Lecture 8 - Atomic Spectra (2/4/99)

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ASTR012 Reading:
Chapter 8-2, 8-3 (ZG4)

pages 27 - 33

? Key Question: What is the origin of spectral lines in stars?
! Key Principle: The Bohr model
# Key Problem: What are the energy levels in hydrogen and how are they related to line series?


  1. The Bohr Atom
  2. Particles and Waves
  3. The Hydrogen Line Spectrum
  4. Ionization and Molecules

Particles and Waves:

Just as electromagnetic waves of light are also particles (photons), all particles such as electrons and protons are waves! It turns out that a particle of momentum p has a wavelength of

 l = h / p = h / mv 

so more massive particles have smaller wavelenghths. This property is used (sort of) in electron microscopes, which act like normal microscopes except using the wave nature of electrons instead of light. Because the wavelength of an electron wave is very very small, it can see smaller things than normal light based microscopes.

When we put an electron into an "orbit" around an atomic nucleus, we must consider the wave-nature. To pack the electron wave into orbit we must put it in as a standing wave with an integer number of wavelengths around the circumference:

These integer numbers of waves cause quantization of the orbits.

The Bohr Atom: Orbital Radii and Energy Levels

The condition that an electron wave not interfere with itself on an orbit means that there must be an integer number, say n, wavelengths on an orbit. The smallest allowed orbit contains a single wavelength. We wish to compute the radius and Energy of the orbits in the hydrogen atom. This was done by Niels Bohr. This gives the relation that

2 r = n e

where lambda_e is the wavelength of the electron and r is the orbital radius.

From the physicist de Broglie, we know that the wavelength of an electron is proportional to the inverse of its momentum:

e = h / m v,

where m is the electron mass and v is the velocity of the electron, and h is Planck's constant. Thus, we find the velocity of the electron to be given by:

v = (h/2) (n/ m r)

or equivalently, the orbital angular momentum

L = m v r = n

where "h-bar" is

= (h/2).

Thus, in the quantum atom, the orbital angular momentum L is quantized, with the quantum of angular momentum being h-bar! This is a profound result, and is as important for atomic physics as the quantization of light E = hf was for blackbody radiation.

Just like when we wanted to derive Kepler's law from the gravitational force we used the centripetal acceleration to give us the velocity, we do the same here. We have the Coulomb force

F = e2/r2

where we have absorbed the usual MKS electrostatic constant into the units for the electron and proton charges +e and -e

e2 / 4 0 --> e2.

The centripetal force is just F = m v^2 / r, so we equate

( e2 / r2 ) = (m v2 / r)

and thus we find:

v2 r = e2 / m

Substituting in our expression for the velocity in the above equation gives

( n /mr )2 r = (e2 / m)


rn = n22 / me2.

If we then define the Bohr radius

a0 = 2 / m e2

then we find the simple relation for the radius of orbit n:

rn = a0 n2

The orbital radii of the hydrogen atom, quantised by the number of wavelengths n:

To find the energy levels, we need to know what the energy of a moving particle under a force is. It turns out it has two terms due to the motion and due to the force:

E = mv2/2 - e2/r

The first term (m v^2/ 2) is called the kinetic energy and is the energy carried in the motion of the electron, like the energy in a thrown baseball which you feel when it hits you. The second term (-e^2/r) is called the potential energy, and it is the energy "contained" in the electric force that you must work with or against to move an electron closer or farther from the nucleus. It has the opposite sign to the kinetic energy because the total energy is conserved, you cant make or destroy energy out of nothing. Thus if you speed up an electron (increase its kinetic energy) you must draw it from the potential energy due to the electric force (decrease its potential energy). If we compute the energy E, we see that since v^2 r = e^2 / m from the centripetal force, then L

E = - e2/2r

Substituting in r_n = a_0 n^2, then easily we find:

E = -e2 / (2a0n2)

If we define the energy at the ground state to be

E0 = e2/2a0

the Rydberg Energy for hydrogen (sometimes we write E_R instead of E_0), then

E = - E0 n-2.

Thus, the energy levels are spaced like the inverse square of the level number n, while the radii are spaced like the square of n.

The energy levels of the hydrogen atom:

From all the constants you can compute the values of the Bohr Radius a_0 = 5.29 x 10^-11 m = 0.053 nm = 0.53 Å, and the Rydberg energy E_0 = 2.18 x 10^-18 Joules = 13.6 eV. We have defined a more convenient unit of energy: the electron volt (eV). This is the energy gained by an electron accelerated through a voltage difference of 1 volt (like from a battery). It is very tiny ( 1eV = 1.602 x 10^-19 J ), but more convenient for the energies encountered in atomic physics.

In this energy level scheme, we see that an electron at rest infintely far away from the proton (in the absence of any other atoms!) has an energy of E = 0 (for n = infinity). The ground state n = 1 has an energy E = -13.6 eV. Thus, it takes an energy of 13.6 eV to remove an electron in the ground state from the hydrogen atom.

Hydrogen Energy Levels and Line Series

The Rydberg formula for the transition wavelengths between two energy levels m and n in hydrogen:

The line series of hydrogen corresponding to transitions between upper levels and lower levels:

In particular, the Rydberg formula gives the spaceing for the hydrogen series:

Jumping between these levels can be initiated by absorption of a photon of the correct energy, but the atom quickly returns to its de-excited state emitting a photon of the same energy:

Electron Orbitals and Quantum Numbers

The quantum numbers of electrons in the atom:

The configuration of electrons in the different l orbitals:

Atomic Spectra in Outline:

  1. The Atom
  2. Coulomb Forces and Electron Orbits
  3. Quantization and Energy Levels

Key Problems:

  1. Why don't electrons radiate away energy and spiral into the nucleus?
  2. What are wavelengths of the Lyman and Balmer series of H I and He II?

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smyers@nrao.edu Steven T. Myers