Question

Find the de Broglie wavelength of relaativistic electrons reaching the anticathode of an `X`-ray tube if the short wavelength limit of the continous `X`-rays spectrum is equal to `lambda_(sh)= 10.0p m`?

Answer 1

For relativistic elelctrons, the formula for the short wavelength limit is `X`-rays will be
`(2pi ħc)/(lambda_(sh))=m_(0)c^(2)((1)/(sqrt(1-beta^(2)))-1)=csqrt(P^(2)+m^(2)c^(2))-mc^(2)`
or `((2pi ħ)/(lambda_(sh))+mc)^(2)=P^(2)+m^(2)c^(2)`
or `((2piħ)/(lamda_(sh)))((2piħ)/(lambda_(sh))+2mc)=P^(2)`
or `p=(2piħ)/(lambda_(sh))sqrt(1+(mc lambda_(sh))/(piħ))`
Hence `lambda_(sh)=lambda_(sh)//sqrt(1+(mc lambda_(s h))/(piħ))-3.29p m`