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Anomalous Mössbauer fraction in small magnetic particles due to magnetostriction
mr
8 M 8
Journal of Magnetism and Magnetic Materials 127 (1993) 346-358 North-Holland
M 8 Anomalous Miissbauer fraction in small magnetic particles due to magnetostriction M-E.Y. Mohie-Eldin
and L. Gunther
Department of Physics and Astronomy, Tufts lJniversi@, Medford, MA 02155, USA
Received 18 February 1992; in revised form 11 January 1993
The biological molecule ferritin and its proven synthetic counterpart polysaccharide iron complex (PIG) have been shown to contain small ( < 100 A in diameter) antiferrimagnetic cores at their centers. Mossbauer studies of these molecules have revealed an anomalous drop in the Mossbauer fraction (f-factor) as the temperature rises above 30 K for mammalian ferritin and 60 K for PIC. Above the blocking temperature, superparamagnetic relaxation results in the disappearance of hyperhne splitting. Data that are treated with FFT procedures to eliminate the thickness effect still exhibit this anomaly. We have investigated the effect of superparamagnetic relaxation on the f-factor. Spin-lattice relaxation was excluded based upon a calculation of the rate of energy transfer from the spin system to the lattice. We have found the following process as a plausible explanation of the anomaly: Superparamagnetic relaxation brings about a dynamical displacement of the Mossbauer nucleus through magnetostriction. These displacements produce a Doppler broadening of the Miissbauer spectrum that reduces the apparent f-factor. The temperature dependence of the theoretically calculated f-factor agrees qualitatively with experiment. Finally, there is semi-quantitative agreement if the as yet unknown dimensionless magnetostriction constant were to be on the order of 10m3.
1. Introduction Ferritin is a ubiquitous protein, widespread among plants, mammals [l], and several bacteria [2,3]. It is designed to maintain iron in an available, non-toxic form. The mammalian ferritin molecule is a roughly spheroidal, 120 A diameter protein with a core that has a capacity for up to about 4500 iron atoms in the 75 A diameter protein interior cavity [4]. The protein shell is composed of 24 nearly identical subunits that are arranged so as to isolate the iron-containing core from the cellular environment. Six hydrophilic and eight hydrophobic channels provide access to the protein interior, presumably the electrons, protons, iron ions, and other small ions and molecules. Correspondence
to: Dr M-E.Y. Mohie-Eldin, Department of Physics and Astronomy, Tufts University, Medford, MA 02155, USA. Tel: + l(617) 628 5361.
The ferritin iron core is a hydrous ferric oxide phosphate with nominal formula (FeOOH), (FeO.H,PO,) and a structure similar to the polycrystalline mineral ferrihydrite, in which Fe(II1) ions have sixfold oxygen coordination and oxygen are hexagonally close-packed [51. Phosphate occurs in disordered regions of the core, possibly at chain ends of the iron polymer and/or at the junction of crystallites with each other or with the protein surface. The protein shell influences the three dimensional structure of the core as well as nucleation of the core and iron release from the core [6,7]. The most important property of ferritin in the context of this work is that it is antiferromagnetic, with a NCel temperature TN of about 174 K [8,9]. Polysaccharide ion complex (PI0 121is a synthetic complex of ferric iron and carbohydrate which is marketed under the name ‘Niferex’ as an oral hematinic by Central Pharmaceuticals, Inc. (Seymour, Indiana). PIC has been shown to con-
0304~8853/93/$06.00 0 1993 - Elsevier Science Publishers B.V. All rights reserved
M-E.Y. Mohie-EMin, L. Gunther /Anomalous
sist of large spheres 30-100 A in diameter, with a high iron content (47.93%) and coated with carbohydrate (20.32%). Investigations [2,10,11] using X-ray diffraction and Miissbauer spectroscopy have shown that the iron core of PIC resembles the core of ferritin [121. As a result PIC has come to be used as a synthetic counterpart of ferritin
Miissbauer fraction in small magnetic particles
347
that is easy to produce and whose core size can be controlled so as to gain a better understanding of ferritin itself. All previously published work on the Mossbauer characteristics of ferritin and PIC that are known to us treats the temperature dependence of the f-factor (spectral area) for temperatures T
(a) 0.99
0.97
0.95
0.93
-10.0
-5.0 Velocity
-10.0
-5.0 Velocity
0.0
5.0
10.0
(mm/set)
0.0
5.0
10.0
(mm/set)
Fig. 1. (a) L-east-squares fit of the PIC Miissbauer spectrum at T = 4.2 K. uc is a velocity cut-off corresponding to a frequency oI used in eq. (10). See text for further details. (b) Least-squares fit of the PIC Miissbauer spectrum at T = 100 K. The solid lines represent the theoretical fits of the spectra.
348
i6E.Y. Mohie-Eldin, L. Gunther / Anomalous Miissbauer fraction in small magnetic particles f-FACTOR
Fig. 2. T’he f-factor as a function of temperature for PIC. The o represent experimental data points fNew(T).The X represent fN&“), eq. (12), and therefore includes the effect of magnetostriction. The solid curve is a fit to experiment of the the f-factor that is obtained from the Debye theory while excluding the effect of magnetostriction
above 80 K, when only quadrupole splitting is observed. One obtains a smooth dependence, with the f-factor saturating at about 100 K. Recently [1,3], measurements of the ferritin and PIC Mijssbauer spectra have been extended to temperatures as low as 4.2 K [1,3]. Hyperfine splitting appears below the blocking temperature around 60 K for PIC and around 30 K for ferritin (see fig. 1 taken from [l]). Coincidentally, the f-factor exhibits an anomalous sharp decrease as the temperature increases above the corresponding temperatures (see fig. 2, taken from [l]). We note that the error bars are quite large. We have, therefore, included in the figure a fit to the data obtained from the debye model alone. We note that portions of the curve fall outside of the error bars.
2. Magnetostriction as a source of the anomalous f-factor In this paper, we present a mechanism involving magnetostriction that is based upon superparamagnetic fluctuations that can account for the observed anomaly. The mechanism results in an anomalous drop in the f-factor as the temperature si increased rather than the alternative viewpoint of an anomalous rise in the f-factor as
the temperature is decreased. The coincidence of this anomaly with the already well-understood disappearance of hyperfine splitting follows from the dependence of both upon superparamagnetic fluctuations. Magnetostriction brings about a strain in the particle that changes the equilibrium position of the nuclei. The strain is often in the direction of the magnetization M and we will assume that this is so in order to simplify our discussion, while noting that our essential results do not depend upon this assumption. If a particle were spherical in the absence of magnetostriction, it will be ellipsoidal in its presence, with the axis of revolution parallel to M. In the presence of switching, M spends essentially all of its time directed along one of the two directions of one of the easy axes. A simple reversal of direction along a single axis does not change the strain, so that in our theory, we must assume the presence of more than one easy axis. Switches from one easy axis direction to that of a second easy axis then bring about displacements of the Miissbauer nucleus, which causes a Doppler broadening of the Mijssbauer spectrum in a manner analogous to that of diffusion in the sudden jump approximation among a local cluster of sites.
3. Switching of magnetization in small antiferromagnetic particles Ferritin and PIC cores are single domain small particles which exhibit superparamagnetic behavior [3,10,13-151. The internal (hyperfine) field at the iron nucleus in these cores depends upon the number of unpaired electrons surrounding the nucleus. In these particles, the Fe3+ ion (with five unpaired electrons) possesses a resultant spin along the direction of quantization. This leads to spin polarization of the s-electrons. Thus an effective field H,, is established at the nucleus by the Fermi contact interaction. The magnetic interaction lifts the degeneracy of the nuclear ground and excited states, leading to a six-line absorption spectrum. Whether or not the internal magnetic hyperfine splitting is observable by Mossbauer spec-
M-E.Y Mohie-Eldin, L. Gunther / Anomalous M6ssbauer fraction in small magnetic particles
troscopy depends upon the relaxation time rs of the unpaired electron spins. If TV3 rL, where 7L is the Larmor precession time of the nuclear spin, a hyperfine split six-line spectrum is observed. If 7s -K rL, the average effective field H,, observed at the nucleus is zero and a two-line spectrum, due to crystal field splitting, is observed. For materials with antiferromagnetic interactions, such as FeOOH, the ferritin and PIC cores in particular, [3,13,16], and for temperatures less than the Neel temperature TN, there exists a critical particle size above which the material exists in an antiferromagnetic state. When fine particles of sizes smaller than the critical size are considered at a temperature less than TN, thermal energy can lead to relaxation of the antiferromagnetically ordered spins from one configuration to another configuration (t 4 i i . . . 1 7 J. t . . . >. This relaxation process is called ‘switching’. In the absence of an applied field, the two states have equal energy. Because the system is paramagnetic, and the spins of each small magnetic particle switch in unison, as a large spin, the system is said to be in a ‘superparamagnetic’ state. Often, the phenomenon of switching is referred to as ‘superparamagnetism’. The relaxation rate of transitions that occur between the two configurations is given by Arrhenius’ law [17]: r, = r, exp( - U/k,T)
clearly have a distribution of blocking temperatures. Note that the blocking temperature depends on the characteristic time interval of the specific experiment. For Miissbauer spectroscopy, that time interval is the lifetime of the excited nuclear state, T,, (1.45 x lo-’ s in the case of 57Fe).
4, Thickness effects One might argue that the observed drop in the f-factor is a trivial result of the thickness effect being enhanced when the spectrum changes from a six-line to a two-line spectrum, and therefore that the drop does not reflect an intrinsic anomaly. This explanation was ruled out as follows: Mijssbauer studies [1,3] were carried out on a series of samples of varying thickness, containing 10, 20, 35, 50, 75, and 100 mg of PIC, respectively. Results showed that the anomaly persists even for the thinnest absorbers. In addition, the thickness effect was removed from the experimental spectra for all samples through deconvolution by the fast-Fourier transform routine of Ure and Flinn [19,20]. The resulting spectra agreed within experimental error. Figure 2 shows the normalized f-factor for the 35 mg sample obtained after deconvolution. We therefore conclude that the anomaly is genuine.
(1)
is Boltzmann’s constant, T is the absolute temperature and r, is a pre-exponential factor that is weakly dependent on T. U is the energy barrier between the two configuration and is strongly dependent upon the particle volume. (Often, U is assumed to be proportional to the volume, though there are clear experimental indications from Miissbauer experiments that this can be far from being so [18]). For a distribution of small particle sizes in a given sample, there will be a distribution of relaxation times at a given temperature. The ‘blocking temperature’ is defined as the temperature below which, during the course of an experiment, switching is too slow to be observed. In a given sample, the individual particles will k,
349
5. Magnetic relaxation effects in Miissbauer spectra One of us has investigated [3] the influence of superparamagnetic relaxation on the f-factor by considering Jones and Srivastava’s [21,22] multistate superparamagnetic model of Miissbauer spectra. Based upon this model, we attempted to obtain superparamagnetic relaxation rates for small antiferromagnetic particles. We felt the anomalous drop of the factor could be produced by a strong (exponential) temperature dependence of the pre-exponential factor in eq. (1) over a specific temperature range, caused by, according to Jones and Srivastava [22], a magnon-phonon interaction that becomes the dominant cause
350
M-E.Y. Mohie-Eldin,L. Gunther / Anomalous Miissbauerfraction in smallmagneticparticles
of the superparamagnetic relaxation over that specific temperature range. However, calculations [3] carried out for an antiferromagnetic superparamagnetic system revealed that the pre-exponential factor did not exponentially depend on the temperature over any relevant temperature range. After eliminating a possible thickness effect origin to the anomalous drop in the f-factor of ferritin and PIC, as stated above, we turned to the possibility that vibrations of the protein core as a whole is the origin of the anomalous drop of the f-factor. However, the relative agreement between the Debye temperatures obtained from the Miissbauer fraction and isomer shift data of PIC [l] eliminated that hypotheses. Note that the fluctuation of the strong hyperfine field experienced by the 57Fe Miissbauer nucleus has drastic effects on the shape of the recoilless absorption spectra as well as the f-factor. This effect operates in pure paramagnetic as well as in magnetically ordered materials [16,23,24]. The question that we asked ourselves at this point was how superparamagnetic relaxation might affect the hyperfine field in such a way as to produce the anomalous drop in the f-factor. The Fe5’ Mijssbauer nucleus has a ground state with I = l/2 and g = 0.18. The relevant 14 keV excited state has I * = 3/2 and g * = - 0.103 and a lifetime 7, = 1.45 X lo-’ s. The latter is subject to an effective hyperfine field H,, that splits the level into four levels, with Zeeman energies c-g *M,*p,H,,) and the ground state is split into two levels, with energies ( -gM#,Z&). Here, & is the nuclear Bohr magneton. Magnetic dipole transitions (AM = 0, f 1) lead to a six-line absorption spectrum. If the field seen by the nucleus is constant, only one condition is required for observing hyperfine splitting of Mossbauer spectrum, namely that the separation between neighboring lines be larger than the natural line width r = r;i N 7 x lo6 s-l. That is,
where
oL = Larmor
frequency
of the nucleus
_ lo8 rad/s. Alternatively, we require that the Larmor period o;i = T,_< 7,. The interaction between the nucleus and the ion spin S is of the form I - S. For simplicity, we neglect other interactions between the nucleus and its surroundings, such as the quadrupole interaction. Before proceeding, it is important to note that the field seen by a given nucleus is determined by a single ionic spin and not, for example, by the magnetization, which is a bulk property. Thus, the origin of H,, at the nucleus in a paramagnetic material is the same as that in a material which exhibits magnetic order. The specific properties of the system of ionic spins enter only indirectly in the hyperfine splitting, that is, only in so far as they determine the time dependence or the average of the single spin. Therefore, any fluctuation in the ionic spin components will be transmitted directly to the nucleus in the form of fluctuations of the corresponding components of HeE. As a magnetic probe, however, the nucleus will respond to such fluctuations only if they are slow compared to the nuclear frequency w,_. Thus, if rs is the correlation time of the fluctuations of a particular component of S, two extreme cases A and B present themselves for paramagnetic systems as well as for magnetically ordered systems. We now consider the various cases: 5.1. Paramagnetic systems [16,23,24] There are two regimes, depending upon the relative magnitude of the Larmor time and the electron spin relaxation time: Condition A: rL s- T$: The nuclear spin senses only the average value of the components of S. For simplicity, consider an electron spin of l/2, with its z-component switching in a random manner from S, = + l/2 to S, = -l/2 and vice versa, such that the two spin states are equally probable. Van Der Woude and Dekker [161 assumed that this spin flip process is produced by the interaction of the spin with its environment (e.g. spin-spin & spin-lattice interactions). They also assumed that this process can be described as a stationary Markoff process and that the
M-E.Y. Mohie-Ekiin, L. Gunther / Anomalous Mssbauer fraction in small magnetic particles
probability that S, = - l/2 at time t + At, given that a time t, S, = + l/2, is given by P(++,
t,-3,
t+At)=fiAt=t, 7s
(3)
in the limit that At + 0. Here R is the spin flip frequency and 7s is, as stated before, the correlation time of the electron spin. The same expression holds, of course, if the spin states are interchanged. Suppose now that the electron is in the state S, = + l/2 and that H,, is in the +z direction. Consider one of six possible nuclear transitions, M,* = - 3/2 to MI = - l/2. Let wr be the corresponding angular frequency of the emitted -y-ray. If the electron spin flips to S, = - l/2, the direction of H,, reverses; that is, H,, and the quantum numbers M,, MI* change sign. Hence the above nuclear transition becomes that of M,* = +3/2 to MI = + l/2. We let oh be the frequency of the corresponding emitted y-ray. The problem of calculating the Miissbauer absorption spectrum corresponding to one of these nuclear transitions (say the wi transition, MI* = - 3/2 to M, = - l/2) is very similar to that of motional narrowing in NMR (first outlined by Abragam [25]) and was carried out by Van Der Woude and Dekker [24]. They found that when the particle is in the paramagnetic state, the nucleus senses only the average value in time of S, and hence of Hefi, which are both zero. As a result, hyperfine splitting is ineffective and the Mossbauer spectrum consists of a single peak at a frequency equal to the average of the two frequencies, w1 and we, namely, (oi + wJ/2. The same holds true for the other two pairs of transitions having the frequency pairs (wZ, 0,) and (03, wJ, respectively. All pair averages are equal: This (oi + we)/2 = co2 + 0,)/2 = (wg + wJ/2. frequency will be set equal to zero as a reference value. Condition B: 7L < TV:The nucleus responds to the instantaneous spin state of the electron. Van Der Woude and Dekker [24] then found that each of the above three pairs of transitions produces two peaks, lying at o = +a,, o = f&, o = f&, where S, = (ua - 0~1, 6, = (w, - w,),
351
and 6, = (04 - ws), respectively. All these frequencies are on the order of the Larmor frequency wL. Thus, in this regime, the hyperfine field is effective in splitting the spectrum into six lines. 5.2. Magnetically ordered systems The anomalous drop in the f-factor for PIC begins at about 60 K, while for mammalian ferritin it begins at about 30 K [3]. Therefore, the relevant temperature range that concerns us is 4.2-150 K, which is well below the NCel temperature. Thus, we are mainly concerned with the antiferromagnetically ordered phase of these materials. As in the case of the paramagnetic state, there are two regimes that obtain: Condition A: rL >> TV: In this regime, the nuclear spin senses only the average value of the electron spin components S,. In the ordered magnetic material, (S,) is not zero and is proportional to the magnetization of its sublattice. The electron spin component S, fluctuates between but now with probabilities +1/2 and -l/2, P[S, = f l/2] = (1 f 77)/2, where 77 is an order parameter proportional to the magnetization with a range 0 I rl I 1. The only difference between the magnetic phase and the paramagnetic phase is that on the average, S, spends more time in one state than the other. There are two switching frequencies 0, a’, corresponding to two relaxation times 7s and ~,1, one for each spin state. The probability of eq. (3) changes when the spin state changes. If the average magnetization of the material is constant, it follows that a(1 + 17)= LY(l - 7). In this case, the Mossbauer intensity distribution [24] corresponding to the nuclear transition from MI* = - 3/2 to MI = - l/2 produces one sharp peak at o = -T&, while the reverse transition produces a peak at o = +$i. Similar results apply to the remaining two pairs of nuclear transitions. Hence in this situation, hyperfine splitting is observed and the average value of the hyperfine field (H,,(T)) as measured by the position of the Mossbauer peaks, is proportional to 17, i.e. the sublattice magnetization. In fact, (H,,(T)) should follow the magnetization up to TN as long as condition A is satisfied.
352
M-E.Y. Mohie-Ekiin, L. Gunther / Anomakws Miissbauer fraction in small magwtic particles
Condition B: 7L ez TV: In this regime, the fluctuations of S, are such that the nucleus responds to the instantaneous value of S,. The result is that the spectrum has six sharp peaks, at o = far, f&, and fi&, regardless of the value of 77, that is, independent of the degree of order. Thus, hyperfine splitting is observed with the hyperfme field, as defined by the position of the Mossbauer peaks, not being proportional to the magnetization. Condition B usually is applicable at temperatures so high that the Weiss molecular field model applies. So far, we have considered the behavior of bulk samples, for which superparamagnetic fluctuations can be neglected. The ferritin core is so small that the sublattice magnetization (m) fluctuates significantly due to thermal activation processes. If &., in eq. (1) is so small that &~r z+-1, the situation described under condition A of the paramagnetic phase, applies. We have a complete destruction of hyperfine splitting even below the NCel temperature, since now the fluctuations of the magnetization imposes fluctuations of the electrons spin. Next, we take into account quadrupole splitting. In the case of ferritin and PIC, the 57Fe Miissbauer nucleus experiences an electric-field gradient (EFG) due to its non-cubic site in the lattice. The interaction of the nuclear quadrupole moment with the EFG splits the I = 3/2 excited state into two substates: MI = f 3/2 and MI = f l/2. Quadrupole splitting is subordinate to the dominant magnetic hyperfine field interaction of the nucleus. When the fluctuations of the hyperfine field become too fast for the Miissbauer nucleus to follow it, the six-line Mossbauer spectrum associated with hyperfine structure is destroyed. Only then is the two-line spectrum associated with quadrupole splitting observable. When the hyperfine field fluctuates slowly, a six-line spectrum obtains, albeit with the position of the peaks affected by the EFG. As long as. superparamagnetic fluctuations are not too great, the only condition that is required in order to observed hyperfine splitting in the Miissbauer spectrum is that the separation between successive lines be larger than the natural linewidth. We have observed [1,3] that the hyper-
fine field (H,,(T)) obtained from the Miissbauer spectrum of PIC is proportional to the observed sublattice magnetization, for temperatures when the hyperfine structure can be observed, that is, below the blocking temperature at 63 K. We are thus led to assume that condition A is valid below the blocking temperature.
6. Theory Our theory is based upon Doppler broadening due to displacements of the Miissbauer nucleus that are brought about by magnetostriction in response to switching. The displacement of a given nucleus is conveniently referred to the center of mass of the particle, so that it is clear that the various nuclei in the particle undergo different magnitudes of displacement, depending upon their respective components of displacement in the direction of M: Suppose that M is in the x-direction, then the displacement will be proportional to the x-component of the position vector of the nucleus from the center of mass. We will present for a single nucleus, recognizing that the actual spectrum is an average of our expression over the various nuclear displacements in the particle. We begin by focusing our analysis on the effect of switching, through magnetostriction, on a single Miissbauer line (the effect of switching on the spectrum as a whole, for example, on line splitting, is set aside, for simplicity). We leave to a future publication a comprehensive treatment of the two effects. Also, for simplicity, we assume that the nucleus has only two possible components of its displacement in the direction of the emitted y-ray, f a, respectively, with respect to an average value ‘. We will refer to these two states as the ‘position states’ of the nucleus. Finally, we assume that the switches of magnetization are Poisson distributed. Then the probability that n = switched take place in a time interval t * Suppose that the actual values with respect to the center of mass of the particle are x1 and x1, respectively. Then only the parameters f(x, - x,)/2 = f a appear in the expression for the spectral intensity.
M-E. Y. Mohie-Eldin, L. Gunther /Anomalous Miissbauer fraction in small magnetic particles
such that a value +a is changed to -a, versa, is given by P(n, t) =
(vt) e-“‘7,
or vice
n
(4)
where v is the switching rate between the two position states, given by
(5) Clearly Y N Z, and v,, N Z, (see eq. (1)). The intensity distribution of a single Mijssbauer absorption line is given by [261: Z(O) = Rei
mdt ;exp(
-iot
-Zt)+(t>.
(6)
The first term is a Lorentzian with the natural line width and a weight (1 -a), where IY= sin’ qu. The second term is a Lorentzian broadened by 21, and has a weight CL The total area under Z(w) in eq. (9) is unity. However, the data are analyzed by necessity with cut-offs of, say, -wy at the lower end of the frequency range and +w, at the upper end. The frequency cut-offs wI and o, are determined by the fixed velocity cut-offs, f 12.00 mm/s, and depend upon the position of a peak along the length of the spectrum. For example, oc= u++,/c, where uc is indicated in figure l(a). or, is the y-ray frequency. Then the total ‘measured’ area under a single line, excluding the DebyeWaller factor exp( - 2W), is given by
Here, A=
4 = (exp(iqx(t))
4(t)
(l-4
exp( -i&O))),
(7) where q is the wavevector of the emitted y-ray (= 7.22 A-‘> and x(t) and x(0) are the projections of the displacements of the Mossbauer nucleus at times 0 and t, respectively, along the direction of the emitted y-ray, and hence are equal to either +a or --c1. Displacements due to vibration about lattice positions are assumed to have been taken into account and included in the Debye-Waller factor exp( - 2W). The quantity q 0[x(t) - x(O)] will have values zero if there is no net change in the position of nucleus, corresponding to an even number of position switches, and +2qu for a switch from -a to +a, or -2qa for a switch from +a to -a, corresponding to an odd number of switches. Thus, =
c P(n, t) evenn ++ C P(n, t)[e2iq” + e-2iqa] evenn =e -“’ cash vt + e-“’ sinh vt cos 2qu = cos2 qu + sin2 qu ev2”‘.
(8)
The resulting single-line spectrum is given by
r/r
Z( co) = cos2qa 02+r2 (r+ 2v)/lr + sin2 qu 02+(r+21g2’
(9)
353
[
Tr
tan-’
a
tan-‘Z+
+; [
t$+,,-1: I UC
2v + tan
-l-
%
r+2v
1* (10)
The experimental data were analyzed by a leastsquares fit using a VAX (5000 series) computer at the Francis Bitter National Magnet Laboratory. The program used for data analysis was originally written by Varret [27], and was modified by Jacques Teillet, currently of the Institut National des Sciences AppliquCes de Rouen and Professor of Physics at the University of Rouen (Rouen, France), who kindly allowed us to use it. Basically, the program computes a theoretical Mossbauer spectrum for the 57Fe nucleus and fits it with the observed data by using a variation of several parameters. However, the program that we used could fit the observed data to: (a) a sum of l-5 magnetic sextets; (b) a sum of l-5 quadrupole split doublets; (c) a broad distribution of magnetic hyperfine fields; (d) a broad distribution of quadrupole split doublets, or a combination of all the above. We note that all individual lines are assumed to have a Lorentzian lineshape. For the best fit of the data, we chose the following combinations: - In the low-temperature range (4.2 K I T I 60 K), we fit the spectra with a distribution of magnetic hyperfine fields plus a central doublet (quadrupole splitting, QS), where QS is one-half
354
M-E.Y. Mohie-Eldin, L. Gunther /Anomalous Miissbauer fraction in small magnetic particles
Ultimately, the observed spectrum in the mid range of temperatures was fitted to a mixture of a six-line spectrum and a two-line quadrupole spectrum, with weights Z’, and Pd, respectively. Both sets are present so as to reflect the varying degree of blocking and switching that is present. Furthermore, the sets of lines of both the sextet and the doublet are fitted with fractional weights xi=1_6 and ~~_i,~, respectively, with Cx, = 1 and Cyj = 1. Thus the total area, excluding the Debye-Waller factor, is given by
the difference in separation of the two highest minus the two lowest-velocity peaks. - In the intermediate-temperature range (60 K < T I 100 K), the spectra were fitted to a sum of l-4 magnetic sextets plus a central doublet. - For the range (100 K< T < 250 K), the spectra were fitted to a sum of l-3 doublets. - In the high-temperature range (250 K I T I 295 K), we fit the spectra with a distribution of quadrupole doublets. For this program, all lines were given the same fixed linewidth so as to give the best fit for all temperatures. The total intensity, which is the sum of the individual line intensities, the isomer shift (IS), the quadrupole splitting (QS) were all chosen so as to fit the experiment. It is important
Ath(T) =Ps t
XiA(oti,
@ui)
i=l
to note here that the anomaly in the f-factor is not dependent in any way on the fitting procedure since the fitting procedure automatically produces a fit that reproduces the observed total intensity.
C YjA(wtj,
+p~
(11)
wuj)*
j=l
The total area, including the Debye-Waller factor and normalized to a temperature of 4.2 K is given by
In fig. 3, we plot the residuals for the fit at to fig. l(a). We note that the plot is featureless and no clustering on one or the other side of the ‘0’ line in the vicinity of any resonant frequency (velocity) occurs. T = 4.2 K, corresponding
exp[ -2W(T)l
Ah(T) fNth(T) = Ath(4.2K)
exp[ -2W(4.2
K)] ’
(12)
FRACTONAL RESIDUALS x 1O-2 0.3
1 .
0.2
. .
. .
.
*
-
.
q
. ..
- . ,. l
. .
-15
.
-
.
*
.
.-
.
..
-5
q
__ .
. ..=. . . .
. .
0 VELOCITY (mm/set)
Fig. 3. Fractional residuals: [(theor. fit - experiment)/theor.
. .
em-
. .*
n
. n
.
l
l
.
l
-10
.
.
m
.
l
:-..--_
l
. .
.
l.
.
.__
-
.
. - .
.
.
. ,
.
.
.
. . . ..
l .
n
n
c
.
_...
.
.
. .
-0.2
.
-
.
.
-0.1 I_
. .
-.
_
.
.
.
.
.
0
-
.
.
n .
0.1
.*
.
l
.
5
. .
10
.
. .
n
15
fit] versus velocity (mm/s) for the PIG absorber’s Mijssbauer spectrum at T = 4.2K.
M-E.Y: Mohie-Eldin, L. Gunther /Anomalous Miissbauer fraction in small magnetic particles
Our goal is to determine the displacement a, the frequency prefactor va, and the barrier energy U from the data and using the model. To do so, we needed to factor out of the function f,,&T), the function A,,(4.2), and the two factors involving the Debye-Waller factor. First we note that at 4.2 K, pS = 1, Pd = 0, and v S r. Then, A,,(4.2K)
= i zxi[tan-r$
+ tan-‘?],
(13) which is directly extractable from the experimental data. In order to determine the Debye-Waller factor as a function of temperature, we used the Debye approximation in our analysis 1281. For temperatures T greater than one-half of the Debye temperature On, W(T) = 3E,T/k,O& where E, is the recoil energy of the Miissbauer nucleus. For temperatures much greater than the blocking temperature, A,,(T) is temperature-independent. Also, Pd = 1 and 2v is much greater than r, tic, and 0,. Thus,
Ath(T) --$ i
YjACotj,
wuj)
j=l
It is clear from the above that for large T,
dln[.fdT)l dT
6ER
-)-ma
(15)
In fig. 4, we plot lnf versus T. We see that the data are consistent with eq. (15). Using the recoil energy E, = 1.9 X 10 -3 eV [14], we obtain a Debye temperature 8, = (248 f 12)K. Hence, the Debye-Waller factor is known for all temperatures [28]. We can also determine the parameter cy= sin’ qu to be equal to 0.08, using egs. (12) and (14). And final1 , given that 4 = 7.22 A-‘, we find that a = 0.04 x which corresponds to the average displacement for nuclei located at various
3.55
distances from the center of mass of the particle. From the value of a we can obtain an estimate of the dimensionless magnetostriction constant A, which is equal to the relative elongation Al/l of the sample. As a rough estimate, we associate our value of a with a nucleus located at a distance of 0.5 of the particle radius r. An analysis of the data provides us with the distribution of particle sizes in a well-established way from the distribution of hyperfine fields [l,lO]. The resulting average particle radius (r) is 3.5 A, which corresponds to an average volume (V) of about 1.8 X lo-i9 cm3. Then A e a/OS(r) = 2.28 X 10m3. This value of A is unusually large, considering that the Mossbauer spectrum of PIC indicates no presence of Fe*+ ions, while Fe3+ ions normally have an isotropic electron distribution. However, small particles have a significant fraction of Fe3+ ions under strain near the surface; these ions have a modified electron distribution among the d-orbitals that might result in a large magnetostriction [29]. It is noteworthy in this regard that the anisotropy energy of small particles can be up to two orders of magnitude larger than that in the bulk; see ref. [30], for example. For a discussion of magnetostriction and anisotropy energy and their connection, see ref. [31]. Now the function A,,(T) can be factored out of the function f,,,(T). The parameters v0 and U were determined so as to fit the observed f,(T), with the result that vi’ = 2.1 x 10-l’ s and U = 5.4 x lo-l4 erg, corresponding to a temperature Up/k, = 390 K. Figure 2 shows the fit of fNth (the ‘X’ points) to the experimental data f,(T). We have also included in the figure a fit to the data (the solid curve) obtained from the Debye model alone (0, = 248 K), i.e. excluding the effect of magnetostriction. We note that portions of the solid curve in the vicinity of the temperature range of the anomaly lie outside of the error bars. In fig. 5, the values of l&(T) (ln[v’= v(T)/IV at T = 60,85, 100, 125 K are shown. The function v’(T) is obtained from the fit of fNt,,(T) to the observed f,(T). We can obtain an estimate of the anisotropy energy per volume, K, by setting the barrier energy equal to K(V). The result is that K = 3 X lo5 erg/cm3, which is somewhat
356
IU-E.Y. Mohie-Eldin,
L. Gunther
/ Anomalous
Miissbauer fraction in small magnetic particles
lnf “3
-0.6
0
100
50
150
200
250
300
350
T (K) Fig. 4. In fNe.&‘)
versus temperature. The slope gives @,, = (248f 12)K.
larger than the value found for ferritin (N lo5 erg/cm31 [lo]. Finally, we can define a blocking temperature as that temperature at which the switching rate is equal to the natural linewidth (7 X lo6 Hz). The result is a temperature of 60 K. This value is very close to the temperature 63 K at which the sextet and doublet fractions P, and Pd, respectively, were found to be equal 111. Before concluding this section we note that a more comprehensive theoretical treatment of this
I
0
v’(T)
0.002
0.005
Inverse
0.007
0.01
Temperature
0.012
0.015
0.017
(l/K)
Fig. 5. In v’(T) versus temperature, obtained from a fit of f Nth, eq. (121, to the experimental f-factor for PIC. Y’ = 1
indicates the blocking temperature.
anomaly would have integrated all mechanisms that affect Miissbauer spectral area of the superparamagnetic particles. We expect that the magnetostriction mechanism is the dominant one in the temperature range of the anomaly. These mechanisms are (1) the superparamagnetic relaxation, (2) the magnetostriction mechanism, (3) the small fluctuations of the particle magnetization. However, our main objective was to obtain a first order approximation of the effect of magnetostriction on the spectral area, where we must note that the ‘full’ spectra area (in the limits u = f co mm/s> is constant and independent of temperature. Our theory focuses on the effect of magnetostriction, which is reflecting superparamagnetism on a single Mossbauer line. The effect of this mechanism is the broadening of the Mijssbauer line, which leads to a loss of area into its wings. Such a loss in area can be measured by determining the width and position of the Mijssbauer line. To the extend that the ‘Teillet’ fit reflects the actual spectrum, our fit takes into account the distribution of hyperfine fields which is produced due to the distribution of particle volumes and the fast fluctuations in the particle’s magnetization. We must also note that the cut-off frequencies, i.e. the limits of the Miissbauer spectra which
M-E.Y: Mohie-Eldin, L. Gunther / Anomalous iU&sbauer fraction in small magnetic particles
were determined with respect to the position of the line peaks, were kept constant for all spectra irrespective of whether the spectrum was a sextet or a doublet. This indicates a possible loss of area into the wings corresponding to the maximum hyperfine field splitting, which occurs for the lowest-temperature (T = 4.2 K) spectrum. This loss in area will decrease with an increase in temperature. This parallels the effect of the distribution of hyperfine fields on the Mijssbauer spectral area, where a distribution of hyperfine fields broadens the Mossbauer line and hence shifts its area into its wings. This occurs far below the blocking temperature. In this temperature range where the spectral area is affected by the distribution of the hyperfine fields, the magnetostriction mechanism is negligible. Therefore, the basic effect of the distribution of hyperfine fields is to reduce the Miissbauer spectral area below the blocking temperature by shifting some of it into the wings. The magnetostriction mechanism according to our theory has the same effect on the area above the blocking temperature. In other words, if there did not exist a magnetostriction mechanism and only a distribution of hyperfine fields, there would be no anomaly or if the width of the distribution of hyperfine fields were very large, a depression in the lower temperature range of the f-factor versus temperature curve would be exhibited contrary to observation. The above argument indicates that the apparent anomaly is a balance of the effects of both mechanisms, where the magnetostriction mechanism has the greater effect on the spectral area in the temperature range of the anomaly. This validates our approach to obtain a first-order approximation of the effect of the magnetostriction mechanism on the spectral area.
7. Conclusions The experimental magnetic [l] and Miissbauer [ll data indicate that the PIC and ferritin molecules possess magnetic anisotropy energy which may not be strictly uniaxial. This is in congruence with our theory. We, therefore, believe that the magnetostrictive mechanism de-
357
scribed above provides a reasonable explanation for the anomalous f-factor and indicates that the anomaly should be present in all superparamagnetic particles which possess (and exhibit) types of magnetic anisotropy more complex than uniaxial anisotropy. The parameters needed to fit the experimental results for PIC are plausible. The anisotropy energy per unit volume obtained for PIC is higher than the value found for ferritin (- lo5 erg/cm3). Such a difference might be due to differences in the chemical structural or surface properties of the core and might also reflect the presence of phosphate in ferritin and its absence in PIC and/or perhaps the poorer crystallinity of the ferritin core with respect to that of PIC. On the other hand, it is interesting that St Pierre et al. [32] found evidence that ferritin’s magnetic anisotropy constant K should be greater than the value reported so far in the literature, based upon his measurements of the Miissbauer spectra of ferritin in large applied magnetic fields. As another test of the validity of our theory, it would be very interesting to perform experiments that yield the magnetostriction constant for PIC and ferritin. Furthermore, it would be interesting to look for this anomaly in hemosiderin, other ferritin-like molecules [33], and, more generally, in other single-domain magnetic particles whose magnetic anisotropy energy is not uniaxially symmetric. Finally, as we alluded to before, it is important to remember that the two phenomena, the f-factor anomaly and the collapse of the hyperfine splitting to a quadrupole doublet, have been treated by our theory separately. Therefore, it is appropriate that the next stage should be a comprehensive theory that integrates the mechanisms of superparamagnetism, magnetostriction, as well as magnetization fluctuations and predicts their effects on the Miissbauer spectra of small magnetic particles with different types of magnetic anisotropy [341.
Acknowledgements We wish to express our gratitude to R.B. Frankel and G.C. Papaefthymiou for the innu-
M-E.Y. Mohie-Ekiin, L. Gunther /Anomalous
358
merable discussions we had with them and their constant encouragement. We are also grateful to John Slonczewski for fruitful discussion regarding magnetostriction in iron compounds. References [l] M-E.Y. Mohie-Eldin [2] [3] [4] [5] [6]
[7] [8] [9] [lo] [ll] [12] [13] [14]
and R.B. Frankel, submitted for publication. K.A. Berg et al., J. Inorg, Biochem 22 (1984) 125. M-E.Y. Mohie-Eldin, PhD Thesis, Tufts University (1992). C. Ford et al., Phil. Trans. R. Sot. Lond. B304 (1984) 551. K. Towe et al., J. Colloid. Interface Sci. 24 (1967) 384. E. Theil, in: Advances in Inorganic Biochememistry, vol 5, ed. Theil, Eighom, and Marzilli (Elsevier, New York, 1983), ch. 1. D. Rice et al., in: Advances in Inorganic Biochememistry, op. cit., ch. 2. A. Blaise et al., Acad. Sci. Paris 261 (1965) 2310. A. Blaise and J.L. Girardet, in: Proc. Int. Conf. on Mag., Moscow (1973). J. Williams et al., Phys. Med. Biol. 23 (1978) 835. T. Shapiro and P. Saltman, Struct. Bonding (Berlin) 6 (1969) 116. P. Murphy et al., J. Colloid Interface Sci. 56 (1976) 312. T.G. St Pierre et al., Hyperfine Interactions 29 (1986) 1427. J.G. Stevens and V.E. Stevens, eds. Mijssbauer Effect Data Index (Plenum, New York, 1979).
Miissbauer fraction in small magnetic particles 1151 J. Webb and H.B. Gray, B&hem.
Biophys. Acta 351 (1974) 224. [161 F. Van Der Woude and A.J. Dekker, Phys. Stat. Solidi, 13 (1966) 181. [171 T. Nakamura et al., Phys. L&t. 12 (1964) 178. t181L. Gunther, unpublished results. 1191M. Ure and P. Flinn, Mijssbauer Effect Methodology, vol 7, ed. I. Gruverman (Plenum, New York, 1971) p. 45. 1201 B. Kolk, Phys. Rev. B12 (1975) 1620. 1211D.H. Jones and KKP. Srivastava, Phys. Rev. B34 (1986) 7542. [221K.K.P. Srivastava and D.H. Jones, ICAME (1987). [231F. Van Der Woude and A.J. Dekker, Phys. Stat. Solidi 9 (1965) 775. [241F. Van Der Woude and A.J. Dekker, Solid State Commun. 3 (1965) 319. 1251A. Abragam, Principles of Nuclear Magnetism, (Oxford Press, Oxford, 1983). [261R. Sack, Molec. Phys. 1 (1958) 163. t271F. Varret, J. Phys. Chem. Solids 37 (1976) 265. [281H. Frauenfelder, Mijssbauer Effect (W.A. Benjamin, New York, 1963). [291We are indebted to G.C. Papaefthymiou for this obsetvation. 1301CM. Schelinger, D. Griscom, G.C. Papaefthymiou and D.R. Veblen, J. Geophys. Res. 93 (1988) 9137. [311 J.C. Slonczewski, J. Appl. Phys. 32 (1961) 253s. [32] T.G. St Pierre et al., J. Mag. Mag. Mater 69 (1987) 276. [331 S.H. Bell, B&hem. Biophys. Acta 787 (1984) 227; E.R. Bauminger, Biochem. Biophys. Acta. 623 (1980) 237. 1341We have since submission produced such a comprehensive theory. See L. Gunther and M-E.Y. Mohie-Eldin, to appear in J. Magn. Magn. Mater.
8 M 8
Journal of Magnetism and Magnetic Materials 127 (1993) 346-358 North-Holland
M 8 Anomalous Miissbauer fraction in small magnetic particles due to magnetostriction M-E.Y. Mohie-Eldin
and L. Gunther
Department of Physics and Astronomy, Tufts lJniversi@, Medford, MA 02155, USA
Received 18 February 1992; in revised form 11 January 1993
The biological molecule ferritin and its proven synthetic counterpart polysaccharide iron complex (PIG) have been shown to contain small ( < 100 A in diameter) antiferrimagnetic cores at their centers. Mossbauer studies of these molecules have revealed an anomalous drop in the Mossbauer fraction (f-factor) as the temperature rises above 30 K for mammalian ferritin and 60 K for PIC. Above the blocking temperature, superparamagnetic relaxation results in the disappearance of hyperhne splitting. Data that are treated with FFT procedures to eliminate the thickness effect still exhibit this anomaly. We have investigated the effect of superparamagnetic relaxation on the f-factor. Spin-lattice relaxation was excluded based upon a calculation of the rate of energy transfer from the spin system to the lattice. We have found the following process as a plausible explanation of the anomaly: Superparamagnetic relaxation brings about a dynamical displacement of the Mossbauer nucleus through magnetostriction. These displacements produce a Doppler broadening of the Miissbauer spectrum that reduces the apparent f-factor. The temperature dependence of the theoretically calculated f-factor agrees qualitatively with experiment. Finally, there is semi-quantitative agreement if the as yet unknown dimensionless magnetostriction constant were to be on the order of 10m3.
1. Introduction Ferritin is a ubiquitous protein, widespread among plants, mammals [l], and several bacteria [2,3]. It is designed to maintain iron in an available, non-toxic form. The mammalian ferritin molecule is a roughly spheroidal, 120 A diameter protein with a core that has a capacity for up to about 4500 iron atoms in the 75 A diameter protein interior cavity [4]. The protein shell is composed of 24 nearly identical subunits that are arranged so as to isolate the iron-containing core from the cellular environment. Six hydrophilic and eight hydrophobic channels provide access to the protein interior, presumably the electrons, protons, iron ions, and other small ions and molecules. Correspondence
to: Dr M-E.Y. Mohie-Eldin, Department of Physics and Astronomy, Tufts University, Medford, MA 02155, USA. Tel: + l(617) 628 5361.
The ferritin iron core is a hydrous ferric oxide phosphate with nominal formula (FeOOH), (FeO.H,PO,) and a structure similar to the polycrystalline mineral ferrihydrite, in which Fe(II1) ions have sixfold oxygen coordination and oxygen are hexagonally close-packed [51. Phosphate occurs in disordered regions of the core, possibly at chain ends of the iron polymer and/or at the junction of crystallites with each other or with the protein surface. The protein shell influences the three dimensional structure of the core as well as nucleation of the core and iron release from the core [6,7]. The most important property of ferritin in the context of this work is that it is antiferromagnetic, with a NCel temperature TN of about 174 K [8,9]. Polysaccharide ion complex (PI0 121is a synthetic complex of ferric iron and carbohydrate which is marketed under the name ‘Niferex’ as an oral hematinic by Central Pharmaceuticals, Inc. (Seymour, Indiana). PIC has been shown to con-
0304~8853/93/$06.00 0 1993 - Elsevier Science Publishers B.V. All rights reserved
M-E.Y. Mohie-EMin, L. Gunther /Anomalous
sist of large spheres 30-100 A in diameter, with a high iron content (47.93%) and coated with carbohydrate (20.32%). Investigations [2,10,11] using X-ray diffraction and Miissbauer spectroscopy have shown that the iron core of PIC resembles the core of ferritin [121. As a result PIC has come to be used as a synthetic counterpart of ferritin
Miissbauer fraction in small magnetic particles
347
that is easy to produce and whose core size can be controlled so as to gain a better understanding of ferritin itself. All previously published work on the Mossbauer characteristics of ferritin and PIC that are known to us treats the temperature dependence of the f-factor (spectral area) for temperatures T
(a) 0.99
0.97
0.95
0.93
-10.0
-5.0 Velocity
-10.0
-5.0 Velocity
0.0
5.0
10.0
(mm/set)
0.0
5.0
10.0
(mm/set)
Fig. 1. (a) L-east-squares fit of the PIC Miissbauer spectrum at T = 4.2 K. uc is a velocity cut-off corresponding to a frequency oI used in eq. (10). See text for further details. (b) Least-squares fit of the PIC Miissbauer spectrum at T = 100 K. The solid lines represent the theoretical fits of the spectra.
348
i6E.Y. Mohie-Eldin, L. Gunther / Anomalous Miissbauer fraction in small magnetic particles f-FACTOR
Fig. 2. T’he f-factor as a function of temperature for PIC. The o represent experimental data points fNew(T).The X represent fN&“), eq. (12), and therefore includes the effect of magnetostriction. The solid curve is a fit to experiment of the the f-factor that is obtained from the Debye theory while excluding the effect of magnetostriction
above 80 K, when only quadrupole splitting is observed. One obtains a smooth dependence, with the f-factor saturating at about 100 K. Recently [1,3], measurements of the ferritin and PIC Mijssbauer spectra have been extended to temperatures as low as 4.2 K [1,3]. Hyperfine splitting appears below the blocking temperature around 60 K for PIC and around 30 K for ferritin (see fig. 1 taken from [l]). Coincidentally, the f-factor exhibits an anomalous sharp decrease as the temperature increases above the corresponding temperatures (see fig. 2, taken from [l]). We note that the error bars are quite large. We have, therefore, included in the figure a fit to the data obtained from the debye model alone. We note that portions of the curve fall outside of the error bars.
2. Magnetostriction as a source of the anomalous f-factor In this paper, we present a mechanism involving magnetostriction that is based upon superparamagnetic fluctuations that can account for the observed anomaly. The mechanism results in an anomalous drop in the f-factor as the temperature si increased rather than the alternative viewpoint of an anomalous rise in the f-factor as
the temperature is decreased. The coincidence of this anomaly with the already well-understood disappearance of hyperfine splitting follows from the dependence of both upon superparamagnetic fluctuations. Magnetostriction brings about a strain in the particle that changes the equilibrium position of the nuclei. The strain is often in the direction of the magnetization M and we will assume that this is so in order to simplify our discussion, while noting that our essential results do not depend upon this assumption. If a particle were spherical in the absence of magnetostriction, it will be ellipsoidal in its presence, with the axis of revolution parallel to M. In the presence of switching, M spends essentially all of its time directed along one of the two directions of one of the easy axes. A simple reversal of direction along a single axis does not change the strain, so that in our theory, we must assume the presence of more than one easy axis. Switches from one easy axis direction to that of a second easy axis then bring about displacements of the Miissbauer nucleus, which causes a Doppler broadening of the Mijssbauer spectrum in a manner analogous to that of diffusion in the sudden jump approximation among a local cluster of sites.
3. Switching of magnetization in small antiferromagnetic particles Ferritin and PIC cores are single domain small particles which exhibit superparamagnetic behavior [3,10,13-151. The internal (hyperfine) field at the iron nucleus in these cores depends upon the number of unpaired electrons surrounding the nucleus. In these particles, the Fe3+ ion (with five unpaired electrons) possesses a resultant spin along the direction of quantization. This leads to spin polarization of the s-electrons. Thus an effective field H,, is established at the nucleus by the Fermi contact interaction. The magnetic interaction lifts the degeneracy of the nuclear ground and excited states, leading to a six-line absorption spectrum. Whether or not the internal magnetic hyperfine splitting is observable by Mossbauer spec-
M-E.Y Mohie-Eldin, L. Gunther / Anomalous M6ssbauer fraction in small magnetic particles
troscopy depends upon the relaxation time rs of the unpaired electron spins. If TV3 rL, where 7L is the Larmor precession time of the nuclear spin, a hyperfine split six-line spectrum is observed. If 7s -K rL, the average effective field H,, observed at the nucleus is zero and a two-line spectrum, due to crystal field splitting, is observed. For materials with antiferromagnetic interactions, such as FeOOH, the ferritin and PIC cores in particular, [3,13,16], and for temperatures less than the Neel temperature TN, there exists a critical particle size above which the material exists in an antiferromagnetic state. When fine particles of sizes smaller than the critical size are considered at a temperature less than TN, thermal energy can lead to relaxation of the antiferromagnetically ordered spins from one configuration to another configuration (t 4 i i . . . 1 7 J. t . . . >. This relaxation process is called ‘switching’. In the absence of an applied field, the two states have equal energy. Because the system is paramagnetic, and the spins of each small magnetic particle switch in unison, as a large spin, the system is said to be in a ‘superparamagnetic’ state. Often, the phenomenon of switching is referred to as ‘superparamagnetism’. The relaxation rate of transitions that occur between the two configurations is given by Arrhenius’ law [17]: r, = r, exp( - U/k,T)
clearly have a distribution of blocking temperatures. Note that the blocking temperature depends on the characteristic time interval of the specific experiment. For Miissbauer spectroscopy, that time interval is the lifetime of the excited nuclear state, T,, (1.45 x lo-’ s in the case of 57Fe).
4, Thickness effects One might argue that the observed drop in the f-factor is a trivial result of the thickness effect being enhanced when the spectrum changes from a six-line to a two-line spectrum, and therefore that the drop does not reflect an intrinsic anomaly. This explanation was ruled out as follows: Mijssbauer studies [1,3] were carried out on a series of samples of varying thickness, containing 10, 20, 35, 50, 75, and 100 mg of PIC, respectively. Results showed that the anomaly persists even for the thinnest absorbers. In addition, the thickness effect was removed from the experimental spectra for all samples through deconvolution by the fast-Fourier transform routine of Ure and Flinn [19,20]. The resulting spectra agreed within experimental error. Figure 2 shows the normalized f-factor for the 35 mg sample obtained after deconvolution. We therefore conclude that the anomaly is genuine.
(1)
is Boltzmann’s constant, T is the absolute temperature and r, is a pre-exponential factor that is weakly dependent on T. U is the energy barrier between the two configuration and is strongly dependent upon the particle volume. (Often, U is assumed to be proportional to the volume, though there are clear experimental indications from Miissbauer experiments that this can be far from being so [18]). For a distribution of small particle sizes in a given sample, there will be a distribution of relaxation times at a given temperature. The ‘blocking temperature’ is defined as the temperature below which, during the course of an experiment, switching is too slow to be observed. In a given sample, the individual particles will k,
349
5. Magnetic relaxation effects in Miissbauer spectra One of us has investigated [3] the influence of superparamagnetic relaxation on the f-factor by considering Jones and Srivastava’s [21,22] multistate superparamagnetic model of Miissbauer spectra. Based upon this model, we attempted to obtain superparamagnetic relaxation rates for small antiferromagnetic particles. We felt the anomalous drop of the factor could be produced by a strong (exponential) temperature dependence of the pre-exponential factor in eq. (1) over a specific temperature range, caused by, according to Jones and Srivastava [22], a magnon-phonon interaction that becomes the dominant cause
350
M-E.Y. Mohie-Eldin,L. Gunther / Anomalous Miissbauerfraction in smallmagneticparticles
of the superparamagnetic relaxation over that specific temperature range. However, calculations [3] carried out for an antiferromagnetic superparamagnetic system revealed that the pre-exponential factor did not exponentially depend on the temperature over any relevant temperature range. After eliminating a possible thickness effect origin to the anomalous drop in the f-factor of ferritin and PIC, as stated above, we turned to the possibility that vibrations of the protein core as a whole is the origin of the anomalous drop of the f-factor. However, the relative agreement between the Debye temperatures obtained from the Miissbauer fraction and isomer shift data of PIC [l] eliminated that hypotheses. Note that the fluctuation of the strong hyperfine field experienced by the 57Fe Miissbauer nucleus has drastic effects on the shape of the recoilless absorption spectra as well as the f-factor. This effect operates in pure paramagnetic as well as in magnetically ordered materials [16,23,24]. The question that we asked ourselves at this point was how superparamagnetic relaxation might affect the hyperfine field in such a way as to produce the anomalous drop in the f-factor. The Fe5’ Mijssbauer nucleus has a ground state with I = l/2 and g = 0.18. The relevant 14 keV excited state has I * = 3/2 and g * = - 0.103 and a lifetime 7, = 1.45 X lo-’ s. The latter is subject to an effective hyperfine field H,, that splits the level into four levels, with Zeeman energies c-g *M,*p,H,,) and the ground state is split into two levels, with energies ( -gM#,Z&). Here, & is the nuclear Bohr magneton. Magnetic dipole transitions (AM = 0, f 1) lead to a six-line absorption spectrum. If the field seen by the nucleus is constant, only one condition is required for observing hyperfine splitting of Mossbauer spectrum, namely that the separation between neighboring lines be larger than the natural line width r = r;i N 7 x lo6 s-l. That is,
where
oL = Larmor
frequency
of the nucleus
_ lo8 rad/s. Alternatively, we require that the Larmor period o;i = T,_< 7,. The interaction between the nucleus and the ion spin S is of the form I - S. For simplicity, we neglect other interactions between the nucleus and its surroundings, such as the quadrupole interaction. Before proceeding, it is important to note that the field seen by a given nucleus is determined by a single ionic spin and not, for example, by the magnetization, which is a bulk property. Thus, the origin of H,, at the nucleus in a paramagnetic material is the same as that in a material which exhibits magnetic order. The specific properties of the system of ionic spins enter only indirectly in the hyperfine splitting, that is, only in so far as they determine the time dependence or the average of the single spin. Therefore, any fluctuation in the ionic spin components will be transmitted directly to the nucleus in the form of fluctuations of the corresponding components of HeE. As a magnetic probe, however, the nucleus will respond to such fluctuations only if they are slow compared to the nuclear frequency w,_. Thus, if rs is the correlation time of the fluctuations of a particular component of S, two extreme cases A and B present themselves for paramagnetic systems as well as for magnetically ordered systems. We now consider the various cases: 5.1. Paramagnetic systems [16,23,24] There are two regimes, depending upon the relative magnitude of the Larmor time and the electron spin relaxation time: Condition A: rL s- T$: The nuclear spin senses only the average value of the components of S. For simplicity, consider an electron spin of l/2, with its z-component switching in a random manner from S, = + l/2 to S, = -l/2 and vice versa, such that the two spin states are equally probable. Van Der Woude and Dekker [161 assumed that this spin flip process is produced by the interaction of the spin with its environment (e.g. spin-spin & spin-lattice interactions). They also assumed that this process can be described as a stationary Markoff process and that the
M-E.Y. Mohie-Ekiin, L. Gunther / Anomalous Mssbauer fraction in small magnetic particles
probability that S, = - l/2 at time t + At, given that a time t, S, = + l/2, is given by P(++,
t,-3,
t+At)=fiAt=t, 7s
(3)
in the limit that At + 0. Here R is the spin flip frequency and 7s is, as stated before, the correlation time of the electron spin. The same expression holds, of course, if the spin states are interchanged. Suppose now that the electron is in the state S, = + l/2 and that H,, is in the +z direction. Consider one of six possible nuclear transitions, M,* = - 3/2 to MI = - l/2. Let wr be the corresponding angular frequency of the emitted -y-ray. If the electron spin flips to S, = - l/2, the direction of H,, reverses; that is, H,, and the quantum numbers M,, MI* change sign. Hence the above nuclear transition becomes that of M,* = +3/2 to MI = + l/2. We let oh be the frequency of the corresponding emitted y-ray. The problem of calculating the Miissbauer absorption spectrum corresponding to one of these nuclear transitions (say the wi transition, MI* = - 3/2 to M, = - l/2) is very similar to that of motional narrowing in NMR (first outlined by Abragam [25]) and was carried out by Van Der Woude and Dekker [24]. They found that when the particle is in the paramagnetic state, the nucleus senses only the average value in time of S, and hence of Hefi, which are both zero. As a result, hyperfine splitting is ineffective and the Mossbauer spectrum consists of a single peak at a frequency equal to the average of the two frequencies, w1 and we, namely, (oi + wJ/2. The same holds true for the other two pairs of transitions having the frequency pairs (wZ, 0,) and (03, wJ, respectively. All pair averages are equal: This (oi + we)/2 = co2 + 0,)/2 = (wg + wJ/2. frequency will be set equal to zero as a reference value. Condition B: 7L < TV:The nucleus responds to the instantaneous spin state of the electron. Van Der Woude and Dekker [24] then found that each of the above three pairs of transitions produces two peaks, lying at o = +a,, o = f&, o = f&, where S, = (ua - 0~1, 6, = (w, - w,),
351
and 6, = (04 - ws), respectively. All these frequencies are on the order of the Larmor frequency wL. Thus, in this regime, the hyperfine field is effective in splitting the spectrum into six lines. 5.2. Magnetically ordered systems The anomalous drop in the f-factor for PIC begins at about 60 K, while for mammalian ferritin it begins at about 30 K [3]. Therefore, the relevant temperature range that concerns us is 4.2-150 K, which is well below the NCel temperature. Thus, we are mainly concerned with the antiferromagnetically ordered phase of these materials. As in the case of the paramagnetic state, there are two regimes that obtain: Condition A: rL >> TV: In this regime, the nuclear spin senses only the average value of the electron spin components S,. In the ordered magnetic material, (S,) is not zero and is proportional to the magnetization of its sublattice. The electron spin component S, fluctuates between but now with probabilities +1/2 and -l/2, P[S, = f l/2] = (1 f 77)/2, where 77 is an order parameter proportional to the magnetization with a range 0 I rl I 1. The only difference between the magnetic phase and the paramagnetic phase is that on the average, S, spends more time in one state than the other. There are two switching frequencies 0, a’, corresponding to two relaxation times 7s and ~,1, one for each spin state. The probability of eq. (3) changes when the spin state changes. If the average magnetization of the material is constant, it follows that a(1 + 17)= LY(l - 7). In this case, the Mossbauer intensity distribution [24] corresponding to the nuclear transition from MI* = - 3/2 to MI = - l/2 produces one sharp peak at o = -T&, while the reverse transition produces a peak at o = +$i. Similar results apply to the remaining two pairs of nuclear transitions. Hence in this situation, hyperfine splitting is observed and the average value of the hyperfine field (H,,(T)) as measured by the position of the Mossbauer peaks, is proportional to 17, i.e. the sublattice magnetization. In fact, (H,,(T)) should follow the magnetization up to TN as long as condition A is satisfied.
352
M-E.Y. Mohie-Ekiin, L. Gunther / Anomakws Miissbauer fraction in small magwtic particles
Condition B: 7L ez TV: In this regime, the fluctuations of S, are such that the nucleus responds to the instantaneous value of S,. The result is that the spectrum has six sharp peaks, at o = far, f&, and fi&, regardless of the value of 77, that is, independent of the degree of order. Thus, hyperfine splitting is observed with the hyperfme field, as defined by the position of the Mossbauer peaks, not being proportional to the magnetization. Condition B usually is applicable at temperatures so high that the Weiss molecular field model applies. So far, we have considered the behavior of bulk samples, for which superparamagnetic fluctuations can be neglected. The ferritin core is so small that the sublattice magnetization (m) fluctuates significantly due to thermal activation processes. If &., in eq. (1) is so small that &~r z+-1, the situation described under condition A of the paramagnetic phase, applies. We have a complete destruction of hyperfine splitting even below the NCel temperature, since now the fluctuations of the magnetization imposes fluctuations of the electrons spin. Next, we take into account quadrupole splitting. In the case of ferritin and PIC, the 57Fe Miissbauer nucleus experiences an electric-field gradient (EFG) due to its non-cubic site in the lattice. The interaction of the nuclear quadrupole moment with the EFG splits the I = 3/2 excited state into two substates: MI = f 3/2 and MI = f l/2. Quadrupole splitting is subordinate to the dominant magnetic hyperfine field interaction of the nucleus. When the fluctuations of the hyperfine field become too fast for the Miissbauer nucleus to follow it, the six-line Mossbauer spectrum associated with hyperfine structure is destroyed. Only then is the two-line spectrum associated with quadrupole splitting observable. When the hyperfine field fluctuates slowly, a six-line spectrum obtains, albeit with the position of the peaks affected by the EFG. As long as. superparamagnetic fluctuations are not too great, the only condition that is required in order to observed hyperfine splitting in the Miissbauer spectrum is that the separation between successive lines be larger than the natural linewidth. We have observed [1,3] that the hyper-
fine field (H,,(T)) obtained from the Miissbauer spectrum of PIC is proportional to the observed sublattice magnetization, for temperatures when the hyperfine structure can be observed, that is, below the blocking temperature at 63 K. We are thus led to assume that condition A is valid below the blocking temperature.
6. Theory Our theory is based upon Doppler broadening due to displacements of the Miissbauer nucleus that are brought about by magnetostriction in response to switching. The displacement of a given nucleus is conveniently referred to the center of mass of the particle, so that it is clear that the various nuclei in the particle undergo different magnitudes of displacement, depending upon their respective components of displacement in the direction of M: Suppose that M is in the x-direction, then the displacement will be proportional to the x-component of the position vector of the nucleus from the center of mass. We will present for a single nucleus, recognizing that the actual spectrum is an average of our expression over the various nuclear displacements in the particle. We begin by focusing our analysis on the effect of switching, through magnetostriction, on a single Miissbauer line (the effect of switching on the spectrum as a whole, for example, on line splitting, is set aside, for simplicity). We leave to a future publication a comprehensive treatment of the two effects. Also, for simplicity, we assume that the nucleus has only two possible components of its displacement in the direction of the emitted y-ray, f a, respectively, with respect to an average value ‘. We will refer to these two states as the ‘position states’ of the nucleus. Finally, we assume that the switches of magnetization are Poisson distributed. Then the probability that n = switched take place in a time interval t * Suppose that the actual values with respect to the center of mass of the particle are x1 and x1, respectively. Then only the parameters f(x, - x,)/2 = f a appear in the expression for the spectral intensity.
M-E. Y. Mohie-Eldin, L. Gunther /Anomalous Miissbauer fraction in small magnetic particles
such that a value +a is changed to -a, versa, is given by P(n, t) =
(vt) e-“‘7,
or vice
n
(4)
where v is the switching rate between the two position states, given by
(5) Clearly Y N Z, and v,, N Z, (see eq. (1)). The intensity distribution of a single Mijssbauer absorption line is given by [261: Z(O) = Rei
mdt ;exp(
-iot
-Zt)+(t>.
(6)
The first term is a Lorentzian with the natural line width and a weight (1 -a), where IY= sin’ qu. The second term is a Lorentzian broadened by 21, and has a weight CL The total area under Z(w) in eq. (9) is unity. However, the data are analyzed by necessity with cut-offs of, say, -wy at the lower end of the frequency range and +w, at the upper end. The frequency cut-offs wI and o, are determined by the fixed velocity cut-offs, f 12.00 mm/s, and depend upon the position of a peak along the length of the spectrum. For example, oc= u++,/c, where uc is indicated in figure l(a). or, is the y-ray frequency. Then the total ‘measured’ area under a single line, excluding the DebyeWaller factor exp( - 2W), is given by
Here, A=
4 = (exp(iqx(t))
4(t)
(l-4
exp( -i&O))),
(7) where q is the wavevector of the emitted y-ray (= 7.22 A-‘> and x(t) and x(0) are the projections of the displacements of the Mossbauer nucleus at times 0 and t, respectively, along the direction of the emitted y-ray, and hence are equal to either +a or --c1. Displacements due to vibration about lattice positions are assumed to have been taken into account and included in the Debye-Waller factor exp( - 2W). The quantity q 0[x(t) - x(O)] will have values zero if there is no net change in the position of nucleus, corresponding to an even number of position switches, and +2qu for a switch from -a to +a, or -2qa for a switch from +a to -a, corresponding to an odd number of switches. Thus, =
c P(n, t) evenn ++ C P(n, t)[e2iq” + e-2iqa] evenn =e -“’ cash vt + e-“’ sinh vt cos 2qu = cos2 qu + sin2 qu ev2”‘.
(8)
The resulting single-line spectrum is given by
r/r
Z( co) = cos2qa 02+r2 (r+ 2v)/lr + sin2 qu 02+(r+21g2’
(9)
353
[
Tr
tan-’
a
tan-‘Z+
+; [
t$+,,-1: I UC
2v + tan
-l-
%
r+2v
1* (10)
The experimental data were analyzed by a leastsquares fit using a VAX (5000 series) computer at the Francis Bitter National Magnet Laboratory. The program used for data analysis was originally written by Varret [27], and was modified by Jacques Teillet, currently of the Institut National des Sciences AppliquCes de Rouen and Professor of Physics at the University of Rouen (Rouen, France), who kindly allowed us to use it. Basically, the program computes a theoretical Mossbauer spectrum for the 57Fe nucleus and fits it with the observed data by using a variation of several parameters. However, the program that we used could fit the observed data to: (a) a sum of l-5 magnetic sextets; (b) a sum of l-5 quadrupole split doublets; (c) a broad distribution of magnetic hyperfine fields; (d) a broad distribution of quadrupole split doublets, or a combination of all the above. We note that all individual lines are assumed to have a Lorentzian lineshape. For the best fit of the data, we chose the following combinations: - In the low-temperature range (4.2 K I T I 60 K), we fit the spectra with a distribution of magnetic hyperfine fields plus a central doublet (quadrupole splitting, QS), where QS is one-half
354
M-E.Y. Mohie-Eldin, L. Gunther /Anomalous Miissbauer fraction in small magnetic particles
Ultimately, the observed spectrum in the mid range of temperatures was fitted to a mixture of a six-line spectrum and a two-line quadrupole spectrum, with weights Z’, and Pd, respectively. Both sets are present so as to reflect the varying degree of blocking and switching that is present. Furthermore, the sets of lines of both the sextet and the doublet are fitted with fractional weights xi=1_6 and ~~_i,~, respectively, with Cx, = 1 and Cyj = 1. Thus the total area, excluding the Debye-Waller factor, is given by
the difference in separation of the two highest minus the two lowest-velocity peaks. - In the intermediate-temperature range (60 K < T I 100 K), the spectra were fitted to a sum of l-4 magnetic sextets plus a central doublet. - For the range (100 K< T < 250 K), the spectra were fitted to a sum of l-3 doublets. - In the high-temperature range (250 K I T I 295 K), we fit the spectra with a distribution of quadrupole doublets. For this program, all lines were given the same fixed linewidth so as to give the best fit for all temperatures. The total intensity, which is the sum of the individual line intensities, the isomer shift (IS), the quadrupole splitting (QS) were all chosen so as to fit the experiment. It is important
Ath(T) =Ps t
XiA(oti,
@ui)
i=l
to note here that the anomaly in the f-factor is not dependent in any way on the fitting procedure since the fitting procedure automatically produces a fit that reproduces the observed total intensity.
C YjA(wtj,
+p~
(11)
wuj)*
j=l
The total area, including the Debye-Waller factor and normalized to a temperature of 4.2 K is given by
In fig. 3, we plot the residuals for the fit at to fig. l(a). We note that the plot is featureless and no clustering on one or the other side of the ‘0’ line in the vicinity of any resonant frequency (velocity) occurs. T = 4.2 K, corresponding
exp[ -2W(T)l
Ah(T) fNth(T) = Ath(4.2K)
exp[ -2W(4.2
K)] ’
(12)
FRACTONAL RESIDUALS x 1O-2 0.3
1 .
0.2
. .
. .
.
*
-
.
q
. ..
- . ,. l
. .
-15
.
-
.
*
.
.-
.
..
-5
q
__ .
. ..=. . . .
. .
0 VELOCITY (mm/set)
Fig. 3. Fractional residuals: [(theor. fit - experiment)/theor.
. .
em-
. .*
n
. n
.
l
l
.
l
-10
.
.
m
.
l
:-..--_
l
. .
.
l.
.
.__
-
.
. - .
.
.
. ,
.
.
.
. . . ..
l .
n
n
c
.
_...
.
.
. .
-0.2
.
-
.
.
-0.1 I_
. .
-.
_
.
.
.
.
.
0
-
.
.
n .
0.1
.*
.
l
.
5
. .
10
.
. .
n
15
fit] versus velocity (mm/s) for the PIG absorber’s Mijssbauer spectrum at T = 4.2K.
M-E.Y: Mohie-Eldin, L. Gunther /Anomalous Miissbauer fraction in small magnetic particles
Our goal is to determine the displacement a, the frequency prefactor va, and the barrier energy U from the data and using the model. To do so, we needed to factor out of the function f,,&T), the function A,,(4.2), and the two factors involving the Debye-Waller factor. First we note that at 4.2 K, pS = 1, Pd = 0, and v S r. Then, A,,(4.2K)
= i zxi[tan-r$
+ tan-‘?],
(13) which is directly extractable from the experimental data. In order to determine the Debye-Waller factor as a function of temperature, we used the Debye approximation in our analysis 1281. For temperatures T greater than one-half of the Debye temperature On, W(T) = 3E,T/k,O& where E, is the recoil energy of the Miissbauer nucleus. For temperatures much greater than the blocking temperature, A,,(T) is temperature-independent. Also, Pd = 1 and 2v is much greater than r, tic, and 0,. Thus,
Ath(T) --$ i
YjACotj,
wuj)
j=l
It is clear from the above that for large T,
dln[.fdT)l dT
6ER
-)-ma
(15)
In fig. 4, we plot lnf versus T. We see that the data are consistent with eq. (15). Using the recoil energy E, = 1.9 X 10 -3 eV [14], we obtain a Debye temperature 8, = (248 f 12)K. Hence, the Debye-Waller factor is known for all temperatures [28]. We can also determine the parameter cy= sin’ qu to be equal to 0.08, using egs. (12) and (14). And final1 , given that 4 = 7.22 A-‘, we find that a = 0.04 x which corresponds to the average displacement for nuclei located at various
3.55
distances from the center of mass of the particle. From the value of a we can obtain an estimate of the dimensionless magnetostriction constant A, which is equal to the relative elongation Al/l of the sample. As a rough estimate, we associate our value of a with a nucleus located at a distance of 0.5 of the particle radius r. An analysis of the data provides us with the distribution of particle sizes in a well-established way from the distribution of hyperfine fields [l,lO]. The resulting average particle radius (r) is 3.5 A, which corresponds to an average volume (V) of about 1.8 X lo-i9 cm3. Then A e a/OS(r) = 2.28 X 10m3. This value of A is unusually large, considering that the Mossbauer spectrum of PIC indicates no presence of Fe*+ ions, while Fe3+ ions normally have an isotropic electron distribution. However, small particles have a significant fraction of Fe3+ ions under strain near the surface; these ions have a modified electron distribution among the d-orbitals that might result in a large magnetostriction [29]. It is noteworthy in this regard that the anisotropy energy of small particles can be up to two orders of magnitude larger than that in the bulk; see ref. [30], for example. For a discussion of magnetostriction and anisotropy energy and their connection, see ref. [31]. Now the function A,,(T) can be factored out of the function f,,,(T). The parameters v0 and U were determined so as to fit the observed f,(T), with the result that vi’ = 2.1 x 10-l’ s and U = 5.4 x lo-l4 erg, corresponding to a temperature Up/k, = 390 K. Figure 2 shows the fit of fNth (the ‘X’ points) to the experimental data f,(T). We have also included in the figure a fit to the data (the solid curve) obtained from the Debye model alone (0, = 248 K), i.e. excluding the effect of magnetostriction. We note that portions of the solid curve in the vicinity of the temperature range of the anomaly lie outside of the error bars. In fig. 5, the values of l&(T) (ln[v’= v(T)/IV at T = 60,85, 100, 125 K are shown. The function v’(T) is obtained from the fit of fNt,,(T) to the observed f,(T). We can obtain an estimate of the anisotropy energy per volume, K, by setting the barrier energy equal to K(V). The result is that K = 3 X lo5 erg/cm3, which is somewhat
356
IU-E.Y. Mohie-Eldin,
L. Gunther
/ Anomalous
Miissbauer fraction in small magnetic particles
lnf “3
-0.6
0
100
50
150
200
250
300
350
T (K) Fig. 4. In fNe.&‘)
versus temperature. The slope gives @,, = (248f 12)K.
larger than the value found for ferritin (N lo5 erg/cm31 [lo]. Finally, we can define a blocking temperature as that temperature at which the switching rate is equal to the natural linewidth (7 X lo6 Hz). The result is a temperature of 60 K. This value is very close to the temperature 63 K at which the sextet and doublet fractions P, and Pd, respectively, were found to be equal 111. Before concluding this section we note that a more comprehensive theoretical treatment of this
I
0
v’(T)
0.002
0.005
Inverse
0.007
0.01
Temperature
0.012
0.015
0.017
(l/K)
Fig. 5. In v’(T) versus temperature, obtained from a fit of f Nth, eq. (121, to the experimental f-factor for PIC. Y’ = 1
indicates the blocking temperature.
anomaly would have integrated all mechanisms that affect Miissbauer spectral area of the superparamagnetic particles. We expect that the magnetostriction mechanism is the dominant one in the temperature range of the anomaly. These mechanisms are (1) the superparamagnetic relaxation, (2) the magnetostriction mechanism, (3) the small fluctuations of the particle magnetization. However, our main objective was to obtain a first order approximation of the effect of magnetostriction on the spectral area, where we must note that the ‘full’ spectra area (in the limits u = f co mm/s> is constant and independent of temperature. Our theory focuses on the effect of magnetostriction, which is reflecting superparamagnetism on a single Mossbauer line. The effect of this mechanism is the broadening of the Mijssbauer line, which leads to a loss of area into its wings. Such a loss in area can be measured by determining the width and position of the Mijssbauer line. To the extend that the ‘Teillet’ fit reflects the actual spectrum, our fit takes into account the distribution of hyperfine fields which is produced due to the distribution of particle volumes and the fast fluctuations in the particle’s magnetization. We must also note that the cut-off frequencies, i.e. the limits of the Miissbauer spectra which
M-E.Y: Mohie-Eldin, L. Gunther / Anomalous iU&sbauer fraction in small magnetic particles
were determined with respect to the position of the line peaks, were kept constant for all spectra irrespective of whether the spectrum was a sextet or a doublet. This indicates a possible loss of area into the wings corresponding to the maximum hyperfine field splitting, which occurs for the lowest-temperature (T = 4.2 K) spectrum. This loss in area will decrease with an increase in temperature. This parallels the effect of the distribution of hyperfine fields on the Mijssbauer spectral area, where a distribution of hyperfine fields broadens the Mossbauer line and hence shifts its area into its wings. This occurs far below the blocking temperature. In this temperature range where the spectral area is affected by the distribution of the hyperfine fields, the magnetostriction mechanism is negligible. Therefore, the basic effect of the distribution of hyperfine fields is to reduce the Miissbauer spectral area below the blocking temperature by shifting some of it into the wings. The magnetostriction mechanism according to our theory has the same effect on the area above the blocking temperature. In other words, if there did not exist a magnetostriction mechanism and only a distribution of hyperfine fields, there would be no anomaly or if the width of the distribution of hyperfine fields were very large, a depression in the lower temperature range of the f-factor versus temperature curve would be exhibited contrary to observation. The above argument indicates that the apparent anomaly is a balance of the effects of both mechanisms, where the magnetostriction mechanism has the greater effect on the spectral area in the temperature range of the anomaly. This validates our approach to obtain a first-order approximation of the effect of the magnetostriction mechanism on the spectral area.
7. Conclusions The experimental magnetic [l] and Miissbauer [ll data indicate that the PIC and ferritin molecules possess magnetic anisotropy energy which may not be strictly uniaxial. This is in congruence with our theory. We, therefore, believe that the magnetostrictive mechanism de-
357
scribed above provides a reasonable explanation for the anomalous f-factor and indicates that the anomaly should be present in all superparamagnetic particles which possess (and exhibit) types of magnetic anisotropy more complex than uniaxial anisotropy. The parameters needed to fit the experimental results for PIC are plausible. The anisotropy energy per unit volume obtained for PIC is higher than the value found for ferritin (- lo5 erg/cm3). Such a difference might be due to differences in the chemical structural or surface properties of the core and might also reflect the presence of phosphate in ferritin and its absence in PIC and/or perhaps the poorer crystallinity of the ferritin core with respect to that of PIC. On the other hand, it is interesting that St Pierre et al. [32] found evidence that ferritin’s magnetic anisotropy constant K should be greater than the value reported so far in the literature, based upon his measurements of the Miissbauer spectra of ferritin in large applied magnetic fields. As another test of the validity of our theory, it would be very interesting to perform experiments that yield the magnetostriction constant for PIC and ferritin. Furthermore, it would be interesting to look for this anomaly in hemosiderin, other ferritin-like molecules [33], and, more generally, in other single-domain magnetic particles whose magnetic anisotropy energy is not uniaxially symmetric. Finally, as we alluded to before, it is important to remember that the two phenomena, the f-factor anomaly and the collapse of the hyperfine splitting to a quadrupole doublet, have been treated by our theory separately. Therefore, it is appropriate that the next stage should be a comprehensive theory that integrates the mechanisms of superparamagnetism, magnetostriction, as well as magnetization fluctuations and predicts their effects on the Miissbauer spectra of small magnetic particles with different types of magnetic anisotropy [341.
Acknowledgements We wish to express our gratitude to R.B. Frankel and G.C. Papaefthymiou for the innu-
M-E.Y. Mohie-Ekiin, L. Gunther /Anomalous
358
merable discussions we had with them and their constant encouragement. We are also grateful to John Slonczewski for fruitful discussion regarding magnetostriction in iron compounds. References [l] M-E.Y. Mohie-Eldin [2] [3] [4] [5] [6]
[7] [8] [9] [lo] [ll] [12] [13] [14]
and R.B. Frankel, submitted for publication. K.A. Berg et al., J. Inorg, Biochem 22 (1984) 125. M-E.Y. Mohie-Eldin, PhD Thesis, Tufts University (1992). C. Ford et al., Phil. Trans. R. Sot. Lond. B304 (1984) 551. K. Towe et al., J. Colloid. Interface Sci. 24 (1967) 384. E. Theil, in: Advances in Inorganic Biochememistry, vol 5, ed. Theil, Eighom, and Marzilli (Elsevier, New York, 1983), ch. 1. D. Rice et al., in: Advances in Inorganic Biochememistry, op. cit., ch. 2. A. Blaise et al., Acad. Sci. Paris 261 (1965) 2310. A. Blaise and J.L. Girardet, in: Proc. Int. Conf. on Mag., Moscow (1973). J. Williams et al., Phys. Med. Biol. 23 (1978) 835. T. Shapiro and P. Saltman, Struct. Bonding (Berlin) 6 (1969) 116. P. Murphy et al., J. Colloid Interface Sci. 56 (1976) 312. T.G. St Pierre et al., Hyperfine Interactions 29 (1986) 1427. J.G. Stevens and V.E. Stevens, eds. Mijssbauer Effect Data Index (Plenum, New York, 1979).
Miissbauer fraction in small magnetic particles 1151 J. Webb and H.B. Gray, B&hem.
Biophys. Acta 351 (1974) 224. [161 F. Van Der Woude and A.J. Dekker, Phys. Stat. Solidi, 13 (1966) 181. [171 T. Nakamura et al., Phys. L&t. 12 (1964) 178. t181L. Gunther, unpublished results. 1191M. Ure and P. Flinn, Mijssbauer Effect Methodology, vol 7, ed. I. Gruverman (Plenum, New York, 1971) p. 45. 1201 B. Kolk, Phys. Rev. B12 (1975) 1620. 1211D.H. Jones and KKP. Srivastava, Phys. Rev. B34 (1986) 7542. [221K.K.P. Srivastava and D.H. Jones, ICAME (1987). [231F. Van Der Woude and A.J. Dekker, Phys. Stat. Solidi 9 (1965) 775. [241F. Van Der Woude and A.J. Dekker, Solid State Commun. 3 (1965) 319. 1251A. Abragam, Principles of Nuclear Magnetism, (Oxford Press, Oxford, 1983). [261R. Sack, Molec. Phys. 1 (1958) 163. t271F. Varret, J. Phys. Chem. Solids 37 (1976) 265. [281H. Frauenfelder, Mijssbauer Effect (W.A. Benjamin, New York, 1963). [291We are indebted to G.C. Papaefthymiou for this obsetvation. 1301CM. Schelinger, D. Griscom, G.C. Papaefthymiou and D.R. Veblen, J. Geophys. Res. 93 (1988) 9137. [311 J.C. Slonczewski, J. Appl. Phys. 32 (1961) 253s. [32] T.G. St Pierre et al., J. Mag. Mag. Mater 69 (1987) 276. [331 S.H. Bell, B&hem. Biophys. Acta 787 (1984) 227; E.R. Bauminger, Biochem. Biophys. Acta. 623 (1980) 237. 1341We have since submission produced such a comprehensive theory. See L. Gunther and M-E.Y. Mohie-Eldin, to appear in J. Magn. Magn. Mater.