To solve this problem, we need to determine the spacing of the interference peaks $\Delta x$ using the de Broglie wavelength of the neutron. Here's the step-by-step solution:

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Step 1: Calculate the de Broglie wavelength of the neutron

The de Broglie wavelength $\lambda$ is given by:

$\lambda = \frac{h}{p}$

where:

• $h = 6.626 \times 10^{-34} \, \text{J·s}$ (Planck's constant)
• $p = \sqrt{2 m K}$ (momentum of the neutron)

Here, $m$ is the mass of the neutron ($m = 1.675 \times 10^{-27} \, \text{kg}$) and $K$ is the kinetic energy of the neutron ($K = 0.026 \, \text{eV}$). Convert $K$ to joules:

$K = 0.026 \, \text{eV} \times 1.602 \times 10^{-19} \, \frac{\text{J}}{\text{eV}} = 4.1652 \times 10^{-21} \, \text{J}$

Now, calculate the momentum $p$:

$p = \sqrt{2 m K} = \sqrt{2 (1.675 \times 10^{-27} \, \text{kg})(4.1652 \times 10^{-21} \, \text{J})}$

$p \approx 3.736 \times 10^{-24} \, \text{kg·m/s}$

Finally, find $\lambda$:

$\lambda = \frac{h}{p} = \frac{6.626 \times 10^{-34}}{3.736 \times 10^{-24}}$

$\lambda \approx 1.774 \times 10^{-10} \, \text{m} \, (\text{or } 0.1774 \, \text{nm})$

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Step 2: Calculate the spacing of the interference peaks $\Delta x$

The formula for the spacing of the interference peaks is:

$\Delta x = \frac{\lambda L}{d}$

where:

• $\lambda = 1.774 \times 10^{-10} \, \text{m}$ (calculated above)
• $L = 1.3 \, \text{m}$ (distance to the screen)
• $d = 0.40 \, \text{mm} = 0.40 \times 10^{-3} \, \text{m}$

Substitute the values:

$\Delta x = \frac{(1.774 \times 10^{-10})(1.3)}{0.40 \times 10^{-3}}$

$\Delta x \approx 5.77 \times 10^{-4} \, \text{m} \, (\text{or } 0.577 \, \text{mm})$

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Final Answer:

$\Delta x = 0.58 \, \text{mm}$

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Let me know if you would like more details or further clarifications!

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Related Questions:

1. How is the de Broglie wavelength related to the kinetic energy of a particle?
2. What happens to the interference pattern if the distance between the slits is reduced?
3. How does the wavelength of a neutron compare to that of a photon of the same energy?
4. Why is the neutron beam considered as a wave in this context?
5. What are some practical applications of neutron interference experiments?

Tip: Always ensure energy units are consistent when using physical formulas. Convert eV to joules if necessary!