Wavelength and frequency

The frequency v of an electromagnetic wave is described in units of cycles per second, or hertz (Hz). In a vacuum, the frequency is related to its wavelength X as

1 It will be our convention on logarithmic plots to label the axis (or axes) as "log y " and to indicate the logarithm of the value rather than the value itself; that is 12 instead of 1012. We also use the conventions log = logio (base 10) and ln = loge (base e).

where c is the speed of light in a vacuum, c = 2.9979 x 108 m/s « 3.00 x 108 m/s

Logj o X (meters)

Radio

Optical %

Log10f (Hz)

30 m 10 MHz

41 neV

1 mm 300 GHz

1.2 meV

900 nm

333 THz 1.4 eV

300 nm

1.0 PHz

13 nm 24 PHz 100 eV

4.9 pm 61 EHz 250 keV

Figure 2.1. The approximate "bands" of the electromagnetic spectrum. The boundaries are not well defined. Radio waves extend further to the left, but the earth's ionosphere is opaque for the most part below about v = 30 MHz. Gamma rays extend to the right by many more factors of 10. Values of the boundaries are given in several units below the figure. Among these, commonly used values are shown in boldface. Note that 1 nm = 10 angstroms.

Wavelength in the visible (optical) range has traditionally been expressed in angstroms (1 A = 10-10 m), but the preferred SI unit is the nanometer (1 nm = 10-9 m) or micrometer (1.0 |m = 10-6 m) traditionally called "microns". Thus,

The human eye is sensitive from about 420 nm (blue) to 680 nm (red). Optical ground-based astronomers work from about 320 to 900 nm.

Frequency-wavelength equivalences over 12 decades of frequency from (1) are given in Table 1. The values given in Table 1 are placed in the context of the observational bands through comparison to Fig. 1. In this text, frequency units will be used extensively.

Photon energy

The energy E (units of joules, J, in the SI system) of a photon is1:

1 We often choose to give SI units (in parentheses) when they are not required, as in (5), to remind the reader of the dimensions. The dimensions often clarify the meaning of the quantity in question.

Table 2.1. Frequency-wavelength correspondence

Frequencya

Wavelengtha

Band

(radio) (microwave) (IR/optical) (x ray) (gamma ray)

a The SI prefixes used here are: m = milli = 10 3, | = micro =

10-6, n = nano = 10-9, p = pico = 10-12,M = mega = 106,G = giga = 109, T = tera = 1012, P = peta = 1015, E = exa = 1018. See complete list in the Appendix, Table A3.

where h is the Planck constant (J s = joule second) and v is the frequency (Hz). The energy of a photon can be given in units of electron volts (eV), where 1 eV is the energy gained by an electron that is accelerated through a 1 volt potential change,

= 1.602 176 x 10-19 (coulombs) x 1.0 (volt) 1 eV = 1.602176 x 10-19 J (electron volt) (2.8)

Astrophysicists often use the energy units of electron volts when dealing with photons and fundamental particles. From the relations (5) and (8), the frequency (Hz) is related to the photon energy (eV) as follows:

E (J) E (eV) x 1.602 176 x 10-19 (J/eV) / E (eV) e v (Hz) =

Thus a photon of energy 1 eV has frequency of 2.4 x 1014 Hz. The rightmost term of (9) leads directly to the relation between the wavelength A,(m) and the photon energy E(eV), c 1 hc

v E (eV) e A.(m) x Ephoton(eV) = hc/e = 1.239 842 x 10-6 A.(nm) x Ephoton(eV) = 1239.842 (2.12)

which tells us that 1 eV corresponds to X = 1.24 x 10-6 m and that radiation in the visible range at X = 620 nm corresponds to photon energies of 2.0 eV. According to (10) or (1), the frequency associated with these 2.0-eV photons is v = 4.8 x 1014 Hz. The reader should remember that optical radiation is characterized by the values ~2 eV and ~5 x 1014 Hz. The x-ray wavelength of X = 1 nm corresponds to 1240 eV or 1.24 keV. From (10), this corresponds to a frequency of 3 x 1017 Hz.

Figure 1 shows frequency increasing to the right whereas wavelength increases to the left, according to (1). The photon energy also increases to the right according to (5). The boundary between the infrared and optical bands is roughly at X = 900 nm or v = 333 THz and the radio/infrared boundary is roughly X = 1 mm or v = 300 GHz. The high frequency end of the radio band is called the microwave band, from about 5 GHz (X = 60 mm).

These names and boundaries of the wavelength bands are historical and not precisely defined. They reflect in part the detection apparatus first used, how the photons are created, or the context of their original discovery. For example, medical radiologists probably call 2-MeV photons "x rays" rather than gamma rays because, I gather, they are generated by energetic electrons crashing into a metal target, but they would call them "gamma rays" if they were emitted by an atomic nucleus (radioactivity). In contrast, astrophysicists would call 2-MeV photons "gamma rays" simply because they are in the gamma-ray frequency range.

The very narrow range of the visible or optical portion of the spectrum is notable. It is only about a factor of three in bandwidth, compared to the overall range of observed photon energies which is more than a factor of 1015! It is a happy circumstance that the peak of the solar radiation falls at the same frequency as a transparent band in the atmosphere (see below). Our eyes evolved, naturally, to operate in this same narrow range. The explosion of astronomical knowledge in this century has been possible, in large part, because we have learned to "see" at wavelengths outside the visible band. The dynamic ranges encountered are so large that many of the figures in this text will show axes in logarithmic units, as in Fig. 1.

Temperature

Photons are often emitted from a body that can be characterized by a temperature T. The SI unit of thermodynamic temperature is the kelvin which refers to the scale known as absolute temperature. In this system, the freezing temperature of water is at T = 273.15 K (0 °C). Temperature can be defined for a gas in thermal equilibrium. In such a gas, the particles have kinetic energies of order kT where k is the Boltzmann constant (units J/K = joules/kelvin), according to the kinetic theory of gases. If these particles emit photons and if the photons are in thermal equilibrium with the particles (i.e., the object is a "blackbody"), the average photon energy hvav will be roughly of order kT, or more precisely,

Ephotons,av = hvav = 2.70kT (thermal equilibrium) (2.13)

k = 1.380 650 x 10-23 J/K (Boltzmann constant) (2.14)

It is thus sometimes convenient, as a reference, to know the relation between frequency and temperature according to the approximation hv ~ kT, kT

v = ^ (2.15) h v(Hz) = 2.084 x 1010 T(K) (for hv = kT) (2.16)

From (16) one finds that a temperature of 10000 K is associated with a frequency of 2 x 1014 Hz. This is in the near-infrared band (meaning near to the optical band). However, since the average photon frequency of a blackbody is actually about a factor of three higher than kT (13), much of the radiant energy from a star of T = 104 K will fall in the optical band where the human eye can sense it. The relation between energy kT (eV) and temperature T(K) is also useful.

Thus our 10 000-K stellar surface contains particles of energy E ~ kT = 0.86 eV. X rays of energy kT ~ 104 eV correspond to temperatures of T ~ 108 K. Do remember that 1 eV corresponds to about 12 000 K.

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