The Lattice Parameter of Gamma Iron and Iron – Chromium Alloys

Home , Ferrite (magnet)

THE LATTICE PARAMETER OF GAMMA IRON AND

IRON-CHROMIUM ALLOYS

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. THESIS OBJECTIVES ...... 29

CHAPTER FOUR. 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. CONCLUSIONS ...... 70

CHAPTER SEVEN. 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°PU step size scanned from 30°PU to 150°P...... 44

Table 4. Data of X-ray diffraction peaks of α-phase pure iron with 0.1 second / degree scan rate and

0.05°PU step size, scanned from 40°PU to 90°P...... 45

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℃PU of AHC 100.29, Ames

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℃PU by Scintag X-1 Cu source.

...... 49

Figure 5-8. The variation of with ...... 50

𝐬𝐬𝐬𝐬𝐬𝐬𝐬𝐬 𝛉𝛉

Figure 5-9. Extrapolation of measured lattice parameters of AHC 100.29 powder ...... 52

IX Figure 5-10. Unit cell structure of Al2O3...... 55

Figure 5-11. Comparison between literature value [23] of c lattice parameter of Al2O3 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 “A3 point” with different chromium content...... 67

Figure 5-19. Binary phase diagram of Fe – Cr system [30]...... 69

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

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.

The Lattice Parameter of Gamma Iron and Iron – Chromium Alloys

ABSTRACT

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

= 0.0021( ) × + 0.3575( ) squares regression analysis yielded: .% . 𝑛𝑛𝑛𝑛 𝑎𝑎 𝑎𝑎𝑎𝑎 𝐶𝐶𝐶𝐶𝐶𝐶 𝑛𝑛𝑛𝑛

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 .

℃

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 and , respectively, the relation between atomic

𝑎𝑎𝛾𝛾 𝑎𝑎𝛼𝛼 radius and the lattice parameters are:

4 = 2 1

𝑟𝑟𝐹𝐹𝐹𝐹−𝛾𝛾 √ 𝑎𝑎𝛾𝛾 4 = 3 2

𝑟𝑟𝐹𝐹𝐹𝐹−𝛼𝛼 √ 𝑎𝑎𝛼𝛼 Using Eq. 1 and the corresponding slopes and intercepts in Table 1, the lattice parameters

of γ Fe and α Fe at 20 P are (0.35696±0.00007) nm and (0.28658±0.00002) nm. The

℃ lattice parameters of γ Fe, , is 1.246 times the value of , which is a little bit larger

𝑎𝑎𝛾𝛾 𝑎𝑎𝛼𝛼 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.

(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 A4 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 A3 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 P, in between. The

℃ 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.

= + T 3

𝑎𝑎 𝑎𝑎0 𝑆𝑆 ∙

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 Value Standard Error

α Intercept (nm) 0.28649 2 × 10 −5 𝑎𝑎0 Slope S (nm/oC) 4.30 × 10 3 × 10 −6 −8 γ Intercept (nm) 0.35679 7 × 10 −5 𝑎𝑎0 Slope S (nm/oC) 8.59 × 10 6 × 10 −6 −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 AxB1-x,

( , ) = + (1 ) 4

𝑎𝑎 𝐴𝐴 𝐵𝐵 𝑥𝑥𝑎𝑎𝐴𝐴 − 𝑥𝑥 𝑎𝑎𝐵𝐵

The lattice constants of α-Fe and α-Cr at 20 P are (0.28662±0.00002) nm and

℃ (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:

( , ) = 0.28662 ( ) + (1 ) 0.2884( ) 5

𝑎𝑎 𝐹𝐹𝐹𝐹 𝐶𝐶𝐶𝐶 𝑥𝑥𝐹𝐹𝐹𝐹 ∙ 𝑛𝑛𝑛𝑛 − 𝑥𝑥𝐹𝐹𝐹𝐹 ∙ 𝑛𝑛𝑛𝑛 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,

= 5.4 × 10 + 0.286620( ) 6 . % −5 𝑛𝑛𝑛𝑛 𝑎𝑎 � � ∙ 𝐶𝐶𝐶𝐶𝐶𝐶 𝑛𝑛𝑛𝑛 𝑎𝑎𝑎𝑎 where is the lattice parameter of the α-phase Fe – Cr alloy and is the composition

𝑎𝑎 𝐶𝐶𝐶𝐶𝐶𝐶 of Cr in the alloy. The standard error of the intercept and slope is 4 × 10 nm and −6 1 × 10 nm / at.%, respectively. By using Eq. 6, the lattice parameters of the α-phase of −6 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

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 /at. %. −5 𝑛𝑛𝑛𝑛 = 9.21 × 10 + 0.28662( ) 7 . % −5 𝑛𝑛𝑛𝑛 𝑎𝑎 � � ∙ 𝐶𝐶𝐶𝐶𝐶𝐶 𝑛𝑛𝑛𝑛 𝑎𝑎𝑎𝑎

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:

= 9.4 × 10 + 0.28662( ) 8 . % −5 𝑛𝑛𝑛𝑛 𝑎𝑎 � � ∙ 𝐶𝐶𝐶𝐶𝐶𝐶 𝑛𝑛𝑛𝑛 𝑎𝑎𝑎𝑎 The slope is 9.4 × 10 /at. %, which is slightly greater than the value reported by −5 𝑛𝑛𝑛𝑛 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

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°P±1 . The 2θ values of the

℃ 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.

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 obeys a linear relationship with temperature, a linear least squares analysis

𝑎𝑎𝛾𝛾 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 Temperature ( )

𝑎𝑎𝛾𝛾 𝑇𝑇 Range ( )

℃ Basinski [7] 916-1388 0.35680(nm) + 8.56751 × 10 ( ) T −6 𝑛𝑛𝑛𝑛 ∙ Goldschmidt [17] 912-1255 0.35753(nm) + 7.80902 × 10 ( ℃ ) T −6 𝑛𝑛𝑛𝑛 ∙ Kohlhaas [18] 950-1361 0.35838(nm) + 7.38413 × 10 ( ℃ ) T −6 𝑛𝑛𝑛𝑛 ∙ Gorton [19] 920-1070 0.35826(nm) + 7.08650 × 10 ( ℃ ) T −6 𝑛𝑛𝑛𝑛 ∙ Onink [20] 907-977 0.35668(nm) + 8.97104 × 10 ( ℃ ) T −6 𝑛𝑛𝑛𝑛 ∙ ℃

Figure 2-8. Literature data of lattice parameter of austenite as a function of temperature [7, 17-20].

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”-Fe16N2

phase, which can be used in magnets for electric vehicles and renewable power

generators. Nitrogen austenite is the precursor of the α”-Fe16N2 phase. In order to obtain

100% α”-Fe16N2 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.

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

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.

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.

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 Al2O3 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, Al2O3, with the accepted values found in the [22. The high

temperature experiment on Al2O3 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 Al2O3 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°P and 150°P, step size was 0.02°P, and the scan rate was 0.04 second / degree.

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°P and 90°P, step size was 0.05°P, and the scan rate was 0.1

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.

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, A1, A2, A3, A4 have been

identified [24]:

723 : A1 point, eutectoid transformation temperature.

℃

769 : A2 point, Currie temperature of α iron.

℃

912 : A3 point, α↔γ phase transition temperature.

℃

1392 : A4 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 A3 and A4. The body-centered cubic

structure of the α and δ phases is stable at temperatures lower than A3 and higher than A4,

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 A3 point, but the

volume of the unit cell decreases by 50% in the transition from γ→δ, i.e. during heating

at the A4 point [7]. However, the volume of the iron sample, bulk or powder, shrinks in

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.

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.

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.

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 2 degree of the ferrite

𝜃𝜃 (110) peak. The mean thermal expansion coefficient of α-Fe from 0 to 916 is

℃ ℃ 14.8 × 10 / , and that of γ-Fe from 916 to 1388 is 24.7 × 10 / [7]. For a −6 −6 ℃ ℃ ℃ ℃ 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.

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

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 = 9 𝜎𝜎 1 = ( 𝑁𝑁) 10 1 √ 2 2 𝜎𝜎 1∙ � 𝑥𝑥𝑖𝑖 − 𝑥𝑥̅ 𝑁𝑁 −= 11

𝑥𝑥̅ ∙ � 𝑥𝑥𝑖𝑖 𝑁𝑁 where is the standard deviation of the measurements, is the number of

𝜎𝜎 𝑁𝑁 measurement, is a random variable and is the mean value. The denominator of Eq.

𝑥𝑥𝑖𝑖 𝑥𝑥̅ 10 is the number of degrees of freedom left after determining from N observations

𝑥𝑥̅ [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

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°P. In another five X-ray measurements, the starting and

stop angles were 40°P and 90°P, respectively. The scan rate was increased to 0.1 second /

degree and the step size was increased to 0.05°P. The faster X-ray scan setting parameters

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°P step size scanned from 30°P to 150°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

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°P step size, scanned from 40°P to 90°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 2 2 , as shown in Figure 5-5 and Figure 5-6. The Bragg’s law and 𝑐𝑐𝑐𝑐𝑐𝑐 θ 𝑐𝑐𝑐𝑐𝑐𝑐 θ 𝑠𝑠𝑠𝑠𝑠𝑠θ θ 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

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.

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

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°P at room temperature by an X-ray diffractometer (Scintag

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]:

n = 2dsin 12

λ θ 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.

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 P by Scintag X-1 Cu source. ℃

The lattice parameter of a cubic lattice is proportional to the interplanar spacing

𝑎𝑎 𝑑𝑑ℎ𝑘𝑘𝑘𝑘 of any particular set of lattice planes:

= 13 + + 𝑎𝑎ℎ𝑘𝑘𝑘𝑘 𝑑𝑑ℎ𝑘𝑘𝑘𝑘 2 2 2 √ℎ 𝑘𝑘 𝑙𝑙 The three integers h, k, l are the Miller indices of a family of lattice planes [28]. After measuring the Bragg angle for this set of planes, d can be determined using Eq. 12,

𝜃𝜃 and thus can be calculated from Eq. 13. However, it is , not , that is included in

𝑎𝑎 𝑠𝑠𝑠𝑠𝑠𝑠𝜃𝜃 𝜃𝜃 the Bragg’s law. Therefore, the precision of and lies in the precision of

49𝑑𝑑 𝑎𝑎 𝑠𝑠𝑠𝑠𝑠𝑠𝜃𝜃

instead of [28]. The value of sin changes slowly with when approaches

𝜃𝜃 θ 𝜃𝜃 𝜃𝜃 (=1.57 radians=90°P), as shown in Figure 5-8, and therefore, in this region, the error in 𝜋𝜋 2 caused by a given error in decreases as increases. Hence, a very accurate

𝑠𝑠𝑠𝑠𝑠𝑠𝜃𝜃 𝜃𝜃 𝜃𝜃 value of sin can be obtained from a measurement of which itself is not perfectly precise, if θ is in the neighborhood of . 𝜃𝜃 𝜋𝜋 𝜃𝜃 2

Figure 5-8. The variation of with .

𝐬𝐬𝐬𝐬𝐬𝐬𝐬𝐬 𝛉𝛉 Unfortunately, the X-ray scan cannot be performed at or close to an angle of because 𝜋𝜋 2 of equipment limitation. The X-ray tube and the detector would collide at a high angle close to , due to the geometry of the conventional Scintag X-ray diffractometer . Since 𝜋𝜋 2 the estimated values of , as well as , approach the true value more closely as

𝑑𝑑 𝑎𝑎 𝜃𝜃 increases, the true value of and can be obtained by simply plotting the measured 50 𝑑𝑑 𝑎𝑎

values against 2 and extrapolating to 2 = , though this curve is not linear. A more

𝜃𝜃 𝜃𝜃 𝜋𝜋 precise method is to plot the measured value of against certain functions of , rather

𝑎𝑎 𝜃𝜃 + than against directly. The function 2 2 , which was first reported by Nelson 𝑐𝑐𝑐𝑐𝑐𝑐 θ 𝑐𝑐𝑐𝑐𝑐𝑐 θ 𝑠𝑠𝑠𝑠𝑠𝑠θ θ and Riley [29𝜃𝜃] in 1944 holds quite accurately down to very low values of and not just

𝜃𝜃 at high angles. The calculated lattice parameter was plotted against the Nelson-Riley

𝑎𝑎 + function 2 2 in Figure 5-9. A linear least squares regression model was utilized 𝑐𝑐𝑐𝑐𝑐𝑐 θ 𝑐𝑐𝑐𝑐𝑐𝑐 θ 𝑠𝑠𝑠𝑠𝑠𝑠θ θ 2 = + = 0 to fit the curve and extrapolate to , i.e. 2 2 . The precise lattice 𝑐𝑐𝑐𝑐𝑐𝑐 θ 𝑐𝑐𝑐𝑐𝑐𝑐 θ 𝑠𝑠𝑠𝑠𝑠𝑠θ θ parameter of AHC 100.29 powder obtained𝜃𝜃 𝜋𝜋 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].

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 Al2O3 is 3 ,

𝑅𝑅�𝑐𝑐 most commonly referred to a hexagonal unit cell. The calculation of the interplanar spacing in hexagonal unit cell is

1 4 + + = × + 14 3 2 2 2 ℎ ℎ𝑘𝑘 𝑘𝑘 𝑙𝑙 2 2 2 𝑑𝑑 𝑎𝑎 𝑐𝑐 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°P to 80°P. Therefore the lattice parameter of

Al2O3 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 Al2O3, 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 Al2O3 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 ( ) + 0.12954 ( ) 15 −6 𝑛𝑛𝑛𝑛 ∙ 𝑇𝑇 𝑛𝑛𝑛𝑛 ℃ and the standard error of the intercept and the slope are, 4 × 10 and 3 × −5 𝑛𝑛𝑛𝑛 10 / respectively. −8 𝑛𝑛𝑛𝑛 ℃ The resulting equation for linear least squares regression analysis performed on Munro’s

data is:

c = 1.342 × 10 + 0.129680(nm) 16 −6 𝑛𝑛𝑛𝑛 � � ∙ 𝑇𝑇 ℃ with a standard error of the intercept and the slope of 8 × 10 and −6 𝑛𝑛𝑛𝑛 6 × 10 / , respectively. −9 𝑛𝑛𝑛𝑛 ℃ Combing Eq. 15 and Eq. 16 to calculate the temperature offset,

1.34 × 10 + 0.12954 (nm) 17 −6 𝑛𝑛𝑛𝑛 � ℃ � ∙ 𝑇𝑇𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 = 1.342 × 10 + 0.129680(nm) −6 𝑛𝑛𝑛𝑛 � ℃ � ∙ 𝑇𝑇𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 within the error of the intercepts and slopes in Eq. 15 and Eq. 16, yields the following

equation for the sample temperature:

= 104.3 ( ) 18

𝑇𝑇𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 𝑇𝑇𝑚𝑚𝑒𝑒𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎 − ℃ The standard error of is ±0.4oC. This means there is a constant 104.3oC offset

𝑇𝑇𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 between and .

𝑇𝑇𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑇𝑇𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡

Figure 5-10. Unit cell structure of Al2O3.

Figure 5-11. Comparison between literature value [23] of c lattice parameter of Al2O3 and experimental values.

5.6 Variation of Lattice Parameter with Temperature

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°P, of both α-phase and γ-phase, are of low intensity even at 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°P and 90°P, step size was 0.05°P, and the scan rate was 0.1 second / degree. Ferrite

(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

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.

( + ) The lattice parameters were then plotted against the function 2 2 , and 𝑐𝑐𝑐𝑐𝑐𝑐 θ 𝑐𝑐𝑐𝑐𝑠𝑠 θ 𝑠𝑠𝑠𝑠𝑠𝑠θ θ 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 P to 931.7 P, and ending at a

℃ ℃ temperature between 954.7 P to 980.7 P, thus the “A3 point” lies in the temperature

℃ ℃ interval between 905.7 P to 931.7 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 P, however, only the strongest peaks, diffracted from (110)

℃ planes in α-Fe, were clearly observable in diffraction patterns obtained at 931.7 P and

℃ 954.7 P. Since the lattice parameter values derived by the 2θ position of the strongest

℃ 57

peak are not necessarily very accurate, the two lattice parameter data points at 931.7 P

℃ and 954.7 P are not in a line with the data points of the α-phase at lower temperatures in

℃ Figure 5-12. The calculation of the lattice parameter of the α-phase did not include the

data points at 931.7 P and 954.7 P due to the lack of accuracy. All the temperatures

℃ ℃ 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.

= + 19

𝑎𝑎 𝑆𝑆 ∙ 𝑇𝑇 𝑎𝑎0 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 P are (0.3572±0.0005) nm and (0.2866±0.0001) nm, respectively in this

℃ 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 S (nm/ P) Standard (nm) Standard

Error for S Error for ℃ 𝑎𝑎0 (nm/ P) (nm)

AHC 100.29 α 4.7 × 10 2 × 10℃ 0.2865 1𝑎𝑎0× 10 −6 −7 −4 γ 8.5 × 10 4 × 10 0.3570 5 × 10 −6 −7 −4 Sutton & α 4.30 × 10 3 × 10 0.2865 2 × 10 −6 −8 −5 Hume-Rothery γ 8.61 × 10 6 × 10 0.3568 7 × 10 −6 −8 −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

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 P was

℃ 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 P, and the α peaks didn’t disappear until 952.7 P.

℃ ℃ Therefore, the phase transformation from α↔γ started somewhere between 917.7 P to

℃ 60

939.7 P, and ended at a temperature between 952.7 P to 980.7 P. The “A3 point” in

℃ ℃ ℃ Ames Fe – Cr powder is thus determined to be between 917.7 P and 939.7 P. A linear

℃ ℃ 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 P is (0.3604±0.0003) nm, and

℃ (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.

Phase S (nm/ ) Standard (nm) Standard

Error for S Error for ℃ 𝑎𝑎0 (nm/ ) (nm)

AMES FeCr α 4.2 × 10 2 × 10℃ 0.2867 1𝑎𝑎0× 10 −6 −7 −4 γ 5.2 × 10 2 × 10 0.3603 3 × 10 −6 −7 −4

5.6.3 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 P was determined to be

℃ (0.28686±0.00001) nm. Twenty-one diffraction patterns of CrA powder were collected at

high temperatures. The data point at 979.7 P is not in a line with the other data points of

℃ 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 P. Therefore, the phase transformation from α↔γ started somewhere between

℃ 941.7 P to 979.7 P, and ended at a temperature between 1005.7 P to 1034.7 P. So the

℃ ℃ ℃ ℃ “A3 point” in Astaloy CrA Fe – Cr powder should lie between 941.7 P to 979.7 P, which 62 ℃ ℃

is higher than in pure iron powder. The increase of the apparent “A3 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.

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 (nm) Standard

Error for S Error for ℃ 𝑎𝑎0 (nm/ P) (nm)

Astaloy CrA α 4.0 × 10 2 × 10℃ 0.2868 2𝑎𝑎0× 10 −6 −7 −4 γ 4.4 × 10 2 × 10 0.3608 3 × 10 −6 −7 −4

Table 9. Summary of lattice parameters of the γ-phase and the α-phase at 20 P of AHC 100.29, Ames Fe – Cr, and Astaloy CrA samples. ℃

Sample Lattice Parameter of Lattice Parameter of

γ-Phase at 20 P (nm) α-Phase at 20 P (nm)

AHC 100.29 0.3572±0.0005℃ 0.2866±0.0001℃

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 P in Table 9 are plotted together with Abrahamson’s data [2] in Figure 5-16. The

℃ 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 P in Table 9 are plotted in Figure

℃ 5-17.

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:

= 0.0021 × + 0.3575 ( ) 20 . % 𝑛𝑛𝑛𝑛 𝑎𝑎 � � 𝐶𝐶𝐶𝐶𝐶𝐶 𝑛𝑛𝑛𝑛 𝑎𝑎𝑎𝑎 The standard error of the slope and the intercept are 7 × 10 / . % and 9 × −4 𝑛𝑛𝑛𝑛 𝑎𝑎𝑎𝑎 10 , respectively. −4 𝑛𝑛𝑛𝑛

Figure 5-18 Variation of temperature interval of the “A3 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 “A3 temperature” decreases with the increase of Cr content. Yet,

Figure 5-18 shows the binary addition of chromium into iron increases the apparent A3

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, , is given by

𝑀𝑀 67

M = exp ( ) 21 𝑄𝑄 𝑀𝑀0 ∙ − = / 𝑅𝑅 𝑅𝑅 22 ∗ 𝑀𝑀0 𝛿𝛿𝑓𝑓 𝑅𝑅𝑅𝑅 = / 23 ∗ 𝑓𝑓 𝑘𝑘𝑘𝑘 ℎ where is a pre-exponential factor, is the atomic diameter, and are the ∗ 𝑀𝑀0 𝛿𝛿 𝑄𝑄 𝑓𝑓 activation energy and frequency for the atoms crossing the interface. In pure iron, with

0.3 , it follows that 0.8 . The interface mobility for Fe – −1 −1 𝛿𝛿 ≈ 𝑛𝑛𝑛𝑛 𝑀𝑀0 ≈ 𝑚𝑚𝑚𝑚𝑚𝑚 ∙ 𝑚𝑚 ∙ 𝐽𝐽 ∙ 𝑠𝑠 Cr alloy with 2 at.% Cr was determined to be (0.37 ± 0.07) , −1 −1 𝑚𝑚𝑚𝑚𝑚𝑚 ∙ 𝑚𝑚 ∙ 𝐽𝐽 ∙ 𝑠𝑠 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.

Figure 5-19. Binary phase diagram of Fe – Cr system [30].

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 Al2O3 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 P to 1300 P were

℃ ℃ 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 P were determined by extrapolating the resulting function to 20 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:

= 0.0021 + 0.3575( ) . % 𝑛𝑛𝑛𝑛 𝑎𝑎 � � ∙ 𝐶𝐶𝐶𝐶𝐶𝐶 𝑛𝑛𝑛𝑛 Table 10. Results of linear least squares regression𝑎𝑎𝑎𝑎 analysis performed on the data of AHC pure iron sample, Ames Fe – Cr sample, and Astaloy CrA Fe – Cr sample.

= ( ) + ( ) 𝑛𝑛𝑛𝑛 Phase S (nm/oC) Standard𝑎𝑎 𝑆𝑆 Error∙ 𝑇𝑇 𝑎𝑎0 𝑛𝑛𝑛𝑛(nm) Standard Error For Sample ℃ for S (nm/oC) (nm) 𝑎𝑎0 0 AHC 100.29 α 4.7 × 10 2 × 10 0.2865 1𝑎𝑎× 10 −6 −7 −4 γ 8.5 × 10 2 × 10 0.3570 5 × 10 −6 −7 −4 AMES FeCr α 4.2 × 10 2 × 10 0.2867 1 × 10 −6 −7 −4 γ 5.2 × 10 2 × 10 0.3603 3 × 10 −6 −7 −4 Astaloy CrA α 4.0 × 10 2 × 10 0.2868 2 × 10 −6 −7 −4 γ 4.4 × 10 2 × 10 0.3608 3 × 10 −6 −7 −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

X-ray measurements were conducted under conditions that deviated from equilibrium.

The temperature intervals where the γ-phase diffraction peaks appeared were 905.7 P to

℃ 931.7 P for AHC 100.29 pure iron powder, 917.7 P to 939.7 P for Ames Fe – Cr

℃ ℃ ℃ powder, and 941.7 P to 979.7 P for CrA Fe – Cr powder. The temperatures at which the

℃ ℃ 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.

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 Al2O3 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.

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

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

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

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

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

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

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