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Dispersion spectrometer

Basic principles

Light source

Sample

Monochromator

Spectrum

Dispersive spectrometers: we use instruments to Disperse energy across a scale appropriate to a chemical property

The Single-Beam Spectrometer

Monochromators

The light sources we use produce continuous emission spectra. But we need single wavelengths, so…

prism based monochromator

This is called the Czerny-Turner setup

Linear time-invariant systems

Consider a black box system with an input and an output:

OR y(t)=Ф{x(t)} (1)

{x(t-t0) } = y(t-t0)

(stars denote complex conjugation) or given in terms of an integral transform With a kernel g (k,t) :

(4)

The response to an arbitrary input function can then be written in terms of responses to either the individual

basis functions or the integral transform kernel:

(5)

A hammer strike amounts to a very short and sharp input signal – a delta-function:

(6)

which looks very similar to Equation (4):

We can now use Equation (5) to write:

(7)

The function h(t) t is called the pulse response of the system. From Equations (7) and (2), the response to any input function x(t) can be calculated

as:

(8)

This integral is known as the convolution integral, and is often abbreviated as:

(9)

So, in some sense, the pulse response h(t) contains complete information about our linear time-invariant black box and allows us to predict its response to an arbitrary input.

Let us do some spectroscopy on

(10)

This demonstrates that exponentials are eigenfunctions of LTI systems. That is, an LTI system cannot shift frequencies, it can only alter their coefficients. Let us feed each frequency in turn:

(11)The function H is called the frequency response of the system.