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# Calculating the Zeeman Effect

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Calculating the Zeeman Effect

 Problem A The Ap star BD+0°4535 has a powerful magnetic field of 2.1 T. Calculate the wavelengths of the  line caused by Zeeman splitting. (Data from Elkin, Kurtz, Nitschelm, and Unda-Sanzana (2010).)   Hints:   Calculate the change in wavelength caused by Zeeman splitting.

Data

Electron Mass in kg

 (2.1)

Electron Charge in J

 (2.2)

Speed of Light in m/s

 (2.3)

Magnetic Field Strength in T

Wavelength of H`[alpha] Line in m

 (2.4)

 (2.5)

 Useful Equation Change in Wavelength caused by Zeeman Splitting

Solution

Calculate the Zeeman split in Hz, using the formula above.

 (4.1)

 (4.2)

The Zeeman shift for this magnetic field is  Hz.

The Hline occurs at 656.281 nm. This corresponds to

 (4.3)

Therefore, the three Zeeman frequencies, in Hz, are

 (4.4)

 (4.5)

 (4.6)

The three Zeeman wavelengths, in m, are

 (4.7)

 (4.8)

 (4.9)

 Problem B Using NASA's Hinode Solar-B telescope, Katsukawa (2011) observed that the iron Fe I line, at  nm, in the umbra of a large sunspot, was split into two components by Zeeman splitting as shown in the figures below (Katsukawa, 2011).   The image shows a sunspot approximately 50,000 kilometres in diameter. The umbra (darkest area) is approximately 15,000 kilometres in diameter. The coloured arrows indicate the magnetic vectors. The letters A, B, C, and D refer to the spectra in the following image.     The depressions on the right indicate the iron line at 630.25 nanometres. The distance between the two Zeeman wavelenths at the umbra (D) is approximately 28 Ångstroms. Therefore, the line spread is approximately 14 Ångstroms.   Find the magnetic field strength of the sunspot's umbra.   Hints: The Fe I line is emitted in a transition from 5D0 to 5P1. Compute the Landé g factors for both orbitals.

Data

Bohr Magneton in ev/G

 (6.1)

Positive Magnetic Spin Number

 (6.2)

Negative Magnetic Spin Number

 (6.3)

Iron FE I Line

 (6.4)

Zeeman Wavelength Shift

 (6.5)

Planck Constant

 (6.6)

Speed of Light

 (6.7)

 Useful Equations Lande g Factor   Energy from Magnetism at an Orbital   Magnetic Field Strength

Solution B

Following n2S+1Lj for the 5P1 level, j = 1, s = 2, and l = 1:

 (8.1)

For the  5D0 level, j = 0, s = 2, and l = 2. It is clear that the Land g factor is undefined. We can, therefore, set it to zero in the calculations.

Find the energy of each level.

 (8.2)

 (8.3)

 (8.4)

 (8.5)

The longest wavelength line (mj = -1/2 → mj = +1/2) is the result of a net energy shift of

 (8.6)

 (8.7)

The total energy difference is

 (8.8)

 (8.9)

The magnetic field is approximately 3000 gauss, which is typical for the umbra of a large sunspot.

 References ------------------------------------------------------------------------   Elkin, V., Kurtz, D., Nitschelm, C. and Unda-Sanzana, E. (2010). The discovery of a 21-kG magnetic field in the Ap star BD+004535. Mon. Not. R. Astron. Soc. 401 (1), L44-L47.   Katsukawa, Y. (2011). Measurement of Solar Magnetic Field. Universe of Spectroscopy. http://prc.nao.ac.jp/extra/uos/en/no06/ (accessed: 2016-08-14.)