 | Society of Amateur Radio Astronomers
|
|---|
Educational Programs
Overview of the Electromagnetic Spectrum
Copyright © 2002 by H. Paul Shuch, Ph.D.
email n6tx @ setileague.org
| Society of Amateur Radio Astronomers
|
|---|
Overview of the Electromagnetic Spectrum
Copyright © 2002 by H. Paul Shuch, Ph.D.
email n6tx @ setileague.org
Upon completion of this lesson, you will demonstrate mastery by:
Introduction: All Waves are Alike
One of the remarkable things about electromagnetic waves is that they all behave fundamentally alike, whether emanating from sunlight, searchlight or satellite. Ultrasonic waves, radio waves, microwaves, infrared, visible light, ultraviolet light, X-rays, gamma rays and cosmic rays all travel through free space basically the same, at the same constant speed, and they all follow the very same Maxwell's Equations. So if you understand one electromagnetic wave, you understand them all. The purpose of this tutorial is to lead you to that universal understanding.
The Spectrum as a Continuum
You are doubtless familiar with the rainbow spectrum obtained when passing sunlight through a prism. It is a continuum; that is, there is no clear and distinct boundary between one color and the next. Rather, they seem to transition uniformly, blending smoothly into one another. However, for convenience, we tend to label the individual colors of the rainbow discretely (red, orange, yellow green, blue, indigo, violet), as though their boundaries were truly distinct. Let's call this process quantizing a continuum; for simplification, we will apply such quantization throughout this tutorial.
The electromagnetic spectrum, like the colors of the rainbow, is in fact such a continuum, extending from DC to daylight, and far beyond. Yet, for convenience, we quantize it into distinct frequency bands, and label each one. Consider, as an analogy, the piano keyboard below:
Each of the piano's 88 keys produces a unique, distinct tone, at a specific frequency somewhere within the 7 1/2 octave span of the instrument's audio range. Now, that audio range is in fact a continuum; yet, we can consider each individual note (e.g., A above middle C) as a discrete signal, representing a unique pitch or frequency.
We can identify each note on the piano keyboard by this unique frequency of vibration. In the case of our A above middle C, that frequency is 440 Hz (where Hz is named for 19th Century radio pioneer Heinrich Rudolph Hertz, and represents cycles per second).
The Visible Light Spectrum
Now, instead of music, let's talk astronomy. Using the same piano keyboard image, it now becomes a metaphor for the whole electromagnetic spectrum (that aggregate of all possible electromagnetic waves, which you'll recall all behave fundamentally alike). The actual piano keyboard produces tones which span the frequency range from around 20 Hz to roughly 4 kHz. But to us, its keys will now represent an entirely different (and much broader) range of frequencies.
Within that broad range, note the six colored keys in the middle of the image above. They represent the entire visible light spectrum (everything the eye can see), all colors red through violet. In our image they are color-coded accordingly. Notice that all frequencies we can see with our Sun-adapted eyes occupy just under one octave on our electromagnetic 'keyboard'. In fact, visible light occupies just under one octave of the electromagnetic spectrum, as well. These few keys represent everything available to optical astronomers, in their quest to understand the cosmos.
Everything to the left of the red key is still 'light', albeit at frequencies which are invisible to the human eye. So is everything to the right of the violet key. Considering how little of the total electromagnetic spectrum the eye can see, one understands why the advent of radio astronomy, X-ray astronomy, and the like have added so much to human knowledge.
Frequency and Wavelength
In the above figure, each key still represents a specific frequency, but we have scaled their values, so the frequency represented by each key may be several billion times higher than that of the corresponding audio note on a real piano. In music, we identified each piano key by a note name and a frequency. In the optical spectrum, we identify each 'note' by its color name (e.g., Red), and a corresponding frequency (such as 400 THz, where T is for Tera, which represents ten to the twelfth power). But each frequency can also be identified by its corresponding wavelength, the longitudinal distance traversed by a single cycle of the wave as it propagates through free space. And the two are inversely proportional; the higher the frequency, the shorter the wavelength.
Note, in the figure below, that the portion of the electromagnetic spectrum which we call visible light spans in color from red to violet; in frequency from 400 THz to 750 THz; and in wavelength from 750 nm to 400 nm.
The Speed of Light
Now, let's play around with those frequency and wavelength numbers a bit. Consider the bottom of the visible spectrum, where the frequency is 400 THz and the wavelength is 750 nm. If we multiply those two values together, thus:
Not coincidentally, if we perform the exact same operation for the upper end of the visible spectrum, where frequency equals 750 THz, and wavelength equals 400 nm, we get a strikingly similar result:
[Equation 1] c = λνwhere:
c = speed of light (3 * 10^8 m/s)
λ (the Greek letter lambda) = wavelength, in meters, and
ν (the Greek letter nu) = frequency, in Hz (cycles per second)
All other frequency and wavelength relationships (including two we will derive below) stem from this basic definition.
Converting Frequency to Wavelength
Since we know the speed of light (it's a constant), and we know that it equals the product of frequency and wavelength, we can do some minor algebraic manipulation to determine wavelength, if we happen to know frequency. Rearranging Equation 1, we get:
[Equation 2] λ = c / νwhere:
λ = wavelength, in meters,
c = speed of light (3 * 10^8 m/s), and
ν = frequency, in Hz (cycles per second)
Let's practice. Consider an orange light source, with a frequency of 500 THz. What is its corresponding wavelength, in meters? Applying Equation 2, we get:
λ = c / ν
λ = (3 * 10^8 meters/sec) / (5 * 10^14 Hz)
λ = 6 * 10^-7 meters = 600 nm
You can check your work by multiplying the known frequency (500 THz) by the wavelength you just computed (600 nm). If the product equals the speed of light, 300 million meters per second, you know you did things right!
Converting Wavelength to Frequency
Well, we still know the speed of light (it's still a constant), and it still equals the product of frequency and wavelength. So let's do som more algebraic magic, to determine frequency, if we happen to know wavelength. Rearranging Equation 1 once more, we get:
[Equation 3] ν = c / λwhere:
ν = frequency, in Hz (cycles per second)
c = speed of light (3 * 10^8 m/s), and
λ = wavelength, in meters
Remember that ν is still the Greek letter nu, and λ is the Greek lambda. So, Equation 3 gives rise to the old engineering school joke (and convenient nmemonic):
"What's new?"
"C over lambda."
Now, let's practice going in that direction. Consider a blue-green light source, with a wavelength of 500 nm. What is its corresponding frequency, in Hertz? Applying Equation 3, we get:
ν = c / λ
ν = (3 * 10^8 meters/sec) / (5 * 10^-7 meters)
ν = 6 * 10^14 Hertz = 600 THz
To check our work, we need merely multiply frequency times wavelength, which takes us back to the speed of light.
Waves as Particles, Planck's Constant and the Energy per Photon
The following section constitutes perhaps the shortest treatment of quantum physics on record. It is introduced here only to show that the electromagnetic spectrum is also an energy continuum.
Up until now, we have been talking about electromagnetic waves. One of the most important realizations of the early Twentieth Century is that light (as well as all the other forms of electromagnetic radiation, which we will introduce shortly) can also accurately be described as particles.
Max Planck had observed in 1900 that radiant energy emitted from vibrating molecules was quantized; that is, it could only take on specific, discrete values. This observation suggested the particle nature of electromagnetic radiation. In 1905, Albert Einstein dubbed these apparent massless packets of pure electromagnetic energy photons.
Putting Planck's and Einstein's observations together allowed certain apparent discrepancies in Maxwell's electromagnetic theory to be reconciled. It also led to the realization that the energy contained in a single photon varies with its frequency, according to this relationship:
[Equation 4] e = h * νwhere :
e is the energy of a single photon, in Joules,
ν (the Greek letter nu) is frequency of the photon's corresponding wave, in Hertz (cycles per second), and
h is a conversion constant derived by Planck, which we call Planck's Constant.
Planck's constant has a value of 6.626 * 10^-34 Joules * seconds. It's one of those numbers you're just going to have to memorize, like p. We can use it to calculate the energy contained in a single photon of light, if the frequency is known. For example:
Consider the lower edge of the visible light spectrum, red light with a frequency of 400 THz. Applying Planck's Constant, we see that the energy in a single photon is
e = h * ν
e = (6.626 * 10^-34 Joules * Sec) x (400 * 10^12 cycles/Sec)
e = 2.65 * 10^-19 Joules
Without belaboring the calculation (which we shall do later anyway, as a review exercise), what can you say about the energy per photon for violet light? Since the upper edge of the visible light spectrum has a frequency just under twice that at the lower edge, and since energy per photon is found by multiplying Planck's constant by frequency, it is reasonable to expect that the energy per photon for vilolet light will be just under twice the value we just calculated for red light.
From this discussion, you can see that the electromagnetic spectrum is not just a frequency continuum or a wavelength continuum. It is also an energy continuum, with the energy per photon increasing as we move up the keyboard.
Spectral Absorption Lines
Notice in our visible spectrum diagram that we have painted the white keys of just under one octave, to correspond to red, orange, yellow, green, blue, and violet, respectively. But the piano keyboard also contains black keys in that range, which in music represent half-steps, or semi-tones. Those do not exactly tie in with our keyboard analogy for the electromagnetic spectrum, but with a little imagination, we can force a fit.
You've no doubt observed the visible light spectrum which is produced by passing sunlight through a prism. You can see the expected colors in spectral order, from red to violet. But if you can look closely enough (with significant magnification), you may also notice some thin black lines running through the rainbow pattern, completely blocking out certain very narrow bands of specific colors. These are called spectral absorption lines, and we will represent them by the black keys on our piano keyboard spectral model.
What causes spectral absorption? Consider first of all that when we walk the Earth, we are in fact slogging along the ocean floor, because we live beneath a fifty-mile thick sea of mostly nitrogen, a little oxygen, and various trace elements which we call the Earth's atmosphere. Now nitrogen, oxygen, and those trace elements are all made up of atoms, and atoms have the unique property of being able to absorb energy at specific levels, depending upon the specific elements in question.
The absorption mechanism has to do with the fact that the electrons in an atom can exist at specific, discrete energy levels (sometimes called orbitals, or shells). To move an electron from its preferred energy level (known as the resting state) to a higher energy level (called an excited state), we need to pump in an exact amount of energy. And, you will recall, each photon contains a specific amount of energy, which we can calculate by multiplying its frequency by Planck's Constant.
Here's where those black lines in the spectrum come from. If we pass sunlight (a broadband frequency source) through the Earth's atmosphere, specific molecules get excited by the energy of the sun. They then absorb energy, but only at those specific colors where the energy per photon corresponds to the energy needed to transition an electron from one orbital to the next. At every frequency (color) for which the Planck energy is exactly correct to excite an atom, the atmosphere sucks out energy, leaving a blank (black absorption line) in the spectrum. In our piano keyboard diagrams, consider the black keys to be those absorption lines, caused by raising the energy level of electrons in specific atoms at their corresponding excitation frequencies.
In fact, this is exactly how spectroscopy works. We know what specific stuff in space is made of, because when we look at the spectra of stars, we can match up the absorption lines with the specific elements which are known to absorb at those particular frequencies.
We now know everything we need to determine the frequency of light needed to excite any atom, if its excitation energy is known. All we need to do is algebraically manipulate Equation 4, thus:
[Equation 5] ν = e / hwhere :
ν is the required excitation frequency, in Hertz (cycles per second),
e is the desired excitation energy, in Joules, and
h is Planck's Constant, 6.626 * 10^-34 Joules * seconds.
For example, let's consider a sample of a gas which we know requires exactly 3.3 * 10^22 Joules of energy to excite. What color of light must we pass through the sample, in order to raise the orbitals of individual atoms into an excited state?
Applying Equation 5, we get:
ν = e / h
ν = (3.3 * 10^-19 J) / (6.626 * 10^-34 J * s)
ν = 498 * 10^12 cycles/second = 498 THz
As it happens, this particular excitation frequency corresponds to orange light, so we can achieve our desired result by pumping orange light into the sample. Similarly, if we pass sunlight through our sample, and then pass the resulting light through a prism, we will detect an absorption line precisely at the particluar shade of orange which corresponds to a ferquency of 498 THz.
The Rest of the Spectrum
So far, all of our exercises have involved light in the visible spectrum. Now, we will consider invisible light. Remember all those notes to the left of our visible octave? They represent electromagnetic waves of lower frequency, hence longer wavelength. Working our way to the left of the keyboard, those bands can be labeled infrared, microwave, radio, and audio waves, respectively (see the Figure below). Similarly, the notes to the right of our visible octave represent higher frequencies than visible light, hence electromagnetic waves of shorter wavelength. Going to the right on our keyboard, we call those bands ultraviolet, X-rays, Gamma rays, and cosmic rays, respectively, as depicted below.
Although their frequencies and wavelengths differ from those of visible light, these classes of invisible light are still electromagnetic waves, behaving exactly the same, and observing exactly the same laws of nature, as do red, orange, yellow, green, blue, etc. And not only do they still propagate through free space at the very same speed of light we computed previously; that speed can still be found by multiplying frequency times wavelength, in any and all of cases.
For each of the spectral segments we have labeled, we can still convert frequency to wavelength, and wavelength to frequency, and determine the energy per photon, by applying the same techniques we have learned for visible light. You will have an opportunity to practice these manipulations in the Review Exercises at the end of this lesson.
Notice that each of the segments of the electromagnetic spectrum occupies several keys on our piano keyboard. We can further subdivide each segment into specific bands, each of which might be represented by a single key. For example, the International Telecommunications Union (ITU) divides the radio spectrum into bands designated LF, MF, HF, VHF, UHF, SHF, and EHF. During the Second World War, the MIT Radiation Laboratory (LabRad) created a cryptic system for the microwave spectrum, naming individual bands L, S, C, X, and Ku. And the fiber-optic industry today identifies specific communications bands within the near-infrared and far-infrared regions. All of these specific band designations are, as we academics like to say, beyond the scope of this course, although interested students will find them detailed in my hypertext-book Conquering Communications, available through Amazon Books.
Incidentally, I think Max Planck would appreciate our piano keyboard analogy. In addition to being a gifted physicist, and one of the fathers of quantum mechanics, it so happens that Planck was trained as a concert pianist.
Spectral Emission Lines
In an earlier section, we saw how absorption lines are formed when atoms (or ions, or compounds, for that matter) transition into an excited state by absorbing energy from photons of the appropriate frequency. The opposite process is also possible. That is, an atom, ion or compound loses energy when electrons fall back from an excited to the ground state. When that occurs, a photon is emitted, the exact frequency of which is determined by the corresponding energy being lost. The result is spectral emission lines, which radiate energy at specific frequencies which are unique to the elements losing energy. These spectral emission lines can be readily detected by telescopes operating in the appropriate wavelength bands, and tell us a great deal about the composition of stars, atmospheres, and the interstellar medium.
Calculating the frequency of a spectral emission line from its corresponding energy level is no different from determining that of a spectral absorption line. Equation 5 still applies. Let us consider, for example, the cluster of microwave spectral emissions from excited hydroxyl ions, when they lose energy. One energy state transition in hydroxyl emits photons with around 1.1 * 10^-24 Joules of energy apiece. What is the corresponding frequency of the resulting emission line? If we apply Equation 5, we get:
ν = e / h
ν = (1.1 * 10^-24 J) / (6.626 * 10^-34 J * s)
ν = 1.66 * 10^9 cycles/second = 1660 MHz
And what is the corresponding wavelength of this hydroxyl emission line? From Equation 2, we find:
λ = c / ν
λ = (3 * 10^8 meters/sec) / (1.66 * 10^9 Hz)
λ = 1.8 * 10^-1 meters = 18 cm
In fact, radio telescopes can easily detect these 18 cm interstellar hydroxyl emission lines, as discussed in the following section.
Astronomy at Low Frequencies
At optical frequencies, astronomy has long shown us what the universe looks like. Radio astronomy has opened up a new window on the heavens, by showing us (through colors of light which are invisible to the human eye) what the universe behaves like. Here we will review briefly a few of the kinds of astronomical observations that can be performed in the low frequency realm. See the SARA Projects page for details on actual observations and equipment.
For the purposes of this discussion, low frequencies will be taken to mean any frequencies below that of the visible light spectrum (that is, to the left of the colored keys on our keyboard). The low frequency spectrum includes audio frequencies (not only those audible to humans, but also subsonic and ultrasonic waves), the radio, microwave, and infrared spectra. Since our electromagnetic spectrum is an energy continuum, with energy per photon increasing directly with frequency, the lower frequency spectrum is a continuum of low-energy photonic phenomena.
Radio astronomy was born in the low-frequency regions of the spectrum, as in fact was electronic communication, and in both disciplines, as our technology has advanced, we have tended to harness photons ever higher in frequency. The very first crude radio telescope was built in the radio spectrum, at a frequency near 20 MHz, in 1932. It was a rotating, directional antenna, designed by Karl Jansky of Bell Laboratories in order to track down a source of electromagnetic interference to transatlantic telephone calls being relayed via short-wave radio. Although his realization that these signals were emanating from space represents the birth of radio astronomy, Jansky himself never followed up on his discovery.
The signal that Jansky detected was thermal radiation from the center of the Milky Way galaxy, and today, most low-frequency radio telescopes still sniff out such thermal signatures. Shortly after Jansky's discovery, Grote Reber, a radio amateur (W9GFZ) in Wheaton IL, built the first modern radio telescope, a parabolic dish antenna with a receiver operating slightly higher in the radio spectrum. With it, he mapped out the Milky Way, producing contour maps similar to the one presented in our own Tutorial 1. Bear in mind that Reber's measurements were still of the thermal continuum; thus, early radio telescopes were (and many still are) basically radiometers.
Spectral emission lines were hypothesized in the 1930s, but not actually measured until 1951, when Ewen and Purcell (see Excercise problem #4, below) first detected microwave emissions from interstellar hydrogen. A decade later, scientists at MIT Lincoln Laboratories measured spectral emission lines from interstellar hydroxyl ions, slightly higher up the microwave band. Today, hundreds of elements, compounds, and even organic molecules have been detected in space by their low-frequency (primarily microwave) emission spectra.
It is important to notice that, while thermal radiation can be detected by instruments operating at any frequency, in order to detect spectral emissions, one must tune one's receiver to a specific, known frequency. Thus, while the distribution of matter in the cosmos may be mapped with a simple radiometer, determining the nature of that matter requires the use of a spectrometer.
If a spectral emission line is amplified in a supporting medium, a natural maser can result. Radio astronomers use microwave spectrometers to seek out natural masers, which emit distinct lines at known transition frequencies of hydrogen, hydroxyl, carbon monoxide, and methanol, to name a few.
Another popular form of low-frequency radio astronomy involves detecting radiation from planets. Jupiter, in particular, is a powerful source of radio emissions from charged particles in its dense atmosphere, interacting with its powerful magnetic field. Jupiter observation with simple, low-cost shortwave radio equipment is an easy and inexpensive introduction to radioastronomy.
Pulsars, first discovered in 1968 by Cambridge University graduate student Jocelyn Bell and her professor Antony Hewish, are rapidly rotating, dense neutron stars. They emit strong, broadband periodic pulses, most readily detected in the radio spectrum, between 100 and a few hundred MHz. Amateur pulsar detection is becoming increasingly popular, as computer algorithms are developed to pull the pulses out from beneath the thermal noise which is prevalent at those frequencies.
At the lowest part of the electromagnetic spectrum, at audio frequencies, amateur radio astronomers are listening to a number of interesting natural phenomena. These include whistlers, bursts of energy produced when charged particles spiral in through the Earth's magnetic field; as well as the detection of solar flares, sunspots, and related solar phenomena. Equipment for such low-frequency astronomy is quite inexpensive, and ideal for school projects.
Several times per year, as the Earth's orbit takes it through the debris field of long-dead comets, small particles of dust shower down on our planet, burning up in our atmosphere. The resulting meteor showers are visible to us on Earth, not only from their visible trails at night, but also through the ionization trails which they produce in our upper atmosphere. Radio observation of meteors involves listening not to the meteors themselves, but rather to signals from distant transmitters, reflected off those ionization trails and into our antennas. Radio amateurs have long used meteor ionization trails as a reflective medium to support long-range, two-way communications. Experimenters also bounce radio and microwave signals off the Moon, and other space debris, and recover the echoes on Earth.
Infrared astronomy is less practiced on Earth than observations at radio and microwave frequencies, partly because our planet's atmosphere blocks much of the infrared energy arriving from space. However, it is gaining popularity as observatories atop tall mountains, and in space, become more common.
There are a number of specific bands in the low-frequency spectrum which have been set aside for radio astronomical observation. Most are in the high radio and low microwave regions. Popular among amateur radio astronomers are the 20 MHz spectrum (for Jupiter observations), bands near 400 and 600 MHz (for thermal continuum measurements and pulsar detection), and the region around 1.4 GHz (for hydrogen-line spectroscopy).
The telecommunications explosion of the last few decades has exploited the radio and microwave segments of the spectrum, making reception equipment (and hence radio astronomy) at these frequencies quite inexpensive. The result is an explosion of interest in radio astronomy, of which you, the reader, are a part.
Astronomy at High Frequencies
For the purposes of this discussion, high frequencies will be taken to mean any frequencies above that of the visible light spectrum (that is, to the right of the colored keys on our keyboard). The high frequency spectrum includes ultraviolet frequencies, as well as X-rays, gamma rays, and cosmic rays. No doubt as human knowledge of the electromagnetic spectrum improves, other segments may be added on new keys, to the right of our existing keyboard. Since our electromagnetic spectrum is an energy continuum, with energy per photon increasing directly with frequency, the higher frequency spectrum is a continuum of high-energy photonic phenomena.
High-energy astronomy is just beginning to come into its own, as we develop technologies for exploiting the higher frequencies. Many of these observations must be made from space, as our ocean of atmosphere both protects us from, and prevents us from learning about, such natural high-energy phenomena as gamma ray bursts, cosmic rays, and X-rays. As a consequence, high-frequency astronomy is an area not yet being exploited by amateurs.
As would be expected, astronomical phenomena observed at the upper reaches of the spectrum are generally cataclysmic, high-energy events, such as supernovae, black holes, star formation, and the collision of galaxies. In many cases (such as X-ray bursts or gamma ray bursters), we are not even sure what mechanism causes the observed phenomenon. It can be expected that, with the continued development of high-energy instruments such as X-ray telescopes in space, our study of high-frequency astronomy will lead to new understandings of the processes that shape the universe.
Review Exercises (solutions appear below)
1. What is the velocity of forward propagation of radiant electromagnetic energy in free space?
2. From memory, what frequencies and wavelengths define the edges of the visible light spectrum?
3. What is Planck's Constant?
4. Interstellar hydrogen, the most abundant element in the universe, emits a strong spectral radiation line at a wavelength of 21 cm. This line, labeled H1, was first detected by physics graduate student Harold Ewen and his professor, Edward Purcell, at Harvard University in 1951. At what frequency did Ewen's receiver operate? What was the received energy per photon?
5. Water vapor in the Earth's atmosphere absorbs microwave radiation at a frequency of 22.235 GHz. What is the corresponding wavelength of this absorption line? How much energy is absorbed by water vapor if it is excited by one million photons?
6. A telescope responds to energetic (6 * 10^-19 Joule) photons. What are the corresponding frequency and wavelength? In which spectral realm does this telescope operate?
7. A radio telescope is used to receive decametric waves from Jupiter, at a frequency of 21 MHz. What is the corresponding wavelength? Would you consider Jupiter's signals to be composed of low or high energy photons? Why are these signals called decametric waves?
8. What is the approximate energy per photon for violet visible light?
9. Radio telescopes operating in the 1 to 30 GHz microwave region are often said to respond to centimeter-waves. Why?
10. What would be the approximate operating frequency region of a millimeter-wave radio telescope? Why?
Exercise Solutions
1. The velocity of forward propagation of radiant electromagnetic energy in free space (more popularly known as the speed of light) equals 3 * 10^8 meters per second, or 300 million meters per second. It can be derived by multiplying any known frequency by its corresponding known wavelength.
2. The visible light spectrum extends from red (400 THz; 750 nm) through violet (750 THz; 400 nm).
3. Planck's Constant, a numeric conversion constant for computing the energy per photon, equals 6.626 * 10^-34 Joules * seconds.
4. To calculate frequency, apply Equation 3:
ν = c / λ
ν = (3 * 10^8 meters/sec) / (2.1 * 10^-1 meters)
ν = 1.42 * 10^9 Hertz = 1420 MHzTo find the energy per photon, apply Equation 4:
e = h * ν
e = (6.626 * 10^-34 Joules * Sec) x (1.42 * 10^9 cycles/Sec)
e = 9.4 * 10^-25 Joules
5. To calculate wavelength, apply Equation 2:
λ = c / ν
λ = (3 * 10^8 meters/sec) / (2.2235 * 10^10 Hz)
λ = 1.35 * 10^-2 meters = 1.35 cmTo find the energy for one photon, apply Equation 4:
e = h * ν
e = (6.626 * 10^-34 Joules * Sec) x (2.2235 * 10^10 cycles/Sec)
e = 1.47 * 10^-23 JoulesThe excitation energy for a million photons is simply the above number multiplied by 10^6:
e = (1.47 * 10^-23 Joules) x (1 * 10^6) = 1.47 * 10^-17 Joules
6. Since we know the energy for a single photon, we can calculate the corresponding frequency from Equation 5:
ν = e / hNow the corresponding wavelength is found from Equation 2:
ν = (6 * 10^-19 Joule) / (6.626 * 10^-34 J * s)
ν = 9 * 10^14 cycles/second = 900 THz
λ = c / ν
λ = (3 * 10^8 meters/sec) / (9 * 10^14 Hz)
λ = 3.3 * 10^-7 meters = 330 nmFinally, from the spectrum chart, we notice that the frequency just computed is a little higher than (and the wavelength is a litle shorter than) that of violet visible light. This signal would thus appear to fall in the ultraviolet spectrum.
7. Given frequency, we calculate wavelength from Equation 2:
λ = c / ν
λ = (3 * 10^8 meters/sec) / (21 * 10^6 Hz)
λ = 1.4 * 10^1 meters = 14 metersSince the above frequency and wavelength are far to the left of our spectrum chart, and energy increases as we work our way up the keyboard, we would consider these to be low-energy photons, even without quantifying them. Electromagnetic emissions with wavelengths in the tens of meters are logically called decametric waves (from the Greek deka, for ten). Don't confuse these with higher frequency decimetric waves, which have wavelengths in the tenths of meters.
8. We note from our spectrum charts that violet light is at the upper region of the visible spectrum, which itself has an upper frequency limit of around 750 THz. From Equation 4:
e = h * ν
e = (6.626 * 10^-34 Joules * Sec) x (750 * 10^12 cycles/Sec)
e = 5 * 10^-19 Joules
9. From Equation 2, we see that the wavelength corresponding to 1 GHz is 0.3 meters (30 cm), while the wavelength corresponding to 30 GHz is 0.01 meters (1 cm). Just as signals with wavelengths in the tens of meters are called decametric waves (see Problem #7 above), so do we label signals with wavelengths in the hundredths of meters as centimeter waves. The most common of all radio telescopes are centimeter-wave instruments, operating in the low end of the microwave spectrum.
10. If centimeter-wave telescopes operate at wavelengths in the hundredths of meters (see problem #9 above), then it is logical that millimeter-wave telescopes operate at wavelengths in the thousandths of meters. From Equation 3, we see that a one millimeter wavelength corresponds to a frequency of 300 GHz. Although there is no hard and fast rule, by convention, radio telescopes operating at frequencies from perhaps 18 GHz to roughly 300 GHz are considered millimeter-wave telescopes.
Completion Standards
If you solved eight or more of the above problems correctly, you have satisfied the objectives of this lesson. If you missed more than two questions, go back and review the material in this tutorial, until you have achieved mastery.
entire website copyright © Society of Amateur Radio Astronomers this page last updated 20 Jun 2009 email the webmaster 
|
 Top of Page |