o (1993 ACTA PHYSICA POLONICA A o 1
ocee i gs o e I I e a io a Sc oo o Mag e ism iałowie a 9
MAG E IC O M AC O MEASU EME S Y O A ISE EU O SCA E I G A ICA IO O EA Y E MIO SU E CO UC O S
K.-U. NEUMANN AND K.R.A. ZIEBECK
e a me o ysics oug o oug U i e si y o ec o ogy oug o oug E11 3 U G ea i ai
e e e ime a ec ique o s i o a ise eu o sca e i g as use i mag e ic o m ac o measu eme s is ese e A i o uc io o e i e e a io a e ca cu a io o mag e ic o m ac o s a mag e i- a io e si ies is gi e e e e ime a ec ique o eu o sca e i g eo y as a ie o e as ic s i o a ise sca e i g e e ime s is ie y i o uce e ca cu a io o e mag e ic o m ac o a e mag e i a- io e si ies a e co si e e o sim e mo e sys ems suc as a co ec io o oca ise mag e ic mome s o a i i e a e ec o sys em e iscussio is i us a e y a e e ime a i es iga io o e mag e ic o m ac o i e ea y e mio su e co uc o s U e13 a U 3 Mag e i a io e si y ma s a mag e ic o m ac o s a e ese e a ei im ica io s o o e ysica qua i ies a e ie y iscusse ACS um e s 75 5 + 71 +
1 I o uc io eu o sca e i g a i a icu a s i o a ise eu o sca e i g is a owe u oo o e c a ac e i a io a i es iga io o ysica o e ies o a a omic sca e e c a ge eu a i y a mag e ic i o e mome o e eu o make i a i ea o e o e i es iga io o mag e ic eg ees o ee om I a i io o usi g eu o sca e i g o i es iga i g e ome a co ec e wi mag e ism ( y e g es a is i g e a u e o mag e ica y o e e s a es e e ime s may e e ise o e c a ac e i a io o e ai s o e ec o ic wa e u c io s Suc a e e ime is ea ise w e s i o a ise eu o s a e use o e e e mi a io o e mag e ic o m ac o a o e mag e i a io e si y Co a y o s ec osco ic ec iques w ic i e e ai s o e wa e u c io y a i es iga io o e e e ime a y e e mi e e e sc eme (e g e c ys a ie e e s o oca ise mome s mag e ic o m ac o measu eme s e e e
( 3 44 K.-U. Neumann, K.R.A. Ziebeck
to here are elastic measurements and therefore these measurements do not involve an energy transfer between neutron and target. Consequently this method does not yield direct information on the energy eigenvalues of the Hamiltonian, that is the energy levels or band structure of the system under investigation. Instead a magnetic form factor investigation can be classified as an investigation of the real space properties of some electronic wave functions. This point will be discussed more fully below, and a clarification will be given as to what exactly is meant by "some electronic wave functions". The investigation of the magnetic form factor by spin polarised neutron scattering is therefore complementary to experiments such as de Haas—van Alphen measurements, which investigate the details in momentum space of energy levels and the Fermi surface. In order to be able to fully appreciate the potential of spin polarised neu- tron scattering for the investigation of magnetization densities and magnetic form factors a brief introduction is given to the experimental technique. First, the basic principles are reviewed of elastic nuclear and magnetic scattering, and a derivation is given for the description of a magnetic form factor measurement. Two simple models, a localised moment system and an itinerant electron model, are consid- ered, and for each of these model systems a magnetic form factor experiment is analysed. In the second section the theory is illustrated using the results of an ex- perimental investigation. A magnetic form factor measurement is considered which was carried out on the heavy fermion compounds UBe1 3 and UPt3. The exper- imental magnetization densities and form factors are presented and discussed in relation to the simple model systems mentioned above. Some consequences of the experimental findings are briefly discussed.
2. M n t f r f t r r nt
Magnetic form factor measurements are usually carried out on single crystals employing spin polarised neutrons. The discussion here will be limited to the case of a paramagnetic material with a crystallographic structure which has a centre of inversion symmetry. An experimental determination of the magnetic form factor comprises of two different measurements. By using unpolarised neutrons, a first experiment establishes the details of the nuclear structure of the crystal under investigation. In a second experiment and by using spin polarised neutrons a quantity known as the flipping ratio is determined for a number of Bragg reflections. In order to establish the details of the nuclear structure it is important to carry out both experiments on the same single crystal and preferably also at the same temperature. The nuclear structure is determined by measuring the inte- grated intensity of Bragg reflections. A subsequent refinement of a crystallographic model and its comparison with the measured Bragg intensities is used to establish important parameters such as lattice site occupations and temperature factors. In this way the nuclear structure factor is determined precisely. In a second experiment the Bragg reflection is measured using spin polarised neutrons and with the crystal in an external magnetic field. Due to the presence of a magnetic field a ferromagnetic moment is induced in the paramagnet. As a result of Magnetic Form Factor Measurements ... 45 the induced ferromagnetic order magnetic Bragg scattering occurs. The magnetic scattering arises in addition to nuclear scattering. As both the magnetic and the nuclear unit cell have the same dimensions, magnetic and nuclear Bragg reflections are superimposed. It turns out that this superposition is more than a mere addition of nuclear and magnetic intensities. By using spin polarised neutrons it is possible to make use of an interference term in the scattering cross-section between nuclear and magnetic scattering. This combination of magnetic and nuclear scattering in the neutron scattering cross-section can be used to precisely determine a quantity known as the flipping ratio. The flipping ratio is a function of the magnetic and nuclear structure factors. If the nuclear structure factor is known, a measurement of the flipping ratio allows a very precise determination of the magnetic structure factor. The physical significance of the entities mentioned here will be discussed more fully below. A more detailed analysis of the theory of neutron scattering can be found in standard text books [1-4].
2.1. Nuclear scattering
For structure determination and for the measurement of the magnetic form factors it suffices to restrict the description of nuclear scattering of neutrons to elastic scattering only. In particular, it is only the integrated Bragg scattered in- tensity which is of relevance here. By definition the Bragg scattering is elastic scattering with no energy transfer between the crystal and the neutron. The dif- ferential neutron scattering cross-section can be written in the form
where ba is the scattering length of the atom i which is located at position k is the neutron scattering vector and dΩ is the infinitesimal solid angle into which the neutrons are scattered. The summation is carried out over all atoms in the crystal. Making use of the translational symmetry of the crystallographic structure, the summation in (1) may be separated into two partial sums
Here the index v sums over all unit cells of the crystal and the summation is carried out over all atoms within the unit cell v. The position of the atoms in each unit cell is given by IZV — IL„ and it is measured relative to the origin of the v-th unit cell as determined by 11 ,. A nuclear structure factor of the unit cell (k may be defined as
where the atomic position Ri is measured with respect to the origin of the unit cell. In general, (k will be a complex number. However, for a crystallographic K -U euma K A ie eck s uc u e wi a ce e o i e sio symme y e uc ea s uc u e ac o may e c ose o e ea ( y aki g e oi o i e sio symme y as e o igi o e u i ce e e y a om a osi io i as i s symme y e a e a om oca e a —I = a wi e a om a i g e same sca e i g e g As a co seque ce e e y co i u io ie (ik • i i (3 is com i e wi i s com e co uga e i e (—ik • a i e summa io e imagi a y a s ca ce I se i g (3 i o (1 a ca yi g ou e summa io o e a u i ce s o e a i es a
e e is e o a um e o u i ce s i e c ys a a δ(k — τ is e " e a u c io " wi r ei g a eci oca a ice ec o e summa io is ca ie ou o e a eci oca a ice ec o s r us sca e i g occu s o sca e i g ec o s k w ic co es o o a eci oca a ice ec o a wi a i e si y w ic is e e mi e y e squa e o e mo u us o e uc ea s uc u e ac o (τ In ac ua e e ime s e agg e ec io wi o e suc a s a eak as i ica e y e " e a u c io " i ( eak oa e i g occu s ue o e eso u io o e i s ume a a so ue o sam e c a ac e is ics (e g as gi e y e s a ia e e o e co e e y sca e i g egio s i e sam e owe e e e ai s o e eak s a e a e o o i e es e e a e e is o ee o u sue is oi a y u e e i eg a e agg i e si y o a agg e ec io c a ac e ise y is gi e y
S(τ is a e ec io e e e sca i g ac o e sca e i g ec o e e e ce o S a ises ue o e me o use o i eg a e e agg eaks e geome ic
co ec io is k ow as e o e co ec io a i is ea i y e imi a e om e measu e i eg a e i e si ies e e emai s a o e a sca e ac o w ic is sca e i g ec o i e e e e sca e ac o is o o io a o e o a um e o u i ce s i e c ys a e o i g e ew sca e ac o (a e e e imi a io o e sca e i g ec o e e e ce y S e e e ime a y e e mi e i eg a e i e si y o e agg e ec io is gi e y
U o a co s a e measu e i e si ies a e e e o e e e mi e y e squa e o e uc ea s uc u e ac o owe e o e esc i io o e uc ea s uc u e e a ue o (τ is equi e As oi e ou a o e o s uc u es wi a i e sio ce e o symme y e uc ea s uc u e ac o may e c ose ea us so i g e uc ea s uc u e e uces o c oosi g o eac e ec io e "co ec " sig i e equa io
I is oi e ou a so i g e a o e ase o em o a e ec io s is equi a e o e e e mi a io o e a omic osi io s i e u i ce is is so ecause a i g o ai e e s uc u e ac o s (τ o a su icie um e o Mag e ic σ m ac σ Measu eme s 7
e ec io s e a omic osi io s may e o ai e y ou ie i e sio W i i g (3 i e o m
i is see a e uc ea s uc u e ac o is o i g e se u e ou ie a s o m o e i e ac io o e ia w ic i u is e e mi e y e osi io s o e a oms a ei sca e i g s e g s e a o e e i a io was ca ie ou usi g some sim i yi g assum io s e a ice as ee ake as e ec a e e ma mo io o e a oms as ee eg ec e e em e a u e mo io a e o oi uc ua io s ca e ake i o accou y e su s i u io w e e W(k is e em e a u e ac o w ic may e a iso o ic e u c io e (—W(k k is k ow as e e ye—Wa e ac o O e e e ime a ac o s suc as e i c io a a so io co ec io s a e a so ea i y a ie [5] a s a a oce u es e is o aki g ese i o accou [ ]
2.2. Magnetic scattering I com a iso o uc ea sca e i g o w ic e i e ac io o e ia e- wee eu o a uc eus is e e mi e y e uc ea o ce mag e ic sca e - i g o eu o s is ase o a e i e y i e e i e ac io mec a ism Mag e ic eu o sca e i g e e e o e e a ises ue o a e ec omag e ic i e ac io e- wee e mag e ic i o e mome s o e eu o a ose o e e ec o s A i o uc io a a com e e e i a io o e asic o mu ae o mag e ic eu o sca e i g ca e ou i s a a e ooks o eu o sca e i g [1 ] e e e i e es is ocuse o e as ic mag e ic sca e i g o y wi e mag- e ic s uc u e ei g es ic e o e omag e ic o e o e u ose o ca yi g ou e eu o sca e i g e e ime i is imma e ia w e e e e omag e ic o e is i i sic o w e e i as ee i uce y e a ica io o a e e a mag e ic ie A mo e e ai e a a ysis (as gi e e g i [1 ] s ows a e s e g o e eu o —e ec o mag e ic mome i e ac io is e e mi e y e o uc o e gy omag e ic a io o e eu o -y (-y = —1 913 a e c assica e ec o a ius wi = 17 5 1 -15 m e o uc M = γr0 = —5 39 1 -15 m e i es a mag e ic sca e i g e g M w ic o i es e u i o measu i g e s e g o mag e ic mome s i mag e ic eu o sca e i g e e ime s e e- o e i e o owi g e mag e ic mome s µ wi a ways e measu e i u i s o M I is a so im o a o o e e e a e mag i u e o M is com a a e wi y ica mag i u es o uc ea sca e i g e g s o a e omag e ica y o e e s a e e ce ime sio s as e i e y e c emica u i ce ł a e o c a ge e ce y summi g o e e same a oms as i (3 a mag e ic s uc u e ac o o e u i ce may e e i e as 48 K.-U. Neumann, K.R.A. Ziebeck
Here µi is the magnetic moment located on the i-th atom, and fi(k) is the atomic magnetic form factor of the atom. The form factor i is normalised according to
The magnetic structure factor as defined in (10) is a vector quantity. The overall strength of the magnetic neutron scattering is determined by the sizes of the mag- netic moments. Atoms with no magnetic moment, µ i = 0, will not contribute to the magnetic scattering. An additional scattering vector dependence is introduced into the structure factor due to the magnetic form factor. In general, the magnetic form factor tends to decrease with increasing magnitude of the scattering vector. As a consequence, the magnetic scattering diminishes for increasing scattering an- gles. This is in contrast to nuclear scattering for which the equivalent nuclear form factor is independent of the scattering vector and equal to one. Similarly to the case of the structure factor for nuclear scattering the mag- netic structure factor can be expressed as the Fourier transform of the interaction potential. For magnetic scattering the interaction potential is given by the mag- netization density within the unit cell
where M(r) is the position dependent magnetization. M(r) is a periodic function which has the periodicity of the nuclear lattice for a ferromagnetically ordered state. For a magnetization distribution which is given as the superposition of lo- calised atomic magnetic moments located at ri within the unit cell the magneti- zation density is given by
In this case the magnetic structure factor in (12) reduces to
A comparison with (10) yields the atomic form factor in the form
as the Fourier transform of the real space atomic magnetization mi(r). The total magnetic moment µ i is defined as the integral of mi(r) according to
This definition together with (15) ensures the normalization of the atomic form factor as defined in (11). For localised moments the integration in (14) and (15) is carried out over the whole of the three-dimensional space. All the various magnetic contributions of each individual atom are subsumed within mi. In general, mi is composed of several magnetic moments, including spin and orbital magnetic contributions and, if necessary, a core polarization as well. For the case of itinerant electron magnetism, for which the electronic wave functions are sufficiently delocalised, the general form (12) has to be used for Mag e ic σ m ac o Measu eme s 49 the description of the magnetic structure factor. In general, it is not possible to uniquely allocate a given magnetization density to an individual atom, and the notion of a magnetic moment located on an atom loses its meaning. It is only mea i g uł o a k a ou e o a mag e ic mome e u i ce M o with the total magnetic moment being defined as
Obviously for the case of localised magnetic moments M o is determined by the sum over all magnetic moments in the unit cell. In this way the magnetic structure factor may be defined for both, localised as well as itinerant electron magnetism. However, for neutron scattering additional complications arise due to the vector form of the magnetic structure factor as compared to an equivalent scalar quantity for nuclear scattering. It turns out [1, 2, 4] that only those components of FM(k) which are per- pendicular to the scattering vector k may give rise to magnetic neutron scattering. The components which are active in the scattering process are defined as
In order'to combine the magnetic scattering contribution as defined by the vector in (18) with the nuclear scattering part as described by the scalar quantity FN(r) the vector FM(k) has to be reduced to a scalar. A more detailed analysis shows that it is the scalar product of F┴M(k)Pwithof the the neutron polarization vector beam that is needed for the neutron scattering cross-section. The scalar magnetic structure factor is defined as
For perfect polarization of the neutron beam P is a unit vector, the direction of which indicates the polarization direction. For a partial polarization of the neutron beam |P| is defined as
with n1, n↓, being the number of spin up or spin down neutrons in the beam. F, F(k) ┴M is the entity which plays the equivalent role to the nuclear structure factor in the formula for the scattering cross-section. For this reason F 1 ( k will be referred to as the magnetic structure factor. (This should not be confused with Fas M(k)┴ defined in (10).) Considering only magnetic scattering, the differential cross-section is determined as
where τM is a reciprocal lattice vector of the magnetic lattice and NM is given by the number of magnetic unit cells in the sample. For ferromagnetic order the chemical and magnetic unit cells are identical. Consequently N11 and τM have the same values as their nuclear counterparts. 50 K.-U. Neumann, K.R.A. Ziebeck
With nuclear and magnetic scattering present both contributions have to be combined. There are the two possibilities of either combining the intensities of nuclear and magnetic scattering or of adding the structure factors first and then inserting the sum into the formula for the differential cross-section. For the first possibility the intensities are summed by adding (4) and (20). This corresponds to an incoherent superposition of the scattered waves from the nuclear and the magnetic subsystems of the target. Here, however, nuclear and ferromagnetic scattering is considered which is superimposed on the same Bragg reflection. Under these conditions it is the sum of the nuclear and the magnetic structure factors as determined by (3), (14) and (19), which has to be evaluated. The underlying assumption is one of a coherent superposition of nuclear and mag- netic scattering contributions. The sum has to be inserted into the expression for the differential cross-section. Assuming perfect polarization of the neutron beam one arrives at
for a neutron spin polarization parallel (a) and antiparallel (b) to the external magnetic field direction. For a partially polarised neutron beam the cross-section is an average of (22a) and (22b) with the weighing factors determined by the proportion of spin up and spin down neutrons in the neutron beam. For this case the cross-section takes the form
The direction of is taken to be parallel to the direction of the external magnetic field. This form of the differential neutron scattering cross-section differs by the presence of an interference term between nuclear and magnetic scattering from the cross-section obtained for an incoherent superposition of the nuclear and magnetic intensities. The incoherent superposition is realized for an unpolarised neutron ex- periment, for which the magnitude of is equal to zero. In this case the differential cross-section takes the form
It is the presence of the interference term between nuclear and magnetic scattering which forms the basis of magnetic form factor measurements. It depends on the polarization of the neutron beam through in (23) or through the change of sign of I in (22a) and (22b). In the next section it will be discussed how this interference is exploited experimentally and how it can be utilised for the investigation of details of the magnetization on an atomic level. Magnetic Form Factor Measurements ... 51
2.3. Experimental e qua i ies w ic a e o e e e mi e y e e ime a e e uc ea a mag e ic s uc u e ac o s ese a e measu e i se a a e e e ime s a y usi g u o a ise a o a ise eu o s es ec i e y e measu eme o e uc ea s uc u e ac o is equi a e o a s uc u e e e mi a io e e ime I e e e ime e i eg a e agg i e si y is e e mi e o a um e o agg e ec io s As iscusse a o e i Sec 1 a s uc u a mo eł may e use o e i e e ee a ame e s o a mo e s uc u e us o ai i g e uc ea s uc u e ac o s FN (r) o a agg e ec io s use i e e i eme e uc ea s uc u e ac o oge e wi o e e e ime a e i ies suc as a ice a ame e e c a e assume k ow w e e s i o a ise eu o e e ime is ca ie ou o e e e mi a io o e mag e ic o m ac o a si g e c ys a o e ma- e ia u e i es iga io is o ie a e i a e e a mag e ic ie i suc a way a e mag e ic mome is a o g e mag e ic ie i ec io is ecessi a es a e mag e ic ie is i ec e a a e o a easy a is o mag e i a io I o - e o ma imi e e mag e ic sig a (see (1 i is e e a e o a ig e mag e ic ie e e icu a o e sca e i g a e (w ic is e i e as e a e s a e y e wa e ec o s o e i comi g a sca e e eu o s wi e sca e i g ec o ei g e i e e ce o ese wo ec o s S i o a ise eu o s a e use i e o m ac o measu eme owe e o o a i a io a a ysis is ee e a e e eu o s a e agg sca e e y e sam e u e i es iga io o a gi e agg e ec io r a assumi g a e ec eu o s i o a - i a io e i e e ia eu o sca e i g c oss-sec io s a e e e mi e o e
e i eg a e i e si ies o is agg e ec io a e gi e y
e e e o es a eu o s i o a i a io a a e o e a ie mag e ic ie w i e I i ica es a a i a a e eu o s i o a i a io y usi g ei e e i e e ia c oss-sec io o e i eg a e agg i e si y e i i g a io R(τ ) is e i e as
E e ime a y i is easie a as e o measu e e e a i e i e si ies o s i u a s i ow eu o s a o y o e oi o e agg e ec io (w ic i e e e ime is ou o e oa e e o a Gaussia eak s a e com a e o 52 K-U. Neumann, K.R.A. Ziebeck a " e a u c io " i ica e i e a o e o mu ae i s ea o a i g o o ai e i eg a e i e si ies o o eu o s i o ie a io s e e o e i e e e ime e c ys a is se so as o ma imi e e cou i g a e o a gi e agg e ec io a e e a i e cou i g a es a e measu e as e e mi e y ( 5a a ( 5 o e e e a e s a es o e uc ea a mag e ic agg e ec io s a e i e ica (τ oes o e e o ow e a io is e i e w e e y usi g e a io o e i eg a e i e si ies o y u i i i g e a io a o y o e oi o e agg e ec io As iscusse i e ai e sew e e [5] o e e e ime a ac o s a e ea i y i c u e i e e i i io o e i i g a io aki g i o accou a a ia o a i a io o e eu o eam e e icie cy e o e eu o s i i e a e a g e O e wee mag e ic mome i ec io a sca e i g ec o e i i g a io akes e o m
e u i g o e e essio o e i i g a io as gi e i ( 7 e equa io is ea i y i e e o yie e a io o e mag e ic o uc ea s uc u e ac o s as a u c io o e i i g a io
e am igui y gi e y e ± √ (τ e m i ( 9 is e a e o e am igui y o w i i g (τ i ( 7 ei e i e ms o ┴Mτo / ( y usi g e i e se a io (τ / ┴MτI ( is e equi a e o e ques io ┴Mτis( / ( o w e e ( a ge o sma e a o e I a ac ua e e ime is oes ge e a y o ese a o em ecause a c oice ca e ma e o ysica g ou s A is oi i is wo s essi g agai a e k ow e ge o e uc ea s uc u e ac o is i a o e e e mi a io o ┴Mτ( e u ce ai y i e mag e ic s uc u e ac o is i s y e e mi e y e e e ime a e o i e measu eme o e i i g a io (τ u seco y a so a mo e im o a y y e u ce ai y i e uc ea s uc u e ac o As see om ( 9 e mag e ic s uc u e ac o is measu e i u i s o e uc ea o e a co seque y a y sys ema ic e o i e e e mi a io o e uc ea s uc u e ac o (as gi e y e eg ec o em e a u e ac o s a so io e i c io e c wi i ue ce e accu acy o e mag e ic s uc u e ac o Some a icu a ea u es o e i i g a io a e wo o i g I is i s o a a qua i y w ic is i e e e o a sca i g ac o y aki g e a io i ( 7 e sca i g ac o s a e ca ce e u e mo e i is a u c io o y o e uc ea a e mag e ic s uc u e ac o s o a ecia e e sig i ica ce o is s a eme co si e e case o weak e omag e ic o e c a ac e ise y
W e e a i g┴M τ / ( ( 7 o ea i g o e i τ e i i g a io is gi e y Mag e ic σ m ac o Measu eme s 53
The deviation of the flipping ratio from one is a direct measure of the magnitude of Fili as measured in units of the nuclear structure factor. As FN(r) is known from a previous structure determination experiment the deviation of the flipping ratio from one immediately yields the value of F ┴Mτbothmagnitude(It(): and itsphase! ( ) should be stressed again that only centrosymmetric structures are considered here. For such structures the phase degree of freedom of the structure factors reduces to a choice of a ± sign in front of the wave function.) In these respects the measurement of the flipping ratio is unusual. It ren- ders possible an absolute measurement without the necessity of a normalization. Furthermore, it allows for a simultaneous determination of the amplitude and the phase of the magnetic scattering contribution. The physical origin of this can be ace ack o e ese ce o e i e e e ce e m i e i e e iał eu o sca e i g c oss-sec io As e i e e e ce e m is o o io a o e o uc o e uc ea a mag e ic s uc u e ac o s e ase o e mag e ic sca e i g ca e e a e o e ase o e uc ea s uc u e ac o e ase o em o e uc ea s uc u e ac o (w ic is e key ques io o a y s uc u e e e - mi a io as ee so e a e ase is a ea y e e mi e (u o a o e a ase ac o commo o a e ec io s us e k ow e ge o e ase o e uc ea s uc u e ac o ca e use o e ac e i o ma io o e ase o e mag e ic s uc u e ac o It is also worth pointing out that due to the interference term the sensitivity is significantly increased. While for an experiment employing unpolarised neutrons and with ferromagnetic order present in the sample the scattered intensity is given by
for an experiment carried out with spin polarised neutrons the intensity change is determined to be given by
For a ratio of F┴M / (τ = 0.1, the changeτ of intensity, δI/I may be compared with and normalized to the intensity I of an identical sample without magnetic order. For an unpolarised neutron experiment δI/I is of order 1%, while a value of . order 20% is found for 81/I for the spin polarised neutron scattering experiment. This increased sensitivity in spin polarised neutron scattering experiments allows for a more accurate determination of the magnetic scattering contribution and, finally, also of the magnetic structure factor. Thus, a measurement of the flipping ratios yields accurate values of the magnetic structure factors. As an experimentally determined entity the magnetic structure factors are model independent. However, the collection of values of mag- netic structure factors is not very revealing if presented on its own and in tabular form. It is more informative to consider the Fourier transform of the magnetic structure factors. As pointed out above the magnetic structure factors are determined by the Fourier transformation of the magnetic interaction potential. According to (12) 54 K.-U. Neumann, K.R.A. Ziebeck
e mag e ic s uc u e ac o M(τ is e e mi e y
The inverse transformation is given by
wi ei g e o ume o e u i ce e summa io is ca ie ou o e a ł eci oca a ice ec o s τ e mag e ic s uc u e ac o s a e e ou ie coe i- cie s i e se ies (3 e ou ie a s o m ack i o ea s ace wi e e o e construct a picture of the real space interaction potential, which amounts here to the magnetization density distribution. In this manner a map of the magnetization density may be constructed and presented in real space as either a cut through the unit cell or as a projection. In order to be able to perform the Fourier transformation by using the exper- imentally determined magnetic structure factors an infinite number of the Fourier components in (34) has to be known. Experimentally this presents a problem, as only a finite number of coefficients can be determined. A Fourier series is an infinite series and it is only complete if all coefficients of the infinite series are included. Therefore, constructing a magnetization map by using a finite subset of coeffi- cients will result in "incomplete" and thus erroneous magnetization density maps. However, it turns out that a magnetization density map constructed by using a finite set of coefficients in (34) may still be meaningful and yields a magnetization density with the important features present. In order to understand the limitations imposed by the finite number of magnetic structure factors and to appreciate the significance of magnetization density maps a more detailed discussion of some of the relevant aspects is helpful. A discussion relating to these questions has been given by Moon [7]. A reflection which is not measurable by neutrons is the Bragg reflection co es o i g o τ = O owe e o is eci oca a ice ec o e e i i io of the magnetic structure factor in (12) reduces to the evaluation of the total magnetic moment in the unit cell as given in (17). While this reflection is not accessible in the neutron scattering experiment, its value may be obtained by magnetization measurements using conventional magnetometers. Its inclusion in the Fourier series and in a magnetization density map adds a scattering vector independent contribution. This Fourier coefficient may thus be considered to fix the zero level in the magnetization density map. A mo e se ious imi a io is e u e cu -o i Iτ i e summa io o e Fourier series. Denote by k, the magnitude of the scattering vector τ beyond which no more experimental values of Fi 1 ( are available. k, is taken to characterise the upper limit of the cut-off in the Fourier series. The coefficients belonging to τ > kc give rise to fast oscillating contributions in the unit cell. These components are needed for the description of the finer details in the magnetization density in real space. By not including these magnetic structure factors in (34) these details are lost. This loss of detailed structure corresponds to a picture where in the M n t r t r M r nt 55 perfect real space magnetization density (which is obtained by including all Fourier coefficients in (34)) a local average is carried out over a small volume element so as to eliminate the fast oscillating contributions within this volume element. In order to estimate the dimension of the volume element and to be able to relate it to k, the following procedure may be used (it suffices to consider the one-dimensional case). The magnitude of the cut-off wave vector kc defines a characteristic length in reciprocal space. Let r, be a characteristic length in real space. A reasonable estimate for the interval of average [0, c] will be given by a value of r, for which the phase factor exp(ik • undergoes several (but at least one) oscillation(s) as r is varied within the limits of the averaging interval and for a scattering vector k > kc . This criterion results in a definition of r, given by k, • r, = 2a. Writing kc in terms of a cut-off scattering angle with a ei g e neutron wavelength employed_ _ in the ex )eriment, the interval of average is defined as For the two- or three-dimensional cases the volume of average may be taken to be a square or a cube with the linear dimensions de. fined by F Thus carrying out a Fourier transform on a limited set of wrier coefficients may still yield a correct magnetization density which, however, is deprived of some details in its structure. The experimentally determined magnetization densities are therefore not sensitive on a length scale defined by r c , with the fine structure being smeared out over the average volume. The coefficients which remain in (34) are all those Fourier components which belong to scattering vectors τ with τ k,. It is obvious that within this limited range the set of coefficients must be complete, or at least as complete as possible, for a meaningful magnetization picture to emerge from the Fourier transformation. As the various Fourier components in (34) are linearly independent a "missing component" in the series cannot be compensated by other ones. A magnetization density map will be severely distorted when constructed with an incomplete finite series with some coefficients missing for a with |n < kc . An example of a magnetization density map with a missing component will be given below. So far the discussion has been restricted to a paramagnetic material with a centrosymmetric crystallographic structure, and the procedures of a magnetic form factor measurement have been developed within this restriction. In order to complete this section, a brief discussion is given of the problems which arise when these restrictions are lifted. First, consider a magnetically ordered state of the sample. A brief but very instructive introduction to the investigation of magnetic order by neutron scatter- ing has been given by Brown [8]. The crystallographic structure is still required to be centrosymmetric. For a ferromagnetically ordered sample a magnetic field is not needed for inducing magnetic moments or aligning disordered ones, because long-range ferromagnetic order is assumed present. However, in order to define a unique direction of the magnetic moment, and thus Fe (τ an external magnetic field has to be applied. This is true even in the case of strong magnetic anisotropy, which tends to align the moments along a given crystallographic axis. Under these ci cums a ces e e e ał mag e ic ie as o emo e e ege e acy co ec e wi e omai s uc u e i e mag e ica y o e e s a e O y i e w o e si - 5 K.-U. Neumann, K.R.A. Ziebeck g e c ys a is a so c a ac e ise y o y o e mag e ic omai e a o ima io s o a y i e e i a io o e i i g a io i ( 7 I as a ea y ee oi e ou a o e a i e case o mag e ic a iso o y e e e a mag e ic ie mus e a ie a a e o a easy a is o mag e i a io A ese ecau io s a e o e ake i o e o u ique y e i e e i ec io o mag e ic mome s i e c ys a u e i es iga io o e case o a a i e omag e ica y o e e s uc u e e measu eme • o e mag e ic o m ac o is com ica e ue o se e a ac o s o ma y a - i e omag e ic s uc u es e mag e ic o e c a ges e si e o e u i ce As a esu mag e ic a uc ea agg e ec io s occu a i e e oi s i eci oca s ace o a mag e ic o m ac o measu eme owe e e ese ce o uc ea a mag e ic sca e i g is ee e a e same eci oca a ice oi I a uc ea eci oca a ice ec o τ uc ea oes o coi ci e wi a τmag e ic e uc ea -mag e ic i e e e ce e m wi e a se om e e as ic eu o sca e i g c oss-sec io us a i e e e ce e m i e e as ic eu o sca e - i g c oss-sec io wi e a se o a i e omag e ic s uc u es wi a i c ease u i ce o ume ue o e mag e ic o e is oes a so i c u e mo e com i- ca e mag e ic s uc u e suc as i comme su a e o e ica o es e e o e o y e case o a a i e omag e wi ou a i c ease u i ce ł ee s o e co si e e u e Suc a a i e omag e ic o e occu s o e am e i a cc s uc u e w e e e mag e ic a oms a osi io s ( a (1/ 1/ 1/ a e ei mag e ic mome s a ig e a i a a e wi es ec o o e a o e e a ious cases o mag e ic o e i g a e es esc i e wi e e o e ime i e sio o e a o Ot . W e ac i g o a mag e ic mome M is o e a o e e ses e mome i ec io acco i g o
I a i io o e ime i e sio o e a o i i is o y e symme y e eme o e i e sio i s ace I, w ic is ee e ou o a e symme y e eme s w ic c a ac e ise e c ys a og a ic s uc u e S ace i e sio oes o c a ge e o ie a io o e mag e ic mome M. ( is is ue o e ac a e s i a o e mag e ic mome is o cou e o e s a ia coo i a es w i e e o i a a o M is o o io a o e seu o ec o r p, w e e r is e osi io o e e ec o u e co si e a io a p is i s i ea mome um e ec o o uc r p oes o c a ge i s o ie a io w e ac e u o y I.) o e omag e ic o e a some a i e omag e s e i e sio symme y o e a o is o com i e wi e ime i e sio As a co seque ce a y a om a osi io i as a equi a e a om oca e a —Ri. us g ou i g ose a oms oge e w ic a e e a e y i e sio symme y; ei co i u io o e mag e ic s uc u e ac o is ou o e o o io a o
As see om (3 u e e a o e assum io s e mag e ic s uc u e ac o is a eał qua i y ( is was assume a o e a s a e wi ou oo o e mag e ic s uc u e ac o o e a amag e i a e e a mag e ic ie Magnetic Form Factor Measurements ... 57
If, however, the antiferromagnetic state is characterised by a combination of operators as given by the product of space inversion times time inversion symmetry operator; the magnetic moment of the symmetry related atom is reversed. This results in a magnetic structure factor proportional to
This magnetic structure factor is purely imaginary. It is phase shifted by 90° with respect to the nuclear structure factor. As a consequence of this phase shift the flipping ratio as defined in (27) is identically equal to one. Denoting the real and imaginary part of the structure factor by Re and Inn, respectively, and under the assumption as stated above that the structure factors are given by
The flipping ratio as defined in (27) is equal to one for all reflections τ because the identity
is fulfilled for all Bragg reflections. Therefore no (τ ≠1 reflection may be found for an antiferromagnetic structure which combines the space inversion element with time inversion. However, for a structure for which the space inversion element is a proper symmetry element both the nuclear and magnetic structure factors are real quan- tities. This is the case for a ferromagnetic and some types of antiferromagnetic structure, for which the assumptions leading to (36) are true. Only for these struc- tures a measurement of the flipping ratio is a possibility. However, in addition to the structural requirements which have to be fulfilled for a nonzero flipping ratio to occur, a further complication arises due to the presence of magnetic domains in the sample. For most magnetic structures and including a coupling to the lattice a small number of magnetic domains is possible. It suffices here to restrict the discussion to two domains, which are related to one another by time inversion symmetry. Such a pair of magnetic domains is always possible. It is appropriate to denote these domains as a 0° and a 180° domain, because the 180° domain is obtained from the 0° domain by an inversion of all magnetic moment directions. It may also be described as obtaining the 180° domain from the 0° one by the action of O ► on the 0° domain. Both domains are energetically degenerate. It may therefore be expected that the two domains occur with the same probability in a sample. Therefore, the magnetic structure factor of the whole sample has to be obtained by averaging over the various magnetic domains. This average, however, will result in a zero average magnetic structure factor. With the definition of the magnetic structure factor as given in (12) and by using an obvious notation to identify the structure factors for the various magnetic domains one obtains 5 K.-U. Neumann, K.R.A. Ziebeck
e mag e ic s uc u e ac o is a ec o e i ec io o w ic is e e mi e y e i ec io o e mag e ic mome s As e mag e ic mome i ec io is e e se y goi g om o e omai o e o e e mag e ic s uc u e ac o c a ges sig e e o e a e o a e age mag e ic s uc u e ac o esu s o a y sam e wi equa amou s o mag e ic omai s ese w ic a e e a e y ime i e sio symme y is a gume a ies equa y we o a e omag e ica y as we as a a i e omag e ica y o e e s uc u e I o e o emo e e ege e acy o e ° a 1 ° mag e ic omai s a mag e ic ie as o e a ie o a e omag e ica y o e e sam e i su ices o a y a omoge eous e e a mag e ic ie is ie emo es e ege e acy o e mag e ic omai s a i o su icie s e g i may s a i i e a si g e mag e ic omai i e w o e c ys a o a a i e omag e e a ie mag e ic ie as o a e e mo u a io o e a i e omag e ic s uc u e i o e o emo e e ege e acy e wa e ec o o e mag e ic mo u a io is o e o e o a gs oms a a ese o e e a y a ie mag e ic ie s a e a ai a e wi suc s o wa e e g s O e e e ime a y co o e a ame e s suc as essu e o o eak ime i e sio symme y a ey a e e e o e o ca a e o emo i g e ege e acy o e mag e ic omai s us o e is e wi e co c usio a o a a i e omag e ic s uc u e a i i g a io may e o se e o ce ai s uc u es a u e e co i io o a im a a ce o mag e ic omai s w ic a e e a e y ime i e sio symme y e a o e iscussio as i us a e e com e i y o e a gume a i as o e ou e e e ime a i icu ies w ic a e o e o e come o wi s a i g ese o ems measu eme s o e i i g a io i a a i e omag e ic s uc u e a e success u y ee ca ie ou [9] e i e es e ea e is e e e o e o igi a u ica io s o a e ai e iscussio o a ious as ec s o e e e ime o a a i e omag e ica y o e e su s a ce a e e a mag e ic ie may e a ie o i uce a e omag e ic com o e is may e ac ie e y i i g e mag e ic mome s ou o ei a i a a e a ig me i o e i ec io o e e e a mag e ic ie e e y c ea i g a e omag e ic com o e is si ua io is i us a e i ig 1 e e omag e ic com o e o e mag e ic s uc u e is c a ac e ise y a wa e ec o k = w i e e a i e omag e ic com o e is esc i e y a wa e ec o q. I is o y e k com o e w ic cou es o a a ie mag e ic ie e co i u io s o o wa e ec o s may e ecou e a co si e e se a a e y o e e omag e ic com o e e eo y a ies as e e o e a o e o e a i e omag e ic com o e a ose com ica io s a ise w ic a e ee co si e e i e ea ie iscussio I e e e ime o e accou as o e ake o o mag e ic com o e s us i a o m ac o measu eme o a a i e omag e ic c ys a (w ic oes o i c ease e ime sio s o u i ce y o e i g a i e omag e ica y a w ic is ace i a e e a mag e ic ie i is o y e e omag e ic com o e w ic co i u es o e i e e e ce e m owe e ue accou as o e ake o e a i e omag e ic sca e i g co i u io o e o a i e si y o e agg eak Mag e ic σ m ac o Measu eme s 59
o c ys a og a ic s uc u es w ic o o ossess a ce e o i e sio symme y e uc ea a mag e ic s uc u e ac o s a e com e qua i ies is im ies a ła ge um e o com o e s o ei c a ac e i a io O e com- e um e o equi a e y wo ea um e s o c a ac e ise ea a imagi- a y a s a e ee e as com a e o o y o e ea um e a a sig i e case o ce osymme ic s uc u es e ge e a i a io o e a o e e i a io o o -ce osymme ic s uc u es wi esu i mo e com ica e e essio s [5] e o i g e eał a imagi a y a s y a " es ec i e y a o i g e sca e i g ec o e e e ce o e u c io s e i i g a io akes e o m
W i e e uc ea s uc u e ac o Fi + iFN" is k ow i su ices o o e a a si g e measu eme o o y o e i i g a io e agg e ec io is o su icie o e ac i g e com e qua i y ┴M(τ A seco measu eme is ee e o w ic e e e ał mag e ic ie i ec io o e o ie a io o e c ys ał is e e se As oi e ou a o e e ea a o e s uc u e ac o is e a e o a a o e mag e i a io w ic is symme ic wi es ec o e i e sio o e a- o e imagi a y a o e o e a is a isymme ic u e s ace i e sio I e e e a mag e ic ie i ec io is e e se e ea a o e mag e ic s uc u e ac o wi o c a ge i s sig w i e e imagi a y a wi us e- e si g e e e a ie i ec io wi yie a i i g a io gi e y
o a o -ce osymme ic s uc u e e i i g a io is i e e o e wo co - 60 K.-U. Neumann, K.R.A. Ziebeck figurations of the magnetic field. These two measurements are in general sufficient to determine both the real and the imaginary part of the magnetic structure fac- tor. For a centrosymmetric structure the second setup will yield the same flipping ratio as the first measurement.
. M n t f r f t r l l t n f r pl d l t
In the previous section the magnetic form factor measurements have been considered from an experimental point of view. The magnetic structure factors and their Fourier transform in the form of magnetization density maps are the experimentally determined quantities. In order to fully interpret the experimental observation an attempt has to be made to quantitatively describe magnetization densities or magnetic structure factors. By using a model, parameters have to be obtained from observation for a description of the magnetization on an atomic level. For the most general case the theory will be complicated, and the interested reader is referred to the literature for details of some theoretical aspects of neutron magnetic form factors (Harmon [10]). It will also not be attempted here to derive the full expressions for the magnetization in real space or of the wave vector de- pendent susceptibility (see e.g. Oh et.al. [11], Cooke et al. [12]). Rather in order to illustrate the power of magnetic form factor measurements in this section atten- tion will be focused on some simple model systems. The first case is characterised by delocalised or itinerant electrons, while a second model system is one with a collection of localised and atomic-like magnetic moments. The theoretical evaluation of M(r) necessitates the calculation of matrix elements of the form where |W) is the many-electron wave function of the crystal under investigation. Operators will here and henceforth be denoted by a — sign in order to distinguish the operator from its expectation value. Depending on the form of the operator it may be more convenient to present the wave function either in momentum space or in position space. These Hilbert spaces are defined such that the state vectors of the Hilbert space are either eigen- s a es o e łi ea mome um o e a o wi or of the position operator r as given by
The momentum space and the position space are assumed to be complete
and the state vectors normalised Mag e ic σ m ac o Measu eme s 1
e a s o ma io w ic yie s 11 as ei e a e ese a io i mome um o i osi io s ace is gi e y
wi e u c io s u(k a u( ei g gi e as
Wi i u(k a u( is co ai e e i o ma io co ec e wi e occu a io o e a ious e e s o e e gy a s o e so i o a com e e esc i io o e qua um mec a ica s a e o e s a e ec o s a e u c io s u(k a u( a e o i c u e a i e σ• o e o e e e ec o ic s i eg ee o ee om e u c io s u(k a u( a e o i e e e om o e a o e e a s o ma io o u(k om a mome um s ace e ese a io o a osi io s ace e ese a io is ac ie e y a ou ie a s o m o e mag e i a io o e a o M( a se a a io is ossi e o e o a mag e ic mome o e a o i o a s i a a o i a co i u io acco i g o
o e s i a e o i a co i u io o e mag e i a io o e a o a e a osi io e e e ce gi e y
o some mag e ic ma e ia s e o i a com o e is e o is may e e case ei e o a mag e ic mome wi = O (wi M ± a G 3+ ei g e am es o i may a ise ue o e que c i g o e o i a mome ue o c ys a i e e ec ic ie s E am es o e a e a e ou amo g e 3 e eme s a ei a oys U e ese co i io s o y e s i a o e mag e ic mome s o e a o is ac i e e o e a o o e s i e si y M ( is e i e as
e summa io is ca ie ou o e a e ec o s i e c ys a e s i o e a o S is e e mi e as
w e e ś↑ a ś↓ a e o e a o s w ic is i guis e wee s i u a s i ow s a es o e i- e ec o s acco i g o
e e I a I 1 e o e e s i s a e o a si g e e ec o e ec o is a u i ec o w ic e i es e a is o qua i a io i e sys em a a u ique i ec io wi es ec o w ic e e ec o s i s a e ei e u o ow K -U Neumann, K.R.A. Ziebeck
Usi g e a o e e i i io s a a s o mi g o a osi io s ace e ese - a io e ma i e eme (5 is e a ua e o
e ma i e eme ( Ms( " is ea i y e a ua e wi e esu
O i se i g (5 i o (55 a i eg a i g e ma i e eme o e mag e i- a io akes e o m
e ysica sig i ica ce o is esu ca e ma e e e mo e a s a e w e e s i a ia e is a o me om S o e u c io ui( is is ac ie e i a simi a ma e as a o e i (53 y se a a i g e s i u a s i ow a s o ui(r) wi e i se io o a s i i e e mag e i a io e si y ca e w i e i e o m
e a e o es e a e age s i u /s i ow e si y o e e ec o s i e sam e wi e a e age ei g o ai e y ca yi g ou Ei a summi g o e a ł e ec o co i u io s is esu shows a e mag e i a io e si y i ea s ace is e e mi e y e e si ies u ↑↓( 1 o s i u a s i ow e ec o s A e mag e i a io a ises a osi io r i e e si y o s i u e ec o s is i e e om e o e o s i ow e ec o s a is oca io i e c ys a Wi i is esc i io o mag e ic co i u io a ises o com e e y i e a s o a omic o i a s us o y ose e ec o ic wa e u c io s may co i u e o e mag e i a io e si y w ic e o g o a ia y i e e ec o ic a s o oca ise e e s a w ic may gi e ise o a e mag e ic mome As a s ecia case a as a i us a io o e iscussio gi e a o e e ec o s a e co si e e i a so i wi ei wa e u c io s ei g gi e y ose o ee e ec o s e qua um um e k o ese e ec o s is a goo qua um um e yie i g e ee e ec o wa e u c io i mome um s ace I is e e o e a o- ia e o e-e a ua e e ma i e eme o e mag e i a io e si y i eci ocał s ace a y maki g use o e i ea mome um e ese a io o W Acco i g o (1 e ou ie a s o m o e mag e i a io e si y i ea s ace is e mag e ic s uc u e ac o us Mag e ic σ m ac σ Measu eme s 3
e i e S o e mag e ic s uc u e ac o i ica es a i is o y e s i a o e o a mag e i a io w ic is i c u e Wi e e o e com- e e ess e a io ( e ma i e eme (k Ms( k" may e e a e o e ma i e eme i osi io s ace as gi e i (5
e ma i e eme o me y e com i a io o a i ea mome um a a a osi io ke ec o is gi e y
I se i g (5 ( a ( 1 i o (59 a maki g use o e " e a u c io " i e o m
e mag e ic s uc u e ac o akes e o m
I e ici o m a i c u i g e s i u a s i ow o a io o u(k e a o e equa io may e w i e as
Equa io s ( 3 a ( a e co o u io i eg a s a o i g e se u e ou ie a s o ms o (57 a (5 e mag e ic s uc u e ac o as gi e i ( is s i ge e a y a i a o ye es ic e o e case o ee e ec o s • e i eg a io i ( ecomes a icu a y sim e o e case o ee e ec- o s ei wa e u c io is gi e y a e wa es wi u(k equa o a co s a e co s a equa s o e o a occu ie s a es w i e e u c io u(k is equał o e o o k s a es w ic a e o occu ie e e e gy is e sio o ee e ec o s is gi e y a a a o a wi ege e a e e e gy a s o s i u a s i ow e ec o s I o e o a e a o e o mag e ic s uc u e ac o a a e e omag e ic mome e e e gy a s o s i u a s i ow e ec o s mus e is ace i e e gy is is i us a e i ig e i i g o e ege e acy may e ei e ue o a e e a mag e ic ie o ue o a i i sic e omag e ic i s a i i y o e co uc io e ec o a e e mi su ace is assume o e a s e e o o s i u a s i ow e ec o s wi e e mi wa e ec o s gi e y a es ec i e y K.-U. Neumann, K.R.A. Ziebeck
e u c io ut (k — k') (ul(k — k')) as a o e o a ue (≡ 1 i a s e e o a ius kF (kF). e ce e o e s e e is oca e a k. us e co o u io i eg a ( e uces o e i eg a io o e i e sec io o wo s e es o equa a ii a wi ei ce es a is a ce k a a is si ua io is i us a e i ig 3 Usi g ese geome ic a gume s a e o i g k y ek i eg a is Magnetic Form Factor Measurements ... 65
The spin down integral can be evaluated in an identical manner, and the magnetic structure factor is obtained by the subtraction of the results of the integration for spin up and spin down electrons. K.-U. Neumann, K.R.A. Ziebeck
e mag e ic s uc u e ac o may e ca cu a e o e wo cases o com- e e a a ia mag e ic o a i a io o e e ec o s as i us a e i ig e esu i g mag e ic s uc u e ac o s a e i us a e i ig w e e e sca e i g ec o e e e ce is o e o e mag e ic s uc u e ac o o ma ise o o e a e o sca e i g ec o y i i i g FsM(k) y e o a mag e ic mome e esu s o e i eg a io i ( s ows a e co i u io o e ee e ec o mag e i a io o e mag e ic s uc u e ac o is imi e o ow sca e i g ec o s A u e imi o e co i u io may e o ai e om ( y co si - e i g e u y o a ise s a e o sca e i g ec o s k wi k > ma (k↑ k↓ e) mag e ic s uc u e ac o is e o o ee e ec o s A y ica a ue o e e mi wa e ec o may e c ose o e A is esu s i a es ima e o 1/π o e u e cu -o i si /a o a ee e ec o co i u io o a a ia y o a ise s a e e u e cu -o wi e e uce a o ia e y e mag e ic s uc u e ac o o ee e ec o s may e use o es ima e e co i u io o co uc io e ec o s W i e co uc io e ec o s i ea so i s a e e oca ise ey may o e e i e y ee ( a is a e a u(k) as gi e a o e u e ee e ec o ic u e may se e as a easo a e gui e o es ima e ei co i u io o e mag e ic s uc u e ac o I " ea " so i s a aki g a e ea a oys as a e am e e mai a o e mag e i a io is ca ie y oca ise mag e ic mome s w ic o igi a e om -e ec o s A o a i a io o e co uc io e ec o s wi gi e ise o a sma ł a i io a mag e ic mome w ic wi c a ge e a ue o e o a mag e ic mome i e u i ce a i ue ce e i s ew agg e ec io s e o a -e ec o mome o ai e y a e a o a io o e mag e ic o m ac o s o mag e ic s uc u e ac o s o ig e sca e i g ec o s o owe k a ues oes o a ways ag ee wi e o a mag e ic mome as o ai e y mag e i a io measu eme s e i e e ce is usua y a i u e o a co uc io e ec o o a - i a io As ei co i u io is co i e o ow sca e i g ec o s i is some imes a o ima e y a " e a u c io " e e a e o y ew agg e ec io s wi i e a ge o a o e o ee e ec io mag e ic s uc u e ac o us e e ai e o m o e mag e ic s uc u e ac o is o ea i y i es iga e e e ime a y y usi g e ec ique iscusse e e I is o y ue o a e ia io o a ew mag e ic s uc u e ac o s om ei e a o a e a ues a ow a so u e sca e i g ec o s (a mos sig i ica y a k = a e co uc io e ec o o a i a io may e e ec e A a o ima io y some we oca ise a a ow u c io ce - e a e o sca e i g ec o seems o e a equa e o ac ie e co sis e cy wi o se a io I is a ge e a ea u e o e ou ie a s o m a a oca ise co i u io i o e s ace wi esu i a ou ie a s o m w ic as a e oca ise c a ac e e i e se is a so co ec As o e case o ee e ec o s ei wa e u c io is e oca ise i ea s ace us gi i g ise o a s a y eake mag e ic s uc u e ac o I o e o a e a mag e ic s uc u e ac o w ic is mo e e e e i eci oca s ace e sou ce o mag e i a io e si y mus e mo e oca ise i eał s ace e o e e eme com a e o e com e e y e oca ise a ee e ec o s Mag e ic σ m ac o Measu eme s 7 is gi e y com e e y oca ise e ec o s Suc cases a e ea i e i a e ea a ac i i e me a s a ei a oys e -e ec o wa e u c io s a e we ł oca ise wi i e e ec o c ou o e ee a om wi a a ius o e -e ec o wa e u c io w ic is muc sma e a e a ii o e ou e s o p e ec o s us i a oms wi a a ia y i e -e ec o s e a e a oye i o a so i ei -e ec o wa e u c io s o o o e a a ey emai o a a ge eg ee a omic- ike ow- e e com ica io s a ise ue o a i e ac io wi co uc io e ec o s esu i g i a ic s ec um o ysica o e ies i c u i g i e me ia e a e cy a ea y e mio e a iou o e u ose o a a ysi g mag e ic o m ac o measu eme s owe e e mos im o a ea u e is e oca i a io o ose e ec o s w ic gi e ise o e mag e i a io i e u i ce Wi e e am e o -e ec o sys ems i mi i is o e e ec e a e mag e i a io o -e ec o s ca e u ique y a u am iguous y assig e o a ce ai a om e o e a i eg a is sma e wee -e ec o wa e u c io s oca e o eig ou i g a oms a e i eg a may e a o ima e y e o o e u ose o ca cu a i g e mag e i a io e si y I su ices o co si e o y e oca ise co i u io s o e mag e i a io e si y wi a wa e u c io Ψ o e w o e c ys a gi e i a ig i i g a o ima- io o sim ici y i is assume a o y o e a om e u i ce is co i u i g o e oca ise mag e i a io e si y Wi i a a ic u e e -e ec o s wi gi e ise o a a ow a is -e ec o a wi e c a ac e ise y o y a e y mo es is e sio ue o e sma ess o e wa e u c io o e a e wee -e ec o a oms o eig ou i g a ice si es o ea so i s e is e sio o e -e ec o a is mai y ue o e y i i a io wi co uc io e ec o s I o e o sim i y e esc i io o y o e a wi e co si e e e wa e u c io Ψ as o ake ue accou o e oca i a io a a e same ime a so o e equi eme o e oc eo em o wa e u c io s i a e io ic a ice o a oca ise mome sys em i a ig i i g a o ima io e s a e ec o o e w o e c ys a may e e a e as
usi g e com e e a o o o ma asis se o oc s a e ec o s Ψ (k As oi e ou a o e o y o e a wi e co si e e as a co seque ce o w ic e a i e wi e i e e a s o ma io o a oc s a e ec o o a ec o i osi io s ace e ese a io is gi e y
u (k, is a e io ic u c io wi e e io ici y o e a ice o e case o o e mag e ic a om e u i ce a wi i a ig i i g a o ima io e u c io u (k, akes e o m
e u c io Φ( — i is a wa e u c io w ic is assume o esc i e e oca ise co i u io a isi g om e i- a om oca e a 1 1 i is a a ice ec o o e K.-U. Neumann, K.R.A. Ziebeck c ys a og a ic a ice e um e N is equa o e um e o u i ce s i e c ys a w ic o e co i io s s eci ie a o e is equi a e o e um e o mag e ic a oms i e c ys a e o e a is assume eg igi e wi
I is o e assume a (i is gi e i e ms o wa e u c io s w ic a e c a - ac e is ic o e ee a om I may e com ose o se e a a s acco i g o
wi a u c io s α( o e e a sio ei g oca e a e osi io o e i- a om is a ows e i i g o ege e acy o a a om i a c ys a i e e i o me as com a e o a ee a om o e ake i o accou o -e ec o sys ems e e uc io i symme y i s e ege e acy o e e e s e e e gy e e s s i i o c ys a i e e ec ic ie e e s wi e ege e acies o e e e s ei g e e mi e y e oi symme y o e a omic osi io e e owe e e s i i g o e -e ec o a o -e ec o e e s wi o e co si e e u e e ma i e eme o e mag e i a io e si y i ea s ace is gi e y
I (71 e mag e i a io o e a o i c u es e s i a o i a co i u io Wi e s a e ec o as s eci ie a o e i ( e ma i e eme o e mag e i a io akes e o m
I se i g e ma i e eme un(k, i ( i o e a o e o mu a M(r) akes e o m
ecause o e assum io o eg igi e o e a e wee eig ou i g a ice si es e summa io o e e i e j wi o y gi e a co i u io o e e m i = j.
us (73 may e w i e as Mag e ic o m ac σ Measu eme s 9
Wi a s i o e o igi y i (i e y a a ia e a s o ma io — i → a wi e e o e " e a u c io "
e mag e i a io e si y i ea s ace is i a y o ai e as
e ac o k α (k 2 is o i g e se u a occu a io ac o w ic i ea s ace co es o s o a a e age occu a io o e e e o e e s i c u e i Φ( ( M( yie s e emai i g e mag e i a io ac o e si y i e ms o e wa e u c io w ic esc i es e oca ise co i u io e occu a io ac o may e a so e i o Φ( w ic e co es o s o a wa e u c io com ose o a sum o e a occu ie s a es wi weig i g ac o s acco i g o e occu a io o e a ious e e s As oi e ou a o e a eque y use a o ima io is o ake e u c- io s Φ( o e gi e y ose o a ee a om is is e easo w y ee a om o m ac o s a e so o e em oye o a e a a io o mag e i a io e si y measu eme s I is wo eca i g e e i i io o e mag e ic o m ac o (k as gi e i (15 w e e (k was e i e as e ou ie a s o m o e mag e i a- io e si y o a si g e a om e esu s o ca cu a io s o a omic mag e ic o m ac o s o a e ea a ac i i e a oms a e ee a u a e i e i e a u e o o o e a i is ic [13] a e a i is ic [1 ] ca cu a io s us e ca cu a io o e mag e i a io e si y i ea s ace as ee e uce o e ca cu a io o e mag e i a io e si y o a ee a om o s i a o i a co i u io s o e mag e i a io a e mo e ea i y ca cu a e o e ee a om com a e o a e e mi a io y a a s uc u e ca cu a io O e i e ac io s suc as s i —o i cou i g a c ys a ie i e ac io s wi e c ys a i e e i o me may a so e i c u e a icu a y o e i e ac io s w ic gi e ise o e c ys a i e e ec ic ie s i is a a a age a o oca ise a a omic- ike wa e u c io s i is ossi e o se a a e e wa e u c io i o a a gu a a a a ia a y a s o mi g o s e ica coo i a es e wa e u c io Φ(r) may e w i e i e o m / ( = R(r) Y(υ, ). e a gu a a o e wa e u c io is ea i y ea e o oug y aki g i o accou e emo a o s e ica symme y a e i i g o ege e acies i e c ys a G ou eo y e ic s e s i i g a e e e ege e acies e emai i g ma i e eme s w ic a e o i e y symme y co si e a io s a e commo y ake as ee a ame e s o e i e y com a iso wi e e ime s A com e e e i a io o e ma ema ics is gi e i [15 1 ] a i o esey [ ] e case o c ys a ie s as ee co si e e y ouc e e [17] e a ia a o e wa e u c io is o e e mi e y symme y a o e u i is a a e com ica e u c io o I s u a sa is ac o y e e mi a io equi es a u ł sca e e a i is ic ca cu a io o e ee a om o go eyo is a o ima io ecessi a es a u ł sca e a s uc u e ca cu a io is owe e is o e o easi e a e a ia e e e ce o a ee a om ca cu a io is use o com a iso wi e e ime 70 K.-U. Neumann, K.R.A. Ziebeck
The problem of calculating the magnetization density in the unit cell has thus been reduced to the calculation of the magnetization of a single atom. Under the assumption that it is adequate to treat the atom in the crystal essentially in the same manner as a free atom all the standard procedures of spectroscopy may be applied. While the mathematical procedure may be involved it is nevertheless straightforward. A complete exposition of the physics of spectroscopy is given in the book by Condon and Odabasi [18], and it should be consulted in addition to the references. cited above. After the discussion of the calculation of the magnetization density in real space an experimental investigation will be considered in the next section. Exper- imental results are presented and the magnetization densities are interpreted and discussed in relation to the models of localised and itinerant electron magnetization densities.
4. Experimental investigation
The investigation of the magnetic form factor and the magnetization densi- ties in a solid commences with a proper characterization of the nuclear structure. Therefore, the first part of this section will be concerned with the determination of the nuclear structure as defined by the identification of the atomic positions. In the second part data will be presented of the magnetic form factor and magnetization density, and an interpretation will be given of the observation. Two substances will be considered here, UBe13 and UPt3, with both com- pounds belonging to the group of heavy fermion superconductors. The properties of heavy fermion systems are of interest in themselves, and their investigation is currently a very active field of research. However, the physical properties of heavy fermions are not of primary interest here, but rather attention is focused on their magnetic form factors only. Therefore, out of the rich spectrum of heavy fermion properties it is only the static magnetic characteristics as seen in an investigation of the magnetization densities which are of interest here.
$.1. Nuclear structure of UBe13 and UPt3
The crystallographic structure of UBe13 and UPt3 is cubic and hexagonal, respectively. The crystallographic space groups proposed in the literature [19] are given as Fm3m for UBe13 and P63/mmc for UPt3. These structures have been confirmed in a detailed neutron scattering experiment, and the free parameters have been determined experimentally for these structures as given by the positional parameters of one Be atom position in UBe13 and the position of the Pt atom in the unit cell of UPt3. Additional information was obtained in these experiments with regard to temperature factors, absorption and extinction. The nuclear structure of UBe13 is presented in Fig. 5 as a projection of the atomic positions onto one of the cubic faces. Each unit cell contains 8 formula units, with 8 U atoms and 104 Be atoms on two crystallographically different positions. Denoting the two Be sites by Bel and Bell, there are 8 BeI and 96 BeII atoms shown in the projection. For clarity a projection is shown in Fig. 5b which contains Mag e ic σ m ac o Measu eme s 71
only the U atom positions. The uranium and BeI atoms each form a simple cubic sublattice within the UBe13 structure. For the projections shown in Fig. 5b there are two U atoms being projected down onto each uranium site. It is worth pointing out some of the features of the projection of the nuclear density in Fig. 5. First of all, it is a map in real space of the neutron—nuclei in- teraction potential. It is obtained by a Fourier transformation using the nuclear structure factors in a manner identical to the procedures described above for the magnetization density. The nuclear density is seen to be smeared out over a small region with the centre of the density being located at the position of the atom. This smearing is a consequence of the finite Fourier series which has been used to obtain the nuclear density. The "real" interaction potential is a "delta function" when a standard description of the interaction potential is used with the Fermi approximation. Thus, the finite width in the nuclear density map is entirely due 72 K.-U. Neumann, K.R.A. Ziebeck
o e um e o e ou ie coe icie s use i e ou ie i e sio I o e o a ow a com a iso wi e mag e i a io e si y ma e same cu -o is use i e ou ie se ies as wi e use a e i e co s uc io o e mag e i a- io e si y I s u yi g e o ec io o e uc ea e si y i s ou a so e ake i o accou a e uc ea e si y is measu e y e s e g o e eu o — uc eus i e ac io e s e g o e i e ac io o e ia is a am- e e i e y e eu o sca e i g e g s bu a bBe [20]. owe e ue o e y simi a a ues o e sca e i g e g s o U a e (bu = 1 1 -15 m bBe = 7.79 x 10' 15 m i is i icu o is i guis e U a e a om osi io s om e a ues o e co ou e e s i ig 5 o U 3 e uc ea s uc u e is s ow i ig as a ee- ime sio a ic u e igu e 7 s ows e osi io o e U a a om o wo cu s oug Mag e ic σ m ac σ Measu eme s 73
e u i ce e u i ce o U 3 co ai s wo o mu a u i s wi e a oms oca e i wo e ago a a es wi z = 1/ a z = 3/ a as s ow i ig 7 o a es a e e a e o o e a o e y ce es o i e sio symme y w ic a e oca e a e o igi o e u i ce ( a a e osi io ( 5 5 5 o uc ea s uc u es a e ce osymme ic as a co seque ce o w ic e uc ea s uc u e ac o s ca e c ose ea e e o e e ase ac o o e uc ea (a a so o e ( e o- mag e ic s uc u e ac o is gi e y ei e +1 o —1 o ma e ia s a e a amag e ic i e em e a u e a ge i es iga e i e o m ac o measu eme o U 3 a i e omag e ic o e is e o e o occu a a o ima e y 5 K wi a ou i g o e u i ce is owe e wi o i e e e wi e o m ac o measu eme s a e e o e e a i e omag e ic o e o U 3 wi o e u sue u e
4.2. Mag e ic σ m ac σ measu eme s i U e13 a U 3 I e s i o a ise eu o sca e i g e e ime e sam es we e ace i a mag e ic ie o es a o e U e13 sam e e mag e ic ie i ec io was a a e o e c ys a og a ic [1 1 ] i ec io w i e o U 3 e mag e ic ie was a o g [ 1 ] o U 3 is co es o s o a co igu a io w e e e ie is i e e ago a a e a e e icu a o e mag e ica y a i ec io w ic is ou o e gi e y e c ys a og a ic c-a is e i i g a ios o a um e o e agg e ec io s a e ee measu e a em e a u es o 1 K o U e13 a o 5 K a 3 K o U 3 As e i i g a ios o U 3 i o a y wi em e a u e e wo a a se s we e me ge i o e o im o e s a is ics I e s i o a ise eu o sca e i g e e ime s e i i g a ios a e ee o ai e o a measu a e agg e ec io s wi a ues o e sca e i g ec o s c a ac e ise y si Θ/λ 5 Ǻ-1 I o e o ese e a a i a co cise a mea i g u ma e o e may assume a e mag e i a io a ises o y om e 5 e ec o s oca e o e u a ium a om ue o e sim ici y o e s uc u es o o U e13 a U 3 i is ossi e o e uce e mag e ic s uc u e ac o o e mag e ic o m ac o o e u a ium a om Wi i is mo eł o o y aki g i o accou e 5 -e ec o mag e i a io a y usi g (1 a (15 e mag e ic s uc u e ac o ca e w i e as
e e e summa io is ca ie ou o e a u a ium a oms wi ei osi- io s gi e y i wi i e u i ce Acco i g o e assum io ma e a o e a o e a oms i e u i ce o o co i u e o e mag e i a io a e e- o e ey ee o e i c u e i e summa io i (77 Mo eo e a u a ium a oms a e o c ys a og a ica y equi a e osi io s a ei i uce mag e ic mome s a e a equa us wi
e mag e ic o m ac o e uces o 7 K.-U. Neumann, K.R.A. Ziebeck
e ac o ∑ e τ i is a k ow e i y as a e a ame e s suc as a omic osi io s a e k ow us is ac o may ea i y e emo e om (79 e e y e mag e ic s uc u e ac o is e uce o e mag e ic o m ac o o e u a ium a om mu i ie y a co s a e mu i ica i e ac o i o o e mag e ic o m ac o is equa o e o a e omag e ic mome oca e o e -u a ium a om e mag e ic o m ac o o e u a ium a om is e ou ie a s o m o e a omic mag e i a io e si y ese e si ies a e ee ca cu a e o e a ious s a es o io i a io o e ee u a ium a om a e e e a u c io s a e ee a u a e i e i e a u e [1 ] Assumi g a ee 5 -e ec o s a e ese o e u a ium a om a a c ys a ie s i i gs a y i i a io e ec s ca e eg ec e e mag e i a io e si y o e u a ium a om e ai s i s s e ica symme y wi i e c ys a i e e i o me A mag e ic o m ac o w ic is e a ua e u e ese assum io s is k ow as a o m ac o i i o e a o ima io Usi g e i o e a o ima io o e 5 -e ec o mag e i a io e e e- mai s o y o e ee a ame e o e e e mi e y com a iso wi e e ime e u k ow e i y is gi e y e si e o e mag e ic mome oca e o e • u a ium a om W i e e sa u a io mag e ic mome o is co igu a io ca e ca cu a e e mag e ic mome a ig e y e e e a mag e ic ie a a e e e a em e a u e may e su s a ia y i e e om e o a magnetic mome o e a om us wi e assum io o a oca ise mag e i a io e si y ce e o e U a ium a om a wi i a i o e a o ima io o e mag e i a io is i u io a o e- a ame e i as o e ca ie ou e esu s o suc a i i g oce u e a e s ow i ig a ig 9 e co i uous i e co es o s o e mag e ic o m ac o o e u a ium a om w i e e a a oi s a e gi e y e measu eme s o U e 13 a U 3 W i e e e e ime ał a ues o e mag e ic o m ac o o U e 13 a e sa is ac o i y e o uce y a o m ac o o e U a ium a om i i o e a o ima io e co es o i g a a oi s o U 3 s ow a sig i ica sca e a ou e cu e o a s e ica y symme ic mag e i a io is i u io I o e o im o e e ag eeme e wee ca cu a io a o se a io o U 3 o e may y o e a e co i io o a s e ica y symme ic mag e i a io is i u io a mo e e o m ac o wi i a c ys a ie mo e owe e e e wi i is ess s i ge mo e e ag eeme o a im o e mo e ca cu a io a e e ime a o se a io is oo I u s ou a e sca e o e a a - oi s i ig 9 is oo a ie o a ow mo e i g usi g o y a uranium a om mag e i a io I o e o o ai mo e i sig i o e easo o suc a isc e a cy a o e a e o i e i y e a ious co i u io s o e mag e i a io i is i o ma i e o s u y e mag e i a io e si y i ea s ace Usi g e measu e mag e ic s uc u e ac o s i e ou ie se ies e ea s ace mag e i a io e si ies a e ee co s uc e o o U e13 a U 3 e mag e i a io e si y i U e13 is s ow i ig 1 as a o ec io o o o e o e aces o e cu ic u i ce e um e o e ou ie coe icie s a e o e co i io s o o ai i g e o a e e same as e o e o e o ec io Magnetic Form Factor Measurements ... 75
o e uc ea e si y i ig 5 I is see om e mag e i a io o ec io a a mag e ic e si y a ises a e osi io s o e u a ium a oms o o e s uc u e o e ai is isi e wi i e u i ce I a icu a e e is o oca ise mag e ic co i u io ce e a e e a om osi io s is esu is o su isi g i iew o e ac a e mag e ic o m ac- o i i ig is a goo esc i io o e e e ime a a a e mag e i a io e si y i ea s ace is e e o e omi a e y e 5 -e ec o mag e i a io ow- e e o ow sca e i g ec o s a sys ema ic e ia io is ou i ig e wee e o se e a a oi s a e ca cu a e o m ac o cu e is e ia io a ow a ues o si /a is a i u e o a o a i a io o e co uc io e ec o s As iscusse i some e ai i Sec 3 e co uc io e ec o co i u io wi e s a y eake a τ = O a quick y a o e o as e si e o e sca e i g ec o i c eases e ou ie a s o m is o e ec e o s ow sig i ica s uc u e I o e o i us a e e e ec s o a i com e e ou ie se ies a mag e i- a io e si y ma is co s uc e wi o e com o e missi g Suc a ic u e is s ow i ig 11 w e e e com o e co es o i g o τ ( as ee 76 K-U. Neumann, K.R.A. Ziebeck
reduced by a factor of two. The nuclear structure factor of this reflection is acci- dentally very small, as a consequence of which it is experimentally more involved to obtain accurate values of the flipping ratio. This is due to experimental difficul- ties and uncertainties which arise in an accurate determination of the background contribution for weak reflections. For reflections with a low nuclear structure factor other contributions such as those due to multiple scattering may seriously influence the intensity of the Bragg reflection. Thus, the flipping ratio of a weak reflection may be systematically estimated as being either too low or too high. For a mag- netization density map which is constructed using this coefficient in the Fourier series the resulting magnetization density may show additional structure which arises from the difference between the correct and estimated magnetic structure factor. Figure 11 shows magnetization density for the magnetic structure factor Magnetic Form Factor Measurements ... 77
with the "correct" coefficient of the (2, 2, 0) reflection being multiplied by 0.5. This reflection and the ones obtained from it by the application of the symmetry elements of the structure contribute a term proportional to cos (4π/α(r + ry )) to the projection of the magnetization density. In order to see whether or not such structure in the magnetization density is a "reap' effect or whether it arises due to some missing or wrong Fourier com- ponents one may calculate the magnetization using a simple model such as the uranium 5f-electron magnetization only and employing the dipole approximation. is mo eł may e e use o co s uc e mag e i a io e si y i ea s ace y a ou ie a s o ma io a wi e same coe icie s i e ou ie a s- o m as use i e e e ime a se ies Suc a e si y ma s ou e s ow o y oca ise mag e ic co i u io s A y e ia io om is mus e a i u e o a e ec o e ou ie se ies A e a i e y o e may s u y e i e e ce ma w ic is o ai e y ou ie a s o mi g e i e e ce o o se e a ca cu- a e mag e ic s uc u e ac o s o oce u es e a e o i e i y e o igi o e s uc u e w ic is ou i e mag e i a io e si y ma s Figure 12 shows the magnetization density for UPt 3 as a cut through the unit cell at a height of z = 1/4. The positions of the atoms in this plane are indicated in Fig. 7. When studying the magnetization density of Fig. 12, the first impression 78 K.-U. Neumann, K.R.A. Ziebeck
is one of a complex magnetization distribution with various contributions which are located at different positions in the unit cell. Around the position of the uranium atom there is indeed a significant and we ł oca ise mag e i a io o e ou e a ia e e o e 5 -e ec o wa e u c io s is co sis e wi e i e mos mag e i a io is i u io owe e i a i io o e mag e i a io o e 5 -e ec o s a mag e i a io co i u io is i e i ie w ic a ises om a 7s electrons of the uranium atom. The sym- metry of the 6d-electron wave functions is reduced due to the point symmetry of the uranium location. The threefold axis of rotation gives rise to the crystal field splitting of the 6d-electron levels. This is the reason for finding the three regions of positive magnetization density at a distance of ≈A1.2of the U atom position. Both the radial extent of the wave function and the fact that it is split into crys- tal field levels uniquely identify this magnetization distribution as arising from 6d electron. For s or p electrons no splitting is expected to occur for the site sym- metry of the uranium atom. The remaining negative magnetization distribution is attributed to a 7s-electron polarization, with the 7s-electron spin being orientated antiparallel to the direction of the 5f-electron magnetic moment. The remaining magnetization distribution in the unit cell arises due to a magnetic polarization of the platinum electrons. The Pt triangle within the unit cell is seen to carry a significant part of the magnetization density. The electrons have to be considered delocalised. The overlap between neighbouring Pt atom wave functions is significant, as a result of which the magnetization distribution is severely distorted. The appearance of the magnetization density does not any more resemble the one expected for atomic wave functions. In order to model the magnetization density for UPt 3 all those electronic con- Mag e ic σ m ac σ Measu eme s 79 tributions identified above have to be included. The model of a localised moment magnetization can be modified to include the localised magnetization contributions of the uranium atom. It suffices to use free atom wave functions. For the mod- elling of the magnetization density within the triangles of Pt atoms the overlap of wave functions has to be included. This requires a band structure calculation. Until now, however, a theoretical prediction of the magnetization density in UPt3 based on the band structure calculation has not been reported in the literature. Bearing these comments in mind a model calculation has been carried out using free atom wave functions for both, uranium and platinum contributions. By using such a model the agreement with observation could be significantly improved. The agreement is considered fair, however, with the main shortcoming of such a calculation being given by the inability to adequately model the delocalised magnetization density around the Pt atoms.
5. Discussion
The results of the form factor measurement of UPt3 illustrate quite nicely the power of the technique of spin polarised neutron scattering. The magnetiza- tion density as presented in Fig. 12 reveals details of the electronic wave functions, their radial extent and the angular dependence. It is therefore a decisive test for any band structure calculation, whether or not such details. of the electronic wave function are reproduced by the calculation. It was pointed out in the introduc- tion that this type of measurement probes the ground state of the system. The perturbation which is introduced by the application of a magnetic field is of the order of 10 K. This energy is much smaller than the energies involved in typi- cal experiments in spectroscopy, or the accuracy of energy values obtained by a band structure calculation. For the purpose of comparing the magnetic form factor measurements with results of band structure calculations the ground state wave function may be used for the calculation of M(r). In general, band structure calculations are aimed at a determination of the eigenvalues of the hamiltonian of the system. This results in the energy spectrum as given by the band structure and the electronic density of states. In order to obtain good estimates for the energy eigenvalues of the system it is generally not required to have the exact wave function which is an eigenstate of the Hamiltonian. A method due to Ritz is a well known procedure in quantum mechanics whereby a trial wave function with some free parameters is used in a variational procedure to obtain estimates of energies for the system under investigation. Quite different wave functions may lead to very similar estimates for the energies. It is not essential to have the correct electronic wave function for obtaining good estimates for the band structure. However, for obtaining a complete solution of a quantum mechanical system the eigenfunctions have to be determined. In this respect a comparison of calcu- ła e wa e u c io s wi e esu s o mag e ic o m ac o measu eme s ca e e u S i o a ise eu o sca e i g e e ime s a e ca a e o yie i g e y e ai e i o ma io o e wa e u c io s o some e ec o s i e sys em I a i io o o i i g a c a e ge o a s uc u e ca cu a io s e mag- K.-U. Neumann, K.R.A. Ziebeck
e i a io e si ies may yie qua i a i e i o ma io o e eg ee o oca i a io o e ec o s i me a s a o c emica o i g i io ic com ou s A a ge um- e o sys ems a e y ow ee i es iga e wi is ec ique e i e es e ea e is e e e o a com i a io o e e e ces o mag e ic o m ac o mea- su eme s ( ouc e e [ 1] w ic co ai s a a ge um e o sys ems wi a ie ysica c a ac e is ics e e ai e i e e a io o e e ime a esu s a so e e s o a a ge eg ee o e sys ems w ic a e co si e e e e o y e e - am e o a me a ic sys em wi oca ise a e oca ise e ec o mag e i a io co i u io s as ee co si e e o a mo e e ai e iscussio e i e a u e [ - ] may e co su e i a i io o e e ooks o eu o sca e i g [1-3]
Co c usio s A i o uc io as ee gi e o e ec ique o s i o a ise eu o sca e i g wi a a ica io o mag e ic o m ac o measu eme s Some e e - ime a ques io s a e ee iscusse o e i e e a io o e mag e i a io e si y wo mo e s a e ee co si e e a oca ise mome sys em a a e o- ca ise e ec o sys em wi e e ec o esc i e y ee e ec o wa e u c io s e a ious oi s aise i ese iscussio s a e ee i us a e wi e e e i- me a i es iga io o e ea y e mio sys ems U e13 a U 3 ese sys ems e i i e u s ec um o e ec o ic mag e i a io is i u io s i me a s o e ec o s wi oca ise c a ac e is ics i i c u es co i u io s om 5 -e ec o s o e u a ium a om wi e mag e i a io a i g s e ica symme y c ys a ie s i co i u io s o -e ec o s a 7s-e ec o mag e i a io wi o a - gu a e e e ce e oca ise e ec o co i u io s a e ou i mag e i a io e si y i ea s ace e e e ime a ec ique o mag e ic o m ac o measu eme s y eu- o sca e i g is a owe u oo w ic may yie e ai e i o ma io co ce i g e ec o ic wa e u c io s is i o ma io is com eme a y o o e e e ime - a ec iques w ic i es iga e e e e gy s ec um a a s uc u e o e so i Mag e ic o m ac o measu eme s a e a so a c a e ge o eo y a a s uc u e ca cu a io s A c ose com a iso o e mag e i a io e si y i ea s ace wi e ec o ic wa e u c io s o ai e i a s uc u e ca cu a io is ig y esi a e
e e e ces
[1] G Squi es Introduction to the Theory of Thermal Neutron Scattering, Cam i ge U i e si y ess Cam i ge 197 [ ] S W o esey Theory of Neutron Scattering from Condensed Matter, C a e o ess O o 19 [3] W G Wi iams Polarised Neutrons, C a e o ess O o 19 [ ] ow i Etectron and Magnetization Densities in Molecules and Crystals, E ecke e um ess ew Yo k 19 55; ow o sy Appl. Phys. , 15 9 (19 [5] o sy i e [ ] 3 3 Magnetic Form Factor Measurements ... 1
[ ] ow C Ma ewma The Cambridge Crystallographic Subroutine Library (A collection of FORTRAN subroutines for the use in crystallography), I e a I e o I G e o e 199 [7] M Moo Int. J. Magn. 1 19 (1971 [ ] ow Physica B 137 31 (19 [9] a a s A A e i S icka ow J. Appl. Phys. 3 11 (19 3 ; A A e i ow a a s S icka Phys. Rev. Lett. 37 (19 [1 ] a mo J. Phys. (France) C7 177 (19 [11] K O a mo S iu S K Si a Phys. Rev. B 1 1 3 (197 [1 ] Cooke S iu A iu Phys. Rev. B 37 9 (19 [13] M ume A eema E Wa so J. Chem. Phys. 37 1 5 (19 [1 ] A eema esc au J. Magn. Magn. Mater. 1 11 (1979 ; C S assis W eckma a mo esc au A eema Phys. Rev. B 15 3 9 (1977 [15] E a ca J. Phys. C, Solid State Phys. 15 1 (1975 [1 ] E imme S W o esey Rep. Prog. Phys. 3 333 (19 9 • [17] ouc e e Gi o Sc wei e J. Phys. (France) C7 199 (19 [1 ] E U Co o O a asi Atomic Structure, Cam i ge U i e si y ess Cam- i ge 19 [19] i a s Ca e Pearson's Handbook of Crystallographic Data for Inter- metallic Phases, Ame ica Socie y o Me a s Wi au (Swi e a 19 5 [ ] Sea s Thermal Neutron Scattering Lengths and Cross-Sections for Condensed Matter Research, C a k i e uc ea a C a k i e (Ca a a 19 [ 1] ouc e e A Compilation of Magnetic Form Factors and Magnetization Den- sities, eu o i ac io Commissio o e I e a U io o C ys a og a y I G e o e 19 [ ] Sc wei e i e [ ] 5 1 [ 3] Sc wei e i e [ ] 79 [ ] ow i e [ ] 7 3 [ 5] o sy i e [ ] 791 [ ] M Moo J. Phys. (France) C7 1 7 (19