Why Does the Area Under a Diffraction Curve Equal This Value?

In summary, the conversation discusses a curve represented by the function f(\theta) = \frac {I_0sin^2(n\theta/2)}{sin^2(\theta/2)} and its area over a cycle from ##0## to ##2π##, which is equal to ##(2πnI_0)##. The question arises whether there is a reason behind this coincidence and if ##I_{max} = 4\pi n I_0## holds true all the time. The expert responds that the maximum value for the function is ##I_{max}=n^2 I_o## and it occurs periodically when both the numerator and denominator approach zero. The use of the letter ## \phi
  • #1
albertrichardf
165
11
Hi,
consider the following curve:
[tex] f(\theta) = \frac {I_0sin^2(n\theta/2)}{sin^2(\theta/2)} [/tex]

When the area over a cycle from ##0## to ##2π## is evaluated it gives ##(2πnI_0)##. This is exactly ##\frac {I_{max} + I_{min}}{2}## , since
##I_{min}## is ##0##. Is this a coincidence, or is there a reason behind the area under the curve is the same as this value?
Thank you for your answers.
 
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  • #2
Does this mean that ##I_{max} = 4\pi n I_0## all the time? If so, then that would be all the coincidence needed.
 
  • #3
## I_{max}=n^2 I_o ##. The maximum occurs periodically when both the numerator and denominator equal (i.e. approach) zero. This is in the limit ## \frac{\theta}{2} \rightarrow m \pi ##. I don't know if you have the integral evaluated correctly for a single cycle. I would need to try to look that one up. I don't even know that it has a simple closed form. ## \\ ## Usually the letter ## \phi ## is used instead of ## \theta ## in this diffraction theory integral for ##N ## slits, where the phase ## \phi=\frac{2 \pi d \sin{\theta}}{\lambda} ##, and the ## N ## is designated with a capital letter.
 
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FAQ: Why Does the Area Under a Diffraction Curve Equal This Value?

What is the area under a diffraction curve?

The area under a diffraction curve is a measure of the total amount of light that is diffracted by a particular object or material. It is represented by the shaded region under the curve in a diffraction pattern.

How is the area under a diffraction curve calculated?

The area under a diffraction curve is typically calculated using mathematical integration techniques. This involves breaking the curve into smaller sections and finding the sum of the areas of each section. Alternatively, it can also be estimated by counting the number of squares or pixels within the shaded region.

What factors affect the area under a diffraction curve?

The area under a diffraction curve can be influenced by several factors, including the wavelength of the incident light, the size and shape of the diffracting object, and the distance between the object and the detector. Additionally, the properties of the material being diffracted, such as its refractive index, can also impact the area under the curve.

Why is the area under a diffraction curve important in scientific research?

The area under a diffraction curve provides valuable information about the physical and chemical properties of a material. By analyzing the shape and size of the curve, scientists can determine the structure and composition of the material, which can be useful in a variety of fields such as materials science, biology, and chemistry.

Can the area under a diffraction curve be used to identify unknown substances?

Yes, the area under a diffraction curve can be used as a tool for material identification. By comparing the diffraction pattern of an unknown substance to a database of known patterns, scientists can determine the composition of the material and potentially identify it. This technique is commonly used in fields such as forensic science and geology.

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