5 Spectra of Alkali Metals

Contents:

1.     Spectra of Alkali Atoms : Introduction

2.     Spectrum of Na : Quantum Mechanical View

3.     Screening Constant

4.     Doublet Structure of Sodium Series Lines (Spin-Orbit interaction)

5.     Spin- orbit interaction energy for non-penetrating orbits

6.     Spin- orbit interaction energy for penetrating orbits

7.     Spectral lines in emission spectra (Intensity rules) Summary

The students will be able to learn about spectra of alkali metals and spectra of sodium element and spin- orbit interaction energy of orbits.

1. Spectra of Alkali Atoms : Introduction

The series of Alkali atoms contain Lithium (Z=3), Sodium(Z=11), Potassium(Z=19), Rubidium(Z=37) and Cesium(Z=55) and their configurations are 1s²2s,1s²2s²2p⁶3s,1s²2s²2p⁶3s²3p⁶4s, 1s²2s²2p⁶3s²3p⁶3d¹⁰4s²4p⁶5s, 1s²2s²2p⁶3s²3p⁶3d¹⁰4s²4p⁶4d¹⁰5s²5p⁶6s

This suggest that an alkali atom consists of one or more closed shell of electron in its ground state and a single valance electron in a new shell in ns orbit.

Now visualizing the two particle system with a lone (valance) electron subjected to Coulomb field of a point charge that is equivalent to proton of the diameter~ 10-15m and is thus the simplest spectra.

The aim is to solve this two-body system exactly to find the wave functions and understand the meaning of the four quantum numbers  and  .

Keeping in mind the shell model to understand the structure of the alkali spectra and It is the fact that the fine structure and magnetic field splitting are smaller than the gross structure energies by a factor of about   Fine structure and the field splitting of the transitions (lines) can be understood after the gross structure of the spectrum.

So far, on the basis of shell model, the atomic states are known.

OR valence electron of sodium is moving in a net field of  charge (due to the core of finite size) apparently like a lone (valence) electron in hydrogen atom, moves around the proton (point charge). Therefore it is assumed that the spectra of sodium atom are expected to be analogous to that of the hydrogen as in both the cases, respective valence electron moves in a net field of  but with the difference that the field is from the finite size core in former whereas from a point charge (proton) field in the later case.

The closed shell has Zero Total angular momentum and Zero spin angular momentum and designated as ¹S0

The valance electron can be excited to various s,p,d,f,…. Orbits those results in doublet terms.

2.     Spectrum of Na: Quantum Mechanical View

As the potential energy in hydrogen atom is spherically symmetric, that is, it depends only on the radial coordinate ‘r’, its solution, like for all spherically symmetric systems, may be expressed as

 

The orbital angular momentum for a completely filled shell, according to Pauli’s principle, is zero and therefore, their charge distribution is spherically symmetric. Accordingly, in the case of sodium atom (for that matter in case of all alkali atoms) charge distribution in  n=1 & 2 shells (completely filled shells) is spherically symmetric and is known as the core of the atom. Valence electron 3s is, therefore, regarded as moving in a central force field (force that depends only on distance and is completely independent of the angular position) of the core.

Therefore, quantum mechanically also spectrum of sodium is expected to be similar to that of the hydrogen atom.

Experimental Observations

The transition of the electron from one energy level to another is governed by the selection rules Λl=±1

Experimental findings reveal that emission spectra of the alkali atoms can be analyzed into four chief series with the peculiarities as given below.

First series consists of doublets; separation (in cm^-1 ) between the doublet components remains constant (say delta v_s ) as far as the series extends. Series is termed as Sharp series

This arises due to transition from 2S states to lowest 2P state. The lines are observed in visible and infrared regions and are quite narrow.

Second series consists of doublets; separation (in  cm^-1) between the doublet components decreases rapidly as the series extends to higher members. Series is termed as Principal series. This arises due to transition from 2P state to 2S state. The lines are intense and are observed in absorption spectrum as most of the atoms are in ground state. The lines of this series are in ultraviolet region except one in visible region.

Third series initially consists of triplets (three components) followed by apparent doublets; separation (in cm^-1  ) between the outer components remains constant (say delta v_d ) as far as the series extends. Series is termed as Diffuse series. This arises due to transition from 2D state to 2P state. The lines lie in the visible and infrared region. The lines of this series are diffused on one end or both the end of line.

Fourth series, termed as Fundamental series, lies in far infra red region and consists of very close lying doublets. This arises due to transition from 2Fstate to the lowest 2D state. The lines of this series are in visible and infrared regions.

Remarks:

• The Sharp, Diffuse and Fundamental series occur in Emission spectra.
• The Sharp and Diffuse series have common limit, whose wave number is equal to the lowest  term (converge to two limits corresponding to doublet components). Further, the difference between the convergence limits of sharp (or diffuse) series and the principal series is equal to wave number of the first member of the latter (Principal) series. This is refereed as Rydberg Schuster law. This forms one of the basis for analysis of atomic spectra.
• The difference between the limit of Diffuse and Fundamental series is equal to the first line of Diffuse series.
• Each series in alkali atoms converge towards shorter wavelength similar to that of hydrogen.
• In absorption (spectrum) at moderately low temperature, only Principal series is observed implying lowest term is ns and the series appears similar to the Lyman series of hydrogen atom.

The figure shows the transitions forming four chief series.

The figure depicts the s, p, d and f series corresponding to various different values of n and l. Transitions are drawn in accordance with the selection rules (that is, principal quantum number,    can  change  by  any  value  whereas   by  unity only, i.e

The alkali spectra comprises of series of lines (doublets) with successive decreasing separation and the intensity as well, in a fashion similar to that of hydrogen spectrum. The transitions forming the four chief series in alkali spectra may, therefore, be summarized as:

Principal quantum number (measure of total energy) for the hydrogen atom is shown on the right hand side of the Fig.

Sodium alkali atom

The Comparison with the hydrogen atom leads to the fact that the energy levels of sodium lie lower on the energy scale. This suggest that the kinetic energy (and therefore, velocity) of exciting (valence) electron in sodium is greater than the corresponding value of the valence electron in hydrogen atom.

On the other hand, velocity of valence electron decreases with increase of its principal quantum number,  and increases with increase of effective charge,  in the field of which valence electron moves around the nucleus. The latter dependence makes it imperative that the valence electron, in case of sodium atom, sees more than +1e charge and this is possible only if the valence electron penetrates core of the atom; that is electron orbit is elliptical.

In addition, deviation of the potential from the Coulomb potential due to a point charge, causes the term value to depend on  and smaller the value of , larger is the eccentricity Thus, the penetration increases with the decrease of  value and hence the electron experiences more nuclear charge during its penetration. Therefore, the electron moves part of time in a field of greater effective charge. Also, close proximity of the electron distorts the core leading to electrostatic polarization.

Both these effects increase the force of attraction and hence lower the total energy. Difference in energy is attributed to the various amounts of penetration into the core. Orbits that penetrate the core are called penetrating orbits and those does not are non-penetrating orbits.

Note: While describing through quantum mechanics, these effects are taken as first and second order perturbations, respectively.

Following Rydberg relation, lines of the sodium can be represented by a general formula:

Conclusions:

1.  For a given value of n , s-levels lie deepest followed by p, d and f levels. And Quantum defect, for a given value of , is a function of l.

2.  For a given value of  n, eccentricity of the elliptical orbit decreases with the increase of the value of l . That is, as  l  approaches n  , the orbit is tending to be circular.

Spectral lines may be rewritten in a general form like

3. Screening Constant

Lowering of term series is explained using the concept of quantum defect that itself measures the extent of penetration of the valence electron.

Valence electron, on penetration faces charge more than unity as it should while the electron moves well outside the atomic core (non-penetrating orbit). This has got the attention of using effective quantum number n* (<n) OR The empirical adaptation of the Balmer formula for the non-hydrogen like spectra can be made by using the effective nuclear charge, z^*, in the numerator of the relation. Thus,

exactly the orbit and the location of the electron, considering the Heisenberg Uncertainty principle.

4. Doublet Structure of Sodium Series Lines (Spin-Orbit interaction)

The splitting of the levels and hence splitting of the lines is due to spin-orbit interaction on the valence electron of alkali atom.

All the energy levels of the valence electron of alkali atoms except l=0 are splitted into two. One level corresponding top a total angular momentum J=l+ ½ and the other J=l– ½ with the second lying lower than the first.

When  measured  from  the  series  limit,  the  term  value  of  any  fine  structure component is given by T=T₀-T , where  T₀ is a hypothetical term for the center of gravity of the doublet and T gives the shift (in units of a) of the component from T₀ . Separation between the doublet components is given by the difference between their T  values.

5. Spin–Orbit interaction energy for non-penetrating orbits

The difference in the term values is attributed to polarization energy of the atomic core for the non-penetrating orbits. This deviation is understood by using the concept of quantum defect. The term values

6. Spin–Orbit interaction energy for penetrating orbits

An electron in the penetrating orbit spends part of its orbital time period inside the core. It means that the orbit may be considered to be consisting of two segments, one well outside the core and the second is within the core.

Interaction energy is calculated, first considering that the electron is completely outside the core where it experiences effective nuclear charge Z₀ ; then separately as if the orbit is completely inside the core and faces altogether different nuclear charge, say Z_i.

Both these interaction energies are combined taking time weighted contribution. As the time spent within the core is just a fraction of the time (say t) required to traverse the whole path of the orbit, one makes an assumption that time required to traverse segment outside the core, to a first approximation, is equal to t .

When the electron is well outside the core, from the above relation

7. Spectral Lines in Emission Spectra (Intensity rules)

Intensity of a spectral line is defined as a measure of the number of photons of exactly the identical energy arriving per second at a point that corresponds to the energy of the photons in the electromagnetic spectrum.

For a change ∆l ≠ 0 in the transition from one term to another in LS coupling the strongest lines are those for which ∆j has the same sign as ∆l. Of these, the intensities of the lines increase as the magnitude of j increases.

For a change ∆1≠ 0, the weakest lines are those for which ∆j and ∆l have opposite sign. The lines with ∆j = 0 are intermediate in intensity.

Let us apply these rules to fine structure transitions of First member of diffuse series. The transition is from ²D_5/2→²P_3/2, ∆j = 0 and ∆l= +1 while for ²D_3/2 →²P_1/2 transition ∆j = 1 and ∆l = +1, that is, ∆j and ∆1 have the same sign.

From this it is clear that the transition ²D_5/2→²P_3/2is stronger than ²D_3/2→²P1/2 as the former involve a larger value of j. The transition ²D_3/2→²P_3/2 is weakest as ∆j = 0 though ∆l = 1.

Quantitative rules for relative intensity from an initial level or to a final level of a multiplet is proportional to the statistical weight, 2j + 1, of that level. The constant of proportionality is common to all levels of a given multiplet.