DIFFERING ROLES OF DISORDER:
NON-FERMI-LIQUID BEHAVIOR IN UCus-,Nix
AND CURIE TEMPERATURE ENHANCEMENT IN UCu2Si2-.Ge.
DANIEL J. MIXSON II
A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY
UNIVERSITY OF FLORIDA
This work is dedicated to my Lord and Savior Jesus Christ. I can do all things
through Christ which strengthens me.-Philippians 4:13
The Lord has placed a lot of special people in my life. Thus, the thoroughness
of the acknowledgments may exceed that of the actual dissertation. I will attempt
to thank everyone in the order in which they were introduced into my life.
My first thank you goes to my parents, Dan and Judy Mixson, for their love
which includes discipline and support. My sister, Venessa, has been very supportive
of all my undertakings. I also want to thank all four of my grandparents who
played and are continuing to play a major role in my life: Jim and Sue Mixson;
Marion and Winonah Pettit.
I need to recognize my high school physics teacher, Mr. Dan Pate, who served
as a mentor to me during my last two years of high school. His inspiration and
encouragement were indispensable.
A broad thank you goes to my undergraduate institution, Mississippi State
University. The academics were good, but the life lessons I learned hopefully will
never be forgotten.
The majority of these acknowledgments are directed to people affiliated with
the Department of Physics at the University of Florida (UF). First, I want to
thank the two graduate coordinators in the Department of Physics during my
time at Florida: Dr. John Yelton and Dr. Mark Meisel. Also, special thanks go
to the Department of Physics Student Services personnel who were affiliated with
graduate student affairs: Mrs. Susan Rizzo and Mrs. Darlene Latimer. I also want
to thank all the faculty who were instructors in the graduate classes I took: Dr.
Pierre Sikivie, Dr. Richard Woodard, Dr. Charles Thorn, Dr. James Fry, Dr.
Charles Hopper, Dr. Fred Sharifi, Dr. James Dufty, Dr. Alan Dorsey, Dr. David
Tanner, Dr. Robert Coldwell, Dr. Andrey Korytov, and Dr. F. Eugene Dunnam.
I also want to acknowledge the personnel/professors I served under during my
"instructor" years: Mr. Greg Martin, Mr. Ray Thomas, Dr. Robert DeSerio, Dr.
Dmitrii Maslov, Dr. John Klauder, Dr. Gary Ihas, and Dr. Andrew Rinzler.
The most influential professor upon my physics career at UF has to be Dr.
Greg Stewart. I want to thank him for allowing me to work in his lab these past
four years. I especially want to thank him for funding me these past two years
[with money from a Department of Energy (DOE) grant] and for allowing me to go
to Los Alamos National Laboratory (LANL) during the Summer of 2004.
There are many people affiliated with Dr. Stewart's lab that have influenced
my research career. First, special thanks go to Dr. Jungsoo Kim for showing me
the "ins and outs" of the lab. I am indebted to the following people who were/are
affiliated with Dr. Stewart's lab: Mr. Josh Alwood, Mr. Patrick Watts, Mr. Adam
Bograd, Mr. Michael Swick, Mr. Tim Jones, and Mr. Don Burnette. I would
also like to acknowledge Dr. Bohdan Andraka and members of his group I have
interacted with in some form: Dr. Richard Pietri and Mr. Costel Rotundu.
I would also like to recognize Mr. Ju-Hyun Park of Dr. Meisel's lab. His
extensive knowledge was very useful for magnetization measurements I took.
I want to thank the following personnel who help make up part of the infras-
tructure in the Department of Physics at UF and who have been more than willing
to assist me: Mrs. Janet Germay, Mr. Pete Axson, Mr. Marc Link, Mr. Edward
Storch, Mr. Bill Malphurs, Mr. John Mocko, Mr. Greg Bennett, Mr. Jay Horton,
Mr. John Graham, Mr. Greg Labbe, Mr. Bryan Allen, and Mr. Brent Nelson.
I am very grateful to the following five members of my dissertation committee
who have taken time out of their busy schedules to serve: Dr. Greg Stewart (chair),
Dr. Bohdan Andraka, Dr. Pradeep Kumar, Dr. Dinesh Shah, and Dr. David
The following people helped me broaden my knowledge of condensed matter
physics at LANL in the Summer of 2004: Dr. John Sarrao, Dr. Eric Bauer, Dr.
Veronika Fritzsch, Mr. Jason Leonard, Dr. Mike Hundley, and Dr. Joe Thompson.
Finally, I would like to recognize people around the world I have had the
chance to collaborate with on the UCus-aNia and UCu2Si2_-Gea systems. Special
thanks go to Dr. E.-W. Scheidt and Prof. W. Scherer at the University of Augs-
burg for their low temperature measurements on the specific heat of the UCu5-asNia
system with a dilution refrigerator. Also, I would like to thank Drs. T. Murphy
and E. Palm at the National High Magnetic Field Laboratory (NHMFL) in Talla-
hassee, FL, for their low temperature resistivity measurements on the UCus-.Nia
system. Also, I would like to thank Prof. A.H. Castro Neto (Boston University)
for his communication on the Griffiths phase disorder model and in trying to help
me understand how the model applies to the UCu_5-,Nia system. Lastly, I am very
grateful to Dr. M.B. Silva Neto and Prof. A.H. Castro Neto for their theoretical
efforts on the UCu2Si2-zGe_ system.
TABLE OF CONTENTS
ACKNOWLEDGMENTS ..... .. ....................
LIST OF TABLES ... .... ............. .... ..... .
LIST OF FIGURES ................................
ABSTRACT ........ ... ........................
1 INTRODUCTION .........
1.1 NFL Behavior in UCus-.Ni,
1.1.1 UCus-.Pd ......
1.2 Curie Temperature Enhancement in UCu2Si2-zGe_.
2 THEORY ...........................
2.1 Non-Fermi-Liquid Theory .............
2.1.1 Introduction ..................
2.1.2 Fermi-Liquids .................
2.1.3 Non-Fermi-liquids ..............
2.2 Curie Temperature Enhancement Theory ......
3 EXPERIMENTAL TECHNIQUES ....................
3.1 Arc-M elting ..............................
3.2 X-Ray Diffraction/Lattice Parameter Determination from X-Ray
D iffraction . . . .
3.3 Magnetic Susceptibility ........................
3.3.1 DC Magnetic Susceptibility .................
3.3.2 AC Magnetic Susceptibility .................
3.4 Cryogenics ...............................
3.5 Probes . . . . .
3.6 DC Resistivity .............................
3.7 Specific Heat ..............................
. . 1
4 UCus-,Ni, RESULTS AND DISCUSSION ................ 74
4.1 Lattice Parameter Values for UCus-sNia ............. 74
4.2 DC Electrical Resistivity Results for UCus5-Ni3 .......... 79
4.2.1 DC Electrical Resistivity Discussion for UCus-Ni 79
4.3 Magnetization Results for UCus-.Nia ............... 85
4.3.1 Magnetization Discussion for UCus5-Ni. .......... 86
4.4 AC Magnetic Susceptibility Results for UCu5-,Ni ........ 89
4.4.1 AC Magnetic Susceptibility Discussion for UCus5-Ni, 89
4.5 DC Magnetic Susceptibility Results for UCu5-sNis ... 92
4.5.1 DC Magnetic Susceptibility Discussion for UCus-.Ni2 95
4.6 UCus-,Ni2 Specific Heat Results . . 102
4.6.1 UCu5-.Nia Specific Heat Discussion ............ 114
5 UCu2Si2-.Ge, RESULTS AND DISCUSSION .............. 136
5.1 UCu2Si2-_Ge. Results ....................... 139
5.2 UCu2Si2-.GeB Discussion ...................... 142
5.2.1 Tc Enhancement ...................... 142
5.2.2 Resistivity displaying electrons in the ballistic or diffusive
regime ........................... 143
6 CONCLUSIONS AND FUTURE WORK ................. 157
6.1 UCus5-Nia Conclusions ........................ 157
6.2 Future Work Derived from UCus5-Ni, Results ........... 159
6.3 UCu2Si2-_Gez Conclusions ..................... 163
6.4 UCu2Si2-,Ge_ Future Work .................... 164
APPENDIX ..................................... 166
A LATTICE PARAMETER GRAPHS ................... 166
B MAGNETORESISTANCE GRAPHS ................... 169
C UCus-5Ni, MAGNETIC SUSCEPTIBILITY GRAPHS .......... 172
D UCu5sNi, SPECIFIC HEAT GRAPHS .................. 177
REFERENCES ................................... 185
BIOGRAPHICAL SKETCH .............................. 195
LIST OF TABLES
4-1 Lattice parameter values for the nine UCus5-Ni, compounds ...... 126
4-2 The low temperature resistivity results for UCu5s-zNi_ samples. 127
4-3 The low temperature resistivity results for UCus5-Nix samples. 128
4-4 Magnetization data for UCus5_Nia annealed 14 days at 750C. 129
4-5 AC susceptibility results for select UCus5-Nia samples. . 130
4-6 DC magnetic susceptibility results for annealed UCu5-sNi.. .. 131
4-7 Fit results for UCus-_Nis dc susceptibility data at 1 kG and 1 T. 132
4-8 Fit results for UCus-aNia dc susceptibility data at 2 T and 3 T. 133
4-9 Fit results for UCu5-zNi. dc susceptibility data at 4 T. . 134
4-10 Specific heat results for annealed UCu5-aNi, samples. ... 135
5-1 Magnetic susceptibility and resistivity results for unannealed
UCu2Si2-zGe_ samples (I) ........................ 153
5-2 Magnetic susceptibility and resistivity results for unannealed
UCu2Si2-,Ge_ samples (II). . . .. 154
5-3 Magnetic susceptibility and resistivity results for annealed
UCu2Si2-Gea samples (I) ........................ 155
5-4 Magnetic susceptibility and resistivity results for annealed
UCu2Si2-zGe_ samples (II). . . .. 156
LIST OF FIGURES
1-1 The conventional unit cell for the AuBe5 crystal structure ....... 3
1-2 The unit cell for the tetragonal ThCr2Si2 crystal structure. .. 18
2-1 The Doniach phase diagram taken from Ref. . . ... 35
2-2 The Griffiths phase diagram taken from Ref. . . .... 41
2-3 Curie temperature predictions by Silva Neto and Castro Neto. 45
3-1 Cross-section of a helium 3 probe. . . ... 58
3-2 Schematic diagram outlining the thermal relaxation method. .. 64
3-3 An overhead and bottom view of the mounted sample platform. 65
4-1 Lattice parameter values for unannealed UCu5s-_Ni, compounds. 75
4-2 Lattice parameter values for annealed UCu5s_.Nia compounds. 76
4-3 Low temperature normalized resistivity for UCus5-Nia samples. 80
4-4 Magnetization for UCus-zNi, samples annealed 14 days at 750*C. 87
4-5 AC susceptibility data for annealed UCu4.5Nio.5 . .. 90
4-6 DC susceptibility for annealed UCu4Ni in different magnetic fields. 93
4-7 Low temperature dc susceptibility for annealed UCus-..Ni, samples. 94
4-8 Semilog plot of dc magnetic susceptibility for UCu4Ni ..... 99
4-9 Log-log plot of dc magnetic susceptibility for UCu4Ni. ... 100
4-10 Specific heat for eight annealed UCus-.Ni, samples. ... 103
4-11 UCus5-Ni, specific heat results on a semilog plot. . ... 105
4-12 UCus-zNia specific heat results on a log-log plot. . ... 106
4-13 Specific heat of UCu4.1Nio.9 in 0, 3, and 6 T. . .. 108
4-14 Specific heat of UCu3.9Ni1.1 in 0, 3, and 6 T. . .. 109
4-15 Specific heat of UCu3.sNi1.2 in 0, 3, and 6 T. . .. 110
4-16 Specific heat of UCu4Ni in 0, 2, 3, and 6 T.
4-17 Specific heat of UCu3.95Nii.o5 in 0, 2, 3, and 6 T. . ... 112
4-18 Phase diagram for UCus-,Nix and UCus-.Pd. . .. 115
4-19 Specific heat data at 6 T for UCus-5Ni, samples. . .... 124
5-1 Tc determination for annealed UCu2Ge2. . ... 138
5-2 Resistivity versus temperature for UCu2SiGe. . .. 140
5-3 Tc phase diagram for UCu2Si2_-Ge . . ... 144
5-4 The reciprocal RRR values for annealed UCu2Si2-_Gex ........ 146
5-5 Phase diagram for unannealed UIrl-aPt.Al system. ... 152
6-1 DC magnetic susceptibility results for UCu5_-Co. . .... 162
A-1 Theoretical and experimental x-ray diffraction patterns for annealed
U Cu4N i . . . . .
A-2 Lattice parameter values versus corresponding error function values
for annealed UCu4Ni ...........................
B-1 Low temperature magnetoresistance measurements of annealed
U Cu4N i . . . . .
B-2 Low temperature magnetoresistance measurements of annealed
UCus.95Ni.o5. .. .. .. .. .. .. .
C-1 Semilog and log-log plots of UCu4.1Nio.9 dc susceptibility. .
C-2 Semilog and log-log plots of UCu3.95Nil.05 dc susceptibility. .
C-3 Semilog and log-log plots of UCu.Ni1.1 dc susceptibility. .
C-4 Semilog and log-log plots of UCus.8Nil.2 dc susceptibility. .
D-1 The determination of TNde for annealed UCu4.4Ni0.6 ..........
D-2 Specific heat of annealed UCu4Ni in 6 and 13 T. . .
D-3 Specific heat of UCu4Ni in 13 T with Castro Neto-Jones fit. .
D-4 UCu4.1Nio.9 specific heat data in 3 T and 6 T on a log-log plot. .
D-5 UCu4Ni specific heat data in 2 T and 3 T on a log-log plot. .
D-6 UCu3.95Nil.05 specific heat data in 2 T on a log-log plot. .
D-7 UCus.8Nil.2 specific heat data in 3 T on a log-log plot. . .
Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy
DIFFERING ROLES OF DISORDER:
NON-FERMI-LIQUID BEHAVIOR IN UCus_-Ni.
AND CURIE TEMPERATURE ENHANCEMENT IN UCu2Si2-.Ge.
Daniel J. Mixson II
Chair: Gregory R. Stewart
Major Department: Physics
Disorder may be created by the substitution of one element for another where
the elements are found on the periodic chart of elements. One compound that has
structural disorder is UCus5-Nix where the transition element, copper, is replaced
with another transition element, nickel. Another compound that has disorder
is UCu2Si2-TGeq, where silicon atoms are substituted with isovalent germanium
elements. The disorder in these two compounds leads to interesting physics at low
UCu5 is antiferromagnetic at around 16 K; however, the antiferromagnetism
may be suppressed by doping nickel atoms onto the copper sites in the unit cell.
The antiferromagnetic transition is suppressed to zero for the UCu4Ni compound,
leading to non-Fermi-liquid (NFL) behavior due to a quantum critical point.
Multiple UCus5-Nia compounds with small changes in the stoichiometry are
synthesized around UCu4Ni in order to investigate if the cause of NFL behavior
crosses over from a quantum critical point to rare strongly coupled magnetic
clusters in a Griffiths phase scenario. The investigation is carried out by performing
the following measurements on UCus5-Nia compounds: 1) x-ray diffraction;
2) direct current (dc) magnetic susceptibility as a function of temperature in
various magnetic fields; 3) alternating current (ac) susceptibility; 4) magnetization
measurements as a function of magnetic fields; 5) low temperature dc resistivity;
and 6) low temperature specific heat in zero and applied magnetic fields (up to 13
UCu2Ge2 and UCu2Si2 are ferromagnets with ferromagnetic ordering temper-
atures, TcuH, occurring at around 108.5 K and 102.5 K respectively. Various com-
pounds lying between the two aforementioned ferromagnets on the UCu2Si2-zGe,
phase diagram were synthesized in order to test theoretical predictions made
concerning the nonmonotonic behavior of Tcurie as a function of the amount of
structural disorder. The nonmonotonic behavior was confirmed by measuring
the dc magnetic susceptibility. As the amount of disorder is varied, the electron
scattering crosses over from a ballistic regime for no structural disorder to a dif-
fusive regime when disorder is present. This crossover is proven by dc resistivity
The chemical substitution of one element for another element has been a
longstanding practice in the condensed matter physics community. In fact, the first
system published that claimed non-Fermi-liquid (NFL) behavior [i.e., deviations
occurred from Fermi liquid behavior: at low temperatures, p (the resistivity) = po
+ AT2, x (magnetic susceptibility) and C/T (specific heat divided by temperature)
approach a constant value as T -* 0 K] was Yi-UsPd3 by Seaman et al. .
This NFL behavior in Yi-U..UaPd3 was attributed to disorder created by the
chemical substitution of yttrium atoms for uranium atoms . Another such alloy
that displays NFL behavior at low temperatures due to disorder is UCus-sPd,.
The characteristics of the UCus-_Pd, system will be discussed and the motivation
behind the study of the isostructural UCus5-Ni, system will be clarified.
A second part of this dissertation concerns the ferromagnetic behavior of the
ternary UCu2Si2-zGez compounds. The driving force behind these compounds
is disorder, specifically the silicon/germanium sublattice disorder introduced by
substituting (or doping) germanium atoms onto silicon atom sites in the unit cell
(the fundamental "building block" of any crystal structure). Silva Neto and Castro
Neto have made theoretical predictions that this sublattice disorder combined
with thermal and quantum fluctuations will enhance Curie temperature values in
quantum ferromagnets . A discussion of why the UCu2Si2-.Ge, system was
chosen to test the theory will be carried out.
1.1 NFL Behavior in UCus-,Ni.
UCus-5Pd, was the first system to display NFL behavior with no dilution of
the f-atom (i.e., the uranium atom) site . UCu5 is an antiferromagnet with a
Neel temperature, TN, around 15 K . The palladium (Pd) atoms are doped
onto the copper (Cu) atom sites and this suppresses the antiferromagnetic ordering
temperature. Andraka and Stewart reported that doping the Cu sites with Pd
suppressed the antiferromagnetism at x z 0.75 and that UCu4Pd displayed NFL
behavior with a power law divergence in the specific heat (C/T ~ T-0-32 for 1-
10 K) and a linear dependence in the resistivity (p = Po a T for 0.3 to 10 K, with
Po = 375 pA2 cm and a = 6.3 Aj cm K-1) . More recent measurements down to
lower temperatures have shown that UCu4Pd orders antiferromagnetically below
TN = 190 milliKelvin (mK) while UCu3.9Pd1. shows no ordering temperature
down to the lowest measurable temperature .
UCu5 crystallizes in the AuBe5 structure with the beryllium sublattice
possessing two inequivalent beryllium sites. A schematic drawing of the AuBe5
crystal structure is shown in Fig. 1-1 on page 3 . The AuBe5 structure is cubic
with twenty-four atoms occupying a unit cell. So, for UCus, there are four uranium
(U) atoms and twenty Cu atoms. The U atoms occupy the face-centered cubic
(fcc) sites (denoted 4a). The Cu atoms are divided up among sixteen smaller sites
(called 16e sites) and four larger sites (called 4c sites). When the Cu atoms are
replaced by Pd atoms in UCu5s-.Pda compounds and since Pd atoms are larger
than Cu atoms, one might expect an ordered model for the UCu4Pd sample: the
U atoms would occupy the 4a sites, the Pd atoms would occupy the 4c sites, and
the Cu atoms would occupy the 16e sites. Some compounds do exhibit this ordered
arrangement in the AuBe5 structure: AgErCu4  and ErMnNi4 . Despite
claims by Chau et al. using a high-intensity powder diffractometer that UCu4Pd
Figure 1-1: The conventional unit cell for the cubic AuBe5 crystal structure. The
4a sites are located on the edges of the cube and in the center of each square face.
The 4c and 16e sites are crystallographically inequivalent sites with the 4c site
larger than the 16e site. The drawing is taken from the paper by Chau et al. .
is chemically ordered , Pd/Cu disorder does exist on the 4c and 16e sites with
experimental evidence coming from MacLaughlin et al. using magnetic resonance
measurements , Booth et al. using extended x-ray-absorption fine-structure
(EXAFS) , and Weber et al. using unannealed and annealed UCus5-Pd, lattice
parameter values . MacLaughlin et al. found that muons are relaxed rapidly
in UCu4Pd and this rapid relaxation comes from a large disorder effect. Booth
et al. revealed that for the unannealed UCu4Pd, 24 3% of the Pd occupies the
majority sites (the 16e sites) instead of all the Pd occupying the minority 4c sites.
Then, Weber et al. found that annealing UCu4Pd decreased the lattice parameters
compared to unannealed UCu4Pd. These results from the lattice parameters
indicate that Pd atoms are rearranged from the 16e to the 4c sites. However, Booth
et al. performed EXAFS measurements on annealed UCu4Pd samples made by
Weber et al. and still found that 19% of the Pd is located on the 16e sites .
Clearly, UCu4Pd shows Cu/Pd disorder on the Cu sublattices.
The disorder in UCu5-,Pd4 leads to unique low temperature properties in
the specific heat, especially for the UCu4Pd compound. Andraka and Stewart
proposed, based upon their results, that UCu4Pd lies near the suppression of a
second order phase transition, TN = 0 . This quantum critical point (QCP) (i.e.,
TN = 0) was thought to be the source of NFL behavior for the thermodynamic
properties at finite temperatures. As mentioned earlier, power law behavior was
found between 1 and 10 K while the lowest temperature C/T data fell under the
power law curve by as much as 16% at 0.34 K . Then Scheidt et al. extended
the specific heat measurements down to lower temperatures and found that for
unannealed UCu4Pd, the specific heat data leveled off at around 0.2 K . The
specific heat of unannealed UCu4Pd showed logarithmic temperature dependence
over a decade of temperature (0.2 K to 2 K) in agreement with the work of
Vollmer et al. . However, there is disagreement concerning the behavior
below 200 mK. Scheidt et al. measured ac susceptibility as a function of frequency
and found a peak at ~ 0.24 K at 95.5 Hz that shifted to 0.27 K at 995 Hz. This
large shift in the ac susceptibility peak with frequency was strong evidence for
superparamagnetism associated with spin clusters . This magnetism at lowest
temperatures was consistent with the theoretical efforts of Castro Neto et al. 
in which the formation of magnetic clusters (i.e., Griffiths phase behavior )
in highly correlated systems is caused by disorder that leads to competition
between the Kondo and Ruderman-Kittel-Kasuya-Yosida (RKKY) interaction.
In addition, unpublished results by Dr. Jungsoo Kim that measured the specific
heat of UCu4Pd in various magnetic fields showed a magnetic field induced peak,
a feature that is indicative of a possible Griffiths phase disorder description. On
the other hand, Vollmer et al. concluded that below 0.2 K, there is spin-glass
freezing of the U magnetic moments in UCu4Pd. Different interpretations have
continued to carry on. K6rner et al. then stated that UCu4Pd is antiferromagnetic
with TN = 190 mK and simultaneously shows evidence of spin glass behavior
below TN from specific heat and ac susceptibility measurements . Then
Weber et al. performed experiments on annealed samples of UCu4Pd . Not
only did they perform lattice parameter measurements as stated earlier, but
also specific heat data at low temperatures were taken. Annealing the UCu4Pd
sample for 14 days at 750 C suppressed the antiferromagnetic transition with
no hint of the transition down to the lowest measured temperature of 80 mK.
Also, the logarithmic temperature dependence of the specific heat now covered
more than two decades of temperature starting from 80 mK. This expansion of
the logarithmic temperature dependence suggested that crystallographic order
could be an alternative tuning parameter for NFL behavior besides pressure ,
doping , or magnetic field . This logarithmic temperature dependence also
suggests that the Kondo disorder model (consisting of a broad distribution P(TK)
of Kondo temperatures, TK, which extends down to very low temperatures [11,
99]) could be a plausible explanation for the NFL behavior in UCu4Pd .
Another interpretation based upon C/T ; log T concerns ordered systems near a
QCP . Unfortunately, these major theories (Griffiths phase, ordered systems,
Kondo disorder model are discussed in greater detail in the "Theory" chapter)
based upon the specific heat data have gaps in their interpretation. For example,
the specific heat data in annealed UCu4Pd are not consistent with the theoretically
predicted power-law behavior, C/T oc T-1+', for the Griffiths phase model 
while ordered systems (e.g., UBe13) near a QCP do not show a magnetic field
induced peak in C/T . The Kondo disorder model is not a valid interpretation
based upon resistivity data discussed below.
The dc electrical resistivity for UCus-sPd, has been measured by several
groups. The first electrical resistivity measurements for UCu4Pd were reported
by Andraka and Stewart . For temperatures lower than 10 K, the UCu4Pd
resistivity was approximately linear in T: p w Po (1 T/To) where Po 375 p cm
and To = 60 K. This linear temperature dependence of p gave experimental
evidence for a non-Fermi-liquid ground state. Chau and Maple then measured
the electrical resistivity for unannealed UCus-,Pd. with concentrations ranging
from x = 0 to x = 1.5 . They found that p(T) increases monotonically with
decreasing T and does not display a peak or downturn at low temperatures.
Experimental evidence also proved that the resistivity of the Pd-substituted
samples increases with increasing Pd concentration. Specifically, Chau and Maple
fit the electrical resistivity data for UCu4Pd to p po (1 T/To) below 10 K and
found Po = 258 Ap cm and To = 63 K, numbers that are comparable to Andraka
and Stewart's results. This linear dependence in the resistivity suggested that
the NFL behavior in UCu4Pd may be explained by a Kondo disorder model. The
results of Weber et al. seem to suggest otherwise . First, annealing UCu4Pd
samples drops the residual resistivity, po, by a factor of about 2.5. Then, the
linear temperature dependence that was reported previously for the unannealed
samples disappears with annealing. The resistivity of the annealed samples has a
Kondo-like minimum around 35 K. However, the resistivity of the 14-day UCu4Pd
annealed sample for 2 K < T < 8 K may be described by a Fermi-liquid expansion
p pr = AT2 + BT4 where pr = 141.5 I cm, A = -0.024 pQ cm K-2, and B = -
0.00013 p4 cm K-4 . Weber et al. note that the sign of A in the expansion
above is unusual for f moments forming a Kondo lattice. Thus, Weber et al.
conclude that the Kondo disorder model is not applicable for annealed UCu4Pd and
suggest it is not the origin for NFL behavior in unannealed UCu4Pd either.
The third major measurement performed on UCus-zPd. samples concerns
dc magnetic susceptibility measurements. Andraka and Stewart reported that dc
magnetic susceptibility of UCu4Pd could be approximated by T-6 at low fields
(100 G) from 10 K down to 1.8 K . This x temperature dependence is consistent,
although coincidental with an overcompensated four-channel magnetic Kondo
model . It will be noted here (although not important to one of the two
subjects of this dissertation) that a spin glass freezing temperature of 2 K was
measured for the UCu3Pd2 composition . Chau and Maple then measured the dc
magnetic susceptibility for UCus-.sPd, and were able to fit the low temperature X
data for UCu4Pd to x(T) = Xoln(T/To) for 2 K < T < 10 K . Plus, Chau and
Maple state that above x = 1.5, the UCus5-Pda system exhibits spin-glass freezing.
They also mention that for the low temperature x data, it is difficult to distinguish
between a logarithmic and a weak-power-law temperature dependence . Vollmer
et al. then measured the magnetic susceptibility down to 0.08 K for UCu4Pd .
The low temperature x data according to Vollmer et al. suggest a spin-glass
transition around 0.2 K due to the splitting between zero-field cooled magnetization
and field-cooled magnetization. K6rner et al. confirmed the evidence of spin
glass behavior below the antiferromagnetic transition temperature for unannealed
UCu4Pd by dc-susceptibility measurements .
All of these interesting results from the specific heat, resistivity, ac susceptibil-
ity, and dc susceptibility prove no current theoretical model is able to explain all
the current elements of the experimental data . New theories are being devel-
oped to try to explain all the different data such as a Kondo/quantum spin-glass
critical point . These results have also led researchers to further explore the role
of disorder in the UCu5s-Pd, system, especially for UCu4Pd.
Expanding upon their research on unannealed UCu4Pd , Booth et al. made
EXAFS measurements on UCu4Pd as a function of annealing time . Their
results show that the main effect of annealing is to decrease the fraction of Pd
atoms on the nominally Cu 16e sites, confirming the conclusions reached by Weber
et al. concerning annealing upon UCu4Pd . Booth et al. also conclude that
disorder must be included in any complete microscopic theory of NFL properties in
the UCus-5Pd, system.
MacLaughlin et al. performed magnetic resonance measurements upon the
UCu5rPda system and also concluded that structural disorder is a major factor
in NFL behavior. The longitudinal-field muon spin rotation (pSR) relaxation
measurements on UCus-_Pd. at low fields reveal a wide distribution of muon
relaxation rates and divergences in the frequency dependence of spin correlation
functions . Interestingly, the divergences seem to be due to slow dynamics
associated with quantum spin-glass behavior, rather than quantum criticality
as in a uniform system (such as CeNi2Ge2 [68, 140], YbRh2Si2 , and doped
alloys that include CeCus.gAuo.1 [158, 157, 3] and Ce(Rul_.Rh,)2Si2 where
x 0.5 [164, 165]) because of the observed strong inhomogeneity in the muon
relaxation rate, and the strong and frequency-dependent low-frequency fluctuation.
In conclusion, the disorder in the UCus5-Pd, system, especially in UCu4Pd, has led
to considerable interest from researchers and a search to look for compounds in the
AuBe5 cubic structure that possesses sublattice disorder.
When searching for compounds that crystallize in the AuBe5 structure and
that display possibly the same behavior as UCus-_Pdx, two factors should be
taken into consideration. The first factor concerns the substitution of transition
metals for Cu using UCu5 as the starting point. For example, the substitution
of Au for Cu (i.e., UCus5-Aua) leads to an initial increase in TN up to x = 2,
but TN decreases above x = 2  while the substitution of Ag for Cu does not
show the same rapid suppression of magnetic ordering as UCus5-Pd, does .
Also, the substitution of Ag or Au for Cu did not result in any NFL behavior.
Chau and Maple point out that the Au and Ag atoms have a completely filled
d-electron shell and the same valence as Cu . There could be a correlation
between substituting atoms with the same valence and lack of NFL behavior since
Pd atoms have a partially filled d shell and a different valence. Since the complete
subshell configuration atoms with d10 show no NFL behavior, this might possibly
extend to the half filled subshell configuration atoms with d5 due to their stability
(or lower total energy). The second factor concerns how much of a dopant atom
may be substituted in the UCu5 cubic cell. For example, when x approaches 2.4 for
the UCus5-Pd, system, a mixture of cubic and hexagonal phases appears and may
run until x = 5 . The supposition concerning the secondary hexagonal phase
may be related to studies that showed the hexagonal CaCu5 type structure may
exist with atomic radii ratios (Ca site/Cu site) ranging from 1.29 to 1.61 and that
below this range, cubic AuBes-type compounds are formed [42, 19]. A possibility is
that the hybridization between the Pd and Cu atoms reduces the apparent radii of
those atoms and thus the atomic radii ratio of some U/Cu-Pd atoms (maybe the
ratio between nearest atoms) now lies above 1.29 while other ratios are below 1.29.
Thus, the mixture of cubic and hexagonal crystal phases.
It has been known that uranium forms the cubic AuBe5 structure only
with Ni, Cu, and Pt atoms . The UCu5 has been well chronicled with its
antiferromagnetic ordering around 15 K [101, 1041. In addition, UCus undergoes
another phase transition around 1 K (in the electrical resistivity and magnetic
measurements) and this was the first example of such a distinct enhancement
effect that occurs in a magnetically ordered material [117, 115]. Then, Chau and
Maple doped UCu5 with Pt and found a region in the phase diagram where the low
temperature physical properties exhibit NFL behavior . This then leaves UCu5
to be doped with Ni since Ni has the same valence as Pd and Pt and the d-electron
shell is not full or half-full. The main difference between the dopant atoms nickel
and palladium is that the replacement Ni atoms are smaller than the Cu atoms
as opposed to the larger Pd atoms replacing Cu atoms. Sublattice order for x ~ 1
should not occur in UCu5-aNi, as happened in UCus5,Pd2. Thus, the answer to
the question of any NFL behavior in UCu5s-Ni2 as TN --+ 0 being due to disorder
or quantum critical behavior should be better differentiable.
The first part of this section will review in chronological order previous works
performed on the UCus-5.Ni2 system. The surname of the first author and the
publication year will serve as the title for each work's review. One will see that the
majority of the literature has mainly centered around UCu4Ni, possibly in hopes of
searching for interesting behavior as found in UCu4Pd.
126.96.36.199 van Daal 1975
The first work performed on the UCus5-Ni, system was done by van Daal
et al. who measured lattice constants for 0 < x < 5 along with the susceptibility,
specific heat, electrical resistivity, and absolute Seebeck coefficients as a function
of temperature . The susceptibility reveals that UNi5 has a U41 (tetravalent)
state while on the Cu-rich end of the system (especially for 0 < x < 1), the
uranium ions are mixed tetravalent and trivalent (U+). van Daal et al. suggest
that this mixed valency is possibly the source of the anomalies observed in the
Cu-rich compounds (0 < x < 1): an extra increase of the lattice constants, an
extremely high electronic specific heat, a resistivity varying in the paramagnetic
range in proportion to logo10 T at high temperatures and to 1 (T/T*)2 at low
temperatures and being extremely sensitive to deviations from stoichiometry, and a
Seebeck coefficient showing large negative peaks at low temperature .
188.8.131.52 Chakravarthy 1991
Chakravarthy et al. performed neutron diffraction measurements on Cu-rich
compounds, in particular, x = 0.0, 0.2, and 0.5 . The conclusions made are
that UCu5 is an itinerant magnet involving a strong hybridization of 5f electrons
with the conduction electrons. For UCu4.gNio.2, the suppression of long-range
magnetic order (LRO) by small substitution of Cu by Ni indicates the enhancement
of the interaction of 5f electrons with the conduction electrons . However, TN
is not significantly affected (12 K for x = 0.2). Thus, the band structure may
be responsible for the LRO . The third composition, UCu4.sNio.5, did not
show LRO down to 1.5 K and the authors suggested that the suppression of the
ordered magnetic moment of the U ion causes the LRO to disappear in the x = 0.5
184.108.40.206 L6pez de la Torre 1998
L6pez de la Torre et al. made electrical resistivity and magnetic susceptibility
measurements down to 0.4 K for UCu4.5Nio.5 and UCu4Ni samples [891. For
UCu4.5Nio.5, the magnetic susceptibility displays an irreversibility below a broad
maximum for a zero-fiell cooled (ZFC) and field-cooled (FC) regime and the
maximum is the "freezing temperature" for a spin glass at about 6 K. For UCu4Ni,
the magnetic susceptibility shows no spin glass or magnetic order feature down to
0.4 K. No saturation in the magnetic susceptibility is apparent at low temperatures
and an asymptotic T1/2 dependence describes quite accurately the magnetic
susceptibility from 0.4 K to 2.5 K . The electrical resistivity measurements for
UCu4Ni showed that all samples had linear temperature dependence from 0.4 K to
~ 30-40 K. Also, annealing did not change the low temperature electrical resistivity
properties significantly; however, the electrical resistivity values did show a large
sample dependence. The authors raise the question of whether or not this NFL
behavior is attributable to a two-channel Kondo effect , a quantum phase
transition at T = 0 [5, 158], or disorder (specifically Kondo disorder) .
220.127.116.11 L6pez de la Torre 2000
L6pez de la Torre et al. try to address the previous question by applying
the Kondo disorder model to the magnetic, electrical, and thermal properties of
UCu4Ni . The claim is that the existence of two nonequivalent copper sites in
the AuBes cubic structure is the origin of the crystallographic disorder resulting
in a distribution of Kondo temperatures [11, 15]. The phenomenological Kondo
disorder model given by Bernal et al.  was used to describe the magnetic
susceptibility data in a 1 T field and the fit described the data over two decades of
temperature. L6pez de la Torre et al. and van Daal et al. both agree that UCu4Ni
is in a mixed valence state between U3+ (J = 3/2 where J is the effective angular
momentum) and U4+ (J = 1). The electronic specific heat measurements for the
UCu4Ni sample show typical NFL log10 T dependence from 0.9 K to 9 K .
A comparison between the experimental specific heat results and a calculation
made using a distribution of Kondo temperatures (similar to Graf et al. )
and applying the resonant-level model for the specific heat of a single Kondo
impurity  show pretty good agreement from about 1 K to 10 K . L6pez de
la Torre et al. observed an upturn in the specific heat around 0.9 K and they think
the origin may be some form of spin-glass-like freezing at temperatures below their
experimental limit (0.4 K). The upturn in the specific heat may also be related to
the inflection in the magnetic susceptibility around 1 K . The conclusion by
L6pez de la Torre et al. is that the Kondo disorder model could play a significant
role in the NFL behavior reported in UCu4Ni.
18.104.22.168 L6pez de la Torre 2003
L6pez de la Torre et al. made electrical resistivity measurements of UCu4.75Nio.25
and UCu4Ni over a wide temperature range (0.4 800 K) . In summary, the
high-temperature electrical properties of UCu4Ni and UCu4.75Nio.25 were explained
in terms of single-impurity Kondo behavior. The interesting part of this paper
concerns the high values of the electrical resistivities (p(0) ~ 440 pI cm and p is
usually well above 100 /M cm over the entire experimental temperature range).
The rather poor electrical conductivity would indicate that the electronic mean
free path is close to the interatomic distance and kF f is of the order unity where
f is the mean free path. This class of strongly correlated electron systems that
approaches the Ioffe-Regel limit for the metallic state includes high-Tc supercon-
ductors, fullerenes, and ferromagnetic perovskites (e.g., SrRuO3), which all have
properties suggesting some sort of NFL behavior [1541. In the results part of this
dissertation, these reported high electrical resistivity values will be revisited.
22.214.171.124 Present Work
One reason for studying the UCu5-zNia system concerns the fact that the
Kondo disorder model may not be the best explanation for the NFL behavior
present in the experimental data. For example, a quantum critical point (QCP)
may also be a possibility since the antiferromagnetic temperature (a second
order phase transition) is suppressed to zero and the QCP has a large influence
on the measured properties at finite temperatures . However, some recent
research on the isostructural UCus-xPdx system has suggested that theories that
limit the cause of NFL behavior to just one phenomena such as disorder or a
quantum critical point may not be the best explanation. In fact, theories that
combine quantum critical points with disorder may explain all the different data
of systems such as UCus-xPdx and UCu5-_Ni. . One such model in which
the magnetic ordering temperature, Tord,, has been suppressed via doping based
on the effects of the accompanying disorder is the Griffiths phase disorder model.
In the Griffiths phase model, disorder leads to tunneling between closely spaced
energy levels in rare strongly coupled magnetic clusters . The Griffiths phase
model depends on the strong magnetic fluctuations produced "near" a QCP. The
Griffiths phase model predicts that X and C/T follow a power law temperature
dependence, T-1+A . One of the unique features of the Griffiths phase model is
that once a crossover magnetic field is determined (from magnetization versus field
measurements), a magnetic field around the crossover field induces a peak in C/T
at low temperatures (C/T z (H2+^/2/T3-A/2) exp(-_eff.H/kBT)) and the induced
peak broadens and moves to higher temperatures for higher fields. Two systems in
which the NFL behavior may be explained by the Griffiths phase model and that
demonstrate field induced peaks are Cel-aThaRhSb  and Cel-zLa.RhIn5 .
The field induced peak is unique to the Griffiths phase model. This should not
be confused with the peaks in magnetic fields for the multichannel Kondo prob-
lem . There are distinct differences. First, the Griffiths phase model predicts
that for zero magnetic field, the specific heat (C) follows T^ behavior as mentioned
earlier. In contrast, the specific heat for a Kondo system like CrCu  shows
a peak in zero magnetic field. The peak in the specific heat for a Kondo system
shifts towards higher temperatures and the height of the peak also grows as the
magnetic field is increased. The Griffiths phase explanation differs since, although
the field induced peak moves up in temperature, it broadens and decreases in size
with increasing field.
For the present work, eight different UCu5-,Ni, compounds were produced
with x ranging from 0.6 to 1.2. The reason for such a large variation in the concen-
trations is to explore whether or not clear distinctions may be made concerning the
cause of NFL behavior at different concentrations. In fact, small variant concentra-
tions (i.e., x = 1.05, 1.1, and 1.2) have been made near UCu4Ni, the concentration
where TN is approximately zero. A possible crossover concentration level might be
found where one concentration's NFL behavior may be explained by a QCP while
increasing the Ni concentration by 5% could cause the NFL behavior to be due to
another model, like a disorder model. Also, the Griffiths phase model is predicated
upon strong magnetic fluctuations "near" a QCP; it would be interesting to see
if "near" could be quantified in terms of the Ni concentrations. The only system
investigated by small variations of doping around the QCP was UCus-5Pdx, where
the quantum critical concentration was x w 1 as discussed earlier. The problem
with UCus5-Pda was the preferential sublattice ordering of the larger Pd atoms
on the minority 4c site (in the AuBe5 structure) occurring in UCu4Pd as discussed
previously . In fact, MacLaughlin et al. concluded that there were still signs
of disorder on annealed UCu4Pd . The UCus-5Ni_ system should not have
to contend with a preferential sublattice ordering since the Ni atoms are smaller
than the Cu atoms. Thus, the smaller Ni atoms will be distributed at slightly less
than the 25% level on the majority 16e sites (i.e., the physically smaller Be site
in the AuBe5 structure) for UCu4Ni, with some small fraction in the energetically
more unfavorable larger 4c sites. This conclusion will be discussed when the lattice
parameter values are presented in the "UCu5-a.Ni, Results and Discussion" chap-
ter. Thus, all concentrations around UCu4Ni should have a significant amount of
In addition to the lattice parameters for all eight concentrations being re-
ported, direct current (dc) resistivity, magnetization as a function of field, al-
ternating current (ac) susceptibility down to 2 K, direct current (dc) magnetic
susceptibility down to 2 K, and heat capacity in zero field along with heat ca-
pacity in magnetic fields for concentrations around the QCP composition will be
reported in the "UCus-,Ni, Results and Discussion" chapter. To date, a complete
and thorough characterization of the UCus-sNia system around the QCP has
not been reported in the literature. The measurements reported here will try to
answer if clear distinctions can be made about the sources of NFL behavior (i.e.,
quantum criticality and disorder) at various Ni concentrations around the QCP
(i.e., UCu4Ni). Also, the measurements will try to clarify the role of Griffiths phase
model for NFL behavior in the UCus-xNix system. An attempt to determine the
concentration range for which the Griffiths phase model applies will be made.
1.2 Curie Temperature Enhancement in UCu2Si2-,Ge,
A Curie temperature, Tc, enhancement as a function of disorder was pre-
dicted by Silva Neto and Castro Neto . The fundamental idea behind this
enhancement effect concerns electrons scattering from the localized moments and
acting as a heat bath for the spin dynamics. Then, the dissipation that arises from
the electronic diffusion in the case of structurally disordered ferromagnets is what
affects the Curie temperature.
The study of ferromagnetism in disordered alloys has received renewed
interest in recent years due to the development of Gal_-MnAs [114, 110, 112]
and In_-..MnaAs  as ferromagnetic semiconductors (with x ; 1-10%). In fact,
there are now reports of ferromagnetism (some at room temperatures and above) in
several magnetically doped semiconductors , e.g., GaMnP , GaMnN ,
GeMn , GaMnSb . However, the observed ferromagnetism in dilute
magnetic semiconductor (DMS) systems like GaMnAs is not just associated with
disorder created by the random positions of the magnetic dopants, but also the
thermal fluctuations of magnetic moments along with impurity band and discrete
lattice effects playing an integral part if the magnetic coupling can be assumed to
be a simple local exchange coupling between local impurity moments and carrier
spins . Thus, trying to isolate the effects of disorder on the ferromagnetism in a
DMS to test the proposed theory by Silva Neto and Castro Neto is impossible due
to the aforementioned complexities in a DMS.
The choice of a ferromagnetic material to test the theory of Silva Neto and
Castro Neto depends upon being able to test the effect of structural disorder on the
ferromagnetic properties without introducing sundry complications . De Long
et al. investigated over 100 metallic Ce and U compounds to look for trends in the
occurrence of ferromagnetism and to consider parameters such as the closest f-atom
separation notatedd by d) . For example, Hill observed that f-state magnetic
order did not occur for d < dH, where dH 3.4 or 3.5 A, the respective "Hill
limits" for f-state localization in Ce- and U-based materials . Thus, when trying
to induce structural disorder upon the selected compound, it is important not to
inadvertently change the f-atom separation because it would be very difficult then
to distinguish the change in Tc between structural disorder and f atom separation.
De Long et al. also mention that hybridization of f-levels with non-f conduction
states has an effect upon the ferromagnetism . This pretty much rules out any
binary ferromagnetic compounds since doping on one of the two atoms not only
would create structural disorder, but also the doping would affect the hybridization.
The two ferromagnets, UCu2Si2 and UCu2Ge2, meet the conditions outlined
above. UCu2Si2 is a ferromagnet at 103 K (with an antiferromagnetic transition
right above at 107 K)  while UCu2Ge2 is a ferromagnet at 107 K (with a
controversial antiferromagnetic transition at around 43 K [78, 47]) . UCu2Ge2
and UCu2Si2 are isostructural, crystallizing in the ThCr2Si2 structure . The
ThCr2Si2 structure is shown in Fig. 1-2 on page 18. In Fig. 1-2, one can see
Figure 1-2: The unit cell for the tetragonal ThCr2Si2 crystal structure. The c-axis
is up in this diagram, running parallel to the longest dimension of the unit cell.
The ThCr2Si2 is layered with Th-Si-Cr2-Si-Th planes stacked along the c axis. This
diagram was taken from the paper by Welter et al. .
that for the UCu2(Si, Ge)2 compounds, the U atoms replace the Th atoms, the
Cu atoms (the transition metal, T) replace the Cr atoms, and the Ge atoms
replace the Si atoms for UCu2Ge2. The shading in Fig. 1-2 also highlights the
tetraderal coordination of the Si (or Ge) around the T ion with small T-Si (Ge)
distances: 2.40 A for UCu2Si2 and 2.42 A for UCu2Ge2 [28). The smallest distance
between two neighboring U atoms is well above the Hill limit for a U-compound
(3.5 A): 3.98 A for UCu2Si2 and 4.05 A for UCu2Ge2 . It was decided that the
structural disorder could be intentionally created by doping Ge onto the Si site in
UCu2Si2 and that 9 compounds with differing Ge concentration levels (x = 0, 0.2,
0.4, 0.6, 1, 1.4, 1.6, 1.8, and 2) would be arc-melted to investigate whether or not
nonmonotonic behavior exists for Tc as predicted by Silva Neto and Castro Neto.
The replacement of Si atoms with Ge atoms should not affect the magnetic
sublattice in the UCu2Si2_-Gez compounds. The structural disorder .will occur on
the p orbitals deep within the isovalent Si and Ge atoms. The structural disorder
will affect the conduction band due to the complex p-d hybridization between the
T ion and Si/Ge as seen in Fig. 1-2. Thus, the strong hybridization between the
T ions and the U ions should not be directly affected by this intentional structural
disorder. This is in agreement with the neutron diffraction and magnetization
measurements of Chehnicki et al. that gave no magnetic moment on a copper
ion . In fact, the explanation by Chelmicki et al. for why the T ion does
not carry a magnetic moment originates from the coordination of a T ion by
silicon/germanium atoms. Four silicon atoms are located at the comers of a
flattened tetrahedron around each T ion as shown in the shaded part of Fig. 1-2
on page 18. The short T-Si/Ge distances amount to less than the sum of atomic
radii of T and Si/Ge . Thus, the overlap (or hybridization) of electronic shells
and electron density transfer from the 3p shell of the silicon/germanium to the 3d
shell of the Cu ion in UCu2Si2-zGe, compounds is what probably vanquishes the
magnetic moment on the Cu ion. M6ssbauer studies on isostructural NpFe2Si2 
and REFe2Si2 (where RE is a rare earth element) intermetallics [51, 108] show that
the T ion does not carry a magnetic moment. The information by Chehnicki et
al. justifies that the Si/Ge exchange creating sublattice disorder will not affect the
long-range magnetic ordering of uranium moments in UCu2Si2-sGe2.
The effective magnetic moments and the isotropic RKKY mechanism de-
termined by Chehnicki et al. were used by Dr.s Silva Neto and Castro Neto to
calculate the nonmonotonic dependence of Tc . The effective magnetic mo-
ment, pieff., for UCu2Si2 was determined to be 3.58 Bohr magnetons (~B) from the
inverse susceptibility versus temperature curve using the Curie-Weiss law . The
3.58/B value is the exact value calculated for a free ion configuration of U4+ as-
suming 3I-4 as a ground state. The Peff. value for the U ion in UCu2Ge2 is 2.40pJB.
Possible reasons for why the effective moment is lower in UCu2Ge2 are crystal
field effects or magnetic moment compensation . The magnetic interactions
of the U ions may be explained by an isotropic RKKY model, in agreement with
neutron diffraction and magnetization data . This model was also successful
in explaining the magnetic interactions in isostructural UPd2Si2, URh2Si%, and
UPd2Ge2 . The details of using this isotropic RKKY mechanism will be
discussed in the "Theory" chapter of this dissertation.
One important point that cannot be emphasized enough concerns the Si/Ge
sublattice disorder that is intentionally created and differs from other alloying
experiments on the UCu2Si2 and UCu2Ge2 compounds. Since Si and Ge are
isovalent, the carrier density is not changed. In contrast, the carrier density
changes in the UNi2-_CuaGe2 system along with the magnetic sublattice since
the hybridization between the U and Cu ions is altered by the substitution of the
Ni atoms. Also studies done on the UNi2-a.CuzGe2 system [82, 79] show that the
ferromagnetic phase only exists for 1.20 < x < 2 in the UNi2-aCu3Ge2 magnetic
phase diagram. In fact, for x > 0.75 (i.e., the samples closest to pure UCu2Ge2),
there is a ferromagnetic-to-commensurate crossover. For example, for x = 0.95,
a To value of 110 K was observed in ac-susceptibility measurements while at
94 K, the sample underwent another transition to an antiferromagnetic phase .
Kuznietz et al. even state that the transition metal sublattices determine the
type of ordering on the uranium sublattice in the UM2X2 (where M = Co, Ni,
Cu and X = Si, Ge) systems . For the UCu2Si2 side of the phase diagram,
neutron-diffraction results by Kuznietz et al. upon UNio.aoCul.moSi2 show that
, 93% of the sample volume orders antiferromagnetically while the other 7%
orders ferromagnetically . It should be clear from the examples above that
it would be very difficult to individually distinguish the effects on Tc between
the disorder acting on the transition metal's conduction electrons and disorder
altering the magnetic sublattice. Thus, familiarity and acknowledgment concerning
the tremendous amount of work on the magnetism of UCu2Si2 [63, 128, 96]
and UCu2Ge2 [23, 102, 45, 148] is revealed, e.g., UCo2_-CuzGe2 [80, 41] and
Ul-aYaCu2Si2 [64, 62]. However, despite all this research, this UCu2Si2-aGez
is unique because it is the first project to try and isolate the effects of disorder
upon Tc by creating structural disorder at the Si/Ge site (i.e., the 4e site of the
ThCr2Si2 crystal structure).
A final factor in determining the Tc enhancement for UCu2Si2-aGea con-
cerns minor changes in the stoichiometry of the UCu2Si2-aGea system. Particular
attention to the stoichiometry in the UCu2Si2-aGea compounds is paid in the "Ex-
perimental Techniques" chapter and the "UCu2Si2-zGe, Results and Discussion"
chapter. Kuznietz et al. point out that small deviations from stoichiometry on the
copper sublattice alters the number of conduction electrons . The effect of small
stoichiometry changes upon the magnetic properties is closely monitored since
predictions by Silva Neto and Castro Neto originate from conduction electrons on
the Cu sublattice scattering from the localized U spins  as discussed earlier.
In fact, multiple samples with the same Si/Ge concentration are synthesized in
order to compare the stoichiometry differences to the experimentally determined
The low temperature resistivity on the UCu2Si2-_Gea compounds is also
measured. Low temperature resistivity predictions for the scattering of conduction
electrons from localized spin fluctuations have the resistivity varying as T2 as
observed in dilute Pd-Ni alloys  and dilute Ir-Fe alloys . This Fermi-liquid
like resistivity behavior is expected for the UCu2Si2-_Gea compounds. Also, the
resistivity measurements are expected to provide insight into the type of dissipation
occurring from the scatter of the conduction electrons. The two main sources
of dissipation are Landau damping for clean magnets and electronic diffusion
for structurally disordered magnets . The electrons in the Landau damping
case have a longer mean free path than the electrons in the diffusive case. Thus,
the resistivity for the Landau damping electrons (i.e., the ballistic electrons) is
expected to be smaller than the diffusive electrons.
2.1 Non-Fermi-Liquid Theory
The discovery of the electron in 1897 by J.J. Thomson has led to many theo-
ries concerning the behavior of electrons in metals. In 1900, P. Drude formulated
his theory of metallic conduction by using a slightly modified method of the kinetic
theory of a neutral dilute gas. Unfortunately, the Drude model was not accurate in
predicting many physical properties, such as the specific heat of a metal. A quarter
of a century passed until Sommerfeld's model solved the problem by using the
Pauli exclusion principle's requirement that the electronic velocity distribution is
the quantum Fermi-Dirac distribution instead of the classical Maxwell-Boltzmann
distribution. After the passing of another quarter century or so, L.D. Landau intro-
duced his Fermi-liquid theory that was able to explain many physical properties in
metals such as Cu and Al. All of the previous models made use of an independent
electron approximation or free electron model.
As one comes to present day, complex metals have Been discovered where
at low temperatures, the electron-electron interactions cannot be simply ignored
or slightly modified with respect to the free electron case. The strong electronic
correlations lead to non-Fermi-liquid behavior. In fact, non-Fermi-liquid behavior
has been observed in metals with disorder, such as UCus-a.Pda . The theory will
begin with some of Landau's arguments for Fermi-liquid theory and then explore
some of the theories characterizing non-Fermi-liquid behavior.
In 1957, L.D. Landau proposed his Fermi-liquid theory. The original intent
of the theory was to explain the liquid state of the isotope of helium of mass
number 3 . However, Fermi-liquid theory is now being applied to the theory of
electron-electron interactions in metals.
Landau's Fermi-liquid theory has two main points. First, the electrons that
are within kBT of the Fermi energy and that do have interactions with each other
do not ruin the success of the independent electron picture in explaining low-
energy metallic properties. Secondly, single electrons are not just being considered
anymore, but rather quasiparticles (or quasielectrons). These quasiparticles share
many of the properties of non-interacting electrons, but quasiparticles are like
electrons that have been perturbed from their non-interacting state by means of
Instead of going through a quantitative analysis as Landau did to obtain
respective thermodynamic and transport properties [83, 84, 85] of a Fermi liquid,
a qualitative analysis is more appropriate for this dissertation. The quasiparticles
that are of most importance are those that are within kBT of the Fermi surface
as stated above. Most quasiparticles in metals are so far buried down below
the Fermi surface that they are unable to obtain the required energy needed to
reach an unoccupied quantum state. Thus, only quasiparticles within kBT of the
Fermi surface can contribute kB to the specific heat and the specific heat grows
linearly with temperature. Likewise, only quasiparticles within IBB (where B is
an external magnetic field) of the Fermi surface can magnetize with a magnetic
moment proportional to pAB leading to a temperature independent magnetic
susceptibility . If the temperature is above absolute zero, then some energy
levels above the Fermi energy will be occupied within a range of kBT of the Fermi
surface. Therefore, the scattering rate for the quasiparticles near the Fermi surface
is proportional to T2. Since the resistivity is proportional to the scattering rate, the
low temperature resistivity goes like T2.
In 1991, Seaman et al. discovered low-temperature measurements of specific
heat, magnetic susceptibility, and electrical resistivity on the Yi-.,UPd3 system
that were contradictory to Landau's Fermi-liquid theory . Specifically, the
compound Yo.8Uo.2Pd3 displayed a linear low temperature behavior in its resistivity
and a logarithmic relationship in its low temperature heat capacity for over a
decade of temperature in both. To date, over fifty systems have been discovered
that do not obey Landau's Fermi liquid theory .
Since the discovery of Yo.8Uo.2Pd3 displaying this non-Fermi-liquid (NFL)
behavior, there has been considerable interest in explaining this NFL behavior
by condensed matter theoreticians. The various theories are segregated into three
general categories: 1) multichannel Kondo models; 2) models where the magnetic
phase boundary lies near 0 K (i.e., the quantum critical point); and 3) models
based on disorder. A synopsis of the various theories will be given while particular
attention will be paid to models that are applicable to UCus5-Niz.
126.96.36.199 Multichannel Kondo Model
The multichannel Kondo effect is based upon single-impurity physics. The
multichannel Kondo effect is described by a quantum impurity spin S that is
coupled antiferromagnetically to n degenerate channels of spin- conduction
electrons . The so-called Kondo lattice Hamiltonian is derived from the Anderson
Hamiltonian to describe the single-impurity multichannel Kondo model :
HK = Ek kamW + J E S a 'teakme, (2.1)
where ek is the conduction-electron dispersion relation, at and am, are creation
and annihilation operators on electrons with momentum k and spin projection
m, J is the antiferromagnetic coupling constant between the localized impurity
spins and the conduction electrons, S is the localized spin impurity, and o,,' are
the Pauli spin matrices. Three distinct cases arrive from the above Hamiltonian
between the n channels (or flavors) of the conduction electrons and the impurity
spin S :
1. If n = 2S, the conduction-electron channels fully screen the impurity
spin channels and a singlet ground state arises. This leads to Fermi-liquid
behavior as described above.
2. If n < 2S, the conduction-electron channels only partially screen the impurity
spin channels and no singlet ground state exists. The result is a new effective
Kondo effect with a net spin S' = S n/2 . The ground state is Fermi-
liquid-like and mirrors the model of Coqblin and Schrieffer  (n = 1, S >
3. If n > 2S, the local spin is "overcompensated" yielding a critical ground
state with a non-Fermi-liquid excitation spectrum . The result of this
overcompensation is power-law or logarithmic behavior in measured physical
quantities like magnetization, resistivity, and specific heat as the temperature
and external field approach zero.
The third case is particularly interesting since the low temperature dependence
of the magnetic susceptibility, specific heat, and resistivity depend upon the
number of channels of conduction electrons, n, and the impurity spin S. For
example, if n = 2 ("two-channel") and S = 1/2, then the low temperature, zero-
field magnetic susceptibility and specific heat divided by temperature, C/T, go
approximately like log (T/TK) . Such logarithmic temperature dependence has
been observed in systems such as Yi-.UPd3  and Thl-.UzPd2Al3 [34, 38].
Also, the low temperature resistivity at zero-field behaves like Po AVT .
Unfortunately, this low-temperature resistivity has not been observed in many NFL
systems. The reason may be that the low-temperature resistivity predictions were
made in the dilute impurity limit while the NFL behavior in systems crops up more
in the concentrated limit.
Cox and Zawadowski  have addressed this concentrated limit. It is a special
case of the multichannel Kondo effect that has a quadrupolar origin. The NFL
behavior in this model results from an effective exchange interaction between
pseudospins and the electric quadrupole moment of the groundstate . However,
no system has met the strict requirements required by this quadrupolar Kondo
The multichannel Kondo model portrays NFL behavior as single-ion in
nature . This means that each impurity moment is treated independently and
concentrated systems might not be of single-ion character since no interaction
between the impurities is taken into effect.
188.8.131.52 Quantum Critical Point
The second general theoretical category describing non-Fermi-liquid behavior
is quantum critical point models. The quantum critical point theories arise from
the critical phenomena that occur at or near a zero temperature phase transition.
Classically, a phase transition would occur at a nonzero temperature and temper-
ature would be the control parameter. The temperature parameter would control
the thermal fluctuations and if the thermal fluctuations have characteristic energies
(hw*, where w* is the frequency associated with the fluctuations) much less than
kBTc (where Tc is the critical temperature of a phase transition), then the fluc-
tuations may be described by classical statistics . However, in 1976, Hertz 
considered the case when the critical temperature is at T = 0. The fluctuations
would have zero thermal energies. Thus, quantum mechanical fluctuations would
arise that could not be controlled by temperature. These fluctuations may be con-
trolled by chemical substitution, external pressure, or magnetic field. An example
of pressure control is CePd2Si2 where a pressure of ~ 28 kbar suppresses antiferro-
magnetism, induces superconductivity at 0.43 K, and shows NFL behavior in the
electrical resistivity (p(T) oc T12) . The idea is that one of the aforementioned
control parameters "tune" a system from an ordered ground state to a non-ordered
state crossing a quantum critical point . The assumption is that at low enough
temperatures, the system's behavior will be dictated by quantum effects despite
not being able to measure thermodynamic properties at T = 0 . At these
low temperatures, non-Fermi-liquid behavior appears at or near a quantum critical
point since usually a magnetic phase transition is suppressed. The quantum critical
point theory is relevant to the UCus5-Nix system because doping UCu5 with Ni
suppresses the long range antiferromagnetic order and TN&I approaches zero.
As mentioned above, measurements at T = 0 cannot be performed. Thus,
in order to confirm whether or not the NFL behavior is due to a quantum critical
point, one must analyze scaling behavior  (either for temperature or frequency)
on finite-temperature properties and compare to predictions made by several
different models. A synopsis of each model with key points is provided below.
The first theoretical model was developed by Tsvelik and Reizer . This
theory matched the low-temperature thermodynamics measured in Uo.2Yo.sPd3 and
UCu3.5Pd.s5 [136, 5, 4]. The theoretical model states that the two previous systems
are on the verge of a phase transition that occurs at zero temperature and results
in a domination by collective bosonic modes. The model also matched the correct
scaling analysis of the low-temperature properties :
magnetization: M = ~f(
specific heat : (HT) (0,T)
T T gT( +)
where 7 = 0.25 0.3; # + 7 = 1.2 1.3; (the model predicted a scaling dimension
of 4/3) and f and g are nonsingular scaling functions. Along with the above
scaling analysis, the specific heat has a logarithmic divergence and the dc magnetic
susceptibility has a divergent form (T-7) at low temperatures.
A second theoretical model investigating quantum critical phenomena was
developed by Hertz  and Millis . A major assumption is made to integrate
out the fermions (i.e., the conduction electrons) and thus the problem is reduced
to the study of an effective bosonic theory describing fluctuations of the ordering
field . Results depend crucially upon the value of d + z, where d is the spatial
dimension and z the dynamic exponent (z=2 for the antiferromagnet and z=3
for the ferromagnet) of the T = 0 transition . All of the cases considered
except for the two-dimensional antiferromagnet have an effective dimension, (d+z),
greater than their upper critical dimension . Hyperscaling (which is used to
derive the scaling analysis mentioned in the previous paragraph) has been shown
not to apply to systems above their upper critical dimension [17, 121]. Phase
diagrams with crossover temperature relationships between various regions have
been obtained for different cases (e.g., two spatial dimensions and dimensions larger
than two ). None of the derived relationships for measured properties in this
model are applicable to the UCu5s-Nia system.
A third theoretical model investigating quantum critical phenomena was
developed by Moriya and Takimoto  and it involves spin fluctuations in heavy
electron systems around their antiferromagnetic instability. This idea is particularly
applicable to the UCus5-Nia system since the heavy electron system UCus has
its Niel temperature suppressed by Ni doping. This theory is based on exchange-
enhanced spin fluctuations playing a major role in displaying various anomalous
properties around the magnetic phase boundary. Moriya and Takimoto addressed
the problem of coupling among the different modes of spin fluctuations in heavy
electron systems . Their answer was to take a phenomenological point of view
by using a sum rule for the local spin fluctuations valid in the strong correlation
SLO + SLT = +L2
where SLo is the mean square local amplitude of the zero point, SLT is the mean
square local amplitude of the thermal spin fluctuations, and SL is the amplitude
of the local spin density that takes a constant value. The sum rule is then used to
calculate the reduced inverse staggered susceptibility in the low temperature limit,
yo, for a nearly antiferromagnetic metal:
Yo = 2TAXQ(0)
where TA is a characteristic energy parameter and XQ(O) is the local dynamical
susceptibility for the antiferromagnetic wave vector Q. This expression has the
same form as in the self-consistent renormalization (SCR) theory for weak itinerant
antiferromagnetism . The parameter yo goes to zero as the magnetic instability
is approached; in fact, at the critical boundary, yo equals zero . Thus, the
value of yo gives a prediction for the proximity to the magnetic instability .
The value of yo is obtained by fitting the specific heat and electrical resistivity
Moriya and Takimoto made predictions for the specific heat and electrical
resistivity due to the spin fluctuations. When yo = 0, the specific heat takes a finite
value, 70, and starts to decrease with increasing temperature proportionally to
T1/2. Immediately after the square root behavior, the specific heat shows a negative
logarithmic behavior in a certain range of temperature (over about 60% of a decade
in temperature above the 7o ATO5 behavior ). Interestingly, as one gets
further from the phase boundary (i.e., yo 0), Fermi-liquid behavior arises and the
range of linear specific heat increases with increasing yo. The electrical resistivity
at the critical boundary is proportional to T3/2 at low temperatures and as the
temperature increases, there is a certain range where the resistivity is almost linear
in T. The electrical resistivity is similar to the specific heat in that as one moves
further away from the antiferromagnetic critical boundary, there exists a normal
Fermi-liquid behavior at low temperatures. Physically, the interpretation of the
departure from the T1/2 behavior to the logarithmic dependence in the specific
heat and crossover from the T3/2 behavior to the linear dependence in the electrical
resistivity is that at the lowest temperatures, the coupling among the different
modes of the spin fluctuations is small in magnitude; however, as the temperature
increases, the coupling also increases . Application of Moriya and Takimoto's
theory will be discussed later on.
Several other theoretical models exist describing quantum critical phenomena.
However, in the interest of relevance to this dissertation, one should be directed to
Dr. Stewart's review article  for a complete summary of all theoretical models
pertaining to the quantum critical point.
The third theoretical category describing non-Fermi-liquid behavior concerns
models based on a disorder. These models originate from the multichannel Kondo
model. It was previously discussed that when the conduction-electron channels
are sufficient to compensate the impurity spin, then Fermi-liquid behavior occurs.
TK is the Kondo temperature below which the conduction electrons fully screen
the local impurity spin for S = 1/2 and n = 1 (the number of conduction-electron
bands). The disorder model has been proposed to reduce TK to lower temperatures
for some of the magnetic impurities and thus, some of the long-range magnetic
order survives leading to non-Fermi-liquid behavior. The Kondo temperature, TK,
is given as follows :
kBTK epF exp-1/N(O)J
where EF is the Fermi energy, N(O) is the density of states at the Fermi energy,
and J is the exchange constant between the local moment and conduction-electron.
Mathematically, if disorder could increase N(O), J, or both, then the Kondo
temperature could be lowered below the average TK. Thus, if the temperature were
at average TK, then not all magnetic impurity spins would be compensated for and
the uncompensated spins would lead to NFL behavior.
The first disorder driven model was constructed from the work of Dobrosavl-
jevic et al.  and Bhatt and Fisher . Then, Bernal et al.  used this
"Kondo disorder" model to explain the large inhomogeneous nuclear magnetic
resonance (NMR) linewidths they observed in UCus5_Pda. The large linewidths
reflected a broad distribution of local uranium spin static susceptibilities that were
considered to be due to a probability distribution, P(TK), of Kondo temperatures.
The distribution of Kondo temperatures was assumed to be due to disorder. The
uniform magnetic susceptibility, x(H,T) M(H,T)/H, was thought of as the aver-
age of x(H, T;TK) over P(TK) and fits on x(H,T) were used to obtain parameters
characterizing P(TK). The NMR linewidths, which come from the distribution of
Knight shifts and measure directly the width of the distribution P(X) of X, were
found to agree well with the Kondo disorder model and no further fitting of P(TK)
was required. The temperature and field dependence of the specific heat agreed
well with the Kondo disorder model.
The Kondo disorder model was extended further by Miranda et al. [98, 100,
991, who focused on non-Fermi-liquid behavior due to the interaction between
disorder and strong electron-electron correlations. Their theory sums up as a
small amount of disorder playing a major role in the low-temperature logarithmic
divergence in magnetic susceptibility and specific heat along with the linear
behavior in the low temperature resistivity .
Not only was the multichannel Kondo model mixed with disorder, but
also spin fluctuations near a quantum critical point were mixed with disorder.
Rosch  explained that the reason why the resistivity varies as TI where a is
between 1 and 1.5 for various systems [68, 95, 53] is due to the interplay between
quantum-critical antiferromagnetic spin fluctuations and impurity scattering in
a conventional Fermi-liquid. In other words, the dependence of the exponent de-
pended on sample quality. The disorder in a sample was quantified by taking the
inverse of the residual resistivity ratio: R(T--0)/R(T-+300 K).
The most relevant disorder model to UCu5-sNi, involves a model proposed
by Castro Neto et al. [21, 20]. The model by Castro Neto et al. takes place in a
disordered environment and features the competition between the Kondo effect
and Ruderman-Kittel-Kasuya-Yosida (RKKY) interaction (i.e., magnetic ordering)
as represented by the Hamiltonian in Eq. 2.1. The problem with studying this
competition is the fact that both the RKKY interaction and the Kondo effect have
origins on the same magnetic coupling between spins and electrons . However,
perturbation theory may be used to treat the RKKY interaction since the RKKY
interaction depends on electronic states on the Fermi-surface and states embedded
inside the Fermi-surface while the Kondo effect just affects the Fermi surface. The
RKKY interaction leads to an order of the magnetic moments while the Kondo
coupling leads to the destruction of long-range order in a magnetically ordered
In the model by Castro Neto et al., disorder also affects the magnetic order
in these systems. For a ligand system (e.g., UCus5-Nix, the metallic Cu atoms
are replaced by Ni atoms), if the metallic atoms are replaced with smaller metallic
atoms (as is the case for UCus-5Nis), then the difference in size between the metal-
lic atoms leads to a local lattice contraction which modifies the local hybridization
matrix elements . Large local effects in the system can occur because the local
hybridization matrix elements are exponentially sensitive to the overlap between
different angular momentum orbitals. Also, the local value of the exchange constant
between the conduction electrons and the localized moments, J in Eq. 2.1, changes
due to the change in the local hybridization matrix elements.
Castro Neto et al. disagree with the interpretations of the Doniach argu-
ment  which are based on homogeneous changes in the exchange constants
between conduction electrons and magnetic moments. It is true that when the
local exchange constant, J(i), is below a critical value Jc, the RKKY energy scale
is larger than the Kondo temperature as shown in Fig. 2-1 on page 35. However,
Castro Neto and Jones argue that since disorder is present in the system, the mo-
ment can locally order with its environment as the system is cooled down . The
Doniach argument differs because it is based on a global change in the exchange
constant, J, instead of just a local change. Doniach's picture predicts a change
in the exchange constant over the entire lattice and the ordering temperature in
Fig. 2-1 vanishes at a quantum critical point where the system goes from ordered
state to fully Kondo compensated state . The picture by Castro Neto and Jones
represents a quantum percolation problem where moments are compensated due to
local effects and that a local change in a coupling constant does not immediately
imply a change of the "average coupling" constant. Thus, even if chemical substi-
tution occurs on a non-magnetic site, individual moments will be compensated for
magnetically due to the distribution of exchange constants in the presence of disor-
der . At some critical doping concentration, long-range magnetic order will be
suppressed and the system will enter a paramagnetic phase. However, since Castro
Neto and Jones argue that the situation is a percolation problem, the paramagnetic
phase can still contain clusters of atoms in a relatively ordered state.
The alloying of a metallic atom with a different metallic atom is what leads
to this percolation problem. In this percolation problem, the interest lies in the
Figure 2-1: The Doniach phase diagram taken from Ref. . The thick dashed
line is the Kondo temperature, TK EFe-/N(o)J. The thin dashed line is the
RKKY temperature, TRKKY oc [N(0)J]2. The continuous line that equals zero at Jc
is the ordering temperature, TN.
paramagnetic phase or when p > pc, where p is a percolation parameter (related
to the density of quenched moments) and pc is the percolation threshold of the
lattice. When p > pc, then there are only finite clusters of magnetic moments
(i.e., magnetic atoms) which are more coupled than the average . Castro Neto
and Jones assumed that these clusters do not have strong interactions amongst
themselves and that to a first approximation, they can be thought of as isolated
and permeated by a paramagnetic matrix. Castro Neto and Jones termed the
behavior of a cluster of N atoms in the presence of a paramagnetic environment as
the N-impurity-cluster Kondo effect.
The N-impurity-cluster Kondo effect involves a cluster of N magnetic moments
close to a quantum critical point (due to the presence of disorder). If N is large,
then the magnetic cluster may be considered as a large magnetic grain. The
ground state is either a full ferromagnet or antiferromagnet. The ground state of
the cluster has to be at least doubly degenerate due to time-reversal symmetry
and the cluster can fluctuate quantum mechanically (i.e., tunnel) between the
two degenerate states in the absence of an applied magnetic field . Since
the magnetic clusters can tunnel in the presence of a metallic environment, this
produces dissipation (due to particle-hole excitations in the conduction band ).
Castro Neto and Jones show that the sources for tunneling depend upon
the symmetry of the system. To summarize, if the magnetic cluster has XYZ
symmetry or very low spin isotropy and Nc is the threshold limit of spins in
a given cluster above which the Kondo effect ceases to occur, then NC is the
maximum number of magnetic moments that can still tunnel due to the anisotropy
in the RKKY interaction. If N > Nc, then there is no Kondo effect and the
tunneling ceases along with cluster motion due to freezing. The other case involves
XXZ or Heisenberg symmetry. For this case, the only source of tunneling is the
Kondo effect. This Kondo effect is due to the dissipative dynamics of states .
Dissipation is vital in this case because it allows tunneling to vanish for a finite
number of spins given by Nc.
As mentioned above, tunneling and dissipation are created when a set of
magnetic impurities interacts through a conduction band. Castro Neto and Jones
also show that three different energy scales are created in the mean-field-like
picture: the ordering temperature, the tunneling energy, and the damping energy
. Castro Neto and Jones explored how these different energy scales would affect
the Kondo lattice. If the previous example of a ligand system having its metallic
atom replaced by a smaller atom is used again, then the lattice locally contracts
resulting in local matrix elements that have exponentially large values and the
local exchange parameter may be much larger than the average exchange in the
lattice . The calculations by Castro Neto and Jones reveal that this problem is
equivalent to an anisotropic d+1 classical Ising model with long-range interactions
in the imaginary-time direction, and short-range interactions in the space direction.
The solution to this problem involves extending the quantum droplet model of Thill
and Huse  in the context of insulating magnets. In the paramagnetic phase,
the quantum droplets are the magnetic clusters. A distribution of the energy scales
(or cluster Kondo temperatures) is fixed by the percolation theory mentioned above
and this distribution arises from a distribution of cluster sizes. The behavior of this
statistical problem reveals that as the critical number of spins, Nc, is finite, the
distribution for the energy levels diverges logarithmically; however, as Nc -* oo,
there is a crossover in the problem to a power-law distribution for the energy
levels. Now that the distribution of the energy scales has been obtained, the actual
physical properties of these magnetic clusters were calculated by Castro Neto and
Castro Neto and Jones considered two distinct domains of temperature for the
physical properties: T < T* and T > T* where T* is a crossover temperature. The
crossover temperature values depend upon the source of tunneling. For example,
if the source is RKKY interaction, then the critical number of magnetic clusters
for which tunneling ceases is very large and the crossover temperature is effectively
zero . For the case where the Kondo effect is the source of tunneling, then the
crossover temperature is on the order of magnitude of about 0.5 K for a ligand
The asymptotic behavior of the magnetic susceptibility and specific heat is
considered for temperatures less than the crossover temperature. Interestingly,
there are no analytic expressions for the magnetic susceptibility and specific
heat of a Kondo system . For T < T*, tunneling ceases and the tunnel
splitting is zero. The spin cluster motion is frozen and dissipation dominates the
spin dynamics. This leads to a divergent behavior in the susceptibility at low
(T) K Tln(W/T) (2.2)
where W s EF, the Fermi energy. Similarly, the specific heat diverges for extremely
low temperatures :
(T) oc T (W/T) (2.3)
Unfortunately, the measurement of temperatures below the crossover temperature
is physically impossible except for a few rare earth cases .
The temperature range above the crossover temperature and below the average
RKKY interaction energy gives rise to physical properties that are indicative
of quantum Griffiths-McCoy singularities in a paramagnetic phase . For
temperatures above the crossover temperature, a power-law behavior is present
for the magnetic susceptibility : x(T) oc T-'+A, where A < 1. In the high
temperature limit, the susceptibility goes like normal Curie behavior: x(T) oc 1/T.
Likewise, power-law behavior is present in the specific heat: Cv(T) oc T-'+A, with
A < 1. The lambda values above do not have to be the same, contrary to an earlier
work . Castro Neto and Jones caution that the temperature above the crossover
temperature should not exceed the order of the ordering temperature of the pure
system because the magnetic clusters will decompose.
Castro Neto and Jones also consider physical quantities with the application of
an external magnetic field, H, and in the temperature regime above the crossover
temperature. The response functions are calculated by knowledge of the dynamical
exponent in the Griffiths-McCoy phase. The first physical quantity to be considered
is the magnetization and Castro-Neto and Jones observed that the magnetization
has a scaling form
M(H, T) = f, (, (2.4)
where fA(x,y) is a simple scaling function and w0 is the attempt frequency of the
cluster. The focus will be on the case of wo > T, EH (where EH is the magnetic
energy of the cluster that is proportional to H); thus, the scaling function above
will just be dependent on H/T. If T > EH (i.e., the low-field limit), then Castro
Neto and Jones calculated the scaling function to be fA(x,0) 1 when x -- 0.
The magnetization has a linear dependence with the magnetic field at constant
temperature or the magnetization diverges at low temperatures like TA-1 for
A < 1 at constant magnetic field. If EH > T, then the high-field limit has
the magnetization scaling like H' and the susceptibility (x = dM/dH) goes like
HA-1 [211. The following conclusion may be drawn about the scaling function:
fA(x,0) ; xA-1 asx -+ o0. One important note that will be further explored
in the "UCus-.Ni. Results and Discussion" chapter is that the lambda in the
high-field magnetization should equal the lambda determined from the low-field,
low-temperature magnetic susceptibility.
Castro Neto and Jones also calculate low-temperature limits in the context of
specific heat. Their calculations yield a scaling form
Cv(H, T) = T^ gx (2.5)
where gA(x,y) is a scaling function. Just like the magnetization case, the scal-
ing function will only be dependent upon one variable, gX (x, 0), since it is as-
sumed wo > EH, T. For low fields, the specific-heat coefficient, Uvv(H, T)/T,
diverges at low temperatures like TA-1. The scaling function behavior is then
g9(x, 0) --+ 1 for x --+ 0. In the case of high fields, the dominant behavior is a little
Cvv(H, T) oc 2A/2 exp (2.6)
The scaling behavior is g9(x, 0) --* e-x x2+A/2 asx -- oo. The more complicated
scaling behavior is from a Schottky anomaly due to the high magnetic field .
The scaling behavior for the magnetization and specific heat is summarized in
Fig. 2-2 on page 41 (taken from Ref. ).
In concluding this theory of Griffiths-McCoy singularities in the paramag-
netic region close to the quantum critical point for magnetic orders of U and
Ce intermetallics, Castro Neto and Jones have made predictions for the behav-
ior of the imaginary part of the average frequency dependent susceptibility, the
nonlinear magnetic susceptibility, the Knight shift measurement in NMR, and
neutron scattering . All the behaviors at low temperatures are based upon the
quantum-mechanical response of magnetic clusters. Experiment has shown that
these magnetic clusters may be the source of NFL behavior in Cel-,LaRhIn5 
and Cel-,ThRhSb .
2.2 Curie Temperature Enhancement Theory
The enhancement (i.e., the unexpected nonmonotonicity) of the Curie tem-
perature, Tc, was predicted by Silva Neto and Castro Neto . Silva Neto and
Figure 2-2: The Griffiths phase diagram taken from Ref. . T-H phase diagram
showing the behavior of the scaling functions for magnetization, fx(H/T), and spe-
cific heat, gA(H/T). The T ~ H line is the crossover behavior between the high and
low field limits. r is the average RKKY interaction in the system and above F, the
magnetic cluster does not exist.
Castro Neto believe that their study in the realm of quantum ferromagnets is
applicable to a wide class of systems:
1. heavy-fermion compounds such as URu2-,RezSi2  and Thi-aUaCu2Si2 ;
2. dilute magnetic semiconductors such as Gai_-MnaAs ;
3. ferromagnetic dichalcogenides such as CeTe2 ;
4. manganites such as Lal-aSr.MnO3 ; and
5. two-dimensional electron systems in the quantum Hall regime .
The Curie temperature enhancement theory starts from the premise that local
moments couple directly to an itinerant electron liquid and this premise can be
expressed mathematically by the Hamiltonian 
H = -J Si. S + JK E Si c 0,o'a o, ci,,j + He (2.7)
where J is a Heisenberg exchange between localized spins Si (total spin S), JK is
an exchange coupling between localized spins and conduction electrons, c!,. (ci,,)
is the creation (annihilation) electron operator, V are the Pauli matrices, and He
describes the conduction electrons. Silva Neto and Castro Neto find for the case
of a metallic system described by the Hamiltonian in Eq. 2.7 that dissipation (due
to disorder) results from electrons scattering off localized moments and acting
as a heat bath for the spin dynamics. The two main sources of dissipation are
Landau damping in the case of clean magnets and electronic diffusion in the case of
structurally disordered magnets [49, 44]. The effects of dissipation upon Tc will be
To begin their study of the dissipative problem, Silva Neto and Castro Neto
considered an action of the form 
So(n) =/ dr /ddx MoA(n) orn + 6 (Vn)2 (2.8)
where /3 = 1/T, n is a N-component vector that represents the local magnetization,
pO is the spin-stiffness, and Mo is the magnetization density in the ground state.
The reason for showing this action is that the first term of the expansion inside the
integral is a topological term (the kinematical Berry phase [123, 59]) described by
the vector potential A of a Dirac monopole at the origin of spin space. This term
signals the presence of localized moments . This term is the main difference
between the action of Silva Neto and Castro Neto and the action used by Hertz
in his famous study of itinerant quantum critical phenomena . Hertz did not
include such a topological term. The topological term, along with dissipation, plays
a vital role in the behavior of the Curie temperature .
Silva Neto and Castro Neto determined the effects of dissipation upon the
Curie temperature. The phase diagram of the dissipative problem is determined
by applying the momentum shell renormalization group and large-N analysis
to the partition function of the system . The dissipation depends upon the
conduction electron dynamics as mentioned previously: Landau damping for a
clean system and electronic diffusion for a dirty system. Silva Neto and Castro
Neto show that if the Curie temperature is a function of dissipation, then the Curie
temperature decreases faster for Landau damping than for electronic diffusion.
The physical interpretation is that the ballistic electrons (from Landau damping)
have an infinite mean free path and strongly scatter from the localized magnetic
moments in the "cleaner" system. This strong scattering leads to a large amount
of thermal and quantum fluctuations in the system. In the other case, the diffusive
electrons (from electronic diffusion) have a finite mean free path and do not
increase the thermal and quantum fluctuations as much. Interestingly, Silva Neto
and Castro Neto predict about a 1% increase for the Curie temperature of a dirty
system with a small amount of dissipation compared to the Curie temperature of a
dirty system without dissipation (i.e., no disorder). No increase is expected for the
Curie temperature of a clean system with a small amount of dissipation compared
to a clean system without any dissipation. The increase of the magnetism due to
diffusion appears to be important in the 2D MOSFET problem also . One final
note concerning the Curie temperature enhancement involves the importance of the
topological term in the competition between the dissipative and topological terms.
If a diffusive system has no topological term, then Tc is rapidly suppressed and
a slight increase in Tc is not predicted. To summarize the behavior of the Curie
temperature between the ballistic/diffusive electrons and the existent/non-existent
topological term, a figure taken from Silva Neto and Castro Neto is provided on
page 45 .
Resistivity measurements should provide insight into whether or not a system
is in the ballistic or diffusive case. Silva Neto and Castro Neto state that ballistic
electrons have an infinite mean free path while diffusive electrons have a finite
mean free path. Thus, b > d where t(b,d) represent the (ballistic, diffusive)
electrons respectively. Since the resistivity, p, is inversely proportional to the mean
free path , then one would expect the resistivity for a diffusive system to be
larger than the resistivity for a ballistic system. Thus, not only will magnetic
susceptibility measurements be performed on UCu2Si2-sGe, compounds, but also
In summary, disorder along with the competition between the RKKY in-
teraction and the Kondo effect may lead to NFL behavior in UCus5-Nia due to
magnetic clusters. On the other hand, disorder may lead to electronic diffusion in
ferromagnetic UCu2Si2-_Ge: and possible Curie temperature enhancement.
0.95 .. :..?'^,, "-
0.1 02 03 0.4 05 0.6 0.7 0.8 0.9
Figure 2-3: Curie temperature predictions by Silva Neto and Castro Neto. The
Curie temperature Tc(qo)/Tc(m0 = 0) as a function of dissipation, N. S is the to-
tal spin. If 6 = 1, then the dissipation is in the ballistic (clean) regime while 6 = 2
gives the diffusive (dirty) regime. The bare topological constant is represented by
Co and as Co -- 0, then the topological term disappears.
All alloys in this dissertation were created by an arc-melting technique. In or-
der to accurately categorize a system or compound, several experimental techniques
must be employed. The first experimental technique is x-ray diffraction which
confirms whether or not the compound formed into the correct crystal structure.
The x-ray pattern is also used to determine the lattice parameter values of the
fundamental "building block" of the compound, the unit cell. After confirmation
of the correct crystal structure, many techniques are used to determine the correct
thermodynamic and transport properties of the particular sample. These tech-
niques (in no particular order) include direct current (dc) magnetic susceptibility,
alternating current (ac) magnetic susceptibility, dc resistivity, and heat capacity
measurements. All techniques and measurements previously mentioned will be
described in further detail below.
The UCu5-.sNia and UCu2Si2-_Gex systems had their own unique preparation
procedures. The general idea behind the arc-melting process will be discussed first,
followed by the preparation of the UCus-5Ni, system and UCu2Si2-.Gea.
The arc-melting process is based upon a large amount of current (with low
voltage) passing through a tungsten electrode and producing an electric arc in Ar
gas that runs from the tungsten tip down to a water cooled copper hearth inside a
vacuum-tight chamber. Arc-melting is ideal for melting constituent elements with
high melting points together. This dc apparatus melts the elements together in a
zirconium-gettered argon atmosphere. The arc strikes the zirconium first because
zirconium has a high absorption rate for certain residual gas components like
oxygen. The argon atmosphere is used because argon is an inert gas and the argon
helps raise the pressure inside the vacuum chamber so that the electric arc may
be struck. Whenever melting elements together with significantly different melting
points, the general principle is to melt the element with the lower melting point
first and then draw the higher melting point elements into the molten element. In
this respect, the mass loss of the lower melting point element is minimized. This
arc-melting process is very similar to tungsten inert gas (TIG) welding.
The arc-melting of the UCus5-Nix involves the melting of three constituent
elements. The uranium and nickel have similar melting points and vapor pressures.
The copper has the lowest melting point and the highest vapor pressure. Thus,
excess copper (~ 0.5% of the total Cu mass) was added to each UCus5-Nia
sample to account for the copper loss. Each UCu5s-Nia sample was arc-melted
three to four times. After each melt, the sample bead was flipped over to insure
The effort involved in the arc-melting of the UCu2Si2-_Ge_ compounds was
a bit more intensive. The dilemma in arc-melting the UCu2Si2-.Ge, compounds
came from the explosive nature of melting two semiconductors together, Si and
Ge, along with the electric arc thermally shocking the formed UCu2Si2-_Gez
lattice. Two techniques were employed to solve these problems. First, the U, Si,
and Ge constituent elements were melted together first and then subsequently
melted together a minimum of at least six times. Second, the copper element was
then melted into the U, Si, and Ge bead. All elements were melted together a
minimum of six times to insure homogeneity. The second technique was employed
on all arc-melts after the initial one. Instead of moving the electric arc directly
from the zirconium bead to the UCu2Si2-,Ge, bead (this tended to blow apart
the UCu2Si2-zGez bead), the electric arc was moved very slowly towards the
UCu2Si2-zGe, bead. Thus, the arc's convection and radiant heat would melt the
UCu2Si2-aGe, bead and, before the arc was directly over the UCu2Si2-zGe, bead,
the bead would be molten. These techniques were used in the production of all
Excess mass amounts of Cu, Si, and Ge were added before arc-melting. U
has a very high melting point and low vapor pressure, so the mass loss of U was
assumed to be negligible. Cu and Ge had almost identical melting points and vapor
pressures, so the mass loss of Cu and Ge was assumed to be equal. The melting
point of Si is about 50% higher than the melting points of Cu and Ge; thus, it was
assumed that the mass loss of Si would be about half the mass loss of Cu or Ge.
The assumptions above were taken into account so that the appropriate extra
masses could be added before arc-melting. Since a minimum of 12 arc-melts (as
discussed previously) were performed on each UCu2Si2-..Ge, sample, all melts
would "blow away" approximately 1.5% of the total Ge mass present, 0.50% of the
total Cu mass, and about 5% of the total Si mass for compounds on the Ge rich
side. Arc-melting compounds on the Si rich side would result in a loss of about 2%
of the total Si mass, 5% of the total Ge mass, and 1% of the total Cu mass. Thus,
depending on which UCu2Si2-,Ge. compound was being synthesized, excess Cu,
Ge, and Si was added according to the above conditions.
One final technique that is related to sample preparation and arc-melting
is the process of annealing. The purpose of annealing is to eliminate as much
disorder as possible. All UCus5-Ni_ and UCu2Si2-a.Gez compounds were annealed.
Annealing involves pieces of sample being wrapped in tantalum foil and the
wrapped sample is sealed in a quartz tube under vacuum. The samples are placed
in ovens for an experimentally determined time such that the mass loss remains
acceptably small (< 0.5% of the total mass) while the order of the crystal lattice is
improved. The order in the lattice may be checked by x-ray diffraction.
3.2 X-Ray Diffraction/Lattice Parameter Determination from X-Ray Diffraction
Lattice parameter values can play an important part in determining the site
occupation of atoms in a crystal structure along with the lattice values telling
the story of whether or not the lattice expands or contracts upon the doping of
a certain element. For example, the UCus5-Pds system, which crystallizes in the
face-centered cubic (fcc) AuBe5 structure, shows that Pd substitution initially goes
onto the larger Be I site until about x = 0.8 and for x > 0.8, the Pd atoms also
start to occupy the smaller Be II sites also. This was seen not only in muon spin
rotation (pSR)  and extended x-ray-absorption fine-structure (EXAFS) 
measurements, but also by a change in slope of the lattice parameter values at
x m 0.8 .
The present study is similar to those on the UCus-5Pdz system. Instead of
doping larger (relative to the Cu atom) Pd atoms on the Cu sites, smaller Ni atoms
will be doped on the Cu sites. The knowledge of the lattice parameter values at
different Ni concentrations is imperative to gain insight on which copper site the Ni
atoms are found (i.e., the Be I or Be II site).
Lattice parameters are calculated from x-ray diffraction patterns. X-ray
diffraction was run on a Phillips XRD 3720 machine at the Major Analytical In-
strumentation Center (MAIC) on the University of Florida campus. The complete
x-ray diffraction process is described elsewhere .
The particular machine on which the x-rays were generated used two different
wavelengths of radiation [copper Kal = 1.54056 Angstrom (A) and copper
Ka2 = 1.54439 Angstrom (A)]. Also, the intensity of the a, beam was twice
as great as the a2 beam. This made resolving two distinct peaks at a particular
crystallographic plane sometimes quite difficult since the ai peak would conceal the
The typical x-ray pattern was generated with a 20 (degrees) domain from
200 to 1200 in 0.020 steps where 20 is the total angle by which the incident X-
radiation beam is deflected by a lattice plane. Almost all measurements had a one
second step time where the detector on the x-ray machine moved continuously
through a 0.020 interval in one second. The reason for such a long total scan time
(~- 90 minutes) was to obtain good resolution around the high angle x-ray peaks
The Phillips XRD 3720 machine operation was controlled with a computer
interface. Once the x-ray process was complete, the computer would generate a
list of peaks from the x-ray pattern. The computer found the peaks at a respective
20 value by finding the minimum of the 2nd derivative of the peak (i.e., where the
change in the slope on the x-ray pattern is least).
Once the peaks along with their particular 20 values are found via the com-
puter, lattice parameter determination may begin in earnest. The easiest way
to explain the lattice parameter calculation  is by using an example. The
UCu5s-Nix system will be the example. Since the UCus-5 Ni, system forms in
the cubic AuBe5 crystal structure, this system has very high symmetry and the
perpendicular distance between two parallel lattice planes, d, may be written in
terms of the Miller indices (hkl) of the plane :
d(hkl) = h2k2 12 (3.1)
h" + k2 +12
where lattice parameter value, a, is the side of the cubic crystal one is calculating
and the Miller indices are defined as the coordinates of the shortest reciprocal
lattice vector normal to a lattice plane, with respect to a specified set of primitive
reciprocal lattice vectors (e.g., a plane with Miller indices h,k,l is normal to the
reciprocal lattice vector hb1 + kb2 + lbs) .
The inter-plane distance, d, is related to half the total angle by which the
incident X-radiation beam is deflected, 0, by the well known Bragg condition :
nA = 2d sin0 (3.2)
where n is a positive integer and A is the wavelength of the incident radiation
beam. If one combines Eq. 3.1 and Eq. 3.2 and assumes that n = 1, then 'a' may
be written in terms of A, 0, and (hkl):
A h2 + k2 + 12
a = 2sin (3.3)
The values obtained from Eq. 3.3 will be the ordinate values for the graphical
extrapolation method. The reason 'a' cannot be obtained directly from Eq. 3.3 is
that Eq. 3.3 would require 0 to equal 900 and 20 would be 180, which is physically
impossible for the XRD machine. If 20 were 1800, this would mean that the
source and detector of the XRD machine were located at the same position. The
detector and source at the same position on the XRD machine is not possible. The
maximum achievable 20 angle by the XRD machine is 1400. This justifies why an
extrapolation method is needed to obtain 'a'.
The abscissa values used in the graphical extrapolation method take into
account some possible sources of error. The function to be used for the x-axis
values is advocated by Nelson and Riley : 1 (cos2 0/0 + cos2 0/ sin 0). This
functional form takes into account possible errors caused by absorption and
divergence of the x-ray beam.
Before the 'a' values can be graphed against the functional x-axis form, it has
to be determined what (hkl) values correspond with peaks in the x-ray pattern.
Thus, a particular 0 value may be linked with a set of (hkl) values. Since it is
known that UCus-5Nix crystallizes in the AuBe5 structure , a theoretical
crystal pattern can be calculated with known information [33, 155] such as a lattice
parameter constant (in this case, used the known UCu5 value: 7.043 A ), space
group number (216 for UCu5 structure), and atomic positions in the unit cell. This
information was used by the PowderCell computer program to provide a theoretical
pattern which included peaks with known (hkl) values. A visual inspection between
the theoretical patterns and the experimental patterns of the UCus5-Nix system
could link peaks with particular (hkl) values as shown in Fig. A-1 on page 167 for
UCu4Ni annealed 14 days at 7500C. A list was generated with (hkl) values and
corresponding 20 values.
Once the (hkl) values and corresponding 20 values are obtained, the 'a'
values and x-axis values may be obtained from Eq. 3.3 and the Nelson-Riley error
function respectively. A graph plotting the 'a' values versus the corresponding
error function values for annealed UCu4Ni is shown in Fig. A-2 on page 168. The
values when plotted should show that the extrapolation line has a negative slope.
The reason behind this negative slope is because all the sources of error lead to
high values of 0 and so to low values of the lattice parameters . It should be
stressed here that it is very important to have as many (hkl) values with 200 values
greater than 800 as possible. The importance comes from the fact that 0 values
close to 900 give Nelson-Riley functional values close to zero, insuring that the
extent of extrapolation is not large .
The graphical extrapolation method is an excellent technique for determining
lattice parameter values on a highly symmetric system such as UCus5-Nia. The
error in the lattice parameter values as will be seen later for the unannealed
and short term annealed UCus5_Ni. compounds is on the order of 10-3 A. This
error may be improved upon by running a cubic Silicon standard along with the
respective experimental compound or by running the Silicon standard alone to
determine the offset in the XRD machine. The latter process was done for this
dissertation. The determination of lattice parameter values has been facilitated
greatly by the advent of computers with software such as Jade.
3.3 Magnetic Susceptibility
Magnetic susceptibility measurements were used to measure the low tem-
perature magnetization of the UCus5-Nix and UCu2Si2-_Ge, samples. The
susceptibility measurements were taken with two superconducting quantum in-
terference devices (SQUIDs): an MPMS-5S [can measure with a magnetic field
from 0 to 5 Tesla(T)] and an MPMS XL (measures up to 7 T) machine, both made
by Quantum Design. Both machines could take magnetization measurements in
the temperature range from 2 K to 300 K. The MPMS XL machine only had the
capability to measure dc magnetic susceptibility while the MPMS-5S machine
could measure both ac and dc magnetic susceptibility. Alternating current and dc
magnetic susceptibility measurements will be discussed below.
3.3.1 DC Magnetic Susceptibility
Direct current magnetic susceptibilities are usually made in 1 kiloGauss (kG)
magnetic fields. If the signal of the sample is comparable to the signal of the
addenda (i.e., the plastic straw that holds the sample), then a 1 T magnetic field
may be used so that the sample's signal is dominant over all other signals.
Direct current magnetic susceptibility measurements are based upon the
principles of Lenz's Law. The sample is magnetized by a 1 kG magnetic field and
the magnetic moment as a function of temperature of the sample is measured .
The magnetic susceptibility is determined by the following ratio: x = M/H. The
magnetic moment is measured by induction techniques. The inductive measure-
ments are done by moving the sample relative to a set of superconducting pickup
coils and the SQUID instruments measure the current induced in superconducting
pickup coils. The sample is typically moved 4 cm through the superconducting
pickup coils while 48 data points are taken during the sample's movement to pro-
duce the magnetization curves. The sample is moved through the coils four times
and the average of the four measurements is reported for a particular temperature.
Not only is the dc magnetization measured at various temperatures in a
constant magnetic field, but also the dc magnetization is measured in different
magnetic fields at a constant temperature. The magnetization versus field curves
are typically generated at the lowest possible temperature (2 K). The MPMS XL is
a particularly good machine for this type of measurement since it has a secondary
impedance line that can maintain low temperatures continuously over an extended
period of time (on the order of about 12 hours).
3.3.2 AC Magnetic Susceptibility
Unlike dc magnetic susceptibility measurements where the sample moment is
constant during measurement time, ac magnetic susceptibility measurements for
the MPMS-5S machine use a small ac drive magnetic field that is superimposed
on the dc magnetic field, causing a time-dependent moment in the sample. Alter-
nating current measurements do not require sample motion (as for the dc case)
since the field of the time-dependent moment induces a current in the supercon-
ducting pickup coils. The measurements are usually made in a narrow frequency
band, the fundamental frequency of the ac drive magnetic field. The ac suscepti-
bility measurements in this dissertation were made at three different frequencies:
9.5 Hertz (Hz), 95 Hz, and 950 Hz. These three frequencies stayed within the
frequency range of the MPMS-5S machine and avoided any integer multiples of the
frequency of a common electrical outlet (60 Hz).
Alternating current magnetic susceptibility measurements yield two quantities:
the magnitude of the susceptibility, X, and the phase shift, 0 (relative to the ac
drive magnetic field). This phase shift comes from the fact that the magnetiza-
tion of the sample may lag behind the ac drive magnetic field . In terms of
complex notation, the ac susceptibility measurements provide an in-phase, or real,
component X' and an out-of-phase, or imaginary, component X". The ac magnetic
susceptibility measurements on certain UCu5-aNix samples were used to determine
the antiferromagnetic phase transition temperature, TN&d, for each compound.
A peak in the real component of the ac magnetic susceptibility indicated the
antiferromagnetic phase transition.
The remaining two experimental techniques, dc resistivity and heat capacity,
are primarily measured in a temperature range from 300 milliKelvin (mK) to about
10 K. In order to achieve such low temperatures, one must make use of liquid
nitrogen and liquid helium.
The probe that is used to measure either low temperature resistivity or specific
heat is first cooled down to the boiling point of liquid nitrogen (, 77.4 K ).
The probe is then quickly lifted out of the liquid nitrogen and placed in a Dewar, a
vacuum insulated flask. The Dewar is then filled with liquid helium, which boils at
4.2 K. After a couple of hours, the inside of the probe is cooled down to 4.2 K.
Helium 4 can be cooled to below its boiling temperature by reducing the
pressure inside the Dewar to below atmospheric pressure. In fact, using the
combination of a large vacuum pump and blower, a pumped bath of liquid helium 4
can cool a probe down to about 1.1 K. If liquid helium 4 is cooled below its
transition temperature, or "lambda point" (, 2.17 K ), then the helium 4 starts
to become a superfluid helium 4 that behaves as if it had no viscosity, i.e., it can
flow through tiny holes. Also, the superfluid helium 4 has no entropy and it flows
into a heated area to cool that area and restore the uniform mixture of normal
and superfluid helium 4. This last physical property of superfluid helium 4 allows
the temperature of the probe to be stable enough such that resistivity or specific
heat may be measured. A full dewar of helium 4 in the lab that has a vacuum
pump/blower attached to it can maintain a low temperature of 1.1 K for about 12
to 16 hours.
In order to achieve a temperature lower than about 1.1 K, helium 3 needs to
be used. Helium 3 boils at 3.2 K [at 1 atmospheric pressure (atm)] . Helium 3
is inserted as a gas into the probe and due to its rarity, helium 3 is trapped inside
the probe so that it may be reused. Once the probe is cooled down to 1.1 K with
the superfluid helium 4, the helium 3 gas is condensed by coming into contact
with the 1.1 K probe and the helium 3 collects in the helium 3 pot. The low
temperatures down around 300 mK are achieved by reducing the vapor pressure
of the collected liquid helium 3 by using the internal sorption pump (i.e., charcoal
on the end of a long rod) . The sorption pump is "turned on" by lowering
the charcoal into the vicinity of the helium 3 pot. This cools down the "warmed"
charcoal to below 20 K since the charcoal was initially at the top of the probe. The
helium 3 liquid is pumped on by the sorption pump while the sample (connected to
the helium 3 pot via a weak thermal link) is cooled down to around 300 mK. The
probe can stay down at such a low temperature for about three to four hours until
all the liquid helium 3 evaporates. The sorption pump is then turned off by raising
the charcoal in the probe so that it is above 20 K (i.e., above the 4He level in the
dewar) which desorbs the helium 3 gas and allows the helium 3 gas condensation
cycle to start over again.
In the above discussion concerning cryogenics, the generic term probe has
been used. In reality, two kinds of probes were used in the lab. One probe had a
helium 4 pot which did not require the entire Dewar to be vacuum pumped upon.
The second type of probe had no helium pot and pumping upon the Dewar was
imperative to achieve a temperature around 1.1 K.
A probe with a helium 4 pot has its cross section inside the vacuum can
shown in Fig. 3-1 on page 58 . The vacuum can is mounted on the flange's
probe with a brass taper joint seal that is lightly coated with silicone high vacuum
grease, preventing the superfluid helium 4 to penetrate. The sample platform at the
bottom is connected to a copper block and a thermally conducting grease is used
so that the temperature gradient between the copper block and sample platform is
minimal. The other side of the copper block is connected to a helium 3 pot with a
brass thermal link and large diameter Cu wires (not shown in Fig. 3-1) run from
the copper block to the helium 3 pot for improved thermal conductivity These
connections allow the sample platform to reach low temperatures of 300 mK once
the sorption pump is turned on by lowering the charcoal inside the pumping line for
the helium 3 pot in Fig. 3-1.
Once the helium 3 cryostat in Fig. 3-1 on page 58 is placed in a Dewar and
allowed to reach normal liquid helium temperature (~ 4.2 K), the helium 4 pot
may be pumped upon, which allows the helium 3 gas to condense. As the probe is
cooling down inside the Dewar, a line connecting the helium 4 pot and the helium 4
bath inside the Dewar is open so that the helium 4 pot may be filled with liquid
helium 4. Once the probe cools down to around 4.2 K, the temperature is lowered
further by closing the capillary linking the helium 4 pot and the helium 4 bath
with a needle valve and then connecting an external vacuum pump to the pumping
line connected to the helium 4 pot. Thus, the liquid helium 4 inside the helium 4
pot is just pumped on instead of the entire helium 4 bath inside the Dewar. The
germanium thermometer close to the sample platform in Fig. 3-1 registers a low
temperature of about 1.6 K after the helium 4 pot has been pumped on for ~ 20-30
The other important aspect to the helium 3 probe concerns the wiring for the
electronics to measure DC resistivity and specific heat. The wires come from the
for 4He Pot
Brass Thermal Link-
-coal sorb trap
.Pumping Line for Vacuum Can
and Tube for Wires
-Copper Heat Sink
4He Pot and 'He Bath
Heat Sinking Pins
-Heat Sinking Copper Block
Figure 3-1: Cross-section of a helium 3 probe. This simplified cross-section con-
tains a helium 4 pot. Such a probe is able to achieve a low temperature around
top of the probe and run down the pumping line for the vacuum can in Fig. 3-1.
The wires from the top of the probe are soldered to the copper block above the
helium 4 pot. The copper block is in temperature equilibrium with the Dewar's
helium 4 bath by the copper heat sink in Fig. 3-1. The wires on the top copper
block in Fig. 3-1 are then usually connected to the copper block right above the
sample platform by more wires wrapped around the helium 3 and helium 4 pots
and secured to these pots with General Electric (GE) varnish 7031 (a good thermal
conductor). The reason for the wires not running from the top of the probe directly
to the copper block above the sample platform concerns the amount of heat that
would be transferred from the top of the probe which is at room temperature. The
intermediate copper block connections allow the heat flow to be minimized.
The actual wires that are used in the helium 3 probe are either #40 gauge
copper or #40 gauge manganin wires. Either type of wire is insulated and requires
friction (from sandpaper or an eraser) to remove the insulation. The difference
between copper and manganin wires is that manganin wire is about 30 times more
electrically resistive than copper wire . The higher electrical resistivity of the
manganin wires is associated with less heat being transferred from the top of the
probe. Thus, manganin wires should be used for low current loads while copper
wires are used for high current loads. The wiring in the probe is composed of pairs
of the same type of wiring twisted around each other so that the electrical noise is
The probe without the helium 4 pot has the exact same configuration as in
Fig. 3-1 except for the helium 4 pot and the pumping line for the liquid helium 4.
The Dewar itself may be thought of as the helium 4 pot for the case of the probe
without the helium 4 pot.
3.6 DC Resistivity
Direct current electrical resistivity is based upon an equation derived from the
Drude model for electrical conduction :
P= T (3.4)
where p is the electrical resistivity that is being solved for, A is the cross sectional
area of the resistivity bar being measured, and L is the distance between the
voltage wires. The electrical resistivity was measured using a standard four wire
technique that is discussed below.
The first part of the four wire technique involves making resistivity bars. The
standard way of making resistivity bars involved cutting the particular sample
with a diamond saw. The diamond saw could cut resistivity bars with rectangular
cross sectional areas that had dimensions on the order of thirty thousandths of an
inch. Another way to produce resistivity bars that avoids possible microcracks in
the bars is the "sucker" method. The sucker method is done while arc-melting.
The idea behind the sucker method is that while the arc has a sample in its
molten state, a pressure difference between the arc-melter chamber and an external
chamber on the sucker apparatus has enough force to push down the molten sample
into a square cross sectional copper area. The copper area is water cooled, so the
molten sample should quench as a square resistivity bar ideally. A sucker produced
resistivity bar on a UCu4Ni sample will be discussed later on in this dissertation.
Once the resistivity bars are made, four platinum wires need to be attached
to each bar. The platinum wires have a diameter of 0.002 inches and a purity of
99.95%. There are two methods to attach these platinum wires to a resistivity
bar. One method is to use EPO-TEK H31LV silver epoxy to "glue" the platinum
wires onto the resistivity bar. The only problem with this method is that extra
resistance is added to the measurement with the resistance of the silver epoxy
and possible oxide barriers on the surface of the sample. The second method of
attachment was used in this dissertation. It involved spot welding the four wires
onto the resistivity bar. Spot welding minimized the contact resistance. The
four wires should be spot welded on the same face of the resistivity bar with two
wires on one end of the bar and the other two wires on the opposite end. The
two outside platinum wires are the current leads while the two inside platinum
wires are the voltage leads. The resistivity bar is then attached to a piece of non-
electrically conducting compensated silicon with GE varnish 7031. The varnish
is a good thermal conductor and a poor electrical conductor. The four platinum
wires are attached to the silicon base with silver epoxy and there is some slack in
the platinum wire between the resistivity bar and silver epoxy to account for the
contraction at low temperatures. The silicon base is attached to a low temperature
probe with a thermally conducting grease such as Apiezon N grease.
The low temperature resistivity measurements gather the resistance of
the sample at a particular temperature. The resistance in Eq. 3.4 is then used
to calculate the resistivity. One can see from Eq. 3.4 that in order for larger
resistance measurements (hence, larger voltage measurements from Ohm's Law),
one should maximize the distance between the voltage leads, L, and minimize the
cross sectional area of the resistivity bar, A. This helps reduce the scatter in the
resistance measurements (another apparatus that would help reduce the scatter is
an ac resistance bridge).
The resistance measurements are automated. An HP 9000/300 series computer
controls the current sources and voltmeters taking the measurements. The scatter
in the resistivity measurements is reduced two ways. One way is to measure the
resistivity in one current direction and then reverse the current to measure the re-
sistivity in the opposite direction. The average of these two absolute values should
eliminate any (e.g., thermoelectric) offset in the voltmeter taking measurements.
A second procedure should eliminate the effect of any drift in the voltmeter. The
computer is programmed with an external and internal loop while taking resistance
measurements. The internal loop reverses the polarity of the current as described
above while the external loop dictates how may times the internal loop should be
performed. The external loop averages the measurements taken by the internal
loop. Most resistivity measurements in this dissertation had an external loop value
of 10 and an internal loop value of 5.
3.7 Specific Heat
Specific heat is the quantity of heat needed to raise a unit mass of sample by
a unit degree of temperature while keeping the property x (in this dissertation,
pressure, p) constant during the rise of temperature:
C = lim. (3.5)
Before 1968, the above definition was used in a technique called the adiabatic
method  to measure specific heat. The adiabatic method added a pulse of
power (dQ) to a sample and the temperature rise (dT) in the sample was noted.
A couple of drawbacks to the adiabatic method are the large sample size needed
to minimize the effects of stray heat leaks and the thermal isolation of the sample
from its surroundings .
In 1968, Sullivan and Seidel published their ac heat capacity technique .
The ac method measures small samples that makes use of a commercially available
lock-in amplifier. The strength of this technique is its ability to detect very small
changes in heat capacity . However, a drawback to this ac temperature
calorimetry is in measuring the absolute value of the specific heat of a sample. This
ac method usually provides only a relative measurement of the specific heat .
The specific heat technique used in this dissertation is the thermal relaxation
method . The thermal relaxation method was constructed from the solution of a
one-dimensional heat-flow equation with appropriate boundary conditions :
P = A + C(T)T-, (3.6)
where P is the power put into the sample, A is the cross sectional area of the wire
linking the sample platform to the copper block, K is the linking wire's thermal
conductivity and the heat flows along the z-axis. The solution for C in Eq. 3.6 is
C = KT1 where T1 is the time constant of
T = To + ATe-1t/' (3.7)
with AT = T To and To is the copper block temperature.
The answer above is interpreted by Fig. 3-2 on page 64. In Fig. 3-2, the
sample along with all the addenda on the sapphire (A1203) platform is heated a
small AT above the copper block temperature To, by means of power, P, flowing
through the platform heater. Once the power is turned off on the heater, the
temperature of the sample and platform decay exponentially as in Eq. 3.7 with
time constant, rl, through the linking wires. The time constant, rI, is determined
using computer analysis of the temperature decay. By knowing the thermal
conductance, K, of the linking wires, one may determine the total heat capacity of
the sample and platform. If the addenda are known as in Fig. 3-2, then one may
determine the heat capacity of the sample by subtracting the addenda from the
total heat capacity. All components of the addenda will be discussed in detail later
on. It should be said that the rise in temperature, AT, should be small enough
so that Tr does not change appreciably between To and To + AT . When
performing low temperature specific heat measurements, AT was usually around
4%. The absolute accuracy of the thermal relaxation method to measure specific
heat is estimated to be 5% , which is verified by measuring the known specific
heat of a high purity standard, e.g., Pt or Au.
To + AT CTotal =
Csample + -
- Ge Chip
- Silver Epoxy
- 1/3 of Au-Cu Wires
- Thermal Grease
Four Au-Cu Wires
Figure 3-2: Schematic diagram outlining the thermal relaxation method. The ther-
mal relaxation method is used for the specific heat measurements. The addenda
is composed of the sapphire, the germanium chip thermometer, EPO-TEK H31LV
silver epoxy, 1/3 of the linking wires, and thermal grease. The linking wires are the
four Au-Cu wires connecting the sample platform to the copper block.
Solder for Au (solid circle)
(44% In, 42% Sn, 14% Cd)
Figure 3-3: An overhead and bottom view of the mounted sample platform. (a.)
The thermal epoxy used was a two component Stycast 1266 epoxy. The copper
block post connects the copper ring to the copper block as shown in Figure 3-1.
The Au-Cu wires connecting the sample platform to the copper ring are Au with
7% Cu to provide the correct magnitude for the thermal link.
(b.) The evaporated 7% Ti and 93% Cr heater is a thin continuous film on the
bottom of the sapphire disk. The Ge chip thermometer is linked to the silver epoxy
with Au-Cu wires.
In Fig. 3-3, two views of the sample platform are shown. The Au-Cu wires
that were used in this dissertation were Au-7% Cu wires. These wires steady the
platform on the sapphire and serve as electrical contacts and thermal links to
the copper ring (hence the copper block). The thermal conductance of the wires
depends on several factors. The first factor has been mentioned indirectly, the
composition of elements making up the wires. For example, Au-1% Cu wires have
a thermal conductance seven times that of Au-7% Cu, while Cu-2% Be has a
thermal conductance five times less than Au-7% Cu. The second factor that varies
the thermal conductance in the wires is the diameter of the wire. The specific
heat measurements in this dissertation used 0.003 inch (in.) diameter Au-7% Cu
wires. The thermal conductance of the 0.003 in. diameter wires is nine times
that of the 0.001 in. diameter wires . The reasoning behind choosing the
0.003 in. diameter wires is that 0.001 in. diameter wires are a little too fragile
while inserting or removing the sample onto or off of the sample platform.
The Au-7% Cu wires are attached to the copper ring in part (a.) of Fig. 3-3
with special solder (44% In, 42% Sn, 14% Cd) on a silver pad. The silver pad
is electrically insulated from the copper ring with a thermal epoxy (in this case,
Stycast 1266 epoxy); however, the thermal epoxy is a good thermal conductor. The
reason for using the special solder is that regular solder (composed of Pb and Sn)
will dissolve the Au-Cu wires.
The sample itself is mounted on the "rough" side of a smaller sapphire disk.
The flat part of a sample (for improved thermal conductivity) is attached to the
sapphire disk with GE 7031 varnish. Then, the "smooth" side of the sapphire with
the sample is mounted on the 3/8 in. sapphire disk in part (a.) of Fig. 3-3 with
thermally conductive grease (in this case, Wakefield grease) . Both sapphire
disks are pressed firmly together to allow for a continuous and even distribution
of thermally conductive grease between the disks. Before each measurement, both
sapphire disks have to have the grease cleaned off of them with trichloroethane.
The mass of all components (sample, GE 7031, sapphire, and grease) has to be
known for accurate specific heat values.
The platform heater in part (b.) of Fig. 3-3 is composed of a thin film
of 7% Ti and 93% Cr on the "rough side" of the 3/8 in. sapphire disk. This
platform heater is created by evaporating a Tio.07Cro.93 sample onto the sapphire
disk with evaporator instrumentation. The resistance of this platform heater is
around 300 Ohms (12)  and the resistance remains fairly constant with varying
The doped germanium chip thermometer (commercially obtained from
Cryocal or Lakeshore Cryotronics) in part (b.) of Fig. 3-3 is used for the sample
platform thermometer. The reason for using doped germanium was that its
resistance is a rapid function of temperature . This sensitivity is vital for
the thermal relaxation method since the increase in the platform temperature at
each data point is not very large (~ 4%). The calibration of the germanium chip
thermometer's resistance versus temperature will be discussed later on.
The key to an accurate measurement of a sample's specific heat is the proper
subtraction of all components (i.e., the addenda) involved in the thermal re-
laxation method. As mentioned previously, some of the addenda are listed in
Fig. 3-2 on page 64. The sapphire disks that serve as the sample platform have
high thermal conductivity (about 1 Watt cm-1 K-1 at T = 4 K) and low spe-
cific heat (2 pJ gram-1 K-1 at T = 2 K and OD w 1035 K) . The platform
heater in part (b.) of Fig. 3-3 that is composed of Cro.07Tio.93, which has a
mass < 0.01 mg, is assumed to have a negligible contribution to the specific heat
of the addenda . The other component on the bottom of the sample platform
next to the platform heater is the germanium resistance thermometer that has a
specific heat value of 0.018 pJ/K at 2 K with a mass of 3.8 mg . The leads
attached by the factory to the germanium resistance thermometer are connected to
the Au-7% Cu wires by means of H31LV silver epoxy. The silver epoxy is a source
of addenda contribution. For example, a mass of 0.63 mg of silver epoxy has a
specific heat value of 0.12 jJ/K at T = 2 K . The GE7031 varnish and Wake-
field grease also add to the addenda. At T = 2 K, the Wakefield grease has a heat
capacity of 0.16 1pJ/K for a mass of 0.12 mg . The final component of the ad-
denda concerns the Au-7% Cu wires. The diameter of the Cu wires not only has to
be known (0.003 in.), but also the length of the four wires (each wire was 0.25 in.
long) as shown in part (a.) of Fig. 3-3 on page 65. Bachmann et al. determined
that 1/3 of the Au-7% Cu wire's heat capacity should be included as addendum .
At T = 2 K, the amount of addenda for 1.5 mg of Au-7% Cu wires is 0.044 AJ/K.
The addenda contribution from the Au-7% Cu wire rises rapidly with temperature
due to the low Debye temperature of the wires (OD ~ 165 K ). The tempera-
ture range (0.3 K to 8 K) at which the specific heat was measured for samples in
this dissertation avoided too large an addenda contribution from the Au-7% Cu
wires. The computer programming language, HPBASIC, used temperature depen-
dent and mass dependent polynomial fits for each addenda contribution mentioned
above in order to provide a realtime subtraction for the heat capacity measured at
a particular temperature. A standard piece of gold was measured using the thermal
relaxation method and it was found that if all the aforementioned addenda was not
subtracted off, then the specific heat of gold at 2 K would be about 10% too high
from its reported standard value (~ 5.001 mJmole-1 K-1 ) .
Once the cryostat with all of its components is assembled, the cryostat parts
mentioned above need to be interfaced with instrumentation (i.e., voltmeters and
current sources) that is controlled by an HP 9000/300 series computer. The block
thermometer (i.e., a resistor) is hooked to a voltmeter and current source so that
resistance (i.e., temperature) values may be obtained. The block heater (usually
made up of a bundle of manganin wire) which controls the temperature of the
block thermometer, the copper ring, and the sample platform is connected to a
current source so that Joule heating may raise the low temperature of the bottom
of the cryostat. The platform heater that provides the small temperature rise to
the sample platform is also connected to a current source. The final component
concerns the germanium chip platform thermometer whose resistance is very
sensitive to small temperature changes. The platform thermometer is half of one
arm of an ac Wheatstone bridge. The other half of the arm is connected to a
variable resistor (i.e., a resistance box). The other arms of the ac Wheatstone
bridge consist of two known resistors (each resistor is 90 ku). The platform
thermometer has a lock-in amplifier connected across it to provide a source of ac
excitation current and to serve as a null detector . The ac excitation current
is limited by the self heating of the platform thermometer. The other reason for
using the lock-in amplifier is to increase the signal-to-noise ratio  by filtering
out noise at other than the measurement frequency (~ 2700 Hz) . The lock-in
amplifier is connected to the HP 9000/300 series computer through an analog-
digital converter. The converter digitizes the signal from the ac Wheatstone bridge
for the computer during specific heat measurements.
Before actual specific heat values are measured, proper calibration of the ger-
manium chip thermometer has to be performed. The ac Wheatstone bridge is used
to measure the platform thermometer resistance at a known block temperature on
the bottom of the cryostat . The reciprocal of the platform temperature, T,
is then fitted to a polynomial as a function of the natural logarithm of platform
thermometer resistance, R:
= Ai (ln R)' (3.8)
where n = 4 was used in the calibration. Equation 3.8 is then used (by the HP
9000/300 series computer) to interpolate the platform thermometer temperature
value using the resistance value experimentally found.
The final calibration concerns measuring the thermal conductivity, K, of the
four Au-7% Cu wires in Fig. 3-3 on page 65. The thermal conductivity, K, may
be written in terms of the power, P, supplied across the platform heater and the
associated temperature rise, AT, in the platform thermometer:
P = KAT. (3.9)
If one starts from a base temperature, To, on the copper ring and the sample
platform in part (a.) of Fig. 3-3, then a small amount of known current is sup-
plied through the platform heater and the corresponding voltage drop across the
heater is measured in order to determine the power, P. The platform thermometer
resistance is determined before and after the power is supplied to the heater so
that To and AT are known from the germanium platform thermometer calibration
discussed previously. The thermal conductance of the Au-Cu wires is then deter-
mined using Eq. 3.9 at a temperature of To + AT/2. The thermal conductance
of the wires divided by temperature, K/T, may be written as a power series of just
temperature or the natural logarithm of temperature
= AiT' (or) = B(ln T)' (3.10)
where n = 4 was usually used by the HP 9000/300 series computer to find an
interpolated conductance value at a particular temperature. The advantage of the
second form in Eq. 3.10 is that the same conductance equation may be used over a
larger temperature range (30 mK to 10 K) while the first form is usually split into
a low temperature equation and a high temperature equation where 1.3 K is the
approximate dividing temperature.
The calibrated thermal conductivity equation is then checked by either
measuring pure palladium, platinum, gold, or copper and comparing to known
standard specific heat values of the aforementioned elements . The standard
sample used in this dissertation that was measured to check the conductance values
was pure platinum (99.9985% purity).
A problem that has to be taken into account when measuring specific heat
values by the thermal relaxation method is known as the "T2 effect." The T2
effect is the thermal lag between the sample and the platform . The T2 effect
comes from large thermal resistance, poor conductivity of the sample, or of the
sample-platform contact and throws the temperature decay off from its exponential
shape . A couple of procedures exist to account for this T2 effect. First, the
temperature decay curve should be composed of the sum of two exponential curves
T(t) = To + A exp (- + Bexp ( (3.11)
where T is the temperature, To is the baseline temperature, t is time, Tr is the time
constant of the sample and addenda, T2 is the second time constant that throws the
exponential decay off, and the coefficients A and B are determined by a graphical
fit using the HP 9000/300 series computer. The second procedure for improving the
T2 effect concerns the lock-in amplifier. The signal integration time should be set
to less than T1/40 for a better T2 correction. Not only is the T2 effect less smeared,
but also a transient, which causes rounding of the exponential decay curve near the
beginning, is eliminated.
When the actual specific heat measurements are taken, a program developed
by Dr. Bohdan Andraka on the HP 9000/300 series computer controls the mea-
surements. Once all known addenda masses are entered into the computer, the
computer first prompts the user for the base temperature, T, at which both the
block thermometer and platform thermometer are stabilized. The base tempera-
ture, T, is determined by varying the resistance in the ac Wheatstone bridge to
match the resistance of the platform thermometer using the lock-in amplifier as
the null detector. The experimentally determined resistance is then entered into
the computer so that the platform thermometer temperature is interpolated from
the calibrated germanium platform thermometer resistance. The computer also
"communicates" with the current source and voltmeter to determine the resistance
of the block thermometer and hence the temperature of the block thermometer
using a calibration curve. The computer then asks for the amount of current to
supply to the platform heater such that the temperature rise, AT, is about 4%
higher than T. Since the time constants that were dealt with in this dissertation
were usually under ten seconds, the new platform temperature usually became
stable in under one minute. The variable resistor was then changed to determine
the new, higher platform temperature. After the higher platform temperature was
determined, the resistance box was manually turned to the average of the two
determined resistance values with the current to the platform heater still on. The
computer then would perform one sweep to determine the temperature decay curve.
This sweep entailed the computer turning off the current to the platform heater
and then signal averaging 4000 points from the output of the lock-in amplifier for
the platform temperature decay curve. The platform temperature decays down to
the base temperature. The 4000 points are plotted on a semilogarithmic graph (the
distance from the decay curve to the base temperature is plotted on the graph)
and a least squares fit is used to determine the time constant value. Multiple
sweeps may be performed automatically by the computer and signal averaged to
improve the signal-to-noise ratio. Also, the computer allows the user the option of
taking into account the r2 effect. After the time constant value is determined, the
computer outputs the correct specific heat value of the sample after subtracting
off the addenda contributions. Then, more current is supplied to the block heater
such that the next specific heat value may be determined at a higher temperature.
A couple of systematic errors one needs to be aware of that might show up as
oscillations in the base temperature concerns changes in block heater current (due
to electrical pickup) and fluctuations in the surrounding liquid helium bath .
Specific heat measurements were also performed in magnetic fields (2 -
13 T) for this dissertation. The thermal conductance of the Au-7% Cu wires
has to be corrected depending upon the amount of magnetic field used. The
thermal conductance of the Au-7% Cu wires decreases approximately linearly
such that at 12 T, the thermal conductance of the Au-7% Cu wires is 3% less
than the 0 T thermal conductance values . Also, calibration curves have
to be reconfigured for the block thermometer. A block thermometer calibration
curve has to be constructed for every magnetic field using a field correction
according to Naughton et al. . Some drawbacks to measuring in magnetic
fields are that the germanium platform thermometer has a large magnetoresistance
p(18 T)/p(0 T) 5.6 at 4.2 K  and a noise problem in magnetic fields .
The ease of specific heat measurements has been facilitated by advancements
in technology. In fact, Quantum Design sells an automated specific heat measure-
ment system that uses thermal relaxation calorimetry and is named the physical
property measurement system (PPMS) . It will be mentioned here that a cou-
ple of UCu2Si2-_zGea samples in this dissertation had their heat capacity values
measured on a PPMS at LANL.
UCu5-_Nix RESULTS AND DISCUSSION
Nine UCus5-Nix compounds were arc-melted and annealed as described in
the "Experimental Techniques" chapter. The nine compositions were x = 0.5,
0.6, 0.75, 0.8, 0.9, 1.0, 1.05, 1.1, and 1.2. The "full spectrum" of measurements
was performed upon the UCu5sNix system. The following sections contain
lattice parameter measurements, dc electrical resistivity measurements, ac and dc
magnetization measurements, and specific heat measurements (in zero magnetic
field and various magnetic fields). The development of the NFL behavior in the
UCus5-Nix system will be investigated through the perspective of the Griffiths-
phase disorder model. All tables in this chapter are found at the end of the chapter
beginning on page 126.
4.1 Lattice Parameter Values for UCus-,Ni,
The lattice parameter values for the nine cubic UCus5-Nix compounds are
found in Table 4-1 on page 126. The unannealed and annealed (for 14 days at
7500C) samples that had their lattice parameter values reported came from the
same arc-melted bead. The arc-melted bead was cut in half with a diamond wheel
saw and half the bead was used as an unannealed sample while the other half was
annealed. All lattice parameters were determined using the graphical extrapolation
method described in the preceding "Experimental Techniques" chapter.
The unannealed and annealed lattice parameter values are shown graphically
in Fig.s 4-1 and 4-2. The unannealed UCu5_-Nix compounds show no signs of
a change in the slope of the lattice parameter value versus Ni concentration as
occurred in UCus-,Pd, . Annealing the UCus5-Ni, compounds also shows
that there is no existence of preferential occupation by Cu or Ni as occurred with
I I I I I I I
6.96 1 1 1 1 1--
0.0 0.2 0.4 0.6 0.8 1.0 1.2
x Ni concentration
Figure 4-1: Lattice parameter values for unannealed UCus_-Ni_ compounds. The
best fit line is for the nine experimentally determined lattice parameters in this
dissertation (represented by hollow squares while the vertical lines are the corre-
sponding error bars). The equation for the best fit line is as follows: a = 7.04521 -
0.06233 x. The UCu5 value (represented by a hollow circle) is taken from litera-
ture  and it was an unannealed sample.
6.99- UCu Ni
6.98- annealed 14 days 750C
0.0 0.2 0.4 0.6 0.8 1.0 1.2
x Ni concentration
Figure 4-2: Lattice parameter values for annealed UCu5s-Nix compounds. The
best fit line is for the nine experimentally determined lattice parameters in this
dissertation (represented by hollow squares while the vertical lines are the corre-
sponding error bars). The equation for the best fit line is as follows: a = 7.03348 -
0.05171 x. The UCu5 value (represented by a hollow circle) is taken from litera-
ture [181 and it was an unannealed sample.
Pd in UCus5-Pd, . In fact, the annealing just sharpens the Vegard's law
behavior (i.e., the linear behavior of the lattice parameters as a function of Ni
doping since all compounds form in the same crystal structure). The difference
in the goodness of fits (i.e., the standard deviations) between the best fit lines for
unannealed (larger standard deviation) and annealed UCu5_-Nia lattice parameters
is greater than a factor of two (the best fit lines are seen in Fig.s 4-1 and 4-2).
Virtually all nine of the experimentally determined lattice parameters for annealed
UCus5-Ni, lie along (within their error bars) a straight line in Fig. 4-2.
Not only do the lattice parameters as a function of annealing show no change
in the order of the UCus_-Nix samples, but also high angle x-ray diffraction lines
provide insight into the order. The (7 3 1) diffraction line occurs at ~ 115.90
on an x-ray diffraction scan as seen in Fig. A-1 on page 167. The line width for
the (7 3 1) peak is the same for the unannealed UCus5-Ni, compounds and the
UCus5-Nia compounds annealed at 7500C for 14 days. Thus, short term annealing
does not increase the order of the UCus5-Nix compounds, contrary to what was
observed for UCus-_Pd, .
The lattice parameters' results also allow one to make a few comments
regarding the occupation of atomic positions in the cubic unit cell. In order to
get a visual representation of the discussion that follows, one may want to refer
back to the conventional AuBe5 crystal structure in Fig. 1-1 on page 3 of the
"Introduction" chapter. The lattice parameter results show that the 4a sites in
Fig. 1-1 are occupied by the U atoms as occurred in UCu5_-Pda. The larger
minority sites inside the unit cell (i.e., the 4c site) are basically occupied by the
larger Cu atoms (relative to the Ni atoms) while the smaller majority sites (the 16e
sites) inside the unit cell have approximately 25% of the Ni concentration located
at one of the four sites in the unit cell. A small percentage of the Ni concentration
does occupy the 4c sites even for the annealed cases. The exact amount would
have to be determined by pSR and EXAFS measurements as was done for the
UCu5sPd. system [15, 93]. This is in stark contrast to UCu5s-Pd- where the
Pd atoms (larger than the Cu atoms) show preferential sublattice ordering as
discussed earlier . To summarize, the 4c sites inside the unit cell for annealed
UCu4Pd have a > 80% occupation by the Pd atoms while the 16e sites have a
less than 5% occupation by Pd . Thus, the partial order present (along with
disorder) in the UCus5-Pdx system is not a concern in the UCus5-Nia system. The
lattice parameters in Fig.s 4-1 and 4-2 reveal this. The UCus5_Nix system has a
higher degree of disorder than the UCus5-Pdx system with the Ni atoms having
a much greater percentage of occupation at the 16e sites as compared to the Pd
atoms (~ 25% versus < 5%). The lattice parameter results also show that clear
distinctions can be made concerning the roles of quantum criticality and disorder in
the NFL behavior of UCus-5Nix without the worry of partial order as occurred in
The lattice parameter values are used to determine the smallest U-U separa-
tion in the lattice. If one goes back to Fig. 1-1 on page 3 in the "Introduction"
chapter, one may determine that the smallest U-U distance is ~ 0.707a (the short-
est distance is a straight line from a corner 4a site to a nearest face-centered 4a site
in Fig. 1-1) where 'a' is the calculated lattice parameter of a particular UCus_5Nix
sample. The smallest lattice parameter value is for unannealed UCu3.8Nil.2 where
a = 6.97107 A in Table 4-1. Therefore, the smallest U-U separation is ~ 4.93 A.
This is not close to the Hill limit of 3.5 A  where if the distance of nearest U-U
separation, du-u, is less than 3.5 A, then the f-electron orbitals of uranium in
a lattice will overlap with those of the neighboring U ions and produce itinerant
f-electron behavior (i.e., non-magnetic ordering). If du-u > 3.5 A, then uranium
f electrons (barring strong hybridization effects) are localized and magnetic .
As one will see in the specific heat section of this chapter, UCus.8Nil.2 does not
display magnetic ordering down to ~ 0.3 K. Thus, there is significant f-electron
hybridization with the d-electron orbitals of Cu and Ni in the UCus_5Nix samples
since none of the samples in this dissertation have du-v < 3.5 A. This is consistent
with the conclusions drawn by Chakravarthy et al. that said hybridization was
responsible for the suppression of the long range magnetic order in UCus5_Nix .
4.2 DC Electrical Resistivity Results for UCus5-Nix
The low temperature resistivity results for the select UCus5-Ni. samples
that were measured are shown in Tables 4-2 and 4-3. All but two samples were
measured down to ~ 0.3 K (using helium 3 gas as described in the "Experimental
Techniques" chapter). The other two samples (UCu4Ni and UCu3.95Nil.05, both
annealed 14 days at 7500C) had their resistivity measured down to 0.060 K using
a dilution refrigerator at the NHMFL in Tallahassee, FL. A graph summarizing
the resistivity for annealed (14 days 750C) UCus-zNix samples in the range
0.9 < x < 1.2 along with unannealed x = 0.75 is shown in Fig. 4-3.
4.2.1 DC Electrical Resistivity Discussion for UCus_-Ni.
The UCus_5Ni. results in Tables 4-2 and 4-3 show large, varying residual
resistivity, Po, values. This was expected since previous literature values for
UCu4Ni were sample dependent with po ranging from 400 to 800 pf cm . A
UCu4Ni resistivity bar was produced using the "sucker" method discussed in the
"Experimental Techniques" chapter to investigate whether or not microcracks in
the sample were the cause for the large po values. An example of microcracks being
the cause of large po values is U2Co2Sn where an arc-melted U2Co2Sn resistivity
bar gave po = 800 jf cm  while a "sucker" produced resistivity bar gave
po = 60 p2 cm . Thus, the sucker method for U2Co2Sn avoided microcracks
and the po values were reduced by a factor of ~ 13. Yet, the quenched UCu4Ni
resistivity bar (i.e., the sucker method) in Table 4-2 (page 127) is only ~ 30%
lower than the arc-melted, unannealed UCu4Ni resistivity bar. Thus, microcracks
annealed 14 days 7500C
Figure 4-3: Low temperature normalized resistivity for UCus-_Ni, samples. The
UCus5_Ni, samples were annealed 14 days at 7500C and have Ni concentrations be-
tween 0.9 and 1.2. Also, the resistivity for unannealed UCu4.25Nio.75 is plotted. The
inset shows the extreme low temperature resistivity values for annealed UCu4Ni,
UCu3.95Ni1.05, and UCu3.gNil.1. The graph legend also applies for the inset. The
solid lines in the inset are high temperature best fit power law lines (that yield
minimum X2 values) showing that the UCu4Ni p data begin to deviate from the
high temperature fit line around 0.992 K while the UCu3.95Ni.0o and UCua3.Nii.1
p data deviate from the lines beginning at 0.834 K and 0.821 K, respectively. The
absolute accuracy of all p data is ~ 5%.
do not appear to be the major cause for the large po values in the UCu5s-Nix
One column in Tables 4-2 and 4-3 that may provide insight into the cause of
the behavior of the resistivity for UCu5_-Nix samples is the RRR (i.e., the residual
resistivity ratio) column. If one compares the RRR values of the UCu5-_Nia
samples (annealed 14 days 7500C) for 0.9 < x < 1.2, one sees a monotonic increase
in RRR as the Ni concentration increases. This trend may be irrelevant since
the RRR values for all short term annealed (i.e., 14 days at 750C) UCus_-Ni.
samples is below 1.0. In Rosch's theoretical work on the interplay of disorder and
spin fluctuations near a QCP , he defined a parameter x that was a measure
of the impurity scattering. 1/x is equivalent to RRR  and this means that
all UCu5_,Nix samples have an x value greater than 1. Rosch stated that x = 0
is a perfectly ordered sample while x > 0.1 is rather disordered . Thus, all
UCus5_Ni. samples that were annealed for 14 days at 750C are significantly
disordered according to their RRR values.
An investigation of the data in Tables 4-2 and 4-3 (on pages 127 and 128)
and the short term annealed data shown graphically in Fig. 4-3 do not indicate
any significant changes brought about by annealing as occurred in UCu4Pd .
Annealing UCu4Pd reduced the residual resistivity by a factor of 2.5  and the
A value (with appropriate units) in the power law form (Po + ATh) was reduced
by a factor of ~ 10 (-6.3 to -0.6 ). Also, a large change in the qualitative
behavior of the resistivity curve was brought about by annealing UCu4Pd. In the
temperature region between 2 K and 8 K, the annealed UCu4Pd resistivity data
was fit with a Fermi-liquid expansion p Pr = AT2 + BT4 while the unannealed
UCu4Pd resistivity data showed NFL behavior with p Pr oc T . The
UCu_5-Nix equivalent to UCu4Pd, UCu4Ni, does not show a significant reduction
in its po value with annealing (only about a 17% decrease). Also, the A values in
the temperature regions of 2 K and 10 K, and 2 K and 12 K for unannealed and
annealed 14 days 750C UCu4Ni do not show a significant difference as occurred in
UCu4Pd (-5.52 for unannealed UCu4Ni and -5.03 for annealed UCu4Ni). It is not
just the UCu4Ni sample that shows this lack of significant change. A perusal of all
UCus5_Nix samples in Tables 4-2 and 4-3 shows that short term annealing causes
no significant changes.
Despite the extreme sample dependence of the UCus5-Ni, system which
makes any trends in po as a function of Ni concentration enigmatic, a few trends
in the short term annealed UCus5_Ni. samples may be observed in Tables 4-
2 and 4-3. First, the absolute values of A for the low temperature fit regions
decrease monotonically as the Ni concentration is increased for the short term
annealed UCu5-_Ni. samples with A = -93.0 IM cm K-0.224 for x = 0.9 and
A = -3.85 p2 cm K-0743 for x = 1.1. The "high" temperature fit forms show
that the resistivity values for 0.9 < x < 1.2 decrease just above a linear rate
with UCU3.gNil.2 decreasing at a rate almost equal to 1 (0.996). Interestingly,
Fig. 4-3 on page 80 shows that unannealed UCu4.25Ni0.75 unannealedd x = 0.75
and x = 0.80 have very similar low temperature curvatures) has upward curvature
over its entire measured temperature range. Yet, this upward curvature present
in unannealed x = 0.75 is not apparent until below 2 K for short term annealed
x = 0.9. This upward curvature is quite apparent for short term annealed x = 1.0
and x = 1.05 at the lowest measurable temperatures. However, the upward
curvature gets less steep with increasing Ni concentration and this is seen in the
inset in Fig. 4-3. The "high" temperature fit lines in the inset do not deviate as
quickly for x = 1.1 and x = 1.05 as occurred for x = 1.0. In fact, this upward
curvature disappears for x = 1.2. The cause of this upward curvature is unknown
and remains an open ended question. This upward curvature in the resistivity
has also been seen in annealed (14 days at 750C) UCu4Pd below 1 K  and
annealed (7 days at 9000C) UCu4Ni .
Figure 4-3, along with Tables 4-2 and 4-3 raise some interesting points
concerning the UCus-aNix resistivity. First, the resistivity of UCu4.25Nio.75 does
not show an antiferromagnetic phase transition at around 1.9 K as was seen in the
specific heat (and will be discussed in the specific heat section below). The most
interesting point concerns a comparison of the resistivity results of two 8 week
850C annealed UCu4Ni samples. The first sample is a UCu4Ni (production date
of 2-27-03 in Table 4-2) resistivity bar that was annealed for 8 weeks at 850C
after the resistivity was measured for the bar when it was unannealed. The po
value decreases by a factor of 20 and the RRR value is ~ 3.18 (very close to the
RRR value of 2.5 for annealed UCu4Pd ). Also, a similar result was obtained
for a UCu3.95Nii.o5 (7-14-03) resistivity bar that had its resistivity measured when
it was an unannealed bar and then was annealed at 850C for 8 weeks. The only
preparation difference between the UCu4Ni entries (in Tables 4-2 and 4-3) that
were annealed for 8 weeks at 850C was that the (2-27-03) entry (RRR ~ 3.18)
had its dimensions sanded really well before annealing so that the spot welded Pt
wires were not incorporated into the sample during annealing. The second sample
in Table 4-3 is a (10-13-03) entry (RRR ~ 0.313) for UCu4Ni that had its bar cut
from the arc-melted bead after the bead had been annealed for 8 weeks at 8500C
with no sanding of the resistivity bar dimensions. These minor differences led to
significant changes in the resistivity behavior.
The sanding of the resistivity bar was not the cause of the significant change
in the resistivity behavior because the long term annealed UCu4Ni sample (prod.
date: 4-13-04 in Table 4-3) was sanded and cleaned right before the Pt wires were
attached while the long term annealed UCu3.95Ni.o05 sample (prod. date: 4-14-04)
was not sanded before attaching Pt leads. Both samples show similar resistivity
behavior and have almost identical RRR values that are below 1. A couple of
possible explanations for the strange resistivity behavior have been formulated.
One explanation concerns the possibility of a copper oxide layer forming during
the 8 week annealing process and that the resistivity of a copper oxide layer
was measured for the resistivity bars that were not sanded. A second possible
explanation comes from Dr. Bohdan Andraka pertaining to small Cu "islands"
possibly existing in the UCus5-Ni, samples and that just pure Cu is present in
the grain boundaries. Thus, the Pt wires were possibly attached to pure Cu with
some impurities mixed in and the resistivity of a sample similar to the classical
examples of a Kondo system (i.e., diluted alloys of Fe and Cr in Cu ) were
measured. Magnetoresistance measurements need to be performed on the same
eight week annealed bars (that showed such odd resistivity behavior) in order to
test the validity of this theory.
Magnetoresistance measurements were taken on the short term (14 days at
7500C) annealed UCu4Ni (2-27-03) and UCu3.95Ni.o5 (7-14-03) samples. The
results are shown in Fig.s B-1 and B-2 of Appendix B. The results in both graphs
were taken by Dr. Jungsoo Kim at the NHMFL. The magnetoresistance low
temperature results show that for UCu3.95Ni.o05, the approximate po values in
zero field and in a 13 T field only have about a 2.6% difference between them
(~ 929 pM cm in 0 T and ~ 905 Ma cm in 13 T). Similarly, the results for UCu4Ni
show that when a 13 T field is applied, the po value is only reduced by about
2.8% (~ 797 ~Q cm in 0 T and ~ 775 Mp cm in 13 T). These magnetoresistance
measurements suggest that the origin of the large po values in the UCus_-Ni,
samples is not of an electronic nature. One last interesting point that is fairly well
seen in both figures of Appendix B is that the upward curvature in the temperature
dependence of the graphs in zero field flattens out with the application of magnetic
field and the temperature dependence remains fairly constant for fields higher than
4 T. Quantitatively, best fit lines (of the power law form po + AT') were fit to each
magnetic field data set in Fig.s B-1 and B-2. The a values obtained for UCu4Ni in
0 T, 4 T, and 13 T fields were 0.432, 0.794, and 0.762 respectively. The a values for
UCu3.95Nil.05 in 0 T, 4 T, and 13 T fields were 0.269, 0.722, and 0.711 respectively.
The physical reason for these magnetoresistance results is at present unexplained.
A final question that needs to be answered concerning these UCus5-Nix
resistivity results is whether or not the short term annealed resistivity provides
insight into the inherent properties of UCus5_Nia, i.e., does quantum criticality or
disorder dominate the behavior in the critical concentration of x = 1.0 (as will also
be discussed in the specific heat section)? The answer is no since the short term
annealing results for UCu4Ni do not vary significantly from the unannealed results
for UCu4Ni as seen in Table 4-2. The resistivity results for UCu4Ni are similar to
the resistivity results for unannealed UCu4Pd . There was a drastic change
in the transport properties, e.g., resistivity, of UCu4Pd after annealing as has
been documented earlier. Thus, the unannealed UCu4Pd gave no insight into the
intrinsic disordered NFL behavior. Likewise, the resistivity for short term annealed
UCu4Ni (or any short term annealed UCus5-Nix sample) may not be connected
to its intrinsic NFL behavior. The resistivity results for the short term annealed
UCu5sNix samples are most likely stemming from an extrinsic effect that arises
from the disorder present in these samples.
4.3 Magnetization Results for UCus_-Nix
The magnetization results for the short term annealed UCus5-Nix samples
around the critical concentration, x = 1.0, are shown in Table 4-4 on page 129 and
Fig. 4-4 on page 87. The numerical information in Table 4-4 is shown graphically
in Fig. 4-4. Figure 4-4 shows the excellent agreement between the fit lines and the
data while the goodness of fits (the standard deviation and the X2 values) in Ta-
ble 4-4 confirm the excellent agreement. The reason for the low field magnetization
being fit to a linear form, M ~ H, and the high field magnetization being fit to a
power law form, M ~ HA, is to investigate whether or not the NFL behavior in the
UCu5-sNia system may be fit to the Griffiths phase disorder model .
4.3.1 Magnetization Discussion for UCu5-_Nix
A couple of trends are readily seen in Fig. 4-4 and Table 4-4. First, the
field range over which the linear magnetization form fits is widened as the Ni
concentration goes from 0.8 to 1.2. The slope of the linear form fit decreases
monotonically with increasing Ni concentration. Likewise, the constant in the front
of the field term for the power law fit decreases monotonically with increasing
Ni concentration. Also, the exponent value, A, in the power law form increases
monotonically for 0.9 < x < 1.2. In fact, the high field magnetization power
law form (for UCu3.sNil.2) is very close to a linear form with A = 0.953 (for
0.9 T < H < 7 T). Also, the unphysical constant offsets in the power law form
(M = A' + B' H^) and linear fit form (M = A + B H) for all Ni concentrations
listed in Table 4-4 are recognized. These offsets also show a trend as they get less
negative with increasing Ni concentration.
The Griffiths phase disorder model of Castro Neto and Jones  makes clear
predictions concerning such magnetization data shown here. As mentioned earlier,
the Griffiths phase disorder model predicts that the magnetization should exhibit
low field behavior (M ~ H) that crosses over to high field behavior (M ~ H^)
at some crossover magnetic field, Hcrossover. In other words, the magnetic spin
clusters should show some saturation behavior at higher magnetic fields when the
Griffiths phase disappears. This is evident in Table 4-4, especially around the
critical concentration of 1.0. The short term annealed UCu3.95Nil.0o sample has
a little higher A value (0.860) than the A values of Ceo.sTh0.2RhSb (0.686 )
and Ceo.o5Lao.95RhIn5 (0.41 ), two known samples that fit predictions made
by the Griffiths phase disorder model. Also, the magnetization results for samples
annealed 14 days 750C
10000 20000 30000 40000 50000 60000 70000
Figure 4-4: Magnetization for UCus-5Nix samples annealed 14 days at 7500C. The
magnetization versus field measurements were taken at a constant temperature
(2 K). The inset shows where the low field linear form (M ~ H) deviates from the
data for each sample while the primary graph shows the excellent fit of the high
field power law form (M ~ HA) to the data.
above the critical concentration (e.g., x = 1.2) do not saturate as quickly as those
samples adjacent to x = 1.0 in the phase diagram. This may suggest that the Ni
concentration range is limited for which the Griffiths phase model applies. The
dc magnetic susceptibility results and the specific heat results are intertwined
to the magnetization results in the Griffiths phase model since the A value from
the magnetization power law form should match the low temperature magnetic
susceptibility exponent, X oc T-'1+, and the exponent in the magnetic field
induced peak form in the specific heat, C/T ~ (H2+A/2/T3-A/2)e-effH/kBT, for
H > Hcrosoer, where HIrosoer is determined from the magnetization data.
Since saturation is present in the magnetization data, it is recognized that
the magnetization and susceptibility of the UCu_5-Nix samples may be fit with
the Kondo disorder model . However, as will be discussed below in the
specific heat section, it is believed that the Kondo disorder model does not apply
to UCu5_-Ni_ in light of the fact that the Kondo impurity model, with its four
fit parameters, was unable to fit the specific heat data in a magnetic field of
isostructural UCus5-Pd.  and in fact does not reproduce the peak in C(H)
observed for UCus-_Ni,.
Finally, the method (as described in the inset of Fig. 4-4) of determining
Hro,,o from the magnetization data seems rather arbitrary. There was no
clear procedure in the literature for describing how Ho,,,e was determined for
Ceo.o5Lao.s9RhIn5 (z 0.8 T ) and Ceo.sTh0.2RhSb ( 1 T ). It was attempted
in this UCus5-Ni. study that if the Griffiths phase disorder model applied to
UCu5-_Ni,, then the rare strongly coupled magnetic clusters (responsible for the
NFL behavior) might possibly show hysteresis at each sample's respective Hcrosaoer-
Thus, the zero field cooled (ZFC) magnetization at T = 2 K was measured from
0 to 7 T and immediately afterwards, the field cooled (FC) magnetization was
measured from 7 to 0 T. This procedure was performed on three of the short term