Generation of Electrons, Electromagnetic Radiation and Neutrons; Absorption, Fluorescence and Detection
JVS 8/23/97revision of 4/2/83 ms. The material on synchrotrons needs upgrading as in Smith and Rivers (1995).
This chapter begins with the theory and practice of the generation of electrons, electromagnetic radiation (principally X-rays), and neutrons.
Electrons are generated by six processes: •decay of a radioactive nucleus giving negative beta particles; •heating of a metal (thermionic emission); •application of a strong electric field to a metal (field emission); •excitation by electromagnetic radiation (photoemission); •and excitation by charged particles including electrons (secondary-electron emission). In the sixth process, ionized atoms release Auger electrons whose kinetic energy is obtained from a radiation-less transition between orbital energy levels. Acceleration and focusing by electromagnetic fields of electrons in a vacuum is exploited in scanning and transmission electron microscopes (SEM, TEM, STEM), in special instruments for low-energy electron diffraction (LEED) and photo-emission electron microscopy (PEEM), in electron microprobes (EMP) for chemical analysis, and in synchrotron storage rings (SSR) for generation of intense infrared, ultraviolet and X-ray beams. Protons are used for chemical analysis by particle-induced X-ray emission (PIXE) in the proton microprobe; deuterons and other heavy ions can also be used.
Portable sources of gamma-rays are useful for some X-ray studies, but most X-rays are generated from electron beams. Deceleration of a free electron results in the emission of a continuous spectrum of electromagnetic radiation (Bremsstrahlung: German for braking radiation) with a limiting frequency n = E/h, where E is the kinetic energy of the electron. The spectrum has a toroidal angular distribution at low electron velocity, but is compressed into a fine jet at high velocity, as is exploited in the continuous spectrum produced by the centripetal force in an SSR. Interactions between electrons and atoms in a conventional Xray tube, an EMP, or an EM, produce a continuous spectrum from deceleration during angular deflection and a characteristic spectrum from electron transitions between quantized energy states in atoms ionized by emission of one or more electrons. Details of the collision-induced X-ray spectra are complex because of absorption and fluorescence of X-rays in the target, and all equations for multi-element targets are merely approximations; theoretical problems are least for the thin-film condition in analytical electron microscopy (AEM). Because a proton is much heavier than an electron, the bremsstrahlung is insignificant for collision with an atom, and characteristic X-rays are emitted without the continuous spectrum.
X-ray fluorescence (XRF) using primary X-ray and gamma-ray sources is important for chemical analysis, especially of trace elements, because of the absence of the continuous spectrum generated from electrons in an EMP. X-ray absorption is exploited in X-ray microscopy (XRM) and X-ray tomography (XRT), electron-energy-loss spectroscopy (EELS), and X-ray absorption spectroscopy (XAS.). Portable alpha-ray sources are also used to generate X-rays for radiography.
The continuous electromagnetic spectrum emitted upon impact of electrons with atoms contains photons with lower energy than X-rays. The visible light is used directly in cathodoluminescence microscopy (CLM), and the ultraviolet and infrared radiation can be examined indirectly with special detectors.
Electrons and electromagnetic radiation can be observed by absorption in the silver emulsion of a photographic film. Individual photons produce ionizations in solid or gas counters which yield a detectable pulse after amplification. The energy of each photon can be estimated either by measuring the wavelength with a diffraction-based crystal spectrometer (WDS) or by direct measurement with a solid-sate detector of the energy of the ionization pulse (EDS).
High-energy neutrons (several MeV) are produced in various nuclear reactions including fission, fusion, and reactions involving incident particles and photons. Slow neutrons with a continuous spectrum peaking near 1Å are produced by multiple scattering down to thermal velocity by collisions with light nuclei in various moderators (e.g. water, graphite). Neutrons are detected by collision with nuclei which then cause ionization in a counter; the wavelength is measured with either a diffraction-based crystal spectrometer or a time-of-flight instrument.
2.1 Generation of Electrons
2.1.1 Radioactive Decay
Negatively-charged beta-rays (electrons, of course) are emitted with kinetic energy from zero up to 10 MeV from certain radioactive nuclei. Each nucleus gives monochromatic beta-rays which can be used in spectroscopy. A table of energy and half-life is given in the American Institute of Physics Handbook.
2.1.2 Thermionic Emission
The emission of electrons into a vacuum from a heated electronic conductor is illustrated in Fig. 2.1. In a cold pure metal (a), all electrons are below the Fermi energy level. Thermal energy allows electrons to form a space cloud in the vacuum (b), and application of an electric field allows the electrons to be collected on an anode; otherwise, an equilibrium is set up between the electrons inside and outside the metal. The Richardson equation for the saturation current density is
I = AT2 exp ( - W/kT) 2.1
where I is the emission current density (amp.m-2), A is a property of the conductor, W is the work function (eV), k is Planck's constant, and T is the absolute temperature (Kelvin). A tungsten wire is used in most X-ray tubes, electron microscopes and electron microprobes to take advantage of the high temperature for melting (3680 K) and evaporation. In a conventional X-ray tube, the wire is a coil approximately 1 cm by 1 mm , and the temperature is adjusted to minimize evaporation of W atoms which slowly contaminate the target. In the EM and EMP, the wire is bent into a V shape so that the maximum temperature and electron emission occurs from the tip of the filament (Fig. 2.2); the area of this tip is then demagnified by lenses to give a fine electron beam on the specimen. For tungsten, the work function between the Fermi level and the vacuum is 4.5 eV and A is about 106 amp.m-2.
Unless an accelerating voltage is applied, there is no emitted current from a hot filament because of the formation of a space charge of electrons near the metal surface. The saturation current is measured by using the metal as a cathode of a vacuum tube and collecting the electrons on an anode which is sufficiently positive to dissipate the space charge. In a conventional X-ray tube, sufficient stability is obtained by regulating the filament voltage (for heating) and the accelerating voltage between cathode and anode. In the EM and EMP, the emission current from the electron ''gun'' is controlled very accurately by enclosing the filament in a cavity using a grid cap with a small hole (Fig. 2.2). As the filament temperature increases (Fig. 2.3), saturation occurs of the space charge in the cavity and the number of escaping electrons remains constant as the temperature is increased further. The number of these electrons is controlled by a bias voltage applied between the filament and the grid cap. After escaping through the hole, the electrons are accelerated towards an anode plate and pass through a hole in it. It is extremely important for filament life and mechanical stability that the filament be heated only up to the beginning of saturation; as the filament resistance increases due to thinning from evaporation the filament heating voltage should be reduced.
2.1.3 Field Emission
Even a cold conductor will emit electrons into a vacuum if a strong enough electric field is applied. The electrons overcome the work function by quantum-mechanical tunnelling; for a work function of 4 eV, a field greater than 109 volt/meter is required for significant emission. A pointed tip is used to obtain very high brightness in some electron microscopes with very high spatial resolution; W and LaB6 are suitable materials, and they must be kept in an ultrahigh vacuum to minimize contamination. The field-emission microscope (1936: E.W. Muller) can provide a magnified image of a metal tip with a resolution of 2 nm from the emitted electrons, or of 2 Å for He ions instead of electrons.
2.1.4 Photoemission
Photoemission of electrons from a material occurs by conversion of energy from incident photons ranging from the infrared to X-ray regions of the electromagnetic spectrum. The electrons are emitted promptly at a rate proportional to the intensity of the incident radiation, and the kinetic energy E of the electrons varies from 0 to (hn - P) where n is the frequency of the radiation and P is the photoelectric threshold energy (Fig. 2.1); in metals, P equals the work function W, whereas in semiconductors it is larger. This Einstein relation is only approximate, and a minor correction is needed for the Compton effect (Chapter 1) which is complicated by the degree of coupling of the electron momentum to the potential of the host material.
The photoelectrons can be removed by an electric field. They are focused and magnified in a photo-emission-electron-microscope (PEEM) to give an image of a surface which depends on variation in the photoelectric threshold energy and the topography (Wegmann, 1970). The photoelectric threshold energy depends on both chemical composition and crystallographic orientation of the surface. Because photoelectrons are absorbed very strongly, the depth resolution of PEEM is about 10 nm for common minerals illuminated by ultraviolet light in a commercial instrument. Figure 2.4 is a photoemission electron micrograph of a peristerite intergrowth of albite (dark) and oligoclase (light) whose details are explained in the legend.
Photoemission is also used in phototubes and photoelectron multipliers which are used as detectors of electromagnetic radiation (section 2.4).
2.1.5 Secondary-electron Emission
Consider a beam of primary electrons (Fig. 2.5; PE) with an energy of (say) 15 to 100 keV entering a material with a flat surface. Most of the electrons will be absorbed in the target after undergoing elastic and inelastic collisions with electrons of the target material. Elastic means no loss of kinetic energy (''billiard-ball'' collision) whereas inelastic means that energy is transferred to the struck electron. Some of the transferred energy goes into X-rays and Auger electrons, but most ends up in lattice vibrations (heat). A small fraction of the electrons escapes back into the vacuum, and the electrons are divided into the back-scattered electrons (BSE) and the secondary electrons (SE). The BSE can be arbitrarily divided into ones which have undergone either low or high energy loss (LL and HL); the former escape after a collision within about 10 nm of the surface, and the latter range down to about 50 eV energy after undergoing multiple collisions as deep as about 1 µm. The SE have an energy up to 50 eV and can be generated either by the primary electrons (PE) or the back-scattered electrons (BSE). These secondary electrons can escape only from a surface layer a few nanometers deep at most.
The intensity of BSE is approximately proportional to the atomic number Z, and is little affected by surface topography for a surface polished with 1 µm abrasive. However, the SE are strongly affected by the topography, and indeed are the key to the success of scanning electron microscopes for imaging of unpolished surfaces. Details are given in Chapter 5, but briefly a focused beam of PE as narrow as 10 nm, or even less in special instruments, is scanned across the surface of a specimen, as in a television receiver. The intensity from an SE detector or a BSE detector is used to modulate the brightness of the image on an oscilloscope whose beam is scanning in synchronism with the PE beam. The BSE detector shows the variation of the mean atomic number Z across a flat specimen, and the SE detector demonstrates the topography without loss of focus as in an optical microscope. Figure 2.6 shows representative photographs of a polished meteorite and a zeolite powder. The surface of an insulator must be coated with a thin metal film. Because the specimen current is reduced by the sum of the BSE and SE, it may be used to give a ''negative'' image of the surface. The spatial resolution is better for the BSE (LL) image than for the SE image, and becomes equal to the diameter of the incident beam for electrons with no significant energy loss.
2.1.6 Ionization, Emission of Auger Electrons and X-rays
Although most of the kinetic energy (>99%) that is transferred to the target from a primary electron ends up on average as heat, an occasional PE ionizes an atom by ejection of an orbital electron (Fig. 2.7a). Each resulting pair of electrons will be involved in interactions with other orbital electrons, and a small fraction will ultimately contribute to the BSE and SE (preceding section). Each hole in an inner shell (e.g. K) of an ionized atom will be filled by an electron which drops down from an outer shell (e.g. L). The released energy will appear either as an X-ray photon (Fig. 2.7c) or as an Auger electron (Fig. 2.7d). The former leaves a singly-ionized atom which can undergo further electronic transitions until electric neutrality is attained. The latter leaves a doubly-ionized atom which will ultimately become neutral. In the process of X-ray-induced ionization (Fig. 2.7b), a singly-ionized atom is produced by energy transfer from an X-ray photon; once the ionization has occurred, the subsequent emission of X-ray photons and Auger electrons is identical to that for electron-induced ionization.
Auger electrons were discovered by P. Auger in 1925 after experiments revealed that insufficient X-rays were being produced by the process of fluorescence (Compton and Allison, l949, p. 479-482). In a Wilson cloud chamber containing argon atoms irradiated with X-rays, a long ionization track was found for each orbital electron knocked out by an X-ray photon (hn1). The kinetic energy E3 of this electron (which is a photoelectron) is equal to hn1 minus the ionization energy. In Figure 2.7b, the K, L and M electron shells are shown schematically as arcs around the atomic nucleus (square), and the ionization energy for a K electron is denoted EK. The ionization energy is the energy required to remove the electron to infinity (i.e. equivalent to the vacuum in Fig. 2.1). Most, but not all (90%), of the long ionization tracks were accompanied by a short irregular track which was attributed to an Auger electron (Fig. 2.7d) with lower kinetic energy than the photoelectron. The single tracks (10%) were explained by emission of a fluorescent X-ray hn2 which did not ionize the argon gas. For both the Auger electron and the fluorescent X-ray, the energy is obtained from the transfer of an outer electron (e.g. L) to fill the hole (e.g. K). The Auger electron escapes from the L shell in Figure 2.7d leaving a doubly-ionized atom, in contrast to the singly-ionized product in Figure 2.7c. Note that the term fluorescence is appropriate only when the ionization is produced by an X-ray.
Conservation of energy requires the following equations:
E1' + E2 = E1 - EK (Fig. 2.7a) 2.2
E3 = hn1 - EK (Fig. 2.7b) 2.3
hn2 = EK - EL (Fig. 2.7c) 2.4
E4 = EK - EL -EL' (Fig. 2.7d) 2.5
where E1...E4 are the kinetic energies mv2/2 of the various electrons, EK and EL are the ionization energies for an electron in the K or L shell of a neutral atom, and EL' is for an L electron of a singly-ionized atom.
The probability that an ionized atom will emit either an Auger electron or an X-ray is given by the fluorescence yields wA and wx where
wA + wx = 1 2.6
For ionization of the K shell, the fluorescence yield of K X-rays increases from 0.02 for sodium (Z = 11) to 0.4 for copper (Z = 29) to 0.8 near Z = 46. Thus only a tiny fraction of isolated Na atoms emit X-rays directly. However, in a solid, all the Auger electrons are absorbed except for those very close to the surface (few nm). Newly ionized atoms emit further X-rays.
Auger electrons can be used for chemical analysis of atoms in a surface layer. Because of absorption before the electrons escape into a vacuum, the energy spectrum for each type of Auger electrons ranges from zero up to the initial kinetic energy determined by the orbital energy levels. In a multielement material, each chemical element will have its characteristic Auger spectrum. Figure 2.8 shows an Auger spectrum for magnesium oxide in which the energy limits for Auger K electrons from Mg and O are displayed.
[The details of Auger electron generation are complex, and depend on an understanding of electronic structure and selection rules. Figure 2.9 summarizes some of the processes. The K and L levels are split into 3 and 5 sublevels labeled with Roman numerals in decreasing order of ionization energy (Figure 2.9a). Filling of a hole in the K shell by an LIII electron will yield either a Ka X-ray (Fig. 2.9b) or an Auger electron (e.g. from the LII shell, Fig. 2.9c). This Auger electron can be imagined to result from auto-ionization by an imaginary X-ray photon. Figure 2.9d shows emission of an MII rather than an LII Auger electron. Transfer of an MI electron to the LIII level in (e) might lead to escape of an MIV electron with production of a triply-ionized atom. The Coster-Kronig transition involves movement of a vacancy between two subshells of the same shell (e.g. LI to LIII) and emission of an Auger electron from an outer shell (e.g. MII; Fig. 2.9f).]
2.1.7 Generation of Electron Beams
The simplest instruments accelerate the electrons in a straight line. For accelerating voltage (V) up to 100 keV used in conventional X-ray tubes and electron microscopes, direct application of voltage between the cathode source and an anode (e.g. Fig. 2.2) is sufficient. For studies in atomic and nuclear physics, various kinds of linear accelerator (LINAC) produce electrons with high energy (e.g. Van der Graaff generator, up to 10 Mev; travelling-wave accelerator, up to 50 GeV); however, they have too high an energy to be directly useful for most techniques involving diffraction and chemical analysis. Rapidly gaining importance as an intense source of ultraviolet and X-radiation is the synchrotron storage ring in which the centripetal acceleration of electrons in circular motion leads to emission of a continuous spectrum of electromagnetic radiation along the forward tangent (next section).
2.2 Generation of Electromagnetic Radiation
2.2.1 General
Although emphasis is placed here on the generation of X-rays, softer radiation is also considered. Three processes can generate electromagnetic radiation: •nuclear reactions, •electronic transitions in excited or ionized atoms, and •acceleration or deceleration of charged particles. Gamma-rays from nuclear reactions provide portable sources of X-rays, but are not used in the region of longer wavelengths.
Visible light, together with some infrared and ultraviolet light, is generally produced by electronic transitions in atoms excited in one of four sources: •a hot tungsten filament in an ordinary light bulb; a metal vapor (e.g. Na, Ne, Hg, Xe) in a gas-discharge lamp; •a phosphor combination in a fluorescent lamp; and an electrically-driven arc (e.g. C). Most of these sources are polychromatic (i.e. have many wavelengths), and monochromatic light is usually selected with a grating monochromator. Gas-discharge lamps emit light with specific frequencies controlled by transitions between outer electrons of atoms, and monochromatic light can be obtained by using appropriate absorbing filters (e.g. Na and Hg light sources used in crystal optics). All the light from the above sources is incoherent. Coherent light from various kinds of gas and solid lasers (Light Amplification by Stimulated Emission of Radiation) has advantages for certain experiments.
X-rays are generally produced by penetration of an electron beam into a target. Ionized atoms undergo electronic transitions which provide a characteristic spectrum of X-rays consisting of a few sharp lines characteristic of the target material. The X-rays can be used for chemical analysis in an electron microprobe (EMP) or an analytical electron microscope (AEM). Atoms can also be ionized by interaction with X-rays, allowing chemical analysis by X-ray fluorescence (XRF). Excitation by protons and alpha particles also allows chemical analysis from the emitted X-rays and the technique is labeled PIXE (Particle- or Proton-Induced X-ray Emission). Electromagnetic radiation of longer wavelength is emitted in all these instruments, but is too weak to be usable in routine studies.
Acceleration of charged particles results in the emission of a continuous spectrum of electromagnetic radiation. Protons and alphaparticles are too heavy to give useful radiation, but electrons with kinetic energy near 1 MeV produce an intense beam when undergoing centripetal acceleration in a circular orbit. Synchrotron storage rings (SSR) of electrons have become important sources of X-ray and ultraviolet light; note that synchrotron scientists may use ''light'' as a shorthand for ''soft electromagnetic radiation'', and do not imply ''visible light.'' Electromagnetic radiation is also emitted by deceleration of electrons in matter, and the characteristic X-ray spectrum mentioned in the preceding paragraph is superimposed on the continuous spectrum of the Bremsstrahlung (German for braking radiation).
Ionized atoms can lose energy by emission of an Auger electron (section 2.1.6) instead of a photon. Some photons and Auger electrons will be captured in the target material and further electronic transitions will be induced in new atoms with concomitant emission of electrons and photons. Each new hole will be filled in turn by an electron of lower energy until finally the atoms of the target become electrically neutral. Insulators must be coated with a conducting film in order to bleed off excess electrons. The various electrons moving through the target will undergo deceleration and contribute to the continuous spectrum. All these processes result in a complex theory for the generation of the combined characteristic and continuous spectrum from electron impact. In the simple theory, it is assumed that photons are emitted only .from singly-ionized atoms, but in the full theory it is necessary to consider weak satellite lines emitted from multiply-ionized atoms. Again in the simple theory, only certain lines are allowed by selection rules, but in the full theory it is necessary to consider weak forbidden lines which result from distortion of electron clouds. The energy levels of orbital electrons close to the nucleus are essentially unaffected by chemical bonding, but the levels of the outer electrons are affected thereby allowing information to be obtained from the wavelength of associated lines. Although the intensities of the spectral lines are regular functions of the atomic number Z, complications arise from absorption and from fluorescence in multi-element targets. Nevertheless X-ray fluorescence analysis and electron microprobe analysis are extremely important analytical tools based on measurement of X-ray spectra excited respectively by X-rays and electrons.
2.2.2 Gamma-rays
Gamma-rays are involved in many nuclear reactions, and are of principal interest here as a portable source of X-rays. A nucleus left in an excited state after alpha-decay, beta-decay, or electron capture can emit a gamma-ray (i. e. an electron) with an energy mostly in the range 0.05 to 3 MeV. Only a soft gamma-ray has a half-life long enough to be of practical use. A disc containing 55Fe atoms is used for testing X-ray detection systems with the Mn Ka rays (half-life 2.7 years); 22 other sources of soft g-rays are listed in Heinrich et al. (1981, p. 28-33). Hard a-rays are useful for radiography of thick metal castings and pipes (e.g. 60Co , 1.2 and 1.3 MeV).
2.2.3 Electronic Structure, Energy Levels, Selection Rules and Transition Probability.
Quantum Numbers. The Bohr-Sommerfeld model of the atom provides a convenient introduction, but it must be emphasized that each fixed electron orbit is merely a convenient average of the probability distribution of an electron cloud (orbital) required by quantum mechanics.
Each electron is specified by a unique set of quantum numbers (Pauli's exclusion principle).
(a) The principal quantum number n specifies the momentum mvr = nh/2_, where m is the electron mass, and r and v are the radius and velocity for the mean orbit. For an atom with atomic number Z, the sum of the kinetic and potential energies for a circular orbit is
E(n) = - 2_2me4Z2/h2n2. 2.7
Each integral value of n corresponds to 2 or more electrons in a shell, each of which inherits the historical labeling of atomic spectra: n = 1, K; 2, L; 3, M; 4, N. In detail, an adjustment is needed in E(n) for the nuclear mass which affects the center-of-mass and produces an isotope effect for light elements. The amount of energy E(n) corresponds to a photon energy hn(n). Spectroscopists use the wave number n(n)/c. An electron transition from ni to nj with i > j releases the following amount of energy which may appear as a photon:
hn = 2_2me4Z2/h2.(nj-2 - ni-2) 2.8
The wave number of the photon is
RZ2(nj-2 - ni-2) 2.9
where the Rydberg constant R is
2_2me4/ch3 = l09,737 cm-1. 2.10
In detail, the outer electrons are shielded by the inner electrons from the electric charge on the nucleus, and the atomic number is reduced to (Z - s) where s is the screening constant which changes with i and j, and also with Z when Z is large. The theoretical and experimental bases for the above equations were established in 1913 by N. Bohr and H.G.J. Moseley; the latter died in World War 1.
(b) The azimuthal quantum number l expresses either the shape of the electron orbit (Bohr-Sommerfeld model) or the probability distribution of an orbital for the wave-mechanical description. Quantization of the angular momentum changes a circular orbit (l = 0) to an increasingly elliptical orbit as the integer l increases from 1 to (n - 1). The orbital energy is almost independent of l, but there is a slight decrease as l increases because of the precession of the elliptical orbits and because of variation in a screening effect (see later). Spectral designations of s (sharp), p (principal), d (diffuse), f are applied to orbits with l = 0, 1, 2, and 3 respectively.
(c) The spin quantum number s expresses the two choices of + 1/2.h/2_ for the angular momentum S of the electron spin. The vector sum of the orbital angular momentum L and the spin momentum S is the total angular momentum J. and the quantum number is denoted j; this summation results from interaction of magnetic forces from the orbiting and spinning motions of the electron. For a single electron with l = 1, the electron spin can be either parallel or antiparallel giving j = 3/2 or 1/2 for the combined angular momentum of the elliptical orbit and the electron spin. The resulting difference in energy gives a fine doublet for a photon produced by an electron transition from a state with l = 1 to a state with l = 0. This simple theory for interaction between L and S must be replaced by the Dirac theory in which the spins depend on [s(s + 1], [l(l + 1)] and [j(j + 1)], and the spin-orbit splitting of the energy (Eso) for a single electron is
Eso = [ - Ra2Z4/2n3l(l + 1/2)(l + 1)].[j(j+ 1) - l(l + 1) - s(s + 1)] 2.11
The fine structure constant a is given by
a = 2_e2/hc. 2.12
(d) The magnetic quantum number m expresses the quantization of the component of J projected along an applied electric or magnetic field; m may take integral values from +l to - l.
Selection Rules. Electron transitions are governed by conservation of parity in the quantum-mechanical theory. Because the dipole electric field associated with a photon has one unit of parity, an uncoupled electron must change its l quantum number by 1 in an allowed transition involving photoemission. There is no parity restriction on the quantum number n. The transitions forbidden for a dipole-controlled event are sometimes observed with a low probability because of effects from electric quadrupoles and magnetic dipoles associated with distorted orbitals. Such forbidden lines are not observed for deep-seated transitions between electrons whose energy is dominated by interaction with the nucleus, and which are strongly shielded from chemical bonds. The orbitals for outer electrons of light atoms are particularly distorted from the ideal shape, and the theory of optical spectra from excited atoms is complicated because of coupling between the orbitals which are strongly shielded from the nuclear charge. A deep-seated orbital close to the nucleus has a nearly ideal probability distribution which is essentially unaffected by the other orbitals, and the only significant coupling is between its own angular momentum and spin momentum (jj coupling). In contrast, the shielded orbitals of an isolated light atom tend to interact so that the individual angular momenta (Li) of i electrons couple together to give a resultant L, while the spins (Si) couple to give a resultant S; then the two resultants couple together to give the total angular momentum J. This ideal LS coupling (known as Russell-Saunders coupling) becomes less important as the atomic number Z increases and the orbitals become deeper seated, and there is a continuous transition to jj coupling. The selection rules for LS coupling are DL = 0, ±1, DJ = 0, ±1, and DS = 0; however, a transition cannot occur between two states with J = 0.
Spectral Terminoloqy. Energy levels could be labeled simply with the quantum numbers, and spectral lines by the labels of the initial and final quantum states of the electron. However, spectroscopists give this information in a way which can be confusing at first. Consider first a single electron with a set of values of l, s and j. The value of l is specified by a capital letter S (for l = 0), P (1), D (2) or F (3), the value of j by a trailing subscript, and the number of ways in which l and s can be added vectorially to give j by a leading superscript. Thus l = 2 and s = l/2 give D from the value of l, there are two ways of combining l and s to give j, and the values of j are 3/2 and 5/2; hence there are two spectral terms 2D3/2 and 2D5/2. For l = 0, the only term is 1S1/2, and for l = 1 the terms are 2P1/2 and 2P3/2. Apparently the leading superscript is redundant, but this is not so when LS coupling is involved.
Consider two p electrons with different n. Both have l = 1 and the resultant l can be quantized as 0, 1 and 2 (Fig. 2.10a: note the equilateral triangle for L = 1). The spins of the p electrons (s = l/2) can be summed to S = 0 and 1 (caution: do not confuse this S with the spectral label S which derives from Sharp). A D spectral term with L = 2 can couple with S = 0 to give J = 2, and with S = 1 to give J = 1, 2 and 3 (note the isosceles triangle in Fig. 2.l0b for J = 2). Because the latter is a triplet, the leading superscript is 3, and the three subterms are 3D1, 3D2 and 3D3. The term for S = 0 and L = 2 has only one choice of J = 2, and it is labeled 1D2. For L = 1, coupling with S = 0 gives 1P1, and with S = 1 gives 3P0, 3P1 and 3P2. Finally for L = 0, coupling with S = 0 gives 1S0, and with S = 1 gives 1S1.
Optical Spectra from Hydrogen. All complications can be ignored for a hydrogen atom whose single electron occupies the state of lowest energy (n = l = 0) unless it is excited into higher states. The energy scale in Figure 2.11 is given directly in eV and indirectly in wave-number (cm-1). The energy levels become closely spaced as n increases because of the l/n2 term in equation 2.7 (Z, of course, is 1 for hydrogen), and a continuum (i.e. for an unbound electron) is reached at about 14 eV from the stable level with n = 1. For convenience, the electrons are plotted in rows and columns according to n and l, and sloping lines show the transitions only for n up to 3. The 2s level with n = 2 and l = 0 is given the spectral term 2S1/2 because j = s = ±1/2; actually this label applies to all the s levels whatever the value of n because there is only one electron, and LS coupling cannot apply. The ± corresponds to the two directions of electron spin which can lead to an energy term depending on the orientation with respect to a magnetic field (anomalous Zeeman effect). For l = 1, j can be either 1/2 or 3/2, and for l = 2, j can be 3/2 or 5/2. To a first approximation, the spectral lines have wave-numbers given by
R(1/n12 - 1/n22) 2.13
where n2 and n1 are the initial and final quantum numbers of the electron (see equations 2.8 and 2.9). Each value of n1 gives a particular series of lines (1, Lyman; 2, Balmer; 3, Paschen) which reaches a limiting energy for infinite n2; this is the ionization energy for n1. To a second approximation, the spin coupling causes energy shifts (Figure 3.11) which result in a fine structure of each line; thus the Ha line from n = 3 to n = 2 should have 7 components. Actually there are only 5 observed lines because of accidental overlaps caused by the Lamb shift, which is a subtle interaction of an electron with the zero-point fluctuations of an electromagnetic field.
There is a hyperfine splitting caused by interaction between the magnetic moments of the orbiting electron and the nuclear magnetic moment (if non-zero); additional lines occur if a chemical element has isotopes with different magnetic moments. Finally, there is a slight change of energy from the mass of the nucleus. The width of spectral lines depends on the lifetime of a particular state, the Doppler shift, and the interaction between adjacent atoms. All these factors have been clearly established for hydrogen gas, but are not so readily evaluated for heavier atoms with several electrons. Fortunately most of them can be neglected for most purposes, especially in the X-ray spectrum; however, the interaction between atoms in a solid leads to serious complications (cf. Fig. 2.1) which can yield information of considerable value to solid-state physicists and chemists.
X-rays and Optical Spectra from Sodium. Going to the next level of complexity, consider a sodium atom whose single electron in the 3s level has some similarity to the single electron in the hydrogen atom. The other ten electrons (Table 2.l) occupy the 1s, 2s and 2p levels. To satisfy the Pauli Exclusion Principle, there are two electrons with s = +1/2 or - 1/2 in each s shell. For the 2p shell with l = 1, j can be either l/2 or 3/2. The j = 1/2 state contains two electrons corresponding to m = 1/2 and - 1/2. The j = 3/2 state contains four electrons corresponding to m = - 3/2, - 1/2, +1/2 and +3/2 when a magnetic field is present. In the absence of a magnetic field, the different values of m do not lead to any energy difference, and the L shell is subdivided into LI for 2s, and LII and LIII for the j = l/2 and 3/2 subdivisions of 2p. The closed shells have no residual electron spin, but the single 3s electron is uncoupled. The LII,III electrons have an ellipsoidal probability distribution whose interaction with the nucleus is less than for the LI electrons, and the energy level is lower for the former by 32 eV; there is only a trivial energy difference for the 2p electrons and a mean value of 31.1 eV above the MI state is given in Table 2.l.
When a sodium vapor is excited in an electric discharge, the 3s electron can move into an excited state, of which the first seven are listed in Table 2.1 and shown in Figure 2.l2. Note that the 4s state appears before the 3d state because the elongated probability distribution results in lower interaction with the nucleus than the isotropic distribution for the 4s state; a similar effect explains the whole sequence 3p, 4s, 3d, 4p, 5s, etc. [Actually the overlap of wave functions must be considered for quantum-mechanical calculations.] In order to return to the 3s ground state, the electron undergoes one or more transitions and emits photons. If in a p state, it can go directly to the 3s state by changing l by 1, but if it is in an s, d or f state, it must go indirectly by way of a p state because l cannot change by any other number than 1 in a dipolar transition. This conclusion is based on jj coupling for undistorted orbitals. For sodium, the outer orbitals are distorted because of interactions, and LS coupling allows weak lines which disobey the Dl = 1 rule for allowed transitions in jj coupling. Only the strong allowed transitions are shown in Fig. 2.l2.
Table 2.1 Energy levels for a sodium atom
Number of Quantum Numbers Energy
Labels Electrons n l s j m eV
excited states
4d 4 2
} } - 4.2
4f 4 3*
5s 5 0 - 4.1
4p 4 1* - 3.6
3d 3 2 - 3.5
4s 4 0 - 3.1
3p 3 1* - 2.1
unexcited atom
3s MI 1† 3 0 1/2 1/2 1/2 0
LIII 4 2 1* ±1/2 ±3/2 ±3/2, +1/2
2p } } 31 1
LII 2 2 1* ±1/2 ±1/2 ±1/2
2s LI 2 2 0 ±1/2 ±1/2 ±1/2 63.3
1s K 2 1 0 ±1/2 ±1/2 ±1/2 1072.1
* States with odd parity in l. †Only 1 out of 2 positions occupied.
The intensity depends on the number of atoms in the initial state, and on the transition probability. The former depends on the excitation conditions, particularly the temperature of the sodium vapor, and the second depends on the interaction between the wave functions of the initial and final state. The latter decreases strongly with Dn and depends less strongly on the values of l. Consequently the strongest lines in the optical spectrum of excited Na vapor are the well-known yellow NaD doublet at 5896 and 5890 cm-1.
An unexcited sodium atom ionized by removal of a K electron can emit a photon (or an Auger electron) as an electron is transferred from an LII or an LIII subshell with Dl = 0. These subshells are so similar in energy that only a single Ka line at 11.9 Å should be observed. The transition M Æ K is forbidden for an isolated atom, but a weak ß line near 11.6 Å can occur as a shoulder on the Ka line for Na atoms in a solid because the M electron actually becomes part of the valence band for which the orbital is not symmetrical with l = 0. Actually the Ka line for Na atoms in a solid is not fixed at 11.9 Å because even the 2p levels are affected by the chemical bonding. This means that standards for electron microprobe and X-ray fluorescence analyses must be chosen to have similar chemical bonding to the unknown.
Periodic Table of Chemical Elements. In unexcited atoms, the Z electrons occupy the lowest energy levels, and the Periodic Table of Chemical Elements can be developed from the Pauli Exclusion Principle and the Aufbau procedure. For low Z, the assignment of electrons is straightforward, but for high Z there are complexities caused by subtle differences in the energy levels. Up to argon (Z = 18), the electrons enter the shells successively in the obvious order 1s, 2s, 2p, 3s and 3p. However, the extra electron in potassium (Z = 19) goes into the ''spherical'' 4s shell instead of the highly ''ellipsoidal'' 3d shell, because of the energy difference described for the excited Na atom. The metals of the first transition series (Sc, 21 to Cu, 29) fill the 3d shell as electrons remain in the 4s shell. In general, this preference for a spherical orbit is maintained throughout the Periodic Table with the following filling sequence in the Aufbau procedure: 4s, 3d, 4p, 5s, 4d, 5p, 6s, 4f, 5d, 6p, 5f. In detail, there are minor anomalies (e.g. Cr has only one 4s electron whereas its neighbors have two) and complications (e.g. high- vs. low-spin state of Fe in which different values of s and j are used). Tables 2.2 and 2.3 show the electronic structures for Fe (Z = 26) and Eu (Z = 63).
X-ray Spectra. The lines of the X-ray emission spectrum are usually given two labels: first, the designation of the shell with the original hole (K, L, M, N), and second, a Greek letter which specifies the electron shell which develops a hole after electron transfer. Unfortunately, the second label is arbitrary (Compton and Allison, 1949, p. 596), and it is advisable to consult a reference table (e.g. American Institute of Physics Handbook). For complete clarity, it is possible to use the designations of the levels of the electron holes: e.g. Kß for sodium is K ¨ M. In routine work, only a few lines are needed: thus the a1, a2 and ß1 lines in the K spectrum of phosphorus result from an electron moving respectively from the LIII, LII and M shells. Figure 2.l3 shows the variation of the wavelength of the four principal X-ray lines or clusters of lines against Z. The plot is almost linear between l-1/2 and Z in accord with Moseley's equation (2.8 with (Z - s ) in place of Z); the non-linear deviation is due to variation of the screening constant s with Z. In the K spectrum, the wavelength changes from 0.24 Å at Z = 70 to 12 Å at Z = 11, whereas in the L spectrum the corresponding wavelengths are 1.7 and 400 Å.
Because of the splitting of energy levels for each principal electron shell (K, L, M, etc.) as l and j are changed, each principal X-ray line becomes split into two or more lines as Z increases. Consider, first, the relatively simple spectrum for Fe (Z = 26) whose electronic structure is given in Table 2.2 with energy levels for an isolated atom
Table 2.2 Energy for quantized levels of orbital electrons of unexcited iron atom
Labels Number of Quantum Numbers Energy
Electrons n l j eV
4s NI 2 4 0 1/2 0
3d MV 2† 3 2 5/2
{ } 3.6
MIV 4 3 2 3/2
3p MIII 4 3 1* 3/2
{ } 54 0
MII 2 3 1* 1/2
3s MI 2 3 0 1/2 92.9
2p LIII 2 2 1* 3/2 708.1
{ }
LII 4 2 1* 1/2 721.1
2s LI 2 2 0 1/2 846.1
1s K 2 1 0 1/2 7112.0
*States with odd parity in l. †Only 2 out of 6 positions occupied.
The iron atom is assumed to be in a low-spin state.
unaffected by chemical bonding. The characteristic X-ray spectrum contains the following lines:
LII Æ K Ka2 1.9400 Å
LIII Æ K Ka1 l.9360
MII,III Æ K Kß1,3 l.7566
MIV,V Æ K Kß5 1.7442
MII,III Æ L4 Lß3,4 l5.6
MI Æ LII Lh 19.7
MIV Æ LII Lß1 17.3
MI Æ LIII Ll 20.1
MIV,V ÆLIII La1,2 17.6
Each wavelength can be calculated from the difference in energy between the final and starting levels: e.g. Ka2 and Ka1
l2,398 / (7112.0 - 721.1) = 1.9400 Å 2.14
12,398 / (7112.0 - 708.1) = 1.9360 Å 2.15
The Ka line is split into a doublet which is resolved in careful work, but is observed as a single line in instruments with low resolution. The Kß should be single because MII and MIII have essentially the same energy level; however a forbidden line Kß5 with Dl= 2 occurs with a low intensity because the 3d orbital is close to the ground state of energy, and is distorted by chemical bonding. The L spectrum contains five lines which merge into a complex line whose mean wavelength and profile depends on the chemical bonding.
The spectrum for europium (Z = 63) is much more complicated, and it is difficult to find a complete set of data from reference tables. The spectral lines listed in the American Institute of Physics Handbook, p. 7-l07 and other tables, are matched with the electron transitions in Table 2.3. Several allowed lines in the M spectrum for transitions to MI, MII and MIII were not listed, but presumably could be found by careful experimental study. A weak line would be expected for Mv Æ LII.
Readers will find that interpretation of X-ray spectra is difficult for atoms with high Z, and for all transitions involving outer electrons whatever is the value of Z. A forbidden line will actually be present for a transition involving an energy level up to ~l00 eV from the ground state. Furthermore any outer energy level can be perturbed by several eV simply because of chemical bonding; this is particularly serious between the metallic state and an ionic state. As a crude rule of thumb, forbidden transitions have trivial intensity for X-rays with wavelength less than 5 Å (2 keV), and become increasingly important for longer wavelengths into the ultraviolet. The perturbation of energy levels follows a similar rule; thus in electron microprobe analysis (EMPA), all iron compounds give essentially the same wavelength for FeKa radiation (1.94 Å), but even the change from 4- to 6-coordination of Al (Ka 8.3 Å) in oxygen compounds produces a wavelength shift which must be compensated in quantitative analysis with a crystal spectrometer.
The relative intensities of the lines in the X-ray spectrum depend on two factors: the population of the initial electron shell which supplies the electron, and the degree of interaction between the wave fuctions for the initial and final states of the electron. The basic theory is inadequate, and the experimental data are imprecise because of complications from absorption in the X-ray source (Compton and Allison, l949, p. 637-654). Because the LIII state has twice as many electrons as the LII state, and because their wave functions are the same except for the weak magnetic characteristic, the Ka1 line is twice as intense as the Ka2 line. All the Kß lines must be weaker than the Ka lines because the M wave functions interact less strongly than the L wave functions with the K wave function. The Kb1 line ranges from one-sixth to one-fourth of the intensity of the Ka1 line as Z increases from 25 to 50, while the Kb2 line which involves the N shell has only one-hundredth (Z = 30) to one-twentieth (Z = 50) of the intensity of the Ka1 line. In the L spectrum, the variation of line intensity is complex. For Ba (Z = 56), only three lines are observed in routine work with the La1 and La2 lines overlapping to give the strongest peak at 2.78 Å while weaker Lb1 (2.57 Å) and Lb2,15 (2.40 Å) give separate peaks. For higher atomic numbers (Z ~80), the approximate intensities in the L spectrum are a1 l00%, a2 11%, b1 50%, b2 25%, b3 5%, b4 4%, b6 l%, g1 l0%, l 3%, various other lines 0.2-1%.
In addition to forbidden lines which result from distortion of the electron orbitals, there are satellite lines which result from electronic transitions in multiply-ionized atoms. Such multiple ionization is particularly frequent in the light elements because of the high probability for emission of Auger elements (loc. cit.) and becomes less frequent for the heavy elements. Because the shielding of the nuclear charge is reduced by absence of one or more electrons, the X-ray photon has higher energy (typically 10 - 30 ev) than for a singly-ionized atom, and the satellite line is always displaced slightly to the high-energy (low-wavelength) side. For Z > 20, it is difficult to detect satellite lines, but they are easily detected for the lighter elements with a wavelength spectrometer. White and Johnson (1972) list some but not all the satellites, and it is wise to run a standard with a similar type of chemically bonding when a complete identification is needed of all spectral lines for an unknown sample. Thus a silicon Kb1 satellite is close to the Sr La1 line, and is a serious problem for analysis of Sr in feldspar minerals (Smith and Ribbe, 1966).
Angular Distribution of Photons. The wave functions for an isolated ionized atom are effectively spherical during emission of a photon, and the radiation pattern for photons is isotropic (i.e. spherical), and there is no polarization. Because the outer electrons of atoms bound into a solid will be affected by an anisotropic type of chemical bonding, photons involving such outer levels wili be emitted anisotropically with some degree of polarization. This effect is pronounced for optical spectra; thus visible light stimulated by electron bombardment (cathodoluminescence) is emitted anisotropically. Twin lamellae in microcline are revealed by a change of intensity of blue light across a twin boundary without using a polarizer in the viewing microscope; use of a polarizer intensifies the contrast (Smith and Stenstrom, 1966). X-ray photons with l < 4 Å can be assumed to be emitted essentially isotropically from ionized atoms because the energy levels are sufficiently deep-seated to be unaffected by chemical bonding, but those with l > 10 Å might be sensibly anisotropic. Note that the emission of characteristic photons from ionized atoms preserves no memory of the ionization event, and that the direction of the incident particle or photon which caused the ionization is quite irrelevant to the angular distribution of the outgoing photon.
2.2.4 Bremsstrahlung (Braking Radiation)
General. Charged particles radiate when accelerated or decelerated, and the resulting Bremsstrahlung or continuous spectrum must be distinguished clearly from the characteristic spectrum produced by electronic transitions in ionized atoms. In a synchrotron storage ring (SSR), electrons undergo centripetal acceleration in the segments passing between bending magnets (Fig. 2.14). In a conventional X-ray tube (Fig. 2.15a) the electrons strike a metal target and X-ray photons are emitted as some but not all electrons are deflected during inelastic collisions (Fig. 2.15c). The maximum amount of available energy is eV (where V is the accelerating voltage) and the photon energy cannot be larger than this limit. Some electrons cause ionization with consequent emission of characteristic X-rays (preceding section) which are superimposed on the continuous spectrum. In contrast, the spectrum from an SSR bending magnet consists solely of Bremsstrahlung. The theory is complex, and the present treatment proceeds from the classical model to the relativistic quantum mechanical treatment. Electrons have only a small mass, and the Bremsstrahlung just extends into the X-ray region for an accelerating voltage as low as 1000 volts. However protons and ionized atoms are so massive that the Bremsstrahlung is very weak and confined to long wavelengths (including optical ones) even for high accelerating voltages.
Classical Model for Electron Acceleration. In the classical Thomson-Lorentz model for electron acceleration based on Maxwell's electrodynamics (Compton and Allison, 1949, p. 56-60), energy is radiated in all directions in the form of a transverse electromagnetic wave with the electric vector changing sinusoidally in the plane containing the acceleration vector and the direction of wave propagation (Fig. 2.16a). For zero velocity, the wave amplitude in a given direction is represented by a chord of a torus (Fig. 2.l6b), and the intensity of the radiation at an angle f is proportional to sin 2f (Fig. 2.16c) where f is the angle between the direction of wave propagation and the acceleration vector. In the classical model, the radiation is emitted uniformly throughout the entire range of frequency (Compton and Allison, p. l02) whatever the kinetic energy of the electron (Fig. 2.l7a). The Larmor formula for the radiated power P from a single electron accelerated by dv/dt is
P = 2/3.e2/c3.|dv/dt|2 2.16
Note that the vertical brackets imply a scalar quantity.
Quantum-Mechanical Model for Electron Acceleration. In this model, the maximum amount of kinetic energy (eV) of an electron which is brought to rest by a head-on collision provides an upper limit to the photon energy (huo). Strictly speaking, the collisions cannot be treated individually when the overall spectrum is being considered, but H. A. Kramer used the correspondence principle to predict that the available energy (Fig. 2.17a) is partitioned equally into the allowed frequency states (0 to uo). Experimental data for thin metal targets are in approximate agreement with this prediction when account is taken of absorption of the softer radiation before escape from the target. Figure 2.l7b shows the spectral distribution when plotted against wavelength instead of frequency. Let In and Il be the amount of energy in the increments dn and dl of frequency and wavelength. Since c = nl,
- dn/dl = n2/c = In/ Il 2.17
This simple theory must be modified for a thick target which absorbs all the incoming electrons. Most of the electrons undergo multiple scattering, and each simple dipole pattern has a different orientation. A qualitative estimate of the total scattering is shown in Fig. 2.16d. The frequency limit is reduced as each electron loses kinetic energy (Fig. 2.17a) and a summation of the decreasing contributions yields an intensity-wavelength plot shaped like a whale's back (Fig. 2.16c).
Experimental measurements showed that the fractional conversion efficiency of kinetic energy (eV) into photon energy (_hn) of the entire continuous spectrum for total stopping of electrons in a thick target of atomic number Z is given by
11 x 10-10ZV 2.18
Thus for a copper target (Z = 29) in an X-ray tube with accelerating voltage of 40,000 volts, the efficiency for generation of the continuous spectrum is only 0.13%. For a tungsten tube (Z = 74) operated at 70 kV, the efficiency is 4.5 times greater.
Relativistic Effect for Conventional X-ray Tube. An electron accelerated by 50 kV is travelling at 27% of the velocity of light, and the Doppler shift associated with transformation of its reference frame to that of the observer of the X-ray photons cannot be ignored. Fig. 2.16c shows how the sin2q distribution of energy for zero velocity is forced into the forward direction as v/c = b increases, where v is the incident velocity of the electron. For b = l/3, the maximum is shifted to q ~65°, and about three-quarters of the energy goes into the forward direction. This phenomenon is very important for the electron microprobe when trace elements are being analyzed since increase of the accelerating voltage reduces the background (and increases the efficiency of generation of the characteristic lines).
Centripetal Acceleration and Relativistic Effect for a Synchrotron Storage Ring. Each bending magnet has a vertical magnetic field strength B which applies a force (evB) towards the center of the SSR on each electron. This force produces a centripetal acceleration v2/r ( = w2r) where v and w are the linear and angular velocities and r is the radius. The rest mass mo must be divided by the relativistic factor (1 - b2)1/2. Equating the electromagnetic force with the product of mass and acceleration gives
w = v/r = eB(l - b2)1/2/m0 2.19
The acceleration of the electron towards the center of the SSR produces a toroidal radiation pattern around the radial vector (Fig. 2.18a) in the reference frame of the moving electron. In the X-ray ring of the National Synchrotron Light Source (Fig. 2.l4), the electrons are accelerated by 2.5 GeV (G means Giga = 109), and b is nearly 1 (>0.999). Transformation to the reference frame of the observer turns the toroid into a narrow tangential jet (Fig. 2.18b).
In practice, it is necessary to insert the electrons into the storage ring as one or more bunches, each of which ''rides'' a radio-frequency pulse. The electron pulses are squeezed into the center of the near-circular orbit by various steering magnets which control various forms of instability; actually, oscillations of the electron orbit are one of the most difficult technical challenges for design of an SSR. Each bunch radiates electromagnetic radiation into a rotating beam which flashes 1/w times a second; the pulse length can be 0.1 to 0.4 nanosecond. Each bending magnet is separated by a straight section from the adjacent magnets, and radiation is generated only within each magnet. Hence the radiation is emitted with a near-rectangular crosssection whose horizontal divergence is l/r where l is the length of the bending magnet, and whose vertical divergence both above and below the median plane is s' [note distinction between primed and unprimed s].
tan s' = sin s/g (b + cos s) 2.20
The quantity s is the angle between the tangential velocity vector and the wave vector of the radiation in the rest frame of the electron, and g = eV/mc2 where eV is the kinetic energy of the electron. Because the maximum value of sin s is 1, for which (b + cos s) is nearly 1, tan s' is approximately 1/g. For a bending magnet of the X-ray storage ring of the NSLS, the horizontal divergence is 15 milliradians. This is typically split up into two beams by a mask yielding divergence 2 milliradians at each of two experimental stations. The vertical divergence of l/g is 0.20 milliradian for V = 2.5 GeV. This equals 0.011 degree both above and below the median plane. X-rays from a conventional X-ray tube are only weakly constrained by the relativistic transformation, and it is impractical to lower the divergence with collimators below 0.05° for routine operation.
The power radiated by electrons in a synchrotron storage ring is very large, and must be replaced during each circuit by a kick from a radiofrequency oscillator. An electron has a kinetic energy E given by
E = m0c2 [(1 - b2) -1/2 - 1] 2.21
where m0 is the rest mass and b = v/c.
The radiated power P is given by
P = 2/3. e2 c/r2. b4 (E/mc2)4 2.22
The energy lost for each circuit is
_E = 4_/3.e2/r.b3(E/mc2)4 2.23
where r is the bending radius.
For E in Giga electron-volt (GeV), r in meter, b in kilogauss for the field strength of the bending magnet, I in ampere, and b near 1,
_E(keV) = 88.5E4/r 2.24
P(kilowatt) = 88.5 E4I/r = 2.65 BE3 I 2.25
For E = 2.5 GeV, I = 0.5 A and r = 8.7 m, P = 200 kW. A conventional X-ray tube operates at about 1 kW, and the efficiency of generation of the braking radiation is not greater than 0.5%. Hence the NSLS produces at least 4 x l04 times as much useful power as the X-ray tube.
The spectral distribution is somewhat similar to the background continuum of a conventional X-ray tube. Fig. 2.l9 shows how the spectrum moves to higher frequencies and total photon intensity from the VUV to the X-ray storage rings of the NSLS. Each spectrum is characterized by a minimum wavelength (lm) and a critical wavelength (lc). The latter divides the spectrum into two parts with equal energy (not equal number of photons), and
lc = 4_r/3g3 = 186/(BE2) 2.26
where lc is in Å, B in kG, and E in GeV. For 2.5 GeV and 12 kG, lc is 2.5 Å. The minimum wavelength lm is also inversely related to the cube of g and is approximately eight times smaller than lc. The intensity decreases rapidly as lc is approached, and the useful limit on the wavelength is about one-fifth of lc where the intensity is down about 25 times over that for lc.
So far it has been assumed that the bending magnets are of the permanent type with a field strength B of ~12 kG (= 1.2 Tesla). Superconducting magnets can give 5 to 6 Tesla with the possibility of a 5-fold increase in power (equation 2.25) and a 5-fold reduction of critical wavelength (equation 2.26); however they are not economic as simple bending magnets. They are being used in special insertion devices (ID) which consist of a series of adjacent bending magnets of alternating polarity fitted into a straight section between two bending magnets. Beamline X17 of NSLS is fitted with a wiggler composed of superconducting magnets, and has produced high-energy beams of great use to diamond-cell and large-volume press pioneers. Beamline X25 has an ID with permanent magnets.
Third-generation synchrotrons now coming into operation were deliberately designed to have a long insertion device between each pair of adjacent bending magnets. There are two types: a wiggler is composed of N bending magnets with opposing polarity so that each electron bunch regains its original trajectory after wiggles. The independent jets from each magnet add up to give an elliptical beam with the same spectral profile as a single magnet, but with N times the flux. An interference wiggler (generally called an undulator) has smaller magnetic deflection. The jets interfere with each other, as in a diffraction grating, to yield a very narrow circular jet with the photons concentrated into narrow energy harmonics. The band width is ~1/N. Variation of magnetic field causes the harmonics to change energy. As the magnetic deflection changes, an undulator can be turned into a wiggler. Insertion devices which can cover the hybrid region are becoming popular because of their great flexibility. Many subtle details are reviewed in Smith and Rivers (1995).
Radiation emitted tangentially from an electron accelerated radially has its electric vector in the plane of the storage ring. Elliptical polarization occurs away from this special geometry, and integration over all angles for a radial electron gives ~75% polarization. Hence the center of the beam from a bending magnet is used to obtain the best linear polarization in the plane. Oscillations made by the electron bunches away from the perfect circle cause reduction in the degree of polarization. The harmonics from an undulator have special polarization properties.
Figure 2.l Energy levels for electrons. For a pure metal at absolute zero (a) all electrons occupy energy levels for the ''electron gas'' which is coupled to the atoms in the crystal structure. The levels above the Fermi level are unoccupied. In a hot metal (b), an equilibrium is set up among electrons moving from below the Fermi level up into the space cloud and back again. The space cloud is stable unless electrons are attracted to an anode by an electric field. In a semiconductor (c), there is a forbidden band of energies. The work function W is greater than the width of the conduction band and less than the photoelectric threshold energy P.
Figure 2.2 Schematic section through an electron gun used in EMs and EMPs. A pointed tungsten wire W is held by a circular insulator I inside a grid cap which is a half-closed cylinder with a circular aperture A1. The filament is heated by a filament voltage across DE, and the number of electrons escaping from the space charge through aperture A1 is controlled by the bias voltage between E and F. Escaping electrons are accelerated towards the anode plate and pass through the aperture A2; the accelerating voltage (EG) is commonly l5 kV for an EMP and 100 kV for an EM.
Figure 2.3 Relation between emission current and filament heating voltage in an electron gun. As the temperature increases, emission begins at A, reaches a maximum at B (''false saturation''), decreases to C, and reaches true saturation at DE. If the gun is slightly out of mechanical alignment, the emission current decreases to E1. A larger bias voltage confines the space charge and reduces the emission current. If the gun is strongly out of alignment, the relation is more complex, and the emission current may be reduced to zero for all values of filament voltage. To prolong filament life it is very important to select D rather than E.
Figure 2.4 Photoemission electron micrograph of peristerite showing an intergrowth of albite (dark) and oligoclase (light). The coarser intergrowth is misleading because the surface was polished almost parallel to a near-lamellar intergrowth whose irregularities are greatly exaggerated. A brittle fracture produced the offset; note the sharp change of emission in the crack. Mechanical deformation produced the twin lamellae (Albite law?) running NNE-SSW in which the albite and oligoclase lamellae are seen end on. From the late F. Laves (pers. comm.). See Smith (l974, v. l, p. 505-507) for other examples. l6mm across.
Figure 2.5 Schematic diagram showing generation of back-scattered electrons (BSE) with high- and low-energy loss (HL, LL) and of secondary electrons (SE) associated with the primary electrons (PE) or the back-scattered electrons (BE).
Figure 2.6 Scanning electron micrographs of (a) the back-scattered electrons from the polished surface of the Murchison meteorite, and (b) the secondary electrons from the surface of a powder mainly of Linde A zeolite. In (a), the atomic-number contrast distinguishes between spinel, perovskite and hibonite. The black areas are holes. From MacPherson et al. (l983). In (b), the intergrown cubes remain in focus from top to bottom, and the topographic contrast from different faces with the same chemical composition is clearly shown. Rosettes of an unidentified phase are present, as well as minute plates and rods on some surfaces. Gold-coated. Ten micrometer scale marker.
Figure 2.7 Ionization of an electron of the K shell by a primary electron (a) or a photon (b), and subsequent emission of a photon (c) or an Auger electron (d). Each atom is shown schematically with a nucleus (square) and three shells (K, L, M) of orbital electrons in sequence of energy. The Fermi Level for a metallic atom (FL) has an energy slightly smaller than that for an electron at infinity or just escaping into a vacuum (cf. Fig. 2.1). The ionization energies for K and L electrons are EK and EL. The energy scale is strongly compressed towards the nucleus to allow room for drafting. Each electron is shown by a dot, and each hole by a circle.
Figure 2.8 Spectrum of Auger electrons emitted from a clean surface of MgO. The energy limits for unabsorbed Auger electrons from the K shells of Mg and O are marked.
Figure 2.9 Details of emission of Auger electrons. A single-ionized atom with a vacancy in the K shell is shown in (a); sub-shells labeled with Roman numerals are shown for the L and M shells. Emission of a Ka2 photon by electron transfer from the LIII to K shell is shown in (b). Four types of Auger electron emission are given in (c)-(f); in the special case of a Coster-Kronig transition, an electron moves in the same shell, here given as LIII to LI.
Figure 2.10 Vector diagrams for coupling of spin and angular momenta. (a) Independent coupling of li and si to give L and S, respectively. (b) Coupling of L and S to give J.
Figure 2.11 Energy levels and allowed transitions in a hydrogen atom. Levels up to n = 5 and l = 4 are shown. The splitting of the 2p, 3p and 3d levels is greatly exaggerated for clarity. The fractions are values of j. Note the reversal of energy scales for eV and wave number (cm-1).
Figure 2.12 Energy levels and allowed transitions for a sodium atom with unexcited and excited levels (Table 2.l). Note change of scale at the zero energy level.
Figure 2.13 Relation between atomic number and wavelength for the principal X-ray lines.
Figure 2.14 The National Synchrotron Light Source at the Brookhaven National Laboratory. Electrons from a source at A are accelerated in a linear accelerator (LINAC) and deflected at B into a circular energy booster. They are extracted at C or D to provide one or more pulses which are stored either in the vacuum ultraviolet or X-ray rings. The VUV storage ring contains 8 bending magnets (thick arcs) separated by straight sections containing steering and focusing elements (dots). Each pie-shaped radiation field is split into two beam lines. Most bending magnets in the X-ray storage ring provide two beam lines each; two are inaccessible because of the booster. The energy loss for each circuit is replaced in a radio-frequency oscillator (square). Inset. Centripetal acceleration and radiation pattern from a bending magnet; splitting of an X-ray beam 15 milliradian wide into two beams 2 milliradian wide.
Figure 2.l5 X-ray generation in a conventional X-ray tube. (a) Cross-section of cylindrical tube mounted in metal housing. Electrons are emitted from a coiled tungsten filament (cathode) 1 cm x 1 mm, and weakly focused by a focusing cap onto a grounded metal target (anode) which is cooled from the back side by a shaped jet of water. X-rays are emitted in all directions from the 1 cm x 1 mm x 5mm target volume, and some escape through 4 beryllium windows set in a glass envelope, of which two are shown. A wheel containing various filters is shown at the right. All X-ray tubes should have fail-safe shutters in the safety shield (not shown here). (b) X-rays are generated from a thin surface layer in the target, and the 1 cm x 1 mm area is viewed at an angle (comm only 6°) so that foreshortening gives either a square cross-section (1 mm x 1 mm) for single-crystal or Debye-Scherrer powder techniques or a line (1 cm x 0.1 mm) for powder diffractometry. (c) The braking radiation is generated by inelastic collisions with atoms which do not become ionized. A ''head-on'' collision gives maximum energy and frequency of the X-ray photon, and a glancing collision gives lower energy. Most X-rays are generated within 3 mm of the surface.
Figure 2.l6 Classical model for braking radiation. (a) The electric vector E varies sinusoidally in the plane of a particular wave vector of the braking radiation and the acceleration vector a of the electron. (b) For vanishingly small electron velocity, the wave amplitude for a particular direction is given by the chord sin f of a torus with unit diameter. (c) The intensity for a particular direction of the wave vector is given by a flattened torus (crosses) whose radius vector is proportional to sin2 f. The dotted line shows the relativistic effect for an electron travelling with one-third the velocity of light. (d) Qualitative estimate of effect of multiple deflections in the target for electron with b = 1/3.
Figure 2.17 Spectral distribution of braking radiation. (a) Distribution of intensity versus frequency. For a thin target with no significant loss of kinetic energy of incident electrons, the quantum-mechanical model gives an even distribution up to a limiting frequency (crosses) whereas the classical model gives an even distribution over all frequencies (crosses and dots). A thick target can be modeled by reducing the frequency limit in steps, and the integrated spectrum is given by the dotted line. Absorption in the target will reduce the intensity especially at low frequency. (b) Distribution of intensity versus wavelength for a thin target corresponding to the crosses in (a). (c) Experimental data for a thick tungsten target for electrons accelerated to various voltages.
Figure 2.l8 Radiation pattern for an electron in a circular orbit for (a) the reference frame of the electron, and (b) the reference frame of the observer. In (a), the toroid is centered around the radius vector. In (b), the toroid is squeezed into a narrow jet.
Figure 2.19 Predicted spectra for the 0.7 and 2.5 GeV storage rings in NSLS with the normal bending magnets, and for a 6-Tesla standard wiggler in the 2.5 GeV ring. The brightness is expressed as the number of photons per second emitted in a milliradian angle for a 1% increment of wavelength. The electron beam current is 1 ampere in the VUV ring and 0.5 ampere in the X-ray ring.
Last revised: February 24, 1998