X-KAY IVnCROSCOPY 



satisfying this condition would be imprac- 

 ticably thin, but Von Borries (1949) and 

 Lenz (1954) have calculated the distance in 

 which an electron is expected to suffer 1 

 elastic collision*, and this is a convenient 

 criterion for our purposes. At 10 kV it is ap- 

 proximately 2 X 10~^ gm/cm^ for all ele- 

 ments, rising to about 8 X 10~^ gm/cm^ at 

 40 kV. As the most probable angle of single 

 scattering is only 0.01-0.1 radian, this cri- 

 terion ensures that the electrons are usually 

 deviated by only quite small amounts from 

 their initial directions. The inelastic en- 

 counters cause even smaller deviations (for 

 example 1.4 X 10~^ radian in carbon at 20 

 kV) and so can be neglected in this connec- 

 tion; and it can be readily shown (e.g., from 

 the data of Lane and Zaffarano, 1954) that 

 the amount of energy lost in this critical 

 thickness if very small, being for example 

 0.04 kV for almninum, for an incident elec- 

 tron energy of 10 keV. At 10 keV the full 

 range is 0.26 mg/cm^ for aluminum, or 130 

 times the Aiifhellungsdicke. The range in- 

 creases slowly with increasing atomic nmn- 

 ber, and from the Bethe-Bloch expression 

 (e.g. Paul and Steinwedel, 1955) can be 

 shown to be greater for gold than for alu- 

 minum by a factor of about 2. 



It is clear that work with thin targets is 

 more fundamental and can be compared 

 more readily with theory, whereas thick tar- 

 gets are of more practical importance in that 

 they provide maximum yields of x-rays. 

 Thick targets can be subdivided into foils 

 which are thin enough to transmit the x-radi- 

 ation produced, as used in the projection 

 x-ray microscope (e.g., Nixon (1955)), and 

 the massive anticathode familiar in conven- 

 tional x-ray tubes. 



We shall consider the production of the 

 continuous and characteristic spectra in thin 

 and thick targets, in the energy region of 

 interest in x-ray microscopy. 



* Termed by them the 'Aufhellungsdicke', in 

 connection with image formation in electron mi- 

 croscopy. 



Continuous Spectrum from Thin Tar- 

 gets 



Spectral Distribution. The spectral 

 distribution from thin targets was first stud- 

 ied by Nicholas (1929) using an electron 

 energy of 45 keV and targets of aluminum 

 and gold. Measurements were made at an- 

 gles of 40, 90 and 140 degrees to the forward 

 direction. In general the data showed that 

 the intensity per unit frequency interval I^ 

 was approximately constant up to the high 

 frequency limit vo , and zero thereafter. A 

 theoretical treatment by Kramers (1923) 

 gave a similar relationship, of the form 



I,dv = —--- dv. 



This expression is not in general true, al- 

 though it is a good approximation when con- 

 sidering radiation emitted at right angles to 

 the incident electron beam, using targets of 

 high atomic number and electrons of low 

 energy. 



The more rigorous quantum mechanical 

 theory of the continuous spectrum is due to 

 Sommerfeld and has been put into numerical 

 form by Kirkpatrick and Wiedmann (1945) 

 who gave theoretical spectral distributions 

 for a wide range of conditions. According to 

 their data, the shape of the spectral distribu- 

 tion depends on the parameter Z^/V, and 

 not upon either of these two variables sepa- 

 rately, except at the extreme low frequency 

 end of the spectrum, where the screening ef- 

 fect of outer electrons, which reduces the 

 x-ray intensity somewhat, depends on Z 

 separately. The spectral shape depends 

 markedly on the angle of observation. Fig. 1 

 (calculated from Kirkpatrick and Weid- 

 mann's data) illustrates the spectral dis- 

 tributions at 0° and 90°, and also averaged 

 over all directions, for an electron energy of 

 10 keV, using targets for which Z = 13 

 (aluminum) and 58 (the highest available 

 from their calculations, at this low energy). 

 The simple expression of Kramers can be 



654 



