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University Physics Volume 3

8.2 Orbital Magnetic Dipole Moment of the Electron

University Physics Volume 38.2 Orbital Magnetic Dipole Moment of the Electron
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  1. Preface
  2. Unit 1. Optics
    1. 1 The Nature of Light
      1. Introduction
      2. 1.1 The Propagation of Light
      3. 1.2 The Law of Reflection
      4. 1.3 Refraction
      5. 1.4 Total Internal Reflection
      6. 1.5 Dispersion
      7. 1.6 Huygens’s Principle
      8. 1.7 Polarization
      9. Chapter Review
        1. Key Terms
        2. Key Equations
        3. Summary
        4. Conceptual Questions
        5. Problems
        6. Additional Problems
        7. Challenge Problems
    2. 2 Geometric Optics and Image Formation
      1. Introduction
      2. 2.1 Images Formed by Plane Mirrors
      3. 2.2 Spherical Mirrors
      4. 2.3 Images Formed by Refraction
      5. 2.4 Thin Lenses
      6. 2.5 The Eye
      7. 2.6 The Camera
      8. 2.7 The Simple Magnifier
      9. 2.8 Microscopes and Telescopes
      10. Chapter Review
        1. Key Terms
        2. Key Equations
        3. Summary
        4. Conceptual Questions
        5. Problems
        6. Additional Problems
    3. 3 Interference
      1. Introduction
      2. 3.1 Young's Double-Slit Interference
      3. 3.2 Mathematics of Interference
      4. 3.3 Multiple-Slit Interference
      5. 3.4 Interference in Thin Films
      6. 3.5 The Michelson Interferometer
      7. Chapter Review
        1. Key Terms
        2. Key Equations
        3. Summary
        4. Conceptual Questions
        5. Problems
        6. Additional Problems
        7. Challenge Problems
    4. 4 Diffraction
      1. Introduction
      2. 4.1 Single-Slit Diffraction
      3. 4.2 Intensity in Single-Slit Diffraction
      4. 4.3 Double-Slit Diffraction
      5. 4.4 Diffraction Gratings
      6. 4.5 Circular Apertures and Resolution
      7. 4.6 X-Ray Diffraction
      8. 4.7 Holography
      9. Chapter Review
        1. Key Terms
        2. Key Equations
        3. Summary
        4. Conceptual Questions
        5. Problems
        6. Additional Problems
        7. Challenge Problems
  3. Unit 2. Modern Physics
    1. 5 Relativity
      1. Introduction
      2. 5.1 Invariance of Physical Laws
      3. 5.2 Relativity of Simultaneity
      4. 5.3 Time Dilation
      5. 5.4 Length Contraction
      6. 5.5 The Lorentz Transformation
      7. 5.6 Relativistic Velocity Transformation
      8. 5.7 Doppler Effect for Light
      9. 5.8 Relativistic Momentum
      10. 5.9 Relativistic Energy
      11. Chapter Review
        1. Key Terms
        2. Key Equations
        3. Summary
        4. Conceptual Questions
        5. Problems
        6. Additional Problems
    2. 6 Photons and Matter Waves
      1. Introduction
      2. 6.1 Blackbody Radiation
      3. 6.2 Photoelectric Effect
      4. 6.3 The Compton Effect
      5. 6.4 Bohr’s Model of the Hydrogen Atom
      6. 6.5 De Broglie’s Matter Waves
      7. 6.6 Wave-Particle Duality
      8. Chapter Review
        1. Key Terms
        2. Key Equations
        3. Summary
        4. Conceptual Questions
        5. Problems
        6. Additional Problems
    3. 7 Quantum Mechanics
      1. Introduction
      2. 7.1 Wave Functions
      3. 7.2 The Heisenberg Uncertainty Principle
      4. 7.3 The Schrӧdinger Equation
      5. 7.4 The Quantum Particle in a Box
      6. 7.5 The Quantum Harmonic Oscillator
      7. 7.6 The Quantum Tunneling of Particles through Potential Barriers
      8. Chapter Review
        1. Key Terms
        2. Key Equations
        3. Summary
        4. Conceptual Questions
        5. Problems
        6. Additional Problems
        7. Challenge Problems
    4. 8 Atomic Structure
      1. Introduction
      2. 8.1 The Hydrogen Atom
      3. 8.2 Orbital Magnetic Dipole Moment of the Electron
      4. 8.3 Electron Spin
      5. 8.4 The Exclusion Principle and the Periodic Table
      6. 8.5 Atomic Spectra and X-rays
      7. 8.6 Lasers
      8. Chapter Review
        1. Key Terms
        2. Key Equations
        3. Summary
        4. Conceptual Questions
        5. Problems
        6. Additional Problems
    5. 9 Condensed Matter Physics
      1. Introduction
      2. 9.1 Types of Molecular Bonds
      3. 9.2 Molecular Spectra
      4. 9.3 Bonding in Crystalline Solids
      5. 9.4 Free Electron Model of Metals
      6. 9.5 Band Theory of Solids
      7. 9.6 Semiconductors and Doping
      8. 9.7 Semiconductor Devices
      9. 9.8 Superconductivity
      10. Chapter Review
        1. Key Terms
        2. Key Equations
        3. Summary
        4. Conceptual Questions
        5. Problems
        6. Additional Problems
        7. Challenge Problems
    6. 10 Nuclear Physics
      1. Introduction
      2. 10.1 Properties of Nuclei
      3. 10.2 Nuclear Binding Energy
      4. 10.3 Radioactive Decay
      5. 10.4 Nuclear Reactions
      6. 10.5 Fission
      7. 10.6 Nuclear Fusion
      8. 10.7 Medical Applications and Biological Effects of Nuclear Radiation
      9. Chapter Review
        1. Key Terms
        2. Key Equations
        3. Summary
        4. Conceptual Questions
        5. Problems
        6. Additional Problems
        7. Challenge Problems
    7. 11 Particle Physics and Cosmology
      1. Introduction
      2. 11.1 Introduction to Particle Physics
      3. 11.2 Particle Conservation Laws
      4. 11.3 Quarks
      5. 11.4 Particle Accelerators and Detectors
      6. 11.5 The Standard Model
      7. 11.6 The Big Bang
      8. 11.7 Evolution of the Early Universe
      9. Chapter Review
        1. Key Terms
        2. Key Equations
        3. Summary
        4. Conceptual Questions
        5. Problems
        6. Additional Problems
        7. Challenge Problems
  4. A | Units
  5. B | Conversion Factors
  6. C | Fundamental Constants
  7. D | Astronomical Data
  8. E | Mathematical Formulas
  9. F | Chemistry
  10. G | The Greek Alphabet
  11. Answer Key
    1. Chapter 1
    2. Chapter 2
    3. Chapter 3
    4. Chapter 4
    5. Chapter 5
    6. Chapter 6
    7. Chapter 7
    8. Chapter 8
    9. Chapter 9
    10. Chapter 10
    11. Chapter 11
  12. Index

Learning Objectives

By the end of this section, you will be able to:
  • Explain why the hydrogen atom has magnetic properties
  • Explain why the energy levels of a hydrogen atom associated with orbital angular momentum are split by an external magnetic field
  • Use quantum numbers to calculate the magnitude and direction of the orbital magnetic dipole moment of a hydrogen atom

In Bohr’s model of the hydrogen atom, the electron moves in a circular orbit around the proton. The electron passes by a particular point on the loop in a certain time, so we can calculate a current I=Q/tI=Q/t. An electron that orbits a proton in a hydrogen atom is therefore analogous to current flowing through a circular wire (Figure 8.10). In the study of magnetism, we saw that a current-carrying wire produces magnetic fields. It is therefore reasonable to conclude that the hydrogen atom produces a magnetic field and interacts with other magnetic fields.

Figure (a) shows a current carrying loop. The loop has current I circulating counterclockwise as viewed from above. A vector mu pointing upward is shown at the center of the loop. Figure (b) shows the hydrogen atom as an electron, represented as a small ball and labeled minus e, making a counterclockwise circular orbit, as viewed from above. A sphere, a vector mu pointing downward, and a vector L pointing upward are shown in the center of the orbit.
Figure 8.10 (a) Current flowing through a circular wire is analogous to (b) an electron that orbits a proton in a hydrogen atom.

The orbital magnetic dipole moment is a measure of the strength of the magnetic field produced by the orbital angular momentum of an electron. From Force and Torque on a Current Loop, the magnitude of the orbital magnetic dipole moment for a current loop is

μ=IA,μ=IA,
(8.13)

where I is the current and A is the area of the loop. (For brevity, we refer to this as the magnetic moment.) The current I associated with an electron in orbit about a proton in a hydrogen atom is

I=eT,I=eT,
(8.14)

where e is the magnitude of the electron charge and T is its orbital period. If we assume that the electron travels in a perfectly circular orbit, the orbital period is

T=2πrv,T=2πrv,
(8.15)

where r is the radius of the orbit and v is the speed of the electron in its orbit. Given that the area of a circle is πr2πr2, the absolute magnetic moment is

μ=IA=e(2πrv)πr2=evr2.μ=IA=e(2πrv)πr2=evr2.
(8.16)

It is helpful to express the magnetic momentum μμ in terms of the orbital angular momentum (L=r×p).(L=r×p). Because the electron orbits in a circle, the position vector rr and the momentum vector pp form a right angle. Thus, the magnitude of the orbital angular momentum is

L=|L|=|r×p|=rpsinθ=rp=rmv.L=|L|=|r×p|=rpsinθ=rp=rmv.
(8.17)

Combining these two equations, we have

μ=(e2me)L.μ=(e2me)L.
(8.18)

In full vector form, this expression is written as

μ=(e2me)L.μ=(e2me)L.
(8.19)

The negative sign appears because the electron has a negative charge. Notice that the direction of the magnetic moment of the electron is antiparallel to the orbital angular momentum, as shown in Figure 8.10(b). In the Bohr model of the atom, the relationship between μμ and LL in Equation 8.19 is independent of the radius of the orbit.

The magnetic moment μμ can also be expressed in terms of the orbital angular quantum number l. Combining Equation 8.18 and Equation 8.15, the magnitude of the magnetic moment is

μ=(e2me)L=(e2me)l(l+1)=μBl(l+1).μ=(e2me)L=(e2me)l(l+1)=μBl(l+1).
(8.20)

The z-component of the magnetic moment is

μz=(e2me)Lz=(e2me)m=μBm.μz=(e2me)Lz=(e2me)m=μBm.
(8.21)

The quantity μBμB is a fundamental unit of magnetism called the Bohr magneton, which has the value 9.3×10−24joule/tesla9.3×10−24joule/tesla (J/T) or 5.8×10−5eV/T.5.8×10−5eV/T. Quantization of the magnetic moment is the result of quantization of the orbital angular momentum.

As we will see in the next section, the total magnetic dipole moment of the hydrogen atom is due to both the orbital motion of the electron and its intrinsic spin. For now, we ignore the effect of electron spin.

Example 8.3

Orbital Magnetic Dipole Moment What is the magnitude of the orbital dipole magnetic moment μμ of an electron in the hydrogen atom in the (a) s state, (b) p state, and (c) d state? (Assume that the spin of the electron is zero.)

Strategy The magnetic momentum of the electron is related to its orbital angular momentum L. For the hydrogen atom, this quantity is related to the orbital angular quantum number l. The states are given in spectroscopic notation, which relates a letter (s, p, d, etc.) to a quantum number.

Solution The magnitude of the magnetic moment is given in Equation 8.20:

μ=(e2me)L=(e2me)l(l+1)=μBl(l+1).μ=(e2me)L=(e2me)l(l+1)=μBl(l+1).
  1. For the s state, l=0l=0 so we have μ=0μ=0 and μz=0.μz=0.
  2. For the p state, l=1l=1 and we have
    μ=μB1(1+1)=2μB μz=μBm,wherem=(1,0,1),so μz=μB,0,μB.μ=μB1(1+1)=2μB μz=μBm,wherem=(1,0,1),so μz=μB,0,μB.
  3. For the d state, l=2l=2 and we obtain
    μ=μB2(2+1)=6μB μz=μBm,wherem=(2,1,0,1,2),so μz=2μB,μB,0,μB,−2μB.μ=μB2(2+1)=6μB μz=μBm,wherem=(2,1,0,1,2),so μz=2μB,μB,0,μB,−2μB.

Significance In the s state, there is no orbital angular momentum and therefore no magnetic moment. This does not mean that the electron is at rest, just that the overall motion of the electron does not produce a magnetic field. In the p state, the electron has a magnetic moment with three possible values for the z-component of this magnetic moment; this means that magnetic moment can point in three different polar directions—each antiparallel to the orbital angular momentum vector. In the d state, the electron has a magnetic moment with five possible values for the z-component of this magnetic moment. In this case, the magnetic moment can point in five different polar directions.

A hydrogen atom has a magnetic field, so we expect the hydrogen atom to interact with an external magnetic field—such as the push and pull between two bar magnets. From Force and Torque on a Current Loop, we know that when a current loop interacts with an external magnetic field BB, it experiences a torque given by

τ=I(A×B)=μ×B,τ=I(A×B)=μ×B,
(8.22)

where I is the current, AA is the area of the loop, μμ is the magnetic moment, and BB is the external magnetic field. This torque acts to rotate the magnetic moment vector of the hydrogen atom to align with the external magnetic field. Because mechanical work is done by the external magnetic field on the hydrogen atom, we can talk about energy transformations in the atom. The potential energy of the hydrogen atom associated with this magnetic interaction is given by Equation 8.23:

U=μ·B.U=μ·B.
(8.23)

If the magnetic moment is antiparallel to the external magnetic field, the potential energy is large, but if the magnetic moment is parallel to the field, the potential energy is small. Work done on the hydrogen atom to rotate the atom’s magnetic moment vector in the direction of the external magnetic field is therefore associated with a drop in potential energy. The energy of the system is conserved, however, because a drop in potential energy produces radiation (the emission of a photon). These energy transitions are quantized because the magnetic moment can point in only certain directions.

If the external magnetic field points in the positive z-direction, the potential energy associated with the orbital magnetic dipole moment is

U(θ)=μBcosθ=μzB=(μBm)B=mμBB,U(θ)=μBcosθ=μzB=(μBm)B=mμBB,
(8.24)

where μBμB is the Bohr magneton and m is the angular momentum projection quantum number (or magnetic orbital quantum number), which has the values

m=l,l+1,...,0,...,l1,l.m=l,l+1,...,0,...,l1,l.
(8.25)

For example, in the l=1l=1 electron state, the total energy of the electron is split into three distinct energy levels corresponding to U=μBB,0,μBB.U=μBB,0,μBB.

The splitting of energy levels by an external magnetic field is called the Zeeman effect. Ignoring the effects of electron spin, transitions from the l=1l=1 state to a common lower energy state produce three closely spaced spectral lines (Figure 8.11, left column). Likewise, transitions from the l=2l=2 state produce five closely spaced spectral lines (right column). The separation of these lines is proportional to the strength of the external magnetic field. This effect has many applications. For example, the splitting of lines in the hydrogen spectrum of the Sun is used to determine the strength of the Sun’s magnetic field. Many such magnetic field measurements can be used to make a map of the magnetic activity at the Sun’s surface called a magnetogram (Figure 8.12).

The figure shows the effect of magnetic field, B sub ext, on two different spectral lines, corresponding to the l=1 to l=0 transition on the left and the l=2 to l=0 transition on the right. The spectra are shown for no external field, for a non zero external field and for a large external field. With no external field, both transitions appear as single lines. In the second case, when magnetic field is applied, the spectral lines split into several lines; the line on the left splits into three lines. The line on the right splits into five. In the third case, the magnetic field is large. The left line is again split into three lines and the right into five, but the split lines are farther apart than they are when the external magnetic field is not as strong.
Figure 8.11 The Zeeman effect refers to the splitting of spectral lines by an external magnetic field. In the left column, the energy splitting occurs due to transitions from the state (n=2,l=1(n=2,l=1) to a lower energy state; and in the right column, energy splitting occurs due to transitions from the state (n=2,l=2)(n=2,l=2) to a lower-energy state. The separation of these lines is proportional to the strength of the external magnetic field.
A magnetogram of the sun, which appears as a gray disc against a black background, with white and black spots scattered on it. Most of the spots are concentrated in the center right part of the image.
Figure 8.12 A magnetogram of the Sun. The bright and dark spots show significant magnetic activity at the surface of the Sun. (credit: NASA, SDO)
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