Buschow K.H.J. (Ed.) Concise Encyclopedia of Magnetic and Superconducting Materials

Подождите немного. Документ загружается.

energy is applied. Particular nuclei absorb this energy

at sharply deﬁned frequencies; this signal is then de-

tected, ampliﬁed, and presented as a frequency spec-

trum. NMR of hydrogen nuclei in bulk materials was

ﬁrst observed in 1946 independently by the research

groups of Bloch at Stanford and Purcell at Harvard,

for which they received the 1952 Nobel Prize for

Physics. It was quickly realized that liquid samples

give much narrower peaks with significantly im-

proved resolution of peaks in the spectrum.

The discovery of the chemical shift, i.e., the de-

pendence of the resonance frequency of a given nu-

cleus on its specific chemical environment in the

molecule, established NMR as a fundamental tool for

structure determination. The three most important

types of information obtained in a high-resolution

NMR spectrum are frequency, area, and multiplicity

of the resonance lines. The exact resonance frequency

of a nucleus is characteristic of its local electronic

environment and is measured against a reference

peak in the sample. The area under the resonance line

can be integrated to give quantitative information

about the relative number of nuclei giving rise to each

peak. Peaks may be split into multiple lines owing to

scalar (spin–spin) coupling with neighboring nuclei.

This interaction occurs through the electrons of the

bonding network and is due to the different combi-

nations of spin orientations that neighboring nuclei

can adopt. The splitting pattern indicates structural

information about the neighboring nuclei.

Advances in instrument hardware since the late

1960s have significantly enhanced the sensitivity and

resolution of NMR spectroscopy. As a result, spectra

may be obtained of solutions of low concentration,

nuclei may be observed that have very low natural

abundance, and resolution of structural features is

greatly improved. Whereas during the ﬁrst 20 years

after its discovery NMR applications were mostly

limited to proton nuclei, modern spectrometers can in

principle observe any nucleus, limited only by re-

strictions of its magnetic properties. Furthermore,

developments in software and RF pulse sequence

programming have dramatically extended the range

and power of NMR experiments. Methods exist to

map out fully by NMR the structure of an organic

molecule for example. Although methods have also

been established to obtain high-resolution NMR

spectra of solid materials, the differences in the tech-

niques are sufﬁcient to require that this be a topic

for a separate article (see Solids, Nuclear Magnetic

Resonance of ).

1. Principles

Both classical and quantum mechanical descriptions

may be used to describe the NMR phenomenon.

Since each approach has certain advantages, some

features of each will be used in this discussion.

While all atomic nuclei possess charge and mass,

some have magnetic moments and angular momen-

tum and therefore behave as spinning bodies. Nuclei

with even mass and atomic number have no nuclear

spin and therefore do not exhibit the NMR phenom-

enon. These include the common isotopes

Si,

and

O. Of the remaining nuclei, those with odd

mass numbers and even nuclear charge numbers have

spin quantum numbers, I, which are odd-integral

multiples of

, whereas those with even mass numbers

and odd nuclear charge have integral values of I. The

nuclear angular momentum is given by I(h/2p), where

h is Planck’s constant. The magnetic moment, m,is

proportional to the angular momentum by a factor g,

called the magnetogyric ratio, which is characteristic

of a given nucleus:

m ¼ ghI=2p ð1Þ

A set of magnetic quantum numbers given by the

series

m ¼ I ; ðI  1Þ; ðI  2Þ; y; I ð2Þ

describes the allowed values of the magnetic moment

vector. Thus, there are (2I þ1) possible spin states for

a given nucleus. For the simplest case of those nuclei

with I ¼

this corresponds to two spin states. These

spin states are degenerate in the absence of a mag-

netic ﬁeld, but they correspond to states of different

potential energy in the presence of a uniform mag-

netic ﬁeld, H

. For spin-

nuclei the two spin states

correspond to the magnetic moment aligned either

with or against the applied magnetic ﬁeld ðm ¼þ

m ¼

Þ. Inducing and detecting transitions between

these two states form the basis of NMR experiments.

The interaction between the applied magnetic ﬁeld,

, and the nuclear magnetic moment, m, causes pre-

cession of the nucleus about the direction of the ap-

plied ﬁeld. The frequency of precession is given by the

Larmor equation:

¼ g=2pH

ð3Þ

where n

is the Larmor frequency of a nucleus and is

linearly dependent on the strength of the magnetic

ﬁeld, H

. If a small magnetic ﬁeld, H

, is applied

perpendicular to H

, and made to rotate at a fre-

quency of n

, energy is absorbed from the H

ﬁeld

into the spin system. This is the resonance condition.

If H

is rotating at any frequency other than n

,no

net absorption of energy occurs. The Larmor fre-

quency is very different for individual nuclei. For ex-

ample, in a magnetic ﬁeld of 11.75 T the Larmor

frequencies are 500.10 MHz for

H, 470.48 MHz

for

F, 125.75 MHz for

C, 50.67 MHz for

99.35 MHz for

Si, and 202.44 MHz for

P. Al-

though there are a few nuclei whose Larmor frequen-

cies are close enough to overlap frequency ranges, in

990

Nuclear Magnetic Resonance Spectrometry

general only one nuclear species at a time is observed

in a given spectrum.

The separation of the energy levels is small, and at

equilibrium there is only a slight excess of nuclei in

the lower energy spin state as given by the Boltzmann

population distribution. Since NMR experiments are

dependent on this small population to produce a sig-

nal, NMR is relatively insensitive compared with

some other spectroscopic techniques. Once energy

transfer occurs into the spin system, the process by

which the nuclei re-establish a Boltzmann population

distribution is called spin–lattice relaxation. This

process is dependent on the presence of ﬂuctuating

local magnetic ﬁelds. Several common mechanisms of

spin–lattice relaxation include interactions with

neighboring magnetic nuclei, free rotation of a func-

tional group or molecule, the presence of significant

anisotropy in the chemical shift (see Sect. 3), and in-

teractions through scalar coupling (see Sect. 4) with

another rapidly relaxing nucleus. Spin–lattice relax-

ation times, T

, for most common nuclei are in the

range 0 1–20 s, but much smaller or greater T

values

are also possible.

The ease of observing NMR signals for different

nuclei is dependent on I, g, and the natural abun-

dance of the isotope. In general, nuclei with I ¼

produce spectra with the narrowest lines. Nuclei with

possess electric quadrupole moments which can

significantly shorten relaxation times and broaden

resonance lines. For this reason these nuclei are

generally less suitable for traditional NMR experi-

ments. The magnetogyric ratio is a fundamental con-

stant of the nucleus, which determines the frequency

and inherent sensitivity of the nucleus to the NMR

experiment. In general, spin-

nuclei with a high g

value and reasonable natural abundance are easily

observed by NMR spectroscopy. Examples of nuclei

with 100% natural abundance which ﬁt this descrip-

tion are

F, and

2. Instrumentation and Experimental Details

Figure 1 shows a block diagram of a typical NMR

spectrometer. In most modern NMR spectrometers

the magnetic ﬁeld is generated by a superconducting

magnet. The sample is placed into the NMR probe

which is located in the center of a homogenous region

of the magnet. The spectrometer console provides at

least three RF channels which are transmitted into

the probe. The observe channel supplies the observe

RF for the nucleus of interest and receives the signal

back from the sample. The decoupler channel is used

to irradiate another nucleus in the sample other than

the one being observed. The lock circuit conducts a

separate NMR experiment that continually measures

another resonance, typically deuterium (

H), and ad-

justs for minor drift in the magnetic ﬁeld. This en-

sures long-term stability of high-resolution NMR

experiments. Modern spectrometers typically employ

two computers, one for data acquisition and process-

ing and another for spectrometer control.

Both liquid and solid samples are normally dis-

solved in a deuterated solvent such as chloroform-d

which provides a signal for the lock circuit. The liquid

sample is placed in a capped, thin-walled glass tube,

which is typically either 5 mm or 10 mm in diameter,

although microsample probes are also available.

Figure 1

Block diagram of a typical NMR spectrometer. The sample is introduced through the upper stack of the magnet into

the probe. The shims are used to optimize the homogeneity of the magnetic ﬁeld around the sample. Instrument

operation is carried out from the NMR console through the user interface computer.

991

Nuclear Magnetic Resonance Spectrometry

3. Chemical Shifts

As mentioned above, nuclei of a given type do not all

exhibit transitions at exactly the same frequency.

Each nucleus is surrounded by an electron cloud, and

the motion of these electrons creates a secondary

magnetic ﬁeld which tends to oppose the applied

magnetic ﬁeld; chemical shift differences arise be-

cause this local electron density varies depending on

the electronegativity of the atoms attached to the

nucleus and the type of bonding in which it is in-

volved (e.g., single vs. multiple bonds). Thus, the

higher the electron density about a nucleus, the

greater the secondary magnetic ﬁeld, or shielding ef-

fect. The effect of the nearest neighboring atoms or

functional groups is usually most important. It is

these small differences in resonance frequency among

the chemically different nuclei in a sample that help

identify the specific structures present.

The peak positions are determined using an inter-

nal standard which may be either a peak from an-

other material added to the sample or from the

deuterated solvent in the case of

Cor

H NMR.

These positions are reported relative to the position

of a reference material. Although NMR data are

measured in frequency units, Hz, they are usually re-

ported in the dimensionless units of ppm which is

independent of the magnetic ﬁeld strength. Since the

resonance frequencies are directly proportional to

magnetic ﬁeld strength, this is achieved by taking the

difference in frequency between the peak and the

standard (in Hz), multiplying by 10

and dividing by

the operating frequency of the spectrometer, n

. This

dimensionless chemical shift is given the symbol d:

d ¼ 10

ðn

sample

 n

ref

Þ=n

ð4Þ

Using these chemical shifts permits comparisons to

be made of data acquired on instruments of different

magnetic ﬁeld strength. As an example of a typical

spectrum, Fig. 2 shows the

H NMR spectrum of

ethylbenzene. It contains four sets of resonance lines

from low ﬁeld to high ﬁeld, or left to right: a singlet, a

quartet, and another singlet. The last singlet is due to

the protons of tetramethylsilane (TMS), (CH

)

Si,

which is commonly used as the standard for

and

Si NMR. By convention, the chemical shift for

the standard is set to 0 ppm.

4. Proton NMR

Consider the

H NMR spectrum shown in Fig. 2. The

dominant features of the spectrum are the presence of

the singlet at d ¼7.2 ppm (phenyl ring protons), tri-

plet at d ¼1.2 ppm (methyl protons), and quartet at

d ¼2.6 ppm (methylene protons). The neighboring

group affects more than just the chemical shift of a

peak: in the case of the methyl group, for example,

the neighboring methylene group not only affects the

position of the resonance line but causes the peak to

be split into a triplet as well. The explanation for this

phenomenon, called spin–spin splitting, lies in the

fact that the neighboring protons may also be in dif-

ferent orientations with respect to the applied mag-

netic ﬁeld. Since the proton is a nucleus with I ¼

two possible spin states exist for each proton. De-

pending on whether the neighboring nuclear spin is

aligned with (parallel) or opposed to (antiparallel) the

proton at resonance, it will either increase or decrease

the net magnetic ﬁeld experienced by the observed

proton. This gives rise to slight differences in energy

levels and thus different resonance lines. In the case

of the methylene protons of ethylbenzene, three com-

binations of the orientations (mm, mk ¼km, kk)of

the two spins are possible for the neighboring nuclei

producing a 1:2:1 triplet for the methyl protons.

Spin–spin splitting is a primary characteristic of

NMR. These interactions are said to be electron cou-

pled since they operate through the bonding network;

the effect is quite strong over the ﬁrst few bonds but

diminishes greatly thereafter.

The number of lines produced by spin–spin cou-

pling is given by

Multiplicity ¼ 2nI þ 1 ð5Þ

where n is the number of protons in the neighboring

group and I is the spin quantum number. The sep-

aration between the lines of a multiplet is referred to

as the coupling constant, J. This value, expressed in

frequency units (Hz), reﬂects the magnitude of the

spin–spin interaction and is independent of the ap-

plied magnetic ﬁeld. Characteristic coupling con-

stants are of great value in structure elucidation.

Factors that inﬂuence the magnitude of the coupling

Figure 2

Proton NMR spectrum (300 MHz) of ethylbenzene.

Assignments to the protons in the structure are shown.

The internal standard is TMS, which is set to a chemical

shift of 0 ppm.

992

Nuclear Magnetic Resonance Spectrometry

constant include bond order, electronegativity of the

substituents on the coupled nuclei, and bond angles.

To determine if two nuclei are coupled to each

other, a double resonance experiment can be per-

formed. In this type of experiment, two RF ﬁelds are

simultaneously applied to the sample: one is used to

observe the spectrum in the normal manner, and the

other is a perturbing ﬁeld applied at the resonance

frequency of another nucleus that is suspected to be

coupled to the one of interest. If the perturbing (or

decoupling) ﬁeld is sufﬁciently strong the nucleus of

interest appears as if it is no longer coupled. In the

spectrum of ethylbenzene, for example, if the decou-

pling ﬁeld is applied at the resonance frequency of

the methylene protons, the methyl proton triplet col-

lapses to a single peak. Similarly, irradiating at the

center of the methyl proton resonance causes the

methylene proton quartet to collapse to a singlet. The

perturbation of coupled spins to determine connec-

tivity in a molecule is a key feature in a number of

more advanced NMR techniques.

5. Carbon-13 NMR

The sensitivity of the

C nucleus is a factor of ap-

proximately 6000 lower than that of

H owing to its

smaller magnetogyric ratio and low natural abun-

dance of 1.1%. Although

C spectra were obtained

as early as the 1950s (Lauterbur 1957), significant

growth in the application of

C NMR and other

low-sensitivity nuclei did not occur until after the in-

troduction of the pulsed Fourier transform method

(Ernst and Anderson 1966).

The resulting dramatic reduction in the amount of

time necessary for data acquisition allowed

C NMR

spectroscopy to become a routine tool for probing

molecular structure. The rare occurrence of the iso-

tope greatly simpliﬁes

C NMR spectra by eliminat-

ing the spin–spin coupling (

C–

C) that is so

dominant in

H NMR spectra.

Further simpliﬁcation of

C spectra is achieved by

removing the scalar coupling between

C and

Hby

double-resonance decoupling over the

H chemical

shift range. In the resultant spectrum, each line cor-

responds to a magnetically unique carbon atom in the

sample. The relatively large

C chemical shift range,

which is about 20 times that of

H, affords excellent

resolution of even structurally quite similar atoms. As

an example of a typical spectrum, Fig. 3 shows the

C spectrum of ethylbenzene. Note that all six struc-

turally unique carbon atoms give resonances that are

well resolved from each other.

The ability to differentiate structures in

CNMR

is facilitated by the fact that the major factors gov-

erning

C nuclear shieldings are well established. Ex-

perimentally, it has been found that substituent effects

C chemical shifts are largely additive. Semiem-

pirical methods, therefore, are usually successful in

predicting these shifts. The largest variation in

chemical shifts occurs with direct substitution by

highly electronegative elements such as oxygen and

ﬂuorine. The hybridization at carbon also strongly

affects the position of the resonance line. Carbon–

carbon double bonds (CQC), for example, typically

appear in the region 90–160 ppm from TMS, whereas

carbon–carbon single bonds (C–C) usually occur in

the range 0–90 ppm from TMS.

6. Other Nuclei

Prior to 1970 the bulk of NMR studies concerned the

proton. The rest of the interest at that time was di-

vided among the high-sensitivity nuclei

P, and

B. Since then, studies involving virtually all observ-

able NMR nuclei have been carried out. Extensive

compilations of chemical shift data can be found for

the nuclei mentioned above as well as

Si,

195

Pt,

119

Sn, among others.

7. Quantitative NMR

NMR can be readily made quantitative since there

are no response factors or matrix effects that need to

be considered. In principle this means that the area

under each peak in a spectrum is proportional to the

number of nuclei of that type that are present in the

molecular structure. In practice, however, several

factors must be taken into account to obtain quan-

titative data.

In the data acquisition parameters, a sufﬁciently

wide spectral width needs to be used to ensure a uni-

form level of RF power has been applied across the

range of interest. Two other factors can distort the

intensities of the peaks in the spectrum. Perturbations

of the populations can occur when nuclei other than

protons are observed and proton decoupling is used.

Figure 3

Carbon-13 NMR spectrum of ethylbenzene with

assignments of the peaks to the carbons in the structure.

Note that each unique carbon atom produces a different

peak in the spectrum.

993

Nuclear Magnetic Resonance Spectrometry

This is called the nuclear Overhauser effect (NOE).

The NOE can be eliminated by turning the proton

decoupler off during the delay between RF pulses

(gated decoupling) to allow the normal equilibrium

populations to be established. The second factor that

must be considered is the spin–lattice relaxation time

of the nuclei. For quantitative analysis it is necessary

to use a sufﬁcient delay between RF pulses to ensure

that all nuclei are fully relaxed before the next RF

pulse is applied. This may be seconds, or even min-

utes, and could hinder rapid acquisition of spectra. A

small amount of a paramagnetic relaxation reagent,

such as tris(acetylacetonato) chromium(III), is added

to shorten the relaxation times of nuclei other than

protons. Relaxation reagents provide an extremely

efﬁcient pathway for nuclear spins to return to the

ground state through electron–nuclear dipole–dipole

relaxation produced by the unpaired electron of the

reagent. Thus, for quantitative analysis by NMR of

nuclei other than protons, a small amount (50–75 mg)

of a relaxation reagent should be added to the solu-

tion if possible to reduce T

, gated decoupling should

be used to eliminate any residual NOE, and a sufﬁ-

ciently long relaxation delay must be employed to

permit complete relaxation of the nuclei. For protons

it is sufﬁcient simply to use a relaxation delay of ap-

propriate length. Frequently the T

values of the nu-

clei of interest are measured and a relaxation delay of

ﬁve times the largest T

is used.

8. Two-dimensional NMR

A major development in NMR was the introduction

of the concept of two-dimensional NMR spectros-

copy (2D NMR) in the early 1970s (Jeener 1971). In

the standard one-dimensional NMR experiment the

application of a RF pulse results in the nuclear mag-

netic moments precessing about the direction of the

magnetic ﬁeld. This response to the RF pulse induces

a signal in the receiver coil of the probe which is

monitored for a period of time (the acquisition time).

Thus, the detected NMR signal (or free-induction de-

cay, FID) is a function of a single time variable.

Fourier transformation of this signal produces the

normal one-dimensional frequency spectrum.

If the spin system is allowed to evolve under an-

other set of experimental conditions (e.g., the length

of time the proton decoupler is on) for a period of

time prior to detection of the FID, and if this evo-

lution period is varied systematically, the NMR signal

is now a function of two time variables. The data are

collected as a series of FIDs, each of which has the

same acquisition time, t

, but a different evolution

period, t

. Two Fourier transformations are per-

formed on this data matrix, one on the rows of the

matrix (t

), and the second on the columns of the data

). The result of this process is now a matrix (the

two-dimensional spectrum) which is a function of two

frequencies, n

and n

, and for which the intensity of

the signal at each pair of frequencies is a third di-

mension. The specific identity of these frequencies

depends on the two-dimensional experiment per-

formed. Some of the applications include heteronu-

clear and homonuclear correlation spectroscopies in

which pairs of coupled nuclei are identiﬁed through

scalar coupling (both short and long range), J-

resolved spectroscopy in which the coupling patterns

and chemical shifts are separated and shown on dif-

ferent axes, and exchange spectroscopy in which nu-

clei involved in chemical exchange can be identiﬁed.

An example of a two-dimensional heteronuclear

(

H–

C) chemical shift correlation (HETCOR) spec-

trum of ethylbenzene is shown in Fig. 4. The proton

spectrum is shown on the y-axis and the carbon

spectrum is shown along the x-axis. The two-dimen-

sional contour plot shows peaks at the intersections

of directly coupled carbon and proton nuclei. Note

that the

C resonance at 145 ppm has no directly

bonded proton so there is no peak at that position in

the two-dimensional plot.

See also: High-temperature Superconductors, Cup-

rate: Magnetic Properties by NMR/NQR; Nuclear

Resonance Scattering; Solids, Nuclear Magnetic Res-

onance of

Acknowledgment

Paul Donahue and Tom Early are gratefully ac-

knowledged for their assistance with data, ﬁgures,

and helpful discussions.

Figure 4

Two-dimensional heteronuclear (

H–

C) chemical shift

correlation (HETCOR) plot showing connectivities

between carbon nuclei and their directly bonded

protons.

994

Nuclear Magnetic Resonance Spectrometry

Bibliography

Bax A 1982 Two-Dimensional Nuclear Magnetic Resonance in

Liquids. Delft University Press, Dordrecht, The Netherlands

Becker E D 1980 High Resolution NMR, Theory and Chemical

Applications, 2nd edn. Academic Press, New York

Berger S, Braun S, Kalinowski H-O 1997 NMR Spectroscopy of

the Non-Metallic Elements. Wiley, Chichester, UK

Breitmaier E, Voelter W 1987 Carbon-13 NMR Spectroscopy,

High-Resolution Methods and Applications, 3rd edn. VCH,

New York

Croasmun W R, Carlson R M K (eds.) 1987 Two-Dimensional

NMR Spectroscopy, Applications for Chemists and Biochem-

ists, Methods in Stereochemical Analysis. VCH, Weinheim,

Germany, Vol. 9

Crutchﬁeld M M, Dungan C H, Letcher J H, Mark V, Van

Wazer J R 1967 Topics in Phosphorus Chemistry: Vol. 5. P

Nuclear Magnetic Resonance. Wiley, New York

Ernst R R, Anderson W A 1966 Rev. Sci. Instrum. 37,93

Gorenstein D G (ed.) 1984 Phosphorus-31 NMR, Principles and

Applications. Academic Press, New York

Harris R K 1983 Nuclear Magnetic Resonance Spectroscopy, A

Physicochemical View. Pitman, Marshﬁeld, MA

Harris R K, Mann B E 1978 NMR and the Periodic Table.

Academic Press, New York

Jeener J 1971 Ampe

re International Summer School II, Basko

Polje, Yugoslavia

Kalinowski H-O, Berger S, Braun S 1988 Carbon-13 NMR

Spectroscopy. Wiley, Chichester, UK

Lauterbur P C 1957 J. Chem. Phys. 26, 217

Quin L D, Verkade J G (eds.) 1994 Phosphorus-31 NMR Spec-

tral Properties in Compound Characterization and Structural

Analysis. VCH, New York

E. A. Williams

General Electric Company, Niskayuna

New York, USA

Nuclear Resonance Scattering

Nuclear resonance scattering (NRS) with synchro-

tron radiation combines the outstanding properties of

ssbauer spectroscopy (see Mo

ssbauer Spectro-

metry) with those of synchrotron radiation. Since its

ﬁrst observation in 1984 (Gerdau et al. 1985) a rapid

development of the technique and its applications

followed which has just been reviewed (Gerdau and

de Waard 2000). Thanks to the outstanding pro-

perties of third generation synchrotron radiation

sources, NRS has nowadays become an established

spectroscopy on an atomistic scale which is element

and even isotope sensitive and nondestructive. Ap-

plications comprise two main ﬁelds, ‘‘hyperﬁne spec-

troscopy’’ and ‘‘structural dynamics.’’ In hyperﬁne

spectroscopy NRS is complementary to other nuclear

techniques and yields useful information on atomic,

magnetic, and electric properties and structures.

Those ﬁelds of applications beneﬁt most which ex-

ploit the specific properties of synchrotron radiation:

applications to high pressure, to grazing incidence

geometry (surfaces and multilayers), to single crys-

tals, and to very small samples. Structural dynamics

on a ps to ms timescale as free or jump diffusion as

well as rotational motions can directly be measured in

the time domain by nuclear quasi-elastic scattering

techniques. On the fast timescale the (partial) density

of phonon states is directly accessible by nuclear in-

elastic scattering.

1. Principles of Nuclear Resonance Scattering

Nuclear resonance scattering is a resonant x-ray scat-

tering technique in the time domain. It is based on the

ssbauer effect and the scattering proceeds via the

ssbauer level whereas Mo

ssbauer spectroscopy is

an absorption spectroscopy and is normally carried

out in the energy domain. Both are, in principle,

connected by the Heisenberg uncertainty principle.

In case of Mo

ssbauer spectroscopy (see Mo

ssbauer

Spectrometry) a radioactive source provides the

g-quanta which are very sharp in energy (neV-meV)

and by varying the energy one measures absorption

spectra behind the sample. In nuclear resonance scat-

tering an x-ray pulse from a synchrotron radiation

source which is very sharp in time (B100 ps) and very

broad in energy (white radiation) excites the nuclear

levels in the sample. The successive decay of these

levels gives rise to an exponential intensity decay in

time. This decay corresponds to the Lorentzian line in

ssbauer absorption spectroscopy.

There are several techniques utilizing nuclear reso-

nance scattering: nuclear forward scattering (NFS),

nuclear quasi-elastic scattering (NQES), nuclear

Bragg diffraction (NBD), nuclear reﬂectometry (NR,

for the investigation of surfaces and multilayers), nu-

clear small angle scattering (NSAS, measuring for ex-

ample, magnetic domain structures), nuclear inelastic

scattering (NIS), and synchrotron radiation-based

perturbed angular correlation (SRPAC). In the fol-

lowing a short introduction will be given to the main

ﬁelds: nuclear forward scattering, nuclear quasi-elastic

scattering, and nuclear inelastic scattering (see Fig. 1).

1.1 Nuclear Forward Scattering

Nuclear forward scattering (and nuclear reﬂectome-

try) has the closest analogy to Mo

ssbauer spectro-

scopy; in fact, it is its scattering variant. The main

domain is the determination of hyperﬁne parameters

such as internal magnetic ﬁelds (magnetic hyperﬁne

ﬁeld), electric ﬁeld gradients, isomer shifts, and the

Lamb-Mo

ssbauer factor (f

The setup for NFS (see Fig. 1) also looks very

similar to a standard set-up for Mo

ssbauer spectro-

scopy (see Mo

ssbauer Spectrometry). However, while

ssbauer spectroscopy is an absorption spectro-

scopy (one g-quantum is absorbed by one nucleus)

995

Nuclear Resonance Scattering

NFS is a scattering spectroscopy. The ‘‘white’’ syn-

chrotron radiation excites all Mo

ssbauer levels in the

sample and creates a ‘‘coherent collective nuclear

state’’. In the static case this nuclear state will decay

in the forward direction giving rise to an excess of

intensity at delayed times (see Fig. 1, NFS). The

timescale is determined by the lifetime t

of the in-

volved nuclear level. Multiple scattering may inﬂu-

ence the measured time response giving rise to

‘‘dynamical beats.’’ Furthermore, in case of split nu-

clear levels due to hyperﬁne interaction (electric,

magnetic) an additional interference pattern, the so-

called ‘‘quantum beat’’ structure is superimposed.

(a) Dynamical beats

For samples with a large effective thickness,

eff

¼n

, with n

the density of Mo

ssbauer

nuclei, d the geometrical thickness, s

the resonant

ssbauer cross section, and f

the Lamb–Mo

ss-

bauer factor, a speed-up effect is observed and dy-

namical beats (Bessel beats) show up in the time

spectra.

In Fig. 2 the situation is shown for Mo

ssbauer and

NFS spectra. For a thin sample, t

eff

¼1, we have a

single Lorentzian line in the Mo

ssbauer spectrum

which corresponds to an exponential decay in the

NFS spectrum (Fig. 2(a),(b), dotted lines). Increas-

ing the effective thickness (t

eff

¼25) the Mo

ssbauer

line becomes much wider and non-Lorentzian

(Fig. 2(a), solid line). In the NFS spectrum (Fig. 2(b),

solid line) we observe two features: (i) a speed-up,

showing up as an increase in intensity at early

times (from 10

to 10

units in the present example)

and (ii) a dynamical beat structure is superimposed

to the decay. From this beat structure the effec-

tive thickness and by this f

can very precisely be

determined.

NFS

NIS

SRPAC

......

...

........

......

...

......

...

Undulator

Storage ring

Monochromator

(HRM)

Sample

Figure 1

Scheme of the setup for some techniques in NRS: NFS, NIS, and SRPAC. Furthermore, the corresponding time and

energy spectra are schematically shown measured by the detectors for NFS, SRPAC, and NIS. The storage ring is

operated in ‘‘few bunch’’ mode with e.g. 176 ns spacing between adjacent buckets (red bullets). The undulator

produces the well-collimated synchrotron radiation which is monochromized at the nuclear resonances by a high

resolution monochromator, HRM (bandwidth DEEmeV). Finally the radiation impinges the sample. Depending on

the scattering process and the scattering geometry different techniques are exploited. Variable sample environments

allow for the combination of high/low temperature, high pressure, and external magnetic ﬁeld.

996

Nuclear Resonance Scattering

(b) Quantum beats

As is well known (see Mo

ssbauer Spectrometry)

hyperﬁne interaction might split the nuclear levels.

In Mo

ssbauer spectroscopy several absorption lines

reveal this splitting whereas in NRS an interference

pattern, the quantum beat structure, shows up.

In case of quadrupolar interaction the excited nu-

clear state (e.g., in case of

Fe) splits to the 73/2 and

71/2 levels. This gives rise to two absorption lines in

ssbauer spectroscopy (see dotted line in Fig. 2(c)).

The corresponding spectrum in NFS shows up as an

interference pattern of these two transitions with a

single frequency O of the quantum beats (see dotted

line in Fig. 2(d)). Due to the thick sample with an

effective thickness t

eff

¼25, the dynamical beat struc-

ture strongly modulates the quantum beats as an en-

velope. It is clearly seen that the quantum beat

structure is equidistantly spaced whereas for the dy-

namical beat structure the distance of the minima

increases with time. This can be described in a good

approximation by

NFS

ðtÞp

eff

t=t

cos



t=t

 J

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

eff

t=t



ð1Þ

The term cos

OtÞ describes the quantum beats

and the Bessel function J

the dynamical beats. For

comparison the unsplit case is shown as solid lines in

the same ﬁgures. Changing the strength of the hyper-

ﬁne interaction will result in a different splitting

and correspondingly in a different quantum beat

frequency O.

In the case of magnetic hyperﬁne interaction, full

splitting of the nuclear levels occurs giving rise to

six nuclear transitions and correspondingly to six

absorption lines in Mo

ssbauer spectroscopy with

Fe (see Mo

ssbauer Spectrometry). In NRS spec-

troscopy a more detailed interference pattern will re-

sult. Contrary to Mo

ssbauer spectroscopy where the

g-rays from the radioactive source are normally un-

polarized now the x-rays from the synchrotron radi-

ation source are highly linearly polarized. This

feature strongly modiﬁes the time spectra which are

displayed in Fig. 3. The important parameters are the

orientation of the three vectors with respect to each

other, the wave vector

k, the polarization vector

and the hyperﬁne ﬁeld vector

If all three vectors are perpendicular to each other

a simple quantum beat pattern with one single fre-

quency and high contrast results originating from the

two Dm ¼0 transitions (Fig. 3a).

A similar spectrum with only one frequency, how-

ever, with less contrast appears when

k. In this

case the two Dm ¼þ1 and the two Dm ¼1 tran-

sitions interfere independently giving rise to left and

right hand circular polarization (Fig. 3c).

100

0 500 1000 1500

Ω

23.25 MHz

Ω

0 MHz

Calc. intensity (a.u.)

Time after excitation (ns)

-30 -20 -10 0 10 20 30

100

eff

/ line=25

Transmission (%)

Relative energy (Γ

)

eff

=25

Figure 2

Simulations of corresponding spectra of Mo

ssbauer spectroscopy (left panel, with G

¼_/t

) and NFS (right panel) for

thin (t

eff

¼1) and thick (t

eff

¼25) samples. For the single lines (panel a) the thin sample leads to an exponential decay

whereas the spectrum from the thick sample is further modulated by the dynamical beats (panel b). In panel c the

energy spectra of a thick sample (t

eff

¼25) with one (solid line) and two (dotted line) transition lines, respectively, are

shown. In panel d the corresponding time spectra are shown. The sample with the two lines shows in addition to the

dynamical beats the fast, equidistant quantum beat structure with frequency O superimposed (from Gru

nsteudel

1998).

997

Nuclear Resonance Scattering

Finally, for

E all Dm ¼71 transitions interfere

giving rise to a more complicated spectrum which is

sigma polarized (Fig. 3b). The slow overall modula-

tion is caused by dynamical beats due to the ﬁnite

effective thickness.

1.2 Nuclear Quasi-elastic Scattering

Nuclear quasi-elastic scattering measures structural

dynamics on a ps to ms timescale. The coherent and

the incoherent channel can be utilized. In the ﬁrst

case, the coherent channel, the Lamb–Mo

ssbauer

factor has to be greater than zero (f

40). The setup

is the same as in NFS (see Fig. 1). The incoming x-ray

pulse creates a coherent collective nuclear state which

decays in the static case in forward direction. Dy-

namics, for example, the jump of a Mo

ssbauer nu-

cleus from one atomic site to another (in space and

angle, see Dattagupta (1976) for example), destroys

this state. As a consequence no x-ray is scattered in

forward direction, i.e., the measured intensity in the

NFS detector is decreased at later times. We will get

an ‘‘accelerated decay’’ or a ‘‘damping’’ of the NFS

intensity I

NFS

(t) which might be described in a sim-

pliﬁed picture by

IðtÞpI

NFS

ðtÞe

2l

l

ð2Þ

The ﬁrst exponential is related to the van Hove self-

intermediate function with l

being the translational

relaxation rate and the second one to the rotational

correlation function with l

being the rotational re-

laxation rate. In the second case, the incoherent

channel, the scattering is independent of the Lamb-

ssbauer factor, f

. The setup (see Fig. 1) is sim-

ilar as in traditional time differential perturbed an-

gular correlation (see Perturbed Angular Correlations

(PAC)) spectroscopy. Therefore the technique is

called synchrotron radiation-based perturbed angular

correlation, SRPAC. The incoming x-ray pulse selec-

tively excites a single nucleus which decays in

the static case with an angular distribution accord-

ing to the anisotropy parameter, A

. Dynamics, for

.....

...

∆m=

m=0∆

∆

σσ

(a)

(b)

(c)

Time (ns)

Figure 3

Left panel: Measured time spectra of iron (NFS) in case of magnetic hyperﬁne interaction for various alignments of

the hyperﬁne ﬁeld

H with respect to the wave vector k

and the polarization

E. Solid lines are ﬁts according to the full

theory. Right panel: nuclear transition lines with their polarization state, ss linearly polarized, 1 and r left- and

right-hand circular polarized, respectively. Dm—change of magnetic quantum number (reproduced with permission

from Leupold et al. (2003); r Kluwer Academic Publisher).

998

Nuclear Resonance Scattering

example, rotational motion monitored by the electric

hyperﬁne interaction _O, changes this distribution

and gives rise to a damping of the intensity signal.

The scattering intensity can be written as

I ¼ I

t=t

f1  A

ðtÞg ð3Þ

with A

the anisotropy and angular term and G

the

perturbation factor. It reduces in the slow approxi-

mation to

ðtÞpe

l

cos



ð4Þ

and in the fast approximation to

ðtÞpe

ðO

Þt

ð5Þ

Combining both techniques the translational and ro-

tational relaxation rates can be separately extracted.

1.3 Nuclear Inelastic Scattering

Nuclear inelastic scattering is a very recent method in

order to measure directly the (partial) density of

phonon states (Seto et al. 1995; Sturhahn et al. 1995;

Chumakov et al. 1995). The principle setup is shown

in Fig. 1. The synchrotron radiation is monochromi-

zed by a scanning high resolution monochromator

(HRM) with (sub-)meV (1 meV ¼0.24 THz) energy

resolution. At resonance the x-ray pulse creates a co-

herent collective nuclear state as in NFS which decays

either in forward direction or incoherently, due to

internal conversion, spin ﬂop etc., in the entire solid

angle of 4p. This feature gives a simple and effective

method at hand to measure the instrumental function

(detector NFS) in parallel with the inelastic spectrum

(detector NIS). While scanning the HRM the nuclear

resonance can only be excited when at the same time

a phonon is excited or annihilated. In NIS this is a

purely incoherent process with the consequence of a

perfect averaging over the momentum

q. The succes-

sive nuclear decay proceeds as a single exponential in

the entire solid angle covered by the ‘‘4p’’-detector

(detector NIS). In practice only a solid angle of about

p can be achieved.

As an example the energy dependence of nuclear

inelastic scattering of synchrotron radiation in a po-

lycrystalline a-iron sample at room temperature is

shown as a function of energy of the incident radi-

ation (Fig. 4). The central peak corresponds to elastic

scattering. The structure beyond the central peak

shows the energy spectrum of inelastic scattering, ac-

companied either by creation (E40) or by annihila-

tion (Eo0) of phonons. At ambient temperature one

may recognize various contributions to the energy

spectrum, which correspond to inelastic scattering

accompanied by excitation or annihilation of differ-

ent number of phonons. The normalized probability

of nuclear inelastic scattering W(E) can be decom-

posed in terms of a multiphonon expansion (Singwi

and Sjo

lander 1960)

WðEÞ¼f

dðEÞþ

n¼1

ðEÞ

ð6Þ

The Dirac d-function d(E) describes the elastic part of

scattering (zero-phonon term), and the nth term of

the series S

(E) represents the inelastic scattering ac-

companied by creation (annihilation) of n phonons.

The one-phonon term is given by

ðEÞ¼

 gð7E7Þ

E ð1  e

bE

ð7Þ

and the subsequent terms under harmonic approxi-

mation may be found through the recursive relation:

ðEÞ¼

N

ðE

ÞS

n1

ðE  E

ÞdE

ð8Þ

Here b ¼(k

1

with k

the Boltzmann constant, T

the temperature, E

¼_

/2M the recoil energy of a

free nucleus, k the wave vector of the x-ray quantum,

M the mass of the atom. The function g(E) is the

normalized density of phonon states

gðEÞ¼V

ð2pÞ

qd½E  _o

qÞ ð9Þ

-100 -50 0 50 100

0-phonon term

1-phonon term

2-phonon term

3-phonon term

Counts

Relative energy (meV)

Figure 4

Expansion of the energy spectrum of nuclear inelastic

scattering of synchrotron radiation in a-iron in

multiphonon terms. The data were taken at room

temperature. Different symbols show the regions of the

spectra, where the corresponding contributions are

dominant. The lines are the calculations according to

Eqns. (6)–(8) and convoluted with the instrumental

function of the monochromator (reproduced with

permission from Chumakov and Sturhahn (1999); r

Kluwer Academic Publisher).

999

Nuclear Resonance Scattering