Spectral Analysis. While studying the spectrum of a gas...

Spectral Analysis. While studying the spectrum of a gas cloud in space, an astronomer magnifies a spectral line that results from a transition from a $p$ state to an $s$ state. She finds that the line at 575.050 nm has actually split into three lines, with adjacent lines 0.0462 nm apart, indicating that the gas is in an external magnetic field. (Ignore effects due to electron spin.) What is the strength of the external magnetic field?

Key Concepts

Zeeman Effect

This phenomenon describes the splitting of atomic or molecular spectral lines into multiple components when subjected to an external magnetic field. The splitting arises because the energy levels of electron orbitals shift due to their interaction with the magnetic field, leading to observable changes in the emitted or absorbed light. It is a fundamental tool in astrophysics and atomic physics for probing magnetic fields and atomic structure.

Bohr Magneton

The Bohr magneton is the physical constant that represents the elementary unit of magnetic moment for an electron due to its orbital or spin angular momentum. It serves as a measure for the magnetic interaction between the electron and an external magnetic field and is critical in quantifying the energy shifts observed in the Zeeman effect.

Wavelength-Energy Relationship

This concept involves the conversion of wavelength differences to energy differences using the relationship E = hc/?, where h is Planck’s constant and c is the speed of light. It is essential for interpreting how small shifts in spectral lines, such as those caused by the Zeeman splitting, correspond to changes in energy levels within the atom.

Transcript

00:01 This question regards the zeman effect or zeman splitting.

00:05 And it asks if we have a gas that is undergoing a p to s transition that emits a certain wavelength plus or minus a little bit, resulting in a three -band wavelength.

00:21 This asks what the magnitude of the external magnetic field is.

00:27 Zeman splitting says that energy levels of electrons split and enter new discrete energy levels under the influence of an external magnetic field.

00:40 So given our initial wavelength of 575 .05 nanometers, that splitting occurs and causes two new wavelengths to come about that are 0 .0462 nanometers away.

00:57 The question again asks, what is the strength of the magnetic field? so to do this, first we need to understand what the energy is of a single photon, and that is going to equal hc over lambda.

01:17 Now, we can see that we can get a few different energies from this problem.

01:22 We have our initial wavelength, and then we have an excited state and a slightly disexcited state.

01:28 The longer wavelength will be more energetic, and that's the one we're going to use to solve the problem, but you could use the other one as well.

01:38 Our initial energy, which we will label ei, using planx constant, which is 4 .14 times 10 to the minus 15 ev and the speed of light, divided by lambda, which is just going to be r575 .05 .05 nanometers.

01:59 If we plug all of that in, we end up getting 2 .1981 electron volts.

02:09 That is the energy of our ground, if you will, state photon.

02:16 The unexcited photon.

02:19 Now, our final energy, we can choose either of the different bands.

02:26 We're going to choose, in this case, the higher wavelength band.

02:29 So adding 0 .0462.

02:32 If we do this, we have a slightly higher wavelength, which ends up being 575.

02:40 575 .0962.

02:48 And if we do the same calculation with h and c, we end up getting 2 .196 electron volts...