12.6 MAGNETIC DIPOLE MOMENTS OF ELEMENTARY PARTICLES 12.6.1 ELECTRON An elementary particle, besides its orbital angular momentum, may also have in-ternal angular momentum, or spin, cf..
Trang 1646 12 The Molecule in an Electric or Magnetic Field
q= x y z, we obtain
μq(t)= μ0 q+
q
αqq
E0
q cos(ωt)
+1 2
q q
βqqq
E0
q cos(ωt)
× E0
q cos(ωt)
+1 6
q q q
γqqqq
E0
q cos(ωt)
E0
q cos(ωt)
× E0
q cos(ωt)
Second (SHG) and Third (THG) Harmonic Generation
After multiplication and simple trigonometry we have
μq(t)= μω =0 q+ μω qcos ωt+ μ2ω qcos(2ωt)+ μ3ω qcos(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
q
αqq(0; 0)E0
2
q q
βqqq(0; 0 0)E0
q E0
q
+1 6
q q q
γqqqq(0; 0 0 0)E0
q E0
q E0
q
+1 4
q q
βq q q(0; −ω ω)Eω
q
+1 4
q q q
γqqqq(0; 0 −ω ω)E0
q Eω
q
μω q=
q
αqq(−ω; ω)Eω
q q
βqqq(−ω; ω 0)Eω
q E0
q
+1 2
q q q
γqqqq(−ω; ω 0 0)Eω
q E0
q E0
q
accom-panied (in parenthesis) by the frequencies ω corresponding to the three directions x y z of the
oscillating electric field.
Trang 2+1 8
q q q
γqqqq(−ω; ω −ω ω)Eω
q Eω
q
μ2ω q=1
4
q q
βq q q(−2ω; ω ω)Eω
q
+1 4
q q q
γqqqq(−2ω; ω ω 0)Eω
q E0
q (12.51)
μ3ω q= 1
24
q q q
γqqqq(−3ω; ω ω ω)Eω
q Eω
q (12.52)
We see that:
• An oscillating electric field may result in a non-oscillating dipole moment related
to the hyperpolarizabilities βq q q(0; −ω ω) and γqq qq(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)
har-monics (SHG and THG, respectively) This is supported by experiment
(men-tioned in the example at the beginning of the chapter), applying incident light of
frequency ω we obtain emitted light with frequencies472ω 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
re-quire 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
Rev Letters 7 (1961) 118.
hyperpolariz-ability What counts is the hyperpolarizability of the crystal unit cell.
Trang 3648 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) mag-netic 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
An elementary particle, besides its orbital angular momentum, may also have in-ternal 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 dipole
moment magnetic moment Mspin el(eq (3.62), p 122):
with the gyromagnetic factor49
γel= −2μB
¯h where the Bohr magneton (m0is the electronic rest mass)
μB= e¯h 2m0c 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
Land the associated magnetic dipole moment
Quantum electrodynamics explains this effect qualitatively – predicting a factor very close to the experimental value50 2.0023193043737, known with the breath-taking accuracy of±00000000000082
gyro-magnetic
factor
related to the resulting magnetic moment.
Trang 412.6.2 NUCLEUS
Let us stay within the Dirac theory If, instead of an electron, we take a nucleus of
charge+Ze and atomic mass51M, then we would presume (after insertion into the
above formulae) the gyromagnetic factor should be γ= 2Z
M
¯h , where μN= e ¯h
(mHdenoting the proton mass) is known as the nuclear magneton.52For a proton nuclear
magneton (Z= 1, M = 1), we would have γp= 2μN/¯h, whereas the experimental value53is
γp= 559μN/¯h What is going on? In both cases we have a single elementary
parti-cle (electron or proton), both have the spin quantum number equal to12, 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
elec-tron 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
va-lence 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
gyromag-netic 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
ma-chinery of the nuclei and vacuum polarization lead to the observed gyromagnetic
coefficients.54Science has had some success here, e.g., for leptons,55but 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:
where IArepresents the spin angular momentum of the nucleus 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.
moment and charge of a particle (the spin is associated with a rotation, while the magnetic moment is
moment similar to that of a proton despite the zero electric charge The neutrino has no charge, nearly
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.
Trang 5650 12 The Molecule in an Electric or Magnetic Field
12.6.3 DIPOLE MOMENT IN THE FIELD
Electric field
The problem of an electric dipole μ rotating in an electric field was described on
p 631 There we were interested in the ground state When the field is switched off (cf p 176), the ground state is non-degenerate (J= M = 0) After a weak electric field (E) is switched on, the ground-state wave function deforms in such a way as to
prefer the alignment of the rotating dipole moment along the field Since we may always use a complete set of rigid rotator wave functions (at zero field), this means
the deformed wave functions have to be linear combinations of the wave functions corresponding to different J.
Magnetic field
Imagine a spinning top which is a magnet If you make it spin (with angular mo-mentum I) and leave it in space without any external torque τ , then due to the fact that space is isotropic, its angular momentum will stay constant, becausedIdt = τ = 0
(τ is time), i.e the top will rotate about its axis with a constant speed and the axis will not move with respect to distant stars, Fig 12.9.a
The situation changes if a magnetic field is switched on Now, the space is no longer isotropic and the vector of the angular momentum is no longer conserved
However, the conservation law for the projection of the angular momentum on the di-rection of the field is still valid This means that the top makes a precession about the
Fig 12.9. Classical and quantum tops (magnets) in space (a) The space is isotropic and therefore the classical top preserves its angular momentum I, i.e its axis does not move with respect to distant stars and the top rotates about its axis with a constant speed (b) The same top in a magnetic field The space
is no longer isotropic, and therefore the total angular momentum is no longer preserved The projection of the total momentum on the field direction is still preserved The magnetic field causes a torque τ (orthogonal to the picture)
the top axis about the direction of the field (c) A quantum top, i.e an elemen-tary particle with spin quantum number
en-ergy eigenstates that correspond to two precession cones, directed up and down.
Trang 6field axis, becausedIdt = τ = 0, and τ is orthogonal to I and to the field, Fig 12.9.b.
In quantum mechanics the magnetic dipole moment M= γI in the magnetic field
H= (0 0 H), H > 0, has as many stationary states as is the number of possible
projections of the spin angular momentum on the field direction From Chapter 1,
we know that this number is 2I+ 1, where I is the spin quantum number of the
particle (e.g., for a proton: I=1
2) The projections are equal (Fig 12.9.c) mI¯h with
mI= −I −I + 1 0 +I Therefore,
the energy levels in the magnetic field
Note, that the energy level splitting is proportional to the magnetic field
intensity, Fig 12.10
If a nucleus has I=1
2, then the energy difference E between the two states
in a magnetic field H: one with mI= −1
2 and the other one with mI=1
2, equals
E= 2 ×1
2γ¯hH = γ ¯hH, and
where the Larmor57frequency is defined as
νL=γH
We see (Fig 12.10) that for nuclei with γ > 0, lower energy corresponds to mI=
1
2, i.e to the spin moment along the field (forming an angle θ= 54◦44 with the
magnetic field vector, see p 28)
Fig 12.10.Energy levels in magnetic field H = (0 0 H)
for a nucleus with spin angular momentum I
is doubly degenerate (b) For γ > 0 (e.g., a proton) I and
di-rection of the magnetic field has lower energy (c) For
γ < 0 I and M have the opposite direction The state with
Trang 7652 12 The Molecule in an Electric or Magnetic Field
Note that there is a difference between the energy levels of the electric dipole moment
in an electric field and the levels of the magnetic dipole in a magnetic field The difference is that, for the magnetic dipole of an elementary particle the states do not have admixtures from the other I values (which is given by nature), while for the electric dipole there are admixtures from states with other values of J
This suggests that we may also expect such admixtures in a magnetic field In fact this is true if the particle is complex For example, the singlet state (S= 0) of the hydrogen molecule gets an admixture of the triplet state (S= 1) in the magnetic field, because the spin magnetic moments of both electrons tend to align parallel
to the field
12.7 TRANSITIONS BETWEEN THE NUCLEAR SPIN QUANTUM STATES – NMR TECHNIQUE
Is there any possibility of making the nuclear spin flip from one quantum state
to another? Yes Evidently, we have to create distinct energy levels correspond-ing to different spin projections, i.e to switch the magnetic field on, Figs 12.10 and 12.11.a After the electromagnetic field is applied and its frequency matches the energy level difference, the system absorbs the energy It looks as if a nucleus absorbs the energy and changes its quantum state In a genuine NMR experiment, the electromagnetic frequency is fixed (radio wave lengths) and the specimen is scanned by a variable magnetic field At some particular field values the energy dif-ference matches the electromagnetic frequency and the transition (Nuclear Mag-netic Resonance) is observed
The magnetic field that a particular nucleus feels differs from the external mag-netic field applied, because the electronic structure in which the nucleus is im-mersed in, makes its own contribution (see Fig 12.11.b,c) Also the nuclear spins interact by creating their own magnetic fields
We have not yet considered these effects in the non-relativistic Hamiltonian (2.1) (e.g., no spin–spin or spin–field interactions) The effects which we are now dealing with are so small, of the order of 10−11 kcal/mole, that they are of no importance for most applications, including UV-VIS, IR, Raman spectra, electronic structure, chemical reactions, intermolecular interactions, etc This time, however, the sit-uation changes: we are going to study very subtle interactions using the NMR technique which aims precisely at the energy levels that result from spin–spin and spin–magnetic field interactions Even if these effects are very small, they can be observed Therefore, we have to consider more exact Hamiltonians First, we have
to introduce
• the interaction of our system with the electromagnetic field,
• then we will consider the influence of the electronic structure on the magnetic field acting on the nuclei
Trang 8Fig 12.11. Proton’s shielding by the electronic structure (a) The energy levels of an isolated proton in
a magnetic field (b) The energy levels of the proton of the benzene ring (no nuclear spin interaction
is assumed) The most mobile π electrons of benzene (which may be treated as a conducting circular
wire) move around the benzene ring in response to the external magnetic field (perpendicular to the
ring) thus producing an induced magnetic field The latter one (when considered along the ring six-fold
axis) opposes the external magnetic field, but at the position of the proton actually leads to an additional
increasing of the magnetic field felt by the proton This is why the figure shows energy level difference
increases due to the electron shielding effect (c) The energy levels of another proton (located along the
ring axis) in a similar molecule This proton feels a local magnetic field that is decreased with respect
to the external one (due to the induction effect).
• and finally, the nuclear magnetic moment interaction (“coupling”) will be
con-sidered
12.8 HAMILTONIAN OF THE SYSTEM IN THE
ELECTROMAGNETIC FIELD
The non-relativistic Hamiltonian58H of the system of N particles (the j-th particleˆ
having mass mj and charge qj) moving in an external electromagnetic field with
supple-mented by the relativistic terms from the Breit Hamiltonian (p 131).
Trang 9654 12 The Molecule in an Electric or Magnetic Field
vector potential A and scalar potential φ may be written as59
ˆ
1 2mj
ˆpj−qj
cAj
2
+ qjφj
where ˆV stands for the “internal” potential coming from the mutual interactions
of the particles, and Aj and φj denote the external vector60and scalar potentials
Aand φ, respectively, calculated at the position of particle j
12.8.1 CHOICE OF THE VECTOR AND SCALAR POTENTIALS
In Appendix G on p 962 it is shown that there is a certain arbitrariness in the choice of both potentials, which leaves the physics of the system unchanged If for
a homogeneous magnetic field H we choose the vector potential at the point indi-cated by r= (x y z) as (eq (G.13)) A(r) =1
2[H × r], then, as shown in
Appen-dix G, we will satisfy the Maxwell equations, and in addition obtain the commonly used relation (eq (G.12)) div A≡ ∇A = 0, known as the Coulombic gauge In this
Coulombic
gauge way the origin of the coordinate system (r= 0) was chosen as the origin of the
vector potential (which need not be a rule)
BecauseE = 0 and A is time-independent, φ = const (p 962), which of course
means also that φj= const, and as an additive constant, it may simply be elimi-nated from the Hamiltonian (12.58)
12.8.2 REFINEMENT OF THE HAMILTONIAN
Let us assume the Born–Oppenheimer approximation (p 229) Thus, the nuclei occupy some fixed positions in space, and in the electronic Hamiltonian (12.58)
we have the electronic charges qj = −e and masses mj = m0= m (we skip the subscript 0 for the rest mass of the electron) Now, let us refine the Hamiltonian
by adding the interaction of the particle magnetic moments (of the electrons and nuclei; the moments result from the orbital motion of the electrons as well as from the spin of each particle) with themselves and with the external magnetic field
We have, therefore, a refined Hamiltonian of the system [the particular terms of
field, after introducing the vector and scalar potentials for particle of charge q we have to replace E by
potentials at the position of the particle For many particles we sum these contributions up and add
the interparticle interaction potential (V ) This is what we wanted to obtain (H Hameka, “Advanced Quantum Chemistry”, Addison-Wesley Publishing Co., Reading, Massachusetts (1965), p 40).
particle moves faster on one side of the vector potential origin and slower on the opposite side.
Trang 10the Hamiltonian correspond61 to the relevant terms of the Breit Hamiltonian62
(p 131)]
ˆ
where (δ stands for the Dirac delta function, Appendix E, N is the number of
electrons, and the spins have been replaced by the corresponding operators)
ˆ
H1=
N
j=1
1 2m
ˆpj+e
cAj
2 + ˆV + ˆHSH+ ˆHIH+ ˆHLS+ ˆHSS+ ˆHLL (12.60) ˆ
H2= γel
N
j=1
A
γA
ˆs
rAj3 − 3(ˆsj· rAj)(ˆIA· rAj)
rAj5
ˆ
H3= −γel
8π 3
N
A
ˆ
A<B
γAγB ˆIA· ˆIB
R3AB − 3(ˆIA· RAB)(ˆIB· RAB)
R5AB
where in the global coordinate system the internuclear distance means the length
of the vector RAB= RB− RA, while the electron–nucleus distance (of the electron
j with nucleus A) will be the length of rAj= rj− RA We have:
• In the term ˆH1, besides the kinetic energy operator in the external magnetic field
[with vector potential A, and the
convention Aj ≡ A(rj)] given by
N
cAj)2, we have the Coulomb potential ˆV of the interaction
of all the charged particles Next, we
have:
– The interaction of the spin magnetic
moments of the electrons ( ˆHSH) and
of the nuclei ( ˆHIH) with the field H
These terms come from the first part
of the term ˆH6 of the Breit
Hamil-tonian, and represent the simple
Zee-man terms:
Pieter Zeeman (1865–1943), Dutch physicist, professor at the University of Amsterdam.
He became interested in the influence of a magnetic field
on molecular spectra and dis-covered a field-induced split-ting of the absorption lines in
1896 He shared the Nobel Prize with Hendrik Lorentz
“ for their researches into the influence of magnetism upon radiation phenomena ” in 1902.
The Zeeman splitting of star spectra allows us to
deter-mine the value of the mag-netic field that was on the star
at the moment the light was emitted!
derived from classical electrodynamics.
nucleus, what is known as the shielding constant (related to the shielding of the nucleus by the electron
cloud) and the internuclear coupling constants The shielding and coupling constants enter in a specific
way into the energy expression Only those terms are included in the Hamiltonian that give non-zero
contributions to these quantities.