112 The Electron Microscope 



flowing in the two coils in opposite directions, so that they pro- 

 duce on the axis magnetic fields Ho of opposite signs. If the 

 field at the right is somewhat weaker than the field at the left, 

 the axial point in which Ho changes its sign will be at the right 

 of the partition. A field line of roughly parabolical shape starts 

 from this point. The cathode is placed at or near to this field 

 line, so that there is no magnetic flux between the cathode and 

 the axis. 



The reason for this particular arrangement of the cathode can 

 be understood from equation (2) in chapter 2, which states that 

 an electron moving in an axially symmetrical magnetic field 

 acquires an angular momentum around the axis proportional to 

 the flux between the circle from which it started and the circle 

 at which it arrives. The angular momentum of an electron 

 which reaches the axis is zero, and for the present we will neglect 

 the initial velocities at the cathode. This means that an electron 

 starting from the cathode can reach the axis only if it has 

 acquired no momentum, that is, if the flux between the cathode 

 and the axis is zero. Collisions between electrons and between 

 electrons and gas molecules would make the condition less 

 stringent, but we will neglect these for the present. 



The boundaries of the space which is accessible to electrons 

 can be easily obtained from the equivalent electrostatic potential, 

 U, which was introduced in chapter 2. Equation (8) was de- 

 rived for exactly the case which applies here, i.e., for electrons 

 which need not cross a magnetic flux to reach the axis. In the 

 accessible space U must be positive, i.e., 



U = </>-:^A2^0 (38) 



We can restrict this space therefore either by making 4> negative, 

 or by making A (which is defined by equation (3) in chapter 2) 

 large. It is suggested to use limitation by strong magnetic fields 

 on the cylindrical part of the boundary, and limitation by nega- 

 tive potentials at the ends. In the middle part where the mag- 

 netic field H is nearly uniform the condition is, as here A ^ ^Hr 



