Información Financiera

For a single domain ferromagnet, it can be seen from (2.38) that for a particular reflection, the magnetic cross-section is proportional to S2 1 q 12. For a multi-domain ferromagnet, the situation is more complex. L e t us assume the magnetization direction for a particular domain in a ferromagnet makes

d i - ( 0,0 ,0 ) ,

(2.46)

evaluation o f G ^ j is simply a m atter o f summing the scattered waves from these two points:-

_ , . . . 2h k 1 V1

Gfcu - 1 + ex p {i(-5 -+ j - f 5 )) . (2.47)

It follows that

T ab le 2.1 R ela tiv e In t— jU— nf M D « lnw-anvle refla rtin n . R E F LE C T IO N (hkl) _{« h k l} 001 0 002 4 008 0 004 4 100 1 101 3 102 1 103 3 104 1 200 1 201 3 202 1 208 3

an angle 0 to the c-axis. Because o f the other domains within the sample, this does not uniquely define the moment orientation within the sample. Each domain will have a different value o f q and that particular value o f q will also depend on the orientation o f the scattering vector for the reflection under consideration. L e t us now evaluate q for two important reflections, the hOO and 001 type reflections. We choose to represent these reflections by the 100 and 002 respectively. Im m ediately we know x 100 lies along a * and x 002 lies along c*. D en otin g the angular displacem ent o f the basal plane component o f

magnetisation from the a * axis by 0 (figure 2.3), and using a variation on the

expansion o f 1 q 12,

l q l 2 > [ l - ( t S ) * ] , (2.49)

for the 100 we have

' l i o o ^ - d - G i o o ^ ) * ] .

■ 1 - cos2« , (2.50)

where a = cos'*( sin© cose).

1 q 10012 ■ 1 - sin2© cos2« . (2.51)

Also, for the 002,

]<loo2|2" 1 - co®2®.

1 q oo212 ■ sm2e • (2.52)

I q I 2 is sim ply the square o f the projection o f the m agnetization onto the scattering plane defined by x, and consequently there w ill be no scattering from a domain whose magnetization vector is parallel to the scattering vector.

For a particular Bragg reflection it is important to average q over the possible domain orientations. For the canted ferromagnetic structure it may be assumed that the domain easy directions lie in two cones centred on ± c * and with a semi-cone angle o f 0. C learly I q 0021 2 w ill remain unchanged after dom ain averagin g, since a ll moments w ill h ave the sam e basal plans component. I q 10012 is, however, highly sensitive to domain orientation. For a uniform distribution o f domains about c*;-

< iq io o |J> - sin2©

coa20 (2.63)

where the triangular brackets denote the average value. T h e fir s t term represents averaging over different basal plane directions and the second over the tw o c-axis directions. In summary, fo r a m ulti-dom ained canted ferromagnetic sample with no applied field:

2JL4 M odulated Structures in Gd-Y and G d-8c A lloys

W e shall now examine in more d eta il the diffraction conditions for modulated structures with propagation directions along the c * axis. Such structures give rise to magnetic Bragg reflections or 'satellites' associated with each reciprocal lattice point (q permitting). Th e satellites are found on lattice rows parallel to the c * axis. The existence o f a single pair o f satellites implies a structure in which the scattering amplitudes o f the atoms are modulated sinusoidally. From the location o f the satellites on lines parallel to e * it m ay be concluded that the wavevector o f the modulation is parallel to that direction. The absence o f satellites o f 001* type implies a uniaxial configuration in which the moments are parallel or anti-parallel to the c-axis. This follows because the neutron senses only the component o f the moment which is perpendicular to the scattering vector.

W ith these points in mind, we d erive th e intensities o f m agnetic reflections for a conical helical structure described by the spin components:-

(2.64)

Neutron Diffraction in the ttudy o f Gd-Y and Gd-Sc Alloys 33

SM » S_i_ coa(na>), Sny » Sj. gin(na)), « S, ,

where n denotes the index o f the layer in which the moment lies. In this configuration, th e re is a ferromagnetic component along the c-axis so that there is magnetic intensity residing on nuclear reciprocal lattice points which are not o f the 001 type. Satellites due to the helical part o f the structure are, however, still observed at 001 lattice points. For a given reflection, the conical configuration g ives rise to intensities:-

( T T o ^ f K ) 2 G ju ^ »in2; (2.66)

(5 “ h l ± “ (TTo)2 « ^ O f « 2 G jy S i J( 1 ♦ CO.2; ) , (2.57)

where £ is the angle between the scattering vector and c *, and f^ftc)* is the form factor evalu ated for each satellite separately. Note that the magnetic cross-section is proportional to (S q)2 in (2.56) and (2.57); it has value S2 sin2£ and S2 j ( 1 ♦ coe2£ ) respectively. I f the periodicity o f the helix is not too short, fjjjix)* may be averaged between the two satellites, or the value at the nuclear peak may be taken. For the 100 and 002 reflections, we have:-

<2-m>

[ £ L r «* * * > -& ■

<Z59)

(2.61)

Diffraction data from a simple basai plane h elix may also be calculated from (2.56) and (2.57). In this case there is no m agnetic contribution to the normal lattice sites (S| -0).