stark effect

8/10/2019 Stark Effect

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Stark Effect in Atomic Spectra

The splitting of atomic spectral lines as a result of an externally applied electric

field was discovered by Stark, and is called the Stark effect. As the splitting of

a line of the helium spectrum shows, the splitting is not symmetric like that of the

Zeeman effect.

The splitting of the energy levels by an

electric field first requires that the field

polarize the atom and then interact with

the resulting electric dipole moment. Thatdipole moment depends upon the magnitude

of Mj, but not its sign, so that the energy

levels are split into J+1 or J+1/2 levels, for

integer and half-integer spins respectively.

The Stark effect has been of marginal benefit in the analysis of atomic spectra,

but has been a major tool for molecular rotational spectra.

Atomic Properties

The electrons associated with atoms are found to have measurable properties

which exhibit quantization. The electrons are normally found in quantized energy

states of the lowest possible energy for the atom, called ground states. The

electrons can also exist in higher "excited states", as evidenced by the line spectra

(e.g. the hydrogen spectrum) observed when they make transitionsback to the

ground states. The existence of these excited states can be demonstrated more

directly in collision experiments like the Franck-Hertz experiment .


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Other properties associated with the electron energy levels such as orbital angular

momentum and electron spin are also quantized and give rise to the quantum

numbers used to characterize the levels. These quantized properties are associated

withperiodic table of the elements, and the requirements of the Pauli exclusion

principle on the quantum numbers can be viewed as the origin of the periodicity.

The periodic table provides a convenient framework for cataloging other physical

and chemical properties of atoms. While the hydrogen electron energy levels are

found to depend only upon the principal quantum number, the energy levels in

other atoms are found to have strong dependence upon the orbital quantum

number. These levels show a smaller amount of dependence upon the total

angular momentum. This dependence may arise from interactions within the atom

such as the spin-orbit interaction or may arise only when external fields are

applied. When magnetic fields are applied, there is splitting of atomic energy

levels from the Zeeman effect, and in response to electric fields there is splitting

called the Stark effect.

For light atoms with multiple electrons

outside a closed shell, we can combine

the orbital angular moments to give a

resultant angular momentum L.

Considering the vector model, we

expect that we can get different values

of L for a given set of individual orbital

angular momenta. It turns out that the

largest possible L value has the lowest

energy. This is sometimes referred to as

the "orbit-orbit effect" (Hund's Rule #2)


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With multiple electrons there is an additional source of splitting of the electron

energy levels which is characterized in terms of another quantum number, the

total anglular momentum quantum number J. The source of the splitting is called

the spin-orbit effect. For light atoms, the spins and orbital angular momenta of

individual electrons are found to interact with each other strongly enough that you

can combine them to form a resultant spin S and resultant orbital angular

momentum L (this is called Russell-Saunders or LS coupling). The S and L are

combined to produce a total angular momentum quantum number J, and it is

found that higher J values lie lower in energy (Hund's Rule #3)

When an external interaction such as a magnetic field is applied, then further

splitting of the energy levels occurs, and that splitting is characterized in terms of

the magnetic quantum number associated with the z-component of angular

momentum. This splitting is called the Zeeman effect. Effects on energy levels

from applied electric fields are called Stark effects.

Orbital Angular MomentumThe orbital angular momentum of electrons in atoms associated with a givenquantum state

is found to be quantized in the form

This is the result of applying quantum theory to the orbit of the electron. The

solution ofthe Schrodinger equation yields the angular momentum quantum number. Even in

the case

of the classical angular momentum of a particle in orbit,


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the angular momentum is conserved. The Bohr theoryproposed the quantization

of theangular momentum in the form

and the subsequent application of the Schrodinger equation confirmed that form

for theorbital angular momentum.

The spectroscopic notation used for characterizing energy levels of atomic

electrons is

based upon the orbital quantum number.

Total Angular MomentumWhen the orbital angular momentum and spin angular momentum are coupled, thetotal

angular momentum is of the general form for quantized angular momentum

where the total angular momentum quantum number is

This gives a z-component of angular momentum

This kind of coupling gives an even number of angular momentum levels, which

is consistent with the multiplets seen in anomalous Zeeman effects such as that of

sodium. As long as external interactions are not extremely strong, the total angularmomentum of an electron can be considered to be conserved and j is said to be a

"good quantum number". This quantum number is used to characterize the


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splitting of atomic energy levels, such as the spin-orbit splitting which leads to the

sodium doublet.

Magnetic Interactions and theLande' g-FactorA magnetic moment experiences a

torque in a magnetic field B. The

energy of the interaction can be

expressed as

Both the orbital and spin angular

momenta contribute to the magnetic

moment of an atomic electron.

.where g is the spin g-factor

and has a value of about 2,

implying that the spin angular

momentum is twice as

effective in producing a magnetic moment. The

interaction energy of an atomic electron can

then be written which when evaluated in terms

of the relevant quantum numbers takes the

form


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Evaluating the Lande' g-Factor

The Lande' g-factor is a geometric factor which arises in the evaluation of the

magnetic interaction which gives the Zeeman effect. The magnetic interaction

energy which is continuous in the classical case takes on the quantum form

which is like a vector operation based

on the vector model of angular

momentum

The problem with evaluating this scalar product is that L and S continually change in

direction as shown in the vector model.

The strategy for dealing with this problem is to use the direction of the total angular

momentum J as a coordinate axis and obtain the projection of each of the vectors in

that direction. This is done by taking the scalar product of each vector with a unit

vector in the J direction. These vector relationships must be evaluated and expressed

in terms of quantum numbers in order to evaluate the

energy shifts. Carrying out the scalar products above

leads to


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"Anomalous" Zeeman EffectWhile 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. Their explanation gave additional insight into the


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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.

For magnetic field in the z-direction this gives

Considering the quantization of angular momentum , this gives equally spacedenergy levels displaced from the zero field level by

This displacement of the energy levels gives the uniformly spaced multiplet

splitting of the spectral lines which is called the Zeeman effect. The magnetic

field also interacts with the electron spin magnetic moment, so it contributes to the

Zeeman effect in many cases. The electron spin had not been discovered at the

time of Zeeman's original experiments, so the cases where it contributed were


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considered to be anomalous. The term "anomalous Zeeman effect" has persisted

for the cases where spin contributes. In general, both orbital and spin moments are

involved, and the Zeeman interaction takes the form

The factor of two multiplying the electron spin angular momentum comes from

the fact that it is twice as effective in producing magnetic moment. This factor is

called the spin g-factor or gyromagnetic ratio. The evaluation of the scalar product

between the angular momenta and the magnetic field here is complicated by thefact that the S and L vectors are both precessing around the magnetic field and are

not in general in the same direction. The persistent early spectroscopists worked

out a way to calculate the effect of the directions. The resulting geometric factor

gL in the final expression above is called the Lande g factor. It allowed them to

express the resultant splittings of the spectral lines in terms of the z-component of

the total angular momentum, mj.


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Term SymbolsThe heirarchy of labels for the electrons of multi-

electron atoms is configuration, term, level, and state.

The term uses the multiplicity 2S + 1, total orbitalangular momentum L, and total angular momentum

J. It assumes that all the spins combine to produce S,

all the orbital angular momenta couple to produce L, and then the spin and orbital

terms combine to produce a total angular momentum J.

Different terms will in general have different energies, and the order of those

energies is usually that given by Hund's Rules, although there are exceptions. The

different terms for a given configuration are obtained by forming the different

combinations of angular momenta for the electrons outside closed shells, making

sure the Pauli Exclusion Principle is obeyed


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Hund's Rules1. The term with maximum multiplicity lies lowest in energy2. For a given multiplicity, the term with the largest value of L lies lowest

in in energy.

3. For atoms with less than half-filled

shells, the level with the lowest value

of J lies lowest in energy.

Hund's rules assume combination to

form S and L, or imply L-S (Russell-

Saunders) coupling. Note: Some

references, such as Haken & Wolf, use

Hund's Rule #1 to apply to the nature

of full shells and subshells. Full shells

and subshells contribute nothing to the

total angular momenta L and S. If you call this Hund'e Rule #1, then the above

rules will be bumped up one in number. I don't know which is the more common

practice.

Hund's Rule #1The term with the maximum multiplicity lies lowest in energy.

Example: In the configuration we expect the orderThe explanation of the rule lies in the effects of the spin-spin interaction. Though

often called by the name spin-spin interaction, the origin of the energy difference

is in the coulomb repulsion of the electrons. It's just that a symmetric spin state

forces an antisymmetric spatial state where the electrons are on average further


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apart and provide less shielding for each other, yielding a lower energy. The

sketches below attempt to visualize why that is so.


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These sketches are conceptual only. No attempt has been made to do any realistic

scaling. Note that the energies we are discussing here are electric potential

energies, so that a negative electron in the vicinity of a positive nucleus will have

a negative energy leading to a bound state. Any force between the electrons will

tend to counter that, contributing a positive potential energy which makes the

electrons less tightly bound, or higher in potential energy.

Hund's Rule #2For a given multiplicity, the term with the largest value of L lies lowest in energy.

The basis for this rule is essentially that if the electrons are orbiting in the same

direction (and so have a large total angular momentum) they meet less often than

when they orbit in opposite directions. Hence their repulsion is less on average

when L is large. These influences on the atomic electron energy levels is

sometimes called the orbit-orbit interaction. The origin of the energy difference

lies with differences in the coulomb repulsive energies between the electrons.


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For large L value, some or all of the electrons are orbiting in the same direction.

That implies that they can stay a larger distance apart on the average since they

could conceivably always be on the opposite side of the nucleus. For low L value,some electrons must orbit in the opposite direction and therefore pass close to

each other once per orbit, leading to a smaller average separation of electrons and

therefore a higher energy.

Hund's Rule #3For atoms with less than half-filled shells, the level with the lowest value of J lies

lowest in energy.

When the shell is more than half full, the opposite rule holds (highest J lies

lowest). The basis for the rule is the spin-orbit coupling. The scalar product SL is


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negative if the spin and orbital angular momentum are in opposite directions.

Since the coefficient of SL is positive, lower J is lower in energy.

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