2 Explain How a Continuous X ray Spectrum is Produced by an X ray Tube Source

X Ray Spectra

NDT Techniques: Radiographic

V.V. Kljuev , in Encyclopedia of Materials: Science and Technology, 2001

3 X-ray Spectrum Analysis

X-ray spectrum analysis is an element analysis of the material on the basis of its x-ray spectrum. A qualitative x-ray spectrum analysis is performed using Moseley's law. Quantitative x-ray spectrum analysis is performed by the intensity of the lines employing a crystal analyzer, scintillation and ionization counter, and coordinate plotter. All elements with atomic number Z9 (sometimes lighter elements) may be determined by x-ray spectrum analysis methods.

Generally, the x-ray fluorescence spectrum (secondary spectrum) of the sample is excited by the primary radiation of the x-ray tube. The limit of element detection by the secondary fluorescence spectrum is ∼10⁻³–10⁻⁴%. When the radiation is excited by protons with energies of 1–2   MeV, the element detection limit decreases to ∼10⁻⁵–10⁻⁶%. A relative preciseness of quantitative analysis beyond the detection limit may reach 1% and less.

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Time resolved energy dispersive X-ray diagnostic for the TCV tokamak

J. Sousa , ... C.A.F. Varandas , in Fusion Technology 1996, 1997

1 INTRODUCTION

Measurement of the X-ray spectrum by means of pulse height analysis is used nowadays by many tokamaks to identify the main high Z plasma impurities, to measure the electron temperature (T_e) and to detect the runaway electrons [1]. This paper describes an energy dispersive X-ray diagnostic that has been developed for the TCV tokamak [2] aiming at performing time resolved measurements in L and H mode regimes [3].

Section 2 contains the general description of the diagnostic, which is based on a liquid Nitrogen cooled solid state Germanium diode. This detector is sensitive to X-rays in the 1-100 keV range, with energy resolution of about 130 eV. Section 3 presents the specially developed electronic unit that is used to post-reconstruct a sequence of groups of X-ray events which arrive during periodic intervals of time. Section 4 describes the installation of the diagnostic on the TCV tokamak and section 5 includes the experimental results.

Finally section 6 contains the conclusions and planned system enhancements.

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Fundamentals of Atomic and Radiation Physics

R.A. EDWARDS M.A. (CANTAB.) , in Physics for O.N.C. Courses, 1970

CHARACTERISTIC X-RAY SPECTRA

Superimposed on the continuous X-ray spectrum described above are sharp intensity peaks occurring at wavelengths which depend on the target material and not on the p.d. across the tube. For most elements commonly used as targets these intensity peaks occur in two groups, one at lower wavelengths called the K-series and a longer wavelength group called the L-series. For elements with atomic number greater than 66 even longer wavelength series, the M- and N-series occur.

In 1913 Moseley investigated these characteristic spectra and discovered a linear relationship between the square root of the wave numbers for the lines of the K-series for various elements and the atomic numbers of the elements. Figure 27.5 shows a plot of $\surd (\overline{v}/R)$ against Z for the longest wavelength member of the K-series (the K_α -line) for a number of elements against the atomic number Z of the element, R being the Rydberg constant. The straight line fits the equation $\overline{v}=R{(Z-1)}^{2}3/4$ , which may be written as

FIG. 27.5.

$\overline{v}=R{(Z-1)}^2[(1/1^2)-(1/2^2)],$

which is similar to the Balmer formula for the visible emission spectrum of hydrogen. This formula provides the clue to the interpretation of characteristic X-ray spectra. If an electron for which n = 1 (the K-shell) is raised, by the energising of the atom, to the L-shell and then falls back to the K-shell, a quantum of radiation of wavelength $\lambda (=1/\overline{v})$ is radiated. This is the K_α -line. The other lines are emitted as a result of energy changes illustrated in Fig. 27.6, the L-series corresponding to the formula $\overline{v}=R{(Z-7ċ4)}^2[(1/2^2)-{(1/3)}^2]$ . It must be understood that characteristic X-rays result from energy changes of the electrons in the inner shells (normally filled) of the atoms of the target material.

FIG. 27.6.

The electron shells originally derived their labels K, L, M, N, etc., as a result of these investigations into X-ray spectra, these letters having originally applied to the X-ray spectral series.

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X-Ray Fluorescence Spectrometers*

Utz Kramar , in Encyclopedia of Spectroscopy and Spectrometry (Second Edition), 1999

Dispersing Crystals

At the X-ray diffraction crystals the X-ray spectrum is dispersed into different wavelengths according to Bragg's law

$n\cdot \lambda =2dsin\text{Θ}$

The wavelengths of characteristic lines used in X-ray spectrometry range from ∼0.03   nm (Ba K) to ∼   10   nm (Be K). This range cannot be covered by use of a single diffraction crystal. The detectable wavelength and high-order reflections are limited by the relation between d and λ.

$n\lambda \le 2d$

For example with LIF 200, the crystal with the broadest application range, the elements with atomic numbers <19 (K) are not detectable. At shorter wavelength, the dispersion between neighbouring lines decreases and they cannot be resolved from each other. The diffraction crystals have to be selected with respect to their reflectivity and to be suitable for the lines to be detected. Commonly used X-ray diffraction crystals and their application range are compiled in Table 3.

Table 3. Diffraction crystals commonly used in wavelength-dispersive X-ray fluorescence

Analyser crystal	Material	2d (nm)	Detectable elements		Efficiency
			K	L
	Topaz	0.27	V–Ta	Ce–U	Average
LIF(220)	Lithium fluoride	0.29	V–Ta	Ce–U	High
LIF(200)	Lithium fluoride	0.4	K–La	Cs–U	High
Ge	Germanium	0.65	P–Zr		Average
PET	Pentaerythrite	0.87	Al–Ti	Kr–Xe	High
AdP	Ammonium dihydrogen phosphate	1.06	Mg		Low
TAP/TlAP	Thallium biphthalate	2.58	F–Na		High
OVO-55	W/Si multilayer	5.5	N–Si	Ca–Br	High
OVO-160	W/C multilayer	16	Be–O		?
OVO-C	V/C multilayer	12	C		?
OVO-B	Mo/B4C multilayer	20	Be–B		?
PbSD	Lead stearate decanoate	10	B–F		Average

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Atomic Data Needs for Understanding X-ray Astrophysical Plasmas

Randall K. Smith , Nancy S. Brickhouse , in Advances In Atomic, Molecular, and Optical Physics, 2014

3.2.4 Other Ions

Other Fe L-shell lines also contribute to the X-ray spectrum, with their counterparts for lower Z L-shell lines extending into the soft X-ray band (e.g., Li et al., 2013). Fe M-shell lines are also important for the longer wavelength LETG on Chandra (e.g., Del Zanna, 2012; Keenan et al., 2006). Most of the interesting line ratios include weak lines for which blending with unknown lines or with lines at incorrect wavelengths remains a concern; thus there are few known astrophysics problems to date. One exception comes from a study the strong coronal source Capella, for which high signal-to-noise grating data exist. Desai et al. (2005) have shown that the EUV/X-ray line ratios for Fe XVIII and Fe XIX are too weak in Capella compared with theory. Whether this is an atomic theory problem or not remains to be determined, though EBIT measurements should be helpful in sorting out various issues (Träbert et al., 2012). A detailed study of the X-ray spectrum of Fe XIX, particularly in the Ne IX diagnostic region, demonstrates the completeness of the collisional excitation data for assessing blending (Ness et al., 2003).

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Application of micro- and nanoprobes to the analysis of small-sized 2D and 3D materials, nanocomposites, and nanoobjects

A.D. Pogrebnjak , ... A.A. Goncharov , in Handbook of Modern Coating Technologies, 2021

5.3.1 Method of characteristic X-ray emission induced by beam ions

The PIXE method is based on atomic ionization in the sample. The observed X-ray spectrum consists of the continuous spectrum induced by bremsstrahlung radiation of secondary electrons and the linear spectrum produced by recombination of ionized atoms and the filling of K, L, and M electron shells. The PIXE method is currently well established; as mentioned above, its advantages are due to the relatively low bremsstrahlung background compared with electron beams in EPMA. Fig. 5–7 represents X-ray spectra obtained for a thick sample using EPMA and PIXE. The comparison shows that the high bremsstrahlung background in EPMA (Fig. 5–7A) prevents detection of small concentrations of certain elements that are readily identified in the PIXE spectrum (Fig. 5–7B).

Figure 5–7. X-ray spectra induced by different types of charged particle beams: (A) electrons, 20   keV and (B) protons, 2.5   MeV [80,81]. Optical image (C) and tomogram (D) of green algae Euglena gracilis. The surface reconstructed from the STIM data is compared with the high-phosphorus region identified from the results of using the PIXE method [82]. PIXE, Particle-induced X-ray emission; STIM, scanning transmission ion microscopy.

Several software packages for quantitative analysis of PIXE spectra are available. These programs were tested independently under the auspices of the IAEA; the results are published in Ref. [83].

Worthy of special mention is the GeoPIXE software package [83] that allows not only quantitative analysis but also two-dimensional mapping of the location of elements for SNMPs, when the data are acquired during scanning in an event by-event mode and each event is marked by three parameters, that is, beam energy and beam positions (x, y) of the PIXE yield. Modification of the PIXE method for the improvement of locality of the analysis implies diminishing the beam spot size on the surface of the object under study. However, it leads to a substantial decrease in the beam current and, therefore, the number of atom ionization events.

The maintenance of the PIXE yield by increasing current density alone when using high-brightness ion sources is somewhat limited by radiation damage to the initial material and defects incorporated into the original sample. Another approach, that is, increasing the solid angle of the detector by enlarging its area, is inefficient because it lowers detector resolution and enhances the superposition effects of event records within a narrow time interval.

One of the ways to solve this problem is the development of matrix detectors with an appropriate controller for synchronizing a set of all events and improving the sensitivity of the PIXE method up to several hundred ppb. The development of such matrix detectors is discussed in Refs. [84,85]; their controllers govern parallel processes and have the following characteristics: a solid angle of 1.2 sr, energy resolution of ~184   eV (Mn Kα), and peak-to-background ratio of ≈10³. A new parallel data acquisition method permits loading up to 10⁸ events per second and practically preventing event overlaps during detection.

The high sensitivity of the PIXE method coupled with submicron spatial resolution of SNMPs is widely used in medical applications, for example, for the elucidation of factors responsible for various diseases and the analysis of the distribution of chemical elements present at low concentrations in arterial walls [86]. The role of iron in the pathogenesis of Parkinson's disease, the subject of a growing interest among neurochemists, was considered in Ref. [87]. It was demonstrated that iron accumulates in abnormally high quantities in the gray matter of Parkinson's patients. The combination of PIXE and STIM makes it possible to reconstruct 3D tomograms illustrating the spatial distribution of microelements. Fig. 5–7C and D shows a cell of the green algae Euglena gracilis for which such a 3D tomogram was obtained.

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X-Ray Astronomy

M.F. Corcoran , in Encyclopedia of Physical Science and Technology (Third Edition), 2003

IV.D.2 RXTE: Fast Variability Studies

The Rossi X-Ray Timing Explorer (RXTE) was launched in 1996. RXTE was designed to provide extremely sensitive, high-time-resolution (to the microsecond level) studies of the variations of X-ray brightness and X-ray spectra from compact objects, active galaxies, and bright stars. A particular strength of RXTE is the capability to schedule coordinated observations with other observatories to allow observations of physical processes in astronomical objects over a wide range of wavelength regions. RXTE's instrumentation includes the Proportional Counter Array (PCA), a set of six collimated proportional counters coaligned to point simultaneously at a specified area of the sky of roughly 1 square degree in size. RXTE also carries the High Energy X-ray Transient Experiment (HEXTE) designed to study source variability in the 15–250   keV range, and an All-Sky Monitor (ASM) designed to study variability of bright X-ray sources in the entire sky. One of the major discoveries of RXTE is the detection of the fastest periodic signals ever observed from an astronomical source. These "millisecond" pulsations (typically there are roughly 1000 X-ray flashes per second) are produced by some neutron star binary systems and are used to probe the condition of material in the extreme gravity very near the surface of the compact object. RXTE also showed that at least some of the hitherto mysterious objects known as soft gamma ray repeaters show X-ray pulsations as well and that these objects are probably types of compact X-ray binary systems with exceedingly strong magnetic fields. RXTE also produced the first detailed measure of the X-ray variability of Eta Carinae, a star that many astronomers think is the most massive star in the Milky Way; these observations showed a substantial, periodic eclipse of the X-ray emission every 5   years, probably produced by the presence of a companion star.

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Ionization and Excitation of Atoms and Molecules

Ilya Obodovskiy , in Radiation, 2019

4.6.2 Coster–Kronig Effect

One of the variants of the Auger effect was Coster–Kronig effect named after the physicists D. Coster and R. Kronig [13], who pointed out that the nonradiative transition L₁  →   L₃ plays an important role when interpreting certain features of X-ray spectra. In this case, the vacancy is filled by an electron from the upper subshell of the same shell. It is obvious that the primary vacancy is formed not lower than on the L₁ subshell because the K-shell does not have subshells. The primary vacancy could be filled with an electron from the L₃ subshell. Thus, the Coster–Kronig transition option can look like a L₁L₃M_4,5 transition.

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SOLID THIN FILMS AND LAYERS

E. Atanassova , T. Dimitrova , in Handbook of Surfaces and Interfaces of Materials, 2001

2.3.2.1 Structure Analysis

To study the structural properties of the Ta₂O₅ films we performed X-ray diffraction measurements on the 2θ – θ mode on a series of samples obtained under nonheated and heated substrates. All of the samples corresponding to certain T _s showed identical X-ray spectra. For the base silicon substrate we obtained a sharp and intense X-ray reflection peak at 2θ = 30.005°, d = 2.976 Å, and three week peaks at 2θ = 23.635°, 39.970°, 48.900° and d = 3.716 Å, 2.254 Å, 1.865 Å, respectively. These peaks correspond to reflections from the bulk silicon crystal. Figure 15 shows the XRD pattern (20) plot of the as-deposited film, and in Table VI the diffraction peaks extracted are listed. The nature of the X-ray spectra indicates an amorphous structure of Ta₂O₅—no diffraction peaks (except for those from the Si substrate and the SiO₂ layer, probably presented at the interface with Si) were observed in the spectra of the as-deposited films. The interplanar spacing showed good agreement with that of a SiO₂-α cristobalite difftactogram. As is seen, two intensive peaks at 2θ = 69.41° and 69.60° for the layers obtained at room temperature are registered. A slight shift of the peaks positioned with respect to the position of SiO₂ [214] is observed. The spectrum corresponding to the layers obtained at T _s = 493 K showed the existence of a sharp reflex at 20 = 56.93° and a smaller one at 58.93°. A detailed study of these two most intense peaks in the diffractogram reveal the same slight difference in the peak position. This can also be seen in Table VI, where the position of the observed peaks is compared with the corresponding reflexes [301] and [222] of SiO₂ [71, card no. 11−695]. Generally it may be concluded that the as-deposited Ta₂O₅ layers have an amorphous structure for the unheated and the heated substrates. At the same time there is strong evidence for the presence of crystalline SiO₂, most likely at the interface with Si. Our electrical and XPS measurements undoubtedly indicated the formation of an ultrathin SiO₂ film at the Si-Ta₂O₅ interface. The formation of an interfacial SiO₂ is obviously an unavoidable process during RF sputtering and is due to the simultaneous action of two factors favorable for its formation, namely, a discharge containing active oxidizing particles and a silicon surface that is easily oxidized. Now the present XRD data are in accordance with the electrical and XPS results. The Ta₂O₅ films annealed at 873 K reveal the same tendency in the diffraction pattern as the above results for the as-deposited samples. The Ta₂O₅ layers are X-ray amorphous; no diffraction peaks, except for those from the silicon substrate and from the intermediate SiO₂ at the interface with Si, were observed in the spectra of the annealed films. It is also in accordance with the results of others [14, 55, 56]. After the annealing of the layers corresponding to the unheated substrate, small-intensity [111], [202], and [220] reflections appear in the spectrum (Fig. 16a, Table VII). A little change in crystal modification of SiO₂ after annealing of the layers with T _s = 493 K is observed (Fig. 16b, Table VII). The annealed films have an amorphous structure of Ta₂O₅, and the interplanar spacings for SiO₂ are very similar to that of crystalline SiO₂ (α-cristobalite). The XRD pattern of the films annealed at 1123 K for 30 min shows diffraction peak characteristics of Ta₂O₅ crystals. The spectra exhibit well-defined peaks of orthorombic phase. It is seen from Figure 17 that Ta₂O₅ has a crystalline structure after annealing at 1123 K—sharp diffraction lines appear in the XRD pattern. The temperature of the phase transition is therefore between 873 and 1123 K. In Table VIII the observed diffraction peaks obtained from a random powder sample [71] are also given for comparison. From the Table one can identify the polycrystalline Ta₂O₅ of the samples as the low-temperature β-Ta₂O₅ modification. As is seen, most of the observed peaks agree well with the orthorombic crystal structure of tantalum oxide films [71, card no. 25−0922]. From the comparison of the measured and the reference spectra it can be seen that they are identical, with the exception of certain peaks in crystal direction, such as [1 12 1]; [3 3 0]. Inasmuch as the latter missing two peaks have a low theoretical intensity, their absence from the spectrum of the samples investigated is not significant. This is why we can conclude that the experimental spectrum for layers obtained at 293 K and annealed at 1123 K are very well fitted with the reference spectrum of orthorombic phase. Other authors also observed a β crystalline phase in Ta₂O₅ after high-temperature annealing, at 973 K [65], 1173 K [65], 1173–1273 K [63], and temperatures higher than 1273 K [72]. The [2 1 0], [0 0 1], and [0 8 1] peaks are strong and the rest are weaker (Fig. 17). The peaks with small intensities appearing at 2θ = 39.375°, 47.600°, 48.400°, and 52.860° can be associated with the crystal phase of SiO₂ [71, card no. 11−695]. The highly intensive peak at 68.610° and the peak at 74.490° are due to the signal from the silicon substrate. The spectrum of the layer deposited at T _s = 493 K also corresponds to the presence of an orthorombic phase of Ta₂O₅, and again the peaks [0 0 1] and [0 8 1] are the strongest ones. A well-pronounced peak appears at [2 1 1], instead of the [2 0 1] peak corresponding to the case where T _s = 293 K (Fig. 17). A number of small peaks present in the spectrum for T _s = 293 K (Fig. 17) disappear in the spectrum corresponding to T _s = 493 K, with the exception of the small peak at 57.215°, which can be associated with a small shift of the peak corresponding to the phase [3 12 1]. So, in general, the spectrum for T _s = 493 K exhibits more pronounced peaks, but their number is smaller in comparison with the spectrum of the sample obtained at 293 K. The peaks at 69.455° and 69.650° also correspond to the silicon substrate. The only peak that could be related to the crystal phase of SiO₂ at 47.180° is not very pronounced, with a significant (three to four times) intensity reduction. This why we interpret the spectrum as not showing crystal SiO₂. Then it can be concluded that the higher substrate temperature obviously stimulates the formation of amorphous SiO₂ (and consequently of better quality) rather than the crystalline one. Then one can expect that the layers obtained at 493 K will have better interface properties. In addition, the substrate temperature during deposition has a negligible effect on the structure of the layers (crystal phases corresponding to heated and unheated substrate in general are the same; the differences are small details).

Fig. 15. XRD spectra of as-deposited Ta₂O₅ layers (d = 27–28 nm). (a) T _s = 293 K. (b) T _s = 493 K.

Table VI. Diffraction Pattern for the As-Deposited Layers

				Reported powder data b
Samples Ts (K)	2Θ, observed (degrees)	d(hkl), Intensity, observed (Å)	Intensity, observed (%) a	2Θ(degrees)	d(hkl)(Å)	Intensity, theoretical (%)	Plane (hkl)
293	69.085	1.3618	1.7	68.674	1.365	4	214
	69.410	1.3563	100	69.418	1.352	4	321
	69.600	1.3530	45.4	69.788	1.346	2	303
493				31.461	2.841	14	102
	33.495	2.6731	16.8
				36.079	2.485	20	200
	56.930	1.6161	100	57.083	1.612	3	301
	58.930	1.5660	37.9	58.868	1.567	2	222

Only peaks with intensity higher than 1 have been reported.

a

Peak intensity, not integrated intensity.

b

Standard Diffraction Powder Data, pp. 11−695.

Fig. 16. XRD spectra of Ta₂O₅ layers after oxygen annealing at 873 K (d = 26–27 nm). (a) T _s = 293 K. (b) 493 K.

Table VII. Interplanar Spacing d (Å) Obtained from the Films Annealed at 873 K for 30 min

				Reported powder data a
Samples Ts (K)	2Θ, observed (degrees)	d(hkl), Intensity, observed (Å)	Intensity, observed (%)a	2Θ(degrees)	d(hkl)(Å)	Intensity, theoretical (%)	Plane (hkl)
293	28.790	3.0984	4	28.438	3.14	12	111
	44.010	2.0558	7	44.842	2.019	4	202
	52.420	1.7440	15	51.939	1.757	2	220
	69.150	1.3574	100	69.418	1.352	4	214
	70.125	1.3409	82	69.788	1.346	2	303
493	69.085	1.35848	2	68.674	1.365	4	214
	69.390	1.3532	100	69.418	1.352	4	321
	69.585	1.3499	44	69.788	1.346	11	303

a

Standard Diffraction Powder Data, pp. 11−695.

Fig. 17. XRD spectra of Ta₂O₅ layers after oxygen annealing at 1123 K (d = 26–27 nm).

Table VIII. Net Plane Spacing for the Ta₂O₅ Layers after Annealing at 1123 K in Comparison with the ASTM X-ray Data for Crystalline Ta₂O₅

				Reported powder data a
Samples Ts (K)	2Θ, observed (degrees)	d(hkl), Intensity, observed (Å)	Intensity, observed (%)	2Θ(degrees)	d(hkl)(Å)	Intensity, theoretical (%)	Plane (hkl)
293	23.070	3.8520	63.0	22.902	3.880	85	00 1
				28.794	3.098	40	200
	28.935	3.0832	100	28.870	3.090	4	2 1 0
	29.435	3.03197	38.3	28.995	3.077	2	08 1
	36.005	2.49235	7.3	36.665	2.449	75	1111
	37.180	2.41624	15.7	37.072	2.423	35	20 1
	37.180	2.41624	15.7	37.072	2.423	35	20 1
				38.234	2.352	2	1 12 1
	39.375	2.28645	8.2
	40.795	2.21008	4.3	40.339	2.234	2	27 1
	41.265	2.18598	4.3	41.324	2.183	2	28 1
	43.195	2.09268	10.1	42.930	2.105	3	1 180
				44.323	2.042	2	330
	46.955	1.93349	15.7	46.686	1.944	25	002
	47.600	1.90878	12.2
	48.400	1.87909	11.1
	52.860	1.73056	4.3
	57.380	1.60452	4.3	57.478	1.602	2	3 12 1
	62.225	1.49071	6.4	62.679	1.481	3	480
	68.610	1.36673	21.2
	74.490	1.27272	10.1
493	23.250	3.82263	25.5	22.902	3.880	85	00 1
	28.995	3.07697	41.6	28.995	3.077	2	08 1
	37.380	2.40377	5.9	37.136	2.419	4	2 1 1
	57.215	1.60875	4.2	57.478	1.602	2	3 12 1
	69.455	1.35214	100
	69.650	1.34883	77.2

a

1995 JCPDS, International Centre for Diffraction Data, p. 25−0922, orthorhombic.

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Solar-terrestrial Magnetic Activity and Space Environment

W.Q. Gan , ... J. Chang , in COSPAR Colloquia Series, 2002

METHOD

We consider two cases to describe the power-law electron spectrum. The case 1 has a sharp cutoff at the lower energy end

(1) $F\left(E_0\right)=\{\begin{array}{cc}\hfill A{E}_0^{-\delta }\hfill & \hfill E_0>E_c\hfill \\ \hfill 0\hfill & \hfill E_0\le E_c\text{,}\hfill \end{array}$

and the case 2, we call a saturation form

(2) $F\left(E_0\right)=\{\begin{array}{cc}\hfill A{E}_0^{-\delta }\hfill & \hfill E_0>E_c\hfill \\ \hfill A{E}_c^{-\delta }\hfill & \hfill E_0\le E_c\text{.}\hfill \end{array}$

Although these two distributions are rather ideal, they may be representative to some degree, and the actual distribution of electrons below the E_c may probably be intermediate. Gan et al. (2001), by using the thick-target bremsstrahlung formula (Brown 1971; Tandberg-Hansen & Emslie 1988), studied the hard X-ray spectra resulted from a power-law electrons with a lower energy cutoff in the form of either (1) or (2). They found that for the photon energy є bigger than E_c , the hard X-ray spectrum presents a usual power-law form with a spectral index δ-1. But for the ϵ smaller than the E_c , the hard X-ray spectrum flattens towards to the lower energies. The degree of the flattening depends on the electron power-law index δ. If we use a double power-law to simulate the calculated hard X-ray spectrum

(3) $I\left(\mathit{ϵ}\right)=\{\begin{array}{cc}\hfill a{\mathit{ϵ}}^{-{\gamma }_1}\hfill & \hfill \mathit{ϵ}<{\mathit{ϵ}}_b\hfill \\ \hfill b{\mathit{ϵ}}^{-{\gamma }_2}\hfill & \hfill \mathit{ϵ}\ge {\mathit{ϵ}}_b\text{,}\hfill \end{array}$

we may establish a theoretical relationship between γ ₁ and γ ₂ (as shown in Fig. 1) as well as the one between ϵ and γ ₂ (Gan et al., 2001). The significance of this relationships is that if the observed hard X-ray spectrum can be simulated with a broken-down double power-law, and the two power-law spectral indices satisfy the relationship shown in Figure 1, it implies that the observed hard X-rays can be explained by a power-law spectrum of electrons with a lower energy cutoff; then from the relationship between ϵ_b and γ ₂ (Gan et al., 2001), one can further get the E_c . Therefore, we have established a quantitative method to deduce the E_c directly from the hard X-ray observations.

Fig. 1. Fitted double power-law indices (γ ₁ vs. γ_s ) for the calculated hard X-ray spectra emitted by a beam of power-law electrons with a sharp lower energy cutoff (case 1, dashed line) and with a flux saturation below a cutoff (case 2, solid line). The points with error bars are the fitted double power-law indices for the observed hard X-ray spectra of the 54 BATSE/CGRO events. The dotted line represents a single power-law.

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