Quantum Numbers

By solving the Schrödinger equation (HΨ = EΨ), we obtain a set of mathematical equations, called wave functions (Ψ). Each wave function is associated with an energy level E, eigenvalue of the Hamilonian operator H. The wave functions describe the spatial probability distribution of the electrons: The probability of finding an electron of a certain energy at some position in space.
A wave function for an electron in an atom is called an atomic orbital; this atomic orbital describes a region of space in which there is a high probability of finding the electron. Energy changes within an atom are the result of an electron changing from a wave pattern with a given energy to a wave pattern with a different energy (usually accompanied by the absorption or emission of a photon of light).
Each electron in an atom is described by four different quantum numbers. The first three (n, l, ml) specify the particular orbital of interest, and the fourth (ms) is related to the spin.

• n = main quantum number (1, 2, 3…). This number specifies the energy of an electron and the orbital size, it is related to the radial part of  Schrödingers equation
• l = azimuthal quantum number (0,..., n-1) (orbital angular momentum) which defines the shape of an orbital with a particular principal quantum number

• l = 0 → S-orbital; S for Sharp
• l = 1 → P-orbital; P for Principle
• l = 2 → D-orbital; D for Difuse
• l = 3 → F-orbital, F for Fundamental

• s = spin quantum number (1/2). This one specifies the orientation of the spin axis of an electron.

j = inner quantum number (l ± s) = (l ± 1/2) > 0) (total angular Momentum

m = magnetic quantum number number indicates the orbital orientation in space of a given energy (n) and shape (l)

                        m_l: +l → -l
                        m_s: +s → -s
                        m_j: +j → -j

 

Element	Be
Single Configuration	1s2 2s2
Configuration Interaction	a(1s2 2s2)+b(1s2 2p2)

Each electron in the atom is given a unique set of 4 quantum numbers. This is called the Single Configuration Approximation. Normally only the first two quantum numbers are shown as these determine the electron energy. For example, the ground state (ie the state with the lowest energy) of Helium, which has two electrons, is 1s², where 1 is the value of n, s the value of l and the superscript ² is the number of equivalent electrons.

A more complete description of the electron states involves Configuration Interaction, ie a correction is included for the distortions of the wave functions caused by the interactions between electrons. An exemple is given in the adcacent table.The ground state of Berylium, having for electrons, is not given by a single configuration, but by a mixture of (1s² 2s²) and (1s² 2p²).This configuration interaction is due to the fact that the 4 electrons do not behave exactly like four independent electrons such but the do interact. As a result, the one electron states, derived for Hydrogen, do not exactly fit the multi electron system.

In spectroscopy, an energy level is represented as ^2S+1L_J (for example ²P_3/2), where 2S+1 is the multiplicity and J the total angual momentum.

Example: LS or Russel - Sounders-coupling
In this case angular momentums (li) of the individual electrons are coupled to form the total orbital angular momentum (L) and the spins (si) of the electrons for the total spin (S). These two angular momenta then couple by spin-orbit interaction to form the total angular momentum (J) of the atom.Depding on the relative orientation of L and S the total angular momentum J may take different values :

J= ‌ L-S ‌ ; ‌ L-S ‌ +1;....; ‌ L+S ‌ 

An other coupling mechanism is the jj – coupling which can be seen at heavy atoms.In this case the orbital angular momentum and the spin of a single electron in the atom will couple to form the total angular momentum of this electron. All the total angular momenta of the electrons in the atom will then couple to form the total angular momentum of the atom.

The intensity of an emission line depends on the number of emitting atoms and the likelihood of the transition, described by its Atomic Transition Probability (ATP). An ATP depends on the angular and radial components of the wave functions involved. No exact solutions exist for multi-electron atoms. The angular components can be solved for many transitions using Racah coefficients with various schemes that approximate the coupling of the angular momenta of the electrons. The most important schemes are LS- or Russell-Saunders Coupling, jj- or Jj-Coupling, and Jl-Coupling. The radial components can be obtained using various approximations such as the Hartree-Fock approach or the hydrogenic approach of Bates and Damgaard.

References:

[1] http://en.wikipedia.org/wiki/Image:Na-lamp-3.jpg
[2] Skriptum Experimentalphysik III, Institut für Experimentalphysik, Auflage WS 2001/2002
[3] http://de.wikipedia.org/wiki/Bild:NaD-terms.png

First published on the web: 15 November 1999 by Richard Payling

Authors of the latest version (March 2008): Lara Lobo Revilla and Karl Preis . The text is based on a lecture given by Prof E.B. Steers at the first Gladnet training course in Antwerp Sept. 2007

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Radiation interaction processes and selection rules and Einstein coefficients

Atoms are excited to higher energy levels by collisions or by the absorption of radiation and can de-excite by ejecting electrons or by emitting energy as discrete quanta called photons. These photons have a characteristic wavelengths which are the inverse of the energy emitted. The optical range of wavelengths of interest in chemical analysis is 100 nm to 900 nm, from the extreme ultraviolet (UV) to the near infrared (IR). The visible light region is approximately 400 nm (blue) to 700 nm (red).. In the thermodynamic equilibrium between matter and interacting electromagnetic radiation, three basic processes have been described by A. Einstein. For electric dipole interaction the following selection rules apply for transitions between the energy levels 0 and 1.

Δ n = 0, ± 1, ± 2, ± 3,...
Δ L = ± 1
Δ J = 0, ± 1
Δ mj = 0, ± 1

Max Planck, summarizing Rutherfords and Bohrs theories, came to the conclusion that electrons turn around their nuclei, following a planetary model, in well defined orbits with discrete energy levels without emitting radiation. But they may pass from an orbit to another one emit light with frequency:

(1.0)

Einstein, in 1917, gave a great contribution to “quantum optics”, deriving Plancks blackbody law from thermo dynamical considerations started by Planck. He described the number of transitions the atoms make per second between an excited state and an another state with less of energy. He also introduced rate constants, now called Einstein coefficients related to three different processes: spontaneous emission, photoabsorption and stimulated emission.

Spontaneous emission: this process occur when the electron is in its upper state (E₂) and no photon present: it can emit a photon spontaneously.

• Scheme of atomic spontaneous emission –

Let us consider an atom in an excited state 2 of energy E₂ can in general make a spontaneous radiative transition to a state 1 of lower energy E₁, with emission of a photon of energy:

(1.1)

corresponding to a spectrum line of wavenumber:

(1.2)

We shall denote by a₂₁ the probability per unit time that an atom in state 2 will make such a transition to the state 1.
For an isolated, field-free atom in a state with total angular momentum J_i, there are:

(1.3)

degenerate quantum states of energy E_i , corresponding to the 2J_i + 1 possible values of the magnetic quantum number Mi.
The Einstein spontaneous emission transition probability rate is defined to be the total probability per unit time of an atom in a specific state j making a transition to any of the g_i states of energy level i:

(1.4)

If at time t there are N₂(t) atoms in state 2, the rate of change of N2 due to spontaneous transitions to all states of the level 1 is :

(1.5)

Atomic absorption or Photoabsorption: this process occurs when the electron is in its lower level (E₁) and n photons are present: it can absorb a photon and make a transition to its upper level (E₂).

• Scheme of photoabsorption-

We can say that transitions may not only occur spontaneously, but may also be induced by the presence of a radiation field. We assume this radiation field to be isotropic and unpolarized and to have energy per unit volume of r(σ) dσ in the wavenumber range dσ. The Einstein coefficients of absorption B₁₂ and of stimulated emission B₂₁ are defined as follow: if r(σ) is essentially constant over the profile of the spectrum line, then absorption by atoms in a state 1 results in transition to states of the level 2 at a rate:

(1.6)

Stimulated emission: this process was invented by A. Einstein for symmetry reasons and to satisfy Plancks blackbody equation. It occur when the electron is in its upper level (E₂) and an electromagnetic radiation at (or near) the same frequency is present: it can emit an additional photon by stimulated emission decaying to the lower energy level (E₁).

- Scheme of atomic stimulated emission-

and atoms in a state 2 are stimulated (or induced) to make radiative transitions to state of the level 1 at the rate.

(1.7)

Values of the three Einstein coefficients are not independent and their mutual relationship may be inferred as follow. We suppose the radiation field and the atoms to be in mutual thermodynamic equilibrium at temperature T. The radiation energy density per unit wavenumber interval is given by Planck’s law

(1.8)

and the relative numbers of atoms in different quantum states are given by the Maxwell-Boltzmann law

(1.9)

According to the law of Detailed Balance, the rate of transition from all states of level 1 to all states of level 2 due to absorption from the radiation field must be equal to the rate of spontaneous plus induced emission from level 2 to level 1:

(1.10)

dividing by N₂ and using (1.9) we obtain

(1.11)

which by comparison with (1.8) implies

g1B12 = g2B21	(1.12)
	(1.13)

Thanks to relation (1.12), we shall drop the subscripts and refer simply to g_A and g_B because g is always the statistical weight of the initial level, i.e., the upper level for emission and the lower level for absorption.
The Einstein transition probabilities are intrinsic physical properties of the atom depending only on the initial and final states, on the intensity of any incident light, how strongly is the interaction between light and atoms and are independent of whether or not a state of thermodynamic equilibrium actually exists.

References:

• H. Haken, Light , North-Holland Physics Publishing, 1981.
• Robert D. Cowan, The Theory of Atomic Structure and Spectra, University of California Press, 1981.
• B. L. van der Waerden, Sources of Quantum Mechanics, Dover Publications, New York, 1968.

First published on the web: 15 November 1999 by Richard Payling

Author of the latest version (May 2008): Elisa Barisone from EMPA in Thun, Switzerland.

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Atomic Spectroscopy – How to describe a line spectrum

G. Kirchhoff, R.W. Bunsen (1859): Every element has its characteristic line spectrum.

If you have a light source (e.g. mercury vapour lamp) and an instrument to separate the light into different wavelengths (e.g. a grating spectrograph), than you will see hopefully a line spectrum. The emitted light comes from electron transitions between different energy levels in the atom.
At the beginning of the 20th century physicists tried to understand this process. First they looked at an atom with only one electron (hydrogen) and made empirical formulas to describe the line series of the hydrogen line spectra (Lyman, Balmer, Paschen, Brakett, Pfund).
Rydberg reformulated Balmer’s formula and found an equation which works also for the other series.

[4] Hydrogen energy levels

The “n” in those formulas is the so called “control variable” Later it turned out to be the principal quantum number.

1911 postulated Rutherford his atomic model in which he said the electron moves around the nucleus (the positive charge being positioned in the centre of the atom) on a circular path.

After this, Niels Bohr postulated in the year 1913:

1. The electron is able to move around the atomic nuclei on specific stable orbits without loosing its energy. With the orbital angular momentum
   r… orbit radius, m…electron mass, ω  … angular frequency,
   
2. If the electron moves from an orbit with high energy to an orbit with lower energy, energy will be emitted.

λ … wavelength
ν … frequency
n… principle quantum number
h … Planck’ s constant (= 6,626 · 10^-34 Js = 4,136 · 10^-15 eVs)
c … speed of light (= 2,998 · 10⁸ m/s)

This model is called the “Bohr’s model”. With this model it is possible to describe the energy of an electron in the coulomb field of the nucleus. And so it is also possible to describe the energy difference (wavelength) of a transition from a higher to a lower energy level (orbit). The energy of an electron moving in the Coulomb field of the nucleus of Z elemental charge units is given by.

 Z … charge of the nucleus (for hydrogen Z=1)
 ε₀ … dielectric coefficient

If the electron moves from orbit b (nb) to orbit a (na) with (nb>na), t hen the frequency of the emitted light is given by:     

Bohr’s theory gives the right values for a one electron system, but without fine structure. Fine structure means that the spectral lines are consisting of several lines (depending on angular momentum and spin). This was the classical approach to describe the energy levels in the atom.
Sommerfeld extended the theory of Bohr and he said that the electron is moving on elliptical orbits around the nucleus. He needed two quantum numbers to describe the energy levels (primary (n) and secondary quantum number (l)).
But this wasn’t satisfying because:

• it is a half classical theory - nearly without quantum mechanics, in particular it does not solve a serious problem. In classical electrodynamics, an electron moving on circles performs an accelerated movement, and consequently must emit radiation and loose its energy;
• the atoms are considered as flat, tiny circles or ellipses;
• the theory works without the electron spin so that the fine structure is false;
• it doesn’t work for other atoms than hydrogen.

           So the solution of all this gave Quantum mechanics (Schrödinger and Heisenberg). The Schrödinger equation for an atom with one electron gives the right energy values for the electron.

After solving the equation you will get for every wave function (depends on n and l) an Eigen value (energy level). 
Experiments (Stern, Gerlach 1921) also have shown that the electron has a spin (magnetic moment).  The orbital angular momentum l of the electron interacts with the electron spin s (ms = ± 1/2). So that the solution of the problem is given by (with fine structure):

α is the so called fine structure constant that arises in the Sommerfeld fine structure formula in which the motion [m = m(v)] of the electron is relativistic treated. 

Now it is obvious that the energy of an electron in a field of the nucleous depends on 3 quantum numbers (n, l, and s). 

First published on the web: 11 March 2008

Authors : Lara Lobo Revilla and Karl Preis . The text is based on a lecture given by Prof E.B. Steers at the first Gladnet training course in Antwerp Sept. 2007

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Hydrogen like atoms : Sodium D line

You can treat sodium as an “one electron” system because of the closed noble gas body “under” the working electron (You will see later!).  Other atoms than hydrogen have more electrons around the nucleus. So the problem is now to solve the Schrödinger equation for more than one electron. The solution is only as approximation available. But the main thing is that the spins and the angular momentum of the electrons interact with each other (depending on the element).

Sodium has only one more electron than a noble gas. Because noble gases have a ”closed” shell, sodium can be considered as a one electron system. Therefore the overall sum of the angular momentums and the spins of the noble gas body is zero. So the working electron is the outside one. Because of that, the sodium spectra looks similar to the hydrogen spectra.
In table 1 you can see the picture of a sodium high pressure lamp which is very often used as street lighting. The light of sodium lamp is dominated by the very bright orange yellow light which comes from the transition.
2S1/2 – 2P1/2 (589.6 nm) and 2S1/2 – 2P3/2 (589.0 nm). 
Note: there are no transitions with n=1, 2 because they are used in the noble gas body (Neon). There are transitions involving electrons in inner shells, but the energies involved are higher, and so the  radiation (absorbed or emitted) is not visible. The term scheme of sodium is shown in table 2.  [2]

Sodium high pressure lamp and its spectrum [1]

The yellow Sodium-“D” Lines (fine structure)[3]

References:

[1] http://en.wikipedia.org/wiki/Image:Na-lamp-3.jpg
[2] Skriptum Experimentalphysik III, Institut für Experimentalphysik, Auflage WS 2001/2002
[3] http://de.wikipedia.org/wiki/Bild:NaD-terms.png

First published on the web: 15 November 1999 by Richard Payling

Authors of the latest version (March 2008): Lara Lobo Revilla and Karl Preis . The text is based on a lecture given by Prof E.B. Steers at the first Gladnet training course in Antwerp Sept. 2007

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