Cristian determines that the spacing between the planes of a crystalline material is 2.46x10-10 m. When a beam of x-rays makes an angle of 1520° with the planes of the crystal, first order constructive interference occurs. Determine the wavelength of the x-rays in nanometers with two decimal places please.

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**Determining the Wavelength of X-rays Using Crystallography** 

Cristian determines that the spacing between the planes of a crystalline material is 2.46 x 10<sup>-10</sup> meters. When a beam of x-rays makes an angle of 15.20° with the planes of the crystal, first order constructive interference occurs. Determine the wavelength of the x-rays in nanometers with two decimal places please.

*Explanation:*

* Given:
  - Spacing between planes (d): \( 2.46 \times 10^{-10} \) meters.
  - Angle of incidence (\( \theta \)): 15.20°
  - First order constructive interference: n = 1

* Formula:
  \( n \lambda = 2d \sin(\theta) \)

  Where:
  - \( n \) is the order of constructive interference
  - \( \lambda \) is the wavelength of the x-rays
  - \( d \) is the spacing between planes
  - \( \theta \) is the angle of incidence

* Steps to Solve:
  - Convert the angle from degrees to radians if necessary.
  - Substitute the given values into the formula and solve for \( \lambda \).
  - Convert the wavelength from meters to nanometers.

* Additional Notes:
  - Ensure precise calculations to obtain the wavelength accurate to two decimal places.
  - Remember that 1 meter = \( 10^9 \) nanometers.
Transcribed Image Text:**Determining the Wavelength of X-rays Using Crystallography** Cristian determines that the spacing between the planes of a crystalline material is 2.46 x 10<sup>-10</sup> meters. When a beam of x-rays makes an angle of 15.20° with the planes of the crystal, first order constructive interference occurs. Determine the wavelength of the x-rays in nanometers with two decimal places please. *Explanation:* * Given: - Spacing between planes (d): \( 2.46 \times 10^{-10} \) meters. - Angle of incidence (\( \theta \)): 15.20° - First order constructive interference: n = 1 * Formula: \( n \lambda = 2d \sin(\theta) \) Where: - \( n \) is the order of constructive interference - \( \lambda \) is the wavelength of the x-rays - \( d \) is the spacing between planes - \( \theta \) is the angle of incidence * Steps to Solve: - Convert the angle from degrees to radians if necessary. - Substitute the given values into the formula and solve for \( \lambda \). - Convert the wavelength from meters to nanometers. * Additional Notes: - Ensure precise calculations to obtain the wavelength accurate to two decimal places. - Remember that 1 meter = \( 10^9 \) nanometers.
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