Atomic physics

of the Quantum Numbers. The Exclusion Principle and the Periodic

Table. More on Atomic Spectra: Visible and X-Ray. Spontaneous and Stimulated Transitions. Lasers.

to quantum mechanics, an atom in a state whose principal

quantum number is n can take on the following discrete values of the magnitude of the orbital angular momentum:

ml
A vector model that describes space quantization for the case

l=2.

ms
Electron spin

follow the same quantum rules as orbital angular momentum. Te

magnitude of the spin angular momentum for the electron is

Physical Interpretation of the Quantum Numbers

The Spin Magnetic Quantum Number ms

atoms with many electrons, it is often easiest to assign

the quantum numbers to the electrons in the atom as opposed to the entire atom. An obvious question that arises here is, “How many electrons can be in a particular quantum state?” Pauli answered this important question in 1925, in a statement known as the exclusion principle:

No two electrons can ever be in the same quantum state; therefore, no two electrons in the same atom can have the same set of quantum numbers.

states must obey both the exclusion principle and Hund’s rule

when an atom has orbitals of equal energy, the order

in which they are filled by electrons is such that a maximum number of electrons have unpaired spins.

elements versus atomic number

does not change are very unlikely to occur and are

called forbidden transitions. (Such transitions actually can occur, but their probability is very low relative to the probability of “allowed” transitions.) The various diagonal lines represent allowed transitions between stationary states. Whenever an atom makes a transition from a higher energy state to a lower one, a photon of light is emitted.

the allowed transitions are
the allowed energies for one-electron atoms and

ions, such as hydrogen and He+ are

This equation was developed from the Bohr theory, but it serves as a good first approximation in quantum theory as well.

when high-energy electrons or any other charged particles bombard a

metal target. The x-ray spectrum typically consists of a broad continuous band containing a series of sharp lines.

X-ray radiation with its origin in the slowing down of electrons is called bremsstrahlung, the German word for “braking radiation.”

The discrete lines called characteristic x-rays.

machine that uses a linear accelerator to accelerate electrons up

to 18 MeV and smash them into a tungsten target. The result is a beam of photons, up to a maximum energy of 18 MeV, which is actually in the gamma-ray range. This radiation is directed at the tumor in the patient.

lines are formed when electrons drop from upper levels to

vacancies other than those in the K shell. For example, L lines are produced when vacancies in the L shell are filled by electrons dropping from higher shells. An Lα line is produced as an electron drops from the M shell to the L shell, and an Lβ line is produced by a transition from the N shell to the L shell.

estimate the energy associated with an electron in the L

shell:

After the atom makes the transition, there are two electrons in the K shell. We can approximate the energy associated with one of these electrons as that of a one-electron atom. (In reality, the nuclear charge is reduced somewhat by the negative charge of the other electron, but let’s ignore this effect.) Therefore,

of !1/l versus Z, where λ is

the wavelength of the Kα x-ray line of the element of atomic number Z.

From this plot, Moseley determined the Z values of elements that had not yet been discovered and produced a periodic table in excellent agreement with the known chemical properties of the elements. Until that experiment, atomic numbers had been merely placeholders for the elements that appeared in the periodic table, the elements being ordered according to mass.

the photon stimulates the atom to make the upward transition.

state, the excited atom can make a transition back to

a lower energy level, emitting a photon in the process. This process is known as spontaneous emission because it happens naturally, without requiring an event to trigger the transition. Typically, an atom remains in an excited state for only about 10-8 s.

occurs. In this process, the incident photon is not absorbed;

therefore, after the stimulated emission, two photons with identical energy exist: the incident photon and the emitted photon.

in these technological applications are the following:

• Laser light is

coherent. The individual rays of light in a laser beam maintain a fixed phase relationship with one another.
• Laser light is monochromatic. Light in a laser beam has a very narrow range of wavelengths.
• Laser light has a small angle of divergence. The beam spreads out very little, even over large distances.

energy transitions either upward (stimulated absorption) or downward (stimulated emission).

The two processes are equally probable. When light is incident on a collection of atoms, a net absorption of energy usually occurs because when the system is in thermal equilibrium, many more atoms are in the ground state than in excited states. If the situation can be inverted so that more atoms are in an excited state than in the ground state, however, a net emission of photons can result. Such a condition is called population inversion. Population inversion is, in fact, the fundamental principle involved in the operation of a laser (an acronym for light amplification by stimulated emission of radiation). The full name indicates one of the requirements for laser light: to achieve laser action, the process of stimulated emission must occur.

must be a buildup of photons in the system. The

following three conditions must be satisfied to achieve this buildup:

• The system must be in a state of population inversion: there must be more atoms in an excited state than in the ground state. That must be true because the number of photons emitted must be greater than the number absorbed.
• The excited state of the system must be a metastable state, meaning that its lifetime must be long compared with the usually short lifetimes of excited states, which are typically 1028 s. In this case, the population inversion can be established and stimulated emission is likely to occur before spontaneous emission.

long enough to enable them to stimulate further emission from

other excited atoms. That is achieved by using reflecting mirrors at the ends of the system. One end is made totally reflecting, and the other is partially reflecting. A fraction of the light intensity passes through the partially reflecting end, forming the beam of laser light.

helium–neon gas laser. Figure is an energy-level diagram for the

neon atom in this system. The mixture of helium and neon is confined to a glass tube that is sealed at the ends by mirrors. A voltage applied across the tube causes electrons to sweep through
the tube, colliding with the atoms of the gases and raising them into excited states. Neon atoms are excited to state E3* through this process (the asterisk indicates a metastable state) and also as a result of collisions with excited helium atoms. Stimulated emission occurs, causing neon atoms to make transitions to state E2. Neighboring excited atoms are also stimulated. The result is the production of coherent light at a wavelength of 632.8 nm.

50 layers of fabric at a time, is one of

the many applications of laser technology.

point of six counterpropagating laser beams along mutually perpendicular axes.

the normal high speeds of the atoms are reduced when

they are restricted to the region of the trap. As a result, the temperature of the collection of atoms can be reduced to a few microkelvins. The technique of laser cooling allows scientists to study the behavior of atoms at extremely low temperatures