646
12. The Molecule in an Electric or Magnetic Field
q = x y z, we obtain
μq (t) = μ0q +
q
+
αqq Eq0 + Eqω cos(ωt)
1
βqq q Eq0 + Eqω cos(ωt) × Eq0 + Eqω cos(ωt)
2 qq
+
1
6
q q q
γqq q q Eq0 + Eqω cos(ωt) Eq0 + Eqω cos(ωt)
× Eq0 + Eqω cos(ωt) + · · · (12.49)
Second (SHG) and Third (THG) Harmonic Generation
After multiplication and simple trigonometry we have
μq (t) = μω=0q + μωq cos ωt + μ2ωq cos(2ωt) + μ3ωq cos(3ωt) (12.50)
where the amplitudes μ corresponding to the coordinate q ∈ x y z and to the
particular resulting frequencies 0 ω 2ω 3ω have the following form46
1 μω=0q = μ0q +
αqq (0; 0)Eq0 +
βqq q (0; 0 0)Eq0 Eq0
2
q
+
+
1
6
q q q
q q
γqq q q (0; 0 0 0)Eq0 Eq0 Eq0
1
βqq q (0; −ω ω)Eqω Eqω
4 qq
+
μωq =
1
4
q
+
q q q
γqq q q (0; 0 −ω ω)Eq0 Eqω Eqω αqq (−ω; ω)Eqω +
1
2
q q q
q q
βqq q (−ω; ω 0)Eqω Eq0
γqq q q (−ω; ω 0 0)Eqω Eq0 Eq0
46 According to convention, a given (hyper)polarizability, e.g., γ (−3ω; ω ω ω), is accomqq q q
panied (in parenthesis) by the frequencies ω corresponding to the three directions x y z of the
incident light polarization (here: q , q and q , preceded by minus the Fourier frequency of the
term, −3ω, which symbolizes the photon energy conservation law). Some of the symbols, e.g.,
γqq q q (−ω; ω −ω ω), after a semicolon have negative values, which means a partial (as in
γqq q q (−ω; ω −ω ω)) or complete (as in βqq q (0; −ω ω)) cancellation of the intensity of the
oscillating electric field.
647
12.5 A molecule in an oscillating electric field
+
1 γqq q q (−ω; ω −ω ω)Eqω Eqω Eqω 8
q q q
μ2ωq =
1
βqq q (−2ω; ω ω)Eqω Eqω
4 qq
+
1 γqq q q (−2ω; ω ω 0)Eqω Eqω Eq0 4
(12.51)
q q q
μ3ωq =
1
24
q q q
γqq q q (−3ω; ω ω ω)Eqω Eqω Eqω (12.52)
We see that:
• An oscillating electric field may result in a non-oscillating dipole moment related
to the hyperpolarizabilities βqq q (0; −ω ω) and γqq q q (0; 0 −ω ω), which
manifests as an electric potential difference on two opposite crystal faces.
• The dipole moment oscillates with the basic frequency ω of the incident light
and in addition, with two other frequencies: the second (2ω) and third (3ω) harmonics (SHG and THG, respectively). This is supported by experiment (mentioned in the example at the beginning of the chapter), applying incident light of
frequency ω we obtain emitted light with frequencies47 2ω and 3ω.
Note that to generate a large second harmonic the material has to have large
values of the hyperpolarizabilities β and γ. The THG needs a large γ. In both
cases a strong laser electric field is necessary. The SHG and THG therefore require support from the theoretical side: we are looking for high hyperpolarizability
materials and quantum mechanical calculations may predict such materials before
an expensive organic synthesis is done.48
MAGNETIC PHENOMENA
The electric and magnetic fields (both of them are related by the Maxwell
equations, Appendix G) interact differently with matter, which is highlighted in
Fig. 12.8, where the electron trajectories in both fields are shown. They are totally
different, the trajectory in the magnetic field has a cycloid character, while in the
electric field it is a parabola. This is why the description of magnetic properties
differs so much from that of electric properties.
47 This experiment was first carried out by P.A. Franken, A.E. Hill, C.W. Peters, G. Weinreich, Phys.
Rev. Letters 7 (1961) 118.
48 In molecular crystals it is not sufficient that particular molecules have high values of hyperpolarizability. What counts is the hyperpolarizability of the crystal unit cell.
648
12. The Molecule in an Electric or Magnetic Field
Fig. 12.8. The trajectories of an electron in the (a) electric field – the trajectory is a parabola (b) magnetic field, perpendicular to the incident velocity – the trajectory is a cycloid in a plane perpendicular
to the figure.
12.6 MAGNETIC DIPOLE MOMENTS OF ELEMENTARY
PARTICLES
12.6.1 ELECTRON
magnetic dipole
moment
An elementary particle, besides its orbital angular momentum, may also have internal angular momentum, or spin, cf. p. 25. In Chapter 3, the Dirac theory led
to a relation between the spin angular momentum s of the electron and its dipole
magnetic moment M spinel (eq. (3.62), p. 122):
M spinel = γel s
with the gyromagnetic factor49
μB
h̄
where the Bohr magneton (m0 is the electronic rest mass)
γel = −2
μB =
eh̄
2m0 c
The relation is quite a surprise, because the gyromagnetic factor is twice as large
as that appearing in the relation between the electron orbital angular momentum
L and the associated magnetic dipole moment
μB
M orbel = −
L
(12.53)
h̄
gyro-magnetic
factor
Quantum electrodynamics explains this effect qualitatively – predicting a factor
very close to the experimental value50 2.0023193043737, known with the breathtaking accuracy of ±00000000000082.
49 From the Greek word gyros, or circle; it is believed that a circular motion of a charged particle is
related to the resulting magnetic moment.
50 R.S. Van Dyck Jr., P.B. Schwinberg, H.G. Dehmelt, Phys. Rev. Letters 59 (1990) 26.
649
12.6 Magnetic dipole moments of elementary particles
12.6.2 NUCLEUS
Let us stay within the Dirac theory. If, instead of an electron, we take a nucleus of
charge +Ze and atomic mass51 M, then we would presume (after insertion into the
eh̄
Z μN
above formulae) the gyromagnetic factor should be γ = 2 M
h̄ , where μN = 2m c
H
(mH denoting the proton mass) is known as the nuclear magneton.52 For a proton
(Z = 1, M = 1), we would have γp = 2μN /h̄, whereas the experimental value53 is
γp = 559μN /h̄. What is going on? In both cases we have a single elementary particle (electron or proton), both have the spin quantum number equal to 12 , we might
expect that nothing special will happen to the proton, and only the mass ratio and
charge will make a difference. Instead we see that Dirac theory does relate to the
electron, but not to the nuclei. Visibly, the proton is more complex than the electron. We see that even the simplest nucleus has internal machinery, which results
in the observed strange deviation. There are lots of quarks in the proton (three valence quarks and a sea of virtual quarks together with the gluons, etc.). The proton
and electron polarize the vacuum differently and this results in different gyromagnetic factors. Other nuclei exhibit even stranger properties. Sometimes we even
have negative gyromagnetic coefficients. In such a case their magnetic moment is
opposite to the spin angular momentum. The complex nature of the internal machinery of the nuclei and vacuum polarization lead to the observed gyromagnetic
coefficients.54 Science has had some success here, e.g., for leptons,55 but for nuclei
the situation is worse. This is why we are simply forced to take this into account
in the present book56 and treat the spin magnetic moments of the nuclei as the
experimental data:
M A = γA I A (12.54)
where I A represents the spin angular momentum of the nucleus A.
51 Unitless quantity.
52 Ca. 1840 times smaller than the Bohr magneton (for the electron).
53 Also the gyromagnetic factor for an electron is expected to be ca. 1840 times larger than that for a
proton. This means that a proton is expected to create a magnetic field ca. 1840 times weaker than the
field created by an electron.
54 The relation between spin and magnetic moment is as mysterious as that between the magnetic
moment and charge of a particle (the spin is associated with a rotation, while the magnetic moment is
associated with a rotation of a charged object) or its mass. A neutron has spin equal to 12 and magnetic
moment similar to that of a proton despite the zero electric charge. The neutrino has no charge, nearly
zero mass and magnetic moment, and still has a spin equal to 12 .
55 And what about the “heavier brothers” of the electron, the muon and taon (cf. p. 268)? For the
muon, the coefficient in the gyromagnetic factor (2.0023318920) is similar to that of the electron
(20023193043737), just a bit larger and agrees equally well with experiment. For the taon we have
only a theoretical result, a little larger than for the two other “brothers”. Thus, each of the lepton
family behaves in a similar way.
56 With a suitable respect of the Nature’s complexity.
nuclear
magneton