Magnetic hyperfine splitting in superparamagnetic particles in external magnetic fields

Journal

160

of Magnetism

and Magnetic

MAGNETIC HYPERFINE SPLITl-ING IN EXTERNAL MAGNETIC FIELDS

IN SUPERPARAMAGNETIC

Steen MIZIRUP, Per H. CHRISTENSEN

’

Laboratory of Applied Physics II, Technical University of Denmark,

DK-2800

Materials 68 (1987) 160-170 North-Holland, Amsterdam

PARTICLES

Lyngby, Denmark

and Bjerne S. CLAUSEN Haldor Topsee Research Laboratories, Received

27 February

DK-2800

Lyngby, Denmark

1987

The magnetic hyperfine splitting in Mossbauer spectra of superparamagnetic particles, induced by an external magnetic field, has been calculated. Numerical results have been obtained both for isolated particles with a finite value of the magnetic analytical approximations are derived. The anisotropy energy constant and for strongly interacting particles. Moreover, theoretical results are compared with results of experimental studies of supported a-Fe particles and magnetite particles in ferrofluids.

1. Introduction Ultrafine particles often have unique structural, chemical and magnetic properties and are thus of great interest both from a scientific and from a technological point of view. Such particles are extensively used in the electronic and the chemical industry, for example in magnetic memories, ferrofluids and catalysts. Mbssbauer spectroscopy is an ideal technique for studies of the magnetic properties of ultrafine particles [l-3]. In the temperature interval between the superparamagnetic blocking temperature and the Curie temperature the magnetic hyperfine splitting collapses becau e of fast superparamagnetic relaxation. Howeve , a magnetic hyperfine splitting can be induced b\ an external magnetic field. From the field dependence of the induced hyperfine splitting the magnetic moment of the particles can be determined, and if the magnetization is known, r Present address: Denmark.

Beta Plan International,

Helsinger,

151: ~(~,=coth(~)-&,

(1)

where p(T) = M(T)T/ is the magnetic moment of the particle, V is the particle volume, M(T) is the magnetization, T is the absolute temperature, k is Boltzmann’s constant and B is the applied magnetic field. For large magnetic fields (p(T)B/kT > 2) the Langevin function can be approximated by:

L{q+l---&.

0 Elsevier Science Publishers B.V. Physics Publishing Division)

0304-8853/87/$03.50

(North-Holland

DK-3000

the particle size can be estimated [1,2,4-61. This method for particle size determination is based on the assumption that the magnetic anisotropy energy is negligible compared to the Zeeman energy of the particle. Moreover, the magnetic interaction with neighboring particles is also neglected. When these conditions are fulfilled, the average magnetic hyperfine splitting of the Miissbauer spectrum is essentially proportional to the Langevin function

(2)

S. Merup et al. / Superparamagnetic

This relation has been used for determination of the particle size of microcrystals of, for example Fe,O, [5], metallic Fe [4,7], Ni [IS], Co [9], Fe-Co alloys [lo] and amorphous Fe-C alloy particles

161

particles

8-(O,O,Eu

Pll.

However, in the case when the magnetic anisotropy energy or the magnetic coupling among the particles is not small compared to &T)B, eqs. (1) and (2) may not be valid approximations. The superparamagnetic behaviour of an aligned assembly of particles having uniaxial anisotropy energy in magnetic field B applied along the axis of symmetry has previously been described [12]. However, a more general description is needed in order to explain the magnetic behaviour of, for example, small particles which are fixed on a support. In such samples the orientation of crystal axes is random. In sections 2 and 3 of the present paper we discuss how the hyperfine field of randomly oriented particles in the presence of an external magnetic field is influenced by the magnetic anisotropy. In section 4 we give the results of calculations of the influence of magnetic interaction among the particles. In all the calculations it is assumed that the superparamagnetic relaxation is fast compared to the time scale of Mijssbauer spectroscopy (= 1O-8-1O-9 s), i.e. the temperature is above the blocking temperature. The results are compared with experimental observations in section 5.

Consider a particle of volume V, with uniaxial anisotropy, and with its easy direction of magnetization defined by the unit vector e. The z-axis is chosen as the direction of the applied magnetic field B, and the xz-plane is determined by the plane (e, B) (see fig. l), i.e. e = (sin 8, 0, cos 8), where 8 = ~(e, B). The energy of the particle is described by: = -KV(

e*u)2

-I.~(T)(B*u),

(3)

where K is the magnetic anisotropy energy constant, and u is a unit vector describing the direction of the magnetization, M(T), in the particle. According to eq. (3), the energy depends on both

“0

lb-e ‘e-bin

O,o,cos 0)

e

X

Fig. 1. Cartesian system used for calculation of b( (Y, p).

the magnitude of B and the angle 8 between B and e. This is illustrated in fig. 2. For calculation of the magnetic properties of a sample of superparamagnetic particles we introduce the partition function: Z= /exp[or(e*u)2

+ $(u*B)]

du,

(4)

where LY= KV/kT and /I = p(T)B/kT. The integration is taken over all directions in space. For a given ferromagnetic particle with easy direction e, the reduced average magnetization (M(T))/ M(T) is equal to the average of u which is given by: (u> = Z-‘Ju

2. Influence of magnetic anisotropy

E(u)

1

exp[a(e*u)2

+ i(u*~)]

du. (5)

Note that (u,,) = 0, i.e. (u) lies in the xz-plane and can be represented by (u) = !$a, P)(sin uo, 0, cos uo),where u. = I, B) and b( a, P) is defined as: b(a,

P) = i(u) I = (m2

+ G4’)“‘.

(6)

The magnetic splitting of the Mijssbauer spectrum of a superparamagnetic particle above the blocking temperature is proportional to the sum of the external magnetic field and the induced average hyperfine field. The latter is proportional to b(cw, p). Thus, the total magnetic field at the

162

//,ll/l~1111111

0"

180’

Il/l/lll/lll/llllllI

360’

0”

I11111111~IIII~C111~

180”

360’

0”

180”

lIIIIfII1111JJ111/1/

360’

0”

180”

Angle between G and E Fig. 2. Magnetic energy of a ferromagnetic microcrystal as a function of the angle between the magnetization applied magnetic field. Results are given for various values of the angle 0 between B and e for /3/a = p(T) B/KV

nucleus may be written:

Bobs

=

B,b(a,/3)-B cosq,,

(7)

where B,, is the saturation hyperfine field, and it is assumed that B -=z B,b(a,j3)and that the hyperfine field is opposite to the magnetization (as is the case for iron). The induced magnetic hyperfine splitting depends on both the magnitude of the applied magnetic field and the angle 8 between the easy direction e and the applied field. This is illustrated in fig. 3 where we have shown the variation of b(a[, p) as a function of the angle 19 for different values of the parameter /I for OL= 2.0. The results were obtained by numerical integration of eq. (5). This dependence of b(a, p) on the angle B implies that the existence of randomly oriented particles in a sample will result in a Miissbauer spectrum which consists of a sum of spectra with different magnetic splittings, i.e. the lines will be broadened.

360’

direction and the = 0, 1, 2,4 and 8.

In a sample with random orientation of the easy directions of magnetization the average value of the reduced hyperfine field, (b( (Y, P)) can be

e (deg ) Fig. 3. Reduced induced hyperfine field, b( a, p), as a function of the angle 8 between B and e. Results are given for a = 2 for the values of p indicated in the figure.

S. h40rup et al. / Superparamagnetic

calculated by averaging b(a, p) over all orientations of e. Values of (b(a, p)) have been obtained by numerical integration of eq. (5) and subsequent averaging over all directions of e. Fig. 4 shows the results plotted as a function of fi-’ for (Y= 0, 4, 8 and 16. For comparison we have also shown the approximation L(p) = 1 - p-’ (eq. (2)) which is valid for LY= 0 and p-’ +z 1. The figure illustrates that for small values of CX(1y5 4) the linear approximation is a reasonably good approximation for large values of /3. Thus, it can be concluded that moderate values of the parameter (Y do not introduce significant errors when the particle size is estimated by use of eq. (2). Finally, it should be emphasized that in the above calculation of b(a, p), particles of only one size and with spherical shape have been considered. If the particles have a distribution in sizes, this will give rise to an additional line broadening in the Mbssbauer spectra, and if the particles deviate from the spherical shape, there will be a distribution in demagnetization fields [1,4] which also contributes to the broadening of the lines.

particles

163

3. Derivation of the high field approximation In certain special cases it is possible to derive simple approximate analytical expressions instead of performing the numerical calculations discussed in section 2. In this section we derive an expression for b(a, /I) which is valid for (Y+z j3 and p z++1. The derivation is based on the low temperature approximation for b(a, p), obtained by Morup [6]: b(a,

P) = 1 -

%‘-

{(i!$)n’+(qJ.(8)

In this expression, E is the total magnetic energy of the microcrystal given by eq. (3), and uX, uY and U, are the direction cosines of the magnetization vector. The z-axis is here defined as the magnetization direction which gives minimum energy, and the x and y directions are chosen such that

The second derivatives are taken at the energy minimum. The Cartesian system used for the present calculations is shown in fig. 5. The x direction is in the plane defined by B and e. We later show (eq. (16)) that this choice

Fig. 4. Average value of the reduced induced hyperfine field (b(a, /3)) as a function of B-’ for a= 0, 4, 8 and 16. The dashed line represents the linear approximation given by eq.

Fig. 5. Cartesian system used for calculation of the high field

(2).

approximation for b( a, 8).

164

S. Morup et al. / Superparamagnetic

satisfies the condition given by eq. (9). For an arbitrary magnetization u,), eq. (3) can be written: (r&P uy, E = -p(u,B,

+ u,B,)

- KV(e,u,

lowest energy can thus be found from the relation: direction,

+ ep,)‘.

(10)

When the thermal energy is small compared to the magnetic energy, the magnetization direction will be close to the z direction. We may then use the approximation: u, =

(1 - a; - U:) 1’2 z 1 - +uf - :a;.

(11)

(12)

- 2KVe,e,,

2a sin u. = -sin P

w0 cos w0

(21)

or cos ug = 1 [

2

2ff B

(1

sidw,

cos2wg

l/2 1 .

(22)

In the following we assume that OL-=xp. Eq. (22) can then be approximated by: 2

After inserting eq. (11) into eq. (lo), we can find the derivatives at the minimum: -pB,

particles

(13)

cos

ug

=

Inserting obtain:

1- 2 s (

sin2w, cos2w0.

1

eq. (23)

into eqs. (18)

(23) and (19)

we

(3,-pB[1+2;(2cos’w,-I)

= pB, + 2KV( ez - ez),

(15)

= pB, + 2KVe/e,2,

(16) Introducing the angles u0 and w,, defined in fig. 5, we find: = pB sin u0 - 2KV sin w,, cos wa,

An expression for b(cz, p) can now be obtained by inserting eqs. (24) and (25) into eq. (8). After performing a Taylor expansion using again the condition that LY-=-CL j3, we finally obtain:

(17) b(cY, p) = 1 -p-l

a2E i-iau:

0

= pB cos u,, + 2KV(2

cos2wo - l),

= pB cos u,, + 2KVcos2w,.

(18)

09)

Since the z-axis was defined as the magnetization direction which gives the minimum energy, we find that

1

aE au,,

-= i

According

0

.

(20)

to eqs. (17) and (20) the direction

of

+2

(

1+ ;(l

- 3 cos2w0)

‘y 2(l-3cos2wo+4cos4wo) (P1

. i (26)

From this expression it is seen that the value of b(a, p) depends significantly on the value of the angle wO. This is in accordance with the numerical results shown in fig. 3. The magnetic splitting of the Mossbauer spectrum is proportional to the total magnetic field at the nucleus. Since the external magnetic field B normally is small compared to the induced field, B,b(cw, p), and because the fields are normally

S. Mwup

antiparallel given by: B ohs = B&Y, - Bdb,

et al. / Superparamagnetic

for 57Fe, we find that the total field is

,8) - B cos u,, P) 2

sin%,

cos2w0

(W The width of the distribution will be of the same order of magnitude. In practice, a microcrystalline sample contains a distribution in particle sizes and magnetic anisotropy energy constants. This will lead to additional broadening of the hyperfine field distribution. In a sample with identical microcrystals and random orientation of the easy directions the average magnetic hyperfine field can be found analytically by averaging b(a, p) (eq. (26)) over all directions of e. It is noteworthy that after averaging, one finds that both the first order and second order terms in LX/P vanish, and thus one obtains: P))

= I - P-’

(29)

which is equivalent to eq. (2). Therefore, when the field dependence of (b( (Y, p)) is used for particle size determination in a sample of randomly oriented microcrystals, the corrections due to the finite value of cr are negligible when (Y-=SC ,8, i.e. when the magnetic anisotropy energy is much smaller than the Zeeman energy

4. Influence ticles

of magnetic

interaction

negligible. However, recent Mossbauer studies of various microcrystalline systems have shown that the magnetic interaction energy may be of the same order as or even larger than the magnetic anisotropy energy [13-191. The contribution to the magnetic energy of a crystallite (i) which arises from the magnetic interaction with the surrounding magnetic crystallites (j) may be written as [6,13]:

1. (27)

In spectra of samples of microcrystals with a random orientation, a distribution in magnetic hyperfine fields will be present. If the values of (Y and p are identical for all particles, the distribution can be calculated from eqs. (26) and (27). The difference in magnetic splitting for particles with e parallel and perpendicular to B is given by:

(b(%

165

panicles

among par-

(30)

Kz is a magnetic coupling constant for the interaction between the crystallites i and j with magnetization vectors Mi( T) and Mj( T), respectively. K$/ may have contributions from both the magnetic dipole interaction between the magnetic moments of the particles, pi and cr,, and the exchange interaction between pairs of surface atoms belonging to the two crystallites. The magnetic behavior of a system of interacting particles with negligible magnetic anisotropy in zero applied magnetic field has been calculated [6,13] using a modified Weiss molecular field theory. Thus in eq. (30), Mj(T) is replaced by the average magnetization and is written: E, = -&P,(T)

l

(M(T)),

(31)

where K,

= XX;.

(32)

Below a certain temperature, Tp, the magnetic interaction between the particles leads to ordering of the magnetic moments. This is the so-called superferromagnetic state [13]. The magnetic properties of the system can conveniently be described by the reduced average magnetization defined by: b(T)

= (M(T))/M(T).

(33)

The transition temperature above which the system is superparamagnetic is given by [6,13]: T

=

P In the above calculations it was assumed that the magnetic interaction among the crystallites is

-CK$~W;(T).M~(T),

Ei=

KmM2RJ 3k *

Below Tp, b(T)

can be derived

(34) from the implicit

166

S. Mwup

et al. /

Superparamagnetic particles

equation: b(T)

=L{

J$??(T,

T,)b(T)),

(35)

where m(T,

T,) =M(T>/M(T,).

(36)

In the following we discuss the magnetic properties of such a superferromagnetic system in the presence of an applied magnetic field. It is assumed that the magnetic anisotropy energy is negligible compared to the magnetic interaction energy. Thus, we consider a system with the magnetic energy given by: E, = -&&f,(T)

l

(M(T))

The average magnetization implicit equation: (M(T))

- VMi(T)

lB.

(37)

is now given by the

0.41

,

,

0.0

,

,

0.2

,

0.4

,

,

0.6

( 0.6

P

Fig. 6. Reduced induced hyperfine field b(P, y) as a function of p-’ for y = 0.0, 0.5, 1.0, 1.5 and 2.0. The dashed line represents the linear approximation given by eq. (2).

= M(T)

which leads to: Introducing b(P, y) = (M(T))/M(T), by combining eqs. (34) and (38): b(P,

u>=~{?+(k

v>+P),

we obtain

(39)

where

Y =

+“(T,

T,).

(40)

Numerical values of b(P, y) have been obtained from eq. (39). Fig. 6 shows plots of b(j?, y) as a function of p-l for y = 0.0, 0.5, 1.0, 1.5 and 2.0. We have also indicated the linear approximation b(j3,O) - 1 - p-i. The magnetic hyperfine splitting of a Mbssbauer spectrum can be found from:

b(P, Y) -

/-dT)B 3k[T-

Tpm2(T,

Eq. (43) is similar to the Curie-Weiss law. The main difference is that Tp is multiplied by the temperature dependent factor m2(T, T,). However, if T is well below the Curie temperature of the material, m2(T, T,) = 1. For large values of the argument of the Langevin function, eq. (39) can be approximated by:

b(B, Y) = l-

1 Yb(B> Y) +

=

W(P,

Y) - B.

(41)

For small arguments of the Langevin function, i.e. for T > Tp and p -=.z1, e.q. (39) can be approximated by: b(P,

Y>-3[Yw,

u>+P]

(42)

(44)

P.

For p z++1 and p x- y, this leads to: 2

b(&y)=l-p-i

1$+--++$ i

B obs

(43)

Tp)] .

(45)

1

which gives the first and second order corrections to eq. (2) when a weak interaction among the particles is present. The magnetic interaction may be described in terms of an effective internal magnetic field B,(T, B) acting at the particle. This effective mag-

S. iUm4p et al. / Superparamagneticparticles

netic field is given by: Bi(T,

B, = ~(~(T)).

(46)

By using eqs. (34) and (36) one finds: B,(T,

B) =

3kTpm2(T, T,)b(P,Y)

(47)

P(T) The ordering pressed by: T

=

temperature

Tp can now be ex-

P(O)Bi(T=O)

P

3km2(0,

(48)

T,)

and eq. (44) can be written:

Reduced

temperature,

t=T/Tp

Fig. 7. Reduced induced hyperfine field b( t, 6) as g function of the reduced temperature I = T/T, for 6 = 0.0,0.02, 0.10 and 0.50.

kT b(T’

B) = ’ - p(T)[

B + B,(T,

B)] *

(49)

It is interesting that the magnetic coupling may be estimated from the field dependence of the magnetic hyperfine splitting above Tp if the magnetic anisotropy is negligible. This can be done, for example, by rewriting eq. (49) as follows: [l-b(T,

B)]-‘=~[B+Bi(T,

B)].

(50)

Thus, a plot of [l - b(T, B)]-’ as a function of B gives a straight line with slope p(T)/kT and intercept p(T)Bi(T, B)/kT. In earlier publications [6,13] we have discussed the temperature dependence of b(j3, y) in zero applied magnetic field. It is easy to extend the calculations to the case when j3 # 0. In the following we assume for simplicity that the temperature is well below the Curie temperature of the.material, i.e. m’(T, Tp) = 1 and p(T) = p(O). Then eq. (39) can be written: b(fi,

S)=+(B,

a)+$

(51)

where

(52) In fig. 7 we show the temperature dependence of b(/3, 6) as a function of the reduced temperature t = T/T, for 6 = 0.0,0.02,0.10 and 0.50.

5. Discussion The results obtained above are useful for the interpretation of Miissbauer spectra of superparamagnetic particles in applied magnetic fields, especially when the spectra are used for particle size determination. As an example of a study of non-interacting superparamagnetic particles with large anisotropy we consider the spectra of carbon-supported &-Fe particles published by Christensen et al. [7]. The analysis of the results, based on eq. (2), yielded a magnetic moment of about 1.4 x 10P2” J T-’ corresponding to a particle size of 2.5 nm if the particles are assumed spherical. The temperature dependence of the zero field spectra indicated a very large magnetic anisotropy energy constant, K= 0.7 X lo6 J rne3, Therefore, tit 80 K we find that (Y= 5 and /3= 12.5 for B = 1.0 T, i.e. a/P = 0.4. Thus, the condition that a/B -K 1 was not fulfilled in this study because. the maximum external field used for the particle size determination was 1.03 T. However, the fact that (Y does not affect the average splitting of a high-field spectrum of randomly oriented particles up to second order in u//3 (see eq. (29)) suggests that the particle size determination may not be seriously affected by the large anisotropy. In fact, it can be seen from fig. 4 that the average splitting is only little affected by a value of cyof the order of 4 for 8-l 5 0.5. The largest value of /3-’ used for the

168

S.Mwup et al. / Superparamagnetic particles

particle size determination was about 0.5 (T = 300 K, B = 0.55 T). It can therefore be concluded that the error in particle size introduced by neglecting the magnetic anisotropy was insignificant in this case. It should also be noticed that the procedure for particle size determination from the magnetic field dependence of Mijssbauer spectra, discussed in this paper, is only applicable when the superparamagnetic relaxation is so fast that the magnetic hyperfine splitting collapses in zero applied magnetic field. This is only the case when (Y is smaller then about 5-10 [6], and for such values of (Y the magnetic anisotropy does not significantly affect the particle size determination. The lines in the magnetically split spectra of the carbon-supported a-Fe particles were substantially broadened, even in an applied field of 1.0 T at 80 K. Part of this broadening can be attributed to the random orientation of the easy directions of magnetization of the particles. According to eq. (28) the width of the distribution in hyperfine fields due to this effect should be of the order of 3.5 T. This corresponding to a line broadening of the outer lines of the order of 1 mm s-l which is of the same order of magnitude as the line width observed in the experimental spectrum. When ultrafine magnetic particles are prepared on a high surface area support, it is often reasonable to assume that they are well separated, and therefore the magnetic interaction can be considered negligible as in the example discussed above. However, if the magnetic particles are unsupported, they may be in close contact, and there may then be a significant magnetic interaction. In previous studies of ultrafine goethite particles [13,15] it was concluded that the exchange interaction between pairs of atoms belonging to neighboring crystallites was responsible for a strong magnetic interaction between the particles. This type of interaction is expected to be insignificant if the particles are separated by more than l-2 nm. However, if the particles are ferro- or ferrimagnetic, the magnetic dipole interaction may still be significant even if the particles are separated by several nanometers. As an example we consider a ferrofluid in which the magnetic particles are covered by

surfactant molecules which prevent close contact between the magnetic surface atoms. The magnetic field, B,, acting on one particle in such a system has been calculated by Kneller [20] who assumed that the particles are spherical and randomly distributed in an elipsoid of the non-magnetic material. At T = 0 'thefield B,(O)is then given by: B, (0) = B - CL~N&M(O) + ~loNt,aM(O),

(53)

where cl0 is the vacuum permeability, Ns is the demagnetizing coefficient of the elipsoid, and (Y is the volume fraction occupied by the ferromagnetic particles. The last term in eq. (53) represents the field from the close environments of the particle. The value of N, cannot be calculated easily because of the random distribution of the particles in the sample, but it is reasonable to assume that it is of the order of l/3. Np is in the range between 0 and 1, depending on the shape of the sample and the magnetization direction. We thus find that B,(T= 0),which is the sum of the last two terms in eq. (53), is of the order of magnitude: 1B,(T=O)I = &.i,,c&(O).

(54)

The packing fraction (Y is given by: CX= V/&,

(55)

where V is the particle volume, and d is the mean distance between the particles. By inserting eq. (55) into eq. (54) we find:

,e,(T=O),=+$ Both theoretical [21,22] and experimental [23,24] studies of ferrofluids have indicated that the particles may not be randomly distributed, but often show a tendency of chain formation. This is in particular the case when a ferrofluid is exposed to an external magnetic field. In this case the value of B,(T) should therefore rather be calculated by using a model in which spherical particles form an infinite linear chain with a distance d between neighboring particles (center to center). Using this model we find: ]B,(T=O)]

=?

f @+?++(3,, n=i (nd)3

(57)

S. Msrup et al. / Superparamagnetic

where .Qx) is the Riemann zeta function for which c(3) = 1.202. The values of B,(T= 0) found from eqs. (56) and (57) differ by less than 15%. The ordering temperature T, of this system can be calculated by inserting Bi( T = 0), estimated from eq. (57) into eq. (48). We then find: (58) As a numerical example we may consider a typical ferrofluid containing 8 nm magnetite particles with a separation d = 12 nm. Inserting these values in eq. (57), we find that B,(T = 0) - 0.036 T. From eq. (58) we find that Tp = 110 K. This value is similar to those found in magnetic susceptibility measurements of ferrofluids which exhibit a Curie-Weiss behaviour [25,26]. At 220 K we find from eq. (40) that y = 1.5, i.e. the dependence of b(@, y) follows the curve (y = 1.5) shown in fig. 6. If the finite value of y is neglected and the magnetic field dependence of b(/3, y) is used for particle size determination, the resulting particle size will be about 6.5% too large. 6. Conclusions The results presented in this paper illustrate how the magnetic field dependence of the induced magnetic hyperfine splitting of Mbssbauer spectra of superparamagnetic particles is affected by a finite value of the magnetic anisotropy energy constant and by a magnetic interaction among the particles. In samples with isolated and randomly oriented particles the average hyperfine field is only little influenced by the magnetic anisotropy when the value of p = p(T)B/kT is large. Therefore, the particle size determined on the basis of eq. (2) is not significantly affected by the magnetic anisotropy. A magnetic interaction among the particles will affect the induced magnetic hyperfine splitting, but at temperatures well above the ordering temperature, Tp, the error introduced by using eq. (2) is small. The results of the calculations seem to agree well with experimental observations.

particles

169

Acknowledgements Support from the Danish Technical Research Council is gratefully acknowledged. The authors are indebted to E. Tijrnqvist and J.E. Knudsen for valuable discussions.

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