Topic name: Atomic spectra,selection rules,energy level diagram of atomic sodium

Atomic spectra

When an atom absorb energy, it get excited from ground state to excited state, the energy absorbed by the atom form absorption spectra while energy released by the atom on returning to ground state form emission spectra.

The range of characteristic frequencies of electromagnetic radiation that are readily absorbed and emitted by an atom. The atomic spectrum is an effect of the quantized orbits of electrons around the atom. An electron can jump from one fixed orbital to another: if the orbital it jumps to has a higher energy, the electron must absorb a photon of a certain frequency; if it is of a lower energy, it must give off a photon of a certain frequency. The frequency depends on the difference in energy between the orbitals. Explaining this phenomenon was crucial to the development of quantum mechanics. The atomic spectrum of each chemical element is unique and is largely responsible for the color of matter. Atomic spectra can also be analyzed to determine the composition of objects, such as stars, that are far away.

Atomic spectroscopy is the determination of elemental composition by using the Electromagnetic Spectrum or Mass Spectrum, respectively to optical or mass spectroscopy. It can be divided by atomization source or by the type of spectroscopy used. In the latter case, the main division is between optical and mass spectrometry. Mass spectrometry generally gives significantly better analytical performance, but is also significantly more complex. This complexity translates into higher purchase costs, higher operational costs, more operator training, and a greater number of components that can potentially fail. Because optical spectroscopy is often less expensive and has performance adequate for many tasks, it is far more common.Atomic absorption spectrometers are one of the most commonly sold and used analytical devices.

When atoms are excited they emit light of certain wavelengths which correspond to different colors. The emitted light can be observed as a series of colored lines with dark spaces in between; this series of colored lines is called a line or atomic spectra. Each element produces a unique set of spectral lines. Since no two

elements emit the same spectral lines, elements can be identified by their line spectrum.

1885 - Johann Jacob Balmer

Analyzed the hydrogen spectrum and found that hydrogen emitted four bands of lightwithin the visible spectrum:

Wavelength (nm)     Color

656.2 red486.1 blue434.0 blue-violet410.1 violet

Balmer found that the data fit to the following equation:

= wavelength (nm) RH = Rydberg's constant = 1.09678 x 10-2 nm-1

n1 = the lower energy level n2 = the higher energy level

For example, to calculate the wavelength of light emitted when the electron in a hydrogen atom falls from the fourth energy level to the second energy level:

  Each series is named after its discoverer.

The Lyman series is the wavelengths inthe ultra violet (UV) spectrum of thehydrogen atom, resulting from electronsdropping from higher energy levels intothe n = 1 orbit.

The Balmer series is the wavelengths inthe visible light spectrum of the hydrogenatom, resulting from electrons falling fromhigher energy levels into the n = 2 orbit.

The Paschen series is the wavelengths in the infrared spectrum of the hydrogenatom, resulting from electrons falling fromhigher energy levels into the n = 3 orbit.

The Brackett series is the wavelengths in the infrared spectrum of the hydrogenatom, resulting from electrons falling fromhigher energy levels

into the n = 4 orbit. The Pfund series is

the wavelengths in the infrared spectrum of the hydrogenatom, resulting from electrons falling fromhigher energy levels into the n = 5 orbit.

How Atoms React when Struck by Light

Electrons can only exist in certain areas around the nucleus called shells. Each shell corresponds to a specific energy level which is designated by a quantum number n. Since electrons cannot exist between energy levels, the quantum number n is always an integer value (n=1,2,3,4…). The electron with the lowest energy

level (n=1) is the closest to the nucleus. An electron occupying its lowest energy level is said to be in the ground state. The energy of an electron in a certain energy level can be found by the equation:

En = -RH/n2 (equation 3)

Where RH is a constant equal to 2.179 x 10 -18 J and n is equal to the energy level of the electron.

When light is shone on an atom, its electrons absorb photons which cause them to gain energy and jump to higher energy levels. The higher the energy of the photon absorbed, the higher the energy level the electron jumps to. Similarly, an electron can go down energy levels by emitting a photon. The simplified version of this principal is illustrated in the figure to the left based on the Bohr model of the Hydrogen atom. The energy of the photon emitted or gained by an electron can be calculated from this formula:

Ephoton =RH (1/ni2 – 1/nf

2) (equation 4)

Where ni is the initial energy level of the electron and n f is the final energy level of the electron.

The frequency of the photon emitted when an electron descends energy levels can be found using the formula:

νphoton = (Ei - Ef)/h (equation 5)

Where Ei is the initial energy of the electron and Ef is the final energy of the electron.

Since an electron can only exist at certain energy levels, they can only emit photons of certain frequencies. These specific frequencies of light are then observed as spectral lines. Similarly, a photon has to be of the exact wavelength the electron needs to jump energy levels in order to be absorbed, explaining the dark bands of an absorption spectra.

Emission Lines

As discussed above, when an electron falls from one energy level in an atom to a lower energy level, it emits a photon of a particular wavelength and energy. When many electrons emit the same wavelength of photons it will result in a spike in the spectrum at this particular wavelength, resulting in the banding pattern seen in atomic emission spectra. The graphic to the right is a simplified picture of

a spectrograph, in this case being used to photograph the spectral lines of Hydrogen. In this spectrograph, the Hydrogen atoms inside the lamp are being excited by an electric current. The light from the lamp then passes through a prism, which diffracts it into its different frequencies. Since the frequencies of light correspond to certain energy levels (n) it is therefore possible to predict the frequencies of the spectral lines of Hydrogen using an equation discovered by Johann Balmer.

ν= 3.2881 x 1015s-1 (1/22 - 1/n2) (equation 6)

Where n must be a number greater than 2. This is because Balmer’s formula only applies to visible light and some longer wavelengths of ultraviolet.

The frequencies in this region of Hydrogen’s atomic spectra are called the Balmer series. The Balmer series for Hydrogen is pictured to the left. There are several other series in the Hydrogen atom which correspond to different parts of the electromagnetic spectrum. The Lyman series, for example, extends into the ultraviolet, and therefore can be used to calculate the energy of to n=1.

Absorption Lines

When an electron jumps from a low energy level to a higher level, the electron will absorb a photon of a particular wavelength. This will show up as a drop in the number of photons of this wavelength and

as a black band in this part of the spectrum. The figure to the right illustrates a mechanism to detect an absorption spectrum. A white light is shone through a sample. The atoms in the sample absorb some of the light, exciting their electrons. Since the electrons only absorb light of certain frequencies, the absorption spectrum will show up as a series of black bands on an otherwise continuous spectrum.

Selection Rules for Electronic Transitions

In spectral phenomena such as the Zeeman effect it becomes evident that transitions are not observed between all pairs of energy levels. Some transitions are "forbidden" ( i.e., highly improbable) while others are "allowed" by a set of selection rules. The number of split components observed in the Zeeman effect is consistent with the selection rules:

These are the selection rules for an electric dipole transition. One can say that the oscillating electric field associated with the transitions resembles an oscillating electric dipole. When this is expressed in quantum terms, photon emission is always accompanied by a change of 1 in the orbital angular momentum quantum number. The magnetic quantum number can change by zero or one unit.

Another approach to the selection rules is to note that any electron transition which involves the emission of a photon must involve a change of 1 in the angular momentum. The photon is said to have an intrinsic angular momentum or "spin" of one, so that conservation of angular momentum in photon emission requires a change of 1 in the atom's angular momentum. The electron spin quantum numberdoes not change in such transitions, so an additional selection rule is:

The total angular momentum may change be either zero or one:

An exception to this last selection rule it that you cannot have a transition from j=0 to j=0; i.e., since the vector angular momentum must change by one unit in a electronic transition, j=0 -> 0 can't happen because there is no total angular momentum to re-orient to get a change of 1

Zeeman Effect in Hydrogen

When an external magnetic field is applied, sharp spectral lines like the n=3→ 2transition of hydrogen split into multiple closely spaced lines. First observed by Pieter Zeeman, this splitting is attributed to the interaction between the magnetic field and the magnetic dipole moment associated with the orbital angular momentum. In the absence of the magnetic field, the hydrogen energies depend only upon the principal quantum number n, and the emissions occur at a single wavelength.

Note that the transitions shown follow the selection rule which does not allow a change of more than one unit in the quantum number ml.

"Anomalous" Zeeman Effect

While the Zeeman effect in some atoms (e.g., hydrogen) showed the expected equally-spaced triplet, in other atoms the magnetic field split the lines into four, six, or even more lines and some triplets showed wider spacings than expected. These deviations were labeled the "anomalous Zeeman effect" and were very puzzling to early researchers. The explanation of these different patterns of splitting gave additional insight into the effects of electron spin. With the inclusion of electron spin in the total angular momentum, the other types of multiplets formed part of a consistent picture. So what has been historically called the "anomalous" Zeeman effect is really the normal Zeeman effect when electron spin is included.

"Normal" Zeeman effect

This type of splitting is observed with hydrogen and the zinc singlet.

This type of splitting is observed for spin 0 states since the spin does not contribute to the angular momentum.

"Anomalous" Zeeman effect

When electron spin is included, there is a greater variety of splitting patterns.

Paschen-Back Effect

In the presense of an external magnetic field, the energy levels of atoms are split. This splitting is described well by the Zeeman effect if the splitting is small compared to the energy difference between the unperturbed levels, i.e., for sufficiently weak magnetic fields. This can be visualized with the help of a vector

model of total angular momentum. If the magnetic field is large enough, it disrupts the coupling between the orbital and spin angular momenta, resulting in a different pattern of splitting. This effect is called the Paschen-Back effect.

In the weak field case the vector model at left implies that the coupling of the orbital angular momentum L to the spin angular momentum S is stronger than their coupling to the external field. In this case where spin-orbit coupling is dominant, they can be visualized as combining to form a total angular momentum J which then precesses about the magnetic field direction.

In the strong-field case, S and L couple more strongly to the external magnetic field than to each other, and can be visualized as independently precessing about the external field direction.

Energy level diagram for sodium atom:

You would observe an intense signal at 589 nm in the emission of the sodium atom. That in fact consists of a pair of lines at 589 and 589.6 nm that could not be resolved since of instrumental limitations. This pair of lines originates from the relaxation of the excited electron in 3p to the 3s level and is responsible for the features yellow glow of the sodium light.

The origin of the signals could be rationalised in terms of the energy level diagram, as given in Figure.

Figure: The energy level diagram for sodium atom

As you have learnt above, within case of atomic emission spectroscopy, an intensity of emission signal depends upon a number of atoms within the excited

state. In case of Na the ground state is 2S1/2 (3s) and the excited states are 2P3/2, 2P1/2 (4p), 2P3/2, 2P1/2 (3p), 2D5/2, 2D3/2 (3d), 2S1/2 (4s); and many more. The intensity of the emission although the excited atoms return to the ground state will be maximum from the first excited state, if this transition is permitted (?s = 0, ?l = ±1). As in case of sodium, 2P3/2 and 2P1/2 states would be most populated and transition from this state to ground state is allowed (as it meets the selection rule given above, that is (?s = 0, ?l = ±1)); the intensity of emission from this state would be maximum. The fraction of atoms within the excited states (corresponding to the prominent spectral line) of different elements at several temperatures.

The Sodium Doublet

The well known bright doublet which is responsible for the bright yellow light from a sodium lamp may be used to demonstrate several of the influences which cause splitting of the emission lines of atomic spectra. The transition which gives rise to the doublet is from the 3p to the 3s level, levels which would be the same in the hydrogen atom. The fact that the 3s (orbital quantum number = 0) is lower than the 3p (l=1) is a good example of thedependence of atomic energy levels on angular momentum. The 3s electron penetrates the 1s shell more and is less effectively shielded than the 3p electron, so the 3s level is lower (more tightly bound). The fact that there is a doublet shows the smaller dependence of the atomic energy levels on the total angular momentum . The 3p level is split into states with total angular momentum j=3/2 and

j=1/2 by the magnetic energy of the electron spin in the presence of the internal magnetic field caused by the orbital motion. This effect is called the spin-orbit effect. In the presence of an additional externally applied magnetic field, these levels are further split by the magnetic interaction, showing dependence of the energies on the z-component of the total angular momentum. This splitting gives the Zeeman effect for sodium.

The magnitude of the spin-orbit interaction has the form In the case zB = BSzLz.

of the sodium doublet, the difference in energy for the 3p3/2 and 3p1/2 comes from a change of 1 unit in the spin orientation with the orbital part presumed to be the same. The change in energy is of the form

E = BgB = 0.0021 eVzB = BSzLz.

where B is the Bohr magneton and g is the electron spin g-factor with value very close to 2. This gives an estimate of the internal magnetic field needed to produce the observed splitting:

BgB = (5.79 x 10-5 eV/T)2B = 0.0021 eV

B = 18 Tesla

This is a very large magnetic field by laboratory standards. Large magnets with dimensions over a meter, used for NMR and ESR experiments, have magnetic fields on the order of a Tesla.

Sodium Spectrum

The sodium spectrum is dominated by the bright doublet known as the Sodium D-lines at 588.9950 and 589.5924 nanometers. From the energy level diagram it can be seen that these lines are emitted in a transition from the 3p to the 3s levels. The line at 589.0 has twice the intensity of the line at 589.6 nm. Taking the range from 400-700nm as the nominal visible range, the strongest visible line other than the D-lines is the line at 568.8205 which has an intensity about 0.7% of that of the strongest line. All other lines are a factor of two or more fainter than that one, so for most practical purposes, all the light from luminous sodium comes from the D-lines.

The illustration at left shows the interference pattern formed by the sodium doublet in aFabry-Perotinterferometer. At right is a sketch of the origin of the sodium doublet.

Sodium Energy Levels:Orbital Dependence

The sodium 3s level is significantly lower than the 3p because of greater penetration past the shielding of the 1s electron.

Both levels penetrate enough to be significantly lower than the n=3 hydrogen energy which they would have if the shielding were perfect.

What Causes Electron Energies to Depend Upon the Orbital Quantum Number?

From the Bohr model or the hydrogen Schrodinger equation, the solution for the electron energy levels gives:

This fits the hydrogen spectrum unless you take a high resolution look at fine structureor the structure produced by external magnetic fields (Zeeman effect), etc.

Hydrogen-like atoms such as lithium and sodium might be expected to exhibit similar energy levels. They consist of closed shells with a single electron outside. Envisioning a Bohr-type shell structure with just a single electron in the outer shell, the net charge inside that shell is just one net positive charge. This leads to the following expectation:

However, when data from spectra are used to build energy level diagrams for these atoms, a strong orbital dependence of the energy is found for the electrons of low angular momentum as shown below.

References

• Donald A McQuarrie,Quantum Chemistry,Viva Books Private Limited,First Edition.

• Atkins P.W,Physical Chemistry,Oxford University Press,sixth edition,1998.

• web.mst.edu/~tbone/.../Atomic_spectra.ppt_files/Atomic_spectra.ppt.ppt

• hyperphysics.phy-astr.gsu.edu/hbase/atomic/grotrian.html en.wikipedia.org/wiki/Selection_rule

• chemwiki.ucdavis.edu/Physical_Chemistry/.../Selection_Rules