Physics

# Spectroscopy

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### Definition:

• Spectroscopy = began as the study of the interaction between matter and electromagnetic radiation, has been extended to include any sort of interaction with radiative energy that can be measued in terms of wavelength and/or frequency

### Three transforms used in most modern forms of spectroscopy

• Exponential FT is Lorentzian:

$g(k) = e^{-k/a}$ ⇒ $\mathcal{F} \{ g(k) \} = f(x) = \sqrt{\frac{2}{\pi}} \frac{(1/a)^2}{x^2 + (1/a)^2}$

• Gaussian FT is Gaussian:

$g(k) = e^{-k^2/a^2}$ ⇒ $\mathcal{F} \{ g(k) \} = f(x) = \frac{a}{\sqrt{2\hbar}} e^{-x^2a^2/4}$

where $\hbar = h/2\pi$

• Square wave FT is the sinc function
• Note: $\mathrm{sinc}(x) = \frac{\sin (x)}{x}$

$g(k) = \frac{1}{a}$ for $-\frac{a}{2} < k < \frac{a}{2}$ ⇒ $\mathcal{F} \{ g(k) \} = f(x) = \frac{1}{\sqrt{2\pi}} \mathrm{sinc}(\frac{xa}{2})$

### Spectral Variables

• Radiation is composed of individual photons
• Each of these photons is entirely characterized by its frequency ($\upsilon$)
• This is an intrinsic property of the photon and cannot be changed over its “lifetime”
• All radiation has a characteristic $\upsilon$
• Spectral variables are all used to characterize/describe radiation particles (e.g., light) and are related to frequency

• Note: for very small particles, $\rho = h/\lambda$ where $\latex \rho =$ the momentum of the particle and $h =$ Planck’s constant

### Spectral Distribution

• Each photon carries a specific amount of energy ($Q_{ph}$)
• $Q_{ph} = h\upsilon_{ph}$ where $\upsilon_{ph}$ = photon frequency
• The energy carried by a photon depends on the photon’s frequency, or equivatlently, its wavelength ($\lambda_{ph} = c/\upsilon_{ph}$ )
• Every dete4ctor of radiation (e.g., the eye) is sensitive to finite wavelength intervals rather than a single value $\lambda$