Electromagnetic Spectrum

# Electromagnetic Spectrum

## Connecting Wavelength, Energy and Time

In the ultraviolet you have 3 to 30 eV energies, in the range of 100 to 1000 eV you have soft x-rays, and beyond that hard x-rays. In the visible spectrum you have wavelengths of a nanometer. But in the soft x-rays you have wavelengths in the order of angstroms. This is why you can use the diffraction of soft x-rays to explore the crystal structure of molecules; the wavelength is in the approximate scale of that of atoms and bonds. In the range of mega eV it is the range of gamma rays.

On the lower energy end there is the infrared. If you take an IR spectrum of a pi conjugated system the bond stretching of a CC double bond is about 1600 wave numbers or about .2 eV and the frequency is in the sub pico second regime. In a pi conjugated molecule in an excited state there will be bond length relaxation in few tenths of a picosecond. When you have motion of molecules as a whole you can have of 20-50 wave numbers in the millivolt scale, and timescales of 10-100 pico seconds.

Beyond that the microwave and radio wave with wavelengths that are very large, and in the mega or kilohertz. It is useful to keep in mind the relations between the wavelengths, spectrum and energies.

1 eV = 1.6 x10-19 J

= 96.5 kJ/mol ~100 kJ/mol

~23kcal/mol

= 8065 cm-1 (wave numbers)

## Visible spectrum

The visible is a very small part of the electromagnetic spectrum, from 700nm on the low energy red side to 400nm on the high energy violet side

The energies of the spectrum vary from about 3eV on the violet side to 1.5eV on the red side in approximate terms. The frequencies are in the mid 1014 scales. This is important because it determines the time scales for events. For example an event such as energy transition in a pi conjugated system which will be 2 or 3eV, that means you will be in the range of the visible spectra frequencies so your timescale for that process will be in the femptoseconds.

The color red green blue (RGB) colors when they are combined create white light. You can also combine magenta and green to create white light.

## Converting from wavelength to eV

It is possible to convert quickly from wavelength to electron volts

For 1 eV:

$h\nu = \frac{hc}{\lambda}\,\!$

V ~2.5 x 1014 Hz and λ ~1240nm

$\frac{1240}{\lambda (nm)} ~ energy (eV)\,\!$

So a light beam at 600 nm you take 1240/600 that is about 2 eV and it is in the yellow portion of the spectrum. Or 400 nm gives 3eV and so on.

## Thermal Energy

kT (300K) is the thermal energy at 300K

~0.025 eV

~2.5 kJ/mol

~0.6 kcal/mol

~200 cm-1 (wave numbers)

This tells you that if you take a crystal of π- conjugated material at room temperature and examine the CC bond stretching that requires 1600 wave numbers, there is not enough energy at room temperature to excite it. On the other hand motions of the whole molecule that only require 50 wave numbers you can achieve this level of excitation at room temperature. These estimations help you determine if you have the right amount of energy and an appropriate timescale for the process that you are working with.

## Uncertainty and precision

The precision of the energy times the precision of the time has to be large h/4π

$\Delta E \cdot \Delta t > \frac{h}{4\pi}\,\!$

$h = 4.14 \times 10^{-15} eV\cdot s\,\!$

$\frac{h}{4\pi}= 3.3 \times 10^-16 eV\cdot S\,\!$

For light:

$E = hv = h/T\,\!$

$E \cdot T = h(>\frac{h}{4\pi})\,\!$

For T= 5 x 10-15 s  :

E~0.8eV

We have now developed lasers with pulses less than a femtosecond. But if you have pulse that is on the order of one femtosecond the precision on the energy can be as large as few eV. The faster the pulse the less the precision of energy. If you are trying to study energy very accurately you don’t want to use the fastest pulses because it hurts the accuracy of your energy measurement. Its better to stay with picosecond laser pulses for this. If on the other hand you are trying follow chemical reactions in real time then you need to go with faster and faster lasers at the expense of energy information precision. This is all a result of Heisenberg's uncertainty principle.

A vibration at 1600 cm-1 (~0.2eV) will be seen in the time domain as an oscillation with a period of ~20 fs.