Zeeman Effect and Lande g-factor

Indian Institute of Science Education and Research, Kolkata 3rd – 10th March 2010

1 | Z e e m a n E f f e c t , E x p t . – 4

Zeeman Effect and Lande g-factor

Harsh Purwar (07MS-76) Amit Nag (07MS-19)

VIth Semester, Integrated M.S. Indian Institute of Science Education and Research, Kolkata

Experiment No.: 4 Condensed Matter Physics Lab (PH – 324)

Objectives of the Experiment: To observe anomalous Zeeman effect in Mercury atoms from the light emitted by the hot mercury

atoms by the application of a strong magnetic field and using a Fabry-Perot interferometer.

To find an estimate of the Lande g-factor 𝑔1 − 𝑔0 for a transition from a state labeled 0 to 1. Here we observe a transition which lies close to 546.1 nm that is from 73𝑆1 to 63𝑃2 in Hg atoms.

Introduction: The Zeeman Effect is the splitting of a spectral line into several components in the presence of a

static magnetic field. This effect is named after the Nobel laureate Dutch physicist Pieter Zeeman (1865-1943), who got Nobel for discovery of this effect. Zeeman observed in 1896 that sodium's spectral pair of D-lines would split when placed in an intense magnetic field in a laboratory. Working under his mentor Lorentz, Zeeman confirmed Lorentz's model for electron orbitals by constructing a model for his observations in the context of electron angular momentum. In addition, Zeeman was able to demonstrate applications of his method such as measuring the magnetic fields around the sun by observation of spectra at different points on the surface. This model was later modified to fit a quantum mechanical model which utilized a meta-state to describe the energy splitting found when placing an atom in a magnetic field.

The Zeeman Effect is very important in applications such as nuclear magnetic resonance spectroscopy (NMR), electron spin resonance spectroscopy (ESR), magnetic resonance imaging (MRI) and Mossbauer spectroscopy. It has also been utilized in Atomic absorption spectroscopy to improve its accuracy.

Theory: As mentioned above Zeeman effect is the breaking of degeneracy in atomic energy levels due to

the interaction between the magnetic moment of an atom and an external magnetic field. The strength of this interaction in each energy state depends upon the total angular momentum of the atom, given by,

𝐽 = 𝐿 + 𝑆

where 𝐿 and 𝑆 are the orbital and spin angular momenta, respectively, and were 𝐽 can take values

from 𝐿 − 𝑆 to 𝐿 + 𝑆 . The z-component of 𝐽 is labeled by the quantum number 𝑚𝐽 , and is restricted

to discrete values, in integer increments, in the range given by

− 𝐽 ≤ 𝑚𝐽 ≤ 𝐽

Thus, there are 2 𝐽 + 1 allowed orientations of 𝐽 . In zero magnetic field there is no energetically

preferred orientation of 𝐽 , and the energy states of the atom are said to be degenerate in 𝑚𝐽 , which means

they are all the same. The presence of an external magnetic field defines a preferred axis, namely the direction of the

field. Each allowed orientation of 𝐽 with respect to this axis is associated with a different energy, thus



breaking the degeneracy. An easily observable result of this is the splitting of a spectral line into several component lines, the number of which depends upon the spin and orbital angular momenta in the initial and final energy states involved in producing the spectral line.

The so-called normal Zeeman effect refers to the splitting of a spectral line into three components - a phenomenon which was explained classically in terms of changes in frequency of orbit of an electron in the presence of a magnetic field. This effect is observed in spectral lines for which the initial and final energy states have zero spin angular momentum.

Figure 1: Normal Zeeman effect – only three lines of different frequencies are observed.

By contrast, the anomalous Zeeman effect refers to the splitting of spectral lines into more than three components. The term `anomalous' was given because the phenomenon, which had been observed since the late 1890s, was unexplained until the discovery of electron spin some thirty years later.

The energy of interaction between a magnetic moment 𝜇 and an applied magnetic field 𝐻 is given by,

𝐸mag = 𝜇 . 𝐻

As customary, we choose the z-axis of our coordinate system to be along the direction of the applied

magnetic field 𝐻 which then gives, 𝐸mag = 𝜇𝑧𝐻

or in terms of the z-component of the angular momentum 𝐸mag = 𝑔𝜇0𝑚𝐽𝐻………… 1

where 𝑔 is the ratio of the magnetic moment of the whole atom to the fundamental constant called the Bohr magneton, 𝜇0. (The minus sign has disappeared since for electrons the direction of the magnetic moment is opposite to the direction of the angular momentum.)

The 𝑔-factor for a free atom in a given state 2𝑆+1 𝐿𝐽 can be calculated according to,

𝑔 = 1 +𝐽 𝐽 + 1 − 𝐿 𝐿 + 1 + 𝑆 𝑆 + 1

2𝐽 𝐽 + 1 ………… 2

As mentioned earlier, spectral emission lines which exhibit the normal Zeeman effect involve

transitions between energy states which have zero spin angular momentum, 𝑆 = 0. Thus, the orbital angular momentum, 𝐿, and the total angular momentum are identical, and we have 𝑔 = 1 for both energy states. The energy shift for a given value of 𝑚𝐽 , given by Equation (1), is the same in both the upper and



the lower energy state. In the electric dipole transition process (which is the dominant process for this source) the change in 𝑚𝐽 is restricted according to the selection rule, Δ𝑚𝐽 = −1,0, +1.

As a result, when a magnetic field is applied the zero field line is observed to split into only three separate components (each of which still have degeneracy). Spectral lines which exhibit the anomalous Zeeman effect involve energy states with non-zero spin angular momentum, 𝑆 ≠ 0. Electric dipole transitions do not involve a change in spin, but do involve a change in orbital angular momentum according to the selection rule, Δ𝐿 = ±1 Because of this, the 𝑔-factors will be different for the upper and lower energy states, and the resulting splitting of the zero field line in the presence of a magnetic field may be more complex.

Figure 2: Anomalous Zeeman effect – 9 lines of different frequencies are observed.

Fabry-Perot Interferometer: The apparatus focuses on the ability to read small changes the energy of photons. This can be

achieved by employing a interferometer, in general a device which utilizes the interference effects of light to measure small changes in the energy per photon of that light. In fact, interferometry is not limited to merely photons, but can be achieved with any phase dependent quantum system such as a mass interferometer which can theoretically measure incredibly small changes of acceleration in a system, better than 10-5 m/s2. To observe the Zeeman effect in mercury, however, we concern ourselves primarily with the 546.1 nm, 73𝑆1 → 63𝑃2 transitions. In this case we employ a Fabry-Perot interferometer to create high-resolution interference patterns suitable for the measurement.

The Fabry-Perot interferometer is a special type of interferometer that utilizes two partially reflective glass plates placed closely together with a layer of air in between. The glass plates may be coated with a thin layer of silver in order to increase reflectivity between the two plates. These plates, when placed exactly parallel to each other, create interference patterns by reflecting light multiple times within the air cavity. Much care must be taken in aligning the plates properly to acquire useful data using a kinematic mount to hold one of the plates. Ideally, when aligned, the interference pattern should have little dependence on the angle at which one views the image.



Experimental Setup: A schematic diagram of the experimental setup to observe Zeeman Effect is shown below.

S: Light Source (mercury lamp in our case). H: Electromagnets properly connected to the power supply. L1, L2: Convex lenses of appropriate focal length so as to obtain an intense & better resolved fringe patterns. P: Polarizer properly set so as to allow only 3 pi lines to pass through. F1: IR filter. F2: Filter, allows only wavelengths close to 546.1 nm (corresponding to 3 sigma lines) to pass through. FP: Fabry-Perot etalon or interferometer with the distance between the reflecting surfaces = 3.995 mm. C: Charge coupled device or CCD or video camera.



Procedure: The magnetic field was first calibrated to determine a relationship between the power supply

current (𝐼) and the magnetic field 𝐵 produced by the electromagnets using the Hall probe.

The various optical components were fitted on an optical bench/train and were aligned properly along a common axis. This could easily be done by turning on the mercury lamp and after a few minutes when it is warmed up (about 5 minutes), the path of the beam is probed with a small piece of white paper/card to check that the beam is passing through the center of each optical component.

After proper alignment and adjustment, circular interference fringes could be seen on the computer screen as shown in Figure 3 using a charge coupled device (CCD) or a sensitive video camera.

Figure 3: Interference pattern with approximately zero magnetic field.

The interference fringe patterns are recorded/saved by varying the magnetic field or more specifically by varying the current through the electromagnets say in steps of 0.5 A.

These fringe patterns for various magnetic fields are then analyzed using a small MATLAB script attached at the end of this report for the values of the following parameters:

o 𝛿𝑎𝑏 is the difference of the squares of the radial distance between the lines corresponding to a fringe. 𝛿𝑎𝑏 = 𝑏2 − 𝑎2

o 𝛿12 is the difference of the squares of the radial distance between the two main fringes. 𝛿12 = 22 − 12

Data collected is listed in a tabular fashion in the following section.



Figure 4: Interference pattern showing the splitting of the main fringes into 3 distinct fringes for magnetic field

corresponding to 3.63 amperes of current flowing through the electromagnets.

Figure 5: Intensity profile of the interference pattern obtained above for magnetic field corresponding to 3.63

amperes of current flowing through the electromagnets.



Observations: Table 1: For calibration of the magnetic field produced by the electromagnets with current passing through them.

Obs. No. Current through the Electromagnets (A) Magnetic Field (Gauss)

1 0 570

2 0.23 1070

3 0.44 1580

4 0.6 1980

5 0.81 2540

6 1.06 3230

7 1.25 3790

8 1.42 4280

9 1.61 4830

10 1.9 5630

11 2.11 6200

12 2.32 6760

13 2.42 7010

14 2.72 7790

15 2.96 8390

16 3.15 8880

17 3.39 9440

18 3.55 9830

Above calibration data was fitted linearly using 𝑦 = 𝑚𝑥 + 𝑐 model to get, 𝑚 = 2672, 𝑐 = 469.3

𝑦 = 2672𝑥 + 469.3



Table 2: For recording the position of the fringes (Attached MATLAB script was used for this) and corresponding magnetic field (H).

S. # Current

(A) Corr. M. Field

(Gauss) Centre

(Pixel #) a

(Pixel #) b

(Pixel #) c

(Pixel #)

𝜹𝒊𝒋

𝒓𝒋𝟐 − 𝒓𝒊

𝟐

𝜹𝒂𝒃

𝒓𝒃𝟐 − 𝒓𝒂

𝟐

𝜹𝒃𝒄

𝒓𝒄𝟐 − 𝒓𝒃

𝟐

1

3.63 10169 358

600 626 646 *** 13260 11120

2 702 719 736 58497 11985 12563

3 781 795 809 60648 12040 12432

4 844 857 870 58032 12805 13143

5 898 911 925 56808 14209 15680

6 954 965 977 62640 13233 14712

Average 59325 12922 13275

1

3.50 9821 351

598 626 645 *** 14616 10811

2 706 721 737 61275 10875 12096

3 778 794 807 59349 13920 11687

4 846 857 867 59787 11011 10220

5 954 962 976 117285 9712 17304

Average 74424 12027 12424

1

3.19 8993 338

443 472 502 *** 6931 8940

2 602 625 642 64413 12673 10047

3 706 718 731 62031 8976 10049

4 843 856 869 123924 13299 13637

5 1049 1058 1068 250076 12879 14500

Average 125111 10952 11435

Figure 6: The graph displays the relation between the strength of the magnetic field and both 𝛿𝑎𝑏 and 𝛿𝑏𝑐 .

𝑦 = 1.366 × 10−4 𝑥 + 1.118

𝑦 = 1.343 × 10−4 𝑥 − 1.102



As mentioned earlier the distance 𝑑 between the two reflecting surfaces in the Fabry-Perot interferometer = 3.995 mm or 𝑑 = 0.3995 cm.

S. # Magnetic Field (H)

(Gauss) 𝛅𝐢𝐣 𝛅𝐚𝐛 𝛅𝐛𝐜

𝚫𝛎𝐚𝐛 = 𝛅𝐚𝐛

𝟐𝐝 𝛅𝐢𝐣

(cm-1)

𝚫𝛎𝐛𝐜 = 𝛅𝐛𝐜

𝟐𝐝 𝛅𝐢𝐣

(cm-1)

1 10169 59352 12922 13275 0.274 0.280

2 9821 74424 12027 12424 0.202 0.210

3 8993 125111 10952 11435 0.110 0.114

From the above fit we have,

Δ𝜈𝑎𝑏 = 1.343 × 10−4 cm−1Gauss−1

Δ𝜈𝑏𝑐 = 1.366 × 10−4 cm−1Gauss−1 So we have,

𝑔1 − 𝑔0 =Δ𝜈𝑎𝑏 + Δ𝜈𝑏𝑐

2= 1.354 × 10−4 cm−1Gauss−1

Whereas the theoretical value of 𝜇0 ℎ𝑐 = 4.669 × 10−5 cm−1.

There is a good theoretical reason for this discrepancy involving the electronic spin in the system. In the approximation given in the lab manual, spin of the electrons was not taken into account, leaving the value S = 0. In reality, the strength of the spin magnetic dipole is of the same order of magnitude as the angular momentum dipole. The net consideration of both spin and angular dipoles is called the Lande g-factor, given by the equation,

𝑔𝐽 𝑛, 𝑙, 𝑠, 𝑗 = 1 +𝐽 𝐽 + 1 − 𝐿 𝐿 + 1 + 𝑆 𝑆 + 1

2𝐽 𝐽 + 1

From this equation, we can make a better approximation for 𝑔1 − 𝑔0 as,

𝑔𝐽 7,0,1,1 − 𝑔𝐽 6,1,1,2 = 1 +2 + 2 − 0

4 − 1 +

2 + 6 − 2

12 =

1

2

Thus, we end up with a factor of (1/2) for the difference in the splitting, giving a more acceptable expected value of 2.335 × 10−5 cm−1Gauss−1. This value therefore is well within the bounds of error of the experimental value.

Conclusions: The anomalous Zeeman effect was observed and from the observations calculated value of the

Lande g-factor was found to be 1.354 × 10−4 cm−1Gauss−1.

References: Supplied User manuals and reading material.

Wikipedia.org (http://en.wikipedia.org)



Designed MATLAB Scripts: Script 1: Following MATLAB script plots the intensity profile of the interference pattern of all the images whose filenames are contained in the variable ‘Fil’ and allows user to determine approximate centre of the circular fringes to calculate the square radius. Fil = ['3.63';'3.50';'3.19';'2.90';'2.60';'2.28';'1.99']; for aa = 1:length(Fil)

im = imread([Fil(aa,:),'.bmp']); im = double(im); data = im(480,:,2); data = smooth(data,3); figure; plot(data); grid; title(['For Current = ',Fil(aa,:)]); l1 = input('Enter the lower limit: '); l2 = input('Enter the upper limit: '); [A,B] = min(data(l1:l2)); fprintf(['Center is at: ',num2str(B+l1),'\n']);

end

Script 2: Following script is a bit complex and can be used for the determination of the centre of the fringes more accurately. It uses the Image Processing Toolbox of MATLAB and determines centre by fitting a circle on the boundary of the fringe. im=imread('<filename>'); gr=rgb2gray(im); the=graythresh(gr); bw=im2bw(gr,the); % bw=-(bw-1); bw=imfill(bw,'holes'); imshow(im); hold on; fprintf('Enter the coordinates of approx. center (determined by eye estimation)\n'); c=input('x-coordinate: '); r=input('y-coordinate: '); init_val=bw(r,c); val=init_val; j=c;i=r; while(val==init_val) i=i-1; val=bw(i,j); end x1=j; y1=i+1; B=bwtraceboundary(bw,[y1,x1],'SE'); x=B(:,2); y=B(:,1); abc=[x,y,ones(length(x),1)]\-(x.^2+y.^2); a=abc(1); b=abc(2); c=abc(3); xc=-a/2; yc=-b/2; plot(xc,yc,'.r'); hold off; fprintf('Center is: (%d,%d)\n',round(xc),round(yc));

Zeeman Effect and Lande g-factor

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