Finding the Temperature Coefficient for Copper Wire and a Semiconducting

Thermistor

Created by Brian Hallee

Partnered by Joseph Oxenham

Performed September 8th, 2010

Historical Background

The idea of a thermistor, or thermally sensitive resistor, has been around for over 150

years. Although one of his lesser-known discoveries, the first documented use of an NTC,

(Negative Temperature Coefficient, something to be returned to in the theoretical basis

section), thermistor came from Michael Faraday in 1833. 1 After the initial discovery, it was

quickly realized that thermistors could be separated entirely into two different categories: NTC

and PTC thermistors. Interestingly, the classification didn’t solely depend on metallurgical

properties due to the fact that at certain temperatures some types can actually switch

categories! Silicon is one such example that exhibits NTC properties until 250K, where a positive

temperature coefficient sets in. 2 All thermistors are made using semi-conducting metallic

compound oxides such as manganese, copper, cobalt, and nickel, as well as single-crystal

semiconductors silicon and germanium.3 Many different types of thermistors exist for different

uses. The coated lens type, while not utilized in this experiment, is one example as seen in Fig. 1

on page 2. While Faraday was first to discover the thermistor properties of semiconductors,

Samuel Ruben was quickest to perfect it and seal it under a U.S. patent. Almost a century after

Faraday’s breakthrough, Ruben released his “Electrical Pyrometer Resistance” findings in which

he used a special technique to “cook” a copper base in an oxidizing atmosphere to create a

cuprous oxide. After being cleansed in hydrochloric and nitric acid, a thin film of this oxide

remained that gave his thermistor a negative temperature coefficient without the drawbacks of

standard semiconducting materials.4 He explains in his patent that as he experimented with

adding heat to the device, its resistance dropped noticeably and reproducibly. As a final notable

mention, Rueben explains similar phenomena occurred when mixing the cuprous oxide with

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cuprous sulphide, or melting antimony sulphide with

cuprous sulphide. The practicality of this intricate

thermistor was widespread, as it drove applications in

voltage protection, temperature control, and calorimitry

to name a few.

The notion of resistance increasing with

temperature in regular conducting materials, however,

is not a new one by any means. A. E. Kennelly and

Reginald A. Fessenden write in 1893 in the Physical Review about the linear relationship

between increased temperature and resistance in a sample of copper. In their testing between

the ranges of -69⁰C and 123⁰C, the same range we have worked in, they explain how copper’s

temperature coefficient is a positive 4.18% per degree Celsius.5 Thus, they seem to have proven

very early on that, unlike semi-conductors, well-conducting metals do not exhibit strong, or

any, fluctuations in R vs. T linearity. We will prove in later sections that this century-old value

for the coefficient of copper (alpha) is quite reproducible.

Although not explored in this lab, it was only a few decades later when physicists made

perhaps the most astonishing discovery relating to electrical conductivity. On April 8, 1911,

Kamerlingh Onnes and his cohorts experimented with vapor pressures of liquid helium to drop

the temperature of mercury to a level where resistance “practically” disappeared.6 Today, we

denote this phenomenon as superconductivity and it involves similar quantum effects explored

in the theory section of this report on thermistors. Thus, historical experimenting has proven to

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Example Coated Lens-Type NTC Thermistors (FIG. 1)

http://www.patarnott.com/atms360/pptATMS360/CircuitLabThermistors.ppt

us that the relationship between resistance and temperature can take wildly different turns

given the circumstances and materials used.

Theoretical Basis

In order to better understand the results presented here, the underlying physical

workings of the materials must be grasped. The idea of a negative or positive temperature

coefficient is simple. It is the deciding factor as to whether resistance increases with increasing

temperature, or decreases. Like Faraday, in this experiment we utilized a negative temperature

coefficient thermistor and attempt to grasp the type of curve drawn out over a 10⁰C-90⁰C

range. We start by assuming that 150+ years of semiconductor engineering is correct in

assuming thermistors exhibit a relationship with temperature as such:

R (T )=k∗T (eqn. 1)

R is naturally the resistance, T is temperature, and k is an unknown “temperature coefficient of

resistance”. This k value can, at face value, be either negative or positive depending on whether

we are using an NTC or PTC. A significant problem with this simple equation is the fact that

most, if not all, thermistors do not wield a simple linear, quadratic, power, or other pre-

determined relationship between its resistance and temperature imposed. Thus, naturally,

equation 1 above breaks down when large temperature ranges are used. To resolve this, two

geophysicists John S. Steinhart and Stanly R. Hart developed a 3rd power logarithmic equation

with four coefficients to solve for the inverse of temperature in all semiconductors. The

equation is as follows: 7

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1T

=a+b∗ln ( R )+c∗ln2 (R )+d∗ln3 (eqn. 2)

The lowercase letters are individual Steinhart-Hart temperature coefficients. While in most

professional cases, a, b, c, and d can be solved using four data points to solve four simultaneous

equations, we can exploit the fact that our thermistor has a negative temperature coefficient

and simplify the coefficients. Both c and d will practically fade into insignificance, and we can

set a= 1To

− 1βln ( Ro ) and b=1

β to arrive at the β parameter equation:

1T

= 1T o

− 1βln ( Ro )+ 1

βln ( R )

Finally, we use logarithmic properties to achieve

1T

=1

T o

+1βln( R

Ro) (eqn. 3)

where β is the new temperature coefficient of resistance. Before moving on, the final equation

used for determining resistance must be derived:

1T

=1T o

+1βln( R

Ro) →To β=Tβ+T oT∗ln( R

Ro)→

β∗(T o−T )=T oT∗ln( RRo

)→ β∗( 1T −1T o

)=ln( RRo

)Place both sides as the exponent of e and multiply by R0 to get resistance as a function of

temperature:

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R (T )=Ro eβ ( 1

T− 1

T o

)(eqn. 4)

Ro=Resistanceat lowest temperatureT o=Lowest temperature data point

e=The Naperian base :2.781

Equation 2 can be deceiving. As I previously mentioned, we have taken the Steinhart-Hart

equation and modified it to work succinctly for negative temperature coefficient thermistors.

Thus, in our case, beta will actually come out to be positive. Now, this experiment enabled us

to compare a thermistor to an actual conducting wire made from copper. The electro-chemical

properties of conducting metals such as copper have been fairly well understood for many

years now. Excluding the strange nature of semi-conductors, most metals exhibit a linear

increase in resistance with increasing temperature. To fully appreciate this fact, we have to

understand copper at the atomic level. Electric current is simply the passage of electrons

through a wire over time somewhat analogously to the flow rate of water passing through a

pipe. When we introduce the concept of resistance, you can picture driving thick nails into the

pipe to impede the flow of water through it. At the atomic level, these “nails” are actually

nuclei vibrating around their mean position. The electrons passing through the wire are either

smashed into these nuclei, impeding their progress, or deliberately thrown off course due to

the strong electric forces present between the positively charged protons and the electrons’

own negative charge. It is fairly simple to visualize and conceive the notion that if these nuclei

were to vibrate back and forth quicker and quicker, and with larger ranges, the electrons’

chances of hitting nuclei increases dramatically. This is precisely the phenomenon that occurs

when the temperature of the wire is increased. Thus, we can appreciate why conducting

metals increase resistance with temperature, and why NTC semiconductors can be labeled as

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“strange”. The math behind regular conducting metals is a bit simpler relative to thermistors.

We expect the temperature coefficient of these metals to be as close to zero as possible, thus

we define it as such:

α= 1Ro

∗( dRdT

) (eqn. 5)

We can easily form a function R(T ) by separating variables and integrating demonstrated in

the following steps:

α∗dT= 1Ro

∗dR → α∫0

T

dT= 1Ro∫R 0

R

dR → αT= 1R0

( R−R0 )

Multiply and subtract by R0 to get resistance as a function of temperature:

R (T )=R0αT +R0→

R (T )=R0(aT+1) (eqn. 6)

The physics behind the strange phenomena exhibited in this experiment on thermistors is

somewhat convoluted. Semiconductors exhibit conduction band energy states at the very high

energy end. Most electrons in semi-conductors are stuck in the low energy band bound by the

valence band. Hence, very few charge carriers are available at low temperatures. However,

when temperature is increased and heat energy is bestowed to the metal, many of these lower

energy electrons begin making the jump to the conducting band and free holes in the valence

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band. This occurrence overrides the fact that atomic vibrations are increasing simultaneously,

and the end result is decreased resistance. 8

Apparatus

This experiment was carried out utilizing a relatively simple set of tools and apparatuses.

The central piece of this experiment was a tripod-supported, insulated canister used to house

the copper coil and thermistor. This canister was electrically supplied by a two-prong wall plug

to operate its internal resistance source. This source involved a simple resistance heater with a

coefficient of performance = 1. It was hand operated and spring loaded to return to the off

state in order to avoid a boiling water or fire hazard. Next, the copper coil and thermistor units

were both coupled to a lid engineered to seal off the canister and minimize heat loss. These

lids came equipped with a sealed access point for a thermometer, and a stirring apparatus to

enable uniform heat dissipation throughout the canister. Lastly, the lids were electrically

coupled to the thermistor and copper wounds and were subsequently equipped with positive

and negative terminals on the top. We utilized these terminals through alligator clips and color-

coated wires (Red: positive, Black: negative) that were fed into a Keithley 197 Amp Multimeter.

This multimeter allowed us to simultaneously feed current into the copper or thermistor coils

and determine the resistance through one device. This multimeter was stated to have an

accuracy of 1mΩ.9 Lastly, we had five medium to large beakers at our disposal in order to

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achieve water just above the freezing point to cool down the canister. As an effort to cut down

on unnecessary reproduction, we utilized an HP Pavilion laptop with Excel formulas pre-loaded

in order to visualize our trend lines on the spot.

Procedure

We began this experiment by acquiring roughly 1/2 pound of ice (1 large beaker) and

mixing it with tap water. The instruments necessary for the procedure were conveniently laid

out beforehand, and thus we were able to begin immediately. Before connecting the canister

to the wall outlet, we brought the metal inside down to roughly 5⁰C by flushing it with ice-cold

water from the large beaker. Another pre-experiment task was to determine the lead

resistance on the wires we had for use. This was determined by inserting them into the

multimeter, ensuring the multimeter was on and reading resistance on a 2Ω scale, and rubbing

the leads together to remove a bit of corrosion. While the actual lead resistance was rather

“jumpy” and hard to measure, we averaged the value over a roughly ten-second interval. Once

satisfied, we left the water inside the canister, plugged in the resistance heater, and opted to

begin with the copper coil apparatus. We placed it snugly onto the canister, inserted the

thermometer, connected the alligator clips to the terminals, and observed the multimeter to

ensure a proper connection had been made. The temperature was brought up to 10⁰C for the

first data point, and the data collection was underway. We divided the remaining tasks as such:

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Joseph Oxenham operated the resistance heater and ensured the temperature was brought up

slowly to the next data point. I maintained the homogeneity of the heat in the liquid via the

stirring apparatus and inserted the data points into Microsoft Excel. As previously mentioned,

the tables, formulas, and graphs were created prior to the experiment. Thus, we were able to

observe the linear relationship between Resistance and Temperature on the fly during the

copper experiment to ensure all equipment and procedures were optimal. We attempted, and

mostly succeeded, in taking data points for every 10⁰C increase in temperature. Therefore, a

total of nine data points were taken ending with a 90⁰C data point. Once this portion of the lab

had concluded, the copper lid was removed and ice-cold water was again flushed in and out of

the canister until the water maintained a temperature of 5⁰C. The thermistor device was

housed by the same type of lid as the copper device, ergo the same procedures applied. Again,

we acquired data points for every 10⁰C increase in temperature beginning with 10⁰C and

culminating with 90⁰C. We were able to observe the relationship between (1/T-1/Td) and

ln(R/Rd) in real-time to ensure the experiment was correct. As the lab concluded, we dumped

out the water, removed all the components, turned off/unplugged electronics, and cleaned

beakers in order to prepare for the next usage.

Results

From our experimentation, we gathered data in the form Ohms (Ω), Temperature (⁰C

and ⁰K), and unit-less values of (1/T-1/Td) and ln(R/Rd) for the thermistor.

Rd = Resistance at lowest temperatureTd = Lowest temperature: In our case 12 C (Thermistor)⁰

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As far as resistance is concerned, we noted both the raw measured resistance and the actual

resistance with the lead resistance taken into account. All of these values can be viewed in the

appendix section of this report. We found the lead resistance to be 0.258Ω, thus the actual

value of resistance was determined viaRmeasured−Rlead=Ractual. With the Ractual and T values in

hand, we were able to make a graph demonstrating the linear relationship between the two in

copper wire demonstrated below.

0 10 20 30 40 50 60 70 80 90 1000

1

2

3

4

5

6

f(x) = 0.0160753098188751 x + 3.39778169685415R² = 0.995952008290898

Resistance Vs. Temperature of Copper Coil

Temperature (⁰C)

Resis

tanc

e (Ω

)

Now, as we derived earlier, eqn. 6 represents the change of resistance vs. temperature for

copper.

R (T )=R0 (aT+1 )→ α= 1T∗( R

Ro

−1)=(R−R0)(T−0)

R0= Sl ope

Intercept

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We can use Excel’s built in algorithms to calculate alpha as follows:

α=SLOPE(C6 :C14 , A6 : A14)

INTERCEPT (C6 :C 14 , A 6 :A 14).10 This automatically gives us the value

α=0.00473112 1℃

. This is acceptable, as it closely matches the value given in the lab handout:

α=0.00433 1℃

.11 However, we wish to verify that excel is indeed performing the calculation

correctly. We take the first two data points (10℃ and 20℃ respectively) and use them

simultaneously to solve for the slope and intercept as such:

3.577Ω=m∗10℃+b∧3.691Ω=m∗20℃+b

Subtract Equations→0.113Ω=m∗(20℃−10℃ ) → m=0.0113 Ω℃

Plug∈m →3.577Ω=0.0113 Ω℃

∗10℃+b →

b=3.577Ω−(0.0113 Ω℃

∗10℃)=3.465Ω

α=Slope

Intercept=0.0113

Ω℃

3.465Ω=0.00329

1℃

.

This value of α is fairly close, so we are comfortable with our calculation. As shown in the graph

above, our coefficient of determination is practically 100%, so we can again be sure our

experimental process was not flawed.

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Next we deal with our calculations pertaining to the thermistor apparatus. We were

informed from our handout that, in this case, R vs. T would not be linear. Instead, what we got

seemed to represent an exponential decay as shown below:

270 280 290 300 310 320 330 340 350 360 3700

10

20

30

40

50

60

Resistance Vs. Temperature of a Thermistor

Temperature (⁰K)

Resis

tanc

e (Ω

)

The important concept to take away from this graph is the notion that resistance did indeed

decline with increased temperature. Temperature was displayed in Kelvins for the reason that

an absolute scale was necessary for generating the logarithmic resistance ratio graph to solve

for β. To achieve this, all Celsius data points were converted using this formula:

T 0K=T℃+273.15

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We previously derived the relationship for resistance vs. temperature of a thermistor

culminating with eqn. 4. Our goal is to solve for β, (the negative temperature coefficient), and

that is found via the following two steps:

ln ( RR0

)=β ( 1T − 1T o

)→ β=ln( R

R0 )( 1T − 1

T o)

Therefore, we need a graph demonstrating the relationship between ln ( RR0

) and ( 1T −1T o

) to

solve for β. The graph below exhibits just that.

-0.0008 -0.0007 -0.0006 -0.0005 -0.0004 -0.0003 -0.0002 -0.0001 0

-3

-2.5

-2

-1.5

-1

-0.5

0f(x) = 3421.98849517027 x + 0.0439590120530753R² = 0.999006386019973

ln(R/Rd) vs. (1/T - 1/Td) of a Thermistor

(1/T-1/Td)

ln(R

/Rd)

Armed with the knowledge that β is a constant, we expect this graph to be perfectly linear (or

close to it). Excel tells us we have a coefficient of determination of practically 100%. Thus, our

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data can be viewed as highly precise. In this case, our slope is our β, and it is rather close to the

lab handout’s expected range of 3530⁰ K ±80⁰ K. It was acquired using the formula

β=SLOPE ( M 6 :M 14 , L6 :L14 ).10 We perform a sample calculation using the first and ninth

data points to ensure the quality of Excel’s algorithms:

β=ln ( 4.132Ω

52.242Ω)

(1

363.150 K−

1

285.150K)= 3,368.27⁰K

This value is very close to our experimental value, so we can ensure the calculation is correct.

Discussion

Overall we are very satisfied with the results of our experimentation. Reviewing our

coefficient of determination value (0.996) acquired for the copper coil (0.996), we had a very

precise set of data points. Knowing the accepted value for copper from the lab handout, we

can calculate the percent approximation error using the following formula:

δ=|α accepted−α expirimental|

α accepted

∗100%=|0.00433 1℃−0.00473112 1

℃|0.00433

1℃

∗100%=9.26%

While small, an almost ten-percent error has to be accounted for. This is most likely due to the

inexperience in working the resistance heater. While it was stressed in class that a gradual

increase in heat over the few degrees preceding the data point would garner the best results,

we may have acted overly ambitious on this first run. Consequentially, we were not allowing

for uniform heat dispersion throughout the entire canister. This, in turn, caused resistances to

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vary throughout the copper coil and subsequently cause our readings to vary. Another problem

we ran into on this first run is variance in the multimeter itself. When the time came to take

note of a resistance, the multimeter would frequently jump over a range of roughly 0.3Ω. Thus,

we were forced to quickly average this value in order to gather a data point. In hindsight, this

too may have been caused by the hastily controlled resistance source and non-uniformly

resisting copper wire. Returning to the work of Kennelly and Fessenden, we did, however,

demonstrate similar values to their experiments dating over a century ago. Their value being

0.004181℃

, we experienced a 13.2% approximation error relative to this α.

We had a bit more luck with in determining the temperature coefficient of the thermistor

apparatus. The lab handout states that the manufacturer of this thermistor already accepts an

absolute error of ±80⁰ K . Thus we can begin by computing the percent error of the

manufacturer’s value, and compare that to our percent approximation error of the

experimental value.

%Relative Error ( Manufacturer )= errormeasurement

∗100%= 800 K35300 K

∗100%=2.27%

δ=¿35300 K−34220K∨ ¿3530⁰ K

∗100%=3.06% error¿

Thus, our experimental error is less than one percent greater than manufacturing error, so we

can be very comfortable with this value acquired. For the error that we did happen to obtain,

this, again, is likely due to ambitious resistance heating and other minute issues such as lead

corrosion, terminal corrosion, etc. We were, however, intrigued by the fact that our β value

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came out to be only two ⁰K off from the attempt from the writers of the lab handout. While

they were forced to utilize a slide-wire Wheatstone bridge and we were blessed with a digital

multimeter, the 0.053% error from their attempt gave us something to further ground our own

attempt.

While most of the theories behind the behavior of conducting and semi-conducting metals have

been well understood for, at the very least, decades, this experiment does its job at solidifying

certain concepts and implications of these technologies. First, copper’s positive temperature

coefficient demonstrates why it makes such a useful tool for widespread use among our

communications and energy infrastructure. It self-protects itself from thermal runaway. In

layman’s terms, copper and other conductors will “sense” that extra or unruly current is passing

through via heat generation and will automatically increase the resistance to this current. While

this is not applicable to, say, replacing a fuse, it does function optimally for the uses for which it

is designed (e.g. local telephone or power cables). On the other hand, NTC thermistors such as

the one experimented on in this lab are vulnerable to such a runaway. However, these units are

not exploited for their load carrying capacities so much as they are for their sensitivity. In this

lab we observed a change in almost fifty Ohms in our thermistor versus roughly 1.2 Ohms in the

copper coil. Our conclusion on this device is that it would be very useful in application settings

centered on temperature control and compensation. When coupled to a simplified Ohmmeter

and computer chip, this devices change in resistance could immediately be relayed to the

computer chip and the temperature could be changed according to the algorithm set in place

by the software engineer. To no surprise, the thermistor is very heavily used in all areas of

temperature control such as thermostats, ovens, refrigerators, A/C units, fire alarms, fever

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thermometers, coffee makers, and more.5 In summary, both conducting wires and semi-

conducting thermistors have very practical uses for their R vs. T properties that will likely leave

them around for decades, if not centuries, to come.

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APPENDIX – RAW DATA

Copper Coilα = 0.00473112 (1/⁰C)

Lead Resistance = 0.258Ω

T (⁰C) R-meas. (Ω) R-s (Ω)

10 3.835 3.577

20 3.949 3.691

30 4.090 3.832

40 4.329 4.071

50 4.487 4.229

60 4.645 4.387

70 4.790 4.532

81 4.942 4.684

90 5.085 4.827Thermistor Apparatus

Lead Resistance = 0.258Ω

β = 3421.988495 (⁰K)

T (⁰C) R-meas. (Ω) T (⁰K) R-s (Ω) (1/T-1/Td) ln(R/Rd)

12 52.500 285.15 52.242 0 0

20 41.347 293.15 41.089 -9.6 x 10-5 -0.24015

30 27.530 303.15 27.272 -0.00021 -0.65003

40 19.300 313.15 19.042 -0.00031 -1.00924

50 13.206 323.15 12.948 -0.00041 -1.39495

60 9.735 333.15 9.477 -0.00051 -1.70702

70 7.390 343.15 7.132 -0.00059 -1.99130

80 5.794 353.15 5.536 -0.00068 -2.24461

90 4.390 363.15 4.132 -0.00075 -2.53713

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APPENDIX – DISC FILES

Root Directory

o Lab Report_Brian Hallee_Lab 1.doc

Official digital copy of “Finding the Temperature Coefficient for Copper

Wire and a Semiconducting Thermistor” lab report.

o Lab1_Temperature Coefficient_9-8-10.xls

Comprehensive spreadsheet containing all raw data entered during the

experiment, all three graphs presented in this report associated with the

copper wire and thermistor, and calculated values for α and β.

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ENDNOTES

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1 Cornerstone Sensors. (n.d.). A Brief History of NTC Thermistors. Retrieved September 18, 2010, from Cornerstone Sensors: http://www.cornerstonesensors.com/?LinkIn=http://www.cornerstonesensors.com/About.asp?PageCode=Brief

2 Radio-Electronics. (n.d.). Thermistor. Retrieved September 18, 2010, from Radio-Electronics.com: http://www.radio-electronics.com/info/data/resistor/thermistor/thermistor.php

3 Arnott, P. (n.d.). Thermistors: Thermal Resistors. Retrieved September 19, 2010, from PatArnott.com: patarnott.com/atms360/pptATMS360/CircuitLabThermistors.ppt

4 Ruben, S. (1930, March 19). Electrical Pyrometer Resistance. Retrieved September 19, 2010, from FreePatentsOnline: http://www.freepatentsonline.com/2021491.pdf

5 A. E. Kennelly and Reginald A. Fessenden. (1893). Some Measurements of the Temperature Variation in the Electrical Resistance of a Sample of Copper. The Physical Review, 260-273.

6 Kes, D.v. (2010). The Discovery of Superconductivity. Physics Today, 38-43.

7 John S. Steinhart, Stanley R. Hart, Calibration curves for thermistors, Deep Sea Research and Oceanographic Abstracts, Volume 15, Issue 4, August 1968, Pages 497-503, ISSN 0011-7471, DOI: 10.1016/0011-7471(68)90057-0.

8 Nave, R. (n.d.). Band Theory for Solids. Retrieved September 18, 2010, from HyperPhysics: http://hyperphysics.phy-astr.gsu.edu/hbase/solids/band.html#c5

9 ValueTronics. (n.d.). Keithley 197A Autoranging Digital Multimeter. Retrieved September 19, 2010, from ValueTronics International Inc.: http://www.valuetronics.com/Used_Keithley_197A.aspx

10 See Lab1_Temperature Coeffieicient_9-8-10.xls on accompanying CD-ROM

11 Instructions for the Use of NO. 2836. Thermistor Temperature-Coefficient Apparatus. (n.d.).