Phase Retrieval and Reconstruction of Coherent Diffraction Imaging

KANG Xu; LIU Jin

doi:10.11858/gywlxb.20190761

Volume 33 Issue 3

Jun 2019

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DONG Qi, WEI Zhuobin, TANG Ting, ZHANG Ning. Influence of Explosion Depth on Bubble Pulsation in Shallow Water Explosion[J]. Chinese Journal of High Pressure Physics, 2018, 32(2): 024102. doi: 10.11858/gywlxb.20170580

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KANG Xu, LIU Jin. Phase Retrieval and Reconstruction of Coherent Diffraction Imaging[J]. Chinese Journal of High Pressure Physics, 2019, 33(3): 030105. doi: 10.11858/gywlxb.20190761

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KANG Xu, LIU Jin. Phase Retrieval and Reconstruction of Coherent Diffraction Imaging[J]. Chinese Journal of High Pressure Physics, 2019, 33(3): 030105. doi: 10.11858/gywlxb.20190761

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Phase Retrieval and Reconstruction of Coherent Diffraction Imaging

doi: 10.11858/gywlxb.20190761

KANG Xu,
LIU Jin^{*
,
,}

Institute of Fluid Physics, CAEP, Mianyang 621999, China

Received Date: 18 Apr 2019
Rev Recd Date: 14 May 2019

Abstract

Abstract

The coherent diffraction imaging (CDI) is an ultra-high resolution imaging technique that is sensitive to the density of the material. Compared to the surface-sensitive imaging methods with ultra-high resolution, the CDI is able to probe the interior of the sample by taking advantages of hard X-rays. According to the imaging layout, the space resolution of CDI is variable and can reach up to an atomic scale. This feature depends on the iterative phase retrieval method that almost becomes the signature of CDI. Based on oversampling a sample in a detected image, the phase and intensity of X-ray beam can be retrieved simultaneously by iterative calculations with constraints, and then are used to reconstruct the sample. Meanwhile, the three-dimensional reconstruction could be realized by combining image orientating and merging techniques. Here we present the imaging theory, phase retrieval and reconstruction methods of the CDI technique, and its diagnostic ability in a variety of reconstruction situations by experimental and simulation examples, to hopefully provide a systematic introduction of its development.
- coherent diffraction imaging,
- phase retrieval,
- two-dimensional reconstruction,
- three-dimensional reconstruction

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M_n+1AX_n相材料是三元层状化合物（简称MAX相材料），其中：M代表Ti、V、Zr等过渡金属元素；A代表A组元素；X代表C或者N；n=1, 2, 3, ···^[1]。MAX相这一概念最早由Barsoum^[2]提出，这类材料普遍具有陶瓷材料和金属材料的双重特性，可在高压、高温、强腐蚀等极端条件下稳定存在^[3-4]，并表现出较好的稳定性和抗氧化性，具有极其重要的研究价值和广阔的发展前景^[5]，因此探究高压等极端状态下的晶体性质变化具有重要意义。

近年来，有关三元层状M_n+1AX_n相材料的研究很多，主要集中在211相、312相和413相^[6]。随着研究的深入，M_n+1A₂X_n双“A”层221相、322相等结构被陆续得到，第一种双“A”层MAX相化合物Mo₂Ga₂C由Hu等^[7]于2015年成功制备，2016年Thore等^[8]通过第一性原理计算预测出V₂Ga₂C的存在，V₂Ga₂C、Ti₃Au₂C₂等双“A”型MAX相的理论预测和实验制备极大地丰富了MAX族化合物^[9]。V₂Ga₂C 是典型的由理论预测得到的新型双“A”层MAX相材料， Thore等^[10]根据声子谱没有虚频判定V₂Ga₂C具有稳定结构，研究发现这类双“A”层的MAX相材料普遍具有较强的金属性，如更好的机械延展性、易于加工等^[11]。目前，常压下V₂Ga₂C的研究日趋丰富，受限于实验条件的复杂性，高压下V₂Ga₂C的结构、电子、弹性等性能研究较为困难，为此基于密度泛函理论的第一性原理计算能够很好地解决这一问题。

本研究通过第一性原理对V₂Ga₂C六方结构的能带结构、态密度等电子结构和弹性性能等力学性质进行计算，根据玻恩稳定准则等相关理论，预测高压状态下V₂Ga₂C结构的力学稳定性，并对高压下V₂Ga₂C的晶体结构、电子结构和弹性性质等进行分析，为新型双“A”型M_n+1A₂X_n相的相关研究提供理论参考。

1. 计算方法与参数

采用第一性原理计算方法，运用基于密度泛函理论的Materials Studio软件中的CASTEP量子力学程序^[12-13]，选用倒易点阵空间表征的Cepeley-Alder超软赝势^[14]。利用总能量的平面波赝势替代离子势，并通过广义梯度近似（Generalized gradient approximation, GGA）中的PBE（Perdew, Burke and Ernzerhof）^[15-16]方法对电子间的相互作用和相关势进行校正。为确保总能量和原子间的作用力最小化，采用Broyden-Fletcher-Goldfarb-Shanno（BFGS）算法，布里渊区K点网格数为17 × 17 × 3，平面波截断能选取550 eV。进行原胞的几何优化（Geometry）时，能量收敛标准为5×10⁻⁶ eV/atom，最大作用力为0.01 eV/Å，应力偏差小于0.02 GPa，自洽场收敛精度为5×10⁻⁷ eV/atom。

2. 结果与讨论

2.1 晶体结构

V₂Ga₂C为六方晶系，空间群是P6₃/mmc，每个晶胞有10个原子，晶体结构见图1。V₂Ga₂C晶体与常见的V₂GaC的结构和性质类似，晶胞键角 $\alpha$ = $\;\beta$ =90°， $\gamma$ =120°，不同的是V₂Ga₂C有双Ga层结构及不同的晶胞键长a（a=b）、c，V₂Ga₂C晶胞中各原子坐标为V（1/3, 2/3, 0.0645）、Ga（1/3, 2/3, 0.6814）、C（0, 0, 0）。经优化计算，得到V₂Ga₂C的晶胞参数为a=b=2.950 Å，c=17.807 Å，与Thore等^[8]计算得到的数据（a=b=3.064 Å，c=18.153 Å）基本一致，即本研究构建的模型是准确可行的。

图 1 V₂Ga₂C的晶体结构

Figure 1. Crystal structure of V₂Ga₂C

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为了研究高压对V₂Ga₂C晶胞结构的影响，在0～70 GPa压强范围内以10 GPa为间隔进行结构优化，得到V₂Ga₂C晶胞的相对晶格参数变化情况，见图2。从图2可以看出，随着压强增大，晶格常数a、c和体积V均有不同程度的减小，同时在压强范围内V₂Ga₂C晶胞表现出较好的可压缩性，其中相对键长比a/a₀和c/c₀从1逐渐减小到0.9019和0.9331，相对晶格参数c/a从0 GPa的6.0362上升到70 GPa的6.2455，c轴较a轴随压强增大收缩得较慢，且键长的减小导致了晶胞体积V的缩小，上述晶胞参数的变化均体现了V₂Ga₂C的各向异性。此外，根据计算得到的V₂Ga₂C晶胞在不同压力下的晶格参数及相对晶格常数a/a₀、c/c₀、c/a和相对晶胞体积V/V₀的变化趋势平缓，可判定在0～70 GPa压力范围内V₂Ga₂C很难发生相变，即本研究利用图1的V₂Ga₂C结构探究压力对其电子性质、弹性性质的影响是合理准确的。

图 2 V₂Ga₂C的相对晶格参数和相对体积随压强的变化

Figure 2. Pressure dependence of relative lattice parameters and relative unit cell volume for V₂Ga₂C

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2.2 高压下的力学稳定性

力学稳定性是晶体材料稳定存在的重要因素。为研究压强对V₂Ga₂C晶胞力学稳定性的影响，从0 GPa开始，以每10 GPa为一个间隔进行结构优化，通过不同压强下的弹性常数预测V₂Ga₂C晶胞的力学稳定性。通过各个压强状态下V₂Ga₂C晶胞的结构优化，得到0～80 GPa不同压强状态下的晶体结构，各压强状态下的弹性常数见表1。

表 1 不同压强下V₂Ga₂C的弹性常数

Table 1. Pressure dependences of elastic constants for V₂Ga₂C

Pressure/GPa	C₁₁/GPa	C₃₃/GPa	C₄₄/GPa	C₁₂/GPa	C₁₃/GPa
0	275.07	309.04	88.92	65.69	48.31
10	325.89	437.85	103.70	79.38	101.23
20	392.18	492.57	112.70	113.70	122.66
30	475.27	582.90	99.80	176.94	174.81
40	462.17	636.80	103.00	156.69	180.10
50	524.87	696.25	51.41	228.45	221.66
60	558.67	743.34	29.53	220.05	219.34
70	640.36	854.90	4.53	280.99	275.36
80	618.66	892.67	−84.84	305.13	305.24

下载: 导出CSV

| 显示表格

弹性常数是晶体对作用力反应最直观的数据体现。根据V₂Ga₂C的晶胞结构，V₂Ga₂C晶体的弹性常数具有对称性，即C₁₁= C₂₂，C₁₃=C₃₁=C₃₂=C₂₃，C₁₂=C₂₁，C₄₄=C₅₅。由表1数据可知，弹性常数C₁₁、C₃₃随压强的增大逐渐增大，C₄₄先增大后逐渐变小直至减小到负数，C₁₂、C₁₃和C₆₆也有不同程度的增大。V₂Ga₂C晶胞为六方晶系，因此可以通过玻恩稳定准则^[17]、正交系统的力学稳定性公式^[18]以及弹性常数的变化规律，预测V₂Ga₂C六方三元层状化合物的力学稳定性。

玻恩稳定准则可写为

$\begin{array}{l} {C_{12}} > 0,\;\;\;{C_{33}} > 0,\;\;\;{C_{44}} > 0,\;\;\; {C_{11}} - {C_{12}} > 0,\;\;\;\left( {{C_{11}} + {C_{12}}} \right){C_{33}} - 2{C^2_{13}} > 0 \end{array}$

(1)

验证V₂Ga₂C晶胞正交系统力学稳定性的公式为

$\begin{array}{l} {C_{ij}} > 0\;{\rm{ }}\left( {i = j, \;\;{\rm{ }}0 \leqslant i \leqslant 6} \right),\;\;\;{C_{11}} + {C_{22}} - 2{C_{12}} > 0,\;\;\;{C_{11}} + {C_{33}} - 2{C_{13}} > 0,\\ {C_{22}} + {C_{33}} - 2{C_{23}} > 0,\;\;\;{C_{11}} + {C_{22}} + {C_{33}} + 2{C_{12}} + 2{C_{13}} + 2{C_{23}} > 0 \end{array}$

(2)

将表1的弹性常数代入式(1)、式(2)，可知V₂Ga₂C晶胞在0～70 GPa符合式(1)，在80 GPa时不符合式(1)。因此六方V₂Ga₂C晶胞的的弹性常数在0～70 GPa压强范围内处于力学稳定状态，80 GPa下V₂Ga₂C晶胞结构不稳定。

2.3 弹性性质

为了研究压强对V₂Ga₂C晶体弹性性质的影响，在不同的压强下对晶胞结构进行优化，在此基础上计算不同压强状态下的弹性常数（见表1）。弹性常数C₁₁、C₂₂和C₃₃分别表示晶胞受压沿a、b和c轴的线性压缩阻力，C₁₁、C₂₂较小而C₃₃最大，说明V₂Ga₂C在a、b轴上容易压缩，在c轴上难压缩；弹性常数C₄₄、C₅₅和C₆₆与材料抗剪切变形能力有关，C₄₄还与硬度有关，随压强增大而减小的C₄₄表明V₂Ga₂C材料抵抗形变的能力一般。

根据Voigt-Reuss-Hill近似理论^[19]，V₂Ga₂C的体积弹性模量B的最大值B_V、最小值B_R和平均值B_H，以及剪切弹性模量G的最大值G_V、最小值G_R和平均值G_H可以通过式(3)～式(8)得到

${B}_{\rm V}=\frac{2\left({C}_{11}+{C}_{12}\right)+{C}_{33}+4{C}_{13}}{9}$

(3)

${B}_{\rm R}=\frac{\left({C}_{11}+{C}_{12}\right){C}_{33}-{2C}_{13}^{2}}{{C}_{11}+{C}_{12}+2{C}_{33}-{4C}_{13}}$

(4)

$B_{\rm H}=\frac{{B}_{\rm V}+{B}_{\rm R}}{2}$

(5)

${G}_{\rm V}=\frac{{C}_{11}+{C}_{12}+2{C}_{33}-4{C}_{13}+12{C}_{55}+12{C}_{66}}{3}$

(6)

${G}_{\rm R}=\frac{5}{2}\frac{\left[\left({C}_{11}+{C}_{12}\right){C}_{33}-{2C}_{13}^{2}\right]{C}_{55}{C}_{66}}{3{B}_{\rm V}{C}_{55}{C}_{66}+\left [ \left({C}_{11}+{C}_{12}\right){C}_{33}-2{C}_{13}^{2}\right]\left({C}_{55}+{C}_{66}\right)}$

(7)

$G_{\rm H}=\frac{{G}_{\rm V}+{G}_{\rm R}}{2}$

(8)

根据Pugh准则^[17]可以鉴别晶体的韧脆性，B_H/G_H < 1.74为脆性材料，相反为韧性材料。根据表1弹性常数和式(3)～式(8)，可以计算体积模量B_H和剪切模量G_H，得到压强与B_H/G_H的关系曲线，见图3。从图3可以明显看出，压强小于20.15 GPa时，V₂Ga₂C表现为脆性材料，压强为20.15～70.00 GPa时表现为韧性材料。此外，通过式(9)可以预测维氏硬度（H_V）的变化（其中K = G_H/B_H，H_V的单位为GPa），维氏硬度随压强的变化见图4。从图4中曲线的变化趋势可以看出，维氏硬度随着压强的增大逐渐变小，原因是V₂Ga₂C晶胞的键长和a、b轴随压强的增大急剧压缩，故维氏硬度随之减小。

图 3 压强与B_H/G_H之间的关系

Figure 3. Pressure dependence of B_H/G_H

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图 4 V₂Ga₂C材料的维氏硬度随压强的变化

Figure 4. Pressure dependence of Vickers hardness for V₂Ga₂C

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${H\rm_V} = 0.92{K^{1.137}}{G_{\rm H}^{0.708}}$

(9)

然而遗憾的是，目前公开发表的有关V₂Ga₂C在高压状态下的力学性能研究报道较少，难以与本计算得到的理论预测进行对比分析。

2.4 电子性质

为探究压强与V₂Ga₂C电子性质的关系，在0～70 GPa的压强范围内通过GGA-PBE密度泛函理论计算，得到V₂Ga₂C的能带结构图、电子总态密度图，其中0 eV处的虚线表示费米能。

下面以0、35和70 GPa的能带结构为例进行分析，如图5所示。从图5可以明显看出，0 GPa下V₂Ga₂C无带隙，35 GPa下仍无带隙，直到接近力学稳定临界状态的70 GPa下仍未产生带隙，总体上能带曲线仅有很小幅度的变化。由此可知，在力学稳定范围内，V₂Ga₂C均无带隙，且压强的增加对能带结构的影响很小，即V₂Ga₂C材料为导体材料且压强对其影响较小或几乎没有影响。

图 5 不同压强下V₂Ga₂C的能带结构

Figure 5. Pressure dependence of electronic band structures for V₂Ga₂C

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电子态密度也是V₂Ga₂C电子性质的重要组成部分，选取0、35和70 GPa状态下的电子态密度分析压强与电子态密度的关系，见图6。由图6可知，随着压强增大，V₂Ga₂C的总态密度在费米能级附近变动较小，对电子性质影响较小。

图 6 不同压强下V₂Ga₂C的总态密度

Figure 6. Pressure dependence of total state density for V₂Ga₂C

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3. 结　论

基于密度泛函理论的第一性原理，研究了压强对V₂Ga₂C晶体的力学稳定性及压强对V₂Ga₂C结构、弹性和电子性质的影响。根据玻恩稳定准则预测了V₂Ga₂C结构稳定存在的压强区间为0～70 GPa，并通过正交系统的力学稳定公式验证结果可靠。同时，研究了0～70 GPa压强下V₂Ga₂C的晶体结构、弹性性质与电子结构，压强使V₂Ga₂C压缩，体积、相对晶胞参数a/a₀、c/c₀等均有不同程度的减小，都体现了V₂Ga₂C具有各向异性；随着压强增大，通过弹性常数可知V₂Ga₂C在a、b轴上较c轴易压缩，且在20.15 GPa时从韧性转变为脆性，其硬度也随之变小；从V₂Ga₂C在各个压强状态下的态密度和能带结构可知，在力学稳定的条件下压强对V₂Ga₂C材料的电子性质影响不大。然而目前有关V₂Ga₂C材料的研究较少，希望本研究结果可以为双“A”型MAX相材料的实验制备和理论研究提供参考。

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