Physical Principles of Electron Microscopy: An Introduction to TEM ...

64 Chapter 3
As seen from the right-angled triangle in Fig. 3-4, the electric field E that
gives a barrier width (at the Fermi level) of w is E = (�/e)/w. Taking w = �
(which allows high tunneling probability) and � = 4.5 eV for a tungsten tip,
so that (�/e) = 4.5 V, gives E = 4.5 / (0.5 × 10 -9 ) � 10 10 V/m. In fact, such a
huge value is not necessary for field emission. Due to their high speed v,
electrons arrive at the cathode surface at a very high rate, and adequate
electron emission can be obtained with a tunneling probability of the order
of 10 -2 , which requires a surface field of the order 10 9 V/m.
This still-high electric field is achieved by replacing the Wehnelt cylinder
(used in thermionic emission) by an extractor electrode maintained at a
positive potential +V1 relative to the tip. If we approximate the tip as a
sphere whose radius r