Extended Essay Theoretical Version

EXTENDED ESSAYIN PHYSICS

Uy Quoc NguyenBerg Videreg̊aende Skole

May 2011

Conductivity of graphite for small temperature intervals

Word Count: 3617

Uy Quoc Nguyen Extended Essay Physics

Abstract.In this extended essay I have studied the conductivity of graphite’s

relation to the temperature. The research question for this ex-tended essay is thus: “What is the relationship between the con-ductivity and temperature of graphite for small temperature in-tervals?”. The approach for this research was intended to bepurely theoretical and so there are no self-conducting experimentsthroughout the essay. However, data from an investigation doneby L. J. Collier and his colleagues was exploited in order providean easy way to derive the final result for the relationship betweenthe conductivity and temperature changes.

Furthermore, theories of conductivity in semi-conductors wereestablished by introducing the properties of metallic-conductorsin order to provide a solid base for understanding the mechanismof freely moving electrons in graphite. That means it started offby introducing the most basic concepts to Ohm’s law and thenproceeded to study the influence of the drift velocity of electrons.Moreover, electronic orbital theories were also applied as it is cru-cial for the understanding of electron and hole interactions.

The conclusion that was finally deduced is that the conductiv-ity of graphite are has a linear relationship with the change intemperature and is given by:

σ(T ) =σ0

1 + α∆T

where α is the temperature coefficient of resistivity. However, itis also concluded that this expression is only valid for small tem-perature changes because it would lead to absurd result for largetemperature intervals. That is graphite becomes a superconductorat 0◦C and so on.

Word Count: 243


Contents

1. Introduction 41.1. Approach 52. Ohm’s Law Re-investigated 62.1. The Drift Velocity 62.2. The Current and Potential Difference 73. Structure of Graphite 93.1. Orbitals and Electron Configurations 93.2. Hybridization of Orbitals 103.3. Delocalization of Electrons and Holes 114. The Conductivity and Temperature Relationship 135. Conclusion and Evaluation 15Bibliography 17


1. Introduction

Ohm’s law is an idealized model, that is the law is not obeyed forevery situation. The law states that: for a constant temperature thepotential difference at the ends of an electrical conductor is directlyproportional with the current flowing through it1. This means thatthe resistance for the electrical conductor, being the constant of theratio between the potential difference and the current, is dependent ontemperature. This law was discovered by the German physicist GeorgSimon Ohm in 1826 and was published in 1827 as a book Die gal-vanische Kette, mathematisch bearbeitet (The galvanic circuit, workedmathematically)2.

Later years, in 1911 the Dutch physicist Heike Kamerlingh Onnesconducted an experiment in Leiden where he cooled down pure mer-cury with liquefied helium. His result showed that the electrical re-sistance in metallic conductors disappeared when the temperature ofthe conductor were below a critical temperature3. Thus, the metallicconductors became perfect conductors also known as superconductors.2 years later the Nobel Prize in Physics was awarded to KamerlinghOnnes for his discovery of this phenomenon4.

Today, the application of the superconductivity for research is cru-cial in physics. One of its most important applications can be foundwith The Large Hadron Collider (LHC) located at Cern in Geneva,Switzerland. LHC, is a particle accelerator with a circumference of26.7km. Its goal in high-energy particle physics research is to collidebeams of hadrons in order to search for new particles and confirm thehypothesized particle, the Higgs Boson5. In addition, it is also used tocollide lead ion beams with energies up to 1150TeV in order to studynew states of matter. The requirement for colliding these high-energybeams of particles is to bend the beams to a fixed radius. This require-ment is met by exploiting high-field superconducting magnets6.

1Oxford Dictionary of Physics: Sixth edition (Oxford University Press, 2009),page 371.

2Georg Simon Ohm: Die galvanische Kette, mathematisch bearbeitet (BerlinT.H. Riemann 1827).

3J.R. Waldram: Superconductivity of Metals and Cuprates (IOP Publishing Ltd1996), page 1-2.

4”The Nobel Prize in Physics 1913”. Nobelprize.org. 20 Nov 2010http://nobelprize.org/nobel prizes/physics/laureates/1913/

5”Scientific American”. Scientificamerican.com. 21 Nov 2010http://www.scientificamerican.com/article.cfm?id=5-goals-for-the-lhc

6”CERN Document Server”. Cdsweb.cern.ch. 21 Nov 2010http://cdsweb.cern.ch/record/473537/files/lhc-project-report-441.pdf

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The focused question for this extended essay will be:

What is the relationship between the conductivity andtemperature of graphite and temperature for small tem-perature intervals?

My main reason for choosing this question is because I wanted tostudy how Ohm’s law is dependent on temperature and find a math-ematical expression for the conductivity of a conductor as a functionwithin a temperature interval. This curiosity was developed when Iwitnessed superconductivity for the first time, many years ago before Istarted the IB programme, in the University of Oslo when they demon-strated how the concept of superconductivity could be exploited inmaglev transportation. Also my supervisor encouraged me to choosea specific conductor to make my focused question more narrowed. Inthis case, I chose graphite as my electrical conductor because it is nota metallic conductor and therefore does not conduct electricity in thesame manner as metals and may be more interesting to study than anymetallic conductors.

1.1. Approach.One problem that occurs when studying this topic is that conductiv-

ity is a quantum mechanical phenomenon. Thus, it is only completelyunderstood using the quantum theory. However, the IB Physics HLsyllabus only offer a brief introduction to quantum physics, the knowl-edge required to study conductivity in the context of quantum physicswas beyond my limit of understanding. Furthermore, the mathematicsrequired to understand the physics on that level is too advanced forthe knowledge acquired in mathematics HL to handle. Even so, thefocused question is going to be approached with the attempt of usingclassical physics to provide arguments to the conclusion.

The essay is going to have a theoretical approach using Ohm’s lawas a reference to compare the outcome of a new derived expression forcurrent. This new expression is going to be derived after discussionof mobile electrons and resistivity within a metallic conductor. Afterthe idea have been established for a metallic conductor, the discussionwill be carried to the structure of graphite considering hybridized or-bitals, delocalization of electrons within the covalent bonds, betweenthe carbon atoms and how graphite conduct electricity through theinteraction between electrons and holes. For both cases, the effect oftemperature is going to be included. To the end of the essay the rela-tionship between the resistivity of graphite and temperature for smalltemperature intervals is elaborated and the expression for that is goingto be derived.

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2. Ohm’s Law Re-investigated

Ohm’s law states that for a constant temperature there is a linear re-lationship between the potential difference V at the ends of a conductorand the current I in the conductor. Hence,

(1) V ∝ I

The proportionality constant is thus the resistance R throughout theconductor and Ohm’s law finally becomes:

(2) V = RI

However, both the resistance and the current are dependent on thetemperature. Thus, the equations can then be written as a function oftemperature:

(3) V = R(T )I(T )

This case is going to be further investigated.

2.1. The Drift Velocity.Take for instance a metallic conductor of length l. Metallic conduc-

tors are basically a composition of metallic elements creating metallicbonds between the atoms with their electrons. These electrons arethus freely to move around with enormous speed due to thermal en-ergy. Because, these free electrons move in random directions there areno net flow of charges through the conductor, consequently there areno current running through the conductor7. If we then apply a con-stant potential difference V from a to b on the conductor, the electricfield created in the conductor will do work on the electrons bringingthem from point a to point b. Hence, the force that is acting on eachindividual electron with charge −e is given by:

(4)−→F = −e

−→E

And thus, there will be a net flow of charges in one direction. Therefore,there will be a current in the conductor.

The electrons which first moved with enormous speed is now movingwith a tremendous reduction of speed. This is due to the increasednumber of collisions between the vibrating ions and the moving elec-trons in the conductor, because it is forced to move in one specificdirection. This new net velocity is called the drift velocity denoted −→vd .If the times between each collision are τ then the acceleration of oneparticular electron is given by the equation:

(5) −→a =d−→vddτ

7Sears & Zemansky’s: University Physics 12th edition with modern physics(Pearson Education, Inc 2008) page 847.

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Thus, solving this differential equation gives the drift velocity −→vd :∫ −→dvd = −→a

∫dτ(6)

−→vd = −→a τ + c(7)

The constant c is the drift velocity when τ = 0 thus v0. Now, if weassume that all the drift velocity is completely destroyed after eachcollision that is v0 = 0 then equation 7 can be rewritten as:

(8) −→vd = −→a τ

By combining equation 4 and 8, and solving for the drift velocity −→vdwe get that:

(9) −→vd = − eτme

−→E

where me is the mass of the electrons.

2.2. The Current and Potential Difference.When a potential difference V is applied over a conductor with length

l an electric field is created inside that conductor. The electrons beingcharged particles is experiencing this electric field and will thus try tomake it zero. Therefore, the electrons are moving with a drift velocityvd given by equation 9 and a current I is created. Current is definedas “the rate of flow of a charge at a particular cross section”8. Thus,

(10) I =dQ

dt

where Q is the charge.Consider any metallic conductor with a cross sectional area A and

length l. A potential difference V is applied of a length l on that con-ductor and the electrons will get a drift velocity vd given by equation9. The number of electrons passing through that cross sectional areaper time is n and after a time dt the electrons have travelled dx = vddt.The amount of charge dQ travelling through the cross sectional area Ais consequently:

(11) dQ = −neAvddt

When dividing dt on both sides of the equation in order to find therate of flow of charges through the cross sectional area A. Hence, weobtain equation 10 which is the definition for current.

(12) I = −neAvdIf we now substitute for the drift velocity vd given by equation 9 anexpression of current with the electric field created by the potential

8Oxford Dictionary of Physics: Sixth edition (Oxford University Press, 2009),page 110

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difference is obtained.

(13) I =ne2τ

me

A−→E

The factor ne2τme

is dependent on the property of the conductor, that ishow many free electrons per ions and the time interval between eachcollisions. This property is known as the conductivity and is denotedσ. If we then substitute the term for conductivity σ we get that thecurrent is given by

(14) I = σA−→E

Since, the electric field−→E created by the potential difference V is

(15)−→E =

V

l

By substituting equation 15 into equation 14 and solving for V willgive us the expression

(16) V =l

σAI

Resistivity ρ is defined as “a measure of a metal’s ability to opposethe flow of an electric current”9. A conductor with small resistivity isconsidered to be a good conductor and vice versa. Hence, the resistivityis given by the reciprocal of conductivity:

(17) ρ =1

σ

Substituting this into equation 16 finally gives

(18) V =ρl

AI

By comparing equation 18 and equation 3 we arrive to the result thatthe resistance is

(19) R =ρl

A

Since, the cross sectional area A and the length l of the conductoris constant, the resistivity ρ is the only factor that is dependent ontemperature. The ions within the conductors vibrate with amplitudeaccording to the temperature. Thus, an increase in temperature willdecrease the time interval between each collisions τ . Thus, the electronswill experience a higher resistivity and the drift velocity vd becomessmaller and the rate of flow of charges dQ

dtthrough a cross sectional

area A will also decrease10.

9Oxford Dictionary of Physics: Sixth edition (Oxford University Press, 2009),page 472

10Sears & Zemansky’s: University Physics 12th edition with modern physics(Pearson Education, Inc 2008), page 852

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3. Structure of Graphite

Graphite is a substance which is composed of layers of graphenesheld together by an intermolecular force called van der Waals’ forces.Moreover, graphenes are again composed of carbons. These carbons arebound together by the intramolecular force: covalent bonds to eachother creating planars of hexagonal nets of carbons11. Hence, eachcarbon is bound to 3 other carbons.

Carbon has 4 valence electrons which can be exploited for covalentbonding to other elements. Thus, due to this fact, if carbon is thenbounded to 3 other carbons then this results to 2 single bonds to 2 ofthe carbons and 1 double bond to the last one. The double bond is ofgreat interest in the context of conductivity for graphite. When orbitaltheory is applied to carbon we find that each carbons in graphene issp2 hybridized12. This means that all the carbons have a resonancestructure in which the electrons are delocalized between the bonds13.Thus, the delocalization of electrons will almost function as free mobileelectrons in a metallic conductor and thus graphite will be able toconduct electricity. These things are going to be further discussed.

3.1. Orbitals and Electron Configurations.The atomic model has an interesting history. The most popular

model which is greatly used in junior- and senior high schools is themodel proposed by the Danish physicist Niels Henrik David Bohr, whowas awarded the Nobel Prize in Physics in 1922 for his work14. How-ever, the atomic model introduced in the middle school is somewhatsimplified. The electron’s orbital around the nucleus is more complex.

The atomic model proposes certain energy levels in which electronscan exist in the atom. The first energy level consists of 2 electronswhile the higher energy levels are able to hold onto 8 electrons andeven more when moving to even higher energy levels. However, eachenergy level consists of sublevels which are considered to be orbitalsdenoted s, p, d and f in which electrons are probable to exist15.

11H. E Hall: Solid State Physics (John Wiley & Sons Ltd. 1974), page 4212H. E Hall: Solid State Physics (John Wiley & Sons Ltd. 1974), page 4213John Green & Sadru Damji: Chemistry 3rd edition (IBID Press, reprinted

2008), page 11914”The Nobel Prize in Physics 1922”. Nobelprize.org. 28 Nov 2010

http://nobelprize.org/nobel prizes/physics/laureates/1922/15Sears & Zemansky’s: University Physics 12th edition with modern physics

(Pearson Education, Inc 2008), page 1406-14079


Figure 1. Orbitals of a hydrogen atom of sublevels s to f

The figure16 above shows how the orbitals of an hydrogen atom isvisualized. Each orbital can only contain a maximum of 2 electronsof opposite spins17. The first energy level only consist of 1s orbitalwhich is why it is only able to hold on to 2 electrons. The secondand the third energy level consist of 1s and 3p (px, py and pz) orbitalseach. Every element has all the orbitals available but only fills thelowest orbitals with electrons. “The arrangement of electrons aboutthe nucleus of an atom” is called for an electron configuration18. Thismodel is also applicable for other elements as well and in this case themodel in “Figure 1” is going to be applied to the carbon atom.

Carbon is the number 6 element in the periodic table. Thus, a neutralcarbon has 2 electrons in the first s-orbital (1s). Moreover, it also have2 electrons in the second s-orbital (2s) and then it has 2 more electronsin the first p-orbital (2p) which is found in the second energy level.Thus, the electron configuration of carbon is written as:

1s22s22p2

Although it is called the orbitals, it is not for certain that the electronsare orbiting the nucleus in that region. In physics the “orbitals” areoften referred to as the probability density for electrons.

3.2. Hybridization of Orbitals.

16”UCDavis Chemwiki” chemwiki.ucdavis.edu 28 Nov 2010http://chemwiki.ucdavis.edu/@api/deki/files/4826/=Single electron orbitals.jpg

17John Green & Sadru Damji: Chemistry 3rd edition (IBID Press, reprinted2008), page 61

18Oxford Dictionary of Chemistry: Sixth edition (Oxford University Press,2008), page 139

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Carbon is an element which tends to form covalent bonds to otherelements. In the case of a carbon to carbon bond, one carbon can eitherbe bound to 4 other carbons due to its 4 valence electrons, or it can bebound to 3 other carbons as in a graphite layer, or to 2 other carbons.Before any element starts to form covalent bonds with other elementsits orbitals usually overlap each other to make a hybrid orbital19 orhybrid wave functions20. This process when orbitals overlap to makehybrids is often referred to as hybridization.

Carbon can form 3 different hybridized orbitals. These three hy-bridizations are directly related to the amount of covalent bonds a car-bon can create with other elements which was mentioned above. Whencarbon overlaps its orbitals to form an hybridized orbital, it “sends”one of the electron from its 2s orbital to a higher orbital, the 2p orbital.Hence, get the electron configuration:

1s22s12p1x2p1y2p

1z

Now, if carbon wants to create 4 single bonds also known as σ-bonds,every orbital from the second energy level that is 2s, 2px, 2py and 2pzoverlap to form 4 hybrid orbitals with 1 electron in each orbital. Thus,each electron can contribute to form 4σ-bond to other elements21 andit is sp3 hybridized. However, if carbon wants to form 2 double bondswhich means that this involves 2σ-bonds and 2π-bonds, it will onlyoverlap 2s orbital with one of the 2p orbitals to make 2 hybrid orbitalsand thus 2σ-bonds. Moreover, 2 of the 2p orbitals is not hybridizedand hence contribute to create 2π-bonds22 and becomes sp hybridized.Thus, in the case of carbon bonds in graphene, it overlaps 2s and 2 ofthe 2p orbitals to form 3σ-bonds and thus leaving the last 2p orbital toform π-bond to one of the other carbons thus becomes sp2 hybridized.

3.3. Delocalization of Electrons and Holes.Since, every electrons are negatively charged, a sp2 hybridized carbon

will have a structure which makes all of its σ-bonds displaced 120◦ fromeach other. Thus, makes the graphene into a planar sheet where all thecarbons are bound together into a hexagonal net23. Furthermore, dueto carbon being sp2 hybridized, there will be a π-bond to one of theother carbons. However, since all of the carbons which are bounded






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together are all sp2 hybridized the π-bond can be equally said to bebound to any carbon that particular carbon is σ-bonded to. Hence, thewave function of the electron in the “leftover” 2p orbital is extendedover the whole graphite layer. Thus, the electron is delocalized24 andcan move freely around the whole graphite layer. Accordingly, if apotential difference is applied on graphene the electron will move inthe opposite direction of the electric field and a current is created andwhich is how graphite are able to conduct electricity.

Suppose all the electrons on graphene are standing still except forone. That one electron is thus bounded to one particular carbon atomand is also forced to stay at rest. This is due to the other electronsbecause there are no “space” in the 2p orbital that the electron canjump into. Then a potential difference is applied over the graphitelayer and the electrons starts to drift in one direction. The electronbehind can then occupy the empty 2p orbital and wait for another 2porbital to be available to move to. Now, the “space” that the electronsare leaving behind are called “holes”25. When a potential differenceis applied to a graphite layer the electrons will start to move in onedirection, and the holes which act as positive charges moves in theopposite directions. Thus, the conductivity of a semiconductor such asgraphite are dependent on the mobility of both the electrons and the“holes” as well. So that the conductivity of a semiconductor is

(20) σ = σe + σh

Where σe is given by the term in equation 14

(21) σ =ne2τeme

+pe2τhmh

where p is the number of holes and mh are their mass. Now, if wedefine mobility µ to be26:

(22) µ =−→vd−→E

Then the conductivity for a semiconductor in this case graphite is givenby27

(23) σ = e (pµh − nµe)This is the intrinsic conductivity which is the conductivity of a puresemiconductor. In the context of conductivity in pure semiconductorsτ has a different meaning. In this case, τ will be the average amount



26H. E Hall: Solid State Physics (John Wiley & Sons Ltd. 1974), page 9827H. E Hall: Solid State Physics (John Wiley & Sons Ltd. 1974), page 98

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of time the electron spend on travelling to each 2p orbitals. τ as wellas ρ in this case will decrease with increasing temperature due to theelectrons undergoes thermal excitations28 and thus become more mo-bile. The conductivity is also very dependent on how many holes perelectrons the conductor has. If there are a large amount of holes inthe semi-conductor the electrons can move more freely and thus thesemi-conductor is able to conduct electricity better.

4. The Conductivity and Temperature Relationship

In 22. August 1938 a report concerning the variation with temper-ature of the electrical resistance of carbon and graphite between 0◦Cand 900◦C were submitted to the Photometry Division of the NationalPhysical Laboratory in Teddington29. It was an experiment conductedby L. J. Collier and his colleagues in order to investigate how the elec-trical conductivity of carbon and graphite is related to the temperature.

In this experiment they used carbon rods which were heated in vac-uum in a nichrome-wound resistor-furnace30. In order to measure theresistance they had a fixed current from a battery. After the pre-workthey started to measure the resistance and temperatures with an in-crement of about 50◦C. This experiment was conducted upon a totalof five carbon rods and two complete sets of observations were madeon each rod except for one. The table below31 shows the data acquiredfrom the experiment. On each rods the ratio between the resistance attemperature T ◦C, RT , and the resistance at temperature 0◦C, R0 werecalculated. Moreover, a mean ratio was also calculated.

Figure 2. Table taken directly from their work.


29L. J. Collier et al 1939 Proc. Phys. Soc. 51 page. 14730L. J. Collier et al 1939 Proc. Phys. Soc. 51 page. 14731L. J. Collier et al 1939 Proc. Phys. Soc. 51 page. 150

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Now from this data we can deduce that the ratio for each rod includingthe mean is decreasing with increasing temperature. This means thatRT < R0 for temperatures T > 0◦C. In this case the resistivity mustbe the one factor that is decreasing according to equation 19.

Now, through this data they found the relation32 of the form:

(24) RT = R0

(1− 3.17× 10−4T + 1.73× 10−7T 2 − 6.3× 10−11T 3

)This is probably the equation for a best fit polynomial with the orderof 3. As we can conclude from equation 24 also is that RT is decreasingfor higher temperatures as the leading coefficient in this third degreepolynomial is negative.

When plotted the ratio RT

R0with temperature T they obtained a graph

that looks like the one below33.

Figure 3. Graph also taken directly from their work

Our main focus on this graph is the slightly curved function and thelinear function which the function is corresponding to the table givenabove, as the curved functions are values that corresponds to anotherexperiment. The function is slightly curved and thus as the range oftemperature here is large we can conclude that for a small temperatureinterval the relationship between RT and T is linear. Hence the ratioRT

R0is proportional to some temperature interval ∆T with an initial

value of 1:

RT

R0

= k∆T + 1(25)

32L. J. Collier et al 1939 Proc. Phys. Soc. 51 page. 15033L. J. Collier et al 1939 Proc. Phys. Soc. 51 page. 151

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Now, k is called the temperature coefficient in physics and is most ofthe time denoted with α. Thus equation 25 can be written as:

RT

R0

= α∆T + 1(26)

So solving for RT which is the R(T ) stated at the beginning of thisessay:

R(T ) = R0 (1 + α∆T )(27)

We can also express this in terms of resistivity:

ρ(T )l

A= ρ0

l

A(1 + α∆T )(28)

ρ(T ) = ρ0 (1 + α∆T )(29)

Now, the relationship between the conductivity σ and resistivity ρ isgiven in equation 17 therefore in terms of σ the equation becomes

1

σ(T )=

1

σ0(1 + α∆T )(30)

σ(T ) =σ0

1 + α∆T(31)

This temperature coefficient of resistivity α for graphite is approxi-mately34 α = −0.0005(◦C)−1 in room temperature. Thus, for increasedtemperature the denominator decreases which again causes an increasein the conductivity of graphite. However, when the temperature in-terval ∆T approaches ≈ 2000◦C the conductivity σ(T ) approachesinfinity. This shows that the equation does not hold for high tem-perature intervals. Moreover, the temperature coefficient of conduc-tivity of graphite might have another value when measured for othertemperature intervals.

5. Conclusion and Evaluation

The conductivity of graphite does show some other interesting prop-erties than metallic-conductors. One of them is that it is a better con-ductor at higher temperatures in contrast to the metallic-conductorsbecomes better conductors at lower temperatures. The relationshipbetween the conductivity of graphite and temperature is thus:

σ(T ) =σ0

1 + α∆T

However, as further investigated this relationship does not hold forevery temperature intervals. As the temperature interval approaches2000◦C graphite becomes a superconductor. Indeed it becomes a su-perconductor at high temperatures however, as far as the expression


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is concerned it can as well as be superconducting at T = 0. There-fore, this expression is again investigated with the purpose of studyingsmall temperature changes. This expression must then be concernedas “classical conductivity” study as superconductivity does require farmore advanced study of quantum mechanics in order to describe thephenomena.

Further study of this topic may be to check how the coefficient isvarying with different temperature intervals. Thus a more general ex-pression may be derived from it in order to expand the limitation ofthe temperature interval.

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Bibliography

• Sears & Zemansky’s: University Physics 12th edition with mod-ern physics (Pearson Education, Inc 2008)• H. E. Hall: Solid State Physics (John Wiley & Sons Ltd. 1974)• J.R. Waldram: Superconductivity of Metals and Cuprates (IOP

Publishing Ltd 1996)• L. J. Collier et al 1939 Proc. Phys. Soc. 51: page 147 - 152

(IOPscience)• John Green & Sadru Damji: Chemistry 3rd edition (IBID Press,

reprinted 2008)• Nobelprize.org: The Nobel Prize in Physics 1913 and The NobelPrize in Physics 1922 (viewed 20. & 28. Nov 2010)• Scientific American: Sidebar: 5 goals for the LHC (viewed 21.

Nov 2010)• CERN Document Server: Superconductivity and Cryogenics Forthe Large Hadron Collider (viewed 21. Nov 2010)• Oxford Dictionary of Physics: Sixth edition (Oxford University

Press 2009)• Georg Simon Ohm: Die galvanische Kette, mathematish bear-beitet (Berlin T.H. Riemann 1827)

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