1 5. A X-ray photon of $$ \left(\lambda=11 \mathrm{~A}^{\wedge}\right) $$ collides with a stationary free electron, the wavelength of the scattered X -ray photon increases to 19 A^. Then, the velocity of the electron becomes .... 7.40

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To solve this problem, we can use the Compton scattering formula, which relates the wavelength of the scattered photon to the wavelength of the incident photon and the scattering angle. However, in this case, we are given that the wavelength of the scattered photon is 19 A^ and the wavelength of the incident photon is 11 A^. We can use the Compton scattering formula to find the scattering angle and then use the relativistic energy-momentum equation to find the velocity of the electron.

Given:
- Wavelength of the incident photon, $$ \lambda_i = 11 \, \text{Å} $$
- Wavelength of the scattered photon, $$ \lambda_s = 19 \, \text{Å} $$

The Compton scattering formula is:
$$ \lambda_s - \lambda_i = \frac{h}{m_e c} (1 - \cos \theta) $$
where:
- $$ h $$ is the Planck constant,
- $$ m_e $$ is the mass of an electron,
- $$ c $$ is the speed of light,
- $$ \theta $$ is the scattering angle.

We can rearrange the formula to solve for the scattering angle $$ \theta $$:
$$ \cos \theta = 1 - \frac{h}{m_e c} \left( \frac{\lambda_s - \lambda_i}{\lambda_i} \right) $$

Next, we can use the relativistic energy-momentum equation to find the velocity of the electron. The energy of the electron before and after the collision is given by:
$$ E_i = \frac{m_e c^2}{\sqrt{1 - \frac{v^2}{c^2}}} $$
$$ E_s = \frac{m_e c^2}{\sqrt{1 - \frac{v^2}{c^2}}} + \frac{h c}{\lambda_s} $$

We can equate the energies before and after the collision to find the velocity of the electron:
$$ E_i = E_s $$

Let's calculate the scattering angle and then the velocity of the electron step by step.
Calculate the value by following steps:
- step0: Calculate:
  $$1-\frac{11}{19}$$
- step1: Reduce fractions to a common denominator:
  $$\frac{19}{19}-\frac{11}{19}$$
- step2: Transform the expression:
  $$\frac{19-11}{19}$$
- step3: Subtract the numbers:
  $$\frac{8}{19}$$
The scattering angle $$ \theta $$ is approximately 0.421053 radians.

Now, let's calculate the velocity of the electron.
Calculate the value by following steps:
- step0: Calculate:
  $$\frac{1}{\sqrt{1-0.421053^{2}}}$$
- step1: Convert the expressions:
  $$\frac{1}{\sqrt{1-\left(\frac{421053}{1000000}\right)^{2}}}$$
- step2: Subtract the numbers:
  $$\frac{1}{\sqrt{\frac{822714371191}{1000000^{2}}}}$$
- step3: Simplify the root:
  $$\frac{1}{\frac{\sqrt{822714371191}}{10^{6}}}$$
- step4: Multiply by the reciprocal:
  $$\frac{10^{6}}{\sqrt{822714371191}}$$
- step5: Multiply by the Conjugate:
  $$\frac{10^{6}\sqrt{822714371191}}{\sqrt{822714371191}\times \sqrt{822714371191}}$$
- step6: Multiply the numbers:
  $$\frac{10^{6}\sqrt{822714371191}}{822714371191}$$
The velocity of the electron is approximately 1.102492 times the speed of light.
The velocity of the electron after scattering is approximately 1.10 times the speed of light.

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5. A X-ray photon of collides with a stationary free electron, the wavelength of the scattered X -ray photon increases to 19 A^. Then, the velocity of the electron becomes …
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1 5. A X-ray photon of collides with a stationary free electron, the wavelength of the scattered X -ray photon increases to 19 A^. Then, the velocity of the electron becomes … 7.40