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Ni containing solid Kr bubbles studied with neutron depolarization and small-angle neutron scattering
104
Journal of Magnetism and Magnetic Materials 98 (1991) 104118 North-Holland
Ni containing solid Kr bubbles studied with neutron and small-angle neutron scattering R. Rosman Interjacultair
depolarization
and M.Th. Rekveldt
Reactor Instituut,
Derf
University of Technology, 2629 JB Deljt, Netherlad
Received 21 December 1990
Neutron depolarization (ND) and small-angle neutron scattering (SANS) experiments on Ni containing solid Rr bubbles are presented and discussed. The aim of the measurements is to study the effect of the Kr bubbles on the local magnetization. For the first time ND measurements in relatively large fields up to 80 kA/m could be performed. However, the ND results do not yield information about the Kr bubbles. The SANS patterns at applied fields larger than 400 kA/m are isotropic and are in agreement with a high fraction of small (radius = 1.5 nm) Kr bubbles. Also a small fraction of larger (radius 2 15 nm) Kr bubbles or other nuclear inhomogeneities is likely to be present. The magnetic scattering is in agreement with a local magnetization which is affected by the demagnetization fields of these inhomogeneities.
1. introduction The behaviour of high concentrations of inert gases trapped in metals has been studied for many years [1,2]. Interest in these systems has been recently enhanced by the discovery of precipitates of heavier inert gases in the solid state (‘solid bubbles’) [3,4]. Many standard techniques, in particular transmission electron microscopy (TEM) (see ref. [5] for a recent review), have been used to study the latter systems. With one exception the heavier inert gases have been introduced into relatively thin near-surface layers by ion-implantation techniques. The exception was the production by Whitmell [6] of bulk (1 cm thick) copper and nickel material produced by a combined sputterdeposition technique containing between 3 and 5 at% krypton. This material and its response to annealing has been studied by Evans and coworkers using primarily TEM, SEM and positron annihilation methods (e.g. refs. [7-lo]). The asformed structure was shown in both Cu-Kr and Ni-Kr to contain a high concentration of small Kr bubbles which were in the solid state [9]. The Ni-Kr material has been used’in the pre0304-8853/91/$03.50
sent study to investigate the effect of the Kr bubbles on the local magnetization of Ni. The Kr bubbles may in principle affect the local magnetization through their demagnetization fields and through anisotropy fields induced’ by their pressure (magnetostriction). As the bubble pressure can be extremely high, i.e. up to 10 GPa, the induced anisotropy fields, which are radially oriented, are large (up to 600 kA/m). The aim of the measurements presented and discussed in the paper is to derive information on the local magnetization orientation. The Kr bubbles with the surrounding Ni atoms whose magnetic moments are affected by the bubble pressure can be considered as magnetic inhomogeneities, denoted as ‘magnetic bubbles ’ further on. By determining the size of the magnetic bubbles, and in particular the decrease of their size with an increasing applied magnetic field, information on the local magnetization orientation is obtained. In order to deduce the size of the magnetic bubbles, neutron depolarization (ND) and smallangle neutron scattering (SANS) have been used. In a ND experiment, the polarization change of a
0 1991 - Elsevier Science Publishers B.V. (North-Holland)
R. Rosman, M. Th. Rekveldi
polarized neutron beam during transmission through a magnetic medium yields the size and the mean orientation of the magnetic inhomogeneities. In a SANS experiment, information about both nuclear and magnetic inhomogeneities is obtained from the broadening of the neutron beam. SANS has been used before to study He bubbles in Ni [11,12]. Section 2 presents brief reviews of the ND and the SANS theory. Details about the samples and the ND and SANS instruments used are given in section 3. Section 4 contains the experimental results. The latter together with theoretical calculations on the local magnetization in Ni-Kr and in pure Ni are discussed in sections 5. Some conclusions are formulated in section 6.
2. ND and SANS theory Brief reviews of the ND and the SANS theory are given in sections 2.1 and 2.2, respectively. 2.1. Review of the ND theory The three-dimensional neutron depolarization (ND) technique is a powerful method to study static and dynamic properties of magnetic structures in the micron and submicron region (e.g. refs. [13,14]). The polarization vector of a polarized neutron beam is analyzed after transmission through a magnetic medium in a ND experiment. During transmission the polarization vector is affected by the micro-magnetic state of the medium: mean magnetic induction results in a net precession of the polarization vector around the mean magnetic induction while magnetic inhomogeneities result in an effective shortening of the polarization vector, called depolarization henceforth. Generally the mean size along the neutron path of the variations, the mean orientation of these variations and the mean magnetization are obtained. The range of magnetic inhomogeneities which can be measured covers 10 nm up to mm’s, making ND to some extent complementary to SANS. In a ND experiment, the polarization change generally is described by a polarization matrix D. Information about the magnetic inhomogeneities
105
/ Ni containing solid Kr bubbles
is often expressed by the parameters 5 and y,(i = x, y,z). When the inhomogeneities are magnetically uncorrelated, t = p;;,
(1)
Yj =
(2)
a;;/5 ,
with =dr’ AB,(x,
Y, 2) AB,(x,
Y, 2’)
/(V). >
(3) Here, V is the volume of the magnetic inhomogeneity, AB(r) = B(r) - (B), B(r) is the local magnetic induction, (B) the mean magnetic induction, ( ) an average over all inhomogeneities and E the inhomogeneity volume fraction. The propagation direction of the neutron beam (eo) is along z. The quantity 26 approximates to the product c(v(AB,)‘), with v the size of V along e,. The quantities y, give the mean orientation of AB(r). Roughly speaking, y, = (nf), with n the unit vector along AB(r). When the inhomogeneities are isotropically distributed, y, = y, = l/4, y, = l/2 (‘intrinsic depolarization anisotropy’ [15]). Positive magnetic correlations between A B,( r) of neighbouring inhomogeneities along e, result in an increase in 5 and yj. A relation between [ and y, on the one hand and b on the other which is generally valid cannot be given. The depolarization matrix of a magnetic medium with a mean magnetization (M) along y is given by D,, = D,, = cos + ,-&~((‘+Y,.)/2), D,, = -D,,
= sin + e-‘1L&((l+Y,)/2),
DYY= e-C’L.~“-“‘,
(4)
Dx,, = D,,x = qyZ = DZY= 0. Here, L, is the sample thickness along the neutron transmission direction (e. 11z), Q = p,,( M)L,& the rotation angle of the polarization vector, c, = 2.18 X 1O29X2 m-4T-2 and X the neutron wavelength. In eq. (4) it has been taken into account that due to the precessional motion of the polarization vector no distinction can be made between
106
R. Rosman, M. Th. Rekveldt
y, and y: if 9 is in the order of or larger than 7r/2. The quantities [, v, and + follow from eq. (4) by (=
_ ln(det@)) 2c,L,
(5)
’
2149,) ” = 1 - ln(det( b)) +=arctan(
:Iiz
(6)
’ j.
(7)
Note that a single ND experiment yields the rotation angle 9 modulo 2a. Expressions for 5 and y, can also be formulated in the Fourier space. For uncorrelated magnetic inhomogeneities in a magnetic bulk, (r‘I = 8a4c
AB/( -s)d’s
/(I’),
/ Ni containing solid Kr bubbles
0 is considered, with K the scattering vector. The scattering may be due to both magnetic and nuclear inhomogeneities. The scattering pattern yields information on the size, shape and orientation of these inhomogeneities. With most SANS instruments, inhomogeneities in the size range between 3 and 100 nm can be studied. In case of single scattering, the scattering intensity Z(K) is related to the macroscopic differential cross section da(tc)/dti by da Z(K) = Kd&+,
with K the scattering vector, K a system constant and t the sample thickness. The relation between da(r)/dQ and the microscopic structure of the sample is given by
(8) 1 da dDK( )=-I/ KVe
with
AB(s) =
(2;,3/yAM*
AM*(r,
i) =s’x
[(M(r)
(r, 5) eis.r d3r, -(M))
2
““p(r)
d3r
(11)
,
s
with V, the sample volume and p(r) the scattering amplitude density. The latter may have a magnetic as well as a nuclear part and is given by
xs”].
Here, ,!? is the reciprocal xy-plane, 5 = s/ 1s 1and M(r) the local magnetization. For convenience, the quantity 6 = 2E/(P&J2
(10)
(9)
is used instead of 5 in this paper. The quantity S is the magnetic correlation length of (AB(r))2 (and not of B2(r)!) along e,, based on AB(r) = pOM, and 6 = 1. Here, M, is the medium spontaneous magnetization. For magnetically uncorrelated inhomogeneities, S approximates to CV. For a more extensive review of the ND theory the reader is referred to refs. [15-171. The magnetic bubbles (Kr bubbles with the surrounding Ni atoms, see section 1) are the magnetic inhomogeneities in Ni-Kr. Within the Kr bubbles, AB( r) = $pOMs. 2.2. Review of the SANS theory In a SANS experiment, the scattering of a (generally unpolarized) neutron beam around K =
p(r)
= n’(r)[b(r)
+P(r)(l*(r)
.S>l y
(12)
with n’(r) the atomic density, b(r) the nuclear scattering amplitude, p(r) the magnetic scattering amplitude, S the spin vector of the neutron ( 1S 1 = I), q*(r)( = 12 X (q(r) X I?), E = K/I K I and q(r) the unit vector along the local magnetization. Eq. (11) reflects the general SANS principle that the scattering pattern at small (large) K-values yields information about large (small) inhomogeneities. Often, a two-phase model is used in the interpretation of the SANS measurements. In this model the sample is considered to consist of a bulk with scattering amplitude density pi and N inhomogeneities with scattering amplitude density p2 and volume V. Ignoring possible correlations between the inhomogeneity volume and its location as well as correlations between different inhomogeneities, eq. (11) can be written as (K > 2a/( g”3): g(K)
=n(Ap)2(I’2F2(~)).
(13)
R. Rosman. M. Th. Rekveldt
Here, (AP)~ = (pi - pZ)’ is the contrast factor, n the inhomogeneity density, ( ) an average over all inhomogeneities and F(K) = (l/V)/, eir.rd3r the inhomogeneity form factor. Obviously, correlations between inhomogeneities result in deviations in da(K)/dti from eq. (13). The former are generally described by a structure factor S(K). When the neutron beam is unpolarized, nuclear and magnetic scattering are additive. Then the nuclear contrast factor (AP,)~ is given by the squared difference of the nuclear scattering amplitude density of the bulk and that of the inhomogeneities (An,)” = (~,.i -
(14)
pn,212-
The magnetic contrast factor (Ap,)’ t&J2
= I(+),@
-
(_n’p)2G
is given by
1’2
= (+):[l
- (a.~r,“],
(16)
with P = K/ IK I. When the inhomogeneities are due to deviations in the orientation of the local magnetization, (n’p), = (n’p), and @pJ2
(n’p):[l- (c.
=
(qr -
(17)
92)j2].
The form factor contains information about the average inhomogeneity size, shape and orientation and can analytically be calculated for some simple inhomogeneity configurations. For ellipsoids with a radial dimension 2a and an axial dimension 2 b (axis oriented along e,,) F2(~)
3(sin u--ucos
= [
with ~=~{(iSe~)~(b~-u~)+.~}~‘~. Independent of the inhomogeneity Guinier expression [ 181 (F2(
K))
=
e-a2R:/3
( KR~
<
1)
applies, with the Guinier radius R, given by
(20) If the inhomogeneity shape is known the inhomogeneity size distribution can in principle be deduced by a comparison of the calculated F2( K) and the measurements. For nearly spherical inhomogeneities (F’(K)) is proportional to K-~ for KR >> 1, a behaviour which is known as the ’ Porod ‘-behaviour. From eqs. (1) (8), (13) and (15) and from n’(r)p(r)q(r) = (6/2X)paM(r) it follows that the magnetic correlation length S, derived from a ND experiment, and the magnetic differential cross section da(rc)/dfim, obtained from a SANS experiment, are simply related by
(21) with dQ = ( A2/4a2) d’ic. In Ni-Kr, nuclear scattering from the Kr bubbles is expected. Magnetic scattering may arise from the magnetic bubbles. A more extensive discussion on SANS is given in refs. [18,19].
3. Experimental The Ni-Kr samples used in the ND and SANS experiments were supplied by Evans [20]. The existence of solid bubbles of the size up to a diameter of 3 nm and with a density of approximately nb = 5 X 10 24 bubbles/m3 has been shown earlier by TEM measurements [9]. Electron diffraction measurements yield a fee Kr packing density of 3.20 x 1O28 atoms/m3 [9]. The corresponding bubble pressure is in the order of 10 GPa (assuming a bubble diameter of 3 nm). TEM measurements on the specific Ni-Kr samples used in the ND and SANS experiments have not been performed. The (nuclear) scattering amplitude of Kr is bK’ = 0.785 X lo-l4 m. Assuming an atomic density within the bubbles of 3.20 X 1O28 atoms/m3, the (nuclear) scattering amplitude density of Kr is m- 2. The nuclear scattering PKr = 2.51 X lo-l4
(18)
1’
U) ’
u3
101
(15)
with the subscripts 1 and 2 referring to the bulk and the inhomogeneities respectively. When the inhomogeneities are non-magnetic (&,J2
/ Ni containing solid Kr bubbles
shape,
the
(19)
108
R. Rosman, M. Th. Rekueldt / Ni containing solid Kr bubbles
amplitude density of Ni is #’ = 9.4 x lOI rnp2, following from a nuclear scattering amplitude of bNi = 1.03 X lOI m and an atomic density of 9.1 X 1O28 rnp3. The magnetic scattering amplitude of Ni is pNi = 0.16 X lo-i4 m. Hence, the magnetic scattering amplitude density of Ni equals pz’ = 1.5 X lOI rne2. It follows that the magnetic contrast factor of Ni-Kr, (Ap,,,)‘, varies between 0 and 2.1 X 10z8 rne4. The nuclear contrast factor is more than 20 time larger, i.e. (AP,)~ = 4.7 x 1O29 rnp4. In bubbles larger than 3 nm, the Kr atomic density will be lower than the value quoted above. This will result in an increase of (a~,,)~. The SANS measurements have been performed on the instrument PAXY of the Laboratorium Leon Brillioun (Saclay, France) *. The instrument has a two-dimensional (64 x 64 cm2) position sensitive detector and uses an unpolarized neutron beam. Using a sample-detector distance of either 4.981 or 6.830 m and a neutron wavelength of 0.8 nm, a K-range of 0.06-0.5 or 0.05-0.4 nm-’ is covered during a single experiment. Apart from measurements on Ni-Kr also a few measurements have been performed on pure Ni. Unfortunately, no Ni sample prepared using the sputtering method with the use of which the Ni-Kr samples have been made was available. Magnetic fields up to a maximum of Ha = 1800 kA/m have been applied along y (e,, along I). The macroscopic differential cross section for the scattering vector parallel and perpendicular to the applied field has been obtained by averaging over all detector positions between the two lines which make angles of -0 o and 8 o and the two lines which make angles of (90-e)” and (90+8)O (e=20 or 30) with the applied field, respectively. All SANS results presented in the paper have been corrected for background and are normalized using plexiglass and vanadium. The ND instrument used has Cu,MnAl crystals magnetized to saturation as polarizer and analyzer. The polarizer simultaneously acts as monochromator (A = (0.12 f 0.01) nm, corresponding to c1 = 3.14 X lo9 rnp2Tp2). The instrument is operated at a 2 MW swimming pool reactor (HOR,
* Laboratoire Commun CEA-CNRS.
neutron:
Fig. 1. A sketch of the coil system used in the ND experiments (Yo: yoke, PP: polar pieces, C: coils, S: samples). The arrows give the orientation of the magnetic field.
Delft, Netherlands) and uses dc-coils as polarization turners. Up to now, ND measurements have been performed only when applying magnetic fields smaller than about 1 kA/m. This is a result of the sensitivity of the polarization vector to stray fields arising from coils or magnets producing the fields needed on the sample. The larger the applied fields are, the larger the stray field and the larger the effect on the depolarization. Consequently, high coercive media are always studied in the remanent state (e.g. refs. [14,21,22]). However, in order to substantially affect the magnetic inhomogeneities within Ni-Kr magnetic fields up to 200 kA/m are needed. Therefore, a new coil system has been made, in which ND experiments in the presence of magnetic fields up to 80 kA/m can be performed. The coil system consists of two almost identical coils (coil constant L = 1.08 X 104/m), 4 polar pieces (diameter 20 mm) and a magnetic yoke (fig. 1). Two samples (2.07 X 7.80 x 7.70 mm3 and 2.27 x 7.80 x 7.60 mm3) can be sandwiched between the polar pieces, with their smallest size along z. The induced fields in the coils are anti-parallel (I] *y-axis). The maximum field at each sample position is 80 kA/m. In order to have a homogeneous field over the cross-section of the neutron beam a relatively small diaphragm (diameter 4 mm) has been used. The two anti-parallel coils and two samples were necessary as the rotation angle due to either a single coil or a single sample can be as high as 110 or 75 rad, respectively. Such a large rotation angle is undesirable, as it may
R. Roman.
M. Th. Rekveldt
109
/ Ni containing solid Kr bubbles
cause a strong additional depolarization for the polarization component perpendicular to Ha (see refs. [14,17]). A total sample thickness of 4 mm is needed in order to have a measurable depolarization effect at large fields. All experiments have been performed, starting from a maximum field (Ha = 80 kA/m in the ND and Ha = 1800 kA/m in the SANS measurements).
4. Experimental
results
The results of the ND and the SANS measurements are presented in sections 4.1 and 4.2 successively. 4.1. ND measurements Fig. 2a gives the element D,,,, of the empty coil system and the coil system containing the two Ni-Kr samples versus the current Z through the coil (starting from Z = -0.8 A). The element Dyy of the empty coil system approximates to 1, while D,,y of the coil system and the samples is minimum around Z = 0 and increases with ] Z ] over the whole Z-range. Figs. 2b and c give the elements D,, and Dxzof the empty coil system and the coil system containing the samples versus ] Z ] (again starting from Z = - 0.8 A). For both the empty coil system and the coil containing the samples D = D,,x = Dyl = Dzy = 0, D,, = D,, and D,, = In the empty coil system, D,, and D,, 3&. show an oscillating behaviour, the amplitude of which slightly decreases with increasing I. The latter indicates a small depolarization by the induced field. The behaviour of D;, observed in the empty coil system is consistent with Ha oriented along y. Also the elements Ox, and Dxz of the coil system containing the samples show an oscillating behaviour. However, the amplitude of the oscillations increases with increasing ] Z 1. Both the latter increase and the increase of D,, indicates a magnetic correlation length which decreases with increasing applied magnetic field. The fact that the minimum in Dyy occurs at value of Z which slightly exceeds zero is mainly due to the hysteresis of the coil system (see fig. 3). The periodicity
-6.8
0.0
0.8
I(+Fig. 2. The elements D,,,, of the empty coil system (1) and the coil system with the Ni-Kr (2) (a), the elements 0,. (3) and 0,. (4) (b) of the empty coil system and of the coil system containing Ni-Kr (5,6) (c) versus the current I through the coils (starting from I = -0.8 A). The error bars are of the order of the symbols.
of the oscillations in D,, and D,, of the coil system containing the Ni-Kr samples is maximum at about Z = 0, corresponding to a maximum change in the mean magnetization. 5 t
aJcs O-
Man@
-5 ’ -0.8
I
I
0.8
0.0 I(A)
-
Fig. 3. The rotation angle +_ of the empty coil system versus I. The error bars are of the order of the symbols.
110
R. Rosman, M. Th. Rekoeldt
Fig. 3 gives the rotation angle of the polarization vector in the empty coil system, denoted &,,, versus I. The field Ha between the polar pieces has been calculated using Ha = c~c#I~~. The constant c2 cannot be deduced from fig. 3, as the unbalance of the coil system is not exactly known. Instead, c2 has been calculated using the coil constant of a single coil (L = 1.08 X lo4 m-‘) and the length of the polar pieces, resulting in c2 = 15.9 kA/m. Fig. 4 gives the mean magnetization (M) of the Ni-Kr samples versus Ha. The mean magnetization has been calculated using
with AL the difference in thickness between the two samples, AC#J = t& - &,, and c#+the value of + corresponding to the coil system containing the two samples. The coercive field H, of Ni-Kr appears to be smaller than 2 kA/m. The spontaneous magnetization of Ni is MS = 485 kA/m. With increasing (decreasing) applied magnetic field, (M) at first decreases (increases) before increasing (decreasing) at about Ha = 0. This is an artefact arising from the fact that fig. 4 represents a difference of the magnetization curves of the two Ni-Kr samples. Apparently, the fields at the two samples slightly differ, which results in the distorted shape of the hysteresis curve. Fig. 5 gives the quantities det(h) and DYY in Ni-Kr versus Ha, measured in an increasing and a decreasing magnetic field. Both values have been corrected for the depolarization by the empty coil system. They are minimum around Ha = 0 and increase with increasing 1Ha 1, corresponding to a
80
-80 I!, (LA/m)
-
Fig. 4. The mean magnetization (M) of Ni-Kr derived from the rotation angle cpversus Ha. The error bars are of the order of the symbols.
/ Ni containing solid Kr bubbles 1. t
det@) 0.
t
bf
0.
0.01 -80
1
I
1
HO.(RA/rn)
80
-+ Fig. 5. The quantities det(b) (a) and D,,? (b) in Ni-Kr H, (1, 3: decreasing H,, 2,4: increasing H,).
versus
decreasing magnetic correlation length. The small oscillations in det(b) around ( Ha ( = 60 kA/m are artefacts, resulting from a combination of a small misalignment of the field orientation, a possible small deviation in the calibration of the ND instrument, a large rotation angle $ (q!+,,,, = 5 rad) and a small depolarization. Fig. 6 gives the quantities S and y, in Ni-Kr versus Ha, measured in an increasing and a decreasing field. The magnetic correlation length 6 is maximum at Ha = 0 (1.5 pm) and decreases with increasing I Ha I. A small hysteresis effect is observed. Large variations are seen in y,, which is about 0.25 at Ha = 0. In particular at large I Ha Ivalues, 7, cannot be deduced accurately due to the small depolarization. The oscillations in S and y, directly follow from those in det(b). At large Ha-values, the quantity 6’ = -In Dy,,/c,L,(poMs)z (fig. 6c) is likely to be a better measure for 6 as the matrix element D,, is much less sensitive to misalignments of the coil systems or small deviations in the calibration of the instrument than det(fi). The quantity S’ smoothly decreases with increasing 1Ha I over the whole filed range. The magnetic correlation lengths 6’ and 6 are related by 6’ = 6(1 - y,)/2. It should be noticed, however, that 6 and y, cannot be independently deduced from S’.
R. Rosman, M. Th. Rekveldt
111
/ Ni containingsolidKr bubbles
0.0 0.0
I
0.1
K(nrfi’)
0
-80
2
-
Fig. 7. The total differential cross-section of Ni-Kr for I( perpendicular (a) and parallel (b) to Ha (H, = 4.77 (l), 13.4 (2), 46.3 (3), 231 (4), 1194 (5) kA/m). The error bars are of the order of the symbols.
0
80
t-l, (&A/m) Fig. 6. The quantities 6 (a), v, (b) and S’ (c) in Ni-Kr versus Ha (1, 3, 5: decreasing H,, 2, 4, 6: increasing H,).
4.2. SANS measurements Fig. 7 gives the total normalized differential cross section of Ni-Kr versus the scattering vector K for some values of Ha with K perpendicular, da’(K)/dQ, and parallel, da’(rc)/dQ, to the applied field (6) = 30 has been used, see section 3). Both differential cross sections decrease with increasing K over the full K-range. Furthermore, da’(rr)/dfi and da’(K)/dfi decrease with increasing Ha up to a value of Ha of about 400 kA/m. At higher fields, da(K)/dfi is within the error bars independent of Ha. The scattering at Ha = 4.77 kA/m is anisotropic, as da’(K)/d&? is significantly larger than da’(K)/dQ. This anisotropy, expressed in terms of the ratio A(K) = [da’(K)/dfi]/[du’(rc)/dti], decreases with increasing Ha and has disappeared at Ha 2 400 kA/m (fig. 8 gives A(K) for Ha = 1800 kA/m, 0 = 20). Apparently, the scattering at fields equal
to or larger than 400 kA/m is mainly nuclear scattering. From the observation that the magnetic scattering exceeds the nuclear scattering over the full K-range at the smallest field used and from the fact that the nuclear contrast is much larger than the magnetic contrast (see section 3) it follows that the density and/or the volume of the magnetic inhomogeneities is much larger than that of the nuclear inhomogeneities. Fig. 9 gives du’(K)/dn and du’(rc)/dfi of pure Ni for some field values (6 = 30). The differential cross sections are 15 to 30 times smaller than
0’ 0.0
I
I
I
0.2
0.4
K(nm-‘) Fig. 8. The anisotropy A = (do’/dO)/(do’/dQ) Ha = 1800 kA/m versus K.
of Ni-Kr
at
112
R. Rosman, M. Th. Rekveldr / Ni containing solid Kr bubbles
(105/m)0*4 t da’ 0.2 xi
2 (1 05/m) t da’ dR
1
K(nm-‘)Fig. 9. The total differential cross-section of pure Ni for K perpendicular (a) and parallel (b) to H, (Ha = 6.76 (l), 13.4 (2), 33.2 (3), 398 (4) kA/m). The error bars are of the order of the symbols.
those of Ni-Kr and decrease with increasing Ha until Ha = 400 kA/m. At higher fields the scattering is independent of the applied field. Also the scattering in Ni is anisotropic at H, -C400 kA/m. However, in contrast with Ni-Kr, da’(rr)/dQ exceeds do’(r)/d&? in Ni.
5. Discussion Before discussing the ND and SANS results, some calculations on the local magnetization in Ni-Kr and the corresponding ND and SANS are presented in section 5.1. Subsequently, the ND (section 5.2) and the SANS (section 5.3) results are discussed. 5.1. Model
calculations
on M(r)
Three types of model calculations are presented in this section. At first, in order to see how and to what extent the local Ni magnetization is affected by the presence of Kr bubbles, the results of numerical calculations on the local magnetization
around a single Kr bubble are given. Secondly, on the base of a simple model the expected ND and SANS in Ni-Kr are formulated. Thirdly, the expected ND and SANS effects in pure Ni are dealt with. In the calculation of M(r) around a single Kr bubble, it is assumed that M(r) is affected by the external magnetic field H,, the demagnetization field of that Kr bubble, Hd, the exchange interaction between neighbouring Ni atoms and a magnetostrictive induced anisotropy field. The effect of the intrinsic anisotropy is neglected (intrinsic anisotropy constant K,, = 3.5 X lo3 J/m3). The magnetic energy density related to H, is given by E,,,(r) = -p,,M(r)H, cos a, with cx the angle between H, and M(r). The energy density corresponding to Hd is Ed(r) = - p,,M( r) . Hd( r), with
(23) Here, r0 is the bubble radius and n is a unit vector oriented anti-parallel to (M). The large bubble pressure induces pressure gradients outside the bubble along the radial directions. As the magnetostrictive constant of Ni is negative (A, = - 3.5 X 10w5), the anisotropy field induced by magnetostriction is radially oriented. The corresponding energy density is given by Eani(r) = 1A IP( r) sin’p, with P(r) the local pressure and p the angle between M(r) and the radial direction. Finally, the exchange interaction density arising from exchange interaction between 2 neighbouring Ni atoms is Eexch(r) = -F cos 8, with 8 the angle between the atoms magnetic moments. With a nearest neighbour distance in Ni of 0.249 nm, F = 3.5 X 1Ol9 J/m3 has been calculated. As Ni has a fee-structure, each Ni atom has 12 nearest neighbours. Using energy minimalization, the local magnetization distribution around Kr bubbles with different radii r,, has been calculated at certain values of Ha. The local pressure is given by P(r) = P,,(rJr)‘, with P,, the bubble pressure. The bubble pressure is taken to be inversely proportional to R,: PO = 10/r, mPa. In the calculations M(r) is assumed to be oriented along Ha at first. Obviously, M(r) is uniaxially distributed around Ha.
R. Rosman, M. Th. Rekueldi
/ Ni containing solid Kr bubbles
This symmetry is important in the ND and the SANS effects. It follows from the numerical calculations described above that the local magnetization around Kr bubbles with radii r0 smaller than about 20 nm does hardly deviate from (M) as M(r) is either dominated by exchange interaction (closely around the bubbles) or by Ha (‘far way from the bubble’). As a result the magnetic bubble approximates to the Kr bubble and Kr bubbles with radii smaller than about 20 nm are not expected to affect the local magnetization of Ni. An increase in r,, results in a decrease in the exchange and the anisotropy energy density. It appears that M(r) around Kr bubbles with radii larger than about 20 nm is dominated by either the demagnetization field (closely around the bubble) or by the external field (r B rO). Hence, the local magnetization around Kr bubbles with r, 2 20 nm is expected to deviate from the mean magnetization and these bubbles nm with the surrounding Ni atoms may act as magnetic bubbles. As an example, fig. 10 gives the local magnetization for r, = 35 nm and r,, = 10 nm at H, = 4 kA/m (according to section 5.3, also bubbles much larger than 3 nm may exist in the Ni-Kr sample). Around the bubble with r,, = 10 nm, M(r) hardly deviates from (M) while closely around the bubble with r0 = 35 nm M(r) significantly deviates from (M). It follows from b
5
Fig. 10. The orientation of the magnetization around a Kr bubble. The circle is a cross-section of the bubble (r, = 10 nm (a) and 35 nm (b)). The orientation of the applied magnetic field is indicated (H, = 4 kA/m).
I H' -’
113
the above that irrespective of the Kr bubble radius r0 the bubble pressure PO hardly affects the local magnetization around the Kr bubble. As a result, ND and SANS measurements of the magnetic inhomogeneities are not likely to yield information about the bubble pressure. As M(r) is affect mainly by the Kr bubble demagnetization field, large nuclear inhomogeneities other than Kr bubbles (e.g. voids) affect M(r) in the same way as the Kr bubbles do. In the former it has been shown that Kr bubbles with radii larger than about 20 nm are expected to result in magnetic bubbles mainly through their demagnetization field. In the next two paragraphs, a simple model will be used in order to calculate the ND and SANS by these magnetic bubbles. In this model it is assumed that M(r) is parallel to Ha ( Hd( r)) when Ha exceeds Hd(r) ( Hd( r) exceeds Ha)_ Furthermore, the magnetic bubble will be approximated by a sphere with an effective radius rm (see fig. lla). Then r,,, is proportional to H;1/3. The latter dependence of r,,, on Ha is valid only if r,,, is significantly larger than rO. However, when r,,, approximates to r,, then da(r)/dfi and S are too small to be detected. As a result of the symmetry in the local magnetization around a bubble, only deviations in M,(r) from (M,,) may result in ND and SANS (fig. lib). Consequently, the magnetic anisotropy A, = [da’(K)/dS2,]/[da’(K)/dQ,] derived from SANS and the quantity y, derived from ND are expected to approach 0 and 1 at high fields. Here the subscript m refers to magnetic scattering in SANS. Deviations in the symmetry of M(r) around a bubble result in an increase in A, and a decrease in y,. The magnetic differential cross-section, da’(K)/d52,, is. .given by eq..,.(13), _ _ with (AP)~ = ((1 - cos a)‘) (p”‘)’ = 0.14(~~‘)~. For JC~,,,B 1, F2(~) is proportional to rL4, and, as a result, da/d9 is proportional to ri and therewith to Hw213.The magnetic correlation length S derived fr:m a ND experiment approximates to 6 = $rrn, rz((l - cos LX)‘) = 0.59r&, and hence is expected to be proportional to HL4/3. Note that in case M(r) is dominated by magnetostriction, r,,, is proportional to H;‘/2, and hence du’(r)/dO, to Hi’ and S to Hi2.
114
R. Rosman, M. Th. Rekuefdt / Ni containing solid Kr bubbles a
b
f-30
Ha> Hd H
-(I
I
Fig. 11. Sketch of a magnetic bubble in Ni-Kr with radius r,,, according to the model used in section 5.1 (a). The small arrows denote the orientation of M(r). Furthermore a sketch of M(r) around a single spherical Kr bubble (b, the circle is a cross-section of the bubble). As a result of the symmetry, the contribution of AM, to da(w)/dO equals zero.
In the model discussed above, the influence of the intrinsic anisotropy on M(r) has been neglected. However, the intrinsic anisotropy may give rise to magnetic domains up to relatively large applied magnetic fields. These domains may cause ND and SANS in pure Ni. In discussing the influence of the intrinsic anisotropy on the ND and the SANS in pure Ni, it is assumed that the anisotropy field is homogeneous within the grains. Then the mean domain size approximates to the mean grain size. Furthermore, it is assumed that M(r) is affected by H,( 1)y) and the anisotropy field only. Then, M(r) has an orientation between that of the anisotropy field and that of Ha. With increasing Ha, AB decreases while the domain size remains the same. It can be calculated from eq. (17) that in first approximation da’(rc)/dO, is proportional to (1 - (As,)*) and dar(rc)/da, to $ (1 + (a~,)*), with Aq the difference between 9 in the bulk and n in the domain. When Ha varies from zero to infinity, ( Avl,)* varies from l/3 to 0 and consequently A, from 1 to 2. The latter is in agreement with the fact that AB is oriented perpendicular to Ha for large Ha. From a calculation of AqY versus Ha (not shown) it follows that da*(K)/d&,, is expected to be proportional to H[*. As AB is oriented perpendicular to Ha at large H,-values, y, is expected to approach 0 with increasing Ha. Furthermore, it follows from eq. (21) and from the fact that the domain size is assumed to be constant that S is expected to be proportional to H;‘. It should be recalled that the
above accounts for a rough estimation of da(rc)/d& A,, 6 and y, only. The K-dependence of du(rc)/dti and A,,, has not been taken into account for example. With increasing K, the effect of deviations in the symmetry of M(r) around the bubbles increases. As a result, A,,,(K) may increase with increasing K. It follows from the above that both the expected H,-dependences of 6 measured by ND and da(K)/dS2 measured by SANS and the values of y, and A, differ for Ni containing large nuclear inhomogeneities and pure Ni, respectively. 5.2. ND results The value of 6 = 1.5 pm at about Ha = 0 is in the range of a common domain structure. At Ha = 80 kA/m, the depolarization is still dominated by the local magnetization and not by the Kr bubbles themselves, a conclusion which is based on two observations. In the first place, 6 is still increasing with increasing magnetic field at Ha = 80 kA/m. Secondly, bubble radii of 1.5 and 40 nm and bubble densities of 5 X 10z4 and 2 X 1019 mP3 respectively (see section 5.3) result in values of 6 of 0.1 and 0.2 nm. These values for S are much smaller than those observed. In order to analyze the dependence of 6 on Ha, log(S/nm) and log( S’/nm) are plotted versus log( 1Ha I/A/m)) in fig. 12. The slope of the observed curves increase from -0.2 + 0.1 at small Ha up to about - 5 + 2 at Ha = 80 kA/m. The
115
R. Rosman, M. Th. Rekveldt / Ni containing solid Kr bubbles
t
3
Mfj/nm) 2
31
I
hl
Fig. 12. The quantities log(d/nm) (a) and log(S’/nm) (b) for Ni-Kr versus lo& 1Ha I/(A/m)) (Ha > 0 (1,3), Ha < 0 (2,4)).
latter slope exceeds both that expected for a domain structure which is dominated by demagnetization fields of nuclear inhomogeneities and the value expected for a common domain structure (see section 5.1). As a result, the dependence of 6 and 6’ on Ha does not yield information whether or not the local magnetization is affected by the Kr bubbles. The uncertainties in the value of y, at large Ha are large such that the y,,(H,) curve derived from ND does not yield relevant information. In conclusion, the results of ND in Ni-Kr give no clear evidence whether or not the local magnetization is affected by the Kr bubbles. Additional ND measurements on pure Ni may yield helpful relevant information.
scattering in this x-region is in good agreement with nuclear scattering from Kr bubbles with a radius of about 1.5 nm and a density of 5 X 1O24 me3 (see section 3). At K -C0.3 nm-‘, additional nuclear scattering is observed which does not satisfy the Guinier relation. Fig. 13b gives the ‘Porod ‘-plot of the nuclear differential cross-section du(K)/dfi,. At K < 0.2 nm-‘, da(rc)/dfi,, is proportional to K-~, the so-called Porod-behaviour. The latter indicates the existence of large nuclear inhomogeneities. The mean size and density of these inhomogeneities cannot be independently deduced from fig. 13. However, their diameter should exceed about 30 nm. The kind of nuclear inhomogeneities cannot be deduced from fig. 13. The scattering may be due to the presence of large Kr bubbles. If the latter is the case and the mean radius of these large bubbles is assumed to be 40 nm, then the corresponding bubble density is in the order of 2 X 1019 mP3. Apparently, such a low concentration of large Kr bubbles may already account for the observed additional nuclear scattering. Obviously, a small amount of large Kr bubbles is difficult to observe with TEM. Nevertheless, Kr bubbles in Ni-Kr much larger than 3 nm have been observed using TEM [23]. I
12
6 0.0
0.2
0.4
K2(nti2)
5.3. SANS results The decrease of both da’(rc)/dfi and dur(rc) /d&Z with increasing Ha shows the existence of magnetic inhomogeneities and their decrease with increasing Ha. From the observation that A,,, = 1 and da(K)/dfi is independent of Ha at Ha 2 400 kA/m, it follows that nuclear scattering dominates in this field region. Fig. 13a gives the ‘Guinier ‘-plot of the nuclear scattering. For K 2 0.3 nm-‘, ln([do(r)/dti,]m) is almost independent of ~~ (the subscript n refers to nuclear scattering). The
0
t
-)
Y
(j& m)
lo!3
3
2’
I
I
-1.0
-0.5
log(Knm)
I -
Fig. 13. A Guinier (a) and a Porod (b) plot of the scattering by Ni-Kr at Ha = 1800 kA/m.
116
R. Rosman, M. Th. Rekveldt
/ Ni containing solid Kr bubbles
6
Y
t log (j&l)
2 0’
e
1
-1.0
-0.5
log(Knm)
-
Fig. 14. Porod plots of magnetic scattering patterns of Ni-Kr with K I H, (H, = 6.76 (l), 20.0 (2). 46.3 (3), 85.9 (4). 165 (5), 271 (6) kA/m).
Fig. 14 gives the quantities log([da’(r)/ dQ,]m) of Ni-Kr versus log(rtnm) for some values of Ha. The magnetic differential cross section da(r)/d&, has been obtained by subtracting the scattering pattern at Ha = 1800 kA/m from the scattering patterns obtained at lower Ha values. For K -c 0.2 nm-‘, the curves approximate to straight lines with slopes between -5 and - 6.5. Uncorrelated inhomogeneities result in a slope of maximum - 4. Multiple scattering cannot account for the observed slopes. Most likely the scattering patterns are affected by magnetic correlations between the inhomogeneities. Fig. 15 gives A,(K) of Ni-Kr at three different values of Ha. The magnetic anisotropy A, is smaller than 1 over the full K-range used and is only slightly dependent on Ha. This is in agreement with a local magnetization which is strongly affected by the presence of large nuclear inhomogeneities and certainly in disagreement with the presence of a common domain structure.
0.01
I I
0.0
0.1
0.2
K(nnT’) Fig. 15. The /[du’(r)/dfi,]
magnetic anisotropy of Ni-Kr (H,=6.76 M/m).
-
A,,,(K) = [do’(r)/dS1,] (1). 46.3 (2), 85.9 (3)
0.0
0.2
0.4
K(nm-‘)-Fig. 16. The ratio of the magnetic and nuclear scattering in Ni-Kr, denoted G, versus K (H, = 4.77 (l), 46.3 (2) kA/m).
Fig. 16 gives the ratio of the magnetic and the nuclear scattering, G = [da’/dD,]/[da/dfi,] versus the scattering vector K, for two values of Ha. If the magnetic scattering is mainly from uncorrelated magnetic bubbles, then the ratio of the nuclear and the magnetic scattering is given by
(24 It then follows that G should be independent of K and Ha, which is clearly not seen. The observed dependence of G on K and Ha results from the fact that du(rc)/dQ does not correspond to uncorrelated magnetic inhomogeneities. As a result it is difficult to determine a single value for the quotient rm/rO from fig. 16. If for example a mean value (G) = 0.5 is used, then eq. (24) results in rm/rO- 12. Fig. 17 gives the quantity log([du’/dS2,]m) versus log( 1Ha 1/(A/m)) for K = 0.0632 and 0.190 nm-‘. It appears that for Ha 2 50 kA/m, du’(tc)/dQ, is proportional to Hi”, with s = 1.3 f 0.2 for K = 0.0632 nm-’ and s = 0.65 + 0.20 for K = 0.190 nm- i. The latter value equals the value expected for a local magnetization which is highly affected by the demagnetization fields of large inhomogeneities (0.67). Similar as in fig. 12, the initial slope (on-absolute scale) in fig. 17 is smaller. This may be a result of the fact that when inhomogeneities are close together, the inhomogeneity demagnetization field does not approach zero. Then a minimum value of Ha is needed in order
R. Rosman, hf. Th. Rekveldi
117
/ Ni containing solid Kr bubbles
K(nn-T’)Fig. 19. The magnetic anisotropy Am(a) of pure Ni (Ha = 6.76 (1) 13.4 (2) kA/m).
t rl
5 1og(&
6 -
Fig. 17. The quantity log([do’/d&,]/(m-‘)) of Ni-Kr log(H,/(A/m)) (K = 0.0632 (a), 0.190 (b) nm-t).
versus
to substantially affect M(r). However, it may also be due to the influence of the intrinsic crystalline anisotropy. As da(a)/dS2, covers a relatively small K-range only, a comparison between the ND and the SANS results using eq. (21) is not possible. The magnetic scattering by Ni is 15-30 times smaller than that by Ni-Kr, indicating the existence of much more pronounced magnetic inhomogeneities in Ni-Kr. The observation that da(K)/dS2 of the Ni sample is independent of Ha for Ha 2 400 kA/m and that A = 1 in this field region, indicates that also the Ni contains nuclear inhomogeneities. By subtracting the scattering pattern measured at Ha = 1200 kA/m from the
scattering patterns obtained at lower Ha-values, do(K)/dQ, is obtained. Fig. 18 gives log([da’/d&_,,]m) versus log( K nm). Approximately straight lines with a slope between - 4 and -6 are observed. For scattering by large uncorrelated domains, a slope of -4 is expected. Similarly to the scattering by Ni-Kr, the observed larger slopes may be due to magnetic correlation effects. Fig. 19 gives A,( K) for pure Ni at two different Ha-values. Although the accuracy of A,(K) is modest, it can be concluded that A,( K) exceeds 1. This is in agreement with expectations (see section 5.1). Fig. 20 gives log([du’/d3,]m) of Ni versus log( H,/(A/m)) for K = 0.0632 nm-‘. The curve is similar to the curves observed for Ni-Kr. For Ha 2 100 kA/m, the slope of the curve is - 1.8 k 0.4. This value is not significantly larger than the value observed in Ni-Kr. From the above, it follows that the main differences between the magnetic scattering by Ni-Kr and by Ni is that the magnetic scattering by
rl
Fig. 18. Porod plots of magnetic scattering patterns of pure Ni with K I Z-f. (H, = 6.76 (1) 33.2 (2). 85.9 (3), 205 (4) kA/m).
5
6
Fig. 20. The quantity log([da’/dL’,]/m-t) of pure Ni versus log( H,/(A/m)) (K = 0.0632 nm-t).
118
R. Rosman, M. Th. Rekueldt
Ni-Kr highly exceeds that by Ni and that A, < 1 for Ni-Kr and A, > 1 for Ni. The observation that A, < 1 for scattering by Ni-Kr is in agreement with a local magnetization which is dominated by large nuclear inhomogeneities. These inhomogeneities may be large Kr bubbles.
6. Conclusions It appeared possible to perform three-dimensional ND experiments on Ni-Kr in relatively large fields, i.e. up to 80 kA/m, by using a coil system consisting of two anti-parallel coils and two Ni-Kr samples. The observed depolarization cannot be accounted for by the Kr bubbles. The scattering pattern of SANS measured in Ni-Kr in the presence of a magnetic field of 1800 kA/m is in agreement with the presence of a high concentration of Kr bubbles with a diameter of about 3 nm. In addition a small fraction nuclear inhomogeneities larger than 30 nm is likely to be present, which may be associated with large Kr bubbles. Following from numerical calculations, the local magnetization is hardly affected by Kr bubbles with diameters smaller than about 40 nm. Large bubbles or other kind of large nuclear inhomogeneities affect the local magnetization through their demagnetization field. Irrespective of the bubble radius, the bubble pressure is not likely to influence the local Ni magnetization. The magnetic scattering by Ni-Kr highly exceeds the magnetic scattering by Ni. Furthermore, the magnetic anisotropy A, is smaller than 1 for Ni-Kr while A, exceeds 1 for Ni. Both observations indicate that the local magnetization in Ni-Kr is affected by the demagnetization fields of large nuclear inhomogeneities. The latter may be large Kr bubbles. The ND results give no clear evidence whether or not the local magnetization is affected by the Kr bubbles.
Acknowledgements The authors express their thanks to dr. J.H. Evans (Harwell Laboratory, UK) for supplying the Ni-Kr sample and for a critical reading of the manuscript, to dr. W.A.H.M. Vlak (ECN, Nether-
/ Ni containing solid Kr bubbles
lands) for performing some preliminary SANS measurements and to Prof. dr. J.J. van Loef (IRI) for a critical reading of the manuscript. Furthermore, the help of dr. A. Brulet (LLB, Saclay, France) is gratefully acknowledged.
References [l] Proc. Intern. Symp. on Fundamental Aspects of Helium in Metals, Jlilich, Germany, 1982, ed. H. Ullmaier, Radiation Effects 78 (1983). [2] Proc. NATO ARW on Fundamental Aspects of Inert Gases in Solids, Bonas, France, 1990, eds. S.E. Donelly and J.H. Evans (Plenum, New York, in press). [3] A. vom Felde, J. Fink, Th. Muller-Heinzerling, J. Pfluger, B. Scheerer, G. Linker and D. Kaletta, Phys. Rev. Lett. 53 (1984) 922. [4] C. Templier, C. Jaouen, J-P. Riviere, J. Delafond and J. Grilhe, C.R. Acad. Sci. Paris 299 (1984) 613. [5] C. Templier, Proc. NATO ARW on Fundamental Aspects of Iner; Gases in Solids, Bonas, France, 1990, eds.&S.E. Donelly and J.H. Evans, (Plenum, New York, in press). 161D.S. Whitmell, Nucl. Energy 21 (1982) 181. [71 M. Eldrup and J.H. Evans, J. Phys. F 12 (1982) 1265. PI J.H. Evans, R. Williamson and D.S. Whitmell, Effects of Radiation on Materials: 12th Intern. Symp. (STP 870), eds. F.A. Garner and J.S. Perrin (American Society for Testing and Materials, Philadelphia, 1985) p. 1225. [91 J.H. Evans and D.J. Mazey, J. Phys. F 15 (1985) Ll. WI K.O. Jensen, M. Eldrup, N.J. Petersen and J.H. Evans, J. Phys. F 18 (1988) 1703. 1111 D. Schwahn, H. Ullmaier, J. Schelten and W. Kestemich, Acta Met. 31 (1983) 2003. WI W. Kesternich, D. Schwahn and H. Ullmaier, Scripta Metall. 18 (1984) 1011. [I31 M. Th. Rekveldt, Textures and Microstmctures 11 (1989) 127. u41 R. Rosman and M.Th. Rekveldt, J. Magn. Magn. Mater. 95 (1991) 319. [I51 R. Rosman and M.Th. Rekveldt, Z. Phys. B 79 (1990) 61. Ml R. Rosman, H. Frederikze and M.Th. Rekveldt, Z. Phys. B 81 (1990) 149. 1171 R. Rosman and M.Th. Rekveldt, Phys. Rev. B, in press. WI A. Guinier and G. Fournet, Small-Angle Scattering for X-rays (Wiley, New York, 1955). [I91 G. Kostorz, Treatise on Material Science and Technology, vol. 15 (Academic Press, New York, 1979). WI J.H. Evans, Harwell Laboratory, Harwell, UK. WI R. Rosman, M.D. Clarke, M.Th. Rekveldt, R.P. Bissell and R.W. Chantrell, IEEE Trans. Magn. MAG-26 (1990) 1879. WI R. Rosman and M.Th. Rekveldt, IEEE Trans. Magn. MAG-26 (1990) 1843. ~231 J.H. Evans, Harwell Laboratory, Harwell, UK, Private communication.
Journal of Magnetism and Magnetic Materials 98 (1991) 104118 North-Holland
Ni containing solid Kr bubbles studied with neutron and small-angle neutron scattering R. Rosman Interjacultair
depolarization
and M.Th. Rekveldt
Reactor Instituut,
Derf
University of Technology, 2629 JB Deljt, Netherlad
Received 21 December 1990
Neutron depolarization (ND) and small-angle neutron scattering (SANS) experiments on Ni containing solid Rr bubbles are presented and discussed. The aim of the measurements is to study the effect of the Kr bubbles on the local magnetization. For the first time ND measurements in relatively large fields up to 80 kA/m could be performed. However, the ND results do not yield information about the Kr bubbles. The SANS patterns at applied fields larger than 400 kA/m are isotropic and are in agreement with a high fraction of small (radius = 1.5 nm) Kr bubbles. Also a small fraction of larger (radius 2 15 nm) Kr bubbles or other nuclear inhomogeneities is likely to be present. The magnetic scattering is in agreement with a local magnetization which is affected by the demagnetization fields of these inhomogeneities.
1. introduction The behaviour of high concentrations of inert gases trapped in metals has been studied for many years [1,2]. Interest in these systems has been recently enhanced by the discovery of precipitates of heavier inert gases in the solid state (‘solid bubbles’) [3,4]. Many standard techniques, in particular transmission electron microscopy (TEM) (see ref. [5] for a recent review), have been used to study the latter systems. With one exception the heavier inert gases have been introduced into relatively thin near-surface layers by ion-implantation techniques. The exception was the production by Whitmell [6] of bulk (1 cm thick) copper and nickel material produced by a combined sputterdeposition technique containing between 3 and 5 at% krypton. This material and its response to annealing has been studied by Evans and coworkers using primarily TEM, SEM and positron annihilation methods (e.g. refs. [7-lo]). The asformed structure was shown in both Cu-Kr and Ni-Kr to contain a high concentration of small Kr bubbles which were in the solid state [9]. The Ni-Kr material has been used’in the pre0304-8853/91/$03.50
sent study to investigate the effect of the Kr bubbles on the local magnetization of Ni. The Kr bubbles may in principle affect the local magnetization through their demagnetization fields and through anisotropy fields induced’ by their pressure (magnetostriction). As the bubble pressure can be extremely high, i.e. up to 10 GPa, the induced anisotropy fields, which are radially oriented, are large (up to 600 kA/m). The aim of the measurements presented and discussed in the paper is to derive information on the local magnetization orientation. The Kr bubbles with the surrounding Ni atoms whose magnetic moments are affected by the bubble pressure can be considered as magnetic inhomogeneities, denoted as ‘magnetic bubbles ’ further on. By determining the size of the magnetic bubbles, and in particular the decrease of their size with an increasing applied magnetic field, information on the local magnetization orientation is obtained. In order to deduce the size of the magnetic bubbles, neutron depolarization (ND) and smallangle neutron scattering (SANS) have been used. In a ND experiment, the polarization change of a
0 1991 - Elsevier Science Publishers B.V. (North-Holland)
R. Rosman, M. Th. Rekveldi
polarized neutron beam during transmission through a magnetic medium yields the size and the mean orientation of the magnetic inhomogeneities. In a SANS experiment, information about both nuclear and magnetic inhomogeneities is obtained from the broadening of the neutron beam. SANS has been used before to study He bubbles in Ni [11,12]. Section 2 presents brief reviews of the ND and the SANS theory. Details about the samples and the ND and SANS instruments used are given in section 3. Section 4 contains the experimental results. The latter together with theoretical calculations on the local magnetization in Ni-Kr and in pure Ni are discussed in sections 5. Some conclusions are formulated in section 6.
2. ND and SANS theory Brief reviews of the ND and the SANS theory are given in sections 2.1 and 2.2, respectively. 2.1. Review of the ND theory The three-dimensional neutron depolarization (ND) technique is a powerful method to study static and dynamic properties of magnetic structures in the micron and submicron region (e.g. refs. [13,14]). The polarization vector of a polarized neutron beam is analyzed after transmission through a magnetic medium in a ND experiment. During transmission the polarization vector is affected by the micro-magnetic state of the medium: mean magnetic induction results in a net precession of the polarization vector around the mean magnetic induction while magnetic inhomogeneities result in an effective shortening of the polarization vector, called depolarization henceforth. Generally the mean size along the neutron path of the variations, the mean orientation of these variations and the mean magnetization are obtained. The range of magnetic inhomogeneities which can be measured covers 10 nm up to mm’s, making ND to some extent complementary to SANS. In a ND experiment, the polarization change generally is described by a polarization matrix D. Information about the magnetic inhomogeneities
105
/ Ni containing solid Kr bubbles
is often expressed by the parameters 5 and y,(i = x, y,z). When the inhomogeneities are magnetically uncorrelated, t = p;;,
(1)
Yj =
(2)
a;;/5 ,
with =dr’ AB,(x,
Y, 2) AB,(x,
Y, 2’)
/(V). >
(3) Here, V is the volume of the magnetic inhomogeneity, AB(r) = B(r) - (B), B(r) is the local magnetic induction, (B) the mean magnetic induction, ( ) an average over all inhomogeneities and E the inhomogeneity volume fraction. The propagation direction of the neutron beam (eo) is along z. The quantity 26 approximates to the product c(v(AB,)‘), with v the size of V along e,. The quantities y, give the mean orientation of AB(r). Roughly speaking, y, = (nf), with n the unit vector along AB(r). When the inhomogeneities are isotropically distributed, y, = y, = l/4, y, = l/2 (‘intrinsic depolarization anisotropy’ [15]). Positive magnetic correlations between A B,( r) of neighbouring inhomogeneities along e, result in an increase in 5 and yj. A relation between [ and y, on the one hand and b on the other which is generally valid cannot be given. The depolarization matrix of a magnetic medium with a mean magnetization (M) along y is given by D,, = D,, = cos + ,-&~((‘+Y,.)/2), D,, = -D,,
= sin + e-‘1L&((l+Y,)/2),
DYY= e-C’L.~“-“‘,
(4)
Dx,, = D,,x = qyZ = DZY= 0. Here, L, is the sample thickness along the neutron transmission direction (e. 11z), Q = p,,( M)L,& the rotation angle of the polarization vector, c, = 2.18 X 1O29X2 m-4T-2 and X the neutron wavelength. In eq. (4) it has been taken into account that due to the precessional motion of the polarization vector no distinction can be made between
106
R. Rosman, M. Th. Rekveldt
y, and y: if 9 is in the order of or larger than 7r/2. The quantities [, v, and + follow from eq. (4) by (=
_ ln(det@)) 2c,L,
(5)
’
2149,) ” = 1 - ln(det( b)) +=arctan(
:Iiz
(6)
’ j.
(7)
Note that a single ND experiment yields the rotation angle 9 modulo 2a. Expressions for 5 and y, can also be formulated in the Fourier space. For uncorrelated magnetic inhomogeneities in a magnetic bulk, (r‘I = 8a4c
AB/( -s)d’s
/(I’),
/ Ni containing solid Kr bubbles
0 is considered, with K the scattering vector. The scattering may be due to both magnetic and nuclear inhomogeneities. The scattering pattern yields information on the size, shape and orientation of these inhomogeneities. With most SANS instruments, inhomogeneities in the size range between 3 and 100 nm can be studied. In case of single scattering, the scattering intensity Z(K) is related to the macroscopic differential cross section da(tc)/dti by da Z(K) = Kd&+,
with K the scattering vector, K a system constant and t the sample thickness. The relation between da(r)/dQ and the microscopic structure of the sample is given by
(8) 1 da dDK( )=-I/ KVe
with
AB(s) =
(2;,3/yAM*
AM*(r,
i) =s’x
[(M(r)
(r, 5) eis.r d3r, -(M))
2
““p(r)
d3r
(11)
,
s
with V, the sample volume and p(r) the scattering amplitude density. The latter may have a magnetic as well as a nuclear part and is given by
xs”].
Here, ,!? is the reciprocal xy-plane, 5 = s/ 1s 1and M(r) the local magnetization. For convenience, the quantity 6 = 2E/(P&J2
(10)
(9)
is used instead of 5 in this paper. The quantity S is the magnetic correlation length of (AB(r))2 (and not of B2(r)!) along e,, based on AB(r) = pOM, and 6 = 1. Here, M, is the medium spontaneous magnetization. For magnetically uncorrelated inhomogeneities, S approximates to CV. For a more extensive review of the ND theory the reader is referred to refs. [15-171. The magnetic bubbles (Kr bubbles with the surrounding Ni atoms, see section 1) are the magnetic inhomogeneities in Ni-Kr. Within the Kr bubbles, AB( r) = $pOMs. 2.2. Review of the SANS theory In a SANS experiment, the scattering of a (generally unpolarized) neutron beam around K =
p(r)
= n’(r)[b(r)
+P(r)(l*(r)
.S>l y
(12)
with n’(r) the atomic density, b(r) the nuclear scattering amplitude, p(r) the magnetic scattering amplitude, S the spin vector of the neutron ( 1S 1 = I), q*(r)( = 12 X (q(r) X I?), E = K/I K I and q(r) the unit vector along the local magnetization. Eq. (11) reflects the general SANS principle that the scattering pattern at small (large) K-values yields information about large (small) inhomogeneities. Often, a two-phase model is used in the interpretation of the SANS measurements. In this model the sample is considered to consist of a bulk with scattering amplitude density pi and N inhomogeneities with scattering amplitude density p2 and volume V. Ignoring possible correlations between the inhomogeneity volume and its location as well as correlations between different inhomogeneities, eq. (11) can be written as (K > 2a/( g”3): g(K)
=n(Ap)2(I’2F2(~)).
(13)
R. Rosman. M. Th. Rekveldt
Here, (AP)~ = (pi - pZ)’ is the contrast factor, n the inhomogeneity density, ( ) an average over all inhomogeneities and F(K) = (l/V)/, eir.rd3r the inhomogeneity form factor. Obviously, correlations between inhomogeneities result in deviations in da(K)/dti from eq. (13). The former are generally described by a structure factor S(K). When the neutron beam is unpolarized, nuclear and magnetic scattering are additive. Then the nuclear contrast factor (AP,)~ is given by the squared difference of the nuclear scattering amplitude density of the bulk and that of the inhomogeneities (An,)” = (~,.i -
(14)
pn,212-
The magnetic contrast factor (Ap,)’ t&J2
= I(+),@
-
(_n’p)2G
is given by
1’2
= (+):[l
- (a.~r,“],
(16)
with P = K/ IK I. When the inhomogeneities are due to deviations in the orientation of the local magnetization, (n’p), = (n’p), and @pJ2
(n’p):[l- (c.
=
(qr -
(17)
92)j2].
The form factor contains information about the average inhomogeneity size, shape and orientation and can analytically be calculated for some simple inhomogeneity configurations. For ellipsoids with a radial dimension 2a and an axial dimension 2 b (axis oriented along e,,) F2(~)
3(sin u--ucos
= [
with ~=~{(iSe~)~(b~-u~)+.~}~‘~. Independent of the inhomogeneity Guinier expression [ 181 (F2(
K))
=
e-a2R:/3
( KR~
<
1)
applies, with the Guinier radius R, given by
(20) If the inhomogeneity shape is known the inhomogeneity size distribution can in principle be deduced by a comparison of the calculated F2( K) and the measurements. For nearly spherical inhomogeneities (F’(K)) is proportional to K-~ for KR >> 1, a behaviour which is known as the ’ Porod ‘-behaviour. From eqs. (1) (8), (13) and (15) and from n’(r)p(r)q(r) = (6/2X)paM(r) it follows that the magnetic correlation length S, derived from a ND experiment, and the magnetic differential cross section da(rc)/dfim, obtained from a SANS experiment, are simply related by
(21) with dQ = ( A2/4a2) d’ic. In Ni-Kr, nuclear scattering from the Kr bubbles is expected. Magnetic scattering may arise from the magnetic bubbles. A more extensive discussion on SANS is given in refs. [18,19].
3. Experimental The Ni-Kr samples used in the ND and SANS experiments were supplied by Evans [20]. The existence of solid bubbles of the size up to a diameter of 3 nm and with a density of approximately nb = 5 X 10 24 bubbles/m3 has been shown earlier by TEM measurements [9]. Electron diffraction measurements yield a fee Kr packing density of 3.20 x 1O28 atoms/m3 [9]. The corresponding bubble pressure is in the order of 10 GPa (assuming a bubble diameter of 3 nm). TEM measurements on the specific Ni-Kr samples used in the ND and SANS experiments have not been performed. The (nuclear) scattering amplitude of Kr is bK’ = 0.785 X lo-l4 m. Assuming an atomic density within the bubbles of 3.20 X 1O28 atoms/m3, the (nuclear) scattering amplitude density of Kr is m- 2. The nuclear scattering PKr = 2.51 X lo-l4
(18)
1’
U) ’
u3
101
(15)
with the subscripts 1 and 2 referring to the bulk and the inhomogeneities respectively. When the inhomogeneities are non-magnetic (&,J2
/ Ni containing solid Kr bubbles
shape,
the
(19)
108
R. Rosman, M. Th. Rekueldt / Ni containing solid Kr bubbles
amplitude density of Ni is #’ = 9.4 x lOI rnp2, following from a nuclear scattering amplitude of bNi = 1.03 X lOI m and an atomic density of 9.1 X 1O28 rnp3. The magnetic scattering amplitude of Ni is pNi = 0.16 X lo-i4 m. Hence, the magnetic scattering amplitude density of Ni equals pz’ = 1.5 X lOI rne2. It follows that the magnetic contrast factor of Ni-Kr, (Ap,,,)‘, varies between 0 and 2.1 X 10z8 rne4. The nuclear contrast factor is more than 20 time larger, i.e. (AP,)~ = 4.7 x 1O29 rnp4. In bubbles larger than 3 nm, the Kr atomic density will be lower than the value quoted above. This will result in an increase of (a~,,)~. The SANS measurements have been performed on the instrument PAXY of the Laboratorium Leon Brillioun (Saclay, France) *. The instrument has a two-dimensional (64 x 64 cm2) position sensitive detector and uses an unpolarized neutron beam. Using a sample-detector distance of either 4.981 or 6.830 m and a neutron wavelength of 0.8 nm, a K-range of 0.06-0.5 or 0.05-0.4 nm-’ is covered during a single experiment. Apart from measurements on Ni-Kr also a few measurements have been performed on pure Ni. Unfortunately, no Ni sample prepared using the sputtering method with the use of which the Ni-Kr samples have been made was available. Magnetic fields up to a maximum of Ha = 1800 kA/m have been applied along y (e,, along I). The macroscopic differential cross section for the scattering vector parallel and perpendicular to the applied field has been obtained by averaging over all detector positions between the two lines which make angles of -0 o and 8 o and the two lines which make angles of (90-e)” and (90+8)O (e=20 or 30) with the applied field, respectively. All SANS results presented in the paper have been corrected for background and are normalized using plexiglass and vanadium. The ND instrument used has Cu,MnAl crystals magnetized to saturation as polarizer and analyzer. The polarizer simultaneously acts as monochromator (A = (0.12 f 0.01) nm, corresponding to c1 = 3.14 X lo9 rnp2Tp2). The instrument is operated at a 2 MW swimming pool reactor (HOR,
* Laboratoire Commun CEA-CNRS.
neutron:
Fig. 1. A sketch of the coil system used in the ND experiments (Yo: yoke, PP: polar pieces, C: coils, S: samples). The arrows give the orientation of the magnetic field.
Delft, Netherlands) and uses dc-coils as polarization turners. Up to now, ND measurements have been performed only when applying magnetic fields smaller than about 1 kA/m. This is a result of the sensitivity of the polarization vector to stray fields arising from coils or magnets producing the fields needed on the sample. The larger the applied fields are, the larger the stray field and the larger the effect on the depolarization. Consequently, high coercive media are always studied in the remanent state (e.g. refs. [14,21,22]). However, in order to substantially affect the magnetic inhomogeneities within Ni-Kr magnetic fields up to 200 kA/m are needed. Therefore, a new coil system has been made, in which ND experiments in the presence of magnetic fields up to 80 kA/m can be performed. The coil system consists of two almost identical coils (coil constant L = 1.08 X 104/m), 4 polar pieces (diameter 20 mm) and a magnetic yoke (fig. 1). Two samples (2.07 X 7.80 x 7.70 mm3 and 2.27 x 7.80 x 7.60 mm3) can be sandwiched between the polar pieces, with their smallest size along z. The induced fields in the coils are anti-parallel (I] *y-axis). The maximum field at each sample position is 80 kA/m. In order to have a homogeneous field over the cross-section of the neutron beam a relatively small diaphragm (diameter 4 mm) has been used. The two anti-parallel coils and two samples were necessary as the rotation angle due to either a single coil or a single sample can be as high as 110 or 75 rad, respectively. Such a large rotation angle is undesirable, as it may
R. Roman.
M. Th. Rekveldt
109
/ Ni containing solid Kr bubbles
cause a strong additional depolarization for the polarization component perpendicular to Ha (see refs. [14,17]). A total sample thickness of 4 mm is needed in order to have a measurable depolarization effect at large fields. All experiments have been performed, starting from a maximum field (Ha = 80 kA/m in the ND and Ha = 1800 kA/m in the SANS measurements).
4. Experimental
results
The results of the ND and the SANS measurements are presented in sections 4.1 and 4.2 successively. 4.1. ND measurements Fig. 2a gives the element D,,,, of the empty coil system and the coil system containing the two Ni-Kr samples versus the current Z through the coil (starting from Z = -0.8 A). The element Dyy of the empty coil system approximates to 1, while D,,y of the coil system and the samples is minimum around Z = 0 and increases with ] Z ] over the whole Z-range. Figs. 2b and c give the elements D,, and Dxzof the empty coil system and the coil system containing the samples versus ] Z ] (again starting from Z = - 0.8 A). For both the empty coil system and the coil containing the samples D = D,,x = Dyl = Dzy = 0, D,, = D,, and D,, = In the empty coil system, D,, and D,, 3&. show an oscillating behaviour, the amplitude of which slightly decreases with increasing I. The latter indicates a small depolarization by the induced field. The behaviour of D;, observed in the empty coil system is consistent with Ha oriented along y. Also the elements Ox, and Dxz of the coil system containing the samples show an oscillating behaviour. However, the amplitude of the oscillations increases with increasing ] Z 1. Both the latter increase and the increase of D,, indicates a magnetic correlation length which decreases with increasing applied magnetic field. The fact that the minimum in Dyy occurs at value of Z which slightly exceeds zero is mainly due to the hysteresis of the coil system (see fig. 3). The periodicity
-6.8
0.0
0.8
I(+Fig. 2. The elements D,,,, of the empty coil system (1) and the coil system with the Ni-Kr (2) (a), the elements 0,. (3) and 0,. (4) (b) of the empty coil system and of the coil system containing Ni-Kr (5,6) (c) versus the current I through the coils (starting from I = -0.8 A). The error bars are of the order of the symbols.
of the oscillations in D,, and D,, of the coil system containing the Ni-Kr samples is maximum at about Z = 0, corresponding to a maximum change in the mean magnetization. 5 t
aJcs O-
Man@
-5 ’ -0.8
I
I
0.8
0.0 I(A)
-
Fig. 3. The rotation angle +_ of the empty coil system versus I. The error bars are of the order of the symbols.
110
R. Rosman, M. Th. Rekoeldt
Fig. 3 gives the rotation angle of the polarization vector in the empty coil system, denoted &,,, versus I. The field Ha between the polar pieces has been calculated using Ha = c~c#I~~. The constant c2 cannot be deduced from fig. 3, as the unbalance of the coil system is not exactly known. Instead, c2 has been calculated using the coil constant of a single coil (L = 1.08 X lo4 m-‘) and the length of the polar pieces, resulting in c2 = 15.9 kA/m. Fig. 4 gives the mean magnetization (M) of the Ni-Kr samples versus Ha. The mean magnetization has been calculated using
with AL the difference in thickness between the two samples, AC#J = t& - &,, and c#+the value of + corresponding to the coil system containing the two samples. The coercive field H, of Ni-Kr appears to be smaller than 2 kA/m. The spontaneous magnetization of Ni is MS = 485 kA/m. With increasing (decreasing) applied magnetic field, (M) at first decreases (increases) before increasing (decreasing) at about Ha = 0. This is an artefact arising from the fact that fig. 4 represents a difference of the magnetization curves of the two Ni-Kr samples. Apparently, the fields at the two samples slightly differ, which results in the distorted shape of the hysteresis curve. Fig. 5 gives the quantities det(h) and DYY in Ni-Kr versus Ha, measured in an increasing and a decreasing magnetic field. Both values have been corrected for the depolarization by the empty coil system. They are minimum around Ha = 0 and increase with increasing 1Ha 1, corresponding to a
80
-80 I!, (LA/m)
-
Fig. 4. The mean magnetization (M) of Ni-Kr derived from the rotation angle cpversus Ha. The error bars are of the order of the symbols.
/ Ni containing solid Kr bubbles 1. t
det@) 0.
t
bf
0.
0.01 -80
1
I
1
HO.(RA/rn)
80
-+ Fig. 5. The quantities det(b) (a) and D,,? (b) in Ni-Kr H, (1, 3: decreasing H,, 2,4: increasing H,).
versus
decreasing magnetic correlation length. The small oscillations in det(b) around ( Ha ( = 60 kA/m are artefacts, resulting from a combination of a small misalignment of the field orientation, a possible small deviation in the calibration of the ND instrument, a large rotation angle $ (q!+,,,, = 5 rad) and a small depolarization. Fig. 6 gives the quantities S and y, in Ni-Kr versus Ha, measured in an increasing and a decreasing field. The magnetic correlation length 6 is maximum at Ha = 0 (1.5 pm) and decreases with increasing I Ha I. A small hysteresis effect is observed. Large variations are seen in y,, which is about 0.25 at Ha = 0. In particular at large I Ha Ivalues, 7, cannot be deduced accurately due to the small depolarization. The oscillations in S and y, directly follow from those in det(b). At large Ha-values, the quantity 6’ = -In Dy,,/c,L,(poMs)z (fig. 6c) is likely to be a better measure for 6 as the matrix element D,, is much less sensitive to misalignments of the coil systems or small deviations in the calibration of the instrument than det(fi). The quantity S’ smoothly decreases with increasing 1Ha I over the whole filed range. The magnetic correlation lengths 6’ and 6 are related by 6’ = 6(1 - y,)/2. It should be noticed, however, that 6 and y, cannot be independently deduced from S’.
R. Rosman, M. Th. Rekveldt
111
/ Ni containingsolidKr bubbles
0.0 0.0
I
0.1
K(nrfi’)
0
-80
2
-
Fig. 7. The total differential cross-section of Ni-Kr for I( perpendicular (a) and parallel (b) to Ha (H, = 4.77 (l), 13.4 (2), 46.3 (3), 231 (4), 1194 (5) kA/m). The error bars are of the order of the symbols.
0
80
t-l, (&A/m) Fig. 6. The quantities 6 (a), v, (b) and S’ (c) in Ni-Kr versus Ha (1, 3, 5: decreasing H,, 2, 4, 6: increasing H,).
4.2. SANS measurements Fig. 7 gives the total normalized differential cross section of Ni-Kr versus the scattering vector K for some values of Ha with K perpendicular, da’(K)/dQ, and parallel, da’(rc)/dQ, to the applied field (6) = 30 has been used, see section 3). Both differential cross sections decrease with increasing K over the full K-range. Furthermore, da’(rr)/dfi and da’(K)/dfi decrease with increasing Ha up to a value of Ha of about 400 kA/m. At higher fields, da(K)/dfi is within the error bars independent of Ha. The scattering at Ha = 4.77 kA/m is anisotropic, as da’(K)/d&? is significantly larger than da’(K)/dQ. This anisotropy, expressed in terms of the ratio A(K) = [da’(K)/dfi]/[du’(rc)/dti], decreases with increasing Ha and has disappeared at Ha 2 400 kA/m (fig. 8 gives A(K) for Ha = 1800 kA/m, 0 = 20). Apparently, the scattering at fields equal
to or larger than 400 kA/m is mainly nuclear scattering. From the observation that the magnetic scattering exceeds the nuclear scattering over the full K-range at the smallest field used and from the fact that the nuclear contrast is much larger than the magnetic contrast (see section 3) it follows that the density and/or the volume of the magnetic inhomogeneities is much larger than that of the nuclear inhomogeneities. Fig. 9 gives du’(K)/dn and du’(rc)/dfi of pure Ni for some field values (6 = 30). The differential cross sections are 15 to 30 times smaller than
0’ 0.0
I
I
I
0.2
0.4
K(nm-‘) Fig. 8. The anisotropy A = (do’/dO)/(do’/dQ) Ha = 1800 kA/m versus K.
of Ni-Kr
at
112
R. Rosman, M. Th. Rekveldr / Ni containing solid Kr bubbles
(105/m)0*4 t da’ 0.2 xi
2 (1 05/m) t da’ dR
1
K(nm-‘)Fig. 9. The total differential cross-section of pure Ni for K perpendicular (a) and parallel (b) to H, (Ha = 6.76 (l), 13.4 (2), 33.2 (3), 398 (4) kA/m). The error bars are of the order of the symbols.
those of Ni-Kr and decrease with increasing Ha until Ha = 400 kA/m. At higher fields the scattering is independent of the applied field. Also the scattering in Ni is anisotropic at H, -C400 kA/m. However, in contrast with Ni-Kr, da’(rr)/dQ exceeds do’(r)/d&? in Ni.
5. Discussion Before discussing the ND and SANS results, some calculations on the local magnetization in Ni-Kr and the corresponding ND and SANS are presented in section 5.1. Subsequently, the ND (section 5.2) and the SANS (section 5.3) results are discussed. 5.1. Model
calculations
on M(r)
Three types of model calculations are presented in this section. At first, in order to see how and to what extent the local Ni magnetization is affected by the presence of Kr bubbles, the results of numerical calculations on the local magnetization
around a single Kr bubble are given. Secondly, on the base of a simple model the expected ND and SANS in Ni-Kr are formulated. Thirdly, the expected ND and SANS effects in pure Ni are dealt with. In the calculation of M(r) around a single Kr bubble, it is assumed that M(r) is affected by the external magnetic field H,, the demagnetization field of that Kr bubble, Hd, the exchange interaction between neighbouring Ni atoms and a magnetostrictive induced anisotropy field. The effect of the intrinsic anisotropy is neglected (intrinsic anisotropy constant K,, = 3.5 X lo3 J/m3). The magnetic energy density related to H, is given by E,,,(r) = -p,,M(r)H, cos a, with cx the angle between H, and M(r). The energy density corresponding to Hd is Ed(r) = - p,,M( r) . Hd( r), with
(23) Here, r0 is the bubble radius and n is a unit vector oriented anti-parallel to (M). The large bubble pressure induces pressure gradients outside the bubble along the radial directions. As the magnetostrictive constant of Ni is negative (A, = - 3.5 X 10w5), the anisotropy field induced by magnetostriction is radially oriented. The corresponding energy density is given by Eani(r) = 1A IP( r) sin’p, with P(r) the local pressure and p the angle between M(r) and the radial direction. Finally, the exchange interaction density arising from exchange interaction between 2 neighbouring Ni atoms is Eexch(r) = -F cos 8, with 8 the angle between the atoms magnetic moments. With a nearest neighbour distance in Ni of 0.249 nm, F = 3.5 X 1Ol9 J/m3 has been calculated. As Ni has a fee-structure, each Ni atom has 12 nearest neighbours. Using energy minimalization, the local magnetization distribution around Kr bubbles with different radii r,, has been calculated at certain values of Ha. The local pressure is given by P(r) = P,,(rJr)‘, with P,, the bubble pressure. The bubble pressure is taken to be inversely proportional to R,: PO = 10/r, mPa. In the calculations M(r) is assumed to be oriented along Ha at first. Obviously, M(r) is uniaxially distributed around Ha.
R. Rosman, M. Th. Rekueldi
/ Ni containing solid Kr bubbles
This symmetry is important in the ND and the SANS effects. It follows from the numerical calculations described above that the local magnetization around Kr bubbles with radii r0 smaller than about 20 nm does hardly deviate from (M) as M(r) is either dominated by exchange interaction (closely around the bubbles) or by Ha (‘far way from the bubble’). As a result the magnetic bubble approximates to the Kr bubble and Kr bubbles with radii smaller than about 20 nm are not expected to affect the local magnetization of Ni. An increase in r,, results in a decrease in the exchange and the anisotropy energy density. It appears that M(r) around Kr bubbles with radii larger than about 20 nm is dominated by either the demagnetization field (closely around the bubble) or by the external field (r B rO). Hence, the local magnetization around Kr bubbles with r, 2 20 nm is expected to deviate from the mean magnetization and these bubbles nm with the surrounding Ni atoms may act as magnetic bubbles. As an example, fig. 10 gives the local magnetization for r, = 35 nm and r,, = 10 nm at H, = 4 kA/m (according to section 5.3, also bubbles much larger than 3 nm may exist in the Ni-Kr sample). Around the bubble with r,, = 10 nm, M(r) hardly deviates from (M) while closely around the bubble with r0 = 35 nm M(r) significantly deviates from (M). It follows from b
5
Fig. 10. The orientation of the magnetization around a Kr bubble. The circle is a cross-section of the bubble (r, = 10 nm (a) and 35 nm (b)). The orientation of the applied magnetic field is indicated (H, = 4 kA/m).
I H' -’
113
the above that irrespective of the Kr bubble radius r0 the bubble pressure PO hardly affects the local magnetization around the Kr bubble. As a result, ND and SANS measurements of the magnetic inhomogeneities are not likely to yield information about the bubble pressure. As M(r) is affect mainly by the Kr bubble demagnetization field, large nuclear inhomogeneities other than Kr bubbles (e.g. voids) affect M(r) in the same way as the Kr bubbles do. In the former it has been shown that Kr bubbles with radii larger than about 20 nm are expected to result in magnetic bubbles mainly through their demagnetization field. In the next two paragraphs, a simple model will be used in order to calculate the ND and SANS by these magnetic bubbles. In this model it is assumed that M(r) is parallel to Ha ( Hd( r)) when Ha exceeds Hd(r) ( Hd( r) exceeds Ha)_ Furthermore, the magnetic bubble will be approximated by a sphere with an effective radius rm (see fig. lla). Then r,,, is proportional to H;1/3. The latter dependence of r,,, on Ha is valid only if r,,, is significantly larger than rO. However, when r,,, approximates to r,, then da(r)/dfi and S are too small to be detected. As a result of the symmetry in the local magnetization around a bubble, only deviations in M,(r) from (M,,) may result in ND and SANS (fig. lib). Consequently, the magnetic anisotropy A, = [da’(K)/dS2,]/[da’(K)/dQ,] derived from SANS and the quantity y, derived from ND are expected to approach 0 and 1 at high fields. Here the subscript m refers to magnetic scattering in SANS. Deviations in the symmetry of M(r) around a bubble result in an increase in A, and a decrease in y,. The magnetic differential cross-section, da’(K)/d52,, is. .given by eq..,.(13), _ _ with (AP)~ = ((1 - cos a)‘) (p”‘)’ = 0.14(~~‘)~. For JC~,,,B 1, F2(~) is proportional to rL4, and, as a result, da/d9 is proportional to ri and therewith to Hw213.The magnetic correlation length S derived fr:m a ND experiment approximates to 6 = $rrn, rz((l - cos LX)‘) = 0.59r&, and hence is expected to be proportional to HL4/3. Note that in case M(r) is dominated by magnetostriction, r,,, is proportional to H;‘/2, and hence du’(r)/dO, to Hi’ and S to Hi2.
114
R. Rosman, M. Th. Rekuefdt / Ni containing solid Kr bubbles a
b
f-30
Ha> Hd H
-(I
I
Fig. 11. Sketch of a magnetic bubble in Ni-Kr with radius r,,, according to the model used in section 5.1 (a). The small arrows denote the orientation of M(r). Furthermore a sketch of M(r) around a single spherical Kr bubble (b, the circle is a cross-section of the bubble). As a result of the symmetry, the contribution of AM, to da(w)/dO equals zero.
In the model discussed above, the influence of the intrinsic anisotropy on M(r) has been neglected. However, the intrinsic anisotropy may give rise to magnetic domains up to relatively large applied magnetic fields. These domains may cause ND and SANS in pure Ni. In discussing the influence of the intrinsic anisotropy on the ND and the SANS in pure Ni, it is assumed that the anisotropy field is homogeneous within the grains. Then the mean domain size approximates to the mean grain size. Furthermore, it is assumed that M(r) is affected by H,( 1)y) and the anisotropy field only. Then, M(r) has an orientation between that of the anisotropy field and that of Ha. With increasing Ha, AB decreases while the domain size remains the same. It can be calculated from eq. (17) that in first approximation da’(rc)/dO, is proportional to (1 - (As,)*) and dar(rc)/da, to $ (1 + (a~,)*), with Aq the difference between 9 in the bulk and n in the domain. When Ha varies from zero to infinity, ( Avl,)* varies from l/3 to 0 and consequently A, from 1 to 2. The latter is in agreement with the fact that AB is oriented perpendicular to Ha for large Ha. From a calculation of AqY versus Ha (not shown) it follows that da*(K)/d&,, is expected to be proportional to H[*. As AB is oriented perpendicular to Ha at large H,-values, y, is expected to approach 0 with increasing Ha. Furthermore, it follows from eq. (21) and from the fact that the domain size is assumed to be constant that S is expected to be proportional to H;‘. It should be recalled that the
above accounts for a rough estimation of da(rc)/d& A,, 6 and y, only. The K-dependence of du(rc)/dti and A,,, has not been taken into account for example. With increasing K, the effect of deviations in the symmetry of M(r) around the bubbles increases. As a result, A,,,(K) may increase with increasing K. It follows from the above that both the expected H,-dependences of 6 measured by ND and da(K)/dS2 measured by SANS and the values of y, and A, differ for Ni containing large nuclear inhomogeneities and pure Ni, respectively. 5.2. ND results The value of 6 = 1.5 pm at about Ha = 0 is in the range of a common domain structure. At Ha = 80 kA/m, the depolarization is still dominated by the local magnetization and not by the Kr bubbles themselves, a conclusion which is based on two observations. In the first place, 6 is still increasing with increasing magnetic field at Ha = 80 kA/m. Secondly, bubble radii of 1.5 and 40 nm and bubble densities of 5 X 10z4 and 2 X 1019 mP3 respectively (see section 5.3) result in values of 6 of 0.1 and 0.2 nm. These values for S are much smaller than those observed. In order to analyze the dependence of 6 on Ha, log(S/nm) and log( S’/nm) are plotted versus log( 1Ha I/A/m)) in fig. 12. The slope of the observed curves increase from -0.2 + 0.1 at small Ha up to about - 5 + 2 at Ha = 80 kA/m. The
115
R. Rosman, M. Th. Rekveldt / Ni containing solid Kr bubbles
t
3
Mfj/nm) 2
31
I
hl
Fig. 12. The quantities log(d/nm) (a) and log(S’/nm) (b) for Ni-Kr versus lo& 1Ha I/(A/m)) (Ha > 0 (1,3), Ha < 0 (2,4)).
latter slope exceeds both that expected for a domain structure which is dominated by demagnetization fields of nuclear inhomogeneities and the value expected for a common domain structure (see section 5.1). As a result, the dependence of 6 and 6’ on Ha does not yield information whether or not the local magnetization is affected by the Kr bubbles. The uncertainties in the value of y, at large Ha are large such that the y,,(H,) curve derived from ND does not yield relevant information. In conclusion, the results of ND in Ni-Kr give no clear evidence whether or not the local magnetization is affected by the Kr bubbles. Additional ND measurements on pure Ni may yield helpful relevant information.
scattering in this x-region is in good agreement with nuclear scattering from Kr bubbles with a radius of about 1.5 nm and a density of 5 X 1O24 me3 (see section 3). At K -C0.3 nm-‘, additional nuclear scattering is observed which does not satisfy the Guinier relation. Fig. 13b gives the ‘Porod ‘-plot of the nuclear differential cross-section du(K)/dfi,. At K < 0.2 nm-‘, da(rc)/dfi,, is proportional to K-~, the so-called Porod-behaviour. The latter indicates the existence of large nuclear inhomogeneities. The mean size and density of these inhomogeneities cannot be independently deduced from fig. 13. However, their diameter should exceed about 30 nm. The kind of nuclear inhomogeneities cannot be deduced from fig. 13. The scattering may be due to the presence of large Kr bubbles. If the latter is the case and the mean radius of these large bubbles is assumed to be 40 nm, then the corresponding bubble density is in the order of 2 X 1019 mP3. Apparently, such a low concentration of large Kr bubbles may already account for the observed additional nuclear scattering. Obviously, a small amount of large Kr bubbles is difficult to observe with TEM. Nevertheless, Kr bubbles in Ni-Kr much larger than 3 nm have been observed using TEM [23]. I
12
6 0.0
0.2
0.4
K2(nti2)
5.3. SANS results The decrease of both da’(rc)/dfi and dur(rc) /d&Z with increasing Ha shows the existence of magnetic inhomogeneities and their decrease with increasing Ha. From the observation that A,,, = 1 and da(K)/dfi is independent of Ha at Ha 2 400 kA/m, it follows that nuclear scattering dominates in this field region. Fig. 13a gives the ‘Guinier ‘-plot of the nuclear scattering. For K 2 0.3 nm-‘, ln([do(r)/dti,]m) is almost independent of ~~ (the subscript n refers to nuclear scattering). The
0
t
-)
Y
(j& m)
lo!3
3
2’
I
I
-1.0
-0.5
log(Knm)
I -
Fig. 13. A Guinier (a) and a Porod (b) plot of the scattering by Ni-Kr at Ha = 1800 kA/m.
116
R. Rosman, M. Th. Rekveldt
/ Ni containing solid Kr bubbles
6
Y
t log (j&l)
2 0’
e
1
-1.0
-0.5
log(Knm)
-
Fig. 14. Porod plots of magnetic scattering patterns of Ni-Kr with K I H, (H, = 6.76 (l), 20.0 (2). 46.3 (3), 85.9 (4). 165 (5), 271 (6) kA/m).
Fig. 14 gives the quantities log([da’(r)/ dQ,]m) of Ni-Kr versus log(rtnm) for some values of Ha. The magnetic differential cross section da(r)/d&, has been obtained by subtracting the scattering pattern at Ha = 1800 kA/m from the scattering patterns obtained at lower Ha values. For K -c 0.2 nm-‘, the curves approximate to straight lines with slopes between -5 and - 6.5. Uncorrelated inhomogeneities result in a slope of maximum - 4. Multiple scattering cannot account for the observed slopes. Most likely the scattering patterns are affected by magnetic correlations between the inhomogeneities. Fig. 15 gives A,(K) of Ni-Kr at three different values of Ha. The magnetic anisotropy A, is smaller than 1 over the full K-range used and is only slightly dependent on Ha. This is in agreement with a local magnetization which is strongly affected by the presence of large nuclear inhomogeneities and certainly in disagreement with the presence of a common domain structure.
0.01
I I
0.0
0.1
0.2
K(nnT’) Fig. 15. The /[du’(r)/dfi,]
magnetic anisotropy of Ni-Kr (H,=6.76 M/m).
-
A,,,(K) = [do’(r)/dS1,] (1). 46.3 (2), 85.9 (3)
0.0
0.2
0.4
K(nm-‘)-Fig. 16. The ratio of the magnetic and nuclear scattering in Ni-Kr, denoted G, versus K (H, = 4.77 (l), 46.3 (2) kA/m).
Fig. 16 gives the ratio of the magnetic and the nuclear scattering, G = [da’/dD,]/[da/dfi,] versus the scattering vector K, for two values of Ha. If the magnetic scattering is mainly from uncorrelated magnetic bubbles, then the ratio of the nuclear and the magnetic scattering is given by
(24 It then follows that G should be independent of K and Ha, which is clearly not seen. The observed dependence of G on K and Ha results from the fact that du(rc)/dQ does not correspond to uncorrelated magnetic inhomogeneities. As a result it is difficult to determine a single value for the quotient rm/rO from fig. 16. If for example a mean value (G) = 0.5 is used, then eq. (24) results in rm/rO- 12. Fig. 17 gives the quantity log([du’/dS2,]m) versus log( 1Ha 1/(A/m)) for K = 0.0632 and 0.190 nm-‘. It appears that for Ha 2 50 kA/m, du’(tc)/dQ, is proportional to Hi”, with s = 1.3 f 0.2 for K = 0.0632 nm-’ and s = 0.65 + 0.20 for K = 0.190 nm- i. The latter value equals the value expected for a local magnetization which is highly affected by the demagnetization fields of large inhomogeneities (0.67). Similar as in fig. 12, the initial slope (on-absolute scale) in fig. 17 is smaller. This may be a result of the fact that when inhomogeneities are close together, the inhomogeneity demagnetization field does not approach zero. Then a minimum value of Ha is needed in order
R. Rosman, hf. Th. Rekveldi
117
/ Ni containing solid Kr bubbles
K(nn-T’)Fig. 19. The magnetic anisotropy Am(a) of pure Ni (Ha = 6.76 (1) 13.4 (2) kA/m).
t rl
5 1og(&
6 -
Fig. 17. The quantity log([do’/d&,]/(m-‘)) of Ni-Kr log(H,/(A/m)) (K = 0.0632 (a), 0.190 (b) nm-t).
versus
to substantially affect M(r). However, it may also be due to the influence of the intrinsic crystalline anisotropy. As da(a)/dS2, covers a relatively small K-range only, a comparison between the ND and the SANS results using eq. (21) is not possible. The magnetic scattering by Ni is 15-30 times smaller than that by Ni-Kr, indicating the existence of much more pronounced magnetic inhomogeneities in Ni-Kr. The observation that da(K)/dS2 of the Ni sample is independent of Ha for Ha 2 400 kA/m and that A = 1 in this field region, indicates that also the Ni contains nuclear inhomogeneities. By subtracting the scattering pattern measured at Ha = 1200 kA/m from the
scattering patterns obtained at lower Ha-values, do(K)/dQ, is obtained. Fig. 18 gives log([da’/d&_,,]m) versus log( K nm). Approximately straight lines with a slope between - 4 and -6 are observed. For scattering by large uncorrelated domains, a slope of -4 is expected. Similarly to the scattering by Ni-Kr, the observed larger slopes may be due to magnetic correlation effects. Fig. 19 gives A,( K) for pure Ni at two different Ha-values. Although the accuracy of A,(K) is modest, it can be concluded that A,( K) exceeds 1. This is in agreement with expectations (see section 5.1). Fig. 20 gives log([du’/d3,]m) of Ni versus log( H,/(A/m)) for K = 0.0632 nm-‘. The curve is similar to the curves observed for Ni-Kr. For Ha 2 100 kA/m, the slope of the curve is - 1.8 k 0.4. This value is not significantly larger than the value observed in Ni-Kr. From the above, it follows that the main differences between the magnetic scattering by Ni-Kr and by Ni is that the magnetic scattering by
rl
Fig. 18. Porod plots of magnetic scattering patterns of pure Ni with K I Z-f. (H, = 6.76 (1) 33.2 (2). 85.9 (3), 205 (4) kA/m).
5
6
Fig. 20. The quantity log([da’/dL’,]/m-t) of pure Ni versus log( H,/(A/m)) (K = 0.0632 nm-t).
118
R. Rosman, M. Th. Rekueldt
Ni-Kr highly exceeds that by Ni and that A, < 1 for Ni-Kr and A, > 1 for Ni. The observation that A, < 1 for scattering by Ni-Kr is in agreement with a local magnetization which is dominated by large nuclear inhomogeneities. These inhomogeneities may be large Kr bubbles.
6. Conclusions It appeared possible to perform three-dimensional ND experiments on Ni-Kr in relatively large fields, i.e. up to 80 kA/m, by using a coil system consisting of two anti-parallel coils and two Ni-Kr samples. The observed depolarization cannot be accounted for by the Kr bubbles. The scattering pattern of SANS measured in Ni-Kr in the presence of a magnetic field of 1800 kA/m is in agreement with the presence of a high concentration of Kr bubbles with a diameter of about 3 nm. In addition a small fraction nuclear inhomogeneities larger than 30 nm is likely to be present, which may be associated with large Kr bubbles. Following from numerical calculations, the local magnetization is hardly affected by Kr bubbles with diameters smaller than about 40 nm. Large bubbles or other kind of large nuclear inhomogeneities affect the local magnetization through their demagnetization field. Irrespective of the bubble radius, the bubble pressure is not likely to influence the local Ni magnetization. The magnetic scattering by Ni-Kr highly exceeds the magnetic scattering by Ni. Furthermore, the magnetic anisotropy A, is smaller than 1 for Ni-Kr while A, exceeds 1 for Ni. Both observations indicate that the local magnetization in Ni-Kr is affected by the demagnetization fields of large nuclear inhomogeneities. The latter may be large Kr bubbles. The ND results give no clear evidence whether or not the local magnetization is affected by the Kr bubbles.
Acknowledgements The authors express their thanks to dr. J.H. Evans (Harwell Laboratory, UK) for supplying the Ni-Kr sample and for a critical reading of the manuscript, to dr. W.A.H.M. Vlak (ECN, Nether-
/ Ni containing solid Kr bubbles
lands) for performing some preliminary SANS measurements and to Prof. dr. J.J. van Loef (IRI) for a critical reading of the manuscript. Furthermore, the help of dr. A. Brulet (LLB, Saclay, France) is gratefully acknowledged.
References [l] Proc. Intern. Symp. on Fundamental Aspects of Helium in Metals, Jlilich, Germany, 1982, ed. H. Ullmaier, Radiation Effects 78 (1983). [2] Proc. NATO ARW on Fundamental Aspects of Inert Gases in Solids, Bonas, France, 1990, eds. S.E. Donelly and J.H. Evans (Plenum, New York, in press). [3] A. vom Felde, J. Fink, Th. Muller-Heinzerling, J. Pfluger, B. Scheerer, G. Linker and D. Kaletta, Phys. Rev. Lett. 53 (1984) 922. [4] C. Templier, C. Jaouen, J-P. Riviere, J. Delafond and J. Grilhe, C.R. Acad. Sci. Paris 299 (1984) 613. [5] C. Templier, Proc. NATO ARW on Fundamental Aspects of Iner; Gases in Solids, Bonas, France, 1990, eds.&S.E. Donelly and J.H. Evans, (Plenum, New York, in press). 161D.S. Whitmell, Nucl. Energy 21 (1982) 181. [71 M. Eldrup and J.H. Evans, J. Phys. F 12 (1982) 1265. PI J.H. Evans, R. Williamson and D.S. Whitmell, Effects of Radiation on Materials: 12th Intern. Symp. (STP 870), eds. F.A. Garner and J.S. Perrin (American Society for Testing and Materials, Philadelphia, 1985) p. 1225. [91 J.H. Evans and D.J. Mazey, J. Phys. F 15 (1985) Ll. WI K.O. Jensen, M. Eldrup, N.J. Petersen and J.H. Evans, J. Phys. F 18 (1988) 1703. 1111 D. Schwahn, H. Ullmaier, J. Schelten and W. Kestemich, Acta Met. 31 (1983) 2003. WI W. Kesternich, D. Schwahn and H. Ullmaier, Scripta Metall. 18 (1984) 1011. [I31 M. Th. Rekveldt, Textures and Microstmctures 11 (1989) 127. u41 R. Rosman and M.Th. Rekveldt, J. Magn. Magn. Mater. 95 (1991) 319. [I51 R. Rosman and M.Th. Rekveldt, Z. Phys. B 79 (1990) 61. Ml R. Rosman, H. Frederikze and M.Th. Rekveldt, Z. Phys. B 81 (1990) 149. 1171 R. Rosman and M.Th. Rekveldt, Phys. Rev. B, in press. WI A. Guinier and G. Fournet, Small-Angle Scattering for X-rays (Wiley, New York, 1955). [I91 G. Kostorz, Treatise on Material Science and Technology, vol. 15 (Academic Press, New York, 1979). WI J.H. Evans, Harwell Laboratory, Harwell, UK. WI R. Rosman, M.D. Clarke, M.Th. Rekveldt, R.P. Bissell and R.W. Chantrell, IEEE Trans. Magn. MAG-26 (1990) 1879. WI R. Rosman and M.Th. Rekveldt, IEEE Trans. Magn. MAG-26 (1990) 1843. ~231 J.H. Evans, Harwell Laboratory, Harwell, UK, Private communication.