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On the first figure, you can see schematic diagram of typical thermionic electron gun from Wiki.with filament voltage(1-2) electrons are emitted and with bias(1-3) - focused. Electron Gun and here is filament voltage VS emitted electrons number, for different bias voltage.

it has a typical exponential behavior, but as you can see, there is kind of bump, the false peak near 3.4 volts. In the text this peak is explaining: „The false peak near 3.4 volts caused by region of filament that reaches emission temperature before tip„. but its still unclear for me. question is - what is the reason of this peak of current?

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  • $\begingroup$ Different values of bias resistance distinguish the two graphs. It would be useful to see these on the circuit diagram. $\endgroup$ – Philip Wood Nov 5 '17 at 8:58
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There is literature which indicates that the false peak arises due to parts of the filament reaching thermionic emission temperature before the tip does. Not much is said beyond this terse explanation.

I would venture a guess that the above is true, and occurs due to space charge effects that suppress passage of electrons areas other than the tip, once it has become the dominant source of electrons. We must also consider that as overall emission rises with heating current, changing bias voltage (which arises anytime electron emission current changes due to voltage drop across the bias resistor) between wehnelt and filament is effectively moving the source image focal plane closer and closer to the filament tip. This is why, in a source image mode, the emission area shrinks as heating current is raised.

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