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Magnetic Form Factor Measurements by Polarised Neutron Scattering: Application to Heavy Fermion Superconductors

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Magnetic Form Factor Measurements by Polarised Neutron Scattering: Application to Heavy Fermion Superconductors o (1993 ACTA PHYSICA POLONICA A o 1 oceeigs o e I Ieaioa Scoo o Mageism iałowiea 9 MAGEIC OM ACO MEASUEMES Y OAISE EUO SCAEIG AICAIO O EAY EMIO SUECOUCOS K.-U. NEUMANN AND K.R.A. ZIEBECK eame o ysics ougooug Uiesiy o ecoogy ougooug E11 3U Gea iai e eeimea ecique o si oaise euo scaeig as use i mageic om aco measuemes is esee A ioucio o e ieeaio a e cacuaio o mageic om acos a magei- aio esiies is gie e eeimea ecique o euo scaeig eoy as aie o easic si oaise scaeig eeimes is iey iouce e cacuaio o e mageic om aco a e mageia- io esiies ae cosiee o sime moe sysems suc as a coecio o ocaise mageic momes o a iiea eeco sysem e iscussio is iusae y a eeimea iesigaio o e mageic om aco i e eay emio suecoucos Ue13 a U3 Mageiaio esiy mas a mageic om acos ae esee a ei imicaios o oe ysica quaiies ae iey iscusse ACS umes 755+ 71+ 1 Ioucio euo scaeig a i aicua si oaise euo scaeig is a oweu oo o e caaceiaio a iesigaio o ysica oeies o a aomic scae e cage euaiy a mageic ioe mome o e euo make i a iea oe o e iesigaio o mageic egees o eeom I aiio o usig euo scaeig o iesigaig eomea coece wi mageism (y eg esaisig e aue o mageicay oee saes eeimes may e eise o e caaceiaio o eais o eecoic wae ucios Suc a eeime is eaise we si oaise euos ae use o e eemiaio o e mageic om aco a o e mageiaio esiy Coay o secoscoic eciques wic ie eais o e wae ucio y a iesigaio o e eeimeay eemie ee sceme (eg e cysa ie ees o ocaise momes mageic om aco measuemes eee (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. Mnt fr ftr rnt 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 k,. 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 46 K-U euma KA ieeck sucue wi a cee o iesio symmey e ucea sucue aco may e cose o e ea (y akig e oi o iesio symmey as e oigi o e ui ce eey aom a osiio i as is symmey eae aom ocae a —I= a wi e aom aig e same scaeig eg As a cosequece eey coiuio ie(ik • i i (3 is comie wi is come cougae i e(—ik • a i e summaio e imagiay as cace Iseig (3 io (1 a cayig ou e summaio oe a ui ces oe aies a ee is e oa ume o ui ces i e cysa a δ(k — τ is e "ea ucio" wi r eig a ecioca aice eco e summaio is caie ou oe a ecioca aice ecos r us scaeig occus o scaeig ecos k wic coeso o a ecioca aice eco a wi a iesiy wic is eemie y e squae o e mouus o e ucea sucue aco (τ In acua eeimes e agg eecio wi o e suc a sa eak as iicae y e "ea ucio" i ( eak oaeig occus ue o e esouio o e isume a aso ue o same caaceisics (eg as gie y e saia ee o e coeey scaeig egios i e same owee e eais o e eak sae ae o o iees ee a ee is o ee o usue is oi ay ue e iegae agg iesiy o a agg eecio caaceise y T is gie y S(τ is a eecio eee scaig aco e scaeig eco eeece o S aises ue o e meo use o iegae e agg eaks e geomeic coecio is kow as e . oe coecio a i is eaiy eimiae om e measue iegae iesiies ee emais a oea scae aco wic is scaeig eco ieee e scae aco is ooioa o e oa ume o ui ces i e cysa eoig e ew scae aco (ae e eimiaio o e scaeig eco eeece y S e eeimeay eemie iegae iesiy o e agg eecio is gie y U o a cosa e measue iesiies ae eeoe eemie y e squae o e ucea sucue aco owee o e esciio o e ucea sucue e aue o (τ is equie As oie ou aoe o sucues wi a iesio cee o symmey e ucea sucue aco may e cose ea us soig e ucea sucue euces o coosig o eac eecio e "coec" sig i e equaio I is oie ou a soig e aoe ase oem o a eecios is equiae o e eemiaio o e aomic osiios i e ui ce is is so ecause aig oaie e sucue acos (τ o a suicie ume o Mageic σm acσ Measuemes 7 eecios e aomic osiios may e oaie y ouie iesio Wiig (3 i e om i is see a e ucea sucue aco is oig ese u e ouie asom o e ieacio oeia wic i u is eemie y e osiios o e aoms a ei scaeig segs e aoe eiaio was caie ou usig some simiyig assumios e aice as ee ake as eec a e ema moio o e aoms as ee egece e emeaue moio a eo oi ucuaios ca e ake io accou y e susiuio wee W(k is e emeaue aco wic may e aisooic e ucio e(—W(kk 2 is kow as e eye—Wae aco Oe eeimea acos suc as eicio a asoio coecios ae aso eaiy aie [5] a saa oceues eis o akig ese io accou [] 2.2.
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