and high temperature tests, respectively.
The variations of the lattice parameter with temperature, from 800℃ to 1300℃ were
P
P
obtained for the three powder samples. Linear relationships between the lattice parameter
and temperature were observed and determined in all three samples. Linear least squares
regression analysis was performed on the data of the three samples in the γ-phase region.
70
The results are summarized in Table 10. And the lattice parameters of the γ-phase of the
three samples at 20℃ were determined by extrapolating the resulting function to 20℃ .
P
P
The values are (0.3572±0.0005) nm, (0.3604±0.0003) nm, and (0.3609±0.0003) nm for
pure iron powder, Fe – Cr powders with 1 at.% Cr and 1.8 at.% Cr, respectively. The
variation of the γ-iron lattice parameter with binary addition of Cr is:
𝑛𝑛𝑛𝑛
𝑎𝑎 = 0.0021 �
� ∙ 𝐶𝐶𝐶𝐶𝐶𝐶 + 0.3575(𝑛𝑛𝑛𝑛)
𝑎𝑎𝑎𝑎. %
Table 10. Results of linear least squares regression analysis performed on the data of AHC pure
iron sample, Ames Fe – Cr sample, and Astaloy CrA Fe – Cr sample.
Sample
AHC 100.29
Phase
S (nm/oC)
α
4.7 × 10−6
2 × 10−7
0.2865
4.2 × 10−6
2 × 10−7
0.2867
2 × 10−7
0.2868
γ
AMES FeCr
α
γ
Astaloy CrA
𝑛𝑛𝑛𝑛
) ∙ 𝑇𝑇 + 𝑎𝑎0 (𝑛𝑛𝑛𝑛)
℃
Standard Error
𝑎𝑎0 (nm)
o
for S (nm/ C)
𝑎𝑎 = 𝑆𝑆(
α
γ
8.5 × 10−6
5.2 × 10−6
4.0 × 10−6
4.4 × 10−6
2 × 10−7
0.3570
2 × 10−7
0.3603
2 × 10−7
0.3608
Standard Error For
𝑎𝑎0 (nm)
1 × 10−4
5 × 10−4
1 × 10−4
3 × 10−4
2 × 10−4
3 × 10−4
In the data points near the α↔γ phase transformation temperature of all the three samples,
the occurrence of the diffraction peaks of both α and γ phase was observed, indicating the
71
X-ray measurements were conducted under conditions that deviated from equilibrium.
The temperature intervals where the γ-phase diffraction peaks appeared were 905.7℃ to
P
931.7℃ for AHC 100.29 pure iron powder, 917.7℃ to 939.7℃ for Ames Fe – Cr
P
P
P
powder, and 941.7℃ to 979.7℃ for CrA Fe – Cr powder. The temperatures at which the
P
P
phase transformations were observed increase with increasing amount of binary addition
of Cr into Fe. This phenomenon indicates a slower ferrite to austenite transformation
kinetics in Fe – Cr alloys than in pure iron, which can be explained by the slower
interface mobility for Fe – Cr than in pure iron.
The determinations of the room temperature lattice parameters of the nitrogen-free
γ-phase in the Fe – Cr samples provide the starting points in the relation between the
lattice parameter of γ-phase with interstitial nitrogen in two Fe – Cr powders.
72
Chapter Seven.
Suggestions for Future Work
In the process of analyzing and interpreting data, some limitations were identified which
present opportunities for future research. Section 5.5 presented the determination of
temperature offset between the thermocouple reading temperature and the real
temperature of the sample surface by comparing the measured lattice parameter of
another material Al 2 O 3 with the theoretical values. If possible, an attempt should be made
to perform the high temperature X-ray experiment using a hot stage with a more accurate
temperature measurements. The error of the results in this work can be reduced with a
more accurate temperature measurement system. Additionally, the hot stage could have
been operated under vacuum by using the turbo pump, and this may allow a longer time
length of X-ray exposure at high temperatures, and therefore result in smaller standard
error.
73
Appendix 1: AHC 100.29 Powder Lattice Parameter Data
Temperature (℃)
Lattice Parameter (nm)
Structure
838
0.29038
body-centered cubic
855
0.29059
body-centered cubic
872
0.29049
body-centered cubic
880
0.29062
body-centered cubic
906
0.29059
body-centered cubic
932
0.29199
body-centered cubic
932
0.36495
face-centered cubic
955
0.29171
body-centered cubic
955
0.36519
face-centered cubic
981
0.36562
face-centered cubic
1007
0.36539
face-centered cubic
1046
0.36585
face-centered cubic
1076
0.36620
face-centered cubic
1086
0.36636
face-centered cubic
1103
0.36648
face-centered cubic
1129
0.36616
face-centered cubic
74
1146
0.36678
face-centered cubic
1176
0.36686
face-centered cubic
1202
0.36723
face-centered cubic
1256
0.36783
face-centered cubic
1278
0.36784
face-centered cubic
1308
0.36833
face-centered cubic
75
Appendix 2: AMES Fe – 1Cr Powder Lattice Parameter Data
Temperature (℃)
Lattice Parameter (nm)
Structure
822
0.29021
body-centered cubic
831
0.29021
body-centered cubic
864
0.29031
body-centered cubic
918
0.29041
body-centered cubic
940
0.29063
body-centered cubic
940
0.36534
face-centered cubic
953
0.29073
body-centered cubic
953
0.36506
face-centered cubic
981
0.36548
face-centered cubic
995
0.36537
face-centered cubic
1026
0.36563
face-centered cubic
1058
0.36571
face-centered cubic
1082
0.36581
face-centered cubic
1094
0.36602
face-centered cubic
1116
0.36619
face-centered cubic
1156
0.36627
face-centered cubic
76
1189
0.36645
face-centered cubic
1235
0.36659
face-centered cubic
1250
0.36673
face-centered cubic
1277
0.36706
face-centered cubic
1304
0.36709
face-centered cubic
77
Appendix 3: Astaloy CrA Fe–Cr Powder Lattice Parameter
Data
Temperature (℃)
Lattice Parameter (nm)
Structure
821
0.29021
body-centered cubic
842
0.29023
body-centered cubic
874
0.29033
body-centered cubic
903
0.29040
body-centered cubic
942
0.29043
body-centered cubic
980
0.29073
body-centered cubic
980
0.36237
face-centered cubic
1006
0.29054
body-centered cubic
1006
0.36510
face-centered cubic
1035
0.36537
face-centered cubic
1062
0.36557
face-centered cubic
1080
0.36565
face-centered cubic
1125
0.36570
face-centered cubic
1165
0.36600
face-centered cubic
1177
0.36603
face-centered cubic
78
1195
0.36604
face-centered cubic
1221
0.36618
face-centered cubic
1244
0.36625
face-centered cubic
1257
0.36640
face-centered cubic
1288
0.36638
face-centered cubic
1303
0.36655
face-centered cubic
79
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