# br A derivative of a cut off curve

A derivative of a cut-off curve gives us the autophagy pathway axial energy distribution in the input beam: N(W∥) = –dS/du. According to Eq. (1), there are three ways to calculate the cut-off function from the experimental data:

If the analyzer operates properly, these three functions must coincide (Fig. 3b). Their substantial divergence would show that the input beam is not the only significant current source in the analyzer, and the registered data should be discarded as dubious.
The cut-off curves S(u) measured with uniform magnetic field distribution in the analysis volume (conserving transverse energy component W⊥) represent axial energy distribution N(W∥ ) at the target. When the built-in analyzer coils are turned on to make the magnetic profile non-uniform, the energy redistributes between the components while the beam moves from the target to the retarding space, which affects the investigated N(W∥) spectra. The magnetic profile distortion degree can be characterized with “magnetization” parameter β = /B0, where B0 and are magnetic induction values for the target plane and a position of the retarding potential minimum (see plot in Fig. 1) respectively. A set of spectra measured for the same beam parameters and different β can yield information on full 2D energy distributions N(W∥, W⊥).
The law of the energy exchange between the components is the simplest for “adiabatic” conditions, when variation of magnetic and electric fields in space is slow:where RL and LL are Larmor parameters of electron trajectory;
is Larmor cyclic frequency; e and m are electron charge and mass.
In adiabatic case, the transverse energy of any electron is proportional to the magnetic field at its current position B, while its full energy W remains constant:

Index 0 marks the values corresponding to a fixed initial axial position of the particle that we choose coinciding with the input aperture of the analyzer and with the target plane, where the magnetic field B0 is the same for all shots of a series.
In theory, axial energy spectrum N(W∥) obtained by derivation of a cut-off curve may be considered also as a result of convolution (integration, projection) of two-dimensional distribution at the spatial position of measurement N(W∥, W⊥) along vertical lines W∥ = const. In the case of uniform magnetic field, no energy transform between components occurs, and such connection can be established between the measured spectra N(W∥) and the 2D energy distribution at the target N(W∥0, W⊥0) characterizing the investigated facility flow (see Fig. 4a). When the field in the analyzer is made non-uniform, the axial energy spectrum N(W∥) is vertical projection of 2D energy distribution in the point of measurement N(W∥, W⊥) (Fig. 4b), transformed according to formulae (4a) and (4b). At the target position, the field does not change, and the energy distribution in (W∥0 , W⊥0) coordinates is the same as that in the uniform field (Fig. 4c).
Approached formally, relations (4a) and (4b) can be interpreted as description of a linear transformation of the coordinate plane (W∥0, W⊥0) → (W∥, W⊥) with β serving as a parameter. This transformation reflects any straight line of (W∥, W⊥ ) onto another straight line crossing the abscissa axis at the same point and having β times greater (for β > 1) angle to positive direction of the abscissa axis (see Fig. 5). The image line will be vertical (α′ = π/2), if the angle of the initial line slope is equal to

Thus, returning to Fig. 4b and c, we can use the fact that integration (projection, convolution) along vertical projection lines at the (W∥, W⊥) plane is equivalent to the integration along straight lines sloped by α in (W∥0, W⊥0) coordinates. Consequently, the axial energy distributions measured for different values of β parameter (they will be denoted as (W∥)) can be considered as parallel projections of 2D energy distribution at the target N(W∥0, W⊥0) under different aspect angles α determined by Eq. (5). In practice, rather broad range of aspect angles may be available: α is small in the case of high magnetization in the analyzer (for instance, α = π/4 for β = 2), and is close to 3π/4 if the built-in coils substantially reduce the external guiding magnetic field (β → 0).