Three-dimensional wave propagation through single crystal
solid–liquid interfaces
Yichi Lua) and Haydn N. G. Wadley
Intelligent Processing of Materials Laboratory, School of Engineering and Applied Science,
University of Virginia, Charlottesville, Virginia 22903
~Received 18 April 1997; accepted for publication 22 October 1997!
Large differences in the ultrasonic velocity of the solid and liquid phases of semiconductors have
stimulated an interest in the use of laser ultrasonic methods for locating and characterizing solid–
liquid interfaces during single crystal growth. A previously developed two-dimensional ray tracing
analysis has been generalized and used to investigate three-dimensional ultrasonic propagation
across solid–liquid interfaces in cylindrical bodies where the receiver is located at an arbitrary
position relative to the source. Numerical simulations of ultrasonic ray paths, wavefronts, and time
of flight have indicated that ultrasonic sensing in the diametral plane is a preferred sensing
configuration since the transmitted, reflected, and refracted rays all propagate in this plane,
significantly simplifying analysis of the results. While other sensing configurations can also provide
information about solid–liquid interfaces, they require a more complicated analysis because the
planes in which reflected and refracted rays propagate are not known a priori, and fewer ray paths
are accessible for interface interrogation because of large refractions. © 1998 Acoustical Society
of America. @S0001-4966~98!00203-3#
PACS numbers: 43.35.Bf @HEB#
INTRODUCTION
The Bridgman growth of single crystal semiconductor
materials with low thermal conductivity often results in poor
crystal quality and a low yield of useful wafers for microelectronic,
optoelectronic,
or
infrared
detector
applications.1–6 The crystal quality is thought to be largely
controlled by the velocity and curvature of the solid–liquid
interface during the growth process. The best crystal quality
usually results from a planar or slightly convex interface
shape that propagates at a uniform slow rate along the axis of
the boule. Thus sensing and control of the solid–liquid interface during crystal growth appears to be a key step toward
improving crystal quality. In most systems of interest, large
decreases in ultrasonic velocity accompany melting. The
large difference in the ultrasonic velocity between the solid
and the liquid phases of most semiconductor materials, together with the recent emergence of noninvasive laser ultrasonics, have stimulated an interest in using ultrasonics to
monitoring solid–liquid interfaces during single crystal
growth.7–9
In previous work,10 the severe ray bending associated
with the large velocity change at liquid–solid interfaces has
been shown to require a detailed ray tracing analysis to use
ultrasonic time of flight ~TOF! projection data for characterizing a solid–liquid interface. Using a two-dimensional
analysis, a detailed study of ray paths, wavefronts, and TOF
of ultrasound signals propagating on either the transverse or
diametral planes of liquid single crystal solid bodies was
reported. Numerical simulations indicated that the magnitude
and direction of the group velocity, the solid–liquid velocity
ratio, and the curvature of the interface together controlled
a!
Now at Southwest Research Institute, NDE Science and Technology Division, 6220 Culebra Road, San Antonio, TX 78228-0510.
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J. Acoust. Soc. Am. 103 (3), March 1998
the ray bending behavior, and thus determined ultrasonic
paths across the interface. These theoretical studies indicated
that when the ray paths through an interface could be computed, the interface location and curvature could be reconstructed from sparse ultrasonic TOF data collected on the
diametral plane. Laser ultrasonic experiments conducted on
benchtop models have recently confirmed that the interface
location and curvature can indeed be reconstructed from the
measured TOF data using a simple nonlinear least squares
algorithm, an axisymmetric interface model, and the ray tracing algorithm, developed in Ref. 10.11
The sensor configurations explored to date have been
constrained by our limited understanding of wave propagation at anisotropic solid–liquid interfaces. The availability of
only 2-D wave propagation algorithms required the placement of sources and receivers on the transverse or diametral
plane where both the transmitted and refracted rays must
propagate. Here, we generalize the 2-D wave propagation
analysis to 3-D, and allow the receiver to be located at arbitrary positions relative to the source. Similar to our previous
work10 we study only the ~fastest! quasilongitudinal wave
mode. The diffracted rays have been studied in our previous
work. We employ a matrix/vector approach for the 3-D ray
tracing analysis that has previously been developed for isotropic materials.12 The key step in analyzing the 3-D wave
propagation in anisotropic materials is shown to be the determination of the usually unknown plane in which both the
reflected and transmitted rays propagate. With this plane determined, the procedure for constructing elastic stiffness tensors and the Christoffel equations that govern wave propation velocity in the local coordinate systems of the interface
then become the same as the 2-D case.10 Numerical simulations of ray paths, wavefronts, and TOF are presented for
both convex and concave interface shapes with the ultrasonic
source/receiver located either above or below the interface. It
0001-4966/98/103(3)/1353/8/$10.00
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1353
F
F
G
0
0
Bs 5
,
2 ~ R c 2h !
c s 5h 2 22R c h,
h.0.
~5!
For concave interfaces,
G
0
0
,
Bs 5
22 ~ R c 1h !
c s 5h 2 12R c h,
h,0.
~6!
In Eqs. ~5! and ~6!, h defines the interface convexity
~Fig. 1!. The interface is flat when h50(c s 50). In terms of
the cylinder radius, R, and the convex interface convexity,
h, the interface radius of curvature R c , can be expressed as:
R c5
FIG. 1. An illustration of 3-D ray tracing in a single crystal solid–liquid
body.
will be shown that rays traveling in nondiametral planes can
be used for interface sensing. However, because the plane in
which the reflected and transmitted rays propagate is not
known beforehand ~and needs to be determined at each intersection point of the ray path with the interface!, analysis
of TOF data is considerably more complex. When diametral
plane access is available, it appears to be the preferred approach for sensing axisymmetric interfaces.
R 2 1h 2
.
2h
~7!
We let S be a prescribed source point, and l be the initial
ray direction vector pointing from the source point, S. P1 and
P2 are then the first and the second intersection points of a
general ray path propagating through the interface to a receiver point R; t1 and r1 are then the transmitted and reflected ray direction vectors at P1 , while t2 is the transmitted
ray direction vector at P2 . Thus
P1 5S1L 1 l,
~8!
where L 1 is the distance from S to P1 .
Since P1 is on the interface, it satisfies Eq. ~3!:
PT1 As P1 1BTs P1 1c s 50.
~9!
Substituting Eq. ~8! into Eq. ~9! yields
I. RAY PATH ANALYSIS
L 21 ~ lT As l! 1L 1 ~ 2ST As l1BTs l! 1 ~ ST As S1BTs S1c s ! 50.
Consider a circular crystal cylinder of radius R ~Fig. 1!.
The axis of the cylinder is chosen to coincide with the x 3
axis of our coordinate system. Following Shah12 the outer
boundary surface of the cylinder, f c , can be expressed as:
Equation ~10! yields two roots for L 1 . When both roots
are complex, the ray path does not intersect the interface;
when both roots are real and positive the smaller one is the
correct solution for L 1 ; when one root is positive and the
other is negative, the positive one is the solution for L 1 ~the
negative root is for the ray traveling in the opposite direction
with respect to l!; when both roots are real and negative, then
the ray travels away from the interface ~it does not intersect
the interface!.
The normal, N1 5(n 1 ,n 2 ,n 3 ), at P1 to the interface can
be obtained from the gradient of the interface:
f c 5xT Ac x2c c 50,
~1!
where x5(x 1 ,x 2 ,x 3 ), and
F G
1
0
0
Ac 5 0
0
1
0 ,
0
0
c c 5R 2 .
~2!
We assume that the solid–liquid interface is a spherical
cap with a radius, R c , centered on the x 3 axis. The intersection of the solid–liquid interface and the cylinder lies in the
x 1 2x 2 plane. Thus the interface is symmetric with respect to
both the x 1 2x 3 and the x 2 2x 3 planes. The interface, f s ,
can be expressed as:
f s 5x
where
T
As x1BTs x1c s 50,
F G
1
0
0
As 5 0
0
1
0 .
1
0
~4!
For convex interfaces,
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~3!
J. Acoust. Soc. Am., Vol. 103, No. 3, March 1998
N1 5
¹fs
A~ ¹ f s ! T ~ ¹ f s !
,
~10!
~11!
where
¹ f s 52As P1 1Bs
~12!
is the gradient of the interface at P1 . The normal of the
interface, N1 , is chosen such that it always points into the
liquid (n 3 .0).
The incident angle a in1 at P1 to the interface can now be
expressed as:
cos a in1 52l–N1 .
~13!
To determine the plane in which reflected and transmitted rays propagate, we notice that when a ray is incident
upon an interface, both the reflected and transmitted rays
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1354
FIG. 3. Transmitted ray paths for a convex interface, h515 mm, s 3 5
215 mm. ~a! The viewing direction is parallel to the x 1 axis; ~b! the viewing direction is parallel to the x 2 axis.
FIG. 2. Ray paths for a convex interface, h515 mm, s 3 515 mm. ~a! Reflected ray paths; ~b! singly transmitted ray paths; ~c! doubly transmitted ray
paths.
propagate in the plane defined by the incident vector, l, and
the normal to the interface, N1 , at the intersection point
P1 . 13–19 If r is an arbitrary vector in the plane, this plane is
given as:
~ r2P1 ! • ~ l3N1 ! 50,
~14!
where (l3N1 ) is the normal to the plane at P1 .
To determine the reflected and transmitted ray paths
emitting from P1 and propagating in the plane defined by Eq.
~14!, we first need to determine an expression for the stiffness tensor in the plane. We introduce a local coordinate
system (x 81 ,x 82 ,x 83 ) in which the x 82 axis is in the N1 direction, the x 83 axis is in the (l3N1 ) direction, and the x 81 axis is
in the direction defined by @ N1 3(l3N1 ) # which is tangential
to the interface. Using the directional cosines between
(x 1 ,x 2 ,x 3 ) and (x 18 ,x 28 ,x 38 ) the expression of the stiffness
tensor in the local (x 18 ,x 28 ,x 38 ) coordinate system can now be
obtained following the procedure described by Auld.10,15
Knowing the plane in which the incident, reflected, and
transmitted rays propagate and the expression of the elastic
stiffness tensor in the local (x 18 ,x 28 ,x 38 ) coordinate system,
the problem of determining the reflected and transmitted ray
paths becomes 2-D, and can thus be solved using the procedures described in a previous publication.10
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J. Acoust. Soc. Am., Vol. 103, No. 3, March 1998
When the source point is in the liquid, the direction vectors of the transmitted rays in the solid, t1 , and reflected rays
in the liquid, r1 , can be shown to be
t1 5
F
G
ng
ng
l1
cos~ a in1 ! 2cos~ a tr1 ! N,
nl
nl
r1 5l12 cos~ a in1 ! N1 ,
~15!
~16!
where v g is the magnitude of the group velocity in the solid,
a tr1 is the refraction angle in the solid, Fig. 1. When the
source point is in the solid, all rays intersecting the interface
are transmitted into the liquid due to the smaller liquid velocity and the smaller refraction angle. The refraction direction vector in the liquid in this case is:
t1 5
F
G
nl
nl
l2
cos~ a in1 ! 2cos~ a tr1 ! N1 .
ng
ng
~17!
For a convex interface and a source point above the
interface, a transmitted ray path may intersect the interface
again at P2 ; the point P2 is obtained in exactly the same way
as P1 . This type of ray is called a doubly transmitted ray.
When the source point is below a concave interface, there is
only one intersection point on the solid–liquid interface due
to the interface curvature and the smaller refraction angle in
the liquid resulting from the smaller liquid velocity. The ray
with a single intersection point on the interface is called singly transmitted ray. Using t1 and r1 , the second intersection
point P2 and the reflected ray point R leaving P1 can be
written as
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1355
FIG. 4. Ray paths for a concave interface, h5215 mm, s 3 515 mm. ~a!
Reflected ray paths; the viewing direction is parallel to the x 1 axis; ~b!
reflected ray paths; the viewing direction is parallel to the x 2 axis; ~c! transmitted ray paths; the viewing direction is parallel to the x 1 axis; ~d! transmitted ray paths; the viewing direction is parallel to the x 2 axis.
P2 5P1 1L 2 t1 ,
R5P1 1L R r1 ,
~18!
where L 2 is the distance from P1 to P2 , L R is the length of
the reflected ray leaving P1 .
When there is a second intersection point P2 , the direction vector t2 for the transmitted ray at P2 is
t2 5
F
G
nl
nl
t 2
cos~ a in2 ! 2cos~ a tr2 ! N2 ,
ng 1 ng
~19!
where a in2 and a tr2 are the angles of the incident and transmitted rays, N2 is the normal to the interface at P2 , Fig. 1.
All rays eventually intersect the cylinder at P3 . The distance
between P2 and P3 when there is a second intersection point
on the solid–liquid interface, or between P1 and P3 when
there is no second intersection point, can be solved by substituting either P2 or P1 into Eq. ~9! where As and c s are
replaced by Ac and c c in Eq. ~2!, and Bs is set to be zero. The
reflected ray path is then given by S, P1 , and R, and the
singly transmitted ray path by S, P1 , P3 , while the doubly
transmitted ray path is given by S, P1 , P2 , and P3 .
II. NUMERICAL IMPLEMENTATION
To numerically simulate wave propagation we use the
material constants of germanium ~Ge! because its elastic
constants as functions of temperature are available for both
the solid and the liquid. The analysis can easily be extended
to any anisotropic material ~cubic or noncubic! with known
elastic properties. We use the values of elastic stiffness constants, c i j , of Ge at 900 °C ~Ref. 20! to represent the solid.
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J. Acoust. Soc. Am., Vol. 103, No. 3, March 1998
FIG. 5. Transmitted ray paths for a concave interface, h5215 mm, s 3 5
215 mm. ~a! The viewing direction is parallel to the x 1 axis; ~b! the viewing direction is parallel to the x 2 axis.
c 115108.45 GPa,
c 125108.45 GPa,
~20!
c 445108.45 GPa.
At this temperature the solid density r 55.26 g/cm3. 21 The
sound velocity in the liquid ~at a temperature just above the
Ge melting temperature of 937 °C! is taken to be 2.71
mm/ms.22 In all cases R537.5 mm.
To analyze a representative convex interface shape, we
take h515 mm. For a source point above the interface at
(0,2R,15 mm), Fig. 2~a!, ~b!, and ~c! shows reflected, singly transmitted, and doubly transmitted ray paths. When the
initial ray directions, l, are in the diametral plane, all ray
paths remain in the plane. When the initial ray directions are
not in the diametral plane the reflected and transmitted rays
travel away from the diametral plane. This arises because
when the source point is in the liquid, the convex liquid–
solid interface acts as a divergent lens to ray paths ~see Fig.
6 of Ref. 10 for rays traveling in the 2-D transverse plane of
a liquid cylinder containing a solid core!. Notice that most
forward propagating doubly transmitted ray paths are in the
diametral plane with a narrow incident angle region; the doubly transmitted rays that do not propagate in the diametral
plane propagate backward due to severe ray bending.
The singly transmitted ray paths for the same convex
interface but with a source located below the interface at
(0,2R,215) are shown in Fig. 3~a! and ~b!. For this configuration, all rays that intersect the interface are refracted
into the liquid due to the smaller liquid velocity. No doubly
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1356
FIG. 6. Wavefronts for a convex interface, h515 mm, s 3 515 mm. ~a! Direct wavefronts; ~b! reflected wavefronts; ~c! singly transmitted wavefront; ~d!
doubly transmitted wavefronts. Thick lines are intersections of wavefronts with either the solid–liquid interface or the ampoule’s cylindrical surface.
transmitted ray paths exist for this configuration. Rays are
seen to again travel in the diametral plane when the initial
ray direction vector is also in the diametral plane. When the
initial ray directions are not in the diametral plane, the ray
paths are bent by the interface toward the diametral plane,
similar to rays traveling in the transverse plane of a solid
cylinder containing a liquid core ~see Fig. 5 of Ref. 10!.
Figure 4~a! and ~b! shows reflected and singly transmitted ray paths, for a concave interface with h5215 mm and
a source point above the interface at (0,2R,15 mm). The
reflected ray paths travel toward the diametral plane due to
the interface curvature, while the transmitted ray paths are
bent away from the diametral plane due to the larger refraction angle in the solid. When the source point is below the
interface at (0,2R,215 mm) @Fig. 5~a! and ~b!# the transmitted ray paths are bent toward the diametral plane due to
the smaller refraction angle in the liquid. There are no dou1357
J. Acoust. Soc. Am., Vol. 103, No. 3, March 1998
bly transmitted rays since all rays that intersect the interface
are refracted into the liquid due to the smaller refraction
angle in the liquid.
With the ray paths fully analyzed, we can draw several
important conclusions concerning optimal laser ultrasonic
sensing configurations. When the initial ray directions are in
the diametral plane and the interface curvature is symmetric
with respect to the plane, rays always travel in the diametral
plane. Thus the plane in which reflected and transmitted rays
travel is known a priori and it is straightforward to then
determine the ray path between prescribed source and receiver points. When the initial ray directions are not in the
diametral plane, the planes in which reflected and transmitted
rays travel are not known beforehand and must be determined in order to obtain the reflected and transmitted ray
paths. The orientation of the plane depends on the interface
curvature and the incident ray direction vector and is not
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1357
20ms of propagation does not touch the interface due to the
screening effect of the interface. Figure 6~b! shows wavefronts for the reflected rays that are reflected by the solid–
liquid interface into the liquid region. Notice that the reflected wavefront at 10 ms intersects the ampoule cylinder at
only two small separate sections. The wavefront for singly
transmitted rays after 10 ms propagation is shown in Fig.
6~c!. It has a ridge shaped area below the source, which is
due to the smaller group velocity on the ~100! crystal plane
of this cubic material. Due to the high velocity in the solid,
the singly transmitted wavefront at 20 ms has propagated out
of the region shown in Fig. 6~c!. The doubly transmitted
wavefronts are shown in Fig. 6~d!. These wavefronts have
larger y values than the corresponding wavefronts for the
reflected rays because the doubly transmitted rays have traveled a significant distance in the solid which has a higher
velocity than the liquid. The relatively thick middle portion
and the thin wings of the wavefronts in Fig. 6~d! indicate that
the region where it is possible to detect the doubly transmitted rays is greater near the diametral plane.
IV. TIME OF FLIGHT PROJECTIONS
To use ultrasonic TOF data for interface characterization, we are interested in ray paths that intersect the solid–
liquid interface and thus carry information about the interface. To illustrate, we study the four following situations.
A. A convex interface, h 515 mm, and source point
above interface at „0,2 R ,15…
FIG. 7. Time of flight projection data for a convex interface, h515 mm,
s 3 515 mm. ~a! Diametral reflected, singly transmitted, and doubly transmitted ray paths as functions of the vertical receiver coordinate, r 3 ; ~b!
circumferential reflected and singly transmitted ray paths as functions of the
receiver angle a.
known a priori. The bending of rays occurs near the solid–
liquid interface and, for interface reconstruction purposes,
this is easier to account for in the diametral plane than in
other planes. Thus for ultrasonic characterization of a solid–
liquid interface, positioning the ultrasonic source and receiver in the diametral plane should always be the preferrable sensor approach. There are no fundamental difficulties
to using ray paths in nondiametral planes for interface sensing; it is just a more complex and time consuming task. In
fact, the very large refractions in nondiametral plane configurations imply a high sensitivity to the exact interface geometry, and so this off-diametral approach might be pursued
if additional curvature reconstruction accuracy were needed.
III. WAVEFRONT SURFACES
Ultrasonic signals can be detected only at points where
the wavefronts intersect the crystal surface. Wavefronts are
obtained by connecting points along ray paths with the same
traveling time. Here we only show wavefronts for a convex
interface shape, h515 mm, where the source point is above
the interface at (0,2R,15). Figure 6~a! shows the wavefronts
for the direct rays in the liquid. The wavefronts are spherical,
centered at the source point since the liquid velocity is isotropic. Notice that the bottom edge of the wavefront after
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J. Acoust. Soc. Am., Vol. 103, No. 3, March 1998
Figure 7~a! and ~b! shows the diametral and transverse
plane TOF projection results, respectively. In Fig. 7~b! the
receiver angle, a, corresponds to the angle of deviation from
the diametral plane projected onto the transverse plan ~see
Fig. 4 of Ref. 10 for additional details!. The reflected, singly
transmitted, and doubly transmitted signals can all be detected in the diametral plane @Fig. 7~a!#. The TOF data for
the reflected ray paths increase with the vertical receiver position due to the longer propagation distance and the smaller
liquid velocity. TOF values of both the singly and doubly
transmitted ray paths increase with the vertical distance between the source and the receiver. The minimum TOF value
for the transmitted ray paths is reached when the receiver
point meets the interface at r 3 50.
Most doubly transmitted rays travel in the diametral
plane @Fig. 2~c!# therefore a circumferential scan of the receiver around the crystal cylinder is unlikely to detect doubly
transmitted signals except near the diametral plane. In the
transverse planes reflected and singly transmitted TOF data
increase with reducing uau because of the longer propagation
distance. TOF values for the reflected ray paths increase as
the receiver point is raised. Those for the singly transmitted
ray paths decrease as the receiver point is moved closer to
the interface.
B. A convex interface, h 515 mm, and source point
below interface at „0,2 R ,215…
Diametral and transverse plane TOF projection data are
shown in Fig. 8~a! and ~b!. These are no doubly transmitted
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1358
FIG. 8. Time of flight projection data for a convex interface, h515 mm,
s 3 5215 mm. ~a! Direct and singly transmitted ray paths as functions of the
vertical receiver coordinate, r 3 ; ~b! singly transmitted ray paths as functions
of the receiver angle a.
ray paths as discussed above. In the diametral plane when the
receiver is below the interface, only the TOF for the direct
ray paths can be detected, and its behavior is fully described
by the group velocity on this $100% plane.10 The TOF values
for the transmitted ray paths increase with the vertical receiver position due to the longer propagation distances and a
smaller liquid velocity. By comparing the diametral TOF
data between Fig. 7~a! and Fig. 8~a! we see that the location
of the convex interface with respect to the source point can
be readily determined by translating the receiver point in the
diametral plane. The transmitted transverse plane TOF data
are similar to those of Fig. 7~b! when the source is above the
interface but the TOF values are smaller.
C. A concave interface, h 5215 mm, and source
point above the interface at „0,2 R ,15…
Diametral and transverse plane TOF projection data are
shown in Fig. 9~a! and ~b!. The diametral reflected rays can
be detected only when r 3 .s 3 due to the interface curvature
@Fig. 4~a!# while the transmitted rays can be detected only
when r 3 ,215 mm due to the large refraction angle in the
solid @Fig. 4~b!#. However, except for the rays in the diametral plane, the transmitted rays intersect the outer sample surface well below the interface ~due to the interfaces curvature!, and so only the circumferential TOF data for reflected
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J. Acoust. Soc. Am., Vol. 103, No. 3, March 1998
FIG. 9. Time of flight projection data for a concave interface,
h5215 mm, s 3 515 mm. ~a! Diametral reflected and singly transmitted ray
paths as functions of the vertical receiver coordinate, r 3 ; ~b! reflected ray
paths as functions of the receiver angle a.
rays are shown in Fig. 9~b!. They have a steep hump near
a 50. It is clear by comparing Fig. 7 with Fig. 9 that the
interface is convexity, or concave can be easily determined
either from the diametral or circumferential TOF data.
D. A concave interface, h 5215 mm, and source point
below interface at „0,2 R ,215…
Diametral and transverse plane TOF projection data are
shown in Fig. 10~a! and ~b!. The diametral plane TOF for the
direct ray paths can be detected only when r 3 .s 3 , due again
to the screening effect of the interface. The diametral TOF of
transmitted ray paths can be detected only when r 3
.36 mm because of the smaller refracted angle in the liquid
@Fig. 5~a!#. By comparing the diametral TOF data in Figs.
8~a! and 10~a!, we see again that the interface convexity can
be determined from the diametral TOF data alone due to the
different functional behavior of the projection data sets at
a50.
In Fig. 10~b! the maximum in the TOF data for the direct ray paths is caused by the anisotrophy of the solid velocity and the change in ray propagation distance as rays are
swept from the 2x 1 to the x 2 direction ~Fig. 1!. The small
differences in the r 3 5220 and 230 mm data are also a
consequence of the solid velocity anisotrophy. By comparing
Fig. 10~b! with Fig. 8~b!, it can again be seen that the interface convexity can be determined from transverse plane projections when the source point is below the interface.
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1359
providing a promising real time direct method for solid–
liquid interface characterization during the vertical Bridgman
single crystal growth.
ACKNOWLEDGMENTS
This work has been performed as a part of the research
of the Infrared Materials Producibility Program conducted by
a consortium that includes Johnson Matthey Electronics,
Texas Instruments, II-VI Inc., Loral. the University of Minnesota, and the University of Virginia. We are grateful for
the many helpful discussions with our colleagues in these
organizations. The consortium work has been supported by
ARPA/CMC under Contract No. MD A972-91-C-0046
monitored by Raymond Balcerak.
M. C. Flemings, Solidification Processing ~McGraw-Hill, New York,
1974!.
2
T. Fu and W. R. Wilcox, ‘‘Influence of insulation on stability of interface
shape and position in the vertical Bridgman–Stockbarger technique,’’ J.
Cryst. Growth 48, 416–424 ~1980!.
3
M. Pfeiffer and M. Mühlberg, ‘‘Interface shape observation and calculation in crystal growth of CdTe by the vertical Bridgman method,’’ J.
Cryst. Growth 118, 269–276 ~1992!.
4
P. Rudolph and M. Mühlberg, ‘‘Basic problems of vertical Bridgman
growth of CdTe,’’ Mater. Sci. Eng. B 16, 8–16 ~1993!.
5
S. Brandon and J. J. Derby, ‘‘Internal radiative transport in the vertical
Bridgman growth of semitransparent crystals,’’ J. Cryst. Growth 110,
481–500 ~1991!.
6
S. Kuppurao, J. J. Derby, and S. Brandon, ‘‘Modeling the vertical Bridgman growth of cadmium zinc telluride. I. Quasi-steady analysis of heat
transfer and convection,’’ J. Cryst. Growth 155, 93–102 ~1995!.
7
R. L. Parker and J. R. Manning, ‘‘Application of pulse-echo ultrasonics to
locate the solid/liquid interface during solidification and melting,’’ J.
Cryst. Growth 79, 341–353 ~1986!.
8
C. K. Jen, Ph. de Heering, P. Sutcliffe, and J. F. Bussiere, ‘‘Ultrasonic
monitoring of the molten zone of single crystal germanium,’’ Mater. Eval.
49, 701–707 ~1991!.
9
F. A. Mauer, S. J. Norton, Y. Grinberg, D. Pitchure, and H. N. G. Wadley,
‘‘An ultrasonic method for reconstructing the two dimensional liquid–
solid interface in solidifying bodies,’’ Metall. Trans. B 22, 467–473
~1991!.
10
Yichi Lu and Haydn N. G. Wadley, ‘‘Two-dimensional wave propagation
in cylindrical single-crystal solid–liquid bodies,’’ J. Acoust. Soc. Am. 98,
2663–2680 ~1995!.
11
D. T. Queheillalt, Y. Lu, and H. N. G. Wadley, ‘‘Ultrasonic sensing of the
location and shape of a solid/liquid interface,’’ J. Acoust. Soc. Am. 101,
843–853 ~1997!.
12
Pravin M. Shah, ‘‘Ray tracing in three dimensions,’’ Geophysics 38, 600–
604 ~1973!.
13
F. I. Fedorov, Theory of Elastic Waves in Crystals ~Plenum, New York,
1968!.
14
M. J. P. Musgrave, Crystal Acoustics ~Holden-Day, San Francisco, 1970!.
15
B. A. Auld, Acoustic Fields and Waves in Solids ~Krieger, Malabar, FL,
1990!.
16
E. G. Henneke, ‘‘Reflection-refraction of a stress wave at a plane boundary between anisotropic media,’’ J. Acoust. Soc. Am. 51, 210–217 ~1971!.
17
E. G. Henneke and G. L. Jones, ‘‘Critical angle for reflection at a liquid–
solid interface in single crystals,’’ J. Acoust. Soc. Am. 59, 204–205
~1976!.
18
A. Atalar, ‘‘Reflection of ultrasonic waves at a liquid–solid interface,’’ J.
Acoust. Soc. Am. 73, 435–440 ~1983!.
19
S. I. Rokhlin, T. K. Bolland, and L. Adler, ‘‘Reflection and refraction of
elastic waves on a plane interface between two generally anisotropic media,’’ J. Acoust. Soc. Am. 79, 906–918 ~1986!.
20
Yu. A. Burenkov and S. P. Nikanorov, ‘‘Elastic properties of germanium,’’ Sov. Phys. Solid State 12, 1940–1942 ~1971!.
21
V. M. Glasov, S. N. Chizhevskaya, and N. N. Glagoleva, Liquid Semiconductors ~Plenum, New York, 1969!.
22
V. M. Glazov, A. A. Aivazov, and V. I. Timoshenko, ‘‘Temperature dependence of the velocity of sound in molten germanium,’’ Sov. Phys.
Solid State 18, 684–685 ~1976!.
1
FIG. 10. Time of flight projection data for a concave interface,
h5215 mm, s 3 5215 mm. ~a! Direct and singly transmitted ray paths as
functions of the vertical receiver coordinate, r 3 ; ~b! direct ray paths as
functions of the receiver angle a.
An inspection of TOF data in Figs. 7, 8, 9, and 10 also
indicates that a greater diversity of ray paths ~i.e., reflected,
singly/doubly transmitted! can be detected near the interface
in the diametral plane than in other nondiametral planes, indicating again that ultrasonic sensing in the diametral plane
is likely to be the preferred sensing configuration.
V. CONCLUSIONS
A method has been developed for the calculation of
three-dimensional ray paths, wavefront shapes, and TOF values during ultrasonic propagation through cylindrical single
crystal solid–liquid bodies. It has been demonstrated that
both the interface location and convexity strongly influence
ultrasonic TOF projection data. Positioning sensors near the
diametral plane is the preferred sensing configuration for
solid–liquid interface sensing because: ~1! the plane in
which reflected and transmitted rays propagate is known beforehand; and ~2! more ray paths ~reflected, transmitted, etc.!
are available in this plane for interface reconstruction purposes. There is no fundamental difficulty in using ray paths
in other nondiametral planes; it is just more complicated and
fewer ray paths are available for interface reconstruction.
The TOF projection data and different interface curvatures
are distinctively different for different interface curvatures,
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J. Acoust. Soc. Am., Vol. 103, No. 3, March 1998
Y. Lu and H. N. G. Wadley: Three-dimensional wave propagation
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