Fundamentals of the Physics of Solids / 03-The Building Blocks of Solids

3.2 The State of Ion Cores

61

be comparable to the paramagnetic contribution. This explains why the e ective magneton number determined from the ground-state configuration does not agree with the measured value.

3.2.8 Electron Spin Resonance

As we have seen, when a paramagnetic ion of angular momentum J is placed in a uniform magnetic field, the initial 2J + 1-fold degeneracy of the ground state – that is due to the 2J + 1 possible values of the quantum number mJ – is split. In the presence of an electromagnetic radiation, transitions between neighboring levels become possible in accordance with the selection rule mJ = ±1. As the energy di erence between two adjacent Zeeman levels is

E = gJ μBB ,

(3.2.89)

atoms can absorb photons of this energy, and thus the electromagnetic field of the corresponding frequency is absorbed in a resonant manner. This phenomenon is called electron spin resonance (ESR) or electron paramagnetic resonance (EPR). In fields on the order of several teslas, this resonant absorption occurs at a frequency of a few GHz (or, in terms of the wavelength, in the centimeter (microwave) range). In experimental studies samples are placed in a microwave field whose frequency is determined by the dimensions of the cavity, and absorption is measured as the strength of the applied field is varied. Then the Landé gJ -factor is determined from the transition frequency. The method is therefore suited to identifying paramagnetic ions within the sample.

To understand resonant absorption, consider the magnetic moment μJ as a classical vector that has an angular momentum J = −μJ /γJ , where

	γJ = gJ	μB	= gJ	e	(3.2.90)
				2me

is the gyromagnetic ratio. For notational simplicity, we shall omit the label J. A magnetic moment μ whose axis makes an angle θ with the direction of the uniform magnetic field B experiences a torque

M = μ × B ,

(3.2.91)

therefore the equation of motion of the magnetic moment is given by

1 dμ	= B × μ .	(3.2.92)
γ dt

Thus the moment precesses about the magnetic field direction at a constant angular frequency ωL = γB – the Larmor frequency. When a weak perpendicular field of angular frequency ω is superimposed on the uniform field, the precessing moment will resonantly absorb energy from the magnetic

62 3 The Building Blocks of Solids

field, provided ω is equal to the precession frequency ωL. For all other frequencies the relative phases shift continuously with time, and so no energy will be absorbed.

For a quantum mechanical derivation of this equation we start with

dμ(t)	=	i	[H(t), μ(t)] ,	(3.2.93)
dt

where the contribution of the magnetic field to the total Hamiltonian is

Hext = −μ · B(t) .

(3.2.94)

According to (3.2.69), the magnetic moment is proportional to the total angular momentum, thus the known commutation relations of the latter lead to

dμ	= γB × μ .	(3.2.95)
dt

When quantum mechanical averages of the operators are taken, classical equations of motions are recovered. In what follows, this average will be meant by

μ.

We shall now consider the previously outlined case. Two fields are applied: a constant field, of magnitude B0, along the z-axis, and a linearly polarized time-dependent one, of amplitude 2B1 and angular frequency ω, along the x-axis:

Bx = 2B1 cos ωt , By = 0 , Bz = B0 .

(3.2.96)

The equation of motion in component form is

dμx		= −γμyB0 ,
dt
dμy		= γμxB0 −	2γμzB1 cos ωt ,	(3.2.97)
dt
dμz		= 2γμyB1 cos ωt .
dt

If the time-dependent field is assumed to be weak compared to the constant field, then the time dependence of the z component of the magnetic moment can be ignored, as it is on the order of B12. The solution of the system of equations is then

μx =	2γB1 ωL μz	cos ωt ,	μy =	2γB1 ω μz	sin ωt ,	(3.2.98)
	ωL2 − ω2			ωL2 − ω2

where ωL = γB0. Besides μx (which is in phase with the perturbing field) a y component (which lags π/2 behind the applied field) is also present. As a generalization of this result the time dependence of the magnetic moment can be written as

			3.2			The State of Ion Cores			63
μ =	μ		+ χ	2B1	cos ωt + χ	2B1	sin ωt ,	(3.2.99)
				μ0		μ0

where the quantities χ and χ are tensors. The term proportional to χ gives the response that is in phase with the applied field, so χ is the usual susceptibility. The term proportional to χ , which is π/2 out of phase with the applied field, can be related to energy absorption, as the energy absorbed per unit time is

Q =	ω	χ B1 .	(3.2.100)
	2μ0 B1

It is customary to consider χ and χ as the real and imaginary parts of a complex susceptibility, χ = χ + iχ . Then

μ = μ + χ	B1	e−iωt + χ	B1	eiωt .	(3.2.101)
	μ0		μ0

The linearly polarized field can be made up of two counterrotating circularly polarized radiations; in this case χ and its complex conjugate correspond to the response to positive and negative polarization, respectively. From the expressions in (3.2.98) it is readily seen that

χ	= χ	=	γμ0ωL μz	=	ωL2		χ	0	,	(3.2.102)
xx	yx		ωL2 − ω2		ωL2 −	ω2

where χ0 = μ0 μz /B0 is the paramagnetic susceptibility.

The above expressions are singular at ω = ωL. The imaginary part of χxx cannot be precisely determined from these equations. One can only say that it vanishes for all frequencies ω = ωL, i.e., energy absorption is possible only at the resonance frequency. The amplitude can nevertheless be determined when one takes into account the interrelatedness of the real and complex parts. To see this, consider the so-called relaxation function, φ(t − t0). This describes how a system in thermal equilibrium in a magnetic field of unit strength will relax toward its new equilibrium state when the magnetic field is suddenly turned o at time t0. Similarly, −φ (t −t0) = −dφ(t −t0)/dt gives the magnetization caused at time t by the application of a delta function pulse at an earlier time t0. For time-dependent magnetic fields integration of the contributions from times prior to t leads to

μ(t) = μ −	t
	φ (t − t0)B(t0) dt0 .	(3.2.103)

−∞

When a harmonic magnetic field B(t) = μ0He−iωt of frequency ω is applied, the time-dependent part of the magnetic moment will vary with the same frequency,

μ(t) = μ	+ μ(ω)e−iωt ,	(3.2.104)

and thus the frequency-dependent magnetic susceptibility is given by

64	3 The Building Blocks of Solids
		t				∞
	χ(ω) = −μ0		φ (t − t0)eiω(t−t0)dt0 = −μ0				φ (t)eiωtdt .	(3.2.105)
	−∞					0
Integration by parts now leads to
				∞
			χ(ω) = χ0 + iω μ0	0	φ(t)eiωtdt .			(3.2.106)

One may, in principle, obtain the relaxation function – and thus the real and imaginary parts of the susceptibility as well – by solving the earlier equations for the temporal variation of the magnetic moment in the special case when the magnetic field is suddenly turned o . Instead of proceeding along this line we shall make use of only one point of the above considerations: that magnetization is the causal response to an applied field. As it will be demonstrated in the Appendix of Volume 3, causality implies that when the frequency variable is extended to the complex plane, the frequency-dependent susceptibility is analytic on the upper half plane – moreover, its real and imaginary parts are related by the Kramers–Kronig relations20

2

∞

(ω )

ω χ

χ (ω) = χ (∞) +

dω ,

(3.2.107-a)

π

ω 2

−

ω2

0

∞

ω 2

ω2 dω .

(3.2.107-b)

χ (ω) = −π

2

ωχ

(ω )

0

−

Using the previously derived form for χxx, we have

χ

=

π

δ(ω

−

ω

L

)ω

L

χ

.

(3.2.108)

2

0

xx

A sharp resonance appears only because we have ignored all mechanisms that can cause the magnetic moment to relax. For an isolated atom the z component of the angular momentum is conserved. However, in condensed matter angular momentum can be transferred to other atoms or the lattice, and this leads to the relaxation of the magnetic moment toward its equilibrium value. This phenomenon is called the spin–lattice relaxation. We shall not discuss these interactions in detail here – nonetheless assume that the relaxation of a magnetic moment can be characterized by a longitudinal or spin–lattice relaxation time T1. Then the equation of motion for the z component of the magnetization is

dμz	= 2γμ B		cos ωt	−	μz − μz	.	(3.2.109)
dt
	y	1			T1

20 H. A. Kramers, 1927, R. Kronig, 1926.

3.2 The State of Ion Cores

65

The new term corresponds to the assumption that without the driving precessing field the system would approach its equilibrium state exponentially, with a characteristic time T1. As the variation of the spin component along the field B0 is accompanied by the variation of the magnetic energy of the ion, during the spin–lattice relaxation process energy has to be transferred to another part of the system: the lattice.

Other ions also influence the temporal variations of the x and y components of the magnetic moment, as the moment is moving in the nonuniform field created by its neighbors. This can be characterized phenomenologically by a transverse or spin–spin relaxation time T2. As no energy transfer occurs during transverse relaxation, spin–spin relaxation is usually several orders of magnitude faster than spin–lattice relaxation: T2 T1. Typical values are T1 10−6 s and T2 10−10 s.

When relaxation is taken into account, the equation of motion reads

dμx		= −γμyB0 −	μx	,
dt			T2
dμy				(3.2.110)
		= γμxB0 − 2γμzB1 cos ωt −			μy
						.
dt					T2

These equations and (3.2.109) are jointly called the Bloch equations.21 When only the constant field is present, the previous equations simplify to

	dμx		= −γμyB0 −				μx			,
	dt						T2
	dμy		= γμxB0 −			μy				(3.2.111)
								,
	dt					T2
	dμz		=	−	μz − μz		.
	dt
					T1
Their solution is
μx = A cos ωLt · e−t/T2 ,
μy = A sin ωLt · e−t/T2 ,										(3.2.112)

μz = μz 1 − Be−t/T1 .

The oscillation of angular frequency ωL is indeed damped exponentially, and also the z component approaches its saturation value exponentially – however, the two relaxation times are di erent.

When precession is maintained via the application of a field of angular frequency ω, a forced oscillation is obtained. To simplify the solution of the

21F. Bloch, 1946. Felix Bloch (1905–1983) shared the Nobel Prize with Edward Mills Purcell (1912–1997) in 1952 “for their development of new methods for nuclear magnetic precision measurements and discoveries in connection therewith”.

66 3 The Building Blocks of Solids

equations of motion, the linearly polarized field of frequency ω is replaced by two fields of amplitude B1 that are circularly polarized and counterrotating in the (x, y) plane. Only the component corotating with the precession direction will have important e ects, the other can be neglected. In the circularly polarized field obtained this way the equations of motion are

dμx

= −γμyB0 + γμz B1 sin ωt −

μx

,

dt

T2

dμy

= γμxB0

− γμz B1 cos ωt −

μy

,

(3.2.113)

dt

T2

dμz

= γμ

B

cos ωt

−

γμ B

sin ωt

−

μz − μz

.

dt

y

1

x

1

T1

As the magnetic moment corotates with the perturbing field, it is convenient to switch to a coordinate system rotating with angular frequency ω. The components in the two frames are related by

μx = μx cos ωt + μy sin ωt ,				μy = μy cos ωt − μx sin ωt		(3.2.114)
and
μx = μx cos ωt − μy sin ωt ,				μy = μy cos ωt + μx sin ωt .		(3.2.115)
Thus, in the rotating reference frame
	d μ		=	dμ		(3.2.116)
					− ω × μ ,
	dt			dt

where ω is a vector of magnitude ω along the constant magnetic field. In this reference frame the equations of motion are

d μx

= −γμy B0 + ωμy −

μx

,

dt

T2

d μy

= γμx B0

− γμz B1 − ωμx −

μy

,

(3.2.117)

dt

T2

d μz

= γμ

B

μz − μz

.

1

dt

y

− T1

It is immediately seen that in the rotating coordinate system the moment precesses as if acted upon by an e ective field Be = B − ω/γ.

Assuming again that the temporal variations of the z component can be neglected, a stationary solution is found in the rotating reference frame:

μ

=

1 + (ω

γB1(ω − ωL)T22

T

μ

,

x

−

ω

)2T 2

+ γ2B2T

z

L

2

1

1

2

μy

=

γB1T2

μz

,

(3.2.118)

−1 + (ω

ω

)2T 2 + γ2B2T

T

−

L

2

1

1

2

μ

=

1 + (ω

1 + (ω − ωL)2T22

T

μ

.

z

−

ω

)2T 2

+ γ2B2T

z

L

2

1

1

2

3.2 The State of Ion Cores

67

Going back to the original coordinate system, we have

T22

−

ωL) cos ωt +

1

1 + (ω − ωL)2T22

+ γ2B12T1T2

T2

μx =

(ω

sin ωt

T22

−

ωL) sin ωt

−

1

1 + (ω − ωL)2T22

+ γ2B12T1T2

T2

μy =

(ω

cos ωt

These yield

χ

=

χ0

ω

T

(ωL − ω)T2

,

2

2 1 + (ω − ωL)2T22 + γ2B12T1T2

xx

L

χ

=

χ0

ω

T

1

.

2

2 1 + (ω − ωL)2T22 + γ2B12T1T2

xx

L

Introducing the notation

2

2

Γ =

"1 + γ2B12T1T2 ≈

,

T2

T2

the real and imaginary parts of susceptibility are found to be

χ

=

χ0

ω

(ωL − ω)

,

2

L (ω − ωL)2 + (Γ/2)2

xx

χ

=

χ0

ω

1

1

.

2

L T2 (ω − ωL)2 + (Γ/2)2

xx

γB1 μz ,

(3.2.119)

γB1 μz .

(3.2.120)

(3.2.121)

(3.2.122)

Figure 3.2 shows the frequency dependence of the real and imaginary parts of susceptibility for typical values of the magnetic field and the relaxation time.

"

L

'

Fig. 3.2. Frequency dependence of the real and imaginary parts of susceptibility

Thus, instead of the infinitely sharp resonance obtained with the neglect of relaxation processes, a Lorentzian form is obtained for the imaginary part. In

68 3 The Building Blocks of Solids

weak perturbing fields the full width at half maximum (FWHM) is given by 1/T2. The line width is therefore a measure of the spin–spin relaxation time. In the relaxation-free limit T2 → ∞ (Γ → 0) (3.2.108) is recovered for the imaginary part – however, the real part is now not precisely (3.2.102), as the field counterrotating with the magnetic moment has been neglected. The two expressions are, nonetheless, identical in the vicinity of the resonance (i.e., for frequencies ω ≈ ωL) indicating that the approximation is justified.

The foregoing analysis showed that for a given external field there is a single resonant frequency. The selection rule mJ = ±1 implies that transition is possible only between adjacent levels. As the energy di erence E = gJ μBB is independent of mJ , the spectrum would consist of a single line for any value of J. In solids, however, this may change. The electrostatic potential due to neighboring atoms modifies the ground state of paramagnetic ions, and in particular, may partially lift the 2J + 1-fold degeneracy. If the resulting energy shift is smaller than that caused by the magnetic field, absorption may occur at several nearby frequencies. This gives the fine structure of paramagnetic resonance. Atomic levels are further split by the hyperfine interaction with the nucleus of magnetic moment μI . This gives rise to the hyperfine structure of the absorption spectrum. The location of the resonance thus indicates the species of the paramagnetic ion as well as the properties of its surroundings, while its line width yields information about the interactions responsible for relaxation.

3.3 The Role of Nuclei

So far we have only discussed the states of core electrons. The nucleus has been assumed to be a pointlike and structureless object whose charge – together with that of the electrons – determines the total charge of the ion. However, the nucleus also possesses a magnetic moment through which it can interact with the electrons’ spin. The role of the nucleus will also be important in the study of the structure and dynamics of solids by neutron scattering. Below we shall only consider the consequences of magnetic interactions.

3.3.1 Interaction with Nuclear Magnetic Moments

Both nucleon types, protons and neutrons, possess a magnetic moment. When expressed in terms of the nuclear magneton μN = e /2mp, the magnetic moment of the proton is μp = 2.793 μN, while that of the neutron is μn = −1.913 μN. Consequently most of the nuclei also possess a magnetic moment. The magnetic moment of a nucleus of angular momentum (nuclear spin) I is customarily written as

μI = gI μNI ,

(3.3.1)

where gI is the g-factor of the nucleus. Protons and neutrons are spin-1/2 particles, thus their g-factors are gp = 5.586 and gn = −3.826. For most

3.3 The Role of Nuclei

69

nuclei, the g-factor is of order unity, and the quantum number I can take integer or half-integer values.

Owing to the great disparity between the proton and electron mass, the nuclear magneton is three orders of magnitude smaller than the Bohr magneton, μN ≈ μB/1836. For this reason one can usually ignore nuclear magnetic moments when studying magnetic properties. However, there are some situations where the magnetic field due to nuclear magnetic moments becomes important. As we have already mentioned, such fields give rise to the hyperfine structure of energy levels.22 Using the Hamiltonian (3.2.19) of an electron in a magnetic field and the Coulomb gauge div A = 0,

Hhf =	e	p · A(r) − B(r) · μs ,	(3.3.2)
	me

gives the nonrelativistic contribution to the hyperfine interaction that is proportional to the nuclear magnetic moment. The field B(r) due to a nucleus of magnetic moment μI is derived from the vector potential

A(r) =	μ0	μI × r	=	−	μ0	μ		1	.	(3.3.3)
	4π r3				4π
							I × r

Here r is the position vector relative to the nucleus. The first term in (3.3.2) represents the spin–orbit interaction between the nuclear spin and the orbiting electron, while the second term is the spin–spin (or dipole–dipole) coupling.

Substitution of the above expression for the vector potential into the Hamiltonian of the hyperfine interaction and rearrangement of the scalar triple product gives the following form for the first term:

(1)	=	eμ0	p · [μI × r] =	eμ0		(3.3.4)
Hhf					l · μI .
		4πmer3		4πmer3

This expression can also be viewed as the energy of the nuclear magnetic moment μI in the magnetic field

	B(r) =	μ0	−e l	(3.3.5)
		4π mer3

created by orbiting electron charges.

In the second term of (3.3.2) B(r) is derived from the previous expression for the vector potential as B = × A. Making use of the vector identity

a × (b × c) = (a · c)b − (a · b)c

(3.3.6)

that also holds when some of the vectors are the operator , we have

22The electric quadrupole moment of nonspherically symmetric nuclei also contributes to the hyperfine structure – but here we are concerned with magnetic interactions alone.

70

3 The Building Blocks of Solids

(2)

μ0

(μI · )(μs · ) − (μI ·

1

Hhf = −

μs) 2!

.

(3.3.7)

4π

r

According to (C.3.8), 2(1/r) vanishes everywhere except at r = 0, thus

with the exception of this point

I · r5

s ·

−

Ir·3 s

,

Hhf

= −4π

3(μ

r)

(μ

(3.3.8)

(2)

μ0

r)(μ

μ

)

which is indeed a dipole–dipole interaction between the nuclear magnetic moment and the electron spin.

At r = 0 the Laplacian is 2(1/r) = −4πδ(r), thus the second term in

Hhf(2) gives a singular contribution on the order of

−	μ0	4πμI · μs δ(r) .	(3.3.9)
	4π

Another singular contribution comes from the first term of (3.3.7). It can be evaluated using the Fourier representation of 1/r, as given in (C.1.65):

lim (μ

)(μ

)

1

lim

1

eiq·r

(μ

q)(μ

q) dq .

I ·

s ·

r

2π2

q2

I ·

s ·

(3.3.10)

r→0

= − r→0

When the parentheses are expanded,

(μI · q)(μs · q) =

μI,αμs,β qαqβ ,

(3.3.11)

αβ

and the limit r → 0 is taken, only the terms α = β contribute to the integral on the right-hand side of (3.3.10):

r→0

e

·

q2

μI,αμs,β dq = δαβ

3 r→0

e

·

μI,αμs,α dq

(3.3.12)

lim

iq

r qαqβ

1

lim

iq

r

= δ

(2π)3

μ

μ

δ(r) ,

αβ

3

I,α

s,α

hence

1

4π

r

=

−

(3.3.13)

0 {(μI · )(μs · )} r

3 δ(r)(μI · μs) .

lim

→

When all these contributions are collected, the interaction between the nuclear magnetic moment μI and the electrons is written as

Hhf

−4π

mer3

· I

3(μs

·

r5

·

−

r·3

(3.3.14)

=

μ0

−e

l

μ +

r)(μI

r)

(μs μI )

8π

+ 3 μs · μI δ(r) .