JPH0358058B2 - - Google Patents

Description

[Detailed description of the invention]

The present invention relates to a method for measuring a deviation angle of a cut plane of a single crystal. Many single crystals exhibit anisotropy as a property. Anisotropy refers to differences in mechanical, electrical, or optical properties depending on crystal orientation. In order to obtain an element that effectively utilizes the physical properties of such a crystal, it is necessary to cut the material at a certain orientation and at a certain angle with respect to a predetermined lattice plane. For example, in the case of a crystal resonator used as a reference oscillator for controlling the transmission frequency in communication equipment or obtaining a time reference, the angle of the cut plane with respect to the predetermined lattice plane or crystal axis is called The name is given as follows. However, in reality, it is difficult to make an ideal cut, and there is a slight deviation from the ideal angle. phenomenon is used. First, with reference to FIGS. 1 and 2, problems with the conventional measurement method will be discussed. In the conventional measurement method, as shown in Figure 1, the crystal is arranged so that the inclination direction of the crystal lattice plane with respect to the cut plane is parallel to the plane (device plane B) perpendicular to the crystal rotation axis that includes the direction of the incident X-ray. The measurement target was limited. In FIG. 1, N→s indicates a unit vector normal to the cut plane, N→p indicates a unit vector normal to the lattice plane, and the lattice plane is indicated by a broken line. That is, in the conventional method, the deviation angle δ was determined by determining the ω rotation angle that causes diffraction only for a sample prepared so that the rotation axis z of the crystal, the rotation axis lattice plane (broken line), and the cut plane are parallel. deviation angle δ
is given by δ=ω−θ. where θ is the known Bragg angle in a given lattice plane. However, as shown in Figure 2, if the lattice plane is not parallel to the crystal rotation axis but tilted by δ ₁ as shown in the figure, the diffraction plane (the plane where the incident X-rays X→i and the diffracted X-rays X→d span)
OPQO′ is no longer parallel to the device plane;
The ω rotation angle that causes diffraction by the lattice plane is ω = θ ₀ + δ ₂
becomes. Here θ ₀ is the apparent Bragg angle, and this θ ₀
and Bragg angle θ have the relationship shown by the following equation. sinθ ₀ = sinθcosδ ₂ /cosδ=sinθ cosδ ₂ √1+ ² ₁ + ² ₂ Therefore, if the lattice plane is not parallel to the crystal rotation axis or is not well arranged in parallel, the diffraction conditions may be changed. If the deviation angle δ is determined by observing the ω rotation angle that satisfies the following, there is a risk that a good product will be treated as a defective product or a defective product will be treated as a non-defective product due to incorrect judgment. Also, from the above formula, δ ₁ , δ ₂
I can't even ask for it. Conventionally, samples in which the direction of inclination of the lattice plane with respect to the crystal surface (cut plane) was unknown were tested using the Laue method. However, the Laue method is a photographic method that requires a considerable amount of exposure time and requires work in a dark room, making it difficult to conduct tests efficiently.
Furthermore, the measurement accuracy is also poor. An object of the present invention is to provide a method for measuring a deviation angle of a cut plane of a single crystal, which solves the measurement problems in the case where there is an inclination as described above. In order to achieve the above object, the method for measuring the deviation angle of a cutting surface of a single crystal according to the present invention includes: rotating a conventional single crystal around an ω rotation axis parallel to its cutting surface using a rotating means; The cut surface is irradiated with X-rays as a narrow beam perpendicular to the ω rotation axis, and the rotation angle (ω) of the crystal at which the intensity of the diffracted X-rays diffracted by the cut surface is maximized by a detector. The angle (φ) of the diffracted X-ray with respect to a plane perpendicular to the ω rotation axis is detected, and the Bragg angle (θ) of the lattice plane that caused the diffraction is calculated as δ=cos. ^-1 (sinθcos ω＋cos θcos ωcos φ) and the deviation angle (δ) between the crystal cutting surface and the lattice plane
It is configured to ask for. According to the above configuration, it is possible to quickly and accurately measure the deviation angle of a crystal plane tilted at a complicated angle, which was impossible using conventional methods, and the object of the present invention can be completely achieved. DESCRIPTION OF THE PREFERRED EMBODIMENTS The method of the present invention and the apparatus for carrying out the method will be described in more detail below with reference to the drawings and the like. FIG. 3A is a perspective view showing the first embodiment of the apparatus for carrying out the method of the present invention. The radiation from the radiation source S is restricted to a point by the collimator 6 and the slit, and is incident on the sample crystal 1. The sample crystal 1 is arranged so that its surface (cut surface) includes the z-axis. If the sample crystal 1 is rectangular, it can be realized by an L-shaped crystal holder block as shown in the figure. With this crystal holder block 2, the bottom surface of the crystal is arranged parallel to the xy plane. Note that when the sample crystal 1 has a cylindrical shape, the crystal is held so that the cut plane coincides with the z-axis and the axial direction of the cylinder is parallel to the xy plane.
By doing this, the X-rays become (x, y, z)
When the coordinates (a, b, c) that are irradiated to the origin of the coordinates and represent the external shape of the crystal are determined (x,
y, z) coordinates. This device is composed of an ω rotation motor 5, a worm 4, and a foil 3, and has a mechanism for rotating the crystal 1 around the z-axis (hereinafter simply referred to as ω rotation mechanism), and the angle ω between the crystal surface and the incident X-ray is can be observed. 2θ in the xy plane relative to the direction of the incident X-ray
X in the angular direction (diffraction angle with respect to a given lattice plane)
By arranging the ray detector 8, it is possible to detect X-rays diffracted by the lattice plane. When the lattice plane is parallel to the z-axis (rotation axis), the direction of the diffracted X-rays is within the xy plane (device plane), but when it is tilted with respect to the z-axis, depending on the magnitude of the tilt, The radiation is radiated along the generatrix of a right circular cone whose half apex angle is 2θ with the direction of the incident X-ray as its axis. The X-ray detector 8 has a size that allows it to detect X-rays, and can detect diffracted X-rays if the X-rays are incident within this range. The slit plate 10 is provided in front of the X-ray detector 8, and is provided with a slot 10a to allow only the X-rays diffracted from a predetermined grating plane to pass through.The slit plate 10 is placed in the yz plane as shown in FIG. When arranged in parallel, the slot 10a has a circular arc (radius =
It is formed by a slot with an appropriate width along Ltan2θ). By doing this, diffraction lines from another lattice plane that are close to the diffraction angle (2θ) with respect to a certain lattice plane do not enter the sensitive area 8a of the X-ray detector 8 shown by the broken line in FIG. 3B. It is blocked by the slit plate 10. Next, depending on where on the X-ray detector 10 the X-rays diffracted by the lattice plane are incident, the xy plane (device plane) and the diffraction plane (the plane where the directions of the incident X-rays and the diffracted ) will be explained about the observation of the angle φ. The device of this embodiment is provided with a sector 7 that rotates around the x-axis for observing the φ. First, the crystal 1 is rotated by the ω rotation mechanism described above, and the crystal 1 is fixed at an angular position of ω, where the intensity of X-rays diffracted from the lattice plane is maximized. During this ω rotation, the sector 7 is removed to a position where it does not block the arcuate groove 10a. Next, move sector 7 to sector motor 9.
The sector 7 is fixed by detecting the rotation angle at which the diffraction spot is halved (the X-ray intensity is halved) by the cut-off surface of the sector 7 which has been rotated. At this time, the angle formed by the y-axis and the lower end surface 7a of the sector 7 shown in the cut-off plane FIG. 3B is φ. The state in FIG. 3B where the blocking plane and the y-axis are parallel is set to φ=0, and from this state, the counterclockwise rotation angle + angle and the clockwise rotation angle are set as one angle. Next, a second embodiment of the apparatus for observing the φ angle mentioned above will be explained with reference to FIG. This embodiment uses a position sensitive proportional counter 11. The same reference numerals are given to parts common to the first embodiment described above. As shown in FIGS. 4B and 4C, the core wire 11a is stretched in an arc shape so that the X-rays diffracted by the lattice plane enter the core wire 11a of the proportional counter tube 11, and the X-ray entrance window 11b It is also provided in an arc shape with an appropriate width so as not to obstruct the incident X-rays. The output of this proportional counter 11 is sent to a position analysis circuit 14 via preamplifiers 12 and 13.
The analysis results are displayed by the intensity and incidence display device 15. In the second embodiment, the crystal 1 is rotated by the ω rotation mechanism in the same manner as described above, and the ω position angle at which the intensity of the diffracted X-rays is maximized is detected. The output of the proportional counter tube 11 is supplied to a preamplifier 12,
13 to the position analysis circuit 14, and calculates the difference between the outputs of both amplifiers to detect the diffracted X-rays.
When the height from the xy plane is detected, the φ angle is determined, and the intensity and position are displayed using 15. If you know the ω angle and φ angle as described above, the second
The angle between the crystal surface and the lattice plane shown in the figure, that is, the deviation angle δ and the direction of inclination 4 can be calculated as described below. Also, when the crystal shape is rectangular,
The angle δ ₁ between the intersection line between the side surface of the crystal and the lattice plane and the crystal surface, and the angle δ ₂ between the intersection line between the bottom plane and the lattice plane and the crystal surface can also be calculated as described below. These series of measurement procedures are performed according to a predetermined program while observing the ω angle and φ angle, and δ, δ ₁ , δ ₂ ,
ψ is automatically calculated and displayed. Automatic selection based on this calculation is also possible. Next, using the ω angle and φ angle mentioned above, δ, δ ₁ , δ ₂ , ψ
Explain what is required. In the following explanation, the symbols have already been explained, but they will be defined and used as follows. (a, b, c); Coordinates representing the external shape of the crystal. The b-axis is parallel to the unit vector N→s perpendicular to the crystal surface. (x, y, z); Coordinates representing the device. x-axis → parallel to incident X-ray z-axis → ω rotation rotational axis X → i; incident X-ray unit vector X → d; diffraction X-ray unit vector. N→p; unit vector normal to the diffraction lattice surface (lattice surface normal) N→s; unit vector normal to the crystal surface (crystal surface normal) When X→i and X→d are given with reference to Figure 5 N→p (lattice plane normal) is determined, and the magnitude and direction of the inclination with respect to N→s (crystal surface normal) are calculated. First, consider the case where the diffraction plane (OPQO') is rotated by φ around the x-axis with respect to the device plane (x, y plane) because the outer diffraction plane is tilted with respect to the crystal surface. . For the orthogonal coordinate system (x, y, z), X→i is parallel to the x-axis and the direction is opposite, so X→i=-1 0 0 ...(1) X→d is relative to X→i Since it is a direction that forms an angle ^of 2θ and is rotated by φ around the x-axis, immediately can be found. N→p must be within the plane (diffraction plane) defined by X→i, X→d, so N→p is X→i, X→
It can be expressed as a linear combination of d. N→p=aX→i+bX→d...(3) Coefficients a and b are determined by the following conditional expressions. −(X→i・N→p)=(X→d・N→p) ...(4) (N→p・N→p)=1 ...(5) When executed, from equation (4) (a+b ) [X→i・X→d+1]]0 Since [ ]≠0, a+b=0 ∴a=−b From equation (5), a ² +b ² +2ab(X→i×X→d)=1 a= -b, (X→i・X→d)=cos2θ 2(1−cos2θ)a ² =1 ∴a=±1/2sinθ, b=〓1/2sinθ (compound same order) a→+, b→ -The sign indicates -N→b, so N→p=-1/2sinθX→i+1/2sinθX
→d = 1/2sinθ−−1 0 0+−cos2θ sin2θ cosφ sin2θ sinφ=sinθ cosθ sinφ cosθ sinφ ……(6) On the other hand, rotate the crystal around the z-axis, and rotate the incident X-ray X→i and the crystal plane ( If the diffraction condition is satisfied when the angle formed with the a-axis is ω, then N→s can be found as follows. (x, y, z) coordinates and (a,
b, c) The relationship between the coordinates is expressed as follows by orthogonal transformation ω. x y z=ωa b c ...(7) ω=cosω sinω0 −sinωcosω0 0 0 1 ...(8) Since N→s is parallel to the b axis, N→s in the (a, b, c) coordinate system _a,b,c =0 1 0 ...(9) Therefore, in the (x, y, z) coordinate system, N→s is N→s=sinω cosω 0 ...(10) N→p and N→s If the angle ^formed by Equation (6) is (x, y,
z) It is inconvenient because it is indicated by coordinates. Therefore, as shown in Figure 6, the coordinate system (a, b,
It is good to express Np by c). As can be easily understood from Figure 6, the ac plane coincides with the surface of the crystal, the b axis perpendicular to this is parallel to Ns (crystal surface normal), ω rotation is performed around the c axis, and N→s is It is given by the following formula. N→s _a,b,c =0 1 0 Inverse transformation of equation (7) is performed. ω ^-1 = cosω sinω 0 −sinω cosω 0 0 0 1 ...(12) N→p _a,b,c =ω ^-1 N→p _x,y,z = sinθ cosω−c
osθsinωcosφ sinθ sinω+cosθcosωcosφ cosθ sinφ (13) Each component of equation (13) indicates the direction cosine of the a, b, and c axis directions and N→p _{a, b, and c,} respectively. Naturally, (N→p _a,b,c・N→s _a,b,c ) is (11)
It comes down to the formula. Equation (13) indicates the magnitude and direction of the inclination of the lattice plane with respect to the crystal surface, but it is not yet practical. Therefore, as shown in Figure 7, the angle ψ between the direction of maximum inclination on the crystal surface and the a-axis, and if the external shape of the crystal is a square parallelepiped as shown in the figure, the intersection line between the side surface and the lattice plane. It is specific and meaningful to determine the angle δ ₁ formed by the crystal surface and the angle δ ₂ formed between the intersection line of the base and the lattice plane and the crystal surface. If N→p _a . _b . _c (a _p , b _p , c _p ), from Figure 7 tanx＝c _p /−a _p ……(14) tanδ ₁ = c _p /b _p ……(15) tan δ ₂ =−a _p /b _p (16) cos δ = b _p (17) δ, δ ₁ , δ ₂ , and ψ are given as follows using equations (13) to (17). δ＝cos ^-1 (sinθcosω＋cosθcosωcosφ)...(11) δ ₁ = tan ^-1 (cosθ sinφ／sinθsinω＋cosθcosωcos
φ)……(18) δ ₂ = tan ^-1 (cosθsinωcosφ−sinθcosω／sinθsin
ω＋cosωcosφ）……(19) ψ＝tan ^-1 (cosθ sinφ／cosθsinωcosφ−sinθcos
ω) = tan ^-1 (tan δ ₁ /tan δ ₂ ) ...(20) However, for the ψ angle, the following formulas are applied depending on the signs of δ ₁ and δ ₂ .

[Table] As explained above, in the present invention, by measuring ω and φ, the angle δ between the lattice plane and the cut surface can be measured. This δ can be applied even when the external shape of the crystal is not a square, such as a cylinder or a sphere, by determining the coordinates representing the external shape as described above. In this case as well, (11), (18), (19),
By the algorithm shown in equation (20), (θ, ω, φ)
It can be immediately calculated and displayed as a function of . Therefore, compared to the above-mentioned Laue method, accurate measurement can be performed in a short time, and it can be suitably used for production control of vibrators.

[Brief explanation of drawings]

FIG. 1 and FIG. 2 are schematic diagrams for explaining the problems of the conventional deviation angle measuring device. FIG. 3 is a diagram showing the first embodiment of the apparatus for carrying out the method of the invention, in which FIG. 3A is a perspective view, FIG. 3B is a front view of the detector, and FIG. FIG. 3 is a plan view showing the relationship. FIG. 4 shows a second embodiment of the apparatus for carrying out the method of the present invention, in which FIG. A is a perspective view, FIG. B is a front view of a detector, and FIG. C is a crystal and a detector. FIG. Figures 5 and 6
7 and 7 are schematic diagrams for explaining that the deviation angle can be calculated from ω and φ, respectively. 1...Sample crystal, 2...Crystal holder block,
3...Warm foil, 4...Worm, 5...
ω rotation motor, 6...Collimator, 7...Sector, 8...X-ray detector, 9...Sector motor, 1
0...Slit plate, 11...Proportional counter tube, 12,
13...Preamplifier, 14...Position analysis circuit, 1
5...Intensity position display device.

Claims (1)

[Claims] 1. Rotating a known single crystal around an ω rotation axis parallel to its cutting surface by means of a rotation means, and converting X-rays from an X-ray source into a narrow beam perpendicular to the ω rotation axis. The cut surface is irradiated, and the rotation angle ω of the crystal when the intensity of the diffracted X-rays diffracted at the cut surface by a detector is maximized, and the angle (φ) of the diffracted X-rays with respect to a plane perpendicular to the rotation axis ), and by substituting the above ω, φ and the Bragg angle (θ) of the lattice plane that caused the diffraction into the following equation, the deviation angle (δ) between the crystal cut surface and the lattice plane is obtained.
A method for measuring the deviation angle of a cut plane of a single crystal configured to find the following. δ＝cos ^-1 (sinθcosω +cosθcosωcosφ)