Vehículo Huinday Matrix

The importance of inelastic neutron scattering comes from the fact th a t the en­ ergy of moderated neutrons (either cold, therm al or hot) are of the same order of magnitude as the elementary excitations in condensed m atter. The two most studied excitations with this technique are phonons which characterise the vibra­ tional modes of atoms in a crystal and magnons which characterised the vibrations of magnetic moments of the atoms in a crystal. Of course, other excitations of magnetic or other origin, such as the crystal field excitations, are also observed by inelastic neutron scattering.

The inelastic coherent scattering is a process in which the momentum and energy conservation laws should apply:

(3.31) k f - ki~^T = q

where q and are respectively, the wavevector and the energy of the quasipar­ ticle describing the interaction considered; ki, kf, Ei and E f are the wavevectors and the energies of the incident and scattered neutron and f a vector of the re­ ciprocal space (nuclear or magnetic). The sign plus corresponds to the creation of a quasiparticle and the minus to its absorption.

W ith inelastic neutron scattering we measure the energy of the excitations as a function of the scattering vector q. The curves E { ^ are called dispersion relations of the excitations considered.

The position of the peaks on the energy distribution of scattered neutrons determines the energy of the absorbed or emitted quasiparticles. Experimentally, it is therefore necessary to have access to any point of the four-dimensional space (Q, Lü). Different spectrometers enable us to study this space [3]. The triple axis spectrometer is the best adapted to the study of elementary excitations in

Chapter 3. Experimental techniques 60

condensed m atter with single crystal samples; indeed the decoupling of the en­ ergy with the three dimensions of the space makes possible the determ ination of dispersion curves of the quasiparticles corresponding to the studied excitations. The time of flight spectrometers are also very im portant: these instrum ents do not decouple the four dimensions but they allow a large energy range to be stud­ ied simultaneously and are better for bigger energy transfer (E > 50meV) and polycristalline samples.

In F ig . 3.6 we see a schematic view of a triple axis spectrometer. A monochro­ m atic beam with wavelength A* = is extracted from the white beam com­ ing from the reactor, after passing through the collimator Ci. Using a Bragg reflection from a crystal M {monochromator) characterised by a distance between crystalline planes d ^ :

Xi = 2dM sin {9m) (3.32)

The Bragg angle 29m is define by the angle between the two collimators Ci and C2, which deflne the divergence of the beam. Therefore the choice of the incident

neutron wavelength is related to the angle 9m-

The neutron scattered by the sample, of wavelength A/ = 27r/kf is again scattered for the third time by another crystal, called the analyser. The positions of the detector and the analyser are deflned by the angle 9a, and allow us to choose

the energy of the detected neutrons.

The basic interest of the triple axis spectrometer is the possibility of perform­ ing scans in (Q, co) space, with Q normally limited to the equatorial scattering plane. Mainly the experiments are performed from scans in energy with Q fixed or from scans in Q with constant energy.

C h apter 3. E xp erim ental techniques

o r

Oelec

W,

Figure 3.6; Schematic view

Chapter 3. Experimental techniques 62

section of equation (3.1) takes the form [17, 18]; / d'^a k ki 27T ~h \d Q d E j - QaQp) (3.33) q/3

The first term in this equation gives the coupling between the neutron and elec­ tron spin and f(Q) is the magnetic form factor of the unpaired electrons. The term (^dap — QaQp^ ensures th at only components of the spin system perpen­ dicular to the scattering vector Q contribute to the scattering cross section. The final term is the Debye- Waller factor. The term

S°‘^{Q,uj) = ^ [ e x p ( - i u j t ) d t ' ^ e x p iQ{fm - f„) x (S ^{ t = 0)S^{t)} m n

(3.34) is the scattering law. The spin correlation function (5 “ (t = O)S^(t)), is a sum of contributions due to dynamic (oscillatory and relaxational) processes and the equilibrium arrangement of the spins.

\ " (Q,cu) + S “'>(Q)S(cv) (3.35) 7T

where the term w) is the imaginary part of the generalised susceptibility due to the spin dynamics and is obtained by applying the fluctuation-dissipation theorem. The term n{uj) 4- 1 = ( 1 — exp{—huj/kT))~^ is known as the detailed

balance factor, and is related to the possible states populated at a given tem pera­ ture. The last term (elastic in energy) is the long-time static magnetic correlation function.

For the special case of the crystal field transition from states F^ scattering law for N identical magnetic ions reduces to;

the

= yV p r.(r„|7 “ | r „ ) ( r „ | + E r . - E r J (3.36) where pr„ is the Boltzmann population factor. From the symmetry relations associated with the m atrix elements we find the cross section:

(Pa dÇîdEi = Nprr 762 1 rUeC^ k ki 2 9 fiQ ) ^ -2Vy(Q) h |( r m |i l r „ ) |'5 ( n u ; + E r „ - E r „ ) (3.37)

The polarisation factor permits a discrimination between transverse (o;=x,y) and longitudinal {a=z) crystal field transitions by measuring at different Q when single crystals are available. For Q \\ c only transverse transitions are observed, whereas for Q J_ c the transverse transitions lose half their intensities, and in addition longitudinal transitions appear. The eigenvalues and eigenfunctions of a particular crystal field level scheme may be tested by a comparison of both the energies {Er^ — Ey^) and the relative intensities of the observed transitions with the values calculated from equation (3.37). The tem perature dependence (arising through ppn) provides an additional check.

The introduction of exchange interactions between ions couples the single-ion crystal field states discussed. Collective excitations are formed from phase-related linear combinations of the single-ion transitions. These exciton states, as they are usually called, display significant dispersion which can be followed throughout the Brillouin zone in many cases [18, 19].

B ibliography

[1] D. Jiles, Introduction to Magnetism and Magnetic Materials (Chapman & Hall Ltd., London, England, 1991).

[2] Volume:! . Theory, Instruments and Methods, Neutron and Synchrotron Ra­

diation for Condensed Matter Studies, edited by J. Baruchel, J. L. Hodeau, M. S. Lehmann, J. R. Regnard, and C. Schlenker (Springer Verlag / Les Editions de Physique, Berlin, Germany / Les Ulis, France, 1993).

[3] G. S. Bauer and W. Biihrer, in Neutron scattering. Lecture notes of the first summer school on neutron scattering, edited by A. Furrer (Paul Scherrer Institute, Villingen, 1993), Vol. 1, pp. 1-50.

[4] Neutron Scattering, Methods of Experimental Physics, edited by D. L. Price and K. Skoeld (Academic Press, New York, USA, 1987).

[5] C. G. Shull and J. S. Smart, Physica B 76, 1256 (1949).

[6] G. H. Lander, W. G. Stirling, S. Langridge, and D. Gibbs, J. Magn. Magn.

Mater. 140-144, 1349 (1995).

[7] G. H. Lander, J. Alloys Comp. 250, 544 (1997).

[8] S. W. Lovesey, Theory of Neutron Scattering from Condensed Matter, Vol. 1

of The International Series of Monographs in Physics (Clarendon Press, Ox­ ford, 1987).

[9] G. L. Squires, Introduction to the Theory of Thermal Neutron Scattering, 2

ed. (Dover Publications, INC., Mineola, New York, USA, 1996). 10] C. G. Shull and Y. Yamada, J. Phys. Soc. Japan 17, 1 (1962). 1 1] M. Sakata and M. Sato, Acta Cryst. A 46, 263 (1990).

1 2] M. Sakata, R. Mori, S. Kumazawa, and M. Takata, J. Appl. Cryst. 23, 526

(1990).

13] R. J. Papoular and B. Gillon, Europhys. Lett. 13, 429 (1990). 14] W. Jauch and A. Palmer, Acta Cryst. A 49, 590 (1993).

15] J. P. Desclaux and A. J. Freeman, J. Magn. Magn. Mater. 8 , 119 (1978).

16] C. H. Lander, M. S. S. Brooks, and B. Johansson, Phys. Rev. B 43, 13672 (1991).

17] L. van Hove, Phys. Rev. 95, 249 (1954).

18] W. C. Stirling and K. A. McEwen, in Magnetic Excitations, Methods of Experimental Physics (Academic Press, New York, USA, 1987), Vol. 23C, pp. 159-237.

[19] B. R. Cooper, in Magnetic Properties of Rare Earths Metals (Plenum Press, New York, 1972).

C hapter 4

U ( P d l _ a ; P t a ; ) 3

On studying the crossover between itinerant and localised behaviour we have chosen a system in which the actinide element is the same, Uranium, and in which by substitution of Pd by isoelectronic F t the distances between neighbouring uranium atoms remain nearly identical: 4.10Â in UPdg and 4.12Â for UPtg [1]; see T a b le 4.1.

Despite these similarities, the low-temperature properties of the compounds UPdg and UPtg are very different. The more external 5d electrons of P t are more likely to hybridise with the 5 / electrons of U than the 4d electrons of Pd. Remember th at in progressing from localised to itinerant magnetism there are three im portant parameters: Actinide element, interactinide distance and actinide environment; see section 1.1.

Table 4.1: Lattice parameters (a and c), neighbouring uranium distance and c/a* {2c j a for UPtg) ratio quite deviated from ideal c / a = 1.633 for hep UPtg (MgCdg type structure) and dhcp UPdg (TiNig type structure) from [1].

a(A) c(A) d[/_[/(A) c/a* UPds 5.757 9.621 4.102 1.671

UPt3 5.752 4.889 4.123 1.700

B A B A A C A B A

Figure 4.1: The hep and dhcp crystal structures.

The main crystallographic difference between these compounds, see Fig. 4.1,

is the stacking of the atomic layers: in UPdg the stacking is ABAC (double hexagonal closed packed, dhcp), in UPts the stacking is ABAB (hexagonal closed packed, hep). In the hep structure only uranium atoms with hexagonal symmetry exist, whereas in the dhcp structure the B and C sites have hexagonal symmetry, while the A sites have local cubic symmetry for an ideal c/a ratio.

All the samples used in this work are single crystals grown by D. Fort at the University of Birmingham by the Czochralski method, using depleted uranium, unless the contrary is specifically stated.

4.1

Literature Survey

UPds and UPtg are two extreme examples of magnetic behaviour in the literature. UPtg is a classical example of an itinerant magnet] it is a heavy fermion, in which unconventional superconductivity (Tg = 0.5K) [2] and antiferromagnetic order (T„ = 5K) with extremely small moments {p = 0.02/ig/U -atom [3]) coexist. UPdg

Chapter 4. U(Pdi^xPC)3 68 6 UPd. H=O.OT UPd. o L a - a x i s H=7.0T O 2 0 2 3 4 5 6 7 8 9 10 11 12 d a x i s .... TEMPERATURE (K) c a x i s -I 5% P t. -I .1 I ... .L J I I I L TEMPERATURE (K)

Figure 4.2: Specific heat and thermal expansion of UPda at zero field (open circles) and H = 6T and H = 7T, respectively (solid squares) The arrows indicate the transition temperatures at zero field. Zero field results of thermal expansion for U (Pdo.9 5Pto.0.5 ) 3 are also shown. From [11, 12].

is a localised magnet^ and most of its features can be explained within the standard

model of the rare-earths [4]. M a g n e t i c e x c it a t i o n s were first observed by Murray

and Buyers [5, 6], who ascribed them to c r y s ta l fi e l d e x c ita tio n s . The finding of crystal field transitions by means of neutron scattering is a clear indication of localised magnetism, and this is the only example in metallic uranium-based compounds for which these transitions are clearly observed. UPdg is also well known by displaying q u a d r u p o la r t r a n s i t i o n s [7, 8]. It also orders magnetically

with a very small moment at T/v = 4.4K.

4.1.1

M agnetic properties of U Pds

The presence of at least two transitions in UPdg was known for a long time: h ea t c a p a c ity [9] and th e r m a l e x p a n s io n m e a s m e m e n t s [10] revealed transitions around 7 and 5K. More recent thermal expansion, specific heat [11], see F ig. 4.2, and

6 0 0 0 • N S F P/Q ^ 4 0 0 0 2000 0 5 15 UPd 4 0 0 0 • NSF 3 0 0 0 P // Q

I

%

I

^ 2000 § 1000 T e tn p e ra tu re /K

F ig u r e 4.3: The peak intensity at Q = ( |,0 ,3 ) and Q — ( |,0 ,4 ) for P || Q as a function of temperature. The lines are guide to the eye. From [15].

inagiietisatioii measurements [13] on single crystals confirmed these transitions and also revealed the presence of a third transition. From these measurements it is now clear that there exist three transitions, at tem peratures (in zero field) of 7.8, 6 . 8 and 4.4K. We shall call these temperatures Tp, T% and T2, respectively.

Murray and Buyers [14] discovered new reflections below T% at positions ( /i+ |,0,/) in reciprocal space and attributed them to a compression and rarefac­

tion of the lattice with three fold symmetry about the c-axis with the form of a triple-q structure. Later Steigenberger et al. [15] found th at the tem perature dependence of reflections such as (|,0,3) and (|,0,4) also showed anomalies at the T2 transition. Using neutrons polarised parallel to the scattering vector it

was found that the scattering was non-spin-flip, see Fig. 4.3, demonstrating that their origin was structural rather than magnetic.

It is the periodic lattice distortion produced by the ordering of the quadrupo­ lar moments at Ti which couple to neutrons and gives rise to the superlattice

Chapter 4. U(Pdi-xPtx)3 70 reflections. Their tem perature dependence also shows a change of intensity at T2,

in this case with a marked different behaviour at I = even and I = odd reflections. Only in the case of (|,0,0) a change in intensity is detected in the spin-flip chan­ nel below T2, showing th a t this transition has a magnetic as well as structural

character.

The group theory analysis of Walker et ai [8] identified the symmetry of the

order param eter as B2g. The doubling of the unit cell means th a t the structure is antiferroquadrupolar (AFQ). The possible components of the structure are a combination of Qx^-y^, Qzx and Qzz quadrupolar moments on the cubic sites and Qzx on the hexagonal sites. The ordered phase was identified as a triple-q, see F ig . 4.4 and the space group of this phase was P3m l.

More recent measurements of these superlattice reflections with applied field [7] shows an enhancement over their zero field values. From these it was clear th a t the scattering at ( | ,0,1) develops below Ti' with a small but distinct anomaly at

T i, whereas the much less intense (|,0,2) peak develops only below T%. In general ( | ,0,/ = odd) reflections appear below Ti/, whereas I = even reflections appear

below T i. The fact th at the intensities are greatly enhanced applying a magnetic field, indicates the presence of magnetic scattering as well as a structural com­ ponent. Measurements with polarised neutrons dem onstrate th at the scattering between Ti/ and T% is entirely non-spin-fiip for ( | ,0,1) with the direction of the

neutron polarisation perpendicular to the scattering plane and parallel to the applied field along the a-direction which implies the moments are lying along the a-axis. For the (|,0 ,/ = even) reflections which develop for T < T i the scattering with applied magnetic field is predominantly spin-flip. From these result together with the field induced reflection ( | ,0,0) it is deduced th a t an antiferromagnetic

structure with moments aligned along the c-axis imply the appearance of Qzy and Qzx quadrupolar moments below T%.

Murray and Buyers interpreted the observed magnetic excitations [5, 6] as

y - f i r j ^ V ^ V p V p y o y ® Q - Q Q - ® & o p V p V p V p V O (P O Ct ® 0 * 0 0 ^ 0 0

*

> 0

p

1 9 p V p V p V p

0* O

0 * 0 0 * 0

0 * 0 0 * 0

P p V P 'O P V p p j o P O 0 * O 0 * O l> O (0 *

Figure 4.4 : Basal plane projection of the antiferroquadrupolar structure on the cubic sites of UPda below T i . The circle and the ellipses represent the uranium ions at the cubic A-layer sites. In the high temperature phase the uranium ions will have non-zero quadrupole moments Qzz corresponding to the uraniums having a spheroidal charge distribution with the c-axis being the axis of revolution. The (^a;2_y2 contribution to the order parameter results in the charge distribution of

an ion being no longer spheroidal, but acquiring the symmetry of an ellipsoid with its three principal axes having unequal lengths; the ellipses indicate the relative orientations of these ellipsoids (one principal axis remains normal to the basal plane) for the different ions in the cubic sites (the central ion remains spheroidal). The effect of the Qzx contribution can be visualised by imagining that the ellipsoidal charge distribution for each ion is rotated by a small angle in the positive direction out an appropriate rotation axis; this rotation axis lies in the basal plane and is indicated with an arrow; the central ion is not rotated. The unit cell of the structure is indicated.

Chapter 4. U(Pdi-xPtx)s 72 (meV) 50 CUBIC-SITES (K) (meV) HEXAGONAL-SITES 40 30 b c W 20 10 ±0.888li2>i0,2211il>i0.3881±4> 50 -0.707l+3>-0.707l-3> 500 0.S90l+4>-0.685kl>-0.428l-2> ₄₀ 0.253Ü4>0.967li2> to.173l+4>-0.409l42>+0,201l+1>-0.655l-1>-0.125l-2>+0,S64l-4> 0.643l+3>-0.41710>-0.643l-3> 400 0.707l+3>+0.7071-3> - - 30 - - 300 20 0.707l+3>-0.707l-3> - - 200 l+1> - - 100 10 -0.967i±4>+0.253H2> - -0.136li2>i0.694l+b+0.707l±4> 0 0 0.295t+3>+0.909l0>-0.295l-3> WAVE-FUNCTION WAVE-FUNCTION 500 400 300 200 100

Figure 4 .5 : Crystal field scheme calculated with the effective charges o f Murray and Buyers [6].

cubic site ions: the higher lying modes (at 15-20meV) arising from transitions on the hexagonal sites, whilst the modes at l-3meV arise from the cubic sites. They calculated a crystal field hamiltonian with neighbouring effective charges, Q„, assuming a 5f^ configuration for the uranium atoms, which was later confirmed by the interm ultiplet transitions observed in high-energy neutron spectroscopy [16]. The crystal field scheme resulting is shown in F ig . 4.5. The excitations at the hexagonal sites which exhibit collective behaviour are explained with interionic isotropic coupling with nearest and next-nearest couplings within and between hexagonal planes which gave a good description of not only energies but also the intensities, see F ig. 4.6. For a review of their studies see [17].

The low energy modes and their tem perature dependence were studied in detail by McEwen et al. [18]. They found four modes below 3meV at a general wavevector at T = 1.8K. Some peaks disappear as the tem perature is raised and

X t“ 4 . 0 >- O z w o 3 5 w E 3 . 0 1 . 5 0 3 1 0 3 . 5 0 3 0 0 3 1 0 3 0 0 4 102 R E D U C E D W A V E V E C T O R Q

Figure 4.6: Dispersion of collective excitations in UPda as in [5].

Little work has been done on the U(Pdi_a;Pta; ) 3 alloys close to the UPdg side.

It is known th a t the transitions in UPdg are extremely sensitive to P t doping and T=0.05 is enough to no longer observe transitions by means of therm al expansion [1 2]. A change in crystal structure is observed with powder X-ray diffraction

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