MODE SELECTIVE ACOUSTIC SPECTROSCOPY OF TRIGONAL CRYSTALS
by
Carlos F. Martino
B.S., University of Houston, 2004
A thesis submitted to the
University of Colorado at Denver
in partial fulfillment
of the requirements for the degree of
Master of Science
Mechanical Engineering
2004
hi
5
This thesis for the Master of Science
degree by
Carlos F. Martino
has been approved
by
Mohsen Tadi
Date
Martino, Carlos F. (M.S., Mechanical Engineering)
Mode Selective Acoustic Spectroscopy of Trigonal Crystals
Thesis directed by Professor John Trapp
ABSTRACT
The project developed a noncontacting electromagnetic-acoustic resonance technique
to determine vibrational mode symmetry. The electric field induces stresses on the
surface of the specimen without mechanical contact. Different geometrical configu-
rations of capacitor plates were constructed with a prescribed symmetry belonging
a particular irreducible representation of the D3<2 point group. Vibrational mode
selectivity can be achieved by changing the symmetry of the field. Mode selective
technique gives an advantage in identifying vibrational modes of measured spectra
and in the analysis of the Ritz method.
This abstract accurately represents the content of the candidates thesis. I rec-
ommend its publication.
Signed
ni
CONTENTS
Figures ................................................................... vi
Tables..................................................................... vii
Chapter
1. Introduction................................................................ 1
2. Ritz Method................................................................. 4
2.1 Variational Formulation................................................... 4
2.2 Cartesian Coordinates..................................................... 5
2.3 Cylindrical Coordinates .................................................. 6
3. Inverse Algorith........................................................... 10
4. Limitations of the Inverse Algorith ....................................... 13
5. Theoretical Tools.......................................................... 15
5.1 Group Theory............................................................. 15
5.1.1 Elements and Representations........................................... 16
5.1.2 Reducibility of Representations........................................ 16
5.1.3 Character.............................................................. 17
5.1.4 D3d Point Group........................................................ 18
5.2 Piezoelectric Constitutive Relation...................................... 20
5.3 Differential Forms....................................................... 21
5.3.1 Tangent Space ......................................................... 22
5.3.2 First Pause, Explanations.............................................. 22
IV
5.3.3 The Differential of a Map........................................... 23
5.3.4 Second Pause, Explanations.......................................... 23
5.3.5 k-Forms and Cotangent Space......................................... 24
5.3.6 Third Pause, Explanations........................................... 25
5.3.7 Exterior Differentiation and Pull-Back.............................. 26
5.3.8 Example............................................................ 27
6. Transducers and Electronics............................................. 29
7. Symmetry Analysis....................................................... 31
8. Measurements............................................................ 35
8.1 Mode-selective Measurements........................................... 37
9. Data Analysis........................................................... 40
10. Conclusion............................................................. 42
References................................................................. 43
v
FIGURES
Figure
5.1 Symmetry and definitions of axes for an object belonging to the D^d point
group................................................................... 18
6.1 Crystal inside cylinder surrounded by capacitor plates .................... 29
6.2 Four configurations for generating A2g and Aiu modes .................... 30
8.1 Resonant spectra for configuration a....................................... 38
8.2 Resonant spectra for configuration b ...................................... 38
8.3 Resonant spectra for configuration c....................................... 39
8.4 Resonant spectra for configuration d,...................................... 39
vi
TABLES
Table
3.1 Calculated resonance frequencies f^i by Ritz algorithm in cylindrical co-
ordinates and mode symmetry................................................ 11
3.2 Calculated resonance frequencies /d* by Ritz algorithm in cylindrical coor-
dinates and measured resonance frequencies by mode selective fms method 12
5.1 Character table for D3d point group...................................... 19
7.1 Symmetry analysis for distinct geometrical configurations of electric po-
tential ................................................................... 34
8.1 Mode symmetry within D3d subset, resonance frequencies /m; calculated
by the Ritz algorithm; resonance frequencies measured by RUS fnus, and
mode selective /ms methods........................................... 36
vii
1. Introduction
Resonant ultrasound spectroscopy (RUS) is a highly recognized technique used to
determine the elastic constants of crystals. However, as with the other conventional
techniques for measuring elastic constants, RUS is in some respects limited in accu-
racy. One limitation stems from the fact that the contacting transducers used in RUS
to excite resonant modes introduce shifts in resonant frequencies due to mechanical
contact. Two other factors which we will consider are related to the Ritz method, an
iterative inverse algorithm that is frequently used to extract elastic constants from
measured spectra. In the inverse method, a set of elastic constants close to the real
values have to be known beforehand in order for the algorithm to converge. This
imposes a limitation a priori; elastic constants for novel material may not be known
with sufficient accuracy and the Ritz method fails. Also, as pointed out by Ogi [11]
in experiments conducted by Maynard, switched assignments of frequencies may oc-
cur during the iteration calculation for finding the best fit which breaks the one to
one correspondence needed of measured and calculated resonance frequencies. As a
result, the inverse algorithm may converge to a false minimum and the outcome is an
incorrect set of elastic constants. The latter limitation is addressed in section 4. The
present study addresses these issues by developing noncontacting techniques to ex-
cite and detect vibrational modes with a prescribed symmetry. The research includes
group theoretical analysis of the symmetry of vibrational modes that are excited with
several transducer configurations.
The crystal under study is quartz in cylindrical geometry with the trigonal axis
1
oriented along the cylinder axis. The specimen is inserted in a hollow cylinder sur-
rounded by capacitor plates where an electric field is applied. The geometry of the
capacitor plates and electric field maintain all the elements of the D3d point group
of the crystal. Stresses are created on the sample as the capacitor plates are driven
by tone-burst excitation. The electromagnetic field couples to the crystal structure
through the piezoelectric coefficients. The same plates are used as a receiver. Modes
are excited which are related to the particular symmetry of the electric field. Since the
geometrical configuration of the electric field is easily changed, vibrational modes be-
longing to a specific group of vibrations can be generated and detected. The purpose
of this study is to demonstrate mode-selective noncontacting transduction methods
and to demonstrate the advantage of our technique for Ritz analysis.
The thesis is organized as follows. The chapter 2 introduces the basic ideas
of Ritz method. The purpose of this section is to present the eigenvalue equation
in matrix form Eqn. 2.11. This equation is derived substituting the approximating
functions Eqn. 2.10 into Hamiltons final variational form in cylindrical coordinates
Eqn. 2.9. The topic of chapters 3 and 4 are the inverse algorithm and its limitations
for determining elastic constants.
Chapter 5 is concerned with theoretical tools used in the analysis of mode sym-
metry in chatper 7. Group representation and applications to crystal structure is
presented in chapter 5.1 through 5.1.4. For clarity and consistent presentation, the
basic equation of piezoelectricity is given in section 5.2. Differential forms, the main
tool in studying the symmetry properties of electric field and strain field, is out-
lined in section 5.3. Pauses have been added in this section in order to facilitate the
understanding of the new terminology.
2
The method of constructing different configurations is described in chapter 6. A
significant contribution to the thesis is the symmetry analysis in chapter 7. In this
section, the tools of differential geometry and group theory are used to determine
the symmetry of electric field and vibrational modes excited with distinct transducer
configurations.
In the chapter. 8, the measurements are presented. The resonance frequencies
measured with RUS technique is shown in table 8.1. The technique of generating and
detecting groups of vibrations belonging to an irreducible representation is described
in Sec. 8.1.
The topic of chapter. 9 is data analysis. In this section, we discuss the method
for determining the modes entered in table 8.1. The possible advantage of the mode-
selective technique is briefly mentioned. In chapter. 10, a summary of the study is
presented.
3
2. Ritz Method
Resonant ultrasonic spectroscopy (RUS) is based on the measurement of vibra-
tional modes of samples. The most common configuration uses a parallelepiped which
is lightly sandwiched between two piezoelectric transducers. One transducer generates
a continuous-wave (cw) driving force and the other detects ultrasonic oscillations. The
frequency of the driving transducer is stepped through a wide range, which includes
resonant vibrational modes of the sample. The response of the sample is detected
by the other transducer. As the driving frequency approaches an eigenfrequency of
the sample, a large response is detected. The measured resonant frequencies are then
used in a variational calculation incorporated in a Levenberg-Marquardt iterative
minimization routine to find the complete set of elastic constants.
2.1 Variational Formulation
Hamiltons variational formulation is presented in this section. Hamiltons prin-
ciple yields the final variational form of the equations of motion. In the Ritz method,
approximate solutions are sought for this equation in the form of power series.
Hamiltons principle for a linear elastic medium free of body forces and ignoring
piezoelectric effects [10] is given by [5]
solid, tk are the components of the specified surface traction, S is the variation, Uj is
the displacement in the j-th direction, and the overdot represents differentiation with
(2.1)
where t is time, V and S are the volume and surface occupied and bounding the
4
respect to time. The strain energy, represented by UQ, is given by
Uq -^Cijklijekl, (2-2)
where the components of the elastic stiffness tensor are denoted by Cijki and e^i are
the components of the strain tensor. Summation over repeated indices is implied.
2.2 Cartesian Coordinates
The relationship between stress and strain is
O'ij Gijkl^kl (2-3)
The constitutive relation (2.3) for the higher trigonal crystal classes reduces in rec-
tangular coordinates to
Cn C\2 C\2 C\4 0 0 Â£1
C\2 C22 C23 C24 0 0 e2
03 G13 C23 C33 0 0 0 Â£3
04 C14 C24 0 C44 0 0 Â£4
05 to 10 0 0 0 0 Â£5
V6 0000 c56c66 e6
where
(2.4)
C22 = C11 (2.5a)
C23 = C13 (2.5b)
C24 C14 (2.5c)
C55 = C44 (2.5d)
C56 = C14 (2.5e)
G&6 =-(Cn ~ C12) (2.5f)
5
The conventional contracted notation [6] (Cmi = Cn, C1234 = Cu, etc.) is used here.
2.3 Cylindrical Coordinates
The strain components in cylindrical coordinates are given by [6]
dur 61 6rr Qr > (2.6a)
dug Ur Â£2 Â£96 + > 06 r (2.6b)
du2 3 = Qz (2.6c)
dug 1 duz 4 lgz r, ) dz r dO (2.6d)
duz dur e5 2erz + 0 dr dz (2.6e)
1 dur dug 6 66 2 erg + Q r dO dr r (2.6f)
The constitutive equation (2.3) must be written in cylindrical coordinates which
involves transforming the components of the stiffness tensor. The components of the
elastic stiffness tensor in cylindrical coordinates, denoted with a bar, are given in
terms of C{j by
C14 C56 C14 cos 30, (2.7a)
C15 = C25 = C46 Ci4sin3# (2.7b)
C24 = C14. (2.7c)
6
The other Qj are unchanged by the transformation. The matrix representation of
the constitutive equation in cylindrical coordinates becomes
' ' Cn Cn Ci 3 C14 C15 0 err
<796 C12 C22 C23 C24 C25 0 09
& zz Ci 3 C'23 C33 000 zz
< > = < >
&9z Cu C24 0 C44 0 C46 C0z
GrQ \ > 0 0 C46 C56 Cqq r$ \ /
The insertion of the strain-displacement and constitutive equation into Hamiltons
principle yields the final variational form [9]:
u
d6ur
dr
+
+
+
+
+
+
+
+
+
i duz dur
Cl5[~fr + 97
85 ur
dr
+
Cl * + ci2('1^? + -
dr
r 86
du.
1 d5ug 5ur
r 86 ^ r
n dlLz .u r {du<> 1 duA r (dUz .
c23-^- + c24^+ -j+C25 \^ + dz
dur (1 dug uT \ duz
85uz
~dz
^13-3---f C23 I H------) + C33-Z
dr \r 86 r J dz
= dur /1 dug Ur\ fdug 1 duz
+ C24 + Â¥) + Caa VÂ¥7 + 7~m
1 d5ug 5ur
r 80 ^ r
{ dSug 1 85uz
V dz +r~dO
( dSug 1 d5uz
. 1 dur dug ug
Ci6 1 r~dO + IT Â¥
c15^+c25 ag+]+Cu
\ dz r 80
duz dur\ / d5uz d5ur
dr ^ dz ) \ dr dz
. 1 dur dug Ug
56 r dO dr r
(d5uz d5uT
\ dr ^
duz
dz
r, 1 dug 1 duz\ (duz dur
CA6[~d7+r~dOj+C56
. 1 dur dug Ug
66 r dO dr r
1 d5ur dSug
5ut
r dO
dr
1 d5ur ^ dSug 5ug ^
r dO
dr
pui2(urSur + ug5ug + uz5uz)}rdrdOdz 0.
(2.9)
This equation is then solved using approximation functions for the displacements.
The approximation functions in cylindrical coordinates used by Heylinger and John-
7
son are of the form
ur(r, 6, z) = ^2 acanj3ra cos (n6)zp
ot,n,j3
+ Â£ sin(m0)zT (2.10a)
K}m, 7
Ug{r, 0,z)=]T bcan/3ra cos(n0)zP
a,n,0
+ Â£ fC,7r',sin(m0)z1' (2.10b)
K,m,7
uz(r, M) = X] cn/3r cos(n0)^
a,n,/3
+ Cv/Ksin(m0)27. (2.10c)
K,m,j
Substitution of the approximation functions into the final variational form of Hamil-
tons principle yields the matrix equation:
[Mn] [0] [0] w [A11] [A12] [A13] '{a}' / {0}
[0] [M22] [0] < {6} > w2 + [A21] [A22] [A23] < > = < {0}
[] [0] [M33] {d} \ > [K31] ^32] ^33] w. {0} V. /
where
/ pSjtfjdV, (2.12a)
Jv
M$ = f fiV^dV, (2.12b)
Jv
M$ = [ p^MdV. Jv (2.12c)
and 'I'} are known functions of the spatial coordinates used in the specific formula-
tion. As presented in [9], the three displacements are approximated by a finite linear
8
(2.13a)
combinations of T) of the form
u1(xi,x2,x3) = ^2aj'Zr1j(xi,x2,xz),
j=i
U2(x i,x2,x3) = '^2lbj^]{x1,x2,xz), (2.13b)
3=1
U3(x1,x2,x3) = Y^dj^3j(x1,x2,x3), (2.13c)
j=i
where the cij, bj, and dj are unknown constants. Knm are coefficient matrices which
are given in the paper by Heylinger and Johnson[9]. As pointed out by Heylinger and
Johnson, the matrices presented in cylindrical coordinates include explicit dependence
on 9.
The eigenvalue problem in Eqn (2.12) can generally be block diagonalized. This
has the following consequence: vibrational modes can be studied based on their gen-
eral symmetry.
9
3. Inverse Algorith
In the forward calculation, also known as the Ritz method, a set of frequencies
are computed from known elastic constants. This was the topic of section 2. These
frequencies are given in order with mode symmetry. For example, table 3.1 shows
calculated frequencies in cylindrical coordinates for the quartz crystal in the study.
The Ritz method is used in iterative inversion algorithms to determine unknown
elastic constants from measured spectra. In the inverse algorithm, the measured
frequencies and symmetries are matched with corresponding calculated frequencies.
This is shown in table 3.2. The task of matching the frequencies is not straightforward.
As a result, one needs a set of elastic constants near the real values in order for the
inverse algorithm to converge.
There is a one to one correspondence between measured and calculated frequen-
cies. This is observed in modes 12 and 15 in table 3.2. The inverse algorithm, which
is similar to a least-squares fitting routine, finds the best fit to the set of known elastic
constants from the ordering of measured and calculated frequencies.
10
Table 3.1: Calculated resonance frequencies fcai by Ritz algorithm in cylindrical
coordinates and mode symmetry
Mode fcal( Mhz) D-id subset
1 .000000000 Mu
2 .7443790856 E o
3 .1180346475 Mg
4 .2194887379 E c
5 .1431850250 Au/
6 .1504337543 E u
7 .2047928435 Aig
8 .2098375310 Eg
9 .2291843921 E[/
10 .2399999364 Eg
11 .2692167178 E u
12 .2801698267 Mg
13 .2807119878 Eg
14 .2858444684 Aig
15 .2906401542 Mu
11
Table 3.2: Calculated resonance frequencies fcai by Ritz algorithm in cylindrical
coordinates and measured resonance frequencies by mode selective /ms method
Mode /cai(mhz) Â£>3d subset fMs (mhz)
1 000000000 A2 u
2 74.43790856 E[/
3 118.0346475 A2G
4 219.4887379 Eg
5 143.1850250 Ait/
6 150.4337543 E u
7 204.7928435 Aig
8 209.8375310 Eg
9 229.1843921 Ey
10 239.9999364 Eg
11 269.2167178 Eg
12 280.1698267 A2 G 280.85411
13 280.7119878 E<3
14 285.8444684 Aig
15 290.6401542 A2 u 290.92855
12
4. Limitations of the Inverse Algorith
A key issue to the successful determination of elastic constants is to have the cor-
rect correspondence mentioned above [11]. However, this task is not straightforward.
Mode ordering becomes increasingly challenging for modes with resonance frequen-
cies higher than 700 khz. This is due in part to frequency shifts in observed spectra
from mechanical contact A mismatched order of frequencies in the inverse algorithm
results in an incorrect set of elastic constants. Additionally, there is the potential is-
sue of switched assignment of frequencies during the iteration process. For example,
consider the case where the difference in frequency from RUS and mode selective
Mode fmjs( khz) fMs{ khz) D3d subset
98 824.727 824.173 Ai u
99 825.345 825.976 A 2g
technique is in the same order of mode spacing. As pointed out by Ogi [11], modes
98 and 99 may be switched. In other words, the iteration process may find the best
fit to the set of known elastic constants by breaking the one to one correspondence.
The resonance frequency obtained from RUS technique, mode 98, may be matched
with the measured frequency from mode-selective technique, mode 99. This causes
the inverse algorithm to converge to a false minimum.
Our study addresses these issues by developing a noncontacting transduction
technique that excites and detects piezoelectrically only a group of vibrations with
13
a prescribed symmetry. This has the following implication: the eigenvalue equation
can be divided into submatrices based on their symmetry. As a result, the inverse
algorithm can be implemented for frequencies belonging to a group of vibrations with
the same symmetry. This eliminates the possible mismatch in ordering of calculated
and observed frequencies.
14
5. Theoretical Tools
This section is concerned with theoretical tools. It is intended to provide funda-
mental concepts relevant to the symmetry analysis. We begin our discussion with the
definition of a group. As an example, we present the general linear group, GL(n),
which serves as a gentle introduction to group representation in section 5.1.1 and its
properties in section 5.1.2 and 5.1.3. The purpose of section 5.1.4 is to give a brief
description of some of the applications of group theory to crystallography. Useful ap-
plications of the character table will be encountered later in the symmetry analysis.
Section 5.2 presents the basic equations of piezoelectricity. The geometry of differ-
ential forms and its applications to continuum mechanics is the topic of section 5.3.
5.1 Group Theory
Consider a finite set G of elements a, b, c..., and define a map such that it takes
elements g,h E G into gh E G with the following properties:
There is an element e E G such that ge equals g for all g E G, (5.1a)
There is an element g~l E G such that gg~l equals e. (5.1b)
Then this set is called a group. As an example, consider GL(V, C) defined to be
GL(V, C) = (a | a : V >-> V, a is linear) (5.2)
An element a of GL(V, C) is a linear transformation from the vector space V into V,
which has an inverse. V is a vector space over the complex numbers. If a set of basis
15
is introduced in V, then each linear transformation a is expressed as a square matrix.
Therefore, GL(V) is identified with the group of invertible square n x n matrices
where n is the dimension of the vector space V.
5.1.1 Elements and Representations
Suppose now G is a finite group. A linear representation is a map p from the
group G into the group GL(V,C) such that p(st) = p(s)p(t) for s,t Â£ G. In order
words, a representation p assigns to each element s Â£ G a linear transformation p(s)
of V such that st Â£ G corresponds to p(s)p(t). When p is given, V is called the
representation space of G.
A representation is unitary provided p(s)* = p(s)~1, where the denotes the
adjoint. In other words, p(s) is unitary with respect to an inner product (,) if,
(p(s)x, p(s)y) = (x, y) for all x,y eV,s eG (5.3)
Given a representation p, a natural question arises: is the representation unitary with
respect to an inner product? The answer is yes for finite groups. The chosen inner
product is
(p(s)x, p(s)y) = ~Y^ P(s)xip(s)y]*, (5.4)
9 sgG
where g is the number of elements in the group. The number of elements in the group
is called the order.
5.1.2 Reducibility of Representations
Let p be a linear representation of G. Let W be a proper subspace of V such
that p{s)W Â£ W for all s Â£ G. Proper subspace means If / 0 and If / V. Then
W is called stable under G and this representation is called reducible. In other words,
16
every linear transformation p maps W into W. Upon introducing a basis ei,..., em
for W, p(s) will be a square matrix of the form
[p(s)]ei,...,em
**
0
(5.5)
In other words, the matrix is upper triangular where the denotes the components.
The subspace W is complemented if there is W1 G V such that V = W Wx and
W1 is stable under G. The linear subspace W1 is the set
W = (y G W | (x,y) = 0,Vx G W)
(5.6)
By introducing basis en, ...,ep for W1- p(s) will be represented by a square matrix
with respect to basis e-y,..., em, en,..., ep in V of the form
\P(*)]
&1
*0
0*
In other words, [p] will be block diagonal.
(5.7)
5.1.3 Character
Let p be a linear representation of a finite group G in the vector space V. For
each element s G G define the complex valued function
X(s) = Trace [p(s)j
This function is called the character of the representation p.
(5.8)
17
5.1.4 D3*
Quartz crystal has the symmetry of one of the higher trigonal crystal classes,*
D3. Since linear elastic vibrations are insensitive to lack of inversion, the inversion
operator can be added to D3 point group. This makes the D3 point group equivalent
to D3d point group [9].
The symmetry a crystal in the form of a cylinder with the trigonal axis oriented
along the cylinder axis belonging to the crystallographic point group D3(* is shown in
Fig. 5.1. A crystallographic point group G is the set of all symmetry operations about
a fixed point that leaves a crystal structure unchanged. The D3d point group contains
2 I
Figure 5.1: Symmetry and definitions of axes for an object belonging to the D3d
point group
the following elements: C3z, C^1; rotations of 2-7r/3 and -27t/3 (three-fold rotations)
about the vertical axis (z), C2x, C2f, C2d\ rotations of 7r (two-fold rotations) about x,
18
/, and d (where d, on the back side of the object, is in the same plane as x and / and
halfway between them), /; inversion, and IC2z, IC^,IC2x, IC2f, IC2d\ the products
of inversion and the rotations.
There are six irreducible representations of the crystallographic point group D3(f,
as shown in table 5.1: four one-dimensional representations, which normally are
labeled Al5, A2g, Alw, A2u, and two two-dimensional representations labeled Eff and
Eu[3]. The subscripts g and u indicate that the corresponding basis functions
are even and odd, respectively, under inversion. The columns in the character table
designate the classes, which are defined to include the following elements [3]:
Table 5.1: Character table for D3^ point group.
Ci c2 c3 C4 c5 C6
A lg 1 1 1 1 1 1
> to to 1 1 -1 1 1 -1
A].** 1 1 1 -1 -1 -1
A2u 1 1 -1 -1 -1 1
E, 2 -1 0 2 -1 0
Eu 2 -1 0 -2 1 0
19
C! = E, (5.9a)
C2 = CZz,CÂ£, (5.9b)
Cz = Czxi Czf, Czdi (5.9c)
III (5.9d)
a=10^,10^, (5.9e)
Cs = ICb,IClf,ICM. (5.9f)
In the remainder of the thesis, I will refer to an element of C\ as cji, an element of C2
as Â£2 and so on.
5.2 Piezoelectric Constitutive Relation
In this section, we present the basic equations of piezoelectricity. It is intended
to make the symmetry analysis self-consistent.
In a solid medium, the electromechanical constitutive equation is
$ij T $ ij klffikl,
(5.10)
where dijki are defined as the piezoelectric strain coefficients, Sij and Tij are the
components of the strain and stress field, and Ek are the components of the electric
field. The piezoelectric strain coefficient matrix for a crystal belonging to DZd point
group in Cartesian coordinates is given by
dn dn 0 du 0 0
0 0 0 0 di4 2dn
0 0 0 0 0 0
(5.11)
The piezoelectric strain coefficients need to be transformed into cylindrical co-
ordinates. After some matrix algebra, we obtain the final form of [d] in cylindrical
20
coordinates:
[d]
duCosZcf) dnCos3(j)Q du 0 2dnsm30
dusinScj) d\\sin3 0 0 d^ 2dncos30
0 0 0 0 0 0
The strain field for an azimuthal electric field is given by Eqn. 5.10:
[5]
duCosScf) dnsm30 0
dxxCos3(j) dusin3 0
0 0 0
dl4 0 0
0 du 0
Ex
Ei
0
(5.12)
(5.13)
2dx\sin3^) 2dncos30 0
Note that the z-component of the electric field does not contribute to the strain field.
The strain field in tensor notation is
Sij = [Ei(xi)du cos 30 E2(xi)dxx sin30]ex e\
+[-Ei(xi)dn cos30 + E2{xi)dn sin30]e2 e2
+[Ex(xi)du]e2 e3 [E^x^dx^ex e3
+[Ei{xi)2dxx sin30 E2(xi)2dn cos30]ei e2. (5-14)
We will use this form in the symmetry analysis as it is the most convenient.
5.3 Differential Forms
It is now accepted that differential geometry is helpful in order to gain a deeper
understanding of elasticity [7]. Tensor analysis is simplified with the introduction of
differential forms in manifolds. The notion of the pull-back of a form will be of crucial
important to the symmetry analysis that is pursued in Sec. 7
21
To explain differential forms, we have to comment on differentiable manifolds
over which they are defined. In brief, a differentiable manifold is a representation of
a figure as a geometric object [1].
Our interest is in studying the action of the group on geometric objects such as
functions, 1-forms, and 2-forms defined at a point q G N. These matters will be
clarified with examples as we proceed. First, several definitions will be given.
5.3.1 Tangent Space
is an n-dimensional vector space. The tangent space to 9ft71 at a point p G 5Rn,
Tp9?n, is a vector space consisting of all tangent vectors to 9in at p. The basis for this
vector space is denoted by
d d
dvi"',dvn
An arbitrary vector v from the point p can be written as a linear combination
(5.15)
V = V
d
dx1
v
7 U
_d_
dxT
(5.16)
5.3.2 First Pause, Explanations
We make the first pause to explain the subtle terminology. Consider 9?2, the real
plane. The set of vectors originating from the origin O is called the tangent space at
the origin, T0$t2. We can also consider TP$R2, the set of vectors originating from the
point p. This space is simply TC5R2 with the origin moved to p.
Eqn. 5.16 tells us to write a vector v G 9?2 in terms of the basis ^. At first,
this notation contradicts everything we have learned in calculus. It needs further
explanation. Let us consider a vector v as a differential operator as in Eqn. 5.16. The
22
operator acts on a function / at a point p 6 3ft2 as
f tv 19f(p) 2 df(p) /K
MM = -^r + (5-17)
This equation is more familiar to us and is no other than the directional derivative of
/ at p in the direction of v. A differentiable assignment of a tangent vector at each
point p in 3ft2 describes a vector field. We now continue.
5.3.3 The Differential of a Map
Let F : 3ftn h* 3ftn be a smooth (infinitely differentiable)map. The differential of
a map F, denoted by T1*, is defined as follows [4]. For x 6 Tp3ftn, Fffx) is the velocity
vector of the image curve at F{p) on 3ftn. In other words,
for x e Tp3ftn, Fk : Tp3ftn Tp(p)3ftn, (5.18a)
Fffx) = vj,wÂ£ Tp(p)3ftn. (5.18b)
5.3.4 Second Pause, Explanations
The definition of the differential of a map will be clear from the geometry of the
Jacobian matrix [4],
Let x1,...xn and y1,...yr be coordinates for 3ftn and 3ftr respectively. Let F : 3ftn
3ftr be a smooth map.
A point y e 3ftr is given by
ya = Fa(x),a=l,...,r. (5.19)
Let v be a tangent vector at x0, v E TXo3ftn. Take any smooth curve xit) such that
x(0) = x0 and ^ = v. The image of this curve under the map F, F(x(t)), has a
23
(5.20)
tangent vector w at y0 given by the usual chain rule
dya . dx'
dxi X dt
w
i=1 i= 1
The assignment v i-* w defines a linear transformation, the differential of F at io,
given by 5.18. -^(0) is the velocity vector at xQ. We note that the differential of F is
simply the Jacobian matrix ^r^o-
5.3.5 k-Forms and Cotangent Space
Let V be a vector space over the reals, Â§ft. A linear functional a on V is a linear
transformation cr.V 3?. The set of all linear functionals a on V forms a new vector
space V* called the dual space to V. In notation,
V* = {a : V 3ft, a is a linear map}. (5.21)
The dual space T*Kn to Tp$tn at the point p is called the cotangent space, dx1,..., dxn
form a basis for the cotangent space T*9T\ Therefore, the most general linear func-
tional is expressed as
a ^^ajdx^, (5.22)
3
where aj = a(ej). The ej form a basis for the vector space V.
Now we have reached the main subject of this section, the notion of 1-form. A
linear functional a : Tp^R.n 5? is called a 1-form. Its important to realize that
Eqn. (5.22) holds in any coordinate system. In general, a k form is a multi linear
real-valued function
a : TpW1 x TpW x ... x TpW1 ^ 3ft. (5.23)
of A; tuples of vectors that is antisymmetric
a(vi,...vr,vs,...,vk) = -a{v1,...vs,vr,...,vk). (5.24)
24
in each of its entries. In notation, we have
a= fh...ik(xi,...xn)dx11 A ... Adxlk, (5.25)
where the coefficients are C functions on $Rn. The symbol A denotes the product
that satisfies the equation
dxl A dxi = dxJ A dxl. (5.26)
The following notation will be used in subsequent section: Ak^Stn) denotes the
set of k forms on 5Rn. In particular, following [1], .4(5?n) = C'(3?n), a differential
form of degree 0 are Cica functions.
5.3.6 Third Pause, Explanations
We have introduced the notion of a dual space without making reference to the
dual basis. If e1}..., en is a basis for the vector space V, the dual basis al,...,an of V*
is defined by putting
= 4 (5.27)
The action of the dual basis on a vector v Â£ V is defined as
u
3 3 3
The dual basis cr*, a linear functional, reads off the i-th component of a vector with
respect to the basis ei,...,en. This is exactly what a dual basis accomplishes, as
known from linear algebra.
The dual basis for the cotangent space was introduced as dx1,dxn. Why do
we have different notation? The cds are the most general form for a dual basis. As
we will see, they have a very special representation.
25
In section 5.3.2, we defined a vector v at a point p as a differential operator on
functions / near p. In notation, we have
(/)(?) = vvf = df(v).
(5.29)
If v vj where tÂ£- is a basis in local coordinates of V, then we have
(vf)(p) = df(y) = df(J2vj-^r) =
(5.30)
3 J 3
In particular, we may consider the differential of a coordinate function, xl
ft
ds'(-zr) = ^-) = *}
K dxj dxj
and
(5.31)
(5.32)
3 ' 3
The linear functional dx% reads off the i-th component of the vector v with respect to
a given basis. In other words, we conclude that
a1 = dxl.
(5.33)
5.3.7 Exterior Differentiation and Pull-Back
Two more operations need to be defined in order to understand the section on
symmetry analysis. The exterior differentiation is a map [1],
d : .Afc(&") ^fc+1(SRn), (5.34)
defined as follows. Let a e Ak($ln), that is a = JT a(xi, ...xn)dxh A ... A dxik, then
da = ^ ^^-^-dxj A dx11 A ... A dx%k. (5.35)
LsJb -j
3 J
26
The pull-back is defined as follows [4]. Let F : M W be a differentiable map of
manifolds. Consider a function f :W ^ lft (0 form on W), its pull-back to M is
(F*f)(x) = (foF)(x). (5.36)
In general, the pull-back of a k form is defined by
F*ak{vu ...vk) = ak(F*v1;.., F*vk). (5.37)
5.3.8 Example
I will conclude this section with an exercise. As a simple example, the following
exercise illustrates the notion of the pull-back. Let F : U i->- 3?3 be usual change of
variable map from spherical coordinates to Cartesian ones,
F{r,d,(f) = (rcos$sin, rsintfsin^, r cos<^>) = (x,y,z). (5.38)
Let Xi,yi be local coordinates for Â£/, 5ft3 respectively. Consider the 3 form w =
dx A dy A dz in 5ft3, that is w G .A3 (3ft3). The pull-back of w to U is
F*w = w(Fkvi, F+V2, F+Vz). (5.39)
Note that the components of F* is simply the Jacobian matrix
[F*] = (
cos d sin
sini9sin0 r cos d sin r sin d cos
cos(f) 0 rsin
Carrying out the calculation and noting that F*(^|:) =
fdyiJL dyrn_d_ dyn d
dVi"drd dy dyn
)dr Add A df>.
(5.40)
(5.41)
27
By multilinearity, further simplification follows as
E
dyi dym dyn
dr dd dcj)
dr A dd A d.
(5.42)
The coefficient is nothing more than the Jacobian determinant. In other words, to
obtain the volume form in spherical coordinates from Cartesian ones, one has to
multiply by the Jacobian determinant.
28
6. Transducers and Electronics
Fig. 6.1 shows the quartz crystal used in this study inside a cylinder on which
capacitor plates are mounted. The crystal is in the form of a cylinder with the
trigonal axis oriented along the cylinder axis. The material of the surrounding cylinder
is a composite of phenol-formaldehyde resin and fabric. The capacitor plates have
dimensions of 8.45 mm by 3.85 mm. The small electrodes are 3.84 mm by 1.50
mm. The inner and outer diameters of the cylinder are 10.96 mm and 13.20 mm
respectively. The shaded regions and the S point have the same meaning as in
Fig. 6.2. The cylindrical crystal has a radius of 5.389 mm and a height of 15.064 mm.
The crystal is loose inside the cylinder and rests on a Teflon washer placed inside the
Figure 6.1: Crystal inside cylinder surrounded by capacitor plates
cylinder. The center of the crystal is vertically aligned with the middle electrodes.
The plates are driven by a sinusoidal tone-burst excitation. Coupling to resonant
vibrations is directly piezoelectric. The reverse mechanism is used in the receiving
process.
29
We have measured the resonant spectrum for the four configurations shown in
Fig. 6.2. Each cylinder consists of capacitor plates with a different arrangement of
Figure 6.2: Four configurations for generating A2g and A\u modes
electric potential. The darker and lightly shaded plates corresponds to positive and
negative voltages respectively. The symbol S between two middle electrodes corre-
sponds to the two-fold symmetry axis of the cylinder. In other words, the arrangement
of capacitor plates with the corresponding voltages are even or odd under two-fold
rotations about point S.
30
7. Symmetry Analysis
The present section is of central importance to the thesis. Its aim is to formulate
the symmetry analysis of the potential, electric field and strain field in a trigonal
crystal of cylindrical form surrounded by each configuration of electrodes that is
shown in Fig. 6.2.
Let us introduce notation. Let M and N be 2-dimensional manifolds, and let
p(gi) : M H JV be a transformation that maps a point p 6 M to a point q G AT;
p(gi) corresponds to the symmetry operations of the Dsd point group. M refers to
the surface of the cylinder in the reference configuration, N is the image of M under
p(gi). Of course, M and N are the same object, but it is useful to keep the distinction
for clarity.
We consider a cylinder with a prescribed electrode geometry which determines the
potential as in Fig. 6.2 c. In this geometry, the upper and lower plates with positive
potential and the middle plates with negative potential, the electric potential has A\g
symmetry.
Let Xi,yj be local coordinates for the cylinder in the reference and deformed
configuration, M,N respectively. We now consider a potential function V{y.j) on
the deformed configuration, N, V G A(N) and want to obtain the corresponding
function in local coordinates of M. This is simply done by applying the pull-back to
V,
9*V = V(gi(Xi)). (7.1)
31
In reference to Fig. 6.2 c for our potential we have
9\V = V<,gi(x)) = V(x) (7.2a)
&V=V(sn(x)) = V(x) (7.2b)
9'3V = V(S,(x)) = V(x) (7.2c)
9lV = V(gt(x)) = V(x). (7-2d)
for the given configuration of electrodes, gi corresponds to symmetry operations of
the point group D3d In other words, the potential V transforms as the 1st irreducible
representation of the D3d point group. For example, giV p(.s)^V. This is clearly
seen from the character table 5.1.
For the same electrode configuration, next we consider the electric field, 1-form
on q E N which is obtained by differentiating V,
t' dy
(7.3)
Following the same argument as in the previous section for changing variables, we
have
dV %
SiE = H 7T7r^-
^ dyj OXi
(7.4)
However, since the pull-back commutes with the exterior derivative, it follows that
gtE = g*dV = d(g;V) = dfc(n)) = d(p(s)"V) = p(sf>d V. (7.5)
In other words, E also transforms as the 1st irreducible representation.
32
Finally, the elements of the group act on the strain given in Eqn. 5.14 as:
giS = S
g2S = [Ei(g2Xi)dn cos 30 E2(g2Xi)dn sin30]ei e1
+[-Ei(g2Xi)dn cos 30 + E2(g2Xi)du sin30]e2 <8> e2
+[El(g2xi)du]e2 e3 [E2(g2Xi)d14]e: e3
+[-E1(g2xi)2dn sin30 E2(g2Xi)2dn cos 30]ei e2 (7.6a)
g2S = S (7.6b)
g3S = [Eifax^dn cos 30 + E2(g4Xi)du sin 30] ex ex
+[-Â£?i(^4a:i)dii cos 30 E2(g'4a;i)(Ziisin30]e2 e2
+[Ei(54Xi)di4]e2 e3 + [E2(g4Xi)diA\ei e3
+[Ei(^4a;j)2d1i sin 30 + E2(g4Xi)2dn cos30]ex e2 (7.6c)
23S = 5 (7.6d)
= [~Ei(g7Xi)dn cos 30 + E2(g7Xi)dn sin30]ex ex
+[E1(g7xi)dn cos 30 E2(g7Xi)du sin30]e2 e2
+[Ei(g7Xi)du]e2 <8> e3 + [E2(g7Xi)di4]e4 e3
+[Ei{g7Xi)2dn sin30 + E2(g7Xi)2dn cos30]ei e2 (7.6e)
9iS = -S. (7.6f)
Clearly, the strain transforms as A\u irreducible representation.
We have considered the transformation properties of the potential, electric field
and strain field for a geometry named configuration c. There are four more distinct
geometries each determining an electric potential. For a concise presentation, the
symmetry analysis for each geometry is summarized in table 7.1.
33
Table 7.1: Symmetry analysis for distinct geometrical configurations of electric
potential
Configuration Electric Potential Electric Field Strain Field
Configuration a a2u a2u A2g
Configuration b a2u to e A2g
Configuration c Alg Alg A-iu
Configuration d A\g Alg Aiu
Configuration e A\u Aiu Alg
So far, we have considered vibrational modes belonging to Aiu, A2g, and Aig
irreducible representations. Another configuration will make it possible for the electric
field to couple to modes belonging to A2u irreducible representation, but these are
not considered in this thesis.
34
8. Measurements
The RUS measurements of the quartz crystal in our study were done Sudook Kim
(NIST). Paul Heylinger from CSU performed the Ritz analysis in cylindrical coordi-
nates. Table 8.1 is produced in the following sequence. Paul Heylinger calculated
the resonance frequencies from known elastic constants using the Ritz algorithm in
cylindrical coordinates. Columns one and two of table 8.1 correspond to the result
of such calculation. Suddok Kim measured resonance frequencies by RUS technique
and performed the inverse calculation to obtained a one to one correspondence to
calculated frequencies. The result is tabulated in column three.
35
Table 8.1: Mode symmetry within D3^ subset, resonance frequencies f^i calculated
by the Ritz algorithm; resonance frequencies measured by RUS faus, and mode
selective $ms methods
D3d subset fcai(mhz) fR[/S(mhz) /jMs(mhz)
A2g .28016982
Al g .28584446
Alg .31939219
Aiu .35905380
Alg .35905380
A2g .36504338
Alg .39182803
Alu .40730615
Alu .42222552
A2g .43033460
Alg .43994784
A2g .48147576
Aiu .48932100
Aiu .53755763
A2g .53791507
Alg .53950823
Alg .56629747
A 2g .57308357
Alg .60194796
Alg .60433499
Aiu .60927481
Alg .62738376
Aiu .63040679
Alg .63683793
Alg .64412775
A 2g .65475916
Aiu .65940491
A2g .67967300
Aiu .68489207
Aiu .69432753
A2g .69847668
.28042149 .28085411
.28618691 ------
.31962320 .31950967
.35981989 .35971750
.35981989 .35971750
.40850267 .40849190
.42314362 .42302235
.43106188 .43098730
.48236368 .48205503
.49018565 .49002648
.53802118 .53779503
.53852784 .53839189
.57349634 .57330146
.60259221 .60227588
.62740372
.63108901 .63077233
.64787068 .64764179
.65539230 .65513295
----- .67956145
.69227358 ------
----- .69762718
36
8.1 Mode-selective Measurements
In our study, we have concentrated on modes belonging to Aiu and A2g irre-
ducible representation. The last configuration in Fig. 6.2 is used to align the sample
in the following manner. The sample is inserted into the cylinder with electrode
configuration e. The orientation of the S point is found by optimizing modes
that are identified in table 8.1. The A\g modes are used to optimize the alignment of
configurations a,b,c, and d as explained below.
The sample is rotated about the vertical axis z until A\g are completely sup-
pressed. The final position of the sample corresponds to the two-fold rotation axis
being aligned at the S point in configuration a and b. Once this alignment is accom-
plish, the position of the crystal is left undisturbed for measurements with configu-
rations a,b,c,d. Only the polarity of the capacitor plates was changed to generate the
different electric field symmetries. Thus, we can easily select the vibrational modes
belonging to an irreducible representation by changing the electrical leads to each
plate.
Resonant spectra for the four configurations a,b,c, and d are presented in Fig. 8.1-
Fig. 8.4.
37
0.7
r1
e
C3
O
1 03'
0.2
0.4 0.6 0.8
Frequency (Mhz)
1.0;
Figure 8.1: Resonant spectra for configuration a
Figure 8.2: Resonant spectra for configuration b
38
0.5 '
g 0.4 -
2
Â£
& 0.3-
0.1 -
0.2
Lj____________,... I i I 1 11 ii I i i I
i 1 I ...... I l
0.4 0.6 0.8 1.0^
Frequency (Mhz)
Figure 8.3: Resonant spectra for configuration c
Figure 8.4: Resonant spectra for configuration d
39
9. Data Analysis
The last column in table 8.1 presents the modes measured by the mode-selective
technique. Each configuration coupled to modes with a particular symmetry, as shown
in the graphs. The strength of the coupling for the different modes in each geometry
varied in amplitude from .003 to .65 (arb. units). However, not all the modes that
appear in the spectra are entered in table 8.1. Some modes with k2g symmetry
were detected with a configuration that does not match the correspondence shown in
table 7.1. The coupling to these modes were very weak ranging from .003 to .007 (arb.
units). Modes with coupling strength lower than .008 (arb. units) were not entered
into the table. For example, configuration c coupled to two A2g modes at .60227588
mhz and .69762718 mhz and configuration d coupled to A2g mode at .43098730 mhz.
The peaks corresponding to these modes are tiny and cannot be identified clearly in
the graphs.
Our technique demonstrates mode selectivity; each configuration generates and
detects a group of vibrations belonging to a particular irreducible representation.
Mode selectivity can make the Ritz algorithm more robust through definite mode
identification, and the eigenvalue matrix can be reduced to submatrices associated
with modes having a specified symmetry.
Table 8.1 does not include modes with frequencies higher than 700 khz since
mode identification becomes significantly more challenging. In this frequency range,
calculated and measured frequencies using our technique can differ by more than
1000 khz for high frequency modes and measured frequencies from RUS and the
40
mode selective technique can differ by 600 khz. The RUS technique introduces shifts
in frequencies due to mechanical contact. These frequencies are higher than the
corresponding ones from the mode selective technique. It is generally the higher-
frequency modes that can demonstrate the advantage of our technique for the Ritz
analysis.
As an example of this advantage, we will consider the following case. There
is a difference in symmetry assignment arising from the RUS and mode selective
technique for a mode near 881.0 khz. The RUS technique through the Ritz analysis
gives the mode 881.013 khz A2g symmetry The mode selective technique indicates
Aiu symmetry: the corresponding peak is absent with configurations a and b and
it is present with configurations c and d. There is a contradictory assignment of
symmetry between RUS and mode selective technique. A misidentified mode in the
inverse algorithm causes the inversion to converge to a false minimum.
41
10. Conclusion
This study developed noncontacting electromagnetic transduction methods to
excite and detect vibrational modes with a prescribed symmetry. Different transducer
configurations were used to couple to modes with symmetry belonging to each of the
three irreducible representations of the D3(/ point group. Group theoretical methods
were used to analyze the symmetry of the induced strain field in terms of the applied
electric field.
The technique demonstrates mode selectivity. Modes belonging to different irre-
ducible representations of the D3Â£j point group were generated and detected based on
the symmetry of the applied electric field. Our technique presents a clear way to iden-
tify mode symmetry. The identification of mode symmetry presents two advantages
in the Ritz analysis: more positive correspondence of measured and calculated fre-
quencies and division of the Ritz eigenvalue problem into submatrices which reduces
the computational time.
42
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[6] B.A. Auld. Acoustic Fields and Waves in Solids (Krieger, Malabar, FL, 1990).
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Hall, New Jersey, 1983).
[8] W. Johnson and P. Hey linger. Symmetrization of Ritz approximation functions
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(2003).
[9] P. Heylinger and W. Johnson. Traction-free vibrations of finite trigonal elastic
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43
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