Calculate the wavelength of the least energetic photon capable of producing an electron-positron pair in meters.
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E = h v
v = λ c
Minimum energy
= 2 m e v 2
= 2 ( 9 . 1 ∗ 1 0 − 3 1 k g ) ( 3 ∗ 1 0 8 m / s ) 2
= 1 . 6 3 8 ∗ 1 0 − 1 3 J
= 1 . 0 2 2 M e V
Minimum frequency
= h 2 m e v 2
= 6 . 6 2 6 ∗ 1 0 − 3 4 J s 2 ( 9 . 1 ∗ 1 0 − 3 1 k g ) ( 3 ∗ 1 0 8 m / s ) 2
= 2 . 4 7 2 ∗ 1 0 2 0 H z
Minimum wavelength
= f r e q u e n c y c
= 2 . 4 7 2 ∗ 1 0 2 0 H z 3 ∗ 1 0 8 m / s
= 1 . 2 ∗ 1 0 − 1 2 m
This is the threshold wavelength of an incoming photon in pair production. Simply use the photoelectric effect formula of hv and set it equal to two electron (neglecting nuclear recoil)...and solve in terms of wavelength! :)
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