Average Temperature Coefficient Of Resistance Research Articles

Dynamic mechanical response of carbon nanotube yarns and their in situ electrical measurements

Carbon nanotube yarns (CNTYs) are outstanding hierarchical fibers with electrical properties that depend on temperature and mechanical stimuli, which makes them attractive as smart materials for sensing applications. In order to better understand the electrical response of CNTYs under a combination of dynamic loading and variations of temperature, their monotonic and cyclic tensile mechanical response, in situ Raman spectroscopy, mechanical hysteresis, and dynamic mechanical analysis (DMA) are investigated herein. The piezoresistivity and thermoresistivity of CNTYs were characterized to correlate the contribution of temperature and strain to the effective electrical response of the CNTY under DMA testing. It was found that the tensile load bearing of the CNTYs is governed mainly by structural changes of its bundles/fibrils, rather than by stretching of the carbon bonds. The large energy dissipation capabilities of the CNTYs arise mainly from friction among their fibrils and bundles, and also due to irreversible morphological changes in their hierarchical structure. The (mechanically constrained) thermoresistive characterization showed that the electrical resistance of the CNTYs decreases with increasing temperature, yielding an average temperature coefficient of resistance of −8.63 × 10−4 K−1. The electrical response of the CNTYs during DMA temperature scans is governed by their thermoresistive response.

Annealing temperature effect on the temperature coefficient of resistance for vanadium oxide (VOx) thin films as bolometer materials

Vanadium oxides (VOx) are representative materials with a high temperature coefficient of resistance (TCR); however, VOx films can have complex phase structures that are dependent on their fabrication method. While past research has focused on the TCR behavior of VOx thin films, this study investigates the TCR of VOx thin films annealed at different temperatures as well as focuses on the relation between the VOx phase, surface morphology, sheet resistance, and TCR. VOx thin films were deposited via radio-frequency magnetron sputtering and annealed at 150 °C–500 °C in 20% O2. Alongside morphological changes, the deposited VOx thin films exhibited phase changes from V2O5 to VO2 with increasing annealing temperature. The VOx thin films annealed at 300 °C and 330 °C showed the lowest and highest average TCR of 1.25%/°C and 3.34%/°C, respectively. Furthermore, a bolometer fabricated using the higher-TCR film showed more than 5-fold infrared responsivity under the same infrared intensity.

Establishment and verification of resistance temperature coefficient model of P-type non-uniformly doped resistance

In this paper, a first-order average temperature coefficient of resistance () calculation model is established and analyzed based on the distribution of piezoresistive doping concentration in bulk silicon. Furthermore, by extracting the experimental results of the first-order of multiple research groups and combining the first-order calculation model, the new mobility model in the concentration range of piezoresistance (1 × 1018–1 × 1020 at cm−3) is obtained by fitting. The first-order of five implantation concentrations under the same process is tested. The results show that the error between the first-order obtained based on the new mobility calculation model and the test results is within 5%. However, the first-order based on the Arora mobility model has a deviation of 104.6% under the implantation condition of 3.75 × 1015 at cm−2. At the same time, the effects of different annealing temperatures and time on the first-order at the implantation concentration of 8.5 × 1013 at cm−2 are compared. The results show that a higher annealing temperature or longer annealing time is not conducive to reducing the first-order , but the difference is small.

Fabrication and Characterization of the Micro-Heater and Temperature Sensor for PolyMUMPs-Based MEMS Gas Sensor.

This work describes the fabrication and characterization of a Micro-Electro-Mechanical System (MEMS) sensor for gas sensing applications. The sensor is based on standard PolyMUMPs (Polysilicon Multi-Users MEMS Process) technology to control the temperature over the sensing layer. Due to its compact size and low power consumption, micro-structures enable a well-designed gas-sensing-layer interaction, resulting in higher sensitivity compared to the ordinary materials. The aim of conducting the characterization is to compare the measured and calculated resistance values of the micro-heater and the temperature sensor. The temperature coefficient of resistance (TCR) of the temperature sensor has been estimated by raising and dropping the temperature throughout a 25–110 °C range. The sensitivity of these sensors is dependent on the TCR value. The temperature sensor resistance was observed to rise alongside the rising environmental temperatures or increasing voltages given to the micro-heater, with a correlation value of 0.99. When compared to the TCR reported in the literature for the gold material 0.0034 °C, the average TCR was determined to be 0.00325 °C and 0.0035 °C, respectively, indicating inaccuracies of 4.6% and 2.9%, respectively. The variation between observed and reported values is assumed to be caused by the fabrication tolerances of the design dimensions or material characteristics.

Monitoring of Curing and Cyclic Thermoresistive Response Using Monofilament Carbon Nanotube Yarn Silicone Composites

The curing process and thermoresistive response of a single carbon nanotube yarn (CNTY) embedded in a room temperature vulcanizing (RTV) silicone forming a CNTY monofilament composite were investigated toward potential applications in integrated curing monitoring and temperature sensing. Two RTV silicones of different crosslinking mechanisms, SR1 and SR2 (tin- and platinum-cured, respectively), were used to investigate their curing kinetics using the electrical response of the CNTY. It is shown that the relative electrical resistance change of CNTY/SR1 and CNTY/SR2 monofilament composites increased by 3.8% and 3.3%, respectively, after completion of the curing process. The thermoresistive characterization of the CNTY monofilament composites was conducted during heating–cooling ramps ranging from room temperature (RT~25 °C) to 100 °C. The thermoresistive response was nearly linear with a negative temperature coefficient of resistance (TCR) at heating and cooling sections for both CNTY/SR1 and CNTY/SR2 monofilament composites. The average TCR value was −8.36 × 10−4 °C−1 for CNTY/SR1 and −7.26 × 10−4 °C−1 for CNTY/SR2. Both monofilament composites showed a negligible negative residual relative electrical resistance change with average values of ~−0.11% for CNTY/SR1 and ~−0.16% for CNTY/SR2 after each cycle. The hysteresis amounted to ~21.85% in CNTY/SR1 and ~29.80% in CNTY/SR2 after each cycle. In addition, the effect of heating rate on the thermoresistive sensitivity of CNTY monofilament composites was investigated and it was shown that it reduces as the heating rate increases.

Micro‐electrodes for in situ temperature and bio‐impedance measurement

AbstractWith fast recovery time and effective in situ tumor tissue killing ability, thermal ablation has become a popular treatment for tumors compared with chemotherapy and radiation. The thermal dose measurement of current technology is usually accompanied by monitoring a large area impedance across two ablation catheters and the localized impedance measurement is difficult to achieve. In this work, thermal‐resistive sensor and impedance sensor are fabricated on the curved surface of a capillary tube with 1 mm outer diameter. The device is applied for real‐time in situ tissue impedance monitoring during thermal ablation. The calibrated thermal‐resistive sensors have an average temperature coefficient of resistance (TCR) of 0.00161 ± 5.9% °C‐1 with an accuracy of ±0.7 °C. By adding electro‐polymerized PEDOT:PSS (poly(3,4‐ethylenedioxythiophene)‐poly(styrenesulfonate)) on the 300 µm diameter gold electrodes, the interface impedance reduces two orders from 408 to 3.7 kΩ at 100 Hz. The Randles equivalent circuit model fittings show a two‐order improvement in the electrode capacitance from 7.29 to 753 nF. In the ex vivo porcine liver laser ablation test, the temperature of the porcine liver tissue can reach 70°C and the impedance would drop by 50% in less than 5 minutes. The integration of laser ablation fiber with the impedance and temperature sensors can further expand the laser ablation technique to smaller scale and for precise therapeutics.

Development of a Fluorinated Graphene-Based Resistive Humidity Sensor

This work presents the development of novel fluorinated graphene (FG)-based resistive humidity sensor. The humidity sensor was fabricated by drop-casting FG suspension, as the humidity sensing material, on silver (Ag)-based interdigitated electrodes (IDEs). The silver-based IDEs were screen printed on a flexible polyimide substrate. The FG suspension was synthesized by uniform dispersion of FG in isopropyl alcohol (IPA), using the ultra-sonication process. The resistive response of the fabricated humidity sensors towards varying relative humidity (RH) levels was investigated, when the RH was varied from 20% to 80%, in steps of 10%, and at a temperature of 24 °C. A relative resistance change of 13.3% was observed when the RH was changed from 20% to 80%, with a sensitivity of 0.22%/%RH for the FG-based humidity sensor. Response time and recovery time of 82 s and 125 s, respectively, was obtained for the fabricated sensor. In addition, the effect of varying operating temperatures on the response of the fabricated humidity sensors was investigated. The average temperature coefficient of resistance of sensors was obtained as approximately -0.3%/°C. A linear relation between the temperature and the relative resistance change of sensors was observed. Further, first-principles study, employing density functional theory calculations, was performed to investigate interactions between the fluorine atom and graphene substrate, as well as humidity sensing behavior of the FG. DFT calculations showed that hydrogen atoms of the water molecule move towards the fluorine atom of the FG during the relaxation process, confirming the hydrogen bonding between FG and water molecules. The Eads of -0.50 eV was calculated for the adsorption of water molecule on the FG, demonstrating the strong humidity sensing property of the FG. The results demonstrate that FG, a highly stable derivative of graphene, is a potential material for humidity sensing applications.

Cyclic Thermoresistivity of Freestanding and Polymer Embedded Carbon Nanotube Yarns

Carbon nanotube yarns (CNTYs) are hierarchical fibers with outstanding electrical properties, and understating the temperature‐dependence of their electrical resistance (thermoresistivity) is essential for sensing applications and development of self‐sensing polymer composites. The cyclic thermoresistive response of individual CNTYs and the effect of embedding the yarn into a polymer are experimentally investigated herein. The effect of confining the CNTY by a thermosetting polymer is addressed by studying the thermoresistive response of CNTY/vinyl ester single‐fiber composites. Heating–cooling cycles ranging from 25 (room temperature [RT]) to 100 °C and 25 to −30 °C are applied to individual CNTYs and to CNTYs embedded into a vinyl ester polymer, while their electrical resistance is simultaneously recorded. Both the CNTY and its single‐fiber composite show a negative dependence of electrical resistance with temperature. For both temperature ranges (above and below RT), the average temperature coefficient of resistance found for individual CNTYs is ≈−9.5 × 10−4 K−1, and its magnitude decreases about ≈30% when the yarn is embedded into the vinyl ester polymer. The hysteresis rendered by the different heating and cooling pathways is small for individual CNTYs, and largely increases when the CNTY is embedded into the polymer.

Balancing Scattering Channels: A Panoscopic Approach toward Zero Temperature Coefficient of Resistance Using High-Entropy Alloys.

Designing alloys with an accurate temperature-independent electrical response over a wide temperature range, specifically a low temperature coefficient of resistance (TCR), remains a big challenge from a material design point of view. More than a century after their discovery, Constantan (Cu-Ni) and Manganin (Cu-Mn-Ni) alloys remain the top choice for strain gauge applications and high-quality resistors up to 473-573 K. Here, an average TCR is demonstrated that is up to ≈800 times smaller in the temperature range 5-300 K and >800 times smaller than for any of these standard materials over a wide temperature range (5 K < T < 1200 K). This is achieved for selected compositions of Alx CoCrFeNi high-entropy alloys (HEAs), for which a strong correlation of the ultralow TCR is established with the underlying microstructure and its local composition. The exceptionally low electron-phonon coupling expected in these HEAs is crucial for developing novel devices, e.g., hot-electron detectors, high-Q resonant antennas, and materials in gravitational wave detectors.

Zinc oxide decorated multi-walled carbon nanotubes: their bolometric properties

We report the synthesis of MWNT/ZnO hybrid nanostructures. A simple, affordable, chemical procedure to functionalize MWNTs with ZnO nanoparticles was performed. A significant portion of the surface of MWNTs was covered with ZnO nanoparticles; these particles formed highly porous spherical nodules of 50–150 nm in diameter, sizes that are an order of magnitude larger than similar ZnO nanonodules reported in the literature. Hence, the self-assembled nanocomposite the ZnO exhibited a large surface-to-volume ratio, which is a very advantageous property for potential catalytic applications. The resultant MWNT/ZnO nanocomposites were characterized by x-ray diffraction, scanning and high-resolution transmission electron microscopy, and UV–vis and Raman spectroscopy. The temperature coefficient of resistance (TCR) of the nanocomposites was measured and reported. The average TCR value goes from −5.6%/K up to −18%/K, over temperature change intervals from 10 K to 1 K. Based on these TCR results, the nanocomposite MWNT/ZnO prepared in this work is a promising material, with potential application as a bolometric sensor.

Research on a micromachined flexible hot-wire sensor array for underwater wall shear stress measurement

This paper reports an innovative flexible hot-wire senor microarray and its experimental studies for underwater wall shear stress measurement. 20 parallel channels of the hot-wire sensor are embedded between two polyimide layers for curved surface applications, which is fabricated based on micromachining processes, including platinum sputtering, nickel electroplating, etc. An accurate method to calculate the conversion factor of the compensation loop is proposed and verified for the temperature-compensated circuit. An average temperature coefficient of resistance (TCR) is measured 2136.7 ppm/°C with linearity better than 0.05 % for platinum thermal resistors. Output instabilities of 0.22 and 0.63 mV are obtained in static flow and moving flow, respectively. The feasibility of the hot-wire sensors for detecting underwater wall shear stress is verified with an average sensitivity of 0.0123 $${\text{V}}^{2} /{\text{Pa}}^{1/3}.$$ According to the experiment results, the flexible sensor microarray is very promising for characterizing underwater wall shear stress distributions in the boundary layer.

Characterization of embedded microheater of a CMOS–MEMS gravimetric sensor device

A CMOS–MEMS device for mass detection has been designed using 2008 CoventorWare software and fabricated using 0.35µm CMOS technology. This paper reports the characterization of the microheater and the temperature sensor embedded in the device. The measured resistances of the microheater and the temperature sensor were found to be close to the modeled values within ~4.2% error. The average temperature coefficient of resistance (TCR) of the temperature sensor of five dies was determined by increasing or decreasing the temperature in a range of 25°C–100°C. The resistance of the temperature sensor was found to increase with either an increase in ambient temperature or the voltage applied to the microheater, with a correlation factor of 0.99. The average TCR was found to be 0.0034/°C for the increasing temperature and 0.0036/°C for the decreasing temperature as compared to 0.0037°C reported in the literature, indicating an error of 8.1% and 3.5%, respectively. These differences between the measured and reported values are believed to be due to fabrication tolerances in the design dimensions or the material properties. The humidity was found to have a negligible effect on the resistance of the temperature sensor for increasing humidity levels from 40% to 90%. The repeatability of the measurements has shown low standard errors, which gives confidence in the reliability of the fabricated device.

Performance assessment of thermal sensors during short-duration convective surface heating measurements

The determination of convective surface heating is a very crucial parameter in high speed flow environment. Most of the ground based facilities in this domain have short duration experimental time scale (~milliseconds) of measurements. In these facilities, the calorimetric heat transfer sensors such as thin film gauges (TFGs) and coaxial surface junction thermocouple (CSJT) are quite effective temperature detectors. They have thickness in the range of few microns and have capability of responding in microsecond time scale. The temperature coefficient of resistance (TCR) and the sensitivity are calibration parameter indicators that show the linear change in the resistance of the gauge as a function of temperature. In the present investigation, three of types of heat transfer gauges are fabricated in the laboratory namely, TFG made out of platinum, TFG made out of platinum mixed with CNT and chromel–alumel surface junction coaxial thermocouple (K-type). The calibration parameters of the gauges are determined though oil-bath experiments. The average value TCR and sensitivity of platinum TFG is found to be 0.0024 K−1 and 465 μV/K, while similar values of CSJT are obtained as, 0.064 K−1 and 40.5 μV/K, respectively. The TFG made out of platinum mixed with CNT (5 % by mass) shows the enhancement of TCR as well as sensitivity and the corresponding values are 0.0034 K−1 and 735 μV/K, respectively. The relative performances of heat transfer gauges are compared in a simple laboratory scale experiment in which the gauges are exposed to a sudden step heat load in convection mode for the time duration of 200 ms. The surface heat fluxes are predicted from the temperature history through one dimensional heat conduction modeling. While comparing the experimental results, it is seen that prediction of surface heat flux from all the heat transfer gauges are within the range of ±4 %.

Effect of Nitrogen Doping on the Electrical Properties of 3C-SiC Thin Films for High-Temperature Sensors Applications

3C-SiC is a promising structural material for piezoresistive sensors used in high-temperature applications. For sensor development, the preparation of sensor materials and study of its electrical properties, such as resistivity, barrier height of grain boundaries, and temperature coefficient of resistivity, are important in addition to structural properties and these have to be optimized. In the present work, 3C-SiC thin film with in situ doping of nitrogen is prepared through low-pressure chemical vapor deposition by using methyl trichloro silane, ammonia, and hydrogen as precursors. Electrical properties of deposited 3C-SiC thin films with varying nitrogen doping concentration through four probe technique are studied. Atomic force microscopy investigations are carried out to study the grain size on and average root-mean-squared roughness 3C-SiC thin films. A decrease in the degree of crystallinity is observed in nitrogen-doped 3C-SiC thin films. The sheet resistivity of nitrogen-doped 3C-SiC thin film is found to decrease with increase in temperature in the range from 303 to 823 K. The sheet resistivity, average temperature coefficient of resistance, and barrier height of the grain boundaries of film doped with 17 at.% of nitrogen are 0.14 Ω cm, −1.0 × 10−4/K, and 0.01 eV, respectively. Comparing all the nitrogen-doped 3C-SiC thin films, the film doped with 17 at.% of nitrogen exhibits an improved structural and electrical properties and it can be used as sensing material for high-temperature applications.

Microstructure and electrical properties of nitrogen doped 3C–SiC thin films deposited using methyltrichlorosilane

The growth, microstructure and electrical properties of in-situ nitrogen doped 3C–SiC (111) thin films for sensor applications are presented in this paper. These thin films are deposited at a pressure of 2.5mbar and temperature of 1040°C on thermally oxidized Si (100) substrates from methyltrichlorosilane (MTS) precursor using a hot wall vertical low pressure chemical vapor deposition (LPCVD) reactor. Ammonia (NH3) is used as the nitrogen doping gas. The sensor response depends on chemical composition, structure, morphology and operating temperature. The above properties are investigated for all in situ nitrogen doped (0, 9, 17 and 30at% of nitrogen) 3C–SiC thin films using X-ray diffraction (XRD), Fourier transform infrared spectroscopy (FTIR), atomic force microscopy (AFM) and four probe method. The XRD patterns of the 3C–SiC thin films show a decrease in the crystallinity and intensity of the peak with increase in dopant concentration from 0 to 17at%. AFM investigations show an improvement in the grain size of the nitrogen doped 3C–SiC thin films with increase in nitrogen concentration from 0 to 17at%. The sheet resistance of nitrogen doped 3C–SiC thin films is measured by the four probe technique and it is found to decrease with increase in temperature in the range of 40–550°C. The resistivity and average temperature coefficient of resistance (TCR) of doped 3C–SiC thin film deposited with 17at% of nitrogen concentration are found to be 0.14Ωcm and −103ppm/°C, respectively and this can be used as a sensing material for high temperature applications.

Wireless Remote Weather Monitoring System Based on MEMS Technologies

This study proposes a wireless remote weather monitoring system based on Micro-Electro-Mechanical Systems (MEMS) and wireless sensor network (WSN) technologies comprising sensors for the measurement of temperature, humidity, pressure, wind speed and direction, integrated on a single chip. The sensing signals are transmitted between the Octopus II-A sensor nodes using WSN technology, following amplification and analog/digital conversion (ADC). Experimental results show that the resistance of the micro temperature sensor increases linearly with input temperature, with an average TCR (temperature coefficient of resistance) value of 8.2 × 10−4 (°C−1). The resistance of the pressure sensor also increases linearly with air pressure, with an average sensitivity value of 3.5 × 10−2 (Ω/kPa). The sensitivity to humidity increases with ambient temperature due to the effect of temperature on the dielectric constant, which was determined to be 16.9, 21.4, 27.0, and 38.2 (pF/%RH) at 27 °C, 30 °C, 40 °C, and 50 °C, respectively. The velocity of airflow is obtained by summing the variations in resistor response as airflow passed over the sensors providing sensitivity of 4.2 × 10−2, 9.2 × 10−2, 9.7 × 10−2 (Ω/ms−1) with power consumption by the heating resistor of 0.2, 0.3, and 0.5 W, respectively. The passage of air across the surface of the flow sensors prompts variations in temperature among each of the sensing resistors. Evaluating these variations in resistance caused by the temperature change enables the measurement of wind direction.

A study of electrical properties and microstructure of nitrogen-doped poly-SiC films deposited by LPCVD

The electrical properties and microstructure of polycrystalline 3C-SiC (poly-SiC) films were studied for different nitrogen doping concentrations. Nitrogen-doped poly-SiC films were deposited by LPCVD (low pressure chemical vapor deposition) at 900 °C and 2 Torr using 100% SiH 2Cl 2 (35 sccm) and 5% C 2H 2 (180 sccm) as the Si and C precursor gases, and 1% NH 3 (20–100 sccm) as the dopant source gas. The resistivity of the poly-SiC films at 30 °C decreased from 1.466 Ω cm to 0.036 Ω cm as the flow of NH 3 dopant source gas increased from 20 sccm to 100 sccm. The surface roughness and crystalline structure of the poly-SiC films did not depend on the dopant concentration. The average surface roughness was 19–21 nm, and the average surface grain size was 165 nm. In XRD spectra, poly-SiC is so highly oriented along the (1 1 1) plane at 2 θ = 35.7° that other peaks corresponding to SiC orientations are not presented. Resistance and temperature coefficient of resistance (TCR) measurements were carried out in the 30–450 °C temperature range. While the resistance decreased with increasing measurement temperature, the magnitude of the resistance change was much larger in the films with lower doping concentration. On the other hand, the linearity of resistance variation was better in the films with higher doping concentration. In case of poly-SiC films deposited with 20 sccm and 100 sccm 1% NH 3, the average TCR is −2050.3 ppm/°C and −1957.0 ppm/°C, respectively.

Electrical properties of nickel oxide thin films for flow sensor application

In this work, Ni oxide thin films, with thermal sensitivity superior to Pt and Ni thin films, were formed through annealing of Ni films deposited by a r.f. magnetron sputtering. The annealing was carried out in the temperature range of 300–500 °C under atmospheric conditions. Resistivity of the resulting Ni oxide films were in the range of 10.5 μΩ cm/°C to 2.84 × 10 4 μΩ cm/°C, depending on the extent of Ni oxidation. The temperature coefficient of resistance (TCR) of the Ni oxide films also depended on the extent of Ni oxidation; the average TCR of Ni oxide resistors, measured between 0 and 150 °C, were 5630 ppm/°C for the 300 °C and 2188 ppm/°C for 500 °C films. Because of their high resistivity and very linear TCR, Ni oxide thin films are superior to pure Ni and Pt thin films for flow and temperature sensor applications.

PTC behavior of polymer composites containing ionomers upon electron beam irradiation

We have prepared polymer composites of low-density polyethylene (LDPE) and ionomers (Surlyn 8940) containing polar segments and metal ions by melt blending with carbon black (CB) as a conductive filler. The resistivity and positive temperature coefficient (PTC) of the ionomer/LDPE/CB composites were investigated with respect to the CB content. The ionomer content has an effect on the resistivity and percolation threshold of the polymer composites; the percolation curve exhibits a plateau at low CB content. The PTC intensity of the crosslinked ionomer/LDPE/CB composite decreased slightly at low ionomer content, and increased significantly above a critical concentration of the ionomer. Irradiation-induced crosslinking could increase the PTC intensity and decrease the NTC effect of the polymer composites. The minimum switching current (Itrip) of the polymer composites decreased with temperature; the ratio ofItrip for the ionomer/LDPE/CB composite decreased to a greater extent than that of the LDPE/CB composite. The average temperature coefficient of resistance (αT) for the polymer composites increased in the low-temperature region.

Resistivity of Antimony-Tin Single Crystals at Low Temperatures

The electrical resistivity of single crystals of pure antimony and alloys of antimony containing small amounts of tin has been determined in the temperature range from 300\ifmmode^\circ\else\textdegree\fi{}K to 4.2\ifmmode^\circ\else\textdegree\fi{}K. In all cases the crystals were orientated such that [111] was perpendicular to the electric field, and various amounts of tin up to about three atomic percent were employed. All the alloys showed normal metallic properties, i.e., the resistivity decreased as the temperature was lowered. However, the average temperature coefficient of resistance between 4.2\ifmmode^\circ\else\textdegree\fi{}K and 77.3\ifmmode^\circ\else\textdegree\fi{}K decreases sharply with increasing tin content and assumes a small constant value at three atomic percent tin. While the addition of tin at room temperature changes the resistivity only slightly, the effect at liquid helium temperature is very large. This residual resistivity is approximately a parabolic function of the tin concentration. Qualitative comparison of these results with the Bloch theory of metals is made.