Energy and Structure of the Terbium Domain Wall

Abstract

The domain wall energy is calculated by the balance between exchange, magnetocrystalline anisotropy and magnetoelastic energy contributions. The described method is theoretical and is based on experimental measurements of neutron inelastic scattering. The domain wall energy is determined by both finding the minimum of energy and deriving the energy and setting it to zero. The determination was undertaken for the discrete case, and this means that the calculation was performed for each plane or atomic layer. This is in contrast with the Bloch wall, which assumes continuum mean. The energy of the Lilley domain wall was discussed. Most of the energy of the Bloch wall was comprised inside the Lilley distance (above 99.9% of the energy). Antiferromagnetic interactions strongly decreased the domain wall energy. The negative terms due to antiferromagnetism must be considered in the Hamiltonian describing the exchange energy terms. The domain wall energy and width of terbium were reassessed. The values varied between 83.7 and 95.2 Kelvin (10.3 to 11.2 ergs/cm²). The domain width was estimated to be 57 Angstroms. It was found that a significant part of the total domain wall energy was concentrated on the planes at the center of the domain wall.

1. Introduction

The published information about the terbium domain wall structure and its energy is very limited. Egami and Graham mention a domain width of 20 atomic layers and domain wall energy of 90 K for terbium [1]. More information was provided inside the unpublished PHD thesis of Egami, where the terbium domain wall energy is given with the slightly higher value of 95.2 K [2]. The question of the narrow domain wall in rare-earths such as terbium and dysprosium has been revisited several times [3,4], but the study of Egami and Graham [1] essentially remains the main reference. For example, one experimental determination of the dysprosium domain wall width in 2007 [4] was compared with the old numbers mentioned by Egami and Graham in 1971 [1].

Although many studies discussed the experimental observation of domains in terbium [5,6,7,8,9,10], the terbium domain width and terbium domain wall energy have not been accessed from these measurements. A possible exception is the PHD thesis of Morrison [11], which gives an upper bound of 120 Angstroms at 150 Kelvin for the terbium domain width, which implies 42 atomic layers of width.

The estimate of Egami and Graham for the terbium domain wall [1,2] used the data of Bjerrum Moller neutron inelastic scattering for the exchange constants [12,13]. These data are obtained from experiments [12,13]. However, different data were published later, by McKintosh [14] and by Lindgard [15]. Here, it is discussed how the values of these authors may affect the terbium domain wall energy and its structure. In neutron inelastic scattering, the wave dispersion is measured and the exchange parameters are later extracted with some theoretical model [16,17]. This means that even the same set of experimental data can produce different exchange data.

In our previous study, the domain wall structure and energy was determined using the Monte Carlo Simulated Annealing method [18]. The MCSA method finds the minimum energy by a trial and error technique. Here, two different numerical methods are applied for solving the set of equations, one by finding the energy minimum [18] and another by deriving the energy and equaling to zero [19]. The use of both methods—at the same time—is better to assure that a solution with less energy is found. In this type of problem, false energy minimums are always a problem to be avoided.

Some theoretical calculations, such as the Kooy–Uenz model [20] make a relevant assumption that the domain wall thickness is negligible. This detail will be discussed in this article. How reasonable is this hypothesis? For hard magnetic materials, with high magnetocrystalline anisotropy, it is presumable that the domain wall is thin. However, where is the spent energy concentrated? How is the energy distributed along the different planes, along a domain wall? This question has been rarely discussed. This detail of the energy distribution inside the domain wall was not addressed in our previous study about domain walls [18]. But now this will be one of main subjects of discussion.

This study directly applies to dysprosium [1,2], and indirectly to cobalt, another hexagonal metal. It can also be extended also to nickel (fcc) and iron (bcc), after properly taking into account the crystalline structure.

2. The Bloch Model

It may be noteworthy to start with the old Bloch wall model [21], Equation (1). Here, γ_Bloch denotes the Bloch domain wall energy, A_Bloch the Bloch stiffness, and K₁ the first order magnetocrystalline anisotropy constant. In Equation (2), the γ_Lilley domain wall energy is defined. From Equations (3) and (4), the Lilley domain width δ is defined [22]. To obtain a simple analytical expression for the domain wall energy, Bloch did some assumptions: Bloch only considers only the next neighbors and assumes the electron as localized, completely neglecting antiferromagnetism, which was only described later by Neel [23]. The knowledge about exchange interactions has improved significantly since the decade of 1930; for example, the exchange constants of gadolinium were recently described up to 26 next neighbors [24] from experimental neutron inelastic scattering data, where many of the interactions were found to be of antiferromagnetic nature. Bloch approximation can be summarized as [25] $\cos \theta =1-2{\sin }^2(\theta /2)\approx 1-2{(\theta /2)}^2$, which is only acceptable if $\sin \theta \approx \theta$. As a consequence of the Bloch approximation, the Bloch wall is infinite, but this is a mathematical artifact of the Bloch model, due to imposing $\theta \approx 0$. To circumvent this problem, several different arbitrary definitions for the Bloch domain wall width have been suggested [26]. The most common is by the slope of the S shape of the Bloch wall, as depicted in Figure 1. This definition could already be found in Becker and Doring’s book [27] and is named the Lilley domain wall width δ by Hubert and Schafer [26]. The energy of the Lilley domain wall is given in Equation (2). Equation (3) comes from the Bloch wall model and it is used for finding Equation (4).

If the quantity ψ = 2arctan(e^π)≈3.055219 replaces θ in Equation (3), the Lilley domain wall width is found, see Equation (4) and Figure 1. The quantity A_Bloch (see Equation (5)) described in the Bloch solution can be more adequately called Bloch stiffness, to avoid confusion with the exchange terms defined in the Hamiltonian given by Equation (6).

$${\gamma }_{Bloch}=2\sqrt{A_{Bloch}{K}_1}{\displaystyle \underset{(\pi -\psi )/2}{\overset{(\pi +\psi )/2}{\int }}\sin \hspace{0.17em}\hspace{0.17em}\theta \hspace{0.17em}\hspace{0.17em}d\theta }=4\sqrt{A_{Bloch}{K}_1} (\mathsf{\psi }=\mathsf{\pi })$$

(1)

$${\gamma }_{Lilley}=2\sqrt{A_{Bloch}{K}_1}{\displaystyle \underset{(\pi -\psi )/2}{\overset{(\pi +\psi )/2}{\int }}\sin \hspace{0.17em}\hspace{0.17em}\theta \hspace{0.17em}\hspace{0.17em}d\theta }={\displaystyle \frac{4e^{\pi }}{\sqrt{1+e^{2\pi }}}}\sqrt{A_{Bloch}{K}_1} (\mathsf{\psi }=2\arctan ({\mathrm{e}}^{\mathsf{\pi }}))$$

(2)

$$x=\sqrt{{\displaystyle \frac{A_{Bloch}}{K_1}}}\ln (\tan {\displaystyle \frac{\theta }{2}})$$

(3)

$$\delta =\pi \sqrt{{\displaystyle \frac{A_{Bloch}}{K_1}}}$$

(4)

$$A_{Bloch}={\displaystyle \frac{{\gamma }_{Bloch}^2}{16\hspace{0.17em}\hspace{0.17em}{K}_1}}$$

(5)

The energy found with ψ = 2arctan(e^π) used in the interval of integration of Equation (2) is slightly lower than that of the ${\gamma }_{Bloch}=4\sqrt{A_{Bloch}{K}_1}$ of the Bloch wall model—where ψ = π—and results in ${\gamma }_{Lilley}=3.9963\sqrt{A_{Bloch}{K}_1}$. This means that most of the exchange energy is spent inside the Lilley distance, in the Bloch wall model. The Lilley domain width δ is much smaller than the Bloch wall, because the Bloch wall is always infinite. The asymptotical behavior of the function θ(x) obtained from Equation (3) is clear in Figure 1. The cosine function is also plotted in Figure 1, which is a measure of energy, because the energy between neighbor spins is given by a scalar product. According to Figure 1, most of the domain wall energy is spent at the center of the wall and this is found even with the rough Bloch model. In the next section. the domain wall energy will be calculated in different ways, with a discrete model presented by Egami and Graham [1,2].

3. The Egami Model

For neighbor atoms with the same spin, the Heisenberg–Dirac–Van Vleck Hamiltonian [28] can be expressed by the scalar product $E=-2IS\cdot S$, which gives $E=-2I{S}^2\cos \theta$, where E is energy, θ is the angle between the neighbor spins, S is the atom spin, I is the interplanar exchange along the c axis of the HCP crystal, and A_n is the exchange constant given by $A_n=-I_n{S}^2$. Different sets of data for the exchange constants A_n =−I_nS² of Terbium are shown in

Table 1. Different sets of data for the interplanar exchange constants along the c-axis of terbium.

Exchange Term An	Moller [12,13]	Mckintosh [14]	Lindgard [15]
A1	127.4	112.4	121.6
A2	31.3	30.1	35.9
A3	2.1	−4.2	−0.4
A4	−14.6	−15.5	−20.0
A5	-	3.3	3.8

. Each A₁, A₂, A₃, A₄, A₅ refers to each −I_nS², according to Equation (6). Positive A_n means ferromagnetic interactions, whereas negative A_n implies antiferromagnetic interactions. It should be noted that, for Moller data [12,13], only the 4th term is negative, whereas for the other sets of data [14,15], the 3rd and 4th terms are negatives. The other sets of data [14,15] include a positive A₅, which is null in Moller data [12,13]. Antiferromagnetic interactions are relevant in terbium and other rare-earths [29,30,31].

$$E_{ex}=-{\displaystyle \sum _{i,k=\pm 1-n}^{ }\hspace{1em}{I}_{i,i+k}{S}_i\cdot S_{i+k}}$$

(6)

To facilitate the comparison with Egami and Graham [1,2], we maintained Kelvin as the unity of energy in Table 1. It should be noted that the energy data in Kelvin may be easily converted in other units, as eV, by taking into account the total angular momentum (spin + orbit), which is 6 for Terbium and 15/2 for Dy. It seems Egami followed Cooper [32], which expressed A₁ and A₂ of dysprosium in Kelvin.

A general expression which takes into account the exchange energy A_n with other neighbor planes up to n can be written as follows (where N is an integer number and denotes the total number of evaluated planes) by Equation (7):

$$E_{ex}=-N{\displaystyle \sum _i{\displaystyle \sum _{k=-n}^{k=+n}{A}_{i,k}\hspace{0.17em}\hspace{0.17em}\cos ({\theta }_i-{\theta }_{i+k})}}$$

(7)

Moreover, the anisotropy term E_an considers the summation of the magnetostriction and magnetocrystalline contributions, respectively, shown by Equation (8). G = K₆⁶ is used as a sixfold basal plane anisotropy constant, following Egami [2]. C is both the 1st order magnetostriction coefficient [14] and the twofold uniaxial single-ion magnetoelastic energy [1].

$$E_{an}=N{\displaystyle \sum _i\left\{C_i\hspace{0.17em}\cos (2{\theta }_i)+G_i\hspace{0.17em}\cos (6{\theta }_i)\right\}}$$

(8)

The wall energy E_wall is assumed as the summation at the equilibrium configurations of the spin orientations of both the exchange and anisotropy contributions, shown by Equation (9).

$$E_{wall}=E_{ex}+E_{an}$$

(9)

The specific modeling approach adopted for these energy contributions strongly affects the final spin equilibrium configuration and hence the prediction of the energy wall. The solution of the Hamiltonian Equation (9) provides both the spin orientation distribution and wall energy.

Egami et al. [1,2] used the following approach with the approximation considering several neighbor planes to predict the energy of the wall modeled by Equations (10) and (11).

$$E_{ex}=-{\displaystyle \sum _i{\displaystyle \sum _k^{ }{A}_{i,k}\left\{\cos ({\theta }_i-{\theta }_{i-k})+\cos ({\theta }_i-{\theta }_{i+k})-2\right\}}}$$

(10)

$$E_{an}={\displaystyle \sum _i\left\{C_i\hspace{0.17em}\hspace{0.17em}\cos (2{\theta }_i)+G_i\hspace{0.17em}\hspace{0.17em}\cos (6{\theta }_i)-1\right\}}$$

(11)

Egami et al. [1,2] proposed the solution of the torque system of each plane obtained by assuming the equilibrium system, shown by Equation (12):

$${\tau }_i=-{\displaystyle \frac{\partial E_{wall}}{\partial {\theta }_i}}\approx 0$$

(12)

This approach has the advantage of the direct solution of the equilibrium spin configuration but needs additional boundary conditions for the planes on the extreme values. These shortcomings are relevant if one uses extended plane interactions that demand additional boundary conditions values. Therefore, alternative methods that include the complete set of unknowns are preferable. This can be handled by using numerical optimization methods [33]. In the present paper, we propose the solution of the spin configuration and the energy of the planes by using two alternative and complementary approaches. This method can be named as the joint—total energy minimization and the torque equilibrium solutions—approaches.

Model Solution

The equilibrium spin configuration and their energy can be obtained by two approaches: (i) solving the total energy minimization problem with the restrictions imposed by the boundaries, (ii) solving the system equation assuming that the torque acting on the atomic plans at the equilibrium conditions are negligible. Thus, these two approaches can be summarized as follows:

Problem 1, Equation (13):

$$Minimize:\hspace{0.17em}\hspace{0.17em}{E}_{wall}=(E_{ex}+E_{an})\hspace{0.17em}=F({\theta }_i)\hspace{0.17em}with\hspace{0.17em}(0\le {\theta }_i\le \pi )$$

(13)

where the objective functions are considered depending on the model assumption for the Hamiltonian, Equation (14).

$$F({\theta }_i)=-N{\displaystyle \sum _i\left\{{\displaystyle \sum _{k=-n}^{k=+n}{A}_{i,k}\hspace{0.17em}\cos ({\theta }_i-{\theta }_{i+k})}+C_i\hspace{0.17em}\cos (2{\theta }_i)+G_i\hspace{0.17em}\cos (6{\theta }_i)\right\}}$$

(14)

Problem 2, Equation (15):

$${\tau }_i=-{\displaystyle \frac{\partial E_{wall}}{\partial {\theta }_i}}=-N{\displaystyle \sum _i{\displaystyle \sum _{k=-n}^{k=+n}2A_{i,k}\sin ({\theta }_i-{\theta }_{i+k})}}+2C_i\sin (2{\theta }_i)+6G_i\sin (6{\theta }_i)\approx 0$$

(15)

We addressed these two problem approaches by using alternative numerical methods for the minimization of the energy functions based on the quasi-Newton methods and Newton–Raphson method for solving the non-linear system of torque equations, respectively.

A general formulation for the quasi-Newton method is based on the local approximation of the gradient of the objective function. This approach was used to solve problem 1, Equations (16) and (17).

$$\nabla F({\theta }_i)={\displaystyle \frac{\partial F({\theta }_i)}{\partial {\theta }_i}}$$

(16)

$$\left[{\theta }_i^{k+1}\right]=\left[{\theta }_i^k\right]-\lambda \nabla F({\theta }_i^k)$$

(17)

Thus, the solution is searching for the optimum value of λ which satisfies the condition $F({\theta }_i^{k+1})\le F({\theta }_i^k)$ until the repeated local evaluation of the gradient approaches zero and simultaneously the decrease rate of the objective function becomes negligible. This iterative procedure is shown in Figure 2. Figure 2 shows the computational steps used to obtain the optimum solution of the spin configuration which minimize the total energy of the system. The algorithm is initialized with the material properties and the model parameters and the minimal conditions are searched for in two main steps: first, there is a local estimation of an optimal search direction, and then in the sequence, there is a search for an optimal step size. The processes are repeated until the convergence criteria are achieved. Finally, the quantities are calculated based on the optimal spin configuration.

The second problem is addressed by iterative solution of the non-linear system of equations represented by Equation (15). The iterative procedure is repeated until the orientation corrections vector, $\Delta \theta$, has become negligibly small (Equations (18)–(20)). τ is torque and J denotes the Jacobian matrix. This iterative procedure is illustrated in Figure 3.

$$\left[J({\theta }_i^k)\right]\left[\Delta {\theta }_i\right]=-\tau ({\theta }_i^k)$$

(18)

$$\left[{\theta }_i^{k+1}\right]=\left[{\theta }_i^k\right]+\left[\Delta {\theta }_i\right]$$

(19)

$$\left[J({\theta }_i^k)\right]={\displaystyle \frac{\partial }{\partial {\theta }_i}}\tau ({\theta }_i^k)$$

(20)

The algorithm is based on the main iterative procedures: (a) local numerical evaluation of the Jacobian matrix of the torque system equations using central finite difference; (b) Jacobian matrix conditioning by pivoting method; (c) solve the non-linear system of spin orientation corrections; (d) check for the convergence and repeat until the residual torque is negligible for all equations simultaneously and the spin corrections are sufficiently small. These steps are detailed in Figure 3.

4. The Terbium Domain Wall

The elastic constants used in the calculation were C = −3.94 K and the basal plane anisotropy G = K₆⁶ = −0.57 K (Rhyne and Clark, [34]). These values are for T = 4 Kelvin [2]. The obtained values using the numbers in Table 1 are presented in

Table 2. Calculated terbium domain wall energy according three different sets of data derived from experiments.

	Moller [12,13]	Mckintosh [14]	Lindgard [15]
Domain wall energy (K)	95.2	83.7	90.9

.

As discussed by Koehler [35], there are some discrepancies between the literature data concerning the magnetoelastic constant C and the magnetocrystalline anisotropy term G. For example, Birss et al. [36] later mentioned that the data of Rhyne and Clark for lower temperatures of G = K₆⁶ need revision and that the values of K₆⁶ near very low temperatures were in fact higher.

Figure 4 presents the spin profiles calculated with the data of Table 1. In spite of significant similarity between the profiles, there is a difference of up 10% concerning the calculated domain wall energy, varying between 83.7 and 95.2 K. The total sum of domain wall energy is presented in Table 2. The domain wall width is estimated in 20 atomic planes, for all three sets of data. The energy per plane calculated with the data of Table 1 is presented in Figure 5. It can be observed that most of the energy difference occurs for some specific planes, which are distant from the center of the wall.

In Figure 5, It should be noted that the energy for the planes of center is very high—for example, in only two planes it is spent above 30 K of energy, which can be compared with ~90 K of energy of the entire domain wall. This means that only two planes concentrate above one-third of the total domain wall energy of terbium.

The effect of the constants C and G is discussed using Moller data—the same previously used by Egami [1,2]. In Figure 6, it is observed that by decreasing C, the domain wall width increases. It is estimated in 26 atomic planes for C = −1.5 K. Figure 7 shows the energy per layer, and it is observed that, by increasing C, the energy of the central planes increases significantly. In Figure 8, it is observed that variation in C can significantly affect the domain wall energy.

G is almost one order of magnitude lower than C, and its effect is smaller. Some observed trends are only partially similar to G when compared with C. The data of Figure 9 suggest that G may affect the domain wall width, but only with values almost one order of magnitude higher (in modulus) than G = K₆⁶. Thus, the domain wall width is still estimated in 20 planes. Nevertheless, when it was assumed G= −3.5 K, the domain wall width is estimated in 16 or 18 atomic planes.

If G is assumed as −3.5 K, almost the same of order of magnitude of C, the energy per layer is principally distributed among the central planes, as can be noted in Figure 10. Thus, it is observed that C and G affect the energy per layer differently. This is noted when Figure 10 is compared with Figure 7. It is found that G did not affect the domain wall width (Figure 7) but significantly affected the energy by plane. In contrast, it was found that by decreasing G, the domain wall energy decreases, as shown in Figure 11.

As can be seen in Figure 12, by decreasing A₂, the domain wall energy decreases, eventually going to zero near A₂ = 2.2 K. Here, negative domain wall energy implies antiferromagnetism. This negative contribution for the energy is clearly caused by the negative A₄ term. This means that other distant next neighbors A_n cannot be neglected. More importantly, especially the negative next neighbors can not be neglected, because they significantly reduce the energy. It should be noted that A₄ is in modulus less than 10% of A₁, and even so it is able to lead to antiferromagnetism, with A₂ and A₃ being small, but positive (see Table 1 for the Moller set of data).

Figure 13 shows that the energy is spent essentially at the center of the domain wall. This supports the models where the domain wall is assumed as very thin, with almost negligible width. The effects of altering A₃ and A₄ of the Moller set of data [12,13] are presented in Figure 14, Figure 15, Figure 16 and Figure 17. The effect of varying A₄ is shown in Figure 16 and Figure 17. Even small variations in A₄ can affect significantly the domain wall energy. On the other hand, as A₃ (in modulus) is smaller than A₂ and A₄, its variation is less significant on the domain wall energy, as shown in Figure 14 and Figure 15.

Even varying A₂, A₃, and A₄, the domain wall width was found to consist of 20 planes (Figure 13, Figure 15 and Figure 17), in general agreement with Egami [1,2]. With c = 0.56966 nm for terbium, this results in 57 Angstroms of width. In phases with high magnetocrystalline anisotropy, such as terbium, which is the subject of the present study, the domain wall thickness is typically very small, of the order of a few interatomic distances. This supports assumptions of negligible domain width. Additionally, the spent domain wall energy is concentrated mostly in a few planes at the center of the wall.

5. Some Remarks

The present study illustrates how difficult it is to calculate the domain wall energy from exchange energy parameters. In practice, the typical experimental method for domain wall energy determination is by domain observation (metallography), for example by the Kerr effect using Equation (21) [37,38,39]. This can be achieved with an optical microscope with polarized light, which is a very simple and cheap experimental apparatus. In Equation (21), D is the domain width, L is the grain size, and Ms is the Magnetization of Saturation. Techniques such as neutron inelastic scattering are very difficult access. This means that it is very difficult to find experimental values for A₁, A₂, A₃, A₄, A₅ (see Table 1) and other distant exchange energy terms in the literature. A few determinations of exchange energy values by neutron inelastic scattering have been reported, as discussed by Lindgard [16]. Scheie et al. recently performed one such determination [24] for gadolinium. These experiments of neutron inelastic scattering have only been undertaken in a handful of laboratories across the world, including in Oak Ridge, Tennessee [24], Riso in Denmark [12], and Chalk River in Canada [17].

The domain wall energy γ can have two definitions: (i) by the balance between magnetostatic and exchange energies, as shown in Equation (21), and (ii) by the balance between magnetocrystaline and exchange energies, which is the case in the Bloch model.

$$\gamma ={\displaystyle \frac{0.85D^2{M}_S^2}{L}}$$

(21)

The Bloch assumption of θ~0 implies an absolute continuum mean. This is in disagreement with the real world, where the matter is divided in atoms. With the Bloch model, it is possible to calculate the energy in a given region (as, for example, the case of Lilley energy) but not the specific energy of a given plane. In contrast, the Egami discrete model enables energy prediction in specific different planes. Thus, the energy in each plane can be found. The quantity A_Bloch is not related to any A₁, A₂, A₃, A₄ of the Moller set of data [12,13]. This is a clear conclusion from Figure 12. The same conclusion also comes from Figure 14 and Figure 16. However, A_Bloch could be inferred from Equation (21), using Equation (5). This A_Bloch, found from Equation (21), is the typical exchange stiffness reported in textbooks, but there are other methods for estimating A_Bloch [8], including one based on the Curie temperature Tc, which is quite inaccurate [38]. Nevertheless, both the Bloch model and the Egami model provide an “S-shape” spin profile (see Figure 1, Figure 4, Figure 6 and Figure 9). Whereas in the Bloch model the domain wall width is in fact infinite, the Egami discrete model directly provides a domain wall width. In this sense, the discrete model is much more accurate.

According to the Egami Hamiltonian, the domain wall energy depends on a complex method of the values of all evaluated parameters, C, G, A₁, A₂, A₃… A_n, and can not be directly related to the exchange stiffness given by Equation (5), and this is in contrast with the Bloch model in Equation (1).

Finally, as the negative terms reduce the energy of the system, they need to be considered in the Hamiltonian. This was also the conclusion of our previous study on spin-glasses [40]: even distant negative exchange terms can reduce the overall energy of the system.

6. Comment on the Bloch Wall

In 1932, Felix Bloch was the first to address the domain wall problem [21]. Bloch assumed exchange interactions only with the first next neighbor and neglected antiferromagnetism. It is interesting to discuss the circumstance where Bloch used the approximation 1 − cos θ~θ²/2 for Equation (6), which results in Equation (22), a completely different expression. Instead of solving Equation (23) or Equation (24) or even Equation (25), Bloch used a trick of algebra to obtain Equation (26). However, Bloch significantly modified the more rigorous formulation Equation (6) to obtain Equation (22), giving origin to several mathematical artifacts. In 1932, analytical equations—even approximated—were the goal of many researchers because computers were not available at that time.

If exchange is assumed as a cosine function, as given in the Heisenberg–Dirac–Van Vleck model [41], Equations (24) and (25) are obtained. However, Equation (24) is much more difficult to solve than Equation (26). Although Equation (24) is difficult to solve, Equation (5) is easily obtained from Equation (26). The Bloch exchange stiffness A_Bloch can be defined in Equation (5), and it should be noted that, in this definition, distant exchange energy terms are neglected.

$$E_{ex}=A{({\displaystyle \frac{d\theta }{dx}})}^2$$

(22)

$$dx={\displaystyle \frac{d\theta }{\arccos \hspace{0.17em}\hspace{0.17em}(\frac{-K\hspace{0.17em}{\sin }^2\theta }{2A})}}$$

(23)

$${\displaystyle \int }dx={\displaystyle \int }{\displaystyle \frac{d\theta }{\arccos \hspace{0.17em}\hspace{0.17em}(\frac{-K\hspace{0.17em}{\sin }^2\theta }{2A})}}$$

(24)

$$2A\hspace{0.17em}\hspace{0.17em}\cos ({\displaystyle \frac{d\theta }{dx}})=-K\hspace{0.17em}{\sin }^2\theta$$

(25)

$$A\hspace{0.17em}{({\displaystyle \frac{d\theta }{dx}})}^2=K\hspace{0.17em}{\sin }^2\theta$$

(26)

The Bloch and Lilley models are frequently found in textbooks. The Bloch domain wall is infinite, as shown in Figure 1. However, the Bloch domain wall energy is finite, given by $\gamma =4\sqrt{AK}$. As mentioned in the beginning of this paper, to overcome the problem of the infinite domain wall, Lilley suggested that the integration should be performed between (π − λ)/2 and (π + λ)/2 [22,42]. In fact, the energy of the Lilley wall is slightly lower than that of the Bloch wall. The energy of the Lilley domain wall is $\gamma =3.9925\sqrt{AK}$. However, the Lilley domain width δ is much smaller because the Bloch wall is infinite. According to Lilley, the domain wall thickness δ is between −π/2 and π/2 of (A/K)^0.5. The asymptotical behavior of the function θ(x) obtained from Equation (3) is clear in Figure 1.

In models evaluating the reversal of magnetization, the first and second derivatives of energy are very important [43,44]. But by replacing the 1 − cos θ by θ²/2, a serious problem happens, whereby the derivatives are different; see the comparison between Equation (27) and Equation (28). This affects all models where the approximation 1 − cos θ~θ²/2 is assumed. By making θ~sinθ, the domain wall becomes infinite, which is a mathematical artifact. Curling [45] comes from Equation (22), which assumes an infinite domain wall and can not be found with Equation (6) [46].

$${\displaystyle \frac{d}{d\theta }}{\displaystyle \frac{{\theta }^2}{2}}=\theta \mathrm{and} {\displaystyle \frac{d^2}{d{\theta }^2}}{\displaystyle \frac{{\theta }^2}{2}}=1$$

(27)

$${\displaystyle \frac{d}{d\theta }}1-\cos \theta =\sin \theta \mathrm{and} {\displaystyle \frac{d^2}{d{\theta }^2}}1-\cos \theta =\cos \theta$$

(28)

If exchange is taken into account as a scalar product (cosine function), then infinities are not possible. According to Bloch approximation, 1 − cos θ ~ θ²/2. However, if θ → ∞, then θ²/2 → ∞. By another hand, 1 − cos θ varies between 0 and 2. The cosine function varies between −1 and 1 and never goes to infinite. Thus, some suggested “infinities” [47] can be attributed to mathematical artifacts (due to replacement of 1 − cos θ by θ²/2). As θ²/2 > 1 − cos θ, then Equation (26) significantly overestimates the exchange energy contribution.

Bloch assumed exchange only with the first next neighbor and that the electron was not itinerant. In contrast, as can be seen in the RKKY model [48], Equation (29), exchange interactions go up to the end of the crystal. In the RKKY function (Equation (29)), exchange energy decreases with 1/x³ instead of increasing with θ² as in Equation (22). Finally, as discussed in the present study, distant negative exchange energy terms need to be considered because they reduce the energy of the system [40,49,50].

$$F(x)=\cos (x)/x^3-\sin (x)/x^4$$

(29)

7. Conclusions

A method is presented to determine the domain wall energy by both finding the minimum energy and deriving the energy and setting it to zero. This method was used to solve the Egami energy Hamiltonian.

The energy of the Lilley domain wall was discussed. Most of the energy of the Bloch wall model is comprised inside the Lilley distance (above 99.9% of the energy).

Antiferromagnetic interactions strongly decrease the domain wall energy. Even distant terms with antiferromagnetic exchange interaction can have significant effects. Negative terms due to antiferromagnetism must be considered in the Hamiltonian describing the exchange energy terms. This means that the exchange energy cannot be truncated at the first term of the Egami Hamiltonian, as it is assumed in the Bloch model. If negative terms are neglected, the whole result is based on a false minimum.

The domain wall energy of terbium was reassessed; the values found with exchange energy data from Mackintosh [14], Lindgard [15], and Moeller [12,13] varied between 83.7 and 95.2 Kelvin (10.3 to 11.2 ergs/cm²). The domain wall thickness was estimated as 57 Angstroms. The results were in excellent agreement with Egami [1,2].

There are several conclusions from this study, and one of them is that the Bloch wall model is inaccurate, as is the Lilley definition of domain wall width. The domain width must be calculated according Equation (6), i.e.,

$$E_{ex}=-{\displaystyle \sum _{i,k=\pm 1-n}^{ }\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{I}_{i,i+k}{S}_i\cdot S_{i+k}}$$