ROLE OF METALLURGY IN THE THERMAL CONDUCTIVITY OF
SUPERCONDUCTING NIOBIUM
By
Saravan Kumar Chandrasekaran

A DISSERTATION
Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of
Mechanical Engineering – Doctor of Philosophy
2013

ABSTRACT
ROLE OF METALLURGY IN THE THERMAL CONDUCTIVITY OF
SUPERCONDUCTING NIOBIUM
By
Saravan Kumar Chandrasekaran
Superconducting radio frequency (SRF) cavities for particle accelerators are fabricated
using high purity niobium to allow for continuous high power operation. A hot-spot generated
at a surface defect during accelerator operation can quench superconductivity and limit its
performance. Increased heat conduction to the liquid helium bath dissipates hot spots,
maintaining performance. Properly treated niobium may exhibit a local maximum (i.e.,
phonon peak) in thermal conductivity at T ≈ 1.8 K, the accelerator operating temperature.
Conductivity with a phonon peak may be more than ten times that without the peak.
Fabrication from ingot to cavity involves substantial processing, which in turn defines the
metallurgical state of the niobium. Understanding the role of metallurgy and processing on
conduction in the phonon regime can lead to improved accelerator performance.
This study reveals the role of metallurgy (e.g., grain size and orientation, imperfection
density, metal purity) and processing (e.g., deformation, heat treatment) on the magnitude
of the phonon peak. This study examines the most comprehensive set of conditions in a
group of related uniaxial tensile specimens of mono- and bicrystal niobium with purity of
100

RRR

200. Measurements of the thermal conductivity of the niobium were made

with a coldest bath temperature of 1.6 K. A novel parameter estimation technique used
data from a range of temperatures to determine parameters in a theoretically based model
of the thermal conductivity of superconducting metals, without resorting to intermediate
approximations.

The grain orientations and imperfection densities of the specimens were examined using
electron backscatter diffraction and high energy X-ray diffraction, respectively. The role of
deformation was examined by uniaxially straining several specimens to 2–38%, typical of
SRF cavity manufacturing. The kinetics of recovering the phonon peak post-deformation
was revealed by heat treating specimens from 140 ‰ for 48 h to 1200 ‰ for 2 h in a highvacuum furnace. Two specimens were saturated with hydrogen to examine its effect on
thermal conductivity.
The as-received specimens displayed no phonon peak due to thermal strain-induced dislocations from ingot production. The recovery of a phonon peak depended sigmoidally on
the heat treatment temperature, with a plateau after 1000 ‰. The dislocation density of
the specimens decreased with increasing heat treatment temperature, but the vacancy concentration increased dramatically. Phonon conductivity appears to be dependent on the
vacancy concentration. In addition, smaller RRR yielded a smaller phonon peak.
Results revealed that the density of dislocations introduced during deformation depends
on both grain orientation and strain level. The recovery of the phonon peak by postdeformation heat treatment at 1000 ‰ for 2 h was complete for specimens with ≤ 74%
reduction in the phonon peak due to strain-induced dislocations. Although hydrogen infused during typical cavity processing steps left the phonon peak unchanged, specimens
saturated with hydrogen displayed a 25% reduction. The concentration of hydrogen was
deduced to be affected by vacancy concentration of the specimens.
These results should assist those designing and manufacturing SRF cavities by guiding them to the processes that may provide the greatest possible thermal conductivity for
niobium cavities operating at T = 1.8 K.

In memory of my beloved grandfather, Mr. T.A. Muthuswamy.

iv

ACKNOWLEDGMENTS

The majority of this work was supported by the U.S. Department of Energy, Office of High
Energy Physics, through Grant No. DE-S0004222. This research was partially supported by
Fermi National Accelerator Laboratory and National Superconducting Cyclotron Laboratory
(NSCL). Use of the Advanced Photon Source was supported by the U.S. Department of
Energy, Office of Science, Office of Basic Energy Sciences, under Contract No. DE-AC0206CH11357. The large grain niobium discs used in this research were generously provided by
Dr. Ganapati Myneni from the Thomas Jefferson National Accelerator Facility, in exchange
for RRR estimates of the discs in their as-received condition.
I wish to express my deepest gratitude to my dissertation adviser, Dr. Neil Wright, for
his excellent guidance, patience, and encouragement. His ability and willingness to tackle
heat transfer problems, both theoretically and experimentally, using his broad knowledge
has guided me through this work. His ability to think from the perspective of the audience
is astounding, and I have benefitted from this on several occasions. I am thankful to him
for demonstrating how to be diplomatic and to look at the larger picture to take balanced
decisions.
I am grateful to Dr. Thomas Bieler for introducing me to the concepts of metallurgy,
and for patiently supporting me while I delved into the gigantic set of X-ray data. I have
greatly benefitted from his open-door policy during this work. His commitment to science
and knowledge is an inspiration, especially for young scientists.
I am thankful to Chris Compton for his technical and logistical support during the entire
period of this research. I truly appreciate his efforts to keep my hopes up during the long
delays in scheduling experiments, especially during the last two years of this work.
v

I would also like to thank Dr. James Beck for his thoughtful comments during this work,
and for guiding me through the intricacies of parameter estimation. His desire to share
knowledge and his willingness to assist in identifying solutions to problems of all magnitudes
is inspiring.
I wish to thank Drs. Farhang Pourboghrat and Craig Somerton for their comments and
suggestions on my work, and for being members on my doctoral guidance committee.
I am grateful to Drs. Terry Grimm and Walter Hartung for introducing me to the world
of accelerator physics. Walter’s patience while clearing my doubts in this field is truly
appreciated.
I could not have completed the cryogenic experiments without the expert assistance of
Steve Bricker, Dave Norton, and Allyn McCartney. Steve’s and Dave’s patience during
the pump down of liquid helium, especially while maintaining steady pressures for constant
temperature calibration data points, is truly appreciated. I am thankful to Allyn for often
going out of the way to ensure that the cryogenic group was able to support my helium
requirements. I also appreciate Dave’s advice and assistance with data acquisition. I am
also grateful to John Bierwagen for his assistance on several occasions, and for always being
willing to help.
I am grateful to Lindsay Dubbs, Kyle Elliott, and Ian Malloch for performing chemical
treatments on my specimens, often at very short notice, while I was safely outside the acid
room. I wish to thank undergraduate students, Aaron Winter, Andrew Rauch, and Joseph
Asciutto, for their assistance in preparing sensors for the experiments.
I am indebted to my fellow graduate student, Di Kang, for performing the majority of the
OIM reported in this work, and for voluntarily training me to use the OIM microscope and the
Laue X-ray diffraction apparatus. I am grateful to Bite Zhou for demonstrating and teaching
vi

me to acquire high energy X-ray diffraction data at the APS at ANL. I wish to thank fellow
graduate students from Mechanical Engineering and NSCL, especially Jeremiah Holzbauer,
Payam Darbandi, Aboozar Mapar, and Derek Baars, for several useful conversations.
I wish to thank the administrative staff of Mechanical Engineering and NSCL, especially
Aida Montalvo, Craig Gunn, and Amanda Alter, for their excellent support and assistance.
The love, support, and encouragement from my parents, L.M. Chandrasekaran and Devi
Chandrasekaran, and sister, Gayathri Chandrasekaran, have been the foundation of my
standing today. The advice and support from my in-laws, Drs. K. Balu and P.S. Meenakshi,
on the personal and professional aspects of graduate life have come in handy several times.
I cannot express enough my appreciation of my best friend and wife, Deebika Balu. Despite
the physical distance between us, her willingness and desire to stand by me and encourage
me during easy and tough times were instrumental in me making it to the other side. Thank
you!

vii

TABLE OF CONTENTS

LIST OF TABLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

xi

LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiii

Chapter 1 Introduction . . . . . . . . . . . . . . . . .
1.1 Background . . . . . . . . . . . . . . . . . . . . . .
1.2 RF cavity basic definitions . . . . . . . . . . . . . .
1.2.1 Superconducting RF cavities . . . . . . . . .
1.3 Niobium SRF cavity limits . . . . . . . . . . . . . .
1.3.1 Magnetic field . . . . . . . . . . . . . . . . .
1.3.2 Thermal breakdown . . . . . . . . . . . . .
1.3.2.1 RF surface defects . . . . . . . . .
1.3.2.2 Global thermal instability . . . . .
1.3.2.3 Multipacting . . . . . . . . . . . .
1.3.2.4 Field emission . . . . . . . . . . . .
1.4 Heat transfer in SRF cavities . . . . . . . . . . . .
1.5 SRF cavity fabrication and processing . . . . . . . .
1.5.1 Deep drawing and electron beam welding . .
1.5.2 Etching and cleaning . . . . . . . . . . . . .
1.5.3 Cavity heat treatments . . . . . . . . . . . .
1.5.3.1 Moderate to high temperature heat
1.5.3.2 Low temperature heat treatments .
1.6 Cavity processing and thermal conductivity . . . .

. . . . . . .
. . . . . . .
. . . . . . .
. . . . . . .
. . . . . . .
. . . . . . .
. . . . . . .
. . . . . . .
. . . . . . .
. . . . . . .
. . . . . . .
. . . . . . .
. . . . . . .
. . . . . . .
. . . . . . .
. . . . . . .
treatments
. . . . . . .
. . . . . . .

.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.

.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.

.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.

.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.

.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.

.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.

.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.

1
1
4
6
8
8
9
9
10
11
11
11
12
16
17
17
17
18
19

Chapter 2 Thermal conductivity . . . . . . . . . . . . . .
2.1 Theory of Thermal conductivity . . . . . . . . . . . . .
2.1.1 Electron thermal conductivity . . . . . . . . . .
2.1.2 Thermal conductivity due to phonons . . . . . .
2.1.3 Thermal conductivity of superconducting metals
2.2 Thermal conductivity of superconducting niobium . . .

.
.
.
.
.
.

.
.
.
.
.
.

.
.
.
.
.
.

.
.
.
.
.
.

.
.
.
.
.
.

.
.
.
.
.
.

.
.
.
.
.
.

.
.
.
.
.
.

.
.
.
.
.
.

.
.
.
.
.
.

.
.
.
.
.
.

.
.
.
.
.
.

25
25
25
28
32
33

Chapter 3 Experimental Methods
3.1 Temperature measurement . . .
3.2 Specimen heaters . . . . . . . .
3.3 Niobium specimens . . . . . . .
3.4 Specimen processing . . . . . .
3.4.1 Heat treatments . . . . .

.
.
.
.
.
.

.
.
.
.
.
.

.
.
.
.
.
.

.
.
.
.
.
.

.
.
.
.
.
.

.
.
.
.
.
.

.
.
.
.
.
.

.
.
.
.
.
.

.
.
.
.
.
.

.
.
.
.
.
.

.
.
.
.
.
.

.
.
.
.
.
.

35
38
39
39
42
42

.
.
.
.
.
.

.
.
.
.
.
.

.
.
.
.
.
.

viii

.
.
.
.
.
.

.
.
.
.
.
.

.
.
.
.
.
.

.
.
.
.
.
.

.
.
.
.
.
.

.
.
.
.
.
.

.
.
.
.
.
.

.
.
.
.
.
.

.
.
.
.
.
.

.
.
.
.
.
.

.
.
.
.
.
.
.
.
.
.
.
.
.

.
.
.
.
.
.
.
.
.
.
.
.
.

.
.
.
.
.
.
.
.
.
.
.
.
.

.
.
.
.
.
.
.
.
.
.
.
.
.

.
.
.
.
.
.
.
.
.
.
.
.
.

Chapter 4 Results: Phonon peak and heat-treatment temperature
4.1 Set 1: Bicrystal specimens from different ingots . . . . . . . . . . . .
4.1.1 Thermal conductivities . . . . . . . . . . . . . . . . . . . . . .
4.2 Set 2: Two hour heat treatment kinetics . . . . . . . . . . . . . . . .
4.2.1 Thermal conductivities . . . . . . . . . . . . . . . . . . . . . .
4.2.2 Dislocation content . . . . . . . . . . . . . . . . . . . . . . . .
4.2.2.1 Dislocation density from kpp . . . . . . . . . . . . .
4.2.2.2 Orientation imaging microscopy . . . . . . . . . . . .
4.2.2.3 X-Ray diffraction . . . . . . . . . . . . . . . . . . . .
4.3 Young’s modulus and the phonon peak . . . . . . . . . . . . . . . . .
4.4 Parameters from thermal conductivity model . . . . . . . . . . . . . .
4.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.
.
.
.
.
.
.
.
.
.
.
.
.

.
.
.
.
.
.
.
.
.
.
.
.
.

.
.
.
.
.
.
.
.
.
.
.
.
.

. 61
. 62
. 62
. 75
. 75
. 76
. 76
. 85
. 85
. 92
. 99
. 106
. 113

Chapter 5 Results: Deformation and the Phonon peak
5.1 Uniaxial straining of specimens . . . . . . . . . . . . . .
5.2 Thermal conductivities of strained specimens . . . . . . .
5.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . .
5.3.1 Deformation on kpp . . . . . . . . . . . . . . . . .
5.3.2 Recovery of kpp . . . . . . . . . . . . . . . . . . .
5.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . .

.
.
.
.
.
.
.

.
.
.
.
.
.
.

.
.
.
.
.
.
.

.
.
.
.
.
.
.

.
.
.
.
.
.
.

.
.
.
.
.
.
.

.
.
.
.
.
.
.

.
.
.
.
.
.
.

.
.
.
.
.
.
.

.
.
.
.
.
.
.

.
.
.
.
.
.
.

114
115
117
127
127
128
134

Chapter 6 Results: Hydrogen and
6.1 Infusing hydrogen into niobium
6.1.1 Hydrogen concentration
6.2 Thermal conductivity . . . . . .
6.3 Discussion . . . . . . . . . . . .
6.4 Conclusion . . . . . . . . . . . .

.
.
.
.
.
.

.
.
.
.
.
.

.
.
.
.
.
.

.
.
.
.
.
.

.
.
.
.
.
.

.
.
.
.
.
.

.
.
.
.
.
.

.
.
.
.
.
.

.
.
.
.
.
.

.
.
.
.
.
.

.
.
.
.
.
.

135
135
136
137
141
143

3.5

3.6

3.4.2 Deformation . . . . . . . . . . . . .
3.4.3 Hydrogen infusion . . . . . . . . . .
3.4.4 BCP etching . . . . . . . . . . . . .
Specimen metallurgy characterization . . .
3.5.1 High energy X-ray diffraction . . .
3.5.2 Electron backscatter diffraction . .
3.5.3 Young’s modulus . . . . . . . . . .
Estimating k(T ) from measurements . . .
3.6.1 Thermal conductivity estimation .
3.6.2 Parameter estimation algorithm . .
3.6.2.1 Gauss minimization . . .
3.6.2.2 Scaled sensitivity analysis
3.6.3 Uncertainty in k . . . . . . . . . .

Chapter 7

.
.
.
.
.
.
.
.
.
.
.
.
.

.
.
.
.
.
.
.
.
.
.
.
.
.

the Phonon
. . . . . . . .
. . . . . . . .
. . . . . . . .
. . . . . . . .
. . . . . . . .

.
.
.
.
.
.
.
.
.
.
.
.
.

.
.
.
.
.
.
.
.
.
.
.
.
.

.
.
.
.
.
.
.
.
.
.
.
.
.

.
.
.
.
.
.
.
.
.
.
.
.
.

Peak
. . . .
. . . .
. . . .
. . . .
. . . .

.
.
.
.
.
.
.
.
.
.
.
.
.

.
.
.
.
.
.

.
.
.
.
.
.
.
.
.
.
.
.
.

.
.
.
.
.
.

.
.
.
.
.
.
.
.
.
.
.
.
.

.
.
.
.
.
.
.
.
.
.
.
.
.

.
.
.
.
.
.
.
.
.
.
.
.
.

.
.
.
.
.
.
.
.
.
.
.
.
.

.
.
.
.
.
.
.
.
.
.
.
.
.

.
.
.
.
.
.
.
.
.
.
.
.
.

44
45
45
45
45
48
51
53
54
56
57
57
59

Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144

ix

BIBLIOGRAPHY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146

x

LIST OF TABLES

Table 3.1

Processing histories of the nine bicrystal and four monocrystal specimens. The hydrogen experiment consisted of infusing hydrogen into
the specimens at 300 ‰ for 1 h, at a pressure of 0.5 atm in a 75 %
nitrogen, 25 % hydrogen atmosphere. These specimens were subsequently heat treated at 800 ‰ for 2 h to degas the hydrogen. The
deformation experiment involved straining the specimens to different nominal engineering strains, and the specimens were later heat
treated at 1000 ‰ for 2 h to examine the recovery of the phonon peak. 41

Table 4.1

Heat treatment histories for the specimens in Set 1. Estimated RRR
values, tantalum contents, and the use of Ti getter during the heat
treatments of the specimens are also tabulated. . . . . . . . . . . . .
Ratios of the phonon conductivity of the specimens in Set 1, after
their two heat treatments. kpp in the ratios refer to the thermal
conductivity at the phonon peak after heat treatments, while k2 refers
to k at T = 2 K in their as-received condition, and k3 refers to k at
T = 3 K after the 1100 ‰ heat treatment. For specimens 7 and 8
after the 140 ‰ for 48 h heat treatment, there was no phonon peak,
and hence the kpp /k2 ratios are not listed. . . . . . . . . . . . . . .

Table 4.2

Table 5.1

63

74

State of specimens prior to deformation. The Euler angles obtained
using orientation imaging microscopy are also listed. Also tabulated
are the Schmid factors for {110} and {112} slip systems, and the
Burger’s vector directions for these specimens. The level of strain
chosen for these specimens based on the Schmid factors and Burger’s
vector directions are also tabulated. . . . . . . . . . . . . . . . . . . 116

xi

Table 5.2

Effect of uniaxial strain on the thermal conductivities of the two
grains in specimens 2, 3, 7 and 13. Tabulated are the ratios of
kpp,2 /kpp,1 , kpp,3 /kpp,2 , and kpp,3 /kpp,1 . The first ratio is a measure
of the decrease in kpp after deformation with respect to kpp before
deformation, the second ratio the increase in kpp after the 1000 ‰
for 2 h heat treatment with respect to the kpp after deformation, and
the third ratio the change in kpp after the 1000 ‰ for 2 h heat treatment with respect to the kpp before deformation. In the ratios, kpp,1
is the k at the phonon peak before deformation, kpp,2 the kpp after
deformation, and kpp,3 the kpp after the 1000 ‰ heat treatment. In
cases where there is no phonon peak, k at 2 K is used. Additionally,
the kpp /k3 values for the specimens after the 1000 ‰ heat treatment
are listed. Also tabulated are the chosen nominal strains along with
the measured engineering strains on the two grains of the specimens. 124

Table 6.1

Hydrogen specimen histories. . . . . . . . . . . . . . . . . . . . . . . 137

Table 6.2

Ratio of k at the phonon peak to that at 3 K for specimens 5 and 8
before hydrogen infusion, after hydrogen infusion, and after hydrogen
degassing heat treatment at 800 ‰ for 2 h. . . . . . . . . . . . . . . 138

xii

LIST OF FIGURES

Figure 1.1

Figure 1.2

A schema of a single-cell elliptical SRF cavity. Charged particles are
accelerated in a vacuum by an RF field confined to the inner volume.
The cavity is submerged in liquid helium. For interpretation of the
references to color in this and all other figures, the reader is referred
to the electronic version of this dissertation. . . . . . . . . . . . . . .

3

One dimensional heat transfer from the inner to the outer wall of a
Nb SRF cavity. Subscripts s, k, h and ∞ refer to the inner surface,
due to conduction, due to convection, and of liquid helium (LHe),
respectively. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

12

Figure 1.3

The steps involved in the fabrication of an elliptical cavity from either
sheet niobium or ingot niobium discs. There is significant processing
done after obtaining the elliptical shape, and is illustrated in Figure 1.4. 14

Figure 1.4

Processing of cavities after obtaining elliptical shape. The steps involved in obtaining the initial elliptical cavity by deep drawing are
presented in Figure 1.3. Fabrication steps for seamless cavities obtained by hydroforming or spinning are different from those presented
in Figure 1.3, but the processing steps are similar. . . . . . . . . . .

15

Figure 1.5

Deep drawing a niobium disk into a half cell. . . . . . . . . . . . . .

16

Figure 1.6

Performance of a niobium cavity before and after an 800 ‰ and a
1400 ‰ heat treatment. After heat treatment, the cavity produces
higher accelerating gradients before quenching, although Q0 is unchanged. Quenching is due to loss of superconductivity initiated at a
local hot spot. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

18

Quality factor as a function of magnetic flux before and after 120 ‰
48 h baking at three different temperatures. . . . . . . . . . . . . . .

19

Effect of room temperature deformation on the thermal conductivity
of superconducting single crystal niobium. The undeformed specimen
has the highest conductivity at 1.8 K, followed by the 4% and 8%
strained specimens, respectively. The vertical dashed line is drawn at
T = 1.8 K. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

21

Figure 1.7

Figure 1.8

xiii

Figure 1.9

Thermal conductivity of a niobium sample after various heat treatments with and without Ti getter. 2000 ‰ for 2 h (squares), 1250 ‰
for 6 h with Ti gettering (cirles), 1300 ‰ for 10 h with Ti gettering
(triangles). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

23

The two major groups of factors, i.e., metallurgy and processing,
that influence the thermal conductivity of polycrystal and large grain
superconducting Nb. . . . . . . . . . . . . . . . . . . . . . . . . . .

24

Ratio of the thermal conductivity of superconducting and normal
conducting tantalum (Ta) and niobium (Nb). . . . . . . . . . . . . .

29

Figure 3.1

An illustration of the experimental insert. . . . . . . . . . . . . . . .

36

Figure 3.2

An illustration of the experimental insert, when lowered into a cryogenic Dewar. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

37

A bi-crystal Nb specimen in the holder. A steady temperature gradient is obtained on the Nb specimen using a heater on one end (top)
while cooling the other side (bottom). The temperature gradient is
measured using four carbon resistors positioned along the length of
the specimen. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

40

A measured temperature profile in the hot zone of the furnace at
Michigan State University during a 1000 ‰ for 2 h heating. The
ramp rate was 10 ‰/min, while the cooling rate was uncontrolled. .

43

Illustration for Bragg’s law for X-ray diffraction. The diffracted Xrays will produce a peak when there is constructive interference in
them. For this to occur, the difference in path lengths of the diffracted
beams (2d sin Θ) should equal an integer (n) multiple of the X-rays’
wavelength (λ). Here, d is the inter-planar spacing and Θ is the angle
of incidence of X-rays. . . . . . . . . . . . . . . . . . . . . . . . . . .

46

An illustration of the specimen-detector set-up at the Advanced Photon Source at Argonne National Laboratory. The diffracted X-rays
were collected using an area detector having a resolution of 5 pixels/mm with 2048×2048 pixels, placed about 1 m from the specimen.
Peaks observed on the detector would have satisfied Bragg’s law. . .

47

Figure 1.10

Figure 2.1

Figure 3.3

Figure 3.4

Figure 3.5

Figure 3.6

xiv

Figure 3.7

A diffracted X-ray peak from a single crystal Nb specimen measured
on an area detector at the APS at ANL. A three dimensional contour
plot for this peak is illustrated in Figure 3.8. The coordinate system
is the same as that represented in Figure 3.6. The origin is at the top
left corner of the detector. . . . . . . . . . . . . . . . . . . . . . . .

49

A diffracted X-ray peak from a single crystal Nb specimen measured
on an area detector at the APS at ANL. A two dimensional contour
plot for this peak, as would be visible on the detector, is illustrated
in Figure 3.7. The coordinate system is the same as that represented
in Figure 3.6. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

50

Measured temperature gradients along the length of a Nb specimen,
for a bath temperature of Tb = 1.6 K. . . . . . . . . . . . . . . . . .

55

Figure 3.10

Scaled sensitivity coefficients for the five βi defined in equation (3.9).

58

Figure 4.1

The thermal conductivities of grains 1 (squares) and 2 (circles), and
across the grain boundary (triangles) of specimen 1 in their as-received
condition (open symbols) and after 600 ‰ for 6 h heat treatment
(half-filled symbols). The Euler angels φ1 , Φ, and φ2 representing
the crystal orientation along the heat flow direction for the two grains
after the 600 ‰ heating, as well as the inverse pole figure illustrating
the orientations, are shown below the thermal conductivity plot. . .

65

The thermal conductivities of the grains 1 (squares) and 2 (circles),
and across the grain boundary (triangles) of specimen 2 in their asreceived condition (open symbols), after 600 ‰ for 6 h heat treatment
(half-filled symbols), and after 1100 ‰ for 4 h (solid symbols). The
Euler angels φ1 , Φ, and φ2 representing the crystal orientation along
the heat flow direction for the two grains after the 1100 ‰ heating, as
well as the inverse pole figure illustrating the orientations, are shown
below the thermal conductivity plot. . . . . . . . . . . . . . . . . . .

66

The thermal conductivities of the grains 1 (squares) and 2 (circles),
and across the grain boundary (triangles) of specimen 3 in their asreceived condition (open symbols), after 750 ‰ for 2 h heat treatment
(half-filled symbols), and after 1100 ‰ for 4 h heat treatment (solid
symbols). The Euler angels φ1 , Φ, and φ2 representing the crystal
orientation along the heat flow direction for the two grains after the
1100 ‰ heating, as well as the inverse pole figure illustrating the
orientations, are shown below the thermal conductivity plot. . . . .

67

Figure 3.8

Figure 3.9

Figure 4.2

Figure 4.3

xv

Figure 4.4

Figure 4.5

Figure 4.6

Figure 4.7

Figure 4.8

The thermal conductivities of the grains 1 (squares) and 2 (circles),
and across the grain boundary (triangles) of specimen 4 in their asreceived condition (open symbols), after 750 ‰ for 2 h heat treatment
(half-filled symbols), and after 1100 ‰ for 4 h heat treatment (solid
symbols). The Euler angels φ1 , Φ, and φ2 representing the crystal
orientation along the heat flow direction for the two grains after the
1100 ‰ heating, as well as the inverse pole figure illustrating the
orientations, are shown below the thermal conductivity plot. . . . .

68

The thermal conductivities of the grains 1 (squares) and 2 (circles),
and across the grain boundary (triangles) of specimen 5 in their asreceived condition (open symbols) and after 800 ‰ for 2 h heat treatment (half-filled symbols). The Euler angels φ1 , Φ, and φ2 representing the crystal orientation along the heat flow direction for the two
grains after the 800 ‰ heating, as well as the inverse pole figure illustrating the orientations, are shown below the thermal conductivity
plot. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

69

The thermal conductivities of the grains 1 (squares) and 2 (circles),
and across the grain boundary (triangles) of specimen 6 in their asreceived condition (open symbols) and after 800 ‰ for 2 h heat treatment (half-filled symbols). The Euler angels φ1 , Φ, and φ2 representing the crystal orientation along the heat flow direction for the two
grains after the 800 ‰ heating, as well as the inverse pole figure illustrating the orientations, are shown below the thermal conductivity
plot. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

70

The thermal conductivities of the grains 1 (squares) and 2 (circles),
and across the grain boundary (triangles) of specimen 7 in their asreceived condition (open symbols) and after 140 ‰ for 48 h heat
treatment (half-filled symbols), and after 1100 ‰ for 4 h heat treatment (solid symbols). The Euler angels φ1 , Φ, and φ2 representing
the crystal orientation along the heat flow direction for the two grains
after the 1100 ‰ heating, as well as the inverse pole figure illustrating
the orientations, are shown below the thermal conductivity plot. . .

71

The thermal conductivities of the grains 1 (squares) and 2 (circles),
and across the grain boundary (triangles) of specimen 8 in their asreceived condition (open symbols), after 140 ‰ for 48 h heat treatment (half-filled symbols), and after 1100 ‰ for 4 h heat treatment
(solid symbols). The Euler angels φ1 , Φ, and φ2 representing the
crystal orientation along the heat flow direction for the two grains after the 1100 ‰ heating, as well as the inverse pole figure illustrating
the orientations, are shown below the thermal conductivity plot. . .

72

xvi

Figure 4.9

Figure 4.10

Figure 4.11

Figure 4.12

Figure 4.13

Figure 4.14

Representative thermal conductivities of the five niobium specimens
in Set 1 that were heat-treated at 1100 ‰ for 4 h. The estimated
RRR for each specimen after the 1100 ‰ heat treatment is listed in
the legend. A dependence of maximum in kpp on RRR is visible. . .

73

Thermal conductivity curves for specimen 9 in its as-received condition (open symbols) and after the 600 ‰ for 2 h heating (half-filled
symbols). The three curves for each condition are the k for each T
sensor pair on the single crystal specimen. The Euler angels φ1 , Φ,
and φ2 representing the crystal orientation along the heat flow direction for the specimen after the 600 ‰ heating, as well as the inverse
pole figure illustrating the crystal orientation, are shown below the
thermal conductivity plot. . . . . . . . . . . . . . . . . . . . . . . .

77

Thermal conductivity curves for specimen 10 in its as-received condition (open symbols) and after the 800 ‰ for 2 h heating (half-filled
symbols). The three curves for each condition are the k for each T
sensor pair on the single crystal specimen. The Euler angels φ1 , Φ,
and φ2 representing the crystal orientation along the heat flow direction for the specimen after the 800 ‰ heating, as well as the inverse
pole figure illustrating the crystal orientation, are shown below the
thermal conductivity plot. . . . . . . . . . . . . . . . . . . . . . . .

78

Thermal conductivity curves for specimen 11 in its as-received condition (open symbols) and after the 1000 ‰ for 2 h heating (half-filled
symbols). The three curves for each condition are the k for each T
sensor pair on the single crystal specimen. The Euler angels φ1 , Φ,
and φ2 representing the crystal orientation along the heat flow direction for the specimen after the 1000 ‰ heating, as well as the inverse
pole figure illustrating the crystal orientation, are shown below the
thermal conductivity plot. . . . . . . . . . . . . . . . . . . . . . . .

79

Thermal conductivity curves for specimen 12 in its as-received condition (open symbols) and after the 1200 ‰ for 2 h heating (half-filled
symbols). The three curves for each condition are the k for each T
sensor pair on the single crystal specimen. The Euler angels φ1 , Φ,
and φ2 representing the crystal orientation along the heat flow direction for the specimen after the 1200 ‰ heating, as well as the inverse
pole figure illustrating the crystal orientation, are shown below the
thermal conductivity plot. . . . . . . . . . . . . . . . . . . . . . . .

80

Comparison of the thermal conductivities of the four niobium specimens in Set 2 in their as-received condition and after 2 h heat treatments. For clarity, only one curve per specimen is plotted. . . . . . .

81

xvii

Figure 4.15

Thermal conductivity ratios kpp /k3 for the specimens in Set 2 (squares)
in their as-received condition, and after 2 h heat treatments. The
solid line represents a sigmoidal curve fit to the kpp /k3 data for the
specimens in Set 2 (squares). The kpp /k3 for specimens in Set 1
heat-treated at 1100 ‰ for 4 h (triangles) is also plotted. . . . . . . 82

Figure 4.16

The estimated dislocation densities for specimens 9 – 12 using equation (4.1). Note that the dislocation density Nd is plotted over Figure 4.15, for comparison with the thermal conductivities of the specimens. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

84

The local average misorientation for the single crystal specimens 9 –
12, in their as-received conditions and after the 2 h heating at temperatures between 600 ‰ and 1200 ‰. . . . . . . . . . . . . . . . . .

86

Stereographic projection for specimen 9 after the 600 ‰ for 2 h heat
treatment, using Euler orientations obtained by EBSD. The stereographic projection was used to identify planes from which X-ray
diffraction peaks could be used to estimate the imperfection density
of the specimen. The grey shaded region represents crystallographic
space which was not visible to the X-rays due to the 45°sweep of
the specimens. The stereographic projection was generated using a
MATLAB algorithm developed by Bieler. . . . . . . . . . . . . . .

88

Stereographic projection for specimen 10 after the 800 ‰ for 2 h heat
treatment, using Euler orientations obtained by EBSD. The stereographic projection was used to identify planes from which X-ray
diffraction peaks could be used to estimate the imperfection density
of the specimen. The grey shaded region represents crystallographic
space which was not visible to the X-rays due to the 45°sweep of
the specimens. The stereographic projection was generated using a
MATLAB algorithm developed by Bieler. . . . . . . . . . . . . . .

89

Stereographic projection for specimen 11 after the 1000 ‰ for 2 h heat
treatment, using Euler orientations obtained by EBSD. The stereographic projection was used to identify planes from which X-ray
diffraction peaks could be used to estimate the imperfection density
of the specimen. The grey shaded region represents crystallographic
space which was not visible to the X-rays due to the 45°sweep of
the specimens. The stereographic projection was generated using a
MATLAB algorithm developed by Bieler. . . . . . . . . . . . . . .

90

Figure 4.17

Figure 4.18

Figure 4.19

Figure 4.20

xviii

Figure 4.21

Figure 4.22

Figure 4.23

Figure 4.24

Stereographic projection for specimen 12 after the 1200 ‰ for 2 h heat
treatment, using Euler orientations obtained by EBSD. The stereographic projection was used to identify planes from which X-ray
diffraction peaks could be used to estimate the imperfection density
of the specimen. The grey shaded region represents crystallographic
space which was not visible to the X-rays due to the 45°sweep of
the specimens. The stereographic projection was generated using a
MATLAB algorithm developed by Bieler. . . . . . . . . . . . . . .

91

Detected peak maximum locations for peaks from {222} planes for
specimen 9 after the 600 ‰ for 2 h heat treatment. The x- and ydistances are with respect to the location of the beam on the detector.
The four peak clusters are the locations of the X-rays diffracted by
the two {111} planes identified in Figure 4.18. Each individual cluster
is a group of {222} peaks from several of the 1620 frames recorded
per specimen. The number listed in each cluster, towards the center
of the figure, is the actual number of frames from which {222} peaks
are detected. The spread of the peaks within each cluster represents
imperfection density in the crystal. . . . . . . . . . . . . . . . . . . .

93

Detected peak maximum locations for peaks from {222} planes for
specimen 10 after the 800 ‰ for 2 h heat treatment. The x- and ydistances are with respect to the location of the beam on the detector.
The four peak clusters are the locations of the X-rays diffracted by
the two {111} planes identified in Figure 4.19. Each individual cluster
is a group of {222} peaks from several of the 1620 frames recorded
per specimen. The number listed in each cluster, towards the center
of the figure, is the actual number of frames from which {222} peaks
are detected. The spread of the peaks within each cluster represents
imperfection density in the crystal. . . . . . . . . . . . . . . . . . . .

94

Detected peak maximum locations for peaks from {222} planes for
specimen 11 after the 1000 ‰ for 2 h heat treatment. The x- and ydistances are with respect to the location of the beam on the detector.
The four peak clusters are the locations of the X-rays diffracted by
the two {111} planes identified in Figure 4.20. Each individual cluster
is a group of {222} peaks from several of the 1620 frames recorded
per specimen. The number listed in each cluster, towards the center
of the figure, is the actual number of frames from which {222} peaks
are detected. The spread of the peaks within each cluster represents
imperfection density in the crystal. . . . . . . . . . . . . . . . . . . .

95

xix

Figure 4.25

Figure 4.26

Figure 4.27

Figure 4.28

Detected peak maximum locations for peaks from {222} planes for
specimen 12 after the 1200 ‰ for 2 h heat treatment. The x- and ydistances are with respect to the location of the beam on the detector.
The four peak clusters are the locations of the X-rays diffracted by
the two {111} planes identified in Figure 4.21. Each individual cluster
is a group of {222} peaks from several of the 1620 frames recorded
per specimen. The number listed in each cluster, towards the center
of the figure, is the actual number of frames from which {222} peaks
are detected. The spread of the peaks within each cluster represents
imperfection density in the crystal. . . . . . . . . . . . . . . . . . . .

96

The normalized standard deviation of the locations of the diffracted
X-ray peak maximums, as defined in equation (4.4). Smaller values
of normalized standard deviation imply reduced dislocation contents.
Specimen 12, heat treated at 1200 ‰ for 2 h, has the smallest deviation values amongst the four specimens, indicating greatest recovery
of dislocations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

97

The average of full width at half maximum (FWHM) of the four
{222} peak clusters for specimens 9 – 12, when measured along the X
and Y directions. A smaller FWHM in both the x- and y-directions
would imply a sharper peak. . . . . . . . . . . . . . . . . . . . . . .

98

An inverse pole figure displaying the crystal orientations of grains 1
(red) and 2 (green) of specimens 1 – 8, and of the single crystal specimens 9 – 12 (red), after their hottest temperature heat treatments. A
shaded grey background, from light to dark grey, is used to represent
the variation in E of Nb from 83.2 GPa to 145.5 GPa, respectively. .

99

Figure 4.29

A comparison of the normalized phonon peak (kpp /k3 ) with E as
estimated using equation (3.4). Only specimens that are heat treated
at Th ≥ 1000 ‰ are used for this purpose. The solid line represents
a least squares fit to the data, and reveals a mild correlation between
kpp /k3 and E. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

Figure 4.30

Comparison of β1 (scattering of normally conducting electrons by impurities) with RRR for the twelve specimens, in their as-received condition and after heat treatments. The inverse proportionality would
be expected from the definition of β1 (= ρ295 /(L RRR)−1 ). . . . . . 102

xx

Figure 4.31

Comparison of β3 (scattering of phonons by normal conducting electrons) with the heat treatment temperature for the twelve specimens.
The dashed line represents the theoretical value for β3 . In these specimens, increased heat treatment temperature brings the measured β3
into agreement with the theoretical value. . . . . . . . . . . . . . . . 103

Figure 4.32

Comparison of β4 (scattering of phonons by lattice defects and boundaries) with the heat treatment temperature for the twelve specimens.
As-received specimens display greater β4 values than those post-heat
treatment. Wide variations in the as-received β4 values may indicate a broad range of process histories. Note again that the 750 ‰
heat treatment was with Ti getter and had a temperature ramp rate
of 5 ‰ per minute, whereas at 800 ‰ there was no getter and the
temperature ramp rate was 10 ‰ per minute. Note also that the
600 ‰ heat treatment for specimens 1 and 2 was for 6 h, while that
for specimen 9 was for 2 h. . . . . . . . . . . . . . . . . . . . . . . . 104

Figure 4.33

Comparison of β5 (condensation of electrons into Cooper pairs) with
the heat treatment temperature. Note that the relative range of the
ordinate is smaller than those for other βi s, indicating that the scatter
is smaller than it appears. . . . . . . . . . . . . . . . . . . . . . . . 105

Figure 4.34

The theoretically calculated equilibrium vacancy concentration for
Nb with respect to temperature. . . . . . . . . . . . . . . . . . . . . 111

Figure 5.1

Painted 2 mm x 2 mm grid on the surface of specimen 2, (a) before
deformation, and (b) after 20% nominal uniaxial tensile strain. This
grid is used to estimate local strains on the two grains (left: grain 1,
right: grain 2) of the bi-crystal specimen. Also pictured are the sideview of the specimen with the grid. The variable effect of deformation
on the thickness of the two grains is visible in the side-view of (b).
Similar grids were painted on the remaining three specimens. . . . . 118

Figure 5.2

Thermal conductivities for specimen 2 after the 1100 ‰ for 4 h heat
treatment (open symbols), after 20% nominal uniaxial tensile strain
(half-filled symbols), and after the 1000 ‰ for 2 h recovery heat treatment (solid symbols). The thermal conductivity across the grain
boundary (triangles) could not be estimated reliably after deformation, due to large variation in the cross sectional areas of the two
grains in the specimen as visible in Figure 5.1(b). The Euler angels
φ1 , Φ, and φ2 as well as the inverse pole figure representing the crystal orientation along the heat flow direction for the two grains after
the 1100 ‰ heating (open symbols) and after deformation (half-filled
symbols) are shown below the thermal conductivity plot. . . . . . . 120
xxi

Figure 5.3

Thermal conductivities for specimen 3 after the 1100 ‰ for 4 h heat
treatment (open symbols), after 20% nominal uniaxial tensile strain
(half-filled symbols), and after the 1000 ‰ for 2 h recovery heat treatment (solid symbols). Similar to specimen 2, the thermal conductivity
across the grain boundary (triangles) for specimen 3 could not be estimated reliably after deformation, due to large variation in the cross
sectional areas of the two grains in the specimen. The Euler angels
φ1 , Φ, and φ2 as well as the inverse pole figure representing the crystal orientation along the heat flow direction for the two grains after
the 1100 ‰ heating (open symbols) and after deformation (half-filled
symbols) are shown below the thermal conductivity plot. . . . . . . 121

Figure 5.4

Thermal conductivities for specimen 7 after the 1100 ‰ for 4 h heat
treatment (open symbols), after 2% nominal uniaxial tensile strain
(half-filled symbols), and after the 1000 ‰ for 2 h recovery heat treatment (solid symbols). The Euler angels φ1 , Φ, and φ2 as well as the
inverse pole figure representing the crystal orientation along the heat
flow direction for the two grains after the 1100 ‰ heating (open symbols) and after deformation (half-filled symbols) are shown below the
thermal conductivity plot. . . . . . . . . . . . . . . . . . . . . . . . 122

Figure 5.5

Thermal conductivities for specimen 13 in its as-received condition
(open symbols), after 2% nominal uniaxial tensile strain (half-filled
symbols), and after the 1000 ‰ for 2 h recovery heat treatment (solid
symbols). The Euler angels φ1 , Φ, and φ2 as well as the inverse
pole figure representing the crystal orientation along the heat flow
direction for the two grains in their as-received condition (open symbols) and after deformation (half-filled symbols) are shown below the
thermal conductivity plot. . . . . . . . . . . . . . . . . . . . . . . . 123

Figure 5.6

The change in kpp after deformation with respect to kpp before deformation (kpp,2 /kpp,1 ) as a function of the engineering strain for the
bicrystal specimens 2, 3, and 7 (different squares) are compared with
the data from single crystal Nb specimens presented by Wasserb¨ch
a
and Singer et al. , and polycrystal Nb presented by Aizaz et al. .
kpp,2 /kpp,1 = 1 implies no change in kpp after deformation. A value
smaller than unity implies reduction in kpp after deformation. The
data from Wasserb¨ch, Singer, and Aizaz are differentiated in the
a
legend with a “W”, “S”, and “A”, respectively. Lines are added to
aid in presentation and do not represent a model. Inserted is the
inverse pole figure with orientations for specimens presented in this
work, along with those for Wasserb¨ch’s specimens. . . . . . . . . . 126
a

xxii

Figure 5.7

Comparison of kpp /k3 , for the three grains with engineering strains
≥5%, with the sigmoidal curve from Figure 4.15, fit to the kpp /k3
data for the 2 h heat-treated specimens of Set 2 in Chapter 4. . . . . 130

Figure 5.8

The change in kpp after deformation with respect to that before deformation (abscissa) is compared with the change in kpp after the
1000 ‰ recovery heat treatment with respect to that after deformation (left ordinate) and the change in kpp after the 1000 ‰ recovery
heat treatment with respect to that before deformation (right ordinate) for specimens 2, 3, and 7. For all three ratios, a value of unity
implies no change in kpp , while a value smaller than unity implies reduction in kpp . An exponential relation is visible between kpp,2 /kpp,1
and kpp,3 /kpp,2 , with greater reduction in kpp after deformation leading to greater recovery of kpp after heat treatment. This implies that
greater dislocation density leads to greater recovery of dislocations
during heat treatments, irrespective of crystal orientations. The solid
lines represent exponential curve fits to the data for specimens 2, 3,
and 7. Since specimen 13 did not have a phonon peak before deformation, the corresponding data points are omitted from this plot. . 132

Figure 5.9

Comparison between the decrease in kpp after deformation (abscissa)
and increase in kpp after the 1000 ‰ recovery heat treatment (ordinate) for specimens 2, 3, and 7. An exponential relation is visible
between the two, with greater reduction in kpp after deformation
leading to greater recovery of kpp after heat treatment. This implies
that greater dislocation density leads to greater recovery of dislocations during heat treatments, irrespective of crystal orientations. The
solid line represents an exponential curve fit to the data for specimens 2, 3, and 7. Since specimen 13 did not have a phonon peak
before deformation, the corresponding data points are omitted from
this plot. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133

xxiii

Figure 6.1

Thermal conductivity curves for specimen 5 after the first 800 ‰ heat
treatment (open symbols), after hydrogen infusion (solid symbols),
and after the hydrogen degassing heat treatment at 800 ‰ for 2 h
(half filled symbols). Conductivity curves estimated from each of
the three temperature sensor pairs are illustrated. After hydrogen
infusion, a 25% reduction in the magnitude of the phonon peak was
observed for sensor pairs 2 and 3, while a 30% reduction was noted
for sensor pair 1. No change was detected in k for T > 3 K. After
the 800 ‰ for 2 h heating to degas hydrogen from the specimen, k
at the phonon peak was recovered to the before state for sensors 2
and 3. For sensor pair 1, the degassing heating recovered the k at
the phonon peak to approximately 9% less than its before hydrogen
infusion condition. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139

Figure 6.2

Thermal conductivity curves for specimen 8 after the 1100 ‰ heat
treatment (open symbols), after hydrogen infusion (solid symbols),
and after the hydrogen degassing heat treatment at 800 ‰ for 2 h
(half filled symbols). Conductivity curves estimated from each of
the three temperature sensor pairs are illustrated. After hydrogen
infusion, a 25% reduction in the magnitude of k at the phonon peak
was observed along with a 10% reduction in k at 4.2 K. The 800 ‰
for 2 h heating to degas hydrogen from the specimen showed no effect
on the conductivity at the phonon peak, though a mild decrease in k
for T > 3 K was noted. . . . . . . . . . . . . . . . . . . . . . . . . . 140

xxiv

Chapter 1
Introduction
Studying the effect of the metallurgical state on the thermal conductivity of superconducting
niobium offers the opportunity to examine the influence of vacancies, dislocations, and grain
boundaries on the phonon contribution to conductivity. The motivation for this study is the
design and manufacture of improved superconducting radio frequency particle accelerator
cavities. First, an overview of superconductivity, as applied to accelerator cavities is given.
Then, heat transfer in accelerator cavities is considered.

1.1

Background

Particle accelerators are used in high-energy physics research, biomedical treatments, and
homeland security applications to accelerate beams of charged atomic or sub-atomic particles.
High energy physics applications often require particle beams to be accelerated to nearly
the speed of light, c. These accelerators are comprised of several components, including
accelerating cavities, magnets, and solenoids. Accelerator cavities impart energy to the
particles being accelerated, while the magnets and solenoids steer the particle beam in the

1

desired direction and keep the beam focused.
To impart the energy, current is applied to the interior of the accelerating cavity. Radio
frequency (RF) alternating current provides energy to each particle in the beam from each
cavity. Synchronizing the current provides repeated acceleration at each cavity. RF cavities
have traditionally been manufactured from metals, such as copper and niobium. Normal
conducting, room temperature cavities have been made of copper due to its ease in formability and due to its high thermal conductivity (400 Wm−1 K−1 ). However, room temperature
cavities have high electrical (RF) losses due to the electrical resistivity of normal conducting
copper, and their accelerating gradients are limited by the capacities of RF power supplies.
They are also limited to pulsed operation as the electrical resistance causes large temperature
excursions.
High-energy physics research and other applications require continuous operation making
superconducting metals the materials of choice for RF cavities. Niobium is widely used to
fabricate superconducting radio frequency (SRF) cavities, as it has the highest superconducting critical temperature (Tc ) for any metal (Tc = 9.25 K). Superconducting metals have zero
DC electrical resistance and a very small RF electrical resistance. Though non-zero, the RF
electrical resistance in superconducting Nb is much smaller than that in normal conducting
copper. Despite the need for constant cryogenic refrigeration of the SRF cavities, the total
power requirements for the operation of SRF cavities is about 200 times smaller than that
required for normal conducting copper cavities [1]. The niobium cavities are submersed in a
liquid helium bath and operate at 1.8 K

T

4.5 K.

A schema of an elliptical single-cell cavity is shown in Figure 1.1. The particle beam
travels through the beam pipe and energy is imparted across the iris by the RF electric
field. The frequency of the electric field is synchronized with the velocity of the particles
2

Equator

Beam
pipe

Particle
beam

Iris

RF field

Figure 1.1: A schema of a single-cell elliptical SRF cavity. Charged particles are accelerated
in a vacuum by an RF field confined to the inner volume. The cavity is submerged in liquid
helium. For interpretation of the references to color in this and all other figures, the reader
is referred to the electronic version of this dissertation.

3

being accelerated, so as to attract the particle toward the iris during the first half of the RF
cycle and repel the particle away from the iris during the second half cycle of the alternating
electric field. For example, in an electron accelerator, the positive half cycle of the RF electric
field would attract the electron toward the iris, and the negative half cycle would repel the
electron away from the iris. The particle beam and the RF electric field are confined to the
inner volume of the cavity, which is maintained under vacuum.
The RF electric field applied on the interior of the cavity induces a magnetic field in the
azimuthal direction. The magnetic field penetrates the inner surface of the cavity up to the
London penetration depth, which is about 40 nm for Nb. The interaction of the RF magnetic
field with Nb results in Joule heating. The presence of local surface defects on the inner
surface of these cavities creates hot spots that reduce cavity performance, and ultimately
lead to the destruction of superconductivity, also known as quench. Heat conduction from
the heat source to the liquid helium bath surrounding the cavity prevents or delays the onset
of a quench. Impurities and crystal imperfection added during metal forming may reduce
the thermal conductivity and thus, the cavity performance.

1.2

RF cavity basic definitions

In RF cavities, the average accelerating electric field (Eacc ) experienced by the particle being
accelerated is used as a measure of performance for the cavity. The average accelerating
electric field is related to the applied accelerating voltage (Vacc ) as

Eacc =

4

Vacc
,
d

(1.1)

where d is the effective accelerating distance within the cavity. In the case of elliptical
cavities such as that illustrated in Figure 1.1, d is the iris-to-iris distance.
The efficiency of a cavity is measured using the quality factor Q, which is defined as the
ratio of the energy stored in the cavity to the energy dissipated in a cycle. It may be written
as
Q=

ωU
,
Pdiss

(1.2)

where ω is the RF angular frequency, U is the total electromagnetic energy stored in the
cavity per RF cycle, and Pdiss is the energy dissipated by the cavity per RF cycle. The
stored energy U in equation (1.4) is defined as

1
U = µ0
|H|2 dV,
2
V

(1.3)

where µ0 is the magnetic permeability of Nb, H is the amplitude of the magnetic field, and
V is the volume of the cavity. The total dissipated power per RF cycle is written as

1
|H|2 dS,
Pdiss = Rs
2
S

(1.4)

where Rs is the surface resistance of the cavity, and S is the surface of the cavity. Hence, Q
can be written as
1 ωµ
|H|2 dV 1
G
2 0 V
Q= 1
·
=
,
Rs
Rs
|H|2 dS
2 S

(1.5)

where G is termed the geometry factor, and is influenced by the shape of the cavity and the
electromagnetic field distribution within the cavity.

5

1.2.1

Superconducting RF cavities

SRF cavities are desirable because these are high Q devices as compared with room temperature devices, and a necessity for high-voltage continuous (CW) applications.
The two-fluid model explains the zero DC electrical resistance of superconductors, for
T < Tc , as a result of the formation of Cooper pairs (fluid 1) in a bath of normal conducting electrons (fluid 2). The Cooper pairs are bound by energy 2∆(0), where ∆(0) is
the superconductor energy gap at T = 0 K. The electrons in Cooper pairs, despite their
similar charge, interact with phonons and remain paired until the energy of the phonon is
greater than 2∆(0). The number of electrons condensed into Cooper pairs is a function of
T . All electrons are Cooper pairs at T = 0 K, and for T > 0 K the Boltzmann function
exp(−∆(T )/kB T ) defines the number of electrons that are condensed into Cooper pairs.
The Cooper pairs in DC superconductors flow without any resistance through the lattice,
and hence the electrical resistance is zero. In RF applications, however, the magnetic field
penetrates the surface of the superconductor. The depth of this penetration is known as the
London penetration depth, and is defined as

m
,
ns e2 µ0

λL =

(1.6)

where m is the mass of an electron, ns the number of superconducting electrons, e the charge
of an electron, and µ0 the magnetic permeability. The temperature dependent expression
for λL , as obtained from the two-fluid model, is
1

λL (T ) = λL (0)
1−

6

T
Tc

4

.

(1.7)

The periodic reversal of the magnetic field in the penetration depth, induces an electrical current that interacts with the normal electrons, and leads to electrical resistance and
dissipation. For T < Tc /2, this electrical resistance was quantified by the BCS theory to be

RBCS (T ) = A

∆(0)
ω2
exp −
T
kB T

,

(1.8)

where A is a constant dependent on several material properties (e.g., the London penetration
depth, mean free path length of electrons, Fermi velocity, coherence length), ω the frequency
in GHz, 2∆(T ) the superconductor energy gap, and kB the Boltzmann constant. Due to
the dependance of A on material properties, especially the mean free path length of the
electrons, it is influenced by the impurity content of Nb. A measure of the impurity content
is the residual resistivity ratio (RRR), defined as
ρ
RRR = 295 ,
ρ4.2

(1.9)

where ρ295 is the room temperature electrical resistivity, and ρ4.2 the normal conducting
electrical resistivity at 4.2 K. Hence, A cannot be easily estimated using standard formulae,
and computer programs written independently by Turneaure [2] and Halbritter [3] can be
used to estimate A.
Below a certain temperature, in addition to RBCS , a residual resistance R◦ has been
observed experimentally [1]. This resistance is a constant and is independent of temperature.
With careful preparation of the RF surface of the cavity, R◦ as low as 1 nΩ had been observed.

7

These resistances act in series, and hence the total surface resistance Rs may be written as

Rs = RBCS (T ) + R◦ = A

1.3
1.3.1

∆(0)
ω2
exp −
T
kB T

+ R◦ .

(1.10)

Niobium SRF cavity limits
Magnetic field

When a DC magnetic field is applied on a superconductor, the superconductor expels the
magnetic field completely. This phenomenon is known as the Meissner effect. When the
applied DC magnetic field is increased above a critical value Hc , the superconductor becomes
normal conducting and allows magnetic field lines to penetrate the conductor. The critical
magnetic field Hc depends on the temperature, and is approximated as

Hc (T ) ≈ Hc (0) 1 −

T 2
,
Tc

(1.11)

where Hc (0) is the critical magnetic field at T = 0 K. Niobium being a Type-II superconductor, has two critical magnetic fields. When the applied DC magnetic field H is greater
than the first critical magnetic field Hc1 , normal conducting magnetic flux vortices penetrate the superconductor. Despite these vortices being normally conducting, the material
around these vortices is still superconducting and can support the flow of Cooper pairs.
Above a second critical magnetic field Hc2 , however, the normal conducting magnetic flux
vortices completely occupy the superconductor, transforming it into a normal conductor.
The phase with partial occupancy of magnetic flux vortices, i.e., for Hc1 ≤ H ≤ Hc2 , is
referred to as the “mixed state”. For Nb at T = 0 K, Hc = 2000 Oe, Hc1 = 1700 Oe, and
8

Hc2 = 2400 Oe [1].
When an RF magnetic field is applied on Type-II superconductors like Nb, the critical
magnetic field is further enhanced to a value known as the “superheated” critical magnetic
field Hsh . Due to the RF period (e.g., t = 10−9 s for 1 GHz cavity) being several orders of
magnitude smaller than the time t ≈ 10−6 s estimated by Flippen [4] for the penetration of
the magnetic flux vortices, the critical magnetic field required for phase transformation to
normal conducting is Hsh > Hc .

1.3.2

Thermal breakdown

The thermal limit for SRF cavities is when the temperature T exceeds Tc . Often times, the
increased temperatures decrease Hc , thereby decreasing performance of the cavity. There
are several factors that can lead to T ≥ Tc , including defects on the RF surface. There
are four dominant causes for thermal breakdown, which can act either individually or in
unison. The four are: defects on the RF surface of the cavity, global thermal instability
(GTI), multipacting, and field emission.

1.3.2.1

RF surface defects

The interaction of the RF magnetic field with the defect on the RF surface leads to Joule
heating and power dissipation. The timely dissipation of this excess heat from the RF
surface, through the Nb bulk, to the liquid helium surrounding the exterior of the cavity
ensures that T at the defect remains cooler than Tc . Depending on the size, and hence
the resistance, of the defect and the RF field being applied, the power loss could increase
substantially to the state where T ≥ Tc at the edge of the defect. This leads to the surface
surrounding the defect to become normal conducting, and thereby increasing the RF power
9

dissipation. For effective containment of T , the thermal conductivity k of Nb and either the
heat transfer coefficient h (for T > 2.17 K) or the Kapitza conductance hK (for T < 2.17 K)
across the Nb-liquid helium interface play a vital role.
The defect-limited maximum of H for a hemispherical defect on the RF surface was
estimated by Padamsee [1] to be

Hd, max =

4k
(Tc − Tb ),
ad Rd

(1.12)

where ad is the radius of the defect, Rd the surface resistance due to the defect, and Tb the
√
temperature of the liquid helium bath. Note that Hd, max ∝ k, hence greater k in the Nb
bulk would increase the maximum defect-limited magnetic field.

1.3.2.2

Global thermal instability

The surface resistance of a SRF cavity is a function of the squared of the frequency of the
cavity and the temperature T at the cavity surface. In defect-free RF cavities, at high
frequency operations, the increase in Rs due to the frequency causes an increase in Pdiss
leading to an increase in Ts . In such situations, if the heat generated cannot be dissipated
to the liquid helium, a thermal runaway can occur until superconductivity is lost leading to
rapid dissipation of the stored energy within the cavity. This phenomenon is known as the
global thermal instability (GTI) [5], and occurs simultaneously in the high magnetic field
regions of the cavity, i.e., near the equator of the cavity.

10

1.3.2.3

Multipacting

Multipacting is a resonant electron phenomenon where an electron strikes the surface of
the cavity, causing secondary electrons to be released from the surface of the cavity. These
secondary electrons are influenced by the RF electric field, and can accelerate and strike the
surface of the cavity near its point of origination, releasing more such secondary electrons
and dumping energy into the surface of the cavity causing a local temperature rise. As
multipacting is a resonant phenomenon, it occurs at a specific frequency for a microwave
cavity.

1.3.2.4

Field emission

Field emission is a non-resonant phenomenon, where electrons from the surface of the cavity
are extracted by the RF electric field. These electrons, influenced by the cavity’s electric
field, are accelerated and dumped on the surface of the cavity at a location other than its
point of extraction. At the location of dump, the energy gained by the electrons from the
electric field are lost as heat.

1.4

Heat transfer in SRF cavities

For increased performance of SRF cavities, the RF surface temperature Ts needs to be as
cool as possible. The heat generated on the RF surface by the various sources listed in
Section 1.3.2 can cause the temperature of Nb to rise and increase the surface resistance
Rs . Due to the interrelation between Ts and Rs , a thermal runaway can occur, leading to
quench. To prevent or delay the onset of such an event, the heat being generated at the
RF surface of the cavity needs to be dissipated away from the RF surface, through the Nb

11

TS = T∞ + ∆Tk + ∆Th
BS sin(ωt)
∆x
∆

q = ½ Rs (BS / µ0)2
∆T
∆ k=q∆ /k
∆x

Nb

∆Th = q / h

LHe, T∞

Figure 1.2: One dimensional heat transfer from the inner to the outer wall of a Nb SRF
cavity. Subscripts s, k, h and ∞ refer to the inner surface, due to conduction, due to
convection, and of liquid helium (LHe), respectively.
bulk and into the liquid helium surrounding the cavity. Hence, the thermal conductivity k
of bulk Nb and the Kapitza conductance hK across Nb-helium interface play a vital role in
the performance and thermal stability of SRF cavities. As is conventional in heat transfer,
the power dissipated at the RF surface Pdiss will be denoted as the heat flux q. Local surface
defects increase Rs , and in turn increase q. In the case of defects on the RF surface, k of Nb
needs to be sufficient to diffuse q away from the defect before Ts gets hotter than Tc .
For heat transfer purposes, the heat dissipation through the Nb wall can be considered
as a one dimensional heat transfer problem [6]. A schematic of the one dimensional heat
transfer problem with heat flow from the RF surface to the liquid helium bath surrounding
the cavity is illustrated in Figure 1.2.

1.5

SRF cavity fabrication and processing

There are several types of SRF cavity designs, each specific to a particular operating frequency range. For near the speed of light applications, high frequency cavities are used.
12

These cavities typically are elliptical, as illustrated in Figure 1.1.
Conventionally, elliptical cavities are fabricated either using polycrystal niobium from
rolled sheets or large grain niobium discs cut from ingots. Regardless of the initial material,
the fabrication of elliptical SRF cavities includes deep drawing of niobium discs into half
cells, electron beam welding of two half cells to form a cavity, and mechanical cavity shape
tuning to attain desired frequency. These steps are often interspersed with one or more
buffer chemical polishing (BCP) or electro-polishing (EP) steps to remove damage layer
and contaminants from the RF surface of the cavity. After fabrication into a cavity, it
undergoes several processes prior to actual application. In addition to BCP or EP, the
processing of these cavities include one or more heat treatments in high to ultra-high vacuum
at temperatures hotter than 600 ‰ to primarily degas hydrogen. Additionally, an in situ
heating at around 120 ‰ – 140 ‰ for 24–48 h is performed, which increases high field cavity
performance in cavities fabricated from polycrystal Nb [7, 8] and large grain or single crystal
Nb [9]. The processing of cavities is empirically optimized to increase cavity performance,
by obtaining smooth RF surfaces for reduced RF losses, and increase bulk properties for
improved heat transfer from the inside RF surface of the cavity to the liquid helium bath
in contact with the outer surface of the cavity [1]. Flow charts summarizing the significant
steps in fabrication and processing of elliptical cavities are illustrated in Figures 1.3 and 1.4,
respectively. Cavities fabricated using alternate techniques, such as hydroforming [10, 11] or
spinning [12, 13], have fabrication paths that are dissimilar to that shown in Figure 1.3, but
have similar processing steps as listed in Figure 1.4.

13

Receive ingot discs

Receive sheet Nb

Flaws?
(Scratches,
inclusions)

No

Yes

Grind

Deep draw half cells, and
trim iris and equator
Degrease and etch ∼ 5 µm

Yes

Flaws?
No
Electron beam weld
iris in high vacuum

Grind iris and etch ∼ 5 µm
Electron beam weld
equator in high vacuum

Weld
flaws?

Yes

Fix flaws (tumbling or centrifugal barrel
polishing)

No
Elliptical cavity
Figure 1.3: The steps involved in the fabrication of an elliptical cavity from either sheet
niobium or ingot niobium discs [1, 14, 15]. There is significant processing done after obtaining
the elliptical shape, and is illustrated in Figure 1.4.
14

Elliptical cavity
Etch 100 µm
Hydrogen degassing at moderate Th in high
vacuum (600 ‰, 10 h or 800 ‰, 2–3 h)
Etch 100 µm
Mechanical tuning of cavity to obtain desired frequency
Etch 20 µm
High pressure rinse
120 ‰, 48 h in situ baking in ultra-high vacuum
Figure 1.4: Processing of cavities after obtaining elliptical shape [1, 14, 15]. The steps
involved in obtaining the initial elliptical cavity by deep drawing are presented in Figure 1.3.
Fabrication steps for seamless cavities obtained by hydroforming or spinning are different
from those presented in Figure 1.3, but the processing steps are similar.

15

Figure 1.5: Deep drawing a niobium disk into a half cell. (Copyright Wiley-VCH Verlag
GmbH & Co. KGaA. Reproduced with permission from [1].)

1.5.1

Deep drawing and electron beam welding

After inspection of the niobium sheets using eddy current scanning or squid scanning, discs
are cut from the sheets to be deep drawn. Two-part aluminium dies are used to form the Nb
into the desired shape, as illustrated in Figure 1.5. To prevent Nb from sticking to the die,
conventional motor oil is used as lubrication. Subsequently, the formed Nb is trimmed at
the equator and iris regions to final half cell shape. The half cells are then ultrasonic cleaned
with a detergent to remove contaminants left over from the deep drawing and trimming
processes, and about 20 µm of material is etched from the cavity prior to welding.
The half cells are electron beam welded in high vacuum. The welds at the iris are
performed first, followed by equator welds to form full cavities. The iris welds are inside
welds, while the equator welds are done from the outside of the cavity. The welds and the
inner surface of the cavity are then inspected for any imperfections. Grinding, tumbling or
centrifugal barrel polishing are often used to get rid of the imperfections [15].

16

1.5.2

Etching and cleaning

The cavity is etched either using BCP or EP to remove the surface damage layer (∼ 100 µm)
formed during deep drawing. The BCP is a mixture of 49% diluted hydrofluoric acid, 69%
diluted nitric acid, and 85% diluted phosphoric acid, in ratios of 1:1:1, 1:1:2 or 1:1:4. To
reduce the absorption of hydrogen by Nb during BCP, the acid mixture is maintained at
10 ‰ [16]. Alternately, instead of BCP, EP is used to etch the cavity. During EP, the cavity
is electrically charged and immersed in a 1:9 mixture of hydrofluoric and sulfuric acids. The
electrical charging preferentially etches surface peaks, leading to a smooth surface.
The final step in the processing of cavities prior to assembling it into the accelerator (or
test apparatus), is a high pressure rinse using ultra pure water [15]. Water jets are sprayed
onto the inner surfaces of the cavity at a pressure of 100 bar, to remove any contaminants
present on the surface.

1.5.3

Cavity heat treatments

1.5.3.1

Moderate to high temperature heat treatments

The performance of SRF cavities was often reduced when a cavity was slow cooled to 4.2 K.
It was linked to the precipitation of hydrogen into hydrides, at temperatures of 77–150 K [16,
17]. An empirical solution was to heat treat the cavity, prior to cool down, at Th ≥ 600 ‰
in high vacuum [17] to sufficiently degas hydrogen from the material and prevent hydride
formation. Additionally, the rate of cool down from 298 K to 4.2 K was increased, to
reduce the time available for the precipitation of hydrides [16]. Figure 1.6 illustrates the
improvement of cavity performance after an 800 ‰ and a 1400 ‰ heat treatment [18].
Hotter temperature heat treatments (Th ≥ 1200 ‰) in high vacuum in the presence of

17

10

after HPP

9

0

5

10

15

20

25

10

30

E
10 11

10

(c)

x -ray s tarts

Q

0

10 10

Q0

quench quench

10

no quench
lim ited b y a m p lifier

10 9

no H T , R R R 400
H T 80 0 C , R R R 40 0
H T 14 0 0 C , R R R 77 0

0

5

10
E

acc

15
20
[MV/m]

25

30

10

Figure in cavity performance due
1.6: Performance
before and after an 800 ‰ and a 1400 ‰
provementtreatment [18]. After of a niobium cavitycavity produces to variousgradients
treatments
heat
heat treatment, the
higher accelerating
before quenching, although Q is unchanged. 800 C to loss of superconductivity
(c) successive application fromQuenching±is dueandCommons Attribution
of [18] under the Creative 1400 ±C heat trea
initiated at a local hot spot. (Reproduced
License 3.0. Copyright 2000 by The American Physical Society.)
daries by additional BCP.
0

a getter material, like titanium or yttrium, were also used to enhance the performance of

cavities [19, 20].
–0.3 mm was In addition [Fig. 16(b)] indicating the Nb to
seen to degassing hydrogen, these heat treatments anneal an
reduce the strain and dislocations in the material. In the absence of a getter material, the
oreignthermal conductivity and RRR ofa higher diffusion of surface charge the
material with Nb reduces due to nuclear impurities into
m. NeutronAtabsorption measurements of 2–3. the
bulk [21].
T ≥ 1200 ‰, with a getter, the RRR increases by a factor at
entrum GKSS in Geesthacht gave no signal,
1.5.3.2 Low temperature heat treatments
at the neutron absorption coefficient of the unAn in situ, relatively low temperature heat treatment at 100 ≥ T ≥ 170 ‰ for t > 24 h
mination found to increase cavity performance at high accelerating fields [7, 8, identiwas similar to that of Nb. The 22]. A comparison
was
e foreign inclusion was finally accomplished
18
luorescence at the Hamburger Synchrotronh

h

Nb
regi
abou
T
pany
10 –
Two
tail

Q0

1011

1010
Q-drop

Quench

109
0

20

40

60

80

100

120

140

Bpeak (mT)

Figure 1.7: Quality factor as a function of magnetic flux before (solid symbols) and after
120 ‰ 48 h baking (open symbols) at three different temperatures: 1.37 K (squares), 2 K
(diamonds), and 2.2 K (circles, triangles). (Reprinted with permission from [23]. Copyright
2004, American Institute of Physics.)
of the effect of the in situ low temperature bake for different operating temperatures is
illustrated in Figure 1.7 [23].

1.6

Cavity processing and thermal conductivity

The thermal conductivity k of Nb and the Kapitza conductance hK across the Nb-liquid
helium interface are vital properties influencing the quench field of a cavity. The induced
RF magnetic field generates heat on the interior surface of the cavity, and sometimes is

19

locally enhanced due to surface imperfections and defects. To keep the Nb superconducting,
this heat must be dissipated to the liquid helium that is in contact with the exterior surface
of the cavity.
The plastic deformation involved during deep drawing Nb into half cells generates dislocations in the material. Excessive dislocations are sources for scattering phonons. Phonons
are thermally activated lattice vibrations, and are a component of thermal conductivity. The
diminishing effect of deformation induced dislocations on the thermal conductivity of Nb was
first observed by Rowell [24], and was confirmed by several researchers [25–28]. The effect of
room temperature deformation on the thermal conductivity of single crystal Nb, for 4% and
8% strain, is redrawn from [26] in Figure 1.8. The local peak in the thermal conductivity at
around 1.8 K is due to the contribution of phonons, and is referred to as the phonon peak.
Figure 1.8 shows that the phonon conductivity can vary by more than an order of magnitude
at 1.8 K, due to deformation by virtue of dislocations.
In deep drawn Nb cavities as well as hydroformed and spun cavities, prior to any heat
treatment, it can be assumed that there is no phonon peak. Although the effect of deformation on the phonon peak can be reversed [25] by annealing Nb close to its melting
temperature (2468 ‰), there is an undesired loss in the mechanical properties even for heat
treatments at 1200 ≤ Th ≤ 1400 ‰ [29]. Additionally, the cost of heat treatments increases
exponentially with temperature, due to the exponential increase in the power requirements
for the furnaces [30].
The commonly used high temperature (Th ≥ 1200 ‰) heat treatments, with getter material, during cavity processing increases the thermal conductivity of undeformed polycrystal
Nb. With a getter, the RRR increases and leads to greater thermal conductivity in the
electron dominated regime (T ≥ 3 K), as illustrated in Figure 1.9 [20]. The corresponding
20

1 0 0 0

1 0 0

T h e rm a l c o n d u c tiv ity (W m

-1

K

-1

)

U n d e fo rm e d
4 % s tra in
8 % s tra in

1 0

1

0 .1
0 .1

1

1 0

T e m p e ra tu re (K )

Figure 1.8: Effect of room temperature deformation on the thermal conductivity of superconducting single crystal niobium. The undeformed specimen has the highest conductivity at
1.8 K, followed by the 4% and 8% strained specimens, respectively. The vertical dashed line
is drawn at T = 1.8 K. (Copyright Taylor & Francis. Redrawn with permission from [26])

21

increase in the phonon dominated regime (T ≤ 3 K) in the polycrystal Nb specimens is
due to the increase in grain size during annealing. In polycrystal Nb, it has been observed
that the phonon peak is limited mainly by grain size (grain boundary scattering), while it is
mainly limited by specimen boundary scattering in monocrystal and large grain Nb.
In addition to the temperature of measurement, the factors influencing the thermal conductivity of polycrystal and large grain superconducting Nb can thus be grouped as metallurgical and processing factors, as illustrated in Figure 1.10. The metallurgical factors are
grain size, crystal orientation, and purity (RRR), while deformation and heat treatments are
the processing factors that affect the thermal conductivity.
Although researchers have looked at the effect of one or more factors on the thermal
conductivity of superconducting Nb, there still is a need for better correlation of the thermal
conductivity with the processing history. Additionally, despite there being separate studies
on the role of deformation induced dislocations on the phonon peak of large grain Nb and
the role of heat treatments on the phonon and electron conductivity of polycrystal Nb, there
is very little knowledge about the role of heat treatments on the thermal conductivity of
deformed Nb. One study [28] examined the effect of moderate temperature (Th = 750 ‰)
and high temperature (Th ≈ 1300 ‰) heat treatments on the thermal conductivity of 3%
strained polycrystal Nb, and noted that high temperature heat treatments were required to
recover the phonon peak after deformation, despite the small strains. No such deformationheat treatment studies exist for greater strains or large grain Nb.
In this study, the recovery of the phonon peak in undeformed mono- and bicrystal Nb
are correlated with the temperature of heat treatment, and the effects of various levels of
deformation and subsequent heat treatments are examined. Additionally, the influence of
RRR, grain orientation, and interstitial hydrogen on phonon conductivity is studied.
22

Figure 1.9: Thermal conductivity of a niobium sample after various heat treatments with
and without Ti getter. 2000 ‰ for 2 h (squares), 1250 ‰ for 6 h with Ti gettering (cirles),
1300 ‰ for 10 h with Ti gettering (triangles). (Reprinted from [20], Copyright 1988, with
permission from Elsevier.)

23

Grain
size

Bath
temp.

k

Crystal
orient.

Heat
treat.

Metallurgy

Processing
Purity

Deform.

Figure 1.10: The two major groups of factors, i.e., metallurgy and processing, that influence
the thermal conductivity of polycrystal and large grain superconducting Nb.

24

Chapter 2
Thermal conductivity

2.1

Theory of Thermal conductivity

The thermal conductivity of metals is comprised of two components. The first is due to the
flow of electrons in the metal (ke ), and the second is due to elastic thermal vibrations of
the lattice, also known as phonon conduction (kg ). These two components are additive, and
hence the total thermal conductivity (k) of any metal can be written as

k = ke + kg .

2.1.1

(2.1)

Electron thermal conductivity

The conductivity due to the flow of electrons in normal conducting metals (ken ) has two
forms of resistances to its flow. The first is from the interaction of electrons with interstitial
and substitutional impurities in the metal (Wei ), and the second is from the momentum
exchange of the electrons with the lattice (Weg ). These resistances act in series, and can be

25

added to get the total resistance to the conduction of electrons as

Wen = Wei + Weg ,

(2.2)

1
1
1
=
+
.
ken
kei keg

(2.3)

or,

The resistance due to scattering of normal electrons by impurities is inversely proportional
to the temperature, due to its value being a constant [31] for a given impurity concentration.
The Wiedeman-Franz law relates the impurity limited electronic thermal conductivity kei
and the electrical conductivity σ of the metal to the temperature as

kei
= L T,
σ

(2.4)

where the constant of proportionality L is the Lorentz constant, and the electrical resistivity
of the metal ρ = σ −1 .
The scattering of normal electrons by the lattice vibrations keg was theoretically defined
in [31, 32], and was simplified in [33, 34] as
2/3

1
95.3 Na
=
keg
k295

T 2
= a T 2,
Θ

(2.5)

where Na is the number of conduction electrons per atom, k295 is the thermal conductivity
at room temperature (295 K), and Θ is the Debye temperature.
With the relations for kei and keg in equations (2.4) and (2.5), respectively, the expression
for the total conductivity due to electrons in normal conducting metals (equation (2.3)) can

26

be rewritten as
ρ295
1
=
+ a T 2.
ken
L RRR T

(2.6)

When superconducting, metals have the advantage of having zero electrical resistance
when the temperature is below the superconducting critical temperature (Tc ). However,
when T < Tc , electrons within the metal begin condensing into Cooper pairs [35] and do
not aid in heat conduction [36]. The number of electrons forming Cooper pairs increases
with decreasing temperature, with all electrons having paired up at T = 0 K [35]. Bardeen,
Rickayzen and Tewordt [36] proposed a model for the thermal conductivity of superconductors based on the BCS theory of superconductivity [35]. To account for the effect of the
condensation of electrons to form Cooper pairs on the electronic thermal conductivity of
superconducting metals, they derived an expression for the ratio of the electronic thermal
conductivity in the superconducting state (kes ) and the electronic thermal conductivity in
the normal state (ken ) as
2F1 (−y) + 2y ln(1 + e−y + y 2 /(1 + ey ))
kes
=
,
ken
2F1 (0)

(2.7)

with
y=

∆(T )
,
kB T

where 2∆(T ) is the superconductor energy gap, and kB is the Boltzmann constant. The
Fermi-Dirac functions of the form Fn (x) in equation (2.7) were evaluated by Rhodes [37],
and were defined as
∞

Fn (x) =
0

zn
dz.
1 + ez−x

Koechlin and Bonin [34], for simplicity, redefined kes /ken = R(y).
27

Hence, for superconducting metals at T < Tc the total thermal conductivity due to the
flow of normal conducting electrons can be written as

kes = R(y)

2.1.2

ρ295
L RRR T

+ aT2

−1

.

(2.8)

Thermal conductivity due to phonons

In normal conducting metals, the conductivity due to phonon motion kgn is usually a negligible fraction of the electron based thermal conductivity due to the scattering of phonons by
normal electrons. However, in the superconducting regime, the pairing of electrons to form
Cooper pairs leads to a reduction in the electron contribution as expressed in equation (2.7)
[36], and hence a reduction in the number of phonons that are being scattered. Experimentally it has been observed in superconducting metals that the phonon based conductivity
kgs becomes larger than kes for T

0.3 · Tc . It has also been noted that at T = 0.2 · Tc

the thermal conductivity in the superconducting metal can be greater than or equal to the
thermal conductivity had the metal been normal conducting. This is depicted in [38] for the
superconducting metals tantalum (Ta) and niobium (Nb). For convenience, the illustration
from [38] is shown in Figure 2.1.
Similar to the electronic thermal conductivity, the phonon conduction in superconducting
metals can also be expressed in terms of the different resistances or scattering mechanisms.
Hulm [33] lists four scattering sources for phonons:
ˆ normal conducting electrons,
ˆ crystal boundaries, and specimen boundaries in the case of monocrystals,
ˆ impurities and crystal imperfections, and

28

Figure 2.1: Ratio of the thermal conductivity of superconducting and normal conducting
tantalum (Ta) and niobium (Nb). Note that ks at T = 0.2 · Tc is primarily by phonons.
(Reprinted from [38], Copyright 1961, with permission from Elsevier.)

29

ˆ Umklapp phonon-phonon scattering.

However, it is also pointed out that Umklapp scattering is negligible at low temperatures [33],
which would be the case for superconducting metals below Tc . Hence, the phonon based
thermal conductivity in superconducting metals can be expressed as

1
1
1
1
=
+
,
+
kgs
kge kgb kgi

(2.9)

where the subscripts s, e, b, and i indicate superconducting, electron limited, boundary
limited, and imperfection limited phonon conduction, respectively.
Bardeen et al. [36] derived an expression for the phonon conductivity limited by electron
scattering in superconducting metals kges . This was simplified in [34] as

kges = exp(−y)

27 I3 (∞) k295
(2 RRR Na Θ)2

T 2 = exp(−y)DT 2 ,

(2.10)

where exp(−y) accounts for the reduction in normal conducting electrons for T < Tc , and
I3 (∞) is the third order Gr¨neisen integral.
u
The phonon conductivity limited by crystal boundary scattering was shown to have a
cubic dependence on the temperature by Casimir [39], and was expressed as

c
kgb = 2.31 × 103 RP A2/3 T 3 ,

(2.11)

where R is the radius for a circular cylindrical specimen, A = Cv /T 3 with Cv being the

30

specific heat per unit volume, and
2 (vl /vt )2 + 1

P =

3

2 (vl /vt ) + 1

2/3

,

where vl and vt are the longitudinal and transverse velocities of sound in the material,
respectively. For a specimen with square cross section d2 , R in the above equation would be
replaced [39] by R = 0.56 ∗ d. For specimens with a rectangular cross section, like the ones
√
presented here, R would be replaced by R = 0.56 Acs , where Acs is the cross sectional area
of the specimen.
Koechlin and Bonin [34] generalized the Casimir [39] expression to be valid for any form
of phonon scattering other than that due to normal electrons expressed in equation (2.10).
The relation given in [34] is

1
1
kgb’ =
+
kgb kgi

−1

3.6 × 109 3
=
lT = BlT 3 ,
a◦ Θ

(2.12)

where a◦ is the lattice parameter in ˚, and l is the phonon mean free length. It should be
A
noted that the mean free length here is limited not only by Casimir boundary scattering,
but also by other non-electron based phonon scattering sources.
Hence, the phonon dominant thermal conductivity of metals for T > Tc can be written
as
kgs =

−1
1
1
+
.
exp(−y)DT 2 BlT 3

31

(2.13)

2.1.3

Thermal conductivity of superconducting metals

The total thermal conductivity of superconducting metals for T < Tc can be written in the
form of equation (2.1), by substituting the electron and phonon terms on the right hand side
with equations (2.8) and (2.13), respectively, as

ks = kes + kgs = R(y)

ρ295
L RRR T

+ aT2

−1

+

−1
1
1
+
.
exp(−y)DT 2 BlT 3

(2.14)

It has been experimentally observed and theoretically explained that there are a few
significant temperatures at which some phenomenological events occur:
1. at T1 ≈ 0.3 ∗ Tc , kes and kgs are similar in magnitude, i.e., kes /kgs ≈1,
2. at T2 ≈ 0.2 ∗ Tc , a maximum in kgs is noticed in “defect-free” metals, and
3. at T3 ≈ 0.01 ∗ Tc , Casimir boundary scattering is the dominant scatterer for kgs .
Given the different scattering mechanisms, the following can be deduced for the conductivity
in the superconducting regime: (1) For T1

T < Tc , k is dominated by normal conducting

electrons and the contribution from phonons is suppressed due to its scattering by the normal
electrons. In this temperature range, the magnitude of kes is limited by impurity scattering.
(2) For T2

T

T1 , electrons and phonons contribute to k, and kgs is still limited by

the normal electrons. Since the number of normal electrons decreases with decreasing T ,
kgs becomes the dominant mode for k in this temperature region. (3) For T3

T

T2 ,

the thermal conduction is almost only by phonons. The mean free lengths of the phonons
are limited primarily by lattice defects, and hence are affected by the different types of
dislocations and their concentration. (4) For T

T3 , kgs is limited primarily by boundary

scattering as calculated by Casimir [39], and has a T 3 dependence.
32

2.2

Thermal conductivity of superconducting niobium

The thermal conductivity of superconducting niobium has been studied with great interest
due to its superconducting critical temperature (Tc = 9.25 K) being the highest for any
pure metal. This high Tc translates to the appearance of the lattice conductivity at a
relatively high temperature (T ≈ 0.3 · Tc = 2.775 K). The reduced electron contribution for
temperatures below T ≈ 0.2 · Tc = 1.85 K in niobium, enabled researchers to examine lattice
defects by studying their effect on phonon conduction [24–26, 40–43]. Such studies often
decouple the electron and phonon components of the thermal conductivity using theoretical
relations presented in the previous sections. To study the effects lattice defects on the
phonon conduction, however, the thermal conductivity needs to be dominated by defectlimited phonon conduction with minimal scattering by electrons. Hence such studies focus
on the response of kgs for undeformed and deformed specimens in the temperature regions
of T3

T

T2 , i.e., 0.1

T

1.85 K.

Straining niobium specimens, and thereby introducing dislocations, has been shown to
gradually change the temperature dependence of the phonon thermal conductivity from T 3
to T 2 depending on the level of deformation [24, 26]. The additional resistance generated
due to the introduction of dislocations (either from uniaxial strain or bending) has been used
to develop theoretical models for the phonon scattering mechanisms involved [25, 26, 40–
47], so as to better understand the mechanics of dislocations and defects. In order to have
negligible effects by the electrons, these studies focused on the conductivity of dielectrics and
the conductivity of superconductors for T < 0.1 ∗ Tc , i.e., T

1 K for Nb. These studies,

despite their depth into scattering mechanisms, have not provided a method to estimate
the thermal conductivity of superconducting Nb after different processing steps and do not

33

examine the recovery of the phonon peak after deformation using heat treatments. The
operation of SRF cavities at temperatures hotter than the T

1 K range, combined with

the need for improved thermal design, necessitated the study to relate processing history
with the thermal conductivity of Nb.

34

Chapter 3
Experimental Methods
An experimental system to measure the steady temperature profile and heat flux on as many
as two specimens has been commissioned by Aizaz [48]. This system consists of an evacuated
tube, one end of which is attached to a stainless steel Dewar lid, while the other end has
two specimen housings. This tube-lid assembly is collectively known as the insert, and is
lowered into a cryogenic Dewar that is then filled with liquid helium (He). The specimens
are mounted on to conflat flanges and placed in evacuated chambers that are surrounded
by liquid He, at the lower end of the evacuated tube. An illustration of the experimental
insert [48] is shown in Figure 3.1. Illustrated are the various air-to-vacuum feed-throughs on
the Dewar lid that are used for instruments such as the He level sensor, He bath pressure
sensor, He bath temperature sensor, and cabling for temperature sensors and heaters on the
Nb specimens. Also shown at the bottom of the vacuum-tube are the specimen housing for
the thermal conductivity specimens. Two identical arms have been added to the system
to measure up to four specimens at a time, doubling throughput. Figure 3.2 illustrates
a two-dimensional CAD rendering of the experimental insert after being lowered into the
Dewar [48], and highlights the structure of the cryogenic Dewar.
35

Figure 3.1: An illustration of the experimental insert [48]. Shown are the various components of the insert, including the air-to-vacuum feed-throughs on the Dewar lid, the thermal
radiation shielding, and the specimen housing. For the current measurements, two identical
specimen holders have been added.

36

Figure 3.2: An illustration of the experimental insert, when lowered into a cryogenic Dewar [48].

37

3.1

Temperature measurement

Up to four Allen-Bradley 100 Ω, 1/8 W carbon resistors are used to measure the temperature
gradient along the length of each specimen. Carbon resistors are favored for low temperature
measurements due to their exponentially increasing resistance with decreasing temperature,
along with them being relatively inexpensive.
The carbon resistors are attached to the Nb specimens using STYCAST

2850 FT, an

electrically insulating epoxy with low coefficient of thermal expansion, mixed with Catalyst 24 LV in a 100:8 ratio. This epoxy mixture has a 30 minute work life and a 16 h curing
time at room temperature. A custom made fixture is used to hold the resistors in place
during curing.
The resistances of the carbon and germanium resistors are measured as described by
Aizaz [48]. Three LakeShore

Model 218 temperature monitors are used to measure the

resistances. Each temperature monitor has eight channels, each capable of the four-lead
measurement method using a 10 µA excitation current. The temperature monitors are
connected to a data acquisition computer via RS-232 serial interfaces. A custom LabVIEW
program reads and records data from the temperature monitors, along with data from the
He level and He bath pressure sensors. The data is recorded at 2 Hz.
The LakeShore

Model 218 temperature monitors are limited to measuring resistances

up to 7,500 Ω. At temperatures colder than 1.9 K, the resistance of the carbon resistors
is greater than 7,500 Ω. A 10 kΩ resistor is attached in parallel to each carbon resistor to
reduce the overall resistance of the measurement circuit by limiting the current going to the
carbon resistor.
The carbon resistors are calibrated for 1.6 K ≤ T ≤ 4.6 K using a factory calibrated

38

germanium resistor, which can be traced back to international standards. At atmospheric
pressure, liquid He has a saturation temperature of 4.2 K, and sub-atmospheric pumping is
required to decrease the pressure, and hence cool the liquid to 1.6 K. The pumping is done
with several intermediate constant pressure periods to accommodate steady T data points
for the calibration.
To reduce uncertainties due to random noise, the recorded resistances for the carbon
resistors are averaged over 30 s (60 data points). This is done during data extraction for
both the calibration and the thermal conductivity measurements.

3.2

Specimen heaters

A 0.3 mm thin foil heater manufactured by Minco

is used to heat one end of the specimen.

It is pasted on a copper fixture to provide homogeneous heat flux to both sides of the
specimen. A BK Precision

1786B 0–32 V, 0–3 A programmable DC power supply was

used to power the heaters attached to the specimens. Two Hewlett-Packard 34401a high
performance bench-top multimeters were used to measure the power to the heaters, one
as a voltmeter and the other as an ammeter. The multimeters had a voltage and current
resolution of 1 mV and 1 µA, respectively.

3.3

Niobium specimens

Thirteen mono- and bicrystal Nb specimens were cut using wire electrical discharge machining (EDM) from several ingot slices. The ingot slices were sourced from several ingots, and
thus had different RRR and tantalum contents. The specimens were shaped as tensile specimens (ASTM E8) to facilitate uniaxial deformation. A bicrystal Nb specimen positioned in
39

Heater
Bi-Crystal Nb
(ASTM E8)

Grain boundary
Temperature
Sensor

Heat Sink

1 cm

Figure 3.3: A bi-crystal Nb specimen in the holder. A steady temperature gradient is
obtained on the Nb specimen using a heater on one end (top) while cooling the other side
(bottom). The temperature gradient is measured using four carbon resistors positioned along
the length of the specimen.
the specimen holder is illustrated in Figure 3.3. A finite electrical power is applied to the
thin foil heater on one end of the specimen while the other end is attached to a cold sink
(conflat flange exposed to liquid He), to obtain a temperature gradient along the length of
the specimen.

40

Table 3.1: Processing histories of the nine bicrystal and four monocrystal specimens. The
hydrogen experiment consisted of infusing hydrogen into the specimens at 300 ‰ for 1 h,
at a pressure of 0.5 atm in a 75 % nitrogen, 25 % hydrogen atmosphere. These specimens
were subsequently heat treated at 800 ‰ for 2 h to degas the hydrogen. The deformation
experiment involved straining the specimens to different nominal engineering strains, and
the specimens were later heat treated at 1000 ‰ for 2 h to examine the recovery of the
phonon peak.
Heat Treatment (HT)
First HT Second HT
Th
t
Th
t
Hydrogen
(‰) (h) (‰) (h) saturated

Spec.

Mono- (M)
or
bicrystal (B)

1

B

600

6

2

B

600

6

1100

4

3

B

750

2

1100

4

4

B

750

2

1100

4

5

B

800

2

6

B

800

2

7

B

140

48

1100

4

8

B

140

48

1100

4

9

M

600

2

10

M

800

2

11

M

1000

2

12

M

1200

2

13

B

41

Deformation
experiment

3.4
3.4.1

Specimen processing
Heat treatments

After baseline measurements, several specimens were heat-treated in a high temperature,
high vacuum furnace [30]. The history of heat treatments for all the specimens are tabulated
in Table 3.1. Specimens 1–8 were used to investigate the response of the phonon peak to
heating protocols frequently used in the SRF community, which ranged from 140 ‰ for 48 h
to 800 ‰ for 2 h. Specimens 2, 3, 4, 7 and 8 were heated a second time at 1100 ‰ for 4 h,
to investigate the effects of prior heat treatments and RRR on the response of the phonon
peak. Specimens 9–12 were prepared to investigate the role of heat treatment temperature
on the thermal conductivity, and were cut from the same grain of an ingot and the crystal
orientation along the heat flow direction were chosen to be the same. These specimens
were heated to different temperatures from 600–1200 ‰, all for the same peak temperature
duration of 2 h.
The 750 ‰ heat treatment was carried out at the Thomas Jefferson National Accelerator
Facility, in the presence of Ti getter at a heating ramp rate of 5 ‰/min. The 140 ‰, 600 ‰,
800 ‰, 1000 ‰, 1100 ‰ and 1200 ‰ heat treatments were performed in a custom built
high-temperature, high-vacuum furnace at Michigan State University [30]. No getter was
used for the 140 ‰ – 800 ‰ heat treatments of specimens 1, 2, and 5–8. For all other heat
treatments, however, Ti getter was used. The temperature ramp rate for all of the specimens
heat treated at Michigan State University was 10 ‰/min. The cooling rate was uncontrolled,
and typically took about 17 h to cool from 1000 ‰ to 50 ‰. A measured temperature profile
for a 1000 ‰ for 2 h heat treatment performed in this furnace is illustrated in Figure 3.4.

42

Temperature (°C)

1 0 0 0

8 0 0

6 0 0

4 0 0

2 0 0

0
0

4

8

1 2

1 6

2 0

T im e (h )

Figure 3.4: A measured temperature profile in the hot zone of the furnace at Michigan State
University during a 1000 ‰ for 2 h heating. The ramp rate was 10 ‰/min, while the cooling
rate was uncontrolled.

43

3.4.2

Deformation

Subsequent to the second heat treatment, specimens 2, 3, and 7 were strained to 20%,
10%, and 2% nominal engineering strains, respectively. Specimen 13, in its as-received
condition, was strained to 2% nominal engineering strain as well. The uniaxial straining was
performed in an Instron

4302 tensile testing machine. The recovery of the phonon peak in

the deformed specimens was examined by subjecting them to a 1000 ‰ for 2 h heating.
The amount of strain for each of the four specimens was chosen based on an analysis of
Schmid factors and Burger’s vector directions. The Schmid factor m is the result of obtaining
the resolved shear stress τ for a crystal with cross-sectional area A being deformed by a force
F applied in the tensile direction. The resolved shear stress is defined as

τ=

F
F
cos(φ) cos(λ) = m,
A
A

(3.1)

where φ is the angle between the direction of F and slip plane normal, and λ is the angle
between F and the slip direction. Although there are several slip planes in a crystal, the
one with the largest Schmid factor will yield first and be dominant. Thus, grains with larger
Schmid factors will deform more readily.
The Burger’s vector, in simple words, represents the direction of dislocation motion in
a crystal. The angle between the Burger’s vector and the tensile direction of the specimen
is used as an alternate measure of dislocation direction. The number of slip systems being
activated during deformation can be inferred from the number of unique Burger’s vector
directions present for the slip planes with the largest Schmid factors.

44

3.4.3

Hydrogen infusion

The interaction of hydrogen with the phonon peak was examined by saturating hydrogen
in specimens 5 and 8, with a subsequent degassing heat treatment at 800 ‰ for 2 h. The
hydrogen was introduced into the specimens in a brazing furnace at 300 ‰ with a 75%
hydrogen, 25% nitrogen gas mixture at 0.5 atm.

3.4.4

BCP etching

The specimens were etched in a 1:1:2 BCP bath at 15 ‰ to remove 15 µm, 20 µm, and 25 µm
of niobium from the surface in their as-received condition, after each processing step, and
after each k measurement, respectively.

3.5
3.5.1

Specimen metallurgy characterization
High energy X-ray diffraction

The imperfection densities of specimens 6–9, subsequent to the 2 h heat treatments, were
qualitatively analyzed using high energy X-ray diffraction.
High energy X-ray diffraction (HEXRD) can be used for volumetric characterization of
defects by using transmission X-ray diffraction [49]. Planes in the specimens that satisfy
Bragg’s law would diffract the X-ray beam to produce a peak on the detector. Bragg’s
law for X-ray diffraction states that for constructive interference to occur in the diffracted
X-rays, the difference in the path lengths of the diffracted beam (2d sin Θ), where d is the
inter-planar spacing and Θ is the incident angle, must equal an integer (n) multiple of the
X-rays’ wavelength (λ). This is illustrated in Figure 3.5. Bragg’s law can thus be written

45

2d sinΘ
Θ

2Θ

d
d sinΘ

d sinΘ

Figure 3.5: Illustration for Bragg’s law for X-ray diffraction. The diffracted X-rays will
produce a peak when there is constructive interference in them. For this to occur, the
difference in path lengths of the diffracted beams (2d sin Θ) should equal an integer (n)
multiple of the X-rays’ wavelength (λ). Here, d is the inter-planar spacing and Θ is the angle
of incidence of X-rays.

46

Diffracted
X-rays

Specimen

2Θ

X-rays

y
z
x

Detector

Figure 3.6: An illustration of the specimen-detector set-up at the Advanced Photon Source at
Argonne National Laboratory. The diffracted X-rays were collected using an area detector
having a resolution of 5 pixels/mm with 2048×2048 pixels, placed about 1 m from the
specimen. Peaks observed on the detector would have satisfied Bragg’s law.
as
nλ = 2d sin Θ.

(3.2)

The imperfection density in specimens 6–9 were estimated by HEXRD using monochromatic X-rays on beamline 6-ID-D at the Advanced Photon Source (APS), a synchrotron
source, at Argonne National Laboratory (ANL). The synchrotron high energy X-rays can
penetrate several mm into the metal [49], thereby making them ideal for the approximately
3 mm thick Nb specimens. An area detector, placed approximately 1 m away from the
47

specimen, was used to collect the diffracted X-rays. The specimen-detector apparatus is
illustrated in Figure 3.6. The 2Θ in Figure 3.6 is equivalent to that in Figure 3.5. The location of the center of the beam on the detector and the distance between the specimen and
the detector were calibrated using a NIST 640c Si powder sample. The X-rays had energy
of about 100 keV (wavelength of approximately 0.12 ˚). The area detector had a resolution
A
of 5 pixels/mm, with 2048×2048 pixels. The detector saturated at intensities greater than
14000 units.
Each of the specimens was examined at 27 locations, with the specimen being rotated
45° (±22.5°) at each location. The diffracted X-ray peak data was recorded in 60 frames
per 45° sweep, i.e., 0.75° per frame, thereby generating 1620 frames per specimen. One
diffracted X-ray peak from a single crystal Nb specimen, as visible on the the area detector,
is illustrated as a two dimensional contour plot in Figure 3.7. The same peak is illustrated in
a three dimensional contour plot in Figure 3.8. Two Gaussian curves are fit to the diffracted
peak in the X- and Y-directions, respectively. The location of the peak maximum and the
calculated full width at half maximum (FWHM) for the detected peaks are used to analyze
the defect concentration in the specimens. Peaks with intensities greater than 14000 units
and intensities less than 3000 units were excluded from the analysis.
The HEXRD of the specimens were performed in collaboration with metallurgists at
Michigan State University.

3.5.2

Electron backscatter diffraction

The crystal orientations of all the specimens in their as-received condition and after some of
the processing were determined using electron backscatter diffraction (EBSD) in a scanning
electron microscope. Diffracted electrons that satisfy the Bragg’s law form Kikuchi patterns
48

Y (c m )
2 5 .6

2 5 .7

2 5 .8

2 5 .9

7 .1

02 .4 0 0 0 . 00
47 28 00 .. 00
91 62 00 .0 0
111 649 842 000
222 416 064 000
23 81 82 00
33 36 60 00
34 08 84 00
44 35 26 00
45 80 04 00
55 25 82 00
56 70 60 00
666 427 842 000
767 294 064 000
77 69 82 00
88 14 60 00
88 86 84 00
99 13 26 00
991 86. 0 40 0 00 8
111 ... 000 538 620
111 ... 111 205 842
11 .. 12 70 60

0

7 .2

In te n s ity

7 .3

EE ++ 00
EE ++ 00
EE ++ 00
EE ++ 00
E + 0

44
44
4

1 2 0 4444 0 0

X

(c m )

2 6 .0

7 .4

7 .5

Figure 3.7: A diffracted X-ray peak from a single crystal Nb specimen measured on an area
detector at the APS at ANL. A three dimensional contour plot for this peak is illustrated in
Figure 3.8. The coordinate system is the same as that represented in Figure 3.6. The origin
is at the top left corner of the detector.

49

1 2 0 0 0
1 0 0 0 0

In te n s ity

8 0 0 0
6 0 0 0
4 0 0 0
2 0 0 0

7 .1
7 .2
0

2 5 .6

7 .3

X

Y (c
m )

(c

7 .4

2 5 .8

m

)

2 5 .7
2 5 .9
2 6 .0

7 .5

Figure 3.8: A diffracted X-ray peak from a single crystal Nb specimen measured on an area
detector at the APS at ANL. A two dimensional contour plot for this peak, as would be
visible on the detector, is illustrated in Figure 3.7. The coordinate system is the same as
that represented in Figure 3.6.

50

on the detector. The process of obtaining Kikuchi patterns and automated indexing of
the patterns to obtain crystal orientation is known as Orientation Imaging MicroscopyTM
(OIM).
The crystal orientation obtained from OIM is represented using Bunge Euler angles,
φ1 , Φ, and φ2 . The Euler angles describe the three angular rotations that are required to
go from the sample reference coordinate system to the crystal coordinate system. For a
sample coordinate system x, y, z, the Bunge method [50] describes the three rotations, using
counter-clockwise rotation as positive, as
1. Rotate φ1 about z-axis, producing new x- and y-axes (x , y ).
2. Rotate Φ about x -axis, to produce new y- and z-axes (y , z ).
3. Rotate φ2 about z -axis, to arrive at the crystal coordinate system.
The OIM measurements of the specimens were performed in collaboration with metallurgists at Michigan State University.

3.5.3

Young’s modulus

The Young’s modulus E is an elastic constant, and is defined as the ratio of the stress σ
to strain

of a material during elastic deformation. According to Hooke’s law, for a body

under stress the strain is reversible provided the applied stress is less than the yield stress,
i.e., elastic deformation, and the strain is proportional to the applied stress. For a material
subjected to uniaxial elastic deformation, Hooke’s law can be written as

= sσ,

51

(3.3)

where s is the elastic compliance constant, and is not isotropic. The Young’s modulus for a
crystal in the direction parallel to the unit vector li , i = 1 . . . 3, was formulated and simplified
for a cubic crystal by Nye [51] as

1
E = s11 − 2 s11 − s12 − s44
2

2 2
l1 l2

2 2
+ l2 l3

2 2
+ l3 l1

−1

,

(3.4)

where s11 , s12 , and s44 were the non-zero components in the compliance constant matrix
sij . Additional non-zero components in the sij matrix were equal to the three compliance
components in equation (3.4). These equal, non-zero components are

s11 = sii , i = 2, 3,
s12 = sij , i = 1, 2, j = 3, and
s44 = sii , i = 5, 6.

The values for s11 , s12 , and s44 for Nb between 4.2 ≤ T ≤ 300 K are tabulated in [52, 53].
For Nb at 300 K [52, 53], s11 = 0.6874 × 10−11 Pa−1 , s12 = −0.2481 × 10−11 Pa−1 , and
s44 = 3.4130 × 10−11 Pa−1 .
Hooke’s law relates E with the speed of sound in the material ν as

ν=

E
,
ρ

(3.5)

where ρ is the density of the material. The Young’s modulus is not isotropic for cubic
materials [51] such as Nb due to the unit vector term in equation (3.4). The smallest value
of E (= 83.24 GPa) is thus in the 111 direction, while the greatest E (= 145.48 GPa) is in

52

the 100 crystallographic direction. Thus, ν is not isotropic and is dependent on the crystal
orientation. The phonon conductivity in insulators can be estimated using the kinetic theory
of gases as
1
kg = Cνl,
3

(3.6)

where C is the specific heat. The thermal conductivity of superconducting niobium for
T

2 K is dominated by phonons with minimal interaction with the electrons, similar to

insulators. The phonon conductivity of superconducting Nb at 1.8 K can thus be related to
E of the material, through equations (3.5) and (3.6), as

1
kg = Cl
3

E
.
ρ

(3.7)

The thermal conductivity at the phonon peak for specimens that are heat treated at
Th ≥ 1000 ‰ are compared with E estimated using equation (3.4).

3.6

Estimating k(T ) from measurements

The temperature and heat flux data from the different Nb specimens are used to estimate
the thermal conductivity of the specimens. For 1-dimensional heat conduction experiments,
k is typically estimated using the Fourier’s law of conduction as

q = −k

dT
,
dx

or,
k=q

53

dx
,
dT

(3.8)

where dx is the distance between two sensors, and dT = T2 − T1 with T1 and T2 being the
measured temperature at sensors 1 and 2, respectively. The negative in equation (3.8) arises
due to T2 > T1 .
The temperature gradients from a Nb specimen during a 1.6 K thermal test is illustrated
in Figure 3.9. The small temperature gradients corresponding to the low heat fluxes are
typical for these bath temperatures. These small temperature gradients lead to larger uncertainties in the estimation of k. Since this study focused on the phonon conductivity of
superconducting Nb, which appears for T

3 K, a novel conductivity estimation technique

had to be developed to reduce the effects of the smaller temperature gradients on k. This
was done by utilizing the theory based model for the thermal conductivity of superconducting metals, which is described in Chapter 2, and using parameter estimation techniques to
obtain parameter groups within the model.

3.6.1

Thermal conductivity estimation

In the past, researchers estimated parameters in equation (2.14) by curve fitting the model
to k data points estimated using Fourier’s law. This method not only contains uncertainties
from T and q measurements, but also has additional uncertainties from the calculation of
k and from the curve fitting.
An alternate method to estimate parameter groups in equation (2.14) was developed as
part of the current study in [54]. This method groups coefficients of T n in the model as

β1 =

ρ295
L RRR

,

β2 = a,

β3 =

54

1
,
D

β4 =

1
,
Bλ

β5 =

∆(T )
,
kB Tc

(3.9)

4 .5

3 2 2
2 8 9
2 4 6
2 0 7
1 7 2
1 3 9
1 0 9
8 3 .3
6 1 .2
4 2 .6
2 7 .3
1 5 .4

T e m p e ra tu re (K )

4 .0
3 .5
3 .0
2 .5
2 .0
1 .5

W /m
W /m
W /m
W /m
W /m
W /m
W /m

2

W /m
W /m
W /m
W /m
W /m

2
2
2
2
2
2
2
2
2
2
2

0

6 .9 1 W /m 2
1 .7 9 W /m 2

1 0

2 0

3 0

4 0

D is ta n c e fro m firs t th e rm is to r (m m )

Figure 3.9: Measured temperature gradients along the length of a Nb specimen, for a bath
temperature of Tb = 1.6 K. Also listed are the respective heat fluxes applied on the specimen
to obtain the temperature gradients. Note the small gradients correspond to the low heat
fluxes, thereby increasing uncertainty in the estimation of k for 1.6 T
2.5 K, which is
the temperature range of interest in this study.

55

to redefine equation (2.14) to be

k(T ) = R(y)

−1
β
β1
β −1
+ β2 T 2
+ −y3 2 + 4
,
T
e T
T3

(3.10)

with
Tc
y = β5 .
T
As described in Chapter 2, each of the five βi can be associated with a particular contribution
to the thermal conductivity: β1 is the contribution of purity of the metal, β2 accounts for the
scattering of electrons by the lattice and phonons, β3 accounts for the scattering of phonons
by normal conducting electrons, β4 accounts for the scattering of phonons by lattice and
specimen boundaries, and β5 is the contribution of the condensation of electrons to k due to
the formation of Cooper pairs.

3.6.2

Parameter estimation algorithm

A novel parameter estimation algorithm was developed to estimate βi s in equation (3.10)
using temperature and heat flux measurements.
Applying boundary conditions of T = T1 at x = x1 and T = T2 at x = x2 for T1 > T2
on the integrated form of the conservation of energy (assuming Fourier’s law), yields

T1 − T2 = q

x2

1
dx .
x =x1 k (T (x))

(3.11)

To estimate the five βi s, the k(T (x)) in equation (3.11) is substituted with the conductivity
model defined in equation (3.10). Then, T1 is assumed to be known and free of error, and

56

equal to the measured temperature, while T 2 is estimated as
x2

1
dx ; i = 1, 2, . . . , 5.
x =x1 k (T (x) , βi )

T 2 = T1 − q

(3.12)

The best estimates of βi for the corresponding temperature and heat flux data are determined
by minimizing a sum of the squares function using Gauss minimization with Box-Kanemasu
interpolation to improve convergence.

3.6.2.1

Gauss minimization

The method of Gauss minimization as defined by Beck and Arnold [55], minimizes a sum of
squares function (S) to yield a minima in the parameter vector (b). This method not only
defines the step size for the correction of b, but also defines a direction for the correction.
However, for the minimization to converge, the initial estimates for b must be in the vicinity
of the minima. Beck and Arnold [55] also state that in the event that there are multiple
local minima, then to estimate the overall minima of b a modification to the Gauss method
of minimization must be implemented. While there are several modifications available to
increase the stability of the Gauss minimization method, the one used in this study is the
Box-Kanemasu interpolation method.

3.6.2.2

Scaled sensitivity analysis

Estimating correlated parameters may result in a nonconverging minimization. Scaled sensitivity coefficients (γi ) indicate correlation between the different parameters, and are defined
as
∂T
; i = 1, 2, . . . , 5.
γi = βi
∂βi
57

(3.13)

S c a le d S e n s itiv ity C o e ffic ie n t (K )

0 .1
0 .0
- 0 .1
- 0 .2
‫1؂‬

- 0 .3
- 0 .4

‫2؂‬

- 0 .5

‫4؂‬

- 0 .6

‫3؂‬

2

‫5؂‬

3

4

5

T e m p e ra tu re (K )

Figure 3.10: Scaled sensitivity coefficients for the five βi defined in equation (3.9) [54].
Scaled sensitivity coefficients for the βi are calculated for the temperature range of the
measurements, 1.6 K ≤ T ≤ 4.6 K and plotted in Figure 3.10.
Curves for β1 , β3 , β4 and β5 in Figure 3.10 are unique and the magnitudes of the maximums are of the order of magnitude of the measured temperatures, hence these parameters
can be estimated [55]. The magnitude of β2 is too small to make reliable estimates within
the measured temperatures. Thus, β2 is not estimated, and its theoretical value [32] is used.

58

3.6.3

Uncertainty in k

The uncertainty in k is estimated using the Kline & McClintock method [56]. In this method
the uncertainty σ for a result R, where R = R(v1 , v2 , v3 ) with v1 , v2 , and v3 being independent variables, can be estimated as

σ=

2
∂R
σv
+
∂v1 1

2
∂R
σv
+
∂v2 2

2
∂R
σv
,
∂v3 3

(3.14)

where σv1 , σv2 , and σv3 are the uncertainties in the independent variables v1 , v2 , and v3 ,
respectively.
The independent variables that implicitly influence k(βi ) in equation (3.11) are T , x, q,
and the cross-sectional area A of the specimen. The uncertainty in k can thus be written as

σk =

2
∂k
σT
+
∂T

2
∂k
σx +
∂x

2
∂k
σq +
∂q

2
∂k
σA ,
∂A

(3.15)

where σT , σx , σq , and σA are the uncertainties in the independent variables T , x, q, and A,
respectively.
The partial derivatives of k, or the sensitivity coefficients, in equation (3.15) difficult
to obtain analytically. The partial derivatives can alternately be estimated numerically by
giving a small perturbation to one independent variable and observing the change in the
dependent variable. For example,

k(T, x, q, A) − k(T + ∆T, x, q, A)
∂k
=
,
∂T
T − (T + ∆T )

where ∆T is a small fraction of T , such as 0.1%.

59

(3.16)

The uncertainty in T is the combined uncertainties of the two temperature resistors that
measure the temperature gradient, and is written as

σT =

2
2
σT + σT ,
1
2

(3.17)

where σT1 and σT2 are the uncertainties of the temperature measurements. The greatest contributor to uncertainty in the temperature measurements are from the interpolation
polynomial used for the calibration [48].
The uncertainties in x and A arise due to length measurements taken on a digital vernier
caliper with a resolution of 0.01 mm. The uncertainty in q is a function of the uncertainties
in the measured current (I) and voltage (V ), as

σq =

2
2
σI + σV ,

(3.18)

where σI and σV are the uncertainties of I and V respectively. The values for σI and σV
are provided by the manufacturer of the multimeters used for measurements.
The uncertainty in k is estimated using data from representative specimens. The average
in the uncertainty in k for the range of temperature measurement is 5%. The maximum in
uncertainty is about 6%.

60

Chapter 4
Results:
Phonon peak and heat-treatment
temperature
The effect of heat treatment temperature on the magnitude of the phonon peak and dislocation content is examined in this chapter. Two sets of niobium specimens were used for this
study. Set 1 comprised of eight bicrystal niobium specimens randomly cut from different
ingot discs. These specimens generate data that account for variables like grain orientation,
purity (RRR), and Ta content, in addition to heat treatment temperature. The specimens
were heated to different temperatures and durations, ranging from 140 ‰ for 48 h to 1100 ‰
for 4 h, covering the range of heat treatments frequently used to heat treat SRF cavities.
Set 2 comprised of four niobium specimens that were cut from one grain of an ingot disc
and had heat transfer directions aligned along the same crystal orientation. These specimens were heated at various temperatures between 600 ‰ and 1200 ‰ for 2 h. Unlike the

61

specimens in Set 1, the specimens in Set 2 reduce the variables in the experiment to be only
the heat treatment temperature.

4.1

Set 1: Bicrystal specimens from different ingots

The specimens in Set 1 were used to explore the effects of several heat treatment parameters in combination with the variation in grain orientation, RRR and Ta contents on the
magnitude of the phonon peak. After thermal measurements in their as-received conditions,
specimens 1 and 2 were heated to 600 ‰ for 6 h, while specimens 3 and 4 were heated to
750 ‰ for 2 h, specimens 5 and 6 were heated to 800 ‰ for 2 h, and specimens 7 and 8
were heated to 140 ‰ for 48 h. After thermal conductivity measurements, specimens 2, 3,
4, 7, and 8 were subsequently heated a second time to 1100 ‰ for 4 h. The specimen heat
treatment conditions, as well as their tantalum contents [57] and RRR estimated from k for
each of the specimens, are tabulated in Table 4.1. All eight specimens presented here have
a gauge length of 40 mm. Specimens 1, 5 and 6 have cross sections of 3.175 × 4 mm2 , while
specimens 2, 3, 4, 7 and 8 have cross sections of 3.25 × 10 mm2 .

4.1.1

Thermal conductivities

The measured thermal conductivities of the eight bicystal specimens listed in Table 4.1, in
their as-received condition and after heat treatments, are plotted as a function of T in Figures 4.1 – 4.8. The thermal conductivities of grains 1 and 2, and k across the grain boundary
(GB) are illustrated in the figures by squares, circles, and triangles, respectively. The Euler
angels φ1 , Φ, and φ2 for the two grains after the hottest temperature heat treatment, as
well as the inverse pole figure illustrating the orientation of the two grains, are shown below

62

Table 4.1: Heat treatment histories for the specimens in Set 1. Estimated RRR values,
tantalum contents, and the use of Ti getter during the heat treatments of the specimens are
also tabulated.

Est.

Ta content

Specimen

RRR

(ppm)

1
2

191
131

1275
668

3
4

190
196

5
6
7
8

Heat Treatment
First heating
Second heating

‰)

Th (

t (h)

Ti

600
600

6
6

N
N

756
756

750
750

2
2

Y
Y

104
143

1322
523

800
800

2
2

1375
704

140
140

48
48

N
N

‰)

N
N

174
200

Th (

t (h)

Ti

1100

4

Y

1100
1100

4
4

Y
Y

1100
1100

4
4

Y
Y

the thermal conductivity plot in the figures. Figure 4.1 shows k as a function of T for both
grains in specimen 1, in their as-received condition (open symbols) and after heat treatment
at 600 ‰ for 6 h (half-filled symbols). The conductivity of specimen 2 in its as-received
condition (open symbols), after the 600 ‰ for 6 h heat treatment (half-filled symbols), and
after the 1100 ‰ for 4 h heat treatment (solid symbols) are illustrated in Figure 4.2. Figures 4.3 and 4.4 show k as a function of T for both grains of specimens 3 and 4, respectively,
in their as-received condition (open symbols), after 750 ‰ for 2 h heat treatment (half-filled
symbols), and after 1100 ‰ for 4 h heat treatment (solid symbols). Figures 4.5 and 4.6 show
k as a function of T for both grains of specimens 5 and 6, respectively, in their as-received
condition (open symbols), and after 800 ‰ for 2 h heat treatment (half-filled symbols). The
conductivities of the two grains in specimens 7 and 8 in their as-received condition (open
symbols), after the 140 ‰ for 48 h heat treatment (half-filled symbols), and after the 1100 ‰
for 4 h heat treatment (solid symbols) are illustrated in Figures 4.7 and 4.8, respectively.
63

These specimens are from different ingots, as suggested by their different values of RRR.
For the 600 ‰ – 800 ‰ heat-treated specimens, there is very little change in the value of
RRR after heat treatment, as might be expected at these relatively moderate heat treatment
temperatures, even with the lack of a getter. After the 1100 ‰ heat treatment, however,
there is a 10% decrease in k at 4.2 K.
A representative set of the measured thermal conductivities of specimens in Set 1 after
the 1100 ‰ heat treatment are plotted as a function of temperature in Figure 4.9. Also
listed in the legend of Figure 4.9 are the estimated RRR values for each of the specimens
after the 1100 ‰ heat treatment.
The magnitude of increase in phonon conduction due to the first heating can be examined
for the specimens by calculating the ratio kpp /k2 , where kpp is k at the phonon peak after
heating and k2 is k at 2 K in the specimens’ as-received condition. To negate the contribution
of electron conduction to the magnitude of the phonon peak, the ratio kpp /k3 is introduced
for the specimens heat treated at 1100 ‰ for 4 h, where k3 is k at 3 k after the heating.
The ratios kpp /k2 and kpp /k3 for the specimens in Set 1 are listed in Table 4.2. Due to the
absence of a phonon peak in k of specimens 7 and 8 after the 140 ‰ for 48 h heating, kpp /k2
are not calculated.

64

1-1 As rec'd
1-1 600 °C

7 0

T h e rm a l c o n d u c tiv ity (W m -1 K -1 )

6 0

1-GB As rec'd
1-GB 600 °C

1-2 As rec'd
1-2 600 °C

5 0
4 0
3 0
2 0
1 0
0

1 .5

1-1
؆1= 277.4°

Φ = 119.5°
؆2= 23.1°

2 .0

2 .5

3 .0

Temperature (K)

3 .5

4 .0

4 .5
[111]

1-2
؆1= 268.2°

Φ = 138.3°
؆2= 374.3°

[001]

[101]

Figure 4.1: The thermal conductivities of grains 1 (squares) and 2 (circles), and across the
grain boundary (triangles) of specimen 1 in their as-received condition (open symbols) and
after 600 ‰ for 6 h heat treatment (half-filled symbols). The Euler angels φ1 , Φ, and φ2
representing the crystal orientation along the heat flow direction for the two grains after
the 600 ‰ heating, as well as the inverse pole figure illustrating the orientations, are shown
below the thermal conductivity plot.

65

2-1 As rec'd
2-1 600 °C
2-1 1100 °C

7 0

T h e rm a l c o n d u c tiv ity (W m -1 K -1 )

6 0

2-GB As rec'd
2-GB 600 °C
2-GB 1100 °C

2-2 600 °C
2-2 1100 °C

5 0
4 0
3 0
2 0
1 0
0

1 .5

2-1
؆1= 9.7°

Φ = 18.4°
؆2= 351.8°

2 .0

2 .5

3 .0

Temperature (K)

3 .5

4 .0

4 .5
[111]

2-2
؆1= 64.7°

Φ = 40.8°
؆2= 261.2°

[001]

[101]

Figure 4.2: The thermal conductivities of the grains 1 (squares) and 2 (circles), and across
the grain boundary (triangles) of specimen 2 in their as-received condition (open symbols),
after 600 ‰ for 6 h heat treatment (half-filled symbols), and after 1100 ‰ for 4 h (solid
symbols). The Euler angels φ1 , Φ, and φ2 representing the crystal orientation along the heat
flow direction for the two grains after the 1100 ‰ heating, as well as the inverse pole figure
illustrating the orientations, are shown below the thermal conductivity plot.

66

3-1 As rec'd
3-1 750 °C
3-1 1100 °C

7 0

T h e rm a l c o n d u c tiv ity (W m -1 K -1 )

6 0

3-GB As rec'd
3-GB 750 °C
3-GB 1100 °C

3-2 As rec'd
3-2 750 °C
3-2 1100 °C

5 0
4 0
3 0
2 0
1 0
0

1 .5

3-1
؆1= 183.2°
Φ = 41.8°
؆2= 135.0°

2 .0

2 .5

3 .0

Temperature (K)

3 .5

4 .0

4 .5
[111]

3-2
؆1= 1.3°

Φ = 13.1°
؆2= 30.0°

[001]

[101]

Figure 4.3: The thermal conductivities of the grains 1 (squares) and 2 (circles), and across the
grain boundary (triangles) of specimen 3 in their as-received condition (open symbols), after
750 ‰ for 2 h heat treatment (half-filled symbols), and after 1100 ‰ for 4 h heat treatment
(solid symbols). The Euler angels φ1 , Φ, and φ2 representing the crystal orientation along
the heat flow direction for the two grains after the 1100 ‰ heating, as well as the inverse
pole figure illustrating the orientations, are shown below the thermal conductivity plot.

67

4-1 As rec'd
4-1 750 °C
4-1 1100 °C

7 0

T h e rm a l c o n d u c tiv ity (W m -1 K -1 )

6 0

4-GB As rec'd
4-GB 750 °C
4-GB 1100 °C

4-2 As rec'd
4-2 750 °C
4-2 1100 °C

5 0
4 0
3 0
2 0
1 0
0

1 .5

4-1
؆1= 2.7°

Φ = 138.7°
؆2= 318.3°

2 .0

2 .5

3 .0

Temperature (K)

3 .5

4 .0

4 .5
[111]

4-2
؆1= 258.0°

Φ = 134.2°
؆2= 298.9°

[001]

[101]

Figure 4.4: The thermal conductivities of the grains 1 (squares) and 2 (circles), and across the
grain boundary (triangles) of specimen 4 in their as-received condition (open symbols), after
750 ‰ for 2 h heat treatment (half-filled symbols), and after 1100 ‰ for 4 h heat treatment
(solid symbols). The Euler angels φ1 , Φ, and φ2 representing the crystal orientation along
the heat flow direction for the two grains after the 1100 ‰ heating, as well as the inverse
pole figure illustrating the orientations, are shown below the thermal conductivity plot.

68

5-1 As rec'd
5-1 800 °C

7 0

T h e rm a l c o n d u c tiv ity (W m -1 K -1 )

6 0

5-GB As rec'd
5-GB 800 °C

5-2 As rec'd
5-2 800 °C

5 0
4 0
3 0
2 0
1 0
0

1 .5

5-1
؆1= 297.7°
Φ = 41.3°
؆2= 37.5°

2 .0

2 .5

3 .0

Temperature (K)

3 .5

4 .0

4 .5
[111]

5-2
؆1= 190.9°
Φ = 7.2°
؆2= 187.6°

[001]

[101]

Figure 4.5: The thermal conductivities of the grains 1 (squares) and 2 (circles), and across
the grain boundary (triangles) of specimen 5 in their as-received condition (open symbols)
and after 800 ‰ for 2 h heat treatment (half-filled symbols). The Euler angels φ1 , Φ, and
φ2 representing the crystal orientation along the heat flow direction for the two grains after
the 800 ‰ heating, as well as the inverse pole figure illustrating the orientations, are shown
below the thermal conductivity plot.

69

6-1 As rec'd
6-1 800 °C

7 0

T h e rm a l c o n d u c tiv ity (W m -1 K -1 )

6 0

6-GB As rec'd
6-GB 800 °C

6-2 As rec'd
6-2 800 °C

5 0
4 0
3 0
2 0
1 0
0

1 .5

6-1
؆1= 52.7°

Φ = 141.1°
؆2= 351.2°

2 .0

2 .5

3 .0

Temperature (K)

3 .5

4 .0

4 .5
[111]

6-2
؆1= 47.3°

Φ = 143.1°
؆2= 0.5°

[001]

[101]

Figure 4.6: The thermal conductivities of the grains 1 (squares) and 2 (circles), and across
the grain boundary (triangles) of specimen 6 in their as-received condition (open symbols)
and after 800 ‰ for 2 h heat treatment (half-filled symbols). The Euler angels φ1 , Φ, and
φ2 representing the crystal orientation along the heat flow direction for the two grains after
the 800 ‰ heating, as well as the inverse pole figure illustrating the orientations, are shown
below the thermal conductivity plot.

70

7-1 As rec'd
7-1 140 °C
7-1 1100 °C

7 0

T h e rm a l c o n d u c tiv ity (W m -1 K -1 )

6 0

7-GB As rec'd
7-GB 140 °C
7-GB 1100 °C

7-2 As rec'd
7-2 140 °C
7-2 1100 °C

5 0
4 0
3 0
2 0
1 0
0

1 .5

7-1
؆1= 157.3°
Φ = 40.3°
؆2= 168.5°

2 .0

2 .5

3 .0

Temperature (K)

3 .5

4 .0

4 .5
[111]

7-2
؆1= 226.2°
Φ = 47.0°
؆2= 172.8°

[001]

[101]

Figure 4.7: The thermal conductivities of the grains 1 (squares) and 2 (circles), and across
the grain boundary (triangles) of specimen 7 in their as-received condition (open symbols)
and after 140 ‰ for 48 h heat treatment (half-filled symbols), and after 1100 ‰ for 4 h
heat treatment (solid symbols). The Euler angels φ1 , Φ, and φ2 representing the crystal
orientation along the heat flow direction for the two grains after the 1100 ‰ heating, as
well as the inverse pole figure illustrating the orientations, are shown below the thermal
conductivity plot.

71

8-1 As rec'd
8-1 140 °C

7 0

T h e rm a l c o n d u c tiv ity (W m -1 K -1 )

6 0

8-GB As rec'd
8-GB 140 °C
8-GB 1100 °C

8-2 As rec'd
8-2 140 °C
8-2 1100 °C

5 0
4 0
3 0
2 0
1 0
0

1 .5

8-1
؆1= 193.7°
Φ = 20.1°
؆2= 146.2°

2 .0

2 .5

3 .0

Temperature (K)

3 .5

4 .0

4 .5
[111]

8-2
؆1= 149.7°
Φ = 33.7°
؆2= 247.1°

[001]

[101]

Figure 4.8: The thermal conductivities of the grains 1 (squares) and 2 (circles), and across the
grain boundary (triangles) of specimen 8 in their as-received condition (open symbols), after
140 ‰ for 48 h heat treatment (half-filled symbols), and after 1100 ‰ for 4 h heat treatment
(solid symbols). The Euler angels φ1 , Φ, and φ2 representing the crystal orientation along
the heat flow direction for the two grains after the 1100 ‰ heating, as well as the inverse
pole figure illustrating the orientations, are shown below the thermal conductivity plot.

72

7 0
R R R
R R R
R R R
R R R
R R R

T h e rm a l c o n d u c tiv ity (W m

-1

K

-1

)

6 0
5 0

= 1 8
= 1 5
= 1 5
= 1 3
= 1 1

8
0
0
4
0

4 0
3 0
2 0
1 0
1 .5

2 .0

2 .5

3 .0

3 .5

4 .0

4 .5

T e m p e ra tu re (K )
Figure 4.9: Representative thermal conductivities of the five niobium specimens in Set 1
that were heat-treated at 1100 ‰ for 4 h. The estimated RRR for each specimen after the
1100 ‰ heat treatment is listed in the legend. A dependence of maximum in kpp on RRR is
visible.

73

Table 4.2: Ratios of the phonon conductivity of the specimens in Set 1, after their two heat
treatments. kpp in the ratios refer to the thermal conductivity at the phonon peak after heat
treatments, while k2 refers to k at T = 2 K in their as-received condition, and k3 refers to
k at T = 3 K after the 1100 ‰ heat treatment. For specimens 7 and 8 after the 140 ‰ for
48 h heat treatment, there was no phonon peak, and hence the kpp /k2 ratios are not listed.
First heating
Specimen
1

2

3

4

5

6

7

8

Second heating

Grain Th (‰) th (h) kpp /k2
1
GB
2

600

6

1.69
1.78
1.58

1
GB
2

600

6

2.20
2.42
-

Th (‰) th (h) kpp /k3

1
GB
2

750

2

2.83
4.99
2.52

1
GB
2

750

2

2.33
2.95
2.69

1
GB
2

800

2

48

48

-

1100

4

2.55
2.52
2.49

1100

4

2.54
2.58
2.62

4

2.41
2.30
2.50

4

2.66
2.68

2.44
2.29
2.24
-

2.53
2.59
2.44

2.34
2.53
2.30

2

4

1100

1
GB
2
1
GB
2
1
GB
2

800

140

140

74

1100

1100

4.2

Set 2: Two hour heat treatment kinetics

The second set of niobium specimens were used to uncover the kinetics of two hour heat
treatments. These four specimens had been cut from one grain of an ingot disc, and had
the same crystal orientation along their heat flow (tensile) directions. Each specimen was
subjected to a unique heat treatment temperature Th , while the duration at constant Th
was 2 h for each specimen. Thermal measurements were performed on all of these specimens
in their as-received conditions and after 2 h heat treatments. The dislocation contents of
these specimens after the 2 h heat treatments were estimated using a theory-based model
for dislocation density, and a qualitative analysis of the imperfection contents was performed
using high energy X-ray diffraction and orientation imaging microscopy.

4.2.1

Thermal conductivities

The measured thermal conductivities of specimens 9–12 in their as-received condition (open
symbols) and after the 600 ‰ – 1200 ‰ for 2 h heat treatments (half-filled symbols) are
plotted as a function of temperature in Figures 4.10–4.13, respectively. The Euler angels
φ1 , Φ, and φ2 of the crystal after the hottest temperature heating, as well as the inverse
pole figure illustrating the crystal orientation, are shown below the thermal conductivity
plot. For comparing the thermal conductivities of the specimens in Set 2 illustrated in
Figures 4.10–4.13, one k curve for each specimen in its as-received condition and after the
2 h heat treatments are plotted as a function of temperature in Figure 4.14. Although
there is a significant increase in k in the phonon dominated regime, there is a 10% and 20%
decrease in k at T = 4.2 K after the 1000 ‰ and 1200 ‰ heat treatments, respectively. This
indicates introduction of some impurities into the specimen during the 1000 ‰ and 1200 ‰

75

heat treatments.
To reduce the influence of the RRR of heat-treated specimens on the perception of their
phonon response, the ratio kpp /k3 is examined for specimens in Set 2. The ratio kpp /k3 for
these specimens in their as-received condition (25 ‰) and after heat treatments (squares) is
plotted in Figure 4.15. A sigmoidal curve was found to best fit the data from specimens in
Set 2, and is illustrated as the solid line in Figure 4.15. Also plotted are the kpp /k3 for the
niobium specimens of Set 1 heat-treated at 1100 ‰ for 4 h (triangles).

4.2.2

Dislocation content

A quantitative comparison of the dislocation contents of specimens 9 – 12 was performed using the magnitude of the phonon peak and the theoretical phonon peak limited by specimen
boundary scattering as defined by Casimir [39]. A qualitative comparison of the concentration of imperfections in the lattice was done using local average misorientation (LAM)
data from the orientation gradient maps for the surfaces of the specimens. Bulk measurements were made using peak spread and full width at half maximum (FWHM) information
from high-energy X-ray diffraction measurements. While the local average misorientation
data represents the surface only, the data from the high-energy X-ray diffraction is a bulk
measurement.

4.2.2.1

Dislocation density from kpp

A method to estimate the density of randomly distributed dislocations Nd in metals using lattice conductivity was presented by Klemens [58], and was simplified for Nb by Wasserb¨ch [26]
a
as
Wgd T 2
= 3.05 × 10−15 m3 K3 /watt,
Nd
76

(4.1)

9-1 As rec'd
9-1 600 °C

7 0

T h e rm a l c o n d u c tiv ity (W m -1 K -1 )

6 0

9-2 As rec'd
9-2 600 °C

9-3 As rec'd
9-3 600 °C

5 0
4 0
3 0
2 0
1 0
0

1 .5

2 .0

2 .5

3 .0

Temperature (K)

3 .5

4 .0

4 .5
[111]

؆1= 87.6°

Φ = 44.8°
؆2= 278.0°

[001]

[101]

Figure 4.10: Thermal conductivity curves for specimen 9 in its as-received condition (open
symbols) and after the 600 ‰ for 2 h heating (half-filled symbols). The three curves for
each condition are the k for each T sensor pair on the single crystal specimen. The Euler
angels φ1 , Φ, and φ2 representing the crystal orientation along the heat flow direction for the
specimen after the 600 ‰ heating, as well as the inverse pole figure illustrating the crystal
orientation, are shown below the thermal conductivity plot.

77

10-1 As rec'd

7 0

T h e rm a l c o n d u c tiv ity (W m -1 K -1 )

6 0

10-2 As rec'd
10-2 800 °C

10-3 As rec'd
10-3 800 °C

5 0
4 0
3 0
2 0
1 0
0

1 .5

2 .0

2 .5

3 .0

Temperature (K)

3 .5

4 .0

4 .5
[111]

؆1= 87.0°

Φ = 44.7°
؆2= 277.9°

[001]

[101]

Figure 4.11: Thermal conductivity curves for specimen 10 in its as-received condition (open
symbols) and after the 800 ‰ for 2 h heating (half-filled symbols). The three curves for
each condition are the k for each T sensor pair on the single crystal specimen. The Euler
angels φ1 , Φ, and φ2 representing the crystal orientation along the heat flow direction for the
specimen after the 800 ‰ heating, as well as the inverse pole figure illustrating the crystal
orientation, are shown below the thermal conductivity plot.

78

11-1 As rec'd
11-1 1000 °C

7 0

T h e rm a l c o n d u c tiv ity (W m -1 K -1 )

6 0

11-2 As rec'd
11-2 1000 °C

11-3 As rec'd
11-3 1000 °C

5 0
4 0
3 0
2 0
1 0
0

1 .5

2 .0

2 .5

3 .0

Temperature (K)

3 .5

4 .0

4 .5
[111]

؆1= 89.3°

Φ = 43.7°
؆2= 278.2°

[001]

[101]

Figure 4.12: Thermal conductivity curves for specimen 11 in its as-received condition (open
symbols) and after the 1000 ‰ for 2 h heating (half-filled symbols). The three curves for
each condition are the k for each T sensor pair on the single crystal specimen. The Euler
angels φ1 , Φ, and φ2 representing the crystal orientation along the heat flow direction for the
specimen after the 1000 ‰ heating, as well as the inverse pole figure illustrating the crystal
orientation, are shown below the thermal conductivity plot.

79

12-1 As rec'd
12-1 1200 °C

7 0

T h e rm a l c o n d u c tiv ity (W m -1 K -1 )

6 0

12-2 As rec'd
12-2 1200 °C

12-3 As rec'd
12-3 1200 °C

5 0
4 0
3 0
2 0
1 0
0

1 .5

2 .0

2 .5

3 .0

Temperature (K)

3 .5

4 .0

4 .5
[111]

؆1= 90.0°

Φ = 43.7°
؆2= 277.4°

[001]

[101]

Figure 4.13: Thermal conductivity curves for specimen 12 in its as-received condition (open
symbols) and after the 1200 ‰ for 2 h heating (half-filled symbols). The three curves for
each condition are the k for each T sensor pair on the single crystal specimen. The Euler
angels φ1 , Φ, and φ2 representing the crystal orientation along the heat flow direction for the
specimen after the 1200 ‰ heating, as well as the inverse pole figure illustrating the crystal
orientation, are shown below the thermal conductivity plot.

80

7 0

9 - As rec'd
9 - 600 °C
10 - As rec'd
10 - 800 °C
11 - As rec'd
11 - 1000 °C
12 - As rec'd
12 - 1200 °C

T h e rm a l c o n d u c tiv ity (W m

-1

K

-1

)

6 0
5 0
4 0
3 0
2 0
1 0
1 .5

2 .0

2 .5

3 .0

3 .5

Temperature (K)

4 .0

4 .5

Figure 4.14: Comparison of the thermal conductivities of the four niobium specimens in
Set 2 in their as-received condition and after 2 h heat treatments. For clarity, only one curve
per specimen is plotted.

81

3 .0

2 .5

k p p / k 3

2 .0

1 .5

1 .0
0

2 0 0

4 0 0

6 0 0

8 0 0

1 0 0 0

Heat Treatment Temperature (°C)

1 2 0 0

Figure 4.15: Thermal conductivity ratios kpp /k3 for the specimens in Set 2 (squares) in their
as-received condition, and after 2 h heat treatments. The solid line represents a sigmoidal
curve fit to the kpp /k3 data for the specimens in Set 2 (squares). The kpp /k3 for specimens
in Set 1 heat-treated at 1100 ‰ for 4 h (triangles) is also plotted.

82

where Wgd is the thermal resistivity of phonons due to dislocations. This resistance can be
separated from the phonon dominated thermal conductivity kg as [26]

−1
kg = kgb + Wgd

−1

,

(4.2)

where kgb is the crystal boundary limited phonon conductivity. For analyzing specimens
presented here, the thermal conductivity at the phonon peak kpp is used for the phonon
conductivity kg . The Casimir [39] relation for k limited by crystal boundaries, defined in
equation (2.11), is used for kgb . For convenience, equation (2.11) is rearranged and presented
here as
c
kgb

=

2.31 × 103 R

Cv 2/3 3
T ,
2/3 T 3

2 (vl /vt )2 + 1
3

2 (vl /vt ) + 1

(4.3)

where R is the radius for a circular cylindrical specimen, Cv the specific heat per unit
volume, and vl and vt are the longitudinal and transverse velocities of sound in the material,
respectively. For a specimen with square cross section d2 , R in the above equation would be
replaced [39] by R = 0.564 ∗ d. For specimens with a rectangular cross sectional area Acs ,
√
like the ones presented here, R would be replaced by R = 0.564 Acs .
The dislocation densities for specimens 9 – 12 obtained using equation (4.1) are illustrated as a function of the heat treatment temperature in Figure 4.16. For perspective and
comparison, Nd is added to the thermal conductivities of specimens 9 – 12 presented in
Figure 4.15. This analysis supports the role of reducing dislocations in the enhancement of
kpp .

83

Set 2 (9, 10, 11, 12)
Set 1 (2, 3, 4, 7, 8)
Dislocation density (theo. est.)
9
8

)

3 .0

1 3

m

7

-2

2 .5

2 .0

D is lo c a tio n d e n s ity (x 1 0

6

k p p / k 3

5
4

1 .5
3
2

1 .0
0

2 0 0

4 0 0

6 0 0

8 0 0

1 0 0 0

Heat Treatment Temperature (°C)

1 2 0 0

1

Figure 4.16: The estimated dislocation densities for specimens 9 – 12 using equation (4.1).
Note that the dislocation density Nd is plotted over Figure 4.15, for comparison with the
thermal conductivities of the specimens.

84

4.2.2.2

Orientation imaging microscopy

The local average misorientation (LAM) for specimens 9 – 12, generated from orientation
gradient maps, in their as-received conditions and after the 2 h heating are illustrated in
Figure 4.17. Specimen 9 in its as-received condition has the smallest and widest LAM, indicating greater crystal imperfections in the specimen. The remaining three specimens have
similar LAM curves in their as-received condition, suggesting similar dislocation content.
After the 2 h heating, specimens 9, 10, and 12, heat treated at 600 ‰, 800 ‰, and 1000 ‰,
respectively, similar LAM curves, with peaks at smaller misorientation angle with narrower
full width at half maximum than that of the as-received state. Specimen 12, heated at
1200 ‰, displays the most LAM peak shift to the left, indicating greater recovery of dislocations in the material. The LAMs for these four specimens, however, do not directly correlate
with the trends of the recovery of the corresponding phonon peaks. This discrepancy may
stem from the thermal conductivity being a bulk measurement, while the LAMs are surface
measurements, and therefore not indicative of the bulk.

4.2.2.3

X-Ray diffraction

High-energy X-Ray diffraction was performed on all four specimens in Set 2 after the 2 h
heat treatment. Each of the specimens was examined at 27 locations, with the specimen
being rotated 45° (±22.5°) at each location. The diffracted X-ray peak data was recorded in
60 frames per 45° sweep, i.e., 0.75° per frame, thereby generating 1620 frames per specimen.
The diffracted peaks corresponding to the planes of interest were identified by comparing
the theoretical d-spacing for the respective planes with that estimated from the radius of the
peaks on the detector with respect to the X-ray beam. The planes to be used to estimate
imperfection density of the specimens were chosen based on the stereographic projections of
85

0 .0 6

9 - As rec'd
9 - 600 °C
10 - As rec'd
10 - 800 °C
11 - As rec'd
11 - 1000 °C
12 - As rec'd
12 - 1200 °C

F ra c tio n

0 .0 4

0 .0 2

0 .0 0
0 .0

0 .5

1 .0

Local average misorientation (deg.)

1 .5

Figure 4.17: The local average misorientation for the single crystal specimens 9 – 12, in their
as-received conditions and after the 2 h heating at temperatures between 600 ‰ and 1200 ‰.

86

these planes.
For specimens 9 – 12, the stereographic projections are similar due to the small variation
in Euler angles, as illustrated in Figures 4.18 – 4.21, respectively. The shaded grey region at
the center of the stereographic projection plot indicates crystallographic space that was not
exposed to the X-rays due to the ±22.5° rotation of the specimen. Hence, only planes that
have plane normal directions in the white space outside of the grey area would be diffracted
onto the detector. For the four specimens illustrated in Figures 4.18 – 4.21, only two of
the four {111} planes with <111> plane normal directions were in the white, detectable
region. Hence, a {222} peak analysis could be used to qualitatively estimate dislocation and
imperfection contents of the specimens.
Since specimens 9 – 12 are single crystals, the diffracted peaks for all 27 locations, i.e.,
all 1620 frames, can simultaneously be used to compare the imperfection densities. The
spread of the diffracted {222} peaks across the 27 locations for each of the four specimens
are illustrated in Figures 4.22 – 4.25. The x- and y-distances are in the figure are with
respect to the location of the beam on the detector. The four peak clusters are the locations
of the X-rays diffracted by the two {111} planes identified in Figure 4.18. Each individual
cluster is a group of {222} peaks from several of the 1620 frames recorded per specimen,
and the actual number of frames from which {222} peaks are detected for each cluster is
listed towards the center of the figure. The normalized standard deviations of the locations
of these peaks are plotted against the temperature of heat treatment in Figure 4.26. The
normalized standard deviation is defined as

norm. σ =

87

2
2
σx + σy ,

(4.4)

1

87.59 44.771 277.95 (−10−1)
Bunge

0.8

(−2−1−1)

(−11−1)

<100>
<110>

(−21−1)
(−12−1)

0.6

(−100)
(−1−10)

0.4

(−2−11)

0.2

(−110)

(−211)

(−111)(−121) (010)
(−1−21)(−1−11) (−101)
(−1−12) (−112)
(011)
(0−11)
(001)
(121)
(112)
(1−12)
(111)

0
−0.2
−0.4
−0.6

Y

−0.8
(101)
−1

Planes
abc 1 1 1
−1

<111>
<112>

X
−0.5

0

0.5

1

Figure 4.18: Stereographic projection for specimen 9 after the 600 ‰ for 2 h heat treatment,
using Euler orientations obtained by EBSD. The stereographic projection was used to identify
planes from which X-ray diffraction peaks could be used to estimate the imperfection density
of the specimen. The grey shaded region represents crystallographic space which was not
visible to the X-rays due to the 45°sweep of the specimens. The stereographic projection
was generated using a MATLAB algorithm developed by Bieler [59].

88

1

87.009 44.684 277.86(−10−1)
Bunge

0.8

(−11−1)
(−21−1)
(−12−1)

(−2−1−1)

0.6

(−100)
(−1−10)

0.4

(−2−11)

0.2

<100>
<110>

(−110)

(−211)

(010)
(−1−21)(−1−11) (−101) (−111)(−121)
(−1−12) (−112)
(011)
(0−11)
(001)
(121)
(112)
(1−12)
(111)

0
−0.2
−0.4
−0.6

Y

−0.8
(101)
−1

Planes
abc 1 1 1
−1

<111>
<112>

X
−0.5

0

0.5

1

Figure 4.19: Stereographic projection for specimen 10 after the 800 ‰ for 2 h heat treatment,
using Euler orientations obtained by EBSD. The stereographic projection was used to identify
planes from which X-ray diffraction peaks could be used to estimate the imperfection density
of the specimen. The grey shaded region represents crystallographic space which was not
visible to the X-rays due to the 45°sweep of the specimens. The stereographic projection
was generated using a MATLAB algorithm developed by Bieler [59].

89

1

89.331 43.702 278.23
Bunge

0.8

<100>
<110>

(−11−1)
(−21−1)
(−12−1)

(−2−1−1)

0.6

(−100)
(−1−10)

0.4
0.2

(−1−21)

0

(−110)

(−2−11) (−211)
(010)
(−111)(−121)
(−1−11)(−101)
(−112)
(011)
(001)

Y

(−1−12)

−0.2

(0−11)

−0.4

(121)

(112)
−0.6

(1−12)

−0.8
−1

(111)
(101)

Planes
abc 1 1 1
−1

<111>
<112>

X
−0.5

0

0.5

1

Figure 4.20: Stereographic projection for specimen 11 after the 1000 ‰ for 2 h heat treatment, using Euler orientations obtained by EBSD. The stereographic projection was used
to identify planes from which X-ray diffraction peaks could be used to estimate the imperfection density of the specimen. The grey shaded region represents crystallographic space
which was not visible to the X-rays due to the 45°sweep of the specimens. The stereographic
projection was generated using a MATLAB algorithm developed by Bieler [59].

90

1

90.018 43.703 277.4
Bunge

0.8

(−2−1−1)

0.6

<100>
<110>

(−11−1)
(−21−1)
(−12−1)

(−100)
(−1−10)

0.4
0.2

(−1−21)

0

(−110)

(−2−11) (−211)
(010)
(−111) (−121)
(−1−11)(−101)
(−112)
(011)
(001)

Y

(−1−12)

−0.2

(0−11)

−0.4

(121)

(112)
−0.6

(1−12)

−0.8
−1

(111)
(101)

Planes
abc 1 1 1
−1

<111>
<112>

X
−0.5

0

0.5

1

Figure 4.21: Stereographic projection for specimen 12 after the 1200 ‰ for 2 h heat treatment, using Euler orientations obtained by EBSD. The stereographic projection was used
to identify planes from which X-ray diffraction peaks could be used to estimate the imperfection density of the specimen. The grey shaded region represents crystallographic space
which was not visible to the X-rays due to the 45°sweep of the specimens. The stereographic
projection was generated using a MATLAB algorithm developed by Bieler [59].

91

where σx and σy are the standard deviations of a peak in the x- and y-directions, respectively.
The primary axes, x- and y-, are defined as illustrated in Figure 3.6. The decreasing dislocation density with increasing increasing heat treatment temperature is clearly illustrated by
the decreasing normalized standard deviation.
The average of the full width at half maximum (FWHM) in x- and y-directions are
plotted against each other for the four specimens in Figure 4.27. The 1000 ‰ heat treated
specimen 11 has the smallest FWHM amongst the four specimens, followed by specimens 12
(green symbols), 10 (red symbols), and 9 (black symbols), heat treated at 1200 ‰, 800 ‰,
and 600 ‰, respectively. A smaller FWHM in both the x- and y-directions is indicative of a
sharper diffraction peak, implying lesser crystal imperfections in the material.

4.3

Young’s modulus and the phonon peak

The Young’s modulus E is estimated for specimens 1 – 12 using equation (3.4) and the
Euler angles for the specimens. A MATLAB

algorithm developed by Bieler [59] is used

to estimate E for specimens 1 – 12 after their hottest temperature heat treatments. The
algorithm calculates E using equation (3.4) and a unit vector parallel to the heat flow direction generated from Euler angles. An inverse pole figure displaying the crystal orientations
of specimens 1 – 12 with a graded background representing the variation in E for Nb is
illustrated in Figure 4.28. The red symbols represent the crystal orientation of grain 1,
while the green symbols represent grain 2 in the bicrystal specimens. For the single crystal
specimens 9 – 12, a red symbol is used. The shades of grey, from light to dark, indicate
the variation in E of Nb from 83.24 GPa to 145.48 GPa, respectively. For the specimens
heat treated at Th ≥ 1000 ‰, the normalized phonon peak (kpp /k3 ) is compared with the

92

Y (c m )
-7 .0

-6 .9

-6 .8

Y (c m )
-6 .7

-6 .6 8 .0

8 .1

8 .2

8 .3

8 .4

-1 1 .5

-1 0 .5
P e a k 3

P e a k 2

-1 1 .3

-1 0 .3
-1 0 .2
2 8

2 9
-1 1 .1

-1 0 .1

9 .7

1 0 .7
2 7

2 4

1 0 .8

9 .9

1 0 .9

1 0 .0

1 1 .0
P e a k 4

X (c m )

9 .8

X (c m )

X (c m )

-1 0 .4

-1 1 .2

X (c m )

-1 1 .4

P e a k 1

1 0 .1

1 1 .1
-8 .4

-8 .3

-8 .2

-8 .1

-8 .0 6 .5

Y (c m )

6 .6

6 .7

6 .8

6 .9

Y (c m )

Figure 4.22: Detected peak maximum locations for peaks from {222} planes for specimen 9
after the 600 ‰ for 2 h heat treatment. The x- and y-distances are with respect to the
location of the beam on the detector. The four peak clusters are the locations of the X-rays
diffracted by the two {111} planes identified in Figure 4.18. Each individual cluster is a
group of {222} peaks from several of the 1620 frames recorded per specimen. The number
listed in each cluster, towards the center of the figure, is the actual number of frames from
which {222} peaks are detected. The spread of the peaks within each cluster represents
imperfection density in the crystal.

93

Y (c m )
-7 .1

-7 .0

-6 .9

Y (c m )
-6 .8

-6 .7 7 .8

7 .9

8 .0

8 .1

8 .2

-1 1 .4

-1 0 .6
P e a k 3

P e a k 2

-1 1 .2

-1 0 .4
-1 0 .3
3 9

4 1
-1 1 .0

-1 0 .2

9 .8

1 0 .6
3 7

3 5

1 0 .7

1 0 .0

1 0 .8

1 0 .1

1 0 .9
P e a k 4

X (c m )

9 .9

X (c m )

X (c m )

-1 0 .5

-1 1 .1

X (c m )

-1 1 .3

P e a k 1

1 0 .2

1 1 .0
-8 .3

-8 .2

-8 .1

-8 .0

-7 .9 6 .6

Y (c m )

6 .7

6 .8

6 .9

7 .0

Y (c m )

Figure 4.23: Detected peak maximum locations for peaks from {222} planes for specimen 10
after the 800 ‰ for 2 h heat treatment. The x- and y-distances are with respect to the
location of the beam on the detector. The four peak clusters are the locations of the X-rays
diffracted by the two {111} planes identified in Figure 4.19. Each individual cluster is a
group of {222} peaks from several of the 1620 frames recorded per specimen. The number
listed in each cluster, towards the center of the figure, is the actual number of frames from
which {222} peaks are detected. The spread of the peaks within each cluster represents
imperfection density in the crystal.

94

Y (c m )
-6 .9

-6 .8

-6 .7

Y (c m )
-6 .6

-6 .5 8 .0

8 .1

8 .2

8 .3

8 .4

-1 1 .6

-1 0 .4
P e a k 3

P e a k 2

-1 1 .4

-1 0 .2
-1 0 .1
2 9

3 0
-1 1 .2

-1 0 .0

9 .7

1 0 .8
3 0

3 0

1 0 .9

9 .9

1 1 .0

1 0 .0

1 1 .1
P e a k 4

X (c m )

9 .8

X (c m )

X (c m )

-1 0 .3

-1 1 .3

X (c m )

-1 1 .5

P e a k 1

1 0 .1

1 1 .2
-8 .5

-8 .4

-8 .3

-8 .2

-8 .1 6 .4

Y (c m )

6 .5

6 .6

6 .7

6 .8

Y (c m )

Figure 4.24: Detected peak maximum locations for peaks from {222} planes for specimen 11
after the 1000 ‰ for 2 h heat treatment. The x- and y-distances are with respect to the
location of the beam on the detector. The four peak clusters are the locations of the X-rays
diffracted by the two {111} planes identified in Figure 4.20. Each individual cluster is a
group of {222} peaks from several of the 1620 frames recorded per specimen. The number
listed in each cluster, towards the center of the figure, is the actual number of frames from
which {222} peaks are detected. The spread of the peaks within each cluster represents
imperfection density in the crystal.

95

Y (c m )
-6 .7

-6 .6

-6 .5

Y (c m )
-6 .4

-6 .3 8 .2

8 .3

8 .4

8 .5

8 .6

-1 1 .7

-1 0 .3
P e a k 3

P e a k 2

-1 1 .5

-1 0 .1
-1 0 .0
3 0

3 2
-1 1 .3

-9 .9

9 .6

1 0 .9
3 5

3 0

1 1 .0

9 .8

1 1 .1

9 .9

1 1 .2
P e a k 4

X (c m )

9 .7

X (c m )

X (c m )

-1 0 .2

-1 1 .4

X (c m )

-1 1 .6

P e a k 1

1 0 .0

1 1 .3
-8 .6

-8 .5

-8 .4

-8 .3

-8 .2 6 .3

Y (c m )

6 .4

6 .5

6 .6

6 .7

Y (c m )

Figure 4.25: Detected peak maximum locations for peaks from {222} planes for specimen 12
after the 1200 ‰ for 2 h heat treatment. The x- and y-distances are with respect to the
location of the beam on the detector. The four peak clusters are the locations of the X-rays
diffracted by the two {111} planes identified in Figure 4.21. Each individual cluster is a
group of {222} peaks from several of the 1620 frames recorded per specimen. The number
listed in each cluster, towards the center of the figure, is the actual number of frames from
which {222} peaks are detected. The spread of the peaks within each cluster represents
imperfection density in the crystal.

96

1 5

N o rm a liz e d S ta n d a rd D e v ia tio n (x 1 0

- 4

m )

Peak 1
Peak 2
Peak 3
Peak 4

1 0

0

5

6 0 0

8 0 0

1 0 0 0

Heat treatment temperature (°C)

1 2 0 0

Figure 4.26: The normalized standard deviation of the locations of the diffracted X-ray peak
maximums, as defined in equation (4.4). Smaller values of normalized standard deviation
imply reduced dislocation contents. Specimen 12, heat treated at 1200 ‰ for 2 h, has
the smallest deviation values amongst the four specimens, indicating greatest recovery of
dislocations.

97

5 .5
0 9 0 9 0 9 0 9 1 0 1 0 1 0 1 0 1 1 1 1 1 1 1 1 1 2 1 2 1 2 1 2 -

F W H M

in Y (x 1 0

- 4

m )

5 .0

4 .5

4 .0

3 .5

3 .0
3 .0

3 .5

4 .0

F W H M

4 .5

in X (x 1 0

5 .0
- 4

P e a
P e a
P e a
P e a
P e a
P e a
P e a
P e a
P e a
P e a
P e a
P e a
P e a
P e a
P e a
P e a

k 1
k 2
k 3
k 4
k 1
k 2
k 3
k 4
k 1
k 2
k 3
k 4
k 1
k 2
k 3
k 4

5 .5

m )

Figure 4.27: The average of full width at half maximum (FWHM) of the four {222} peak
clusters for specimens 9 – 12, when measured along the X and Y directions. A smaller
FWHM in both the x- and y-directions would imply a sharper peak.

98

[111]

123 3063 3063 3073
456 0630 0630 0730
83.24
789 63063 63063 73073
0 0 0

00 .. 00
000 ... 000
00 .. 00
00 .. 011
00 .. 11
00 .. 11
00 .. 11
00 .. 22
00 .. 22
00 .. 22
00 . 223
00 .. 33
00 .. 33
00 .. 33
00 .. 34
00 .. 44
00 .. 44
00 .. 44
00 .. 455
00 .. 55
00 .. 55
00 .. 55
00 . 66
00 .. 66
00 .. 66
00 .. 667
00 .. 77
00 .. 77
00 .. 77
00 .. 78
00 .. 88
000 .. 888
00 .. 889
00 .. 99
00 . 99
00 .. 99
01 .. 90

0 06 6 6 7
012 6306
53467 3036036
890 0306
6
123 3063
456 0603
789 60633

7307
307
3073
0730
7307
3
0730
7307
3

012 0630 0730
4356 6360360 7307307
789 3063 3073
012 0630 7030
345 63063 73730
678 0630 0730
901 63306 73073
234 0630 0730
567 66330 73073
890 0630 0730
123 66303 73073
456 0630 0730
789 06363 73073
012 0630 0730
345 30663 73073
678 0630 0730
901 36063 73073
234 0630 0730
567 36063 37073
890 0063 0730
123 63063 73073
456 0063 0730
789 63063 73073
0 0

E (GPa)

145.48

[101]

[001]

1
2
3
4
5
6
7
8
9
10
11
12

600 °C, 10 h
1100 °C, 4 h
1100 °C, 4 h
1100 °C, 4 h
800 °C, 2 h
800 °C, 2 h
1100 °C, 4 h
1100 °C, 4 h
600 °C, 2 h
800 °C, 2 h
1000 °C, 2 h
1200 °C, 2 h

Red: Grain 1
Green: Grain 2

Figure 4.28: An inverse pole figure displaying the crystal orientations of grains 1 (red) and
2 (green) of specimens 1 – 8, and of the single crystal specimens 9 – 12 (red), after their
hottest temperature heat treatments. A shaded grey background, from light to dark grey, is
used to represent the variation in E of Nb from 83.2 GPa to 145.5 GPa, respectively.
Young’s modulus in Figure 4.29. The solid line represents a least squares fit to the data,
and reveals a mild correlation between kpp /k3 and E.

4.4

Parameters from thermal conductivity model

Of the five βi s listed in equation (3.9), four (i = 1, 3, 4, 5) are estimated while reducing
the T and q data from the measurements of k. The estimated β1 s (scattering of normally

99

2 .8

2: 1100 °C, 4 h
3: 1100 °C, 4 h
4: 1100 °C, 4 h
7: 1100 °C, 4 h
8: 1100 °C, 4 h
11: 1000 °C, 2 h
12: 1200 °C, 2 h

N o rm a liz e d p h o n o n p e a k k

p p

/k

3

3 .0

2 .6

2 .4

Red: Grain 1
Green: Grain 2

2 .2

2 .0
8 0

9 0

1 0 0

1 1 0

1 2 0

1 3 0

Young's modulus E (GPa)

1 4 0

1 5 0

Figure 4.29: A comparison of the normalized phonon peak (kpp /k3 ) with E as estimated
using equation (3.4). Only specimens that are heat treated at Th ≥ 1000 ‰ are used for
this purpose. The solid line represents a least squares fit to the data, and reveals a mild
correlation between kpp /k3 and E.

100

conducting electrons by impurities) for the twelve specimens, in their as-received condition
and after heat treatments, are compared with RRR in Figure 4.30. The inverse proportionality between β1 and RRR is evident, and is as expected from the definition of β1 (=
ρ295 /(L RRR)−1 ).
The estimated β3 (scattering of phonons by normal conducting electrons) for the twelve
specimens, in their as-received condition and after heat treatments, are compared with the
heat treatment temperature and shown in Figure 4.31. The dashed line represents the
theoretical value for β3 . The average of β3 for each Th begins to center around the theoretical
value with increased heat treatment temperature.
The estimated β4 (scattering of phonons by lattice defects and boundaries) for the twelve
heat-treated specimens, in their as-received condition and after heat treatments, are compared with the heat treatment temperature and are illustrated in Figure 4.32. A strong
dependence of β4 on heat treatment temperature is noted, with all of the as-received specimens displaying greater values than any of the heat treated values.
The estimated β5 (condensation of electrons into Cooper pairs) for the twelve specimens,
in their as-received condition and after heat treatments, are compared with the heat treatment temperature and shown in Figure 4.33. Note that the relative range of the ordinate is
much smaller than those of other βi s.

101

0 .0 5

β1 ( Ω m ⋅ K 2 W

-1

)

0 .0 4

0 .0 3

0 .0 2

0 .0 1
5 0

1 0 0

1 5 0

2 0 0

2 5 0

R R R
Figure 4.30: Comparison of β1 (scattering of normally conducting electrons by impurities)
with RRR for the twelve specimens, in their as-received condition and after heat treatments.
The inverse proportionality would be expected from the definition of β1 (= ρ295 /(L RRR)−1 ).

102

6 0 0

1
3
5
7
9
11

4 0 0

β3 ( m ⋅ K

3

W

-1

)

5 0 0

2
4
6
8
10
12
Theory

3 0 0

2 0 0
0

2 0 0

4 0 0

6 0 0

8 0 0

1 0 0 0

Heat Treatment Temperature (°C)

1 2 0 0

1 4 0 0

Figure 4.31: Comparison of β3 (scattering of phonons by normal conducting electrons) with
the heat treatment temperature for the twelve specimens. The dashed line represents the
theoretical value for β3 . In these specimens, increased heat treatment temperature brings
the measured β3 into agreement with the theoretical value.

103

1 .0

1
3
5
7
9
11

0 .6

β4 ( m ⋅ K

4

W

-1

)

0 .8

2
4
6
8
10
12

0 .4

0 .2

0 .0
0

2 0 0

4 0 0

6 0 0

8 0 0

1 0 0 0

Heat Treatment Temperature (°C)

1 2 0 0

1 4 0 0

Figure 4.32: Comparison of β4 (scattering of phonons by lattice defects and boundaries) with
the heat treatment temperature for the twelve specimens. As-received specimens display
greater β4 values than those post-heat treatment. Wide variations in the as-received β4
values may indicate a broad range of process histories. Note again that the 750 ‰ heat
treatment was with Ti getter and had a temperature ramp rate of 5 ‰ per minute, whereas
at 800 ‰ there was no getter and the temperature ramp rate was 10 ‰ per minute. Note also
that the 600 ‰ heat treatment for specimens 1 and 2 was for 6 h, while that for specimen 9
was for 2 h.

104

2 .2

1
3
5
7
9
11

β5

2 .1

2
4
6
8
10
12

2 .0

1 .9

1 .8
0

2 0 0

4 0 0

6 0 0

8 0 0

1 0 0 0

Heat Treatment Temperature (°C)

1 2 0 0

1 4 0 0

Figure 4.33: Comparison of β5 (condensation of electrons into Cooper pairs) with the heat
treatment temperature. Note that the relative range of the ordinate is smaller than those
for other βi s, indicating that the scatter is smaller than it appears.

105

4.5

Discussion

The as-received specimens in Set 1 display a variety of k for T > 3 K, indicating a variation
in RRR. The k for T > 3 K for the as-received specimens in Set 2 have similar magnitudes,
indicating similar RRR, as expected for specimens from the same grain of the same ingot
disc. For T

3 K, however, the as-received specimens in both Sets 1 and 2 display little

or no phonon peak irrespective of their crystal orientations, suggesting the presence of dislocations from residual stresses during cooling of the ingot. Except specimen 3, illustrated
in Figure 4.3, grains 1 and 2 of the as-received specimens in sets 1 and 2 have similar k
curves, suggesting similar RRR and dislocation contents. Although specimens 3 and 4 were
cut from the same ingot disc, they have different as-received k. While the two grains in
specimen 4 (Figure 4.4) display similar k for T > 3 K, indicating similar RRR, the k for
T > 3 K of grain 1 in specimen 3 is 20% smaller than that of grain 2 in specimen 3 and
those in specimen 4. This suggests that there are variations of RRR within a grain in the
same ingot disc. After the 750 ‰ for 2 h heat treatment, however, k at 4.2 K for grain 1 of
specimen 3 and across the GB increases to that of grain 2, indicating removal of impurities
within the grain. Since grain 2 had no change in k at 4.2 K, the change in k of grain 1 can
be attributed to the degassing of gaseous impurities, such as hydrogen.
Although several benefits have been observed for the surface layer of Nb after heat treatment at 100 ‰ – 150 ‰ for 48 h [23], no benefit to the thermal conductivity was observed
for specimens 7 and 8 after a 140 ‰ for 48 h heat treatment. This confirms that the benefits
of the 100 ≤ Th ≤ 150 ‰ heat treatments are limited to the surface of Nb, and has no effect
on the crystallography of the bulk.
An 800 ‰ heat treatment for 2 h in the absence of a titanium getter or a 750 ‰ for

106

2 h heat treatment with a titanium getter resulted in an average 140% increase in k at
the phonon peak. The similarity in kpp /k2 values for 750 ‰ and 800 ‰ heat treatments,
tabulated in Table 4.2, may be due to the small difference in the heat treating temperatures,
combined with the presence of titanium getter at 750 ‰, and the slower temperature ramp
rate of 5 ‰ per minute for the 750 ‰ heat treatment in contrast with the 10 ‰ per minute
ramp rate for the 800 ‰ heat treatment. Note that in the ratio kpp /k2 , kpp is k at the
phonon peak after heat treatment, while k2 is k at 2 K in the as-received condition. The
600 ‰ heat treatment for 6 h in the absence of a titanium getter resulted in an average
100% increase in k at the phonon peak. The 17% difference in the magnitudes of the phonon
peaks for specimens 1 and 2 after the 600 ‰ heat treatment may be due to the difference
in their cross sectional areas. Despite their shortest dimension being similar, their second
dimension in the cross section differs by 60%, which may affect the magnitude of scattering
of some phonon modes.
The anomalous kpp /k2 = 4.9 for specimen 3 across the GB is not well understood. Although, greater recovery of dislocations due to the grain boundary acting as a sink for dislocations might explain this result. Similar trends, albeit lower in magnitude than specimen 3,
are seen across the grain boundary for all the specimens in Set 1 after the 600 ‰ – 800 ‰
heat treatments, except specimen 6. Specimen 2 displays greater kpp across the GB even
after the 1100 ‰ for 4 h heat treatment, while specimens 3, 4, 7, and 8, that were also
heat-treated at Th = 1100 ‰, do not show such differences and have overlapping k curves
for both the grains and across the grain boundary.
When compared with the RRR of the specimens after the 1100 ‰ heat treatment, the
kpp for specimens in Set 1 illustrated in Figure 4.9 show a monotonic dependence on the
RRR. This implies that the dominant phonon scattering mechanism is due to impurities, and
107

no longer due to dislocations formed during the differential cooling of the ingot. This result
is in good agreement with the findings of Wasserb¨ch [45] on the dependence of conductivity
a
on the purity of superconducting tantalum (80 ≤ RRR ≤ 3000) and niobium (RRR=300).
Examining the kpp /k3 ratio in Table 4.2 for the 1100 ‰ heat-treated specimens, thereby
suppressing the effect of RRR (and ke ) on kpp , it is evident that despite the differences in
the magnitudes of kpp the phonon contribution to k is almost a constant. The same ratios
for specimens in Set 2, illustrated in Figure 4.15, also indicate the presence of a constant
phonon contribution to k for Th > 1000 ‰. The coupling of k in the phonon dominant regime
(T ≤ 3 K) with that in the electron dominant regime (T ≥ 3 K) implies that the overall
thermal performance of cavities will be greatest for Nb with the greatest RRR.
The ratio kpp /k3 for the single crystal specimens in Set 2, with the same crystal orientations, displays a sigmoidal dependence on heat treatment temperature, with a plateau
above 1000 ‰. This suggests that for 2 hour heat treatments on undeformed specimens there
will be little improvement in kpp /k3 for heat treatment temperatures hotter than 1000 ‰.
The calculated dislocation densities for these four specimens using the method described in
Section 4.2.2.1 show an inverse sigmoid dependence on Th , with a plateau for Th > 1000 ‰,
and displaying inverse proportionality to kpp /k3 .
No clear correlation is visible between the variation in crystal orientation, as illustrated
through the inverse pole figures, and the thermal conductivities of the specimens. Examining
the kpp /k3 ratio with respect to E, as illustrated in Figure 4.29, for those specimens that
had been heat treated at temperatures greater than 1000 ‰ reveals that there does exist
a mild correlation between them. But the large scatter in data and the small slope of the
least squares fit line indicates that crystal orientation has little influence on the thermal
conductivity at the phonon peak.
108

In their as-received conditions, specimens 10 – 12 have similar LAM curves, implying
similar dislocation contents. Although cut from an adjacent location in the same grain as
specimens 10 – 12, specimen 9 has a smaller peak maximum and wider LAM curve than the
other three specimens, indicating greater initial dislocation content than the other specimens
in the set. After the 2 h heat treatments, specimens 9 – 11 have similar LAM curves, and
are narrower than their as-received conditions. Specimen 12, heat treated at 1200 ‰, has an
LAM curve that has the most shift in the peak maximum.
Specimen 9 has the most spread in the diffraction peaks, as illustrated in Figure 4.22,
while specimen 12 with the 1200 ‰ heat treatment, plotted in Figure 4.25, has the least
spread in the peaks. This implies that the hotter heat treatment temperature increases
recovery in the crystal, and consequently reduces dislocation density in the specimens. This
result is further highlighted when the normalized standard deviations for these peaks are
compared, as shown in Figure 4.26. The average of the normalized standard deviation for
specimen 12 is 66% smaller than that of specimen 9, while the same for specimens 10 and
11 are 14% and 24%, respectively.
The average of the full width at half maximum (FWHM) of each peak in the four clusters
is an indication of crystal imperfection density, including dislocations and vacancies. The
1000 ‰ heat treated specimen 11 displays the least scatter in FWHM, implying least imperfection density. Although the 1200 ‰ heat-treated specimen had the greatest kpp /k3 ratio
and least normalized standard deviation, implying the least number of phonon scattering
sites, it displayed a wider FWHM than the 1000 ‰ heat-treated specimen.
To better understand the reason for the presence of a minimum in the FWHM at 1000 ‰,
the equilibrium concentration of vacancies dependent on T are estimated. An Arrhenius type
equation can be used to estimate the number of vacancies per unit volume (Nv ) [60], such
109

as
Nv
= exp
N

−Ev
kB T

,

(4.5)

where Ev is the vacancy formation energy (2.65 eV for Nb [61]), kB the Boltzmann constant
(8.617 eV/K), T the temperature in K, and N the number of potential vacancy sites in the
unit volume, calculated as
N
N = ρ A,
M

(4.6)

where ρ is the density (8.57 × 106 g/m3 for Nb), NA the Avogadro’s number (6.022 × 1023 ),
and M the molar mass (92.9 g/mol for Nb). The equilibrium vacancy concentration as a
function of T is shown in Figure 4.34. As per the definition, Nv increases exponentially with
T , and is evident in the plot. At Th = 1200 ‰, Nv is about an order of magnitude greater
than that at Th = 1000 ‰.
As the thermal measurements for these specimens are performed at T < 4.2 K and
since the specimens remain at room temperature most of the time, the estimate of Nv
gives the upper limit of vacancy concentration for the specimens. Since vacancies need grain
boundaries or dislocation cores to dissolve into, Nv for the single crystal specimens presented
here can be expected to be mostly preserved even at cooler temperatures.
The inverse proportionality of the dislocation density with Th , combined with the exponential dependence of the vacancy concentration on Th leads to a local minimum in the
total imperfection concentration. Therefore, according to the FWHM analysis, it is counter
productive to heat treat undeformed Nb at Th > 1000 ‰. This result, in combination with
kpp /k3 ratios and normalized standard deviations, also leads to the conclusion that phonon
transport is affected by vacancies.
Comparing β1 (scattering of normally conducting electrons by impurities) with the esti110

E q u ilib riu m

v a c a n c y c o n c e n tra tio n (m

-3

)

1 0 20
1 0 19
1 0 18
1 0 17
1 0 16
1 0 15
1 0 14
1 0 13
1 0 12

6 0 0

8 0 0

1 0 0 0

Temperature (°C)

1 2 0 0

Figure 4.34: The theoretically calculated equilibrium vacancy concentration for Nb with
respect to temperature.

111

mated RRR, for the twelve specimens in their as received condition and after heat treatments,
results in an inverse relation between the two as shown in Figure 4.30. This is as expected
from the definition of β1 (= ρ295 /(L RRR)−1 ). The inverse proportionality provides confidence in the parameter estimation technique used, since β1 is a free parameter during
estimation calculations.
The term for the scattering of phonons by normal conducting electrons β3 , illustrated in
Figure 4.31, shows a mild dependence on the heat treatment temperature. The as-received,
600 ‰ for 6 h, 750 ‰, and 1000 ‰ heat-treated specimens have mean β3 values that are
significantly larger than the theoretical value. The 600 ‰ for 2 h, 800 ‰, 1100 ‰, and
1200 ‰ heat-treated specimens have β3 that are centered about the calculated theoretical
value. The cause for this deviation is not yet understood.
The term β4 illustrated in Figure 4.32, which can be associated with the scattering of
phonons by the lattice defects, shows a strong dependence on heat treatment temperature
[62], with β4 decreasing to about 20% of its as-received value for specimens after Th >
1000 ‰ heat treatments. This again suggests a reduction in phonon scattering sites after
heat treatment. The difference in β4 between the 800 ‰ and the 1000 ‰ – 1200 ‰ heat
treatments is not as dramatic as the difference between the as-received β4 and the 800 ‰
heat treatments. This suggests that β4 has an asymptotic dependence on the heat treatment
temperature. Hence, heat treating at temperatures hotter than 1000 ‰ might not further
decrease β4 . The small values of β4 after 1000 ‰ – 1200 ‰ heat treatments imply that the
lattice scattering sites formed during the production of the ingot might be annihilated and
that β4 might be reaching a limit due to the inability to remove geometrically necessary
dislocations and boundary scattering.
The term associated with the condensation of electrons into Cooper pairs, β5 , shows little
112

dependence on the temperature of heat treatment as shown in Figure 4.33. This might be
expected due to the definition of β5 , which includes the superconductor energy gap ∆(T ),
the Boltzmann constant kB , and the superconducting critical temperature Tc . The value for
β5 from BCS theory is 1.76, though measurements of ∆(T ) have shown departures from this
predicted value [63–66].
Based on the k and βi data from Sets 1 and 2, no readily apparent effect was found of
tantalum content on the thermal conductivity of niobium for concentrations between 523
and 1322 ppm.

4.6

Conclusion

The recovery of the phonon contribution to the phonon peak, by virtue of dislocation density,
will have a sigmoidal dependence on the heat treatment temperature. The best values for the
thermal conductivity at the phonon peak for 2 h heat treatments will be obtained around
1000 ‰. Although hotter heat treatment temperatures result in some improvement, the
vacancy concentration also increases. The magnitude of the phonon peak is affected by
vacancies generated during heat treatments. Smaller RRR will result in a smaller phonon
peak. Crystal orientation plays little role in the variation of the thermal conductivity.

113

Chapter 5
Results:
Deformation and the Phonon peak
The role of deformation and dislocation content on the magnitude of the phonon peak is
examined using four bi-crystal Nb specimens. The specimens were uniaxially strained to
different levels, with individual grains having different local engineering strains ranging from
2% to 38%. The corresponding change in the phonon peak was correlated with the strain
and grain orientation for the individual grains.
The k at the phonon peak of deformed large grain Nb is known to recover to its undeformed value [25] subsequent to heat treatments at temperatures close to the melting
temperature of Nb (2468 ‰). For the production of SRF cavities, however, heat treatment
temperatures (Th ) need to be as small as possible, so as to not compromise their mechanical
properties. The recovery of k at the phonon peak in the strained Nb specimens is examined
here by heat treating the specimens for 2 h at a relatively colder temperature of 1000 ‰.

114

5.1

Uniaxial straining of specimens

Four bi-crystal Nb specimens, shaped according to the ASTM E-8 tensile standards, were
subjected to different magnitudes of uniaxial tensile strain to observe the effect of dislocations on the magnitude of the phonon peak. Specimens 2, 3, 7, and 13 were subject to
nominal engineering strains of 20, 10, 2, and 2 percent, respectively. Prior to deformation,
specimens 2, 3, and 7 were heat treated at 1100 ‰ for 4 h, while specimen 13 was in its
as-received condition. Using Euler angles obtained from orientation imaging microscopy, the
Schmid factors and the direction of the Burger’s vector were estimated using a MATLAB
code developed by Baars [67]. These parameters are listed in Table 5.1 for each specimen.
A larger Schmid factor indicates that a grain will more readily deform. Based on these estimated Schmid factors and Burger’s vector directions, the level of strain for each specimen
was chosen. In addition, the characteristics of the recovery of the phonon peak with a 1000 ‰
for 2 h heat treatment subsequent to the deformation was also examined. The 1000 ‰ for
2 h heating was chosen as the recovery heat treatment based on the results presented in
Chapter 4, where a plateau was observed in the recovery of kpp for temperatures hotter than
1000 ‰.

115

Table 5.1: State of specimens prior to deformation. The Euler angles obtained using orientation imaging microscopy are also
listed. Also tabulated are the Schmid factors for {110} and {112} slip systems, and the Burger’s vector directions for these
specimens. The level of strain chosen for these specimens based on the Schmid factors and Burger’s vector directions are also
tabulated.

Spec. Grain

Euler angles
in degrees
φ1
Φ
φ2

2

1
2

3

1
2

183.2 41.8
1.3 13.1

7
13

* bold:

9.7 18.4 351.8
64.7 40.8 261.2

Largest Schmid factors*

Angle bet. Burger’s vector
and tensile direction
in degrees

-0.493
-0.469

0.484 -0.474
-0.449 -0.365

-0.469
-0.364

135.1
53.8

135.0
30.0

0.387
-0.481

0.350 -0.320
-0.469 -0.387

0.316
-0.363

64.6
50.1

64.6
50.1

1
2

157.3 40.3 168.5
226.2 47.0 172.8

0.365
-0.373

-0.318
-0.337

0.314 -0.293
0.325 -0.309

66.6
114.2

66.6
114.2

1
2

62.6 51.8 266.3
64.4 51.6 314.5

-0.439
-0.446

-0.425 -0.400
-0.429 -0.379

{110} slip system; italics: {112} slip system.
116

0.385
0.352

120.9
57.4

138.4 138.4
53.8 155.9

Chosen
nominal
strain (%)

135.1
53.8

20

70.2
50.1

10

66.6 108.0
69.6 114.2

2

60.7 120.9 60.7
57.4 114.9 114.9

2

64.6
154.6

To better estimate the local strain on the two grains of each bi-crystal specimen, a
2 mm × 2 mm square grid was painted onto the surface of the specimens. The grid on
specimen 2 in its as-received condition and after 20% nominal uniaxial tensile strain is shown
in Figure 5.1. The location of the intersection of the grid lines before and after deformation
was used to estimate the local strain. Also pictured is the side-view of the specimen before
and after deformation.
Due to the differences in the Schmid factors of the two grains, the level of deformation
is not the same for the two grains. This is shown in Figure 5.1(b). The grain with the
greater Schmid factor has the softer orientation, and hence will yield more readily. Contrary
to this rule of thumb, grain 2-1 (left grain in Figure 5.1(b)) despite having a greater Schmid
factor than grain 2-2, has yielded less. The resulting local engineering strains for grains 2-1
and 2-2 were estimated to be 5% and 38%, respectively. This discrepancy in the expected
softer orientation based on the Schmid factor was noted only for specimen 2. This may be
explained by considering the number of slip systems being activated during deformation;
this can be inferred from the number of unique Burger’s vector directions tabulated for
each grain in Table 5.1. Despite grain 2-1 having greater Schmid factors, it has two unique
Burger’s vector directions, while grain 2-2 has only one unique direction. This implies that
even though grain 2-1 will yield before grain 2-2, it will also work harden faster, and thus it
will deform less.

5.2

Thermal conductivities of strained specimens

The thermal conductivities for specimen 2 after the initial 1100 ‰ for 4 h heat treatment
(open symbols), after 20% nominal uniaxial tensile strain (half-filled symbols), and after the

117

1 cm
(a) Before deformation.

1 cm
(b) After 20% nominal uniaxial tensile strain.
Figure 5.1: Painted 2 mm x 2 mm grid on the surface of specimen 2, (a) before deformation,
and (b) after 20% nominal uniaxial tensile strain. This grid is used to estimate local strains
on the two grains (left: grain 1, right: grain 2) of the bi-crystal specimen. Also pictured
are the side-view of the specimen with the grid. The variable effect of deformation on the
thickness of the two grains is visible in the side-view of (b). Similar grids were painted on
the remaining three specimens.

118

1000 ‰ for 2 h recovery heat treatment (solid symbols) are illustrated in Figure 5.2. The
Euler angles and an inverse pole figure illustrating the crystal orientations of the two grains
are shown below the thermal conductivity plot in the figure. The thermal conductivities for
specimen 3 after the initial 1100 ‰ for 4 h heat treatment (open symbols), after 10% nominal
uniaxial tensile strain (half-filled symbols), and after the 1000 ‰ for 2 h heat treatment (solid
symbols) are illustrated in Figure 5.3. The thermal conductivities for specimen 7 after the
initial 1100 ‰ for 4 h heat treatment (open symbols), after 2% nominal uniaxial tensile
strain (half-filled symbols), and after the 1000 ‰ for 2 h heat treatment (solid symbols) are
illustrated in Figure 5.4.
A summary of the k results for the four specimens used in the deformation experiments
is presented in Table 5.2. The results are tabulated as ratios of k at the phonon peak for
the specimens before deformation (kpp,1 ), after deformation (kpp,2 ), and after the 1000 ‰
for 2 h heat treatment (kpp,3 ). The ratio of kpp,2 /kpp,1 presents the decrease in kpp after
deformation, while kpp,3 /kpp,2 presents the increase in kpp after the recovery heat treatment
with respect to kpp after deformation, and kpp,3 /kpp,1 presents the relative change in kpp
after the 1000 ‰ heating with respect to kpp before deformation. For cases where there
was no phonon peak, such as specimen 13 in its as-received condition and after deformation,
the k at 2 K is used. Also tabulated are the kpp /k3 ratios for the four specimens after the
1000 ‰ heat treatment. Additionally, the chosen nominal strains of the specimens and the
measured engineering strains on the two grains of the specimens are also tabulated.
Previous work by Wasserb¨ch [26, 41, 42] and Singer et al. [27] consider the effects of
a
deformation on kpp of single crystal Nb, while Aizaz et al. [28] considered deformation on
polycrystal Nb. Although there are several other publications in the literature discussing
deformation and k in Nb, these five publications are chosen here due to the presence of
119

2-1 1100 °C
2-1 5%
2-1 1000 °C

7 0

T h e rm a l c o n d u c tiv ity (W m -1 K -1 )

6 0

2-GB 1100 °C

2-2 1100 °C
2-2 38%
2-2 1000 °C

5 0
4 0
3 0
2 0
1 0
0

1 .5

2 .0

2 .5

2-1
؆1= 9.7°

2-2
؆1= 64.7°

2-1 5%
؆1= 355.8°

3 .0

Temperature (K)

3 .5

4 .0

4 .5
[111]

2-2 38%
؆1= 92.2°

Φ = 18.4°
؆2= 351.8°

Φ = 20.0°
؆2= 4.37°

Φ = 40.8°
؆2= 261.2°

Φ = 35.1°
؆2= 233.1°

[001]

[101]

Figure 5.2: Thermal conductivities for specimen 2 after the 1100 ‰ for 4 h heat treatment
(open symbols), after 20% nominal uniaxial tensile strain (half-filled symbols), and after the
1000 ‰ for 2 h recovery heat treatment (solid symbols). The thermal conductivity across
the grain boundary (triangles) could not be estimated reliably after deformation, due to
large variation in the cross sectional areas of the two grains in the specimen as visible in
Figure 5.1(b). The Euler angels φ1 , Φ, and φ2 as well as the inverse pole figure representing
the crystal orientation along the heat flow direction for the two grains after the 1100 ‰
heating (open symbols) and after deformation (half-filled symbols) are shown below the
thermal conductivity plot.
120

3-1 1100 °C
3-1 0%
3-1 1000 °C

7 0

T h e rm a l c o n d u c tiv ity (W m -1 K -1 )

6 0

3-GB 1100 °C

3-2 1100 °C
3-2 15%
3-2 1000 °C

5 0
4 0
3 0
2 0
1 0
0

1 .5

2 .0

2 .5

3-1
؆1= 183.2°

3-2
؆1= 1.3°

3-1 0%
؆1= 184.5°

3 .0

Temperature (K)

3 .5

4 .0

4 .5
[111]

3-2 15%
؆1= 323.4°

Φ =41.8°
؆2= 135.0°

Φ = 41.0°
؆2= 130.7°

Φ = 13.1°
؆2= 30.0°

Φ = 6.8°
؆2= 68.2°

[001]

[101]

Figure 5.3: Thermal conductivities for specimen 3 after the 1100 ‰ for 4 h heat treatment
(open symbols), after 20% nominal uniaxial tensile strain (half-filled symbols), and after the
1000 ‰ for 2 h recovery heat treatment (solid symbols). Similar to specimen 2, the thermal
conductivity across the grain boundary (triangles) for specimen 3 could not be estimated
reliably after deformation, due to large variation in the cross sectional areas of the two
grains in the specimen. The Euler angels φ1 , Φ, and φ2 as well as the inverse pole figure
representing the crystal orientation along the heat flow direction for the two grains after the
1100 ‰ heating (open symbols) and after deformation (half-filled symbols) are shown below
the thermal conductivity plot.
121

7-1 1100 °C
7-1 2%
7-1 1000 °C

7 0

T h e rm a l c o n d u c tiv ity (W m -1 K -1 )

6 0

7-GB 1100 °C
7-GB 2%
7-GB 1000 °C

7-2 1100 °C
7-2 2%
7-2 1000 °C

5 0
4 0
3 0
2 0
1 0
0

1 .5

2 .0

2 .5

7-1
؆1= 157.3°

7-2
؆1= 226.2°

7-1 2%
؆1= 158.6°

3 .0

Temperature (K)

3 .5

4 .0

4 .5
[111]

7-2 2%
؆1= 229.2°

Φ = 40.3°
؆2= 168.5°

Φ = 40.8°
؆2= 167.1°

Φ = 47.0°
؆2= 172.8°

Φ = 48.3°
؆2= 169.8°

[001]

[101]

Figure 5.4: Thermal conductivities for specimen 7 after the 1100 ‰ for 4 h heat treatment
(open symbols), after 2% nominal uniaxial tensile strain (half-filled symbols), and after the
1000 ‰ for 2 h recovery heat treatment (solid symbols). The Euler angels φ1 , Φ, and φ2
as well as the inverse pole figure representing the crystal orientation along the heat flow
direction for the two grains after the 1100 ‰ heating (open symbols) and after deformation
(half-filled symbols) are shown below the thermal conductivity plot.

122

13-1 As rec'd
13-1 2%
13-1 1000 °C

7 0

T h e rm a l c o n d u c tiv ity (W m -1 K -1 )

6 0

13-GB As rec'd
13-GB 2%
13-GB 1000 °C

13-2 As rec'd
13-2 2%
13-2 1000 °C

5 0
4 0
3 0
2 0
1 0
0

1 .5

2 .0

2 .5

13-1
؆1= 62.6°

13-2
؆1= 64.4°

13-1 2%
؆1= 63.1°

3 .0

Temperature (K)

3 .5

4 .0

4 .5
[111]

13-2 2%
؆1= 68.1°

Φ = 51.8°
؆2= 266.3°

Φ = 51.7°
؆2= 265.5°

Φ = 51.6°
؆2= 314.5°

Φ = 50.1°
؆2= 311.3°

[001]

[101]

Figure 5.5: Thermal conductivities for specimen 13 in its as-received condition (open symbols), after 2% nominal uniaxial tensile strain (half-filled symbols), and after the 1000 ‰ for
2 h recovery heat treatment (solid symbols). The Euler angels φ1 , Φ, and φ2 as well as the
inverse pole figure representing the crystal orientation along the heat flow direction for the
two grains in their as-received condition (open symbols) and after deformation (half-filled
symbols) are shown below the thermal conductivity plot.

123

Table 5.2: Effect of uniaxial strain on the thermal conductivities of the two grains in specimens 2, 3, 7 and 13. Tabulated are the ratios of kpp,2 /kpp,1 , kpp,3 /kpp,2 , and kpp,3 /kpp,1 .
The first ratio is a measure of the decrease in kpp after deformation with respect to kpp before
deformation, the second ratio the increase in kpp after the 1000 ‰ for 2 h heat treatment
with respect to the kpp after deformation, and the third ratio the change in kpp after the
1000 ‰ for 2 h heat treatment with respect to the kpp before deformation. In the ratios,
kpp,1 is the k at the phonon peak before deformation, kpp,2 the kpp after deformation, and
kpp,3 the kpp after the 1000 ‰ heat treatment. In cases where there is no phonon peak,
k at 2 K is used. Additionally, the kpp /k3 values for the specimens after the 1000 ‰ heat
treatment are listed. Also tabulated are the chosen nominal strains along with the measured
engineering strains on the two grains of the specimens.
Chosen
nominal
strain (%)

Measured
engg.
strain (%)

kpp,2
kpp,1

kpp,3
kpp,2

kpp,3
kpp,1

After
1000 ‰
kpp /k3

Specimen

Grain

2

1
2

20

5
38

0.17
0.08

4.63
7.19

0.78
0.59

1.87
1.57

3

1
2

10

0
15

0.98
0.26

1.10
3.60

1.07
0.94

3.02
2.61

7

1
GB
2

2

2
2
2

0.63
0.54
0.58

1.77
2.00
1.82

1.11
1.07
1.05

2.60
2.47
2.59

13

1
GB
2

2

2
2
2

0.78
0.88
0.79

4.08
3.88
3.60

3.18
3.39
2.84

2.46
2.42
2.46

124

before-deformation k in them for the corresponding strained Nb specimens. Wasserb¨ch also
a
considered the effect of the temperature at which deformation occurred (Tdef ), by straining
several specimens at temperatures ranging between 77 ≤ Tdef ≤ 470 K. Additionally, the
specimens were strained in several steps before reaching the final strain condition, with intermediate k measurements at each strain level. The specimens strained by Singer [27] and
Aizaz [28], and the specimens presented in this work, were all strained at Tdef = 295 K. A
comparison of the change in kpp after deformation with respect to kpp before deformation
(kpp,2 /kpp,1 ) as a function of engineering strain for the data from literature and those presented in this work are illustrated in Figure 5.6. Inserted in the figure is the inverse pole figure
with orientations for specimens presented in this work, along with those for Wasserb¨ch’s
a
specimens. A kpp,2 /kpp,1 value of unity implies no change in kpp after deformation, while a
value smaller than unity implies reduction in kpp after deformation.

125

L ite ra tu re
W - 7
W - 1 9
W - 1 9
W - 2 9
S - 2 9
A - 2 9
W - 3 7
W - 4 7

1 .0
[1 1 1 ]

0 .8

[1 0 1 ]

[0 0 1 ]

0 .4

k

p p ,1 : b e f o r e d e f o r m .
k p p ,2 : a f te r d e f o r m .

k

p p ,2

/ k

p p ,1

0 .6

0 .2

0 .0
0

1 0

7 K
5 K
5 K
5 K
5 K
5 K (p x l)
0 K
0 K

2 0

3 0

T ru e s tra in (% )

4 0

T h is
(2 9
2 2 3 3 7 7 7 -

w o rk
5 K )
1
2
1
2
1
G B
2

Figure 5.6: The change in kpp after deformation with respect to kpp before deformation
(kpp,2 /kpp,1 ) as a function of the engineering strain for the bicrystal specimens 2, 3, and 7
(different squares) are compared with the data from single crystal Nb specimens presented
by Wasserb¨ch [26, 41, 42] and Singer et al. [27], and polycrystal Nb presented by Aizaz et
a
al. [28]. kpp,2 /kpp,1 = 1 implies no change in kpp after deformation. A value smaller than
unity implies reduction in kpp after deformation. The data from Wasserb¨ch, Singer, and
a
Aizaz are differentiated in the legend with a “W”, “S”, and “A”, respectively. Lines are
added to aid in presentation and do not represent a model. Inserted is the inverse pole figure
with orientations for specimens presented in this work, along with those for Wasserb¨ch’s
a
specimens.

126

5.3
5.3.1

Discussion
Deformation on kpp

In Figure 5.6, at first glance, the amount of reduction in kpp after deformation for Wasserb¨ch’s
a
specimens seem to display a dependence on Tdef . Closer examination by considering the crystal orientation reveals that the three data points with anomalously high kpp,2 /kpp,1 values,
Wasserb¨ch’s 77 K specimen along with grains 2-2 and 3-2, have crystal orientations that
a
lie at the center of the inverse pole figure, indicating single slip in these specimens. The
remaining specimens have duplex slip, and generate more dislocations during deformation
due to the interaction of the slip systems.
The specimens with duplex or multiple slip display greater reduction in kpp after deformation. Strains as small as ≈ 2% and 5% results in decreases in kpp of more than 37% and
83%, respectively. The conductivity ratio quantifying the decrease in kpp after deformation
kpp,2 /kpp,1 , for specimens with multiple slip, asymptotically converges to values smaller than
0.05 (greater than 95% reduction in kpp ) for strains greater than

≈ 15%. These results

indicate that dislocations generated during deformation are major phonon scattering sites.
Depending on the crystal orientation of adjacent grains in large grain Nb, by virtue
of the Schmid factors and number of slip systems, deformation can result in preferential
thinning of grains such as that illustrated in Figure 5.1(b). Such differences in the thickness
is undesirable for SRF cavities, where surface smoothness is vital. Therefore, to estimate
the effects of deformation of large grain Nb, it is vital to know not only the level of strain
endured by large grain material but also the Schmid factors of the adjacent grains and the
number of primary slip systems that will be activated during deformation.

127

5.3.2

Recovery of kpp

The four single crystal specimens (Set 2) presented in Chapter 4, heat-treated for 2 h each
at a temperature between 600 ‰ – 1200 ‰, displayed a plateau in the recovery of kpp
for temperatures hotter than 1000 ‰. Based on this result, a 1000 ‰ for 2 h heating was
performed on the deformed specimens, to observe the recovery of the phonon peak.
After the 1000 ‰ for 2 h recovery heat treatment, kpp for both grains and across the grain
boundary in the 2% strained specimen 7 recovered to at least 5% greater than the phonon
peak before deformation. The kpp for the grains in the 2% strained specimen 13 after the
1000 ‰ heat treatment increased 300%, and had an average kpp /k3 ratio of 2.45. This value
of kpp /k3 is consistent with those for the specimens in Sets 1 and 2 from Chapter 4, heat
treated at Th ≥ 1000 ‰, indicating the attainment of the maximum in k at the phonon peak
for this specimens’ purity.
The 1000 ‰ for 2 h heat treatment caused a 10% increase in kpp for the 0% strained
grain 3-1 of specimen 3. The kpp for grain 3-2 with 15% engineering strain, after the 1000 ‰
heat treatment, recovered to 94% of the before deformation kpp . Though no change was
observed in k for T
T

3.5 K for the two grains after deformation, a 15% decrease in k for

3.5 K was observed for both grains after the 1000 ‰ for 2 h recovery heat treatment.

This decrease in k for T

3.5 K combined with the 10% increase in kpp for grain 3-1, led to

the increase of kpp /k3 from 2.55 before deformation (after 1100 ‰ heat treatment), to 3.02
after the 1000 ‰ heat treatment. This value of kpp /k3 is the greatest for all the specimens
and conditions, including the 1100 ‰ for 4 h heating.
The 5% and 38% strained grains 2-1 and 2-2, respectively, did not display complete
recovery in kpp after the 1000 ‰ for 2 h recovery heat treatment. Grain 2-1, which had a

128

5% engineering strain and an 83% decrease in kpp after deformation, recovered partially to
78% of the undeformed kpp . On the other hand, after deformation, grain 2-2 had a 38%
engineering strain with a 92% decrease in kpp . After the 1000 ‰ for 2 h heat treatment, the
phonon peak recovered to 59% of kpp in the undeformed condition. No significant change
was observed in k for T

3.5 K for grain 2-1 after deformation and after the 1000 ‰ heating.

Grain 2-2 also had no change in k for T

3.5 K after deformation. After the 1000 ‰ heating,

however, an 8% decrease in k at 4.2 K was observed for Grain 2-2, which may be attributed
to introduction of small quantities of impurities from the furnace.
The kpp /k3 ratios after the 1000 ‰ heat treatment for the grains with engineering
strains ≥ 5%, i.e., grains 2-1 (square), 2-2 (triangle), and grain 3-2 (circle), are compared in
Figure 5.7 with the sigmoidal curve fit to the kpp /k3 data for the 2 h heat-treated specimens
of Set 2 in Figure 4.15 of Chapter 4. While the 15% strained grain 3-2, after the 1000 ‰
recovery heat treatment, displays a kpp /k3 value comparable with the undeformed values
for this heat treatment temperature, the 5% strained grain 2-1 has a kpp /k3 value that is
28% smaller than that of grain 3-2. Grain 2-2, deformed to 38% engineering strain, has a
kpp /k3 value that is 40% smaller than that of grain 3-2. Grains 2-1 and 2-2 clearly display
incomplete recovery.
Plotting the ratios of kpp,2 /kpp,1 and kpp,3 /kpp,2 with each other, illustrated as the
squares in Figure 5.8, uncovers an exponential relation between the two. In the ratios, kpp,1
is the kpp before deformation, kpp,2 the kpp after deformation, and kpp,3 the kpp after the
1000 ‰ for 2 h heat treatment. A smaller kpp,2 /kpp,1 ratio indicates greater reduction in kpp
after deformation, while a smaller kpp,3 /kpp,2 ratio indicates no recovery of kpp due to the
1000 ‰ heat treatment. For both ratios, a value closer to unity implies no change in kpp . The
exponential relation between the decrease in kpp after deformation and increase in kpp after
129

3 .0

2 .5

Undeformed
2-1: 5%
2-2: 38%
3-2: 15%

k p p / k 3

2 .0

1 .5

1 .0
0

2 0 0

4 0 0

6 0 0

8 0 0

1 0 0 0

Heat Treatment Temperature (°C)

1 2 0 0

Figure 5.7: Comparison of kpp /k3 , for the three grains with engineering strains ≥5%, with
the sigmoidal curve from Figure 4.15, fit to the kpp /k3 data for the 2 h heat-treated specimens
of Set 2 in Chapter 4.

130

the 1000 ‰ heat treatment is highlighted by plotting 1 − (kpp,3 /kpp,2 ) · 100 as a function
of (kpp,3 /kpp,2 ) − 1 · 100, as shown in Figure 5.9. Since the magnitude of the phonon peak
is inversely proportional to the dislocation density, the exponential correlation illustrated in
Figure 5.9 implies greater initial dislocation contents lead to greater recovery of dislocations
during heat treatments, irrespective of crystal orientations. Data from specimen 13 were
omitted from this correlation since it did not have a phonon peak in the before-deformation
condition. Also compared in Figure 5.8 are the relation between kpp,2 /kpp,1 and kpp,3 /kpp,1
(triangles), where kpp,3 /kpp,1 is a measure of how much kpp has recovered with respect to
kpp before deformation. Examining these correlations, it can be inferred that although there
is greater recovery after the introduction of more dislocations, the recovery is incomplete
for the larger dislocation concentrations. For SRF cavities, hence, from the perspective
of obtaining greater kpp and due to the economics of heat treatments, it is beneficial to
minimize the number of dislocations generated during fabrication and processing.

131

8

1 .2

6

1 .0
pp,2: after def.

pp,3: after 1000 °C

/ k

0 .8

k

k

p p ,3

/ k

k

p p ,3

p p ,2

k
4

p p ,1

pp,1: before def.

k

2

0 .6

0

0 .4
0 .0

0 .2

0 .4

k

0 .6

pp,2 / k pp,1

0 .8

1 .0

Figure 5.8: The change in kpp after deformation with respect to that before deformation
(abscissa) is compared with the change in kpp after the 1000 ‰ recovery heat treatment
with respect to that after deformation (left ordinate) and the change in kpp after the 1000 ‰
recovery heat treatment with respect to that before deformation (right ordinate) for specimens 2, 3, and 7. For all three ratios, a value of unity implies no change in kpp , while
a value smaller than unity implies reduction in kpp . An exponential relation is visible between kpp,2 /kpp,1 and kpp,3 /kpp,2 , with greater reduction in kpp after deformation leading
to greater recovery of kpp after heat treatment. This implies that greater dislocation density leads to greater recovery of dislocations during heat treatments, irrespective of crystal
orientations. The solid lines represent exponential curve fits to the data for specimens 2, 3,
and 7. Since specimen 13 did not have a phonon peak before deformation, the corresponding
data points are omitted from this plot.

132

7 0 0

pp,1: before def.

k

pp,2: after def.

k
k

5 0 0

pp,3: after 1000 °C

4 0 0
3 0 0
2 0 0

[(k

p p ,3

/k

p p ,2

) -1 ] x 1 0 0 (% )

6 0 0

1 0 0
0
0

2 0

[1- (k

4 0

pp,2/k pp,1)] x

6 0

100 (%)

8 0

1 0 0

Figure 5.9: Comparison between the decrease in kpp after deformation (abscissa) and increase in kpp after the 1000 ‰ recovery heat treatment (ordinate) for specimens 2, 3, and 7.
An exponential relation is visible between the two, with greater reduction in kpp after deformation leading to greater recovery of kpp after heat treatment. This implies that greater
dislocation density leads to greater recovery of dislocations during heat treatments, irrespective of crystal orientations. The solid line represents an exponential curve fit to the data for
specimens 2, 3, and 7. Since specimen 13 did not have a phonon peak before deformation,
the corresponding data points are omitted from this plot.

133

5.4

Conclusion

Uniaxial engineering strain as small as

≈ 2% is sufficient to reduce the phonon peak by

more than 37%. The magnitude of decrease in k at the phonon peak is not only related
to the level of deformation, but is also strongly influenced by the grain orientation along
the direction of deformation, by virtue of the number of slip systems being activated during
deformation. A 5% strain with duplex slip during deformation causes the phonon peak to
reduce by 83%, while a 15% strain with single slip reduces the phonon peak by just 72%.
The recovery of k at the phonon peak, by virtue of dislocation recovery, is proportional
to the magnitude of the dislocations introduced during deformation. A 1000 ‰ for 2 h heat
treatment is sufficient to recover the phonon peak in grains with up to ≤ 74% reduction in
the phonon peak due to strain-induced dislocations.

134

Chapter 6
Results:
Hydrogen and the Phonon Peak
During fabrication of SRF cavities, BCP or EP is regularly used to etch the interior (RF) surface of the cavity to remove surface irregularities. This step is known to introduce hydrogen
from the acid into the bulk due to the removal of the impermeable niobium pentoxide layer
from the surface, leading to “Q-disease”. The effect of hydrogen on the thermal conductivity
of superconducting niobium is being studied here.

6.1

Infusing hydrogen into niobium

Three bicrystal niobium specimens were infused with hydrogen in a 75% H2 , 25% N2 environment at 0.5 atm (PH2 = 0.375 atm). The hydrogen infusion was carried out in a custom
built brazing furnace at Michigan State University. The coolest temperature that can be
stably maintained in the furnace is 300 ‰, while the maximum pressure is limited to 0.5
atm. For hydrogen to flow into Nb, the impermeable niobium-pentoxide (Nb2 O5 ) layer has
135

to be dissolved. It has been observed [68] that the Nb2 O5 layer dissolves within minutes at
T ≥ 250 ‰, hence the 300 ‰ is an optimum temperature to introduce hydrogen into the
material.
The characteristic distance of diffusion xd was estimated as

xd =

√
Dt,

(6.1)

where D is the diffusion coefficient, and t the time. For the hydrogen-niobium system at
300 ‰, D ≈ 5.5 × 10−5 cm2 s−1 [69]. For a duration of 1 hour, xd = 4.45 mm, estimated to
be sufficient to diffuse hydrogen through the 3 mm thickness of the specimens.
Of the three specimens examined, one was used to measure the concentration of hydrogen,
while the remaining two (specimens 5 and 8) were used to measure the thermal conductivity.
Prior to H2 infusion, specimen 5 had been heat treated at 800 ‰ for 2 h in a high-temperature,
high-vacuum furnace at Michigan State University [30]. Specimen 8 had been heat treated
twice before the H2 treatment, first at 140 ‰ for 48 h, and second at 1100 ‰ for 4 h.
The RRR and processing histories of these two specimens prior to H2 infusion are listed in
Table 6.1, along with the H2 infusion parameters and the H2 degassing heat treatment data.

6.1.1

Hydrogen concentration

The concentration of hydrogen in specimens 5 and 8 were estimated using the temperature
and duration during infusion, pressure of hydrogen during the infusion, and a survey of
experimentally obtained solubility curves for hydrogen in niobium [70, Chapter 2]. For
a temperature of 300 ‰ (573 K) and pressure of 0.375 atm, yields the concentration of
136

Table 6.1: Hydrogen specimen histories.
Heat Treatment (HT)
First HT Second HT
Th
t
Th
t
(‰) (h) (‰) (h)

Hydrogen Treatment
H2 infusion
Recovery HT
TH2 tH2 PH2 Th,H2 tH2
(‰) (h) (atm) (‰)
(h)
300

1

0.375

800

2

300

1

0.375

800

2

Spec.

Est.
RRR

5

104

800

2

8

200

140

48

1100

4

hydrogen in niobium (ratio of the number of atoms of hydrogen to that of niobium) to be
∼0.7, or 41.2 at. % of hydrogen.
Depending on the concentration of hydrogen in niobium and the temperature of the
specimen, several niobium-hydrogen phases have been identified, and are represented as the
niobium-hydrogen phase diagram [71, 72]. After the hydrogen infusion at 300 ‰ the Nb-H
system would have been in the α phase for specimens 5 and 8. Consequently, during cooling
the Nb-H system would have transformed from α to α + β at ∼115 ‰, and to α + β at
∼85 ‰.

6.2

Thermal conductivity

The thermal conductivity for specimen 5 after the first 800 ‰ heat treatment (open symbols),
after hydrogen infusion (solid symbols), and after the 800 ‰ hydrogen degassing heating
(half filled symbols) are illustrated in Figure 6.1. Curves for the estimated k, as a function
of temperature, for the two grains and across the grain boundary are illustrated. The
results indicate that hydrogen can be a significant scattering source for phonons. Figure 6.2
illustrates the thermal conductivity for the two grains and across the grain boundary of

137

Table 6.2: Ratio of k at the phonon peak to that at 3 K for specimens 5 and 8 before
hydrogen infusion, after hydrogen infusion, and after hydrogen degassing heat treatment at
800 ‰ for 2 h.
kpp /k3
Spec.

Before H2

With H2

After 800 ‰

5-1
5-2
5-3

2.47
2.28
2.45

1.81
1.86
1.88

2.27
2.57
2.52

8-1
8-2
8-3

–
2.66
2.68

2.23
2.27
2.22

2.19
2.40
2.29

specimen 8 after the 1100 ‰ heat treatment (open symbols), after hydrogen infusion (solid
symbols), and after the 800 ‰ for 2 h heating to degas hydrogen (half filled symbols). The
ratios of k at the phonon peak kpp and that at 3 K k3 for the two hydrogen specimens
in their pre-hydrogen infusion state, with hydrogen in them, and after the 800 ‰ for 2 h
degassing heat treatment are listed in Table 6.2.

138

5-1 800 °C
5-1 H2

7 0
6 0

5-GB 800 °C
5-GB H2
5-GB 800 °C

5-2 800 °C

T h e rm a l c o n d u c tiv ity (W m

-1

K

-1

)

5-1 800 °C

5-2 800 °C
5-2 H2

5 0
4 0
3 0
2 0
1 0
0

1 .5

2 .0

2 .5

3 .0

3 .5

Temperature (K)

4 .0

4 .5

Figure 6.1: Thermal conductivity curves for specimen 5 after the first 800 ‰ heat treatment
(open symbols), after hydrogen infusion (solid symbols), and after the hydrogen degassing
heat treatment at 800 ‰ for 2 h (half filled symbols). Conductivity curves estimated from
each of the three temperature sensor pairs are illustrated. After hydrogen infusion, a 25%
reduction in the magnitude of the phonon peak was observed for sensor pairs 2 and 3, while
a 30% reduction was noted for sensor pair 1. No change was detected in k for T > 3 K. After
the 800 ‰ for 2 h heating to degas hydrogen from the specimen, k at the phonon peak was
recovered to the before state for sensors 2 and 3. For sensor pair 1, the degassing heating
recovered the k at the phonon peak to approximately 9% less than its before hydrogen
infusion condition.

139

8-GB 1100 °C
8-2 1100 °C
8-1 H2

6 0

8-GB H2

K

-1

)

7 0

T h e rm a l c o n d u c tiv ity (W m

-1

8-2 H2

5 0

8-1 800 °C
8-GB 800 °C
8-2 800 °C

4 0
3 0
2 0
1 0
1 .5

2 .0

2 .5

3 .0

3 .5

Temperature (K)

4 .0

4 .5

Figure 6.2: Thermal conductivity curves for specimen 8 after the 1100 ‰ heat treatment
(open symbols), after hydrogen infusion (solid symbols), and after the hydrogen degassing
heat treatment at 800 ‰ for 2 h (half filled symbols). Conductivity curves estimated from
each of the three temperature sensor pairs are illustrated. After hydrogen infusion, a 25%
reduction in the magnitude of k at the phonon peak was observed along with a 10% reduction
in k at 4.2 K. The 800 ‰ for 2 h heating to degas hydrogen from the specimen showed no
effect on the conductivity at the phonon peak, though a mild decrease in k for T > 3 K was
noted.

140

6.3

Discussion

The thermal conductivity at the phonon peak for specimen 5, grain 1, showed a 30% decrease
after the introduction of hydrogen. Similarly, the kpp of Grain 5-2, 5-GB, and the k at the
phonon peak of both grains and across the grain boundary of specimen 8 displayed a 25%
reduction due to the presence of hydrogen. These suggest that the excess hydrogen acts
as a scattering source for phonons, by probably precipitating into hydrides. No change
was observed in k for T > 3 K for specimen 5, implying that the excess hydrogen is not
a major scattering source for electrons. Contrary to this, k for T > 3 K for specimen 8
after hydrogen infusion displayed a 10% reduction. Note that the RRR, prior to hydrogen
infusion, for specimens 5 and 8 were 104 and 200, respectively. This indicates that for the
estimated concentration of hydrogen, there is sufficient hydrogen present to reduce the RRR
of the higher RRR material and thus caused the reduction in k for T > 3 K, but insufficient
to become a major scattering source of electrons in the lower RRR material.
After the 800 ‰ for 2 h heating to degas specimen 5 of hydrogen, the conductivity at
the phonon peak for grain 5-2 and 5-GB recovered to the before hydrogen state. However,
k at the phonon peak for grain 5-1, despite having a recovery after the 800 ‰ heating,
recovered to 91% of the kpp before hydrogen infusion. The 800 ‰ for 2 h heating to degas
hydrogen from specimen 8 had no effect on the conductivity at the phonon peak, though a
mild decrease in k for T > 3 K was noted.
In the α and α phases, the hydrogen atoms are disordered, and can be considered to
behave like a gas [70, Chapter 2]. In the β phase, however, hydrogen is ordered with well
defined spacing [70, Chapter 2], behaving like a solid. Hence, the mobility of the hydrogen
atoms in the β phase is reduced, and the mobility has been observed to be about two orders

141

of magnitude less than that in the α phase [73–75].
The concentration of hydrogen in these two specimens was estimated to be 41.2 at. % of
hydrogen, which is the α + β phase at room temperature. The additional solid-like β phase
in the bulk of the specimens could be the source of the additional resistance to the flow of
phonons, which led to the reduction of the phonon peak by 25% in both specimens, as seen
in Figures 6.1 and 6.2. However, the reason for the reduction in k for T > 3 K in specimen 8,
or the lack there of in specimen 5, is not clear.
The difference in the ratio of vacancies to dislocations in the two specimens could explain
the extra resistance in specimen 8. Prior to the infusion of hydrogen, specimen 5 had been
heat treated at 800 ‰ for 2 h, while specimen 8 had been heat treated at 1100 ‰ for 4 h.
Through thermal conductivity measurements at the phonon peak [76], hotter temperature
heat treatments have been shown to decrease the number of dislocations in the niobium
specimen. The vacancy concentration, on the other hand, increases exponentially with temperature, as was illustrated in Figure 4.34. Hence, the ratio of the number of vacancies to
dislocations of specimen 8 would be greater than that of specimen 5. Since there was a reduction in k in the electron dominated regime as well for specimen 8, it implies that specimen 8
had greater concentration of hydrogen than the estimated concentration. This suggests that
the concentration of hydrogen is a function of the vacancy concentration. This may have led
hydrogen concentration of specimen 8 to be in the β phase at room temperature, and ε or
ε+λ phase at cryogenic temperatures. This does not, however, readily explain the absence of
recovery of k after the 800 ‰ for 2 h heat treatment of specimen 8. It may be of importance
to note that specimen 5, which displayed complete recovery after the 800 ‰ for 2 h heating,
had been heat treated at 800 ‰ for 2 h prior to the hydrogen infusion. Specimen 8, however,
had been heat treated at 1100 ‰ for 4 h prior to hydrogen infusion, followed by the 800 ‰
142

for 2 h heating. This may indicate the presence of an as yet unclear phenomenon that could
be correlated with the specimens’ condition before infusion of hydrogen.

6.4

Conclusion

The amount of hydrogen infused into niobium during controlled SRF cavity processing has
negligible effect on the thermal conductivity for 1.6 ≤ T ≤ 4.2 K. When saturated with
hydrogen, however, the phonon peak is reduced by 25% due to hydride-phonon scattering.
The concentration of hydrogen is affected by the vacancy concentration.

143

Chapter 7
Conclusions
The thermal conductivity of superconducting niobium is a function of grain orientation, purity, deformation, and heat treatments. Large grain niobium is a better performing alternative to conventional polycrystalline niobium sheets for SRF cavity manufacture. Accelerators
requiring continuous high power operation seek to reduce RF losses by taking advantage of
small surface resistances (< 5 nΩ) in Nb at temperatures of about 2 K to increase efficiency.
Accelerator cavities operating at T ≈ 1.8 K may additionally benefit from increased thermal
stability by virtue of a phonon peak in Nb at 1.8 K. Phonon conductivity in the superconducting regime in polycrystal Nb is significantly limited by grain size. The reduced number
of grain boundaries in large grain Nb enhances the phonon peak.
Niobium ingot production introduces thermal strains in the material, which generate
dislocations and destroy potential phonon peaks. Uniaxial engineering strain as small as
≈ 2% is sufficient to reduce the phonon peak by more than 37%. The magnitude of decrease
in k at the phonon peak is also strongly influenced by the grain orientation along the direction
of deformation. If there are several slip systems being activated during deformation along a
given crystal direction, then a relatively small magnitude of deformation may produce the
144

same degradation on the phonon peak as a much larger strain operating on a single slip
system. A 1000 ‰ for 2 h heat treatment was sufficient to recover the phonon peak after
a strain as large as 15%. Depending on the number of slip systems being activated during
deformation, however, the phonon peak may not be completely recovered even for strains as
small as 5%.
The maximum in the phonon peak in large grain Nb also has a monotonic dependence on
the RRR. The role of RRR on conduction has generally been considered for T > 3 K. The
presence of the phonon peak is greatly influenced by deformation and heat treatments, by
virtue of dislocation concentration. Separating the phonon contribution to the phonon peak
by examining the ratio kpp /k3 reveals that the greatest phonon contribution is obtained
for heat treatment temperatures of about 1000 ‰. Hotter heat treatment temperatures
show some improvement in the phonon peak, however, the concentration of vacancies also
increases. Although greater vacancies did not affect the thermal conductivity, they may
aid in the formation of hydrides and can adversely affect accelerator performance. The role
of saturated hydrogen on the thermal conductivity at the phonon peak appears to result
from the formation of these hydrides. Hydrogen introduced into Nb through typical cavity
processing steps has no effect on the phonon conductivity. While RRR influences the phonon
peak, the substitutional impurity tantalum, for 520–1375 ppm in Nb, has no effect on phonon
conductivity.
This study has shown that for cavities fabricated using large grain Nb, heat treatment is
the single most important factor for producing a phonon peak in the thermal conductivity.
Additionally, the results provide guidance for choosing heat treatment parameters by presenting their benefits with respect to the phonon peak, while also uncovering the drawbacks
of the heating on other factors, such as the increase in vacancy concentration.
145

BIBLIOGRAPHY

146

BIBLIOGRAPHY
[1] H. Padamsee, J. Knobloch, and T. Hays, RF Superconductivity for Accelerators. WileyInterscience Publication, 1998.
[2] J. P. Turneaure, Microwave Measurements of the Surface Impedance of Superconducting
Tin and Lead. PhD Dissertation, Stanford University, 1967.
[3] J. Halbritter, “Comparison between Measured and Calculated RF Losses in the Superconducting State,” Z. Phys. A Hadron Nucl., vol. 238, no. 5, pp. 466–476, 1970.
[4] R. B. Flippen, “The Radial Velocity of Magnetic Field Penetration in Type II Superconductors,” Phys. Lett., vol. 17, no. 3, pp. 193 – 194, 1965.
[5] J. H. Graber, High Power RF Processing Studies of 3 GHz Niobium Superconducting
Accelerator Cavities. PhD Dissertation, Cornell University, May 1993.
[6] A. Gurevich, “Multiscale Mechanisms of SRF Breakdown,” Physica C, vol. 441, pp. 38–
43, 2006.
[7] J. P. Charrier, B. Coadou, and B. Visentin, “Improvements of Superconducting Cavity
Performances at High Accelerating Gradients,” Proc. 6th European Particle Accelerator
Conference, pp. 1885–1887, 1998.
[8] B. Visentin, “Q-Slope at High Gradients: Review of Experiments and Theories,” Proc.
11th Workshop on RF Superconductivity, pp. 74–77, 2003.
[9] P. Kneisel, G. R. Myneni, G. Ciovati, J. Sekutowicz, and T. Carneiro, “Performance of
Large Grain and Single Crystal Niobium Cavities,” Proc. 12th International Workshop
on RF Superconductivity, pp. 134–142, 2005.
[10] W. Singer, H. Kaiser, X. Singer, G. Weichert, I. Jelezov, T. Khabibuline, A. Skasyrskaia,
P. Kneisel, T. Fujino, and K. Saito, “Hydroforming of Superconducting TESLA Cavities,” Proc. 10th Workshop on RF Superconductivity, pp. 170–176, 2001.
[11] W. Singer, “Seamless/Bonded Niobium Cavities,” Proc. 12th International Workshop
on RF Superconductivity, pp. 143–147, 2005.
[12] V. Palmieri, “Seamless Cavties: The Most Creative Topic in RF Superconductivity,”
Proc. 8th Workshop on RF Superconductivity, pp. 553–589, 1997.
[13] V. Palmieri, “Advancements on Spinning of Seamless Multicell Reentrant Cavities,”
Proc. 11th Workshop on RF Superconductivity, pp. 357–361, 2003.
[14] H. Padamsee, RF Superconductivity: Science, Technology, and Applications. WileyVCH Verlag GmbH & Co. KGaA, 2009.
147

[15] P. Kneisel, “Cavity preparation/assembly techniques and impact on Q, realistic Qfactors in a module, review of modules,” Nucl. Instrum. Meth. A, vol. 557, pp. 250–258,
2006.
[16] B. Bonin and R. W. R¨th, “Q-Degradation of Niobium Cavities due to Hydrogen Cono
tamination,” Proc. 5th Workshop on RF Superconductivity, pp. 210–244, 1991.
[17] C. Z. Antoine, B. Aune, B. Bonin, J. M. Cavedon, M. Juillard, A. Godin, C. Henriot,
P. Leconte, H. Safa, and A. Veyssiere, “The Role of Atomic Hydrogen in Q-Degradation
of Niobium Superconducting RF Cavities: Analytical Point of View,” Proc. 5th Workshop on RF Superconductivity, pp. 616–634, 1991.
[18] B. Aune, R. Bandelmann, D. Bloess, B. Bonin, A. Bosotti, M. Champion, C. Crawford,
G. Deppe, B. Dwersteg, D. A. Edwards, H. T. Edwards, M. Ferrario, M. Fouaidy, P.-D.
Gall, A. Gamp, A. G¨ssel, J. Graber, D. Hubert, M. H¨ning, M. Juillard, T. Junquera,
o
u
H. Kaiser, G. Kreps, M. Kuchnir, R. Lange, M. Leenen, M. Liepe, L. Lilje, A. Matheisen,
W.-D. M¨ller, A. Mosnier, H. Padamsee, C. Pagani, M. Pekeler, H.-B. Peters, O. Peters,
o
D. Proch, K. Rehlich, D. Reschke, H. Safa, T. Schilcher, P. Schm¨ser, J. Sekutowicz,
u
S. Simrock, W. Singer, M. Tigner, D. Trines, K. Twarowski, G. Weichert, J. Weisend,
J. Wojtkiewicz, S. Wolff, and K. Zapfe, “Superconducting TESLA cavities,” Phys. Rev.
ST Accel. Beams, vol. 3, no. 9, 2000.
[19] H. Padamsee, “A New Purification Technique for Improving the Thermal Conductivity of Superconducting Nb Microwave Cavities,” IEEE Trans. Magn., vol. MAG-21,
pp. 1007–1010, 1985.
[20] P. Kneisel, “Use of the titanium solid state gettering process for the improvement of the
performance of superconducting rf cavities,” J. Less-Common Met., vol. 139, pp. 179–
188, 1988.
[21] H. Lengeler, W. Weingarten, G. M¨ller, and H. Piel, “Superconducting Niobium Caviu
ties of Improved Thermal Conductivity,” IEEE Trans. Magn., vol. MAG-21, pp. 1014–
1017, 1985.
[22] B. Visentin, J. P. Charrier, B. Coadou, and D. Roudier, “Cavity Baking: A Cure for
the High Accelerator Field Q0 Drop,” Proc. 9th Workshop on RF Superconductivity,
pp. 198–202, 1999.
[23] G. Ciovati, “Effect of Low-Temperature Baking on the Radio-Frequency Properties of
Niobium Superconducting Cavities for Particle Accelerators,” J. Appl. Phys., vol. 96,
pp. 1591–1600, 2004.
[24] P. M. Rowell, “The Thermal Conductivity of Some Superconductors,” Proc. R. Soc.
Lond. A, vol. 254, pp. 542–550, 1960.
[25] A. C. Anderson and S. C. Smith, “Effect of Dislocations on the Lattice Thermal Conductivity of Superconducting Niobium,” J. Phys. Chem. Solids, vol. 34, pp. 111–122,
1973.
148

[26] W. Wasserb¨ch, “Low-Temperature Thermal Conductivity of Plastically Deformed Nioa
bium Single Crystals,” Philos. Mag. A, vol. 38, pp. 401–431, 1978.
[27] W. Singer, A. Brinkmann, A. Ermakov, J. Iversen, G. Kreps, A. Matheisen, D. Proch,
D. Reschke, X. Singer, M. Spiwek, H. Wen, and H. G. Brokmeier, “Large Grain Superconducting RF Cavities at DESY,” Proc. Linear Accelerator Conference, 2006.
[28] A. Aizaz, T. L. Grimm, and N. T. Wright, “Thermal design studies in superconducting rf
cavities: Phonon peak and Kapitza conductance,” Phys. Rev. ST Accel. Beams, vol. 13,
p. 093503, 2010.
[29] M. G. Rao and P. Kneisel, “Mechanical Properties of High RRR Niobium at Cryogenic
Temperatures,” Adv. Cryog. Eng., vol. 40, pp. 1383–1390, 1994.
[30] S. K. Chandrasekaran, “Design and Fabrication of a High-Temperature High-Vacuum
Furnace to Heat Treat Niobium for Superconducting Radio Frequency Cavities,” MS
Thesis, Michigan State University, May 2010.
[31] A. H. Wilson, The Theory of Metals. Cambridge University Press, 1936.
[32] R. E. B. Makinson, “The Thermal Conductivity of Metals,” Proc. Cam. Phys. Soc.,
vol. 34, pp. 474–497, 1938.
[33] J. K. Hulm, “The Thermal Conductivity of Tin, Mercury, Indium and Tantalum at
Liquid Helium Temperatures,” Proc. R. Soc. A, vol. 204, pp. 98–123, 1950.
[34] F. Koechlin and B. Bonin, “Parametrization of the Niobium Thermal Conductivity in
the Superconducting State,” Supercond. Sci. Technol., vol. 9, pp. 453–460, 1996.
[35] J. Bardeen, L. N. Cooper, and J. R. Schrieffer, “Theory of Superconductivity,” Phys.
Rev., vol. 108, pp. 1175–1204, 1957.
[36] J. Bardeen, G. Rickayzen, and L. Tewordt, “Theory of the Thermal Conductivity of
Superconductors,” Phys. Rev., vol. 113, pp. 982–994, 1959.
[37] P. Rhodes, “Fermi-Dirac Functions of Integral Order,” Proc. R. Soc. Lon. A, vol. 204,
pp. 396–405, 1950.
[38] K. Mendelssohn and H. M. Rosenberg, “The Thermal Conductivity of Metals at Low
Temperatures,” Solid State Phys., vol. 12, pp. 223–274, 1961.
[39] H. B. G. Casimir, “Note on the Conduction of Heat in Crystals,” Physica, vol. 5,
pp. 495–500, 1938.
[40] A. C. Anderson and S. G. O’Hara, “The Lattice Thermal Conductivity of Normal and
Superconducting Niobium,” J. Low Temp. Phys., vol. 15, pp. 323–333, 1974.
[41] W. Wasserb¨ch, “Phonon-Dislocation Scattering and Low Temperature Lattice Thera
mal Conductivity of Niobium and Tantalum Single Crystals Deformed at Low Temperatures,” Phys. Status Solidi B, vol. 128, pp. 453–466, 1985.
149

[42] W. Wasserb¨ch, “Lattice Thermal Conductivity and Phonon-Dislocation Interaction in
a
Niobium and Tantalum Single Crystals Plastically Deformed at Intermediate Temperatures,” Philos. Mag. A, vol. 52, pp. 225–241, 1985.
[43] W. Wasserb¨ch, “Low-Temperature Thermal Conductivity of Plastically Deformed Nioa
bium and Tantalum Single Crystals of [001] Orientation,” Phys. Status Solidi A, vol. 128,
pp. 55–68, 1991.
[44] A. Connolly and K. Mendelssohn, “Thermal Conductivity of Tantalum and Niobium
below 1◦ K,” Proc. R. Soc. Lond. A, vol. 266, pp. 429–439, 1962.
[45] W. Wasserb¨ch, “Low Temperature Thermal Conductivity of Pure and Impure Niobium
a
and Tantalum Single Crystals,” Phys. Status Solidi B, vol. 84, pp. 205–214, 1977.
[46] E. P. Roth and A. C. Anderson, “Interaction Between Thermal Phonons and Dislocations,” Phys. Rev. B, vol. 20, pp. 768–775, 1979.
[47] E. J. Cotts, D. M. Miliotis, and A. C. Anderson, “Thermal Transport in Deformed
Lithium Flouride,” Phys. Rev. B, vol. 24, pp. 7336–7341, 1981.
[48] A. Aizaz, Thermal Design Studies in Niobium and Helium for Superconducting Radio
Frequency Cavities. PhD Dissertation, Michigan State University, 2006.
[49] M. R. Daymond and P. J. Withers, “A Synchrotron Radiation Study of Transient Internal Strain Changes During the Early Stages of Thermal Cycling in an Al / SiCw
MMC,” Scripta Mater., vol. 35, no. 10, pp. 1229–1234, 1996.
[50] H.-J. Bunge, Texture Analysis in Materials Science. Butterworths, 1982.
[51] J. Nye, Physical Properties of Cristals: Their Representation by Tensors and Matrices.
Oxford University Press, 1985.
[52] K. Carroll, “Elastic Constants of Niobium from 4.2◦ to 300◦ K,” J. Appl. Phys., vol. 36,
pp. 3689–3690, 1965.
[53] G. Simmons and H. Wang, Single Crystal Elastic Constants and Calculated Aggregate
Properties. A Handbook. The MIT Press, second ed., 1971.
[54] S. K. Chandrasekaran, J. V. Beck, T. R. Bieler, C. C. Compton, and N. T. Wright,
“Parameter Estimation of the Thermal Conductivity of Superconducting Niobium,”
Therm. Cond., vol. 30, pp. 742–749, 2010.
[55] J. V. Beck and K. J. Arnold, Parameter Estimation in Engineering and Science. Wiley,
1977.
[56] S. J. Kline and F. A. McClintock, “Describing uncertainties in single-sample experiments,” Mech. Eng., vol. 75, pp. 3–8, 1953.
[57] G. R. Myneni private communication, 2010.
150

[58] P. G. Klemens, “The Scattering of Low-Frequency Lattice Waves by Static Imperfections,” Proc. Phys. Soc. A, vol. 68, pp. 1113–1128, 1955.
[59] T. R. Bieler private communication, 2012.
[60] P. Moeck, “Physical Metallurgy for Engineers.” Portland State University Lecture, 2012.
[61] H.-E. Schaefer, “Investigation of Thermal Equilibrium Vacancies in Metals by Positron
Annihilation,” Phys. Status Solidi A, vol. 102, pp. 47–65, 1987.
[62] S. K. Chandrasekaran, T. R. Bieler, C. Compton, W. Hartung, and N. T. Wright,
“Comparison of the Role of Moderate Heat Treatment Temperatures on the Thermal
Conductivity of Ingot Niobium,” AIP Conf. Proc., vol. 1352, pp. 131–141, 2011.
[63] P. L. Richards and M. Tinkham, “Far-Infrared Energy Gap Measurements in Bulk
Superconducting In, Sn, Hg, Ta, V, Pb, and Nb,” Phys. Rev., vol. 119, pp. 575–590,
1960.
[64] M. D. Sherrill and H. H. Edwards, “Superconduting Tunneling on Bulk Niobium,” Phys.
Rev. Lett., vol. 6, pp. 460–461, 1961.
[65] P. Townsend and J. Sutton, “Investigation by Electron Tunneling of the Superconductivity Energy Gaps in Nb Ta Sn and Pb,” Phys. Rev., vol. 128, pp. 591–595, 1962.
[66] D. Bonnet, S. Erlenkamper, H. Germer, and H. Rabenhorst, “A New Mmeasurement
of the Energy Gap in Superconducting Niobium,” Phys. Lett., vol. 25A, pp. 452–453,
1967.
[67] D. Baars private communication, 2011.
[68] K. Kowalski, A. Bernasik, W. Singer, and X. Singer, “In Situ XPS Investigtaion of the
Baking Effect on the Surface Oxide Structure Formed on Niobium Sheets Used for Superconducting RF Cavity Production,” Proc. 11th Workshop on RF Superconductivity,
pp. 610–613, 2003.
[69] J. V¨kl and H. Wipf, “Diffusion of Hydrogen in Metals,” Hyperfine Interact., vol. 8,
o
pp. 631–637, 1981.
[70] G. Alefeld and J. V¨kl, eds., Hydrogen in Metals II: Application-Oriented Properties.
o
Springer-Verlag, 1978.
[71] F. Manchester and J. Pitre, “Part I: The Incoherent Phase Diagram,” Phase Diagrams
of Binary Hydrogen Alloys, pp. 115–137, 2000.
[72] “Evaluation of the Propensity of Niobium to Absorb Hydrogen During Fabrication of
Superconducting Radio Frequency Cavities for Particle Accelerators,” J. Res. Natl. Inst.
Stand. Technol., vol. 115, pp. 353–371, 2010.
[73] J. F. Smith, “The H-Nb (Hydrogen-Niobium) and D-Nb (Deuterium-Niobium) Systems,” Bulletin of Alloy Phase Diagrams, vol. 4, pp. 39–46, 1983.
151

[74] M. A. Pick and R. Bausch, “The Determination of the Force-Dipole Tensor of Hydrogen
in Niobium,” J. Phys. F, vol. 6, pp. 1751–1763, 1976.
[75] J.-M. Welter and F. J. Johnen, “Superconducting Transistion Temperature and Low
Temperature Resistivity in the Niobium-Hydrogen System,” Z. Phys. B Con. Mat.,
vol. 27, pp. 227–232, 1977.
[76] S. K. Chandrasekaran, T. R. Bieler, C. Compton, and N. T. Wright, “Effect of Heat
Treatment Temperature on the Thermal Conductivity of Large Grain Superconducting
Niobium,” Proc. 15th International Conference on RF Superconductivity, pp. 593–596,
2011.

152