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As Fig. 2 and Table 1 illustrate, this relaxation

effect can have a significant effect on the observed

enhancement. It follows from Eqn. (3) that maximum

enhancement is at a CP time of





Besides this potential enhancement in magnetization,

and therefore signal strength, CP offers another ben-

eﬁt: the CP process can be repeated after the abun-

dant spin relaxes, irrespective of the state of the rare

spin magnetization and/or relaxation rate. Of course,

this is because the rare spin magnetization is totally

derived from the abundant spins. Pines et al. (1973)

also showed that a significant improvement in reso-

lution could be achieved by decoupling the abundant

spin during the acquisition of the rare spin signal.

Schaefer et al. (1975) showed that if the CP exper-

iment is performed while the solid sample is spinning

at a sufﬁciently high rate (several thousand revolu-

tions per second) at an orientation where Eqn. (1) is

zero (magic-angle spinning, MAS), resolution ap-

proaching that of liquid samples can be obtained. A

sample that is spinning at a rate larger than the CSA

causes the CSA of each individual spin to be averaged

to the isotropic chemical shift value. If the spinning

rate is less than the CSA, spinning sidebands are ob-

served. Figure 3 shows the effect of sample spinning

rate on the

C cross-polarization magic-angle spin-

ning (CPMAS) spectrum of General Electric Lexan

(bisphenol A polycarbonate) resin. The top spectrum

was obtained at high sample spinning and has min-

imal spinning sidebands. The bottom spectrum was

obtained without sample spinning and shows broad,

overlapped resonances. It can be seen from the mid-

dle spectrum, obtained at moderate spinning speeds,

that the intensities of the spinning sidebands approx-

imately mimic the shape of the nonspinning line

shapes.

Figure 1

The theoretical CP dynamics are shown. Curve (a)

illustrates the expected rare spin magnetization as a

function of CP time in the absence of relaxation effects.

The single exponential curve is characterized by a time

constant, T

, the CP time. Curve (b) represents spin

(either rare or abundant) relaxation. The exponential

curve is characterized by a time constant, T

, which in

this specific example is eight times the CP time constant.

Curve (c), which is the expected rare spin magnetization

as a function of CP time, is the combination of curves

(a) and (b). The time axis is in units of T

while the

enhancement axis is in units of g

, the theoretical

maximum. Notice that for this example, maximum

enhancement is achieved at a CP time of about

2.38 T

and is about 74% of the theoretical

maximum.

Figure 2

A family of curves showing the CP magnetization as a

function of CP time. In each curve T

is constant and

is varied. The ratio T

for each curve is: (a) 1.1,

(b) 2.0, (c) 4.0, (d) 8.0, and (e) 16.0. Axis units are as in

Fig. 1.

Table 1

The time of occurrence and the value of the maximum

enhancement of CP for various values of T

max

Enhancement

(%)

1.1 1.05 38

2.0 1.39 50

4.0 1.85 63

8.0 2.38 74

16.0 2.96 83

a The enhancement is a percentage of the theoretical maximum, g

1090

Solids, Nuclear Magnetic Resonance of

These important discoveries and the work on

which they are based provide the foundation for the

most popular form of high-resolution solid-state

NMR, the CPMAS experiment. Conditions for the

high-resolution observation of abundant spins (i.e.,

protons) were worked out later and combined MAS

with multiple pulses during the acquisition of the

signal (homonuclear decoupling) and will not be dis-

cussed in this article.

1. General Instrumentation

While the requirements for CPMAS NMR are

similar to those for liquid NMR, there are several

important differences. Superconducting magnets pro-

vide stable, high-ﬁeld, high-homogeneity applied

ﬁelds for the NMR experiment. In fact, the quality

of modern superconducting magnets is such that

ﬁeld/frequency stabilization (NMR lock) is not re-

quired. Room temperature access for the CPMAS

probe is typically larger than for liquids NMR be-

cause of the special geometry and the higher RF

power handling requirements, requiring physically

larger electronic components. Also, due to the limited

resolution of the typical CPMAS spectrum and the

bulky and unusual geometry of the probe, room

temperature homogeneity shim coils have only a

modest impact on spectra quality and have been

omitted on some solids-only NMR spectrometers.

RF ﬁeld strength requirements for CPMAS NMR

are typically 1–2 orders of magnitude higher than for

liquid NMR resulting in power requirements ranging

from almost 100 W to several kilowatts depending on

the probe efﬁciency. The CPMAS probe is the most

unique part of a solids NMR spectrometer. Several

different designs are commercially available and all

share the following features:

A sample spinning system consisting of a rotor

(spinner) made of a high-strength material to hold the

sample. Turbine-like impellers are part of the rotor so

that it may be spun by pressurized gas ﬂowing over

these features.

A stator (spinner housing) which provides pres-

surized gas through a bearing surface in which the

rotor will spin. The stator is mounted in the probe so

that the sample spins at the magic angle (about 54.71)

relative to the applied magnetic ﬁeld.

A double-tuned RF coil (usually part of the

stator) with suitable tuning circuitry to couple efﬁ-

ciently the high-power transmitters to the coil, re-

sulting in the highest possible RF ﬁeld strengths.

In addition, the probe may have other features

such as spin rate detectors (analog or digital), spin

rate controllers, or variable temperature control.

2. Sample Requirements

Spectra of solid samples from a variety of sources can

be successfully observed by CPMAS NMR. Obvious

choices are crystalline, polycrystalline, and amor-

phous solids as well as plastics and liquid crystalline

materials. Clathrates, inclusion complexes, compos-

ites, mixtures, melts, and other compounded materi-

als are also possible. Powdered or granular material is

easiest to spin, although anything which ﬁts into the

rotor will usually work. Physically large samples need

to be reduced in size until particles are smaller than

the inner dimensions of the rotor (internal diameters

are typically 2–20 mm). Large particles are usually

more difﬁcult to spin and sometimes require some

signal-free balancing material. With some constraints

and care, spectra of highly viscous ﬂuids and gels

Figure 3

C CP spectra of a sample of GE Lexan resin are shown

spinning at three different rates. All spectra were

obtained with a 2.5 ms CP with g

¼g

¼55kHz.

With the sample spinning at 7940 Hz, the top spectrum

shows only minor spinning sidebands. For the center

spectrum, the sample was spun at 3460 Hz and shows

many sidebands. The chemical shift anisotropies of the

aromatic resonances at 150–100 ppm are much larger

than the aliphatic resonances at 50–20 ppm. The lower

spectrum was obtained without spinning, showing

overlapping resonances severely broadened by chemical

shift anisotropy.

1091

Solids, Nuclear Magnetic Resonance of

(e.g., concentrated biological ﬂuids, solutions and

suspensions, and petrophysical ﬂuids) can be run. In

cases where spinning might eliminate liquid from the

sample, they can be spun in special sealed or lined

rotors. Materials that conduct electricity are usually

not suitable for MAS. However, in some cases these

samples can be spun by diluting the material with an

appropriate insulator. In addition, samples which

efﬁciently turn RF energy into heat present special

problems. Because of the high RF power levels

involved, an extremely low RF duty cycle is used

and this usually limits the resolution of the detected

signal.

3. Applications

In the simplest CPMAS experiment, chemical shift

information of the rare spin nuclei is obtained. Ale-

many et al. (1983a) have shown that for most simple

organic compounds, a 2.25 ms CP contact time will

result in near-quantitative

C NMR intensities,

making it routine to obtain reliable integration da-

ta. Quantitative results are often difﬁcult or impos-

sible to obtain in the case of multiphase compounds.

This is especially true for engineered plastics that

often are made up of at least two different compo-

nents that have different physical properties, blended

or bonded together to achieve a desirable physical

performance.

For example, consider a silicone heat-cured

elastomer that is made up of a very-long-chain,

low-modulus, lightly cross-linked dimethylsiloxane

polymer and a high-modulus fumed silica ﬁller.

Chemically, the fumed silica ﬁller is submicrometer-

sized amorphous silicon dioxide particles with the

surface of the particles covered with hydroxide

groups. The

Si CPMAS spectrum of a heat-cured

elastomer is shown at the top of Fig. 4. The dime-

thylsiloxane polymer is observed as the sharp reso-

nance at 23.5 ppm. The fumed silica is observed as

two broad resonances below 100 ppm. The reso-

nance at 113 ppm is from the amorphous fumed

silica and the resonance at 103 ppm is from the

surface silicon, directly bonded to the surface hydro-

xyl groups. Broad peaks at –21.5, 35.8, 57.2, and

66.8 ppm are due to various special methyl silo-

xanes found on the surface of the fumed silica. The

bottom spectrum of Fig. 4 shows what is observed

with a simple Bloch-decay experiment, i.e., not using

CP. Because the

Si nuclei relax very slowly, this

spectrum was acquired with very long relaxation

times between acquisitions. The bottom Bloch-decay

experiment more accurately represents the quanti-

tative distribution of silicon in the sample. For this

nearly quantitative spectrum, only the dimethyl-

siloxane polymer and fumed silica peaks are ob-

served. The fumed silica peaks more quantitatively

represent the distribution of silicon atoms between

the surface and core of the fumed silica particles, with

the surface peak just observable as a shoulder to the

dominant core silica resonance. Owing to the limited

number of acquisitions for this spectrum, the special

surface methyl siloxanes cannot even be observed.

The CP intensity for the silicon in the dime-

thylsiloxane polymer and the silica particles is not

quantitative for different reasons. The polymer un-

dergoes labile motion, interfering with CP from the

methyl

H nuclei to the

Si nuclei resulting in a very

long T

. Thus, only a very weak signal is observed

from the polymer. The silica is immobile owing to the

very high molecular weight and the three-dimensional

structure of the resin making up the particle and

should cross-polarize readily. However, the core silica

resonance at 113 ppm cross-polarizes poorly be-

cause it does not have any

H nuclei nearby to pro-

vide polarization. It is not possible to obtain a

quantitative CPMAS spectrum by lengthening the CP

contact time. While a longer CP time would result in

a stronger, more quantitative polymer resonance, the

relatively short T

of the silica would result in near-

zero CP intensity from the silica. These kinds of

problems are typically encountered when trying to

obtain quantitative CP spectra of multiphase sam-

ples. Nonetheless, CPMAS spectra can be quite

useful for these kinds of samples. In the case of the

heat-cured elastomer illustrated here, the CPMAS

Figure 4

A comparison of

Si spectra is illustrated with and

without CP for a heat-cured elastomer. The top

spectrum is the CPMAS spectrum obtained with a 5 ms

CP time, acquiring 10

scans with a 2 s relaxation time.

The bottom spectrum was obtained with a simple

901 pulse. A 25-fold expansion of the high-ﬁeld region is

shown in the center. As in the CPMAS spectrum, high-

powered proton decoupling was applied during the

acquisition of the signal and the sample was spun at the

magic angle at 7200 Hz. For this Bloch-decay spectrum,

1388 scans were acquired with a 150 s relaxation time.

Each spectrum was acquired over several days.

1092

Solids, Nuclear Magnetic Resonance of

spectrum is very sensitive to the surface of the fumed

silica and is a valuable research tool for understand-

ing polymer–ﬁller interactions at or near the ﬁller

surface.

While typical CPMAS resolution routinely ob-

tained is still a couple of orders of magnitude poorer

than that obtained in liquid NMR, it is usually pos-

sible to identify most chemical types of the rare spins.

However, it is usually impossible to identify a unique

chemical type in the presence of other similar chem-

ical types due to the limited resolution. Successful

strategies for the analysis of chemical processes in

solids usually involve a comparison of the CPMAS

spectra of related materials. For example, researchers

have been able to deduce the chemical mechanism of

the curing process of urea–formaldehyde (Maciel

et al. 1983) and phenolic (Bryson et al. 1983) resins by

comparing

C CPMAS spectra of resin samples sub-

jected to different curing times. In the case of poly-

mers, crystalline–amorphous morphology may be

discernable through an analysis of observed line

shapes. A variety of relaxation measurements can be

performed in order to obtain information about mo-

lecular motion, micro- and macrostructure, and mor-

phology. The abundant spin as well as the rare spin

dynamics can be observed through the rare spin

magnetization, along with the dynamics of magnet-

ization transfer. Especially sensitive to lower frequen-

cy motions, measurement of T

has been useful in

the study of macromolecules (Belﬁore et al. 1983).

Homonuclear decoupling technology has made

available special decoupling sequences that selective-

ly disable homonuclear dipole–dipole interactions.

Applied to the abundant spins, several new solids

techniques were developed. If the abundant spin di-

pole–dipole interaction is sufﬁciently weak, applica-

tion of homonuclear decoupling makes it possible to

observe (and use) scalar coupling between the rare

and abundant spins. For example, Zilm and Grant

(1982) have observed

C—

H scalar coupling in

adamantane and hexamethylbenzene. In another ex-

ample, Terao et al. (1982) have used

C—

H scalar

coupling in camphor to obtain j-resolved

C spectra.

While these and other similar results are very inter-

esting, most solid samples do not have the required

weak dipole–dipole interaction to allow the scalar

coupling to be observed and/or exploited. In another

more useful application of homonuclear decoupling,

a two-dimensional experiment has been developed in

which each rare spin becomes a sensitive probe of its

local dipolar environment (Munowitz et al. 1981,

Schaefer et al. 1983). In this experiment, multiple

data sets are collected. In each sequential data set an

evolving period of homonuclear decoupling is applied

to the abundant spins before the detection of the rare

spin signal. Projecting the processed data along the

evolution axis shows a normal CPMAS spectrum.

Slices along the evolution direction show the local

dipolar ﬁeld for each of the resolved nuclei. Because

of Eqn. (1) and because of the rare spin resolution

typically achievable, these spectra can be rich in dis-

tance and geometry information.

4. Limitations

While CPMAS NMR can approach the resolution

of the NMR of liquids, it is typical for a CPMAS

spectrum resolution to be limited to several ppm.

Amorphous materials and polymers may have a dis-

tribution of isotropic chemical shifts for individual

carbons due to a variety of carbon environments in

the solid matrix. It has been shown that polymers

that have undergone severe mechanical stress have

broadened CPMAS spectra. VanderHart et al. (1981)

have published a comprehensive study of resolution

limitations of CPMAS NMR. The idea of applying

continuous, unmodulated decoupling to the abun-

dant spins during the acquisition of the NMR signal

is to remove the dipolar effect of those nuclei, leaving

only CSA as a source of broadening. This is difﬁcult

to achieve in practice as there is always some nonzero

residual dipolar coupling. Bennett et al . (1995) de-

scribed a simple phase modulation of the decoupling

RF ﬁeld that shows significant improvement over

unmodulated decoupling.

The CPMAS technique relies upon the premise that

equilibrium magnetization of the abundant spins is

many times larger than that of the rare spins and

those rare spins do not interact with each other. Sys-

tems containing rare spins that share a chemical bond

present new problems. For example, a sample that

contains a high enrichment of

C will show partic-

ularly broad carbon resonances as they interact with

one another through their dipolar ﬁelds. Another

more common situation that can limit resolution is

the presence of quadrupolar nuclei. These nuclei ori-

ent their quadrupole moments in their molecular en-

vironment and can generate a powerful magnetic ﬁeld

along their quadrupole moments. This additional ﬁeld

adds to the applied ﬁeld and modiﬁes other spin in-

teractions, resulting in line broadening (Naito et al.

1981). This effect can be minimized by increasing the

applied ﬁeld strength which decreases the relative

contribution of the ﬁeld generated by the quadrupolar

nucleus (Maciel et al. 1983). Of course, this solution is

not always available if no higher ﬁeld NMR spectro-

meter with solid-sample capability is accessible. Even

if higher ﬁeld instruments are available, other com-

plications can arise in very-high-ﬁeld CPMAS NMR.

As the applied ﬁeld is increased, so does CSA, re-

quiring higher sample spinning rates to obtain pure

isotropic resonances and a minimum of spinning side-

bands. More sidebands make spectral interpretation

more difﬁcult. When spinning rates begin to approach

the magnitude of the Hartmann–Hahn RF ﬁeld, this

mechanical motion will begin to interfere with the CP

dynamics, resulting in lower CP efﬁciency, until ﬁnally

1093

Solids, Nuclear Magnetic Resonance of

no signal can be observed (Wind et al. 1988). In these

cases, obtaining a spectrum with sample spinning

sidebands is usually preferred. Alternatively, Dixon

(1981) has devised a CPMAS NMR experiment to

separate spinning sidebands from isotropic reso-

nances, although this experiment may not always

produce the desired results, sometimes distorting

intensities due to relaxation effects.

Finally, in some cases, CP spin dynamics may not

be suitable for a successful experiment. As shown

in Fig. 2, if the T

ratio is too small, CP may

become difﬁcult or impossible to detect (Alemany

et al. 1983b).

5. Summary

While

C and

Si CPMAS NMR was the focus of

this discussion, other nuclei can be observed, includ-

ing

119

Sn, and

P. It is also possible to generate

CP intensity from nuclei other than

H. For example,

C—

F CPMAS spectra can be obtained for ﬂuor-

inated polymers. In a number of respects CPMAS

NMR is experimentally more demanding than con-

ventional liquid NMR. However, CPMAS NMR is

invaluable in the study of solids, providing both static

(structural) and dynamic (motional) information.

While there are some samples which cannot be stud-

ied by CPMAS NMR, this technique can be applied

to a wide variety of materials, regardless of purity,

crystallinity, or other physical attributes.

Acknowledgments

E. A. Williams provided helpful discussions and

P. Donahue helped proof-read the manuscript.

Ye-Feng Wang of GE Silicones, Waterford, NY,

provided the heat-cured elastomer.

See also: High-temperature Superconductors, Cup-

rate: Magnetic Properties by NMR/NQR; Nuclear

Magnetic Resonance Spectrometry; Nuclear Reso-

nance Scattering

Bibliography

Alemany L B, Grant D M, Pugmire R J, Alger T D, Zilm K W

1983a Cross polarization and magic angle sample spinning

NMR spectra of model organic compounds: 1. Highly pro-

tonated molecules. J. Am. Chem. Soc. 105, 2133–41

Alemany L B, Grant D M, Pugmire R J, Alger T D, Zilm K W

1983b Cross polarization and magic angle sample spinning

NMR spectra of model organic compounds: 2. Molecules of

low or remote protonation. J. Am. Chem. Soc. 105, 2142–7

Belﬁore L A, Henrichs P M, Massa D J, Zumbulyadis N,

Rothwell W P, Cooper S L 1983 Molecular dynamics of po-

lycarbonate–diluent systems: applications of high-resolution

carbon-13 solid-state NMR. Macromolecules 16, 1744–53

Bennett A E, Reinstra C M, Auger M, Lakshmi K V, Grifﬁn R

G 1995 Heteronuclear decoupling in rotating solids. J. Chem.

Phys. 103, 6951–8

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experiment. Phys. Rev. 70, 474–85

Bryson R L, Hatﬁeld G R, Early T A, Palmer A R, Maciel G E

1983

C NMR studies of solid phenolic resins using cross

polarization and magic-angle spinning. Macromolecules 16,

1669–72

Dixon W T 1981 Spinning-side band-free NMR spectra.

J. Magn. Reson. 44, 220–3

Hartmann S R, Hahn E L 1962 Nuclear double resonance in

the rotating frame. Phys. Rev. 128, 2042–53

Maciel G E, Szeverenyi N M, Early T A, Myers G E 1983

NMR studies of solid urea–formaldehyde resins using cross

polarization and magic-angle spinning. Macromolecules 16,

598–604

Munowitz M G, Grifﬁn R G, Bodenhausen G, Huang T H

1981 Two-dimensional rotational spin-echo nuclear magnetic

resonance in solids: correlation of chemical shift and dipolar

interactions. J. Am. Chem. Soc. 103, 2529–33

Naito A, Ganapathy S, McDowell C A 1981 High resolution

solid state

C NMR spectra of carbons bonded to nitrogen

in a sample spinning at the magic angle. J. Chem. Phys. 74,

5393–7

Pines A, Gibby M G, Waugh J S 1973 Proton-enhanced NMR

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sorption by nuclear magnetic resonance. Phys. Rev. 69, 37–8

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spin-echo

C NMR of polymers. J. Magnetic Resonance 52,

123–9

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carbon-13 nuclear magnetic resonance study of some solid,

glassy polymers. Macromolecules 8, 291–6

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of carbon-13 spectra by using J-resolved NMR spectroscopy

in solids. J. Am. Chem. Soc. 104, 5228–9

VanderHart D L, Earl W L, Garroway A N 1981 Resolution in

C NMR of organic solids using high-power proton decou-

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361–401

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C/MAS and

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General Electric Company, Niskayuna

New York, USA

Spin Fluctuations

1. Introduction

In general, the term spin ﬂuctuations refers to ﬂuc-

tuations of the magnetic (electron spin) moment in

magnetic systems, both ordered and paramagnetic

ones. In the ground state (zero temperature, zero

1094

Spin Fluctuations

applied magnetic ﬁeld H ¼0) of a ferromagnetic sys-

tem of local moments, for instance, all moments are

oriented in the same direction. Above a critical tem-

perature, the Curie temperature T

, the system is

paramagnetic. At low temperatures, the excitations

from the ground state can well be described as spin

waves or magnons, that is, as independent collective

excitations obeying Bose–Einstein statistics. With in-

creasing temperature (T), the absolute value of the

‘‘long-time averaged’’ magnetization (M) decreases as

M ¼ M

 aT

3=2

(Bloch’s T

3/2

law). The local, mo-

mentary magnetization, M

(r), i.e. the short-time av-

erage over a small volume around r, is written as

ðrÞ¼M þm

ðrÞ, where m

(r), or simply m(r), rep-

resents the random, time-dependent deviation, or

ﬂuctuation. The components along the direction of

M, chosen as the z direction, m

(r), are referred to as

‘‘longitudinal’’ ﬂuctuations, the components in the

xy-plane, m

(r), m

(r), as ‘‘transverse’’ ones. In this

temperature range, the transverse ﬂuctuations can be

described either as spin ﬂuctuations or equivalently as

magnons. The length scale, over which averaging can

be performed without blurring out the deviations, is

determined by the so-called ‘‘correlation length.’’

With increasing temperature, apart from the fact that

the spin waves can no longer be considered as inde-

pendent excitations, the magnitude and the correla-

tion length of the ﬂuctuations increase. In the

neighborhood of the transition temperature, the crit-

ical ﬂuctuations are dominant, and the correlation

length tends to inﬁnity. In this ‘‘critical region,’’ ‘‘re-

normalization group’’ techniques can be applied,

leading to the so-called ‘‘scaling laws.’’ At still higher

temperatures, the correlation length gradually de-

creases down to an atomic scale. The system then

behaves as a collection of independent local atomic

moments.

For some specific models (e.g., the Ising model),

spin ﬂuctuations can be visualized by computer sim-

ulation (see, e.g., http://www.physics.cornell.edu/sss/

Ising/Ising.html).

The behavior of spin ﬂuctuations in an itinerant-

electron band system is qualitatively the same. In the

paramagnetic state, the ﬂuctuations (heavily damped

collective excitations) were originally called ‘‘par-

amagnons.’’ A large effect of spin ﬂuctuations is

expected for strongly enhanced paramagnets (on the

verge of becoming magnetically ordered, see next

section). If, moreover, the number of electrons (or

holes) in the itinerant band is low, at high temper-

atures the ﬂuctuations may effectively cause the

(local) band to be (momentarily) fully polarized.

Then, the system even roughly behaves as a system of

local moments. This is illustrated in Fig. 1, showing

the resistivity of the strongly enhanced paramagnets

Pd, Np, and Pu, as analyzed by Jullien et al. (1974).

The spin ﬂuctuation contribution is obtained by

subtracting the resistivity of a suitable, ‘‘normal’’

metal (Ag in the Pd case, Th for Np and Pu). In local-

moment systems, the ‘‘spin disorder scattering’’ is

saturated at high temperatures. Evidently, such sat-

uration may occur in itinerant-electron systems as

well. Jullien et al. ﬁtted the calculated spin ﬂuctuation

contribution by ﬁtting the spectrum of the ﬂuctua-

tions with wave vector q and frequency o, i.e. of the

dynamical susceptibility w(q,o), applying the ‘‘ﬂuctu-

ation – dissipation’’ theorem.

This theorem is applied in most of the sophisticated

spin ﬂuctuation models, indeed. Each mode, (q,o),

behaves as a harmonic oscillator obeying Bose–Ein-

stein statistics. Consequently, one may distinguish

zero-point ﬂuctuations (or quantum ﬂuctuations) and

thermal ﬂuctuations. The different models are based

on different approximations for the quantum ﬂuctu-

ations (as being dependent on the magnetization, for

instance), for the change-over from spin wave be-

havior at low temperatures to ﬂuctuating behavior at

higher temperatures, for the necessary cut-off wave

vector in the spectrum, for the difference between

longitudinal and transverse ﬂuctuations, and so on.

The occurrence of spin ﬂuctuations is by no means

restricted to simple ferromagnetic materials. They

play an important role in antiferromagnetic and

ferrimagnetic materials as well. The interplay of local

(rare-earth) moments and an itinerant-electron

matrix may enhance the effect of spin ﬂuctuations

(see, e.g., Duc and Brommer (1999) for details). In

0 200 400 600 800 1000

T (K)

100

150

200

250

 (µΩ cm)

Figure 1

The resistivity of Pd, Np, and Pu as a function of

temperature. The solid lines through the data points

show the sum of the calculated spin ﬂuctuation

contribution and an experimentally determined, suitable

‘‘nonmagnetic’’ background (Ag for Pd and Th for Np

and Pu; notice the difference in scales, chosen such that

the background contributions coincide). For data and

details, see Jullien et al. (1974).

1095

Spin Fluctuations

magnetic perovskites (manganites, cobaltates) mag-

netic order may compete with charge order, giving

rise to a very rich phase diagram. Spin ﬂuctuations

can play a decisive role in this competition. In high-

superconductors, spin ﬂuctuations are supposed

to provide the necessary ‘‘glue’’ for electron pairing,

playing the role of the phonons (lattice distortions,

electron–phonon coupling) in the more conven-

tional superconductors. In strongly correlated

electron systems and Kondo systems, at low temper-

atures a collective (excited) state may be formed,

giving rise to spin ﬂuctuations too. At the so-called

spin ﬂuctuation temperature, the collective state is

destroyed and the particular ﬂuctuations do not play

a role any more.

Nevertheless, here attention will be focused on

itinerant-electron systems, mainly because the interest

in spin ﬂuctuations started with a long-standing con-

troversy about the relative importance, in these sys-

tems, of, on the one hand, collective spin ﬂuctuations

and, on the other hand, independent single-particle

excitations.

2. Free-energy Considerations

In the Landau theory of second-order phase transi-

tions (see Phase Transitions, Landau Theory of), the

free energy is expanded as a function of the order

parameter, here the magnetic moment M:

magn

þ ?

þ ? ð1Þ

In the Stoner–Edwards–Wohlfarth (SEW) model for

itinerant electrons (see, e.g., Brommer and Franse

1990, and references therein), one takes A ¼ A

 l,

where A

, like B and C, depends on the band pa-

rameters, and where l represents the exchange energy

(1/2)lM

. The equilibrium condition @F

magn

=@M ¼

H leads to

H ¼ AM þ BM

þ CM

þ ? ð2aÞ

or, written in the ‘‘Arrott–Kouvel’’ form

H=M ¼ A þ BM

þ CM

þ ? ð2bÞ

2.1 Arrott Plots

For very weak itinerant ferromagnets (VWIF), the

term is ignored (B40, for stability). As an ex-

ample, some M

versus m

H/M plots, so-called Arrott

plots, are shown for Ni

42.9

57.1

in Fig. 2. The devi-

ations from the straight lines, at low M

and m

H/M

values, are ascribed to concentration variations (see

Brommer and Franse 1990). Such straight Arrott

plots were often considered as proof for a negligible

spin ﬂuctuation contribution.

Spontaneous order (Ma0 for H ¼0) occurs when

the Landau coefﬁcient A is negative, with M

¼A/

B40. When A is positive, the system is paramagnetic,

and A is the inverse initial susceptibility. Applying

Fermi–Dirac statistics for the independent, single-

electron excitations, the temperature dependence of

the ‘‘Landau co-efﬁcients’’ A, B, C, and so on, can be

determined. In the simplest model (see, e.g., Shimizu

1981 or Brommer and Franse 1990 for a much more

elaborate approach), only A

is taken to be temper-

ature dependent:

ðTÞ¼A

ð0Þð1 þ T

Þð3Þ

For a simple itinerant-band structure with Fermi

level E

, T

is a temperature in the order of the Fermi

temperature E

2.2 Stoner Enhancement

The exchange interaction enhances the paramagnetic

susceptibility by the ‘‘Stoner enhancement’’ factor S,

deﬁned as

S ¼ A

ð0Þ=Að0Þ¼A

ð0Þ=½A

ð0Þl¼1=ð1 

IÞð4Þ

The last equality expresses the enhancement factor

conventionally as a function of the Stoner parameter

Ið¼ l=A

ð0Þ¼NðE

ÞIÞ, where N(E

) is the density

of states at the Fermi level of the itinerant band

(recall that the Pauli susceptibility A

(0)

1

is propor-

tional to this density of states). I is again a measure

for the exchange interaction. Since the unenhanced

01020304050



H/M (T f.u./

)

0.2

0.4

0.6

[10

−2

(

/f.u.)

]

4.2 K

10.2

16.5

22.5

27.4

30.0 K

0.43

0.57

Figure 2

Arrott plots for Ni

0.43

0.57

. The straight lines represent

the equation of state (Eqn. (2b)) m

H/M ¼A þBM

(for

data and details see Brommer and Franse 1990, p. 352).

1096

Spin Fluctuations

inverse (Pauli) susceptibility A

is positive, the system

can only become ferromagnetic for a positive l of

sufﬁcient magnitude. Note that for an itinerant ferro-

magnet S is negative (

I41: the Stoner criterion).

Since A

(T) increases with increasing temperature,

the ﬁrst Landau coefﬁcient in the Stoner model,

ðTÞ¼½A

ðTÞl¼A

ð0Þð1 

I þ T

Þ will

eventually change sign, and the system becomes par-

amagnetic (see also Eqn. (7), next section).

2.3 Incorporating Spin Fluctuations

In a simpliﬁed version of the Moriya (1985) ‘‘self-

consistent renormalization’’ (SCR) treatment of the

spin ﬂuctuations, the terms M

, in Eqn. (1), are re-

placed by /7M

ðrÞ7

S ¼ /7M þ mðrÞ7

S. Taking

the spatial average (/S) in the homogeneous case,

and omitting higher-order terms, one ﬁnds for the

coefﬁcient of the M

term:

AðTÞ¼A

ðTÞþBð3m

þ m

¼ A

ðTÞþð5=3ÞBm

ðTÞ

ð5Þ

The second equality holds for an isotropic system

near T

, with m

¼ m

ðTÞ=3.

In the VWIF system under consideration, the spin

ﬂuctuation term in Eqn. (5) is positive, and conse-

quently the change of sign will occur at a lower

(Curie) temperature. This explanation for the appar-

ent low Curie temperatures of the itinerant-electron

systems was a great success of the spin ﬂuctuation

theory (see next section).

Band parameters (and thus the Landau co-efﬁ-

cients A

, B,y) can be deduced from band calcula-

tions with still increasing accuracy. The necessity to

consider also the spin ﬂuctuations is clearly demon-

strated by Aguayo et al. (2004) in a comparison of the

‘‘prototype’’ itinerant-electron systems Ni

Al and

Ga. Since both systems are Ni-based, the ex-

change parameter l or I will be about the same for

these compounds. The density of states is larger in

Ga. Hence, one would expect Ni

Ga to be the

more enhanced (or more ferromagnetic) compound

(because

I ¼ NðE

ÞI is larger in Eqn. (4)). The op-

posite is true. Now, according to their calculations

the q-dependence of the dynamic susceptibility is

weaker in Ni

Ga. Consequently, there are more spin

ﬂuctuations, and the positive contribution to A( T)

will be larger (see Eqn. (5)). In fact, Ni

Ga even re-

mains paramagnetic down to 0 K!

2.4 Magnetovolume Effects

Differentiating the magnetic free energy (Eqn. (1))

yields the ‘‘magnetic pressure’’ P

þ ?

where, A

¼ @ ln AðV; TÞ=@ ln V, and analogously.

Henceforth, the volume dependence of B (and higher

Landau co-efﬁcients) will be ignored. Incorporating

the spin ﬂuctuations, one ﬁnds the (relative) ‘‘mag-

netic volume’’ o

¼ V

=V as

¼kP

¼

ðkA

St;o

=VÞ½M

þ m

ð6Þ

where, k is the compressibility. The magnetic volume

is proportional to [M

þm

], so there will be a con-

tribution even in the paramagnetic state (M ¼0).

2.5 Metamagnetic Transitions

In the Laves phase RCo

type compounds (R: rare

earth, Y), the Co 3d-subsystem is a metamagnetic,

itinerant-electron system. For YCo

, an applied mag-

netic ﬁeld exceeding B70 T induces a jump of the Co

moment up to B0.5 m

/Co. Substitution of Al for Co

stabilizes this ‘‘high-spin’’ state. Y(Co

1x

) is par-

amagnetic for x ¼0.075, but the moment jumps to

B0.6 m

/Co at the metamagnetic transition ﬁeld of

33 T (see upper panel of Fig. 5). This behavior can be

described by the free-energy expression, Eqn. (1),

when the Landau coefﬁcient A is positive, whereas B

is negative, and C is again positive for stability (see

dashed ﬁtting curve in Fig. 5; see also further discus-

sion in Sect. 4).

According to Eqn. (5), in such a system with Bo0,

the spin ﬂuctuation contribution in A(T) would be

negative, so would promote the transition. Stability

criteria have been discussed extensively by Yamada

et al. (2003). The order parameter (M) in Eqn. (1)

can be taken to be the amplitude of the relevant

antiferromagnetic ordering mode (staggered magnet-

ization). So, the formalism can be applied to anti-

ferromagnetic systems too. For T4T

(E100 K),

YMn

is an itinerant-electron paramagnet in which

the Mn atoms have no magnetic moment. For

ToT

, the Mn subsystem becomes antiferromag-

netic, with a large moment (2.7 m

/Mn) and a large

volume increase (5%). On the basis of their band

structure calculations, and including magnetovolume

coupling terms in the free energy expression, Nakada

et al. (2004) conclude that the Mn subsystem in itself

would not order at all: spin ﬂuctuations are necessary

to ‘‘stabilize’’ the ordered system (see also Sect. 4).

2.6 Beyond SCR

There have been many attempts to develop the spin

ﬂuctuation theory beyond the Moriya SCR level.

Noteworthy are the papers by Takahashi and col-

leagues, e.g., Takahashi (2001), in which the impor-

tant role of the zero-point or quantum ﬂuctuations is

stressed. Kaul and co-workers claim to have found a

solution for the transition from the low-temperature

spin wave regime to the critical regime around T

1097

Spin Fluctuations

and provide and elaborately analyze experiments on

mainly Ni

Al and related compounds (see, e.g., Kaul

1999 and Semwal and Kaul 2002).

3. The Curie Temperature of a Very Weak

Itinerant Ferromagnet

According to the Landau theory for second-order

phase transitions, the ordering transition occurs when

A(T) changes sign (see Eqn. (1), case B40). For an

itinerant ferromagnet, AðTÞ½¼A

ðTÞ is given by

ðTÞ¼A

ð0Þð1 

I þT

¼ðA

ð0Þ=7S7Þð1 þ 7S7T

Þð7Þ

(T) vanishes at the ‘‘Stoner temperature’’

¼ T

=7S7

1=2

. When spin ﬂuctuations are taken

into account, A(T) is given by Eqn. (5). In the simple

treatment here, the temperature dependence of B is

neglected. Moreover, in the discussion on the Curie

temperature only the behavior of m

(T) in the neigh-

borhood of T

is needed. Here, the relation

(T)BT

4/3

is generally accepted. Then, with Eqns.

(3) and (5), one may write

AðTÞ¼Að0Þþa

ðT=T

þ a

ðT=T

4=3

ð8Þ

where T

is an arbitrary reference temperature, for

ferromagnetic materials often chosen equal to T

.As

an example, Fig. 3 shows an analysis of the temper-

ature dependence of the Landau parameter A(T) for a

series of (Ni,Pd)

Al compounds (data taken from ﬁg.

5 in Brommer and Franse 1990). In fact, in order to

obtain a consistent ﬁt for all the curves, a term BT

has been added and the contribution of the spin

ﬂuctuation (i.e., the coefﬁcient a

in Eqn. (7)) was

assumed to be the same for all compounds. Accord-

ing to this analysis, the contributions of the Stoner T

term and the ‘‘spin ﬂuctuation’’ T

4/3

term are of equal

importance. It is easy to imagine that, for this kind

of experimental results, reasonable ﬁts could be

obtained for a pure Stoner model as well as for a

pure Moriya (spin ﬂuctuation) model. Historically, in

the debate on the importance of spin ﬂuctuations

versus Stoner excitations, it is understandable that

the analysis of such measurements never convinced

an opponent (for some more details see Brommer and

Franse 1990).

0 20 40 60 80 100

T (K)

−100

100

200

300

A (T Ni/



)

75−x

14.9

11.2

7.58

5.2

2.21

0.00

Figure 3

Analysis of the ﬁrst Landau coefﬁcient A(T) for some

(Ni,Pd)Al

compounds. Extending Eqn. (7), the solid

lines represent ﬁtting results according to:

A(T) ¼A(0) þa

2St

(T/T

)

þa

4St

(T/T

)

þa

(T/T

)

4/3

With, arbitrarily, T

¼40 K the ﬁtting results for

are: A(0) ¼12.25 TNi/m

, a

2St

¼25.6 TNi/

; a

4St

¼0.37 TNi/m

, a

¼27.36 TNi/m

, giving

directly the relative importance of these contributions in

the total value of 65.6 TNi/m

at T ¼T

¼40 K.

01234

t [= T/T

]

A(T )/A(0) = M

p = 0

p = 0.08

p = 0.5

Stoner

Spin fluct.

Total

T = T

Figure 4

Schematic illustration of the reduction of the Curie

temperature: A(T)/A(0) [ ¼M

] is plotted vs. t ¼

T/T

. The solid lines represent A

(T)/A(0) ¼1T

(Stoner model). The skew dashed line represents the

contribution of the spin ﬂuctuations, (5/3)Bm

(T)/

7A(0)7 ¼(1p)t

4/3

, for p ¼0.5. Here, p is the relative

contribution of the Stoner excitations in A(T

). p ¼0:

Moriya model (Stoner excitations completely ignored),

p ¼0.08, value quoted for MnSi; p ¼0.5: possible value

for ZrZn

and Ni

Al (see Brommer and Franse (1990),

table 5).

1098

Spin Fluctuations

A striking success of the spin ﬂuctuation approach

was the explanation of the reduction of the Curie

temperature with respect to the Stoner temperature.

This is shown schematically in Fig. 4. With respect

to the Stoner temperature, the Curie temperature is

reduced by a factor t

¼ T

½¼ 1=Op, which

is inﬁnite for the pure Moriya model, 12.5 for

MnSi (p ¼0.08) and perhaps about a factor O2 for

ZrZn

and Ni

Al, for which the value p ¼0.5 is

adopted here.

4. Volume Effects

4.1 Separation of Volume Effects and Spin

Fluctuation Contributions

Many itinerant-electron systems are based on transition

metal ions, which are known to exhibit large magne-

toelastic effects (see Eqn. (6)). The volume change may

inﬂuence the properties of the material, for instance,

the resistivity. Then, it may be difﬁcult to distinguish

the volume effect from the spin ﬂuctuation effect.

Figure 5 shows an attempt to separate the effects for

two Y(Co,Al)

compounds. Y(Co

1x

)isferromag-

netic for x ¼0.145. The magnetic moment increases

gradually from B0.1 m

/CouptoB0.5 m

/Co when

the applied ﬁeld is increased from 0 T to 40 T. It ex-

hibits a positive magnetoresistance, proportional to

, and thus interpreted as a (or the) volume effect.

For the metamagnetic compound (x ¼0.075), the sharp

drop in the magnetoresistance, accompanying the com-

parable, but now jump-like, increase of the Co moment,

is ascribed to the suppression of the spin ﬂuctuations,

evidently exceeding the concurrent volume effect.

4.2 Huge Thermal Expansion in YMn

and

(Nd,Lu)Mn

Magnetovolume effects are very clear in YMn

and

related Laves phase compounds (already discussed

010203040

B (T)

−0.1

0.1

0.2

0.4

0.6

M (



/Co)

0 0.1 0.2

[(

/Co)

]

x = 0.145

0.075

x = 0.145

0.075

Y(Co

1−x

Alx)

∆/(0)

∆/(0) (%)

Figure 5

Magnetic moment per Co (upper panel) and the relative

magnetoresistivity for the ferromagnetic (stabilized)

compound Y(Co

0.855

0.145

)

and the metamagnetic

compound Y(Co

0.925

0.075

)

. The dashed line in the

upper panel represents the ﬁtting with Eqn. 2(a) for

x ¼0.075, with A40, B o 0, C40. The other lines are

guides to the eye only. The inset in the lower panel

shows the proportionality Dr/rBM

for x ¼0.145, i.e.,

for the stabilized compound.

0 100 200 300

T (K)

V/ V (%)

1−x

x = 0

x = 0.10

x = 0.30

Figure 6

Thermal expansion of some Nd

1x

compounds

(x ¼0.00, 0.10, and 0.30). The solid line indicates the

background thermal expansion (electronic and lattice

contributions). The excess volume is assumed to be due

to the magnetic Mn moment. For T4T

, the excess

thermal expansion is ascribed to spin ﬂuctuations.

1099

Spin Fluctuations