What will the fringe spacing be if the electrons are replaced by neutrons with t

Question

What will the fringe spacing be if the electrons are replaced by neutrons with the same speed?
What should the speed of the neutrons be to produce interference fringes 1.5 mm apart after passing through the double slit apparatus. Electrons with a speed of 2.0 10° m/s pass through a double-slit apparatus. A detector positioned after the double-slit apparatus measures interference fringes with a fringe spacing of 1.5 mm. 1. What electric potential would be needed to accelerate the electrons to this speed? 2. What will the fringe spacing be if the electrons are replaced by neutrons with the same speed? 3. What should the speed of the neutrons be to produce interference fringes 1.5 mm apart after passing through the double-slit apparatus?

Explanation / Answer

Given

Speed of electron v = 2.0 x 106 m/s

Fringe spacing d = 1.5 mm

Solution

1.

Potential energy and speed relation is

v = {2qV/m}

Here,

v is the velocity of electron,

q is the charge of the electron

V is the potential difference

m is the mass of electron

v = {2qV/m}

v2 = 2qV/m

V = mv2/2q

V = 9.1 x 10-31 x (2.0 x 106)2 / 2 x 1.6 x 10-19

V = 1137.5 V or 1.1375 kV

2)

De Broglie wavelength of electron

e = h/p = h/mv

e = 6.626 x 10-34/9.1 x 10-31 x 2 x 106

e = 0.364 x 10-9 m

De Broglie wavelength of neutron

n = h/mnv-

n = 6.626 x 10-34/1.67 x 10-27 x 2 x 106

n = 1.98 x 10--13 m

the fring spacing x is directly proportional to the wavelength

xe e

xn n

xn / xe = n / e

xn / 1.5 x 10-3 = 1.98 x 10—13 / 0.364 x 10-9

xn = 8.16 x 10-7 m

3)

this spacing can be achieved when the neuton have the same de Broglie wavelength as electron

n = e

h/mnvn = 0.364 x 10-9

6.626 x 10-34 / 1.67 x 10-27 x vn = 0.364 x 10-9

vn = 1.09 x 103 m/s

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