Helmholtz-Lagrange relation

The requirements of the Helmholtz-Lagrange relation and further constraints on the properties of a lens (for example, a constant distance d between the object and the image which is of great practical importance) do not allow an arbitrary choice of values for the magnification and/or acceleration/retardation of the electrons. Hence, in order to obtain freedom in the selection of certain properties, a lens system must have a certain number of elements. For example, if the electron energy is to be varied, keeping the distance d constant, at least two free parameters are... 

Due to the Helmholtz-Lagrange relation, no greater flexibility in the parameters determining the image is possible. However, lenses with even more elements than four can have other advantages, such as lower aberration or a more extended operation range. (See, for example, the movable electrostatic lens in [Rea83].)... 

Of course, the actual values of Mlin, M , and A/r have to fulfil the Helmholtz-Lagrange relation. 

Liouville theorem and related forms The Helmholtz-Lagrange relation given in equ. (4.46) is related to many other forms which all state certain conservation laws (the Clausius theorem, Abbe s relation, the Liouville theorem). The most important one in the present context is the Liouville theorem [Lio38] which describes the invariance of the volume in phase space. The content of this theorem will be discussed and represented finally in a slightly different form which allows a new access to the luminosity introduced in equ. (4.14). 

The applications of the Helmholtz-Lagrange relation given in equ. (4.46) to an arbitrary conjugate object-image pair characterized by left (/) and right (r), respectively, leads to... 

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