#### Line Profiles, Physics tutorial

Transition:

Transition takes place when electron jumps from one energy level to another. When transition happens, a spectral line is produced. In case of strong spectral lines, source function may depart considerably from Planck's function, resembling more direction averaged mean intensity Jv that is provided by:

Jv = 1/4π∫Iv

For such spectral lines, replace conditions of local thermodynamic equilibrium (LTE) by supposition of statistically steady state in every atomic energy level. Spectral line is produced whenever electron jumps from the higher energy level to lower energy level. On the other hand, absorption line is produced when electron jumps from lower energy level to higher energy level. Consider the radiating atom. Assume that a line is created by transition between two bound levels i and j , where j > i. Let k signify continuum. For the optically thin emission line (that is a case where τv is small), emergent intensity Iji of such an emission line (after passing through a plasma) is given by

Iji = njAijvjidV

Where nj is population of upper atomic level from where electrons jump to lower level i during emission of the quanta (photons) hvji, and V is volume of the plasma radiating the line vji.

Population nj of energy level j in the atom (or ion) relies on

i) Ratio, nj/nion where nion is total number of atoms (or ions) in that specific ionization state;

ii) Ionization equilibrium , nion/nelem where nelem is elemental abundance;

iii) Abundance of element relative to hydrogen, nelem/nH where nH is number density of hydrogen; and

iv) Number density of hydrogen, nH.

Each energy level in the atom (or ion) is characterized by four quantum numbers:

(i) Principal quantum number n,

(ii) Spin quantum number s,

(iii) Angular momentum quantum number l, and

(iv) Total angular momentum quantum number j.

Photon is emitted when the electron jumps from the higher energy level characterized by (n, s, l, j) to lower energy level characterized by (n', s', l', j'). On the other hand, photon is absorbed when the electron jumps from lower energy level (n', s', l', j') to the higher energy level (n, s, l, j).  In this situation, electrons are arranged in shells. According to Pauli's exclusion principle, no two electrons may have similar set of quantum numbers. Therefore, energy level of whole atom may be symbolized by (n, S, L, J) whereas energy level of any single electron may be signified by (n, s, l, j). Principal quantum number is always signified by lower-case letter n.

Assume that electrons jump from upper level j signified by (n, S, L, J) to lower level i signified by (n', S', L', J'). Electrons which jump from level j to level i will emit the spectral line of frequency vji provided by

vji = (Ej - Ei)/h

Here Ej is energy of upper level j and Ei is the energy of lower level i. Conversely, absorption line is created when electrons jump from lower level i to upper level j. In case of absorption, photon is absorbed by electron and energy of the photon is utilized to raise electron to the higher energy level.

Transition Rules:

Transition Rules in Absence of the Magnetic Field:

In spectroscopy, it is fairly possible to forecast transitions which will take place under given condition. In this manner, frequencies of resulting set of emission (or absorption) lines can also be forecasted.

This is possible as quantum numbers (n, S, L, J) obey certain transition rules. Especially, electron can jump, only if te change in quantum numbers is steady with the given rules:

(a) S' = S

S'-S = 0 => Δs = 0

(b) L' = L or L' = L ±1

=> L' - L = 0 => L'-L = ±1

=>ΔL = 0 => ΔL = ±1

Combining the two results, we get

ΔL = 0 or ±1 therefore ΔL =-1, 0, +1

(c) J' = j or J' = j±1

=>J'-J = 0 => J'-J = ±1

=>ΔJ = 0 => ΔJ = ±1

Combining two results, we get

ΔJ = 0 or ±1

Therefore ΔJ = -1,0,+1

Degeneracy of Atomic Levels:

Transition rules given above are satisfactory in absence of magnetic field. If the magnetic field is applied, though, total angular momentum J may be pointed only in definite quantized directions with respect to direction of magnetic field. In this situation, J may be pointed in such a way that projection along applied magnetic field can only take one of the given values: J, J-1, J - 2,..., -J. This provides a total of 2J+1 sub-levels. These 2J+1 sub-levels are frequently indistinguishable.

Projection of J along magnetic field is known as magnetic quantum number denoted by Mj. It is fifth quantum number. Magnetic quantum number Mj indicates that level is made up of 2J + 1 sub-levels. Such a level is said to be degenerate. Degree of degeneracy is number 2J + 1. Degree of degeneracy is also known as statistical weight g of that specific level (i.e. g = 2J + 1).

Statistical weight g represents number of electrons which can occupy given level without violating Pauli's exclusion principle. When there is transition among one degenerate state A and another degenerate state B, there will be transition among definite sub-levels of state A and certain sub-levels of state B. Though, there will be no transition between few sub-levels of state B and few sub-levels of state B. Magnetic quantum number Mj shall communicate with each other only when specific rules for Mj are observed. Such rules for Mj are given below:

M'j = Mj or M'j = Mj ±1,

=>M'j - Mj = 0 => M'j - Mj = ±1

=>ΔMj = 0 => ΔMj = ±1

Combining the two results, we get

ΔMj = 0 or ±1

=> ΔMj = 0, ±1

=> ΔMj = -1, 0, +1

Forbidden Transitions:

A series expansion of radiation field can be achieved. Dominant term is the electric dipole contribution. In quantum theory, dipole term matches to electron transitions. It is administered by quantum number rules of equations. In classical treatment, there are other less significant terms like electric Quadrupole term and higher (electric and magnetic) terms. In quantum picture, there are forbidden transitions that don't obey rules set down. These transitions still give rise to observable spectral lines. Under low density conditions, spectral lines resulting from forbidden transitions become fairly important. Therefore, transitions that give rise to magnetic dipole radiation should occur between levels within one and same term

Line Profile:

Absorption Lines:

Light from source can be dispersed in the spectrum by means of prism or diffraction grating. Different sources have different spectra. For example, spectra of the quasar contain continuous spectrum (or continuum) having narrow spectral lines superimposed. These lines are generally absorption (Fraunhofer) lines, but at times bright emission lines, are also seen. Continuous spectrum comes from hot surface of source nucleus. Atoms in the atmosphere absorb definite characteristic wavelengths of radiation leaving dark gaps at corresponding points in spectrum.

The equation is as follows:

hvji = Ej - Ei, Ej>Ei

If energy levels Ej and Ei were sharply defined, infinitely narrow spectral line would be emitted when the electron jumps from energy level j to energy level i. This would match up to infinitely long lifetime of electron in that energy level. Frequency spread δω and lifetime δt are associated by expression:

δωδt ≈ 2π or

Dividing both sides of by 2π, we get

(δω/2π)δt ≈ 1

But, by definition of angular frequency, we know that:

ω = 2πv so that δω = 2πδv and δω/2π = δv

Equation reduces to (2πδv/2π)δt ≈ 1

In order that δvδt ≈ h But E = hv so that δE = hδv. Therefore

δEδt ≈ h

Here, δE is the measure of sharpness of the energies; δt is the measure of electron lifetime; h is Planck's constant; δv signifies spread of spectral line of frequency v.

The Lorentzian Line Profile:

Clearly, lifetime of the electron in any energy level is finite. Thus, energy level should also have finite width. This fact was stated quantum mechanically by weisskopf and Wigner in given probability distribution law:

P(Ej)dE = (Aj/h)(dE/(1/4)A2j + 4π2/h2(E-Ej)2)

Here Ej is mean energy of energy level j and Aj is probability of transition from level j to all other possible levels E. When the electron jumps from level j to level i, intensity of resulting spectral line is provided by

Iji = hvjiAji(Aj + Ai)/[(1/4)(Aj + Ai)2 + 4π2/h2[E - (Ei - Ej)2]]

Here Aji is probability of transition from level j level i.

Equation can be simplified by setting τ = Aj + Ai and noting that E = hv while Ei - Ej = Ej - Ei = hvji.

In order that Iji = EjiAjiτ/[τ2/4 + 4π2(v-vji)2]

For the constant decay probability, Aji Equation may be simplified further by setting

EjiAji = 1 and Iji = Φv so that Φv = (τ/4π2)[1/(4π2(v-v0)2 + (τ/4π)2)]

Here, Φv is normalized profile and τ = /T0 where T0 is average time interval between successive collisions of the radiating particles. Line shape provided by Equation is known as Lorentzian profile and parameter τ is known as Lorentzian width. Damping constant τ Aj + Ai is sum of decay constants of two energy levels involved in electron jump.

Wings of the spectral line will be weak whereas core of spectral line stays strong. Line broadening is measure of spread of frequency v of emitted spectral line on either side of core. Line broadening can be separated in three classes:

3) Collisional (or pressure) line broadening.

Natural line broadening is always there. Doppler line broadening is because of motion of observed atoms in different directions with different velocities. Resulting Doppler shift generates spread in frequency of observed line. Collisional (or pressure) line broadening is because of effects of other particles on radiating atom. Doppler broadening is always proportional (in wavelength units), to wavelength of centre line. Natural and collisional broadenings don't demonstrate any systematic trend with centre wavelength. Though, collisional broadening is always proportional to number of colliding particles per unit volume (and, by extension to pressure). Consider Doppler line broadening and collisional (or pressure) line broadening.

Doppler broadening is because of motion of individual atoms in the hot gas. Thermal motions of atoms (or ions) generate particles that are moving away from observer, and particles moving toward observer. This case gives rise to Doppler Effect. Frequency v of the spectral line emitted by atom moving with velocity vr along line of sight is provided by:

Δv = v-v0 = -(vr/c).vo,

Here v0 is rest frequency of spectral line and c is speed of light. Evidently, Δv is positive when vr is negative. Also, Δv is negative when vr is positive. Negative value of Δv signifies that radiating atom is moving away from observer (i.e. vr is positive). This gives rise to the shift in wavelength of visible line towards red end of optical spectrum. In this situation, a red shift is said to have occurred. Observed frequency is lower than rest frequency of emitted line (i.e. observed wavelength is longer than rest wavelength of spectral line).

When the Doppler broadening of the spectral line is completely caused by thermal motion of radiating particles, width of such a line is known as Doppler width ΔvD

In terms of frequency, the Doppler width ΔvD is provided by ΔvD = v0/c(2kT/m)1/2

Here v0 is frequency of undisturbed line centre, T is temperature, m is mass of particle and k is Boltzmann's constant.

In terms of the wavelength, Doppler width ΔλD is provided by

ΔλD = λ0/c(2kT/m)1/2

Here λ0 is wavelength of undisturbed line centre, ΔλD = λ - λ0, and profile Φλ is now normalized with respect to Δλ.

Gaussian Profile:

Thermal motions of individual atoms aren't the only motion of interest here. At times, radiating atoms may have thermal velocity component and a non-thermal (turbulent) velocity component. In this situation, Doppler width is altered to accommodate non-thermal component in given way:

ΔλD = λ0/c(2kT/m + ξ2)1/2

Profile Φv of Doppler broadened line is then provided by Φv

Φv = (1/√πΔvD)exp[-(Δv/ΔvD)2]

Here Doppler width is again expressed as:

ΔvD = (v0/c)√2kT/m

In terms of wavelength, profile assumes form

Φλ = (1/√πΔλD)exp[-(Δλ/ΔD)2]

This is bell-shaped profile known as Gaussian profile. Gaussian profile falls to half its maximum height (or depth) when exponential term is equivalent to half i.e. when

exp[-(Δλ/ΔD)2] = 1/2

In this situation,

-(Δλ/ΔλD)2 = loge(1/2)

(Δλ/ΔλD)2 = loge2 - loge1

(Δλ/ΔλD)2 = loge(2/1)

(Δλ/ΔλD) = √loge2 and Δλ = ΔλD√loge2

Full width at half height (FWHH) of line profile is stated as wavelength separation between points when profile drops to half its maximum height (or depth) on either side of line centre. On one side of line centre, Δλ = ΔλD√loge2. On the other side of line centre, Δλ = ΔλD√loge2 as well. Therefore full width at half height is provided by sum of two equal values of Δλ i.e.

FWHH = ΔλD√loge2 + ΔλD√loge2 or FWHH = 2ΔλD√loge2

Substituting value of √loge2 we get:

FWHH = 1.667ΔλD

This result is appropriate to both absorption and emission line profiles. Though, actual line will suffer radiative transfer effects (such as saturation) that may reduce dip in the absorption line and therefore increase observed FWHH.

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