The Zeeman Eect. March 12, 2007

Transcription

1 The Zeeman Eect Jasper Palfree Nigel Thorpe March 12, 2007 Abstract The Zeeman eect is investigated by exposing a neon lamp to a magnetic eld of up to ± 0.006) T with a large electromagnet. The light passes through a collimator and a Lummer-Gehrcke interferometer, and the energy split between the satellite dots of the Å line is calculated and plotted against the applied magnetic eld. The slope of this plot is 9.5±0.2) J/T ), which agrees within experimental error with the theoretically predicted value of this slope, the Bohr magneton, of J/T. The Anomalous Zeeman eect is also investigated by observing the energy split between satellite dots of the Å, Å, and Å lines. Comparing the ratio between the observed energy split and that predicted by the Normal Zeeman eect yields part of an Anomalous Zeeman pattern for each line which agrees roughly with accepted patterns. 1 Introduction The splitting of spectral lines due to the inuence of a magnetic eld was rst observed in 1896 by Pieter Zeeman [3]. This phenomenon rightfully dubbed, the Zeeman Eect) can be considered a cornerstone in the birth of modern quantum mechanics. At the time of its discovery, the Zeeman eect was explained using classical mechanics which give identical results to the corresponding quantum mechanical explanations which were to come later on. This way of explaining the Zeeman eect was acceptable until the discovery of a deviation in the predictions, this was dubbed the Anomalous Zeeman eect. What Prof. Zeeman had actually discovered was the eects of intrinsic spin on the electron state transitions which produce spectral lines. We intend to investigate the Zeeman eect by capturing photographs of the spectral lines from a Neon light under the inuence of magnetic elds of differing strengths. The light source is passed through a collimator, a Lummer-Gehrcke plate interferometer, and a constant deviation prism before being captured by a CCD camera with a powerful zoom lens. The resulting image appears as spectral lines split vertically into many dots which is analyzed with computer imaging software. Increasing the magnetic eld causes the observed dots to split, representing transitions between degenerate electron energy states in the sodium source, the degeneracy being broken by the applied eld. We also intend to investigate the eects of intrinsic spin on the spectral pattern the Anomalous Zeeman eect) and also use these results to calculate the value of the Bohr Magneton µ B ). 2 Theory 2.1 Atomic Spectra The orbital electrons of an atom reside in a discrete number of orbitals. The result of this is that orbital electrons are restricted to having a xed energy for a given orbital. When an electron undergoes a transition between orbitals, there must either be energy lost or gained by the electron in order to make this jump. This energy is emitted or absorbed in the form of a photon light). As the electrons make different jumps between energy levels, light of dierent frequencies are emitted producing emission spectra. The frequency of this photon ν) is related to its energy E) by, ν = E h where h is Plank's Constant [3]. 2.2 Orbital Electron States 1) Like energy, the angular momentum of an orbital electron must also be quantized. The electron wavefunction is an eigenstate of the angular momentum component of the hamiltonian with eigenvalue ll + 1) 2 corresponding to its orbital angular momentum. The quantum number m corresponds to 1

2 thing to do is to nd the hamiltonian for an electron. This can be easily done by beginning with a lagrangian of the form L = T V = 1 2 m ev 2 + e Φ A v). 4) The hamiltonian can be derived directly by applying the legendre transform, H = p ṙ L, where p = L ṙ, and rewriting the lagrangian in Einstein notation [2], L = 1 2 m eṙ i ṙ i + e Φ A i ṙ i ) 5) p i = L ṙ i = m e ṙ i ea i 6) Figure 1: Depiction of electron transitions between l = 2 and l = 1 states. Showing σ transitions corresponding to m = ±1 and the π transition corresponding to m = 0 [3]. the projection of the electron's orbital angular momentum along the z axis. The two quantum numbers l and m are called the angular momentum quantum number and the magnetic quantum number, respectively. They are both integers and additionally the value of m must neither be greater than the value of l nor less than the value of l [5]. In the neon atom, electrons may reside in higher energy excited) states with angular momentum quantum number l = 2 and lower energy states with l = 1. This means that in the excited states, the electrons may take on values of m given by and in lower energy states, m = { 2, 1, 0, 1, 2} 2) m j = { 1, 0, 1}. 3) The transitions between these states are depicted in gure 2.2) [3]. All orbital transitions correspond to changes in m of either m = ±1 called σ transitions) or m = 0 called π transitions). Now, in addition to taking some of the electron's energy, the photon emmited from a transition will acquire an angular momentum opposite to that of the change in the electron's angular momentum. The reason for this is simply that angular momentum must be conserved. Therefore, photons resulting from σ transitions will be circularly polarized [3]. 2.3 The Regular Zeeman Eect Now, we wish to investigate the eect of a magnetic eld on the emission spectrum of an atom. The rst H = p i r i L 7) ) pi + ea i = p i = = 1 2 m e m e ) ) pi + ea i pi + ea i m e m e )) pi + ea i e Φ A i m e 8) 1 p i + ea i ) p i + ea i ) + eφ 9) p + ea)2 eφ 10) 11) Now, in order to abide by the rules of quantum theory, the hamiltonian must be converted into an operator. This is done by replacing the momentum with it's operator form, p = i [3] eφ shall be abbreviated as V for simplicity). 1 H = i + ea) 2 + V 12) = e A + A) i + e2 A 2 + V 13) 14) By applying this to an arbitrary state, one may rewrite this hamiltonian as H = e A p + e m e i A + e2 A 2 + V. 15) Now, recall that the magetic eld B) can be expressed in terms of the vector potential A) with, B = A. 16) 2

3 One is free to choose the magnetic eld to be constant in the z direction, so we may choose the vector potential to be, A = B z 2 y, B ) z 2 x, 0. 17) Now, the hamiltonian reduces to [3] H = 2 2 e + B z x i y y ) x + e2 B 2 z 8m e x 2 + y 2) + V 18) Now one may change to polar coordinates, and realize that the rst term and very last term of equation 18) is simply the unperturbed hamiltonian. So now, x y y x = φ 19) V = H o 20) so that the hamiltonian now becomes e H = H o + B z i φ + e2 B 2 z 8m e r 2 r 2 cos φ) ) 21) If the magnetic eld is suciently small, last term of equation 21) is negligible [3] and the energy can be calculated by simply using the Schrödinger equation and replacing the wavefunction with, ψr) = R n,l r)e imφ P m l cos θ) 22) The energy is simply E = E 0) n + B z e m, l m l 23) where E n 0) is the unperturbed energy, and the term e / ) is remarkably the Bohr magneton µ B ) [3]. So the energy will, in fact, depend on the magnetic quantum number. In terms of an emission spectrum, this will correspond to spectral lines splitting since σ transitions will now create photons with energies diering by, E = ±B z µ B 24) which, of course, corresponds to the frequency change, ν = ± eb z 4πm e 25) 2.4 The Anomalous Zeeman Eect The term Anomalous is somewhat misleading in this context. Historically it was called this because at the time, physicists had not discovered intrinsic spin or spin-orbit coupling. The Anomalous Zeeman eect is perhaps more normal than the Regular Zeeman eect. In order to take intrinsic spin and spin-orbit coupling into eect, one must add two terms to the hamiltonian in equation 18). To take into account the eect of intrinsic spin in a magnetic eld [3], and for spin orbit coupling, e m e ŝ B 26) Ze 2 µ ) o ˆl ŝ 8πm 2 or 3. 27) The new terms here are the spin operator ŝ, the permeability constant in vacuum µ o and the nuclear charge Z. In the absence of a magnetic eld, since the hamiltonian will contain operators ˆl 2, ŝ 2 and ĵ 2 = ˆl 2 + ŝ 2 it is a good idea to construct wavefunctions that depend on the quantum numbers associated with these operators. The associated quantum numbers are: ĵ 2 j ˆl2 l ŝ 2 s ĵ z m j. The wave functions will now be composed of two seperable components, a radial contribution and an angular-spin contribution [3], ψ = ψ n,j,mj,l,s = Rr)Y Angle and Spin). 28) Now again, considering the total hamiltonian under the inuence of a small magnetic eld in the z direction with vector potential, A = 1 2 B y, 1 ) 2 B x, 0 29) = ˆl z, the hamiltonian now be- and replacing i comes [3], φ H = V + Ze2 µ o 8πm 2 or 3 ˆl ŝ ) + e Bˆl z + e m e ŝ z B. 30) 3

4 The last two terms may be rewritten as eb ) ˆlz + 2ŝ z = eb ĵz + ŝ z ). 31) In order to determine how this term acts on a wavefunction, consider this: ŝ z ĵ 2 = 1 ĵ2 ˆl 2 + ŝ 2) 32) 2ĵz ) = ŝ z ĵ2 x + ĵy 2 + ĵz 2 33) ) { ) = ĵ z ŝ ĵ + ŝ z ĵ x ĵ z ŝ x ĵx ) } + ŝ z ĵ y ĵ z ŝ y ĵy 34) The term in curly braces {} can be ignored for small magnetic elds [3]. Acting ŝ z ĵ 2 on an arbitrary wavefunction ψ yields, ŝ z ĵ 2 ψ = ŝ z 2 j j + 1) ψ 35) = 1 2ĵz 2 j j + 1) l l + 1) from which one may infer that ŝ z = +s s + 1))ψ 36) j j + 1) l l + 1) + s s + 1) ĵ z. 37) 2j j + 1) Now the energy shift can be determined to be [3], E = µ B B g m j 38) j j + 1) l l + 1) + s s + 1) g = ) 2j j + 1) which depends not only on m j but also the quantum numbers j, l and s. Fortunately measuerement of this energy is not necessary. Anomalous Zeeman patterns observed may be compared with predetermined patterns for dierent wavelengths Comparison of Anomalous Zeeman Patterns Patterns created by the Anomalous Zeeman eect have been recorded in a somewhat cryptic but informative way for various wavelengths. The Anomalous Zeeman eect results in σ or π transitions having various components shifted away from the Regular Zeeman transition dots [1]. The distances of these shifts from the Regular π transition dot are recorded in a numerical code. The rules of this numerical code are as follows: 1. The global denominator reprisents the reciprocal of the fractional unit used in measurement. Figure 2: Depiction of a hypothetical Anomalous Zeeman splitting corresponding to the pattern 0)1)2) Regular Zeeman transitions are labeled full dots and shifts are faded dots at multiples of 1/4 the Regular Zeeman split distance away from the π transition dot original). 2. The numerators represent the distance of the shift from the Normal π transition dot in the fractional units. 3. Numerators contained in brackets ) represent shifts in the π transition, numbers without represent shifts in the σ transitions. 4. If the original transitions are present, they are included. An example is the best way to explain the usage of this code. Figure 2) depicts a hypothetical Anomalous Zeeman splitting. The π transition has two shifts on either side and the σ transitions have one each on the far side. The numerical code for this pattern would be 0)1)2) ) since each split is a multiple of 1/4 the distance between the Regular σ and π transitions. 2.5 The Lummer-Gehrcke Plate Interferometer The Lummer-Gehrcke Plate Interferometer is essentially a moderately thick glass plate with a glass triangular prism attached to one end. The sides of the glass plate are perfectly parallel. Light is directed into this plate through the prism at one end and then undergoes total internal reection as it travels through the plate see gure 3)). At every internal reection some light escapes the prism. With propper tuning, the angle of this escaped light can be made to be in phase with the previous beam of escaped light. This results in the intensication of beam fringes of equal inclination [4]. 4

5 Figure 3: Diagram of the Lummer-Gehrcke Plate Interferometer, showing the angle of incidence of a lightray φ), total internal reection angle of incidence θ), and the thickness t Order of the Spectrum If the thickness of the plate is t and its refractive index is µ, then maxima occur when [1] nλ = µ AB + BC ) AD 41) = 2tµ sec θ 2t tan θ sin φ 42) = 2tµ cos θ 43) = 2t µ 2 sin 2 φ ; n Z 44) The order of the spectrum n may then be found given µ, λ and t are given Seperation of the Orders The way to nd the separation between the orders is rst by nding the change in angle of emergence φ between the n th and n + 1) th orders. Beginning with equation 44), squaring both sides and then differentiating with respect to n one obtains [1], n 2 λ 2 = 4t 2 µ 2 sin 2 φ ) 45) 2nλ 2 = 4t 2 sin φ cos φ dφ 46) dn nλ 2 dφ = 2t 2 dn 47) sin 2φ) δφ = λ µ 2 sin 2 φ t sin 2φ) ; n = 1 48) Determining the Change in Wavelength Dierentiating equation 44) with respect to λ this time gives, ) )) φ n 2 λ = 2t 2µ 2 sin 2φ) 49) λ ) φ 4t2 µ λ = n 2 λ 2t 2 50) sin 2φ) ) 4t 2 µ n 2 λ φ = 2t 2 λ 51) sin 2φ) inserting equation 48) results in [1] nλ 2 = λ m = ) ) 4t 2 µ n 2 λ λ 52) nλ 2 n 2 λ 4t 2 µ Resolving Power ) 53) The resolving power of the L-G interferometer can be easily found by using the angular seperation between two lines just clearly resolved, φ = λ L cos φ and substituting it into equation 51) to give [1], R = λ L λ = µ 2 sin 2 φ µλ λ sin φ 54) )). 55) Determining Wavelengths of Satellite Lines The main use of the L-G plate is to determine the wavelengths of so called satellite lines from the wavelength of a known line. If m is the distance between two observed orders of a known line and s is the measured distance of a satellite from the main line, one may write [1], λ s λ 2 = s λ m m λ 2 56) This result is useful for determining the value of the Bohr magneton from Regular Zeeman splitting. 2.6 Determination of the value of the Bohr Magneton Taking the value of the energy of a photon in terms of wavelength and dierentiating it with respect to time results in, E = hc λ 57) de = hc λ 2 58) E = hc λ s λ 2. 59) The value of the energy shift due to Regular Zeeman splitting may now be inserted from equation 5

6 angle and inclination are adjusted until the spectral lines can be seen. The light is redirected through a deviation prism 6), which can be rotated with the milled ring 7) until spectral lines are seen by the telescope 8). Callibrations are rst done to the deviation prism with the L-G plate removed and the slit widely open. When spectral lines are seen, the L-G plate is replaced and adjusted until the spectral lines can be seen again. When this is done, the slit width is narrowed until the lines become point-like and then the telescope is replaced with the camera. Figure 4: text) [1]. The experimental setup description in 24) along with the value for λ s from combining equations 53) and 56). B z µ B = hc λ s λ 2 60) = hc s nλ 2 ) λ 2 61) m n 2 λ 4t 2 µ = s m hc n ). 62) n 2 λ 4t 2 µ Now using the expression for n from equation 44), B z µ B = = ) shc 2t λ µ 2 sin 2 φ m 2t λ µ 2 sin 2 φ) 2 λ 4t 2 µ shc µ 2 sin 2 φ 2tm µ 2 sin 2 φ µλ 3 Setup and Procedure 63) ) )) 64) 3.1 Initial Conguration Using the Telescope The experimental setup is shown in gure 4). Major calibrations are done while viewing the spectral lines with the telescope. Firstly, the light from a neon lamp 1) was placed in the center of an electromagnet 2) using electrical tape to fasten it securely. The light from the neon lamp travels through the slit into the spectrometer's collimator 3). The collimator focuses the light into an L-G plate interferometer 5) 3.2 Image Capture Data obtained of the Zeeman splitting is captured by the camera 9 in gure 4), whose telemetric photo objective 10) is set to maximum and focused at in- nity. The computer 11) is then used to analyze the resulting pictures using a versatile open-source program called ImageJ. A raw image contains a large ammount of light pollution which originates mainly from the computer monitor. This is dealt with by rst taking a background image with the neon lamp o, which is then subtracted from the succeeding image with the neon lamp on. Once an acceptable image is taken, the experimentalists cheer, and then subsequent images are taken with varying magnetic eld intensity. 3.3 Calibration of the Magnet The magnetic eld's intensity is adjusted by its power generator 12 in gure 4), from which the voltage is measured by a voltmeter 13). The rst step is to determine the relation between the voltage supplied to the electromagnet coils and the resultant magnetic eld. This is required since it is prohibitively inconvenient to have a Hall probe alongside the neon lamp at all times throughout the experiment. Before inserting the lamp between the magnet coils we insert a Hall probe and measure the magnetic eld at several dierent supplied voltages, giving a linear relation allowing us to calculate the magnetic eld that the neon lamp is exposed to at any given driving voltage. The plot is given in gure gure 5). A linear t was applied to the data, giving the relation: B = ± )V ± 0.004)65) With this established, from here on we will only refer to the magnetic eld present when an image was taken rather than the voltage. 6

7 Figure 5: Magnet Calibration; Magnetic eld strength as a function of supplied voltage Figure 6: Refractive index of the Lummer-Gehrcke plate as a function of wavelength 3.4 Determining Index Of Refraction Of The Lummer-Gehrcke Plate The equation we will use to calculate the value of the Bohr magneton requires the interferometer's index of refraction µ as a function of wavelength. A table of values for the refractive index is given, but not at the exact wavelength of the neon emission lines we wish to study. We therefore plot the values given over the wavelength region of interest and t it to a second order polynomial, as shown in gure 6). The curve t gives us the result: µλ) = ± 0.005) λ ± 0.05) 10 5 λ From which directly follows: ± 0.002) 66) = ± 0.005) 1011 λ 1.66 ± 0.05) ) Figure 7: Theoretical Neon Emission Spectrum [Modied from All measures in Å ) 3.5 Classifying The Experimental Emission Spectrum Using the ImageJ software a spectrum is obtained for the neon lamp. This is qualitatively compared to a theoretical neon spectrum as shown in gure 7). The experimentally determined spectrum corresponds remarkably well with the theoretical, and it is given with the correspondingly identied emission lines in gure 8). Using this graph it is easy to identify which lines in our images correspond to which wavelengths by simply counting lines from the readily identiable Å or Å lines. 4 Data And Analysis 4.1 Determining The Value Of The Bohr Magneton The equation used to determine the Bohr magneton was derived in the theory as equation 64), and it is repeated here for convenience: 7

8 Figure 8: Experimental Emission Spectrum All measures in Å ) B z µ B = shc µ 2 sin 2 φ 2tm µ 2 sin 2 φ µλ )) 68) For this section the Å emission line is used as it exhibits the Normal Zeeman eect. That is, its Zeeman pattern is given by: 0)1 69) 1 The appropriate line is identied by comparison with gure 8), and an example image with the appropriate line identied is given in gure 9) From the theory the value of s appearing in equation 68) is the distance between 2 satellite dots and the main dot. Unfortunately the main dot was never visible in any of our images, so the value of s was determined by taking one half of the distance between the two satellites, in eect assuming that the main dot would appear directly in between them. The value used in calculations was taken to be the average of this distance for the multiple pairs of dots in the image. The measurement itself was done by again using the ImageJ software, which allows ne measuring of the distance between intensity peaks in an intensity line prole of an image, an example of which is given in gure 10). The value of m is similarly taken to be the average of the distance between two main dots as though they were directly between the satellites. It is worth noting that an important source of error is due to the fact that these distances can naturally only be measured to the nearest pixel, so our accuracy will necessarily be constrained by the camera's resolution. Plugging the average values for s/m as determined from the images taken at dierent B elds into equation 68) gives values for E which can be plotted Figure 9: Sample image taken at B = ± 0.005) T accenting the line of interest Figure 10: A sample of an intensity line prole to determine distances s and m on the Å line 8

9 B T) Average S/M E J) ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± 0.1 Table 1: Data Values For Plot To Calculate Bohr Magneton Figure 11: Split in energy against applied magnetic eld to determine the value of the Bohr magneton against the magnetic eld strength B. Table 1) tabulates the relevant data. This is then plotted and a linear t is applied, as in gure 11). The linear t gives: the other two lines. The analysis proceeds similarly to that used in the preceeding section, though a value for E need only be determined at one B value for comparison with the corresponding Normal Zeeman value. It is clearly desirable to perform the analysis at the highest B value possible so that the split between satellites is more apparent, which may vary depending on the line being analyzed. However, it is vitally important to ensure that in the image being analyzed the splitting pair has not crossed over the satellite splitting from the adjacent order. It is easy to fall into the trap of thinking that there are triplets or other interesting Zeeman patterns present when really the satellites from adjacent orders are merely overlapping, as this will give eroneous values. The observed values for s/m at each of the three wavelengths, along with their corresponding observed and Normal Zeeman eect E values, observed Zeeman patterns, and theoretical Zeeman patterns are given in table 2). The observed Zeeman pattern is found by simply taken the ratio of the observed E to that observed for the Normal Zeeman eect. The observed patterns match the theoretical patterns fairly well; the satellite observed for appears to be the expected π dot, the rst satellite for Å appears to be the rst σ dot and the second appears to be the rst π. The satellite observed for the Å line is the anomoly of the group, and may in fact be one of the σ dots expected at 24/30 or 34/30. E = 9.5 ± 0.2) B 70) The Bohr magneton is simply the slope of this plot as per equation 68), and the experimentally determined value of 9.5 ± 0.2) J/T ) agrees with the accepted value of J/T within experimental error. 4.2 The Anomalous Zeeman Eect Many lines were observed to split in the images obtained. However, of those only three are consistently discernable and have an anomalous Zeeman pattern suciently dierent from the normal one to have a noticable eect. These are the lines corresponding to wavelengths of Å, Å, and Å. In the case of Å, two pairs of satellites were observed to split from the central point, whereas there was only one splitting pair for 9

10 B T) Λ nm) Satellite Average S/M E Observed J) E Normal J) Observed Pattern Theoretical Pattern[1] ± π1.294 ± ± ± ± 1) 30 0)15) ± σ1.332 ± ± ± ± 1 22 ± 1) ) π1.155 ± ± ± ± π1.228 ± ± ± 0.1 Table 2: Experimental and theoretical anomalous Zeeman eect patterns 29 ± )10)

11 5 Conclusion We studied the Zeeman eect by analyzing images taken of a neon spectrum in a varying magnetic eld after it passed through an L-G interferometer. The resulting energy split was measured for the Å line and was plotted against magnetic eld strength to obtain a value for the Bohr magneton of 9.5 ± 0.2) J/T ), in agreement with the accepted value of J/T. In these splitting patterns the main dots were conspicuously absent even though theory predicts their presence. This is likely due to limitations of the current apparatus. We also investigated the Anomalous Zeeman eect by comparing the energy split in the satellites of the Å, Å, and Å lines to the energy split predicted by the Normal Zeeman effect. The resulting patterns agreed relatively well with those that were expected. While performing the experiment there were a few things that were noted. First, when viewing captured images on the PC in the lab room only a few spectral lines appeared to be visible. The images did not appear to be poor, but openning the same image les on other machines, particularly with decent LCD monitors, revealed much more detail than was displayed on the lab PC's monitor. It came as a bit of a shock to see that our data was actually much better than we had thought. A side-by-side comparison of the author's laptop and the lab machine both displaying the same image through the same software reveals the dierence dramatically. Another thing worth noting was the discovery of ImageJ, a surprisingly powerful, cross-platform open-source image analysis program. Although EyeSpy is needed for image capture, we nd this free alternative superior for analyzing the images once they have been taken. 7 Bibliography References [1] Zeeman lab manual. [2] Herbert Goldstien, Charles Poole, and John Safko. Classical Mechanics. Addison Wesley, [3] H. Haken and H.C. Wolf. The Physics of Atoms and Quanta: Introduction to Experiments and Theory. Springer Berlin Heidelberg New York, [4] R.S. Longhurst. Geometrical and Physical Optics. Longmans, Green and Co Ltd, [5] John S. Townsend. A Modern Approach to Quantum Mechanics. University Science Books, Acknowledgements For helping us with many real and virtual technical diculties we would like to thank Saverio Biunno. For his guidance and experience we would like to thank Fritz Buchinger. We would also like to thank Roland Bennewitz for his theoretical knowledge and suggestion of an excellent reference book. Last but not least, we would like to thank our amazingly helpfull T.A., Roxanne, for getting us through this experiment. 11

12 A Image Data negatives) No Magnetic Field 6.00 Volts Progressively Increasing Magnetic Feild 7.99 Volts 1.99 Volts 4.00 Volts Volts 12

13 14.00 Volts Volts Volts 13