ABSTRACT
Title of dissertation: ELASTICITY IN FERROMAGNETIC
SHAPE MEMORY ALLOYS
Liyang Dai, Doctor of Philosophy, 2004
Dissertation directed by: Professor Manfred Wuttig
Department of Materials Science and Engineering
Ferromagnetic shape memory alloys (FSMAs) are a new class of active materi-
als, which combine the properties of ferromagnetism with those of a diffusionless,
reversible martensitic transformation. These materials are technologically inter-
esting due to the possibility of inducing large shape changes with an external
applied magnetic field; either inducing the austenite/ martensite transformation
or rearranging the martensitic variant structure with an applied field will induce a
reversible shape change. The dependence of a solid?s elastic properties on temper-
ature in the vicinity of a structural transformation provides insight into the nature
of the transition. Therefore, the elasticity of Ni2MnGa and Fe3Pd were studied.
ThetemperaturedependenceoftheelasticconstantsoftheausteniticNi2MnGa,
Ni0.50Mn0.284Ga0.216 and Ni0.49Mn0.234Ga0.276, were studied by an ultrasonic contin-
uous wave method. Anomalous behavior in austenite was observed, which indicates
a premartensitic transition. The temperature dependence of the elastic constants
in martensitic Ni0.50Mn0.284Ga0.216 indicates a structural phase change from the
tetragonal to a second phase at lower temperature. Modeling this phase change
as a reentrance transition reproduces the major aspects of the temperature depen-
dence of the shear elastic constant, (C11-C12)/2.
The elasticity as a function of temperature and magnetic field of Fe3Pd was
studied as well. An abrupt change of the elastic constants at around 45?C indicates
a possible premartensitic transformation. The magnetic field dependence of elastic
constants also indicates a probably magnetic field induced transition.
ELASTICITY IN FERROMAGNETIC
SHAPE MEMORY ALLOYS
by
Liyang Dai
Dissertation submitted to the Faculty of the Graduate School of the
University of Maryland, College Park in partial fulfillment
of the requirements for the degree of
Doctor of Philosophy
2004
Advisory Committee:
Professor Manfred Wuttig, Chairman/Advisor
Adjunct Professor James R. Cullen
Professor George E. Dieter
Professor Alison Flatau
Dr. Kristl B. Hathaway
Professor Alexander Roytburd
Professor Ichiro Takeuchi
c?Copyright by
Liyang Dai
2004
DEDICATION
for my family.
ii
ACKNOWLEDGEMENTS
I would like to begin my thanking by thesis advisor, Professor Manfred Wut-
tig, who introduced me to the exciting field of magnetism, magnetic materials and
ferromagnetic shape memory alloys. This research work would not have been re-
alized without his academic and financial support over the years of my graduate
study. His enthusiasm and collaborative spirit helped to create a great work en-
vironment, and his amazingly broad knowledge and experience gave my research
both direction and inspiration.
I am most grateful to Professor James Cullen, who advised me with the riches
of his theoretic knowledge. Particular gratitude goes to Dr. Jun Cui, who have
helped me learn various aspects of this fascinating field. I am truly grateful for
these valuable experiences.
Iwouldespeciallyliketothankallthemembers ofmycommitteefortheirefforts
on my behalf: Professor Manfred Wuttig, Professor James Cullen, Professor Ichiro
Takeuchi, Dr. Kristl Hathaway, Professor Alexander Roytburd, Professor Alison
Flatau and Professor George Dieter.
Particular and heartfelt thanks go to those friends and colleagues who selflessly
helped me in these years of study. I apologize in advance to those whose important
contributions have been overlooked here because of my own unfamiliarity with
them or my inability to fully appreciate their significance.
iii
TABLE OF CONTENTS
List of Tables vi
List of Figures vii
1 Introduction 1
1.1 Shape Memory Alloys . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2 Magnetostriction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.3 The Free Energy of Ferromagnetic, Shape Memory and Ferromag-
netic Shape Memory Materials . . . . . . . . . . . . . . . . . . . . . 11
1.4 Ferromagnetic Shape Memory Alloy . . . . . . . . . . . . . . . . . . 14
1.4.1 Magnetic Field Induced Strain . . . . . . . . . . . . . . . . . 15
1.4.2 Ferromagnetic Shape Memory Alloys Ni2MnGa . . . . . . . 18
1.4.3 Ferromagnetic Shape Memory Alloys Fe3Pd . . . . . . . . . 29
2 Elastic Constants And The Experimental Method 36
2.1 Elastic Constants . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
2.2 Experimental Technique . . . . . . . . . . . . . . . . . . . . . . . . 38
2.2.1 Elastic Wave Modes in Crystals . . . . . . . . . . . . . . . . 40
2.2.2 Principle of Ultrasonic Continuous Wave Technique . . . . . 46
2.2.3 Experimental Set Up and Arrangement . . . . . . . . . . . . 48
3 Elasticity of Ni2MnGa 52
3.1 Sample Description . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
3.2 Elasticity of Ni2MnGa in the High Temperature Phase . . . . . . . 55
3.2.1 Elastic Constants of Ni0.50Mn0.284Ga0.216 High Temperature
Phase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
3.2.2 Elastic Constants of Ni0.49Mn0.234Ga0.276 High Temperature
Phase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
3.2.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
3.3 Elasticity of Ni2MnGa Low Temperature Phase . . . . . . . . . . . 71
3.3.1 Elastic Constants of Ni0.50Mn0.284Ga0.216 in Low Tempera-
ture Phase . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
3.3.2 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
iv
3.4 Model for the Elastic Behaviors Near Intermartensitic Transitions . 77
3.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
4 Elasticity of Fe3Pd 87
4.1 Sample Information . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
4.2 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . 88
4.3 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
5 Summary 99
A Publications 102
Bibliography 104
v
LIST OF TABLES
1.1 Overview of Crystalline Metallic Alloys with Magnetostriction ?100 . 10
1.2 Single-crystal samples of Ni-Mn-Ga with different martensitic struc-
ture at room temperature[53] . . . . . . . . . . . . . . . . . . . . . 22
1.3 Crystal Structures and parameters of the phases in Fe-Pd Alloy
System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
1.4 Lattice parameters of the phases in Fe-Pd Alloy System[47] . . . . . 33
2.1 Elastic wave modes in cubic crystal. The wave propagation and
polarization directions are represented by ?W and ?P, respectively.
The L and T denote the longitudinal wave and transverse wave
mode, respectively. . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
2.2 Elastic wave modes in tetragonal crystal. The magnetic field, wave
propagation and polarization directions are represented by ?H, ?W
and ?P, respectively. The L and T denote the longitudinal wave
and transverse wave modes. The a, b and c are three constant
coefficients, which are different combinations of elastic constants as
shown in the table. They equal to each other in cubic crystal which
yields the Table 2.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
vi
LIST OF FIGURES
1.1 Microscopic Diagram of the Shape Memory Effect . . . . . . . . . . 5
1.2 Austenitic cubic lattice and martensitic tetragonal lattice in 2D with
two variants. The twin planes in the reference cubic lattice and in
the deformed tetragonal lattice are shown by the dotted lines. . . . 5
1.3 Schematic representation of the relationship between the magnetiza-
tion and magnetostriction. The dashed line is the magnetostriction.
The black solid line is the magnetization hysteresis. . . . . . . . . 7
1.4 Schematic of the free energy of FSMA at T > T0, T = T0 and
T < T0, where T0 is the phase transition temperature. When
T < T0, the free energy has two energy wells located at ?0 =
?
radicalBig
(C +?M2 +?2/D)/? correspondingtothetwomartensiticvari-
ants, ?(?xx ? ?yy). The direction of the magnetization is directed
along the major or minor axes of the low temperature phase. . . . . 13
1.5 Cartoon of the phase transition process and the twin structure in
the Ferromagnetic Shape Memory alloy. . . . . . . . . . . . . . . . . 16
1.6 Comparing the two different process of magnetization by magnetic
anisotropy. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
1.7 The Heusler L21 lattice structure in Ni2MnGa. . . . . . . . . . . . . 20
1.8 Temperaturedependenceoflow-fieldacmagneticsusceptibilitymea-
sured during cooling (solid line) and heating (dashed line) of the
Ni48.8Mn29.7Ga21.5 alloy. Arrows mark phase transformations. C:
ferromagnetic cubic phase, Or: orthorhombic seven-layered phase,
T: tetragonal nonmodulated phase. The inset shows the scattering
intensity distribution in reciprocal space between (400) and (620)
nodes in the orthorhombic phase of the Ni48.8Mn29.7Ga21.5 alloy. Ar-
rows mark additional peaks connected with seven-layered modula-
tion of the lattice. [53] . . . . . . . . . . . . . . . . . . . . . . . . . 23
1.9 Tetragonal variants structure and associated magnetizations. The
cubic axes are denoted using basis hatwideei, and the martensitic variants
and associated magnetizations by Ei and mi, where the compression
axis along hatwideei uniquely defines each variant. . . . . . . . . . . . . . . 25
vii
1.10 Schematically shows the M-H curves parallel and perpendicular to
the compression axis (right). The fixture (left) was constructed from
a non-magnetic CuBe spring alloy and sized to fit in the cooling
fixture used with the VSM [57] . . . . . . . . . . . . . . . . . . . . 27
1.11 This graph shows the M-H curve along c-axis (easy axis) and a-
axis (hard axis) determined at ?17?C with 1.9 MPa applied stress.
The results show that the easy axis within a single variant is the c-
axis as expected, and the anisotropy is uniaxial in nature. Uniaxial
anisotropy constant KU was calculated from the areas between these
2 curves and found to be KU = 2.4?106ergs/cc.[57] . . . . . . . . 28
1.12 Phase diagram of Fe-Pd alloy system. . . . . . . . . . . . . . . . . . 30
1.13 Schematic drawing of A1, L10 and L12 crystal structures. The radii
of the atoms are exaggerated for clarity. The structure of ?-(Fe,Pd),
?1-FePd, and ?2-FePd3 are A1, L10 and L12, respectively. [12] . . . 30
1.14 Phase transformation temperature as function of palladium concen-
tration. [12] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
1.15 Schematically shows the relation between martensite fraction and
temperature. (a) No overlapping case where reversible actuation
is possible between As and M?f and (b) overlapping case where
reversible actuation is not possible. [36] . . . . . . . . . . . . . . . . 34
1.16 Metallographic observation of magnetic field-induced reverse trans-
formation (martensite to austenite) in Fe-30.5% Pd at 2?C. (a)
martensite plates under no magnetic field. (b) Some pleats dis-
appear when a field of 1 T was applied [36] . . . . . . . . . . . . . 35
2.1 Effective elastic constants for the three modes of elastic waves in the
principal propagation directions in cubic crystals. The two trans-
verse modes are degenerate for propagation in the [100] and [111] di-
rections(K: wave vector; L: longitudinal wave; T: transverse wave) [32] 41
2.2 Magnetic field de-twinned the martensitic Ni2MnGa to obtain single
variant tetragonal martensite with c/a<1. . . . . . . . . . . . . . . 43
2.3 All the applicable elastic wave modes in the tetragonal phase. H
and P present the magnetic field direction and wave polarization
direction, respectively. The wave propagation direction is along the
directions normal to the faces of transducers. . . . . . . . . . . . . . 44
2.4 Cartoon shows the different harmonic order standing waves . . . . . 47
2.5 Experimental set up of the continuous ultrasonic wave technique . . 49
2.6 Sample holder of the experimental set up of the continuous ultra-
sonic wave technique. . . . . . . . . . . . . . . . . . . . . . . . . . . 50
2.7 Labview program for automatically controlling measurement process. 51
viii
3.1 SQUID results of Ni0.5Mn0.284Ga0.216 clearly show the Curie Tem-
perature is around 95?C and the martensitic transformation tem-
perature is around 42?C. . . . . . . . . . . . . . . . . . . . . . . . . 53
3.2 DSC measurement of Ni0.50Mn0.284Ga0.216 . . . . . . . . . . . . . . . 54
3.3 SQUID result of Ni0.49Mn0.234Ga0.276 clearly shows the Curie tem-
perature is around 95?C and the martensitic transformation tem-
perature is around 3?C. . . . . . . . . . . . . . . . . . . . . . . . . . 55
3.4 The elastic constants, C11, C44 and Cprime, of Ni0.50Mn0.284Ga0.216 vs.
temperature. The propagation and polarization directions used to
obtain the data are shown on the right. . . . . . . . . . . . . . . . . 58
3.5 The elastic constant, C11, is sensitive to the direction of applied
magnetic field: The change of C11 is plotted vs. temperature. Mea-
surements were taken with a longitudinal wave along ?001? with
applied a 0.8 Tesla magnetic field along ?001? (solid circle ) and
?100?(solid square). . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
3.6 C11 measured by a longitudinal wave along [001]. The temperature
dependence of the C11 were studied with magnetic field parallel and
perpendicular to the wave propagation direction. This plot clearly
shows a premartensitic transition around 60?C. . . . . . . . . . . . 61
3.7 CL measured by a longitudinal wave along ?110?. The temperature
dependence of the CL were studied with magnetic field parallel and
perpendicular to the wave propagation direction.This plot clearly
clearly shows a premartensitic transition around 60?C. . . . . . . . 62
3.8 C measured by a transverse wave along?110?with?110?polarization
direction. The temperature dependence of the C? were studied with
magnetic field parallel and perpendicular to the wave propagation
direction.This plot clearly clearly shows a premartensitic transition
around 60?C. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
3.9 C44 measured by a transverse wave along [100] with ?110? polariza-
tion direction. The temperature dependence of the C44 were studied
with magnetic field parallel and perpendicular to the wave propa-
gation direction. The temperature dependence of C44 has a large
different between with and without magnetic field. This plot clearly
clearly shows a premartensitic transition around 60?C. . . . . . . . 64
3.10 3D view of the temperature dependent X-ray diffraction < 220 >
spectrum and the temperature dependence of the elastic constant,
C44. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
3.11 Temperature dependence of the relative velocity (a), VL V44 and
V?, and the relative attenuation (b) associated with the elastic con-
stant [65] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
ix
3.12 Schematic representation of the martensitic transformation, pre-
martensitic transformation, and Curie temperature of Ni2MnGa al-
loys vs. concentration of valence electrons (e/a). The crystals in-
vestigated have concentrations of e/a = 7.636 and 7.366, which are
lower or slightly higher than 7.60. It therefore falls into an interme-
diate region, where the premartensitic transformation temperature
has merged with the martensite, which is still lower than the Curie
temperature. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
3.13 Temperature dependence of six elastic constants, C11, C33, C44, C66,
C*and C? of Ni0.50Mn0.284Ga0.216 below the first martensitic trans-
formation temperature. The C* is defined in the Table 2.2. The
data points below the inter-martensitic transformation only show
the continuous change of the wave velocities. The elastic constants
notation of the tetragonal, c/a<1 phase, is not suitable anymore.
Since we propose the second martensite to be the tetragonal, c/a>1
phase, those data points merely show some combination of the true
elastic constants. The propagation and polarization directions used
to obtain the data are shown on the right. The field directions used
to create the single-variant states are also shown. . . . . . . . . . . 73
3.14 Schematic representation of structural and magnetic transforma-
tions in NiMnGa alloys vs. concentration of valence electrons (e/a).
The different transformation temperatures, namely Curie temper-
ature, Martensitic Transformation, Cubic to Orthorhombic trans-
formation and inter-Martensitic transformation are depicted by TC,
TM, TI and TInter-M, respectively. The crystal investigated in
this work has a concentration of e/a= 7.636, as marked. The dia-
gram was assembled with the help of the results obtained on three
other crystals of similar composition, whose e/a values are also
marked. According to our elastic constant data, we found the inter-
martensitic transformation at around 220K. The tetragonal non-
layered phase has c/a>1. The dash-dot line was drawn so to allow
the sample with e/a equal 7.74 to be in the non-layered tetragonal
phase at room temperature consistent with earlier work . . . . . . . 75
3.15 The elastic constants, C and C* are plotted vs. temperature from
380K (above the Curie point) to 200K. Measurements were taken
with transverse waves along [110] with polarization [110]. C? was de-
termined with magnetic field along [001] (solid square); and C* was
determined with H parallel to [100] (open square). In the austenite,
which is at the temperature higher than 318K, the two modes are
degenerate. Notice how the C? mode drops rapidly below 233K, so
that the two modes again degenerate at around 220K . . . . . . . . 76
x
3.16 A Strain-space map showing the possible phase equilibrium and dis-
tortionsfromoneofthem. Theradiusofthedottedcircleisthevalue
of ?, the C? and C* are the moduli corresponding to orthorhombic
and tetragonal distortions, respectively. ? is the angle between the
structure?s position and ?3 axis. If ? equals zero, C? and C* corre-
spond to the deformation along ?2 and ?3, respectively. The center
shows the free energy contour with respect to different sign of b. . . 80
3.17 Shows at different sets of ? value, there are different types of tetrag-
onal with respect to the c/a ratio. . . . . . . . . . . . . . . . . . . 81
3.18 This graph shows that the result of the model fit the experimen-
tal data reasonably well. The elastic constant C has a nine times
jump at the cubic-tetragonal boundary, which was experimentally
proved. The calculation also shows that the C and C* degenerate
at the boundary, which also was observed in the experiments. Re-
sult of modeling reproduces the major aspects of the temperature
dependence of the shear elastic constant C. . . . . . . . . . . . . . . 84
4.1 Sample orientation of FePd30?0.5at% . . . . . . . . . . . . . . . . . . 89
4.2 The wave spectrum of longitudinal elastic wave along [001] crys-
talline direction, which is for conducting the elastic constant, C11,
in ferromagnetic shape memory alloy, Fe3Pd. The spectrums show
significant changes around 45?C . . . . . . . . . . . . . . . . . . . . 91
4.3 Magnetic field dependence of Elastic constant, C11, in ferromagnetic
shape memory alloy, Fe3Pd. . . . . . . . . . . . . . . . . . . . . . . 92
4.4 The wave spectrum of the temperature dependence of the shear
elastic wave with polarization direction along [110] and propagation
direction along [110], which is for conducting the elastic constant,
C?, in ferromagnetic shape memory alloy, Fe3Pd. The spectrums
show significant changes around 45?C . . . . . . . . . . . . . . . . . 93
4.5 The wave spectrums of longitudinal elastic wave along [001], which is
for conducting the elastic constant, C11 at 70?C with various mag-
netic field from 0 to 1T. The spectrums show significant changes
around 0.25T . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
4.6 Magnetic field dependence of elastic constant, C11, in ferromagnetic
shape memory alloy, Fe3Pd. . . . . . . . . . . . . . . . . . . . . . . 95
4.7 The wave spectrum of the magnetic field dependence of the shear
elastic wave with polarization direction along [110] and propagation
directionalong[110], whichisforconductingtheelasticconstant, C?.
The spectrums show significant changes with an external magnetic
field around 0.25T . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
xi
4.8 The wave spectrum of the magnetic field dependence of the shear
elastic wave with polarization direction along [110] and propagation
direction along [100], which is for conducting the elastic constant,
C44. The spectrums show significant changes with a external mag-
netic field around 0.25T . . . . . . . . . . . . . . . . . . . . . . . . 98
xii
Chapter 1
Introduction
Smart materials have one or more properties that can be dramatically altered
by external fields. They are distinctly different from structural materials. Their
physical and chemical properties are sensitive to a change in the environment
such as temperature, pressure, electric field, magnetic field, optical wavelength,
adsorbed gas molecules, and pH. Smart materials are the components of a smart
system.
A smart system is a multifunctional structure with the capabilities of sens-
ing, processing, actuating, feedback, self-diagnosing, and self-recovery. All these
characteristics are based on the functionality of the materials composing the sys-
tem and their intercommunication. The system uses the functional properties of
advanced materials to achieve high performance with capabilities of recognition,
discrimination, and adjustment in response to a change of its environment. Each
component of this system must have functionality, and the entire system is inte-
grated to perform a self-controlled smart action, similar to a living creature that
can think, judge, and act. A smart system can be considered as a design philoso-
phy that emphasizes prediction, adaptation, and repetition. Each smart material
1
in the smart system is required to have a specific functionality, such as sensing a
change in its environmental conditions, actuating a mechanical or electric response
simultaneously, automatically modifying one or more of its property parameters,
and quickly recovering its original status once the environmental conditions are
changed. The materials have the ability to modify their intrinsic structures, the
band structure or the domain structure, in response to an external change, such
as temperature, electric field, stress field, or magnetic field. In other words, smart
materials have large mechanical response coefficients in the external fields.
Smart materials can be divided in to three categories according to the different
external fields: Ferromagnetic materials, Ferro-electrical materials, Shape Mem-
ory Alloys (SMA). In these three kinds of smart materials the mechanical response
is magnetostrictive, piezoelectric (electro-strictive) or thermo-strictive (thermo-
mechanical) coefficient, respectively. The mechanical response of each material
can be written as: epsilon1 ? f(H,E,?,T), where H: magnetic field, E: electrical field, ?:
Stress field, and T: temperature. Three of the most widely used smart materials
are piezoelectric, magnetostrictive and the shape memory alloy. These materials
can be taken as typical examples to briefly illustrate structural characteristics of
the smart materials. Recently many new materials with multiple functionalities
have been found, such as FE-FM, FESMA, FMSMA. In this work, we concerned
with the ferromagnetic shape memory materials, which display a large magnetic
field induced strain and martensitic transformation. To date, various ferromag-
netic shape memory alloys, e.g. Ni2MnGa [62], FePd [30], FePt [67], NiCoGa [67],
NiCoAl [67], have been investigated. They have both properties of ferromagnetic
and shape memory alloys. Hence, they display a giant magnetostriction that con-
sists really of a magnetically induced domain redistribution. The understanding
2
of this mechanism requires knowledge of the elastic properties of these materials.
This thesis is directed at the elasticity of Ni2MnGa and Fe3Pd alloys
1.1 Shape Memory Alloys
Shape memory alloys (SMA) are quite fascinating materials characterized by a
shape memory effect and super-elasticity, which are based on the diffusionless and
reversible martensitic transformation [40]. They are now being practically used as
functional alloys in many areas. They exhibit large work output and can be used
as working parts in various actuator systems.
Most shape memory effects are related to the martensitic phase transforma-
tion, which is diffusionless and involves a crystallographic shearing deformation.
The parent phase of this transformation is called austenite; the product phase is
called martensite. There are several ways to transform from austenite to marten-
site, which accommodates the shape change. If the transformation is induced
by temperature, the material has no preference for any of the variants because
they all have the same free energy. If the transformation is induced by stress,
only certain variants will appear because of local energy minimization.[27] If the
martensitic material is also ferromagnetic, there exists the possibility of inducing
the martensitic transformation or of rearranging the variants of martensite with a
simultaneously magnetic field.
Martensite is the relatively soft and easily deformed phase of shape memory
alloys, which exists at lower temperatures. The structure of this phase is twinned
which is the configuration shown in the middle of Figure 1.1. Upon deformation
this phase takes on the second form shown in Figure 1.1 on the bottom right.
Austenite, the stronger phase of shape memory alloys, occurs at higher tempera-
3
tures. The structure of the Austenite is cubic, shown on the upper part of Fig-
ure 1.1. The un-deformed Martensite phase is the same size and shape as the cubic
Austenite phase on a macroscopic scale, so that no change in size or shape is visible
in shape memory alloys until the Martensite is deformed. Therefore this transition
is a thermo-elastic martensitic transformation. Such structural transformations are
diffusion-less, reversible and occur between a high temperature phase and the low
temperature phase. The high temperature phase has high symmetry whereas the
low temperature phase has lower symmetry. Shape memory materials are unique
in their ability to accommodate large strains without plastic deformation. When
the martensitic transformation occurs, a microstructure of symmetry related and
compatible variants is formed, in which two variants meet at a well defined inter-
face called a twin plane as shown in Figure 1.2. The martensite phase has multiple
variants, which are symmetry related configurations of the same atomic lattice.
An external field could rearrange the different variants in the martensite phase
inducing a large strain between the twinned and detwinned structure.
Pseudo-elasticity occurs in shape memory alloys when the alloy is completely
composed of Austenite (temperature is greater than Af). Unlike the shape mem-
ory effect, pseudo-elasticity occurs without a change in temperature. The load
on the shape memory alloy is increased until the Austenite becomes transformed
into Martensite simply due to the loading. The loading is absorbed by the softer
Martensite, but as soon as the loading is decreased the Martensite begins to trans-
form back to Austenite since the temperature of the SMA is still above Af, and the
material returns to its original shape. The most effective and widely used alloys
include NiTi, CuZnAl, and CuAlNi, etc.
Shape memory alloys have the large mechanical response to the stress field or
4
Figure 1.1: Microscopic Diagram of the Shape Memory Effect
Figure 1.2: Austenitic cubic lattice and martensitic tetragonal lattice in 2D with
two variants. The twin planes in the reference cubic lattice and in the deformed
tetragonal lattice are shown by the dotted lines.
5
temperature field. Another type of smart material is the ferromagnetic material,
which displays a large strain by applying magnetic field. This strain is equivalent
to magnetostriction.
1.2 Magnetostriction
When the magnetization state of a ferromagnetic material is changed by an ap-
plied field, there is an accompanying small change in its dimensions. This is called
magnetostriction, which is one of the important properties of ferromagnetic ma-
terials. Magnetostriction represents the balance of the magnetization, crystalline
anisotropy and the elastic energy. It, ?, was defined as:
? = ?ll (1.1)
where ?l denotes the deformation of the specimen in the magnetic field and l is
the length of the specimen[14].
There are two types of magnetostriction. One is the shape magnetostriction,
which is also called Joule magnetostriction. It causes the different crystalline
directions of the sample to have different strain, however the total volume of the
sample will keep the same. Another is the volume magnetostriction, which causes
an equal expansion or contraction in all directions. Volume magnetostriction is a
very small effect and has no bearing on the behavior of practical magnetic materials
in ordinary fields. Therefore in this work, only shape magnetostriction will be
considered
The process of magnetization is then one of converting the specimen from a
multi-domain state into one in which it is a single domain magnetized in the
same direction as the applied filed. Figure 1.3 schematically shows the relationship
6
Figure 1.3: Schematic representation of the relationship between the magnetization
and magnetostriction. The dashed line is the magnetostriction. The black solid
line is the magnetization hysteresis.
between the magnetization and magnetostriction.
Magnetostriction is the shape change of a ferromagnetic material changes dur-
ing the process of magnetization, so it is the elastic response of a crystal to the
spontaneous magnetization.[2] Although the strain ?l/l resulting from magne-
tostriction is small, there exists an inverse effect, which causes many material?s
properties, such as permeability and the size of the hysteric loop, to be highly de-
pendent on stress. Hence magnetostriction has many practical consequences, and
considerable research has been devoted to it.
When a ferromagnetic material is subjected to an applied field, the observed
magnetization depends on both the magnitude of the field and the crystallographic
direction along which it is applied. The dependence of magnetic properties on
7
crystallographic direction is known as magnetocrystalline anisotropy, which is of
fundamental importance in determining the magnetic properties of ferromagnetic
materials.
In a sufficiently large field the magnetization will reach the saturation value,
which is the same for all crystallographic directions of one crystal. Magnetization
in the various crystallographic directions has an associated energy, and it is the
difference in this energy between different directions that determines the magnetic
behavior. The crystallographic direction for which the magnetization reaches sat-
uration in the lowest applied field is known as the easy axis of magnetization. The
work required to rotate the magnetization away from one of the easy axes with an
applied field is described by the magnetocrystalline anisotropy energy. For cubic
crystals the energy of magnetization is given by
?Anis = K0 +K1(?21?22 +?22?23 +?23?21) +K2(?21?22?23) (1.2)
where ?i are the direction cosines of the magnetization with respect to the cubic
axes and ?21 + ?22 + ?23 = 1. The values of K0, K1 and K2 are material constants
associated with the easy, medium and hard axes of magnetization, and have units
of ergs/cm3[14].
The elastic energy density in the approximation of Hooke?s law is a quadratic
function of the strains. In a cubic crystal, the elastic energy can be represented as:
?el = 12C11(?2xx+?2yy+?2zz)+12C44(?2xy+?2yz+?2zx)+C12(?xx?yy+?yy?zz+?zz?xx) (1.3)
where C11, C44 and C12 are the elastic moduli, ?xx, ?yy, ?zz, ?xy, ?yz and ?zx are
the strain tensor components[14].
The magneto-elastic interaction energy is a function of the inter-atomic dis-
tance, because the magnetostriction originates in the interaction between the
8
atomic magnetic moments. In a simple cubic lattice, the magnetoelastic energy is
presented as:
?magel = B1
bracketleftbigg
?xx
parenleftbigg
?21 ? 13
parenrightbigg
+?yy
parenleftbigg
?22 ? 13
parenrightbigg
+?zz
parenleftbigg
?23 ? 13
parenrightbiggbracketrightbigg
+B2 (?xy?1?2 +?yz?2?2 +?zx?3?1) (1.4)
where B1 and B2 are magnetoelastic coupling constants[8].
When a martensitic ferromagnetic material is exposed to a magnetic field, the
magnetostriction produces an elastic strain in the materials that either increases
or decreased the magnetoelastic energy, depending on the conditions. Thus the
magnetoelastic energy depends on the lattice strain and the direction of domain
magnetization. The interaction between an applied stress and magnetostriction is
an important effect, and may have a marked influence on the field-dependent mag-
netic properties. Since the magnetoelastic energy is a linear function with respect
to the strains, the crystal will be deformed without limit unless it is counterbal-
anced by the elastic energy, which is represented by Equation (1.3).
From above discussion, the total energy which relate to the crystal?s magneti-
zation status can be expressed as:
?Total = ?Anis + ?el + ?magel (1.5)
The equilibrium direction of MS is that, which renders ? a minimum and this direc-
tion is a complicated function of K, ? and ?. In cubic crystal the magnetostriction
could be calculated at the equilibrium state, ??/?? = 0.
?100 = ? 2B13(C
11 ?C12)
?100 = ? B23C
44
(1.6)
9
Alloy Magnetostrictive Explanation Reference
Strain ?100 [10?6]
Fe 21 Joule [14]
Mangetostriction
Ni -46 Joule [14]
Mangetostriction
Fe84Al16 86 Joule [11]
Mangetostriction
Fe50Co50 140 Joule [21]
Mangetostriction
Fe83Ga17 207 Joule [10]
Mangetostriction
Co0.8Fe2.2O4 -590 Joule [3]
Mangetostriction
Terfenol-D 1,600 Joule [9]
(FeTbDy) Mangetostriction
Fe3Pd 10,000 Reversible Strain [13]
resulting from
rearrangement of variants
Ni2MnGa 100,000 Reversible Strain [53]
resulting from
rearrangement of variants
Table 1.1: Overview of Crystalline Metallic Alloys with Magnetostriction ?100
10
An overview of magnetostrictive materials is listed in Table 1.1. It clearly shows
that the magnetically induced strain in FSMAs, Fe3Pd and Ni2MnGa, are much
larger than the general magnetostrictive ferromagnetic materials.
Since the spontaneous strain in martensitic materials is commonly one order of
magnitude larger than that of giant magnetostrictive materials, the field-induced
strain available from a material that is both martensitic and ferromagnetic is much
larger than that of magnetostrictive materials.
1.3 The Free Energy of Ferromagnetic, Shape
Memory and Ferromagnetic Shape Memory
Materials
Materials scientists and physicists have spent one decade trying to understand the
phase transitions in the Ferromagnetic Shape Memory Alloys (FSMA), particularly
the relationship between the microstructure and the magnetic properties in FSMA
systems. Here, we introduce the evolution of the SMA and FSMA eigen-strain on
the basis of the simplest Landau energy possible. It will be seen that the magnetic
anisotropy automatically leads to elastic domains that are differently magnetized.
The magnetoelasticity in ferromagnetic alloys may be formally described by
the Landau expansion ?ferro(?,M) = ???M2 + C?2/2?aM2/2 + bM4/4 where
the quantity M is the spontaneous magnetization, coefficient a is related to the ex-
change energy, ? is the magnetoelastic coupling constant, C is the elastic constant
and b is a constant coefficient of forth order magnetization term. This free energy
yields the equilibrium magnetoelastic strain and magnetization in ferromagnetic
materials, ?0 = ?M2/C and M0 = ?
radicalBig
(2???a)/b
11
The free energy of the low temperature phase of a SMA may be written as
?SMA(?) = ??4/4?C?2/2, where ? is a constant coefficient and C designates the
formally negative shear elastic constant, that relates to the shear strain ?xx ??yy.
The spontaneous equilibrium strain is now | ?xx ??yy |0= ?
radicalBig
?/C. It designates
the two variants of the 2D low temperature phase.
The free energy of a FSMA can be represented by the sum of the Landau
energies of SMA and ferromagnetic solids. Then it could be presented in 2D
dimension as:
? = ??2?2M2 ? C2 ?2 ? a2M2 + ?4?4 + b4M4 (1.7)
Introducing the magnetic anisotropy results in martensitic phase is coupled with
the directions of the magnetization. Then the energy is:
?FSMA = ??2?2M2 ? C2 ?2 ? a2M2 + ?4?4 + b4M4 ???m + 12Dm2 (1.8)
where ? is the magnetoelastic coupling constant and D is the constant coefficient
of M2 term, ? = ?xx??yy, m = M2x ?M2y and M2 = M2x +M2y. Expanding around
the equilibrium position at meq = ??/D and M2eq = (a + ??2)/b = M0 + ??2/b
yields
?? = ???M = ?12C??2 + 14???4 (1.9)
where ?M = ?aM20/2+bM40/4 is the magnetization energy, C*= ?M0+C??2/D
is the new elastic constant with magnetization effect and ?? = ? ? ?2/b is the
new coefficient for the forth order term of ?. The equilibrium strain is then ?eq =
?
radicalBig
(C +?M2 +?2/D)/?, which is clearly larger than the equilibrium strain of
SMA, ?
radicalBig
?/C. The results of the free energy in FSMA are schematically shown
in Figure 1.4.
As can be seen in Figure 1.4, this free energy has two energy wells located
at ?0 = ?
radicalBig
(C +?M2 +?2/D)/? corresponding to the two martensitic variants,
12
Figure 1.4: Schematic of the free energy of FSMA at T > T0, T = T0 and T < T0,
where T0 is the phase transition temperature. When T < T0, the free energy
has two energy wells located at ?0 = ?
radicalBig
(C +?M2 +?2/D)/? corresponding to
the two martensitic variants, ?(?xx ??yy). The direction of the magnetization is
directed along the major or minor axes of the low temperature phase.
13
?(?xx??yy). Since (M2x ?M2y) ? (?xx??yy), the direction of the magnetization is
directed along the major or minor axes of the low temperature phase depending on
the sign of ?, see Figure 1.4. It follows that the martensitic variants in FSMAs are
simultaneously elastic and magnetic variants. Their population could be changed
by an elastic and, this is the novel feature of this material, magnetic fields. The
curvature of the minimum energy of a FSMA represents the elastic constant, which
is the topic of this thesis.
1.4 Ferromagnetic Shape Memory Alloy
Ferromagnetic shape memory alloys (FSMA) are a new class of active materi-
als, which combine the properties of ferromagnetism with those of a diffusionless,
reversible martensitic transformation. The term ?ferromagnetic shape memory ef-
fect? is used to describe the reversible change in shape caused by either inducing
the austenite/martensite transformation or rearranging the martensitic variants
with an applied magnetic field. The magnetic field rearranges the variants result-
ing in a giant strain. The variant rearrangement phenomenon is a complicated one
that depends on twin boundary mobility, the magnetic properties of the marten-
site, proper biasing of the initial microstructure, proper specimen orientation, and
specimen shape, among others.
Rearranging martensite variants by a magnetic field is a novel idea and received
attention only in recent years. In 1996, Vasil?ev et al. [62] measured the magne-
tostriction in the ?110? directions of a Ni2MnGa single crystal, and suggested the
existence of field-induced shape memory effect. In the same year, Ullakko et al.
[60] demonstrated the field-induced variant rearrangement by measuring a 0.2%
field-induced strain in an unstressed Ni2MnGa single crystal. In the experiments,
14
the field applied to induce the shape change was 8 kOe, which is considerably
lower than the field commonly used to study the effect of magnetic field (40-350
kOe) on phase transformation temperatures. In 1998, a larger reversible field-
induced strain of 0.6% under cyclic fields of 10 kOe has been achieved in Fe70Pd30
[30]. In 1999, Tickle et al. [58] reported the largest field-induced strain of 4.5% in
Ni2MnGa. In their experiments, the evidence of field-induced variants rearrange-
ment, the movement of twin interfaces by the applied magnetic field, was directly
observed. Theoretical work regarding the field induced variant rearrangement was
carried out by Wuttig, James and Desimone [30, 17], and O?Handley [44].
1.4.1 Magnetic Field Induced Strain
A ferromagnetic material in the demagnetized state is divided into a number of do-
mains. Each domain is spontaneously magnetized to the saturation value MS, but
the directions of magnetization of the various domains are such that the specimen
as a whole has no net magnetization, as shown in Figure 1.6. If the martensitic
transformation is induced by temperature, the material has no preference for any
of the variants because they all have the same free energy. When stress is applied
to this mixture of variants, only certain variants will appear due to local energy
minimization. The process of magnetization is then one of converting the specimen
from a multi-domain state into one in which it is a single domain magnetized in
the same direction as the applied magnetic filed. The process of magnetization
occurs by two mechanisms, domain-wall motion and domain rotation; most of the
magnetostrictive change in length usually occurs during domain rotation.
The magnetic field induced strain in ferromagnetic shape memory alloy is dif-
ferent from the ordinary magnetostriction. Magnetostriction is the spontaneous
15
Figure 1.5: Cartoon of the phase transition process and the twin structure in the
Ferromagnetic Shape Memory alloy.
16
deformation of a solid in response to its magnetization. Martensitic transforma-
tion produces a spontaneous deformation of a solid to form a martensite twin
structure, upon lowering the temperature. If the martensitic material is also ferro-
magnetic, there exists the possibility of inducing the martensitic transformation or
of rearranging the variants of martensite to yield large strain by applying a mag-
netic field. Since the spontaneous strain in martensitic materials is commonly one
order of magnitude larger than that of giant magnetostrictive materials, the field-
induced strain available from a martensitic or ferromagnetic material is potentially
much larger than giant magnetostrictive materials.
The recent progress in designing of a new class of magnetic shape memory alloys
is based on the physical mechanism of the martensite-martensite twin boundary
motion driven by magnetic field in a martensitic state. There exist three possible
mechanisms that can produce the shape change. First, applying a magnetic field
induces a structural transition, as in Fe-Ni alloys.[51] This requires very large field,
about 10 tesla for a 20?C shift. Secondly, a magnetic field rotates the spontaneous
magnetization with respect to the crystal lattice, as in Terfonol-D.[9] This requires
negligible magneto-crystalline anisotropy, so that there is little or no energy cost
for rotating the magnetization. Thirdly, a magnetic field rearranges the martensite
variants, as in Ni2MnGa [57] and Fe70Pd30 [30]. This third mechanism could
result in large strains compared to the other two, and requires high twin interface
mobility, large magnetization, and large magnetic anisotropy. The large anisotropy
essentially pins the spontaneous magnetization along the crystalline easy axis,
ensuring a significant driving force on the twin interfaces.
Figure 1.6 shows schematically the different process between magnetostriction
and magnetic field induced strain. Consider a rectangular bar specimen with ?100?
17
Figure 1.6: Comparing the two different process of magnetization by magnetic
anisotropy.
edges with two variants of martensite. A uniaxial layered magnetic substructure
is present within each variant as shown at zero applied field. When a field is
applied along the long axis of the bar, the variant with horizontal magnetization
is favored and its volume fraction increases at the expense of the other variant.
For large enough fields, the specimen may be completely detwinned, resulting in
a large shape change as shown. The high magnetic anisotropy is one of the most
important properties of FSMA. It decides which magnetic process would require
the lower energy.
1.4.2 Ferromagnetic Shape Memory Alloys Ni2MnGa
The recent progress in designing of a new class of Ferromagnetic Shape Memory
Alloys (FSMA) is based on the physical mechanism of the martensite?martensite
18
twin boundary motion driven by magnetic field. The main thermodynamic driving
forces in this case have been magnetic in nature and are connected with high
magnetization anisotropy and significant differences in magnetization free energies
for different twin variants of martensite. By now, the largest magneto-strain effects
were achieved in NiMnGa ferromagnetic shape memory alloys.
Heusler alloys Ni2MnGa have two transformations: a ferromagnetic transition
and a martensitic transformation. Depending on how the Curie temperature and
the martensitic transformation temperature are ordered, the material could be
expected to have different behaviors. The elasticity yields the information con-
cerning the intrinsic properties of the materials for understanding the nature of
these transformations.
Phase Transitions in Ni2MnGa
Heusler alloy Ni2MnGa belongs to the family of ternary intermetalic compounds,
with X2YZ composition and L21 lattice structure. The unit cell of Ni2MnGa is
shown in Figure 1.7. In Heusler alloys, the manganese atoms mainly carry the
magnetic moment. Neutron diffraction experiments performed by Webster et al.
(1984) [64] confirmed the ordered L21 structure in Ni2MnGa, and measured a
moment of 4.17?B for the manganese atoms, with a small moment, < 0.3?B,
possibly associated with the Ni atoms. They also reported a martensitic transfor-
mation temperature of ?71?C with a small thermal hysteresis of less than 5?C.
In 2002, several kinds of martensitic phase have been found in nonstoichiometric
Ni2MnGa alloys in numerous studies. The structural, magnetic, and mechanical
properties important for a giant magnetic field induced strain response in these
alloys were reported. [54] Several research groups have succeeded in observation
19
Figure 1.7: The Heusler L21 lattice structure in Ni2MnGa.
and detailed the investigation of super large 6% magneto strain effect found in
the five-layer modulated tetragonal martensitic phase[58, 38, 45, 53]. The Sozi-
nov group reported the much larger, up to 10%, magnetic field induced strain
in seven-layer modulated orthorhombic martensite[53]. The important properties
of martensitic phases in Ni2MnGa with different compositions are listed in Table
1.2. The properties of Ni2MnGa are very sensitive to the composition. One of
the crystal structures of the martensite phase was identified as tetragonal, which
also has five-layer shu?ing type modulation, 5M, and a tetragonality aspect ratio
c/a < 1. [37, 34] Depending on the chemical composition and temperature there
are also two other kinds of martensitic phases in Ni2MnGa that have orthorhombic
seven-layered and tetragonal non-layered crystal structures with c/a > 1 [54]. A
(110)[110] shu?e (or modulation) is observed in both 5 M and 7 M phases. [53, 54].
The X-ray studies [54] showed that the martensitic transformation in Ni2MnGa is
20
actually quite complex and exhibits a number of stress induced phases in addition
to the thermal martensite formed upon cooling.
Phase transitions in Ni2MnGa seven?layered martensite were studied by Sozi-
nov. Figure 1.8 shows the temperature dependence of the low-field ac mag-
netic susceptibility of the alloy during heating and cooling. The abrupt change
in the value of the susceptibility at 367? 369 K is attributed to paramagnetic?
ferromagnetic transformation of the L21 cubic phase. The Curie temperature is
about TC = 368K. During the cooling?heating cycle a sequence of martensitic
and intermartensitic transformations was also observed. The martensitic trans-
formation starts at 337 K (Ms) and is completed at 333 K (Mf) during cooling.
At lower temperatures, starting at 245 K, the value of the magnetic suscepti-
bility increased, indicating an intermartensitic transformation. During heating
both structure transformations occur in reverse order and show some temperature
hysteresis. In particular, the reverse intermartensitic transformation takes place
during heating at 306?316K. The reverse martensitic transformation from the or-
thorhombic to the cubic parent phase upon heating takes place between As = 338
K and Af = 342 K.
Moreover, Ni2MnGa exhibits an intermediate, pre-martensitic state, in the tem-
perature region between TC and Ms. The Ni-Mn-Ga alloys with specific composi-
tions show interesting precursor phenomena at temperatures above the martensitic
transitions, which develops a micro-modulated structure. [6, 7, 20, 41, 43] This pre-
martensitic transition is found to occur only for alloys with an electrons to atoms
ration, e/a, smaller than 7.60. The two Ni2MnGa crystals we studied have e/a of
about 7.636 and 7.366, which are either lower or slightly higher than 7.60. There-
fore we did anticipate finding a premartensitic transition in them. In addition to
21
Martensite Type 5M T 7M
(c/a < 1) (c/a > 1) (a > b > c)
Composition, Ni49.2 Ni52.1 Ni48.8
at.%(?0.5%) Mn29.6 Mn27.3 Mn29.7
Ga21.2 Ga20.6 Ga21.5
Lattice a = 0.594 a = 0.546 a = 0.619
Parameters, nm b = 0.594 b = 0.546 b = 0.580
c = 0.559 c = 0.658 c = 0.553
Cell volume, nm3 0.197 0.196 0.199
?0 = (1?c/a)% 5.89 20.5 10.66
?s, MPa 1 12-20 1.1
?f, MPa 2.1 12-20 1.9
(?0Ms), T 0.6?0.05 0.6?0.05 0.6?0.05
Magnetic anisotropy, 1.45(Ku) -2.01(Ku) 1.6(Ka)
105J/m3 0.7(Kb)
MSM strain, % 5.8 Less than 0.02 9.4
Valance electron 0.7628 0.7739 0.7604
ratio, e/a
Table 1.2: Single-crystal samples of Ni-Mn-Ga with different martensitic structure
at room temperature[53]
22
Figure 1.8: Temperature dependence of low-field ac magnetic susceptibility mea-
sured during cooling (solid line) and heating (dashed line) of the Ni48.8Mn29.7Ga21.5
alloy. Arrows mark phase transformations. C: ferromagnetic cubic phase, Or: or-
thorhombic seven-layered phase, T: tetragonal nonmodulated phase. The inset
shows the scattering intensity distribution in reciprocal space between (400) and
(620) nodes in the orthorhombic phase of the Ni48.8Mn29.7Ga21.5 alloy. Arrows
mark additional peaks connected with seven-layered modulation of the lattice. [53]
23
the temperature near MS, anomalous behavior of elastic moduli is expected to oc-
cur near the Curie temperature TC. In previous elastic modulus studies, the lattice
instability of the parent phase when approaching transformation temperature has
been observed [19, 56], which is a manifestation of precursor events. An almost
linear softening of the elastic modulus in the paramagnetic state may also be as-
sociated with it. Both ultrasonic investigations and neutron scattering studies of
polycrystalline Ni2MnGa alloys indicate that the slope of E(T), the Young?s modu-
lus as function of temperature, changes near the Curie point. [7]. According to our
study on the elastic constants temperature dependence on the single crystalline
Ni2MnGa, the bulk elastic softening event is more pronounced in single crystals.
Magnetic Properties in Ni2MnGa
The ferromagnetic shape memory effect, which differs from the ordinary magne-
tostriction by its mechanism, is driven by the magnetic anisotropy energy. The
effect is based on the magnetic-field-induced redistribution of twin variants in the
martensitic phase. When Ni2MnGa alloy is cooled down, it undergoes a structural
transformation from a cubic, austenitic phase, to a tetragonal structure, marten-
sitic phase. In the cubic structure there are three equivalent crystallographic di-
rections for the tetragonal deformation. Therefore, the martensitic phase consists
of twin variants, which have different deformation directions and are separated by
well-defined boundaries. Naturally, the local crystallographic structure determines
the easy axis of magnetization, so that different variants have a different global
direction for the easy axes i.e., the local magnetic moments in the different variants
have different directions in the absence of an external magnetic field. Figure 1.9
represents the three variants that arise on cooling below martensitic transforma-
24
Figure 1.9: Tetragonal variants structure and associated magnetizations. The
cubic axes are denoted using basis hatwideei, and the martensitic variants and associated
magnetizations by Ei and mi, where the compression axis along hatwideei uniquely defines
each variant.
25
tion temperature, Ms. The lattice deformation of a simple unit cell associated
with each variant is shown with the strain exaggerated for clarity. Note that the
magnetization could be also able to point in the ? directions of the mi shown in
Figure 1.9. When an external magnetic field is applied to the sample, the local
magnetizations try to align with the external field. For fields below saturation and
samples with a large enough magnetic anisotropy energy, it will be energetically
favorable to redistribute the twin variants: instead of rotating the magnetizations
with respect to their local crystal structure, the twin boundaries will move and
the easy axes align with the field. This movement leads to the large shape changes
observed in the Ni2MnGa. The mechanism of the large strain is based on the mag-
netic field induced rearrangement of the crystallographic twin variants that lower
magnetization energy. Macroscopically this strain appears as a magnetostrictive
strain. The magnetic anisotropy energy, elastic energy and their association, mag-
netoelastic energy, are very important in FSMA. Figure 1.6 shows a schematic
representation of the field-induced variant rearrangement process.
The martensitic anisotropy of a single variant of Ni2MnGa martensite was
determined by Tickle.[57] The specimen was compressed into a single variant mi-
crostructure using the small fixture shown to the right of Figure 1.10, and M-H
curves parallel and perpendicular to the compression axis as shown were generated.
The applied stress in the single variant state was required to prevent the nucleation
of a second variant for applied fields perpendicular to the compression axis, and
also allowed for a determination of the ordinary magnetostriction constant from
stress anisotropy effects. Figure 1.11 presents the M-H curve of Ni2MnGa. It shows
the easy axis within a single variant is the c-axis, short axis, and the anisotropy is
uniaxial in nature.
26
Figure 1.10: Schematically shows the M-H curves parallel and perpendicular to the
compression axis (right). The fixture (left) was constructed from a non-magnetic
CuBe spring alloy and sized to fit in the cooling fixture used with the VSM [57]
27
Figure 1.11: This graph shows the M-H curve along c-axis (easy axis) and a-
axis (hard axis) determined at ?17?C with 1.9 MPa applied stress. The results
show that the easy axis within a single variant is the c-axis as expected, and the
anisotropy is uniaxial in nature. Uniaxial anisotropy constant KU was calculated
from the areas between these 2 curves and found to be KU = 2.4?106ergs/cc.[57]
28
1.4.3 Ferromagnetic Shape Memory Alloys Fe3Pd
Iron palladium alloys at certain compositions are ferromagnetic shape memory ma-
terials with high coercivity. The research in FePd was motivated by the potential
of its use as a permanent magnet and in magnetic recording media. Alloys with
composition near Fe70Pd30 undergo a highly reversible FCC to FCT transforma-
tion, and have high saturation magnetization, approximately 2/3 to that of Fe at
room temperature. Therefore, they are good candidates for a ferromagnetic shape
memory material.
The phase diagram of Fe-Pd is shown in Figure 1.12. The high temperature
FCC phase, ?Fe, Pd, is a continuous solid solution. At lower temperatures, there
exists ?1-FePd and ?2-FePd. These two ordered phases,?1-FePd and ?2-FePd3
are formed at about 790?20?C and 820?20?C, respectively [35]. Their structural
properties are listed in Table 1.3 and depicted in Figure 1.13. Notice that L10
has a tetragonal structure with alternate layers of atoms, i.e., Fe or Pd atoms;
L12 has a cubic structure with one kind of atom (Fe) at one corner, and the other
kind of atom (Pd) at the face center. The first attempt to determine the lattice
parameters vs. composition was made by Raub et al. in 1963 [47]. The results
are listed in Table 1.4. Due to the limitation of the equipment at that time, the
accuracy of the data can only be ?0.001?A.
In 2002, Cui [12] investigated a phase diagram to show the phase transfor-
mation temperature as function of palladium concentration as Figure 1.14. The
phase diagram shows that for palladium concentration less than 29.25 at%, the
phase transformation is FCC to BCT. The transition temperature increases as
the concentration of palladium decreases. For palladium concentration more than
29.25 at%, the phase transformation is FCC to FCT, followed by FCT to BCT.
29
Figure 1.12: Phase diagram of Fe-Pd alloy system.
Figure 1.13: Schematic drawing of A1, L10 and L12 crystal structures. The radii
of the atoms are exaggerated for clarity. The structure of ?-(Fe,Pd), ?1-FePd, and
?2-FePd3 are A1, L10 and L12, respectively. [12]
30
Pearson Space Strukturbericht Lattice
Phase Symbol Group Designation Parameters
?-Fe cB2 Fm?3m A2 2.866?A at 0%Pd [29]
2.879?A at 2.4 at.%Pd
?-(Fe, Pd) cF4 Fm?3m A1 (Cu) a=3.770-3.867?A [47]
?1-FePd tP2 P4/mmm L10 (CuAu) a=3.87-3.88?A,
c/a=0.95-0.96 [47, 28]
?2-FePd3 cP4 Pm?3m L12 (CuAu3) a=3.837-3.853?A [47, 28]
Table 1.3: Crystal Structures and parameters of the phases in Fe-Pd Alloy System
Both of the transition temperatures decrease as the palladium concentration in-
creases, but the decrease of the FCC to the FCT is slower than that of the FCT to
the BCT. If the palladium concentration exceeds 29.7at%, only the FCC to FCT
transition exists in the phase diagram.
Three phase transformations exist in Fe70Pd30. They are FCC to FCT, FCT
to BCT, and FCC to BCT. The microstructures of FCT phases are twin bands
and that of BCT have the shape of a spear-head. The phase transformation tem-
peratures are sensitive to the composition. The phase diagram shows that the
Fe-Pd alloys with palladium content ranges from 29.5 to 30.5 at.% are suitable for
the study of ferromagnetic shape memory effect. Both x-ray analysis and DSC
measurements show that the FCC to FCT transformation is a first order tran-
sition [12]. The average lattice parameters are a=3.8220 ?A and c=3.6298 ?A for
the martensite, and a0=3.7557 ?A for the austenite. The latent heat is 1.244 J/g
(10.698 J/cm3) for the cooling and 1.267 J/g (10.896 J/cm3) for the heating. The
Curie temperature equals 300?C. [12]
31
Figure 1.14: Phase transformation temperature as function of palladium concen-
tration. [12]
32
?-(Fe, Pd) ?1-FePd ?2-FePd3
At.%Pd a (?A) At.%Pd a (?A) c/a At.%Pd a (?A)
30 3.770 48 3.87 0.95 65 3.837
40 3.781 52 3.88 0.95 67 3.840
48 3.800 58 3.88 0.96 70 3.845
52 3.800 60 3.88 0.96 75 3.853
58 3.821
60 3.826
70 3.845
75 3.853
80 3.859
85 3.867
Table 1.4: Lattice parameters of the phases in Fe-Pd Alloy System[47]
Ferromagnetic shape memory alloys are currently studied for possible appli-
cations of actuation controlled by magnetic field. One application focuses on
the martensitic variants rearrangement induced by an external magnetic field, as
mentioned in the previous discussion. The other utilizes the forward and reverse
martensitic transformation controlled by the magnetic field. The following dis-
cussion is concerned with the latter subject. Figure 1.15 depicts a case close to
actual transformation, which has a temperature range of the transformation and
hysteresis. The definition of the solid and dotted line in Fe-30.5% is that the solid
line refers to an applied magnetic field whereas the dotted line to the state without
magnetic field. This is consistent with the magnetization of the alloy in austenite
and martensite.[36] Austenite is more stable under a magnetic field. To conduct
a simple examination, the amount of the forward and reverse transformation is
assumed to linearly depend on temperature change from the start of forward or
reverse transformation. If an alloy belongs to category (a), the reversible magnetic
actuation is possible. If an alloy has the transformation characteristic of (b), the
reversible magnetic actuation is not achieved. FePd belongs to case (a), thus the
33
Figure1.15: Schematicallyshowstherelationbetweenmartensitefractionandtem-
perature. (a) No overlapping case where reversible actuation is possible between
As and M?f and (b) overlapping case where reversible actuation is not possible. [36]
reversible magnetic actuation is achievable with properly selected compositions.
Figure 1.16 shows that Fe-30.5% Pd belongs to the category described by Fig-
ure 1.15(a). When a specimen was slowly heated under no magnetic field after
the completion of the transformation and the reverse transformation partially pro-
gressed as shown in Figure 1.16(a), a magnetic field of 1T was applied at 2?C.
Then some martensite plates disappeared as shown in Figure 1.16(b). When the
field was removed, the same martensite plates reappeared as in (a). The reversible
change between Figure 1.16(a) and (b) was confirmed by Liang etc. [36]
34
Figure 1.16: Metallographic observation of magnetic field-induced reverse trans-
formation (martensite to austenite) in Fe-30.5% Pd at 2?C. (a) martensite plates
under no magnetic field. (b) Some pleats disappear when a field of 1 T was applied
[36]
35
Chapter 2
Elastic Constants And The
Experimental Method
2.1 Elastic Constants
Elastic properties govern the behavior of a material subjected to stress over a
region of strain where the materials behave reversibly. A solid material changes
its shape when subjected to stress. Provided the stress is below the elastic limit,
the strain is recoverable. The body returns to its original shape when the stress is
removed. Hooke?s law states that for sufficiently small stresses the amount of strain
is directly proportional to the applied stress, so that the strain components are
linear functions of the stress components as Equation (2.1). The proportionality
constant relating stress and strain in the region of elastic behavior is the elastic
constant, Cij.
?i =
6summationdisplay
i,j=1
Cijepsilon1j (2.1)
where ? is the stress and epsilon1 is the strain. There are three conditions defining ideal
elastic behavior, which are implicit in Equation (2.1): (a) that the strain response
36
to each level of applied stress (or vice verse) has a unique equilibrium value; (b)
that the equilibrium response is achieved instantaneously; (c) that the response is
linear. There are six equations as when Equation (2.1) written in expanded form,
which results in a 6?6 Cij matrix. In this thesis, the work was conducted with
cubic and tetragonal crystals. These two simplified matrices take the form as listed
below.
?
??
??
??
??
??
??
??
??
??
?
C11 C12 C12 0 0 0
C11 C12 0 0 0
C11 0 0 0
C44 0 0
C44 0
C44
?
??
??
??
??
??
??
??
??
??
?
(Cubic)
,
?
??
??
??
??
??
??
??
??
??
?
C11 C12 C13 0 0 0
C11 C13 0 0 0
C33 0 0 0
C44 0 0
C44 0
C66
?
??
??
??
??
??
??
??
??
??
?
(Tetragonal)
(2.2)
Still further simplifications of Hooke?s law for crystals can be effected if, instead
of the usual components of stress and strain, six independent linear combinations of
the usual components were chosen, which possess certain fundamental symmetry
properties associated with the crystal in question. These linear combinations,
which are known as symmetry coordinates of stress or stain, or as symmetrized
stress and strain, are obtained by means of group theory [42].
The complete matrices of symmetrized elastic constants and compliances for
the cubic and tetragonal are listed below.
C11 = S11+S12(S11?S12)(S11+2S12)
C12 = S12(S11?S12)(S11+2S12)
C44 = 1S44
?
???
???
?
???
???
?
?
?
??
??
??
?
1
3(C11 + 2C12)
1
2(C11 ?C12)
C44
?
??
??
??
?
(2.3)
37
C11 +C12 = S33S33(S11+S12)?2S2
13
, C44 = 1S44
C13 = S13S33(S11+S12)?S2
13
, C11 ?C12 = 1(S11?S12)
C33 = (S11+S12)S33(S11+S12)?2S2
13
, C66 = 1S66
?
???
????
???
???
?
?
?
??
??
??
??
??
??
??
?
C11 +C12 ?2C13
?2C
13 C33
C11 ?C12
C66
C44
?
??
??
??
??
??
??
??
?
(2.4)
The main purpose of this thesis work is to investigate the elasticity of ferro-
magnetic shape memory alloys. Elasticity always relates to the intrinsic properties
of the materials. An accurate and proper method to study the elasticity is very
important. Ultrasonic measurements determined the elastic constants of single
crystals by using the Ultrasonic Continuous Wave (UCW) technique.[16]
2.2 Experimental Technique
The ultrasonic continuous wave (UCW) method is one of the most versatile modu-
lus measurements and nondestructive evaluation techniques available. It is applica-
ble to a wide variety of materials and environments. The physics and technology
of ultrasonic measurements and testings, which were used for basic research or
routine materials characterization, have always required more than a little sophis-
tication in their instruments and procedures. The UCW technique is certainly an
easier method of making such measurements.
In 1936, Quimby used continuous wave resonance technique to measure Young?s
38
moduli and torsion moduli of cylindrical rods of single crystals at low frequencies,
of 100kc [52]. Bolef in 1960 [1] described the UCW resonance technique for mea-
surement of the velocity of sound in solids. Bolef also discussed the calculation of
the correction, which is due to the presence of the transducer and coupling film.
In 1999, Haluska, et. al., studied the elastic constants of C60 near its TC by the
UCW technique.[22]
The advantages of the UCW technique made it to be one of the most impor-
tant ultrasound measurements methods. The most important advantage of this
technique is the good accuracy. Since the measurements use continuous waves the
correction factor for end effects can easily be calculated. The factors in the cor-
rection terms involve only the length and density of the specimen, coupling film,
and transducers. All of the quantities in the corrected expression for the velocity
of sound can be measured to better than one part in 103 [1]. Therefore, the over
all accuracy of the continuous wave method is comparable to the best methods
now available. Another very important advantage to use ultrasonic method is that
ultrasonic vibrations can be produced in any sort of materials, and they are conve-
nient to process with a various external fields, as temperature field, magnetic field
or stress field, et al., which is very hard to achieve by other methods. In addition,
it has the virtue of simplicity in instrumentation.
In an anisotropic medium there are only certain directions along which elastic
waves can propagate in pure longitudinal and transverse modes. It is desirable to
know these modes. Brugger, in 1965, determined the pure modes for all crystal
point groups belonging to the orthorhombic, tetragonal, cubic, rhombohedral, and
hexagonal systems. [4].
39
2.2.1 Elastic Wave Modes in Crystals
The purpose of this work is to determine the different elastic constants of the
ferromagnetic shape memory materials by measuring the velocities of different
elastic wave modes. The structures of the crystals are mainly cubic or tetragonal
in our work. Therefore the following discusses details of the elastic wave modes
corresponding to the elastic constants of cubic and tetragonal single crystals.
In an anisotropic medium small amplitude elastic plane waves can propagate
along a given direction in three distinct, mutually orthogonal modes. These modes
are generally neither longitudinal nor transverse. The knowledge of these special
directions and modes is very important for the determination elastic coefficients
from sound velocity measurements in a crystalline media. A complete discussion
of the pure mode directions of the point groups belonging to the orthorhombic,
tetragonal, cubic, rhombohedral, and hexagonal system is provided by Brugger
[4]. This thesis work mainly concerns the crystalline structures of Ni2MnGa and
FePd, which are mostly cubic and tetragonal. Cuboidal samples have four < 110 >
normal faces and two < 100 > normal faces. This orientation ensures enough wave
modes to determine all the independent elastic constants in cubic and tetragonal
crystals. Table 2.1 and Table 2.2 list all the elastic wave modes in the cubic and
tetragonal phases, including the relationship between the wave propagation and
polarization directions. The quantities ?H, ?W, ?P, denote the magnetic field, wave
propagation and wave polarization directions, respectively. Figure 2.1 shows the
applicable effective elastic constants for the three modes of elastic waves in the
principal propagation directions in cubic crystals.
Because the crystalline orientation of the samples is four < 110 > normal faces
and two < 100 > normal faces, not all the elastic wave modes are practicable,
40
Figure 2.1: Effective elastic constants for the three modes of elastic waves in the
principal propagation directions in cubic crystals. The two transverse modes are
degenerate for propagation in the [100] and [111] directions(K: wave vector; L:
longitudinal wave; T: transverse wave) [32]
41
Modes ?W ?P Elastic Constants: C = ?V 2
L [100] [100] C11
T [001],[010] C44
L [110] [110] C11 +C12 + 2C442
T1 [001] C44
T2 [110], [010] C11 ?C122
L [111] [111] C11 + 2C12 + 4C443
T [111], [110] C11 ?C12 +C443
Table 2.1: Elastic wave modes in cubic crystal. The wave propagation and polar-
ization directions are represented by ?W and ?P, respectively. The L and T denote
the longitudinal wave and transverse wave mode, respectively.
especially the waves propagating along < 111 > could not be studied. There are
five wave modes for determining the three independent elastic constants in cubic
and eleven wave modes for determining the six independent elastic constants in
tetragonal as shown in Table 2.1 and 2.2. Moreover there are different wave modes
to determine the same elastic constants, C44. The elastic constants could thus be
and were checked self consistently. Therefore the elastic constants in both cubic
and tetragonal systems are over-determined.
The key to conducting the measurements of elastic constants in tetragonal
martensitic phase is to obtain martensitic single variants. In our experiments, we
use an external magnetic field to bias the FSMA samples into their various single
variant states. The magnetic easy axis of Ni2MnGa is on the short axis as shown in
Figure 1.9 [57]. Thus after applying saturation magnetic field, the [001] direction of
the specimen, the short axis, will align with the direction of the external magnetic
42
Figure 2.2: Magnetic field de-twinned the martensitic Ni2MnGa to obtain single
variant tetragonal martensite with c/a<1.
field. By changing the direction of the magnetic field, the sample can switch from
one variant to other as shown in Figure 2.2, such that a sound wave can travel
along different directions and various modes can be realized. The detail of all the
available elastic wave modes are depicted in Figure 2.3 and listed in Table 2.2.
In tetragonal crystals there are eleven of fourteen wave modes for determining
the six independent elastic constants with the orientation of the sample. Moreover
there are three different wave modes to conduct the same elastic constants, C44.
The elastic constants were also checked self consistently in tetragonal system.
43
Figure 2.3: All the applicable elastic wave modes in the tetragonal phase. H and P
present the magnetic field direction and wave polarization direction, respectively.
The wave propagation direction is along the directions normal to the faces of
transducers.
44
Modes ?H ?P ?W Elastic Constants: C = ?V 2
L [001] [001] [001] C33
T [100] C44
L [001] [111] [111] c(C11 + 2C66 +C12) + (2a?b)(2C44 +C13)2a?b+ 2c
T1 [110] c(C11 ?C12) + (2a?b)C442a?b+ 2c
T2 [101] 2cC44 + (2a?b)(C33 ?C44 ?C13)2a?b+ 2c
L [100] [110] [110] [cC11 +a(2C44 +C13)]/(a+c)
T1 [001] [cC66 +aC44]/(a+c)
T2 [110] C? = (cC44 +a(C33 ?C44 ?C13))(a+c)
L [001] [110] [110] (C11 + 2C66 +C12)/2
T1 [110] Cprime = (C11 ?C12)/2
T2 [001] C44
L [100] [001] [001] C11
T1 [010] C66
T2 [100] C44
a = C11 ?2C44 ?C13;b = C11 ?2C66 ?C12;c = C33 ?2C44 ?C12
Table 2.2: Elastic wave modes in tetragonal crystal. The magnetic field, wave
propagation and polarization directions are represented by ?H, ?W and ?P, respec-
tively. The L and T denote the longitudinal wave and transverse wave modes.
The a, b and c are three constant coefficients, which are different combinations of
elastic constants as shown in the table. They equal to each other in cubic crystal
which yields the Table 2.1
45
2.2.2 Principle of Ultrasonic Continuous Wave Technique
A sound wave is the motion of material in response to a periodically varying force.
The speed with which it travels is related to the speed with which the atoms in
the material are able to move back and forth in response to an applied force. This
depends on the elastic constants, which indicate the stiffness of the material, the
density of the atoms, and the type of sound wave being propagated. The elastic
constants were determined from the measurements of the velocity of sound in the
single crystals by an ultrasonic continuous wave resonance technique:
C = V 2 ?? (2.5)
where V represents the sound wave velocity and ? means the sample density and
C is the elastic constant. This technique depends upon setting up standing waves
in a composite oscillator consisting of transducers, coupling film, and specimen,
which is shown in the cartoon of Figure 2.4. The velocity is then obtained in terms
of length, density, and frequency.
The sound velocity is the product of the wave frequency and wavelength as:
Vn = ?n ??n (2.6)
where, Vn, ?n and ?n are the velocity, the frequency and wave length of the nth
order harmonic wave, respectively.
The wave length of the nth order harmonic wave is related to the sample length
and the order level as shown in Figure 2.4.
?n = 2ls/n, (2.7)
where ls is the samples length. Substitute Equation (2.7) into the Equation (2.6)
to get the relation of the velocity, sample?s length, harmonic order and resonant
46
Figure 2.4: Cartoon shows the different harmonic order standing waves
47
frequency :
Vn = ?n2lsn (a)
Vn+1 = ?n+1 2lsn+ 1 (b) (2.8)
The wave velocity is unique with different harmonic orders, yielding Vn = Vn+1.
Then the frequency can be written as the function of the different resonant fre-
quencies between two adjacent harmonic waves, ??, as:
?? = ?nn (2.9)
Substitute Equation (2.9) into Equation (2.8) (a), the velocity of the wave can be
presented as:
Vn = 2ls ??? (2.10)
The waves velocity can be converted into elastic constants by substituting Equa-
tion (2.10) as:
C = V 2n ?? = (2ls ???)2 ?? (2.11)
Above all, the experimental principle of determining elastic constants using
continuous wave method is two-fold. First, velocity of the sound wave is the prod-
uct of samples length and the difference of the two adjacent natural frequencies
of harmonic standing waves; second, elastic constant is the product of the square
of the velocity of sound waves and density of the specimen. Therefore, to mea-
sure elastic constants is to measure the different resonant frequency between the
adjacent harmonic standing waves of different elastic wave modes.
2.2.3 Experimental Set Up and Arrangement
The elastic constants of FSMA are determined applying a ultrasonic continuous
wave method by measuring the velocities of different elastic waves modes. The
48
Figure 2.5: Experimental set up of the continuous ultrasonic wave technique
experimental setup is schematically shown in Figure 2.5. An electro-magnet is used
to bias the specimen into different single variant states. The function generator
and lock-in amplifier represent the electronics to generate and analyze sound wave
signals through the transducers. Commercially available PZT based pressure and
sheartransducerswitharesonancefrequencyof1.5MHzarecoupledtothesamples.
They are used to generate and detect the desired standing waves. A thin layer
of vacuum grease is applied between the sample and the transducers for better
connection. In addition, a 2mm thick rubber bands were fitted in the spaces
between the transducers and the sample holder for applying proper pressure. The
vacuum grease and rubber band were used to improve the coupling between the
transducers and sample. This arrangement allows to make the measurements over a
range of temperatures including those, in which large deformations are expected. A
picture of the sample holder is shown as Figure 2.6. According to the experimental
principle, an automatic system was designed to measure the elastic constants. It
49
Figure 2.6: Sample holder of the experimental set up of the continuous ultrasonic
wave technique.
is shown in Figure 2.6 and Figure 2.7. The system is automatically controlled by
a Labview program. The platform of this program is shown in Figure 2.7.
This chapter represents the experimental principle of ultrasonic continuous
wave technique. It described the automatic system for measuring the elastic con-
stants by UCW technique with applicable external fields. The following chapters
discuss the elasticity study of ferromagnetic shape memory alloys, Ni2MnGa and
Fe3Pd, by this experimental method.
50
Figure 2.7: Labview program for automatically controlling measurement process.
51
Chapter 3
Elasticity of Ni2MnGa
Ferromagnetic shape memory alloy Ni2MnGa has a large magnetic field induced
strain, which results in a large shape change. The elastic energy is a crucial pa-
rameter for this effect. The main purpose of this work is to study the elasticity
of the FSMA, especially the temperature dependence of elastic constants. The
dependence of a solid?s elastic properties on temperature in the vicinity of a struc-
tural transformation provides the nature of the transition and has been the focus
of many studies.
3.1 Sample Description
The present experiments were conducted with Ni2MnGa single crystals, which were
prepared at Adaptive Materials Technology Ltd. (Yritysiha 5, FIN-00390 Helsinki,
Finland) Two samples, Ni0.5Mn0.284Ga0.216 and Ni0.49Mn0.234Ga0.276, were studied.
The cuboidal samples, 10mm long on each side, are oriented so that four opposing
faces have a ?110? normal, whereas the other two faces have a ?100? normal. The
samples were cut by the company. In the martensitic state of these samples the c-
52
axis is the easy magnetization axis, which turns to align with the external magnetic
field. Pressing the materials with a low load, 2 ? 3 MPa reorients the c-axis along
the pressing direction. When a martensitic sample of these materials is put with
its c-axis perpendicular to the external magnetic flux path, the c-axis reorients to
the direction of the field and the sample expands perpendicular to the magnetic
field. Both samples have the tetragonal martensitic structure with 0.94 aspect
ratio, c/a, and their Curie temperatures are both around 95?C, which are shown
by the SQUID (Super-conducting Quantum Interference Device) results as Figure
3.1 and 3.3.
Figure 3.1: SQUID results of Ni0.5Mn0.284Ga0.216 clearly show the Curie Temper-
ature is around 95?C and the martensitic transformation temperature is around
42?C.
53
Figure 3.2: DSC measurement of Ni0.50Mn0.284Ga0.216
Ni0.5Mn0.284Ga0.216 has the martensite start temperature around 42?C. Fig-
ure 3.1 depicts the results of the SQUID measurements, which confirmed the mag-
netic and structural transformation temperatures. Figure 3.2 shows the results
of DSC (Differential Scanning Calorimetry) measurements of Ni0.5Mn0.284Ga0.216,
which indicates the martensitic transformation temperature is around 42?C and
Austenitic transformation temperature is around 50?C. Ni0.49Mn0.234Ga0.276 has
martensite start temperature around 3?C. Figure 3.3 shows the SQUID results,
which indicates the martensitic transition temperature and the Curie tempera-
ture.
54
Figure 3.3: SQUID result of Ni0.49Mn0.234Ga0.276 clearly shows the Curie temper-
ature is around 95?C and the martensitic transformation temperature is around
3?C.
3.2 Elasticity of Ni2MnGa in the High Temper-
ature Phase
The high temperature phase in Ni2MnGa is the austenitic cubic phase. It is often
found that one of the shear moduli, most often C11?C12, tends to decrease as the
martensitic start temperature, MS, is approached from above, in some cases becom-
ing as small as one or two GPa at the transformation temperature. This softening
of the modulus is the result of the tendency of the crystal lattice to transform to a
new structure that is obtainable from the austenite by a continuous deformation of
55
the lattice. In certain NiMnGa alloys, the Martensitic transformation is preceded
by a transition that freezes in place a shu?ing of atomic planes, with a period of
three or more lattice spaces. These transitions can have a significant effect on the
elastic constants.
We investigated the elastic constants in high temperature phase of two single
crystals, Ni0.50Mn0.284Ga0.216 and Ni0.49Mn0.234Ga0.276. Anomalies in the temper-
ature dependences of all the constants were observed at below TC, as well as in
the vicinity of MS. In what follows, we describe our findings and discuss possible
interpretations of the anomalous behavior of the elastic constants at and below
TC.
3.2.1 Elastic Constants of Ni0.50Mn0.284Ga0.216 High Tem-
perature Phase
ThissectionpertainstothepresentstudyofthehightemperaturephaseofNi0.50Mn0.284
Ga0.216. The ultrasonic continuous wave method [16, 1, 15] was used to determine
the velocities of [100] and[110] longitudinal, and [100][001] (T1) and [110][110](T2)
transverse elastic waves, as shown in Figure 2.1. The experimental set up was de-
tailed in Chapter 2. All measurements were conducted in a magnetic field of 0.8T,
sufficiently large to create single domain conditions in the austenite. Two different
field orientations were used, [001] and [100]. The measurements were taken as the
temperature was lowered from 428K down to 268K in the presence of the field.
The results of the temperature dependence of all the elastic constants are de-
picted in Figure 3.4. Anomalous behavior of the elastic constants was observed,
especially an abrupt 15% softening of C11 at the Curie temperature, as shown in
Figure 3.5. The latter anomaly was found to be strongly influenced by the presence
56
and orientation of applied magnetic fields.
The measurements described above and exhibited in Figure 3.4 and Figure 3.5
may be summarized as follows: Figure 3.4 shows the temperature dependence of
the directly measured elastic constants in austenitic Ni0.50Mn0.284Ga0.216. All four
elastic constants of the cubic phase of the alloy have been determined from 428K
down to and just below the onset of the Martensitic phase transition. The three
symmetry constants, C11+2C12, C11-C12 and C44 show anomalous behavior near
the Curie temperature, to varying extent. The shear constants undergo changes
in slope right at TC, while the C11 shows two very different behaviors in the two
magnetic field orientations as shown in Figure 3.5. The softening of C?, (C11-
C12)/2, as the martensitic temperature is approached has been observed in prior
studies in other NiMnGa alloys. Similar shear softening phenomena has also been
found in other ferromagnetic alloys, such as CoNiGa and NiFeGa [55, 65, 66]. The
anomalies in C? and C44 are small in comparison with the variation in C11 at TC,
as can be seen in Figure 3.4. This combination of behaviors is somewhat similar
to that often observed in Invar alloys like Fe-Ni[23, 25] and Fe-Pt[25]. They differ
in that in the latter, C? shows the normal increase with decreasing temperature
above the Curie point, leading to a maximum at TC, while in the present study,
C? decreases monotonically as the temperature is lowered, showing only a slight
inflection at TC.
The largest effect of the onset of magnetic order occurs in the temperature
dependence of C11, which contains both tetragonal shear and volume responses.
However, it is the volume contribution that is most strongly effected at TC. This
becomes evident when the bulk modulus is calculated from the measured C11 and
C12. When the field was parallel to the propagation axis, there was a smooth
57
Figure 3.4: The elastic constants, C11, C44 and Cprime, of Ni0.50Mn0.284Ga0.216 vs.
temperature. The propagation and polarization directions used to obtain the data
are shown on the right.
58
Figure 3.5: The elastic constant, C11, is sensitive to the direction of applied mag-
netic field: The change of C11 is plotted vs. temperature. Measurements were
taken with a longitudinal wave along ?001? with applied a 0.8 Tesla magnetic field
along ?001? (solid circle ) and ?100?(solid square).
59
variation in the bulk modulus through TC. With the field applied perpendicular
to the propagation direction, this modulus drops abruptly at TC, and continues
to decrease until the fall is interrupted at the Martensite temperature, MS, as
shown in Figure 3.5. The latter is similar to the anomalous behavior of the bulk
modulus found in some invar alloys, such as FeNi[23, 59, 24, 26, 56], FePt[59, 26],
FeMn[23], and MnNiCr [23]. In those alloys, the phenomena were explained as
being due to volume magnetostriction. That mechanism, however, cannot explain
the dependence on magnetic field orientation observed in our experiments. Instead,
the orientation dependence leads us to propose that the Curie point also marks a
transition between a cubic and a premartensitic phase, i.e., a modulated phase[7,
6, 56]. The large difference of the C11 mode with the field along two different
?100? directions when the temperature was lower than TC can be accounted for
by assuming that the field reorients the modulation direction, in effect creating a
tetragonal state. Thus the cubic C11 mode splits at the transition into two modes,
similar to the splitting observed in that mode at the martensitic transition.
For the purpose of checking the experimental data, (C11 + C12 + 2C44) was
obtained from the longitudinal mode velocity for waves propagating along the
?110? direction. The same quantity was also extracted from the data in Figure 3.4
using the equation
(C11 +C12 + 2C44) = 2C11 ?(C11 ?C12) + 2C44 (3.1)
The measured and calculated combination (C11 + C12 + 2C44) agreed within 2%
over the entire temperature range of our measurements.
60
3.2.2 Elastic Constants of Ni0.49Mn0.234Ga0.276 High Tem-
perature Phase
The experimental method and set up of the elastic constants in high temperature
phase of Ni0.49Mn0.234Ga0.276 is similar with Ni0.50Mn0.284Ga0.216. This sample has
273K martensitic transformation temperature, which is lower than the 318K of
Ni0.50Mn0.284Ga0.216. All measurements were conducted in a magnetic field of 0.8T.
Two different field orientations were used, [001] and [100]. The measurements were
taken as the temperature was lowered from 428K down to room temperature in
the presence of the field. The results of the elastic constants of Ni0.49Mn0.234Ga0.276
high temperature phase are presented in Figures 3.6, 3.7, 3.8, 3.9.
Figure 3.6: C11 measured by a longitudinal wave along [001]. The temperature
dependence of the C11 were studied with magnetic field parallel and perpendicu-
lar to the wave propagation direction. This plot clearly shows a premartensitic
transition around 60?C.
61
Figure 3.7: CL measured by a longitudinal wave along ?110?. The temperature
dependence of the CL were studied with magnetic field parallel and perpendicular
to the wave propagation direction.This plot clearly clearly shows a premartensitic
transition around 60?C.
62
Figure 3.8: C measured by a transverse wave along ?110? with ?110? polarization
direction. The temperature dependence of the C? were studied with magnetic
field parallel and perpendicular to the wave propagation direction.This plot clearly
clearly shows a premartensitic transition around 60?C.
63
Figure 3.9: C44 measured by a transverse wave along [100] with ?110? polarization
direction. The temperature dependence of the C44 were studied with magnetic field
parallel and perpendicular to the wave propagation direction. The temperature
dependence of C44 has a large different between with and without magnetic field.
This plot clearly clearly shows a premartensitic transition around 60?C.
64
All these results clearly show a premartensitic transition around 60?C. There
is almost no change with different orientations of external magnetic field. This
discrepancy with our expectation might be caused by the different compositions
of samples. The properties of Ni2MnGa Heusler alloys are sensitive to the com-
position. The temperature dependence of C44 has a large difference between with
and without magnetic field, as shown in Figure 3.9. The elastic constants could
not satisfied the equation (C11 + C12 + 2C44) = 2C11 ?(C11 ?C12) + 2C44. This
anomaly could not be completely understood. In order to look more into this dis-
crepancy, X-ray diffraction of the high temperature phase of Ni0.49Mn0.234Ga0.276
was studied.
Austenitic Ni2MnGa has a fcc L21 Heusler structure (space group Fm3m, No.
225). If the Mn and Ga sites are not distinguished, the structure can be viewed
as being composed of eight CsCl-type units (aCsCl=afcc/2), with Ni at the cube
center and Mn and Ga at the corners. Three types of Bragg reflections exist: (1)
the order-independent principal bcc reflections with h + k + l = 4n (structure
factor F = 4(2bNi +bMn +bGa) ? 9.66 ? 10?12cm); (2) reflections of type h +
k + l = 4n + 2 (F = 4(2bNi ?bMn ?bGa) ? 6.82?10?12cm) corresponding to a
B2 (Cs-Cl type) structure; (3) the fcc superlattice reflections with h, k, l all odd
(F = 4(bMn ?bGa) ??4.41?10?12cm).
Figure 3.10 shows the 3D view of the temperature dependent X-ray diffraction
spectrums and the temperature dependence of the elastic constant, C44. The in-
tensity of the Xray peaks decrease rapidly from 60?C to 80?C. The X-ray results
indicate the transition at the temperature higher than the martensitic transfor-
mation. Nevertheless, the temperature dependence of the lattice parameter only
shows the ordinary thermal expansion. The nature of the transition is not clear.
65
Figure 3.10: 3D view of the temperature dependent X-ray diffraction < 220 >
spectrum and the temperature dependence of the elastic constant, C44.
This would be an interesting question for further study.
3.2.3 Discussion
The temperature dependence of the elastic constants of both of the samples clearly
shows there is a transition before the martensitic phase transformation.
There is experimental evidence of a number of precursor phenomena when bcc
solids approach a martensitic transformation.[31, 56, 5, 7] These precursors have
been observed by many different experimental techniques such as x-ray, electron,
66
and neutron scattering and ultrasonic measurements among others[61, 55, 56].
In recent years much attention has been give to precursor effects where acoustic
anomalies, phonon dispersion curves, and diffuse elastic scattering in the parent
phase give an indication of the eventual martensitic structure.[69, 41] Some of these
phenomena, such as the observation of an intermediate tweed structure [48, 50]
are not common to all systems transforming martensitically but others, which are
intimately related to the transformation mechanism, have been reported to occur
in all materials investigated so far. This is the case of the anomalous dynamical
response of the lattice to some specific displacements; all bcc austenitic materials
exhibit a low TA2, [110] propagation and [110] polarization, phonon branch which
is accompanied by a low value of the elastic constant C?, (C11-C12)/2.[41] Both the
elastic constant and the phonon branch soften on approaching the transition.
Heusler alloys undergo a martensitic transformation. Precursor phenomena
have been recently reported for this alloys system. Inelastic neutron-scattering
experiments have evidenced a well-defined dip in the TA2 phonon branch close to
q=0.33.[70, 68] The existence of these soft phonon modes gives rise to diffuse x-ray
[18] and electron scattering [70] at temperatures above the martensitic transforma-
tion temperature, MS. When the sample is cooled down, this dip becomes more
pronounced, but the softening is always incomplete and below a given tempera-
ture still above MS the frequency of the soft modes starts to increase again.[70]
The prototype alloy system where similar softening effects have been observed is
Ni-Al [49]; however, the amount of softening measured in Ni2MnGa is larger and
in Ni-Al there is not an increase of the frequency below a given temperature. Also,
measurements by some of the present authors have shown that[33], at the temper-
ature of phonon condensation the thermal expansion and elastic modulus versus
67
temperature curves exhibit a minimum, which is accompanied by an increase in
the internal friction. Moreover a latent heat around 9 J/ mol was measured using
a highly sensitive calorimeter.[46] All these experimental evidences suggest that
the appearance of the mciro-modulated structure is a true phase transition.
The elastic properties in the vicinity of the premartensitic transformation were
investigated by Trivisonno?s group with ultrasonic study.[65] The temperature de-
pendence around the transition temperature of the longitudinal, fast shear and
slow shear are shown in Figure 3.11(a). The martensitic transformation tempera-
ture of this sample is around 260K. In all the modes, the velocity began to decrease
with temperature in a dramatic way before the martensitic transformation. This
behavior is more consistent with the results obtained in the electromagnetic gen-
eration (EMG) study by Vasil?ev et al.[63]. All the ultrasonic studies show that
the velocity decreases dramatically as the intermediate phase transformation is
approached. This decrease in the velocity for all three modes above the transfor-
mation temperature is clearly the signature of an incipient structural instability.
Ultrasonic attenuation measurements provide additional information about the
transformation. Figure 3.11(b) also shows the temperature dependence of the
attenuation for the longitudinal and shear modes. A large increase in the atten-
uation occurs along with a more rapid decrease in velocity as the intermediate
transformation temperature, TI, is approached. Ultrasonic attenuation and veloc-
ity measurements associated with the C? mode clearly are very sensitive probes
of the structural instability, which is not surprising this mode is anomalously low.
Premartensitic transformation effects in the slow shear velocity and attenuation
are clearly observed 20K above the martensitic transformation.
It was shown that Ni-Mn-Ga alloys with specific compositions develop a mod-
68
Figure 3.11: Temperature dependence of the relative velocity (a), VL V44 and V?,
and the relative attenuation (b) associated with the elastic constant [65]
ulated structure [46, 43, 41, 20, 6, 7] at temperatures above the martensitic tran-
sition. This premartensitic transition is found to occur only for alloys with an
electron to atom ration, e/a smaller than 7.60. Figure 3.12 presents the martensitic
transformation, premartensitictransformation, andCurietemperatureofNi2MnGa
alloys vs. concentration of valence electrons (e/a). The crystal we studied, Ni0.50
Mn0.284Ga0.216, Ni0.49Mn0.234Ga0.276 have e/a about 7.636 and 7.366, respectively.
They are either near or below 7.60. We therefore did not anticipate finding a pre-
martensitic transition in it. In addition to the temperature near MS, anomalous
behavior of elastic moduli is expected to occur near the Curie temperature, TC.
Both ultrasonic investigations and neutron scattering studies of polycrystalline Ni-
Mn-Ga alloys[7] indicate that the slope of E(T), the Young?s modulus as function
of temperature, changes near the Curie point.
Our elastic constants study and said previous investigations clearly show that
69
Figure 3.12: Schematic representation of the martensitic transformation, pre-
martensitic transformation, and Curie temperature of Ni2MnGa alloys vs. concen-
tration of valence electrons (e/a). The crystals investigated have concentrations of
e/a = 7.636 and 7.366, which are lower or slightly higher than 7.60. It therefore falls
into an intermediate region, where the premartensitic transformation temperature
has merged with the martensite, which is still lower than the Curie temperature.
70
in certain Ni2MnGa alloys the martensitic transformation is preceded by a pre-
martensitic transition. However the nature of this transition is not completely
understood yet.
3.3 ElasticityofNi2MnGaLowTemperaturePhase
3.3.1 Elastic Constants of Ni0.50Mn0.284Ga0.216 in Low Tem-
perature Phase
ThetemperaturedependenceoftheelasticconstantsinNi0.50Mn0.284Ga0.216 marten-
sitie were studied in the temperature range from 318K down to 200K. Measure-
ments were conducted using the ultrasonic continuous wave method in a 0.8 T
magnetic field.
In a previous section, we reported our elastic constants measurements in the
austenite phase of Ni0.50Mn0.284Ga0.216. The same cuboidal sample, 10mm long
on each side, had been oriented so that four opposing faces have a ?110? normal,
whereas the other two faces have a ?100? normal. The experimental method and
setup were as the same in the high temperature phase study. All measurements
were conducted in a magnetic field, ?H of 0.8T. The magnetic field was applied
at 420K and remained on as the sample was cooled down to 190K. This was
done in an attempt to create single variant conditions in the non-cubic phases[62].
In anticipation of the formation of two different orientations of a field-induced
tetragonal crystal two field orientations were used, [100] and [001] as shown in
Figure 2.3. According to the sample?s orientation and the elastic wave propagation
( ?W) and polarization (?P) directions, there are five modes with ?H along [001] and six
modes with ?H [100] available to measure[39]. For example, as shown in Table 2.2,
71
waves propagating in [110] with [110] polarization directly determine C? with [001]
?H and C* with [100] ?H. The latter is a combination of the usual tetragonal constants
as shown in Table 2.2. For the purpose of checking the experimental data, we have
eleven modes to measure six independent elastic constants; therefore the elastic
constants were over determined. Moreover there are three different elastic wave
mode to measure the same elastic constants, C44, which was shaded in Figure 2.1.
They gave the same result within 2%. These conditions allowed us to check for
consistency in the determination of several of the constants.
The measurements described above, the elastic constants in tetragonal marten-
sitic state, may be summarized as follows. Velocities of eleven elastic waves were
measured and from them the six independent elastic constants of the tetragonal
phase were determined consistently. Figure 3.13 shows the temperature depen-
dence of six of the modes so determined in the temperature range below the 318K
transition. C66 is very near C44 in that temperature range and C33 shows similar
behavior to C11 as expected. All six elastic constants show the abrupt change
around 220 K, clearly indicating that there is a transition from the tetragonal
phase to another phase. It indicates a structural phase change from the tetragonal
(5M) martensite to a second, probably tetragonal, phase (non-layered) at lower
temperature. We have no direct evidence for the nature of the new phase. How-
ever from our data we can give some insight into what it might be. Moreover we
made a Landau model to understand the anomalous behavior of the small elastic
constants in the inte-rmartensitic transformation, which agrees the experimental
data of the small elastic constants, C?, reasonably well.
72
Figure 3.13: Temperature dependence of six elastic constants, C11, C33, C44, C66,
C*and C? of Ni0.50Mn0.284Ga0.216 below the first martensitic transformation tem-
perature. The C* is defined in the Table 2.2. The data points below the inter-
martensitic transformation only show the continuous change of the wave velocities.
The elastic constants notation of the tetragonal, c/a<1 phase, is not suitable any-
more. Since we propose the second martensite to be the tetragonal, c/a>1 phase,
those data points merely show some combination of the true elastic constants. The
propagation and polarization directions used to obtain the data are shown on the
right. The field directions used to create the single-variant states are also shown.
73
3.3.2 Discussion
Figure 3.14 displays a temperature vs. valence electron concentration (e/a) phase
diagram, which we developed on the basis of these results and those of earlier
work[53, 54]. The most important part of this diagram is in the regime of e/a
around 7.636, because it shows that the plausible low temperature phase at this e/a
ratio is the tetragonal phase with c/a greater than 1, while c/a is known to be less
than one in the first tetragonal phase. The elastic constants in the low temperature
phase make this a reasonable picture. To do this we refer to Figure 3.15, where we
show the two smallest elastic constants over the entire temperature range starting
from 380K. In the cubic phase they are degenerate and equal to C?, and below
318K they are split into two by the tetragonal distortion. The upper mode is
the tetragonal C? mode, describing the stiffness of the tetragonal distortion in the
xy plane. The lower, C*, is actually a combination of the five elastic constants.
Physically it represents the stiffness for further tetragonal distortion in the xz or
yz planes. As the 220 K transition is approached, C? again goes soft and becomes
degenerate with C*. If only C? were getting soft in the tetragonal phase, one would
expect the low temperature phase to be orthorhombic. However the fact that the
C* is also very small, indicates a reentrant behavior, that is, a return to the cubic
phase (c/a=1) and then further distortion to a c/a>1 phase at around 220K.
This section presents our study of the temperature dependence of the elastic
constants of martensitic Ni0.50Mn0.284Ga0.216 from 428K down to 200K. All the
elastic constants have been measured as shown in Figure 3.13. Abrupt changes
of all the elastic constants were found at around 220K. These changes indicate a
structural phase change in the martensitic state. According to the previous in-
vestigations and the data of elastic constants, we found there is inter-martensitic
74
Figure 3.14: Schematic representation of structural and magnetic transformations
in NiMnGa alloys vs. concentration of valence electrons (e/a). The different trans-
formation temperatures, namely Curie temperature, Martensitic Transformation,
Cubic to Orthorhombic transformation and inter-Martensitic transformation are
depicted by TC, TM, TI and TInter-M, respectively. The crystal investigated in
this work has a concentration of e/a= 7.636, as marked. The diagram was as-
sembled with the help of the results obtained on three other crystals of similar
composition, whose e/a values are also marked. According to our elastic constant
data, we found the inter-martensitic transformation at around 220K. The tetrag-
onal non-layered phase has c/a>1. The dash-dot line was drawn so to allow the
sample with e/a equal 7.74 to be in the non-layered tetragonal phase at room
temperature consistent with earlier work
75
Figure 3.15: The elastic constants, C and C* are plotted vs. temperature from
380K (above the Curie point) to 200K. Measurements were taken with transverse
waves along [110] with polarization [110]. C? was determined with magnetic field
along [001] (solid square); and C* was determined with H parallel to [100] (open
square). In the austenite, which is at the temperature higher than 318K, the two
modes are degenerate. Notice how the C? mode drops rapidly below 233K, so that
the two modes again degenerate at around 220K
76
transformation in NiMnGa alloys and developed a reasonable schematic phase di-
agram as shown in Figure 3.14. The degeneracy of C* and C? at around 220K, as
depicted in Figure 3.15, indicates the possibility that the tetragonal (5M) marten-
site returns to the cubic-like phase and then further distorts to the tetragonal
(non-layered) phase at lower temperature.
3.4 Model for the Elastic Behaviors Near Inter-
martensitic Transitions
The Heusler alloy Ni2MnGa is a class of materials which undergoes martensitic
transitions at temperatures which vary from 220K for stoichiometric samples to
room temperature and above as the stoichiometry changes.[54] The alloys are fer-
romagnetic at room temperature; TC is around 380K and varies only slightly as
the composition changes.[61] Many of these alloys display a shape memory effect
which can be activated by the application of a magnetic field. This property,
called magnetic shape memory, has motivated a considerable amount of research
into the physical properties of these materials. Much of this work has been cen-
tered on the cubic to tetragonal or orthorhombic transitions. Lately, however, there
have been reports of intermartensitic transformations[54, 53]. Transitions between
different martensitic states have been observed in Ni50Mn28.4Ga21.6 using elastic
constants measurements [16], and in Ni48.8Mn29.7Ga21.5 using X-ray and DSC mea-
surements [53]. The elastic constant measurements are particularly interesting.
Transitions between different martensitic states of Ni50Mn28.4Ga21.6 have been
observed using elastic constants measurements [16], and in Ni48.8Mn29.7Ga21.5 us-
ing X-ray, DSC measurements [53]. The constant, C?, which softened in the
77
austenitic phases, split into two at the MS, 318K, temperature. One of the two
new shear modules stiffens before softening again at the lower transformation MS2,
220K. The other modulus was shown to remain very soft, around 5GPa, from MS
to MS2. The standard Landau theory of structural transitions, where strain is the
order parameter, assumes that the appropriate elastic modulus softens to zero as
a function of temperature while the coefficients of the higher order terms in the
expression of the free energy are independent of temperature. This model cannot
describe the observed re-entrant behavior of C?. Here we assume that the coeffi-
cient of the third-order term also has significant temperature dependence. This
assumption results in a C? vs. temperature in good agreement with observation.
The temperature behavior of the other soft modulus, C*, is not as well described in
this model. The model and possible modifications to it are discussed and compared
to the elastic constant data.
The standard Landau theory of structural transitions, where strain is the or-
der parameter, assumes that the appropriate elastic modulus softens to zero as
a function of temperature while the coefficients of the higher order terms in the
expression of the free energy are independent of temperature. Here, we assume
that the third-order term also varies with temperature significantly. In terms of
the strain order parameters the free energy function has the general form:
? = 12(C11+2C12)?21+Cprime(?22+?23)+14C44(?24+?25+?26)+13b?3(?23?3?22)+14d(?22+?23)2
(3.2)
where ?1 = (?xx +?yy +?zz)/?3, ?2 = (?xx ??yy)/?2
?3 = (2?zz ??yy ??xx), ?4 = ?xy,?5 = ?yz,?6 = ?zx
Cprime = (C11 ?C12)/2, b = (C111 ?C112 +C123)/6?6
d = (C1111 +C1112 ?3C1122 ?8C1123)/48.
78
C? is taken to depend linearly on temperature, as usual. However, we will also
take b to be temperature dependent in what follows. For the purpose of this study
the significant terms of the free energy in Equation (3.2) are:
? = Cprime(?22 +?23) + 13b?3(?23 ?3?22) + 14d(?22 +?23)2 (3.3)
The first and third terms of the Equation (3.2) were ignored, because C11+ 2C12
and C44 are much bigger than C? in either phases, austenite or martensite.
It is convenient to introduce polar coordinates for ?2 and ?3. Take ?2 =
?sin?,?3 = ?cos?. (cos3? = cos3? ? 3sin2?cos?), ? > 0. If bnegationslash=0, the low en-
ergy tetragonal states are those that minimize b?3cos3?. For b < 0, the choice
of ? = 0, 2pi3 4pi3 ,... gives the lowest free energy. When ? negationslash= 0 , this corresponds to
a tetragonal state with c/a < 1. For b > 0, another set, ? = pi3pi, 5pi3 ,..., would
result in the lowest free energy and correspond to c/a > 1. The possible states
are schematically shown in Figure 3.16 and Figure 3.17. Either sign of b gives the
same result for the effective modulus in the martensite.
For b< 0, we focus on the ? = 0 solution. Then the Equation (3.3) simplifies
as:
?(?)??0 = Cprime?2 ? 13b?3 + 14d?4 (3.4)
We take d>0 and rescale ? by dividing it by d, therefore b ? bd, Cprime ? Cprimed . Further
minimization with respect to ? gives the stable solutions:
?
???
?
???
?
?0 = 0 cubic phase, Cprime > 19b2
?+ = b2c +
radicalBiggparenleftbigg
b2cparenrightbigg2 ?2Cprime tetragonal phase, Cprime < 19b2 (3.5)
There is a first order transition, ?0 to ?+ at the Cprimeboundary. The elastic constant,
C?, at the cubic to tetragonal transition boundary was calculated as:
Cprimeboundary = 19b2 (3.6)
79
Figure 3.16: A Strain-space map showing the possible phase equilibrium and dis-
tortions from one of them. The radius of the dotted circle is the value of ?, the C?
and C* are the moduli corresponding to orthorhombic and tetragonal distortions,
respectively. ? is the angle between the structure?s position and ?3 axis. If ? equals
zero, C? and C* correspond to the deformation along ?2 and ?3, respectively. The
center shows the free energy contour with respect to different sign of b.
80
Figure 3.17: Shows at different sets of ? value, there are different types of tetragonal
with respect to the c/a ratio.
81
Therefore the tetragonal state has the same energy as the cubic phase at Cprime = b29 ,
where ?+ = b2 +
radicalbigg
(b2)2 ?2Cprime = 23b.
For b > 0, the energy favorable angle is: ? = pi3,pi, 5pi3 ,.... The equivalent result
for b < 0 is:
?
???
?
???
?
?0 = 0 cubic phase, Cprime > 19b2
?+ = b2c +
radicalBiggparenleftbigg
b2cparenrightbigg2 ?2Cprime tetragonal phase, Cprime < 19b2 (3.7)
To find expressions for the two lowest moduli in the tetragonal phase, we expand
the free energy around ? = ?+, ? = 0, to second order in ?2 and ?3 and obtain the
energy formula:
? = ?(?+) + 12Cprimetet(?2 ?0)2 + 12C?(?3 ??+)2 (3.8)
Substitute ?3 = ?+ +??3 into Equation (3.3):
? = Cprime
parenleftBig
?22 + (?+ +??3)2
parenrightBig
?13b(?+ +??3)
parenleftBig
(?+ +??3)2 ?3?22
parenrightBig
+ 14d
parenleftBig
?22 + (?+ +??3)2
parenrightBig2
(3.9)
Assuming ????
3
= 0 and neglecting the terms higher than the 3th order. Equa-
tion (3.9) is simplified to
???0 = Cprime?22 +b?+?22 + 12c?2+?22 +Cprime (??3)2 ?b?+ (??3)2 + 32d?2+ (??3)2 (3.10)
The two shear elastic constants are then found to be:
(Cprime)Tet = 3b2
?
??b
2 +
radicaltpradicalvertex
radicalvertexradicalbtparenleftBiggb
2
parenrightBigg2
?2Cprime
?
?? (3.11)
C? = ?2Cprime + b2
?
??b
2 +
radicaltpradicalvertex
radicalvertexradicalbtparenleftBiggb
2
parenrightBigg2
?2Cprime
?
?? (3.12)
82
From our previous discussion that C? at cubic-tetragonal boundary is equal to b29 .
Therefore the C? in tetragonal around the cubic to tetragonal transition boundary
is:
(Cprime)Tet = 3b2
?
??b
2 +
radicaltpradicalvertex
radicalvertexradicalbtparenleftBiggb
2
parenrightBigg2
?2Cprime
?
?? = b2 (3.13)
C* is the elastic constant for ??3. Therefore at cubic phase C*=C?cub, which equals
b29 at the cubic-tetragonal boundary approached from the cubic side. The value of
C* at the boundary from the tetragonal side could be calculated as:
(C?)Tet = ?2Cprime + b2
?
??b
2 +
radicaltpradicalvertex
radicalvertexradicalbtparenleftBiggb
2
parenrightBigg2
?2Cprime
?
?? = 1
9b
2 (3.14)
Therefore the elastic constant, C?, has a nine times jump at the cubic-tetragonal
boundary. This phenomenon was clearly experimentally proved in Figure 3.18.
The calculation also shown that the C? and C* are degenerate at the boundary as
shown experimentally.
This model also yields a C?tet vs. temperature dependence that is in good
agreement with observations. The theory was compared to the experimental data
by choosing suitable coefficients b = b0T ?225225 and Cprime = Cprime0T ?300300 ,where b0 was
chosen to set the overall scale to coincide with the data and C?0 was chosen to fit
C? in the austenitic phase. The choices will not affect the shape of the curve. The
experimental results and the model results of C? are depicted in Figure 3.18. In the
first tetragonal phase, C? rises with decreasing temperature, reaches a maximum,
then decreases to zero at the second transition. Moreover C?tet increases by a factor
of nine at the first martensitic transformation, which agrees with the experimental
result very well. The temperature behavior of the other soft modulus, C*, is not
as well described in this model. It remains positive. We have studied the effects
of an applied magnetic field on both moduli, showing that C? is not altered, while
83
Figure 3.18: This graph shows that the result of the model fit the experimental
data reasonably well. The elastic constant C has a nine times jump at the cubic-
tetragonal boundary, which was experimentally proved. The calculation also shows
that the C and C* degenerate at the boundary, which also was observed in the
experiments. Result of modeling reproduces the major aspects of the temperature
dependence of the shear elastic constant C.
84
C* is softened by its presence, due to magneto elastic coupling. In the followed
section, the possible modifications to it will be discussed to find the modified term
of C?, which make the model fit the experiment data better.
According to the discussion above, the temperature dependence of effect mod-
ulus, C?, in the model with only considered the elastic energy was not fitted with
the experimental data well. Therefore in order to correct the calculation of C?, we
took into account of the magnetic field effect. The free energy of the system with
considering the magnetic field effects could be expressed as following:
?(??) = ?(?) + 12?bardbl?(??)2 + 12????2 + ?me + 12??1 (?Mz)2 (3.15)
where ?me = ?BM3 (?+??)
parenleftBig
1? ?22
parenrightBig
= ?2BNMs?Mz (?+??)
Mz = Ms ??Mz, M3 = N
parenleftBig
M2z ?M2x ?M2y
parenrightBig
, M2z = M2s ?2Ms?Mz
?3 = ?+??, ? = 0 +?, cos? = 1? ?
2
2 (3.16)
Substitute Equation (3.16) into Equation (3.15) with adding a stress, ?, for the
strain, ?, yields free energy ? = ?(??)??(?) as following:
? = 12?bardbl?(??)2 + 12??1 (?Mz)2 ?2BNMs?Mz???? ??? (3.17)
?
???
?
???
?
??
?(?Mz) = ?
?1 (?Mz)?2BNMs?? = 0 (a)
??
?(??) = ?bardbl????2BNMs?Mz ?? = 0 (b)
(3.18)
From the Equation (3.18) (a) the ?Mz could be obtained as the following expres-
sion:
?Mz = 2BNMs????1 (3.19)
Substituting Equation (3.19) into Equation (3.18) (b) yields the expression for ??
as:
?bardbl???? 4B
2N2M2
s ???
??1 ?? = 0 ? ?? =
?
?bardbl?? 4(BNMs)
2
??1
(3.20)
85
Since the elastic constant, C?, is the stiffness according to the deformation ??.
Therefore the C? with magnetic field effect could be represented as:
C? = ?bardbl?? 4B
2N2M2
s
??1 = ?2C
prime + b
2
?
??b
2 +
radicaltpradicalvertex
radicalvertexradicalbtparenleftBiggb
2
parenrightBigg2
?2Cprime
?
??? 4B2N2M2s
??1 (3.21)
The calculation above results in the corrected effective elastic constant, C?, with a
correction term, 4B
2N2M2
s??1 . This term decreases the calculated C? in the model,
so that the theoretic prediction agrees more closely with the experimental results.
3.5 Conclusion
In this chapter, the elasticity of ferromagnetic shape memory alloys, Ni2MnGa,
was studied in the austenitic and martensitic phases. Elasticity was investigated by
the ultrasonic continuous wave method. According to the temperature dependence
of elastic constants in Ni2MnGa, the martensitic transformation, intermartensitic
transition and premartensitic transitions were observed. A Landau model was de-
signed to understand the reentry phenomenon of the small elastic constants at the
intermartensitic transition. The model agrees our experimental results reasonably
well.
86
Chapter 4
Elasticity of Fe3Pd
In this chapter, the results of the magnetic field and temperature dependence of
elastic constant in Fe3Pd were studied. A magnetic field induced transformation
was found by the study of the magnetic field dependence of elastic constants. It is
possible that this phenomenon relates to the magnetic filed induced reversible ac-
tuation [36]. A premartensitic transformation was investigated by the temperature
dependence of the elastic constants.
4.1 Sample Information
In this section the ferromagnetic shape memory alloy, Fe3Pd was grown Bridgman
method by Dr. Cui [12]. Both x-ray analysis and DSC measurements show that
the FCC-FCT transformation is a weak first order thermoelastic transition. The
average lattice parameters are a=3.822 ?A and c=3.6298 ?A for the FCT martensite,
and a0=3.7557 ?A for the cubic austenite. The latent heat is 10.79?11 J/cm3. The
Curie temperature is 300?C. The saturation magnetization is ms=1217 emu/cm3
for the martensite and ms=1081 emu/cm3 for the austenite in Fe3Pd alloy; the easy
87
axis of the austenitic Fe3Pd is [111], where in martensitic Fe3Pd [001] evidently is
the hard axis and [100] and [010] are equally to be the easy axis, which is inverse
with Ni2MnGa. The magnetic anisotropy is -4.8?103 erg/cm3 for the austenite at
60?C, and it is 3.46?105 erg/cm3 for the martensite at -20?C. [12] The maximum
ferromagnetic strain in this material is about 0.9%. The blocking stress is about
-4 MPa. Furthermore, stress has large effect on the phase transformation temper-
ature, for the compressive stress, it is 0.7?C/MPa. [12] The phase transformation
temperature can also be changed by applying a magnetic field during cooling or
heating. The cuboidal sample with 10mm on each side had been oriented so that
six opposing faces have a ?100? normal. In order to measure all the elastic con-
stants with different wave modes, two corners, as shown in Figure 4.1 by dot line,
of the sample were cut to obtain two ?110? normal faces. The sample?s orientation
is shown in Figure 4.1.
4.2 Results and Discussion
All the elastic constants of the Fe3Pd were measured by the ultrasonic continuous
wave technique. The detail of this method and all the elastic wave modes were
discussed in Charpter 2.
In order to study the temperature and magnetic field dependence of elastic
constants, two types of measurements were designed for the study of the elasticity
in Fe3Pd. First is to measure the elastic constants by a various temperature range
from room temperature to 80?C. The other type of measurement is at 70?C with
applying magnetic field from 0 to 10000 Oe.
First, the temperature dependence of the elastic constants was studied. Fig-
ure 4.2 shows the resonance frequencies of longitudinal wave along < 100 >, which
88
Figure 4.1: Sample orientation of FePd30?0.5at%
89
yields C11. The peaks shows significant difference around 45?C. The results of C11,
converting from these spectrums, were depicted in Figure 4.3.
The temperature dependence of C?, which is conducted by the shear wave with
[110] polarization and [110] propagation direction was also studied. Figure 4.4
shows the spectrums with different temperatures. The peaks lost the periodic
resonance frequencies around 45?C. And the spectrums shows the tendency to
getting clearer with increasing temperature higher than 70?C. Even the peaks
were hard to derive the elastic constant, C?, it clearly shows a transition around
45?C, which is consistent with the result of the temperature dependence of C11.
The temperature dependence of C11 and C? clearly shows a transformation
around 45?C. The elastic constants results indicate probably a premartensitic
transformation prior to the martensitic transformation.
The magnetic field dependence of the elastic constants was also studied. All
the elastic constants were measured in 70?C, which is surely in the austenitic
state, with applying magnetic field from 0 to 1T. All the elastic constants showed
a magnetic field induced transformation. This transition might be the magnetic
field induced reverse transition as the previous discussion. Moreover, the result
shows reverse hysteresis of the C11.
Figure 4.5 listed the spectrums of [001] longitudinal wave at 70?C with applying
different magnetic field. The data was converted into elastic constant, C11, which
was depicted in Figure 4.6.
The magnetic field dependence of the shear wave with [110] polarization and
[110] propagation direction was also depicted in Figure 4.4, which shows the spec-
trums with different magnetic field. It also shows that the peaks of the resonance
frequencies were hard to analyze when the field is larger than 0.25T. Even it is
90
Figure 4.2: The wave spectrum of longitudinal elastic wave along [001] crystalline
direction, which is for conducting the elastic constant, C11, in ferromagnetic shape
memory alloy, Fe3Pd. The spectrums show significant changes around 45?C
91
Figure 4.3: Magnetic field dependence of Elastic constant, C11, in ferromagnetic
shape memory alloy, Fe3Pd.
92
Figure 4.4: The wave spectrum of the temperature dependence of the shear elastic
wave with polarization direction along [110] and propagation direction along [110],
which is for conducting the elastic constant, C?, in ferromagnetic shape memory
alloy, Fe3Pd. The spectrums show significant changes around 45?C
93
Figure 4.5: The wave spectrums of longitudinal elastic wave along [001], which is
for conducting the elastic constant, C11 at 70?C with various magnetic field from
0 to 1T. The spectrums show significant changes around 0.25T
94
Figure 4.6: Magnetic field dependence of elastic constant, C11, in ferromagnetic
shape memory alloy, Fe3Pd.
95
hard to directly show the magnetic field dependence of C?, the spectrum indicate a
magnetic field induced transition around 0.25T and it indicate the tendency of the
change of C?. Another proof to confirm this field induced transition is the magnetic
field dependence of shear wave propagating along [001] with [110] polarization di-
rection, which correspond to the elastic constant, C44. The spectrums were listed
in Figure 4.8. The result is consistent with the magnetic field dependence of C11
and C?, which indicates the magnetic field inducing transition.
The results of the magnetic field dependence of the elastic constants clearly
show there is a transition with an external field around 0.25 T.
4.3 Conclusion
This section studied the elasticity of ferromagnetic shape memory alloy, Fe3Pd.
All the elastic constants in its austenitic state were measured by the ultrasonic
continuous wave method with external temperature and magnetic field. The mag-
netic field and temperature effects of the elastic constants were investigated and
discussed. The temperature dependence of the elastic constants shows a proba-
bly premartensitic transformation around 50?C. And the magnetic field inducing
a transformation was observed in the elastic constants results.
96
Figure 4.7: The wave spectrum of the magnetic field dependence of the shear elastic
wave with polarization direction along [110] and propagation direction along [110],
which is for conducting the elastic constant, C?. The spectrums show significant
changes with an external magnetic field around 0.25T
97
Figure 4.8: The wave spectrum of the magnetic field dependence of the shear elastic
wave with polarization direction along [110] and propagation direction along [100],
which is for conducting the elastic constant, C44. The spectrums show significant
changes with a external magnetic field around 0.25T
98
Chapter 5
Summary
Ferromagnetic shape memory alloys (FSMA) have been scientifically interesting
due to the possibility of inducing the shape memory effect with an external ap-
plied magnetic field. They can produce very large repeatable strains and can
produce large work output against smaller applied loads. The loads and magnetic
field can be both used to bias the microstructure. The variants rearrangement
phenomenon is a complicated one that depends on twin boundary mobility, the
magnetic properties of the materials, proper biasing of the initial microstructure,
proper specimen orientation, and specimen shape, among others.
The main purpose of this thesis study was centered on the elasticity study,
which could provide insight into the nature of the materials. Since the elastic
constants are very sensitive, a good experimental method was chosen as ultrasonic
continuous wave method. An automatic control system was design and set up for
all these elastic constants measurements with capability of applying magnetic field
and wide temperature range. The results of all the elastic constants of FSMAs,
Ni2MnGa and Fe3Pd, were reported and discussed in Chapter 3 and Chapter 4,
which were briefly concluded as:
99
1. Thedependenceofasolid?selasticpropertiesontemperatureofNi0.5Mn0.284Ga0.216,
Ni0.49Mn0.234Ga0.276 and Fe3Pd were studied. Measurements were conducted
by the ultrasonic continuous wave method.
2. The temperature dependences of the elastic constants of the austenitic state
in these two Ni2MnGa were studied. Anomalous behavior of the elastic
constants temperature dependence in austenitic Ni0.5Mn0.284Ga0.216 was ob-
served, especially an abrupt 15% softening of C11 at the Curie temperature.
The latter anomaly was found to be strongly influenced by the presence
and orientation of applied magnetic fields. This anomalous phenomenon
indicates a premartensitic transition. Moreover in the austenitic phase of
Ni0.49Mn0.234Ga0.276, the elastic constants also showed a transition before
martensitic phase transformation.
3. The temperature dependence of the martensitic phase of Ni2MnGa was in-
vestigated. The key to conduct those ultrasonic continuous wave technique
measurements in tetragonal martensite is using the magnetic field to bias
the crystal to be single variant. In martensitic Ni0.5Mn0.284Ga0.216, the tem-
perature dependences of the velocities of all eleven elastic wave modes had
abrupt changes at the temperature far below the martensitic phase transfor-
mation. The results indicate a structural phase change from the tetragonal to
a second phase at lower temperature. It leads to propose an intermartensitic
phase transformation.
4. The C? constant, which softened in the austenitic phase, splits into two at
the MS temperature. One of the two new shear moduli stiffens before soft-
ening again at the lower transformation MS2. The other modulus, C*, was
100
shown to remain very soft, from MS to MS2. The standard Landau theory
of structural transitions, where strain is the order parameter, assumes that
the appropriate elastic modulus softens to zero as a function of temperature
while the coefficients of the higher order terms in the expression of the free
energy are independent of temperature. This model cannot describe the ob-
served re-entrant behavior of C?. Here we assume that the coefficient of the
third-order term also has significant temperature dependence. This assump-
tion results in a C? vs. temperature in good agreement with observation. The
temperature behavior of the other soft modulus, C*, is not as well described
in this model.
5. The temperature and magnetic field effects in Fe3Pd by the elastic constants
measurements were investigate by the ultrasonic continuous wave method.
The abrupt change of the elastic constants at around 45OC indicate a possi-
ble premartensitic transformation. The magnetic field dependence of elastic
constants shows a probably magnetic field induced transition.
101
Appendix A
Publications
1 Liyang Dai, James Cullen, and Manfred Wuttig, ?Model for the Elastic Be-
havior Near Intermartensitic Transitions?, J. Appl. Phys., Accepted. (2004)
2 Liyang Dai, James Cullen, and Manfred Wuttig, ?Intermartensitic Trans-
formation in a NiMnGa Alloy?, J. Appl. Phys., Vol95, no.11, 6957-9 (June
2004)
3 Liyang Dai, James Cullen, Jun Cui, Manfred Wuttig ?Elasticity study in
Ferromagnetic shape memory alloys?, Mat. Res. Soc. Symp. Proc., 785,
D2.2.1(invited paper)(2004)
4 Liyang Dai, J. Cui and M. Wuttig, ?Elasticity of austenitic and martensitic
NiMnGa,? in Proceedings of SPIE Vol. 5053 Smart Structures and Materi-
als 2003: Active Materials: Behavior and Mechanics, edited by Dimitris C.
Lagoudas, (SPIE, Bellingham, WA, 2003) 595-602
5 Liyang Dai, J. Cullen and M. Wuttig, ?Magnetism, Elasticity and Magne-
tostriction of FeCoGa Alloys?, J. Appl. Phys., 93, no.10, 8627-9 (2003)
102
6 M. Wuttig, Liyang Dai, J. Cullen, Elasticity and Magnetoelasticity of Fe-Ga
Solid Solutions. Applied Physics Letters, 80, no.7, 1135-7 (Feb. 2002.)
103
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