Domain walls, narrow

Above Eth. the field is sufficiently strong to cause limited translation of the domain wall without disturbing to any significant extent the overall domain structure. This process is described as reversible (more correctly as nearly reversible ) to distinguish it from the very hysteretic and clearly irreversible process evidenced by the hysteresis loop (Fig. 2.46). In this regime the P-E characteristic is a narrow loop, the Rayleigh loop referred to above (c.f. Fig. 6.9). 

The wall-width parameter 80 varies from about 1 nm in extremely hard materials to several 100 nm in very soft materials. It determines the thickness Sb = nSa and energy yw = AK S0 of Bloch domain walls [13, 14, 97, 98], and describes the spatial response of the magnetization to local perturbations [95], Essentially, the thickness of the walls is determined by the competition between exchange, which favors extended walls, and anisotropy, which favors narrow transition regions. 

Typical domain walls are smooth and extend over many interatomic distances. However, deviations from this continuum picture occur in very hard materials (narrow walls), at grain boundaries and in the case of geometrical constraints. Narrow-wall phenomena, which have been studied for example in rare-earth cobalt permanent magnets [189] and at grain boundaries [95, 96], involve individual atoms and atomic planes and lead to comparatively small corrections to the extrinsic behavior. 

As this expression only rehes on the temperature dependence of the magnetic correlations, Eq. 44 is valid for any Ising-Hke chain independently of the domain wall structure. More specifically, in the case of narrow domain walls and for single-ion anisotropy this expression becomes (using the notations of Eqs. 7a and 23) ... 

More than two decades ago Pople and Walmsley (1962) discussed the existence of an unpaired ir electron in the trans-(CH)x as shown in Fig. 12a, which accompanies a localized nonbonding MO level. This has been actually confirmed by ESR analyses (Shirakawa et al., 1978) and ENDOR (Kuroda and Shirakawa, 1982) observations. The lineshape of the ESR shows the motional narrowing from a highly mobile tt electron in the (CH), chain even down to 10 K (Goldberg et al., 1979 Weinberger et al., 1980). It has been concluded that there is one unpaired electron per approximately 3000 carbon atoms. This unpaired tt electron in the trans-(CH), chain is frequently referred to as the bond alternation domain wall, phase kink, or soliton. 

The three intermediate phases II, III, and IV of thiourea are believed to be incommensurate, and the incommensurately modulated structures in phase II and rv have been analyzed [50]. Another commensurate phase which is stable between phase I and phase II in a narrow temperature region of about 2 K has been recognized many years ago [27c,/], and was refined at 170 K by Tanisaki et al. [27b]. This ninefold superstructure of thiourea is characterized by a rotation of the (NH2)2CS molecule along the c axis coupled with a displacement of the center of mass in a plane perpendicular to the axis. Around mirror planes (y = 1/4 and 3/4) the local structure is isostructural with phase I. Therefore the superstructure is constructed of alternately polarized layers which are sandwiched by domain walls (or discom-mensurations) whose local structure is that of the paraelectric room temperature phase (V) around y = 0 and 1/2. 

In references 21 and 74—77, experiments were carried out using the anisotropic dark-held microscopy (ADFM) method, which is a modihcahon of the convenhonal dark-held method wherein a narrow beam is cut out from the dark-held illuminahon cone. In references 74 and 75, the light incidence plane was normal to the domain wall planes (Figure 7.11). 

The continuous change of the bromide coverage with potential upon the transition c(.y2 X 2. 2) R45° —> c(. 2 x p)R45 can be approximated by the power law 6 = 0.122 (E - Ecf + 0.50 with p = 0.40 and Ec = 0.375 V. The measured exponent is smaller than the theoretical value p = 0.50 predicted by Pokrovsky and Talapov for the C UIC transition within a model of noninteracting domain walls [70]. The theoretical prediction is only supposed to apply close to the transition, where the domain walls are narrow relative to their separation. The experimental precision does not permit to quantitatively extract the exponent close to the transition, in which the theoretical prediction could be unambiguously tested. At higher incommensurabilities the bromide coverage is not only determined by the proximity to the C/UIC phase transition, but rather by the compressibility of the bromide adlayer, which may explain the deviations in the exponent. The width of the X-ray diffraction (XRD) peaks in the incommensurate direction scale quadratically with the wave vector component in this direction, and increase continuously by an order of magnitude as the C/UIC transition is approached from the UIC side. 

Dieguez, 0., Meyer, B. and Vanderbilt, D. Reported narrow domain walls, on the order of one to two lattice constants. Energy for 180° walls is 132 mJ/m, for 90° walls, 35 mJ/m Based on defect free single crystal model. 7, 8,9... 

The temperature dependence of remanence, and critical field reflects the temperature dependence of the anisotropy. The temperature Tk below which the anisotropy rises is often considerably smaller than Tq, being about 0.3 Tq. The difficulty of differentiating between critical fields due to metamagnetism and critical fields due to narrow domain walls was evident for a long time, as exemplified by the investigations upon the R3AI2 compounds (Barbara et al., 1968), where some compounds were considered to be metamagnetic. 

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