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Magnetic Resonance
Chapter 21
Magnetic Resonance Chapter Outline 21.1 Nuclear Magnetic Moment 21.2 Zeeman Effect 21.3 Relaxation Phenomena 21.3.1 Spin-Lattice Relaxation 21.3.2 Spin-Spin Relaxation 21.4 Equation of Motion 21.5 Magnetic Resonance in the Absence of Relaxation Phenomena 21.6 Bloch Equations 21.6.1 Free Precession in Static Magnetic Field 21.7 Magnetic Broadening of Resonance Lines
445 446 448 448 450 450 452 454 455 457
21.8 21.9 21.10 21.11 21.12
Effect of Molecular Motion on Resonance Electron Spin Resonance Hyperfine Interactions Knight Shift Quadrupole Interactions in Magnetic Resonance 21.12.1 Nuclear Quadrupole Resonance 21.13 Ferromagnetic Resonance 21.14 Spin Wave Resonance 21.15 Antiferromagnetic Resonance Reference Suggested Reading
457 458 459 460 461 462 464 464 464 464 464
In solids, both the electrons and nuclei possess intrinsic spin angular momenta, which give rise to spin magnetic dipole moments. We have seen in the previous chapters that a static magnetic field produces magnetization, which yields important information about solids. A great deal of information about magnetic solids can be obtained from the nuclear magnetic resonance (NMR) technique in which an alternating magnetic field is also applied. The basic principle of an NMR method is to apply both dc and rf magnetic fields perpendicular to one another. As a result, the nuclear energy states split up into substates. When the frequency of the applied rf field is equal to the natural frequency in the presence of external magnetic field (which corresponds to the frequency of gyroscopic precession of the magnetic moments), then resonance takes place. There are different resonance techniques with which one can study a number of properties, such as the fine structure of nuclear levels, the electronic structure of an impurity, the interaction between magnetic dipoles, the interaction between magnetic dipoles and the lattice, and the line widths of nuclear transitions. The basic ideas behind nuclear resonance techniques will be described in this chapter.
21.1 NUCLEAR MAGNETIC MOMENT In analogy with the magnetic moment of an electron, the magnetic moment of a nucleus with nuclear spin I is defined as ! mI
¼ gI ħI
(21.1)
Here gI is the ratio of the magnetic moment to the angular momentum and is the gyromagnetic ratio. In the case of a nucleus, gI satisfies the relation gI ħ ¼ gI mBp
(21.2)
where mBp is the Bohr magnetron of a proton defined by Eq. (2.8) and gI is the nuclear spectroscopic splitting factor. By substituting Eq. (21.2) into Eq. (21.1), one gets ! mI
¼ gI mB p I
(21.3)
! Eq. (21.3) shows that, for a nucleus, m I
and I are in the same direction, in contrast to an electron. Although the gyromagnetic ratio is a more useful quantity for discussing magnetic resonance in many respects, nuclear magnetic moments are usually tabulated in terms of mBp, a natural unit with a value of 5.05 1024 erg/gauss (see Chapter 2). One can also define the
Solid State Physics. https://doi.org/10.1016/B978-0-12-817103-5.00021-9 © 2019 Elsevier Inc. All rights reserved.
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!
!
magnetic moment of a nucleus m I with nuclear spin I as the vector sum of the magnetic moments of the nucleons m n that are present in it, that is, X! ! mn (21.4) mI ¼ n
21.2
ZEEMAN EFFECT !
The Hamiltonian of a nucleus with magnetic moment m I in an external dc magnetic field H is _
!
H M ¼ m I H ¼ gI ħI H
(21.5)
If the magnetic field is applied in the z-direction, that is, H ¼ ^i3 H, the Hamiltonian becomes _
H M ¼ gI ħHIz ¼ gI mBp HIz
(21.6)
The energy of a nucleus in the state j mIi is given by _
EmI ¼ hmI jH M jmI i ¼ gI ħHmI ¼ gI mBp H mI
(21.7)
where mI is the eigenvalue of Iz defined as hmI j Iz jmI i ¼ mI
(21.8)
Here mI has 2 I + 1 values given as I, I 1, … 1, 0, 1, … (I 1), I. Hence, each nuclear state splits up into 2 I + 1 substates in the presence of a magnetic field and they are equally spaced. Transitions between the substates are allowed only if the magnetic quantum number mI changes by unity, that is, DmI ¼ 1
(21.9)
ħo ¼ Em0 EmI ¼ gI ħH DmI ¼ gI mBp H DmI
(21.10)
mI m0I ¼ DmI
(21.11)
Using Eq. (21.7), we can write I
where
Using the selection rule (21.9) in Eq. (21.10), the transition frequency is given by oL ¼ gI H ¼ gI mBp H=ħ
(21.12)
Eq. (21.12) yields a single resonance frequency, usually called the Larmor frequency oL, which gives the fundamental condition for magnetic resonance absorption. The radiation produced is circularly polarized in a plane perpendicular to the dc magnetic field H. Therefore, in order to excite such transitions, it is necessary to supply radiation with a magnetic vector circularly polarized in a plane perpendicular to the dc magnetic field. One can interpret the above result in the fol! lowing manner: The magnetic dipole moment m I , when placed in a dc magnetic field H, starts precessing about the applied field with the Larmor frequency (see Fig. 21.1). As an example, consider a solid with I ¼ 1/2. In the presence of a magnetic field, mI ¼ 1/2, which gives the energy of the two states as E ¼ (1/2) gI ħ H (see Fig. 21.2). If ħ oL denotes the energy difference between the two energy states, then oL ¼ gI H ¼ gI mBp H=ħ
(21.13)
which is the same as Eq. (21.12). The Zeeman splitting for a solid with I ¼ 3/2 is also shown in Fig. 21.2. Problem 21.1
Let the population of a Zeeman state NmI for a particular value of mI be given as NmI ∝ exp EmI =kB T
(21.14)
Here kB and T are the Boltzmann constant and temperature, respectively. Prove that the average magnetic moment of a nucleus hmIi in the direction of the magnetic field is given by y 2I + 1 2I + 1 1 coth y coth (21.15) hmI i ¼ gI ħ 2 2 2 2 where y¼
gI ħH kB T
(21.16)
Magnetic Resonance Chapter
21
!
FIG. 21.1 Precession of the magnetic moment m I about the applied magnetic field H.
FIG. 21.2 Zeeman splitting in a solid with nuclear spin (A) I ¼ 1/2 and (B) I ¼ 3/2.
Problem 21.2
Using hmIi given by Eq. (21.15), prove that the nuclear magnetic susceptibility in the limit of very high values of T is given by wM ¼ nnuc g2I ħ2 where nnuc is the number of nuclei per unit volume.
I ðI + 1Þ 3kB T
(21.17)
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Solid State Physics
21.3
RELAXATION PHENOMENA
In a paramagnetic substance each atom possesses a finite spin, which gives rise to a finite atomic spin magnetic moment. In the absence of an applied magnetic field the magnetic dipole moments are randomly oriented, giving rise to zero magnetization. With the application of a magnetic field, magnetization appears, which takes some finite time to reach its equilibrium value. When the magnetic field is switched off, the magnetization decreases and goes to zero again in a finite time. Both the increase and decrease in the magnetization are caused by relaxation phenomena and the time taken to reach the equilibrium value is called the relaxation time. In order to have insight into relaxation phenomena, let us consider a system of free magnetic dipoles oriented randomly. If an external magnetic field is applied, the atomic magnetic dipoles start to precess about it, as shown in Fig. 21.1, but the component of magnetization along the direction of H remains unchanged. Therefore, no magnetization will appear. For a finite magnetization to appear, the interaction of an atomic magnetic dipole with other atomic dipoles or with its surroundings is required. Consider a paramagnetic material in which the atoms (magnetic dipoles) form a lattice. The magnetic ! ! field at the position ri of the ith magnetic dipole m i due to the jth dipole m j at rj is given by ! ! m j ri j ri j m j r2ij (21.18) Hi j ¼ r5ij !
The total magnetic field at the position of m i due to all of the other magnetic dipoles is obtained by summing over j, that is, ! ! X X m j ri j ri j m j r2ij Hi j ¼ (21.19) Hi ¼ r5ij j j Hi is called the internal magnetic field experienced by an atomic magnetic dipole and is on the order of mI/a3 where a is on the order of a few angstroms. For example, in iron alum j Hi j 550 gauss. In analogy with the local electric field, Hi is called the local magnetic field and is represented by Hloc(ri). In a magnetic solid there can be two types of relaxation phenomena depending on the value of the applied magnetic field.
21.3.1
Spin-Lattice Relaxation
Spin-lattice relaxation occurs when the applied dc magnetic field H is much stronger than the local magnetic field Hloc. Therefore, a small increase in H will change the magnitude of the total magnetic field H + Hloc, but will not change its direction by very much. In this case the spin interacts with the lattice and exchanges energy with it. The lattice interaction may cause some of the magnetic dipoles to slip from the antiparallel direction to the parallel one, thereby producing a finite magnetization M defined as M ¼ ^i1 M1 + ^i2 M2 + ^i3 M3
(21.20)
The time taken by the material to reach the equilibrium value M0 is called the spin-lattice relaxation time tsl and the phenomenon is called the spin-lattice relaxation. The time tsl is also sometimes called the longitudinal relaxation time. When the magnetic field is switched off the material returns to its initial unmagnetized state. Let the magnetic field H be applied in the z-direction, that is, H ¼ ^i3 H with M0 as the equilibrium magnetization. Further, let Mz(t) be the magnetization at any time t in the z-direction, which is different from M0. If the rate of change of Mz(t) is assumed to be proportional to its deviation from M0, then one can write dMz M0 Mz ¼ dt tsl
(21.21)
where 1/tsl is the constant of proportionality. The above equation can be written as M ðz
ðt dt
(21.22)
h i Mz ðtÞ ¼ M0 1 et=tsl
(21.23)
0
Integrating the above equation, one gets
dMz 1 ¼ M0 Mz tsl
0
Magnetic Resonance Chapter
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FIG. 21.3 Spin-lattice and spin-spin relaxation processes. In the spin-lattice relaxation process, the magnetization M approaches the equilibrium value M0 along the z-direction. In the spin-spin relaxation process, the magnetization M0 (in the xy plane) approaches zero, the equilibrium value, in the presence of magnetic field H0.
Fig. 21.3 shows how a precessing magnetization M (about the z-direction) acquires the equilibrium value of magnetization M0 in the spin-lattice relaxation phenomenon. The variation of Mz(t) with time is shown in Fig. 21.4. The spin-lattice relaxation is measured in strong fields but at low frequencies. It is important to point out that tsl is strongly temperature dependent and decreases with increasing temperature. This variation is due to an increase in the mobility of the magnetic dipoles with an increase in temperature. The time tsl, due to the nuclear moments in solids, may have a value from a few seconds to hours. In solids in which the paramagnetism is due to the electrons, tsl varies between 1011 and 106 s at room temperature. FIG. 21.4 The variation of the z-component of magnetization Mz(t) with time t in the spin-lattice relaxation phenomenon.
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Solid State Physics
21.3.2
Spin-Spin Relaxation
The nuclear spins also interact among themselves via dipole-dipole interactions. In a paramagnetic solid, each spin experiences a local magnetic field Hloc due to the neighboring magnetic dipoles. The direction and magnitude of Hloc may differ from nucleus to nucleus depending on the relative positions of the neighboring nuclei. If the external magnetic field H is much smaller than the local magnetic field Hloc, that is, H ≪ Hloc, then its effect will be to slightly change the direction of the field seen by a dipole, but its magnitude will remain essentially unaltered. The magnetic dipoles will thus precess about a slightly different direction and, as a result, finite magnetization occurs in the direction of H. In this situation the magnetic force due to the external magnetic field is insufficient to cause the magnetic dipoles to flip over and, therefore, there will be no exchange of energy between the spins and the lattice. In other words, spin-lattice relaxation will be absent. On the other hand, there is a finite energy exchange between the spins, giving rise to what is called the spin-spin relaxation process. It is convenient to introduce the spin-spin relaxation time tss to describe the lifetime of a nuclear spin state. The different values of Hloc at different nuclei cause local irregularities in the magnetic field. Further, the different values of Hloc make different nuclei precess with different frequencies. So, the precessing nuclei will go out of phase with each other in a time on the order of tss, yielding zero as the equilibrium values of Mx and My. Let an rf magnetic field H0 be applied in the xy-plane, which is defined as H0 ¼ ^i1 Hx + ^i2 Hy
(21.24)
Here Hx and Hy are the x- and y-components of the magnetic field. The magnetization produced in the xy-plane M0 is written as M0 ¼ ^i1 Mx + ^i2 My
(21.25) 0
where Mx and My are the nonequilibrium values of the magnetizations in the x- and y-directions. H0 and M are called the transverse magnetic field and transverse magnetization, respectively. The rate of change of Mx and My are given as dMy My dMx M ¼ x, ¼ dt tss dt tss
(21.26)
The solution of these differential equations is simple. Solving, we find Mx ¼ M0 x et=tss , My ¼ M0 y et=tss
(21.27)
Therefore, Mx and My approach zero exponentially with time. Fig. 21.3 shows the variation of M0 with time, which ultimately goes to zero (equilibrium value) in the presence of the magnetic field H0. The time tss is measured with small magnetic fields that have very high frequencies, on the order of many Mc/s. At such high frequencies the spin-lattice relaxation process is absent. The relaxation time tss is on the order of 1010 s, which is smaller than the spin-lattice relaxation time tsl. Further, the spin-spin relaxation process is independent of temperature, which makes tss independent of temperature.
21.4
EQUATION OF MOTION !
Consider a nucleus with spin I and magnetic moment m I placed in a constant magnetic field H. The rate of change of the ! nuclear angular momentum I ħ gives the torque τ acting on the nucleus, that is, d ! ðIħÞ ¼ τ dt
(21.28)
with !
!
τ ¼ mI H
(21.29)
Using Eqs. (21.1), (21.29) in Eq. (21.28), one can write !
1 dm I ! ¼ mI H gI dt
(21.30)
Magnetic Resonance Chapter
21
451
!
We have already seen that m I will precess about the magnetic field H with the Larmor frequency oL. Consider now a magnetic material with weakly interacting nuclear spins. The magnetization M is a vector sum of the nuclear magnetic moments per unit volume. If ri is the number of nuclei per unit volume in the ith state, then from Eq. (21.30) one can write dM ¼ gI M H dt where M¼
X
!
ri m Ii
(21.31)
(21.32)
i !
Here m I i is the nuclear magnetic moment in the ith state of the nucleus having spin I. The solution of Eq. (21.31) for a constant magnetic field is the same as that for Eq. (21.30), that is, M will precess about H with the Larmor frequency oL. If M is along the z-direction, then in the equilibrium state, Mz ¼ M0 and Mx ¼ My ¼ 0 and the magnetic susceptibility w0 satisfies the Curie law, that is, M 0 ¼ w0 H
(21.33)
where w0 ¼
CM T
(21.34)
Here CM is the Curie constant. Hence, in a constant magnetic field H, one observes a resonant Larmor frequency nL ¼ oL/2 p that is circularly polarized. For H 104 gauss one obtains nL ranging from 1–50 Mc/s. To observe the resonance absorption of rf energy, one has to apply two magnetic fields: 1. A dc magnetic field H, which can be varied along the z-direction. 2. An rf magnetic field of amplitude H0 (≪H) perpendicular to the dc magnetic field. The reason for the application of the rf field can be understood from the following. Consider a magnetic system with nuclear ! spin I ¼ 1/2, which gives two energy levels (see Fig. 21.2A). The orientation of the magnetic moment m I in the lower energy state is shown in Fig. 21.5 and it precesses about H with the Larmor frequency oL. For resonance absorption to take place, a
!
FIG. 21.5 Precession of the magnetic moment m I about the dc magnetic field H in the presence of rf magnetic field H0.
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Solid State Physics
FIG. 21.6 Schematic representation of the experimental setup used for studying the magnetic resonance in a solid. The dc magnetic field is applied in the z-direction, but the rf magnetic field is applied in the x-direction.
transition must occur from the lower state to the higher state (see Fig. 21.2A), which involves a change in spin from ½ to ½. The transition will take place if H0, applied perpendicular to H, causes the absorption of rf energy at oL ¼ gI H. Further, H0 reverses the tip of the dipole from parallel (mI ¼ 1/2) to antiparallel (mI ¼ 1/2) alignment and vice versa. Let the rf magnetic field be applied in the x-direction. Then, one can write Hx ¼ 2H0 cos o t Hy ¼ Hz ¼ 0
(21.35)
But we know that in the Zeeman effect, the resonance frequency emitted or absorbed is circularly polarized. Therefore, it is convenient to represent the rf field as a combination of two circularly polarized magnetic fields in the xy-plane: one rf field rotates clockwise and the other counterclockwise. These two rf fields can be written as Clockwise : Hx ¼ H0 cos ot, Hy ¼ H0 sin ot
(21.36)
Counter-clockwise : Hx ¼ H0 cos ot, Hy ¼ H0 sin ot
(21.37)
At oL, one of the rotating fields will follow the precessing magnetic dipole and exert a constant torque on it. As a result, the magnetic dipole tips from parallel (antiparallel) to antiparallel (parallel) alignment. Therefore, the other rotating field becomes redundant. Hence, in order to observe resonance absorption of rf energy by the spin system, it is essential that the lower level be more heavily populated than the upper level, which is indeed the case at thermal equilibrium due to the Boltzmann distribution. A schematic diagram of the nuclear resonance experimental setup is shown in Fig. 21.6.
21.5
MAGNETIC RESONANCE IN THE ABSENCE OF RELAXATION PHENOMENA
In the absence of relaxation phenomena, the equation of motion is given by Eq. (21.31), in which the three components of magnetization become h i dMx ¼ gI My Hz Mz Hy (21.38) dt dMy ¼ gI ½Mz Hx Mx Hz (21.39) dt h i dMz ¼ gI Mx Hy My Hx (21.40) dt Let the magnetic field in the xy-plane H0 (with components Hx and Hy) be very small in comparison with the dc magnetic field H ¼ ^i3 H. As a result, the components Mx and My are very small in comparison with Mz. To a first-order approximation, Eqs. (21.38)–(21.40) reduce to h i dMx ¼ gI My H Mz Hy (21.41) dt
Magnetic Resonance Chapter
21
453
dMy ¼ gI ½Mz Hx Mx H dt
(21.42)
dMz ¼0 dt
(21.43)
The oscillating components of the magnetic field Hx and Hy give rise to oscillating components of magnetization Mx and My, which can be represented as Hx ¼ H0x ei o t ,Mx ¼ M0x ei o t
(21.44)
Hy ¼ H0y e
(21.45)
iot
,My ¼ M0y e
iot
Substituting Eqs. (21.44), (21.45) into Eqs. (21.41), (21.42), one gets ioMx gI HMy + gI Mz Hy ¼ 0
(21.46)
gI HMx + ioMy gI Mz Hx ¼ 0
(21.47)
Multiplying Eq. (21.46) by gI H and Eq. (21.47) by i o and subtracting, one gets My ¼
gI Mz o H + ioH L y x o2L o2
(21.48)
Similarly, one can obtain Mx from Eqs. (21.46), (21.47). We find Mx ¼
gI M z o H ioH L x y 2 oL o2
(21.49)
Eqs. (21.48), (21.49) can be written in matrix form as 0
Mx
1
0
o L gI M z B C B o2L o2 B C B C¼ B B B C @ iogI Mz @ A o2L o2 My
1 0H 1 iogI Mz x B C o2L o2 C B CB C CB C C o L gI M z A @ A o2L o2 Hy
(21.50)
In matrix form the magnetic susceptibility can be written as 0 @
Mx My
1
0
A¼@
xy wxx M wM
10 A@
yy wyx M wM
Hx
1 A
(21.51)
Hy
Comparing Eqs. (21.50), (21.51), one can write yy wxx M ¼ wM ¼ yx wxy M ¼ wM ¼
oL gI Mz o2L o2
(21.52)
iogI Mz o2L o2
(21.53)
The transverse magnetization M0 and the transverse magnetic field H0 given by Eqs. (21.25), (21.24) exhibit some interesting features. M0 and H0 are not in the same direction, rather M0 lags behind H0. This can be seen by putting Hx ¼ 0 in Eqs. (21.48), (21.49) and noticing the phase relation and amplitudes of Mx and My. The second interesting feature is that the components of the susceptibility tensor (Eqs. 21.52, 21.53) and magnetization (Eqs. 21.48, 21.49) become infinite at o ¼ oL, the natural frequency of vibration. But actually, the susceptibility and magnetization become very large at oL because, in this state, the applied rf field is synchronous with the precessional motion. This is the condition for electron paramagnetic resonance. The infinite values of susceptibility and magnetization suggest that some kind of relaxation phenomenon exists, which makes them finite but large in value.
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21.6
Solid State Physics
BLOCH EQUATIONS
Bloch derived equations of motion for the magnetic response taking account of the spin-lattice and spin-spin relaxation phenomena. From Eqs. (21.21), (21.40) the rate of change of Mz with time is given by h i M M dMz z ¼ gI Mx Hy My Hx + 0 dt tsl
(21.54)
Similarly, from Eqs. (21.38), (21.39), (21.26) one can write h i M dMx ¼ gI My Hz Mz Hy x dt tss
(21.55)
dMy My ¼ gI ½Mz Hx Mx Hz dt tss
(21.56)
Eqs. (21.54)(21.56) are called Bloch equations. Substituting the values of Hx and Hy from Eq. (21.37) and putting Hz ¼ H in Eqs. (21.54)–(21.56), one gets h i M dMx ¼ gI My H + Mz H0 sin ot x (21.57) dt tss dMy My ¼ gI ½Mz H0 cos ot Mx H dt tss h i M M dMz z ¼ gI Mx H0 sin ot + My H0 cos ot + 0 dt tsl Eqs. (21.57)–(21.59) can be solved for Mx, My, and Mz and the final result is given as " # 1 2H0 tss ðoL o Þ cos ot + 2H0 sin ot Mx ¼ w0 oL tss 2 1 + ðoL oÞ2 t2ss + g2I H20 tsl tss " # 1 2H0 cos ot 2H0 tss ðoL o Þ sin ot My ¼ w0 oL tss 2 1 + ðoL oÞ2 t2ss + g2I H20 tsl tss " M z ¼ w0 H
1 + ðoL oÞ2 t2ss
(21.58) (21.59)
(21.60)
(21.61)
# (21.62)
1 + ðoL oÞ2 t2ss + g2I H20 tsl tss
In actual experiments the linearly polarized rf magnetic field is applied only in one direction, say in the x-direction (see Fig. 21.6). In this case, Hx ¼ 2 H0 cos o t and Hy ¼ 0. For a linearly polarized rf magnetic field, Eqs. (21.60)– (21.62) reduce to " # 1 2H0 tss ðoL o Þ cos ot (21.63) Mx ¼ w0 oL tss 2 1 + ðoL oÞ2 t2ss + g2I H20 tsl tss " # 1 2H0 cos ot My ¼ w0 oL tss 2 1 + ðoL oÞ2 t2ss + g2I H20 tsl tss " M z ¼ w0 H
1 + ðoL oÞ2 t2ss 1 + ðoL oÞ2 t2ss + g2I H20 tsl tss
(21.64)
# (21.65)
The dynamical magnetic susceptibility can be written as wM ðoÞ ¼ w0M ðoÞ + i w00M ðoÞ
(21.66)
Magnetic Resonance Chapter
w0M(o) and w00M(o) are the real and imaginary parts, which from Eqs. (21.63), (21.64) are given by " # Mx Mx 1 ðoL o Þ tss 0 ¼ w o t wM ðoÞ ¼ ¼ Hx 2H0 cos ot 2 0 L ss 1 + ðoL oÞ2 t2ss + g2I H20 tsl tss " # My My 1 1 00 ¼ w o t ¼ wM ð o Þ ¼ Hx 2H0 cos ot 2 0 L ss 1 + ðoL oÞ2 t2ss + g2I H20 tsl tss
21
455
(21.67)
(21.68)
In the denominators of Eqs. (21.63)–(21.65) and Eqs. (21.67)–(21.68), the term g2I H20 tsl tss is a dimensionless constant that determines the degree of saturation. If H0 is very small, then saturation is not reached and g2I H20 tsl tss ≪1 This term can be neglected in the denominators of Eqs. (21.63)–(21.65) and Eqs. (21.67)–(21.68). In this approximation the static magnetic susceptibility from Eq. (21.65) becomes w0 ¼
Mz H
The real and imaginary parts of the dynamical magnetic susceptibility, from Eqs. (21.67), (21.68), become " # 1 ðoL o Þ tss 0 wM ðoÞ ¼ w0 oL tss 2 1 + ðoL oÞ2 t2ss " # 1 1 w00M ðoÞ ¼ w0 oL tss 2 1 + ðoL oÞ2 t2ss
(21.69)
(21.70)
(21.71)
The susceptibilities w0M(o) and w00M(o) are plotted in Fig. 21.7 as a function of (oL o)tss. Fig. 21.7A shows that w00M(o) exhibits resonance absorption at o ¼ oL and the peak has Lorentzian shape. Eq. (21.71) can be written as w00M ðoÞ ¼
ð1=2Þw0 oL =tss ðoL oÞ2 + 1=t2ss
(21.72)
From the above equation we see that the halfwidth Do of resonance at half the height is given by Do ¼
1 tss
(21.73)
The real part of the dynamical magnetic susceptibility w0M(o), shown in Fig. 21.7B, exhibits dispersion, which accompanies the absorption.
21.6.1
Free Precession in Static Magnetic Field
One can solve the Bloch equations for the free precession of the spin system in a static (dc) magnetic field H ¼ ^i3 H. In a static magnetic field the equilibrium magnetization is Mz ¼ M0. In this case the Bloch equations (21.54) – (21.56) reduce to dMx M ¼ gI HMy x dt tss
(21.74)
dMy My ¼ gI HMx dt tss
(21.75)
dMz ¼0 dt
(21.76)
For Eqs. (21.74)–(21.76), we seek damped-oscillator-like solutions of the form 0
Mx ¼ M et=t cos ot 0
My ¼ M et=t sin ot
(21.77) (21.78)
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Solid State Physics
FIG. 21.7 Schematic representation of the real part w0M(o) and imaginary part w00M(o) of the dynamical magnetic susceptibility wM(o) as a function of the frequency o in a solid.
where t0 is the damping time. Substituting Eqs. (21.77), (21.78) into Eqs. (21.74)–(21.76), one can write 1 1 cos ot ¼ ðoL oÞ sin ot t0 tss 1 1 sin ot ¼ ðoL oÞ cos ot t0 tss
(21.79)
(21.80)
Eqs. (21.79), (21.80) will be valid only if t0 ¼ tss
(21.81)
o ¼ oL
(21.82)
and
Magnetic Resonance Chapter
21
457
Eq. (21.81) shows that the damping time is equal to the spin-spin relaxation time. Further, using the analogy of the precession of spins in a static magnetic field with the motion of a damped harmonic oscillator suggests that the spin system exhibits resonance absorption of energy near the Larmor frequency. The frequency width of the response of the system to the driving field will be Do ¼ 1/tss.
21.7 MAGNETIC BROADENING OF RESONANCE LINES It has been observed that resonance absorption of an rf magnetic field is not very sharp, but rather exhibits a peak with finite width called the linewidth. There are a number of physical phenomena that contribute to the linewidth of resonance absorption. 1. In a solid, an inhomogeneous magnetic field gives rise to a broadening of the resonance line. Inhomogeneity in a magnetic field may arise either because of the presence of impurities or due to different values of Hloc at different nuclei (crystal structure effect). 2. If a nucleus possesses a finite magnetic quadrupole moment, it gives rise to what are known as quadrupole effects. These effects destroy the degeneracy of resonance frequencies between nuclear levels with different mI values (Zeeman effect). Quadrupole effects may give rise to resolved or unresolved splitting of a resonance line. If the quadrupole effects are very small (which may be the case in a number of solids), one gets unresolved splitting of the resonance line, which amounts to broadening. 3. Spin-lattice interactions also give rise to a broadening of the resonance lines. This is because the spin-lattice interactions produce an equilibrium population by balancing the rates of transitions. This puts a limit on the lifetime of the Zeeman states and hence results in a broadening of the resonance lines. The coupling between the nuclear magnetic moments in a solid makes a major contribution to the broadening of resonance lines and is considered here. In addition to the applied magnetic field H, a local magnetic field Hloc(ri) is produced at the position ri of a nuclear magnetic moment (see Eq. 21.19). In a crystalline solid Hloc(ri) is different at different lattice positions, yielding an inhomogeneous total magnetic field Htot given by Htot ¼ H + Hloc
(21.83) ! Hloc m I =r31 ,
where r1 is the 1NN disConsidering the interactions between the 1NNs, it is evident from Eq. (21.19) that ˚ , one obtains Hloc ffi 1 gauss. Therefore, if Htot is tance. For a nuclear magnetic moment mI ¼ 1023 erg/gauss and r1 ¼ 2 A different at different nuclei, then the various nuclei will exhibit resonance at different frequencies. If DHloc is the spread in the value of Hloc seen by the nuclei, then it will also be the spread in Htot. Thus, the spread in the resonance frequency Do, that is, oL ! oL + Do, is given by Do ¼ gI DHloc
(21.84)
Knowing the values of Do and DHloc, one can also find the spin-spin relaxation time tss. For example, in CaF2, DHloc ¼ 2 gauss and Do ¼ 5 104 s1, therefore, tss ¼
1 1 ¼ 2 105 s ¼ Do 5 104
(21.85)
21.8 EFFECT OF MOLECULAR MOTION ON RESONANCE The broadening of resonance lines depends on the nature of the material. The linewidth in the case of liquids is very small compared with solids: the linewidth in a liquid can be as narrow as 0.5 cps, but in solids it is on the order of 5000 cps or more at room temperature. It has further been observed experimentally that the resonance peaks are narrower for a solid in which the atoms (nuclei) are in rapid motion. Further, with an increase in the temperature T of a solid, the motion of the atoms becomes faster and the resonance peaks become narrower. The effect of the motion of the atoms on the width of resonance peaks is usually called motional narrowing. This effect is much more pronounced in liquids as the atoms can move much more quickly in random directions. The motional narrowing of resonance peaks can be explained in terms of the local magnetic field Hloc as follows. It has already been explained that the width of a resonance peak arises due to the inhomogeneous nature of Hloc, the crystal structure effect. All of the measurements in a laboratory are made at finite temperatures. At finite temperatures, an
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atom jumps from one lattice position to another, a process similar to the diffusion of atoms. Let us suppose that an atom remains at a single lattice position for an average time of τm seconds. The atoms in a solid move randomly in different directions, as a result of which Hloc at a particular lattice point changes very quickly. The time-averaged value of the local magnetic field hHloci becomes nearly constant and with much smaller spread. In other words, one can say that the random motion of atoms in different directions makes the solid, and hence hHloci, nearly homogeneous, yielding only a very small spread in Hloc. The decrease in the spread of Hloc yields a smaller width of the resonance absorption line. With an increase in temperature, the motion of the atoms increases, yielding a smaller value of DHloc, which in turn makes the resonance lines narrower. In other words, one can say that an increase in the motion (velocity) of atoms decreases the value of τm, which causes a narrowing of the resonance peaks. Measurements of the spin-lattice relaxation time tsl are based on the competition between resonance absorption (which tends to equalize the population of different levels) and the spin-lattice interaction (which tends to maintain the Boltzmann distribution). The values of tsl obtained experimentally vary between 105 104 s: 104 s is the value obtained for ice. The value of tsl is larger for solids than for liquids or gases. For solids, tsl is rarely smaller than 102 s. Rather, it may be very large at low temperatures. On the other hand, for liquids, tsl may be as short as 102 103 s and rarely exceeds a few seconds. This shows that as the atoms in a solid become freer the relaxation time tsl decreases. It is interesting to note that tsl may reduce strongly when paramagnetic ions are present; these ions have an effective magnetic moment that is 103 times as large as the nuclear magnetic moments and they form a very efficient medium for establishing heat contact between the nuclear spins and their surroundings. On the other hand, at low temperatures, there is no variation in the spin-spin relaxation time tss. With an increase in temperature, tss increases and, at a certain temperature, tsl and tss may become approximately equal.
21.9
ELECTRON SPIN RESONANCE
In an atom, both the nucleus and the electrons possess intrinsic spin and hence magnetic moment. Let the magnetic moment of a nucleus with spin I be mI and that of an electron mB. When a magnetic field H is applied to a solid, it splits the energy levels of both the nuclei and the electrons. The splitting of the energy levels of the nuclei gives rise to NMR and that of the electrons gives rise to electron spin resonance (ESR). One should note that there are some solids that exhibit ESR. Consider a solid in which each atom contains only one electron. Let a magnetic field H be applied to the solid in the z-direction, that is, H ¼ ^i3 H. Further, an rf magnetic field H0 is applied perpendicular to H (say in the x-direction). The magnetic part of the Hamiltonian due to the atomic electron is given by _
!
H ¼ m s H ¼ gs mB s H ¼ gs mB Hsz
(21.86)
Here we have used Eq. (18.24) for the spin magnetic moment of an electron. The energy of the electron states is obtained by taking the expectation value of the Hamiltonian as _
Ems ¼ hms jH jms i ¼ gs mB Hms
(21.87)
According to the above equation the electron states get split up into two substates (see Fig. 21.8). The energy difference between the two states becomes DE ¼ 2mB H
(21.88)
Eq. (21.88) gives the condition for a resonance. If an rf field is applied in the xy-plane, then the resonance frequency o0 is given by ħo0 ¼ 2mB H
(21.89)
FIG. 21.8 Splitting of an electron energy state with sz ¼ 1/2 into two substates in the presence of a dc magnetic field H applied in the z-direction.
Magnetic Resonance Chapter
21
459
FIG. 21.9 The energy absorption at the resonance frequency n0, which broadens into a resonance peak.
So, resonance absorption of the rf field takes place at frequency o0. At resonance, the electrons in energy state E1/2 absorb energy and make a transition to the higher energy state E1/2. Note that the frequency o0 is twice the Larmor frequency oL. If we apply H ¼ 1 weber/m2, then o0 30000 Mc/s. In the discussion above, only a single electron per atom is assumed, but in general there are a number of electrons in an atom of a solid. Therefore, the resonance frequency is affected by the following interactions: 1. Electron-electron interactions modify the resonance process. 2. The interaction of nuclear magnetic moments with the electronic magnetic moments affects the electron energy levels. One can say that the nuclear spin can take 2 I + 1 orientations in the magnetic field and each of these alters the electron energy levels by different amounts (hyperfine interaction). These interactions, therefore, broaden a resonance line into a resonance peak, as shown in Fig. 21.9. If the electron-electron and electron-nucleus interactions are small compared with the splitting of the energy levels of a single electron, then they can be treated as a perturbation to find the resultant electron states. Thus, one can find the width of the resonance peak. Other effects, such as the inhomogeneous local electric fields arising from impurities and other crystal defects, also change the position of the resonance lines. ESR is observed only in those solids in which each atom has unpaired electrons so as to yield finite electron spin (e.g., paramagnetic substances) because the paired electrons in the valence band produce no effect. Therefore, ESR is often called electron paramagnetic resonance (EPR).
21.10
HYPERFINE INTERACTIONS
So far, we have studied NMR in which the nuclear energy levels are calculated in the presence of mutually perpendicular dc and rf magnetic fields. But in a solid there are electrons revolving around every nucleus that possess finite magnetic moment. Therefore, additional interactions exist between the magnetic moments of electrons and nuclei, usually called hyperfine interactions, which cause further splitting of the nuclear energy levels. The motion of electrons around the nucleus produces a finite magnetic field with the following contributions: 1. The electrons revolving around the nucleus are equivalent to current loops, which produce a finite magnetic field at the nuclear site. 2. The spinning motion of electrons yields spin current about the nucleus. The spin current produces what is called a contact hyperfine interaction. Even if quenching of the orbital angular momentum, and hence of the orbital magnetic moment, occurs, there will be a finite contact hyperfine interaction. Therefore, the contact hyperfine interaction is of special importance in such solids. The net magnetic field arising from these two contributions interacts with the nucleus to produce the hyperfine interaction. If the hyperfine interaction is small compared with the splitting of the nuclear_energy levels, it may be treated as a perturbation in the nuclear resonance phenomenon. The interaction Hamiltonian H sI between the nuclear magnetic moment and the magnetic moment of the electron can be represented as _
H sI ¼ asI s I
(21.90)
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Solid State Physics
FIG. 21.10 Splitting of an electron state, in a nucleus with spin I ¼ ½, into four states in the presence of hyperfine interactions. Two transitions with frequencies o1 and o2 are allowed by the selection rules.
where I and s are the nuclear and electron spins and asI is the hyperfine constant. The energy eigenvalues of Eq. (21.90) are given by (21.91) EsI ¼ asI mI ms The electron spin quantum number ms has two values, 1/2. For a nucleus with I ¼ 1/2 the quantum number mI also has two values 1/2. According to Eq. (21.91), each electron state gets split up into two states with mI ¼ 1/2, giving rise to four states as shown in Fig. 21.10. The transitions between the four states take place by satisfying the following selection rules DmI ¼ 0, Dms ¼ 1
(21.92)
The allowed transitions yield two resonance lines, as shown in Fig. 21.10.
21.11
KNIGHT SHIFT
Consider a diamagnetic material with nuclear spin I to which the magnetic field H is applied in the z-direction, that is, H ¼ ^i3 H. The magnetic interaction Hamiltonian is given by _
H M ¼ gI ħHIz
(21.93)
When an rf magnetic field H0 is applied in the direction perpendicular to H, nuclear resonance absorption is observed at the Larmor frequency oL ¼ gI H. Now consider a metallic solid with the same nuclear spin I to which the same magnetic field H0 is applied in the xy-plane. It is found that nuclear resonance absorption in the metallic solid is observed at a slightly different value of the dc magnetic field. This effect is known as the Knight shift and it yields knowledge about the distribution of conduction electrons in a metal. In a metallic solid there exists, in addition, the hyperfine interaction given by _
H sI ¼ asI Iz hsz i
(21.94)
where hszi is the average value of the conduction electron spin in a metal. The total interaction Hamiltonian in a metallic solid is the sum of the two Hamiltonians given by Eqs. (21.93), (21.94), that is, _
_
_
H ¼ H M + H sI ¼ ½gI ħH + asI hsz i Iz
(21.95)
hszi is related to the Pauli spin susceptibility wP of the conduction electrons as Mz ¼ Dns gs mB hsz i ¼ wP H
(21.96)
where Dns is the difference of the up and down spin densities per unit volume. From the above equation hszi is given by wP H (21.97) hsz i ¼ Dns gs mB Substituting Eq. (21.97) into Eq. (21.95), we obtain _
H ¼ gI ħH ½1 K Iz
(21.98)
Magnetic Resonance Chapter
21
461
where K¼
DH asI wP ¼ H gI ħDns gs mB
(21.99)
The energy corresponding to the Hamiltonian given by Eq. (21.98) becomes E ¼ gI ħH ½1 K mI
(21.100)
Here the constant K is called the Knight shift and gives the fractional change in magnetic field due to the presence of conduction electrons in a metal. It is also a measure of the shift in energy and hence the resonance frequency. The constant asI depends on the electron density at the nuclear site, that is, j c(0) j2 where jc(r)i is the electron wave function and the nucleus is assumed to be situated at the origin. Therefore, by making a reasonable estimate of the hyperfine coupling constant asI, one can get information about the conduction electron density at the nuclear site. The Knight shift K can be measured experimentally and from it one can find wP (see Eq. 21.99).
21.12
QUADRUPOLE INTERACTIONS IN MAGNETIC RESONANCE
If a nucleus is not spherical in shape, which is the case in general, it possesses a multipole character, that is, it has a dipole moment, quadrupole moment, etc. A nucleus having spin I greater than or equal to one possesses a quadrupole moment whose ab-component can be defined as ð 3 ra rb r2 dab rN ðrÞ d3 r (21.101) Qab ¼ where rN(r) is the nuclear charge density. The diagonal components of the quadrupole moment tensor are defined as ð ð Qxx ¼ 3x2 r2 rN ðrÞd3 r,Qyy ¼ 3y2 r2 rN ðrÞd3 r (21.102) ð Q ¼ Qzz ¼
2 2 3 z r rN ðrÞ d3 r
(21.103)
The quadrupole moment of a nucleus Q is usually defined by the z-component of the quadrupole tensor. Problem 21.3
Prove that the quadrupole moment of a nucleus with spherical shape is zero.
Consider a nucleus, situated at the origin, in a solid that experiences crystal potential V(r). If the energy of the nucleus is expanded in terms of multipoles, we get X 1X pa Ea ð0Þ + Q V ð 0Þ + ⋯ (21.104) E ¼ qVð0Þ 6 a, b ab ab a This equation shows that the nuclear charge interacts with the crystal potential, the nuclear dipole moment interacts with the derivative of the crystal potential (electric field E ¼ r V), and the nuclear quadrupole moment interacts with the double derivative of the crystal potential Vab(0). The quantity Vab(0) is usually called the electric field gradient. From the equation above the quadrupole interaction energy is given by 1X Q V ð 0Þ (21.105) EQI ¼ 6 a, b ab ab _
The quantum mechanical expression for the Hamiltonian of the quadrupole interactions H QI in terms of the principal components of the electric field gradient , that is, Vaa(0), is given by (Galsin, 2002) _
H QI ¼
h i eQ Vzz 3I2z I2 + Vxx Vyy I2x I2y 4I ð2I 1Þ
(21.106)
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Solid State Physics
If, for simplicity, the electric field gradient is assumed to be cylindrically symmetric, then the field gradient eq is defined as eq ¼ Vzz , Vxx ¼ Vyy
(21.107)
_
Therefore, for a cylindrically symmetric field gradient H QI becomes _
H QI ¼
e2 qQ 2 2 3Iz I 4I ð2I 1Þ
(21.108)
The total Hamiltonian of a nucleus in the presence of both a magnetic dipole and a quadrupole moment becomes _
_
_
H ¼ H M + H QI ¼ gI mBp I H +
e2 qQ 2 2 3Iz I 4I ð2I 1Þ
(21.109)
In a number of metals and their alloys, the Zeeman splitting is large, at least 10 times larger than the quadrupole splitting. In such crystalline solids, the quadrupole interactions can be treated as a perturbation. If the magnetic field is applied along the z-direction, that is, along the direction of the principal component of the electric field gradient, then Eq. (21.109) can be written as _ 3e2 qQ I2 2 (21.110) I H ¼ gI mBp HIz + 4I ð2I 1Þ z 3 The energy eigenvalue of the nucleus corresponding to the Hamiltonian given by Eq. (21.110) is EmI ¼ E0mI + E1mI
3 e2 qQ 1 2 m I I ð I + 1Þ ¼ gI mBp H mI + 4I ð2I 1Þ 3
(21.111)
The first term in Eq. (21.111) gives the equally spaced 2 I + 1 nuclear sublevels due to the Zeeman splitting. The second term gives the first-order contribution to the energy due to the quadrupole interactions and further shifts the energy levels. For a nucleus with I ¼ 1/2, there are two Zeeman levels with mI ¼ 1/2. The quadrupole interaction term is zero for mI ¼ 1/2. Hence, for a nucleus with I ¼ 1/2, there is no splitting due to the quadrupole interactions and the transition frequency remains unchanged, that is, nL. For a nucleus with I ¼ 3/2, Eq. (21.111) gives e2 qQ 5 m2I (21.112) EmI ¼ gI mBp H mI + 4 4 The term (m2I 5/4) is 1 for mI ¼ 1/2 and +1 for mI ¼ 3/2; therefore, Eq. (21.112) can be written as EmI ¼ gI mBp H mI + ð1ÞjmI j + 1=2
e2 qQ 4
(21.113)
The magnetic sublevels for I ¼ 1/2, 3/2 in the presence of quadrupole interactions are shown in Fig. 21.11. From the figure it is evident that for I ¼ 3/2 the spectrum becomes asymmetrical about the centroid due to the presence of quadrupole interactions.
21.12.1
Nuclear Quadrupole Resonance
If the applied magnetic field is very small, then the Zeeman splitting is quite small compared with the quadrupole splitting. Therefore, the Hamiltonian corresponding to the Zeeman energy can be treated as a perturbation with respect to the Hamiltonian for the quadrupole interactions. In the limit of zero magnetic field, the total Hamiltonian reduces to the quadrupole interaction Hamiltonian HQI and it is called pure nuclear quadrupole resonance (NQR). So, for NQR the Hamiltonian, from Eq. (21.109) becomes _
_
H ¼ H QI ¼
e2 qQ 2 2 3Iz I 4 I ð I + 1Þ
(21.114)
Magnetic Resonance Chapter
21
463
FIG. 21.11 Quadrupole interaction (QI) in the presence of a magnetic interaction (MI) in a nuclear system with symmetry axis parallel to the applied magnetic field H for (A) I ¼ 1/2 and (B) I ¼ 3/2.
The energy eigenvalues for the Hamiltonian in Eq. (21.114) are given by EmI ¼ hnmI ¼
e2 qQ 2 3mI I ðI + 1Þ 4I ð2I 1 Þ
(21.115)
Eq. (21.115) exhibits twofold degeneracy of energy levels due to the quadrupole interactions: energy levels with Iz ¼ mI have the same energy. The quadrupole splitting of the energy levels for nuclei with I ¼ 1/2, 3/2, and 5/2 is shown in Fig. 21.12. It is evident from the figure that there is no quadrupole splitting in a nucleus with I ¼ 1/2. It becomes finite if I is equal to or greater than one. Further, if the magnetic field is finite but very weak, the doubly degenerate levels split up but lie very close to each other. Transitions between the split nuclear levels occur according to the selection rule DmI ¼ 1. These transitions can be induced by applying an rf magnetic field of the correct frequency. There are a number of other resonance processes also, such as ferromagnetic resonance, spin wave resonance, and antiferromagnetic resonance.
FIG. 21.12 Energy level splitting in the pure quadrupole interaction (QI) for (A) I ¼ 5/2, (B) I ¼ 3/2, and (C) I ¼ 1/2.
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Solid State Physics
21.13
FERROMAGNETIC RESONANCE
In a ferromagnetic solid the magnetization and susceptibility are large. In these solids the shape and structure of the specimen is important in determining the magnetization or the internal magnetic field and hence the resonance frequency. For example, in a solid (having cubic symmetry) in the form of a sphere, the resonance frequency o0 is given by o 0 ¼ gI H
(21.116)
If the solid is in the form of a plate with an applied magnetic field H perpendicular to the surface of the plate, then one gets o0 ¼ gI ðH 4pMÞ
21.14
(21.117)
SPIN WAVE RESONANCE
Consider a thin film of a ferromagnetic solid as in the previous section. The condition for a spin wave in the thin film can be achieved if the spins on the surface of the film experience different anisotropic fields than the spins within the film. Suppose a uniform magnetic field is applied perpendicular to the surface of the film, then the spins on the surface are pinned by surface anisotropic interactions. The field will excite the waves with an odd number of half wavelengths within the thickness of the film. This is called the spin wave resonance (SWR). The condition for the SWR is obtained by adding the exchange contribution, represented as D K2, to Eq. (21.117) to write o0 ¼ gI ðH 4pMÞ + DK2
(21.118)
Here K ¼ n p/L for the mode of a wave with n half wavelengths.
21.15
ANTIFERROMAGNETIC RESONANCE
In an antiferromagnetic solid, one can use the two-sublattice model to study the antiferromagnetic resonance. If the magnetic field is applied in the z-direction (HA is the magnetic field for sublattice A and HA for sublattice B), then the resonance condition becomes o20 ¼ g2I HA ðHA + 2HE Þ
(21.119)
where HE is the exchange magnetic field given by HE ¼ lAA MA
(21.120)
MA is the magnetization in the sublattice A.
REFERENCE Galsin, J. S. (2002). Impurity scattering in metallic alloys. New York: Kluwer Academic/Plenum Publishers.
SUGGESTED READING Bovey, F. A. (1969). Nuclear magnetic resonance spectroscopy. New York: Academic Press. Harris, R. K. (1986). Nuclear magnetic resonance spectroscopy. London: Longmans. Knight, W. D. (1956). Electron paramagnetism and nuclear magnetic resonance in metals. F. Seitz, & D. Turnbull (Eds.), Solid state physics (pp. 93–136). Vol. 2(pp. 93–136). New York: Academic Press. Low, W. (1960). Paramagnetic resonance in solids. In Solid state physics. New York: Academic Press. (Suppl. 2). Pake, G. E. (1956). Nuclear magnetic resonance. F. Seitz, & D. Turnbull (Eds.), Solid state physics (pp. 1–91). Vol. 2(pp. 1–91). New York: Academic Press. Poole, C. P., & Farach, H. A. (1972). The theory of magnetic resonance. New York: J. Wiley & Sons. Slichter, C. P. (1980). Principles of magnetic resonance. Berlin: Springer-Verlag. Winter, J. (1971). Magnetic resonance in metals. London: Oxford University Press.
Magnetic Resonance Chapter Outline 21.1 Nuclear Magnetic Moment 21.2 Zeeman Effect 21.3 Relaxation Phenomena 21.3.1 Spin-Lattice Relaxation 21.3.2 Spin-Spin Relaxation 21.4 Equation of Motion 21.5 Magnetic Resonance in the Absence of Relaxation Phenomena 21.6 Bloch Equations 21.6.1 Free Precession in Static Magnetic Field 21.7 Magnetic Broadening of Resonance Lines
445 446 448 448 450 450 452 454 455 457
21.8 21.9 21.10 21.11 21.12
Effect of Molecular Motion on Resonance Electron Spin Resonance Hyperfine Interactions Knight Shift Quadrupole Interactions in Magnetic Resonance 21.12.1 Nuclear Quadrupole Resonance 21.13 Ferromagnetic Resonance 21.14 Spin Wave Resonance 21.15 Antiferromagnetic Resonance Reference Suggested Reading
457 458 459 460 461 462 464 464 464 464 464
In solids, both the electrons and nuclei possess intrinsic spin angular momenta, which give rise to spin magnetic dipole moments. We have seen in the previous chapters that a static magnetic field produces magnetization, which yields important information about solids. A great deal of information about magnetic solids can be obtained from the nuclear magnetic resonance (NMR) technique in which an alternating magnetic field is also applied. The basic principle of an NMR method is to apply both dc and rf magnetic fields perpendicular to one another. As a result, the nuclear energy states split up into substates. When the frequency of the applied rf field is equal to the natural frequency in the presence of external magnetic field (which corresponds to the frequency of gyroscopic precession of the magnetic moments), then resonance takes place. There are different resonance techniques with which one can study a number of properties, such as the fine structure of nuclear levels, the electronic structure of an impurity, the interaction between magnetic dipoles, the interaction between magnetic dipoles and the lattice, and the line widths of nuclear transitions. The basic ideas behind nuclear resonance techniques will be described in this chapter.
21.1 NUCLEAR MAGNETIC MOMENT In analogy with the magnetic moment of an electron, the magnetic moment of a nucleus with nuclear spin I is defined as ! mI
¼ gI ħI
(21.1)
Here gI is the ratio of the magnetic moment to the angular momentum and is the gyromagnetic ratio. In the case of a nucleus, gI satisfies the relation gI ħ ¼ gI mBp
(21.2)
where mBp is the Bohr magnetron of a proton defined by Eq. (2.8) and gI is the nuclear spectroscopic splitting factor. By substituting Eq. (21.2) into Eq. (21.1), one gets ! mI
¼ gI mB p I
(21.3)
! Eq. (21.3) shows that, for a nucleus, m I
and I are in the same direction, in contrast to an electron. Although the gyromagnetic ratio is a more useful quantity for discussing magnetic resonance in many respects, nuclear magnetic moments are usually tabulated in terms of mBp, a natural unit with a value of 5.05 1024 erg/gauss (see Chapter 2). One can also define the
Solid State Physics. https://doi.org/10.1016/B978-0-12-817103-5.00021-9 © 2019 Elsevier Inc. All rights reserved.
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!
!
magnetic moment of a nucleus m I with nuclear spin I as the vector sum of the magnetic moments of the nucleons m n that are present in it, that is, X! ! mn (21.4) mI ¼ n
21.2
ZEEMAN EFFECT !
The Hamiltonian of a nucleus with magnetic moment m I in an external dc magnetic field H is _
!
H M ¼ m I H ¼ gI ħI H
(21.5)
If the magnetic field is applied in the z-direction, that is, H ¼ ^i3 H, the Hamiltonian becomes _
H M ¼ gI ħHIz ¼ gI mBp HIz
(21.6)
The energy of a nucleus in the state j mIi is given by _
EmI ¼ hmI jH M jmI i ¼ gI ħHmI ¼ gI mBp H mI
(21.7)
where mI is the eigenvalue of Iz defined as hmI j Iz jmI i ¼ mI
(21.8)
Here mI has 2 I + 1 values given as I, I 1, … 1, 0, 1, … (I 1), I. Hence, each nuclear state splits up into 2 I + 1 substates in the presence of a magnetic field and they are equally spaced. Transitions between the substates are allowed only if the magnetic quantum number mI changes by unity, that is, DmI ¼ 1
(21.9)
ħo ¼ Em0 EmI ¼ gI ħH DmI ¼ gI mBp H DmI
(21.10)
mI m0I ¼ DmI
(21.11)
Using Eq. (21.7), we can write I
where
Using the selection rule (21.9) in Eq. (21.10), the transition frequency is given by oL ¼ gI H ¼ gI mBp H=ħ
(21.12)
Eq. (21.12) yields a single resonance frequency, usually called the Larmor frequency oL, which gives the fundamental condition for magnetic resonance absorption. The radiation produced is circularly polarized in a plane perpendicular to the dc magnetic field H. Therefore, in order to excite such transitions, it is necessary to supply radiation with a magnetic vector circularly polarized in a plane perpendicular to the dc magnetic field. One can interpret the above result in the fol! lowing manner: The magnetic dipole moment m I , when placed in a dc magnetic field H, starts precessing about the applied field with the Larmor frequency (see Fig. 21.1). As an example, consider a solid with I ¼ 1/2. In the presence of a magnetic field, mI ¼ 1/2, which gives the energy of the two states as E ¼ (1/2) gI ħ H (see Fig. 21.2). If ħ oL denotes the energy difference between the two energy states, then oL ¼ gI H ¼ gI mBp H=ħ
(21.13)
which is the same as Eq. (21.12). The Zeeman splitting for a solid with I ¼ 3/2 is also shown in Fig. 21.2. Problem 21.1
Let the population of a Zeeman state NmI for a particular value of mI be given as NmI ∝ exp EmI =kB T
(21.14)
Here kB and T are the Boltzmann constant and temperature, respectively. Prove that the average magnetic moment of a nucleus hmIi in the direction of the magnetic field is given by y 2I + 1 2I + 1 1 coth y coth (21.15) hmI i ¼ gI ħ 2 2 2 2 where y¼
gI ħH kB T
(21.16)
Magnetic Resonance Chapter
21
!
FIG. 21.1 Precession of the magnetic moment m I about the applied magnetic field H.
FIG. 21.2 Zeeman splitting in a solid with nuclear spin (A) I ¼ 1/2 and (B) I ¼ 3/2.
Problem 21.2
Using hmIi given by Eq. (21.15), prove that the nuclear magnetic susceptibility in the limit of very high values of T is given by wM ¼ nnuc g2I ħ2 where nnuc is the number of nuclei per unit volume.
I ðI + 1Þ 3kB T
(21.17)
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21.3
RELAXATION PHENOMENA
In a paramagnetic substance each atom possesses a finite spin, which gives rise to a finite atomic spin magnetic moment. In the absence of an applied magnetic field the magnetic dipole moments are randomly oriented, giving rise to zero magnetization. With the application of a magnetic field, magnetization appears, which takes some finite time to reach its equilibrium value. When the magnetic field is switched off, the magnetization decreases and goes to zero again in a finite time. Both the increase and decrease in the magnetization are caused by relaxation phenomena and the time taken to reach the equilibrium value is called the relaxation time. In order to have insight into relaxation phenomena, let us consider a system of free magnetic dipoles oriented randomly. If an external magnetic field is applied, the atomic magnetic dipoles start to precess about it, as shown in Fig. 21.1, but the component of magnetization along the direction of H remains unchanged. Therefore, no magnetization will appear. For a finite magnetization to appear, the interaction of an atomic magnetic dipole with other atomic dipoles or with its surroundings is required. Consider a paramagnetic material in which the atoms (magnetic dipoles) form a lattice. The magnetic ! ! field at the position ri of the ith magnetic dipole m i due to the jth dipole m j at rj is given by ! ! m j ri j ri j m j r2ij (21.18) Hi j ¼ r5ij !
The total magnetic field at the position of m i due to all of the other magnetic dipoles is obtained by summing over j, that is, ! ! X X m j ri j ri j m j r2ij Hi j ¼ (21.19) Hi ¼ r5ij j j Hi is called the internal magnetic field experienced by an atomic magnetic dipole and is on the order of mI/a3 where a is on the order of a few angstroms. For example, in iron alum j Hi j 550 gauss. In analogy with the local electric field, Hi is called the local magnetic field and is represented by Hloc(ri). In a magnetic solid there can be two types of relaxation phenomena depending on the value of the applied magnetic field.
21.3.1
Spin-Lattice Relaxation
Spin-lattice relaxation occurs when the applied dc magnetic field H is much stronger than the local magnetic field Hloc. Therefore, a small increase in H will change the magnitude of the total magnetic field H + Hloc, but will not change its direction by very much. In this case the spin interacts with the lattice and exchanges energy with it. The lattice interaction may cause some of the magnetic dipoles to slip from the antiparallel direction to the parallel one, thereby producing a finite magnetization M defined as M ¼ ^i1 M1 + ^i2 M2 + ^i3 M3
(21.20)
The time taken by the material to reach the equilibrium value M0 is called the spin-lattice relaxation time tsl and the phenomenon is called the spin-lattice relaxation. The time tsl is also sometimes called the longitudinal relaxation time. When the magnetic field is switched off the material returns to its initial unmagnetized state. Let the magnetic field H be applied in the z-direction, that is, H ¼ ^i3 H with M0 as the equilibrium magnetization. Further, let Mz(t) be the magnetization at any time t in the z-direction, which is different from M0. If the rate of change of Mz(t) is assumed to be proportional to its deviation from M0, then one can write dMz M0 Mz ¼ dt tsl
(21.21)
where 1/tsl is the constant of proportionality. The above equation can be written as M ðz
ðt dt
(21.22)
h i Mz ðtÞ ¼ M0 1 et=tsl
(21.23)
0
Integrating the above equation, one gets
dMz 1 ¼ M0 Mz tsl
0
Magnetic Resonance Chapter
21
449
FIG. 21.3 Spin-lattice and spin-spin relaxation processes. In the spin-lattice relaxation process, the magnetization M approaches the equilibrium value M0 along the z-direction. In the spin-spin relaxation process, the magnetization M0 (in the xy plane) approaches zero, the equilibrium value, in the presence of magnetic field H0.
Fig. 21.3 shows how a precessing magnetization M (about the z-direction) acquires the equilibrium value of magnetization M0 in the spin-lattice relaxation phenomenon. The variation of Mz(t) with time is shown in Fig. 21.4. The spin-lattice relaxation is measured in strong fields but at low frequencies. It is important to point out that tsl is strongly temperature dependent and decreases with increasing temperature. This variation is due to an increase in the mobility of the magnetic dipoles with an increase in temperature. The time tsl, due to the nuclear moments in solids, may have a value from a few seconds to hours. In solids in which the paramagnetism is due to the electrons, tsl varies between 1011 and 106 s at room temperature. FIG. 21.4 The variation of the z-component of magnetization Mz(t) with time t in the spin-lattice relaxation phenomenon.
450
Solid State Physics
21.3.2
Spin-Spin Relaxation
The nuclear spins also interact among themselves via dipole-dipole interactions. In a paramagnetic solid, each spin experiences a local magnetic field Hloc due to the neighboring magnetic dipoles. The direction and magnitude of Hloc may differ from nucleus to nucleus depending on the relative positions of the neighboring nuclei. If the external magnetic field H is much smaller than the local magnetic field Hloc, that is, H ≪ Hloc, then its effect will be to slightly change the direction of the field seen by a dipole, but its magnitude will remain essentially unaltered. The magnetic dipoles will thus precess about a slightly different direction and, as a result, finite magnetization occurs in the direction of H. In this situation the magnetic force due to the external magnetic field is insufficient to cause the magnetic dipoles to flip over and, therefore, there will be no exchange of energy between the spins and the lattice. In other words, spin-lattice relaxation will be absent. On the other hand, there is a finite energy exchange between the spins, giving rise to what is called the spin-spin relaxation process. It is convenient to introduce the spin-spin relaxation time tss to describe the lifetime of a nuclear spin state. The different values of Hloc at different nuclei cause local irregularities in the magnetic field. Further, the different values of Hloc make different nuclei precess with different frequencies. So, the precessing nuclei will go out of phase with each other in a time on the order of tss, yielding zero as the equilibrium values of Mx and My. Let an rf magnetic field H0 be applied in the xy-plane, which is defined as H0 ¼ ^i1 Hx + ^i2 Hy
(21.24)
Here Hx and Hy are the x- and y-components of the magnetic field. The magnetization produced in the xy-plane M0 is written as M0 ¼ ^i1 Mx + ^i2 My
(21.25) 0
where Mx and My are the nonequilibrium values of the magnetizations in the x- and y-directions. H0 and M are called the transverse magnetic field and transverse magnetization, respectively. The rate of change of Mx and My are given as dMy My dMx M ¼ x, ¼ dt tss dt tss
(21.26)
The solution of these differential equations is simple. Solving, we find Mx ¼ M0 x et=tss , My ¼ M0 y et=tss
(21.27)
Therefore, Mx and My approach zero exponentially with time. Fig. 21.3 shows the variation of M0 with time, which ultimately goes to zero (equilibrium value) in the presence of the magnetic field H0. The time tss is measured with small magnetic fields that have very high frequencies, on the order of many Mc/s. At such high frequencies the spin-lattice relaxation process is absent. The relaxation time tss is on the order of 1010 s, which is smaller than the spin-lattice relaxation time tsl. Further, the spin-spin relaxation process is independent of temperature, which makes tss independent of temperature.
21.4
EQUATION OF MOTION !
Consider a nucleus with spin I and magnetic moment m I placed in a constant magnetic field H. The rate of change of the ! nuclear angular momentum I ħ gives the torque τ acting on the nucleus, that is, d ! ðIħÞ ¼ τ dt
(21.28)
with !
!
τ ¼ mI H
(21.29)
Using Eqs. (21.1), (21.29) in Eq. (21.28), one can write !
1 dm I ! ¼ mI H gI dt
(21.30)
Magnetic Resonance Chapter
21
451
!
We have already seen that m I will precess about the magnetic field H with the Larmor frequency oL. Consider now a magnetic material with weakly interacting nuclear spins. The magnetization M is a vector sum of the nuclear magnetic moments per unit volume. If ri is the number of nuclei per unit volume in the ith state, then from Eq. (21.30) one can write dM ¼ gI M H dt where M¼
X
!
ri m Ii
(21.31)
(21.32)
i !
Here m I i is the nuclear magnetic moment in the ith state of the nucleus having spin I. The solution of Eq. (21.31) for a constant magnetic field is the same as that for Eq. (21.30), that is, M will precess about H with the Larmor frequency oL. If M is along the z-direction, then in the equilibrium state, Mz ¼ M0 and Mx ¼ My ¼ 0 and the magnetic susceptibility w0 satisfies the Curie law, that is, M 0 ¼ w0 H
(21.33)
where w0 ¼
CM T
(21.34)
Here CM is the Curie constant. Hence, in a constant magnetic field H, one observes a resonant Larmor frequency nL ¼ oL/2 p that is circularly polarized. For H 104 gauss one obtains nL ranging from 1–50 Mc/s. To observe the resonance absorption of rf energy, one has to apply two magnetic fields: 1. A dc magnetic field H, which can be varied along the z-direction. 2. An rf magnetic field of amplitude H0 (≪H) perpendicular to the dc magnetic field. The reason for the application of the rf field can be understood from the following. Consider a magnetic system with nuclear ! spin I ¼ 1/2, which gives two energy levels (see Fig. 21.2A). The orientation of the magnetic moment m I in the lower energy state is shown in Fig. 21.5 and it precesses about H with the Larmor frequency oL. For resonance absorption to take place, a
!
FIG. 21.5 Precession of the magnetic moment m I about the dc magnetic field H in the presence of rf magnetic field H0.
452
Solid State Physics
FIG. 21.6 Schematic representation of the experimental setup used for studying the magnetic resonance in a solid. The dc magnetic field is applied in the z-direction, but the rf magnetic field is applied in the x-direction.
transition must occur from the lower state to the higher state (see Fig. 21.2A), which involves a change in spin from ½ to ½. The transition will take place if H0, applied perpendicular to H, causes the absorption of rf energy at oL ¼ gI H. Further, H0 reverses the tip of the dipole from parallel (mI ¼ 1/2) to antiparallel (mI ¼ 1/2) alignment and vice versa. Let the rf magnetic field be applied in the x-direction. Then, one can write Hx ¼ 2H0 cos o t Hy ¼ Hz ¼ 0
(21.35)
But we know that in the Zeeman effect, the resonance frequency emitted or absorbed is circularly polarized. Therefore, it is convenient to represent the rf field as a combination of two circularly polarized magnetic fields in the xy-plane: one rf field rotates clockwise and the other counterclockwise. These two rf fields can be written as Clockwise : Hx ¼ H0 cos ot, Hy ¼ H0 sin ot
(21.36)
Counter-clockwise : Hx ¼ H0 cos ot, Hy ¼ H0 sin ot
(21.37)
At oL, one of the rotating fields will follow the precessing magnetic dipole and exert a constant torque on it. As a result, the magnetic dipole tips from parallel (antiparallel) to antiparallel (parallel) alignment. Therefore, the other rotating field becomes redundant. Hence, in order to observe resonance absorption of rf energy by the spin system, it is essential that the lower level be more heavily populated than the upper level, which is indeed the case at thermal equilibrium due to the Boltzmann distribution. A schematic diagram of the nuclear resonance experimental setup is shown in Fig. 21.6.
21.5
MAGNETIC RESONANCE IN THE ABSENCE OF RELAXATION PHENOMENA
In the absence of relaxation phenomena, the equation of motion is given by Eq. (21.31), in which the three components of magnetization become h i dMx ¼ gI My Hz Mz Hy (21.38) dt dMy ¼ gI ½Mz Hx Mx Hz (21.39) dt h i dMz ¼ gI Mx Hy My Hx (21.40) dt Let the magnetic field in the xy-plane H0 (with components Hx and Hy) be very small in comparison with the dc magnetic field H ¼ ^i3 H. As a result, the components Mx and My are very small in comparison with Mz. To a first-order approximation, Eqs. (21.38)–(21.40) reduce to h i dMx ¼ gI My H Mz Hy (21.41) dt
Magnetic Resonance Chapter
21
453
dMy ¼ gI ½Mz Hx Mx H dt
(21.42)
dMz ¼0 dt
(21.43)
The oscillating components of the magnetic field Hx and Hy give rise to oscillating components of magnetization Mx and My, which can be represented as Hx ¼ H0x ei o t ,Mx ¼ M0x ei o t
(21.44)
Hy ¼ H0y e
(21.45)
iot
,My ¼ M0y e
iot
Substituting Eqs. (21.44), (21.45) into Eqs. (21.41), (21.42), one gets ioMx gI HMy + gI Mz Hy ¼ 0
(21.46)
gI HMx + ioMy gI Mz Hx ¼ 0
(21.47)
Multiplying Eq. (21.46) by gI H and Eq. (21.47) by i o and subtracting, one gets My ¼
gI Mz o H + ioH L y x o2L o2
(21.48)
Similarly, one can obtain Mx from Eqs. (21.46), (21.47). We find Mx ¼
gI M z o H ioH L x y 2 oL o2
(21.49)
Eqs. (21.48), (21.49) can be written in matrix form as 0
Mx
1
0
o L gI M z B C B o2L o2 B C B C¼ B B B C @ iogI Mz @ A o2L o2 My
1 0H 1 iogI Mz x B C o2L o2 C B CB C CB C C o L gI M z A @ A o2L o2 Hy
(21.50)
In matrix form the magnetic susceptibility can be written as 0 @
Mx My
1
0
A¼@
xy wxx M wM
10 A@
yy wyx M wM
Hx
1 A
(21.51)
Hy
Comparing Eqs. (21.50), (21.51), one can write yy wxx M ¼ wM ¼ yx wxy M ¼ wM ¼
oL gI Mz o2L o2
(21.52)
iogI Mz o2L o2
(21.53)
The transverse magnetization M0 and the transverse magnetic field H0 given by Eqs. (21.25), (21.24) exhibit some interesting features. M0 and H0 are not in the same direction, rather M0 lags behind H0. This can be seen by putting Hx ¼ 0 in Eqs. (21.48), (21.49) and noticing the phase relation and amplitudes of Mx and My. The second interesting feature is that the components of the susceptibility tensor (Eqs. 21.52, 21.53) and magnetization (Eqs. 21.48, 21.49) become infinite at o ¼ oL, the natural frequency of vibration. But actually, the susceptibility and magnetization become very large at oL because, in this state, the applied rf field is synchronous with the precessional motion. This is the condition for electron paramagnetic resonance. The infinite values of susceptibility and magnetization suggest that some kind of relaxation phenomenon exists, which makes them finite but large in value.
454
21.6
Solid State Physics
BLOCH EQUATIONS
Bloch derived equations of motion for the magnetic response taking account of the spin-lattice and spin-spin relaxation phenomena. From Eqs. (21.21), (21.40) the rate of change of Mz with time is given by h i M M dMz z ¼ gI Mx Hy My Hx + 0 dt tsl
(21.54)
Similarly, from Eqs. (21.38), (21.39), (21.26) one can write h i M dMx ¼ gI My Hz Mz Hy x dt tss
(21.55)
dMy My ¼ gI ½Mz Hx Mx Hz dt tss
(21.56)
Eqs. (21.54)(21.56) are called Bloch equations. Substituting the values of Hx and Hy from Eq. (21.37) and putting Hz ¼ H in Eqs. (21.54)–(21.56), one gets h i M dMx ¼ gI My H + Mz H0 sin ot x (21.57) dt tss dMy My ¼ gI ½Mz H0 cos ot Mx H dt tss h i M M dMz z ¼ gI Mx H0 sin ot + My H0 cos ot + 0 dt tsl Eqs. (21.57)–(21.59) can be solved for Mx, My, and Mz and the final result is given as " # 1 2H0 tss ðoL o Þ cos ot + 2H0 sin ot Mx ¼ w0 oL tss 2 1 + ðoL oÞ2 t2ss + g2I H20 tsl tss " # 1 2H0 cos ot 2H0 tss ðoL o Þ sin ot My ¼ w0 oL tss 2 1 + ðoL oÞ2 t2ss + g2I H20 tsl tss " M z ¼ w0 H
1 + ðoL oÞ2 t2ss
(21.58) (21.59)
(21.60)
(21.61)
# (21.62)
1 + ðoL oÞ2 t2ss + g2I H20 tsl tss
In actual experiments the linearly polarized rf magnetic field is applied only in one direction, say in the x-direction (see Fig. 21.6). In this case, Hx ¼ 2 H0 cos o t and Hy ¼ 0. For a linearly polarized rf magnetic field, Eqs. (21.60)– (21.62) reduce to " # 1 2H0 tss ðoL o Þ cos ot (21.63) Mx ¼ w0 oL tss 2 1 + ðoL oÞ2 t2ss + g2I H20 tsl tss " # 1 2H0 cos ot My ¼ w0 oL tss 2 1 + ðoL oÞ2 t2ss + g2I H20 tsl tss " M z ¼ w0 H
1 + ðoL oÞ2 t2ss 1 + ðoL oÞ2 t2ss + g2I H20 tsl tss
(21.64)
# (21.65)
The dynamical magnetic susceptibility can be written as wM ðoÞ ¼ w0M ðoÞ + i w00M ðoÞ
(21.66)
Magnetic Resonance Chapter
w0M(o) and w00M(o) are the real and imaginary parts, which from Eqs. (21.63), (21.64) are given by " # Mx Mx 1 ðoL o Þ tss 0 ¼ w o t wM ðoÞ ¼ ¼ Hx 2H0 cos ot 2 0 L ss 1 + ðoL oÞ2 t2ss + g2I H20 tsl tss " # My My 1 1 00 ¼ w o t ¼ wM ð o Þ ¼ Hx 2H0 cos ot 2 0 L ss 1 + ðoL oÞ2 t2ss + g2I H20 tsl tss
21
455
(21.67)
(21.68)
In the denominators of Eqs. (21.63)–(21.65) and Eqs. (21.67)–(21.68), the term g2I H20 tsl tss is a dimensionless constant that determines the degree of saturation. If H0 is very small, then saturation is not reached and g2I H20 tsl tss ≪1 This term can be neglected in the denominators of Eqs. (21.63)–(21.65) and Eqs. (21.67)–(21.68). In this approximation the static magnetic susceptibility from Eq. (21.65) becomes w0 ¼
Mz H
The real and imaginary parts of the dynamical magnetic susceptibility, from Eqs. (21.67), (21.68), become " # 1 ðoL o Þ tss 0 wM ðoÞ ¼ w0 oL tss 2 1 + ðoL oÞ2 t2ss " # 1 1 w00M ðoÞ ¼ w0 oL tss 2 1 + ðoL oÞ2 t2ss
(21.69)
(21.70)
(21.71)
The susceptibilities w0M(o) and w00M(o) are plotted in Fig. 21.7 as a function of (oL o)tss. Fig. 21.7A shows that w00M(o) exhibits resonance absorption at o ¼ oL and the peak has Lorentzian shape. Eq. (21.71) can be written as w00M ðoÞ ¼
ð1=2Þw0 oL =tss ðoL oÞ2 + 1=t2ss
(21.72)
From the above equation we see that the halfwidth Do of resonance at half the height is given by Do ¼
1 tss
(21.73)
The real part of the dynamical magnetic susceptibility w0M(o), shown in Fig. 21.7B, exhibits dispersion, which accompanies the absorption.
21.6.1
Free Precession in Static Magnetic Field
One can solve the Bloch equations for the free precession of the spin system in a static (dc) magnetic field H ¼ ^i3 H. In a static magnetic field the equilibrium magnetization is Mz ¼ M0. In this case the Bloch equations (21.54) – (21.56) reduce to dMx M ¼ gI HMy x dt tss
(21.74)
dMy My ¼ gI HMx dt tss
(21.75)
dMz ¼0 dt
(21.76)
For Eqs. (21.74)–(21.76), we seek damped-oscillator-like solutions of the form 0
Mx ¼ M et=t cos ot 0
My ¼ M et=t sin ot
(21.77) (21.78)
456
Solid State Physics
FIG. 21.7 Schematic representation of the real part w0M(o) and imaginary part w00M(o) of the dynamical magnetic susceptibility wM(o) as a function of the frequency o in a solid.
where t0 is the damping time. Substituting Eqs. (21.77), (21.78) into Eqs. (21.74)–(21.76), one can write 1 1 cos ot ¼ ðoL oÞ sin ot t0 tss 1 1 sin ot ¼ ðoL oÞ cos ot t0 tss
(21.79)
(21.80)
Eqs. (21.79), (21.80) will be valid only if t0 ¼ tss
(21.81)
o ¼ oL
(21.82)
and
Magnetic Resonance Chapter
21
457
Eq. (21.81) shows that the damping time is equal to the spin-spin relaxation time. Further, using the analogy of the precession of spins in a static magnetic field with the motion of a damped harmonic oscillator suggests that the spin system exhibits resonance absorption of energy near the Larmor frequency. The frequency width of the response of the system to the driving field will be Do ¼ 1/tss.
21.7 MAGNETIC BROADENING OF RESONANCE LINES It has been observed that resonance absorption of an rf magnetic field is not very sharp, but rather exhibits a peak with finite width called the linewidth. There are a number of physical phenomena that contribute to the linewidth of resonance absorption. 1. In a solid, an inhomogeneous magnetic field gives rise to a broadening of the resonance line. Inhomogeneity in a magnetic field may arise either because of the presence of impurities or due to different values of Hloc at different nuclei (crystal structure effect). 2. If a nucleus possesses a finite magnetic quadrupole moment, it gives rise to what are known as quadrupole effects. These effects destroy the degeneracy of resonance frequencies between nuclear levels with different mI values (Zeeman effect). Quadrupole effects may give rise to resolved or unresolved splitting of a resonance line. If the quadrupole effects are very small (which may be the case in a number of solids), one gets unresolved splitting of the resonance line, which amounts to broadening. 3. Spin-lattice interactions also give rise to a broadening of the resonance lines. This is because the spin-lattice interactions produce an equilibrium population by balancing the rates of transitions. This puts a limit on the lifetime of the Zeeman states and hence results in a broadening of the resonance lines. The coupling between the nuclear magnetic moments in a solid makes a major contribution to the broadening of resonance lines and is considered here. In addition to the applied magnetic field H, a local magnetic field Hloc(ri) is produced at the position ri of a nuclear magnetic moment (see Eq. 21.19). In a crystalline solid Hloc(ri) is different at different lattice positions, yielding an inhomogeneous total magnetic field Htot given by Htot ¼ H + Hloc
(21.83) ! Hloc m I =r31 ,
where r1 is the 1NN disConsidering the interactions between the 1NNs, it is evident from Eq. (21.19) that ˚ , one obtains Hloc ffi 1 gauss. Therefore, if Htot is tance. For a nuclear magnetic moment mI ¼ 1023 erg/gauss and r1 ¼ 2 A different at different nuclei, then the various nuclei will exhibit resonance at different frequencies. If DHloc is the spread in the value of Hloc seen by the nuclei, then it will also be the spread in Htot. Thus, the spread in the resonance frequency Do, that is, oL ! oL + Do, is given by Do ¼ gI DHloc
(21.84)
Knowing the values of Do and DHloc, one can also find the spin-spin relaxation time tss. For example, in CaF2, DHloc ¼ 2 gauss and Do ¼ 5 104 s1, therefore, tss ¼
1 1 ¼ 2 105 s ¼ Do 5 104
(21.85)
21.8 EFFECT OF MOLECULAR MOTION ON RESONANCE The broadening of resonance lines depends on the nature of the material. The linewidth in the case of liquids is very small compared with solids: the linewidth in a liquid can be as narrow as 0.5 cps, but in solids it is on the order of 5000 cps or more at room temperature. It has further been observed experimentally that the resonance peaks are narrower for a solid in which the atoms (nuclei) are in rapid motion. Further, with an increase in the temperature T of a solid, the motion of the atoms becomes faster and the resonance peaks become narrower. The effect of the motion of the atoms on the width of resonance peaks is usually called motional narrowing. This effect is much more pronounced in liquids as the atoms can move much more quickly in random directions. The motional narrowing of resonance peaks can be explained in terms of the local magnetic field Hloc as follows. It has already been explained that the width of a resonance peak arises due to the inhomogeneous nature of Hloc, the crystal structure effect. All of the measurements in a laboratory are made at finite temperatures. At finite temperatures, an
458
Solid State Physics
atom jumps from one lattice position to another, a process similar to the diffusion of atoms. Let us suppose that an atom remains at a single lattice position for an average time of τm seconds. The atoms in a solid move randomly in different directions, as a result of which Hloc at a particular lattice point changes very quickly. The time-averaged value of the local magnetic field hHloci becomes nearly constant and with much smaller spread. In other words, one can say that the random motion of atoms in different directions makes the solid, and hence hHloci, nearly homogeneous, yielding only a very small spread in Hloc. The decrease in the spread of Hloc yields a smaller width of the resonance absorption line. With an increase in temperature, the motion of the atoms increases, yielding a smaller value of DHloc, which in turn makes the resonance lines narrower. In other words, one can say that an increase in the motion (velocity) of atoms decreases the value of τm, which causes a narrowing of the resonance peaks. Measurements of the spin-lattice relaxation time tsl are based on the competition between resonance absorption (which tends to equalize the population of different levels) and the spin-lattice interaction (which tends to maintain the Boltzmann distribution). The values of tsl obtained experimentally vary between 105 104 s: 104 s is the value obtained for ice. The value of tsl is larger for solids than for liquids or gases. For solids, tsl is rarely smaller than 102 s. Rather, it may be very large at low temperatures. On the other hand, for liquids, tsl may be as short as 102 103 s and rarely exceeds a few seconds. This shows that as the atoms in a solid become freer the relaxation time tsl decreases. It is interesting to note that tsl may reduce strongly when paramagnetic ions are present; these ions have an effective magnetic moment that is 103 times as large as the nuclear magnetic moments and they form a very efficient medium for establishing heat contact between the nuclear spins and their surroundings. On the other hand, at low temperatures, there is no variation in the spin-spin relaxation time tss. With an increase in temperature, tss increases and, at a certain temperature, tsl and tss may become approximately equal.
21.9
ELECTRON SPIN RESONANCE
In an atom, both the nucleus and the electrons possess intrinsic spin and hence magnetic moment. Let the magnetic moment of a nucleus with spin I be mI and that of an electron mB. When a magnetic field H is applied to a solid, it splits the energy levels of both the nuclei and the electrons. The splitting of the energy levels of the nuclei gives rise to NMR and that of the electrons gives rise to electron spin resonance (ESR). One should note that there are some solids that exhibit ESR. Consider a solid in which each atom contains only one electron. Let a magnetic field H be applied to the solid in the z-direction, that is, H ¼ ^i3 H. Further, an rf magnetic field H0 is applied perpendicular to H (say in the x-direction). The magnetic part of the Hamiltonian due to the atomic electron is given by _
!
H ¼ m s H ¼ gs mB s H ¼ gs mB Hsz
(21.86)
Here we have used Eq. (18.24) for the spin magnetic moment of an electron. The energy of the electron states is obtained by taking the expectation value of the Hamiltonian as _
Ems ¼ hms jH jms i ¼ gs mB Hms
(21.87)
According to the above equation the electron states get split up into two substates (see Fig. 21.8). The energy difference between the two states becomes DE ¼ 2mB H
(21.88)
Eq. (21.88) gives the condition for a resonance. If an rf field is applied in the xy-plane, then the resonance frequency o0 is given by ħo0 ¼ 2mB H
(21.89)
FIG. 21.8 Splitting of an electron energy state with sz ¼ 1/2 into two substates in the presence of a dc magnetic field H applied in the z-direction.
Magnetic Resonance Chapter
21
459
FIG. 21.9 The energy absorption at the resonance frequency n0, which broadens into a resonance peak.
So, resonance absorption of the rf field takes place at frequency o0. At resonance, the electrons in energy state E1/2 absorb energy and make a transition to the higher energy state E1/2. Note that the frequency o0 is twice the Larmor frequency oL. If we apply H ¼ 1 weber/m2, then o0 30000 Mc/s. In the discussion above, only a single electron per atom is assumed, but in general there are a number of electrons in an atom of a solid. Therefore, the resonance frequency is affected by the following interactions: 1. Electron-electron interactions modify the resonance process. 2. The interaction of nuclear magnetic moments with the electronic magnetic moments affects the electron energy levels. One can say that the nuclear spin can take 2 I + 1 orientations in the magnetic field and each of these alters the electron energy levels by different amounts (hyperfine interaction). These interactions, therefore, broaden a resonance line into a resonance peak, as shown in Fig. 21.9. If the electron-electron and electron-nucleus interactions are small compared with the splitting of the energy levels of a single electron, then they can be treated as a perturbation to find the resultant electron states. Thus, one can find the width of the resonance peak. Other effects, such as the inhomogeneous local electric fields arising from impurities and other crystal defects, also change the position of the resonance lines. ESR is observed only in those solids in which each atom has unpaired electrons so as to yield finite electron spin (e.g., paramagnetic substances) because the paired electrons in the valence band produce no effect. Therefore, ESR is often called electron paramagnetic resonance (EPR).
21.10
HYPERFINE INTERACTIONS
So far, we have studied NMR in which the nuclear energy levels are calculated in the presence of mutually perpendicular dc and rf magnetic fields. But in a solid there are electrons revolving around every nucleus that possess finite magnetic moment. Therefore, additional interactions exist between the magnetic moments of electrons and nuclei, usually called hyperfine interactions, which cause further splitting of the nuclear energy levels. The motion of electrons around the nucleus produces a finite magnetic field with the following contributions: 1. The electrons revolving around the nucleus are equivalent to current loops, which produce a finite magnetic field at the nuclear site. 2. The spinning motion of electrons yields spin current about the nucleus. The spin current produces what is called a contact hyperfine interaction. Even if quenching of the orbital angular momentum, and hence of the orbital magnetic moment, occurs, there will be a finite contact hyperfine interaction. Therefore, the contact hyperfine interaction is of special importance in such solids. The net magnetic field arising from these two contributions interacts with the nucleus to produce the hyperfine interaction. If the hyperfine interaction is small compared with the splitting of the nuclear_energy levels, it may be treated as a perturbation in the nuclear resonance phenomenon. The interaction Hamiltonian H sI between the nuclear magnetic moment and the magnetic moment of the electron can be represented as _
H sI ¼ asI s I
(21.90)
460
Solid State Physics
FIG. 21.10 Splitting of an electron state, in a nucleus with spin I ¼ ½, into four states in the presence of hyperfine interactions. Two transitions with frequencies o1 and o2 are allowed by the selection rules.
where I and s are the nuclear and electron spins and asI is the hyperfine constant. The energy eigenvalues of Eq. (21.90) are given by (21.91) EsI ¼ asI mI ms The electron spin quantum number ms has two values, 1/2. For a nucleus with I ¼ 1/2 the quantum number mI also has two values 1/2. According to Eq. (21.91), each electron state gets split up into two states with mI ¼ 1/2, giving rise to four states as shown in Fig. 21.10. The transitions between the four states take place by satisfying the following selection rules DmI ¼ 0, Dms ¼ 1
(21.92)
The allowed transitions yield two resonance lines, as shown in Fig. 21.10.
21.11
KNIGHT SHIFT
Consider a diamagnetic material with nuclear spin I to which the magnetic field H is applied in the z-direction, that is, H ¼ ^i3 H. The magnetic interaction Hamiltonian is given by _
H M ¼ gI ħHIz
(21.93)
When an rf magnetic field H0 is applied in the direction perpendicular to H, nuclear resonance absorption is observed at the Larmor frequency oL ¼ gI H. Now consider a metallic solid with the same nuclear spin I to which the same magnetic field H0 is applied in the xy-plane. It is found that nuclear resonance absorption in the metallic solid is observed at a slightly different value of the dc magnetic field. This effect is known as the Knight shift and it yields knowledge about the distribution of conduction electrons in a metal. In a metallic solid there exists, in addition, the hyperfine interaction given by _
H sI ¼ asI Iz hsz i
(21.94)
where hszi is the average value of the conduction electron spin in a metal. The total interaction Hamiltonian in a metallic solid is the sum of the two Hamiltonians given by Eqs. (21.93), (21.94), that is, _
_
_
H ¼ H M + H sI ¼ ½gI ħH + asI hsz i Iz
(21.95)
hszi is related to the Pauli spin susceptibility wP of the conduction electrons as Mz ¼ Dns gs mB hsz i ¼ wP H
(21.96)
where Dns is the difference of the up and down spin densities per unit volume. From the above equation hszi is given by wP H (21.97) hsz i ¼ Dns gs mB Substituting Eq. (21.97) into Eq. (21.95), we obtain _
H ¼ gI ħH ½1 K Iz
(21.98)
Magnetic Resonance Chapter
21
461
where K¼
DH asI wP ¼ H gI ħDns gs mB
(21.99)
The energy corresponding to the Hamiltonian given by Eq. (21.98) becomes E ¼ gI ħH ½1 K mI
(21.100)
Here the constant K is called the Knight shift and gives the fractional change in magnetic field due to the presence of conduction electrons in a metal. It is also a measure of the shift in energy and hence the resonance frequency. The constant asI depends on the electron density at the nuclear site, that is, j c(0) j2 where jc(r)i is the electron wave function and the nucleus is assumed to be situated at the origin. Therefore, by making a reasonable estimate of the hyperfine coupling constant asI, one can get information about the conduction electron density at the nuclear site. The Knight shift K can be measured experimentally and from it one can find wP (see Eq. 21.99).
21.12
QUADRUPOLE INTERACTIONS IN MAGNETIC RESONANCE
If a nucleus is not spherical in shape, which is the case in general, it possesses a multipole character, that is, it has a dipole moment, quadrupole moment, etc. A nucleus having spin I greater than or equal to one possesses a quadrupole moment whose ab-component can be defined as ð 3 ra rb r2 dab rN ðrÞ d3 r (21.101) Qab ¼ where rN(r) is the nuclear charge density. The diagonal components of the quadrupole moment tensor are defined as ð ð Qxx ¼ 3x2 r2 rN ðrÞd3 r,Qyy ¼ 3y2 r2 rN ðrÞd3 r (21.102) ð Q ¼ Qzz ¼
2 2 3 z r rN ðrÞ d3 r
(21.103)
The quadrupole moment of a nucleus Q is usually defined by the z-component of the quadrupole tensor. Problem 21.3
Prove that the quadrupole moment of a nucleus with spherical shape is zero.
Consider a nucleus, situated at the origin, in a solid that experiences crystal potential V(r). If the energy of the nucleus is expanded in terms of multipoles, we get X 1X pa Ea ð0Þ + Q V ð 0Þ + ⋯ (21.104) E ¼ qVð0Þ 6 a, b ab ab a This equation shows that the nuclear charge interacts with the crystal potential, the nuclear dipole moment interacts with the derivative of the crystal potential (electric field E ¼ r V), and the nuclear quadrupole moment interacts with the double derivative of the crystal potential Vab(0). The quantity Vab(0) is usually called the electric field gradient. From the equation above the quadrupole interaction energy is given by 1X Q V ð 0Þ (21.105) EQI ¼ 6 a, b ab ab _
The quantum mechanical expression for the Hamiltonian of the quadrupole interactions H QI in terms of the principal components of the electric field gradient , that is, Vaa(0), is given by (Galsin, 2002) _
H QI ¼
h i eQ Vzz 3I2z I2 + Vxx Vyy I2x I2y 4I ð2I 1Þ
(21.106)
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Solid State Physics
If, for simplicity, the electric field gradient is assumed to be cylindrically symmetric, then the field gradient eq is defined as eq ¼ Vzz , Vxx ¼ Vyy
(21.107)
_
Therefore, for a cylindrically symmetric field gradient H QI becomes _
H QI ¼
e2 qQ 2 2 3Iz I 4I ð2I 1Þ
(21.108)
The total Hamiltonian of a nucleus in the presence of both a magnetic dipole and a quadrupole moment becomes _
_
_
H ¼ H M + H QI ¼ gI mBp I H +
e2 qQ 2 2 3Iz I 4I ð2I 1Þ
(21.109)
In a number of metals and their alloys, the Zeeman splitting is large, at least 10 times larger than the quadrupole splitting. In such crystalline solids, the quadrupole interactions can be treated as a perturbation. If the magnetic field is applied along the z-direction, that is, along the direction of the principal component of the electric field gradient, then Eq. (21.109) can be written as _ 3e2 qQ I2 2 (21.110) I H ¼ gI mBp HIz + 4I ð2I 1Þ z 3 The energy eigenvalue of the nucleus corresponding to the Hamiltonian given by Eq. (21.110) is EmI ¼ E0mI + E1mI
3 e2 qQ 1 2 m I I ð I + 1Þ ¼ gI mBp H mI + 4I ð2I 1Þ 3
(21.111)
The first term in Eq. (21.111) gives the equally spaced 2 I + 1 nuclear sublevels due to the Zeeman splitting. The second term gives the first-order contribution to the energy due to the quadrupole interactions and further shifts the energy levels. For a nucleus with I ¼ 1/2, there are two Zeeman levels with mI ¼ 1/2. The quadrupole interaction term is zero for mI ¼ 1/2. Hence, for a nucleus with I ¼ 1/2, there is no splitting due to the quadrupole interactions and the transition frequency remains unchanged, that is, nL. For a nucleus with I ¼ 3/2, Eq. (21.111) gives e2 qQ 5 m2I (21.112) EmI ¼ gI mBp H mI + 4 4 The term (m2I 5/4) is 1 for mI ¼ 1/2 and +1 for mI ¼ 3/2; therefore, Eq. (21.112) can be written as EmI ¼ gI mBp H mI + ð1ÞjmI j + 1=2
e2 qQ 4
(21.113)
The magnetic sublevels for I ¼ 1/2, 3/2 in the presence of quadrupole interactions are shown in Fig. 21.11. From the figure it is evident that for I ¼ 3/2 the spectrum becomes asymmetrical about the centroid due to the presence of quadrupole interactions.
21.12.1
Nuclear Quadrupole Resonance
If the applied magnetic field is very small, then the Zeeman splitting is quite small compared with the quadrupole splitting. Therefore, the Hamiltonian corresponding to the Zeeman energy can be treated as a perturbation with respect to the Hamiltonian for the quadrupole interactions. In the limit of zero magnetic field, the total Hamiltonian reduces to the quadrupole interaction Hamiltonian HQI and it is called pure nuclear quadrupole resonance (NQR). So, for NQR the Hamiltonian, from Eq. (21.109) becomes _
_
H ¼ H QI ¼
e2 qQ 2 2 3Iz I 4 I ð I + 1Þ
(21.114)
Magnetic Resonance Chapter
21
463
FIG. 21.11 Quadrupole interaction (QI) in the presence of a magnetic interaction (MI) in a nuclear system with symmetry axis parallel to the applied magnetic field H for (A) I ¼ 1/2 and (B) I ¼ 3/2.
The energy eigenvalues for the Hamiltonian in Eq. (21.114) are given by EmI ¼ hnmI ¼
e2 qQ 2 3mI I ðI + 1Þ 4I ð2I 1 Þ
(21.115)
Eq. (21.115) exhibits twofold degeneracy of energy levels due to the quadrupole interactions: energy levels with Iz ¼ mI have the same energy. The quadrupole splitting of the energy levels for nuclei with I ¼ 1/2, 3/2, and 5/2 is shown in Fig. 21.12. It is evident from the figure that there is no quadrupole splitting in a nucleus with I ¼ 1/2. It becomes finite if I is equal to or greater than one. Further, if the magnetic field is finite but very weak, the doubly degenerate levels split up but lie very close to each other. Transitions between the split nuclear levels occur according to the selection rule DmI ¼ 1. These transitions can be induced by applying an rf magnetic field of the correct frequency. There are a number of other resonance processes also, such as ferromagnetic resonance, spin wave resonance, and antiferromagnetic resonance.
FIG. 21.12 Energy level splitting in the pure quadrupole interaction (QI) for (A) I ¼ 5/2, (B) I ¼ 3/2, and (C) I ¼ 1/2.
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Solid State Physics
21.13
FERROMAGNETIC RESONANCE
In a ferromagnetic solid the magnetization and susceptibility are large. In these solids the shape and structure of the specimen is important in determining the magnetization or the internal magnetic field and hence the resonance frequency. For example, in a solid (having cubic symmetry) in the form of a sphere, the resonance frequency o0 is given by o 0 ¼ gI H
(21.116)
If the solid is in the form of a plate with an applied magnetic field H perpendicular to the surface of the plate, then one gets o0 ¼ gI ðH 4pMÞ
21.14
(21.117)
SPIN WAVE RESONANCE
Consider a thin film of a ferromagnetic solid as in the previous section. The condition for a spin wave in the thin film can be achieved if the spins on the surface of the film experience different anisotropic fields than the spins within the film. Suppose a uniform magnetic field is applied perpendicular to the surface of the film, then the spins on the surface are pinned by surface anisotropic interactions. The field will excite the waves with an odd number of half wavelengths within the thickness of the film. This is called the spin wave resonance (SWR). The condition for the SWR is obtained by adding the exchange contribution, represented as D K2, to Eq. (21.117) to write o0 ¼ gI ðH 4pMÞ + DK2
(21.118)
Here K ¼ n p/L for the mode of a wave with n half wavelengths.
21.15
ANTIFERROMAGNETIC RESONANCE
In an antiferromagnetic solid, one can use the two-sublattice model to study the antiferromagnetic resonance. If the magnetic field is applied in the z-direction (HA is the magnetic field for sublattice A and HA for sublattice B), then the resonance condition becomes o20 ¼ g2I HA ðHA + 2HE Þ
(21.119)
where HE is the exchange magnetic field given by HE ¼ lAA MA
(21.120)
MA is the magnetization in the sublattice A.
REFERENCE Galsin, J. S. (2002). Impurity scattering in metallic alloys. New York: Kluwer Academic/Plenum Publishers.
SUGGESTED READING Bovey, F. A. (1969). Nuclear magnetic resonance spectroscopy. New York: Academic Press. Harris, R. K. (1986). Nuclear magnetic resonance spectroscopy. London: Longmans. Knight, W. D. (1956). Electron paramagnetism and nuclear magnetic resonance in metals. F. Seitz, & D. Turnbull (Eds.), Solid state physics (pp. 93–136). Vol. 2(pp. 93–136). New York: Academic Press. Low, W. (1960). Paramagnetic resonance in solids. In Solid state physics. New York: Academic Press. (Suppl. 2). Pake, G. E. (1956). Nuclear magnetic resonance. F. Seitz, & D. Turnbull (Eds.), Solid state physics (pp. 1–91). Vol. 2(pp. 1–91). New York: Academic Press. Poole, C. P., & Farach, H. A. (1972). The theory of magnetic resonance. New York: J. Wiley & Sons. Slichter, C. P. (1980). Principles of magnetic resonance. Berlin: Springer-Verlag. Winter, J. (1971). Magnetic resonance in metals. London: Oxford University Press.