THE QUANTUM PHYSICS UNDERLYING REMOTE SENSING; USE OF SPECTROSCOPY IN DETERMINING QUANTUM LEVELS

This page is a supplement to your reading of page I-2. The page here was reconstructed (some changes and additions) from the relevant text in the Landsat Tutorial Workbook.

The physical phenomena most frequently sampled in remote sensing are photon energy levels associated with the Electromagnetic Spectrum. The Electromagnetic Spectrum is discussed at some length in this Introduction on page I-4. We reproduce an EM Spectrum Chart here in a rather unusual version in which the plot from left to right is "geared" to Frequency rather than the more conventional Wavelength spread (shown here at the bottom; both Frequency and Wavelength are treated below). The terms for the principal spectral regions are included in the diagram:

The Electromagnetic Spectrum is a plot of Electromagnetic Radiation (EMR) as distributed through a continuum of photon energies. Thus EMR is dynamic radiant energy made evident by its interaction with matter and is the consequence of changes in force fields. Specifically, EMR is energy consisting of linked electric and magnetic rields and transmitted (at the speed of light) in packets, or quanta in some form of wave motion. Quanta, or photons (the energy packets first identified by Einstein in 1905), are particles of pure energy having zero mass at rest. The demonstration by Max Planck in 1901, and more specifically by Einstein in 1905, that electromagnetic waves consist of individual packets of energy was in essence a revival of Isaac Newton's (in the 17th Century) proposed but then discarded corpuscular theory of light. Until the opening of the 20th Century, the question of whether radiation was merely a stream of particles or was dominantly wave motion was much debated.

The answer which emerged early in the 1900s is that light, and all other forms of EMR, behaves both as waves and as particles. This is the famous "wave-particle" duality enunciated by de Broglie, Heisenberg, Born, Schroedinger, and others mainly in the 1920s. They surmised that atomic particles, such as electrons, can display wave behavior, for example, diffraction, under certain conditions and can be treated mathematically as waves. Another aspect of the fundamental interrelation between waves and particles, discovered by Einstein between 1905-07, is that energy is convertible to mass and that, conversely, mass is equivalent to energy as expressed by the famed equation E = mc². In one set of units, m is the mass in grams, c is the speed of EMR radiation in a vacuum in centimeters per second (just under 3 x 10¹⁰ cm/s), and E is the energy in ergs. (Other combinations of units are also used; thus, in high energy processes at high temperatures the electron-volt is the most convenient expression of E.)

EMR waves oscillate in harmonic patterns, of which one common form is the sinusoidal type (read the caption for more information); as seen below, this form is also a plot of the equation for a sine wave.^*

In the physics of wave movement, sinusoidal (also called periodic) types are known as transverse waves because particles within a physical medium are set into vibrational motion normal (at right angles) to the direction of propagation. EMR can also move through empty space (a vacuum) lacking particulates in the carrier medium, so that the EMR photons are the only physical entities. Each photon is surrounded by an electric field (E) and a magnetic field (H) expressed as vectors oriented at right angles to each other. The behavior of the E and H fields with respect to time is expressed by the Maxwell equations (4 needed to completely describe the behavior of the radiation). These equations include the terms μ₀ (permeability of the electric field in a vacuum; Ampere's Law) and ε₀ (permittivity of the magnetic field; Coulomb's Law). An important relationship between these terms and the speed of light is: c = 1/(μ₀ε₀) 1/2. The two fields oscillate simultaneously as described by covarying sine waves having the same wavelength λ (distance between two adjacent crest [trough] points on the waveform) and the same frequency ν (number of oscillations per unit time); ν is the reciprocal of λ, i.e., ν = 1/λ. The equation relating λ and ν is c (speed of light) = λν since light speed is constant, as λ increases (decreases, ν must decrease (increase) the proper amount to maintain the constancy. When the electric field direction is made to line up and remain in one direction, the radiation is said to be plane polarized. The wave amplitudes of the two fields are also coincident in time and are a measure of radiation intensity (brightness).

Units for λ are usually specified in the metric system and are dependent on the particular point or region of the EM Spectrum being considered. Familiar wavelength units include the nanometer; the micrometer (micron now obsolete); the meter; and the Angstrom (10^-8 meters.

A fixed quantum of energy E (in units of ergs, joules, or electron volts) is characteristic of any photon transmitted at some discrete frequency, according to Planck's quantum equation: E = hν = hc/λ. From the Planck equation, it is evident that waves representing different photon energies will oscillate at different frequencies. It follows, too, that the shorter (longer) the wavelength, the greater (lesser) the energy of the photon(s) involved.

How is EMR produced? Essentially, EMR is generated when an electric charge is accelerated, or more generally, whenever the size and/or direction of the electric (E) or magnetic (H) field is varied with time at its source. A radio wave, for example, is produced by a rapidly oscillating electric current in a conductor (as an antenna). At the highest frequency (highest energy) end of the EM spectrum, gamma rays result from decay within the atomic nucleus. X-rays emanate from atoms within the source that are bombarded by high energy particles that cause the source electrons to move to an outer orbit and then revert to one further end (back to the ground state). (This process will be described later in this page.) In stars, high temperatures can bring about this electron transition, generating not only gamma rays and X-rays but radiation at longer wavelengths. Radiation of successively lower energy involves other atomic motions as follows: UV, Visible: transitions of outer electrons to a higher metastable energy level; Infrared: inter- or intra-molecular vibrations and rotations; Microwave: molecular rotations anf field fluctuations.

The interaction of EMR with matter can be treated from two perspectives or frameworks. The first, the macroscopic view, is governed by the laws of Optics, and is covered elsewhere in the Tutorial. More fundamental is the microscopic approach, which works at the atomic or molecular level. This is being considered on this page.

The basis for how atoms respond to EMR is found in the Schroedinger equation (not stated here because of its complexity and the need for background information that appears towards the bottom of the Preface in Section 20). This equation is the quantum analog to the Newtonian force equation (simplest form: F = ma) in classical mechanics. A key term in the Schroedinger equation is E, a characteristic energy value (also known as the "eigenvalue'). For many atomic species, there are a number of possible energy states (eigenstates) or levels within any speciies. For different species , the characteristic states are different and possess unique values (e.g., energies expressed in electron-volts) diagnostic of each particular element. There are, therefore, certain allowable levels for each atomic species whose discrete eigenvalues satisfy the Schroedinger equation. These levels are related to acceptable solutions for determination of the wave function associated with the Schroedinger equation.

Under the influence of external EMR (or thermal inputs), an atomic species can undergo a transition from one stationary state or energy level to another. This occurs whenever the oscillating EMR field (normally, but not necessarily, the electric field) disturbs the potential energy of the system by just the right amount to produce an allowable transition to a new eigenstate (new energy level in the quantum sense). The change in eigenvalue is given by ΔE = E₂ - E₁ = hν, the expression for Planck's Law, in which h = the Planck constant (see Section 20-Preface) and ν is the wave frequency. This transition to the new energy state is evidenced by a reversion to a ground (starting) state or to an intermediate state (which may decay to the ground state) with a consequent output of radiation that oscillates sinusoidally at some specific frequency (wavelength) determined by the exact change in energy (ΔE) associated with the permissable transition. For most atomic species, a large amount of energy input can produce multiple transitions (to various quantum energy levels designated by n = 1, n = 2, ...) and thus multiple discrete energy outputs each characterized by a specific ν (λ).

A convenient way to appreciate this process is to examine several of the principal transitions of atomic hydrogen that has been excited by external energy. Consider these two illustrations:

In the upper diagram circles are used to represent the energy levels, the lowest being the ground state. In the Bohr atom model, these levels relate to orbits (orbitals) occupied by electrons (hydrogen has 1 electron in n = 1; helium has two electrons in n = 1; lithium has these two, plus 1 electron in the next orbit (or shell, but here called "Period", n = 2, and the next element in the Periodic Table, beryllium has 2 electrons in n = 2, then boron with 3 electrons in n =2 until that Period has a maximum number of 8 electrons (the element is helium, with a total of 10 electrons [and 10 protons as these two particles are always equal in the normal {unionized} state]. There are 7 Periods in all, with some containing up to 8 and others more (comprising series, built up from rules not addressed in this review). Currently, there are 103 elements (each with its specific number of protons), some of the higher numbered ones are known only or mainly from high energy physics "atom smashers".

The single hydrogen electron in n = 1 can be transferred into higher energy states within Periods 2 through 6. On decay (transition back to a lower energy level), from any of these levels back to n = 1, discrete energies are released for each level transition, according to the Planck equation. These are expressed here in nanometers but that unit can be converted to micrometers by multiplying its number by 10^-3. The first group of excited states, starting from n = 1, comprises the Lyman series (here, series is not used in the sense of the Period series mentioned in the previous paragraph). The δE for each gives rise to spectra that fall within the Ultraviolet region of the spectrum. The electron may be placed in the n = 2 and n = 3 states and then jumped to higher states. The results are two more series, the Balmer (Visible Range) and the Paschen series (Infrared). Each transition shown on the diagram has a specific wavelength (frequency) representing the energy involved in the level changes.

The lower diagram shows much the same information but with some different parameters. Thus, the energy level for each n is given as a specific value in electron-volts (eV). At the top of this diagram is a black band (part of the spectrum) representing a range of wavelengths, with thin colored lines fixing the location of each δE (as λ) for certain transitions.

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Optional reading: As an aside, the writer (NMS) wants to relate how spectral information about atomic species is obtained: The following pertains to the 1950s version of the Optical Emission Spectroscope, the instrument I used during my Ph.D. research project. This type of instrument was responsible for acquiring data that helped in the fundamental understanding of EMR and the EM Spectrum.

The spectroscope I used works this way. A single- or multi-element sample is placed usually in a carbon electrode hollowed to hold the material at one end; a second opposing pointed electrode helps to establish a small gap (arc) across which an electric current flows. The sample, subjected to a high voltage electric current, experiences high temperatures which vaporizes the material. The excited sample, which in effect "burns", has its material dissociated into constituent atoms, ionizing the elements involved. The excitation produces a series of emission spectra resulting from electron transitions to higher states and almost simultaneous reversions to lower states. There can be hundreds of diffracted spectra formed in the bright white light in the arc gap. Since the spectra relate to different elements, each species producing spectra of discrete, usually unique, wavelengths, this white light must be spread out into its constituent wavelengths. This occurs after part of the radiated light first passes through a narrow aperture or collimating slit and then onto the dispersing component (prism or grating) that accomplishes the spreading (next paragraph). The spread spectra are recorded as images of the slit; an image is reproduced for each wavelength involved. The resulting series of multiple images are extended one-dimensionally by the spreading onto unexposed photographic film (either in a long narrow strip or, in the thesis case, a 4 x 10 inch glass plate with a photo-emulsion coating on one side) which becomes exposed and then is developed to make visible the multiple images (in a film negative, appearing as thin black lines making up a spectrogram).

The spreading is accomplished by passing the light onto a glass (or quartz) prism (which bends the light because the glass refractive index varies with wavelength causing refractive separation) or (in my case) a diffraction grating (thousands of thin ruled parallel lines per linear centimeter on a glass or metal plate). Each given wavelength - representing some exitation state associated with the particular energy transition that an element attains under the burn conditions - is the main factor that determines how much the light is bent by refraction (prism) or diffraction (grating). The equation that describes the angle of spread (relative to the direction of the initial light beam reaching the dispersing prism) as a function of wavelength is: nλ = b sinΘ, where b is the width of the narrow collimating slit, Θ is the angle of spread, and n (0, 1, 2, 3,...) accounts for multiple orders of dispersal. For a diffraction grating b is replaced by d, the reciprocal of the number of diffraction lines (parallel to each other) per centimeter. Each diffracted line (an image of the narrow slit) can be indexed to a particular wavelength. The lines can be identified with multiple elements (different species) whose excited states from ionization give off characteristic ΔE's; this involves painstaking inspection of the recording film (often consuming hours). The appearance of the resulting spectrogram is one of a multitude of irregularly-spaced lines appearing dark (where the film was exposed) against a lighter background. Each line must be measured to obtain its Θ value which serves to determine the wavelength (and its causative element) in a lookup table. Below is a color version of spectrograms. The one at the top is for the spectrum of emitted solar radiation; others are for individual elements occurring in the Sun (the solar spectrum is repeated for reference). The brightness (or darkness, if a film negative is use) is a function of line intensity that in turn depends on the amount of that element present.

A more modern strip chart record is shown below; the height of each line (which widens by dispersion) is a measure of the quantity of the element involved (in the above spectrogram, measurements of photographic density (relative brightness) accompishes this).

Present day emission spectrometers can record the data (the dispersed wavelengths and the intensity of the radiation associated with each) electronically (light sensors) and can digitize these discrete radiation values to facilitate automated species recognition and even amounts of each element (percent or parts per million [ppm]) present.

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Returning to the Hydrogen diagrams, the situation for this element s modes of excitation is relatively "simple". For higher number atomic species, the transitions may be more complex and only approximate solutions to the Schroedinger equation (which can be used instead of the Planck equation to approach the energy changes differently) may result. Other factors enter into the determination of the energy level changes: the nature of the bonding, the coordination of the atoms in molecular or ionic compounds, the distribution of valency electrons in certain orbitals or conduction bands, et al. Without further pursuing these important considerations, we concentrate on this aspect: The transitions relate to three types of non-nuclear energy activity - electronic, vibrational, and rotational. Analysis of each type not only can identify the elements present, but can reveal information about state of matter, crystal structure, atomic motions, etc.

Electronic energy transitions involve shifts of electrons from one quantum level to another according to precise rules. Any allowable transfer is determined from four quantities called the quantum numbers for that atomic system. These are (1) the principal quantum number n; (2) the angular momentum quantum number L; (3) the magnetic quantum number m; and (4), for polyelectronic atoms, the spin quantum number m_s. Electronic transitions occur within atoms existing in all familiar states of matter (solid; liquid; gas; plasma). Elements such as iron (Fe)(also Ni, Co, Cr, V and others) can undergo many transitions (yielding multiple wavelength lines in different parts of the spectrum), each influenced by valence state (degree of oxidation), location in the crystal structure, symmetry of atomic groups containing the element, and so forth. For compounds dominated by ionic bonding, the calculations of permissible transitions are influenced by crystal field theory. (Organic compounds, which usually are held together by covalent bonds, can be studied with a different approach.) Electronic transitions are especially prevalent in the UV, Visible, and Near IR parts of the spectrum.

Vibrational energy is associated with relative displacements between equilibrium center positions within diatomic and polyatomic molecules. These translational motions may be linear and unidirectional or more complex (vary within a 3-axis coordinate system). Specific transitions are produced by distortions of bonds between atoms, as described by such terms as stretching and bending modes. There is one fundamental energy level for a given vibrational transition, and a series of secondary vibrations or overtones at different, mathematically related frequencies (yielding the n orders mentioned above), as well as combination tones (composed of two or more superimposed fundamental or overtone frequencies). A tonal group of related frequencies comprises a band. Again, vibrational energy changes are characteristic of most states of matter (excluding the nucleus). Because these changes require less energy to initiate, the resulting ΔE's tend to occur at lower frequencies located at longer wavelengths in the Infrared and beyond (Microwave).

Rotational energy involves rotations of molecules, These take place only in molecules in the gaseous state and are described in terms of three orthogonal axes of rotation about a molecular center and by the moments of inertia determined by the atomic masses. This type of shift is relevant to the action of EMR on atomspheric gases. Being lower level energy transitions, the resulting emissions are at longer wavelengths. During excitation of gaseous molecules, both vibrational and rotational energy changes can occcur simultaneously.

The net energy detected as evidence of electronic, vibrational, and rotational transitions over a range of wavelengths (tied to one or multiple element species in the sample) is, in another sense, a function of the manner in which energy is partitioned between the EMR source and its interactions with the atoms in the material being analyzed and identified. The EMR (in remote sensing usually solar radiation) may be transmitted through any material experiencing this radiation, or absorbed within it, or reflected by atoms near the surface, or scattered by molecules or particulates composed of groups of atoms, or re-radiated through emission, or, as is common, by some combination of all these processes. There are three prevalent types of spectra associated with a material/object (m/o) being irradiated (as by sunlight): 1) absorption (or its inverse, transmission), 2) reflection, and 3) emission. For absorption spectra, the m/o lies between EMR source and the radiation detector; for reflection spectra, the source and detector are positioned outside of the reflecting surface of the m/o at an angle less than 180°; for emission spectra, the immediate source is located within the m/o although an external activating source (such as the temperature-changing electrical current in the carbon electrode spectrometer described above) is needed to initiate the emission.

Further information on Spectroscopy is presented on page 13-6 ff.

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^* The sine wave or sinusoid is a function that occurs often in mathematics, signal processing, alternating-current power engineering, and other fields. Its most basic form is:

y = A sin(ωt - φ)

which describes a wavelike function of time (t) with: peak deviation from center = A ( amplitude); angular frequency given by ω (radians per second); initial phase (t = 0) = -φ (φ is also referred to as a phase shift, e.g., when the initial phase is negative, the entire waveform is shifted toward future time (i.e. delayed); the amount of delay, in seconds, is φ/ω)

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