ELECTKON MICKOSCOPY 



Using this definition, the total current, i, 

 in the spot will be 



i = 0.62 — • Ja amp 



4 



where the numerical factor 0.62 adjusts for 

 the Gaussian distribution of intensity and 

 applies only to this particular definition of 

 do gi\'en above. 



„ -n-do^ . eV , 



i = 5.65 X 10' - rfoW 



considering the case for the corrected lens, 

 equation 7 becomes 



1 C.2 



c/2 = ri.l77iV - X 10'" + (1.22X)2 



a2 4 



and d is a minimum when a = ctopi given 

 by 



1/8 



....... 09 (-) (7.92X10.,- + .) 



/ J> \3/8 



( 7.92 X lO^j - + 1 j 



d„.ia = 1.29C,"4X5" 



(8) 



(9) 



In terms of electron wavelength 



i = 8.48 X 10-" 4- doW 

 1 \' 



(6) 



Equation 6 may be transposed for do and 

 combined with equation 4 to give: 



d2 = 



1.177fX2 -7 X W + (1.22X)2 

 3 



C 2 

 4 



(7) 



Thus, for a fixed value of spot current, the 

 contribution of the first term in l/a^ is to 

 reduce the diameter of the spot with increase 

 of a while the contribution of the remaining 

 terms is to increase the diameter with in- 

 crease of OL. For any given current, there will 

 be an optimum aperture setting and a cor- 

 responding optimum setting of do (deter- 

 mined by equation 6) at which the total 

 spot diameter will be a minimum. This mini- 

 mum may be derived from equation 7 as it 

 stands, but the resulting expressions are less 

 cumbersome if two cases are considered, (a) 

 for an objective corrected for astigmatism 

 and (b) for an uncorrected objective. In the 

 latter case, astigmatism will in general pre- 

 dominate and the spherical aberration term 

 may be neglected. However, since correction 

 for astigmatism is relatively easily accom- 

 plished, spherical aberration imposes the 

 more fundamental limitation. Therefore, 



It may be shown from the foregoing equa- 

 tions that the Gaussian spot diameter, f/o , 

 must be set to approximately y/Ji dmin for 

 the optimum conditions. 



Expressions equivalent to 8 and 9 may be 

 derived for the case of an uncorrected objec- 

 tive in which astigmatism is predominant. 



Equation 9 shows clearly the hmitations 

 imposed by spherical aberration, diffraction, 

 and by the electron gun. If only very small 

 spot currents are required, the first term in 

 the brackets may become insignificant, in 

 which case the optical aberrations become 

 the ultimate limitation. For the usual values 

 of filament temperature and emission exist- 

 ing in the electron microscope, this occurs at 

 spot currents of approximately 10"^^ amp. 

 Currents somewhat greater than this are re- 

 quired in the scanning microscope. 



Limitations Imposed by Fluctviations 

 in the Electron Beam Current. Because 

 of the particulate nature of an electron 

 beam, the illumination of an object in the 

 electron microscope is a random process. A 

 mean number of electrons n, falling on an 

 element of the object, has associated with 

 it a fluctuation of r.m.s. magnitude \/w. 

 The basic signal-to-noise ratio is then 

 nl\Jn = \/^- If the secondary emission 

 ratio at the specimen is not less than unity 

 and no further noise is introduced during 

 interaction with the specimen, a change An 

 in the number of electrons emitted from the 

 surface as the beam is scanned from one 



244 



