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5 Inelastic Nuclear Resonant Scattering

Soon after the discovery of the Mossbauer effect it became clear that nu-clear resonant absorption would be a very sensitive tool for the investiga-tion of atomic motion. The influence of lattice dynamics on the absorptionof -quanta by nuclei was extensively studied by Singwi & Sjolander [1].Visscher predicted that lattice vibrations should manifest as sidebands ofthe absorption line [2]. However, in a conventional Mossbauer experiment itis very difficult to observe such vibrational sidebands. One reason is thatDoppler velocities of hundreds of m/s are necessary to tune the energy of theprobing photon by typical phonon energies1. Another limit results from thebroadening of the resonance line due to the finite lifetime of the vibrationalexcitations. This reduces the peak absorption cross section by 6 to 7 ordersof magnitude compared to the elastic peak. Despite of these difficulties a firstexperiment confirming the ideas of Visscher was performed in 1979, wherethe phonon spectrum and localized modes in TbOx were measured [4, 5, 6]by using the 58-keV radiation of the 159Tb Mossbauer isotope. Doppler shiftsof up to 30 meV were achieved by employing high-speed rotational motionof the radioactive source. The phonon density of states of TbOx obtainedby this method is shown in Fig. 5.1. However, due to the low count ratesin such experiments, this technique could not compete with the upcomingmethod of inelastic neutron scattering and the idea was neglected for a longtime. Instead, lattice dynamics using nuclear resonant absorption was stud-ied indirectly via the Lamb-Mossbauer factor and the second-order Dopplershift, see e.g. [7, 8]. The situation changed drastically with the advent ofa) synchrotron radiation sources that surpassed the brilliance and spectralflux of radioactive sources by several orders of magnitude and b) x-ray opticsthat allowed for an energy tuning around the resonance with a resolutionin the meV range. It was in 1995 when three groups almost simultaneouslyreported the first phonon spectra recorded by inelastic nuclear resonant ab-sorption [9, 10, 11]. The experiments were performed at undulator-basedbeamlines with energy resolutions in the range of 6meV. Since then the tech-nique has made an impressive progress and nowadays phonon spectra areroutinely recorded with sub-meV energy resolution [12]. It soon also became

1 Such high-speed motion was indeed employed in the pre-Mossbauer era to over-come recoil energies and achieve nuclear resonant absorption [3].

Ralf Rohlsberger: Nuclear Condensed Matter Physics with Synchrotron RadiationSTMP 208, 181231 (2004)c Springer-Verlag Berlin Heidelberg 2004

182 5 Inelastic Nuclear Resonant Scattering

2.7 10 4

resbarn

meV0E E50

60

40

20

010 15 20 25

W

W0

1

W (n > 1)n

Fig. 5.1. Phonon spectrumof 159Tb recorded with theradiation from a radioactivesource (Figure adopted fromWeiss et al. [5])

clear that the enormous brilliance of undulator radiation at third-generationsynchrotrons renders this technique sensitive to very small sample volumes.In the following, the basic features of inelastic nuclear resonant scattering willbe described and applications of this method to study vibrational propertiesof thin films and nanostructures will be demonstrated.

5.1 Inelastic Nuclear Resonant Absorption

Resonant nuclei in condensed matter provide a very accurate energy refer-ence with a resolution that is limited only by the natural linewidth of thetransition. If the energy of the incident radiation is off resonance, excitationof nuclei may be assisted by the creation or annihilation of phonons in thesample. Resonance excitation takes place if the energy of the photon plus theenergy transfer involved in the interaction equals the resonance energy E0.This is shown schematically for the case of an Einstein solid in Fig. 5.2. Thephonon energy spectrum of the sample is obtained by tuning the incidentphoton energy relative to the nuclear resonance2 and recording the yield ofnuclear decay products like fluorescence photons or conversion electrons. Thisyield gives a direct measure of the number of phonon states in the samplefrom which the phonon density of states can be derived [11]. A synopsis ofthe experimental method is shown in Fig. 5.3. The pulsed time structureof synchrotron radiation permits the use of timing techniques to efficientlydiscriminate the weak delayed nuclear response from the intense electronicscattering.2 In contrast, in conventional inelastic x-ray scattering [13, 14] one tunes the inci-

dent photon energy relative to the energy of the scattered photon.

5.1 Inelastic Nuclear Resonant Absorption 183S

(E)

2Eph Eph 2Eph E E0Eph

Eph

E0

1phonon 2phononelastic

0

Fig. 5.2. Excitation processesupon resonant scattering froman Einstein solid and the cor-responding excitation probabil-ity density S(E). Direct exci-tation of the transition withresonance energy E0 leads tothe central elastic line in S(E).Additional transitions (dashedlines) include the creation or an-nihilation of one phonon withthe energy Eph. Multiphonontransitions involve excitation ofmore than one phonon like the2-phonon transitions shown asdotted lines. The strength of thevibrational transitions is deter-mined by the number of phononstates at a given energy (DOS)and their thermal occupationnumber

An excited nucleus may decay via two channels: Radiative decay andinternal conversion. The relative probability of these channels is 1/(1 + )and /(1 + ), respectively, where is the internal conversion coefficient.Since for most Mossbauer isotopes > 1, the dominating channel is internalconversion. Depending on the experimental situation, the following decaychannels can be used as the inelastic signal:

Nuclear resonant fluorescenceThe incident photon is absorbed and a photon is reemitted.

Conversion electron emissionThe excitation energy of the nucleus is transferred to the electron shell, sothat an electron is emitted. Due to the small escape depth of the electrons,this method has a potentially high surface sensitivity.

Atomic fluorescence following internal conversionAfter emission of a conversion electron, the electronic recombination leadsto emission of a fluorescence photon3.

The outstanding brilliance of modern synchrotron radiation sources allowsone to probe the dynamical properties of very small sample volumes. In thefollowing some quantitative aspects of this method will be elaborated.

In an incoherent scattering process the total yield of delayed K-fluorescencephotons is given by3 If the nuclear transition energy is below the K-edge, K-fluorescence is not possi-

ble, as it is the case for 119Sn [15, 16]. Since L-fluorescence photons are often toolow in energy to be detected, nuclear resonant fluorescence has to be used then.

184 5 Inelastic Nuclear Resonant Scattering

APD

sample

0

APDHRM

(a)

(b)

(delayed fluorescence)

time after excitation

log

(int

ensi

ty)

Nonresonant scattering (~10 MHz)

Resonant scattering (~10 Hz)

Fig. 5.3. Synopsis of themethod of inelastic nuclearresonant scattering: (a) Timediscrimination of the delayednuclear resonant scatteringafter excitation at t = 0.The ratio of the nonreso-nant prompt photons andthe resonant delayed countsis about 106. (b) Scheme ofthe experimental setup withhigh-resolution monochroma-tor (HRM) and detection offluorescence quanta over thesolid angle with a large-areaavalanche photodiode (APD).The detector in forward di-rection monitors the energydependence of the elasticallyscattered quanta which givesthe resolution function of thesetup

I(E) = I0 KK0

(E) , (5.1)

where I0 is the incident flux, the effective area density of the nuclei, and Kthe K-fluorescence yield. The ratio of the nonradiative (internal conversion)linewidth K to the natural linewidth 0 of the nuclear transition is given by

K0

=K

1 + , (5.2)

where and K are the total and partial internal conversion coefficients,respectively. (E) is the cross section for nuclear resonant absorption of aphoton with energy E:

(E) =

2(E0)0 S(E E0) , (5.3)

where (E0) = 0 is the nuclear absorption cross section at the resonanceenergy E04. S(E) is the normalized probability of absorption per unit energyinterval at the energy E that depends also on the incident wavevector k0.According to Singwi & Sjolander [1] it can be written as:

S(k0, E) =12

0

dt eiEt0 t/2 F (k0, t) , (5.4)

4 0 is tabulated for all Mossbauer isotopes in Table A.1 of the appendix.

5.1 Inelastic Nuclear Resonant Absorption 185

with

F (k, t) = eikr(0) eikr(t) T . (5.5)The angular brackets indicate thermal averaging over all initial lattice states.The function F (k, t) is often referred to as the self-intermediate scatteringfunction. It describes the correlation between the positions of one and thesame nucleus at different moments separated by the time interval t. Theconcept of the self-intermediate scattering function provides a very generaldescription of the influence of lattice dynamics on resonant absorption.

It should be noted that S(k0, E) does not depend on the direction of thewavevector kS of the scattered photon and is thus independent of the mo-mentum transfer q = kS k0. However, inelastic nuclear resonant absorp-tion depends on the direction of the incident wavevector k0 and is thereforeanisotropic in general [17, 18]. The anisotropy vanishes only in the sphericalsymmetric limit, i.e., for cubic crystals and polycrystalline samples. In thiscase the true phonon density of states can be determined from the measuredS(E).