7.4.4. Atomic Spectra

Table of Contents

Seeing atoms through light

Atomic spectra are the patterns of light that atoms emit or absorb. Instead of producing every possible color, an atom usually interacts with only certain specific wavelengths. This makes the spectrum look like a set of distinct lines rather than a smooth rainbow. These lines are one of the strongest clues that energy inside atoms is quantized, which means it can take only particular allowed values.

When an atom gains energy, one of its electrons can move to a higher energy state. Later, the electron may return to a lower energy state and release the extra energy as light. The light comes in packets called photons. The energy of each photon is exactly equal to the difference between the two atomic energy levels.

The basic rule of atomic spectra is
$$E_{\gamma} = E_i - E_f = hf = \frac{hc}{\lambda}$$
where $E_i$ is the initial energy level, $E_f$ is the final energy level, $h$ is Planck's constant, $f$ is frequency, and $\lambda$ is wavelength.

This is why spectral lines are so important. By measuring the wavelength of the emitted or absorbed light, we can determine the energy differences inside the atom.

Emission and absorption spectra

There are two main kinds of atomic spectra. In an emission spectrum, excited atoms give off light at certain wavelengths. In an absorption spectrum, atoms absorb certain wavelengths from light passing through them.

A hot, low density gas often produces bright spectral lines on a dark background. This is an emission spectrum. A cooler gas in front of a continuous light source often removes particular wavelengths from that light, creating dark lines in a continuous spectrum. This is an absorption spectrum.

The wavelengths involved in absorption are the same as those involved in emission for the same atom, because both depend on the same energy level differences.

Type of spectrum	What happens	Appearance
Emission spectrum	Atom emits photons	Bright lines on dark background
Absorption spectrum	Atom absorbs photons	Dark lines on bright continuous background

Why spectra are discrete

If electrons in atoms could have any energy at all, the emitted light would form a continuous range of wavelengths. But atoms do not behave that way. Electrons are restricted to certain allowed energy levels. Because of this, only certain transitions are possible, and only certain photon energies appear.

Suppose an atom has energy levels $E_1$, $E_2$, and $E_3$. Then possible emitted photons might have energies

$$E_3 - E_2,\quad E_3 - E_1,\quad E_2 - E_1$$

Each energy difference gives one spectral line. Different atoms have different sets of allowed levels, so each element has its own unique spectral fingerprint.

Spectral lines and transitions

A spectral line corresponds to one transition between two atomic states. If the electron drops from a higher level to a lower one, a photon is emitted. If the electron absorbs a photon and jumps upward, the photon energy must match the gap exactly.

This is why atoms do not absorb just any light. They absorb only photons whose energies match allowed transitions.

Rendering scientific sketch…

Atomic transitions and spectral lines

In this diagram, each downward arrow represents emission of a photon. The larger the energy drop, the larger the photon energy and the higher the frequency.

What a spectrum looks like

A spectrum is often displayed as intensity versus wavelength. Each line appears at a particular wavelength, and its brightness depends on how strongly that transition occurs and how many atoms are making it.

Rendering scientific sketch…

Simplified line spectrum

Real spectra can contain many lines, some very close together. Careful measurement of these lines gives detailed information about atomic structure.

Spectroscopy

The study of spectra is called spectroscopy. In practice, light from a source is separated by wavelength using a prism or a diffraction grating, and then the positions of the lines are measured.

Spectroscopy lets physicists identify elements even when only tiny amounts are present. Because each element has a characteristic pattern of spectral lines, observing the spectrum is like reading a barcode.

This method is especially important in astronomy. Light from stars and gas clouds can be analyzed to determine what elements are present, even though we cannot touch those objects directly.

What spectral lines tell us

Atomic spectra reveal several things at once. First, they show that atoms have internal structure with quantized energy levels. Second, they allow us to identify elements. Third, they provide numerical values for energy differences inside atoms.

If a wavelength $\lambda$ is measured, the corresponding energy difference is

$$\Delta E = \frac{hc}{\lambda}$$

Shorter wavelengths correspond to larger energy differences, and longer wavelengths correspond to smaller ones.

Important connection between wavelength and energy:
$$\Delta E = hf = \frac{hc}{\lambda}$$
So, as $\lambda$ decreases, photon energy increases.

Atomic spectra as evidence for quantization

Atomic spectra were historically very important because they showed that classical physics could not fully describe atoms. A classical picture would suggest a continuous range of possible energies, but the observed line spectra are discrete. This pointed toward the quantum description of matter.

Each observed line is evidence that only specific transitions are allowed. The regularity of these patterns led to models of the atom and later to quantum mechanics.

A simple numerical example

Suppose an atom emits a photon with wavelength $500 \, \text{nm}$, which is $500 \times 10^{-9} \, \text{m}$. The energy difference between the two levels is

$$\Delta E = \frac{hc}{\lambda}$$

Using $h = 6.63 \times 10^{-34} \, \text{J s}$ and $c = 3.00 \times 10^8 \, \text{m/s}$,

$$\Delta E = \frac{(6.63 \times 10^{-34})(3.00 \times 10^8)}{500 \times 10^{-9}}$$

$$\Delta E \approx 3.98 \times 10^{-19} \, \text{J}$$

This is the energy lost by the atom and carried away by the photon.

Continuous spectra versus line spectra

Not all light sources produce line spectra. A dense hot object, such as a glowing solid, usually produces a continuous spectrum. A low density gas of atoms often produces line spectra. This difference is important because it helps us understand the physical state of the source.

Source type	Typical spectrum
Hot dense solid or liquid	Continuous spectrum
Hot low density atomic gas	Emission line spectrum
Cool gas in front of continuous source	Absorption line spectrum

Final perspective

Atomic spectra are one of the clearest windows into the atom. They show that atoms can emit and absorb only particular photon energies, which means atomic energies are quantized. By studying the positions and strengths of spectral lines, physicists can identify elements and measure energy differences within atoms.

Key idea:
Atomic spectra consist of discrete lines because electrons in atoms can occupy only specific energy levels, and photons are emitted or absorbed only when
$$\Delta E = hf = \frac{hc}{\lambda}$$

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