Atomic Physics Chapter 4 Fine structures of Atoms 11/20/1316/11/2018 Jinniu Hu 1 !1 7.4 Magnetic Effects on Atomic Spectra—The Normal Zeeman Effect 253 7.4 Magnetic Effects on Atomic Spectra— The Normal Zeeman Effect It was shown as early as 1896 by the Dutch physicist Pieter Zeeman that the spec- tral lines emitted by atoms placed in a magnetic field broaden and appear to split. The splitting of an energy level into multiple levels in the presence of an external magnetic field is called the Zeeman effect. When a spectral line is split into three lines, it is called the normal Zeeman effect. But more often a spectral line is split into more than three lines; this effect is called the anomalous Zeeman effect. The normal Zeeman effect, discussed here, can be understood by consid- ering the atom to behave like a small magnet. We will return to our discussion of the anomalous Zeeman effect, which is more complicated, in Section 8.3. By the 1920s considerable fine structure of atomic spectral lines from hydrogen and other elements had been observed. Fine structure refers to the splitting of a spectral line into two or more closely spaced lines. Meggers AIP/Emilio Collection. Segrè Visual F. Archives, W. As a rough model, think of an electron circulating around the nucleus as a circular current loop. The current loop has a magnetic moment m ϭ IA where A The Dutch physicist Pieter is the area of the current loop and the current I ϭ dq/dt is simply the electron Zeeman (1865– 1943) studied at ϭ Ϫ the University of Leiden under the charge (q e) divided by the period T for the electron to make one revolution famous physicists H. Kamerlingh (T ϭ 2pr/v). Thus Onnes and H. A. Lorentz and 2 received his degree in 1890. While q Ϫe pr Ϫe r v e at Leiden he showed that atomic m ϭ IA ϭ A ϭ ϭ ϭϪ L (7.25) T 2pr /v 2 2m spectral lines were split under the 1 2 influence of an applied magnetic where L ϭ mvr is the magnitude of the orbital angular momentum. Both the field. After this discovery he left magnetic moment m and angular momentum L are vectors so that Leiden in 1897 to go to the Univer- sity of Amsterdam, where he re- e mained until 1935. He shared the m ϭϪ L (7.26) 1902 Nobel Prize for Physics with 2m his mentor Lorentz. The relationship between m and L is displayed in Figure 7.4. In the absence of an external magnetic field to align4.1Magnetic them, the magneticmoment of electron moments m of atoms point in random directions. In classical electromagnetism, As a rough model, think of an electron circulating around if a magnetic dipole having a magnetic moment m is placed in an external mag- netic field, the dipole will experience a torque t ϭ m ϫthe B tendingnucleus asto aligna circular the current loop. The current loop has a magnetic moment dipole with the magnetic field. The dipole also has a potential energy VB in the L field given by q µ = IA = A T VB ϭϪm # B (7.27) e⇡r2 erv = − = − Proton According to classical physics, if the system can change its potential2⇡r/v energy,2 the magnetic moment will align itself with the external magnetic fielde to minimize L e ؊ = energy. −2m m Note the similarity with the case of the spinning top in a gravitational field. A child’s spinning top is said to precess about the gravitationalwhere L=mvrfield; thatis the is, magnitudethe of the orbital angular Figure 7.4 Representation of axis (around which the spinning top is rotating) itself rotatesmomentum. about the direction the orbital angular momentum L of the gravitational force (vertical). The gravitational field is not parallel to the and magnetic moment m of the 16/11/2018 Jinniu Hu angular momentum, and the force of gravity pulling down on the spinning top hydrogen atom due to the elec- results in a precession of the top about the field direction. Precisely the same thing tron orbiting the proton. The di- happens with the magnetic moment of an atom in a magnetic field. The angular rections of L and m are opposite momentum is aligned with the magnetic moment, and the torque between m because of the negative electron and B causes a precession of m about the magnetic field (see Figure 7.5), not an charge. 03721_ch07_241-271.indd 253 9/29/11 4:44 PM Magnetic moment of electron Both the magnetic moment and angular momentum are vectors so that e µ~ = L~ −2m Gyromagnetic ratio e γ = 2m In classical electromagnetism, if a magnetic dipole having a magnetic moment, is placed in an external magnetic field, the dipole will experience a torque ~⌧ = µ~ B~ ⇥ which aligns the dipole with the magnetic field. 16/11/2018 Jinniu Hu Magnetic moment of electron The dipole also has a potential energy in the field given by V = µ~ B~ B − · According to classical physics, if the system can change its potential energy, the magnetic moment will align itself with the external magnetic field to minimize energy. Precession in gravitational field 16/11/2018 Jinniu Hu 254 Chapter 7 The Hydrogen Atom alignment. The magnetic field establishes a preferred direction in space along which we customarily define the z axis. Then we have e U m ϭ m ϭϪm m (7.28) z 2m / B / Bohr magneton where mB ϭ e U/2m is a unit of magnetic moment called a Bohr magneton. Be- 1 cause of the quantization of Lz and the fact that L ϭ / / ϩ 1 U Ͼ m/ U, we cannot have Έm Έ ϭ m ; the magnetic moment cannot align itself exactly in the z z 1 2 direction. Just like the angular momentum L, the magnetic moment m has only certain allowed quantized orientations. Note also that in terms of the Bohr magneton, m ϭϪmBL / U. Magnetic moment of electron EXAMPLE 7.5 An atom with magnetic moment in an external magnetic field has Larmor precession. Determine the precessional frequency of an atom having B z magnetic moment m in an external magnetic field B. This precession is known as the Larmor precessionPrecessional. frequency m u dφ 1 dL ! = = Strategy We have already seen that the torqueL t is dtequal L sin ✓ dt dL = L sin ✓dφ to m ϫ B, but we also know from classical mechanics that the torque is / . The torque in Figure 7.5 is perpendicular to dL dt Newton’s second law in angular form u m, L, and B and is out of the page. This must also be the dL~ L direction of the change in momentum dL as seen in Figure~⌧ = 7.5. Thus L and m precess about the magnetic field. The dt Larmor frequency vL is given by df/dt. So L sin u f e eB ! = µB sin ✓ = dL df L 2mµ sin ✓ 2m Solution The magnitude of dL is given by L sin ✓u df (see ◆ Figure 7.5), so vL is given by 16/11/2018 Figure 7.5Jinniu An Hu atom having magnetic moment m feels a torque df 1 dL t ϭ m ϫ B due to an external magnetic field B. This torque must v ϭ ϭ (7.29) L dt L sin u dt also be equal to dL/dt. The vectors m and L are antiparallel, so the vector dL/dt must be perpendicular to m, B, and L. As shown in We now insert the magnitude of L ϭ 2mm/e from Equation the figure, dL/dt requires both m and L to precess (angle f) about (7.26). The value of dL/dt, the magnitude of m ϫ B, can be the magnetic field B. determined from Figure 7.5 to be mB sin u. Equation (7.29) becomes e eB vL ϭ mB sin u ϭ (7.30) 2m m sin u 2m a b What about the energy of the orbiting electron in a magnetic field? It takes work to rotate the magnetic moment away from B. With B along the z direction, we have from Equation (7.27) VB ϭϪmzB ϭϩmBm/B (7.31) The potential energy is thus quantized according to the magnetic quantum number m/; each (degenerate) atomic level of given / is split into 2/ ϩ 1 different energy states according to the value of m/. The energy degeneracy of a given n/ level is 03721_ch07_241-271.indd 254 9/29/11 4:44 PM Magnetic moment of electron Orbit angular momentum operator Lˆ =ˆr pˆ ⇥ The components of orbit angular moment ˆ @ @ Lx =ˆypˆz zˆpˆy = i~ y z − − @z − @y ✓ ◆ ˆ @ @ Ly =ˆzpˆx xˆpˆz = i~ z x − − @x − @z ✓ ◆ ˆ @ @ Lz =ˆxpˆy yˆpˆx = i~ x y − − @y − @x ✓ ◆ The square of orbit angular momentum operator ˆ2 ˆ2 ˆ2 ˆ2 L = Lx + Ly + Lz 16/11/2018 Jinniu Hu Magnetic moment of electron Eigenstate and eigenvalue Qfˆ = λf f is the eigenstate and ! is the corresponding eigenvalue. The eigenstate and eigenvalue of orbit angular moment ˆ2 2 2 L Ylm = L Ylm = l(l + 1)~ Ylm ˆ LzYlm = ml~Ylm l is the quantum number of orbit angular momentum ml is the magnetic quantum number 16/11/2018 Jinniu Hu 7.3 Quantum Numbers 251 Magnetic Quantum Number mO The orbital angular momentum quantum number / determines the magnitude of the angular momentum L, but because L is a vector, it also has a direction.