3. Computations and Supplementary Information

3.1 Analysis of Zeeman Splitting Data

The same general procedure applies to analysis of the data taken in parts (2), (3), and (5). Fit a "best" straight line to the data points on the plot of Δλ versus B according to one or the other (or you can investigate both) of the following procedures:

I. Fit a beat straight line to the actual data points and to an implied data point at Δλ = B = 0.

II. Ignore the implied point at the origin and fit a best straight line to the actual data points corresponding to B ≠ 0.

Supplementary sheets show that for both the normal and anomalous Zeeman effects, one may write

Eq. (1)

The slope of the straight line that best fits the data will be used with the formula written in the following form:

Eq. (2)

3.2 Analysis of Normal Zeeman Effect

For the normal Zeeman effect, the splitting of the displaced lines is given, with f(g₁,g₂) = 1, as

Eq. (3)

Determine experimental values for μ_B, from your plots of parts (2 - viewed perpendicular to B) and (3 - viewed parallel to B).

3.3 Analysis of Anomalous Zeeman Effect

For the case (or cases) studied in part (5), determine the product f(g₁,g₂)μ_B from your plot.

Compute g₁ and g₂ for the states involved in the transition that yields the observed Hg green line. With attention to the relative intensities of the components of the displaced "lines", estimate a reasonable value for f(g₁,g₂) that should be applicable to your observations. Then compute an expected value for the product f(g₁,g₂)μ_B, for comparison with your experimentally determined value.

3.4 Verification of Formula

Given Δλ = c/f - cf₀ and f = f₀ + ΔE/h, show that |Δλ| ≈ E λ₀²/hc, where λ₀ = c/f₀. The formula is valid provided ΔE << hf₀. Use the binomial expansion for (1 + ΔE/hf₀)^-1.

3.5 Discussion

Your discussion should address such topics as the nature of the normal and anomalous Zeeman effects and the electronic structure of the Cd and Hg atoms as they relate to the observed transitions.

3.6 Supplementary Information

Index of Refraction for Quartz

Figure 3: n vs. λ for Quartz (18°C)

Obersvations of Zeeman Splitting with Lummer-Gehrcke Plate

Figure 4: Appearance of three adjacent fringer.

Eq. (4)

Figure 5: Multiple reflection between the surfaces of a Lummer-Gehrcke plate. d for the plate in the lab is 4mm.

The Lummer-Gehrcke plate consists of an accuratly plane-parallel plate of glass or quartz 10 to 20 cm long, 1 or 2 cm wide, and a few millimeters thick. A prism is cemented on one end (P in Figure 4) so that the light man enter the plate at the proper angle without excessive loss of intensity by reflection. This angle is such that the angle of incidence on the inner surface is slightly less then the critical angle of total reflection. Thus at each reflection a ray of light leaves the surface at a nearly grasing angle. These rays are parallel, and are brought to a focus by the lens on the screen AD.

The fringes observed with the Lummer-Gehrcke plate are Haidinger fringes observed at an angle i near 90°, instead of near 0° as in the Fabry-Perot instrument. High reflecting power with consequent sharpness of the fringes is attained by the fact that near the critical angle the reflection is very strong. On the screen one observes two sets of fringes, one from each side of the plate.

Zeeman Patterns

View	Polarization	Normal Zeeman Effect (Cadmium red line)	Anomalous Zeeman Effect (Mercury green line)
B	PP || B (ΔM=0)
B	PPB (ΔM=±1)
|| B	(ΔM=1)
|| B	(ΔM=-1)

Length of lines shows approximate relative intensities of transitions. (Ref. Condon & Shortley, "The Theory of Atomic Spectra")

Normal Zeeman Effect for Cd Red Line

• Energy shift due to B calculated from ΔE = Mμ_BB, where Bohr magneton μ_B = eh / 2m_e = 9.273x10^-24 [J/T]
  
  
• Frequency corresponding to E₂₀ - E₁₀ is υ_o = (E₂₀ - E₁₀) / h.
  λ_o = c / υ_o = 643.85 nm.
  
  
• Wavelength shift from Δλ = (c / υ) - (c / υ_o) with υ = υ_o + ΔE/h.
  With the assumption that ΔE << hυ_o (equivalent to |Δλ| << υ_o),
  |Δλ| = (ΔE/hc) λ_o² = μ_Bλ_o² / hc.
  B in tesla [T = N/Am],

Polarization viewed || to B, the ΔM=0 transitions are not seen. The ΔM=+1 and ΔM=-1 transitions are circularly polarized, in opposite senses. Viewed B, lines are plane polarized; ΔM=0 transitions || B, ΔM=+1 transitions B.

Anomalous Zeeman Effect for Hg Green Line

• g = 1 + [J(J+1)+S(S+1)-L(L+1)] / [2J(J+1)]
  Note that for S=0; then J=L, and g=1. This gives the normal Zeeman effect.
  
  
• For |Δλ| << λ_o, |Δλ| = ΔEλ_o² / hc
  or write in the form |Δλ| = f(g₁,g₂)μ_BBλ_o² / hc
  For the normal Zeeman effect f(g₁,g₂) = 0 or 1

Transition ΔM = 0. ±1 only			Energy Shift from undisplaced line (E2 - E1) - (E20 - E10) ≡ ΔE	Polarization
				Viewed B	Viewed || B
M2	M1	M2-M1
1	0	+1	g2 μBB	PPB	CP
0	-1		g1 μBB	PPB	CP
-1	-2		(2g1 - g2) μBB	PPB	CP
1	1	0	(g2 - g1) μBB	PP || B	X
0	0		0	PP || B	X
-1	-1		(g1 - g2) μBB	PP || B	X
1	2	-1	(g2 - 2g1) μBB	PPB	CP
0	1		-g1 μBB	PPB	CP
-1	0		-g2 μBB	PPB	CP

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