Magnetic dynamics of fine particles studied by inelastic neutron scattering

Journal of Magnetism and Magnetic Materials 221 (2000) 10}25

Magnetic dynamics of "ne particles studied by inelastic neutron scattering M.F. Hansen *, F. B+dker , S. M+rup , K. Lefmann
, K.N. Clausen
, P.-A. Lindga rd
Department of Physics, Building 307, Technical University of Denmark, DK-2800 Lyngby, Denmark
Department of Condensed Matter Physics and Chemistry, Ris~ National Laboratory, DK-4000 Roskilde, Denmark

Abstract We give an introduction to inelastic neutron scattering and the dynamic scattering function for magnetic nanoparticles. Di!erences between ferromagnetic and antiferromagnetic nanoparticles are discussed and we give a review of recent results on ferromagnetic Fe nanoparticles and canted antiferromagnetic a-Fe O nanoparticles.  2000 Elsevier   Science B.V. All rights reserved. PACS: 75.50.Tt; 76.60.Es; 76.50.#g; 78.70.Nx Keywords: Fine particles; Magnetic resonance; Relaxation e!ects; Inelastic neutron scattering

1. Introduction The magnetic properties of "ne particles (&10 nm) are interesting from a basic research point of view and due to the numerous present and potential future technological applications [1]. When a particle is su$ciently small, it consists of a single magnetic domain with the magnetic moment l"M (¹)<, where M (¹) is the saturation   magnetisation at the temperature ¹ and < is the particle volume. The direction of l is governed by the magnetic anisotropy energy, which is usually

* Correspondence address: University of California, San Diego, Center for Magnetic Recording Research, 0401, 9500 Gilman Drive, La Jolla CA 92093-0401, USA; Fax: #1-858534-2720. E-mail address: [email protected] (M.F. Hansen).

approximated by E "K





dl dl "c l; !B !g #h(t) ,   dt dt

0304-8853/00/$ - see front matter  2000 Elsevier Science B.V. All rights reserved. PII: S 0 3 0 4 - 8 8 5 3 ( 0 0 ) 0 0 3 7 2 - 3

(2)

M.F. Hansen et al. / Journal of Magnetism and Magnetic Materials 221 (2000) 10}25

which was "rst used by Brown [3]. Here, c "gk / is the electron gyromagnetic ratio,  B ,*E /*l is the e!ective anisotropy "eld, g is   the damping constant and h(t) is a random noise term accounting for the e!ect of thermal agitation. At low temperatures, the magnetic moment will mainly be con"ned near the energy minima. The average time between jumps over the energy barrier (the superparamagnetic relaxation time) is given by




E q"q exp
,  k¹

(3)

where ¹ is the temperature and k is Boltzmann's constant. For E /k¹'2.5, the prefactor q can be
 expressed as

 




 


M (¹)  k¹  (p M (0)< 1   #g q "  M (0)  4 E c g E
  
k¹ , (4) ; 1# E
where g ,gc M (0) is the dimensionless damping    factor [4]. The variation of q with temperature is  small compared to that of the exponential function and q is therefore often considered as a constant.  From this expression is seen that large values of the damping factor (g +1) correspond to small values  of q .  According to Eq. (2), the magnetic moment performs a damped precession in the e!ective anisotropy "eld between jumps over the energy barrier. The life time of a precession state is related to the damping constant, and the precession frequency (+c 2E /M (0)<) is comparable to the value of 
 q\. These transverse #uctuations of the magnetic  moment near an energy minimum are also called collective magnetic excitations [5,6]. A measurement of the magnetic properties of "ne particles will be sensitive to #uctuations of the magnetic moment on a characteristic observational time scale t . Therefore, the most reliable dynamic 
 characterisation of "ne particles is obtained by combining experimental techniques with di!erent time scales, such as DC and AC magnetisation measurements (t &10}10\ s), muon spin re
 laxation (t &10\}10\ s), MoK ssbauer spectro
 scopy (t &10\}10\ s) and inelastic neutron 





11

scattering (t &10\}10\ s). Inelastic neutron 
 scattering (INS) has only been used for studying the dynamics of "ne particles in very few cases. The goal of the present paper is therefore to give an introduction to INS on "ne particles in order to illustrate the information that can be obtained by this technique. Such studies are of fundamental interest because INS can be sensitive to #uctuations on the time scale of q and therefore it may  be possible to study the superparamagnetic relaxation and the damped precession simultaneously. The paper is organised as follows. In Section 2, we give a brief introduction to the theory of neutron scattering. In Section 3, we discuss special features of the scattering function for magnetic nanoparticles. For simplicity, we mainly discuss the theory for ferromagnetic particles with uniaxial magnetic anisotropy. Di!erences between ferroand antiferromagnetically ordered particles are highlighted. In Section 4 are given a few experimental considerations concerning the sample amounts and composition and the choice of experimental parameters. In Section 5, we discuss experimental studies of ferro- and antiferromagnetically ordered "ne particles, and "nally we give some conclusions and outlook in Section 6.

2. Basic principles of neutron scattering Neutrons can be scattered by both nuclei and magnetic moments. A neutron experiment is performed by sending neutrons with the initial momentum p " k into the sample and detecting scattered neutrons with the "nal momentum p " k . Experimentally, this can be realised dir  ectly in a triple-axis spectrometer or indirectly in a time-of-#ight spectrometer (see, e.g. Ref. [7]). The scattering process is characterised by the transfer of momentum, q, (k !k ), and energy, e, 

u, (k!k)/2m , where m is the neutron    mass. The intensity of scattered neutrons is given by the folding I(q,u)JS(q,u)R(q,u),

(5)

where S(q,u) is the scattering function and R(q,u) is the instrumental resolution function. General introductions to the theory of neutron scattering

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M.F. Hansen et al. / Journal of Magnetism and Magnetic Materials 221 (2000) 10}25

can be found in Refs. [7}9]. The scattering function is essentially the Fourier transform in time and space of the appropriate nuclear and magnetic correlation functions. Time-independent contributions to the scattering function give rise to elastic (u"0) contributions that are mostly related to structural scattering. Dynamic processes give rise to inelastic contributions to the scattering function, due to scattering by, e.g. phonons or spin waves (magnons). For nuclear scattering, the correlation between the scattering nuclei mainly gives rise to scattering when q coincides with a reciprocal lattice vector. The magnetic contribution to the scattering function is essentially given by



S (q,u)J





¹? exp(iR ) q!iut) , G \ G?

;1s? (0)s?(t)2dt,  G

(6)

where R is the position of the ith moment s , G G a denotes the x, y or z component, 122 denotes a statistical average and ¹? "1!(q?/"q"). The , function ¹? accounts for the fact that only compo, nents of s perpendicular to q can contribute to the magnetic scattering of the neutrons. For a polycrystalline sample with randomly oriented crystals ¹? averages to a constant (") and will not be ,  important for the relative intensities of the magnetic contributions in zero applied "eld, which will be discussed below. We therefore omit this factor in the following. Below, we discuss the correlation functions for magnetic nanoparticles and evaluate the corresponding scattering functions.

3. The scattering function for magnetic nanoparticles 3.1. Coherent rotation and spin-wave gap The magnetisation reversal of a single domain particle may take place via di!erent modes. For large particles, it can be energetically favourable to perform an incoherent rotation via the curling mode [10]. In the present work, however, we will only consider coherent dynamics of the particle spin-systems, i.e. during a magnetisation reversal

the relative directions of the spins in a particle will be "xed. This is the most favourable reversal mode for very small particles. Before we discuss the dynamic scattering function of nanoparticles, it is appropriate to consider the e!ect of the "nite particle size on the phonon and magnon dispersion relations. In bulk materials, the dynamic scattering function can have contributions from scattering by phonons and magnons giving rise to inelastic peaks at u"$e (q ) and 

u"$e (q ), respectively, where the wave vector  q is the representation of q in the "rst Brillouin zone (for notational simplicity we consider only one dimension below). The low-energy excitations due to phonon scattering are given by the acoustic dispersion relation, u (q )" c q , where c is the sound    velocity. A typical sound velocity in solids is c +5  km/s ( c +33 meV As ). For ferromagnetic (FM)  materials, the low-energy spin-wave dispersion curve is quadratic in q , i.e. u$+(q )"C#Dq . For  metallic iron, Shirane et al. [11] have obtained C"0.15 meV and D"281 meV As . For antiferromagnetic (AFM) materials, the corresponding spin-wave dispersion is linear in q , i.e.

u$+(q )+ c q . For hematite (a-Fe O ), Sam    uelsen and Shirane [12] have obtained c +25}30  km/s ( c +165}200 meV As ) depending on the  direction relative to the crystal [111] axis. The described dispersion relations are illustrated in Fig. 1. For an isolated magnetic nanoparticle with diameter d, the "nite size results in an upper limit j +2d on the wavelength of spin waves and

 phonons. Hence, there is a cut-o! in the dispersion curves at q +2p/j [13], below which it is not



 possible to excite or de-excite spin waves and phonons. This cut-o! is illustrated in Fig. 1 by the vertical dotted line for d"16 nm. In the studies performed so far, this implies that the excitation energies for phonons and magnons are outside the experimental energy window. Hence, the particle can be considered as a `super spina, S" s , with G G constant length S. Here we have neglected thermal high-energy magnetic excitations at larger q-values that reduce S, but if the experiment is performed at low temperatures compared to the Curie temperature, this assumption is valid. Furthermore, the value of j related to phonons is well de"ned for

 a powdered sample, but it may not be well de"ned

M.F. Hansen et al. / Journal of Magnetism and Magnetic Materials 221 (2000) 10}25

Fig. 1. A typical low-energy phonon dispersion curve for a solid and the low-energy magnon dispersion curves for hematite (c , and c are the perpendicular and longitudinal spin-wave vel, ocities with respect to the [111] axis) and metallic iron. The vertical dashed line corresponds to q "q for a particle dia  meter of d"16 nm.

for particles embedded in a solid matrix because the phonons may then extend beyond the particles. The spin-wave gap implies that the low-energy dynamic magnetic scattering is insensitive to small changes of q. Therefore, the magnetic scattering function for a system of ferromagnetic magnetic nanoparticles can be separated in a structural part and a dynamic part [9] S (q,u)"D(u) ) u s?(q) ) F?(u), (7)

 ?VWX where D(u)"[1!exp(! u/k¹)]\ is the detailed balance factor, s?(q) are the magnetic susceptibilities and F?(u) are the spectral functions, F?(u),(2p)\1S?2\





1S?(0)S?(t)2e\ SR dt, (8)

\ ful"lling > F?(u) du"1. For uk¹ (1 \ K & 0.09 meV), the detailed balance factor ful"lls

uD(u)+k¹. For `narrowa spectral functions this implies that the energy integrated intensity of the dynamic scattering function is S(q)" S \ (q,u) du+k¹ ) s?(q),k¹ ) s(q). If the experi? mental energy integration is su$cient, S(q) is the measured intensity in a di!raction experiment con-

13

taining information about the magnetic ordering inside a particle (Bragg scattering), the shape and size of a particle (form factor of a particle) and the correlations of the particles in the sample. Below, we discuss the shape and temperature dependence of the dynamic magnetic scattering function for ferromagnetic particles in zero external "eld. This has been treated by a number of authors starting from Eq. (2) [14}17] and more recently using quantum mechanics [18]. At low temperatures (k¹E ), there are two distinctly di!erent
types of dynamics, namely superparamagnetic #uctuations (longitudinal dynamics) and collective magnetic excitations (transverse dynamics). In this case, it is therefore convenient to consider the longitudinal and transverse spectral functions, de"ned by F(u),FX(u) and F,(u),[FV(u)#FW(u)]/2. It should be noted that the energy integrated dynamic intensities, I and I,, of the full scattering O O functions are proportional to the average square values of the longitudinal and transverse components of the superspin, respectively. This is comes from the fact that the scattered intensity is proportional to the autocorrelation function of the superspin (cf. Eq. (6)). At high temperatures (k¹E ), the
two types merge into a single type (isotropic #uctuations). 3.2. Longitudinal dynamics We "rst consider a simple classical picture of the longitudinal dynamics of the particle moment. Given an ensemble of particles with the easy direction along the z-direction and writing the initial con"guration as SX(0)"S , the statistical ensemble  average of SX(0)SX(t) has decreased at a time t*0 later due to superparamagnetic relaxation according to 1SX(0)SX(t)2"S exp(!t/q). By Fourier  transformation of this expression, we obtain 1 C F(u)" , p u#C

(9)

where C"1/q is the HWHM of the Lorentzian pro"le. For a uniaxial ferromagnet, the energy integrated intensity is given by I "C ) 1S 2"C ) S1cosh2, $+ 

(10)

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M.F. Hansen et al. / Journal of Magnetism and Magnetic Materials 221 (2000) 10}25

where C is a proportionality constant and the subscript FM stands for q"0 or that q is at a FM Bragg re#ection. Hence, the correlations in the zdirection give rise to a quasielastic peak (centred at u"0) with a width inversely proportional to the superparamagnetic relaxation time. The area of this peak is proportional to the average value of the squared projection of the particle magnetic moment onto the easy axis. Eq. (9) also enables us to understand the importance of the instrumental resolution, C , for the characteristic observation time  of INS . If qC\, the broadening due to magnetic  #uctuations cannot be resolved and the component is called `resolution limiteda. In contrast, for #uctuations with time scales comparable to C\ a con  siderable broadening can be detected. The longitudinal dynamic scattering function has also been calculated in a more sophisticated theoretical framework using the appropriate Fokker}Planck equation for the dynamics of the magnetic moment. These more detailed calculations only give small corrections (maximum 7%) to the Lorentzian line shape derived above [16]. 3.3. Transverse and isotropic dynamics For the transverse dynamics, it is relevant to consider the correlation function 1S>(0)S\(t)2, where S!(t),SV(t)$iSW(t). In the classical picture (Eq. (2)), the transverse motion of the magnetic moment is a (damped) precession. The frequency of this precession motion depends on the precession angle h and for a ferromagnetic material with uniaxial anisotropy it is given by u "c B ,    where 2K B " cos h,B cos h. )  M 

(11)

The precession motion of the magnetic moment in the anisotropy "eld is analogous to that of a single-domain ferromagnetic particle with zero anisotropy in an externally applied "eld and it is therefore often referred to as intrinsic ferromagnetic resonance. Neglecting the damping of the motion, we can write S\(t)"S exp($iu t), where ,  S ,[S>(0)S\(0)] and the plus and minus in the , exponential function refer to clockwise or counter-

clockwise precession. Then, the correlation function becomes 1S>(0)S\(t)2"S exp ($iu t). ,  After Fourier transformation, two inelastic peaks are obtained at u "$c B with an integrated    intensity proportional to the square of the average transverse component of the particle magnetic moment. Note that the value of u decreases with  increasing temperature due to the increase of the average value of h. The scattering cross-section given above is analogous to that of a spin wave with in"nite wavelength (because the dynamics is coherent). In this qualitative treatment, we have neglected the "nite life time of a precession state (damping) and we have not considered the distribution of precession frequencies. The transverse dynamic scattering function, including the thermal distribution of precession frequencies, has been calculated analytically by several authors for the case of zero damping [15}17] and numerically for a "nite damping constant [17]. The zero-damping result corresponds to the spectral function (valid for "u")u ,c B ) )  )






1 u u F,(u)" 1! exp p , N(p)u u u ) ) )

(12)

where p,E /k¹ and the normalisation factor is
N(p)"(2#p\) exp(py) dy!p\exp(p). The  spectral function has been plotted in Fig. 2 for typical experimental parameters. It has a peak at u"u [1!p\], and for larger values of ) p a FWHM of u 2 [p!p]\. For the case of ) a "nite damping constant, there is no analytical expression for F,(u). For small damping, a folding of Eq. (12) with a Lorentzian may be appropriate. A more phenomenological approach is to use the transverse spectral function for a damped harmonic oscillator [19] 1 2cu  F,(u)" . p (u!u )#4cu 

(13)

For cu , this expression corresponds to two  Lorentzian lines positioned at $u with HWHM  equal to c. For a uniaxial ferromagnet, the energy integrated intensity of the transverse scattering function is given by I, "C ) 1S 2"C ) S1sinh2, $+ ,

(14)

M.F. Hansen et al. / Journal of Magnetism and Magnetic Materials 221 (2000) 10}25

Fig. 2. The transverse dynamic scattering function for zero damping (Eq. (12)) plotted as a function of u/u for K
where C is the same proportionality constant as before. The smearing of the spectral function due to the angle-dependent anisotropy "eld is called inhomogeneous smearing and gives rise to an asymmetric broadening of the inelastic peaks towards zero energy transfer. The broadening due to the "nite lifetime of the precession states is called homogeneous and gives rise to a symmetrical smearing of the inelastic lines. For many practical cases, the homogeneous smearing is small compared to the inhomogeneous smearing, and in these cases Eq. (12) is a good approximation to the exact line shape [17]. The homogeneous broadening increases with temperature [17,18] and for any material there will be a cross-over at a temperature ¹H, where the homogeneous broadening begins to dominate over the inhomogeneous broadening. In that case, Eq. (12) is clearly not applicable and it may be better to use Eq. (13). It should be noted that this cross-over may occur for k¹H(E if the
damping constant is large, but it can also occur for k¹H'E if the damping constant is small [17].
This could suggest that it may be possible to observe two distinct types of dynamics, even for k¹'E . According to Raikher and Stepanov's
calculations [17], the linewidth initially increases with temperatures ¹)E /k up to a value of
0.7 u . For temperatures E /k(¹(¹H, the )


15

linewidth is almost constant, and for ¹'¹H, it increases linearly with ¹. The calculations show that ¹H&10 E /k for g &10\ and that ¹H&
 E /k for g &10\.
 The existence of a value of ¹H larger than E /k
has been criticized by WuK rger [18], who "nds that thermal #uctuations make this behaviour impossible. He considers the homogeneous broadening as the main source of the smearing of the inelastic lines. According to his quantum mechanical calculations, the homogeneous broadening diverges at ¹"E /k and the two types of dynamics merge
into a single isotropic dynamics, approximately characterised by a Gaussian line shape centred at

u"0 with a width given by C "2\ u % ) (1!E /k¹)\ (HWHM " (2 ln 2) ) C ) [18].
% Hence, according to his calculations, there will be no increase of the linewidth with increasing temperatures, when k¹'E .
It should be noted that a particle size and shape distribution may give rise to additional broadening due to variations of M and K and also that  a broadening may result from inter-particle interactions giving rise to an interaction "eld that needs to be added to the anisotropy "eld. It is not possible to say if these e!ects mainly give rise to homogeneous (symmetric) or inhomogeneous (asymmetric) broadening of the lines. Other types of anisotropy than uniaxial or other types of magnetic ordering will change the details of Eq. (12), but the qualitative behaviour will be essentially the same. 3.4. Antiferromagnetic and ferrimagnetic materials We now discuss the description for antiferromagnetic and ferrimagnetic materials. For simplicity, we restrict ourselves to magnetic systems that are described by two sublattices a and b with essentially collinear sublattice magnetisations M "gk <\S and M "gk <\S .

It is only possible for the Fourier sum in Eq. (6) to attain large values near AFM re#ections and FM (" structural) re#ections (including q"0). Below, we consider the energy integrated intensities, S(q), at these re#ections for the longitudinal and transverse #uctuations. For simplicity, we restrict ourselves to the case of zero external "eld. When FM and AFM re#ections are considered,

16

M.F. Hansen et al. / Journal of Magnetism and Magnetic Materials 221 (2000) 10}25

this corresponds to calculating the sum in Eq. (8) for the e!ective superspins S (t),S (t)#S (t) $+
and S (t),S (t)!S (t), respectively. We de"ne $+
the longitudinal and transverse components of the superspins as earlier (S ,SX(0) and S ,  , [S>(0)S\(0)], where S!(t),SV(t)$iSW(t)). It is also convenient to introduce r by the de"nition M ,r ) M and de"ne S,"S "#"S ".

The derivation of the longitudinal dynamics is essentially the same (Eq. (9)). The integrated intensities at FM and AFM re#ections are proportional to 1S 2 and 1S 2, respectively. Simple calcu$+ $+ lations give that I "C ) (1#r)S1cos h 2, (15) $+ I "C ) (1!r)S1cos h 2, (16) $+ where h is the angle between S and an easy direction. In order to compare the intensities with those of a pure ferromagnet, it is convenient to use S"S (1#"r") in the equations given above. It is seen that r"1 (pure ferromagnet) leads to the expected I "C ) S1cos h2 and I "0, and $+ $+ that r"!1 (pure antiferromagnet) leads to I "0 and I C ) S1cos h2. $+ $+ The treatment of the transverse yuctuations is much more involved due to the fact that the resonance equation (Eq. (2)) has to be replaced by an equation for each of the magnetic sublattices, where the anisotropy "eld in each of the equations is replaced by an e!ective "eld including the coupling between the sublattices. The resulting two dynamic modes of the electron magnetic resonance equations generally correspond to precession motions of the sublattices, possibly with slightly di!erent opening angles (see, e.g. Refs. [20,21]). These are generally characterised by S>(t)"p ) S>(t), (17)
where p is real constant that can attain both positive and negative values. The integrated intensities, I, and I, , at FM and AFM re#ections are $+ $+ proportional to 1S 2 and 1S 2, respectively. $+, $+, By simple calculations, we "nd that I, "C ) (1#p)S1sin h 2, $+ I, "C ) (1!p)S1sin h 2. $+

(18) (19)

Note the special cases, p"!1 and p"#1, where all the inelastic magnetic intensity is concentrated at AFM and FM re#ections, respectively. For the case where the angle between the sublattice magnetisations is constant (e.g. p"!1 and r"!1), the amplitude S, of the oscillations is independent of the exchange coupling and is simply determined by the anisotropy energy, i.e. I, "C ) S1sinh2. Modes with p'0 involve $+ #uctuations of the angle between the sublattice magnetisations. These so-called exchange modes have higher resonance frequencies and the square amplitude of the oscillations is very small due to the fact that the exchange coupling energy is much larger than the anisotropy energy. This generally makes the experimentally observed intensity for these modes negligible. We now brie#y consider the longitudinal and transverse dynamic scattering functions for di!erent simple two-sublattice magnetic structures. For a simple two-sublattice antiferromagnet, r"!1 and all the magnetic intensity due to the longitudinal component of the superspin is found at the AFM re#ections (cf. Eqs. (15) and (16)). By solving the electron magnetic resonance equations at low temperatures, it has been shown (see, e.g. Ref. [20]) that the two transverse resonance modes are degenerate with u +$c [2B B ], where   ) # B "K/M , M is the sublattice saturation mag)   netisation, and B is the exchange "eld. Note that # this resonance frequency is considerably higher than the pure FM resonance frequency c B due to  ) the fact that B B . As for ferromagnetic mater# ) ials, u will decrease with increasing temperature.  It has been shown that the two modes correspond to p+!1$[2B /B ] [20]. This implies that ) # most of the transverse dynamic magnetic intensity is found at AFM re#ections. Insertion of p in Eqs. (18) and (19) gives that I, /I, "B /B (1) in $+ $+ # ) agreement with more detailed calculations [9,22]. For ferrimagnetic materials, the magnetic intensity related to the longitudinal component of the sublattice magnetisations is observed at both AFM and FM re#ections because rO!1 (cf. Eqs. (15) and (16)). The treatment of the transverse dynamics is more complex. In the "rst resonance mode, which is similar to FM resonance, the sublattice magnetisations are collinear (same opening angle, h) and

M.F. Hansen et al. / Journal of Magnetism and Magnetic Materials 221 (2000) 10}25

the precession frequency is given by u "  c B , where c and B are e!ective values     that are independent of the exchange "eld. For the simple case of c "c "c , the resonance fre
 quency is u"c (B !rB )/(1#r) (see, e.g. Ref.  ) )
[21]). For this mode p"r ((0) and the resonance frequency is slightly enhanced compared to that of a FM material. From a comparison of Eqs. (18) and (19) and Eqs. (15) and (16) is seen that the dynamics of this mode is equivalent to that of a ferromagnet, except that the dynamic intensities are distributed among the FM and AFM re#ections. In the second resonance mode, the precession angles are appreciably di!erent and the precession frequency is almost exclusively determined by the exchange coupling. The value of p is positive (exchange mode) and can change signi"cantly with the sublattice gyromagnetic ratios and saturation magnetisations (see, e.g. Ref. [21]). However, as discussed in the beginning of this section, the amplitude of the #uctuations is small leading to a negligible experimentally observed intensity. Finally, we consider the canted antiferromagnetic structure, which is relevant for the discussion of hematite nanoparticles. The magnetic structure of hematite is essentially antiferromagnetic (r+!1), but the anisotropic Dzialoshinskii exchange interaction energy introduces a slight canting (+0.13) of the sublattice magnetisations with respect to the collinear antiferromagnetic structure. This canting is so small that essentially all the intensity due to scattering on the longitudinal component of the sublattice magnetisations is at the AFM re#ections. In nanoparticles, the anisotropy can be dominated by a large negative uniaxial anisotropy K (easy  plane), corresponding to the anisotropy "eld B , ) and a uniaxial anisotropy K in the easy plane,  corresponding to the anisotropy "eld B . The "rst ) solution of the resonance equations * the so-called low-frequency (LF) mode * is equivalent to the solution for the pure antiferromagnet, i.e. the frequency is given by u +c [2B B ] [23], *$  # ) p+!1 and the ratio of the integrated intensities for this mode at the AFM and FM re#ections is I, /I, +B /"B " (1) [22]. The second $+*$ $+*$ # ) solution * the so-called high-frequency (HF) mode * has the frequency u +c [2B (B # &$  # ) "B ")#B ], where B is the Dzialoshinskii ) " "

17

exchange interaction "eld [23]. The HF mode has p"1 and involves #uctuations of the angles between the sublattice magnetisations (exchange mode). As discussed in the beginning of this section, this makes the amplitude of the #uctuations very small. The integrated intensity is therefore much smaller than that of the LF mode. If observable, this mode should be seen at both AFM and FM re#ections [22]. Insertion of typical values for hematite [23,24] in the expression for the precession frequencies gives that u +0.3 meV and *$

u +0.6 meV. &$ From the discussion given above, it is seen that a rich variety of phenomena can arise in materials with collinear or canted antiferromagnetic structures. For such materials, it is therefore important to consider the possible resonance modes in order to make a reliable physical interpretation of experimental observations. It should be noted that the resonance frequencies are usually enhanced in these materials due to the occurrence of the exchange "eld in the resonance equations. This enhancement makes the separation between the two inelastic peaks related to the transverse magnetic dynamics larger and hence more easily separated from the longitudinal magnetic dynamics than in FM materials.

4. Experimental considerations The requirements for sample amounts are larger for neutron scattering than for most other experimental techniques. For powdered samples, the optimum sample volume is typically several cm. One also needs to pay attention to sample constituents that give rise to large unwanted scattering. A primary concern is to avoid the presence of hydrogen (e.g. in water), which has a large incoherent scattering cross-section, that signi"cantly reduces the signal-to-noise ratio for nanoparticle samples. One also has to choose experimental parameters carefully to maximize the intensity due to magnetic scattering. For ferromagnetic particles, the magnetic intensity is located at and near the structural Bragg peaks and near q"0. If the particle moments are uncorrelated, the magnetic intensity near q"0 is given by the square of the magnetic form

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M.F. Hansen et al. / Journal of Magnetism and Magnetic Materials 221 (2000) 10}25

factor of a particle (Eq. (6) with t"0 and summing over the atomic magnetic moments of a single particle), which extends out to about q&2p/d. Hence, by choosing an appropriate (low) value of q, it may be possible to have a signi"cant magnetic contribution to the scattered intensity. For AFM particles one has to choose a q-vector corresponding to an AFM re#ection, since practically all the magnetic intensity is found there. For ferrimagnetic particles, it is possible to use both AFM re#ections and structural re#ections. The AFM re#ections have the advantage that the structural contributions to the scattering, including possible inelastic contributions from phonons, are avoided. As discussed earlier, the interpretation of the experimental observations becomes more involved for non-ferromagnetic substances as one has to consider the relevant magnetic resonance modes in order to "nd the intensities and resonance frequencies. One should also have in mind that the Bragg re#ections are size-broadened and that this also reduces the signal-to-noise ratio.

5. Experimental studies To our knowledge, studies of mainly two types of magnetic nanoparticles have been published so far, namely ferromagnetic nanoparticles of Fe prepared by co-sputtering with Al O and canted antifer  romagnetic nanoparticles of a-Fe O prepared by   thermal decomposition of Fe(NO ) ) 9H O. Be  low, we brie#y discuss the observations on these two systems with a focus on the dynamic scattering functions. 5.1. Fe nanoparticles in Al O   The investigations of Fe-nanoparticles have mainly been focused on one sample, which has been studied by small-angle X-ray and neutron scattering (SAXS/SANS) [25}28], conventional inelastic neutron scattering [26,29,30] and recently neutron spin echo (NSE) [31,32]. SAXS and SANS studies have shown that the Fe particles have a diameter of about 2 nm and an average centre-to-centre separation of about 4 nm corresponding to particle volume fraction of 20%. SANS studies and INS

Fig. 3. The SANS intensity of 2 nm Fe particles for di!erent temperatures after correction for substrate scattering. (Reproduced with permission from Ref. [28].)

studies using a large energy window have shown that the particles consist of a ferromagnetic core with a structure similar to that of bulk a-Fe and an amorphous shell with a mixture of Fe and Al O   or iron oxides. The spins of the Fe-atoms in this shell progressively align with the ferromagnetic core, when the temperature is lowered, i.e. the magnetic volume of the particles is temperature dependent [28,30]. The q-dependence of the SANS intensity is shown in Fig. 3 for di!erent temperatures. The temperature dependence of the dynamic scattering function has been studied in a wide temperature range from 25 to 500 K using INS with energy resolutions (FWHM) of 0.013 and 0.020 meV corresponding to the observation times 1;10\ and 7;10\ s, respectively. These measurements were performed at the time-of-#ight spectrometer IN5 at Institut Laue-Langevin (France) and at the triple-axis spectrometer 4F at  laboratoire LeH on Brioullin (France), respectively. The data were obtained at small angles, where there is a signi"cant magnetic intensity due to the particle form factor. The analysis was performed using

M.F. Hansen et al. / Journal of Magnetism and Magnetic Materials 221 (2000) 10}25

Fig. 4. The Gaussian linewidth, C , of the quasielastic feature as % a function of temperature at di!erent q values for 2 nm Fe particles. The line is a guide to the eye. (Reproduced with permission from Ref. [29].)

the combination of an elastic (resolution limited) component and a single broad quasielastic component of Lorentzian [26] or Gaussian [29] line shape. The observed behaviour can be divided into two di!erent characteristic temperature regimes: high temperatures (250 K)¹(500 K) and low temperatures (¹)250 K). In the high-temperature regime, all the magnetic intensity is in the quasielastic component. The Gaussian linewidth, C , which is shown as a func% tion of the temperature in Fig. 4, is essentially constant and equal to about 0.016 meV. Both the models by WuK rger [18] and Raikher and Stepanov [17] predict a quasi-constant linewidth in some temperature regime well above E /k and they ar
rive at the same relation, C +0.7 u (Section % ) 3.3). From this relation is found B "c\ )  u +0.20 T and using M +7.6;10 Am\ [29] )  in Eq. (11), we "nd K+7.5;10 Jm\. With d"2 nm, this gives the particle energy barrier E /k+23
K. Thus, it appears that the highest measuring temperature corresponds to k¹/E +20. The pre
diction of constant linewidth by WuK rger is independent of the damping constant, while the prediction by Raikher and Stepanov requires a damping constant, g , smaller than 0.01 consistent with typical  values for ferromagnets. If the model by Raikher and Stepanov is correct, an increase of the line width is expected if the temperature is elevated even further [17].

19

When the temperature is decreased below 250 K, part of the magnetic intensity becomes resolution limited, i.e. the relaxation becomes anisotropic. The resolution-limited component is attributed to the longitudinal, slowly #uctuating component of the magnetic moments and the quasielastic feature is attributed to the transverse #uctuations of the magnetic moments, but the inelastic double peak expected for the precession modes of the Gilbert equation (cf. Section 3.3) is not observed. In addition to the occurrence of a resolution limited component, the width of the quasielastic component increases. This increase is attributed to growing inter-particle correlations that increase the e!ective "eld in which the magnetic moment #uctuates. For ¹(100 K, this interaction "eld increases signi"cantly leading to weak short-range ferromagnetic correlations between the moments of neighbouring particles [28] and a decrease of the intensity of the quasielastic component (the magnetic moments freeze). Recently, this sample was also studied by neutron spin echo (NSE) [31,32], which probes the correlation function 1SX(0)SX(t)2 directly on observation times in the range from about 10\ s to 10\ s [33]. Above 100 K, the time-dependent longitudinal correlation function can be described by a single exponential decay (cf. Section 3.2). Below 100 K, a distribution of energy barriers (and hence decay times) is needed in order to describe the data. The relaxation time of a particle with median size can be described by q"q exp[E (¹)/k¹], with 
q +2;10\ s and E (¹)/k+100 K#a/¹, 
where a+40}50 K is a phenomenological constant that depends slightly on the q-value of the experiment. It is suggested that the temperature-dependent contribution to the energy barrier is due to the inter-particle interaction renormalisation of the energy barrier suggested by Dormann et al. [34] (see also Refs. [35,36]). In our opinion, this is questionable, as the observed behaviour is a complex mixture of the interaction e!ects and the changes of the saturation magnetisation and magnetic anisotropy due to the progressive alignment of the surface spins with those of the core. The magnetic dynamics of a similar Fe sample with a larger average particle size of 5 nm has also been studied by INS at small angles using an energy resolution (FWHM) of 0.020 meV [37]. For

20

M.F. Hansen et al. / Journal of Magnetism and Magnetic Materials 221 (2000) 10}25

Fig. 5. INS spectra for 5 nm Fe particles at ¹" 76 and 299 K. The solid line is the sum of the dynamic scattering contributions and the dashed line is a "t to a single Lorentzian (a) and two Lorentzian (b) lines. (1 THz + 4.1 meV) (Reproduced with permission from Ref. [37].)

this sample three distinct temperature regimes are observed. Inelastic data obtained at two temperatures are shown in Fig. 5. Parameters from "ts obtained as described below are shown in Fig. 6. In the high-temperature regime (¹'200 K) are observed an elastic signal attributed to the longitudinal component of the magnetic moments and a broad quasielastic peak of Lorentzian shape attributed to the transverse component of the magnetic moments. The Lorentzian line width is roughly temperature independent and has the HWHM value 0.013 meV. Assuming the validity of Raikher and Stepanov's calculations [17] and g (10\ (see Section 3.3), we estimate C " P % 0.011 meV, u "0.016 meV and B "0.14 T. ) ) This corresponds quite well to the result for the 2 nm particles, when it is taken into account that K usually increases with decreasing particle size [38]. From the value of B using the same value of ) M as for the 2 nm particles, we estimate the par ticle energy barrier E /k+250 K. At intermediate


Fig. 6. Temperature variation of the transverse susceptibility and the HWHM values obtained from "ts of INS spectra of 5 nm Fe particles using a single quasielastic Lorentzian line and two Lorentzian lines (at $u ). (Reproduced with permission  from Ref. [37].)

temperatures (100 K(¹) 200 K), which are below the estimated value of E /k, the broad
quasielastic peak changes into a double peak ("tted by a sum of two Lorentzians positioned at $u )  associated with the precession states of the Gilbert equation (cf. Section 3.3). The value of u increases  with decreasing temperature up to about 0.03 meV at ¹"76 K, which compares reasonably well with the estimated value of u at high temperatures. ) The linewidth of the inelastic peaks is almost constant in this regime. At low temperatures (¹)100 K), the value of u decreases drastically  while the linewidth increases, and below 50 K, the data are better described using a single Lorentzian for the energy broadened part of the magnetic intensity. This change of behaviour is attributed to strong competing inter-particle interactions that destroy the coherence of the precession states [37].

M.F. Hansen et al. / Journal of Magnetism and Magnetic Materials 221 (2000) 10}25

The results obtained for the two studied samples are quite similar. At high temperatures compared to the energy barrier, most of the magnetic intensity is in a quasielastic component with a width related to the anisotropy "eld. When the temperature is decreased below a certain value, an increasing fraction of the magnetic intensity becomes resolution limited. For the 2 nm particles, this seems to take place at thermal energies well above the anisotropy energy barrier. For the 5 nm particles, the resolution-limited component has signi"cant intensity at all the investigated temperatures the double peak expected for the transverse #uctuations of the precession modes is only observed at intermediate temperatures. At the lowest temperatures a re-entrant behaviour is seen with the loss of coherence of the precession states. This is frustrated due to strong inter-particle interactions. In addition to the interaction e!ects, the behaviour in both samples is also signi"cantly a!ected by a surface component of the particles, that makes the magnetic parameters (moment and anisotropy) of a particle strongly temperature dependent. As the resulting behaviour is a result of the combination of all these e!ects, it is di$cult to make reliable quantitative analyses of the experimental data except at high temperatures where the interaction e!ects are minimal. 5.2. a-Fe O nanoparticles   We have prepared hematite nanoparticles with an average size of 16 nm by thermal decomposition of Fe(NO ) ) 9H O [24]. The resulting sample is   denoted the `as-prepared samplea. The preparation method leads to the formation of a small amount (+10%) of a poorly crystalline ferrihydrite-like impurity phase in addition to the hematite nanoparticles. This impurity phase can be removed by treatment with oxalate under acidic conditions. A second sample has been prepared by "rst treating the as-prepared sample with oxalate and then coating the resulting particles with surfactant molecules (oleic acid). This sample, denoted as the `coated samplea, has been subject to a detailed study by MoK ssbauer spectroscopy, X-ray di!raction, transmission electron microscopy and magnetisation measurements [24]. This study has shown that the

21

particles have the canted antiferromagnetic structure at least down to 5 K with a magnetisation similar to that of weakly ferromagnetic bulk hematite. The magnetic anisotropy energy can be described by an easy plane (K <(0) with a uniaxial  in-plane anisotropy (K <) ful"lling "K <"   K <, i.e. the relaxation of the sublattice mag netisations is e!ectively two-dimensional over an energy barrier E "K < [24]. From an analysis
 of a temperature series of MoK ssbauer spectra, it was found that the relaxation time of the particles can be described by the Arrhenius law (Eq. (3)), with the median energy barrier E /k+ 600 K and q +
 2;10\ s [24]. Comparisons of X-ray di!raction spectra and transmission electron micrographs of the as-prepared and coated samples have revealed that the average particle sizes are indistinguishable and series of MoK ssbauer spectra obtained as a function of temperature indicate that there are no signi"cant di!erences between the dynamic behaviours of the hematite nanoparticles in the two samples. The `as-prepared samplea has been subject to studies by neutron scattering [39,40] at the RITA (re-invented triple axis) spectrometer at Ris+ National Laboratory (Denmark). The presence of the impurity phase in this sample and water adsorbed onto the particle surfaces seem to strongly reduce inter-particle interactions. Due to the disordered nature of the impurity phase, the inelastic scattering from this phase at a given q-value is negligible. The presence of hydrogen in adsorbed water and in the impurity phase mainly gives rise to an incoherent elastic background, which has been determined from energy scans at q-values away from the re#ections of hematite. In the INS study was used an energy resolution (FWHM) of 0.070 meV [39]. The powder sample was mainly studied near the strongest antiferromagnetic re#ection at q"1.37 As \ corresponding to the rhombohedral (111) re#ection. The studies were performed both in zero "eld as a function of temperature (50}325 K) and in applied magnetic "elds (B "0}6 T) at a constant  temperature of 268 K. Representative data are shown in Fig. 7. We "rst give a qualitative discussion of the observed behaviour. It is clearly seen that the zero"eld INS spectra consist of a narrow and a broad

22

M.F. Hansen et al. / Journal of Magnetism and Magnetic Materials 221 (2000) 10}25

Fig. 7. Representative INS data obtained on a-Fe O nanopar  ticles at q " 1.37 As \, (a) in zero applied "eld at the indicated temperatures, and (b) in the indicated applied "elds at ¹ " 268 K. The lines correspond to "ts to the model described in the text. (Reproduced with permission from Ref. [39].)

component. The relative area of the broad component increases with increasing temperature. In addition, it is seen that the broad component becomes more narrow with increasing temperature. At all temperatures it is observed that the broad component is #at on the top indicating that it is in fact a smeared double peak. These features suggest that the broad component is related to the precession dynamics of the transverse component of the sublattice magnetisations. If so, the splitting of the two peaks giving rise to the broad feature is proportional to the internal resonance "eld. An externally applied "eld superposes on this "eld and gives rise to an increase of the e!ective resonance "eld, i.e. an increase of the splitting of the two peaks. This is con"rmed experimentally by the in-"eld data shown in Fig. 7b, which thus support the interpretation of the broad component in terms of the precession modes of the sublattice magnetisations. The magnetic contribution to the narrow component is due to scattering by the longitudinal component of the sublattice magnetisations. Small changes of the experimental q-value near q " 1.37 As \ did not change the shape of the INS

spectra * only the intensities were a!ected, i.e. no energy dispersion was present. This indirectly con"rms the existence of the energy gap in the dispersion relation discussed in Section 3.1. To our knowledge, this energy gap has not yet been measured directly. The inelastic intensity was also studied at room temperature at the "rst structural (FM) re#ection at q " 1.7 As \. Within the experimental uncertainty, only a resolution limited component was present in the INS spectrum [40] as expected from the discussion in Section 3.4. It therefore seems that the assumptions made in Section 3.1 and the discussion of hematite in Section 3.4 are valid and hence we can apply the proposed models. The broadening of the inelastic peaks is signi"cant even at low temperatures. This can be due to damping, weak inter-particle interactions or a distribution of parameters because of a size and shape distribution of particles. We therefore used the damped harmonic oscillator description as a "rst approximation. Hence, the temperature series of INS spectra obtained in zero "eld was analysed using the sum of a Lorentzian (Eq. (9)) and the transverse scattering function for the damped harmonic oscillator (Eq. (13)). Note that the transverse scattering function is related to the low-frequency electron magnetic resonance mode of hematite (Section 3.4). In addition, it was necessary to include incoherent elastic and inelastic backgrounds. In lack of the present detailed knowledge of the anisotropy energy from Ref. [24], we used a magnetic anisotropy energy of three-dimensional uniaxial form in Ref. [39]. We also remark that there is a deviation of a factor of two in the relation between C and q in Ref. [39] and the correct expression, C" /q, derived in Section 3.2. The quasielastic HWHM linewidth, C, is shown as function of the temperature in Fig. 8. The solid line in the "gure is a "t to C" /q using the Arrhenius relation, q"q exp(E /k¹), with the result 
q +1.4;10\ s and E /k+500$200 K in 
good agreement with the results obtained by MoK ssbauer spectroscopy [24]. The use of the simpli"ed Arrhenius law has been criticized [41], but to our knowledge no satisfying theoretical prediction of the value of q has yet been made for antiferromag

M.F. Hansen et al. / Journal of Magnetism and Magnetic Materials 221 (2000) 10}25

Fig. 8. The quasielastic HWHM vs. temperature. The solid line is a "t obtained using the Arrhenius law for the superparamagnetic relaxation time.

netic materials. In our opinion, one needs to consider the two coupled Gilbert equations (one for each sublattice) and solve the resulting Fokker} Planck equations in order to "nd the real solution of the problem. As for the case of electron magnetic resonance, the solution for antiferromagnetic materials is likely to di!er signi"cantly from that for ferromagnetic materials. Hence, at present, it is di$cult to relate the damping constant to the exponential prefactor of the superparamagnetic relaxation law. The temperature variation of the positions, e " u , of the inelastic peaks is shown in Fig. 9.   The width, c, of the inelastic peaks (not shown) varied from about 0.18 meV at 50 K to 0.11 meV at 325 K. The value of e for antiferromagnets has  been shown in a quantum mechanical calculation [42] to be given by e +gk (s/s ) (2B B ) S /S,  ) # X

(20)

where s is the spin per atomic site (" for high spin Fe(III)) and s"s!. In Fig. 9 are shown "ts  of e (¹) to Eq. (20) with S /S"1cos h2 calculated  X 2 by Boltzmann statistics. The dashed and full lines are obtained using 3d and 2d uniaxial anisotropy, respectively. The "t using 3d uniaxial anisotropy, originally used in Ref. [39], gave the energy barrier E /k"1000$400 K. The re"ned "t using 2d an
isotropy ( justi"ed by the studies in Ref. [24])

23

Fig. 9. The positions of the inelastic peaks vs. temperature. The full and dashed lines are "ts obtained using Eq. (20) with 2d and 3d Boltzmann statistics, respectively.

gave the uniaxial in-plane energy barrier E /k"
600$200 K. Both models resulted in e (¹"0K)"  0.26$0.02 meV. Using Eq. (20) at ¹"0 K (S /S"1) with the bulk values B "900 T and X # M "9;10 JT\m\ [23] and d"16 nm, we  "nd E /k+500 K. The values of E derived from

the broadening of the quasielastic peak, the zerotemperature position of the inelastic peaks and the temperature variation of the positions of the inelastic peaks are in good agreement. Hence, it seems that we can describe the magnetic dynamics of the hematite nanoparticle sample at least semiquantitatively. Finally, we consider the positions of the inelastic peaks at a constant temperature in applied magnetic "elds. Writing the peak positions in zero "eld as e "$gk B , where B0 is along the sublattice   magnetisation directions, the positions in an externally applied magnetic "eld, B , are at e "$   gk "B #B ". Assuming a full alignment of the   canted magnetic moments along the direction of the applied "eld, we approximately have that B NB and hence that e (B )+gk (B #B .       A "t of this expression to the experimentally observed peak positions gives gk "0.12$ 0.01 meV/T in excellent agreement with the theoretical value expected for Fe>, gk "0.116 meV/T (g " 2). If the dominant contribution to the magnetic moment were due to uncompensated spins at the

24

M.F. Hansen et al. / Journal of Magnetism and Magnetic Materials 221 (2000) 10}25

surface of the particles, the low-"eld behaviour would di!er from that observed above. In addition to the shifted peaks, some intensity remains in a broad feature near e "0. The origin and  intensity of this feature is currently not fully understood. In summary, the antiferromagnetic nanoparticles have the advantage over ferromagnetic particles that the separation of the inelastic peaks is larger and that it is possible to have purely magnetic re#ections. In addition, the inter-particle dipole interactions are usually negligible. The magnetic scattering of hematite nanoparticles has been investigated both near an AFM re#ection and at an FM re#ection. The observed behaviour is consistent with that expected for isolated hematite nanoparticles and the parameters can be consistently analysed with values agreeing well with those obtained by, e.g., MoK ssbauer spectroscopy. Some of the e!ects observed in the present study are on the limit of the experimental energy resolution. We have therefore performed a new study at the IRIS time-of-#ight spectrometer at the Rutherford Appleton Laboratory (England) using an FWHM energy resolution of 0.015 meV. This study enables a more detailed analysis of the broadening of the quasielastic peak and of the shape of the inelastic peaks in addition to a more detailed study of the q-dependence of the scattering. The analysis is currently in progress and will be reported elsewhere.

6. Conclusions We have given a brief introduction to the theory for neutron scattering and have shown how inelastic neutron scattering can be applied to obtain information about the dynamics of magnetic nanoparticles on short time scales. Di!erences between ferromagnetic and antiferromagnetic systems have been discussed and the two main studies of nanoparticles performed so far have been reviewed. We have demonstrated that much information can be obtained by use of this technique, which is relevant for a deeper understanding of the dynamics of magnetic nanoparticles, but also that there are still some open questions and problems to resolve.

For ferromagnetic nanoparticles there is the question of inter-particle dipole interactions. It would be interesting to be able to disentangle the e!ects of inter-particle interactions from single particle effects by studying more ideal systems with di!erent volume fractions of particles and without a strong temperature dependence of the properties of the isolated particles. For antiferromagnetic nanoparticles, it would be interesting to study more monodisperse systems and a greater variety of systems. We believe that such studies combined with techniques such as magnetisation measurements, MoK ssbauer spectroscopy (if applicable) and magnetic resonance measurements will form a reference for the theoretical work on the dynamics of antiferromagnetic nanoparticles.

Acknowledgements Financial support from The Danish Natural Science Reseach Council and The Danish Technical Research Council is gratefully acknowledged.

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