Addendum to “Phenomenological Theory of Multiple Spin Density Waves in fcc Transition Metals”
Addendum to “Phenomenological Theory of Multiple Spin Density Waves in fcc Transition Metals”
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“FCC 过渡金属中多重自旋密度波的唯象理论”附录
DOI：
10.1143/jpsj.76.087001
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发表时间：
2007
期刊：
影响因子：
--
通讯作者：
Y. Kakehashi
中科院分区：
文献类型：
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作者：
T. Uchida;Y. Kakehashi
在我们最近的论文中，1）我们在Ginzburg-Landau（GL）型自由能的基础上，研究了面心立方过渡金属中各种自旋密度波（SDW）结构的相对稳定性。我们证明了当存在3Q态的解时，3Q态与相应的2 Q态和1 Q态相比总是稳定的。这种关系导致磁相图，表明具有线性和螺旋极化的相称3 Q态、不相称3 Q多自旋密度波（MSDW）态的可能性。我们建议，这些3Q态是可能的基础上，与以前的基态计算的变化体积和实验结果的Fe沉淀在Cu。图1和图2所示的磁相图。1-4在文献1中，也给出了2 Q和1 Q态，但对它们的稳定性讨论不多。然而，由于四阶GL理论的局限性，这些状态的稳定性应仔细解释。下面我们就其稳定性提出一些意见。图1示出了对应于参考文献1中图4的无公度SDW结构的磁相图，其借助于线性极化的1 Q、2 Q和3Q态的局部稳定性条件来确定（以下称为1 Q、2 Q和3Q状态）和具有螺旋偏振的那些（以下简称1 QH、2 QH和3QH态），并在四阶GL理论的基础上对这些态进行了简单的能量比较。因此，例如，图1中的2 QH相应被解释为2 QH、1 QH、2 Q、1 Q和P（顺磁）状态中2 QH状态稳定的区域。在这里，重要的是要注意，2 QH相作为亚稳态存在于目前的四阶GL理论中。为了理解这一点，考虑图1中彼此相邻的一对2 QH和3QH相。如果我们从3QH相开始，磁矩振幅M3 QH <$1/2 <$3AQ =<$B1Q <$B1QQ <$B2QQH <$1 = 2在2 QH-3QH边界B1 QQ = B1 Q <$1/2处发散到无穷大（所用符号与参考文献1中的符号相同）。注意，在图1中考虑了B2 QQH = B1 Q 1/4 1的情况。这意味着，在2 QH-3QH边界附近和之外，基态不能在四阶GL理论中确定，因此2 QH相是亚稳态。类似地，所有的2 Q和2 QH态，以及部分的1 Q、1 QH、3Q和3QH态都是亚稳态。为了更清楚地说明这一点，我们在图1中用括号表示了这种亚稳态，并将不稳定线附近的区域表示为灰色区域。同样的评论适用于图1A和图1B。1-3在参考文献1中，因此，在图1中用四阶GL理论确定的稳定磁相是3Q、3QH、1 Q和1 QH相。这使我们得出了参考文献1的主要结论，即存在线性和螺旋3 Q MSDW态变得稳定的区域，并且在这些区域中3 Q态的能量总是低于相应的2 Q和1 Q态。
In our recent paper, 1) we have investigated the relative stability among various spin density wave (SDW) structures in fcc transition metals on the basis of the Ginzburg–Landau (GL) type of free energy with the terms up to the fourth order in magnetic moments. We have shown that the 3Q state is always stable compared with the corresponding 2Q and 1Q states when there is a solution of the 3Q state. This relation leads to magnetic phase diagrams that indicate the possibility of the commensurate 3Q state, incommensurate 3Q multiple spin density wave (MSDW) states with linear and helical polarizations. We have suggested that these 3Q states are possible in-Fe on the basis of the comparison with the previous ground-state calculations varying volume and the experimental results for-Fe precipitates in Cu. The magnetic phase diagrams presented in Figs. 1–4 in ref. 1 display also the 2Q and 1Q states, the stability of which was not discussed much in the paper. The stability of these states, however, should be interpreted carefully because of the limitation of the fourth-order GL theory. Below we note some comments on their stability. Figure 1 shows a magnetic phase diagram for incommensurate SDW structures corresponding to Fig. 4 in ref. 1, which was determined by means of the local stability conditions of the 1Q, 2Q, and 3Q states with linear polarization (hereafter referred to as 1Q, 2Q, and 3Q states) and those with helical polarization (hereafter referred to as 1QH, 2QH, and 3QH states), and by means of the simple energy comparison among these states on the basis of the fourth-order GL theory. Accordingly, the 2QH phase in Fig. 1, for example, should be interpreted as the region in which the 2QH state is stable among the 2QH, 1QH, 2Q, 1Q, and P (paramagnetic) states. Here, it is important to notice that the 2QH phase exists as a metastable state in the present fourth-order GL theory. In order to understand this, consider a pair of 2QH and 3QH phases adjacent to each other in Fig. 1. If we start from the 3QH phase the magnetic moment amplitude M3QH ¼ ½À3AQ= ðB1Q þ B1QQ þ B2QQHÞ 1= 2 diverges to infinity at the 2QH–3QH boundary B1QQ= B1Q ¼ À2 (The notations used are the same as those in ref. 1. Note that the case B2QQH= B1Q ¼ 1 is considered in Fig. 1). This means that in the vicinity of and beyond the 2QH–3QH boundary, the ground state cannot be determined within the fourth-order GL theory, thus the 2QH phase is a metastable state. Similarly, all the 2Q and 2QH states, and parts of the 1Q, 1QH, 3Q, and 3QH states are metastable. In order to make clearer this point, we have indicated such metastable states in Fig. 1 with parentheses, and represented the regions in the vicinity of the instability lines as gray zones. The same remarks apply to Figs. 1–3 in ref. 1, Figs. 1 and 2 in ref. 2, and Fig. 1 in ref. 3.In consequence, the stable magnetic phases in Fig. 1 determined within the fourth-order GL theory are the 3Q, 3QH, 1Q, and 1QH phases. This leads us to the main conclusion of ref. 1 that there are regions in which the linear and helical 3Q MSDW states become stable, and in such regions the 3Q state is always lower than the corresponding 2Q and 1Q states in energy.