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By Sean Gallagher, William S. Marras, Alan S. Litsky, Velimir Matkovic, Deborah Burr, John Landoll

Study Design. Survival analysis techniques were used to compare the fatigue failure responses of elderly motion segments to a middle-aged sample. Objectives. To compare fatigue life of a middle-aged sample of lumbosacral motion segments to a previously tested elderly cohort. An additional objective was to evaluate the influence of bone mineral content on cycles to failure. Summary of Background Data. A previous investiga¬tion evaluated fatigue failure responses of 36 elderly lumbosacral motion segments (average age, 81 ± 8 years) subjected to spinal loads estimated when lifting a 9-kg load in 3 torso flexion angles (0°, 22.5°, and 45°). Results demonstrated rapid fatigue failure with increased torso flexion; however, a key limitation of this study was the old age of the specimens. Methods. Each lumbosacral spine was dissected into 3 motion segments (L1-L2, L3-L4, and L5-S1). Motion segments within each spine were randomly assigned to a spinal loading condition corresponding to lifting 9 kg in 3 torso flexion angles (0°, 22.5°, or 45°). Motion segments were statically loaded and allowed to creep for 15 minutes, then cyclically loaded at 0.33 Hz. Fatigue life was taken as the number of cycles to failure (10 mm displacement after creep loading). Results. Compared with the older sample of spines, the middle-aged sample exhibited increased fatigue life (cycles to failure) in all the torso flexion conditions. Increased fatigue life of the middle-aged specimens was associated with the increased bone mineral content (BMC) in younger motion segments (mean ± SD, 30.7 11.1 g per motion segment vs. 27.8 ± 9.4 g). Increasing bone mineral content had a protective influence with each additional gram increasing survival times by approximately 12%. Conclusion. Younger motion segments survive considerably longer when exposed to similar spine loading conditions that simulate repetitive lifting in neutral and flexed torso postures, primarily associated with the increased bone mineral content possessed by younger motion segments. Cycles to failure of young specimens at 22.5° flexion were similar to that of older specimens at 0° flexion, and survivorship of young specimens at 45° flexion was similar to the older cohort at 22.5°. Key words: biomechanics, low back disorders, fatigue failure, age, motion segments, vertebral endplate fractures, torso flexion, lifting, bone mass. Low back disorders (LBDs) are a major cause of both short- and long-term occupational disability in the United States, and it is clear that workers in certain occupations are more susceptible to LBD.2-5 Specifically, epidemiologic studies have shown that jobs involving heavy physical demands such as construction,2,3 mining,2,4 and farming,5 engender increased risk of back pain. These occupations are thought to experience high LBD rates as a result of their extreme postural and manual lifting demands. Studies have shown that jobs involving significant lifting 6-9 and jobs that involve frequent bending10-12 are associated with increased LBD risk. Workers in the mining industry, in particular, often have to lift heavy materials in restricted workspaces that compel torso flexion. An understanding of the impact of repeated loading of the spine in flexion on fatigue failure of spinal tissues is thus a critical issue for our research agency, which is concerned with reducing the pain and disability associated with LBD in the mining industry. It is well accepted that loads experienced by the spine during manual lifting tasks are sufficient to cause fractures in the endplates of lumbar vertebrae, particularly on repeated loading. 13-15 Endplate fractures may not be painful in and of themselves 16; however, evidence suggests that the process of internal disc disruption and disc degeneration may be initiated via endplate fractures. 16,17 It has been shown that endplate damage will alter the distribution of stress in the disc, resulting in buckling of the lamellas of the internal anulus.17 Once this process has been initiated, repeated spine loading will cause additional disruption of the anular fibers, resulting in formation of significant fissures or tears. 11-21 If these tears occur near or extend close to the outer portions of the anulus, a wound healing process is initiated, which involves infiltration of mast cells, macrophages, and re¬lease of cytokines and growth factors. 21 The presence of these infiltrates has been strongly linked to discogenic low back pain. 21,22 Indeed, for patients experiencing chronic low back pain, the prevalence of internal disc disruption is at least 39%,19 which represents the most

By D. A. Wilson

A laboratory method for processing lead battery scrap has been developed that eliminates virtually all of the S02 emission that normally is produced when a reverberatory or blast furnace is used. The method employs a hydrometallurgical step to convert the PbS04 to PbO and CaS04 by mixing the battery scrap with a slurry of Ca(OH)2. The reaction is exothermic and goes to completion in 2 to 5 minutes at room temperature. The dried product is mixed with a KC1-NaC1 flux and carbon and reduced at 6500 C. A minimum charge ratio of 3 parts of carbon to 100 parts of battery scrap is required for reduction. The process produces lead reductions and recoveries as high as 97.0 and 94.9 percent, respectively. The KCI-NaCl flux is recovered using a standard counter-current water leach followed by evaporation.

Jan 1, 1976

By Douglas Milne

Most empirical and numerical approaches to design in rock mechanics incorporate rock mass classification. Numerical design methods generally use classification values to calculate input parameters for stress-based failure criteria. Empirical methods use classification to allow comparisons between similar rock mass conditions, generally based on a graphical design technique that differentiates stable and failed opening geometries. Classification systems are the best tool available for assessing rock mass properties; however, there are problems with classification systems that should be high-lighted. Rock mass performance can only be realistically estimated by coupling a unique description of the rock mass with known loading conditions. Current classification systems cannot provide a unique classification value. The weightings applied to quantify rock mass properties for classification can result in significantly different rock masses having the same classification values. These weightings have been proven effective for tunnel design and support, but classification systems are now used for many more applications. Rock classification systems evolved from a quick and easy field tool for estimating tunnel stability and support requirements. The need for a rapid field tool means that rock mass classification is relatively insensitive to improved methods of measuring rock mass properties. Problems with classification systems and their application are highlighted in this paper. These problems must be recognized and documented before improvements can be made. An understanding of the evolution of classification systems and their application for both numerical and empirical design approaches is invaluable in highlighting current shortcomings.

Jan 5, 2007

By F. E. Brantley, D. M. Yergensen, D. A. Martin

Fluidized - bed roasting was investigated as a means of decomposing gypsum and gypsum mixtures to produce a gas rich in sulfur dioxide and a lime or portland cement residue . Ninety- eight percent desulfurization of gypsum in a 10 - inch fluidized- bed reactor took 1.5 hours at an operating temperature between 1,200 ° and 1,225 ° C. A cement- type residue was obtained but had to be refired at 1,450 ° C before mortar prepared from it met the physical specifications for cement .

Jan 1, 1963

By D. R. Babich, E. R. Bauer

Noise-induced hearing loss (NIHL) in mine workers, especially underground coal miners, continues to be a problem in the mining industry. The recently enacted MSHA noise standard, Part 62 – Occupational Noise Exposure (Federal Register, 1999), is aimed at reducing NIHL in the mining industry. The U.S. National Institute for Occupational Safety and Health (NIOSH) is conducting a cross-sectional survey of noise sources and worker noise exposures to address NIHL in various aspects of coal mining including surface mining, coal preparation plants and underground mining. In underground bituminous coal mining, noise surveys, consisting of full-shift worker noise exposure, time-motion studies and equipment noise profiling, have been completed on longwall and continuous miner sections in six underground coal mines. The studies revealed widely varying shift-to-shift worker exposures, with workers in certain occupations routinely exposed to potentially hazardous levels of noise. A summary of these studies is reported, including worker dose, source/dose relationships and equipment noise profiles.

By J. Chilton

The fact that methane ignitions continue to occur at the mining face indicates that monitoring with machine-mounted methanometers does not always indicate the presence of high methane concentrations. Methane concentrations at the face change quickly due to changes in airflow. By measuring these changes in airflow, it may be possible to predict changes in face methane levels more quickly. Currently there are no techniques or instruments to accurately measure airflow in by the mouth of the ventilation curtain or tubing. Tests conducted in the NIOSH ventilation test gallery compare the data provided by one-, two-, and three-axis ultrasonic anemometers. The effects of changes in airflow direction and turbulence on instrument readings are discussed, and guidelines are given for selecting the type of instruments to be used for monitoring airflow near the mining face.

By J. Shawn Peterson, David S. Yantek, Adam K. Smith

Occupational noise-induced hearing loss continues to be one of the most pervasive health problems in the mining industry, despite over 25 years of regulation. One of the loudest pieces of equipment used in underground mining is the continuous mining machine. Noise sample data collected by the Mine Safety and Health Administration indicate that 42% of noise overexposures between 2000 and 2005 involved continuous mining machine operators. Previously conducted field and laboratory tests have determined that the continuous mining machine conveyor system is a dominant noise source. Loud impacts occur as the conveyor chain flight bars, used to move the mined aggregate, traverse their path from the top to the underside of the conveyor deck. Various noise control treatments have been applied to abate noise caused by the conveyor system. A durable polyurethane coating has been developed for the conveyor flight bars, resulting in a time-weighted average reduction of 3 dB(A) for an eight-hour work shift. In an attempt to further reduce continuous mining machine operator overexposures, a similar urethane coating has been applied to the tail roller component of the conveyor system. Laboratory results showed a 2 dB(A) reduction in sound power levels, but the component failed during underground durability testing. An outer steel sleeve has been added to the urethane coating of the tail roller to enhance wear resistance during mining. The urethane jacketed tail roller is the latest effort, combined with previous noise treatments, to bring the continuous mining machine into compliance with federal noise regulations.

By Oliver Bowles

The following discussion of waste rock utilization constitutes the fifth paper of the current series of reports on lime- plant quarry problems . As here tofore lime manufacturers are invited to cooperate to the fullest extent by supplementing the Bureau's information with the results of their own experiences and observations . Waste utilization is the key to success in many enterprises , and with a broader knowledge of this branch of the lime industry more economical operation is made possible . Your assistance in compiling additional data on this subject is solicited , and such assistance may be rendered by writing answers to the questions on the final page of this report .

Apr 1, 1923

By W. R. Reed, G. J. Joy

The personal dust monitor (PDM) is a new coal mine respirable dust sampling instrument that has the ability to provide accurate end-of-shift and real-time respirable dust exposure data. A hypothetical cost comparison analysis of the PDM with the current MSHA-required coal mine dust personal sampling unit (CMDPSU) was conducted. Some simplifying assumptions were made to create hypothetical cases for analysis. This analysis was limited to respirable dust sampling of one designated operator (DO) and one designated area (DA) for one mechanized mining unit (MMU) which operated one shift per day. Only sampling intervals of the minimum sampling as required by federal regulations and continuous sampling were analyzed. This cost analysis from the company perspective is only one part of an overall cost/benefit analysis that should be conducted at a later date. In this analysis, company-associated costs were examined for both sampling systems. Results show that the PDM, while it has high initial costs of approximately $12,600, may be a cost-effective sampling system for measuring coal mine respirable dust. The estimated present worth cost of conducting the minimum required dust sampling over a five-year life ranges from $5,850 to $19,000 for the CMDPSU, with the higher present worth cost including costs of potential violations incurred. For the PDM, the comparable present worth cost is $14,900 to $19,300, with the lower present worth cost representing a lower capital cost of $8,150 per unit if more than 850 units are produced per year, while the higher present worth cost reflects the result with the higher $12,600 capital cost.

By Mark S. Redfern, Kim M. Corneliusa, Christopher A. Hamrick, Sean Gallaghera

Coal miners often handle heavy electrical power cables, weighing up to 10 kg per meter. These cables are manually lifted and attached to the mine roof to prevent damage from mobile underground equipment. Data suggest that workers who commonly perform cable handling tasks experience a high rate of lost time back injuries. In this study, six male underground miners performed a total of 12 cable-hanging tasks in standing, stooping, and kneeling postures, during which kinematic and ground reaction force data were collected. Reductions in vertical work space were found to result in a linear increase in the peak moment experienced by the lumbarspine (p<0.05). In restricted postures, peak moments were not significantly different in stooping vs. kneeling postures (p>0.05). Average lumbopelvic flexion during the tasks was highest in stooping conditions, followed by standing and kneeling exertions (p<0.05). Implications of this data with respect to design of cable handling tasks are presented and discussed. Keywords: Biomechanics, posture, low back pain, restricted workspace, mining

By Pakalnis Rimas, Lyndon Clark, Tom Brady

A major focus of ground control research presently being conducted by the Spokane Research Laboratory of the National Institute for Occupational Safety and Health (NIOSH) is to incorporate data on weak rock masses into existing design relationships, with an emphasis on updating the span design curve for manned entries and the overbreak curve for longhole entries. Both curves were originally developed at the University of British Columbia, Vancouver, BC. The original database has been augmented by information from mines throughout the United States, Canada, Australia, and Europe. The common factor in all these mines is the presence of a weak back and/or walls. In most cases, the ore zone is the weakest rock unit and must be stabilized so that the mineral-bearing rock can be extracted safety. The current NIOSH research attempts to provide rock mechanics tools to assist a mine operator in making economic decisions that will also ensure a safe working environment. This paper documents the Nevada database with a special emphasis on Nevada underground gold mines.

By Thomas O. Glover

The mining industry of Cameroon continued to be dominated by crude petroleum. Crude petroleum production has become the most important facet of Cameroon&apos;s economic development. Oil production was estimated to be 6.3 million tons in 1983 compared with 5.2 million tons in 1982 and 4.3 million tons in 1981. Over two-thirds of its export income was attributed to crude oil. The country&apos;s only refinery, dedicated in 1981, has a production capacity of 2 million tons. Cameroon has become self- sufficient in petroleum products since commissioning the refinery. The oil sector contributed approximately 13.5% to Cameroon&apos;s gross domestic product in 1982-83. Oil production was expected to rise from 125,000 barrels per day in 1983 to 156,000 barrels per day in 1985 when output peaks. Cameroon&apos;s ultimate recoverable reserves have been calculated at 100 million tons, or 750 million barrels of oil.

Jan 1, 1985

By E. C. Strycula

Solid-state equipment offers a number of potential advantages in the coal mining industry. These advantages can only be realized, however, through careful and judicious design of the control and protection circuitry. Problems relating to the interaction of solid-state equipment on the electrical system and the special design considerations which apply to solid-state circuitry are examined. Design guidelines are provided for solid-state applications in the areas of: (a) equipment operated from a DC trolley wire, (b) drive systems for surface mining equipment, (c) AC motor starters, and (d) electromagnetic interference.

Jan 1, 1981

By Gerrit V. R. Goodman

In This Chapter [Design and operation of machine-mounted scrubbers Dust control with scrubbers and blowing ventilation Dust control with scrubbers and exhaust ventilation Dust control with exhaust ventilation and no scrubber Dust control methods common to all continuous miner sections Dust control for roof bolters] This chapter explains how to control dust at continuous miner sections in coal mines where the main dust sources are continuous miners and roof bolters. In relation to dust, there are three categories of continuous miner faces depending on the type of ventilation and whether or not a machine-mounted dust scrubber is used. These are— 1. Mining machines with dust scrubbers used with blowing face ventilation 2. Mining machines with dust scrubbers used with exhaust face ventilation 3. Mining machines without scrubbers used with exhaust face ventilation The approach to dust control is somewhat different in all three of these. However, there are many dust control features (such as the need to provide adequate airflow) common to all continuous miner sections. For workers at roof bolter faces, there are two dust sources: 1. Dust from the continuous miner when it is upwind. 2. A malfunctioning dust collector on the bolter, which allows dust to escape DESIGN AND OPERATION OF MACHINE-MOUNTED SCRUBBERS [Almost all new continuous miners are equipped with scrubbers. When the dust is excessive, it is possible that the scrubber needs some maintenance.]

Jan 6, 2003

By Fred N. Kissell, Andrew B. Cecala

In This Chapter [Where methane is emitted at longwall faces Where methane accumulates at longwall faces Using the modified shearer-clearer to eliminate ventilation eddy zones Using a walkway curtain to reduce methane buildup during the headgate cutout Control of frictional ignitions and The best location for the methane monitor] The methane released along a longwall face represents only 10%–20% of the total methane emitted from the entire longwall panel. Nevertheless, in very gassy coal seams, this methane released at the face can pose a problem because the shearer is a ready ignition source. Preventing methane ignitions at longwall faces requires four actions. The first is to provide better ventilation around the shearer to eliminate the ventilation eddy zones at the drums where methane builds up. These eddy zones are eliminated by mounting additional water sprays on the shearer to direct air into them. The second action to prevent methane ignitions is to install a water spray behind each cutter bit and regularly replace worn bits. Water sprays behind each cutter bit act to quench the hot metal streak that follows a worn bit when it strikes rock. The third is to ensure that no ventilation eddy zones are inadvertently created by poor placement of water sprays. The fourth is to ensure that the methane monitor on the shearer is in the best location to detect methane accumulations. ADDRESSING METHANE ACCUMULATIONS AT LONGWALL FACES3 Cecala et al. [1985a, 1989] and Denk and Wirth [1991] studied methane emission and ventilation patterns at longwall faces to find where methane accumulations are most likely. Although not always the case, the major source of methane at longwall faces is usually the breakage of coal by the shearer. Stress-related fracturing of the coal seam at the face, called bumps or bounces, can cause the release of additional gas (Figure 4–1).

Jan 6, 2006

By Fred N. Kissell

In This Chapter [The explosibility of methane gas mixtures Effect of pressure and temperature on explosibility Less common sources of methane ignitions The amount of methane stored in coal Forecasting the methane emission rate Layering of methane at the mine roof When the recirculation of mine air is hazardous The importance of higher air velocity in preventing methane explosions and Mine explosions, barometric pressure, and the seasonal trend in explosions] Dealing with methane in mines and tunnels requires knowledge of the circumstances under which dangerous accumulations of methane are likely to occur. This knowledge involves the properties of the gas itself, an awareness of where these accumulations are likely to occur, and facts on how methane mixes safely into the mine air. The other chapters in this handbook address the handling of methane under a variety of specific circumstances, such as at continuous miner faces or coal storage silos. This chapter addresses some broad concepts that serve as a foundation for the suggestions provided in other chapters. THE EXPLOSIBILITY OF METHANE GAS MIXTURES Methane entering a mine or tunnel often enters as a localized source at high concentration. Figure 1–1 depicts a cloud of methane being diluted into a moving air stream. In this illustration, methane enters the mine from a crack in the roof. As the methane emerges from the crack, it progressively mixes with the ventilation air and is diluted. In the event that this progressive dilution reduces the concentration from 100% to 1%,2 as shown in Figure 1–1, the methane passes through a concentration range of 15% to 5%, known as the explosive range. In the explosive range, the mixture may be ignited. Above 15%, called the upper explosive limit (UEL), methane-air mixtures are not explosive, but will become explosive when mixed with more air.

Jan 6, 2006

"The U.S. Bureau of Mines established on August 15, 1983, The Generic Mineral Technology Center for Respirable Dust (Dust Center) within the Mining and Mineral Resource Research Institutes (MRIs) at The Pennsylvania State University and West Virginia University, and in association with participating MRIs at Massachusetts Institute of Technology and the University of Minnesota. In 1986, Michigan Technological University (MTU) was awarded a grant through the Dust Center. The center's research program has been developed with recognition of the stated objective of the Federal Mine Safety and Health law, which is ""... (to) permit each miner the opportunity to work underground during the period of his entire adult working life without incurring any disability from pneumoconiosis of any other occupation-related diseases..."" The Dust Center's program is designed to foster an accelerated attack through fundamental research on problems confronting the control of respirable dust in mines. The interactions of the universities and the research program components in the characterization of coal seams, respirable dust particles and the workers lungs are depicted in the logo of the Dust Center.The primary goal of the Dust Center is to reduce the incidence and severity of respirable dust disease through advancing fundamental understanding of all aspects of respirable dust associated with mining and milling, and the interaction of dust and lungs. The Dust Center's research program explores these concerns with the objective of refining existing strategies and developing new respirable dust control techniques that are consistent with the fundamental dust-lung interaction processes that lead to mine worker disability. The work concentrates on: (1) control of dust generation; (2) dilution, dispersion and collection of dust in mine airways; (3) characterization of dust particles; (4) interaction of dust and lungs; and (5) the relationship of the mine environment, geology, and seam characteristics to dust generation and mobility. The fundamental aspects of this work are applicable to the control of respirable dust problems in both hard rock mines and coal mines, and to other dusts such as those generated by diesel equipment. Achieving the Dust Center's goals also involves the training of engineers and scientists, graduate students, and undergraduate students through their respective institutions, and technology transfer to industry. The distribution of personnel in the various universities are presented in Table 1. A directory of Dust Center principal investigators is presented in Appendix 1. A summary of degrees granted and mineral sector employment is presented in Table 2. The subjects of theses developed under the various research projects are presented separately in Volume 7. The center research is fully compatible and complementary with the existing and ongoing U.S. Bureau of Mines activities (see Volume 2). Additionally, the expertise and facilities of the NIOSH--Division of Respirable Disease studies are available as a result of the existing relationship between NIOSH and West Virginia University. Figure 1, Respirable Dust Center Projects and Activities, presents the various projects in the five major areas of investigation and the scope of characterized dust samples, the suite of tests, the sampling and dust generation methods, and the size, chemical and mineralogical analysis performed in the center. Table 3 presents a summary distribution of the 95 publications and presentations that have emanated from the center. In addition there are more than thirty presentations made at each annual Dust Center review meeting. The titles, authors and location of presentation for publications supported by the Dust Center are presented in separate publication volumes for 1984 (Volume 3), 1985 (Volume 4), 1986 (Volume 5) and 1987 (Volume 6)."

Jan 1, 1988

By R. V. Ramani, S. Shankar

"Available literature on the effect of re-entrainment on overall airborne dust concentration in mine airways or the fundamental mechanisms of coal dust re-entrainment in mine atmosphere is scant (Shankar and Ramani, 1991 ). Therefore, as part of the Respirable Dust Center's ongoing research, the following three-part study was undertaken:• a study of the re-entrainment phenomena of anthracite and bituminous coal dust particles in a wind tunnel under the influence of varying air velocities;• a study of the same in the Lake Lynn Laboratory Mine of the USBM with both air flow and walking by personnel causing re-entrainment; and• the mathematical modeling and computer simulation of the re-entrainment phenomenon.In this paper, results from the re-entrainment studies in the wind tunnel and the Lake Lynn Laboratory mine are presented. Comparing the results from the two studies,it was observed that increasing the air velocity increased the rate of re-entrainment ( Shankar and Ramani, I 994a) and the rate of re-entrainment decreased with the time of exposure of the source dust to the flowing air. In addition, when the source dust was composed of particles of different sizes, the airborne dust concentrations due to re-entrainment were higher than when the source dust was composed of particles of the same size. The contribution of airborne dust concentration due to air flow was negligible compared to the contribution due to walking. IntroductionIn underground coal mines, dust is produced primarily during the process of mining and during such activities as pillar spall, loading, belt transfer and roof fall. The generated dust is entrained and then collects on surfaces through dispersion, diffusion and eventual deposition. Removal forces, such as flowing ventilation air or turbulence caused by walking and moving vehicles, can re-entrain the deposited dust (Hall, 1955). A portion of the re-entrained dust remains airborne and increases the concentration of airborne dust in mine airways.The presence of airborne coal dust in mine airways in large concentrations can be hazardous. Health effects include coal workers' pneumoconiosis (CWP) and silicosis, and it can be a nuisance affecting visibility. In addition, as is common during methane explosions, if the concentration of airborne coal dust becomes significant, there could be a coal-dust explosion (Singer, Cook and Grumer, 1972; Singer, Harris and Grumer, 1974; Edwards and Ford, 1988). Through concerned research efforts over the last decade, significantprogress has been made in reducing the airborne dust concentration in mine atmospheres. However, secondary sources, such as re-entrainment, have not been studied in-depth. It is important to understand the phenomenon of re-entrainment and its potential contribution to the airborne dust concentration in mine airways for further decreasing airborne dust hazards in mines."

Dec 1, 1996

By S. H. Lorain

The Minah silver-lead mine in the Wickes district, Jefferson County, Mont., produced substantial quantities of silver-lead ores in the 1890&apos;s. The mine has been abandoned since the "silver panic" of the early 1900&apos;s. As a part of the Bureau of Mines: program to develop new sources of lead to overcome the growing shortages of that metal, a development project was started at the Minh mine in April 1946 and continued until June 1947, during which 5,600 feet of angledozer trenching and. 1,093 feet of core drilling were completed on the main vein and nearby parallel veins: As a result of this work, interest in the property was stimulated and private operators are now engaged in reopening the mine. This report discusses the past history, geologic setting, and the recent and early operations of the mine.

Jan 1, 1948

By R. Barany

In this paper, closed expressions have been obtained for the frequency spectrum of a Born-Karman square lattice in terms of elliptic and hyperelliptic integrals. This work is part of continuing research by the Bureau of Mines with the thermodynamic properties of metals and minerals. Although other workers have obtained various types of approximations for the general case of the frequency spectrum of a square lattice, no one previously has obtained closed analytic expressions for this spectrum. To understand the optical and thermodynamic properties of crystals, several investigators have studied the distribution of frequencies of normal modes of vibration of the crystal lattice. Born and Karman3 laid the foundation for the theory of lattice dynamics and obtained detailed results for the case of a one-dimensional lattice. Blackman4 used numerical techniques to estimate the frequency distribution of a two-dimensional square lattice. Mbntroll5 analytically determined the frequency distribution of a square lattice for a particular value of the force constants. He also studied the more general problem and obtained some analytic approximations, using the method of moments. However, the calculations by this method were so difficult that it was not always practicable to carry them out. Van Hove6 and Montroll7 studied the nature of the singularities in the frequency spectrum of both a two and

Jan 1, 1962