Atomic Spectra notes

MDCAT Physics

This chapter explains the origin of atomic spectra, the structure of hydrogen energy levels, Bohr’s model, de Broglie waves, spectral series, lasers and X-rays. It also relates wavelength, frequency and photon energy to electronic transitions in atoms.

Atomic Spectra and Electromagnetic Radiation

An atomic spectrum is the pattern of radiation emitted or absorbed by an atom. When the spectrum is viewed through a spectroscope, it consists of definite lines rather than a continuous range of colours. Each element has a characteristic spectrum because its electrons can occupy only certain energy levels.

A spectral line represents a definite amount of absorbed or emitted energy. The pattern of spectral lines acts like a fingerprint of the element. Emission spectra are produced when excited atoms release energy, while absorption spectra are produced when atoms absorb selected wavelengths from continuous radiation.

A spectrum is the arrangement of radiation according to wavelength or frequency.
An emission spectrum consists of bright lines on a dark background.
An absorption spectrum consists of dark lines in a continuous spectrum.
Spectral lines are like a fingerprint pattern of absorbed or emitted energy.
The energy of a photon is E = hf = hc/λ.
A photon with higher frequency has greater energy and shorter wavelength.
During an electronic transition, the atom emits or absorbs a photon, not a y-ray.
The frequency of emitted radiation is f = ΔE/h, where ΔE is the energy difference between two levels.

Rutherford Model and Its Limitations

Rutherford’s nuclear model showed that an atom has a small, dense, positively charged nucleus. Electrons revolve around the nucleus. However, according to classical electromagnetic theory, a revolving charged particle should continuously radiate energy.

As an electron loses energy, its orbit should become smaller and the electron should fall into the nucleus. This does not happen in stable atoms. Rutherford’s model therefore could not explain the stability and behaviour of electrons or the line spectra of atoms.

The nucleus contains almost all the mass of an atom and has positive charge.
Electrons are negatively charged and are present outside the nucleus.
A charged particle moving in a circular path is accelerating, even if its speed is constant.
According to classical theory, an accelerating electron should emit radiation continuously.
Rutherford’s model could not explain why electrons do not spiral into the nucleus.
Rutherford’s model could not explain the discrete line spectrum of hydrogen.
Bohr improved Rutherford’s model by introducing fixed energy levels.

Bohr’s Atomic Model

Bohr proposed that electrons move around the nucleus in certain permitted circular orbits called stationary orbits. An electron in a stationary orbit does not radiate energy. Each orbit has a definite energy and is represented by the principal quantum number n.

Radiation is emitted or absorbed only when an electron changes from one permitted orbit to another. A downward transition releases a photon. An upward transition requires absorption of energy.

The allowed orbits are called stationary states or energy levels.
The principal quantum number n has values 1, 2, 3, and so on.
The first orbit, n = 1, is the ground state.
The higher orbits, n = 2, 3, 4 and so on, are excited states.
Angular momentum is quantized: mvr = nh/2π.
For hydrogen-like atoms, the radius of the nth orbit is rn = n2r1.
For hydrogen, the speed in the first Bohr orbit is 2.19 × 10^6 m s-1.
The speed of an electron decreases as the principal quantum number increases.
A transition from a higher level to a lower level releases energy.
A transition from a lower level to a higher level absorbs energy.

Energy Levels of Hydrogen Atom

The energy of an electron in the nth orbit of a hydrogen atom is En = -13.6/n2 eV. The negative sign shows that the electron is bound to the nucleus. The energy becomes zero when the electron is completely free and infinitely far from the nucleus.

The ground state of hydrogen has n = 1 and energy -13.6 eV. To ionize hydrogen from the ground state, 13.6 eV must be supplied. From an excited level, less energy is needed because the electron is already farther from the nucleus.

Energy of hydrogen electron: En = -13.6/n2 eV.
Ground-state energy of hydrogen is -13.6 eV.
For n = 2, the energy is -3.4 eV.
For n = 3, the energy is approximately -1.51 eV.
Ionization means removing an electron completely from the atom.
Ionization energy from the ground state of hydrogen is 13.6 eV.
Energy required to remove the electron from n = 2 is 3.4 eV.
The energy difference between two levels is ΔE = Eupper - Elower.
A transition to a lower energy level gives a negative change in the atom’s energy, but the emitted photon has positive energy.
As n increases, adjacent energy levels become closer together.
At n = infinity, the electron is free and its energy is taken as zero.

de Broglie Waves and Matter Waves

Louis de Broglie proposed that moving matter has wave-like properties. The wavelength associated with a moving particle is called the de Broglie wavelength. It is given by λ = h/p = h/mv for a non-relativistic particle.

The wavelength is appreciable for very small particles such as electrons. For large objects, the wavelength is extremely small and cannot normally be observed. In atomic physics, the wave nature of electrons helps explain why only certain orbits are allowed.

de Broglie wavelength is λ = h/mv.
Momentum p is equal to mv for a non-relativistic particle.
The de Broglie wavelength decreases when momentum increases.
Matter waves are associated with moving particles.
In the elementary atomic-spectrum treatment, moving charged particles such as electrons are considered in relation to de Broglie waves.
A stationary orbit is allowed when the circumference contains an integral number of de Broglie wavelengths: 2πr = nλ.
The wave nature of electrons supports the quantization of Bohr orbits.
The value of Planck’s constant is h = 6.626 × 10-34 J s.

Hydrogen Spectrum and Spectral Series

The hydrogen spectrum contains several series of lines. Each series is produced when electrons from higher levels fall to a particular lower level. The wavelength of a hydrogen spectral line is calculated by the Rydberg formula.

The general formula is 1/λ = RH(1/n1^2 - 1/n2^2), where n2 is greater than n1. Here RH is the Rydberg constant, approximately 1.097 × 10^7 m-1. The series is identified by the final orbit, not by the initial orbit.

Lyman series: all transitions end at n = 1 and lie in the ultraviolet region.
Balmer series: all transitions end at n = 2 and lie mainly in the visible region.
Paschen series: all transitions end at n = 3 and lie in the infrared region.
Brackett series: all transitions end at n = 4 and lie in the infrared region.
Pfund series: all transitions end at n = 5 and lie in the infrared region.
For the Paschen series, 1/λ = RH(1/3^2 - 1/n2), where n = 4, 5, 6 and so on.
The Balmer series is visible because every transition ends at n = 2.
The longest wavelength in a series is produced by the smallest energy transition.
The shortest wavelength in a series is produced when the electron falls from n = infinity to the final level.
For the longest wavelength of the Paschen series, the transition is n = 4 to n = 3.
The longest Paschen wavelength is approximately 1.875 × 10-6 m, or 0.000001875 m.
The highest frequency in the Lyman series is produced by the transition from n = infinity to n = 1.
Higher frequency means greater photon energy and shorter wavelength.

Wavelength Ratios and Electronic Transitions

The wavelength of a spectral line depends on the energy difference between the two levels. In the Balmer series, the longest wavelength is produced by the transition from n = 3 to n = 2. The shortest wavelength is produced by the limiting transition from n = infinity to n = 2.

Using the Rydberg formula, the ratio of minimum wavelength to maximum wavelength in the Balmer series is 5:9. If the ratio is requested in the reverse order, maximum wavelength to minimum wavelength, it is 9:5.

Balmer longest wavelength: transition 3 to 2.
Balmer shortest wavelength: transition infinity to 2.
Minimum wavelength to maximum wavelength in the Balmer series is 5:9.
Maximum wavelength to minimum wavelength in the Balmer series is 9:5.
A transition from the 3rd orbit to the 1st orbit releases energy.
A transition from a higher orbit to a lower orbit produces an emission line.
A transition from a lower orbit to a higher orbit produces an absorption line.
A larger energy difference produces higher frequency and shorter wavelength.
A smaller energy difference produces lower frequency and longer wavelength.
A red line has lower frequency and energy than a blue line when both are in the visible region.

Lasers and Population Inversion

LASER means Light Amplification by Stimulated Emission of Radiation. In a laser, atoms are first raised to excited states. Under suitable conditions, an incoming photon causes an excited atom to emit a second photon having the same frequency, phase, direction and polarization.

For laser action, more atoms must be present in the excited state E2 than in the lower state E1. This condition is called population inversion. A metastable state helps atoms remain excited long enough for stimulated emission to occur.

Population inversion occurs when the number of atoms in E2 is greater than the number in E1.
Stimulated emission produces photons identical to the stimulating photons.
Laser light is highly monochromatic, meaning it has a narrow range of frequency.
Laser light is coherent, meaning its waves have a constant phase relationship.
Laser light is highly directional and has very little divergence.
Spontaneous emission occurs without an external photon and produces random-phase radiation.
Stimulated emission is the basic process responsible for laser amplification.
The three basic processes are absorption, spontaneous emission and stimulated emission.

X-Rays and Their Spectra

X-rays are electromagnetic radiations with very short wavelengths and high frequencies. In an X-ray tube, electrons are accelerated through a high potential difference and strike a metal target. Their sudden deceleration produces continuous X-rays, while inner-shell electronic transitions produce characteristic X-rays.

A target material must withstand intense heating and produce useful radiation. A high atomic number target is suitable because it gives more intense characteristic X-rays. Tungsten is commonly used as an X-ray tube target because of its high melting point and suitable atomic number.

X-rays are produced when high-speed electrons bombard a metal target.
The characteristic X-ray spectrum is due to bombardment of the target by electrons followed by inner-shell transitions.
Continuous X-rays are produced by the rapid deceleration of incident electrons near target nuclei.
Characteristic X-rays have definite wavelengths that depend on the target element.
A target for an X-ray tube should have a high atomic number.
The target should also have a high melting point to tolerate heating.
When an electron fills a vacancy in the K shell, a K-series photon may be emitted.
Kα radiation is produced when an electron falls from the L shell to the K shell.
Kβ radiation is produced when an electron falls from the M shell to the K shell.
Kα is generally more intense than the other characteristic K-series lines.
In the usual naming of characteristic series, K radiation has shorter wavelength than L radiation, and L radiation has shorter wavelength than M radiation.
An M-series X-ray photon has a longer wavelength than a K-series or L-series photon from the same naming scheme.

Key terms

Atomic spectrum: The pattern of wavelengths or frequencies of radiation emitted or absorbed by an atom.
Emission spectrum: A spectrum of bright lines produced when excited atoms emit radiation.
Absorption spectrum: A spectrum containing dark lines formed when atoms absorb selected wavelengths.
Spectral line: A definite wavelength produced by a particular electronic transition.
Photon: A packet of electromagnetic radiation having energy E = hf.
Stationary orbit: An allowed Bohr orbit in which an electron does not continuously radiate energy.
Ground state: The lowest energy state of an atom, corresponding to n = 1 for hydrogen.
Excited state: Any state of an atom with an electron at a higher energy level than the ground state.
Ionization: The process of removing an electron completely from an atom.
de Broglie wavelength: The wavelength associated with a moving particle, given by λ = h/mv.
Rydberg constant: The constant RH used in the hydrogen spectral formula, approximately 1.097 × 10^7 m-1.
Lyman series: The hydrogen spectral series produced by transitions ending at n = 1.
Balmer series: The hydrogen spectral series produced by transitions ending at n = 2.
Paschen series: The hydrogen spectral series produced by transitions ending at n = 3.
Population inversion: The condition in which more atoms occupy an excited laser level than a lower level.
Stimulated emission: Emission caused by an incoming photon that produces an identical photon.
Characteristic X-rays: X-rays of definite wavelengths produced by inner-shell electronic transitions in a target atom.
Kα radiation: Characteristic X-radiation produced when an electron falls from the L shell to the K shell.

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Comparing distance and displacement

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