Chapter 4: Atomic Spectra

Question 1

Give an example of an atomic two-level system.

Answer 1

The 1s and 2p energy levels in the H atom form a two-level system. If narrow bandwidth laser radiation is used to excite a H atom from the ground 1s state n = 2, the selection rules for electric dipole transitions restrict the excitation to the 2p state. After it is excited this state has only one possible decay pathway. It will decay by spontaneous emission to the 1s state.

Question 2

What effect does stimulated emission have on a two-level system prepared in the excited state?

Answer 2

Stimulated emission de-excites a two-level system prepared in the excited state back to the ground state. This will result in the emission of a photon that propagates in the same direction, has the same frequency, and is in phase with the photon that stimulated the system to emit.

Question 3

In a two-level system how does the Einstein A-coefficient relate to the lifetime of the excited state?

Answer 3

In a two-level system the lifetime of the excited state is τ₂ = 1/A₁₂.

Question 4

Give an expression for the lifetime of the 4p state in the H atom in terms of the relevant Einstein A coefficients.

Answer 4

Question 5

What is meant by the expression metastable state? Give examples of metastable states in the H atom, and in the He atom.

Answer 5

Question 6

What is a population inversion and why is it required for the operation of a laser?

Answer 6

An ensemble of two-level systems exhibits a population inversion when the fraction in the excited state is greater than the fraction in the ground state. For the operation of a laser stimulated emission in the lasing medium must be more likely to occur than spontaneous emission. To ensure that stimulated emission dominates, more of the quantum systems that make up the medium must be in the upper (excited) state associated with the lasing transition than in lower (ground) state. This situation represents a population inversion.

Question 7

Why can a population inversion not be generated in a two-level system?

Answer 7

• If population is intially at level 1, absorption most likely to occur.
• Population is then transferred yo upper (excited) level, meaning population of excited level will increase until both lv1 and lv2 are equal.
• From here, the likelihood of absorption and stimulated emission are both equal.

Question 8

Why is it difficult to construct a laser that operates directly in the X-ray region of the electromagnetic spectrum?

Answer 8

For the operation of a laser the rate of stimulated emission must exceed the rate of spontaneous emission from the excited state. Since the rate of spontaneous emission, the Einstein A-coefficient, depends on the cube of the transition frequency, ν_if , it is difficult to ensure that stimulated emission dominates for very high transition frequencies, i.e., at short wavelengths in the X-ray region of the electromagnetic spectrum.

Question 9

Is the bandwidth of a nanosecond pulsed laser typically larger or smaller than that of a femtosecond pulsed laser?

Answer 9

Question 10

What is the momentum associated with a photon with a frequency of 5.63 × 10¹⁴ Hz?

Answer 10

Photon momentum pλ = h/λ = hν/c. Therefore the momentum associated with a photon of 5.63 × 10¹⁴ Hz is 1.2 × 10⁻²⁷ kg m s−1 .

Question 11

Explain Absorption

Answer 11

This involves the system, initially level 1, absorbing a photon and being excited to level 2. The rate of absorption is directly proportional to the no. of atoms in level 1 and the energy density of the electromagentic field.

Question 12

Explain stimulated emission

Answer 12

A photon stimulates the system, initially at level 2, to decay to level 1 with the emission of another photon.

The photon emitted has the same frequency, is incoherent with (electromagnetic field is oscillating in phase with, and propgating) the photon that stimulated it.

Question 13

Explain spontaneous emission

Answer 13

Rate of emission is directly proportional to he number of atoms in level 2.

Question 14

Einstein coefficients

Answer 14

A=Einstein A coefficient for spontaneous emission

B=Einstein B coefficient for stimulated emission

Question 15

Effects of finite excited state lifetimes

Answer 15

Transistions to states with long life times have narrow spectral width therefore when atom spontaneously decays, cannot be precisely measured, but frequency will be precise.

And vice versa

Question 16

Explain metastable states

Answer 16

If the spintaneous decay of an excited state atom is forbidden by selection rules, then excited state is said to be metastable. However can decay through higher order processes but takes ages.

Question 17

Explain three level lasers

Answer 17

A) Excite atoms from level 1 to 2 using photons of frequency v₁₂.

B) Fast spontaneous decay (internal conversion) to level 3 which is metastable leading to a population inversion between level 3 and 1.

C) Lasing can then occur on the transition from level 3 to 1 at frequency v₃₁. Because level 3 is long lived (metastable) it is possible to ensure that stimulated emission from level 3 to 1 is faster than spontaneous emission.

Question 18

Explain 4 level lasers

Answer 18

A: The first step of the process of generating a population inversion involves exciting, ’pumping‘, atoms from level 1 to level 2. This is sometimes achieved optically, but in the case of the He-Ne laser involves the generation of an electric discharge in a mixture of He and Ne gas.

B: Excitation of level 2 is then followed by rapid energy transfer to level 3. In the case of the He-Ne laser this is achieved via collisions between the excited He atoms and ground state Ne atoms in the gas mixture.

C: Lasing occurs on the transition from level 3 to level 4 at frequency ν34.

D: Level 4 is selected to rapidly decay by spontaneous emission to the ground state, level 1, such that following stimulated emission from level 3 the atoms in the system do not remain for a long time in level 4. Consequently, reabsorption of the amplified radiation cannot occur.