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Observation of growth defects in synthetic quartz crystals by light-scattering tomography
Journal of Crystal Growth 44 (1978) 53—60 © North-Holland Publishing Company
OBSERVATION OF GROWTH DEFECTS IN SYNTHETIC QUARTZ CRYSTALS BY LIGHT-SCATTERING TOMOGRAPHY Kazuo MORIYA and Tomoya OGAWA Department of Physics, Gakushuin University, Mejiro, Tokyo-i 71, Japan Received 31 October 1977; manuscript received in final form 16 January 1978
Laser beam scattering tomography using an arrangement similar to an ultra-microscope was applied to characterization of synthetic quartz crystals and showed optical inhomogeneities whose dimensions were of the same order of magnitude as the wavelength. These inhomogeneities in optical index are caused by aggregation of hydrogen bonded OH in the crystals but the reason why such large aggregation occurs is still a problem to be solved. Using light scattering tomography, we found line defects which are dislocations. This tomography is very suitable for giving a bird’s eye view of crystal imperfections without slicing the crystals, by immersing them in a transparent liquid whose refractive index is little different from that of the crystals.
I. Introduction
with observation by X-ray topography and the extinction coefficient in the 3500 cm 1 region, because X-ray topography and infrared absorption techniques are~themost reliable and widely used techniques to characterize quartz crystals. However the X-ray and infrared techniques have their own weakness, that is, the topographic contrast of X-rays depends upon the thickness of crystalline specimens as well as the structure of defects in them and speciments must almost always be sliced to get good contrast before they are examined, because X-ray beams can not transmit through a thick specimen. The infrared absorption topography technique is less preferable and also needs specimens to be sliced before examination. From the view points of non-destructive inspection and thickness of specimens to be examined, an optical technique is much better than the X-ray and infrared ones, because light beams can transmit through the quartz crystals without appreciable attenuation and be focussed by lens systems to make images. Light scattering tomography using He—Ne laser beams has the above-mentioned merits and also has very good resolution because optical irregurarities due to defects and/or inclusions will act as scatterers even if they are smaller than the wavelength of the incident optical beam (Rayleigh scattering due to small particles) [13]. By this light-scattering tomography, we have
Since quartz crystals are very important in electronic engineering, a lot of quartz crystals are commercially synthesized by the hydrothermal method. The characterization of synthetic quartz crystals is necessary to get good oscillators and has been studied by many authors [1 12]. Dodd and Fraser [1] found good correlation between an elastic loss and extinction coefficient in the 3500 cm~ range, which was confirmed by Rudd et al. [2], Sawyer [3] and others [4,5]. They concluded that the infrared absorption bands are due to inclusions of hydrogen bonded OH [1,6 8], which produce milkiness in a crystal after heat treatment [1,9] Yoshimura and Kohra [10] studied growth defects in synthetic quartz crystals by X-ray diffraction topography and found distortions and inclinations of the lattice planes around the sector boundaries. Iwasaki [11] studied line defects in crystals by etching and X-ray techniques and found that etch tunnels are caused by the strain field along line defects. Homma and Iwata [12] found some impurity segregations in crystals by Lang topography and EPMA (electron probe micro-analyzer). In this paper growth defects in synthetic quartz crystals are studied tomograpKically by light (He Ne laser beam) scattering and discussed in comparison -
53
54
K. Moriya, T. Ogawa
/ Observation of growth defects in synthetic quartz crystals 3900CM’
35O~u’
~ed~
05
Fig. 1. Schematic figure of a Y-cut plate from a synthetic process crystal. are indicated by which the +X,are X, Z~,Z by , Z~ Z quartz Regions separated the and growing sectors.
b— ~
2
(
3900CM
3500CM
-
~
-
)
A~~F
A B C
~ [C
observed a bird’s-eye view of imperfections in a synthetic quartz crystal, that is, qualitative difference of hydrogen bonded OH content among the +X, X and Z growth sectors (fig. 1), and some dislocations. “Breathing of crystal growth” which is caused by fluctuations of temperature and/or solute flows, is also clearly observed. These imperfections in as-grown crystals can be observed without cutting or polishing when they are immersed in a transparent liquid whose refractive index is little different from the index of the crystals.
2. Experimental procedures and results To get information about inclusions and defects in connection with growth conditions, a specially grown crystal was hydrothermally prepared in a Na2CO3 solution at two growth rates: one of them was about 0.4 mm/day in the z direction on each side of the seed crystal corresponding with the region between G and H and between M and N in fig. 2c and the other one was about 0.5 mm/day in the region between H and I or between L and M in the figure. The extinction coefficients at 3500 and 3900 cm 1 in this crystal were measured along three typical traverses, represented by the straight lines AF, A’F’ and KJ in figs. 2a, 2b and 2c, respectively. The coefficients are shown in figs. 2 as a function of position. A Y-cut plate from the synthetic quartz crystal was studied in an ultra-microscope arrangement using a cw He Ne laser, which was similar to that used by *
Vand et al. described the condenser lens as an objective lens because they used an objective of a microscope as a condenser. Here we use the “objective lens” in an ordinary sense.
(b)
______________________________________________ C) K—
900 CM 0
~Th
z
~ L z
Z K-><
w
K 1—
‘3 K
2-—________________________________ _______________________ (C)
Fig. 2. Extinction coefficient at 3500 and 3900cm ~. The trace of the infrared spot is shown by a straight line in each figure.
Vand et al. [14]. The laser beam was focussed by a lens (its focal length: 25 mm) into about a 20 pm diam and traversed horizontally to take photographs of a section through the crystal. To make a fine beam is necessary for good resolution along the thickness direction. Its scanning speed in the specimen was about 100 pm/mm using a Kodak Tri-X film and I mW cw He Ne laser. All the photographs shown in this article were taken by a single scanning of the light pencil. To remove random reflections at surfaces of the quartz specimen, it was immersed in a mixture of 26% dichloronaphthalene and 76% dimethylphthalate (in volume ratio) whose refractive index was little different from the refractive index of the crystal. For ordinary observations it is not necessary to polish the surfaces of the crystals when they are immersed in the mixture but it is advantageous to have a polished surface for the exit scattered light, i.e., on the side of lens ~ A light scattering tomograph of this crystal (fig. 3)
the objective
K. Moriya, T. Ogawa / Observation of growth defects in synthetic quartz crystals
5mm Fig. 3. Light scattering topograph using a
shows a good correlation between the light scattering power at 6328 A or brightness in fig. 3 and the extinction coefficient at 3500 cm shown in figs. 2a 2c After heating at 500°Cfor 60 h in air, milky bands appeared in the crystal, as shown in fig. 4 which was taken by the light scattering technique. However, in this case, the light scattering power was about 20 times larger than before heat-treatment,
cw
55
-1 -
1 mW He Ne laser (before heat-treatment).
Comparing fig. 3 and fig. 4, we find a very good correlation between them in every detail of their shapes but there are some discrepancies between their contrast. Using Ag Kcs radiation, Lang topographs were taken by (2110) and (0003) reflections in a Y-cut plate (its thickness: 1 mm) after 30 sec etching in hydroflouric acid (figs. 5a and 5b), where very many dislocations were observed.
5mm Fig. 4. Light scattering topograph after heating at 500°C for 60 h in air.
56
K. Moriya, T. Ogawa / Observation of growth defects in synthetic quartz crystals
~ ~ JTh~ ~—
-~
-~w~_ ~-°--‘—--
(a)d
5mm~~-~ —
g
~
~
-~
E~
Fig. 5. Lang topographs taken with a Y-cut plate of 1 mm thickness (before heat treatment): (a) (2110) reflection; (b) (0003) reflection.
3. Discussion 3.1. Light scattering from an inhomogeneous distribution ofimpurity atoms Light scattering from small particles is well known as Rayleigh scattering. In the Rayleight theory, the scatterer is treated as a metallic sphere or a discrete sphere with a refractive index different from its ambient medium; but an inhomogenity in a quartz
crystal, or a local change of refractive index in the crystal, will be less abrupt than a Rayleigh scatterer because an inhomogenity will be caused by segregation of impurity atoms or strain due to defects. The refractive index of a volume of matter will be determined by the polarizability of its outer-shell electrons which contribute to chemical bonds. These have much smaller binding energies than those of inner electrons which contribute to X-ray scattering. The scattering factor of optical waves from these electrons
K. Moriya, T. Ogawa / Observation of growth defects in synthetic quartz crystals
is given by
The second term ofeq. (1) is given by
exp(i ~! s’ r) dr
ffp(r)
f~=
where p(r) is the probability that the electrons will be found in an element of volume dr whose mid-point is defined by the vector r and s~= 2 sin 6, where 20 is the angle between incident and scattered beams. In order to survey the nature of scatterers and get their detectable size, we assume that the density of the electrons is p(r)
Po + A sin(j3r)/(3r,
(2)
where Po is the homogeneous component of the electron distribution and the second term is due to an inhomogenity whose size is given by a parameter and its magnitude by A. The scattering factor will be given by
a
f=
sinpr
I” 2 ,4irrp
0
dr
..
fsin(i~r)sinO1r) dr
-~-
4irA p sin(31) cos(pl) fl cos(J31) sin(pl) 2 ~p ~32 p Al / sin(j.zl)\ = 2rr —i 2 1 2irAl/p2 (j3 = p). (Sb) p This term shows that an aggregation of impurity atoms or an irregular arrangement of atoms scatters the laser beam, if j3 p, f2 will be proportional to —
—
—
)
~‘
irA /132p, that is, the product of the cross-sectional area of the irregularity ir/J32 and its magnitude A. On the other hand, when the inhomogeneity is vague and (3! ~ 1, f~ will be given by 4 A 1 = .i_ far sin(pr) dr o 3A ~ [sin(pI) (p1) cos(pl)] —
/47 1
57
—
,
+A fslnGir) 4~’r2sm(pr) dr,
(3)
0
where p = 4ir sin 0/X. Here the integral of eq. (1) should be taken throughout the whole volume of the light pencil in the crystal but / will be good enough if it is of the order of magnitude of the laser beam radius. The first term of eq. (3) is given by
‘
1
(p1)3 ~irl3A Up!). (6) In this extreme case the ingomogeneity causes no =
light scattering even if A is large, that is, the light beam is deflected by this inhomogeneity. The intensity of scattered light I is given by the incident beam intensity I~as j q2 w2 2 1 ÷cos2 20 ~=(~_~ 2 (7) 2f2) 2R where q and m are the charge and mass of the electron, c is the light velocity, R is the distance between a scatterer and observing point, w resonant frequency of bonding electrons and 0w isis the frequency of the laser beam. Usually, w 0 is in the ‘
4lTPo
fi
fr sin(pr) dr
=
o
p
3 =
3 4ir1
3 [sin(pl) 0
—
(pl) cos(pl)]
(p!)3
(4~ ‘
“ ‘
range of ultra-violet light and much higher than It can be seen from this analysis that we can observe fluctuation of refractive index dueand to adue segregation and aggregation of impurity atoms to strains by lattice defects. The detectable size of optical fluctuation is dependent on its magnitude but will be smaller than the wavelength of the laser beam. From a view point of characterization of synthetic quartz crystals, light scattering tomography by a He—Ne laser is compared with the other methods in table 1. ~.
In this equation the term 3 3[sin(pl) (p/) cos(pl)]/(pl) is a Dirac ~ function and has a value only when p =0, if 1 is fairly large. Therefore the homogeneous distribution of the electrons does not make any contribution to light scatteringand then the laser beam propagates along its forward direction without attenuation due to scattering, —
58
K. Moriya, T. Ogawa
/ Observation of growth defects in synthetic quartz crystals
Table 1 Comparison of five methods for characterization of synthetic quartz crystals Item
Method Light scattering tomograph using a laser
Lang topograph
Scanning spot method of 3500 cm 1 radiation
Non-destructive
Yes
Yes
Size of specimen
As-grown crystals immersed in the mixture
Necessary to slice thickness: 1 mm
Physical property detected by the
Inhomogeneity of optical index caused by aggregated gated impurity atoms and irregular arrangement of atoms due to defects
Strain gradient caused by dislocations and other defects
Less than 1 pm
About 1 pm
method
Resolution in plane along
thickness direction Correlation with mechanicalQ
Heat-treatment at 500°C
y-irradiation
Yes
No
No?
Necessary to slice thickness: 1 mm
Sliced specimens are recommended
Stretching vibration of hydrogen bonded OH
Inclusions of ordered water
Diameter of light pencil —20 pm
Spot size slit of IR spectrometer, 2 2 X 0.5 mm Thickness of specimens <1 mm <10mm
Should be good a
Questionable
Perfect (see refs.
Substitional Al atoms
Depends upon optical resolving
power of detection system Within depth of focus of objective lens Good
No
[1] [31)
a Because the light scattering power is proportional to the extinction coefficient at 3500 cm 1 as shown in figs. 2 and 3, and Dodd and Fraser [9] have also indicated a one to one correspondence between the variation in refractive index and the changes in the extinction coefficient at 3500 cm
3.2. Light scattering tomographs and crystal growth Many light scattering tomographs were taken by
3500 cm 1 shown in fig. 2 have a strong correlation with scattering powers for the laser beam, that is, the brightnesses in figs. 3 and 6. Therefore the scatterers
changing the direction of incident beam in a Y-plane of a synthetic quartz, but similar patterns were obtained. This fact means that striations and the other patterns in fig. 3 are caused by dotted irregularities in the crystal, because, if these irregularities are constructed by continua with different refractive indices such as multilayer films of an optical inter-
in the crystal will be aggregated sedimentations of hydrogen bonded OH, which will grow during the heat treatment to much larger dotted clusters which act as more effective scatterers. It is a problem to understand why impurities aggregate to a size which act as light scatterers during growth. However, the striations indicate that the
ference filter, the scattered patterns due to such optical irregularities will be changed by the direction of incident beam due to diffraction and interference, Scattering powers from the X, +X and Z sectors are so different from each other that the photographs with best contrast were taken and shown in figs. 6a 6d. The milky bands due to heat-treatment at 500°C for 60 h in air (fig. 4) are very similar to the patterns before heating (fig. 3). The extinction coefficients at
crystal does not grow at a constant velocity but has an irregularity which may be described as “breathing of crystal growth”. This “breathing” will be caused by fluctuations of heat and material flows. When
solute deposits on a growing surface, the heat of crystallization is emitted and the temperature of the surface will be slightly mcreased, to oppose the flow of the solute [151.The decrease of deposition rate of solute decreases the temperature of the growing surface and then the supersaturation on the surface is
K. Moriva, T. Ogawa / Observation of growth defects in synthetic quartz crystals
iI~
~
(a) ~
1 mm
_____
_______
____
(b)
:~
~mm~
-
::,
___
1(c)
-
d) ~
2mm
Fig. 6. Light scattering topographs (before heat-treatment): (a) near seed; (b) the Z sector; (c) the +X sector; (d) the
increased. The dotted lines observed in fig. 6, especially in fig. 6d, are dislocations which may be decorated by impurities. The number of these lines in fig. 6 is fewer than the number of dislocations seen in fig. 5. One reason for this is that the light scattering tomograph can observe only dislocations within the plane whose thickness is the diameter of the light pencil (about 20 Jzm) and is about 1/50 of the thickness of the specimen used to take the Lang topographs of figs. ~ We still have many problems to be solved as to whether dislocations can be observed by this light scattering method or not, and are studying these in theoretical and experimental ways.
59
X sector.
Acknowledgements The authors wish to express their cordial thanks to Dr. F. Iwasaki (Kinsekisha Laboratory) for measurements of the infrared absorption coefficients in fig. 2 and helpful discussions.
References [1] D.M. Dodd and D.B. Fraser, 3. Phys. Chem. Solids 26 (1965) 673. [2] D.W. Rudd, E.E. Houghton and W.J. Carroll, Western Electric Engineer 1(1966) 22.
60
K. Moriya, T. Ogawa / Observation of growth defects in synthetic quartz crystals
[3] B. Sawyer, IEEE Trans. Sonics Ultrasonics SU-19 [4] [5] [6] [7] [8] [9]
(1972)41. J. Ashihara and S. Taki, Proc. 26th F.C.S. (1972) 93. 3. Ashihara, K. Takazawa, E. Yazaki, 3. Okuda and N. Asanuma, Proc. 28th F.C.S. (1974) 117. DL. Wood, 3. Phys. Chem. Solids 13 (1960) 326. PA. Staats and O.C. Kopp, J. Phys. Chem. Solids 35 (1974) 1029. B.D. Saksena, 3. Phys. Chem. Solids 26 (1965) 1929. D.M. Dodd and D.B. Fraser, Am. Mineralogist 52 (1967) 149.
[10] J. Yoshimura and K. Kohra, 3. Crystal Growth 33 (1976) 311. [11] F. Iwasaki, 3. Crystal Growth 39 (1977) 291. [12] S. Homma and M. Iwata, J. Crystal Growth 19 (1973) 125. [13] M. Tajima and T. lizuka, Japan. J. Appi. Phys. 15 (1976) 651. [14] V. Vand, K. Vedam and R. Stein, J. AppL Phys. 37 (1966) 2551. [15] T. Ogawa, Japan. J. Appl. Phys. 16 (1977) 689.
OBSERVATION OF GROWTH DEFECTS IN SYNTHETIC QUARTZ CRYSTALS BY LIGHT-SCATTERING TOMOGRAPHY Kazuo MORIYA and Tomoya OGAWA Department of Physics, Gakushuin University, Mejiro, Tokyo-i 71, Japan Received 31 October 1977; manuscript received in final form 16 January 1978
Laser beam scattering tomography using an arrangement similar to an ultra-microscope was applied to characterization of synthetic quartz crystals and showed optical inhomogeneities whose dimensions were of the same order of magnitude as the wavelength. These inhomogeneities in optical index are caused by aggregation of hydrogen bonded OH in the crystals but the reason why such large aggregation occurs is still a problem to be solved. Using light scattering tomography, we found line defects which are dislocations. This tomography is very suitable for giving a bird’s eye view of crystal imperfections without slicing the crystals, by immersing them in a transparent liquid whose refractive index is little different from that of the crystals.
I. Introduction
with observation by X-ray topography and the extinction coefficient in the 3500 cm 1 region, because X-ray topography and infrared absorption techniques are~themost reliable and widely used techniques to characterize quartz crystals. However the X-ray and infrared techniques have their own weakness, that is, the topographic contrast of X-rays depends upon the thickness of crystalline specimens as well as the structure of defects in them and speciments must almost always be sliced to get good contrast before they are examined, because X-ray beams can not transmit through a thick specimen. The infrared absorption topography technique is less preferable and also needs specimens to be sliced before examination. From the view points of non-destructive inspection and thickness of specimens to be examined, an optical technique is much better than the X-ray and infrared ones, because light beams can transmit through the quartz crystals without appreciable attenuation and be focussed by lens systems to make images. Light scattering tomography using He—Ne laser beams has the above-mentioned merits and also has very good resolution because optical irregurarities due to defects and/or inclusions will act as scatterers even if they are smaller than the wavelength of the incident optical beam (Rayleigh scattering due to small particles) [13]. By this light-scattering tomography, we have
Since quartz crystals are very important in electronic engineering, a lot of quartz crystals are commercially synthesized by the hydrothermal method. The characterization of synthetic quartz crystals is necessary to get good oscillators and has been studied by many authors [1 12]. Dodd and Fraser [1] found good correlation between an elastic loss and extinction coefficient in the 3500 cm~ range, which was confirmed by Rudd et al. [2], Sawyer [3] and others [4,5]. They concluded that the infrared absorption bands are due to inclusions of hydrogen bonded OH [1,6 8], which produce milkiness in a crystal after heat treatment [1,9] Yoshimura and Kohra [10] studied growth defects in synthetic quartz crystals by X-ray diffraction topography and found distortions and inclinations of the lattice planes around the sector boundaries. Iwasaki [11] studied line defects in crystals by etching and X-ray techniques and found that etch tunnels are caused by the strain field along line defects. Homma and Iwata [12] found some impurity segregations in crystals by Lang topography and EPMA (electron probe micro-analyzer). In this paper growth defects in synthetic quartz crystals are studied tomograpKically by light (He Ne laser beam) scattering and discussed in comparison -
53
54
K. Moriya, T. Ogawa
/ Observation of growth defects in synthetic quartz crystals 3900CM’
35O~u’
~ed~
05
Fig. 1. Schematic figure of a Y-cut plate from a synthetic process crystal. are indicated by which the +X,are X, Z~,Z by , Z~ Z quartz Regions separated the and growing sectors.
b— ~
2
(
3900CM
3500CM
-
~
-
)
A~~F
A B C
~ [C
observed a bird’s-eye view of imperfections in a synthetic quartz crystal, that is, qualitative difference of hydrogen bonded OH content among the +X, X and Z growth sectors (fig. 1), and some dislocations. “Breathing of crystal growth” which is caused by fluctuations of temperature and/or solute flows, is also clearly observed. These imperfections in as-grown crystals can be observed without cutting or polishing when they are immersed in a transparent liquid whose refractive index is little different from the index of the crystals.
2. Experimental procedures and results To get information about inclusions and defects in connection with growth conditions, a specially grown crystal was hydrothermally prepared in a Na2CO3 solution at two growth rates: one of them was about 0.4 mm/day in the z direction on each side of the seed crystal corresponding with the region between G and H and between M and N in fig. 2c and the other one was about 0.5 mm/day in the region between H and I or between L and M in the figure. The extinction coefficients at 3500 and 3900 cm 1 in this crystal were measured along three typical traverses, represented by the straight lines AF, A’F’ and KJ in figs. 2a, 2b and 2c, respectively. The coefficients are shown in figs. 2 as a function of position. A Y-cut plate from the synthetic quartz crystal was studied in an ultra-microscope arrangement using a cw He Ne laser, which was similar to that used by *
Vand et al. described the condenser lens as an objective lens because they used an objective of a microscope as a condenser. Here we use the “objective lens” in an ordinary sense.
(b)
______________________________________________ C) K—
900 CM 0
~Th
z
~ L z
Z K-><
w
K 1—
‘3 K
2-—________________________________ _______________________ (C)
Fig. 2. Extinction coefficient at 3500 and 3900cm ~. The trace of the infrared spot is shown by a straight line in each figure.
Vand et al. [14]. The laser beam was focussed by a lens (its focal length: 25 mm) into about a 20 pm diam and traversed horizontally to take photographs of a section through the crystal. To make a fine beam is necessary for good resolution along the thickness direction. Its scanning speed in the specimen was about 100 pm/mm using a Kodak Tri-X film and I mW cw He Ne laser. All the photographs shown in this article were taken by a single scanning of the light pencil. To remove random reflections at surfaces of the quartz specimen, it was immersed in a mixture of 26% dichloronaphthalene and 76% dimethylphthalate (in volume ratio) whose refractive index was little different from the refractive index of the crystal. For ordinary observations it is not necessary to polish the surfaces of the crystals when they are immersed in the mixture but it is advantageous to have a polished surface for the exit scattered light, i.e., on the side of lens ~ A light scattering tomograph of this crystal (fig. 3)
the objective
K. Moriya, T. Ogawa / Observation of growth defects in synthetic quartz crystals
5mm Fig. 3. Light scattering topograph using a
shows a good correlation between the light scattering power at 6328 A or brightness in fig. 3 and the extinction coefficient at 3500 cm shown in figs. 2a 2c After heating at 500°Cfor 60 h in air, milky bands appeared in the crystal, as shown in fig. 4 which was taken by the light scattering technique. However, in this case, the light scattering power was about 20 times larger than before heat-treatment,
cw
55
-1 -
1 mW He Ne laser (before heat-treatment).
Comparing fig. 3 and fig. 4, we find a very good correlation between them in every detail of their shapes but there are some discrepancies between their contrast. Using Ag Kcs radiation, Lang topographs were taken by (2110) and (0003) reflections in a Y-cut plate (its thickness: 1 mm) after 30 sec etching in hydroflouric acid (figs. 5a and 5b), where very many dislocations were observed.
5mm Fig. 4. Light scattering topograph after heating at 500°C for 60 h in air.
56
K. Moriya, T. Ogawa / Observation of growth defects in synthetic quartz crystals
~ ~ JTh~ ~—
-~
-~w~_ ~-°--‘—--
(a)d
5mm~~-~ —
g
~
~
-~
E~
Fig. 5. Lang topographs taken with a Y-cut plate of 1 mm thickness (before heat treatment): (a) (2110) reflection; (b) (0003) reflection.
3. Discussion 3.1. Light scattering from an inhomogeneous distribution ofimpurity atoms Light scattering from small particles is well known as Rayleigh scattering. In the Rayleight theory, the scatterer is treated as a metallic sphere or a discrete sphere with a refractive index different from its ambient medium; but an inhomogenity in a quartz
crystal, or a local change of refractive index in the crystal, will be less abrupt than a Rayleigh scatterer because an inhomogenity will be caused by segregation of impurity atoms or strain due to defects. The refractive index of a volume of matter will be determined by the polarizability of its outer-shell electrons which contribute to chemical bonds. These have much smaller binding energies than those of inner electrons which contribute to X-ray scattering. The scattering factor of optical waves from these electrons
K. Moriya, T. Ogawa / Observation of growth defects in synthetic quartz crystals
is given by
The second term ofeq. (1) is given by
exp(i ~! s’ r) dr
ffp(r)
f~=
where p(r) is the probability that the electrons will be found in an element of volume dr whose mid-point is defined by the vector r and s~= 2 sin 6, where 20 is the angle between incident and scattered beams. In order to survey the nature of scatterers and get their detectable size, we assume that the density of the electrons is p(r)
Po + A sin(j3r)/(3r,
(2)
where Po is the homogeneous component of the electron distribution and the second term is due to an inhomogenity whose size is given by a parameter and its magnitude by A. The scattering factor will be given by
a
f=
sinpr
I” 2 ,4irrp
0
dr
..
fsin(i~r)sinO1r) dr
-~-
4irA p sin(31) cos(pl) fl cos(J31) sin(pl) 2 ~p ~32 p Al / sin(j.zl)\ = 2rr —i 2 1 2irAl/p2 (j3 = p). (Sb) p This term shows that an aggregation of impurity atoms or an irregular arrangement of atoms scatters the laser beam, if j3 p, f2 will be proportional to —
—
—
)
~‘
irA /132p, that is, the product of the cross-sectional area of the irregularity ir/J32 and its magnitude A. On the other hand, when the inhomogeneity is vague and (3! ~ 1, f~ will be given by 4 A 1 = .i_ far sin(pr) dr o 3A ~ [sin(pI) (p1) cos(pl)] —
/47 1
57
—
,
+A fslnGir) 4~’r2sm(pr) dr,
(3)
0
where p = 4ir sin 0/X. Here the integral of eq. (1) should be taken throughout the whole volume of the light pencil in the crystal but / will be good enough if it is of the order of magnitude of the laser beam radius. The first term of eq. (3) is given by
‘
1
(p1)3 ~irl3A Up!). (6) In this extreme case the ingomogeneity causes no =
light scattering even if A is large, that is, the light beam is deflected by this inhomogeneity. The intensity of scattered light I is given by the incident beam intensity I~as j q2 w2 2 1 ÷cos2 20 ~=(~_~ 2 (7) 2f2) 2R where q and m are the charge and mass of the electron, c is the light velocity, R is the distance between a scatterer and observing point, w resonant frequency of bonding electrons and 0w isis the frequency of the laser beam. Usually, w 0 is in the ‘
4lTPo
fi
fr sin(pr) dr
=
o
p
3 =
3 4ir1
3 [sin(pl) 0
—
(pl) cos(pl)]
(p!)3
(4~ ‘
“ ‘
range of ultra-violet light and much higher than It can be seen from this analysis that we can observe fluctuation of refractive index dueand to adue segregation and aggregation of impurity atoms to strains by lattice defects. The detectable size of optical fluctuation is dependent on its magnitude but will be smaller than the wavelength of the laser beam. From a view point of characterization of synthetic quartz crystals, light scattering tomography by a He—Ne laser is compared with the other methods in table 1. ~.
In this equation the term 3 3[sin(pl) (p/) cos(pl)]/(pl) is a Dirac ~ function and has a value only when p =0, if 1 is fairly large. Therefore the homogeneous distribution of the electrons does not make any contribution to light scatteringand then the laser beam propagates along its forward direction without attenuation due to scattering, —
58
K. Moriya, T. Ogawa
/ Observation of growth defects in synthetic quartz crystals
Table 1 Comparison of five methods for characterization of synthetic quartz crystals Item
Method Light scattering tomograph using a laser
Lang topograph
Scanning spot method of 3500 cm 1 radiation
Non-destructive
Yes
Yes
Size of specimen
As-grown crystals immersed in the mixture
Necessary to slice thickness: 1 mm
Physical property detected by the
Inhomogeneity of optical index caused by aggregated gated impurity atoms and irregular arrangement of atoms due to defects
Strain gradient caused by dislocations and other defects
Less than 1 pm
About 1 pm
method
Resolution in plane along
thickness direction Correlation with mechanicalQ
Heat-treatment at 500°C
y-irradiation
Yes
No
No?
Necessary to slice thickness: 1 mm
Sliced specimens are recommended
Stretching vibration of hydrogen bonded OH
Inclusions of ordered water
Diameter of light pencil —20 pm
Spot size slit of IR spectrometer, 2 2 X 0.5 mm Thickness of specimens <1 mm <10mm
Should be good a
Questionable
Perfect (see refs.
Substitional Al atoms
Depends upon optical resolving
power of detection system Within depth of focus of objective lens Good
No
[1] [31)
a Because the light scattering power is proportional to the extinction coefficient at 3500 cm 1 as shown in figs. 2 and 3, and Dodd and Fraser [9] have also indicated a one to one correspondence between the variation in refractive index and the changes in the extinction coefficient at 3500 cm
3.2. Light scattering tomographs and crystal growth Many light scattering tomographs were taken by
3500 cm 1 shown in fig. 2 have a strong correlation with scattering powers for the laser beam, that is, the brightnesses in figs. 3 and 6. Therefore the scatterers
changing the direction of incident beam in a Y-plane of a synthetic quartz, but similar patterns were obtained. This fact means that striations and the other patterns in fig. 3 are caused by dotted irregularities in the crystal, because, if these irregularities are constructed by continua with different refractive indices such as multilayer films of an optical inter-
in the crystal will be aggregated sedimentations of hydrogen bonded OH, which will grow during the heat treatment to much larger dotted clusters which act as more effective scatterers. It is a problem to understand why impurities aggregate to a size which act as light scatterers during growth. However, the striations indicate that the
ference filter, the scattered patterns due to such optical irregularities will be changed by the direction of incident beam due to diffraction and interference, Scattering powers from the X, +X and Z sectors are so different from each other that the photographs with best contrast were taken and shown in figs. 6a 6d. The milky bands due to heat-treatment at 500°C for 60 h in air (fig. 4) are very similar to the patterns before heating (fig. 3). The extinction coefficients at
crystal does not grow at a constant velocity but has an irregularity which may be described as “breathing of crystal growth”. This “breathing” will be caused by fluctuations of heat and material flows. When
solute deposits on a growing surface, the heat of crystallization is emitted and the temperature of the surface will be slightly mcreased, to oppose the flow of the solute [151.The decrease of deposition rate of solute decreases the temperature of the growing surface and then the supersaturation on the surface is
K. Moriva, T. Ogawa / Observation of growth defects in synthetic quartz crystals
iI~
~
(a) ~
1 mm
_____
_______
____
(b)
:~
~mm~
-
::,
___
1(c)
-
d) ~
2mm
Fig. 6. Light scattering topographs (before heat-treatment): (a) near seed; (b) the Z sector; (c) the +X sector; (d) the
increased. The dotted lines observed in fig. 6, especially in fig. 6d, are dislocations which may be decorated by impurities. The number of these lines in fig. 6 is fewer than the number of dislocations seen in fig. 5. One reason for this is that the light scattering tomograph can observe only dislocations within the plane whose thickness is the diameter of the light pencil (about 20 Jzm) and is about 1/50 of the thickness of the specimen used to take the Lang topographs of figs. ~ We still have many problems to be solved as to whether dislocations can be observed by this light scattering method or not, and are studying these in theoretical and experimental ways.
59
X sector.
Acknowledgements The authors wish to express their cordial thanks to Dr. F. Iwasaki (Kinsekisha Laboratory) for measurements of the infrared absorption coefficients in fig. 2 and helpful discussions.
References [1] D.M. Dodd and D.B. Fraser, 3. Phys. Chem. Solids 26 (1965) 673. [2] D.W. Rudd, E.E. Houghton and W.J. Carroll, Western Electric Engineer 1(1966) 22.
60
K. Moriya, T. Ogawa / Observation of growth defects in synthetic quartz crystals
[3] B. Sawyer, IEEE Trans. Sonics Ultrasonics SU-19 [4] [5] [6] [7] [8] [9]
(1972)41. J. Ashihara and S. Taki, Proc. 26th F.C.S. (1972) 93. 3. Ashihara, K. Takazawa, E. Yazaki, 3. Okuda and N. Asanuma, Proc. 28th F.C.S. (1974) 117. DL. Wood, 3. Phys. Chem. Solids 13 (1960) 326. PA. Staats and O.C. Kopp, J. Phys. Chem. Solids 35 (1974) 1029. B.D. Saksena, 3. Phys. Chem. Solids 26 (1965) 1929. D.M. Dodd and D.B. Fraser, Am. Mineralogist 52 (1967) 149.
[10] J. Yoshimura and K. Kohra, 3. Crystal Growth 33 (1976) 311. [11] F. Iwasaki, 3. Crystal Growth 39 (1977) 291. [12] S. Homma and M. Iwata, J. Crystal Growth 19 (1973) 125. [13] M. Tajima and T. lizuka, Japan. J. Appi. Phys. 15 (1976) 651. [14] V. Vand, K. Vedam and R. Stein, J. AppL Phys. 37 (1966) 2551. [15] T. Ogawa, Japan. J. Appl. Phys. 16 (1977) 689.