2 Radiation Physics
Concepts, quantities and units of expression
Abstract
This chapter discusses radiation-physics and optics from the point of view of biological systems and their scientific study and practical management. It provides the foundation for subsequent chapters in the book.
First edition authors
Andreas Albert, Lars Olof Björn, Lasse Yliantilla
I have retained the use of quantity symbols and names from the first edition. The SI system was updated in 2019 and the IUPAC Gold Book in 2020. I/we have to check that the symbols and nomenclature we use are up-to-date with these updates.
2.1 Introduction
Most if not all organisms are in a way or another directly dependent on light (Björn 2015). For humans and most animals, vision is necessary for navigation and communication. Body colour can both enhance visibility and serve as camouflage. For photoautotrophs like plants, light is a source of energy for growth when captured through photosynthesis, but also when sensed through photoreceptors, a source of information. Flower colours and visual patterns are a means of light-mediated communication (Figure 2.1).
As a source of energy, short wavelength radiation, if in excess, can damage unprotected tissues through uncontrolled photochemical reactions, i.e., by UV-B exposure breaking bonds in molecules (e.g., Rousseaux et al. 1999). However, these short wavelengths together with longer wavelengths make organisms and objects warmer when they are absorbed. Thus, radiation-driven overheating can also damage tissues and cause the death of organisms, including plants (e.g., xxxxx?).
Depending on the intensity and timing, radiation of any wavelength from UV-B to thermal infrared can be damaging or beneficial to organisms, including plants. For a given exposure to radiation, the optical properties of vegetation, plants, individual organs such as leaves, tissues, cells and even organelles modulate the fate of the incident energy: radiation that is not absorbed or that is safely dissipated after absorption does not cause damage. White reflective waxes, pubescence, paraheliotropy, and chloroplast accumulation decrease radiation absorption (Figure 2.2), while screening pigments in the epidermis, transpiration, and small-sized leaves contribute to safe dissipation of energy. For radiation that reaches into chloroplasts, biochemical mechanisms for the dissipation have evolved, such as carotenoid cycling.
Sensing of light through photoreceptors as well as its conversion into chemical energy by photosynthesis, start at a photochemical reaction where one (or occasionally two or more) photons transfer energy to a pigment molecule to excite it. Quantum physics helps explain phenomena observable at larger scales, such as action spectra for whole-plant responses.
In addition to plant’s optical properties, in photobiology research, optics and radiation physics play a crucial role in the design of experiments and reporting of treatment conditions. For example, when studying photosynthesis using chlorophyll fluorescence, the correct interpretation of the measured yield must be based on the absorbed photons, not the incident ones. Radiation needs to be quantified in a way relevant to the object of study. The optical properties of experimental set-ups, need to be adjusted to the aims of each experiment and, crucially, have to be accurately described. This makes it necessary for those doing research in photobiology to have a good command of how UV, visible and infrared radiation interact with different objects and organisms. The aim of this chapter is to provide such a foundation as support for later chapters in the book.
2.2 Wavelength and Colour
Radiation, or radiant energy, can be described as waves, with wavelength or frequency as properties, or based on quantum physics, as photons of specific energy. This “double personality” of light is important as each approach provides a simpler explanation than the other one for different specific phenomena, such as refraction and reflection at boundaries between materials and absorption of radiation by pigments and other molecules. From the first perspective, ultraviolet (UV), visible (VIS), and infrared (IR) radiation are electromagnetic waves and are described by the Maxwell’s equations.1 From the second perspective, Plank’s law describes the energy of a quantum as inversely proportional to wavelength.
The wavelength ranges of visible radiation, described by their colour based on human vision’s response to monochromatic light, are shown in Figure 2.3, topmost bar of wavebands. The colour ranges indicated in Figure 2.3 are one approximation out of several available. We follow the ISO and CIE standards when applicable. The electromagnetic spectrum is continuous with no clear boundaries between one colour and the next. Colours are the result of the interaction of different wavelengths of radiation with the photoreceptor pigments in the eye and the processing of the stimuli by the brain. Alternative definitions based on other sensory systems or energy capture systems are available, some of them applicable to plants are shown in Figure 2.3, two lowermost bars of wavebands.
Different regions or wavebands of UV and IR radiation are also shown in Figure 2.3, but only in the proximity of visible radiation. In Figure 2.4 they are shown for a wider range of wavelengths. It is important to keep in mind that the ISO and CIE definitions of the VIS and UV wavebands overlap. Thus, it is incorrect to sum VIS and UV irradiances, while the definitions of photosynthetically active radiation (PAR) and UV do not overlap. Especially in the IR region, the subdivision is somewhat arbitrary and the boundaries used in the literature vary, with differences even between ISO and CIE standards.
The energy of a quantum of radiation in a vacuum, depends on the wavelength, \(\lambda\), or frequency2, \(\nu\),
\[q = h \cdot \nu = h \cdot \frac{c}{\lambda}\]
with the Planck constant \(h=6.626\times 10^{-34}\,\mathrm{J\,s}\) and speed of light in vacuum \(c=2.998\times 10^{8}\,\mathrm{m}\,\mathrm{s}^{-1}\). When dealing with numbers of photons, the equation above can be extended by using Avogadro’s number \(N_\mathrm{A}=6.022\times 10^{23}\) mol\(^{-1}\). Thus, the energy of one mole of photons, \(q^\prime\), is
\[q^\prime = h^\prime \cdot \nu = h^\prime \cdot \frac{c}{\lambda}\] with \(h'=h\cdot N_\mathrm{A}=3.990\times 10^{-10}\,\mathrm{J\,s\,mol}^{-1}\).
The two equations above, are valid for all kinds of electromagnetic waves.
examples
Example 1: red light at 600 nm has about 200 kJ mol\(^{-1}\), therefore, 1 \(\mu\)mol photons has 0.2 J. Example 2: radiation at 300 nm has about 400 kJ mol\(^{-1}\), therefore, 1 \(\mu\)mol photons has 0.4 J.
2.3 Angle of Incidence
Most frequently we describe the flux of radiation on a plane, usually, but not always a horizontal plane (Figure 2.5). This flux is called irradiance and expressed per unit area and unit time. The area used as a reference is that of the receiving plane. Fluence rate is measured on the surface of a sphere instead of a plane. In this case, the plane is located in three-dimensional (3D) space. When considering a point source of radiation, the angle, \(\alpha\), between the light beam and the receiving plane or surface affects the received irradiance: the shallower the angle of incidence the lower the irradiance from the same beam of light. Both irradiance and fluence rate are fluxes of radiation, expressed per unit area and unit time (Figure 2.5). In the first case the area is that of the illuminated planar surface and in the second case of an spherical surface.
Sensors and their shape
There are, in principle, two possible approaches to measuring radiation. The first is to observe light from one specific direction or viewing angle, which is the radiance \(L\). The second is to use a detector, which senses radiation from more than one direction and measures radiation impinging it from all directions from an enclosing sphere or hemisphere. The relation between irradiance \(E\) and radiance \(L\) at wavelength \(\lambda\) is given by integrating incoming photons over these directions.
\[\begin{aligned} E[0](\lambda) & = \int_\Omega L(\lambda,\Omega) {\rm d}\Omega \label{equ_E0} \\ E(\lambda) & = \int_\Omega I(\lambda,\Omega) |\cos\alpha| {\rm d}\Omega \label{equ_E} %\end{eqnarray} \end{aligned}\]
The shape of a detector entrance-optics, planar or spherical, determines the measured physical quantity, and the weight given to photons from different directions (Figure 2.5). Collection of light on a flat surface (or equivalently through an small opening in an integrating sphere) yields irradiance, energy irradiance, \(E\), or photon irradiance, \(Q\) depending on the spectral response of the detector. A spherical collecting surface, yields fluence rate (also called scalar irradiance) \(E_0\), \(Q_0\). In practice, all spherical sensors have a blind stop due to necesary support and connection to a readout device.
The sun, can be assumed to be a point source (the solar disk is \(\approx 1 ^\circ\) in diameter when seen from Earth). Changes in distance that are of interest when studying plants are extremely small relative to the total Sun to Earth distance and can be ignored. Thus, in the case of direct sunlight, we have to consider only the angle of incidence. For a given light source the maximum irradiance is that received on a plane normal to the light beam (Figure 2.5). As the angle between the beam and the surface becomes shallower, the irradiance measured on the plane decreases, reaching zero, when the beam is parallel to the surface. The relationship between the normal area and that at a shallower angle is given by the cosine of the angle \(\alpha\). When considering the sun, if the receiving plane is horizontal, the angle of interest is the zenith angle (1 - solar elevation angle) (Figure 2.6). If the plane is not horizontal, the angle between the plane and the light beam depends both the the zenith angle and the azimuth angle of the sun’s position, and the effective angle needs to be computed in three dimensions.
Obviously, the discussion above ignored scattered radiation from the sky. One approach to the simulation of the light field when multiple sources or sources with a large emitting area are present, is to use ray tracing, which is essentially a divide and conquer approach.
When we considered a point source like a bare LED chip or light from a small light bulb at a relatively short distance from the receiving plane, distance cannot be ignored as done above for the sun, as the illuminated area increases four times for each doubling in the distance. Thus, each doubling in distance decreases irradiance by 75%.
In the case of light sources with a large surface relative to the illuminated area or light beams focused by a lens or reflector, the rate of increase in area is smaller than for a point source. The extreme case is a highly collimated laser beam, for which the illuminated area increases very slowly with increasing distance. The irradiance, in these cases, is frequently not uniform within the illuminated area, with in most cases higher iradiance at the centre than at the edges of the illuminated area.
Tip
Using a lens or reflector to make a light beam narrower makes it possible to increase irradiance by concentrating the light energy onto a smaller illuminated area, and this effect can be quantitatively large. Alternatively, the narrower beam makes it possible to move the light source away from the illuminated object maintaining the same irradiance. This can provide additional space for other equipment, or when using selective reflectors that transmit infrared radiation and only reflect VIS and/or UV radiation reduce the heat load on the illuminated object.
In a perfectly collimated beam, the photons follow parallel trajectories. In a beam from a ideal point source the trajectories of the photons are equally probable in all directions (away from the centre of an imaginary sphere). Most real light sources behave somewhere in between these extremes.
Angle of Incidence in 3D
While in Figure 2.6, we had the sensor or observer at the centre, and the light source located on an imaginary sphere, here, instead, we locate the point light source at the centre of the sphere. So, when a beam or the radiation passing into a space or sphere is analysed, two important parameters are necessary: the distance to the source and the position of the measurement plane—i.e., if the receiving surface is perpendicular to the beam or at a shallower angle. The geometry is illustrated in Figure 2.7. The radiation is received at distance \(r\) by a surface of area d\(A\), tilted by an angle \(\alpha\) to the unit sphere’s surface element, a solid angle, d\(\Omega\), which is a two-dimensional angle in a space. The relation between d\(A\) and d\(\Omega\) in spherical coordinates is geometrically explained in Figure 2.7.
The solid angle is calculated from the zenith angle \(\theta\) and azimuth angle \(\phi\), which denote the direction of the radiation beam
\[{\rm d}\Omega = {\rm d}\theta\cdot\sin\theta{\rm d}\phi \label{equ_dsolidangle}\] The area of the receiving surface is calculated by a combination of the solid angle of the beam, the distance \(r\) from the radiation source and the angle \(\alpha\) of the tilt:
\[{\rm d}A = \frac{r{\rm d}\theta}{\cos\alpha}\cdot r\sin\theta{\rm d}\phi\] which can be rearranged to
\[\Rightarrow {\rm d}A = \frac{r^2}{\cos\alpha}\quad{\rm d}\Omega\] Thus, the solid angle is given by
\[\Omega = \int_A \frac{{\rm d}A\cdot\cos\alpha}{r^2} \label{equ_solidangle}\] The unit of the solid angle is a steradian (sr). The solid angle of an entire sphere is calculated by integration of the equation above over the zenith (\(\theta\)) and azimuth (\(\phi\)) angles, \(0\le\theta\le\pi(180^\circ)\) and \(0\le\phi\le2\pi(360^\circ)\), and is \(4\pi\) sr.
2.3.1 Radiation Quantities
The amount of radiation can be quantified in different ways. Each quantity is useful in different circumstances. We have already used some of these quantities above, here we Quantities can describe flows (energy per unit time), flux rates (energy per unit time and unit area), exposure (energy per event and unit area), radiant flux (energy per unit time) and radiance (energy per unit area and solid angle). The shape of the receiving surface can be a plane, a sphere, a hemisphere or even a cylinder. In the case of radiance, the angle of acceptance of the sensor can also vary but is usually narrow (\(\leq 10^\circ\)). These definitions are independent of wavelengths, but the quantities are in most cases used to describe energy integrated over a certain range of wavelengths based on the “named” wavebands shown in Figure 2.3 and Figure 2.4. The definitions and symbols of several physical quantities used to describe the “amount of radiation” are given in Table 2.1.
| Symbol | Unit | Description |
|---|---|---|
| \(\Phi=\frac{\partial e}{\partial t}\) | W = J s\(^{-1}\) | Radiant flux: absorbed or emitted energy per time interval |
| \(H=\frac{\partial e}{\partial A}\) | J m\(^{-2}\) | Exposure: energy towards a surface area. (In plant research this is called usually dose (), while in Physics dose refers to absorbed radiation.) |
| \(E=\frac{\partial \Phi}{\partial A}\) | W m\(^{-2}\) | Irradiance: flux or radiation towards a surface area, radiant flux density |
| \(I=\frac{\partial \Phi}{\partial\Omega}\) | W sr\(^{-1}\) | Radiant intensity: emitted radiant flux of a surface area per solid angle |
| \(\epsilon=\frac{\partial \Phi}{\partial A}\) | W m\(^{-2}\) | Emittance: emitted radiant flux per surface area |
| \(L=\frac{\partial^2 \Phi}{\partial \Omega (\partial A\cdot \cos\alpha)}=\frac{\partial I}{\partial A\cdot\cos\alpha}\) | W m\(^{-2}\) sr\(^{-1}\) | Radiance: emitted radiant flux per solid angle and surface area depending on the angle between radiant flux and surface perpendicular |
Taking into account Figure 2.7, the corresponding equations and assuming a homogenous flux, the relation between irradiance \(E\) and intensity \(I\) is given by \[E = \frac{I \times \cos\alpha}{r^2} \label{equ_r2law}\]. The irradiance decreases by the square of the distance to the source and depends on the tilt of the detecting surface area, \(\alpha\). This is valid only for theretical point sources, and a good approximation for small ones. For outdoor measurements the sun can be assumed to be a point source. For artificial light sources simple LEDs (light-emitting diodes) without optics on top are also effectively point sources. However, LEDs with optics—and other artificial light sources with optics or reflectors designed to give a narrower light beam—deviate to various extents from the rule of a decrease of irradiance proportional to the square of the distance from the light source.
When we are interested in photochemical reactions, the most relevant radiation quantities are those expressed in photons. The reason for this is that, as discussed in ?sec-radiation-photochemistry-principles, molecules are excited by the absorption of certain fixed amounts of energy or quanta. The surplus energy “decays” by non-photochemical processes. When studying photosynthesis, where many photons of different wavelengths are simultaneously important, we normally use photon irradiance to describe amount of . The name photosynthetic photon flux density, or , is also frequently used when referring to photon irradiance. When dealing with energy balance of an object instead of photochemistry, we use (energy) irradiance. In meteorology both and visible radiation, are quantified using energy-based quantities. When dealing with photochemistry as in responses mediated by UVR8, a photoreceptor, the use of quantum quantities is preferred. Photon based radiation quantities are listed in Table 2.2, the symbol \(Q\) (“quanta”) is used for photon irradiance.
| Symbol | Unit | Description |
|---|---|---|
| \(\Phi_\mathrm{p}\) | s\(^{-1}\) | Photon flux: number of photons per time interval |
| \(Q=\frac{\partial \Phi_\mathrm{p}}{\partial A}\) | m\(^{-2}\) s\(^{-1}\) | Photon irradiance: photon flux towards a surface area, photon flux density (earlier frequently called photon flux density and abbreviated as PFD) |
| \(H_p=\int_t\;Q\;\mathrm{d}t\) | m\(^{-2}\) | Photon exposure: number of photons towards a surface area during a time interval, photon fluence |
These quantities can be also used based on a ‘chemical’ amount of moles by dividing the quantities by Avogadro’s number \(N_A=6.022\times 10^{23}\) mol\(^{-1}\). To determine a quantity in terms of photons, an energetic quantity has to be weighted by the number of photons, i.e., divided by the energy of a single photon at each wavelength as defined in equation XXXX. This yields, for example, \[\Phi_\mathrm{p} = \frac{\lambda}{h\;c}\times\frac{\partial q}{\partial t}\hspace{1cm}\mathrm{and}\hspace{1cm}Q(\lambda) = \frac{\lambda}{h\;c}\times E(\lambda)\] When dealing with bands of wavelengths, for example an integrated value like from 400 to 700 nm, it is necessary to repeat these calculations at each wavelength and then integrate over the wavelengths. For example, the photon irradiance or in moles of photons is obtained by \[Q_\mathrm{PAR} = \frac{1}{N_\mathrm{A}}\int_{400\;\mathrm{nm}}^{700\,\mathrm{nm}}\frac{\lambda}{hc}\;E(\lambda)\;\mathrm{d}\lambda\] For integrated values of or radiation the calculation is done analogously by integrating from 280 to 315 nm or 315 to 400 nm, respectively.
If we have measured (energy) irradiance, and want to convert this value to photon irradiance, the exact conversion will be possible only if we have information about the spectral composition of the measured radiation. Conversion factors at different wavelengths are given below in Table 2.3. For “average daylight” in the wavelength band 400 to 700 nm the conversion factor is approximately \(4.6\,\mathrm{\mu mol\,J^{-1}}\). This is exact at a wavelength at 550 nm, and will deviate depending on the shape of the spectrum.
| \(\lambda\) (nm) | \(q\) \(\mathrm{(eV)}\) | \(k, I_\lambda \to Q_\lambda\) \((\mathrm{\mu mol\,J^{-1}})\) | \(k, Q_\lambda \to I_\lambda\) \((\mathrm{J\,\mu mol^{-1}})\) |
|---|---|---|---|
| 250 | 4.96 | 2.09 | 0.479 |
| 300 | 4.13 | 2.51 | 0.399 |
| 350 | 3.54 | 2.93 | 0.342 |
| 400 | 3.10 | 3.34 | 0.299 |
| 450 | 2.76 | 3.76 | 0.266 |
| 500 | 2.48 | 4.18 | 0.239 |
| 550 | 2.25 | 4.60 | 0.218 |
| 600 | 2.07 | 5.02 | 0.199 |
| 650 | 1.91 | 5.43 | 0.184 |
| 700 | 1.77 | 5.85 | 0.171 |
| 750 | 1.65 | 6.27 | 0.160 |
| 800 | 1.55 | 6.69 | 0.150 |
| 850 | 1.46 | 7.11 | 0.141 |
| 900 | 1.38 | 7.52 | 0.133 |
2.3.2 Effective radiation quantities
Besides the physical quantities used for all electromagnetic radiation, there are also equivalent spectrally weighted quantities used to assess the effect of radiation on sensory systems, photochemical reactions and photobiological responses.
The most widely used biologically effective quantities are photometric quantities, describing the apparent brightness of light to the human eye. Although quantitatively irrelevant to systems other that human vision, they are commonly used by lamp manufacturers to describe light output of artificial light sources. Photometric sensors are also the most widely available. Photometric or illumination quantities are expressed in different units than those of physical quantities.
Photometric quantities
In contrast to (spectro-)radiometry, where the energy of any electromagnetic radiation is measured in terms of absolute power (\(J\,s^{-1} = W\)), photometry measures light as perceived by the human eye. Therefore, radiation is weighted by a luminosity function or visual sensitivity function describing the wavelength dependent response of the human eye. Due to the physiology of the eye, having rods and cones as light receptors, different sensitivity functions exist for the day (photopic vision) and night (scotopic vision), \(V(\lambda)\) and \(V'(\lambda)\), respectively (Figure 2.8). The maximum response during the day is at \(\lambda=555\) nm and during night at \(\lambda=507\) nm. Both response functions (normalised to their maximum) are shown in the figure below as established by the Commission Internationale de l’Éclairage (CIE, International Commission on Illumination, Vienna, Austria) in 1924 for photopic vision and 1951 for scotopic vision (Schwiegerling 2004). The data are available from the Colour and Vision Research Laboratory. The photopic response, \(V(\lambda)\), is used as the basis of all photometric measurements. However, for very low irradiance, it is possible to compute similar quantities using the scotopic response \(V'(\lambda)\).
Note: It is important to be aware that these response curves are based on a small number of subjects and, due to variation, not necessarily representative of any individual person, or even of persons of different age or of other ethnicities than those in the study in which the data for the curves were obtained.
If we convolute (multiply wavelength by wavelength) the photopic response spectrum with a spectral irradiance we obtain a new curve (Figure 2.9). The area under this curve, multiplies by a constant, is a measure of the perceived brightness of the light by a “typical” human observer, in this case, \(E_\mathrm{v} = 103\,\mathrm{klx}\).
Corresponding to the physical quantities of radiation summarized in Table 2.1 above, the equivalent photometric quantities are listed in the Table 2.4 below and have the subscript \(\mathrm{v}\). The ratio between the (physiological) luminous flux \(\Phi_\mathrm{v}\) and the (physical) radiant flux \(\Phi\) is the (photopic) photometric equivalent \(K(\lambda)=V(\lambda)\times K_m\) with \(K_m=683\) lm W\(^{-1}\) (lumen per watt) at 555 nm. The dark-adapted sensitivity of the eye (scotopic vision) has its maximum at 507 nm with 1700 lm W\(^{-1}\). The base unit of luminous intensity is candela (cd). One candela is defined as the monochromatic intensity at 555 nm with \(I=\frac{1}{683}\) W sr\(^{-1}\). The luminous flux of a normal candle is around 12 lm. Assuming a homogeneous emission into all directions, the luminous intensity is about \(I_v=\frac{12\;\mathrm{lm}}{4\pi\;\mathrm{sr}}\approx 1\) cd.
| Symbol | Unit | Description |
|---|---|---|
| \(e_\mathrm{v}\) | lm s | Luminous energy or quantity of light |
| \(\Phi_\mathrm{v}=\frac{\partial e_\mathrm{v}}{\partial t}\) | lm | Luminous flux: absorbed or emitted luminous energy per time interval |
| \(I_\mathrm{v}=\frac{\partial \Phi_\mathrm{v}}{\partial\Omega}\) | cd = lm sr\(^{-1}\) | Luminous intensity: emitted luminous flux of a surface area per solid angle |
| \(E_\mathrm{v}=\frac{\partial \Phi_\mathrm{v}}{\partial A}\) | lux = lm m\(^{-2}\) | Illuminance: luminous flux towards a surface area |
| \(\epsilon_\mathrm{v}=\frac{\partial \Phi_\mathrm{v}}{\partial A}\) | lux | Luminous emittance: luminous flux per surface area |
| \(H_\mathrm{v}=\frac{\partial e_\mathrm{v}}{\partial A}\) | lux s | Light exposure: quantity of light towards a surface area |
| \(L_\mathrm{v}=\frac{\partial^2 \Phi_\mathrm{v}}{\partial \Omega (\partial A\cdot \cos\alpha)}=\frac{\partial I_v}{\partial A\cdot\cos\alpha}\) | cd m\(^{-2}\) | Luminance: luminous flux per solid angle and surface area depending on the angle between luminous flux and surface perpendicular |
Photosynthetically active radiation (PAR), is also an effective radiation quantity, proposed in the early 1970’s (McCree 1972). PAR is expressed in physical units as a flux of photons of wavelengths between 400 and 700 nm. Before its widespread acceptance and the widespread availability of PAR or “quantum” light sensors, both energy and photometric quantities were frequently used in research on plants and vegetation. Very long term series of global radiation are available, while those for PAR are shorter and fewer.
In the case of UV radiation, different biologically effective weightings are used, mostly to describe damage or inhibition. These quantities are normally expressed using energy flux units. However, these are weighted values that do not describe an energy flux but instead a value assumed to be proportional to a response such as growth inhibition (thus based on the same approach as used for photometric quantities).
Photosynthetically active radiation, Caldwell’s Generalized Plant Damage spectrum or GPAS-based exposures, and several other “effective” quantities are in common use in plant research and plant production. These are described in detail in Chapter XXXXX.
2.4 Further Reading
The book The Optics of Life: A Biologist’s Guide to Light in Nature (Johnsen 2012) gives a detailed account of the different aspects of optics as the relate to organisms. Although animales are emphasized more than plants in the examples, overall it is highly relevant plant biologists.
The classic book Biophysical Ecology (Gates 2012) covers a broader range of subjects and discusses implications for Ecology. It is relevant to the current and to the following chapter.
The book Photobiology: The Science of Light and Life (Björn 2015) covers widely photobiology of animals and plants and includes introductory chapters on the physics of light.
2.5 Terminology and standards
The Compendium of Chemical Terminology (Kaiser and Chalk 2020) (“IUPAC Gold Book”) provides up-to-date guidelines on the use of terminology and symbols used in chemistry including photochemistry.
NIST publication 330 (Newell and Tiesinga 2019) describes the International System of Units (SI) in great detail, and together with the IUPAC Gold Book can help clarify the use of units in publications.
The most recent as well as previous official brochures describing The International System of Units (SI) are available at the BIPM website.
These equations are a system of four partial differential equations describing classical electromagnetism.↩︎
Wavelength and frequency are related to each other by the speed of light, according to \(\nu = c / \lambda\) where \(c\) is speed of light in vacuum. Consequently there are two equivalent formulations for the equations.↩︎