KJ3055-Chapter 8 (x-Ray Spectrometry)

 

Bremsstrahlung (Braking radiation)

 

 

 

Fig. 1

 

 

            Bremsstrahlung (from German bremsen (to brake) and Strahlung (radiation)) (Fig. 1) is X ray radiation (yellow) emitted by charged particles, such as electrons (blue), which are braking around other charged particles, such as an atom nucleus (red). It forms the continuum component of the x-ray spectrum generated by an x-ray tube.

            The spectral distribution of the braking radiation is given by Kramer formula

 

(1)                                                

 

where I is radiation intensity, k1 is an empirical constant, i is the tube current, Z is the atomic number of the target element, is the wavelength and is the cut off wavelength (i.e. the wavelength at which I = 0).

            Electron energy is eU, where e is the electron charge and U is the tube voltage[i]. If U is in kV and e = 1, energy value in keV is:

 

(2)                                                        

 

When an electron hits the target, it slows down to a lower velocity and its kinetic energy decreases as:

 

(3)                                              

 

Here, m and v are electron mass and velocity, respectively.

            The energy lost by the colliding electron turns into radiant energy associated to a photon:

 

(4)                                                  

 

Where h is the Planck constant, c is light velocity in vacuum,  is the frequency and  the wavelength of the associated wave.

            The most energetic photon results when the electron brakes to zero velocity in one single step and its total energy is transferred to a photon, i.e. . In this case we have:

 

(5)                                                 

 

Therefore, the wavelength of the resulting photon assumes the cut off value () according to Duane-Hunt equation:

 

(6)                                                         

 

For a specific U, no radiation with a shorter wavelength is emitted (Fig. 2). Using numerical values for h and c, and with  in nm, it results:

 

(7)                                                       

 

Fig. 2

 

            According to Kramer formula, at a given radiation intensity increases proportionally with i (Fig. 3 a) and Z.

            An increase in U also bring about a rise in intensity, but, at the same time, the cut off wavelength shifts to lower values (Equation ) and the maximum on the curve also sifts in the same direction (Fig. 2 and 3b). The maximum intensity is given by Ulrey formula:

 

(8)                                                      

 

where k2 is an empirical constant.

            Kramer’s formula is an approximation of the spectral distribution. Its derivation ignores the self-absorption of x-rays and electron backscattering effects.

 

Fig. 3. Effect of changing X-ray tube current (a), and accelerating potential (b) on the continuous spectrum. Source

 

In plasma the free electrons are constantly producing Bremsstrahlung in collisions with the positive ions. This contributes to the background signal in plasma emission spectrometry (Chapter 6).

 

Literature

1. H. Ebel, X-Ray Spectrometry 1999, 28, 255.

2. E. Haug, W. Nakel , The elementary process of Bremsstrahlung, World Scientific, River Edge, 2004.  ISBN 9812385789

3. B. Beckhoff, N. Langhoff, B. Kanngiefer, R. Wedell, H. Wolff,  Handbook of Practical X-ray Fluorescence Analysis, Springer, 2006

 

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F. G. Banica, 09-0-25

 


[i] If e = 1and U is in V, the energy is in electronvolt (eV).1 eV is the kinetic energy acquired by an electron when moving across a potential difference of 1 V. The usual energy unit in x-ray physics is keV (1 keV = 1000 eV = 1.60219 x 10-16 J).