Direct Energy, 2018a

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13 THOMAS FERMI ANALYSIS 295 Symbol Quantity SI Units S/V/C Comments E <strong>Energy</strong> J = Nm S −→ E Electric eld V m V intensity E f Fermi energy level J S Also called Fermi level ⃗ k Wave vector m −1 V k f Fermi wave vector m −1 S m Mass kg S M Generalized momentum * S Many authors use p −→ kg·m M Momentum s V Many authors N v (Total) number of electrons per atom Voltage (AC or time varying) electrons atom ⃗v Velocity m s V V Voltage (DC) V S V Volume m 3 S μ chem Chemical potential J atom S ρ ch Charge density C m 3 S V Table 13.1: Variable list. S S use −→ p
296 13.3 Derivation of the Lagrangian 13.2.3 Reciprocal Space Concepts The idea of reciprocal space was introduced in Section 6.4 in the context of crystalline materials. We can describe the location of atoms in a crystal, for example, as a function of position where position −→ r is measured in meters. In this chapter, we are interested in individual atoms instead of crystals composed of many atoms. We can plot quantities like energy E( −→ r ) or voltage V ( −→ r ) as a function of position. Figure 6.11, for example, plots energy versus position inside a diode. In Section 6.4, the idea of wave vector −→ k in units of m −1 was introduced. Wave vector represents the spatial frequency. We saw that we could plot energy or other quantities as a function of wave vector, and Fig. 6.8 is an example of such a plot. We will need the idea of wave vector in this chapter because we describe a situation where we do not know how the energy varies with position, but we do know something about how the energy varies with wave vector. 13.3 Derivation of the Lagrangian The purpose of this chapter is to nd the voltage V (r) and the charge density ρ ch (r) around an atom, and we will use calculus of variations to accomplish this task. We need to make some rather severe assumptions to make this problem manageable. Consider an isolated neutral atom with many electrons around it. Assume T ≈ 0 K, so all electrons occupy the lowest possible energy levels. Assume the atom is spherically symmetric. All of the quantities we encounter, such as voltage, charge density, and Lagrangian, vary with r but do not vary with θ or φ. We will use spherical coordinates with the origin at the nucleus of the atom. While quantities vary with position, assume no quantities vary with time. The charge density ρ ch (r) tells us where the electrons are most likely on average to be found. It is related to the quantummechanical wave function, ψ, by ρ ch = −q ·|ψ| 2 (13.17) where q is the magnitude of the charge of an electron. Assume that all of the electrons surrounding the atomare distributed uniformly and can be treated as if they were a uniformelectron cloud of some charge density. Pick one of the electrons of the atom, and consider what happens when the electron is moved radially in and out. Figure 13.1 illustrates this situation. As the electron moves, energy conversion occurs. The goal of this section is to write down the Hamiltonian and Lagrangian for this energy conversion process. We write these quantities in the units of energy per unit volume per valence electron under consideration.
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Direct Energy Conversion by Andrea
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CONTENTS i Contents Contents i 1 In
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CONTENTS iii 6.3.2 Energy Levels in
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CONTENTS v 9.6 Fuel Cells . . . . .
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Acknowledgements I would like to th
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2 1.2 Preview of Topics to connect
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4 1.2 Preview of Topics Math throug
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6 1.2 Preview of Topics Process Spo
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8 1.2 Preview of Topics drodynamic
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10 1.4 Measures of Power and Energy
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12 1.5 Properties of Materials 1.5
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14 1.5 Properties of Materials most
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16 1.6 Electromagnetic Waves −→
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18 1.6 Electromagnetic Waves Relate
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20 1.6 Electromagnetic Waves A unif
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22 1.7 Problems
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24 2.2 Capacitors 2.2 Capacitors 2.
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26 2.2 Capacitors 2.2.3 Permittivit
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28 2.2 Capacitors â z â y â x No
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30 2.2 Capacitors Figure 2.3: Natur
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32 2.3 Piezoelectric Devices Piezoe
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34 2.3 Piezoelectric Devices Lattic
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36 2.3 Piezoelectric Devices Herman
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38 2.3 Piezoelectric Devices Figure
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40 2.3 Piezoelectric Devices c a β
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42 2.3 Piezoelectric Devices In cer
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44 2.3 Piezoelectric Devices made m
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46 2.3 Piezoelectric Devices tions.
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48 2.4 Problems 2.5. A piezoelectri
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50 2.4 Problems 2.13. Consider a pi
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52 2.4 Problems 2.16. The gure belo
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54 3.2 Pyroelectricity Material Che
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56 3.3 Electro-Optics KH 2 PO 4 [25
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58 3.3 Electro-Optics The rst two t
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60 3.3 Electro-Optics used as an el
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62 3.4 Notation Quagmire Notation i
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64 3.5 Problems 3.5 Problems 3.1. F
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66 3.5 Problems −→ DinC/m 2 unp
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68 4.1 Introduction Center-fed half
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70 4.2 Electromagnetic Radiation In
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72 4.2 Electromagnetic Radiation ar
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74 4.3 Antenna Components and Denit
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76 4.4 Antenna Characteristics ante
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78 4.4 Antenna Characteristics Impe
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80 4.4 Antenna Characteristics Azim
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82 4.4 Antenna Characteristics 4.4.
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84 4.4 Antenna Characteristics 5 Li
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86 4.5 Problems Figure 4.6: A snow
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88 4.5 Problems 4.5. Match the foll
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90 4.5 Problems 4.8. Radiation patt
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92 5.2 Physics of the Hall Eect The
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94 5.2 Physics of the Hall Eect The
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96 5.3 Magnetohydrodynamics This am
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98 5.5 Applications of Hall Eect De
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100 5.6 Problems 5.6 Problems 5.1.
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102 6.2 The Wave and Particle Natur
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104 6.3 Semiconductors and Energy L
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120 6.4 Crystallography Revisited L
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122 6.5 Pn Junctions E Indirect Sem
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124 6.5 Pn Junctions an excess of n
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126 6.5 Pn Junctions V x - + Energy
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128 6.6 Solar Cells to day and loca
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130 6.6 Solar Cells dot based mater
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132 6.7 Photodetectors Solar Panel
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134 6.7 Photodetectors 6.7.2 Measur
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6 PHOTOVOLTAICS 137 6.3. The gure i
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7 LAMPS, LEDS, AND LASERS 139 7 Lam
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7 LAMPS, LEDS, AND LASERS 141 Photo
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7 LAMPS, LEDS, AND LASERS 143 Absor
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7 LAMPS, LEDS, AND LASERS 145 We ca
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7 LAMPS, LEDS, AND LASERS 147 If we
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7 LAMPS, LEDS, AND LASERS 149 tube
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7 LAMPS, LEDS, AND LASERS 151 easil
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7 LAMPS, LEDS, AND LASERS 153 Power
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7 LAMPS, LEDS, AND LASERS 155 Two L
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7 LAMPS, LEDS, AND LASERS 157 but n
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7 LAMPS, LEDS, AND LASERS 159 of ec
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7 LAMPS, LEDS, AND LASERS 161 the o
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7 LAMPS, LEDS, AND LASERS 163 Ti:Sa
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7 LAMPS, LEDS, AND LASERS 165 the b
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7 LAMPS, LEDS, AND LASERS 167 Gas D
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7 LAMPS, LEDS, AND LASERS 169 categ
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7 LAMPS, LEDS, AND LASERS 171 7.8.
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8 THERMOELECTRICS 173 8 Thermoelect
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8 THERMOELECTRICS 175 Symbol Name V
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8 THERMOELECTRICS 177 per unit volu
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8 THERMOELECTRICS 179 kPa, and the
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8 THERMOELECTRICS 181 Seebeck Effec
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8 THERMOELECTRICS 183 gradient acro
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8 THERMOELECTRICS 185 thermocouples
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8 THERMOELECTRICS 187 The gure of m
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8 THERMOELECTRICS 189 or mass invol
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8 THERMOELECTRICS 191 temperatures?
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8 THERMOELECTRICS 193 Figure 8.4: L
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8 THERMOELECTRICS 195 8.9 Problems
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8 THERMOELECTRICS 197 8.8. A thermo
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8 THERMOELECTRICS 199 Figure 8.5 8.
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202 9.2 Measures of the Ability of
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212 9.3 Charge Flow in Batteries an
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214 9.3 Charge Flow in Batteries an
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216 9.4 Measures of Batteries and F
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224 9.5 Battery Types specic energi
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226 9.5 Battery Types at the anode
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228 9.5 Battery Types Figure 9.8: T
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230 9.6 Fuel Cells Fuel cell compon
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232 9.6 Fuel Cells 9.6.3 Practical
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234 9.7 Problems 9.7 Problems 9.1.
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236 9.7 Problems A E - + I B D H 2
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238 10.3 Radiation Detectors high e
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240 10.5 Resistive Sensors capacito
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242 10.6 Electrouidics However, ene
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Page 257 and 258: 246 11.2 Lagrangian and Hamiltonian
Page 259 and 260: 248 11.3 Principle of Least Action
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Page 269 and 270: 258 11.6 Capacitor Inductor Example
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Page 273 and 274: 262 11.7 Schrödinger's Equation 11
Page 275 and 276: 264 11.8 Problems 11.4. Figure 11.2
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Page 299 and 300: 288 12.6 Problems 12.6 Problems 12.
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Page 303 and 304: 292 13.1 Introduction • Describe
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Page 353 and 354: 342 Appendices Symbol Quantity Unit
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346 Appendices Symbol Quantity Unit
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348 Appendices Symbol Quantity Unit
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350 Appendices Prex name Symbol Val
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352 Appendices for voltage. (As an
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354 Appendices Material or Device S
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356 REFERENCES [15] E. C. Jordan an
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358 REFERENCES [45] V. K. Tikhomiro
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360 REFERENCES [73] M. G. Thomas, H
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362 REFERENCES [100] A. Ishibashi,
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364 REFERENCES [125] D. Wright, Req
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366 REFERENCES [150] J. P. Owejan,
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368 REFERENCES [175] G. K. Woodgate
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Index Absorption, 139 Action, 247 A
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372 INDEX Nernst equation, 220 Neur
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