D. E McMORROW
X- RAY SCATTERING STUDIES OF LANTHANIDE MAGNETISM 19
I000 9OO 8OO 7oo
o6oo
41800
, j \
4oK ( E )
<Z) (Z>
2/12 2/12 ~
,t60K
65)
II
2/w 2/90 d D
11 [/ 20K<T<I52K T < 2 0 K
/ t H /~1oo K ,120K I
• 130 K
I I I I I I ~ I I I I I I I I I I /
42100 4.2200 42500 42400 42500 42600 42700 42800 42900 ,~(UNITS OFC °)
Fig. 5. The temperature dependence of the Ho (0,0,4+Z'm) magnetic satellite taken with non-resonant synchrotron radiation. (The temperature increases from right to left.) The lines are drawn to guide the eye. (From Gibbs
et al. 1985.)
to correlation lengths o f several thousand A, or greater. More recent measurements made at the National Synchrotron Light Source on samples with larger surface areas and under better optimized conditions routinely give signals o f several hundred counts s -1 at essentially the same energy on backgrounds o f about 1 s 1 (Gibbs et al. 1988).
Although absolute measurements have not been made o f the non-resonant signal rates for X-ray magnetic scattering, the signal rates in these experiments were reduced from the corresponding charge scattering by o f order 1 x 106, which is consistent with a simple estimate o f the non-resonant cross-section for Ho (eq. 7).
The temperature dependence o f rm obtained in these early experiments by both X-ray (open circles) and neutron diffraction (solid circles) is shown in fig. 6. It is clear that in the temperature range above 20 K the wave vector rm determined by X-rays has preferred, commensurable values, whereas in the lower resolution neutron data there is a continuous variation o f the wave vector with temperature. Other noteworthy features o f the X-ray data include the appearance o f an inflection point near ~m = g at around 70 K, thermal 2 hysteresis below 50 K, and coexistence among phases with differing wave vectors. At the lowest temperatures, there is a first-order transition between two commensurable wave 1 , and there is an indication o f a lock-in vectors, namely, Tm = 2 c * and rm = gc ,
transformation at 5 c * . The inset in fig. 6 shows the variation o f rm during several cycles o f the temperature between 25 K and 13 K. The data suggest a clustering o f the wave vectors around l~m = ~ c * and r m = 5 c * .
We now consider the apparent discrepancies in fig. 6 between the values o f rm determined by X-ray and neutron scattering. This m a y be understood in part from the higher Q resolution o f the former. Another factor is that the X-ray penetration depth in
2 0 D . E M c M O R R O W et al.
.2700 A .2600
b_ o .2500
P-
.2400 rr .2500
R
,,, .2200
>
1.1.1
.2100
Z _o .2OOO
J .I 9 0 0
r ~ 0
.1800 .1700
• (002) -+, (004) -+ NEUTRON SCATTERING 0 (002)+, (004) + X-RAY SCATTERING
./
/ / / ° /
I I
I / I
..,.L_o~i _ _ _ _ 2/12
I I I I I
I0 20 :50 4 0 50
. /
/ / / / / /
/ / / / /
219 / ~ / / /(~.//
I,-
. 1 9 0 0 ~ ~9o o o _ _ 5 / 2 7
cJ O O 0 0 , - m q Cl
~o ~ - 2111
.1800~ I I
IOK 20K 5OK
I I I I I i I I
6 0 70 80 9 0 I00 I10 120 150 TEMPERATURE (KELVIN)
Fig. 6. Temperature dependence of the Ho modulation wave vector r obtained with both synchrotron X-ray (open circles) and neutron (solid circles) scattering. The wave vectors obtained by neutron scattering in the coexistence region below 20 K are the result of fits to the first harmonic. The fine lines across the hysteresis loop indicate the results of cycles of the temperature below 50 K. Inset: Plot of the wave vectors obtained from several cycles of the temperature between 13 and 24.5 K, obtained with X-ray scattering. (From Gibbs et al.
1985.)
Ho at 8 keV is ~1 ~m, whereas neutrons sample the bulk, and it has b e c o m e clear in these and subsequent studies that the surface preparation can alter the magnetic structure within the first several gms. It was also found b y Koehler et al. (1966) that not all b u l k crystals displayed the same wave vectors, and this was ascribed to a variation in the impurity content between the different samples. In fig. 7 the temperature dependence o f Tm is shown for two different samples o f Ho, and also for a thin film o f Ho grown by molecular b e a m epitaxy. It is clear that large differences in rm can be observed depending in the case o f b u l k crystals on surface treatment, impurity content, etc., and in the case o f thin films mainly on the strain resulting from the lattice m i s - m a t c h between the film and substrate (the latter is discussed more fully in section 5).
In addition to the existence o f preferred commensurable magnetic wave vectors in Ho, there was a second distinctive feature o f the X-ray diffraction patterns that led
C.)
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rr"
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>
LIJ
X
Z _o
_1 n
o 0.21
0 . 2 0
0.19
0.18
0.17
( I r I I ... 1
HOLMIUM
0 0
@ @ 0 • • @ • • 0
o 1/5 "2-2
0
o • o o °
X O
~m x o
• X O O
X
X O
4/21 . ~ . 5 . ~ 5•27 - 4 . 4 - 4 . 4 - 4 . 4 2/11 ,5
~ < ~ oo I / 6 6
O.16 I t I I I I I
I0 2 0 3 0 4 0 5 0 6 0 7 0 TEMPERATURE (K)
Fig. 7. Temperature dependence of the modulation wave vector T~ for three different Ho samples: Ho(1) (crosses), Ho(2) (open circles), and thin film (solid circles). The spin-slip structure for the simply commensurate wave vectors are shown on the right. The data for ~, = ~ in Ho(1) were obtained from neutron scattering. Note
also the appearance o f two new lock-in wave vectors at ~ and 4 (From Gibbs 1989.) 1
directly to a spin-slip description of the magnetic structure. Figure 8 shows the detailed evolution of the X-ray scattering as the temperature is lowered from around 25 K to 17K. As the temperature is reduced a second, initially broad peak appears at larger wave-vector transfers. This peak arises from charge scattering, as discussed below. With decreasing temperature, both the magnetic peak and the charge peak shift to lower wave-vector transfers, although the charge scattering shifts considerably farther than the
22 D.E McMORROW et al.
1000
750
500
250
2 250
>_ 0
I - -
~ 2 5 0
I--- Z
o t
250
25C
I i I I [ I l I I [ I I I I I I I I I l I I I I I I I I I
rm= 5/27 Ho
I
l l l t [ l l l l l t i l l 1 t l I l l t 1 1 1 1 I I I 1
4.t5 4.20 4.25 4.30 4.55 WAVE VECTOR (units of c*)
Fig. 8. Synchrotron X-ray diffraction patterns o f the satellite above the (004) Bragg point o f Ho, studied for decreasing temperature from T = 2 5 K to 17K. Note that in addition to the sharp magnetic satellite, a second broad satellite appears at the slip position
% = 12r m 2. When the temperature is lowered and Vm approaches z~ then the lattice modulation peak sharpens up and approaches
~'s = ~c*. (From Bohr et al. 1986.)
/
(~ POLN ANALYZER . . . ~1-J I;
A k ~
k 0 SAMPLE
Fig. 9. A schematic of the X-ray diffractometer showing the polarization analyscr. (From Gibbs et al. 1991.)
magnetic scattering for the same change in temperature. When the magnetic peak reaches Tm = ~7 c*, the radial width of the charge scattering narrows and grows in intensity. As the temperature is lowered still further, the charge peak broadens and disappears.
To establish that the second peak does indeed arise from charge scattering, polarization analysis was performed on both peaks. As may be seen from eq. (20), for linearly o- polarized radiation incident upon a magnetic spiral, there is a rotated or Jr-polarized component of the magnetic scattering, which varies in the non-resonant limit as the sum of the orbital and spin magnetization densities. (There is also a rotated component in the resonant limit, however, in the present case, with an incident photon energy of 7500 eV, the cross-section may be safely analyzed in the non-resonant regime). In contrast, charge scattering does not rotate the linearly a-polarized incident beam. Thus, it is possible to distinguish magnetic scattering from charge scattering in a spiral magnetic structure by the existence of a ~-polarized component in the scattered beam. Polarization analysis was accomplished by placing a second crystal after the sample on the 20 arm of the spectrometer, and scattering in the diffraction plane (see fig. 9). When the incident photon energy is appropriately tuned, the scattering angle for the second crystal is 90 ° . Consequently, one linear component of the beam scattered from the sample is reflected into the detector, while the other is suppressed. By rotating the second crystal by 90 ° around the axis o f the scattered beam (¢poh~ in fig. 9), the former is suppressed and the latter is reflected. In this way, the intensities of the linearly a- and Jr-polarized components of the scattered beam may be measured independently. (For detailed discussions of the operation of the polarization analyser see Gibbs et al. (1991)).
24 D.E McMORROW et al.
4 0 0
3 0 0 0
q - 0")
o 2 0 0
Oo o o
o
"o o =.
°
• o o o e o
© it)
o o
o • g O
oo o
Ho(oo~)
* FILTER IN o FILTER OUT
%
~o
%
oOo o
o o o ~ 0go
4 0 0
- 3 0 0 d 0 rO O9 0 2 0 0
I 0 0 0 O 0 0 0 0 0
0 •
1 0 0 . o l t ~ o o • "
• o q k e O o • o o o
• • ~ qp, O o ~ e
" I, /27 t2/9
4.1800 4.2000 4.2200 4.2400 2 (UNITS OF C*)
Fig. 10. X-ray scattering data from Ho. Open circles: scan of the Ho(004) + magnetic (% = ~vc*) and (T~ = 2c*) satellites taken at 17 K. Solid circles: the same scan, but with the polarization analyses in place. (From Gibbs
et al. 1985.)
Two radial scans taken through the magnetic and charge satellites o f the (0, 0, 4) reflection at 17 K are shown in fig. 10. At this temperature the magnetic satellite is located 2 . The open circles show the scan at T m = 5 ¢ * while the charge peak is located at ~c .
performed without polarization analysis. The solid circles show the scan taken with the polarization analyzer in place and oriented to pass the ~-polarized component o f the scattering, but suppress the o-polarized component. As m a y be seen in this figure, the 2 is suppressed entirely. This establishes that the peak at ~7 is passed while the peak at
2 arises predominantly from charge scattering, while the peak at 2~ is magnetic peak at
in origin.
3.3. The spin-slip model
A unified model o f the magnetic structure o f Ho m a y be derived by making use o f the concept o f discommensurations, as originally proposed for structurally incommensurable materials by McMillan (1976) and discussed in the context o f lanthanide metals by Vigren (1976), Greenough and Blackie (1981), and Bak (1982). During the last twenty years, investigations o f a fascinating variety o f incommensurable systems have revealed
X-RAY SCATTERING STU~)IES OF LANTHANIDE MAGNETISM 25