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为什么在微磁学中，交换长度很小，往往只有10-100nm，但形成的畴往往比交换长度大很多呢？

Created on: October 17, 2025

Answered using GPT-5 Thinking by Chat01

Question

为什么在微磁学中，交换长度很小，往往只有10-100nm，但形成的畴往往比交换长度大很多呢？

Thought

Explaining micromagnetics in Chinese

We need to explain a physics concept in Chinese regarding micromagnetics, focusing on exchange length and domain size. The relationship involves exchange interaction vs anisotropy, scale of magnetization variation, and domain wall width. Domain size is influenced by magnetostatic energy, and equilibrium values are described by formulas like the Kittel law. Both exchange and magnetostatic energies contribute to determining the domain structure.

Explaining domain size and exchange length

The exchange length determines local gradients and domain wall width, not the domain size. Domains span multiple exchange lengths because the cost of creating numerous walls outweighs stray field reduction until a certain scale is reached. Domain size results from a balance of wall energy and magnetostatic forces.

Clarifying domain wall and period size

The width of the domain wall, δ, is proportional to the square root of the anisotropy to exchange energy ratio. The domain period connects to the wall width via a constant. For stripe ferromagnets, the domain period follows Kittel's law. There's a relationship between wall energy and magnetic properties affecting width.

Explaining domain width in thin films

For thin films with up/down stripes, the domain width ww is proportional to the square root of the product of wall energy and film thickness, divided by magnetization squared. With specific material properties (like the exchange constant AA and anisotropy constant KK), domain size increases significantly compared to exchange length.

Thought for 34s

Answer

简要说：交换长度 lexl_{\rm ex} 决定的是磁化在本地能多快变化（比如畴壁、涡核的尺度），而畴的尺寸是由远程的磁偶极（退磁）能和畴壁能之间的竞争决定的，是一个全局最优化结果。两者物理角色不同，所以“畴往往比交换长度大很多”并不矛盾。

关键尺度各管各的

  • 交换长度（常见两种定义） lK=AK,ld=2Aμ0Ms2l_K=\sqrt{\frac{A}{K}},\qquad l_d=\sqrt{\frac{2A}{\mu_0 M_s^2}} 它给出磁化允许的“曲率半径”。典型取值 3–100 nm，直接控制畴壁宽度 δ∼πlK\delta\sim \pi l_K（几十纳米量级）和涡核半径（∼ld\sim l_d）。
  • 畴宽/畴周期来自把“多建几堵墙”的代价与“减少远场磁荷”的收益平衡：
    • 每增加单位面积的畴壁要付出畴壁能密度 σw≈4AK\sigma_w\approx 4\sqrt{AK}；
    • 但能降低与体积相关的退磁能（量级 ∼μ0Ms2\sim \mu_0 M_s^2）。

在许多薄膜/条纹畴的模型（Kittel 型估算）中，最优畴宽 ww 近似为

w  ∼  σw tμ0Ms2    ≈    C t δ(C∼若干的常数)w \;\sim\; \sqrt{\frac{\sigma_w\, t}{\mu_0 M_s^2}} \;\;\approx\;\; C\,\sqrt{t\,\delta}\quad (C\sim \text{若干的常数})

其中 tt 是样品厚度、δ∼πA/K\delta\sim\pi\sqrt{A/K} 是畴壁宽度。
要点：畴宽随着厚度 tt 的平方根增大，因此很容易做到远大于 lexl_{\rm ex}。

直观图像

  • 交换能偏好“尽量平滑/一致”的磁化（抑制急剧转折），因此只在畴壁附近发挥主导作用 → 设定“墙有多厚”（∼lex\sim l_{\rm ex}）。
  • 退磁能是长程相互作用，它驱动系统把磁通闭合起来、减少表面磁荷 → 需要分畴。但每多一个畴就多一堵墙，墙多了又贵。
  • 两者拉扯的结果：建少量的、相对宽大的畴更划算；墙保持只有几十纳米厚，但畴本身可以到几百纳米、微米，远超 lexl_{\rm ex}。

一个数值感觉

取 A=10−11 J/mA=10^{-11}\,\text{J/m}, K=105 J/m3K=10^{5}\,\text{J/m}^3（⇒δ≈πA/K≈30 nm\Rightarrow \delta\approx \pi\sqrt{A/K}\approx 30\,\text{nm}），薄膜厚度 t=100 nmt=100\,\text{nm}。
则 w≈Ctδ∼(2 ⁣− ⁣5)×100×30 nm≈100 ⁣− ⁣300 nmw\approx C\sqrt{t\delta}\sim (2\!-\!5)\times \sqrt{100\times 30}\,{\rm nm}\approx 100\!-\!300\,\text{nm}。
你会看到：畴壁仍是几十纳米，但畴宽已是数百纳米——显著大于交换长度。

结论：交换长度控制的是“磁化能拐多快”（局部结构，如畴壁宽度），而畴大小来自与样品尺寸相关的远程退磁能与畴壁能的折中，因此通常远大于交换长度。

Question

那我能不能得出一个结论，如果这个体系没有各向异性，或者很弱，那么这个体系的畴可能只有交换长度那么长了

Thought

Explaining anisotropies and domain walls

In zero anisotropy, exchange length l_K diverges, but other lengths, like l_d, remain finite. Even without anisotropy, uniform magnetization still leads to domain formation. Domain walls become diffuse, with energy density approaching zero, but their width grows infinitely. However, demagnetizing energy still influences domain structure at typical lengths of 5-10 nm, balancing exchange and dipolar effects.

Calculating exchange length

With a typical Ms around 1e6 A/m, the exchange length l_d computes to about 4 nm. This sets the vortex/core size and Néel caps, but not domain size, which depends on sample geometry and competing energy considerations. As anisotropy K approaches zero, wall energy diminishes, but exchange effects still play a role.

Explaining the ground state with K=0

When K=0, exchange alone can't form well-defined walls; instead, the magnetization spreads, reducing energy density. The result is a vortex or flux closure, minimizing magnetostatic energy. Domain sizes depend on sample geometry — thinner films create in-plane magnetization due to shape anisotropy, influencing domain structures, not just exchange length.

Considering domain wall behavior with weak K

When crystalline anisotropy K approaches zero, the exchange length l_K grows large, affecting domain wall width. As K weakens, domain walls become broader, leading to an increase in domain sizes. However, if K is very small, wall definition blurs, and domain sizes may increase due to weaker wall energy penalties.

Clarifying domain wall width formulas

There’s confusion about the dependence of domain wall width w on anisotropy K. Using Kittel's law, w should increase with K, meaning w ∝ K^{1/4}. However, there's a contradiction with another formula, where w seems to decrease with K. The δ used in Kittel's law should not be the Bloch wall width, but an effective exchange length.

Clarifying domain behavior with weak anisotropy

When anisotropy is weak or absent, domain sizes do not shrink to the exchange length. In fact, the ground state may exhibit magnetization curling or closure domains, where the domain size can be influenced by sample geometry and other factors, not just the exchange length. Domains are often single-domain in-plane when shape anisotropy dominates.

Thought for 1m 43s

Answer

不太对。**“各向异性弱/没有 ⇒ 畴只有交换长度那么大”**这个推论一般是错的。

要点速览

  • “交换长度”有两种常用的：
    lK=A/Kl_K=\sqrt{A/K}（跟晶体各向异性 KK 相关）和 ld=2A/(μ0Ms2)l_d=\sqrt{2A/(\mu_0 M_s^2)}（跟磁偶极相互作用相关）。
    它们控制局部结构（畴壁厚度、涡核大小），而不是整块畴的宽度。
  • 若晶体各向异性 K→0K\to 0：
    • lK→∞l_K\to\infty：畴壁会变得非常宽，而不是“只有交换长度那么窄”。
    • 畴壁能密度 σw≈4AK→0\sigma_w\approx 4\sqrt{AK}\to 0：建墙几乎不“花钱”，经典的“薄而清晰的墙 + 条纹畴”的近似失效，磁化倾向于连续弯曲的闭合磁路（vortex/closure），而不是被限制在 lexl_{\rm ex} 的尺度上。
  • 真正决定是否分畴以及畴多大的是退磁（形状）能 vs. 畴壁/交换代价的全局平衡，典型尺度依赖样品几何尺寸，尤其是厚度 tt。

分三种常见情形看

  1. 薄膜且没有垂直易轴（K≈0K\approx 0，Q 因子很小）
    定义 Q=2K/(μ0Ms2)Q=2K/(\mu_0 M_s^2)。当 Q≪1Q\ll 1 时，所谓形状各向异性（退磁场）强烈偏好面内磁化。面内单畴已经几乎没有磁荷 → 不需要分畴，更谈不上“只有交换长度那么大”。很多软磁薄膜（Permalloy 等）正是这种单畴或大尺度闭合态。

  2. 有（哪怕很弱的）垂直易轴 & 薄膜足够厚，会出现条纹畴
    在“壁可视为相对清晰”的范围内，Kittel 型估算给

    w  ∼  σwKd t ,Kd=μ0Ms22,  σw≈4AK.w \;\sim\; \sqrt{\frac{\sigma_w}{K_d}\,t}\,,\qquad K_d=\frac{\mu_0 M_s^2}{2},\ \ \sigma_w\approx 4\sqrt{AK}.

    因此 w∝(AK)1/4tw \propto (AK)^{1/4}\sqrt{t}。各向异性越弱（K↓K\downarrow），按这个模型反而给出更窄的条纹。但当 KK 小到使畴壁宽度 δ∼πA/K\delta\sim\pi\sqrt{A/K} 与 ww 同量级时，“清晰畴壁”近似崩溃，体系会转向连续闭合纹理，尺度随 tt 与几何走，而不是被 lexl_{\rm ex} 钉死。

  3. 微/纳米尺度器件里的涡旋与闭合畴
    在软磁微盘、纳米条里常见涡旋/漩涡闭合态：

    • 涡核半径 ∼ld=2A/(μ0Ms2)\sim l_d=\sqrt{2A/(\mu_0 M_s^2)}（几个到十来纳米，确实是“交换长度级别”）；
    • 但整个涡旋/闭合畴的尺寸由器件几何决定（直径、厚度），往往是百纳米到微米，远大于任何 lexl_{\rm ex}。

小结（直接回答你的结论）

  • 不能得出“没有/很弱各向异性 ⇒ 畴只有交换长度那么长”的结论。
  • 没有各向异性时：lKl_K 不是变小而是发散；体系通常走向面内单畴或大尺度闭合磁路。
  • 只有畴壁厚度与涡核半径这类局部结构才会落在交换长度量级；畴的宽度/周期主要由退磁能与样品几何设定，常常远大于交换长度，或者在软磁薄膜里干脆不需要分畴。

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