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The formula given for compton scattering shows that when x-ray of one specific wavelength hits carbon or some materials, emitted x-ray will be of one new specific wavelength.

However, according to scattered x-ray intensity (y-axis) vs wavelength (x-axis) graph, it shows that there are multiple wavelengths for each scattering angle.

So, what is the wrong part of my thoughts? How should compton scattering formula be interpreted? And how can one explain that scattered x-ray frequency can be higher than original x-ray frequency?

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2 Answers 2

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Found this for you:

The formula for Compton scattering assumes a free electron initially at rest. In reality, electrons have thermal motion, so only the peak value corresponds to the formula. In addition, electrons are bound by electromagnetic force. Electrons in the inner shell will have a larger effective mass, because the whole atom will recoil with them. This corresponds to other peaks.

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I would add this as a comment to the previous answer, but my starter-reputation precludes me from doing this. So, here is my refinement of Karsus Ren's response.

Compton's formula (and 1923 paper) were about x-ray scattering from free electrons. In 1928, DuMond published a paper showing that for Mo K-alpha x-rays scattered off of beryllium, in addition to the Compton shift, there was a broadening of the scattered spectrum. He (correctly) guessed that this was due to Doppler broadening. The scattering electrons were not at rest, but had a broad, finite momentum distribution. He further calculated what the broadening would look like for a few different model systems. The broadening was much larger than would be expected for either atomic electrons or a classical gas of electrons. It was much better described by a gas of free electrons that obey the Pauli exclusion principle. Since no two fermions can occupy the same quantum state, electrons in a metal occupy states with much larger momentum than would otherwise be expected (the "Fermi sea").

Note that this is not thermal motion, but is instead ground-state quantum motion. That being said, this effect is currently being used to measure temperatures in laser-shock compressed matter such as that in inertial-confinement fusion experiments, where temperatures reach well above 10$^5$ K (where the thermal broadening is measurable on top of the Pauli broadening).

Other structure in the scattering spectrum comes from processes that are not Compton scattering.

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