Single crystals exhibit different properties, depending on the material and their orientation. These properties include:
- Optical (variation of indexes of refraction and spectral transmission)
- Mechanical (i.e., hardness, potential for cleavage, elastic moduli, solubility, and radiation resistance)
- Thermal (conductivity, thermal expansion, specific heat, and melting temperature)
- Electrical (dielectric constant and piezoelectricity).
Differences in orientation can affect the outcome of an experiment. When an instrument part does not function as expected, it is possible that the orientation is not suitable for the application. It is important that both technical and nontechnical users of single-crystal materials understand that the variation in properties of crystalline materials can have a significant influence on the experiment.
Most of the available literature on single crystals describes a specific material for a single application. When ordering a single-crystal material, the orientation should always be specified, even if that specification is “none required” or “random.” When a particular orientation is required, either the Miller or Bravais index should be denoted, as should its tolerance. In addition to the Miller or Bravais index, a letter such as x, y, z, or c may also be used.
Often, orientation is not needed when the single-crystal material has a cubic structure. Other crystal structures do not require orientation if they are being used only for their extended spectral range characteristics or mechanical strength/hardness or chemical resistance; orientation may not be necessary in cases in which the introduction of or change in polarization will not be problematic.
Orientation is usually required when: 1) the thermal expansion, thermal coefficient, and index of refraction have to be the same in all directions; 2) the birefringence must be either at a maximum or minimum; 3) polarization effects must be considered; or 4) the user is looking for a cleavage plane or trying to stay away from one. These are a few scenarios in which orientation may be required. Most frequently, but certainly not always, when orientation is necessary, it is with a noncubic crystal.
Methods and limitations of single-crystal orientation
The four methods used to orient single-crystal materials follow, listed from least accurate to most accurate:
- Crystal morphology (natural facets or cleavage): If crystal facets are present, the accuracy of using them to determine crystal orientation generally is on the order of, at best, 15 min (this is rare and depends on the material) to the more likely result of 5–15° with surface morphology and less than 6 min for cleavage. The method is best used when trying to get an approximation of the orientation before turning to the more accurate technique of X-ray goniometry.
- Optical polariscopy: The accuracy of the optical polariscope method is on the order of 1.0° to greater than 3.0°. The method will give results only on materials that are birefringent, and then only on some of the crystalline axes. Another caveat is that the material must be transparent in the visible wavelengths. For orientations that are not on the optical axis (0001) or orthogonal to it, there will be no meaningful results. A polariscope is a very simple device to make; all that is needed is a box containing a light source (40-W incandescent or equivalent compact fluorescent lamp [CFL]), two pieces of polarizing sheet, and some way to hold them. A better option is to use a visible laser reflecting off a large, polished steel ball or convex mirror (approx. 0.5 in. radius or larger) to disperse the beam, followed by two pieces of polarizing sheet. The image from this polarizer can be projected on a wall or screen. The additional benefit of the polariscope is that in transparent materials, a determination can be made as to the homogeneity of the material.
- X-ray back-reflection Laue method: The X-ray back-reflection Laue method yields an accuracy of 0.50–1.50°. In addition to the X-ray machine, a film plate and instant film are needed. This will yield a series of spots on the film called a Laue pattern. The ability to recognize the scatter pattern and its distance from the center will provide the crystal orientation and the tolerance. Back-reflection Laue is the most difficult to interpret; it requires the highest technical skill and is the most time consuming. The advantage of this method is that it generates a photographic record of the orientation and its tolerance.
- X-ray goniometry: The most accurate and fastest method is X-ray goniometry. It is employed at Meller Optics, Inc. (Providence, RI) and most other companies working with single-crystal materials. The accuracy range of the method is on the order of 0.5–3 min. As with most measurements, the more often they are performed, the greater the accuracy achieved.
The X-ray diffraction analysis system consists of five major parts:
- High-voltage generator: The high-voltage generator typically generates up to at least 35,000–50,000 V and 40–50 mA. The voltage can be adjusted smoothly from approximately 5000 V to the maximum that the generator can produce, and the current from approximately 5 mA to about 40 or 50 mA.
- X-ray tube assembly: The tube that produces the X-rays as well as a water-cooling circuit that prevents the tube from overheating are part of the X-ray tube assembly. The X-ray wavelength often used for measuring crystal orientations is 1.5405 Å (ka1, copper). The tube assembly usually contains a pair of slits that act as a collimator so that the X-ray emits a narrow vertical slit image. Finally, a shutter stops the X-rays from exiting the slit until it is time to take the measurement.
- Goniometer and sample holder assembly: The goniometer and sample holder assembly has θ and 2θ goniometer arms that rotate on a common axis. The 2θ arm holds the part to be measured, and the θ arm holds the detector. On some instruments a digital rotary encoder is mounted to the 2θ arms to make the measurements easier and more accurate.
- X-ray detector, power supply, and readout meters: The detector itself is located on the 2θ arm. Three types of detectors are used: Geiger counter, proportional counter, and scintillation counter. The scintillation counter is the most efficient for converting X-ray photons to an electrical signal, while proportional counters seem to have the highest resolution. Each detector is suitable for single-crystal orientation. The detector power supplies have provisions for increasing or decreasing sensitivity, adjusting the time constant, and changing from linear to log scales. Some detector power supplies have an audible feature that increases in volume and frequency as the user gets close to an orientation peak. In the case of backscatter Laue measurements, the instant film acts as the detector.
- Enclosure: The enclosure covers the entire instrument (except for the high-voltage generator and detector power supply), and is usually made from an X-ray-absorbing transparent plastic that is used to contain direct or stray X-ray radiation. For safety reasons, the enclosure is an item that is often mandated by the state. Access to the X-ray instrument is via doors outfitted with safety interlocks so that the X-ray shutter cannot be opened while the doors are open.
The following information must be known before taking a measurement: 1) the name of the material; 2) the crystal class of the material (cubic, hexagonal, etc.); 3) the orientation, using the Miller or Bravais index, and the tolerance; 4) “a” and “c” values; 5) “d” spacing; and 6) θ and 2θ angle values.
Ultimately, the θ value is all that is needed. The 2θ value can be found by doubling the θ value. Knowing the name of the material and using the published series of handbooks, Standard X-ray Diffraction Powder Patterns (published by the U.S. Department of Commerce/National Institute of Science and Technology), the crystal class, “a” and “c” values, and (in most cases) the “d” value can be obtained. When the crystal class, “a” and “c” values, and orientation are known, the “d” spacing can be calculated; the θ and 2θ angles can then be calculated from the “d” spacing. If using the Miller index, the user can enter the values for h, k, and l. If using the Bravais index, the first, second, and fourth digit values can be entered as h, k, and l. The following sign conventions must be applied in these equations:
Examples of Miller and Bravais indexes:
Miller index: h, k, l = 1, 0, –1
Bravais index: h, k, i, l = 3, 1, –4, 0
For cubic crystals, these expressions can be entered directly as Microsoft® (Redmond, WA) Excel™ equations:
“d” spacing = 1/((h^2+k^2+l^2)/a^2)^0.5
θ = Degrees (ASIN(0.77025/“d” spacing))
Most of the crystalline halides (fluorides, chlorides, bromides, and iodides) are cubic in nature, much softer, and a great deal more soluble than other crystals. Some cubic materials—for example, diamond, spinel, and silicon—are on the other end of the hardness and solubility spectrum. The cubic crystal structure is the easiest by far to visualize and understand.
For hexagonal and trigonal crystals,
“d” spacing = (1/((4/3*(h^2+h*k+k^2)/a^2)+((l/c)^2)))^0.5
θ = Degrees (ASIN(0.77025/“d” spacing))
The classes that are not cubic are hexagonal, trigonal, tetragonal, rhombohedral, and orthorhombic, with hexagonal and trigonal being the orientations most frequently requested. Crystal quartz and sapphire are hexagonal crystals. Sapphire is often used as a window that is highly resistant to influences of temperature, pressure, and corrosion, and as a substrate for blue light-emitting diodes (LEDs). Crystal quartz is most often used in polarization-sensitive optical applications, and in frequency oscillators to take advantage of its piezoelectric properties. These materials (sapphire and crystal quartz), as well as silicon, can be used in a spectral region that has only recently found commercial application in the terahertz wavelength region.
The X-ray generator is turned on and the voltage and current are set to the appropriate values. These values should be set as low as possible to obtain a good reading from the detector. Keeping the voltage and current as low as possible will help to minimize any stray radiation and extend the lifetime of the tube. The detector power supply should also be turned on.
The actual process in making a measurement starts with setting the θ angle on the detector goniometer arm and locking it down. Next, the sample is placed on the 2θ arm at the measuring position and the shutter is opened. The 2θ arm is rotated about its axis slowly and the detector is monitored for a peak reading. When a peak is found, the user should make sure that he/she is at the maxima. An angle reading is taken on that arm and the value is recorded. The desired value is 2θ. The difference between the 2θ and the reading taken by the user is the orientation error. The shutter is then closed. The sample should be turned upside down, the measurement repeated, and the value recorded. One-half of the difference between the two angles is a more accurate measure of the error.
Unless the laboratory orients crystals on a regular basis, it may not be cost effective to purchase the equipment required. As an alternative, there are facilities that work with and orient single crystals to specifications. In either case, X-ray goniometry makes it possible to orient a single crystal to suit any application.
Mr. Kappler is President, Waveplates, Inc., 36 Brimstone Way, Ashland, MA 01721, U.S.A.; tel.: 508-881-5886; fax: 508-881-7797, e-mail: firstname.lastname@example.org. Mr. Lydon is President, and Mr. Turnquist is Vice President and Manufacturing Manager, Meller Optics, Inc., Providence, RI, U.S.A.