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Journal of Tsinghua University(Science and Technology) >

2025 , Vol. 65 >Issue 5: 983 - 991

DOI: https://doi.org/10.16511/j.cnki.qhdxxb.2024.21.026

Electrical Engineering

Analysis and measurement of dimensional changes in the bronze-processed Nb3Sn conductor during heat treatment

  • Weican HUANG 1 ,
  • Xiaohua JIANG 1 ,
  • Guolin CHAI 1 ,
  • Ye LI , 2, *
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  • 1. Department of Electrical Engineering, Tsinghua University, Beijing 100084, China
  • 2. Shenzhen Institutes of Advanced Technology, Chinese Academy of Sciences, Shenzhen 518055, China

Received date: 2024-06-05

  Online published: 2025-04-15

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Abstract

Objective: The dimensional expansion of bronze-processed Nb3Sn wires during heat treatment presents difficulties in the design and fabrication of high-field superconducting magnets, especially for ultrahigh-field magnetic resonance imaging (MRI) magnets at 14 T and above. Thus, the characteristics of the dimensional changes in the bronze-processed Nb3Sn wires during heat treatment must be determined. This study considered bronze-processed Nb3Sn wires reinforced with NbTi-CuNi in a 14 T animal MRI magnet as an example to analyze its dimensional change during heat treatment. The dimensional change rates of the Nb3Sn wire were used as a basis to discuss the influence of the dimensional expansion of the Nb3Sn coil during heat treatment on the electromagnetic properties of the 14 T animal MRI magnet. Methods: The volume ratio of each component in the Nb3Sn wire was analyzed before and after heat treatment. Based on the material properties of each component of the Nb3Sn wire, the rate of change in length during the heating and cooling stages of heat treatment was calculated using an established finite element model. The sum represents the rate of length change during heat treatment. Based on the phase transformation mechanism of Nb3Sn wires during heat treatment, the ratio of voids in the Nb3Sn wire during heat treatment was calculated, and that of each component of the wire was added to determine the range of changes in the cross-sectional area of the Nb3Sn wire during heat treatment. An originally designed experimental apparatus was built to measure the change rate of the circumference of the Nb3Sn single-layer solenoid coil during heat treatment. The measurement results for the single-layer solenoid coil and straight wire were compared. In addition, the measurement findings were compared with the values obtained through calculation. Based on the average dimensional change rates of Nb3Sn wires, we calculated the dimensional expansion of the Nb3Sn coil in a 14 T animal MRI magnet during heat treatment and determined the magnetic field homogeneity and radial Lorentz force along the eccentric direction after heat treatment. Results: The calculation results indicate that the wire had a length change rate of 0.5% during heat treatment and a cross-sectional area change ranging between 0%-4.7%. According to the measurement results, the length change rates of the single-layer coil and straight wire during heat treatment were 0.55% and 0.52%, respectively, whereas the cross-sectional area change rates of the wires were 1.98% and 2.22%, respectively. The expansion of the inner diameter, outer diameter, and axial length of the Nb3Sn coil in the 14 T animal MRI magnet during heat treatment reached 1.19, 2.04, and 7.44 mm, respectively. The magnetic field homogeneity of the magnet changed from 1.1×10-6@6 cm DSV to 31×10-6@6 cm DSV and 45×10-6@6 cm DSV in the cases of zero radial eccentricity and a radial eccentricity of 0.595 mm in the Nb3Sn coil. In addition, the radial eccentricity of the Nb3Sn coil will produce a 1.9×104 N Lorentz force along the eccentricity direction. Conclusions: As the dimensional change rates of the single-layer coil and straight wire during heat treatment were the same, the effect of coil winding on the dimensional change of the Nb3Sn wires during heat treatment was negligible. Moreover, the measured change rate of the wire length was similar to the calculation result, and the measurement finding of the change rate of the wire cross-sectional area fell within the calculation range.

Key words： bronze-processed Nb3Sn; heat treatment expansion; superconducting magnets; magnetic resonance imaging

Cite this article

Weican HUANG , Xiaohua JIANG , Guolin CHAI , Ye LI . Analysis and measurement of dimensional changes in the bronze-processed Nb3Sn conductor during heat treatment[J]. Journal of Tsinghua University(Science and Technology), 2025 , 65(5) : 983 -991 . DOI: 10.16511/j.cnki.qhdxxb.2024.21.026

随着大规模应用的超导磁体不断向更高场强的方向发展，低温超导体中技术相对成熟的NbTi导线已逐渐无法满足需求。14 T及以上场强的磁体通常需使用具有更高临界磁场的超导材料如Nb3Sn[1-2]。Nb3Sn导线通常采用3种制造工艺：青铜法(bronze)、内锡法(internal tin, IT)和粉末套管法(powder in tube, PIT)。其中，青铜法Nb3Sn导线的丝径最小，具有较好的稳定性，在磁共振成像、核磁共振及热核聚变磁体等领域得到广泛应用[3-5]。
Nb3Sn是一种脆性化合物，由Nb和Sn在650~800℃的高温下反应生成。在制作Nb3Sn线圈时，为避免Nb3Sn损坏，通常采用先绕制后进行热处理的工艺[6]，即先使用由Nb丝、Sn(或青铜)和铜构成的具有较好韧性的Nb3Sn“生”线来绕制线圈，然后对线圈进行热处理，使Nb3Sn“生”线在高温下发生相变，生成具有超导性能的Nb3Sn线圈。经过热处理，Nb3Sn导线会出现2%~4%的体积膨胀[7-8]，使热处理后Nb3Sn线圈的尺寸和位置与热处理前相比产生较大偏差。对于磁共振成像和核磁共振磁体而言，这会导致磁体的磁场均匀度严重恶化及Nb3Sn线圈的应力和应变难以准确计算等问题[9-10]。
青铜法Nb3Sn导线的热处理过程主要可分为3个阶段：首先是由室温到热处理温度的升温阶段，接着是在热处理温度下的相变阶段，最后是从热处理温度到室温的降温阶段。文[11]测量了青铜法和RRP法Nb3Sn导线在热处理相变阶段的长度变化，结果显示，相变阶段长度变化小于0.02%。此外，相变阶段前后Nb3Sn导线中的Nb和Sn变成Nb3Sn，材料组分发生了改变。文[7, 12-13]认为Nb3Sn导线热处理后长度发生变化主要是由于热处理升温阶段导线的膨胀率和降温阶段不一样。针对Nb3Sn导线建立有限元仿真模型，可以研究Nb3Sn导线在升温阶段和降温阶段的膨胀率[12]。在相变阶段，导线中的Nb和Sn通过空位扩散朝对方移动，在接触面反应生成Nb3Sn。Nb和Sn原子的移动伴随着空位朝相反方向的移动，由于Sn的扩散速度高于Nb的，将产生朝着青铜方向的净空位移动，进而在青铜一侧生成Kirkendall空隙[14]，空隙的存在导致Nb3Sn导线体积膨胀，膨胀率取决于Sn和Nb原子在Nb3Sn层中的相对扩散速度[15-17]。
文[18]测量了不施加外力下1种青铜法Nb3Sn圆导线热处理后的直径和长度变化率，结果显示，直径变化率为0.8%左右，而长度变化率约为0.33%。文[11]测量了不施加外力下青铜法和内锡法Nb3Sn圆导线热处理后的直径和长度变化率，结果显示，不同的Nb3Sn导线制作工艺会影响尺寸的变化率。
为得到一个14 T动物MRI磁体中NbTi-CuNi加强型青铜法Nb3Sn导线热处理后尺寸变化的规律，本文首先分析了NbTi-CuNi加强型青铜法Nb3Sn导线在热处理前后各组分的体积分数，根据热处理升温和降温阶段Nb3Sn导线中各组分材料参数，建立有限元模型计算导线在热处理升温和降温阶段的长度变化率，进而得到其热处理后的长度变化率；然后根据热处理相变时Sn与Nb原子在Nb3Sn层中的扩散机理，计算相变后Nb3Sn导线中空隙的体积分数，将Nb3Sn导线中各组分体积分数相加，得到Nb3Sn导线热处理相变后横截面积变化的范围。为测量热处理后Nb3Sn线圈的尺寸变化，设计并制作了一套测量Nb3Sn单层螺线管线圈尺寸变化率的实验装置，将使用NbTi-CuNi加强型青铜法Nb3Sn导线绕制的单层螺线管线圈和直导线的测量结果进行对比，评估线圈绕制对Nb3Sn导线热处理后尺寸变化的影响，并将测量结果与理论预测结果进行对比。最后，根据NbTi-CuNi加强型青铜法Nb3Sn导线热处理后的平均尺寸变化率，计算了14 T动物MRI磁体中Nb3Sn线圈热处理后的尺寸膨胀量，讨论了热处理导致的Nb3Sn线圈尺寸膨胀对磁体电磁设计的影响。

1 Nb3Sn导线热处理后尺寸变化分析

1.1 组分变化

本文所使用的Nb3Sn导线为日本古河(Furukawa)生产的NbTi-CuNi加强型青铜法Nb3Sn导线。未热处理的NbTi-CuNi加强型青铜法Nb3Sn导线由Nb丝、Nb阻隔层、青铜(Sn质量分数14.2%)、铜和加强材料NbTi-CuNi合金组成[19]，横截面如图 1a所示。古河推荐的NbTi-CuNi加强型青铜法Nb3Sn导线的热处理过程的温度变化曲线如图 2所示。在670℃的热处理环境下，Nb丝与青铜中的Sn反应生成Nb3Sn，青铜中Sn的质量分数由14.2%[19]降至2.5%左右[6, 12]，同时NbTi-CuNi合金中的Cu与Ti反应生成CuTi化合物，Ni与Ti反应生成NiTi化合物[20]。热处理过程中，青铜的金属相始终为α相[6]。热处理后的Nb3Sn导线由Nb3Sn、Nb丝、Nb阻隔层、青铜(Sn质量分数2.5%)、铜和CuTi-NiTi等组成，横截面如图 1b所示。
图 1 NbTi-CuNi加强型青铜法Nb3Sn导线横截面示意图
图 2 NbTi-CuNi加强型青铜法Nb3Sn导线的热处理过程
本文认为热处理相变前后Nb3Sn导线中Sn原子的质量不变，可通过热处理相变前后青铜中Sn的质量分数变化计算得到热处理后Nb3Sn、Nb和青铜的质量；根据Nb3Sn、Nb及青铜的密度，则可得到其体积。假设NbTi-CuNi在670℃下反应生成CuTi-NiTi等后体积不变，表 1为热处理前后各组分体积相对热处理前Nb3Sn导线体积的分数。将表 1中热处理相变后各组分的体积分数相加，其结果为0.983，这主要是由于Nb3Sn的密度大于Nb和Sn的加权平均密度。
表 1 热处理相变前后Nb3Sn导线各组分体积分数
材料 热处理相变前 热处理相变后
Nb3Sn 0 0.175
Nb 0.2 0.072
青铜(Sn质量分数14.2%) 0.46 0
青铜(Sn质量分数2.5%) 0 0.396
铜 0.21 0.21
NbTi-CuNi 0.13 0
CuTi-NiTi等 0 0.13

1.2 长度变化

针对热处理升温和降温过程的Nb3Sn导线建立有限元模型以计算其升降温过程的长度变化，Nb3Sn导线的结构模型如图 3所示。模型中各组分分数如表 1所示。由于Nb3Sn和加强材料NbTi-CuNi合金、CuTi-NiTi化合物的强度较高，因此可认为其弹性模量不变[21]。同时，由于Nb、青铜和铜在热处理过程中的应力-应变特性呈非线性，采用不同温度下的应力-应变曲线来描述其机械特性。根据文献[12, 20, 22-24]，可得到热处理相变前后Nb3Sn导线中各组分材料参数如表 2所示。根据文[12, 21]，Nb、青铜和铜的应力-应变曲线如图 4所示。
图 3 Nb3Sn导线结构模型
表 2 热处理相变前后Nb3Sn导线各组分材料参数
材料 弹性模量/GPa Poisson比 平均热膨胀系数/K-1
Nb3Sn 165 0.358 8.23×10-6
Nb — 0.38 7.74×10-6
青铜(Sn质量分数14.2%) — 0.33 19.69×10-6
青铜(Sn质量分数2.5%) — 0.33 19.30×10-6
铜 — 0.33 19.22×10-6
NbTi-CuNi 50 0.33 18.80×10-6
CuTi-NiTi等 50 0.33 9.50×10-6
图 4 Nb、青铜和铜在不同温度下的应力-应变曲线
由于青铜和铜的再结晶温度约为200℃[25]，在升温和降温过程中也考虑蠕变导致的应力释放[26]。青铜和铜在高温下蠕变导致的应变速率由Norton-Arrhenius方程进行描述[26]：
$\begin{equation*}\dot{\varepsilon}=\gamma \sigma^{\beta} \exp \left(-\frac{Q}{R T}\right) . \end{equation*}$
其中：γ和β为常数，σ为应力(MPa)，Q为蠕变激活能(kJ·mol-1)，T为温度(K)，R为通用气体常数8.314 J·mol-1·K-1。对于青铜和铜，γ=38.8 MPa-β，β=4.8，Q=197 kJ·mol-1[27]。
在青铜法Nb3Sn导线的制作过程中，铜和青铜在每次冷拉拔后均进行退火处理，因此认为热处理前Nb3Sn导线无内部应力[11]。在670℃的相变过程中，Nb转变为Nb3Sn且青铜中Sn的质量分数改变。假设在670℃的相变过程中由升温导致的所有组分的内应力都得到释放，重新回到应变为0的状态[12]。则对于降温阶段的Nb3Sn结构模型，各组分在670℃下的热应变均为0。
根据图 2的温度曲线对Nb3Sn结构模型施加升温和降温过程的温度载荷，计算其长度变化率，可得升温和降温过程的长度变化率分别为1.16%和-0.66%。根据文[7, 11-13]，认为相变过程中Nb3Sn导线的长度几乎不变，热处理后导线的长度变化率约为升温过程和降温过程的长度变化率之和0.5%。由于Nb3Sn导线在升温过程和降温过程中的体积变化率之和较小，在本文中不予考虑。

1.3 横截面积变化

在670℃相变时，Nb3Sn导线中的Sn和Nb原子均采用空位扩散方式进行迁移[17]。假设相变时Sn和Nb原子在Nb3Sn层中的平均扩散速度分别为fSn和fNb(扩散速度定义为单位时间内穿过Nb3Sn层的原子数量)，Sn和Nb原子的原子体积分别为VSn和VNb，相变过程中参与反应的Sn原子的总体积为V0，相变总时长为t。由于采用空位扩散的方式，当Sn或Nb原子朝一个方向扩散时皆有同体积的空位往相反方向移动，如图 5所示。Nb3Sn可在Nb3Sn与Nb的交界面生成，也可在Nb3Sn与青铜的交界面生成[16]，前一部分Nb3Sn的生成速率主要取决于穿越Nb3Sn层的Sn原子的扩散速度，后一部分Nb3Sn的生成速率主要取决于穿越Nb3Sn层的Nb原子的扩散速度。根据相变过程中Sn原子的体积守恒，有
$\begin{equation*}\left(f_{\mathrm{Sn}} V_{\mathrm{Sn}}+\frac{f_{\mathrm{Nb}} V_{\mathrm{Sn}}}{3}\right) t=V_{0} . \end{equation*}$
图 5 青铜法Nb3Sn导线中原子及空位扩散示意图
假设fSn和fNb之间的比值为kf，VSn和VNb之间的比值为kv，由于Sn的扩散速度大于Nb的[28]，根据Kirkendall效应[17]，相变后在青铜一侧将出现空隙，空隙的体积Vvoid为Sn原子与Nb原子扩散导致的空位体积之差[15-16]，即
$\begin{equation*}V_{\text {void }}=\left(f_{\mathrm{Sn}} V_{\mathrm{Sn}}-f_{\mathrm{Nb}} V_{\mathrm{Nb}}\right) t=\frac{k_{\mathrm{f}} k_{\mathrm{v}}-1}{k_{\mathrm{v}}\left(k_{\mathrm{f}}+1 / 3\right)} V_{0} . \end{equation*}$
当确定参与反应的Sn原子的体积分数后，即可通过Sn和Nb原子间扩散速度的比值以及原子体积的比值来计算反应后空隙的体积分数。根据Sn和Nb的相对原子质量及密度，可得kv=1.51，由表 1可得相变过程中参与反应的Sn原子所占的体积分数为6.4%。所以，相变后空隙体积分数和Sn与Nb原子间扩散速度的比值的关系曲线可由式(3)计算得到，如图 6所示。根据图 6，若Sn与Nb原子的扩散速度相等，则空隙的体积分数约为1.6%；若Sn的扩散速度远大于Nb，则空隙体积分数接近于由于相变而在青铜中损失的Sn原子的体积分数6.4%。由于Sn的扩散速度大于Nb，因此热处理相变后Nb3Sn导线中空隙的体积分数在1.7%~6.4%之间。将表 1中热处理相变后各组分的体积分数与空隙的体积分数相加，即可得到热处理相变后NbTi-CuNi加强型青铜法Nb3Sn导线体积的膨胀率为0~4.7%。由于Nb3Sn导线热处理后长度发生变化主要是由于热处理升温阶段导线的膨胀率和降温阶段不一样，相变时Nb3Sn导线的长度几乎不变[7, 11-13]，相变导致的体积膨胀基本体现为横截面积膨胀，即横截面积的膨胀率在0~4.7%范围内。
图 6 热处理相变后Nb3Sn导线中空隙体积分数和Sn与Nb原子间的扩散速度比值的关系曲线

2 Nb3Sn螺线管线圈热处理后的尺寸测量

2.1 线圈周长测量装置

为测量热处理后Nb3Sn线圈的尺寸，设计单层螺线管线圈周长测量装置如图 7所示，包括线圈骨架、压力传感器安装条、卡盘和弧形承压板等。
图 7 单层线圈周长测量装置的剖视图
单层线圈周长测量原理如图 8所示，其中，红线部分为单层线圈，灰色部分为304不锈钢线圈骨架，蓝色部分为弧形承压板。承压板沿线圈骨架周向对称布置了3块。承压板可与线圈骨架分离，以便在热处理后撑紧线圈，进而测量线圈的周长膨胀量。3块承压板相对骨架在径向上凸出部分的高度一致，称为承压板的凸起高度，当承压板凸起高度为0时，承压板底部与骨架切口底部贴合，承压板底部与骨架切口底部间的距离h2近似与热处理前承压板凸起高度h1相等(两者之间的相对误差不超过1%)。
图 8 单层线圈周长测量原理图
设承压板的宽度为W，线圈的周向张力为Fw，承压板的支撑力为FN，此时线圈的周长可表示为
$L_{\mathrm{w}}=3\left[\left(R+h_{1}\right)(\theta+2 \sin \alpha)+R\left(\frac{2 \mathsf{π} }{3}-\theta-2 \alpha\right)\right].$
其中：R为线圈骨架半径、α为线圈与承压板切向间的夹角，θ为弧形承压板相对线圈骨架的圆心角，且
$\begin{align*}\cos \alpha & =\frac{R}{R+h_{1}}, \end{align*}$
$\sin \frac{\theta}{2}=\frac{W}{2\left(R+h_1\right)}.$
考虑超导线在Fw作用下产生了相应的形变，那么，线圈在没有Fw作用时的周长则为
$\begin{equation*}L_{\mathrm{w} 0}=L_{\mathrm{w}}\left(1-\frac{F_{\mathrm{w}}}{S G}\right) . \end{equation*}$
其中：S为单层线圈导线的总截面积，G为导线沿长度方向的弹性模量。
经过热处理，Nb3Sn线圈周向膨胀，施加在线圈上的Fw消失。热处理后，在承压板下方设置压力传感器，然后操作卡爪顶起承压板进而撑紧线圈。通过传感器测量承压板上的支撑力F′N，可得对应的线圈周长为
$L_{\mathrm{w}}^{\prime}=3\left[\left(R+h_{1}^{\prime}\right)\left(\theta^{\prime}+2 \sin \alpha^{\prime}\right)+R\left(\frac{2 \mathsf{π} }{3}-\theta^{\prime}-2 \alpha^{\prime}\right)\right].$
其中h′1、α′和θ′均为热处理后的值。利用游标卡尺测量承压板底部与骨架切口底部间的距离h′2，有h′1≈h′2，进而可分别由式(5)和(6)计算得到α′和θ′。Nb3Sn线圈此时的周向张力为
$\begin{equation*}F_{\mathrm{w}}^{\prime}=\frac{F^{\prime}{ }_{\mathrm{N}}}{2 \sin \left(\frac{\theta^{\prime}}{2}+\alpha^{\prime}\right)} . \end{equation*}$
所以，线圈在没有F′w作用时的周长为
$\begin{equation*}L_{\mathrm{w} 0}^{\prime}=L_{\mathrm{w}}^{\prime}\left(1-\frac{F_{\mathrm{w}}^{\prime}}{S G}\right) .\end{equation*}$
比较Lw0和L′w0，可以得到线圈的周长膨胀率为
$\begin{equation*}\varepsilon=\frac{L_{\mathrm{w} 0}^{\prime}}{L_{\mathrm{w} 0}}-1 .\end{equation*}$

2.2 测量样品

本文所测量的样品为NbTi-CuNi加强型青铜法Nb3Sn方导线绕制的单层螺线管线圈和直导线。热处理前Nb3Sn导线含绝缘的尺寸为1.51 mm×1.07 mm，不含绝缘的尺寸为1.35 mm×0.9 mm，而单层螺线管线圈的内径为220 mm，线圈轴向长度为100 mm，线圈层数和匝数分别为1和65，匝间距为0.028 mm。直导线的长度取1 m，以减少测量误差。

2.3 测量结果

在氩气保护下，依据图 2的热处理曲线对测量样品进行热处理。所用的热处理炉内温度最大值与最小值之间不超过5℃，以保证热处理过程的温度均匀性。热处理后，单层线圈和直导线尺寸的测量结果如表 3所示。可以看出，单层线圈和直导线热处理后的尺寸变化率基本一致，取其平均值可得：导线长度、宽度和厚度的平均膨胀率分别约为0.54%、1.1%和1.0%。导线横截面积的平均膨胀率约为2.1%，对应的空隙体积分数约为3.8%。由图 6可得，热处理相变时NbTi-CuNi加强型青铜法Nb3Sn导线中Sn与Nb原子在Nb3Sn层中的扩散速度比值约为2。
表 3 样品尺寸变化测量结果
样品类别 尺寸 热处理前 热处理后 膨胀率/%
单层线圈 导线宽度/mm 1.350 1.363 0.96
导线厚度/mm 0.900 0.909 1.00
导线长度/mm 690.80 694.60 0.55
导线横截面积/mm2 1.215 1.239 1.98
直导线 导线宽度/mm 1.350 1.366 1.19
导线厚度/mm 0.900 0.909 1.00
导线长度/mm 1 001.90 1 007.12 0.52
导线横截面积/mm2 1.215 1.242 2.22

3 14 T动物MRI磁体中Nb3Sn线圈尺寸膨胀对电磁设计的影响

14 T动物MRI磁体的线圈电磁结构如图 9所示，各线圈尺寸如表 4所示。磁体的运行电流为230 A，磁体在6 cm球形成像空间(diameter of spherical volume，DSV)内的磁场均匀度为1.1×10-6，简写为1.1×10-6@6 cm DSV。
图 9 14 TMRI磁体线圈电磁结构示意图
表 4 14 T动物MRI磁体线圈尺寸
绕组 内径/mm 外径/mm 长度/mm 线圈轴向间隔/mm 层数 匝数
Nb3Sn主线圈 220 305.12 676.26 — 38 442
NbTi主线圈 345.12 423.03 1 048.38 — 26 606
NbTi补偿线圈1 443.03 510.65 80.52 — 32 66
NbTi补偿线圈2和3 443.03 523.32 214.72 268.32 38 176
NbTi屏蔽线圈 1 007.72 1 049.99 201.3 481.6 20 165
磁体中Nb3Sn螺线管线圈采用NbTi-CuNi加强型青铜法Nb3Sn导线绕制。若Nb3Sn线圈热处理时以2.3节测得的尺寸平均变化率进行膨胀，也即线圈在径向、轴向和周向的膨胀率分别为1.0%、1.1%和0.54%，以表 4中Nb3Sn线圈的尺寸为热处理前的尺寸，则热处理后Nb3Sn线圈尺寸计算结果如表 5所示。由于线圈沿周向膨胀，线圈内径增加了1.19 mm，线圈内表面与内支撑筒间会产生缝隙。
表 5 热处理后Nb3Sn线圈尺寸及膨胀量
项 内径/mm 外径/mm 轴向长度/mm
热处理后数值 221.19 307.16 683.7
膨胀量 1.19 2.04 7.44
如果NbTi线圈尺寸保持不变，且Nb3Sn线圈中心轴保持与NbTi线圈中心轴重合，则Nb3Sn线圈热处理后的尺寸膨胀将导致磁体磁场均匀度由1.1×10-6@6 cm DSV变化为31×10-6@6 cm DSV。若热处理后Nb3Sn线圈水平放置，Nb3Sn线圈可能在重力作用下产生0.595 mm的径向偏心。Nb3Sn线圈偏心后在6 cm球形成像空间表面产生的磁场谐波如图 10所示(图中x和y方向均代表径向，z方向代表轴向，本文中线圈径向偏心假设发生在x方向)，Nb3Sn线圈热处理后尺寸膨胀和偏心将导致磁场均匀度由1.1×10-6@6 cm DSV变化为45×10-6@6 cm DSV，磁场均匀度严重恶化。同时，Nb3Sn线圈径向偏心也将导致其受到沿偏心方向的Lorentz力，其值为1.9×104 N。
图 10 Nb3Sn线圈偏心后成像空间表面磁场谐波

4 结论

本文进行了青铜法Nb3Sn线热处理后尺寸变化分析和测量。针对一个14 T动物MRI磁体采用的NbTi-CuNi加强型青铜法Nb3Sn导线，计算了热处理后的长度变化率和横截面积变化范围，测量了单层螺线管线圈和直导线热处理后导线的尺寸变化率。测量结果表明，单层线圈和直导线热处理后的尺寸变化率基本一致，并且长度变化率的测量结果与计算结果相近，而导线的横截面积变化率的测量结果也在计算结果范围内。将单层线圈和直导线的测量结果平均之后作为NbTi-CuNi加强型青铜法Nb3Sn导线热处理后的尺寸变化率，并据此计算了14 T动物MRI磁体中Nb3Sn线圈热处理后的尺寸膨胀量，讨论了该尺寸膨胀对磁体电磁的影响。

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