Chapter 4.2. X-rays

Characteristic X-ray emission originates from the radiative decay of electronically highly excited states of matter. We are concerned mostly with excitation by electron bombardment of a target that results in the emission of spectral lines characteristic of the target elements. The electronic states occurring as initial and final states of a process involving the absorption of emission of X-rays are called X-ray levels. Levels involving the removal of one electron from the configuration of the neutral ground state are called normal X-ray levels or diagram levels.

Table 4.2.1.1 shows the relation between diagram levels and electron configurations. The notation used here is the IUPAC notation (Jenkins, Manne, Robin & Senemaud, 1991), which uses arabic instead of the former roman subscripts for the levels. The IUPAC recommendations are to refer to X-ray lines by writing the initial and final levels separated by a hyphen, e.g. Cu K-L₃ and to abandon the Siegbahn (1925) notation, e.g. Cu Kα₁, which is based on the relative intensities of the lines. The correspondence between the two notations is shown in Table 4.2.1.2. Because this substitution has not yet become common practice, however, the Siegbahn notation is retained in Section 4.2.2, in which the wavelengths of the characteristic emission lines and absorption edges are discussed.

Table 4.2.1.1| top | pdf | Correspondence between X-ray diagram levels and electron configurations; from Jenkins, Manne, Robin & Senemaud (1991), courtesy of IUPAC Level Electron configuration Level Electron configuration Level Electron configuration K N1 L1 N2 L2 N3 L3 N4 M1 N5 M2 N6 M3 N7 M4         M5

Table 4.2.1.2| top | pdf | Correspondence between IUPAC and Siegbahn notations for X-ray diagram lines; from Jenkins, Manne, Robin & Senemaud (1991), courtesy of IUPAC Siegbahn IUPAC Siegbahn IUPAC Siegbahn IUPAC                     Siegbahn IUPAC In the case of unresolved lines, such as and , the recommended IUPAC notation is .

4.2.1.1.1. The intensity of characteristic lines

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The efficiency of the production of characteristic radiation has been calculated by a number of authors (see, for example, Dyson, 1973, Chap. 3). For a particular line, it depends on the fluorescence yield, that is the probability that the decay of an excited state leads to the emission of a photon, on the statistical weights of the X-ray levels involved, on the effects of the penetration and slowing down of the bombarding electrons in the target, on the fraction of electrons back-scattered out of the target, and on the contribution caused by fluorescent X-rays produced indirectly by the continuous spectrum. The emerging X-ray intensity is further affected by the partial absorption of the generated X-rays in the target.

Dyson (1973) has also reviewed calculations and measurements made of the relative intensities of different lines in the K spectrum. The ratio of the to intensities is very close to 0.5 for Z between 23 and 48. The ratio of to rises fairly linearly with Z from 0.2 at Z = 20 to 0.4 at Z = 80 and that of to is near zero at Z = 29 and rises linearly with Z to about 0.1 at Z = 80. Relative intensities of lines in the L spectrum are given by Goldberg (1961).

Green & Cosslett (1968) have made extensive measurements of the efficiency of the production of characteristic radiation for a number of targets and for a range of electron accelerating voltages. Their results can be expressed empirically in the form where is the generated number of Kα photons per steradian per incident electron, N₀ is a function of the atomic number of the target, E₀ is the electron energy in keV and is the excitation potential in keV. It should be noted that decreases with increasing Z.

For a copper target, this expression becomes or where is the number of Kα photons per steradian per second per milliampere of tube current.

These expressions are probably accurate to within a factor of 2 up to values of of about 10. Guo & Wu (1985) found a linear relationship for the emerging number of photons with electron energy in the range .

To obtain the number of photons that emerge from the target, the above expressions have to be corrected for absorption of the generated radiation in the target. The number of photons emerging at an angle to the surface, for normal electron incidence, is usually written where (Castaing & Descamps, 1955). Green (1963) gives experimental values of the correction factor f(χ) for a series of targets over a range of electron energies. His curves for a copper target are given in Fig. 4.2.1.1 . It will be noticed that the correction factor increases with increasing electron energy since the effective depth of X-ray generation increases with voltage. As a result, curves of as a function of have a broad maximum that is displaced towards lower voltages as decreases, as shown in the experimental curves for copper K radiation due to Metchnik & Tomlin (1963) (Fig. 4.2.1.2 ). For very small take-off angles, therefore, X-ray tubes should be operated at lower than customary voltages. Note that the values in Fig. 4.2.1.2 agree to within ∼40% with those of Green & Cosslett. f(χ) at constant increases with increasing Z, thus partly compensating for the decrease in , especially at small values of . A recent re-examination of the characteristic X-ray flux from Cr, Cu, Mo, Ag and W targets has been carried out by Honkimaki, Sleight & Suortti (1990).

Figure 4.2.1.1| top | pdf | f(χ) curves for Cu K-L3 at a series of different accelerating voltages (in kV). From Green (1963).

Figure 4.2.1.2| top | pdf | Experimental measurements of for Cu K-L3 as functions of the accelerating voltage for different take-off angles. From Metchnik & Tomlin (1963).

The shape of the continuous spectrum from a thick target is very simple: , the energy per unit frequency band in the spectrum, is given by the expression derived by Kramers (1923): where Z is the atomic number of the target and A and B are constants independent of the applied voltage . B/A is of the order of 0.0025 so that the term in can usually be neglected (Fig. 4.2.1.3 ). is the maximum frequency in the spectrum, i.e. the Duane–Hunt limit at which the entire energy of the bombarding electrons is converted into the quantum energy of the emitted photon, where Using the latest adjusted values of the fundamental constants (Cohen & Taylor, 1987): Equation (4.2.1.5) can be rewritten in a number of forms. If is the number of photons of energy E per incident electron, where  photons eV⁻¹  electron⁻¹, and is known as Kramer's constant.

Figure 4.2.1.3| top | pdf | Intensity per unit frequency interval versus frequency in the continuous spectrum from a thick target at different accelerating voltages. From Kuhlenkampff & Schmidt (1943).

From (4.2.1.7), it follows that the total energy in the continuous spectrum per electron is Since the energy of the bombarding electron is , the efficiency of production of the continuous radiation is Crystallographers are more accustomed to thinking of the spectrum in terms of wavelength. Equation (4.2.1.7) can be transformed into which has a maximum at . In practice, the emerging spectrum is modified by target absorption, which is greatest for the longer wavelengths and moves the maximum more nearly to .

It is of interest to compare the X-ray flux in a narrow wavelength band selected by an appropriate monochromator with the flux in a characteristic spectral line, in order to examine the practicability of XAFS (X-ray absorption fine-structure spectroscopy) or optimized anomalous-dispersion diffractometry experiments. For these purposes, the maximum permissible wavelength band is about 10⁻³ Å. From equation (4.2.1.10), we see that, for a tungsten-target X-ray tube operated at 80 kV, is about 1.1 × 10⁻⁵ photons with the Kα energy electron⁻¹ steradian⁻¹ (10⁻³ δλ/λ)⁻¹ for an X-ray wavelength in the neighbourhood of 1.5 Å. By comparison, from equation (4.2.1.2), a copper-target tube operated at 40 kV produces about 5 × 10⁻⁴ Kα photons electron⁻¹ steradian⁻¹. In spite of this shortcoming by a factor of about 45, laboratory XAFS experiments are sufficiently common to have merited at least one specialized conference (Stern, 1980; see also Tohji, Udagawa, Kawasak & Masuda, 1983; Sakurai, 1993; Sakurai & Sakurai, 1994).

The use of continuous radiation for diffraction experiments is complicated by the fact that the radiation is polarized. The degree of polarization may be defined as where and are the intensities of radiation with the electric vector parallel and perpendicular to the plane containing the incident electrons and the direction of the emitted photons. For an angle of π/2 between the electrons and the emitted beam, p varies smoothly through the spectrum; it is negative for the softest radiation, approximately zero at and reaches values between +0.7 and +0.9 near the Duane–Hunt limit (Kirkpatrick & Wiedmann, 1945). Since practical use of white radiation is likely to be in the vicinity of , the effect is not a large one.

It should also be noted that the spatial distribution of the white spectrum, even after correction for absorption in the target, is not isotropic. The intensity has a maximum at about 50° to the electron beam and non-zero minima at 0 and 180° to that beam (Stephenson, 1957).

The commonest source of X-rays is the high-vacuum, or Coolidge, X-ray tube, which may be either demountable and pumped continuously when in operation or permanently sealed after evacuation. The vacuum tube contains an electron gun that incorporates a thermionic cathode, which produces a well defined electron beam that is accelerated towards the anode or target, formerly often called the anticathode. In most X-ray tubes intended for crystallographic purposes, the anode is massive, i.e. its thickness is large compared with the range of the electrons; it is usually water-cooled and its surface is normal to the incident electron beam. Usually, it is desirable for the X-ray source to be small (between 25 μm and 1 mm square) and for the X-ray intensity from the tube to be the maximum possible for the amount of power that can be dissipated in the target. These objectives are best achieved by designing the electron gun to produce a line focus, that is the electron focus on the target face is approximately rectangular with the small dimension equal to the desired effective source size and the large dimension about 10 to 20 times larger. The focus is viewed at an angle between about 2 and 5° to the anode surface to produce an approximately square foreshortened effective source; and the X-ray windows are so positioned as to make these take-off angles possible. For some purposes, very fine line sources are required and windows may be provided to allow the focus to be viewed so as to foreshorten the line width. Higher power dissipation is possible in X-ray tubes in which the anode rotates: the line focus is now usually on the cylindrical surface of the anode with its long dimension parallel to the axis of rotation.

For focal-spot sizes down to about 100 μm, an electrostatic gun is adequate; this consists of a fine helical filament and a Wehnelt cathode, which produces a demagnified electron image of the filament on the anode. For most purposes, the Wehnelt cathode can be at the same potential as the filament but cleaner foci and adjustment of the focal spot size are possible when this electrode is negatively biased with respect to the filament. The filament is nearly always directly heated and made of tungsten. Lower filament temperatures, and smaller heating currents, could be achieved with activated heaters but the vacuum in high-power devices like X-ray tubes is rarely hard enough to permit their use since they are easily poisoned. However, Yao (1992) has reported successful operation of a hot-pressed polycrystalline lanthanum hexaboride cathode in an otherwise unmodified RU-1000 rotating-target X-ray generator.

Very fine focus tubes, with foci in the range between 25 and 1 μm, require magnetic lenses. At one time, the all-electrostatic X-ray tube of Ehrenberg & Spear (1951), which achieved foci between 20 and 80 μm, was very popular.

Sealed-off X-ray tubes for crystallographic use are nowadays made in the form of inserts containing a target of one of a range of standard metals to produce the desired characteristic radiation. A series of nominal focal-spot sizes, shown in Table 4.2.1.3, is commonly available. The insert is mounted inside a standard shield that is radiation- and shock-proof and that is fitted with X-ray shutters and filters and often also with a standardized track for mounting X-ray cameras. The water-cooled anode is normally at ground potential and the negative high voltage for the cathode, together with the filament supply, is brought in through a shielded shock-proof cable. The high voltage is nowadays generally of the constant-voltage type, that is, it is full-wave rectified and smoothed by means of solid-state rectifiers and capacitors housed in the high-voltage transformer tank, which also contains the filament transformer. The high tension and the tube current are frequently stabilized. Only the simplest X-ray generators now employ an alternating high tension that is rectified by the self-rectifying property of the X-ray tube itself.

A demountable continuously pumped form of construction is nowadays adopted mainly for rotating-anode and other specialized X-ray tubes. The pumping system must be capable of maintaining a vacuum of better than 10⁻⁵ Torr: filament life is critically dependent upon the quality of the vacuum.

Rotating-anode tubes have been reviewed by Yoshimatsu & Kozaki (1977). The first successful tube of this type that incorporated a vacuum shaft seal was described by Clay (1934). Modern tubes mostly contain vacuum-oil-lubricated shaft seals of the type due to Wilson (1941) and are based on, or are similar to, the rotating-anode tubes described by Taylor (1949, 1956). In some tubes, successful use has been made of ferro-fluidic vacuum seals (see Bailey, 1978). The main problems in the operation of rotating-anode tubes is the lifetime of the seals and of bearings that operate in vacuo. In successful tubes, e.g. those manufactured by Enraf, Rigaku-Denki, and Siemens, these lifetimes are about the same as the lifetime of the filament under good vacuum conditions, that is, of the order of 1000 h.

Phillips (1985) has written a review article on stationary and rotating-anode X-ray tubes that contains many important practical details.

4.2.1.3.1. Power dissipation in the anode

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The allowable power loading of X-ray tube targets is determined by the temperature of the target surface, which must remain below the melting point. Müller (1927, 1929, 1931) first calculated the maximum loading both for stationary and for rotating anodes. His calculations were refined by Oosterkamp (1948a,b,c) who considered, in particular, targets of finite thickness, and who also treated pulsed operation of the tube. For normal conditions, Oosterkamp's conclusions and those of Ishimura, Shiraiwa & Sawada (1957) do not greatly differ from those of Müller, which are in adequate agreement with experimental observations.

For an elliptical focal spot with axes and , Müller's formula for the maximum power dissipation on a stationary anode, assumed to be a water-cooled block of dimensions large compared with the focal-spot dimensions, can be written where K is the specific thermal conductivity of the target material in W mm⁻¹, is the maximum temperature at the centre of the focal spot on the target, that is, a temperature well below the melting point of the target material, and is the temperature of the cold surface of the target, that is, of the cooling water. The function μ is shown in Fig. 4.2.1.4 . For copper, K is 400 W m⁻¹ and, with = 500 K, For f₂/f₁ = 0.1, and μ = 0.425, this equation becomes In these last two equations, f₁ is in mm.

Figure 4.2.1.4| top | pdf | The function μ in Müller's equation (equation 4.2.1.12) as a function of the ratio of width to length of the focal spot.

For a rotating target, Müller found that the permissible power dissipation was given by where f₂ is the short dimension of the focus, assumed to be in the direction of motion of the target, v is the linear velocity, ρ is the density of the target material, and C is its specific heat.

For a copper target with f₁ and f₂ in mm and v in mm s⁻¹, Equation (4.2.1.16) shows that for very narrow focal spots rotating-anode tubes give useful improvements in permissible loading only if the surface speed is very high (see Table 4.2.1.3). The reason is that with large foci on stationary anodes the isothermal surfaces in the target are planar; with fine foci, these surfaces become cylindrical and this already makes for very efficient cooling without the need for rotation. Rotating anodes are thus most useful for medium-size foci (200 to 500 μm) since for the larger focal spots it becomes very expensive to construct power supplies capable of supplying the permissible amount of power.

Table 4.2.1.3 shows the recommended loading for a number of commercially available X-ray tubes with copper targets, which will be seen to be in qualitative agreement with the calculations. Some of the discrepancy is due to the fact that the value of for the copper–chromium alloy targets used in actual X-ray tubes is appreciably lower than the value for pure copper used here. To a good approximation, the permissible loading for other targets can be derived by multiplying those in Table 4.2.1.3 by the factors shown in Table 4.2.1.4. It is worth noting that the recommended loading of commercial stationary-target X-ray tubes has increased steadily in recent years. This is largely due to improvements in the water cooling of the back surface of the target by increasing the turbulence of the water and the effective surface area of the cooled surface.

Table 4.2.1.4| top | pdf | Relative permissible loading for different target materials Cu Cr Fe Co Mo Ag W 1.0 0.9 0.6 0.9 1.2 1.0 1.2

In considering Table 4.2.1.3, it should be noted that the linear velocities of the highest-power X-ray-tube anode have already reached a speed that Yoshimatsu & Kozaki (1977) consider the practical limit, which is set by the mechanical properties of engineering materials. It should also be noted that much higher specific loads can be achieved for true micro-focus tubes, e.g. 50 kW mm⁻² for a 25 μm Ehrenberg & Spear tube and 1000 kW mm⁻² for a tube with a 1 μm focus (Goldsztaub, 1947; Cosslett & Nixon, 1951, 1960).

Some tubes with focus spots of less than 10 μm utilize foil or needle targets. These targets and the heat dissipated in them have been discussed by Cosslett & Nixon (1960). The dissipation is less than that in a massive target by a factor of about 3 for a foil and 10 for a needle, but, in view of the low absolute power, target movement and even water-cooling can be dispensed with.

Radioactive sources of X-rays are mainly of interest to crystallographers for the calibration of X-ray detectors where they have the great advantage of being completely stable with time, or at least of having an accurately known decay rate. For some purposes, spectral purity of the radiation is important; radionuclides that decay wholly by electron capture are particularly useful as they produce little or no β or other radiation. In this type of decay, the atomic number of the daughter nucleus is one less than that of the decaying isotope, and the emitted X-rays are characteristic of the daughter nucleus. In some cases, the probability of electron capture taking place from some shell other than the K shell is very small and most of the photons emitted are K photons. The number of photons emitted into a solid angle of 4π, uncorrected for absorption, is given by the strength of the source in Curies (1 Curie = 3.7 × 10¹⁰ disintegrations s⁻¹), since each disintegration produces one photon. A list of these nuclei (after Dyson, 1973) is given in Table 4.2.1.5.

Table 4.2.1.5| top | pdf | Radionuclides decaying wholly by electron capture, and yielding little or no γ-radiation Nuclide Half-life X-rays Remarks Element (keV) 37Ar 35 d Cl 2.622 – 51Cr 27.8 d V 4.952 γ at 320 keV 55Fe 2.6 a Mn 5.898 – 71Ge 11.4 d Ga 9.251 – 103Pd 17 d Rh 20.214 Several γ's; all weak 109Cd 453 d Ag 22.16 γ at 88 keV 125I 60 d Te 27.47 γ at 35.4 keV 131Cs 10 d Xe 29.80 – 145Pm 17.7 a Nd 37.36 γ's at 67 and 72 keV 145Sm 340 d Pm 38.65 γ's at 61 keV; weak γ at 485 keV 179Ta 600 d Hf 55.76 – 181W 140 d Ta 57.52 γ at 6.5 keV; weak γ's at 136, 153 keV 205Pb 5 × 107 a Tl L only (Lα1 = 10.27 keV) –

Useful radioactive sources are also made by mixing a pure β-emitter with a target material. These sources produce a continuous spectrum in addition to the characteristic line spectrum. The nuclide most commonly used for this purpose is tritium which emits β particles with an energy up to 18 keV and which has a half-life of 12.4 a.

Radioactive X-ray sources have been reviewed by Dyson (1973).

The growing importance of synchrotron radiation is attested by a large number of monographs (Kunz, 1979; Winick, 1980; Stuhrmann, 1982; Koch, 1983) and review articles (Godwin, 1968; Kulipanov & Skrinskii, 1977; Lea, 1978; Winick & Bienenstock, 1978; Helliwell, 1984; Buras, 1985). Project studies for storage rings such as the European Synchrotron Radiation Facility, the ESRF (Farge & Duke, 1979; Thompson & Poole, 1979; Buras & Marr, 1979; Buras & Tazzari, 1984) are still worth consulting for the reasoning that lay behind the design; the ESRF has, in fact, achieved or even exceeded the design parameters (Laclare, 1994).

A charged particle with energy E and mass m moving in a circular orbit of radius R at a constant speed v radiates a power P into a solid angle of 4π, where The orbit of the particle can be maintained only if the energy lost in the form of electromagnetic radiation is constantly replenished. In an electron synchrotron or in a storage ring, the circulating particles are electrons or positrons maintained in a closed orbit by a magnetic field; their energy is supplied or restored by means of an oscillating radio-frequency (RF) electric field at one or more places in the orbit. In a synchrotron, designed for nuclear-physics experiments, the circulating particles are injected from a linear accelerator, accelerated up to full energy by the RF field and then deflected into a target with a cycle frequency of about 50 Hz. The synchrotron radiation is thus produced in the form of pulses of this frequency. A storage ring, on the other hand, is filled with electrons or positrons and after acceleration the particle energy is maintained by the RF field; the current ideally circulates for many hours and decays only as a result of collisions with remaining gas molecules. At present, only storage rings are used as sources of synchrotron radiation and many of these are dedicated entirely to the production of radiation: they are not used at all, or are used only for limited periods, for nuclear-physics collision experiments.

In equation (4.2.1.17), we may substitute for the various constants and obtain for the radiated power where E is in GeV (10⁹ eV), I is the circulating electron or positron current in milliamperes, and R is in metres. Thus, for example, at the Daresbury storage ring in England, R = 5.5 m and, for operation at 2 GeV and 200 mA, P = 51.5 kW. Storage rings with a total power of the order of 1 MW are planned.

For relativistic electrons, the electromagnetic radiation is compressed into a fan-shaped beam tangential to the orbit with a vertical opening angle , i.e. ~0.25 mrad for E = 2 GeV (Fig. 4.2.1.5 ). This fan rotates with circulating electrons: if the ring is filled with n bunches of electrons, a stationary observer will see n flashes of radiation every 2πR/c s, the duration of each flash being less than 1 ns.

Figure 4.2.1.5| top | pdf | Synchrotron radiation emitted by a relativistic electron travelling in a curved trajectory. B is the magnetic field perpendicular to the plane of the electron orbit; ψ is the natural opening angle in the vertical plane; P is the direction of polarization. The slit S defines the length of the arc of angle Δθ from which the radiation is taken. From Buras & Tazzari (1984); courtesy of ESRP.

The spectral distribution of synchrotron radiation extends from the infrared to the X-ray region; Schwinger (1949) gives the instantaneous power radiated by a monoenergetic electron in a circular motion per unit wavelength interval as a function of wavelength (Winick, 1980). An important parameter specifying the distribution is the critical wavelength : half the total power radiated, but only ∼9% of the total number of photons, is at (Fig. 4.2.1.6 ). is given by from which it follows that in Å can be expressed as where B (= 3.34 E/R) is the magnetic bending field in T, E is in GeV, and R is in metres.

Figure 4.2.1.6| top | pdf | Synchrotron-radiation spectrum: percentage per unit wavelength interval (a) of power of total power and (b) of number of photons of total number of photons at wavelengths greater than λ versus λ/λc. Note that half the power but only 9% of the photons are radiated at wavelengths less than λc; courtesy of H. Winick.

Synchrotron radiation is highly polarized. In an ideal ring where all electrons are parallel to one another in a central orbit, the radiation in the orbital plane is linearly polarized with the electric vector lying in this plane. Outside the plane, the radiation is elliptically polarized.

In practice, the electron path in a storage ring is not a circle. The `ring' consists of an alternation of straight sections and bending magnets and beam lines are installed at these magnets. So-called insertion devices with a zero magnetic field integral, i.e. wigglers and undulators, may be inserted in the straight sections (Fig. 4.2.1.7 ). A wiggler consists of one or more dipole magnets with alternating magnetic field directions aligned transverse to the orbit. The critical wavelength can thus be shifted towards shorter values because the bending radius can be made small over a short section, especially when superconducting magnets are used. Such a device is called a wavelength shifter. If it has N dipoles, the radiation from the different poles is added to give an N-fold increase in intensity. Wigglers can be horizontal or vertical.

Figure 4.2.1.7| top | pdf | Main components of a dedicated electron storage-ring synchrotron-radiation source. For clarity, only one bending magnet is shown. From Buras & Tazzari (1984); courtesy of ESRP.

In a wiggler, the maximum divergence 2α of the electron beam is much larger than ψ, the vertical aperture of the radiation cone in the spectral region of interest (Fig. 4.2.1.5). If and if, in addition, the magnet poles of a multipole device have a short period , the device becomes an undulator: interference will take place between the radiation emitted at two points apart on the electron trajectory (Fig. 4.2.1.8 ). The spectrum at an angle to the axis observed through a pin-hole will consist of a single spectral line and its harmonics of wavelengths (Hofmann, 1978). Typically, the bandwidth of the lines, δλ/λ, will be ∼0.01 to 0.1 and the photon flux per unit band width from the undulator will be many orders of magnitude greater than that from a bending magnet. Existing undulators have been designed for photon energies below 2 keV; higher energies, because of the relatively weak magnetic fields necessitated by the need to keep λ₀ small [equation (4.2.1.21)], require a high electron energy: undulators with a fundamental wavelength in the neighbourhood of 0.86 Å are planned for the European storage ring (Buras & Tazzari, 1984).

Figure 4.2.1.8| top | pdf | Electron trajectory within a multipole wiggler or undulator. λ0 is the spatial period, α the maximum deflection angle, and θ the observation angle. From Buras & Tazzari (1984); courtesy of ESRP.

The wavelength spectra for a bending magnet, a wiggler and an undulator for the ESRF, are shown in Fig. 4.2.1.9 . A comparison of the spectra from an existing storage ring with the spectrum of a rotating-anode tube is shown in Fig. 4.2.1.10 .

Figure 4.2.1.9| top | pdf | Spectral distribution and critical wavelengths for (a) a dipole magnet, (b) a wavelength shifter, and (c) a multipole wiggler for the proposed ESRF. From Buras & Tazzari (1984); courtesy of ESRP.

Figure 4.2.1.10| top | pdf | Comparison of the spectra from the storage ring SPEAR in photons s−1 mA−1 mrad−1 per 1% passband (1978 performance) and a rotating-anode X-ray generator. From Nagel (1980); courtesy of K. O. Hodgson.

The important properties of synchrotron-radiation sources are:

Table 4.2.1.6 (after Buras & Tazzari, 1984) compares the most important parameters of the European Synchrotron Radiation Facility (ESRF) with a number of other storage rings. In this table, BM and W signify bending-magnet and wiggler beam lines, respectively, and is the source divergence; the flux is integrated in the vertical plane. The ESRF is seen to have a higher flux than other sources; even more impressive by virtue of the small dimensions of the source size and divergence are its improvements in spectral brightness (defined as the number of photons s⁻¹ per unit solid angle per 0.1% bandwidth) and in spectral brilliance (defined as the number of photons s⁻¹ per unit solid angle per unit area of the source per 0.1% bandwidth). In comparing different synchrotron-radiation sources with one another and with conventional sources (Fig. 4.2.1.10), the relative quantity for comparison may be flux, brightness or brilliance, depending on the type of diffraction experiment and the type of collimation adopted. Table 4.2.1.7 (due to Farge & Duke, 1979) attempts to compare intensity factors for a number of typical experiments. In general, a high brightness is important in experiments that do not embody focusing elements, such as mirrors or curved crystals, and a high brilliance in those experiments that do.

Table 4.2.1.6| top | pdf | Comparison of storage-ring synchrotron-radiation sources; the parameters were correct in 1985 and, for some sources, may be substantially different from those at earlier or later periods; after Buras & Tazzari (1984), courtesy of ESRP Storage ring Source type No. of poles I (mA) E (GeV) R (m) (mm)† (mm)† (mrad)† (Å) Ec (keV) at at 1.54 Å (1) ESRF BM – 100 5.0 20.0 0.092 0.100 0.008 0.9 14 8 × 1012 1 × 1013 (2) ESRF W 30 100 5.0 11.56 0.062 0.040 0.016 0.5 24 2.4 × 1014 3 × 1014 (3) ADONE (Frascati) BM – 100 1.5 5.0 0.8 0.4 0.04 8.0 1.5 2.4 × 1012 5 × 1010 (4) ADONE (Frascati) W 6 100 1.5 2.6 1.4 0.24 0.08 4.3 3 1.4 × 1013 3.4 × 1012 (5) SRS (Daresbury) BM – 300 2.0 5.56 2.7 0.23 0.05 4.0 3 1 × 1013 3 × 1012 (6) SRS (Daresbury) W 1 300 2.0 1.33 5.3 0.17 0.05 0.9 13 1 × 1013 1.2 × 1013 (7) DCI (Orsay) BM – 250 1.8 3.82 2.72 1.06 0.06 3.6 3.4 7 × 1012 2.4 × 1012 (8) DORIS (Hamburg) BM – 100 3.7 12.22 1.0 0.3 0.05 1.3 9.2 6 × 1012 6.4 × 1012 (9) DORIS (Hamburg) BM – 40 5.0 12.22 1.3 0.65 0.065 0.55 23 3 × 1012 4.4 × 1013 (10) DORIS (Hamburg) W 32 100 3.7 20.57 1.5 0.4 0.033 2.3 5.5 1.9 × 1014 1.3 × 1014 (11) CESR (Cornell) BM – 40 5.5 32.0 1.44 1.0 0.065 1.0 11.5 3.5 × 1012 4 × 1012 (12) CESR (Cornell) W 6 40 5.5 13.2 1.9 1.2 0.05 0.4 28 2 × 1013 3 × 1013 (13) NSLS X-ray (Brookhaven) BM – 300 2.5 6.83 0.25 0.1 0.01 2.4 5 1 × 1013 8 × 1012 (14) SPEAR BM – 100 3.0 12.7 2.0 0.28 0.05 2.7 5 5 × 1012 3 × 1012 (15) SPEAR W 8 100 3.0 5.57 3.2 0.15 0.03 1.0 10 3.8 × 1013 4.5 × 1013 (16) SPEAR W 54 100 3.0 8.36 3.2 0.15 0.03 1.7 7 2.6 × 1014 2.4 × 1014 (17) Photon Factory (Tsukuba) BM – 150 2.5 8.66 2.2 0.6 0.14 3.0 4 6 × 1012 3 × 1012 (18) Photon Factory (Tsukuba) W 3 150 2.5 1.85 1.9 0.7 0.18 0.7 19 1.8 × 1013 2.5 × 1013 (19) VEPP-3 BM – 100 2.2 6.15 6.15 0.08 0.02 3.0 4 3.5 × 1012 1.5 × 1012 †One standard deviation of Gaussian distribution.

Table 4.2.1.7| top | pdf | Intensity gain with storage rings over conventional sources; from Farge & Duke (1979), courtesy of ESF   GX6 rotating-anode tube 2.4 kW (Cu Kα emission) DCI 1.72 GeV and 240 mA ESRF 5 GeV and 565 mA Small-angle scattering with a double monochromator ×500 to 1000 ×15000 to 3000 Protein crystallography with a single-focus monochromator       1 mm3 samples ×50 to 160 ×900 to 1800   Small samples ×30 to 60 ×650 to 1300 Diffuse scattering (wide angles, low resolution and large samples) with a curved graphite monochromator ×20 to 40 ×160 to 320 Non characteristic wavelength (continuous background) EXAFS experimental set-up with a 100 kW rotating anode ×104 ×105

Many surveys of existing and planned synchrotron-radiation sources have been published since the compilation of Table 4.2.1.6. Fig. 4.2.1.11 , taken from a recent review (Suller, 1992), is a graphical illustration of the growth and the distribution of these sources. An earlier census is due to Huke & Kobayakawa (1989). Many detailed descriptions of beam lines for particular purposes, such as protein crystallography (e.g. Fourme, 1992) or at individual storage rings (e.g. Kusev, Raiko & Skuratowski, 1992) have appeared: these are too numerous to list here and can be located by reference to Synchrotron Radiation News.

Figure 4.2.1.11| top | pdf | The evolution of storage-ring synchrotron-radiation sources over the decades, as illustrated by their increasing number and range of machine energies (based on Suller, 1992).

Plasma sources of hard X-rays are being investigated in many laboratories. Most of the material in this section is derived from publications from the Laboratory for Laser Energetics, University of Rochester, USA. Plasma sources of very soft X-rays have been reviewed by Byer, Kuhn, Reed & Trail (1983).

The peak wavelength of emission from a black-body radiator falls into the ultraviolet at about 10⁵ K and into the X-ray region between 10⁶ and 10⁷ K. At these temperatures, matter is in the form of a plasma that consists of highly ionized atoms and of electrons with energies of several keV. The only successful methods of heating plasmas to temperatures in excess of 10⁶ K is by means of high-energy laser beams with intensities of 10¹² W mm⁻² or more. The duration of the laser pulse must be less than 1 ns so that the plasma cannot flow away from the pulse. When the plasmas are created from elements with 15 Z 25, they consist mainly of ions stripped to the K shell, that is of hydrogen- and helium-like ions. The X-ray spectrum (Fig. 4.2.1.12 ) then contains a main group of lines with a bandwidth for the group of about 1%; the band is situated slightly below the K-absorption edge of the target material. The intensity of the band drops with increasing atomic number. For diffraction studies, Forsyth & Frankel (1980, 1984) and Frankel & Forsyth (1979, 1985) used a multi-stage Nd³⁺:glass laser (Seka, Soures, Lewis, Bunkenburg, Brown, Jacobs, Mourou & Zimmermann, 1980), which was able to deliver up to 220 J per pulse of width 700 ps. They obtained  photons pulse⁻¹ for a Cl¹⁵⁺ plasma with a mean wavelength of about 4.45 Å and about  photons pulse⁻¹ for a Fe²⁴⁺ plasma at about 1.87 Å (Yaakobi, Bourke, Conturie, Delettrez, Forsyth, Frankel, Goldman, McCrory, Seka, Soures, Burek & Deslattes, 1981). More recently, the laser was fitted with a frequency conversion system that shifts the peak power of the laser light from the infrared (1.054 µm) to the ultraviolet (0.351 µm) (Seka, Soures, Lund & Craxton 1981). This led to a more efficient X-ray production, which permitted a more than twofold increase in X-ray flux, even though the maximum pulse energies had to be reduced to ∼50 J to prevent damage to the optical components (Yaakobi, Boehli, Bourke, Conturie, Craxton, Delettrez, Forsyth, Frankel, Goldman, McCrory, Richardson, Seka, Shvarts & Soures, 1981). Forsyth & Frankel (1984) used the plasma X-ray source for diffraction studies with 4.45 Å X-rays with a focusing collimation system that delivered up to 10¹⁰ photons pulse⁻¹ to the specimen over an area approximately 150 µm in diameter. More recently, by special target design (Forsyth, 1986, unpublished), fluxes have been increased by factors of 2 to 3 without altering the laser output. Other plasma sources have been described by Collins, Davanloo & Bowen (1986) and by Rudakov, Baigarin, Kalinin, Korolev & Kumachov (1991).

Figure 4.2.1.12| top | pdf | X-ray emission from various laser-produced plasmas. From Forsyth & Frankel (1980); courtesy of J. M. Forsyth.

The cost of plasma sources is about an order of magnitude greater than that of rotating-anode generators (Nagel, 1980). Their use is at present confined to flash-diffraction experiments, since the duty cycle is a maximum of one flash every 30 min. Attempts are being made to increase the laser repetition rate; a substantial improvement could lead to a source that would rival storage-ring sources.

Parametric X-ray generation can be described as the diffraction of virtual photons associated with the field of a relativistic charged particle passing through a crystal. These diffracted photons appear as real photons with an energy that satisfies Bragg's law for the reflecting crystal planes, so that the energy can be tuned between 5 and 45 keV by rotating the mosaic graphite crystal. Linear accelerators with an energy between 100 and 500 MeV produce the incident relativistic electron beam (Maruyama, Di Nova, Snyder, Piestrup, Li, Fiorito & Rule, 1993; Fiorito, Rule, Piestrup, Li, Ho & Maruyama, 1993).

Transition-radiation X-rays with peak energies between 10 and 30 keV are produced when electrons from 100 to 400 MeV strike a stack of thin foils (Piestrup, Moran, Boyers, Pincus, Kephart, Gearhart & Maruyama, 1991). Quasi-monochromatic X-rays result from a selection of target foils with appropriate K-, L- or M-edge frequencies (Piestrup, Boyers, Pincus, Harris, Maruyama, Bergstrom, Caplan, Silzer & Skopik, 1991).

Channelling radiation, resulting from the incidence of electrons with an energy of only about 5 MeV on appropriately aligned diamond or silicon crystals hold out the hope of producing a bright tunable X-ray source.

One or more of these methods may, in the future, be developed as X-ray sources that can compete with synchrotron-radiation sources.

Wavelength tables in previous editions of this volume (Rieck, 1962; Arndt, 1992) were mainly obtained from the compilations prepared in Paris under the general direction of Professor Y. Cauchois (Cauchois & Hulubei, 1947; Cauchois & Senemaud, 1978). A separate effort by the late Professor J. A. Bearden and his collaborators (Bearden, 1967) has been widely used in other aggregations of tabular data and was made available for some time through the Standard Reference Data Program at the National Institute of Standards and Technology (NIST). For simplicity in the following discussion, we use the Bearden database as a frame of reference with respect to which our current, and rather different, approach can be compared. Although a detailed comparison of the historical databases may be of some interest, the result would have only very small influence on the outcome presented here. To specify this framework, we begin with a brief description of the procedures used in establishing this reference database.

Bearden and his collaborators remeasured a group of five X-ray lines (Bearden, Henins, Marzolf, Sauder & Thomsen, 1964), with the remaining entries in the wavelength table coming from a critically reviewed, and re-scaled, subset of earlier measurements (Bearden, 1967). Line locations were given in Å* units, a scale defined by setting the wavelength of W Kα₁ = 0.2090100 Å*. It was Bearden's intention that, for all but the most demanding applications, one could simply assign Å*/Å = 1, with an uncertainty arising from the fundamental physical constants, particularly N_A and hc/e, combined with uncertainties arising from the measurement technology (Bearden, 1965). Not long after the publication of the final compilation (Bearden, 1967), it became clear that the fundamental constants used in defining Å* needed significant revision (Cohen & Taylor, 1973), and that there were some inconsistencies in the metrology (Kessler, Deslattes & Henins, 1979).

Aside from the particular issues noted above, all previous wavelength tables had certain limitations arising from the procedures used in their generation. In particular, except for a small group of five spectra (Bearden, Thomsen et al., 1964), the Bearden tables relied entirely on data previously reported in the literature. Both of the other tabulations also proceeded using only reported experimental values (Cauchois & Hulubei, 1947; Cauchois & Senemaud, 1978). In the Bearden compilation process, available data for each emission line were weighted according to claimed uncertainties, modified in certain cases by Bearden's detailed knowledge of the measurement practices of the major sources of experimental wavelength values. The complete documentation of this remarkable undertaking is, unfortunately, not widely accessible. Our evident need to understand the origin of the `recommended' values has been greatly aided by the availability of a copy of the full documentation (Bearden, Thomsen et al., 1964).

The actual experimental data array from which the previous tables emerged is not complete, even for the prominent (`diagram') lines. In the cases where experimental data were not available [as can be seen only in the source documentation (Bearden, Thomsen et al., 1964)], the gaps were filled by interpolated values based on measurements available from nearby elements, plotted on a modified Moseley diagram in which the term dependence is taken into account (Burr, 1996). In the end, such a smooth scaling with respect to nuclear charge suppresses the effects of the atomic shell structure, a practice that must be avoided in order to obtain the significant improvement in the database that we hope to provide. Also obscured in smooth Z scaling are detectable contributions arising from the fact that nuclear sizes do not change smoothly as a function of the nuclear charge, Z.

There are several possible approaches to generating an improved, `all-Z' table of X-ray wavelengths. These range from the option of conducting a massive measurement campaign to populate more fully the currently available tabular array to a large computational endeavor that might purport to carry out multiconfiguration, relativistic wavefunction calculations for the entire Periodic Table. It seems evident to us that there is little interest in, and even less support for, mounting the large effort needed to realize an improved tabulation of X-ray wavelengths by purely experimental means, while the possibility of proceeding in an entirely theoretical mode is not consistent with the evident need that at least some wavelengths be reported with uncertainties that approach the limit of what can be obtained from the naturally occurring X-ray lines. The actual location of any useful feature of a line is influenced not only by the physical and chemical environment of the emitting atom but also by inevitable multi-electron excitation processes that perturb the entire spectral profile. Calculation of such complexities currently lies beyond the limits of practicality, eliminating the option of proceeding without strong coupling to experimental profile locations, at least for crystallographically important X-ray lines. Similar considerations apply a fortiori to those lines needed as reference wavelengths for exotic atom measurements, such as those leading to masses of elementary particles and tests of basic theory [see e.g. Beyer, Indelicato, Finlayson, Liesen & Deslattes (1991)].

In constructing the accompanying tables, we have chosen a new procedure that differs from those described above, and accordingly requires some detailed commentary. We begin with the presently available network of well documented experimental measurements, originally established to provide a test bed for the theoretical methods developing at that time (Deslattes & Kessler, 1985). This modest network was the first compilation to make use of the, then newly available, connection between the X-ray region and the base unit of the International System of measurement (the SI) based on optical interferometric measurement of a lattice period as revealed by X-ray interferometry. Details of the generation of this network and its subsequent expansion will be given below. Using this network as a test set gave clearer suggestions as to specific limitations of the theoretical modelling than had been evident from using other, less selective, experimental reference compilations available at that time. Extensive theoretical developments before and, especially, after the appearance of this new experimental reference set have shown a steady convergence toward these critically evaluated data. Following this evolution further, our long-term plan is to use these new theoretical calculations to provide a more structured and accurate interpolation procedure for estimating the spectra of elements lying between those for which we have accurate measurements, or spectra well connected to a directly established reference wavelength. The present table provides experimental and theoretical values for some of the more prominent K and L series lines and is a subset of a larger effort for all K and L series lines connecting the n = 1 to n = 4 shells. The more complete table will be published elsewhere and be made available on the the NIST Physical Reference Data web site. In addition, experimental values for the K and L edges are provided. Although the reference data are inadequate in both low and high ranges of Z, the general consistency of theory and experiment through the region 20 Z 90 for the strong K-series and L-series lines suggests that, in the absence of good reference measurements, the uncorrected theoretical values should be considered for applications not requiring the highest accuracy.

Historically, from the first realizations of refined spectroscopy in the X-ray region (ca 1915–1925) up to the period 1975–1985, the best measured X-ray wavelengths had to be expressed in some local unit, most often designated as the xu (x unit) or kxu (kilo x unit). Uncertainty in the conversion factor between the X-ray and optical scales was the dominant contributor to the total uncertainties in the wavelength values of the sharper X-ray emission lines, such as those most frequently used in crystallography. (For a discussion of the present values in relation to previously assigned numerical values on the various scales, see Subsection 4.2.2.14.) This local unit was, for most of the time, `officially' defined by assigning a specific numerical value to the lattice period of a particular reflection from `the purest instance' of a particular crystal. Originally this was rocksalt; later it was calcite. In practice, most work used as de facto standards certain values for Cu and Mo whose inconsistency, though noted by some crystallographers earlier, was seriously addressed by Bearden and co-workers only in the 1960's (Bearden, Henins et al., 1964). This early history was summarized in 1968 (Thomsen & Burr, 1968). Connection of the X-ray wavelength scale to the primary realizations of the length (wavelength) unit in the `metric system' was primarily (at least after about 1930) through ruled grating measurements of longer-wavelength X-ray lines such as Al .

The remainder of the X-ray wavelength database was derived from relative measurements using crystal diffraction spectroscopy. Unfortunately, even the most refined among the ruled grating measurements did not give accuracies comparable to the precision accessible by relative wavelength measurements (Henins, 1971). As noted above, in connection with establishing the previous wavelength table, Bearden introduced a new local unit, the Å*, based on an explicit value for the wavelength of W , chosen to give a conversion factor near unity. This transitional period will not be treated further in the present documentation since, to a substantial extent, developments described in the following paragraphs have effectively eliminated the need for a local scale for X-ray wavelength metrology.

Following the demonstration of crystal lattice interferometry in the X-ray region (Bonse & Hart, 1965a), efforts to combine such an X-ray interferometer with various optical interferometers were undertaken in several (mostly national standards) laboratories. Although the earliest of these, carried out at the National Bureau of Standards (NBS) (now the National Institute of Standards and Technology, NIST) (Deslattes & Henins, 1973) was, in the end, found to be burdened by a serious systematic error (1.8 × 10⁻⁶) in later work at the Physikalisch Technishe Bundesanstalt (PTB) (Becker et al., 1981; Becker, Seyfried & Siegert, 1982), it was clear that accuracy limitations associated with ruled grating measurements no longer dominated the metrology of X-ray wavelengths. The origin of the systematic error in the early NBS measurement was subsequently understood (Deslattes, Tanaka, Greene, Henins & Kessler, 1987), and, more recently, excellent results were obtained in Italy (Basile, Bergamin, Cavagnero, Mana, Vittone & Zosi, 1994, 1995) and Japan (Fujimoto, Fujii, Tanaka & Nakayama, 1997). In all cases, the goal was to obtain an optical measurement of a crystal lattice period (thus far, only Si 220) and to use the calibrated crystals in diffraction spectrometry to establish optically based X-ray wavelengths. Such exercises have been undertaken for several X-ray lines, but the most detailed and well documented results to date were obtained in Jena (Härtwig, Hölzer, Wolf & Förster, 1993; Hölzer, Fritsch, Deutsch, Härtwig & Förster, 1997), where the Kα and Kβ spectra of the elements Cr to Cu were evaluated using silicon crystals well connected with the crystal spacings measured at the PTB.

In addition to the Jena measurements noted above, a number of characteristic X-ray lines were measured on the optically based scale at NBS/NIST. The set of directly measured reference wavelengths is given in Table 4.2.2.1 in bold type. Most of the originally published (NBS/NIST) values were burdened by the 1.8 × 10⁻⁶ error in the silicon lattice period, as noted above. These have been corrected in the numerical results summarized in the table. The directly measured elements and lines appearing in this table were often chosen to meet the need for specific reference values in locations near those of certain optical transitions in highly charged ion spectra or spectra from pionic atoms. In addition, early NBS measurements specifically addressed the lines most often used in crystallography and the W Kα transition. In response to the needs of electron spectroscopy, Al and Mg K spectra were also determined (Schweppe, Deslattes, Mooney & Powell, 1994). In this case, the original lattice error had been previously recognized, so that no rescaling was required. The remaining directly measured entries were obtained as opportunities to do so emerged.

Table 4.2.2.1| top | pdf | K-series reference wavelengths in Å; bold numbers indicate a directly measured line Numbers in parentheses are standard uncertainties in the least-significant figures. Z Symbol A References 12 Mg   9.89153 (10) 9.889554 (88)     (a) 13 Al   8.341831 (58) 8.339514 (58)     (a) 14 Si   7.12801 (14) 7.125588 (78)     (b) 16 S   5.374960 (89) 5.372200 (78)     (b) 17 Cl   4.730693 (71) 4.727818 (71)     (b) 18 Ar   4.194939 (23) 4.191938 (23)     (c) 19 K   3.7443932 (68) 3.7412838 (56)     (d) 24 Cr   2.2936510 (30) 2.2897260 (30) 2.0848810 (40) 2.0848810 (40) (e) 25 Mn   2.1058220 (30) 2.1018540 (30) 1.9102160 (40) 1.9102160 (40) (e) 26 Fe   1.9399730 (30) 1.9360410 (30) 1.7566040 (40) 1.7566040 (40) (e) 27 Co   1.7928350 (10) 1.7889960 (10) 1.6208260 (30) 1.6208260 (30) (e) 28 Ni   1.6617560 (10) 1.6579300 (10) 1.5001520 (30) 1.5001520 (30) (e) 29 Cu   1.54442740 (50) 1.54059290 (50) 1.3922340 (60) 1.3922340 (60) (e) 31 Ga   1.3440260 (40) 1.3401270 (96) 1.208390 (75) 1.207930 (34) (b), (f) 33 As   1.108830 (31) 1.104780 (12) 0.992689 (79) 0.992189 (53) (b), (f) 34 Se   1.043836 (30) 1.039756 (30) 0.933284 (74) 0.932804 (30) (b), (f) 36 Kr   0.9843590 (44) 0.9802670 (40) 0.8790110 (70) 0.8785220 (50) (b) 40 Zr   0.7901790 (25) 0.7859579 (27) 0.7023554 (30) 0.7018008 (30) (b) 42 Mo   0.713607 (12) 0.70931715 (41) 0.632887 (13) 0.632303 (13) (d), (f) 44 Ru   0.6474205 (61) 0.6430994 (61) 0.5730816 (42) 0.5724966 (42) (d), (f) 45 Rh   0.6176458 (61) 0.6132937 (61) 0.5462139 (42) 0.5456189 (42) (d), (f) 46 Pd   0.5898351 (60) 0.5854639 (46) 0.5211363 (41) 0.5205333 (41) (d), (f) 47 Ag   0.5638131 (26) 0.55942178 (76) 0.4976977 (60) 0.4970817 (60) (d), (f) 48 Cd   0.5394358 (46) 0.5350147 (46) 0.4757401 (71) 0.4751181 (71) (d), (f) 49 In   0.5165572 (60) 0.5121251 (46) 0.4551966 (41) 0.4545616 (41) (d), (f) 50 Sn   0.4950646 (46) 0.4906115 (46) 0.4358821 (51) 0.4352421 (51) (d), (f) 51 Sb   0.4748391 (45) 0.4703700 (45) 0.4177477 (41) 0.4170966 (31) (d), (f) 54 Xe   0.42088103 (71) 0.4163508 (14) 0.3694051 (13) 0.3687346 (13) (d) 56 Ba   0.38968378 (74) 0.38512464 (84) 0.3415228 (11) 0.34082708 (75) (d) 60 Nd   0.3248079 (59) 0.3201648 (59) 0.283634 (59) 0.282904 (44) (d), (f) 62 Sm   0.31369830 (79) 0.30904506 (46) 0.273764 (30) 0.273014 (30) (d), (f) 67 Ho   0.26549088 (84) 0.2607608 (42) 0.230834 (30) 0.230124 (30) (f), (g) 68 Er   0.2571133 (11) 0.25237359 (62) 0.2234766 (14) 0.22269866 (72) (d) 69 Tm   0.24910095 (61) 0.24434486 (44) 0.216366 (30) 0.21559182 (57) (f), (h) 74 W   0.21383304 (50) 0.20901314 (18) 0.18518317 (70) 0.1843768 (30) (d), (f) 79 Au   0.18507664 (61) 0.18019780 (47) 0.1598249 (13) 0.15899527 (77) (d) 82 Pb   0.17029527 (56) 0.16537816 (38) 0.1468129 (10) 0.14596836 (58) (d) 83 Bi   0.1657183 (20) 0.1607903 (46) 0.142780 (11) 0.1419492 (54) (f), (g) 90 Th 230 0.13782600 (31) 0.13282021 (36) 0.11828686 (78) 0.11740759 (59) (d) 91 Pa 231 0.1343516 (29) 0.1293302 (27) 0.1152427 (21) 0.1143583 (21) (i) 92 U 238 0.13099111 (78) 0.12595977 (36) 0.11228858 (66) 0.11140132 (65) (d) 93 Np 237 0.1277287 (39) 0.1226882 (36) 0.1094230 (39) 0.1085265 (28) (i) 94 Pu 239 0.1245782 (15) 0.11952120 (69)     (h) 94 Pu 244 0.1245705 (25) 0.1195140 (23) 0.1066611 (18) 0.1057595 (18) (i) 95 Am 243 0.1215158 (24) 0.1164463 (33) 0.1039794 (17) 0.1030803 (17) (i) 96 Cm 248 0.1185427 (23) 0.1134635 (21) 0.1013753 (17) 0.1004708 (16) (i) 97 Bk 249 0.1156630 (54) 0.1105745 (49) 0.0988598 (55) 0.0979514 (54) (i) 98 Cf 250 0.1128799 (82) 0.1077793 (75)     (i) References: (a) Schweppe et al. (1994); (b) Mooney (1996); (c) Schweppe (1995); (d) Deslattes & Kessler (1985); (e) Hölzer et al. (1997); (f) Bearden (1967); (g) Borchert, Hansen, Jonson, Ravn & Desclaux (1980); (h) Borchert (1976); (i) Barreau, Börner, Egidy & Hoff (1982).

The optically based data set was expanded by noting that several groups of accurate (relative) measurements in the literature either contained one of the directly measured lines (bold type) in Table 4.2.2.1 or were explicitly connected to one of them. Most often this situation was realized in reports that indicated a specific reference value, i.e. where it was stated numerical values are based on a scale where, for example, the wavelength of Mo was taken as 707.831 xu. In such cases, and where other indicators of good measurement quality are presented, it is easy to re-scale the data reported so that it is consistent with the optically based data. This procedure was followed for important groups of measurements from earlier work by Bearden and co-workers, and from the X-ray laboratory at Uppsala. The rescaled numerical results are included in Table 4.2.2.1 in normal type along with the specific literature citations. The indicated uncertainties are standard uncertainties as defined by ISO (Taylor & Kuyatt, 1994; Schwarzenbach, Abrahams, Flack, Prince & Wilson, 1995).

To date, only a very limited number of L-series emission lines have been directly measured on an optically based scale. Wavelengths for directly measured L-series lines are reported in Table 4.2.2.2 along with the literature citations. In the future, we hope to expand this limited data set by including other lines and elements that are well connected to these reference lines in the literature.

Table 4.2.2.2| top | pdf | Directly measured L-series reference wavelengths in Å Numbers in parentheses are standard uncertainties in the least significant figures. Z Symbol References 36 Kr 7.82032(13) 7.82032(13) 7.574441(98) (a) 40 Zr 6.07710(48) 6.070250(79) 5.836214(76) (a) 54 Xe 3.025940(22) 3.016582(15) 2.806553(19) (b) 60 Nd 2.38079(52) 2.370526(16) 2.167008(19) (a) 62 Sm 2.210430(24) 2.199873(13) 1.998432(30) (a) 67 Ho 1.856472(15) 1.845092(17) 1.647484(32) (a) 68 Er 1.795701(45) 1.784481(20) 1.587466(86) (a) 69 Tm 1.738003(19) 1.7267720(70) 1.5302410(70) (a) References: (a) Mooney (1996); (b) Mooney et al. (1992).

Only a small number of absorption-edge locations have been directly measured to high accuracy using the currently acceptable protocols. Some of the available data were obtained in order to provide wavelength determinations for spectra from highly charged and/or exotic atoms (Bearden, 1960; Lum et al., 1981). This small group was, however, significantly expanded very recently by an important set of new measurements, extending to Z = 51, that are well coupled to the optical wavelength scale (Kraft, Stümpel, Becker & Kuetgens, 1996) The resulting experimental database is summarized in Table 4.2.2.3. The effort that would be needed to expand the experimental database in a systematic way is quite large. Thus, we make use of a procedure, not previously used for this purpose, that combines available electron binding energy data with emission-line locations from our expanded reference set of emission-line data and emission lines that have been rescaled to be consistent with the optically based scale. At the same time, calculation of the location of absorption thresholds within the theoretical framework (see below) has been undertaken and will be made available in the longer publication and on the web site.

Table 4.2.2.3| top | pdf | Directly measured and emission + binding energies (see text) K-absorption edges in Å Numbers in parentheses are standard uncertainties in the least significant figures. Z Symbol Directly measured Emission + binding energies References 23 V 2.269211(21) 2.26893(11) (a) 24 Cr 2.070193(14) 2.07014(17) (a) 25 Mn 1.8964592(58) 1.896457(42) (a) 26 Fe 1.7436170(49) 1.743589(98) (a) 27 Co 1.6083510(42) 1.60836(17) (a) 28 Ni 1.4881401(36) 1.48823(25) (a) 29 Cu 1.3805971(31) 1.38060(16) (a) 30 Zn 1.2833798(40) 1.28338(15) (a) 39 Y 0.7277514(21) 0.727750(23) (a) 40 Zr 0.6889591(31) 0.688946(30) (a) 41 Nb 0.6531341(14) 0.653112(29) (a) 42 Mo 0.61991006(62) 0.619906(64) (a) 45 Rh 0.5339086(69) 0.533951(10) (a) 46 Pd 0.5091212(42) 0.509156(11) (a) 47 Ag 0.4859155(57) 0.4859168(91) (a) 48 Cd 0.4641293(35) 0.464135(12) (a) 49 In 0.4437454(48) 0.443740(11) (a) 50 Sn 0.4245978(29) 0.424590(13) (a) 51 Sb 0.4066324(27) 0.406612(12) (a) 68 Er 0.2156801(75) 0.2156762(50) (b) 82 Pb 0.1408821(74) 0.1408836(11) (c) References: (a) Kraft et al. (1996); (b) Lum et al. (1981); (c) Bearden (1960).

The feature of absorption spectra customarily designated as `the absorption edge' has been variously associated with: the first inflection point of the absorption spectrum; the energy needed to produce a single inner vacancy with the photo-electron `at rest at infinity'; or the energy needed to remove an electron from an inner shell and place it in the lowest unoccupied energy level. A general discussion of this question has been given by Parratt (1959). If we choose the second alternative, then it is easy to see that, with some care for symmetry restrictions, one can estimate the absorption-edge energy by combining the binding energy for any accessible outer shell with the energy of an emission line for which the transition terminus lies in the same outer shell. Of course, this procedure does not focus on the details of absorption thresholds, the locations of which are important for a number of structural applications. On the other hand, our choice gives greater regularity with respect to nuclear charge and facilitates use of electron binding energies, since they are referenced to the Fermi energy or the vacuum.

Electron binding energies have been tabulated for the principal electron shells of all the elements considered in the present table (Fuggle, Burr, Watson, Fabian & Lang, 1974; Cardona & Ley, 1978; Nyholm, Berndtsson & Mårtensson, 1980; Nyholm & Mårtensson, 1980; Lebugle, Axelsson, Nyholm & Mårtensson, 1981; Powell, 1995). The number of values available offers the possibility of consistency checking, since the K and L shells are connected by emission lines to several final hole states, each of which has (possibly) been evaluated by photoelectron spectroscopy. For each of the elements for which well qualified reference spectra are available, we evaluated edge location estimates using several alternative transition cycles and used the distribution of results to provide a measure of the uncertainty. Comparison of edge estimates obtained by this procedure with experimental data provides a quantitative test of the utility of the chosen approach to edge location estimation. In Table 4.2.2.3, the numerical results in the column labelled `Emission + binding energies' were obtained by combining emission energies and electron binding energies using all possible redundancies. The estimated uncertainties indicated were obtained from the distribution of the redundant routes. As can be seen, the results are in general agreement with the available directly measured values. Accordingly, we have used this protocol to obtain the edge locations listed in the summary tables below.

Only recently has it become possible to understand the relativistic many-body problem in atoms with sufficient detail to permit meaningful calculation of transition energies between hole states (Indelicato & Lindroth, 1992; Mooney, Lindroth, Indelicato, Kessler & Deslattes, 1992; Lindroth & Indelicato, 1993, 1994; Indelicato & Lindroth, 1996). To deal with those hole states for atomic numbers ranging from 10 to 100, one needs to consider five kinds of contributions, all of which must be calculated in a relativistic framework, and the relative influence of which can change strongly as a function of the atomic number:

Such an undertaking, although much more advanced than any other done in the past, still suffers from severe limitations that need to be understood fully to make the best use of the table. The main limitation is probably that most lines are emitted by atoms in an elemental solid or a compound, while the calculation at present deals only with atoms isolated in vacuum. (A purely experimental database would have a similar limitation.) The second limitation is that it is not possible at present to include the coupling between the hole and open outer shells. Coupling between a , or hole and an external 3d or 4f shell can generate hundreds of levels, with splitting that can reach an eV. One then should calculate all radiative and Auger transition probabilities between hundreds of initial and final states. (The Auger final state would have one extra vacancy, leading eventually to thousands of final states.) Such an approach would give not only the mean line energy but also its shape and would thus be very desirable, but is impossible to do with present day theoretical tools and computers. We have thus limited ourselves to an approach in which one computes the weighted average energy for each hole state, and ignores possible distortion of the line profile due to the coupling between inner vacancies and outer shells.

Since we want to have good predictions for both light and heavy atoms, we have to include relativity non-perturbatively. To get a result approaching 1 ×  10⁻⁶ for uranium Kα by applying perturbation theory to the Schrödinger equation, for example, one would need to go to order 22 in powers of Zα = v/c. The natural framework in this case is thus to do a calculation exact to all orders in Zα by using the Dirac equation. We thus have used many-body methods, based on the Dirac equation, in which the main contributions to the transition energy are evaluated using the Dirac–Fock method. We use the Breit operator for the electron–electron interaction, to include magnetic (spin–spin, spin–other orbit and orbit–orbit interactions in the lower orders in Zα and (v/c)² retardation effects. Higher-order retardation effects are also included. Many-body effects are calculated by using relativistic many-body perturbation theory (RMBPT). Since inner vacancy levels are auto-ionizing, one must include shifts in their energy due to the coupling between the discrete levels and Auger decay continuua.

In the following subsections, we describe in more detail the calculation of the different contributions.

The first step in the calculation, following Indelicato and collaborators (Indelicato & Desclaux, 1990; Indelicato & Lindroth, 1992; Mooney et al., 1992; Lindroth & Indelicato, 1993; Indelicato & Lindroth, 1996) consists in evaluating the best possible energy with relativistic corrections, within the independent electron approximation, for each hole state (here , , , , for K, L_II, L_III, M_II, M_III, respectively). Such a calculation must provide a suitable starting point for adding all many-body and QED contributions. We have thus chosen the Dirac–Fock method in the implementation of Desclaux (1975, 1993). This method, based on the Dirac equation, allows treatment of arbitrary atoms with arbitrary structure and has been widely used for this kind of calculation. We have used it with full exchange and relaxation (to account for inactive orbital rearrangement due to the hole presence). The electron–electron interaction used in this program contains all magnetic and retardation effects, which is very important to have good results at large Z. The magnetic interaction is treated on an equal footing with the Coulomb interaction, to account for higher-order effects in the wavefunction (which are also useful for evaluating radiative corrections to the electron–electron interaction). All these calculations must be done with proper nuclear charge models to account for finite-nuclear-size corrections to all contributions. For heavy nuclei, nuclear deformations must be accounted for (Blundell, Johnson & Sapirstein, 1990; Indelicato, 1990). For all elements for which experiments have been performed, we used experimental nuclear charge radii. For the others we used a formula from Johnson & Soff (1985), corrected for nuclear deformations for Z 90. Contribution of deformation to the r.m.s. radius (the only parameter of importance to the atomic calculation) is roughly constant (0.11 fm) for Z 90. There is an unknown region, between Bi and Th (83 Z 90), where deformation effects start to be important, but for which they are not known. When experiments are done for a particular isotope, we calculated separately the energies for each isotope.

As mentioned in the introduction, there are special difficulties involved when dealing with atoms with open outer shells (obviously this is the most common case). Computing all energies for total angular momentum J would be both impossible and useless. The Dirac–Fock method circumvents this difficulty. One can evaluate directly an average energy that corresponds to the barycentre of all with weight (). There are still a few cases for which the average calculation cannot converge (when the open shells have identical symmetry). In that case, the outer electrons have been rearranged in an identical fashion for all hole states of the atom, to minimize possible shifts due to this procedure.

Once the Dirac–Fock energy is obtained, many-body effects beyond Dirac–Fock relaxation must be taken into account. These include relaxation beyond the spherical average, correlation (due to both Coulomb and magnetic interaction), and corrections due to the autoionizing nature of hole states (Auger shift). Since the many-body generalization of the Dirac–Fock method, the so-called MCDF (multiconfiguration Dirac–Fock), is very inefficient for hole states, we turned to RMBPT to evaluate those quantities. These many-body effects contribute very significantly to the final value. Coulomb correlation is mostly constant along the Periodic Table (at the level of a few eV). Magnetic correlations are very strong at high Z. Auger shift is very important for p states. The interested reader will find more details of these complicated calculations in the original references (Indelicato & Lindroth, 1992; Mooney et al., 1992; Lindroth & Indelicato, 1993; Indelicato & Lindroth, 1996). As these calculations are very time consuming, they are performed only for selected Z and interpolated. Since the Auger shifts do not always have a smooth Z dependence, care has been taken to evaluate them at as many different Z's as practical to ensure a good reproduction of irregularities.

The QED corrections originate in the quantum nature of both the electromagnetic and electron fields. They can be divided in two categories, radiative and non-radiative. The first one includes self-energy and vacuum polarization, which are the main contributions to the Lamb shift in one-electron atoms. These corrections scale as (n being the principal quantum number) and are thus very important for inner shells and high Z. The second category is composed of corrections to the electron–electron interaction that cannot be accounted for by RMBPT or MCDF. These corrections start at the two-photon interaction and include three-body effects. The two-photon, non-radiative QED contribution has been calculated recently only for the ground state of two-electron ions (Blundell, Mohr, Johnson & Sapirstein, 1993; Lindgren, Persson, Salomonson & Labzowsky, 1995) and cannot be evaluated in practice for atoms with more than two or three electrons.

The radiative corrections split up into two contributions. The first contribution is composed of one-electron radiative corrections (self-energy and vacuum polarization). For the self-energy and , one must use all-order calculations (Mohr, 1974a,b, 1975, 1982, 1992; Mohr & Soff, 1993). Vacuum polarization can be evaluated at the Uehling (1935) and Wichmann & Kroll (1956) level. Higher-order effects are much smaller than for the self-energy (Soff & Mohr, 1988) and have been neglected. The second contribution is composed of radiative corrections to the electron–electron interaction, and scales as . Ab initio calculations have been performed only for few-electron ions (Indelicato & Mohr, 1990, 1991). Here we use the Welton approximation which has been shown to reproduce very closely ab initio results in all examples that have been calculated (Indelicato, Gorceix & Desclaux 1987; Indelicato & Desclaux 1990; Kim, Baik, Indelicato & Desclaux, 1991; Blundell, 1993a,b).

Table 4.2.2.4 summarizes the theoretical and experimental results for prominent K-series lines and the K-absorption edge. For the emission lines, the upper number (in italics) is the theoretical estimate for this line and the lower number is the experimentally measured value (1) from Table 4.2.2.1 or (2) from the Bearden database or a reference that appeared after the Bearden database corrected to the optically based scale. For the K absorption edge, the upper number (also in italics) was obtained by combining emission lines and photoelectron spectroscopy (see Subsection 4.2.2.7), and the lower number is the experimentally measured value (1) from Table 4.2.2.3 or (2) from the Bearden database or a reference that appeared after the Bearden database corrected to an optically based scale. For the experimental emission and absorption entries, bold type is used for wavelengths directly measured on an optically based scale. The numerical values for wavelengths in angstrom units (1 Å = 0.1 nm) are given to a number of significant figures commensurate with their estimated uncertainties, which appear in parentheses after each theoretical and experimental value.

Table 4.2.2.4| top | pdf | Wavelengths of K-emission lines and K-absorption edges in Å; see text for explanation of typefaces Numbers in parentheses are standard uncertainties in the least significant figures. Z Symbol A K abs. edge 10 Ne   14.6020(93) 14.6006(93)         14.2391(26)   14.6102(44) 14.6102(44) 14.4522(74) 14.4522(74)     14.30201(15) 11 Na   11.9013(59) 11.8994(59)         11.4784(16)   11.9103(13) 11.9103(13) 11.5752(30) 11.5752(30)     11.5692(15) 12 Mg   9.8860(39) 9.8840(39)         9.4479(10)   9.89153(10) 9.889554(88) 9.5211(30) 9.5211(30)     9.51234(15) 13 Al   8.3372(27) 8.3349(26) 7.9412(49)       7.89928(67)   8.341831(58) 8.339514(58) 7.9601(30) 7.9601(30)     7.948249(74) 14 Si   7.1269(19) 7.1208(19) 6.7317(26)       6.70091(46)   7.12801(14) 7.125588(78) 6.7531(15) 6.7531(15)     6.7381(15) 15 P   6.1587(14) 6.1539(14) 5.7834(16) 5.7914(27)     5.75537(33)   6.1601(15) 6.1571(15) 5.7961(30) 5.7961(30)     5.7841(15) 16 S   5.3742(10) 5.3701(10) 5.0202(12) 5.0246(15)     4.99591(24)   5.374960(89) 5.372200(78)   5.03168(30)     5.01858(15) 17 Cl   4.72993(80) 4.72560(77) 4.39810(99) 4.40038(77)     4.37679(18)   4.730693(71) 4.727818(71) 4.40347(44) 4.40347(44)     4.39717(15) 18 Ar   4.19448(62) 4.19162(60) 3.88506(71) 3.88486(70)     3.86552(14)   4.194939(23) 4.191938(23) 3.88606(30) 3.88606(30)     3.870958(74) 19 K   3.74352(50) 3.74055(48) 3.45189(69) 3.45216(58)     3.42856(11)   3.7443932(68) 3.7412838(56) 3.45395(30) 3.45395(30)     3.43655(15) 20 Ca   3.36223(39) 3.35911(38) 3.08855(45) 3.08827(45)     3.061828(87)   3.361710(44) 3.358440(44) 3.08975(30) 3.08975(30)     3.07035(15) 21 Sc   3.03479(33) 3.03129(31) 2.77919(50) 2.77809(49)     2.754176(71)   3.0344010(63) 3.030854(14) 2.77964(30) 2.77964(30)     2.7620(15) 22 Ti   2.75272(27) 2.74886(26) 2.51445(43) 2.51262(45)     2.490681(59)   2.7521950(57) 2.7485471(57) 2.513960(30) 2.513960(30)     2.497377(74) 23 V   2.50798(23) 2.50383(21) 2.28567(37) 2.28332(40)     2.263194(49)   2.507430(30) 2.503610(30) 2.284446(30) 2.284446(30)     2.269211(21) 24 Cr   2.29428(19) 2.29012(18) 2.08702(32) 2.08478(35)     2.067898(41)   2.2936510(30) 2.2897260(30) 2.0848810(40) 2.0848810(40)     2.070193(14) 25 Mn   2.10635(16) 2.10210(15) 1.91175(28) 1.90960(31)     1.892275(36)   2.1058220(30) 2.1018540(30) 1.9102160(40) 1.9102160(40)     1.8964592(58) 26 Fe   1.94043(14) 1.93631(13) 1.75784(25) 1.75617(27)     1.739918(31)   1.9399730(30) 1.9360410(30) 1.7566040(40) 1.7566040(40)     1.7436170(49) 27 Co   1.79321(12) 1.78919(11) 1.62166(22) 1.62039(24)     1.605127(27)   1.7928350(10) 1.7889960(10) 1.6208260(30) 1.6208260(30)     1.6083510(42) 28 Ni   1.66199(10) 1.658049(96) 1.50059(19) 1.49964(21)     1.485300(24)   1.6617560(10) 1.6579300(10) 1.5001520(30) 1.5001520(30)     1.4881401(36) 29 Cu   1.544324(93) 1.540538(85) 1.39246(17) 1.39201(18)     1.379448(23)   1.54442740(50) 1.54059290(50) 1.3922340(60) 1.3922340(60)     1.3805971(31) 30 Zn   1.438963(84) 1.435151(74) 1.29544(17) 1.29506(16)     1.282346(20)   1.439029(12) 1.435184(12) 1.295276(30) 1.295276(30) 1.283739(30) 1.283739(30) 1.2833798(40) 31 Ga   1.343987(72) 1.340095(65) 1.20821(13) 1.20774(14) 1.195547(25)   1.194711(18)   1.3440260(40) 1.3401270(96) 1.208390(75) 1.207930(34) 1.196018(30) 1.196018(30) 1.19582(15) 32 Ge   1.257998(65) 1.254054(58) 1.12924(13) 1.12877(13) 1.116387(37)   1.115585(16)   1.258030(13) 1.254073(13) 1.12938(13) 1.128957(30) 1.116877(30) 1.116877(30) 1.116597(74) 33 As   1.179921(57) 1.175932(52) 1.05774(11) 1.05724(11) 1.044699(56) 1.044836(20) 1.043925(16)   1.179959(17) 1.17595600(90) 1.057898(76) 1.057368(33) 1.045016(44) 1.045016(44) 1.04502(15) 34 Se   1.108801(52) 1.104778(47) 0.992646(96) 0.992152(95) 0.979618(57) 0.979716(26) 0.978818(15)   1.108830(31) 1.104780(12) 0.992689(79) 0.992189(53) 0.979935(74) 0.979935(74) 0.979755(15) 35 Br   1.043841(47) 1.039785(42) 0.933275(87) 0.932768(84) 0.920344(49) 0.920390(28) 0.919501(13)   1.043836(30) 1.039756(30) 0.933284(74) 0.932804(30) 0.920474(30) 0.920474(30) 0.92041(15) 36 Kr   0.984347(42) 0.980267(38) 0.878967(81) 0.878495(75) 0.866209(36) 0.866169(35) 0.865324(12)   0.9843590(44) 0.9802670(40) 0.8790110(70) 0.8785220(50) 0.86611(15) 0.86611(15) 0.865533(15) 37 Rb   0.929713(39) 0.925597(35) 0.829174(71) 0.828681(67) 0.816459(33) 0.816408(33) 0.815270(12)   0.929704(15) 0.925567(13) 0.829222(44) 0.828692(30) 0.816462(44) 0.816462(44) 0.815552(74) 38 Sr   0.879444(36) 0.875298(32) 0.783413(63) 0.782911(58) 0.770774(33) 0.770718(20) 0.769359(11)   0.879443(15) 0.875273(15) 0.783462(44) 0.782932(30) 0.770822(44) 0.770822(44) 0.769742(74) 39 Y   0.833059(32) 0.828875(29) 0.741232(58) 0.740716(53) 0.728801(27) 0.728663(21) 0.727270(10)   0.833063(15) 0.828852(15) 0.741271(44) 0.740731(30) 0.728651(59) 0.728651(59) 0.7277514(21) 40 Zr   0.790181(30) 0.785960(27) 0.702296(53) 0.701766(48) 0.690079(28) 0.689895(21) 0.6884893(99)   0.7901790(25) 0.7859579(27) 0.7023554(30) 0.7018008(30) 0.689940(59) 0.689940(59) 0.6889591(31) 41 Nb   0.750448(28) 0.746189(25) 0.666266(49) 0.665721(44) 0.654328(31) 0.654078(22) 0.652890(93)   0.750451(15) 0.746211(15) 0.666350(44) 0.665770(30) 0.654170(59) 0.654170(59) 0.6531341(14) 42 Mo   0.713612(25) 0.709328(22) 0.632900(44) 0.632345(38) 0.621162(35) 0.620941(21) 0.6196481(87)   0.713607(12) 0.70931715(41) 0.632887(13) 0.632303(13) 0.620999(30) 0.620999(30) 0.61991006(62) 43 Tc   0.679318(24) 0.675017(21) 0.601881(40) 0.601318(35) 0.590423(40) 0.590231(22) 0.5889852(84)   0.679330(44) 0.675030(44) 0.601889(59) 0.601309(59) 0.590249(74) 0.590249(74) 0.589069(15) 44 Ru   0.647415(22) 0.643088(20) 0.573053(37) 0.572478(32) 0.561748(44) 0.561587(22) 0.5603122(81)   0.6474205(61) 0.6430994(61) 0.5730816(42) 0.5724966(42) 0.561668(44) 0.561668(44) 0.560518(15) 45 Rh   0.617652(21) 0.613305(18) 0.546191(34) 0.545606(29) 0.535110(48) 0.534977(22) 0.5337192(74)   0.6176458(61) 0.6132937(61) 0.5462139(42) 0.5456189(42) 0.535038(30) 0.535038(30) 0.5339086(69) 46 Pd   0.589822(20) 0.585459(18) 0.521117(29) 0.520514(27) 0.510283(46) 0.510177(51) 0.5090158(75)   0.5898351(60) 0.5854639(46) 0.5211363(41) 0.5205333(41) 0.5102357(59) 0.5102357(59) 0.5091212(42) 47 Ag   0.563804(18) 0.559420(17) 0.497673(29) 0.497069(25) 0.487060(55) 0.487019(38) 0.4857609(74)   0.5638131(26) 0.55942178(76) 0.4976977(60) 0.4970817(60) 0.4870393(59) 0.4870393(59) 0.4859155(57) 48 Cd   0.539426(18) 0.535020(15) 0.475739(27) 0.475124(23) 0.465335(62) 0.465346(28) 0.4640026(71)   0.5394358(46) 0.5350147(46) 0.4757401(71) 0.4751181(71) 0.465335(10) 0.465335(10) 0.4641293(35) 49 In   0.516551(17) 0.512124(15) 0.455178(25) 0.454552(22) 0.445014(58) 0.445011(27) 0.4435977(70)   0.5165572(60) 0.5121251(46) 0.4551966(41) 0.4545616(41) 0.445007(15) 0.445007(15) 0.4437454(48) 50 Sn   0.495060(16) 0.490612(14) 0.435878(24) 0.435241(20) 0.426120(12) 0.425928(26) 0.4244611(68)   0.4950646(46) 0.4906115(46) 0.4358821(51) 0.4352421(51) 0.425921(12) 0.425921(12) 0.4245978(29) 51 Sb   0.474840(15) 0.470373(13) 0.417736(22) 0.417089(19) 0.408017(57) 0.408004(25) 0.4064886(65)   0.4748391(45) 0.4703700(45) 0.4177477(41) 0.4170966(31) 0.4079791(74) 0.4079791(74) 0.4066324(27) 52 Te   0.455795(14) 0.451310(13) 0.400664(21) 0.400008(18) 0.391161(56) 0.391135(27) 0.3895899(64)   0.4557908(44) 0.4513018(44) 0.4006650(59) 0.4000010(44) 0.3911079(89) 0.3911079(89) 0.389746(15) 53 I   0.437834(13) 0.433330(12) 0.384576(20) 0.383910(17) 0.375286(54) 0.375234(29) 0.3736775(61)   0.437836(10) 0.4333245(74) 0.3845698(59) 0.3839108(59) 0.375236(30) 0.375236(30) 0.373816(15) 54 Xe   0.420879(42) 0.416358(40) 0.369407(40) 0.368730(38) 0.360326(72) 0.36034(12) 0.358683(27)   0.42088103(71) 0.4163508(14) 0.3694051(13) 0.3687346(13) 0.360265(44) 0.360265(44) 0.35841(74) 55 Cs   0.404848(13) 0.400310(11) 0.355067(17) 0.354385(16) 0.346197(49) 0.346102(37) 0.3444778(59)   0.4048411(59) 0.4002960(59) 0.3550553(59) 0.354369(10) 0.346115(30) 0.346115(30) 0.344515(15) 56 Ba   0.389684(12) 0.385129(11) 0.341517(16) 0.340826(15) 0.33285(14) 0.332728(12) 0.3310639(56)   0.38968378(74) 0.38512464(84) 0.3415228(11) 0.34082708(75) 0.332775(15) 0.332775(15) 0.331045(15) 57 La   0.375320(11) 0.370748(10) 0.328692(16) 0.327993(14) 0.32023(13) 0.320101(11) 0.3184025(55)   0.3753186(30) 0.3707426(30) 0.3286909(59) 0.3279879(44) 0.320122(10) 0.320122(10) 0.318445(74) 58 Ce   0.361685(11) 0.3570964(97) 0.316507(15) 0.315795(14) 0.30827(12) 0.308131(10) 0.3065382(54)   0.3616884(30) 0.3570974(30) 0.3165248(59) 0.3158207(30) 0.308165(15) 0.308165(15) 0.306485(74) 59 Pr   0.348755(10) 0.3441494(94) 0.304970(14) 0.304249(13) 0.29694(11) 0.2967952(99) 0.2952418(53)   0.3487542(30) 0.3441452(30) 0.3049796(74) 0.3042656(59) 0.296794(30) 0.296794(30) 0.295184(74) 60 Nd   0.336473(10) 0.3318514(91) 0.294021(13) 0.293290(13) 0.28619(10) 0.2860408(94) 0.2845288(52)   0.33647921(73) 0.33185689(62) 0.2940366(40) 0.2933086(40) 0.28610(15) 0.28610(15) 0.284534(74) 61 Pm   0.3247982(98) 0.3201607(88) 0.283620(13) 0.282880(12) 0.275992(98) 0.2758335(91) 0.2743634(53)   0.3248079(59) 0.3201648(59) 0.283634(59) 0.282904(44) 0.27590(15) 0.27590(15) 0.274314(74) 62 Sm   0.3136913(94) 0.3090384(84) 0.273732(12) 0.272984(12) 0.266459(91) 0.2661277(87) 0.2647027(51)   0.31369830(79) 0.30904506(46) 0.273764(30) 0.273014(30) 0.26620(15) 0.26620(15) 0.264644(74) 63 Eu   0.3031139(91) 0.2984457(81) 0.264322(12) 0.263567(11) 0.257069(85) 0.2569028(81) 0.2555123(51)   0.3031225(30) 0.2984505(30) 0.2643360(74) 0.2635810(74) 0.256927(12) 0.256927(12) 0.255534(15) 64 Gd   0.2930400(89) 0.2883568(79) 0.255371(11) 0.254610(11) 0.248289(23) 0.2481186(76) 0.2467265(48)   0.2930424(30) 0.2883573(30) 0.255344(30) 0.254604(30) 0.248164(44) 0.248164(44) 0.246814(15) 65 Tb   0.2834212(86) 0.2787234(76) 0.246818(11) 0.246054(11) 0.239916(21) 0.2397496(75) 0.2384335(49)   0.2834273(30) 0.2787242(30) 0.246834(30) 0.246084(30) 0.23970(30) 0.23970(30) 0.238414(15) 66 Dy   0.2742462(84) 0.2695341(74) 0.238671(10) 0.237902(10) 0.231955(12) 0.2318190(53) 0.2304867(46)   0.2742511(30) 0.2695370(30) 0.238624(30) 0.237884(30) 0.23170(30) 0.23170(30) 0.230483(15) 67 Ho   0.2654851(81) 0.2607589(72) 0.230896(10) 0.230122(10) 0.224320(18) 0.2241536(66) 0.2229099(45)   0.26549088(84) 0.2607608(42) 0.230834(30) 0.230124(30) 0.22410(30) 0.22410(30) 0.222913(15) 68 Er   0.2571059(79) 0.2523659(71) 0.2234662(97) 0.2226875(98) 0.217046(16) 0.2168806(64) 0.2156719(45)   0.2571133(11) 0.25237359(62) 0.2234766(14) 0.22269866(72) 0.21670(30) 0.21670(30) 0.2156801(75) 69 Tm   0.2490952(77) 0.2443415(68) 0.2163665(94) 0.2155833(95) 0.210099(15) 0.2099331(62) 0.2087587(44)   0.24910095(61) 0.24434486(44) 0.216366(30) 0.21559182(57) 0.20980(30) 0.20980(30) 0.208803(74) 70 Yb   0.2414274(75) 0.2366603(67) 0.2095741(93) 0.2087863(95) 0.203456(14) 0.2032912(59) 0.2021481(43)   0.2414276(30) 0.2366586(30) 0.20960(15) 0.208843(30) 0.20330(30) 0.20330(30) 0.202243(74) 71 Lu   0.2340857(73) 0.2293053(65) 0.2030802(88) 0.2022872(90) 0.191017(13) 0.1969329(58) 0.1957973(42)   0.2340845(30) 0.2293014(30) 0.203093(59) 0.202313(44) 0.19690(30) 0.19690(30) 0.195853(74) 72 Hf   0.2270507(72) 0.2222572(64) 0.1968603(86) 0.1960622(88) 0.191017(13) 0.1908468(56) 0.1897176(42)   0.2270274(44) 0.2222303(44) 0.196863(59) 0.196073(44) 0.19080(30) 0.19080(30) 0.189823(74) 73 Ta   0.2203039(70) 0.2154977(63) 0.1908986(83) 0.1900954(86) 0.185143(12) 0.1849702(54) 0.1838657(41)   0.2203083(59) 0.2155002(59) 0.1908929(30) 0.1900919(59) 0.185191(13) 0.185014(12) 0.183943(15) 74 W   0.2138327(69) 0.2090134(61) 0.1851834(81) 0.1843751(83) 0.179595(12) 0.1794215(52) 0.1783098(41)   0.21383304(50) 0.20901314(18) 0.18518317(70) 0.1843768(30) 0.179603(15) 0.179424(10) 0.178373(15) 75 Re   0.2076150(67) 0.2027835(60) 0.1796955(79) 0.1788824(81) 0.174234(11) 0.1740571(51) 0.1729509(40)   0.2076141(15) 0.2027840(30) 0.1796997(44) 0.1788827(44) 0.174253(15) 0.1740566(89) 0.173023(15) 76 Os   0.2016443(66) 0.1968007(59) 0.1744279(77) 0.1736101(79) 0.169085(11) 0.1689066(50) 0.1678092(40)   0.2016420(30) 0.1967970(30) 0.1744336(44) 0.1736136(44) 0.169103(15) 0.1689085(89) 0.167873(15) 77 Ir   0.1959045(65) 0.1910492(57) 0.1693667(75) 0.1685444(77) 0.164150(11) 0.1639697(51) 0.1628853(39)   0.1959069(30) 0.1910499(30) 0.1693695(30) 0.1685445(30) 0.164152(15) 0.163958(10) 0.162922(15) 78 Pt   0.1903859(61) 0.1855187(55) 0.1645026(72) 0.1636756(74) 0.1593872(99) 0.1592048(46) 0.1581346(38)   0.1903839(59) 0.1855138(59) 0.1645035(44) 0.1636775(44) 0.159392(15) 0.159202(15) 0.158182(15) 79 Au   0.1850702(64) 0.1801914(57) 0.1598202(73) 0.1589887(75) 0.1548206(99) 0.1546363(48) 0.1535699(40)   0.18507664(61) 0.18019780(47) 0.1598249(13) 0.15899527(77) 0.154832(30) 0.154620(13) 0.1535953(74) 80 Hg   0.1799628(61) 0.1750720(54) 0.1553217(69) 0.1544857(72) 0.1504204(94) 0.1502334(46) 0.1491786(38)   0.1799607(44) 0.1750706(44) 0.1553233(44) 0.1544893(44) 0.150402(30) 0.150202(30) 0.149182(15) 81 Tl   0.1750380(60) 0.1701355(53) 0.1509866(68) 0.1501462(70) 0.1461874(92) 0.1459989(77) 0.1449460(37)   0.1750386(30) 0.1701386(30) 0.1509823(89) 0.1501443(74) 0.146142(15) 0.145952(15) 0.144952(15) 82 Pb   0.1702924(59) 0.1653781(53) 0.1468107(67) 0.1459663(68) 0.1421118(88) 0.1419201(75) 0.1408707(37)   0.17029527(56) 0.16537816(38) 0.1468129(10) 0.14596836(58) 0.142122(30) 0.141912(15) 0.1408821(74) 83 Bi 209 0.1657170(58) 0.1607911(52) 0.1427865(65) 0.1419372(66) 0.1381841(87) 0.1379910(72) 0.1369439(37)   0.1657183(20) 0.1607903(46) 0.142780(11) 0.1419492(54) 0.138172(15) 0.137972(15) 0.136942(15) 84 Po 209 0.1613031(58) 0.1563656(51) 0.1389056(63) 0.1380520(65) 0.1343966(85) 0.1342012(69) 0.1331589(36)   0.161302(15) 0.156362(15) 0.138922(30) 0.138072(30) 0.134382(30) 0.134182(30)   85 At 210 0.1570444(56) 0.1520953(50) 0.1351623(62) 0.1343044(63) 0.1307448(83) 0.1305470(67) 0.1295098(36)   0.157052(30) 0.152102(30) 0.135172(59) 0.134322(59) 0.130722(59) 0.130522(59)   86 Rn 222 0.1529334(56) 0.1479727(49) 0.1315499(61) 0.1306882(61) 0.1272218(79) 0.1270211(66) 0.1259898(35)   0.152942(44) 0.147982(44) 0.131552(74) 0.130692(74) 0.127192(74) 0.126982(74)   87 Fr 223 0.1489599(56) 0.1439878(50) 0.1280599(60) 0.1271937(61) 0.1238183(79) 0.1236157(63) 0.1225852(36)   0.148962(44) 0.143992(44) 0.128072(74) 0.127192(74) 0.123792(74) 0.123582(74)   88 Ra 226 0.1451209(54) 0.1401373(48) 0.1246890(58) 0.1238185(59) 0.1205312(77) 0.1203271(60) 0.1192985(35)   0.145119(20) 0.140132(19) 0.124689(15) 0.123815(15) 0.120535(14) 0.120320(14)   89 Ac 227 0.1414083(54) 0.1364131(47) 0.1214301(57) 0.1205554(58) 0.1173552(73) 0.1171477(59) 0.1161246(34)   0.141412(30) 0.136419(12) 0.121432(30) 0.120552(30) 0.117322(30) 0.117112(30)   90 Th 232 0.1378266(53) 0.1328194(47) 0.1182861(56) 0.1174071(56) 0.1142910(71) 0.1140810(57) 0.1130642(34)   0.13782600(31) 0.13282021(36) 0.11828686(78) 0.11740759(59) 0.114262(15) 0.114042(13) 0.113072(15) 91 Pa 231 0.1343514(52) 0.1293324(46) 0.1152364(55) 0.1143530(55) 0.1113088(69) 0.1110964(56) 0.1101087(34)   0.1343516(29) 0.1293302(27) 0.1152427(21) 0.1143583(21) 0.111292(30) 0.111072(30)   92 U 238 0.1309879(52) 0.1259572(46) 0.1122860(53) 0.1113979(54) 0.1084449(67) 0.1082301(54) 0.1072452(33)   0.13099111(78) 0.12595977(36) 0.11228858(66) 0.11140132(65) 0.108372(15) 0.108182(15) 0.107232(15) 93 Np 237 0.1277298(51) 0.1226871(45) 0.1094299(52) 0.1085378(53) 0.1056621(66) 0.1054450(53) 0.1044744(33)   0.1277287(39) 0.1226882(36) 0.1094230(39) 0.1085265(28) 0.105670(31) 0.105457(31) 0.1044605(62) 94 Pu 244 0.1245763(50) 0.1195212(45) 0.1066627(51) 0.1057661(52) 0.1029688(64) 0.1027494(52) 0.1017982(33)   0.1245705(25) 0.1195140(23) 0.1066611(18) 0.1057595(18) 0.1029724(26) 0.1027429(26)   95 Am 243 0.1215172(50) 0.1164501(45) 0.1039811(51) 0.1030805(51) 0.1003579(63) 0.1001364(51) 0.0991999(33)   0.1215158(24) 0.1164463(33) 0.1039794(17) 0.1030803(17) 0.1003537(24) 0.1001357(24)   96 Cm 248 0.1185536(49) 0.1134742(44) 0.1013837(50) 0.1004790(50) 0.0978295(63) 0.0976059(50) 0.0966801(33)   0.1185427(23) 0.1134635(21) 0.1013753(17) 0.1004708(16) 0.0978355(23) 0.0975952(15)   97 Bk 249 0.1156777(49) 0.1105860(43) 0.0988636(48) 0.0979546(49) 0.0953724(61) 0.0951469(49) 0.0942405(32)   0.1156630(54) 0.1105745(49) 0.0988598(55) 0.0979514(54)   0.0942501(50)   98 Cf 250 0.1128873(48) 0.1077832(43) 0.0964130(47) 0.0955000(48) 0.0929867(61) 0.0927593(48) 0.0918695(32)   0.1128799(82) 0.1077793(75) 0.0963915(83) 0.0954860(90) 0.0929715(82) 0.0927508(84) 0.091862(10) 99 Es 251 0.1101788(47) 0.1050620(43) 0.0940403(46) 0.0931231(47) 0.0906838(60) 0.0904543(47) 0.0895840(32)   0.1102072(98) 0.1050554(89) 0.094036(14) 0.093090(14)     0.0895878(97) 100 Fm 254 0.1075497(47) 0.1024201(42) 0.0917379(45) 0.0908165(46) 0.0884443(59) 0.0882127(45) 0.0873575(32)   0.107514(14) 0.102386(13) 0.091715(10) 0.0907943(98) 0.0884212(100) 0.0881872(99) 0.0873356(80)