Physics

Problem

To check the consistency of their apparatus, a laboratory group measures the scattered wavelength

\lambda_{s}

of X-ray photons that initially have wavelength

\lambda_{0}=0.0710\,\text{nm}

. Their results are shown in the table. Angle data for a single run

$\theta$ (deg)	$\lambda_{s}$ (nm)
0	0.0710
30	0.07133
60	0.07223
90	0.07344
120	0.07468
150	0.07558

Using only the information provided in the table, answer the following. (a) For each angle, calculate the wavelength shift

\Delta\lambda

and record your values in an expanded version of the table. (b) On appropriately scaled axes of your own, sketch a graph of

\Delta\lambda

(vertical axis, nm) versus

(1-\cos\theta)

(horizontal axis, dimensionless). Draw a best-fit straight line through your data points and determine its slope, including units. (c) Use the slope you found in part (b) to obtain an experimental value for Planck’s constant. Clearly state the additional physical constants you require and any assumptions you make. (d) Compare your value from part (c) with the accepted value of Planck’s constant. Discuss whether your data support the particle model of light and identify one random error and one systematic error that most plausibly affected the outcome.

Intuitive explanation

This is Compton scattering, where photons collide with electrons and lose energy—the key is that the wavelength shift depends only on the scattering angle $\theta$ , not on the initial wavelength. By plotting $\lambda_s$ versus $\theta$ (or the wavelength shift $\Delta\lambda = \lambda_s - \lambda_0$ versus $\sin^2(\theta/2)$ ), you can extract the Compton wavelength $\lambda_C$ from the slope, which is a fundamental constant. This lets you verify whether the apparatus gives consistent physics without needing any external reference.

Intuitive explanation

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