Freebound boundfree and freefree transitions

On the atomic level, continuum spectra can arise from free-bound or from bound-free transitions of the atom. The former occurs when an electron makes a transition from an unbound (free) state to a bound state, thereby emitting a photon with the energy of the transition. Since the unbound states are a continuum of energy levels, the energies, or frequencies, of the emitted photons are not restricted to well-separated discrete levels. Thus a continuum of photon frequencies (a continuum spectrum) is observed in emission.

The bound-free transitions are the reverse process wherein an energetic photon is absorbed and an electron is ejected into the continuum of free states. These are the photoelectric reactions discussed in Sections 10.2 and 10.5. Since photons at any energy above the ionization energy can be so absorbed, a continuum of absorption in the photon spectrum results.

Free-free transitions occur when an electron has a near collision with an atom but never becomes bound. If the electron loses energy, a photon is emitted; if it gains energy a photon is absorbed. Again, since the energy levels are not discrete, the emitted or absorbed photons exhibit a continuum emission or absorption spectrum.

An example of the free-bound and bound-free transitions occurs in the photosphere of the sun where a second electron can attach itself to the hydrogen atom to form a negative hydrogen ion H-. As might be expected, the second electron is very weakly bound to the atom; it takes only 0.75 eV to separate it from its parent atom. This is to be compared to the 13.6-eV binding energy of the ground state. These (second) electrons can easily be detached by visible photons, which have energies well in excess of 0.75 eV (Fig. 2.1).

Most of the light from the sun arises from these ions which are constantly being formed (free-bound) and dissociated (bound-free) in the hot photosphere. The result is a continuum spectrum, except for the absorption lines that arise from the temperature gradient in the solar photosphere (see below).

Optically thin thermal bremsstrahlung

A common mechanism for the emission of photons is thermal bremsstrahlung from an optically thin plasma. A plasma is a cloud of ionized atoms or molecules. Astrophysical plasmas will typically have a mix of elements, e.g., the solar system abundances (Table 10.2). The dominant component, hydrogen, will be ionized, for the most part, into its constituent protons and electrons. The ions and electrons are typically assumed to be in approximate thermal equilibrium; their speeds follow the Maxwell-Boltzmann distribution. A thermal plasma must be sufficiently hot for its components to remain ionized at least in part. The term "optically thin" means that any emitted photon is very likely to escape the cloud without being absorbed by another atom; the optical depth of the cloud is small, t ^ 1; see (10.30).

Radiation from a hot plasma In a plasma cloud, the radiation arises when electrons are accelerated in near collisions with ions and thereby emit photons. The term "bremsstrahlung" means "braking radiation" in German. In an electron-ion near collision, the less-massive electron undergoes a large acceleration due to the mutual Coulomb force. An accelerated electric charge will always radiate away some of its energy. The radiated photons constitute the observed radiation. Bremsstrahlung gives rise to the x rays in your dentist's x-ray tube. High-energy electrons beamed into a metal target are rapidly braked to a stop by Coulomb interactions. The deceleration results in an intense beam of x rays.

The dependence of the volume emissivity j (W m-3 Hz-1) of emitted radiation upon temperature T and frequency v may be approximately obtained in a semi-classical derivation. This derivation (not worked out here) takes into account only the free-free collisions of electrons and ions. It thus yields only a continuum spectrum. A complete solution taking into account free-bound and bound-free transitions would yield strong emission lines.

The resultant continuum spectrum is j(v,T) a g(v,T,Z)Z2neniT-1/2e-hv/kT (Wm-3Hz-1) (11.17)

where h and k are the Planck and Boltzmann constants respectively (h = 6.63 x 10-34 J s; k = 1.38 x 10-23 J K-1), ne and ni are the number densities (m-3) of the electrons and ions respectively, and Z is the atomic number (charge number) of the ions. The Gaunt factor g(v,T, Z) is a slowly varying term (decreasing with frequency) that takes into account the quantum mechanical effects in the freefree collisions, such as screening by nearby electrons. In the approximation, g = constant, the dependence of j on frequency at a fixed temperature is a simple exponential. The exponential reflects the thermal Maxwell-Boltzmann distribution of electron speeds.

Plasma parameters determined The quantity that one can measure from afar is the specific intensity I which is related to j according to (8.50),

1 fA

I = — I j (r) dr (Specific intensity; hydrogen plasma; (11.18)

4n J0 oil

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