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Crystal Structure Analysis by Neutron Diffraction II.

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phosphorus in silylphosphines or monogermylphosphines [1053.
Similar contradictions can be found for organogermyl-,
organostannyl-, and organoplumbylphosphines. The
course of the oxidation and the unsuccessful attempts
a t specific syntheses of organostannylphosphine oxides
or organostannylphosphonium salts lead one to attribute the stability of the tin-phosphorus bond to a
necessary (p-dx contribution, since the Sn-P bond
is broken as soon as the lone pair of electrons on the
phosphorus is engaged. The 1,2-dipolar additions of
carbon disulfide and phenyl isocyanate also support
this hypothesis, since Sn[P(C6H&]4 is accessible to
these reactions, whereas no electrophilic attack of
these 1,Zdipoles on [(C6H&Sn]3P is observed.
This interpretation of the covalent bond appears to be
opposed by the vibration spectra, which can be interpreted only in terms of a pyramidal structure of the
compounds, though it should be pointed out that it is
by no means certain that (p+d), double-bond components must be associated with a planar structure of
[99] E . A . V. Ebsworth, Chem. Commun. 1966, 530.
[lo01 E . W. Randall and J . J . Zuckerman, G e m . Commun.
1966, 132.
[loll H . Schumann, I. Schumann-Ruidisch, and M. Schmidt in
A . Sawyer: Organotin Chemistry. M. Dekker, New York, in
press.
[I021 H. Schumann, 0. Stelzer, and H . Rosch, J. organometallic
Chem., in press.
[lo31 H. Schumann and 0. Stelzer, unpubtished.
[lo41 E . W. Abel, R . Honigschmidt-Grosich, and S . M . Illingworth,
.I
chem.
.
SOC. (London), A 1968, 2623.
I1051 J . E . Drake and C . Riddle, J. chem. SOC.(London) A
1968, 1675.
the molecule [99,1001. The reactions of the tris(trimethylmetal)-substituted phosphines with carbonyl
compounds of the transition metals also show very
clearly that it is possible to use the lone pair of electrons on the phosphorus for coordinate bonding
without breaking the original P--MIV bonds.
As regards the stability of compounds of elements of
groups IVB and VB, the investigations carried out so
far show clearly that the most stable molecules are obtained with approximately equal covalent radii of the
bonding partners [84,8537,1011.
In the light of our present knowledge, the hypothesis
of the participation of the lone pair of electrons of
phosphorus, arsenic, antimony, and bismuth in the
bond with silicon, germanium, tin, or lead seems quite
feasible. The final decision should be left to detailed
physicochemical studies and structure analyses.
The author is grateful to Dip1.-Chem. Ulrike Arbenz,
Dr. H . Benda, Dr. P. Jutzi, Dr. H . KOpA U.Niederreuther, Fraulein Thea Ostermann, Dr. A . Roth, L. Rosch,
Fraulein Elke Schauer, Dr. P. Schwabe, Dr. 0. Stelzer,
and DipLChem. A . Yaghmai for their assistance in
his own work in this field. Thanks are also due to Prof.
Dr. M. Schmidt for valuable discussions, to Badische
Anilin- urrd Soda-Fabrik, Farbenfabriken Bayer, Farbwerke Hoechst, Werk Gendorf, and the Union Mini2i-e
du Haut Katanga for the free supply of valuable starting
materials, and to the Fonds der Chemischen Industrie
and the Deutsche Forschungsgemeinschaft for financial
support.
Received: December 18, 1968
[A 732 IE]
German version: Angew. Chem. 81, 970 (1969)
Translated by Express Translation Service, London
Crystal Structure Analysis by Neutron Diffraction IF**]
By G. Will[*]
This second part of the article “Crystal Structure Analysis by Neutron Diffraction” deals
with the diffraction of neutrons by magnetically ordered crystals. Neutron diffraction is at
present the only reliable method for the determination of the magnitude, direction, and
spatial distribution of magnetic moments in crystalline substances. Since the magnetic
moments are essentially due to the unpaired electrons, the distribution of these electrons
in the crystal can be measured in this way.
1. Introduction
An important application of neutron diffraction is the
determination of magnetic spin structures in crystals
in the magnetically ordered state. The neutron, or
[*I Priv.-Doz. Dr. G. Will
Abteilung fur Kristallstrukturlehre und Neutronenbeugung
Mineralogisch-PetrologischesInstitut und Museum
der Universitat
53 Bonn, Poppelsdorfer Schloss (Germany)
[**I Part 11: Neutron Diffraction and Magnetic Structures. Part I: Neutron Diffraction at Atomic Nuclei 111.
950
rather its magnetic moment, acts as a probe, by means
of which the spatial electron spin distribution in the
crystal can be measured in atomic dimensions. Neutron
diffraction is at present the only method that allows
the reliable determination of the magnetic environmental relations between the individual atoms or ions
and of the long-range order of the spin. The investigation of magnetism and the use of constantly improving
magnetic materials has now become so important
that the chemist engaged in the preparation and
analysis of such compounds must also be familiar with
the magnetic properties and their investigation.
Angew. Chem. internat. Edit.
/ Yof. 8 (1969)/ No. 12
2. Physical Principles
The neutron has a magnetic moment of pn = -1.91 319
nuclear magnetons ( p ~ ) and
,
can therefore enter into
magnetic dipole-dipole interaction with the magnetic
moments of the atoms. The interaction with the
magnetic moments of shell electrons that are situated
in incomplete shells, and which therefore produce a
resultant magnetic moment, is particularly important.
The interaction with the magnetic moments of the
atomic nuclei, on the other hand, is weaker by a factor
of about 2000, and can be detected only under certain
conditions; this interaction can be disregarded in the
present connection.
The elastic scattering of the thermal neutrons by
crystals consists essentially of two components having
approximately equal orders of magnitude (scattering
lengths between 10-12 and 10-13 cm):
1. nuclear scattering due to interactions between the
neutron and the atomic nuclei [I];
2. magnetic scattering due to a classical dipole interaction between the magnetic moment of the neutron
and the magnetic moments of the shell electrons.
The observed intensity of a diffracted neutron beam
from a crystal is made up additively from these two
components.
Since the intensities in general are proportional to the
squares of the crystallographic structure factors, we
can write
q = magnetic interaction vector (see eq. (6)),F = crystallographic structure factor.
Fnuc1
=,
c
b. e2xi(hxj
1 '
Fmagn
=
,
e2xi(hxj
=
e,m
pn
Sj
=
=
=
0.539 . 10--12 . Sj . f ( k )cm.
charge and mass of the electron
magnetic moment of the neutron (in p ~ )
spin quantum number of t h e atom j
.fj(k)= corresponding magnetic form factor
A
E = scattering vector (unit vector)
2@
=
scattering angle (0 = Bragg angle)
Where t h e orbital moment cannot be neglected, eq. (4a)
becomes
with LJ a n d SJ, the projections of these q u a n t u m numbers
on the vector J.
jj(k) are now magnetic form factors, which describe the
spatial distribution of the unpaired electrons. When the
orbital moments are not negligible, separate form
factorsfL and f S must be used for the orbital and spin
moments; this is so e . g . for lanthanides.
The magnetic scattering of neutrons is particularly
interesting when the spins, in other words the individual magnetic moments, undergo a transition into
an ordered state, i. e. when collective magnetism arises.
Neutron diffraction is at present the only method by
which the long-range order of the magnetic moments
and their relative and absolute orientation can be
determined. The neutron diffraction experiments provide information about:
1. the existence of collective magnetism;
+ kyi + fzj)
2. the transition temperature (Curie or Nee1 temperature, Tc or TN);
+ kyj + fzj)
3. the nature of the magnetism: paramagnetism, ferromagnetism, antiferromagnetism, or ferrimagnetism;
1
2 pj
by TrammelIr31 and BIurnec41. According to these
authors the magnetic scattering amplitude pj is given
by
i
The structure factor in X-ray diffraction, for comparison, is
4. the size of the magnetic unit cell;
5. the distribution of the magnetic atoms (or ions) in
the sublattice and their relative orientation;
6. the absolute orientation of the moments with
respect to the crystal axes;
where f p are the X-ray atomic scattering factors, which
describe the spatial distribution of all the electrons
around the atomic nuclei. The nuclear scattering
amplitudes, which are independent of the scattering
angle, are represented by bj; pj are the angle-dependent
magnetic scattering amplitudes.
The magnetic scattering of neutrons was discussed by
Halpern and Johnson Q1, and later in an extended form
[l] G . Will, Angew. Chem. 8Z, 307 (1969); Angew. Chem. internat. Edit. 8, 356 (1969).
121 0. Halgern and M . H. Johnson, Physic. Rev. 55, 898 (1939).
Angew. Chem. internat. Edit. VoE. 8 (1969) I No. 12
7. the magnitude of the magnetic moments of the
individual atoms or ions;
8. the magnitude and direction of the internal fields
(by application of an external magnetic field);
9. the temperature dependence of the magnetization;
10. the local distribution of the magnetic moment
density in the crystal, i. e . in the case of 3d electrons,
the local distribution of the 3d electron density (covalency effects).
~-
[3] G . T . Trammel/, Physic. Rev. 92, 1387 (1953).
141 M . Blume, Physic. Rev. 130,1670 (1963); 133, A 1366 (1964).
95 1
3. Examples of Applications
3.1. Antiferromagnetic Structure of MnO
The method and the derivation of the results is best
illustrated with the aid of examples. MnO was the
first substance whose antiferromagnetic structure was
established by neutron diffraction 151. This experiment
provided evidence of the very existence of antiferromagnetic order. The same magnetic order as in MnO
has also been found in NiO [61 and in EuTe 171.Though
FeOC6l and C00[6~*1assume an order with the same
spin-sign sequence as MnO, the moments have a
different orientation in relation to the crystal axes
(Table 1).
The nuclear scattering lengths are b~~ = -3.7 f and
bo = 5.77 f (1 f =: 1 fermi = 1 0 - 1 3 cm; cf. also [I]).
When MnO powder is cooled in a cryostat, new reflections that point to magnetic order of the spins are
observed below TN = 122°K. These new reflections
can be indexed as h/2, k / 2 , 112 where all the indices
h, k , I are odd. Integer Miller indices are obtained by
multiplication by 2; this means that to describe the
magnetic arrangement of the spins in the crystal, the
unit cell dimensions must be doubled in all three
directions. The dimensions of the magnetic unit cell
are thus double the dimensions of the chemical unit
cell (see Fig. 1):
amagn = 2 . anucl
bmagn = 2 - bnucl
Cmagn = 2 . c n u c ~
Table 1. Magnetic properties of 3d metal oxides and Eu-chalcogenides.
I MnO
I FeO
TN ( "K)
Symmetry
below
122
rhombohedral
a < 60"
Direction of
magnetic
moment
(111)
1 COO 1 NiO
198
rhombohedral
a > 60"
11111
I EuO
I EuS
29 1
tetragonal
cla < 1
1ii71
I EuSe
523
rhombohedral
a < 60"
(111)
I EuTe
Magnetic order
T c resp. TN ( OK)
10
Ip73111
[a] According to G . Busch, P. Junod, R . G . Morris, and J . Muheim,
Helv. physica Acta 37, 637 (1964). Considerable uncertainty still exists
concerning the magnetic behavior and even the transition temperatures.
Thus Busch et al. found a second transition at 3.8 OK, which could be
confirmed by recent neutron diffraction experiments.
Magnetic order manifests itself in the appearance of
coherent scattering. Collective magnetism can therefore be recognized immediately if two neutron diffraction diagrams are recorded, one above and one below
the Curie or Nee1 temperature, the difference diagram,
representing the magnetic scattering, then being found
by subtraction.
Figure 1 shows a recent neutron diffraction diagram
of MnO recorded by the author, and Figure 2 shows
the antiferromagnetic structure derived from this
diagram. MnO crystallizes in a face-centered cubic
cell with the NaCl structure. The space group extinction rules for the nuclear scattering are therefore
bo
5.77
=
2.07 f
h,k,l = odd: F = b m - b o = -3.7 - 5.77
=
-9.41 f
h,k,l = even: F = b M n +
=
-3.7+
[5] C. G. Shull and J . S. S m a r t , Physic. Rev. 76, 1256 (1949).
161 C. G . ShuII, W. A. Strausser, and E. 0. Wollan, Physic. Rev.
83, 333 (1951).
171 G. Will, S. J . Pickart, H . A. Alperin, and R. Nathans, J.
Physics Chem. Solids 24, 1679 (1963).
181 B. van Laar, Physic. Rev. 138, A 584 (1965); B. van Laor,
J . Schweizer, and R . Lemaire, ibid. 141, 538 (1966).
20
-
30
281")
40
50
Fig. 1. Neutron-diffraction diagram of antiferrornagnetically ordered
MnO recorded at 4.2 OK. Owing to the doubling of the unit cell in the
magnetically ordered state, all the reflections of the magnetic scattering
have odd indices; all the nuclear scattering reflections have even indices.
The reflections (222) and (622) (= h, k , I odd x 2) are strong, and the reflections (400)and (440) (/?, k, I even x 2) are weak, corresponding to the
intensity conditions for a face-centered cubic lattice (NaCl structure) and
considering the negative sign of the nuclear scattering length of manganese.
The magnetic phase transition is associated with a crystallographic phase transition. Below 122OK, in the antiferromagnetic range, MnO crystallizes with a rhombohedra1 unit
cell, though the angle a-differs only very slightly from 90".
At 4.2'K, anuc1 = b = c = 4.415 A and a = 90.6'.
The magnetic order can now be deduced from the size
of the magnetic unit cell together with the observed
extinction rules for the magnetic reflections (MnO:
h, k , 1 even not observed). For MnO, this gives an
antiparallel orientation of the spins along the
Since only the relative
three crystal axes:
orientation is expressed, the situation can be described
by a sign sequence: +-+-+-. . .. This gives the antiferromagnetic structure as shown in Figure 2. It consists of ferromagnetic (111) planes in which all the
spins are parallel, with an antiferromagnetic sequence
of such (111) planes normal to the plane in the [111]
direction.
t-lt4T-l.
The absolute orientation and the magnitude of the
magnetic moments can be determined from the
intensities of the individual reflections. The quantity
q 2 from eq. (2) is used for this purpose.
q is known as the magnetic interaction vector, and
represents a relation between the direction of the
Angew. Chem. internat. Edit. J Vol. 8 (1969) J No. 12
2
t
absolute scale is rather easy in neutron diffraction
studies, since the always-present nuclear scattering
reflections can be used as an internal standard.
0
02'
@
Mn2'
3.2. Investigations on Europium Chalcogenides
+Y
a""d
>
c
1173121
hag"
Fig. 2. Antiferromagnetic structure MnO. The spins lie in the (111)
planes, and all the spins are parallel within the same (1 11) plane. The
same magnetic structure is also found in NiO and EuTe.
magnetic moments, described by a unit vector x , and
the orientation of the reflecting crystal plane h k 1,
described by the scattering vector E (unit vector),
which is normal to the plane hkl (cf. Fig. 3).
q
= E ( E
x)-x
(6)
Multiplication gives
q 2 = 1 - - (E . x)2 =
sin*cc
(7)
where cc is the angle between
E
and x.
Reflection
/ted
E
Maonetization
Scattering
vector
vector
The special importance of the chalcogenides EuO,
EuS, EuSe and EuTe is that they allow the direct
study of the magnetic exchange forces as a function
of the interionic distance in a series of simple compounds. They are of general interest because they
represent the first ferromagnetic insulators that have
been found which are optically transparent in the
visible part of the spectrum. The compounds crystallize in the simple, parameter-free NaCl structure,
which is particularly convenient for theoretical considerations. The magnetic properties of these compounds are shown in Table 1, and Figure 4 shows the
exchange interactions as a function of the lattice constants. Since the Eu-Eu distances are directly reflected in the lattice constants, it is possible to combine
the lattice constants and the magnetic behavior and
estimate from them the exchange interactions. The
strong ferromagnetic interaction J1 between the
nearest-neighbor Eu ions in EuO and EuS is responsible for the ferromagnetic behavior of both compounds. In EuTe, the antiferromagnetic interaction 52
between next-nearest neighbors is distinctly greater
than J1; EuTe therefore behaves antiferrornagnetically. Our neutron diffraction studies [7J have shown
an antiferromagnetic order of the second kind in
EuTe, which means that the compound has the same
spin configurations as MnO (Fig. 2).
beam
rn
Fig. 3. Vector relation between incident neutron beam. diffracted
beam, orientation of the reflecting plane, and the magnetizationdirection.
The scattering intensity is proportional to sin%, i.c. the intensity has a
maximum when the magnetic moments lie in the reflecting plane hkl.
Since the orientation of the various crystallites in a powder
is random, limits are placed on the determination of the
angle a from powder measurements. Thus i t can be shown
= constant = 2/3, and
that for cubic crystal symmetry, <q2>
is thus independent of the orientation. In a cubic crystal
therefore, the orientation of the spins in relation to the
crystal axes cannot be determined from powder measurements. In crystals with rhombohedral, tetragonal, or hexagonal symmetry, the orientation of the spins can be given in
relation to the unique axis of the crystal system, i.e. in
relation to [111] in rhombohedra1 crystals and in relation to
the c axis in tetragonal or hexagonal crystals.
For MnO, and also for EuTe, the measured intensity
ratios show that the magnetic moments lie in the (111)
planes as shown in Figure 2. Within the (111) plane
the orientation cannot be determined from powder
data.
The magnitude of the magnetic moment at the individual ions can be calculated from the absolute intensities using eq. (4a) or (4b). The determination of the
Angew. Chem. inrernaf. Edit.
/ Vol. 8 (1969) / No.
12
4 1
e -10
5.0
5.5
6.0
Lattice constant
Fig. 4. Plot of the magnetic interactions J I and
genides as a function of the lattice constants.
J2
181-
65
in europium chalco-
The results for the magnetic properties of EuSe are
contradictory. The magnetic structure may be influenced by small quantities of impurities or Eu3+
ions. A neutron diffraction study by Pickart and At-
953
showed that the sample used by these authors
had a complicated antiferromagnetic structure, but
that this structure could be changed into a ferromagnetic order by even a small external magnetic
field of about 1700 Oe. EuSe was therefore not taken
jnto account for the dependence of the exchange
forces Jl and 52 in Figure 4.
perinL9l
3.2.1. S u b l a t t i c e M a g n e t i z a t i o n a n d E x c h a n g e
Jnteractions
I n a face-centered cubic lattice, each ion has twelve
nearest neighbors (nn) at a distance 1/2 /2 a0 along the
face diagonals [110] and six next-nearest neighbors
(nnn) at a distance a0 along the axes [loo]. In the order
of the second kind, as the antiferromagnetic order
observed in MnO is called, each Mn2+ ion has six
parallel and six antiparallel nearest neighbors and six
antiparallel next-nearest neighbors (Table 2).
Table 2. Neighborhood relations in face-centered
cubic lattices.
Neighbois
I
ferromagnetic
1
antiferromagnetic
1st kind
1
2nd kind
nn
nnn
When the magnetic order and the transition temperatures
have been determined they can be used together with Weiss's
molecular field theory to derive the Heisenberg exchange
interactions. Heisenberg's view of ferromagnetism is based
o n the idea that the exchange interactions between two
atoms i and j are responsible for the orientation of the
magnetic moments. The magnetic exchange energy is described by a Hamiltonian operator 8.
and Sj are spin operators Ji,j is the exchange integral
between the atoms i and j. Since the theory of exchange interactions is based on ad-hoc assumptions and the exchange
integrals themselves cannot be calculated J must be determined experimentally if one wishes to understand the
magnetic behavior of the substances. However, other interactions such as magnetic dipole-dipole coupling or the
effects of anisotropy energies must generally be taken into
account in addition to the exchange interactions. Measurements o n simple structures such as the parameter-free
structures of the NaCl type are therefore of great interest for
the theory of magnetism. The series MnO, FeO, COO, NiO
and EuO, EuS, EuSe, EuTe have been extensively investigated; the oxides of the 3d transition metals are all antiferromagnetic, while a transition from ferromagnetism to
antiferromagnetism is observed in theeuropiumchalcogenides
(cf. Table 1).
The exchange parameters J can be derived from the experimental data by means of relations from Weiss's molecular
field theory 1101. The following relations are valid:
T c is the ferromagnetic Curie temperature; for antiferromagnetic crystals, 0 is the paramagnetic or asymptotic
Curie-Weiss temperature, which follows from the Curieand
Weiss law. For EuTe it was found that 0 = -7.5'K,
this together with the experimentally found Nee1 point
T N=
~ 7.8OKr71 gives eq. 9a and 9b (order o f the second
kind):
Jz
=
-(0.12 i 0.01)
O K ;
(0.025
Jj
.I
0.025)"K
(Energies are obtained by multiplication by k = 1.38~10-16
erg "K-1).
EuTe thus has a negative nnn exchange interaction, which is
responsible for the antiferromagnetic orientation o f the
spins, and which is about five times as great as the positive
direct nn exchange interaction.
Important information about the sublattice magnetization is
obtained from the temperature behavior of the magnetic
scattering. Figure 5 a shows the temperature dependence of
the magnetic scattering for the (111) reflection of EuTe. I t is
found from this that TN = 7.8 "K. For comparison with the
temperature dependence of magnetization as given by Brillouin, the observed intensity Irel is normalized to I/Io and
plotted against the normalized temperatures T/TN. These
normalized values are shown in Figure 5b; i t can be seen that
the experimental values do not agree with the theoretical
Brillouin curve for a system having spin 7/2.
Both Mnz+ (3d5) in MnO and Euzc (4f7) in EuTe are in pure S
states. There are therefore no orbital moment components to
be taken into account. The dipole interactions are also
negligibly small in these crystals, and the magnetic anisotropy
energy is large only within the (111) planes while it is small
perpendicular to these planes, i.e. in the direction of the antiferromagnetic coupling. The relations are therefore simple
and the deviation of the experimental points from the
Brillouin curve is surprising a t first.
a1
Si
0
[A
iLm
0.2
06
01
TIT,
0.8
1.0
--j
Fig. 5. Temperature dependence of the (strongest) magnetic reflection
(1 11) in EuTe.
(a) Dependence of the intensity on temperature: The ( 1 1 1 ) reflection
disappears at TN = 7.8"K; EuTe is in a paramagnetic state above
this temperature.
191 S. J . Pickart and H . A . Alperin, J. Physics Chem. Solids 29,
414 (1968).
1101 J . S. Smart, Physic. Rev. 86, 968 (1952).
954
(b) Intensity and temperature normalized to 10 and TN ' I = f(T) thus
becomes universal, and should follow the Brillouin function for spin
S = 112 (continuous curve). The observed dependence, on the other
hand, corresponds to a curve (broken) that was calculated by insertion
of a biquadratic term of about 1 % into the Hamiltonian exchange
function.
Angew. Chem. internal. Edit. 1 Vol. 8 (1969) J No. I 2
To explain the discrepancy, Rodbell, Jacobs, 0wen, and
HarrisrI'l introduced a biquadratic contribution into the
normally linear exchange interaction.
If this biquadratic term is taken into account in the calculation of the Brillouin function, a very satisfactory agreement
with the experimental point (broken curve in Fig. 5b) is
obtained. Similar deviations of the experimental points from
the Brillouin curve were found for MnO and NiO, and a
biquadratic exchange term could also be detected for these
crystals .
The biquadratic component amounts to about 1 % of J, and
is therefore very small(inEuTe,.j was found to be N 0.001 OK).
It is only rarely of such importance as in MnO and EuTe,
and is generally overshadowed by the much larger exchange
contributions of the dipole interaction and the magnetic
anisotropy. It would therefore be wrong to try to explain
every deviation from a Brillouin function by a biquadratic
component of the exchange interaction.
3.3. Ferrirnagnecic Structure of Ferrites
Reference was made in the first part of this article to
the important role of neutron diffraction in the investigation of ferrites having the general composition
[X2+(Y3+)2(02-)4][I]. Since X and Y are normally
adjacent elements in the periodic system they can be
difficult or even impossible (e. g. in the case of
MnFe204) to distinguish by X-rays. The neutron
scattering powers of elements of adjacent atomic
numbers on the other hand are generally quite distinct;
neutron diffraction thus permits a definite identification of the ions.
X and Y in the ferrites are usually 3d transition elements. The analysis of the magnetic scattering of
neutrons below the ordering temperature TN also gives
the ionization states of the ions and their distribution
over the lattice sites. A classical example is the investigation of magnetite Fe304 = [Fe2+(Fe3+)2(02->4].
Fez+ and Fe3+ differ in their spin quantum numbers,
S=2 and S=5/2 respectively, and consequently also
have different magnetic scattering powers, p = 1.09
and 1.36~10-12cm respectively. Neutron scattering
at room temperature 1121 gave an inverse spinel structure for magnetite with Fe3+ in the tetrahedral (or A)
sites while the octahedral (or B) sites are uniformly
occupied by Fe2+ and Fe3+. The neutron diffraction
data also show that the moments of the ions in the A
and B sites are antiparaIIe1 to one another, so that the
resultant magnetic moment corresponds to the moment of one Fez+ ion per Fe304 formula unit. Fe304
is an example of ferrimagnetism postulated by N&eZ[131.
The magnetic structure is shown in Figure 6; the
magnetic unit cell has the same dimensions as the
chemical unit cell.
1111 D. S. Rodbell, I. S . Jacobs, J. Owen, and E . A . Harris,
Physic. Rev. Letters I 1 , 10 (1963).
1121 C. G. Shull, E. 0. Wollan, and W. C. Koehler, Physic. Rev.
84,912 (1951).
[I31 L. NPel, Ann. Physik 3, 137 (1948)
Angew. Chem. internat. Edit.
Vol. 8 (1969) / N o . 12
A-sites ltetrahedral Fe3'l
B-sites loctahedral Fe3*/Fe2'i
1173161
Fig. 6. Magnetic structure of magnetite at room temperature
[Fe2+(Fe3+)2(OZ-)4].Magnetite crystallizes with the spinel structure.
One quarter of the unit cell is shown.
Magnetite exhibits a second magnetic phase transition, which was elucidated by an elegant experiment.
Below 110-120 "K (the exact temperature depends on
impurities in the crystals) the crystals exhibit a sudden
change in electric conductivity and in magnetization.
To explain this, Verwey et al. "41 postulated a crystallographic phase transition from the cubic spinel structure to an orthorhombic structure, combined with
ordering of the Fez+/Fe3+ ions on the octahedral sites.
Hunzilton[*sJ was able to confirm this hypothesis by
neutron diffraction with a synthetic Fe304 single
crystal. In this crystal, ordering of the iron ions on the
octahedral sites (now in an orthorhombic unit cell)
takes place at 119°K. The Fe3f ions form simple
chains parallel to the a axis, while the Fez+ ions are
arranged perpendicular to these in chains parallel to
the 6 axis. The orthorhombic u and 6 axes correspond
to the cubic [ l l O ] face diagonals (Fig. 7). All the
magnetic moments in the structure below 119 "K are
parallel or antiparallel to the orthorhombic c (= cubic)
axis. The experimental detection of this order is based
Fig. 7. Distribution of Fez+ ions (large spheres) and Fe3+ ions (small
spheres) in magnetite on the octahedral sites below the second magnetic
transition a t 119 OK. The ions, which are randomly distributed above
this temperature (Fig. 6), now form mutually perpendicular ordered
chains.
1141 E. J . W. Yerwej, P. W. Haayman, and 1.Ronegn, J. chern.
Physics 15, 181 (1947).
[15l W. Hamilton, Physic. Rev. 110, 1050 (1958).
955
on the appearance of the (002) reflection on neutron
diffraction; this reflection is proportional to the
difference in the scattering powers p(Fe3+)-p(Fe2+),
and is forbidden for the cubic random distribution.
3.4. Magnetic Behavior of Solid Solutions in the
Ternary System Fe203-Cr203-Al203
The sesquioxides u-FezO3, CrzO3, TizO3, VzO3 and
A1203 all crystallize hexagonally in the corundum
structure, but only the crystals of cc-FezO3 and CrzO3
exhibit magnetic long-range order of the spins; no
long-range order of the magnetic moments could be
detected by neutron diffraction in Ti203 and V2O3,
despite the observed anomalies in the temperature
dependence of the magnetic susceptibility.
the sign sequence (+-+-). (When Fez03 is cooled
below about -23 OC, the spins flip from the (111) plane
into the [111] direction. This phase transition has not
however been found in any solid solutions; if it does
exist it does so only in a very narrow range around
pure FezO3.)
Since the oxides discussed here form solid solutions,
information on the magnetic interactions, and hence
on the conditions for the various magnetic structures,
should be obtainable from the magnetic properties of
the solid solutions. Thus Cox et al.
found that in
the system Fe203-V203, the magnetic structure of
a-FezO3 persists up to a VzO3 content of 60 %. A particularly interesting phase in the system Fe203-Ti203
is a solid solution having the the composition FeTiO3,
in which there is a completely ordered distribution of
the Fez+ and Ti4+ ions over the A and B sitesr191.
P
cl
bl
Fig. 8. Crystalline and magnetic structures of some sesquioxides.
a) Corundum structure - rhombohedral cell. Large circles = metal ions,
small circles = oxygen ions.
b) Corundum structure
ions, are shown.
-
hexagonal representation. Only the metal
c) Magnetic structure of a-FezO, at 25 “ C .
d) Magnetic structure of Cr20,.
dl
el
FeTi03 has an antiferromagnetic structure, in which
ferromagnetic (111) planes form an antiparallel sequence in the [1111 direction. These ferromagnetic
planes are separated by layers of non-magnetic Ti4+
ions. The sign sequence can be given purely formally
as (+ 00); the spins themselves are oriented parallel
and antiparalle1 to [lll].
~
e) Magnetic structure of FeTiO,. Open circles - Fez+, shaded circles =
Ti4+.
a-FezO3 and CrzO3 have the same crystal structure,
but quite different spin structures r16,17J (Fig. 8). In
Fe203, the spins lie in the hexagonal basal plane at
25°C; the spins are parallel within a plane, and the
planes are arranged in antiparallel sequence along the
hexagonal c-axis. (In a rhombohedral description of
the unit cell, we have ferromagnetic (111) planes in an
antiferromagnetic sequence along [111I.) The structures are described by the atomic positions Al, A2, B1,
and B2. The spins in these positions in a-FezO3 then
have the sign sequence (+ + - -).
In Cr2O3, on the other hand the spins are directed
parallel to rhombohedral [llll; an antiferromagnetic
~
.,
arrangement exists within the individual (111) Planes.
In the rhombohedral [lll] direction, the spins have
____
[16] C. G. ShuN, W. A . Strausser, and E . 0.WuNan, Physic. Rev.
83, 333 (1951).
1171 B. N . Brockhouse, J. chem. Physics 21, 961
956
Complicated magnetic behavior is expected, and
observed 1201, in the solid solutions of the system
Fe203-Cr203, since here we have interaction of two
magnetic structures with opposite spin directions and
different orders. A neutron diffraction study showed
that the magnetic structure from 100 to 35 mole- % of
Fez03 is that of a-Fe203, while at 10 to 0 mole-% of
Fez03 it is that of Cr2O3. The transition temperature
decreases with increasing Cr content: Fe203:T~ =
965 OK; Fez03-Crz03 (40:60 mole- %): TN = 400 OK.
Between 35 and 10 mole- % of Fe2O3, there is a helical
structure described as an “umbrella structure”, i. e. an
arrangement in which the spins lie on cones, adjacent
spins being turned through an angle u that is different
(1953).
[18] D. E . cox, W. J . Takei, R . C. Miller, and G. Shirane, J.
Physics Chem. Solids 23, 863 (1962).
[19] G. Shirane, S. J . Pickart, R . Nathans, and Y . Ishikawa,
J. Physics Chem. Solids 10,35 (1959).
[20] D. E . Cox, W. J . Takei, and G. Shirane, J. Physics Chem.
Solids 24, 405 (1963).
Angew. Chem. infernat. Edit. / Vof. 8 (19691 / No. 12
by 180" in relation to one another. Both M. and the
apex angle of the cone vary with the Cr content. The
orientation of the axis of the cone in the crystal also
changes. The axis of the cone is initially (above 35 %
Fe203) perpendicular to the c axis, i.e. parallel to the
collinear moments in Fe203, the (half) apex angle of
the cone being 19 ". The apex angle increases to I1
at 20 % FezO3, and then decreases again. The orientation of the cone switches into the direction of the c
axis at about 15 % FezO3; it then corresponds to the
collinear moments in CrzO3. The investigations were
recently carried on by Scharenberg [211 and extended
to include the presence of non-magnetic aluminum
ions. Dilution of the magnetic lattice in this way
yields valuable information concerning the range of the
exchange forces and the stability of the magnetic
structures. The results of these measurements
can be seen in the magnetic phase diagram of
Fez03-Crz03-Alz03 (Fig. 9), from which the ranges
of the various structure types are immediately recognizable.
all doubly occupied by electrons. Only the higher antibonding orbitals are singly occupied, and are thus responsible for the magnetic behavior. We follow the
procedure used by Hubbard and Marshallr221 and
consider first a simple antiferromagnetic chain of p
and d orbitals as shown in Figure 10.
P"
L"
d"
P
d
P'
L
M
L'
em + M
('
Ypin
tt
Fig. 10. Schematic diagram of the orbitals of a n antiferromagnetic
chain and the overlap of the form factors of 3d electrons. In the case of
antiferromagnetic orientation of the spins, interference extinction occurs
in the neutron diffraction in the region of the ligands.
L = ligand ion, M
- metal ion. Owing to the antiferromagnetic order, the spins, and hence also the spin
densities and the form factors, have opposite signs in
M and M".
The magnetically active orbital of M can no longer be
described simply by d orbitals, but we have to take
into account the admixture of p orbitals (covalency).
A are the admixture parameters and N is a normalization factor. S, is the overlap integral between p and d
l/N2
= 1 -4
A
.
S o + 2 A2
(12)
orbitals. This can be used to find the moment density
D(r) around M; if overlap between p and p' is neglected, we obtain for D(r)
Fig. 9. Magnetic phase diagram of the ternary system
F e 2 0 ~ - C r ~ 0 ~ - A I 2 0In~ the
.
regions with double shading, the crystals
have t w o magnetic transitions: paramagnetic state - T N ~(magnetic
collinear structure) - T N ~(magnetic spiral structure).
3.5. Determination of Covalent Bond Components in
Antiferromagnetic Salts from Neutron Diffraction Data
In most antiferromagnetic salts of the 3d transition
metals, e.g. LaCr03 or KMnF3, the 3d metal ion is
surrounded octahedrally by six ligands. The orbitals
of the ligands are of s or p character, and give bonding
and antibonding orbitals with the five d orbitals of the
central ion. Only the antibonding orbitals are of
interest for the consideration of the magnetic properties. The bonding orbitals have such low energy levels
in the complexes discussed here that these levels are
The first term containing d2(r) is confined to the
magnetic ion M. The second term describes the overlap
density between p and d orbitals, and is also confined
to the immediate neighborhood of M. The third term,
on the other hand, describes a density distribution,
which lies exclusively at the position of the ligands;
the distance from L to the neighboring metal ion M"
is the same as the distance from M. The analogous
density due to the ion M" at the position L is
roughly equal to the contribution due to M, but is
of opposite sign in antiferromagnetic substances.
This means that &p*(r) and Azp'z(r) cancel out with
the analogous contributions from M and M". It is
1221 3. Hubbard and W. Marshall, Proc. physic. S O C . 86, 561
[21] W . Scharenberg, Dissertation, Universitat Bonn 1968.
Angew. Chem. internat. Edit.
1 Vol. 8
(1969) 1 No. 12
(1 965).
957
clear therefore that in a neutron diffraction experiment
on an antiferromagnetically ordered crystal, only the
contributions of the first two terms in eq. (13) are observable in the Bragg maximum.
Thus whereas the integration of eq. (13) gives 1,
integration of eq. (14) merely gives
Calculation of the magnetic moments Fobs (e.g. according to eq. (4)) from the measured diffraction intensity gives a value that is smaller by 2A2 than the true
moment PO.
theretore laborious and require a number of precautions to eliminate possible sources of error such as,
for example, ‘942 contamination” of the neutron beam
(cf. q.
We have carried out neutron diffraction experiments
of this type on LaCrO3, LaFe03, and LaMn03 as well
as on a series of fluorites having the composition
KMF3 (M = 3d transition elements), and have actually
been able to detect this effect in LaCrOJ and
LaFe03 [231. Figure I1 shows the neutron diffraction
diagrams of LaFe03 and LaCr03.
On application to the three-dimensional crystal structure, considerations analogous to those used for equations (11) to (15) lead to the following relations (A =
covalency coefficient) [22J.
It is therefore possible, in principle at least, to determine the covalent part in antiferromagnetic complexes
of this type from neutron diffraction data. The effects
are naturally very small, and the measurements are
The evaluation of our measurements on powder
specimens is concentrated mainly on the first magnetic
reflection (111) at small scattering angles, where reliable theoretical assumptions regarding the form factor
can be made. This means that the observed deviations
in the values of the form factor can be entirely attributed to the covalency of the bond, and not e.g. to a
possible asymmetric charge distribution. The measurements were hampered by a very small coherent nuclear
scattering contribution to the magnetic scattering.
This nuclear scattering component is the result of the
distortion of the perovskite-like crystal structure from
the ideal (cubic) perovskite structure; it must be
determined separately by a crystal structure analysis 1241. However, this contribution can only increase
the effect and cannot decrease it. LaCrO3 and LaFe03
crystallize in a perovskite-like structure, in which each
cation is octahedrally surrounded by antiparallel
oriented neighbors. The magnetic structure of LaMn03
is different: each Mn3+ ion has two parallel and four
antiparallel neighbors. The unfavorable ratio 2 : 4 of
parallel to antiparallel neighbors decreases the effect;
we feel, that for this reason, among others, we were
unable to detect the covalency parameters in LaMn03.
Table 3 also shows the results of measurements on
NiO 1251, MnO [261, and MnF2 1271. Results of nuclear
magnetic resonance measurements 1281 are given for
comparison.
The values for Fe3+ and Mn2+ both of which have the
3d5 configuration are particularly interesting. It is
I231 R. Nathans, G. Will, and D . E . Cox, Proc. int. Conf.
Magnetism, Nottingham 1964, p. 327.
[24] G. Will and D. E . Cox, Acta crystallogr., in press.
[25] H . A . Alperin, Physic. Rev. Letters 6, 55
L
10
20
28 r1+
30
LO
Fig. 11. Neutron diffraction diagram of a) LaFeOz and b) LaCrO,,
recorded at 4.2 “K in the antiferromagnetic state.
958
(1961).
[26] G. Will, D . E. Cox, and R. Nathans, unpublished results.
[27] R . Nathans, H . A . Alperin, S . J . Pickart, and P. J . Brown,
J . appl. Physics 34, 1182 (1963).
[28] R . G. Shulman and S. Surano, Physic. Rev. 130, 506 (1963);
R. G. Shulman and K . Knox, Physic. Rev. Letters 4 , 603 (1960),
T. P . P . Hall, W. Hayes, R . W . H . Stevenson, and J . Wilkens;
J . chem. Physics 38,1977 (1963).
Angew. Chem. internat. Edit. / Vol. 8 (1969)
1 No. 12
Table 3. Covalency parameters in some antiferromagnetic salts of 3d
elements.
I
1
From N M R measurements
1
Ion
Compound
__
Ni2+
cr3+
Fe3+
Mn2+
KNiFs
K2NaCrF6
Fe3+ in
KM~FJ
KMnF,
Covalency
coefficients ( %)
A&
+ AH
=
4.32
n = 4.1
A2
A$-
A$ = 3.3
A: = 0.76
A&-A$ = 0.18
A: - 0.52
Mn3i
From neutron diffraction
measurements
NiO
LaCrOa
LaFeOs
A& -A
i
:
A; -1-
=
6.0
A$ = 4.6
2A$ + A:
A& i- 2 A k A;
~
MnF2
MnO
LaMnO3 n o effect found
7
=
10.0
3.3
5.5
possible to determine A: and A: (in %) individually
by a combination of N M R and neutron diffraction
data
Fe3 +:
A&
=
Mn2+ in MnF2(MnO): A&
=
5.4
1.2 (1.8)
A& = 2.1
A:,
=
1.0 (1.6)
3.6. Determination of the Local 3d Spin Density
Distribution in Crystal~[~9J
The magnetic form factor f c a n be deduced from the
measured magnetic scattering with the aid of eq. (4).
It is then possible by Fourier transformation of the
diffraction data to calculate the spatial distribution of
the magnetic moments in the crystal. The result is a
diagram of the density distribution of the magnetic
moments around the centers of the atoms similar to
the electron density distribution obtained in X-ray
structure analyses. If we confine our considerations to
the 3d elements, where the crystal fields are strong
compared with the spin-orbital coupling, so that
the orbital moments are generally suppressed, then
the magnetic moment density from the neutron diffraction data directly describes the spin density, and
hence the density of the unpaired (outer) electrons,
which are responsible for the chemical bonding.
Since the outer electrons are on average farther from
the nucleus than the mean total electron density that
determines the X-ray diffraction, the magnetic form
factor, which is a shape factor in the present example
of the 3d electrons, falls off much more rapidly than
the X-ray form factor. This is illustrated in Figure 12
for Mn2+ (the magnetic form factor of Gd3+ is also
shown for comparison). In the lanthanides the unpaired electrons (4f electrons) are on average denser
at the nucleus; the fall-off of the form factor is therefore rather less. Mn2+ and Gd3+ are in an S state with
electronic configurations 3d5 and 4f7 respectively, and
the electron distribution is spherically symmetrical.
Whereas the electron distribution around the nucleus
can be taken as spherically symmetrical on the whole
in X-ray structure analysis, the distribution of the
outer bonding electrons, which are “seen” by the
neutrons, is generally not spherically symmetrical.
Consequently, the “form factor curves” for neutron
diffraction are not smooth curves; the experimental
points are scattered more or less widely about an
average curve. Figure 1 3 shows form factor values for
0.
0
-t
0
-0
0.2
0.6
0.4
_sine
08
1.0
1.2
,p]
_ j
h
Fig. 13. Experimental magnetic form factor values (circles) found for
elementary iron with polarized neutrons, compared with theoretical
curves for various electronic configurations.
-01’
I
0
02
OL
06
08
10
12
14
h
Fig. 12. Form factor curves of the 3d electrons in Mnzf and of the 4f
electrons in Gd3+ for the magnetic scattering of neutrons [the farm
factor curve of MnZ+(with all electrons) for the diffraction of X rays is
also given for comparison].
[29] For a review see e.g. G. WiN, 2. angew. Physik 24, 260
(1 968).
Angew. Chem. internat. Edit. J Vol. 8 (1969) J No. 12
elementary iron in the ferromagnetic state measured
with polarized neutrons, together with the theoretical
spherically symmetrical electronic configurations for
comparison. The deviations of the experimental points
from the spherical symmetry curves are real (e.g.
outside the error limits), and are a direct measure of
the asymmetric distribution of the 3d electrons (the
size of the circles corresponds approximately to the
experimental accuracy).
959
T h e resolution in t h e density d i a g r a m
t o depend on t h e number of reflections
a n d is determined b y t h e sin@/h range. A
o f 1 8, corresponds roughly to sin@/?,
0.4 w - 1 (1 A 0.66 h/2 s i n e ) ; t o obtain a
-
is found
measured,
resolution
= 0.3 t o
resolution
of 0.25 A, therefore, measurement up to sin@/), =
1.2 8,-1 is necessary. The deviations of t h e electron
density distribution from spherical symmetry are
particularly evident in t h e region around 0.8 A-1;
on t h e o t h e r hand, t h e magnetic form factors are very
small between sin@/h= 0.8 and 1.2 K-1 so’ t h a t t h e
determination of t h e 3d electron distribution necessitates t h e accurate measurement of very small quantities.
I t is interesting t o n o t e t h e c h a n g e in t h e sign of t h e
form factor curves from +f t o - f a t a b o u t sin@/h =
0.8 8,-1 as a result of t h e limited spread of t h e 3 d
electrons (the maximum is situated a b o u t 0.35 t o
0.40 8, from t h e center of t h e nucleus).
The usual technique of diffraction of unpolarized
neutrons is generally inadequate for t h e reliable determination of t h e form factors and hence t h e 3d electron
densities. Much more a c c u r a t e measurements are
possible with polarized neutrons.
The neutron has a spin of 1/2, and can occur in the energetically different spin states + 1/2 and -1’2. Both spin states
normally occur with equal probability in a neutron beam
(unpolarized neutrons). With a suitable experimental arrangement (Fig. 14), however, it is possible to obtain a
neutron beam that contains only neutrons in a single spin
state (polarized neutrons). A polarized neutron beam of this
type is obtained e . g . by Bragg reflection from a “polarization
monochromator” (nowadays nearly always a single crystal
having the composition Coo.92Feo.08 which is magnetized in
an external field). Approximately 99 % of the neutrons in the
resulting beam are in the same spin state. The direction of
the spin vector is described by a unit vector
A, which is
referred to the external magnetic field. The neutron beam
can be converted from one spin state into the other by superposition of a low frequency field on the polarized beam
(“flipper”).
\
kb.
H=3kOe
I
H = 18 kOe
H = L O Oe
T
‘49
jp7311LI
Fig. 14. Schematic representation of the experimental arrangement for
measurements with polarized neutrons. A: Polarization monochromator
(Coo.q*Feo.os); B: “flipper”; C: test crystal; D: counter.
The conditions for the polarization of the neutrons are
obtained by insertion of eq. (3a) and (3b) in eq. (2). The
structure factor F can then be expressed in the general form
960
The vectors + A and -A are present in equal proportions in
a beam of unpolarized neutrons; the scattered intensity is
proportional to F* and therefore to b2 + p2; the cross term
(qA) thus becomes zero [eq. 21. For polarized neutrons, o n
the other hand, we have e . g . + A = 1 and -A = 0. Eq. (16)
now leads to a phase dependence of the nuclear scattering
on the magnetic scattering. The resultant intensity is proportional to (b + p)2:
polarized neutrons:
I
unpolarized neutrons: I
- (b + p)’
- bz+ p2
The requirement for the polarization monochromator is
therefore simply b = p. In this case we have for an unpolarized neutron beam I(+x) = (b p)2 = 2b2 and I(-i) =
(b- p)2 = 0.
+
To detect very small scattering contributions p, therefore, it
would be necessary with unpolarized neutrons to measure a
term p2 whose square is small and a term b2 whose square is
large, whereas for polarized neutrons there is a linear relation
between p and b. Another advantage of polarized neutrons
lies in the measuring technique itself. For the accurate determination of p one need only determine accurately the ratio
p: b; the method is therefore o n a relative scale, and is thus
independent of systematic errors (such as the determination
of an absolute scale or of the temperature factor, which is
known to cause a decrease in the scattering intensity with
increasing scattering angle, similar to the decrease in the form
factor itself.) In measurements with polarized neutrons, one
simply determines the polarization ratio P by recording first
I+ with neutrons having A = +1, and then I - with neutrons
having A = -1.
This leads to very accurate values of p. and hence off:
The limits of the method can also be found very readily from
eq. (17). To obtain a difference between If and I - , coherent
overlap of neutron scattering and magnetic scattering must
occur in the reflections. Investigations so far have been
concentrated on ferromagnetic crystals, where this condition
is always satisfied. This is not generally the case in antiferromagnetic crystals, and measurement with polarized neutrons
is fundamentally impossible for e . g . MnO or NiO, where the
neutron scattering reflections and the magnetic reflections
are rigorously separated because the magnetic unit cell is
twice as large as the chemical unit cell, and the crystallographic extinction rules separate the two sets of reflections
exactly.
Detailed investigations have been carried o u t on the
ferromagnetic elements iron 1301, nickel 1311, a n d hexagonal c o b a l t 1327, as well as on Pd-0.01 Fe [331, Fe3Al1341,
and some Fe-Rh alloys 1351. A Fourier synthesis with
t h e experimental data as amplitudes gives t h e electron
density distribution of t h e unpaired electrons. If only
t h e difference values of t h e experimental points from
t h e spherically symmetrical average curve are used as
amplitudes, the deviation of the electron distribution
[30] C. G . Shull and Y. Yamada, J. physic. SOC.Japan, Suppl.
B-I11 17, 1 (1962); C. G . Shull and H . A . Mook, Physic. Rev.
Letters 16, 184 (7966).
1311 H . A . Mook and C.G . Shu//,J. appl. Physics 37,1034(1966).
H. A . Moak, Physic. Rev. 148, 495 (1966).
[32] R . M . Moon, Physic. Rev. 136, A 195 (1964).
[33] W. C. Philips, Physic. Rev. 138, A 1649 (1965).
1341 S. J . Pickart and R. Nafhans, Physic. Rev. 123, 1163 (1961).
[35J G . Shirane, R . Nathans, and C . W.Chen, Physic. Rev. 134,
A 1547 (1964).
Angew. Chein. internat. Edit. / Vol. 8 (1969)1 No. 12
from spherical symmetry is obtained directly. Figure
1 5 shows the result of such a difference Fourier synthesis using the values for iron from Figure 13. An
interesting feature of this spin density distribution is
the extended negative areas, which represent a density
distribution of spins of opposite sign, i.e. antiferromagnetic orientation. The negative polarization is
assigned to the 4s conduction electrons that uniformly
fill the entire crystal with the opposite spin direction.
distribution for spherical symmetry is 40% eg, 60%
The electron distribution obtained by analysis of
the neutron diffraction data are: 3dspin - 0 . 6 5 6 ~ ~ .
3dorblta1 = +0.055 VB; for 4s conduction electrons,
one finds -0.105 p ~ .
tZg).
Table 4. Character of the electron distribution in some 3d elements
and 3d-4d alloys.
3d configuration
Composition
eg (%)
Fe-nucleus
a0
-
2
t2g ( %)
Theoretical values for spherical
symmetry (cubic)
40
60
Fe (body-centered cubic)
Ni (face-centered cubic)
Co (doped with Fe)
(body-centered cubic)
V (body-centered cubic)
53
19
21
47
81
79
19
81
47
53
48
52
28
60
48
40
52
Fen.s~Rho.48
(ferromagnetic)
Fen. sRho. 5
(antiferromagnetic)
Fen.n13Pdn.987
Fe,AI : FeI
FeII
72
Theoretical values for spherical
symmetry (hexagonal)
C o (hexagonal)
39.4
41.6
1
19.0
3.7. The Magnetic Structure of Oxygen
2
Fig. IS. Deviation of the spin density distribution from spherical symmetry in a n iron crystal (difference Fourier diagram). The extended
negative regions are due to the negatively polarized 4s electrons which
form a uniform antiferromagnetic background in the otherwise ferromagnetic crystal (3d electrons). The maxima therefore essentially denote
an additional 3d electron density; a distinct concentration of the 3d
electrons along the crystal axes is evident. This distribution is characteristic of the eg symmetry of the 3d electrons.
The eg or tZg character of the electron distribution can
be deduced from the observed concentration of the
electron density in certain directions in the crystal.
Iron crystallizes in a body-centered cubic lattice. The
five-fold degenerate 3d energy levels are split by the
crystal field into two states with eg symmetry and
three states with tzg symmetry. Owing to the octahedral symmetry of the environment of the iron atoms,
the energies of the eg levels are higher than those of
the t Z g levels. The observed electron concentration
along the cubic [loo] axes (Fig. 15) thus points to eg
symmetry of the electron distribution. Careful numerical evaluation of the experimental data 1301, in which
a small component of the orbital moment and 4s electron contributions were taken into account, gave +2.39
p~ for positive magnetization and -0.21 VB for negative magnetization. Assuming that Hund's rule is
valid, therefore, there are 7.6 electrons in the 3d band
and 0.4 electrons in the conduction band. The electronic configuration in ferromagnetic elementary iron
should thus be 3d7-6 4s 0.4. Of the 7.6 electrons, 53 %
are in an eg symmetry state (along the cube axes) and
47 % in a tZg symmetry state (along the cube diagonals).
Exactly the opposite situation is found in nickel, which
crystallizes in a face-centered cubic lattice; 1 9 % of the
3d electrons are in eg levels and 81 % in t2g levels (the
Angew. Chem. internat. Edit.
Vol. 8 (1969) / No. 12
Finally, brief mention should be made of a neutron
diffraction study on solid oxygen, which showed an
antiferromagnetic order [361. Crystalline oxygen exists
in three phases:
a-02 (0-24 OK); p - 0 2 (24-44 OK); y-02 (44-54 OK).
T h e y phase has 8 molecules in a cubic unit cell. The p
phase has one molecule in a rhombohedra1 cell, the
molecule probabIy being oriented along the trigonal
axis. The structure of the a phase is unknown. The
symmetry is presumably monoclinic. Magnetic measurements on gaseous, liquid, and solid y oxygen
indicate two unpaired electrons in a 3S1 state [371; the
susceptibility indicates an anti-ferromagnetic order at
low temperatures. In the neutron diffraction experiments on solid 8- and a-oxygen, the diagram of a-02
did in fact contain two additional lines at small
scattering angles, which must be attributed to an antiferromagnetic order in a - 0 2 . Since the magnetically
ordered spins are due to unpaired p-electrons which
have a relatively large spread, the magnetic form
factor falls off very rapidly, with the result that the
detection of magnetic reflections become difficult
(f := 0.5 for sin@/h = 0.2 k 1 ) . No magnetic longrange order was observed for P-0,. The magnetic and
neutron diffraction data (maximum of the diffuse
scattering at sin@/h = 0.1 kl),
however, indicate an
antiferromagnetic short-range order in p-02.
Received: December 11, 1968
[A 731 IE]
German version: Angew. Chem. 81, 984 (1969)
Translated by Express Translation Service, London
[36] M . F. CoNins, Proc. physic. SOC.89, 415 (1966).
[37] E. Kanda, T . Hasedo, and A . Otsubo, Physica (Utrecht) 20,
131 (1954); A . S. Borovic-Roinanov, M . P. Orlova, and P. C.
Strelkov, Doklady Akad. Nauk SSSR 99, 699 (1954).
961
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crystals, structure, diffraction, analysis, neutron
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