Fundamentals of THERMOPHOTOVOLTAIC ENERGY CONVERSION
Donald L. Chubb
NASA Glenn Research Center
21000 Brookpark Road, MS 302-1 Cleveland, OH 44135 USA
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Acknowledgements
I began writing this text in 1996 while on a sabbatical leave at the Space Power Institute at Auburn University. M. Frank Rose was director of the Space Power Institute at that time. I thank him for giving me the opportunity to begin writing the text during my stay at Auburn.
I owe special thanks to two people: Barbara Madej for the very tedious job of typing the text. And my wife, Nancy, for proofreading the text and correcting my grammatical errors.
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Preface
Although energy conversion is a technology that is vital for modern life, few textbooks have been written on the fundamentals of the various energy conversion systems. Thermophotovoltaics (TPV) is a simple energy conversion concept well suited for description in a fundamental text. TPV is a static conversion system with no moving parts and only three major parts: an emitter heated by a thermal energy source, and optical cavity for spectral control, and a photovoltaic (PV) array for converting the emitted radiation to electricity.
The book has been written as a text rather than as a review of current TPV research. However, there is some mention of that research.
Most of the material is introduced at the level where physics becomes engineering. For example, radiation transfer theory, which can be considered an engineering discipline, is used throughout the book. The theory is based upon the radiation transfer equation, which originates from basic physics. All the theoretical developments are as self-contained as possible.
The text begins with a chapter that introduces several topics from electromagnetic wave propagation and radiation transfer theory. This is the background material necessary to describe the emitter and optical cavity of a TPV system. Chapter 2 uses a simplified model of a TPV system to illustrate several important properties of the TPV energy conversion concept. In the next three chapters, the theory necessary to describe the performance of emitters, optical filters used for spectral control, and photovoltaic arrays is presented. The final three chapters deal with the performance of the whole TPV system.
Five appendices are included. They provide background material for the main text. Also, Mathematica computer programs for calculating the optical properties of interference filters and for calculating the performance of a planar TPV system are included on a CD-ROM disk. Problem sets are included at the end of each chapter.
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1.6
1.5.1
1.5.2
1.5.3
1.5.4
Table of Contents
1.6.2
1.6.3
2.2
2.3
2.5
2.6
Chapter 3 – Emitter Performances
3.1
3.2
3.3
3.4
3.5
3.6
3.7
3.7.1
3.8
3.8.1
3.8.2
3.8.3
3.8.4
3.9
3.10.4
3.11
3.12
Chapter 4 – Optical Filters for Thermophotovoltaics
4.1 Symbols........................................................................................................179
4.2 Filter Performance Parameters.....................................................................181
4.3 Interference Filters........................................................................................183
4.3.1 Introduction....................................................................................183
4.3.2 Interference....................................................................................183
4.3.3 Interference Filter Model...............................................................185
4.3.4 Reflectance, Transmittance, and Absorptance...............................193
4.3.5 Single Film System........................................................................198
4.3.6 Many Layer System for i = N or i = N /2 and N is an Odd Integer.................................................................210
4.3.7 Equivalent Layer Procedure...........................................................214
4.3.8 Interference Filter with Embedded Metallic Layer........................222
4.3.9 Interference Filter Performance for Angles of Incidence Greater than Zero...........................................................................229
4.4 Plasma Filters...............................................................................................232
4.4.1 Drude Model..................................................................................232
4.4.2 Reflectance, Transmittance, and Absorptance of a Plasma Filter..................................................................................244
4.4.3 Efficiency and Total Transmittance, Reflectance, and Absorptance of a Plasma Filter......................................................249
4.5 Combined Interference-Plasma Filter...........................................................253
4.6 Resonant Array Filters..................................................................................260
4.6.1 Transmission Line Theory.............................................................261
4.6.2 Transmission Line Equivalent Circuit for Resonant Array Filter .....................................................................................266
4.6.3 Metallic Mesh Filter.......................................................................270
4.7 Spectral Control Using a Back Surface Reflector (BSR).............................277
4.7.1 Efficiency of a Back Surface Reflector (BSR) for Spectral Control........................................................................277
4.8 Summary......................................................................................................283
5.1 Symbols........................................................................................................291
5.2 Energy Bands (Kronig-Penney Model) and Current in Semiconductors........................................................................................294
5.3 Density of Electrons and Holes and Mass Action Law................................302
5.4 Transport Equations......................................................................................310
5.5 Generation and Recombination of Electrons and Holes...............................313
5.5.1 Generation of Electrons and Holes.................................................313
5.5.2 Recombination of Electrons and Holes..........................................316
5.6 p-n Junction..................................................................................................320
5.7 Current-Voltage Relation for an Ideal Junction in the Dark.........................323
5.7.1 Assumptions for Ideal p-n Junction................................................324
5.7.2 Current-Voltage Relation for Infinite Neutral Regions..................326
5.7.3 Current-Voltage Relation for Finite Neutral Regions....................330
5.7.4 Depletion Region Contribution to Current and High-Injection Effects.............................................................336
5.8 Ideality Factor and Empirical Current-Voltage Relation of p-n Junction in the Dark...........................................................................338
5.9 Current-Voltage Relation for an Ideal p-n Junction Under Illumination.......................................................................................339
5.9.1
5.9.2
5.9.3
5.9.4 Current-Voltage Relation...............................................................359
5.10 Quantum Efficiency and Spectral Response.................................................362
5.11
6.1
6.2
6.2.1
6.2.2
6.2.3
6.2.4
6.3
6.4
6.5
6.5.1
6.5.2
6.5.3
6.5.4
6.6
7.1
7.2
7.3 Cavity
7.4 Cavity
7.4.1
7.4.2
7.4.3
8.1
8.4
8.5
8.6
8.5.1
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Chapter 1 Introduction
The opening chapter defines the thermophotovoltaic (TPV) energy conversion process, presents a short history of TPV research with possible TPV applications and introduces several concepts from electromagnetic wave propagation and radiation transfer theory. These concepts will be used throughout the text.
1.1 Symbols
A surface area, m2
a absorption coefficient, cm-1
B magnetic induction, Weber/m2
c speed of light in material other than a vacuum, m/sec
co speed of light in a vacuum (2.9979 108 m/sec)
D electric displacement, Coul/m2
E photon energy, J or electric field, V/m
e spectral emissive power, W/m2/ m
eT total emissive power, W/m2
Fo– T fraction of total blackbody intensity or emissive power lying in region 0 – T
H magnetic field, Amp/m
h Plank’s constant (6.6262 10–34 J.sec)
I time average of the Poynting vector for plane waves, W/m2
i radiation intensity, W/m2/ m/steradian
j 1
K extinction coefficient, cm-1
k Boltzmann constant (1.3806 10–23 J/K)
k wave vector, m–1
n index of refraction
Q radiant energy per unit time per unit wavelength, W/ m
Q radiant energy per unit time, W
q radiant energy per unit time per unit wavelength per unit area, W/ m/m2
q radiant energy per unit time per unit area, W/m2
R reflectivity at an interface
r Fresnel reflection coefficient
S source function, W/m2/ m, steradian
T transmissivity at an interface or temperature, K
absorptance
scattering albedo
emittance or electric permittivity, Farad/m ( o = vacuum permittivity = 8.855 10–12 Amp2sec4/kg m3)
complex dielectric constant
reflectance or charge density, coul/m3
conductivity, –1 m –1
s scattering coefficient, cm-1
transmittance or Fresnel transmission coefficient
magnetic permeability, Henry/m ( o = vacuum permeability = 1.2556 10-6 mkg/sec2Amp2)
Wavelength, nm
sb Stefan Boltzmann constant (5.67 10-8 W/m2/K4)
Subscripts
b blackbody
g corresponding to PV cell bandgap energy
R real number
I imaginary number
i incident radiation
o outgoing radiation
Figure 1.1 Thermophotovoltaic (TPV) energy conversion concept.
1.2 Thermphotovoltaic (TPV) Energy Conversion Concept
Similar to all energy conversion concepts, TPV energy conversion is a method for converting thermal energy to electrical energy. The concept is illustrated in Figure 1.1. Thermal energy from any of the thermal sources shown in Figure 1.1 is supplied to an emitter. Radiation from the emitter is directed to photovoltaic (PV) cells where the radiation is converted to electrical energy. In order to make the process efficient, the energy of the photons reaching the PV array must be greater than the bandgap energy of the PV cells. Shaping of the radiation or spectral control is accomplished in the following ways. One method is to use a selective emitter that has large emittance for photon energies above the bandgap energy of the PV cells and small emittance for photon energies less than the bandgap energy. Similar results can be accomplished by using a gray body emitter (constant emittance) and a band pass filter. The bandpass filter should have large transmittance for photon energies above the PV cells bandgap energy and large reflectance for photon energies below the PV cells bandgap energy. A back surface reflector on the PV array can also be used to reflect the low energy photons back to the emitter. It is also possible to combine all three of these methods.
1.3 A Short History of TPV Energy Conversion
Robert E. Nelson, who has done significant early research on rare earth selective emitters for TPV and is familiar with early TPV work, related the following story about the invention of TPV energy conversion. In the mid-1950’s, Henry H. Kolm at
Massachusetts Institute of Technology’s (MIT) Lincoln Laboratory, received a phone call from someone in the defense department who informed Kolm that Russians had developed a method for generating electricity from a lantern flame. He then asked Kolm if he could do the same thing. Kolm’s solution was to place a silicon solar cell, which was in the earliest stage of development at Bell Laboratories, next to the mantel of a Coleman Lantern. He measured the electrical power output and published the result in a Lincoln Laboratory progress report [1] in May 1956. The irony is that the Russian method was thermoelectric, not TPV.
During a technical meeting in Europe in 1960, Kolm informed Pierre Aigrain of France of his experiment. Aigrain, who was a science advisor to Charles de Gaulle, immediately began to work on the concept. Aigrain is generally considered to be the first to have presented the concept in a series of lectures he gave at MIT in 1960. Early papers by D.C. White, B.D. Wedlock and J. Blair [2] and B.D. Wedlock [3] and [4] present the earliest theoretical and experimental TPV research results. The first term used to describe the concept was thermal-photo-voltaic, which then became thermophoto-voltaic, and finally, thermophotovoltaic. General Motors (GM) also had an early interest in TPV.
At GM, TPV work was also begun in 1960 [5] through [7]. In 1963 Werth [6] demonstrated TPV conversion using a propane-heated emitter and a germanium (Ge) PV cell. The emitter temperature was approximately 1700 K. Shockley and Queisser [8] had shown that for a blackbody radiation source, the efficiency of a PV cell was strongly dependent upon the ratio of the cell bandgap energy, Eg, to the radiation source temperature, TE. In fact, Eg/kTE 2.0 to obtain maximum efficiency for a blackbody source. In order to obtain maximum efficiency using silicon (Si), which was the most efficient PV cell available, the emitter temperature would have to be approximately 6000 K.
No workable materials exist at these temperatures. Therefore, for efficient TPV energy conversion at reasonable temperatures, a blackbody source cannot be used. As a result, TPV research has been directed at tailoring the emitter radiation to the PV cell bandgap energy. In other words, by producing radiation within a narrow energy band that lies just above the bandgap energy of the PV cell (where PV cell is most efficient), it is possible to have an efficient system. In addition, just as for the blackbody emitter, there will be an optimum value for Eg/kTE. For reasonable emitter temperatures (TE 2000 K) achieving this optimum condition requires that Eg 1.0 eV. Therefore, low bandgap energy PV cells are required for high efficiency in addition to radiation matched to the PV cell bandgap energy.
The early TPV research of the 1960’s and 1970’s was confined to using either silicon (Eg = 1.12 eV) or germanium (Eg = 0.66 eV) PV cells. However, silicon requires a large emitter temperature (TE 2000 K) for an efficient TPV system, and Ge cells were of low efficiency, although having reasonable bandgap energy for TPV
conversion. Thus, lack of suitable PV cells plus unsuccessful attempts at obtaining radiation matched to the PV cell bandgap energy resulted in a loss of interest in TPV energy conversion. However, in the 1980’s new efficient PV cell materials such as gallium antimonide (GaSb, Eg = 0.72 eV) and indium gallium arsenide (InxGa1-xAs, where 0.36 Eg 1.42 eV depending on value of x), became available. In addition, new selective emitters and filters that can produce the bandgap matched radiation were being developed. As a result, beginning in the late 1980’s, a resurgence of interest in TPV energy conversion occurred.
An extensive bibliography of all TPV research papers has been complied by Lars Browman [9]. Other excellent references for TPV research results are the proceedings of the National Renewable Energy Laboratory (NREL) Conferences on Thermophotovoltaic Generation of Electricity.
1.4 TPV Applications
Besides the development of new PV cell and emitter technology, the many applications for TPV energy conversion are also a driving force in the resurgence of TPV interest. The simplicity and potential high efficiency of TPV conversion are the two attractive features that lead to many potential applications. Since TPV is a direct energy conversion process, the only moving parts in the system are fans or pumps that may be used for cooling the PV cells. The components of the system are the thermal source, the emitter (and possibly a filter), the PV cells, and the waste heat rejection system. Each of these components is in the solid state with only the PV cells and possibly the filter being a somewhat complex solid state device. In addition to simplicity and potential high conversion efficiency, TPV can be easily coupled to any thermal source.
Applications for TPV exist where the thermal source may be solar, nuclear, or combustion. At first it would appear that a solar TPV (STPV) system would have no advantage over a conventional solar PV system. What can be gained by adding an emitter and possibly a filter to the conventional solar PV system? By adding a selective emitter or thermal emitter plus filter, the solar spectrum, which corresponds approximately to a 6000 K gray body emitter, can be shifted to match the bandgap energy of the PV cells. As a result, the STPV system will yield higher efficiency than the conventional solar PV systems. Besides efficiency, an even more important reason for interest in STPV energy conversion is the ability to use thermal energy storage and combustion energy input so that the system can operate when the sun is not available. With the capability of 24-hour-a-day operation, STPV is viable for electric utility use.
A TPV system powered by radioisotope decay (RTPV) is a potential power system for deep space missions where the solar radiation energy density is too low for a conventional PV power system to be used. The first radioisotope decay power systems
used thermoelectric energy converters (RTG). However, TPV has the potential for higher efficiency than thermoelectrics; therefore it is being considered as a replacement.
Combustion driven TPV has many potential commercial applications. For natural gas-fired appliances such as furnaces and hot water tanks, TPV can be added for the cogeneration of electricity. In such applications, attaining high TPV efficiency is not essential since the waste heat for the TPV conversion process is completely utilized. Portable power supplies for both commercial and military use is another important combustion TPV application. An important TPV advantage over existing internal combustion-driven applications is quiet operation. This is especially true in military missions that require the mission to be undetected. Another combustion-driven TPV application with commercial potential is the power supply for hybrid electric vehicles. In such an application the TPV system is sized to provide enough power for operation at cruise speed and battery charging. For acceleration power the batteries are utilized.
One important advantage of all TPV applications is that they are environmentally benign. The TPV system is nearly silent and emits no pollutants. This is obvious in the case of a solar driven system. For a nuclear powered system there are no combustion products; however, care must be taken to insure that no radioactive material is released. Atmospheric pressure burning occurs in the combustion-driven TPV systems. Therefore, the combustion temperature can be well controlled so that the production of toxic nitrous oxides (NOX) is small.
Although there appears to be unlimited potential TPV applications, whether or not they are feasible, will depend upon their cost. At this stage of development, it is impossible to say which applications will be cost effective.
1.5 Propagation of Electromagnetic Waves
Energy transfer between the components of a TPV system is mainly by radiation that consists of propagating electromagnetic waves. The derivation of the equation describing radiation transfer does not require electromagnetic wave propagation theory. However, electromagnetic wave theory is required to obtain equations for the optical properties reflectivity and transmissivity at an interface. Values for these properties are required in analyzing a TPV system. Also, wave propagation must be considered in analyzing the various optical filters that can be used in a TPV system.
1.5.1 Plane Wave Solution to Maxwell’s Equations
Much of the material to be covered in this section can be found in the classic optics text by Born and Wolf [10]. As is well known, Maxwell’s equations describe the propagation of all electromagnetic waves. For TPV applications the electromagnetic waves of interest are mostly in the infrared region ( > 800 nm) of the spectrum. In the International System of Units (SI), Maxwell’s Equations for the electric field, E , and magnetic field, H , are the following [11],
where J is the conduction current, D is the electric displacement, B the magnetic induction and is the electric charge density. In addition to Maxwell’s Equations socalled constitutive relations for the media are required.
Appearing in equation (1.5) is the electric dipole moment per unit volume, P , which for most media depends linearly upon E . Therefore, the second two forms for D result, where o is the vacuum permittivity, is the electric susceptibility and o is the electric permittivity. Similar to D , the magnetic field, B , has two parts: H and the magnetic dipole moment per unit volume, M . For most media M is a linear function of H , which results in the second two forms for B in equation (1.6). Here o is the vacuum permeability, m is the magnetic susceptibility, and om(1) is the magnetic permeability. The properties of the medium ( , , and ) appearing in equations (1.5) to (1.7) are scalar quantities and therefore independent of direction. Thus the media is called isotropic. Also, D , B , and J are linear functions of E and H so the media is called a linear isotropic media. In general, the properties may depend upon direction (anisotropic) so that they are tensor quantities. And E and H can appear as higher order terms in the constitutive relations so that the
Ampere's Law
Law
media may be nonlinear. Finally, for plane electromagnetic waves the properties must be independent of time (stationary media) and space (homogeneous media).
The quantum mechanical theory necessary to determine the medium properties ( , , ) is beyond the scope of this text. However, the discussion of optical filters in Chapter 4 uses a macroscopic model called the Drude model to calculate .
The wave equation for E is obtained by first taking the curl ( ) of equation (1.2) and making use of the vector identity ( A ) = ( • A ) – 2 A , where 2 = •
Then using equations (1.1), (1.3), and the constitutive relations, equations (1.5) to (1.7), the following result is obtained.
So far, the only approximation that has been made is that the media is linear and isotropic. Now assume the media is stationary so that properties are independent of time, t, and that = constant. Therefore, equation (1.8) becomes the following.
Solution of equation (1.9) depends upon the properties ( , , ) and the boundary conditions. There is no indication that wave solutions follow naturally from this equation. No wave vector, k , or frequency, , appears in equation (1.9). Experimentally it has been shown that light has wave-like behavior. Therefore, a wave solution should satisfy equation (1.9) if Maxwell’s equations apply to electromagnetic waves of all frequencies including light. The simplest wave solution is the so-called harmonic plane wave,
or using complex notation,
where E o is a constant and can be complex and equation (1.10a) is the real part of equation (1.10b). Note that jkxt in the exponential of equation (1.10b) yields the same result for equation (1.10a) as jtkx . The spatial and time variations all enter in the exponential term. And k , the wave vector that points in the direction of wave propagation, can be complex but must be independent of x and t. The frequency, , is a real number and also must be independent of x and t. In using complex notation, equation (1.10b), care must be exercised when considering products, since Re[A]Re[B] Re[AB].
If k is real, then E will remain a constant in a plane where the phase of the wave, tkx , remains a constant. Thus, equation (1.10) is called a plane wave. The velocity of this plane is in the direction of the wave propagation ( k direction) and has the magnitude, v , which can be derived as follows.
For constant E ,
and since the phase velocity is in the direction of k ,
the index of refraction, n, is defined as follows,
where co is the speed of light in vacuum (3 108 m/sec) and = 2 co/ is the wavelength in vacuum. As will be shown shortly,
where o is the vacuum permittivity and o is the vacuum permeability. Obviously, if k is complex, n will also be complex. It should be noted that if k and n are complex, then the phase velocity, which is always real, is no longer related to n and k by equation (1.13).
Now consider how k and are related to the media properties for plane waves. This relation, called the dispersion relation, is obtained by substituting equation (1.10b) in equation (1.9). Using the operators = –j k and / t = j that apply for plane waves (problem 1.1) the following result is obtained.
For the vector equation (1.15) to be satisfied E o and k must be either in the same direction or perpendicular to each other. If they are in the same direction, then the physically impossible result = j / is obtained. Therefore, k and E must be perpendicular. This same result for plane waves can be obtained using equations (1.1), (1.5) and (1.7) (problem 1.2). In that case equation (1.15) becomes the following.
linear, homogeneaous, stationary, isotropic medium kj0 (1.16)
Several important conclusions can be drawn from equation (1.16). First, since k and are constants for plane waves, the medium properties must also be constants for the plane wave solution to apply. Therefore, for a linear isotropic medium, plane waves only apply when the medium is also homogeneous and stationary as stated earlier. However, if the medium is linear, stationary, homogeneous but anisotropic it is still possible to have plane wave solutions [12]. The second conclusion drawn from equation (1.16) is that k must be complex for a conductive ( 0) medium. However, if 0 (dielectric), it is still possible for a component of xyz ˆˆˆ kkikjkk to be purely imaginary and equation (1.16) to still be satisfied since kkkk xyz 2222 In fact, that is the situation for total internal reflection at a boundary between two dielectrics. Finally, for the plane wave solution, equation (1.10) to apply E0 and the charge density, , must vanish (problem 1.3). If k has an imaginary part, then equation (1.10b) shows that Ee kx I ~ . Obviously, kx I must be negative to insure that E decays rather than grows exponentially.
If E0 , equation (1.9) becomes the following
homogeneous, stationary, isotropic, medium with no space change
This equation applies when the medium is linear, homogeneous and isotropic, and when no space charge exists. Under the same conditions a similar equation can be obtained for H (problem 1.4). Therefore, plane wave solutions apply for both E and H under the medium conditions just described. However, the condition = 0 is not required to obtain the H wave equation.
If a complex wave vector and index of refraction is defined as follows,
where s is a real unit vector pointing in the direction of wave propagation, then the dispersion relation, equation (1.16), can be solved for the real, kR, and imaginary, kI, parts of k. A negative sign appears before the imaginary parts of k and n to ensure Eand H decay with distance. Substituting equation (1.18) in equation (1.16) and equating real and imaginary parts yields the following.
Obviously, for a dielectric ( = 0), kI = 0 and
and,
where r = / o is the dielectric constant and r = / o is the relative permeability. For a vacuum, ooo vc1/ . The fact that k has no imaginary part if = 0 seems to contradict the discussion proceeding equation (1.17). It was stated that k could have a component that is purely imaginary for a dielectric ( = 0). In that case however, the definition for k given by equation (1.18) does not apply. Equation (1.18) defines the magnitude of k as being complex and the unit vector, s , as being real. However for the case of a dielectric ( = 0), the magnitude of k is real but the unit vector, s , can be complex. As already stated, this is the case for total internal reflection at a boundary between two dielectrics. This will be discussed in detail in Section 1.5.4.
The wave number, k, and therefore the index of refraction, n, are given in terms of the real properties, , , and by equations (1.21) and (1.22). However, referring to the dispersion relation, equation (1.16), it yields,
if a complex dielectric constant is defined as follows,
where the real part of ,
(1.28a) is proportional to the displacement current
and the imaginary part,
is proportional to the conduction current E . Thus for a plane wave in a nonmagnetic material ( r = 1), the electrical properties of the media can be combined in a single parameter; the complex dielectric constant, . This is convenient since the theoretical models for the electrical properties in time-varying fields, such as the Drude model ([13], pg. 225-226), lead to a complex dielectric constant that conforms to equation (1.27).
By defining a complex dielectric constant the governing equation for E or H , equation (1.17), reduces to the standard wave equation.
Using equation (1.28a) and equation (1.28b) in equations (1.21) and (1.22) results in the following relations between k, n, and .
For a nonmagnetic ( r = 1), dielectric I 0 material, equations (1.30) become
Notice that if R 0 , which is the case for many metals at high frequency, RR IR k0k0 and IR RR k0k0 .
When k has an imaginary part then the plane wave will be attenuated in the direction of kI. This can be seen by considering the exponential term in equation (1.10b) (problem 1.5).
So far only the dispersion relation for plane waves and resulting relationships between the wave vector, k , index of refraction, n, and dielectric constant r have been discussed. There are several other plane wave properties that will be used in later chapters.
1.5.2 Energy Flux for Plane Electromagnetic Waves
As already shown in obtaining the disperison relation, k and E are perpendicular. Also, since H satisfies the same wave equation as E , equation (1.17), it is also given by a plane wave solution.
Therefore, from equations (1.4) and (1.6) for plane waves in a homogeneous media,
so that k and H are perpendicular. Also, from equation (1.2)
and since kE,
The quantity, Y, is called the optical admittance of the medium. The three vectors, k , E , and H form an orthogonal system, where EH is in the direction of k (problem 1.6) as shown in Figure 1.2. The magnitude of the electric and magnetic fields varies in the k direction. However, their directions are perpendicular to k . Such a wave is called transverse. If the directions are parallel to k , the wave is called longitudinal
The energy flux associated with plane waves is given by the Poynting vector, S .
As already mentioned, k is in the direction of EH so that it will also point in the direction of energy flow. Since S is a product, the complex representation of E and H cannot be used but Re[ E ] and Re[ H ] must be used in equation (1.36) to calculate Re[ S ]. It is the time average of S , which is defined as the intensity, that is used in calculating the optical properties. It should be pointed out that the intensity, I , defined below is not the same as the radiation intensity, i, that will be used later in radiation transfer theory. The magnitude of I (=I) is the energy flux (W/m2) of the electromagnetic (or radiation) field at a particular frequency, (or wavelength, ). In radiation transfer theory the intensity, i, is the energy flux (W/m2) per solid angle, , per frequency, , (or wavelength, )
SEH (1.36)
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not be unacquainted with that excellent Poem of Wordsworth's, —“The Excursion, being a Portion of the Recluse.”—Ifany knownot thewisdomcontainedinit,forthwithletthemstudyit!—Acquainted with it or not, it is Betty Yewdale that is described in the following lines, as holding the lanthorn to guide the steps of old Jonathan, her husband, on his return from working in the quarries, if at any time he chanced to be beyond his usual hour. They are given at length;— for who will not be pleased to read them deciesrepetita?
Much was I pleased, the grey-haired wanderer said, When to those shining fields our notice first You turned; and yet more pleased have from your lips, Gathered this fair report of them who dwell In that Retirement; whither, by such course Of evil hap and good as oft awaits A lone wayfaring Man, I once was brought. Dark on my road the autumnal evening fell While I was traversing yon mountain pass, And night succeeded with unusual gloom; So that my feet and hands at length became Guides better than mine eyes—until a light High in the gloom appeared, too high, methought, For human habitation, but I longed To reach it destitute of other hope. I looked with steadiness as sailors look, On the north-star, or watch-tower's distant lamp, And saw the light—now fixed—and shifting now Not like a dancing meteor; but in line Of never varying motion, to and fro. It is no night fire of the naked hills, Thought I, some friendly covert must be near. With this persuasion thitherward my steps I turn, and reach at last the guiding light; Joy to myself! but to the heart of Her Who there was standing on the open hill, (The same kind Matron whom your tongue hath praised) Alarm and disappointment! The alarm Ceased, when she learned through what mishap I came, And by what help had gained those distant fields. Drawn from her Cottage, on that open height, Bearing a lantern in her hand she stood Or paced the ground,—to guide her husband home, By that unwearied signal, kenned afar; An anxious duty! which the lofty Site Traversed but by a few irregular paths,
Imposes, whensoe'er untoward chance
Detains him after his accustomed hour
When night lies black upon the hills. ‘But come, Come,’ said the Matron,—‘to our poor abode; Those dark rocks hide it!’ Entering, I beheld
[Miss Sarah Hutchinson, Mrs. Wordsworth's sister, and Mrs. Warter took down the story from the old woman's lips and Southey laid it by for the Doctor, &c. She then lived in a cottage at Rydal, where I afterwards saw her. Of the old man it was told me —(for I did not see him)——“He is a perfect picture,—like those we meet with in the better copies of Saints in our old Prayer Books.”
A blazing fire—beside a cleanly hearth
Sate down; and to her office, with leave asked, The Dame returned.—Or ere that glowing pile Of mountain turf required the builder's hand Its wasted splendour to repair, the door
There was another comical History intended for an Interchapter to the Doctor, &c. of a runaway match to Gretna Green by two people in humble life,—but it was not handed over to me with the MS. materials. It was taken down from the mouth of the old woman who was one of the parties—and it would probably date back some sixty or seventy years.]
Opened, and she re-entered with glad looks, Her Helpmate following. Hospitable fare, Frank conversation, make the evening's treat: Need a bewildered Traveller wish for more?
But more was given; I studied as we sate By the bright fire, the good Man's face—composed Of features elegant; an open brow Of undisturbed humanity; a cheek
Suffused with something of a feminine hue; Eyes beaming courtesy and mild regard; But in the quicker turns of his discourse, Expression slowly varying, that evinced A tardy apprehension. From a fount
CHAPTER CCIX.
EARLY APPROXIMATION TO THE DOCTOR'S THEORY.—GEORGE FOX. ZACHARIAH BEN MOHAMMED.—COWPER. INSTITUTES OF MENU.—BARDIC PHILOSOPHY. MILTON. SIR THOMAS BROWNE.
Lost, thought I, in the obscurities of time, But honour'd once, those features and that mien May have descended, though I see them here, In such a Man, so gentle and subdued, Withal so graceful in his gentleness, A race illustrious for heroic deeds, Humbled, but not degraded, may expire. This pleasing fancy (cherished and upheld By sundry recollections of such fall
There are distinct degrees of Being as there are degrees of Sound; and the whole world is but as it were a greater Gamut, or scale of music.
NORRIS.
Certain theologians, and certain theosophists, as men who fancy themselves inspired sometimes affect to be called, had approached
From high to low, ascent from low to high, As books record, and even the careless mind Cannot but notice among men and things,) Went with me to the place of my repose.
BOOK V. THE PASTOR.
so nearly to the Doctor's hypothesis of progressive life, and propensities continued in the ascending scale, that he appealed to them as authorities for its support. They saw the truth, he said, as far as they went; but it was only to a certain point: a step farther and the beautiful theory would have opened upon them. “How can we choose, said one, but remember the mercy of God in this our land in this particular, that no ravenous dangerous beasts do range in our nation, if men themselves would not be wolves and bears and lions one to another!” And why are they so, observed the Doctor commenting upon the words of the old Divine; why are they so, but because they have actually been lions and bears and wolves? why are they so, but because, as the wise heathen speaks, more truly than he was conscious of speaking, subhominumeffigielatetferinus animus. The temper is congenital, the propensity innate; it is bred in the bone; and what Theologians call the old Adam, or the old Man, should physiologically, and perhaps therefore preferably, be called the old Beast.
That wise and good man William Jones of Nayland has in his sermon upon the nature and œconomy of Beasts and Cattle, a passage which in elucidating a remarkable part of the Law of Moses, may serve also as a glose or commentary upon the Doctor's theory.
“The Law of Moses, in the xith chapter of Leviticus, divides the brute creation into two grand parties, from the fashion of their feet, and their manner of feeding, that is, from the parting of the hoof, and the chewing of the cud; which properties are indications of their general characters, as wildor tame. For the dividing of the hoof and the chewing of the cud are peculiar to those cattle which are serviceable to man's life, as sheep, oxen, goats, deer, and their several kinds. These are shod by the Creator for a peaceable and inoffensive progress through life; as the Scripture exhorts us to be shod in like manner with the preparation of the Gospel of Peace. They live temperately upon herbage, the diet of students and saints; and after the taking of their food, chew it deliberately over again for better digestion; in which act they have all the appearance a brute
can assume of pensiveness or meditation; which is, metaphorically, called rumination,1 with reference to this property of certain animals.
1 Pallentes ruminatherbas.
VIRGIL. Dum jacet, et lentè revocatas ruminatherbas.
OVID.
It were hardly necessary to recal to an English reader's recollection the words of Brutus to Cassius,
Till then, my noble friend, chewupon this,—
JULIUS CÆSAR
.
or those of Agrippa in Antony and Cleopatra,
Pardon what I have spoke; For 'tis a studied, not a present thought, By duty ruminated.
“Such are these: but when we compare the beasts of the field and the forest, they, instead of the harmless hoof, have feet which are swift to shed blood, (Rom. iii. 15.) sharp claws to seize upon their prey, and teeth to devour it; such as lions, tygers, leopards, wolves, foxes, and smaller vermin.
“Where one of the Mosaic marks is found, and the other is wanting, such creatures are of a middle character between the wild and the tame; as the swine, the hare, and some others. Those that part the hoof afford us wholesome nourishment; those that are shod with any kind of hoof may be made useful to man; as the camel, the horse, the ass, the mule; all of which are fit to travel and carry burdens. But when the foot is divided into many parts, and armed with claws, there is but small hope of the manners; such creatures being in general either murderers, or hunters, or thieves; the
malefactors and felons of the brute creation: though among the wild there are all the possible gradations of ferocity and evil temper.
“Who can review the creatures of God, as they arrange themselves under the two great denominations of wild and tame, without wondering at their different dispositions and ways of life! sheep and oxen lead a sociable as well as a peaceful life; they are formed into flocks and herds; and as they live honestly they walk openly in the day. The time of darkness is to them, as to the virtuous and sober amongst men, a time of rest. But the beast of prey goeth about in solitude; the time of darkness is to him the time of action; then he visits the folds of sheep, and stalls of oxen, thirsting for their blood; as the thief and the murderer visits the habitations of men, for an opportunity of robbing, and destroying, under the concealment of the night. When the sun ariseth the beast of prey retires to the covert of the forest; and while the cattle are spreading themselves over a thousand hills in search of pasture, the tyrant of the desert is laying himself down in his den, to sleep off the fumes of his bloody meal. The ways of men are not less different than the ways of beasts; and here we may see them represented as on a glass; for, as the quietness of the pasture, in which the cattle spend their day, is to the howlings of a wilderness at night, such is the virtuous life of honest labour to the life of the thief, the oppressor, the murderer, and the midnight gamester, who live upon the losses and sufferings of other men.”
But how would the Doctor have delighted in the first Lesson of that excellent man's Book of Nature,—a book more likely to be useful than any other that has yet been written with the same good intent.
THE BEASTS.
“The ass hath very long ears, and yet he hath no sense of music, but brayeth with a frightful noise. He is obstinate and unruly, and will go his own way, even though he is severely beaten. The child,
who will not be taught, is but little better; he has no delight in learning, but talketh of his own folly, and disturbeth others with his noise.
“The dog barketh all the night long, and thinks it no trouble to rob honest people of their rest.
“The fox is a cunning thief, and men, when they do not fear God, are crafty and deceitful. The wolf is cruel and blood-thirsty. As he devoureth the lamb, so do bad men oppress and tear the innocent and helpless.
“The adder is a poisonous snake, and hatha forked double tongue; and so men speak lies, and utter slanders against their neighbours, when thepoisonofaspsisundertheirlips. The devil, who deceiveth with lies, and would destroy all mankind, is the old serpent, who brought death into the world by the venom of his bite. He would kill me, and all the children that are born, if God would let him; but Jesus Christ came to save us from his power, and to destroy the worksoftheDevil.
“Lord thou hast made me a man for thy service: O let me not dishonour thy work, by turning myself into the likeness of some evil beast: let me not be as the fox, who is a thief and a robber: let me never be cruel, as a wolf, to any of thy creatures; especially to my dear fellow-creatures, and my dearer fellow Christians; but let me be harmless as the lamb; quiet and submissive as the sheep; that so I may be fit to live, and be fed on thy pasture, under the good shepherd, Jesus Christ. It is far better to be the poorest of his flock, than to be proud and cruel, as the lion or the tiger, who go about seeking what they may devour.”
THE QUESTIONS.
“Q. What is the child that will not learn?
A. An ass, which is ignorant and unruly.
Q. What are wicked men, who hurt and cheat others?
A. They are wolves and foxes, and bloodthirsty lions.
Q. What are ill-natured people, who trouble their neighbours and rail at them?
A. They are dogs, who bark at every body.
Q. But what are good and peaceable people?
A. They are harmless sheep; and little children, under the grace of God, are innocent lambs.
Q. But what are liars?
A. They are snakes and vipers, with double tongues and poison under their lips.
Q. Who is the good shepherd?
A. Jesus Christ.”
There is a passage not less apposite in Donne's Epistle to Sir Edward, afterwards Lord Herbert of Cherbury.
Man is a lump where all beasts kneaded be; Wisdom makes him an Ark where all agree. The fool in whom these beasts do live at jar, Is sport to others and a theatre; Nor 'scapes he so, but is himself their prey, All that was man in him is ate away; And now his beasts on one another feed, Yet couple in anger and new monsters breed. How happy he which hath due place assign'd To his beasts, and disaforested his mind, Empaled himself to keep them out, not in; Can sow and dares trust corn where they have been, Can use his horse, goat, wolf and every beast, And is not ass himself to all the rest.
To this purport the Patriarch of the Quakers writes where he saith “now some men have the nature of Swine, wallowing in the mire: and some men have the nature of Dogs, to bite both the sheep and one another: and some men have the nature of Lions, to tear, devour and destroy: and some men have the nature of Wolves, to tear and devour the lambs and sheep of Christ: and some men have the nature of the Serpent (that old destroyer) to sting, envenom and poison. He that hath an ear to hear , let him hear, and learn these things within himself. And some men have the natures of other beasts and creatures, minding nothing but earthly and visible things, and feeding without the fear of God. Some men have the nature of an Horse, to prance and vapour in their strength, and to be swift in doing evil. And some men have the nature of tall sturdy Oaks, to flourish and spread in wisdom and strength, who are strong in evil, which must perish and come to the fire. Thus the Evil is but one in all, but worketh many ways; and whatsoever a Man's or Woman's nature is addicted to that is outward, the Evil one will fit him with that, and will please his nature and appetite, to keep his mind in his inventions, and in the creatures from the Creator.”
To this purport the so-called Clemens writes in the Apostolical Constitutions when he complains that the flock of Christ was devoured by Demons and wicked men, or rather not men but wild beasts in the shape of men,
by Heathens, Jews and godless heretics.
With equal triumph too did he read a passage in one of the numbers of the Connoisseur, which made him wonder that the writer from whom it proceeded in levity should not have been led on by it to the clear perception of a great truth. “The affinity,” says that writer, who is now known to have been no less a person than the author of the Task, “the affinity between chatterers and monkeys, and praters and parrots, is too obvious not to occur at once. Grunters and growlers may be justly compared to hogs. Snarlers are curs that continually shew their teeth, but never bite; and the spitfire passionate are a sort of wild cats, that will not bear stroking, but will purr when they are pleased. Complainers are screech-owls; and story-tellers always repeating the same dull note are cuckoos. Poets that prick up their ears at their own hideous braying are no better than asses; critics in general are venomous serpents that delight in hissing; and some of them, who have got by heart a few technical terms without knowing their meaning, are no better than magpies.”
So too the polyonomous Arabian philosopher Zechariah Ben Mohammed Ben Mahmud Al Camuni Al Cazvini. “Man,” he says, “partakes of the nature of vegetables, because like them he grows and is nourished; he stands in this farther relation to the irrational animals, that he feels and moves; by his intellectual faculties he resembles the higher orders of intelligences, and he partakes more or less of these various classes, as his inclination leads him. If his sole wish be to satisfy the wants of existence, then he is content to vegetate. If he partakes more of the animal than the vegetable nature, we find him fierce as the lion, greedy as the bull, impure as the hog, cruel as the leopard, or cunning as the fox; and if as is
