Similar to electron diffraction, neutron diffraction microscope is also used for the determination of the structure of molecules. If the wavelength used here is 800 pm, calculate the characteristic velocity associated with the neutron.

To calculate the characteristic velocity associated with a neutron when given its wavelength, we can use the de Broglie wavelength formula. This principle relates the wavelength of a particle to its momentum, which in turn can help us find its velocity. The de Broglie wavelength (\(\lambda\)) is given by the equation:

Understanding the Relationship

The de Broglie wavelength is defined as:

\(\lambda = \frac{h}{p}\)

Where:

• \(\lambda\) is the wavelength of the particle (in meters).
• h is Planck's constant, approximately \(6.626 \times 10^{-34} \, \text{Js}\).
• p is the momentum of the particle (in kg·m/s).

For a neutron, the momentum can also be expressed in terms of its mass and velocity:

p = mv

Where:

• m is the mass of the neutron, approximately \(1.675 \times 10^{-27} \, \text{kg}\).
• v is the velocity of the neutron (in m/s).

Calculating the Velocity

We can combine these equations to find the velocity:

\(\lambda = \frac{h}{mv}\)

Rearranging this gives us:

v = \frac{h}{m\lambda}

Now, substituting the known values:

• Planck's constant, \(h = 6.626 \times 10^{-34} \, \text{Js}\)
• Mass of the neutron, \(m = 1.675 \times 10^{-27} \, \text{kg}\)
• Wavelength, \(\lambda = 800 \, \text{pm} = 800 \times 10^{-12} \, \text{m}\)

Plugging these values into the equation:

v = \frac{6.626 \times 10^{-34}}{(1.675 \times 10^{-27})(800 \times 10^{-12})}

Performing the Calculation

Now, let's calculate the denominator:

1.675 \times 10^{-27} \times 800 \times 10^{-12} = 1.340 \times 10^{-18} \, \text{kg·m}

Now substituting back into the velocity equation:

v = \frac{6.626 \times 10^{-34}}{1.340 \times 10^{-18}} \approx 4.94 \times 10^{4} \, \text{m/s}

Final Result

The characteristic velocity associated with the neutron, given a wavelength of 800 pm, is approximately 49,400 m/s. This high velocity indicates that neutrons, despite their relatively large mass compared to electrons, can still exhibit significant wave-like behavior due to their small wavelength in this context.