76:'s original theoretical treatment, and the multislice method. This article focuses on the multislice method for simulation of dynamical diffraction, including multiple elastic scattering effects. Most of the packages that exist implement the multislice algorithm along with Fourier analysis to incorporate electron lens aberration effects to determine electron microscope image and address aspects such as phase contrast and diffraction contrast. For electron microscope samples in the form of a thin crystalline slab in the transmission geometry, the aim of these software packages is to provide a map of the crystal potential, however this inversion process is greatly complicated by the presence of multiple elastic scattering.
3689:
code accompanies an excellent text of the same name by
Kirkland which describes the background theory and computational techniques for simulating electron micrographs (including multislice) in detail. The main C/C++ routines use a command line interface (CLI) for automated batching of many simulation. The ACEM package also includes a graphical user interface that is more appropriate for beginners. The atomic scattering factors in ACEM are accurately characterized by a 12-parameter fit of Gaussians and Lorentzians to relativistic Hartree–Fock calculations.
1926:
3725:) is a package is written by Laurence Marks of Northwestern University. It uses a command line interface (CLI) and is based on UNIX. A structure file must be provided as input in order to run use this code, which makes it ideal for advanced users. The NUMIS multislice programs use the conventional multislice algorithm by calculating the wavefunction of electrons at the bottom of a crystal and simulating the image taking into account various instrument-specific parameters including
17:
3111:
2507:
864:
44:(HREM) micrographs, and serves as a useful tool for analyzing experimental images. This article describes some relevant background information, the theoretical basis of the technique, approximations used, and several software packages that implement this technique. Some of the advantages and limitations of the technique and important considerations that need to be taken into account are described.
3823:. The software comprises a graphical user interface version for direct visualization of STEM image calculations, as well as a bundle of command-line modules for more comprehensive calculation tasks. The programs have been written using Visual C++, Fortran 90, and Perl. Executable binaries for Microsoft Windows 32-bit and 64-bit operating systems are available for free from the
1607:
1327:
1544:
465:
3122:(FFT) and multiplying each by a phase grating term. The wave is then multiplied by a propagator, inverse Fourier transformed, multiplied by a phase grating term yet again, and the process is repeated. The use of FFTs allows a significant computational advantage over the Bloch Wave method in particular, since the FFT algorithm involves
2240:
3202:
reason for this due to aliasing effects which occur due to wraparound errors in FFT calculations. The requirement is to add additional “padding” to the unit cell has earned the nomenclature “super cell” and the requirement to add these additional pixels to the basic unit cell comes at a computational price.
629:
3697:
This package was released from the
National Center for High Resolution Electron Microscopy. This program uses a mouse-drive graphical user interface and is written by Roar Kilaas and Mike O’Keefe of Lawrence Berkeley National Laboratory. While the code is no longer developed, the program is available
3201:
The most important step in performing a multislice calculation is setting up the unit cell and determining an appropriate slice thickness. In general, the unit cell used for simulating images will be different from the unit cell that defines the crystal structure of a particular material. The primary
91:
A summary of the development of a computer algorithm from the multislice theory of Cowley and Moodie for numerical computation was reported by
Goodman and Moodie. They also discussed in detail the relationship of the multislice to the other formulations. Specifically, using Zassenhaus's theorem, this
79:
The first description of what is now known as the multislice theory was given in the classic paper by Cowley and Moodie. In this work, the authors describe scattering of electrons using a physical optics approach without invoking quantum mechanical arguments. Many other derivations of these iterative
3752:
and convergence. This program is good to use if one already has structure files for a material that have been used in other calculations (for example, Density
Functional Theory). These structure files can be used to general X-Ray structure factors which are then used as input for the PTBV routine in
294:
In 1957, Cowley and Moodie showed that the Schrödinger equation can be solved analytically to evaluate the amplitudes of diffracted beams. Subsequently, the effects of dynamical diffraction can be calculated and the resulting simulated image will exhibit good similarities with the actual image taken
68:
pattern is relatively well understood and documented. However, the reverse mapping from electron micrograph images to the crystal structure is generally more complicated. The fact that the images are two-dimensional projections of three-dimensional crystal structure makes it tedious to compare these
3802:
This is a code written by
Vincenzo Grillo of the Institute for Nanoscience (CNR) in Italy. This code is essentially a graphical frontend to the multislice code written by Kirkland, with more additional features. These include tools to generate complex crystalline structures, simulate HAADF images
3688:
This software is developed by Earl
Kirkland of Cornell University. This code is freely available as an interactive Java applet and as standalone code written in C/C++. The Java applet is ideal for a quick introduction and simulations under a basic incoherent linear imaging approximation. The ACEM
290:
3554:
There are several software packages available to perform multislice simulations of images. Among these is NCEMSS, NUMIS, MacTempas, and
Kirkland . Other programs exist but unfortunately many have not been maintained (e.g. SHRLI81 by Mike O’Keefe of Lawrence Berkeley National Lab and Cerius2 of
3545:
Additional practical concerns are how to effectively include effects such as inelastic and diffuse scattering, quantized excitations (e.g. plasmons, phonons, excitons), etc. There was one code that took these things into consideration through a coherence function approach called Yet
Another
2941:
is uniform within the slice. As a result, it is evident that this method in principle will work for any system. However, for aperiodic systems in which the potential will vary rapidly along the beam direction, the slice thickness has to be significantly small and hence will result in higher
2899:
1921:{\displaystyle {\begin{aligned}\phi ({\mathbf {r} })=1-i{\frac {\pi }{E\lambda }}\int \int \limits _{z'=-\infty }^{z'=z}V({\mathbf {X'} },z')\phi ({\mathbf {X'} },z'){\frac {1}{i\lambda (z-z')}}\exp \left(ik{\frac {|{\mathbf {X-X'} }|^{2}}{2(z-z')}}\right)d{\mathbf {X'} }dz'\end{aligned}}}
3761:
This software is specifically developed to run in Mac OS X by Roar Kilaas of
Lawrence Berkeley National Laboratory. It is designed to have a user-friendly user interface and has been well-maintained relative to many other codes (last update May 2013). It is available (for a fee) from
2186:
69:
projections to all plausible crystal structures. Hence, the use of numerical techniques in simulating results for different crystal structure is integral to the field of electron microscopy and crystallography. Several software packages exist to simulate electron micrographs.
104:-Whelan equations, 3. Sturkey's scattering matrix method. 4. the free-space propagation case, 5. The phase grating approximation, 6. A new "thick-phase grating" approximation, which has never been used, 7. Moodie's polynomial expression for multiple scattering, 8. The
3778:. The source code was published in the book of Electron Microdiffraction. A comparison between multislice and Bloch wave simulations for ZnTe was also published in the book. A separate comparison between several multislice algorithms at the year of 2000 was reported.
2903:
Hence, the iterative application of the aforementioned procedure will provide a full interpretation of the sample in context. Further, it should be reiterated that no assumptions have been made on the periodicity of the sample apart from assuming that the potential
3326:
2069:
is the interaction constant. So far we have set up the mathematical formulation of wave mechanics without addressing the scattering in the material. Further we need to address the transverse spread, which is done in terms of the
Fresnel propagation function
1073:
3803:
and model the STEM probe, as well as modeling of strain in materials. Tools for image analysis (e.g. GPA) and filtering are also available. The code is updated quite often with new features and a user mailing list is maintained. Freely available on their
307:
1337:
2502:{\displaystyle {\begin{aligned}\phi ({\mathbf {X} },z_{n+1})=\int p({\mathbf {X} }-{\mathbf {X'} },z_{n+1}-z_{n})\phi ({\mathbf {X} },z_{n})\exp \left(-i\sigma \int \limits _{z_{n}}^{z_{n+1}}V({\mathbf {X'} },z')dz'\right)dX'\end{aligned}}}
2632:
859:{\displaystyle {\begin{aligned}\Psi ({\mathbf {r} })=\exp(i{\mathbf {k\cdot r} })-{\frac {m}{2\pi \hbar ^{2}}}\int {\frac {\exp(ik\cdot {\mathbf {|r-r'|} })}{\mathbf {|r-r'|} }}V({\mathbf {r'} })\Psi ({\mathbf {r'} })dr'.\end{aligned}}}
3205:
To illustrate the effect of choosing a slice thickness that is too thin, consider a simple example. The Fresnel propagator describes the propagation of electron waves in the z direction (the direction of the incident beam) in a solid:
129:
1027:
2075:
2067:
1597:
2759:
2712:
2764:
2521:
2245:
2080:
2032:
1612:
1342:
1078:
634:
619:
312:
134:
3542:
greater than the lattice parameter (or half the lattice parameter in the case of perovskites) for a multislice simulation would result in missing atoms that should be in the crystal potential.
4163:
Koch, C. and J.M. Zuo, “Comparison of multislicecomputer programs for electron scattering simulations and the Bloch wavemethod”, Microscopy and Microanalysis, Vol. 6 Suppl. 2, 126-127, (2000).
3211:
1322:{\displaystyle {\begin{aligned}\phi ({\mathbf {r} })&=1-{\frac {m}{2\pi \hbar ^{2}}}\int {{\frac {\exp}{|{\mathbf {r-r'} }|}}V({\mathbf {r'} )\phi ({\mathbf {r'} })}dr'}\end{aligned}}}
302:
The following section will include a mathematical formulation of the multislice algorithm. The Schrödinger equation can also be represented in the form of incident and scattered wave as:
295:
from a microscope under dynamical conditions. Furthermore, the multislice algorithm does not make any assumption about the periodicity of the structure and can thus be used to simulate
80:
equations have since been given using alternative methods, such as Greens functions, differential equations, scattering matrices or path integral methods, see for instance the book by
2233:
509:
460:{\displaystyle {\begin{aligned}\Psi ({\mathbf {r} })&=\Psi _{0}({\mathbf {r} })+\int {G({\mathbf {r,r'} })V({\mathbf {r'} })\Psi ({\mathbf {r'} })d{\mathbf {r'} }}\end{aligned}}}
2022:
2752:
1060:
2983:
558:
2939:
955:
3790:
in Germany. Allows simulation of HAADF, ADF, ABF-STEM, as well as conventional TEM and CBED. The executable and source code are available as a free download on the Koch group
3407:
1966:
3774:
This is a software for multislice simulation was written in FORTRAN 77 by J. M. Zuo, while he was a postdoc research fellow at Arizona State University under the guidance of
3517:
1539:{\displaystyle {\begin{aligned}{\mathbf {k} }\cdot ({\mathbf {r-r'} })&=k(z-z')\\|{\mathbf {r-r'} }|&\approx (z-z')+({\mathbf {X-X'} })^{2}/{2(z-z')}\end{aligned}}}
3430:
3350:
531:
2191:
The thickness of each slice over which the iteration is performed is usually small and as a result within a slice the potential field can be approximated to be constant
909:
3540:
3453:
3370:
3149:
3750:
3176:
2516:
3479:
3706:
can be included in as a parameter to account for diffuse scattering, although the accuracy is unclear (i.e. a good guess of the Debye-Waller factor is needed).
3196:
120:
The form of multislice algorithm presented here has been adapted from Peng, Dudarev and Whelan 2003. The multislice algorithm is an approach to solving the
285:{\displaystyle {\begin{aligned}-{\frac {\hbar ^{2}}{2m}}{\frac {\partial ^{2}\Psi (x,t)}{\partial x^{2}}}+V(x,t)\Psi (x,t)&=E\Psi (x,t)\end{aligned}}}
3815:
Multi-slice image simulations for high-resolution scanning and coherent imaging transmission electron microscopy written by Juri Barthel from the
2181:{\displaystyle {\begin{aligned}p({\mathbf {X} },z)={\frac {1}{iz\lambda }}\exp \left(ik{\frac {{\mathbf {X} }^{2}}{2z}}\right)\end{aligned}}}
960:
2894:{\displaystyle {\begin{aligned}q_{n}({\mathbf {X} })=\exp\{-i\sigma \int \limits _{z_{n}}^{z_{n+1}}V({\mathbf {X} },z')dz'\}\end{aligned}}}
2027:
1551:
3966:
4063:
4027:
2639:
3881:
J. M. Cowley and A. F. Moodie (1957). "The Scattering of Electrons by Atoms and Crystals. I. A New Theoretical Approach".
566:
296:
3321:{\displaystyle {\tilde {\phi }}(\mathbf {u} ,z)={\tilde {\phi }}(\mathbf {u} ,z=0)\exp(\pi i\lambda \mathbf {u} ^{2}z)}
4022:. Monographs on the physics and chemistry of materials (1. publ. in paperback ed.). Oxford: Oxford Univ. Press.
97:
41:
3555:
Accerlys). A brief chronology of multislice codes is given in Table 2, although this is by no means exhaustive.
4184:
72:
There are two widely used simulation techniques that exist in literature: the Bloch wave method, derived from
3787:
1971:
2717:
1032:
2194:
882:
We then choose the coordinate axis in such a way that the incident beam hits the sample at (0,0,0) in the
3519:. For small angles this approximation holds regardless of how many slices there are, although choosing a
2953:
472:
112:. The relationship between algorithms is summarized in Section 5.11 of Spence (2013), (see Figure 5.9).
2907:
85:
3375:
1934:
914:
3484:
3820:
3101:
Table 1 - Computational efficiency of Discrete Fourier Transform compared to Fast Fourier Transform
536:
57:
3791:
3786:
The Quantitative TEM/STEM (QSTEM) simulations software package was written by Christopher Koch of
3412:
3333:
514:
3703:
4080:
885:
121:
93:
3983:
3119:
511:
is the Green's function that represents the amplitude of the electron wave function at a point
3522:
3435:
3355:
2627:{\displaystyle {\begin{aligned}\phi _{n+1}=\phi ({\mathbf {X} },z_{n+1})=*p_{n}\end{aligned}}}
36:
with matter, including all multiple scattering effects. The method is reviewed in the book by
3836:
3125:
40:, and also the work by Ishizuka. The algorithm is used in the simulation of high resolution
3728:
3198:
is the number of atoms in the system. (See Table 1 for comparison of computational time).
3154:
65:
8:
4179:
3458:
3151:
steps compared to the diagonalization problem of the Bloch wave solution which scales as
53:
3546:
Multislice (YAMS), but the code is no longer available either for download or purchase.
3912:
3816:
3181:
1332:
Now we make substitutions with regards to the coordinate system we have adhered, i.e.,
29:
3835:
OpenCL accelerated multislice software written by Adam Dyson and Jonathan Peters from
4151:
4118:
4100:
4096:
4059:
4023:
3932:
3775:
3699:
3599:
61:
4092:
3995:
3924:
3372:
is the wavelength of the electron wave (related to the wave vector by the relation
101:
4017:
37:
3928:
3722:
3594:
4173:
4104:
3999:
3936:
3118:
The basic premise is to calculate diffraction from each layer of atoms using
33:
96:, 2. Darwin's differential equations, widely used for diffraction contrast
4053:
81:
16:
3753:
NUMIS. Microscope parameters can be changed through the MICROVB routine.
3609:
109:
3859:
for multislice simulations developed by the group of Stephen Pennycook.
3352:
is the reciprocal lattice coordinate, z is the depth in the sample, and
3804:
73:
1022:{\displaystyle \Psi (r)=\phi (\mathbf {r} )\exp(i\mathbf {k\cdot r} )}
3852:
3840:
2511:
We can therefore represent the modulation function in the next slice
3824:
2062:{\displaystyle {\begin{aligned}\sigma =\pi /E\lambda \end{aligned}}}
1592:{\displaystyle {\boldsymbol {X}}={\begin{pmatrix}x\\y\end{pmatrix}}}
108:
path-integral formulation, and 9. relationship of multislice to the
3714:"NUMIS" redirects here. For the British financial institution, see
4140:(Ph.D.). Vom Fachbereich Physik Technischen Universitat Darmstadt.
3763:
3622:
3110:
105:
2235:. Subsequently, the modulation function can be represented as:
1068:) becomes then an equation for the modulation function, i.e.,
4042:
P. Goodman and A. F. Moodie, Acta Crystallogr. 1974, A30, 280
3880:
3715:
3698:
through the Electron Direct Methods (EDM) package written by
3971:. Commission of the European Communities. pp. 397–552.
3856:
3721:
The Northwestern University Multislice and Imaging System (
92:
paper gives the mathematical path from multislice to 1.
1568:
917:
4081:"The Calculation of Electron Diffraction Intensities"
3731:
3525:
3487:
3461:
3438:
3415:
3378:
3358:
3336:
3214:
3184:
3157:
3128:
2956:
2910:
2762:
2720:
2707:{\displaystyle p_{n}=p({\mathbf {X} },z_{n+1}-z_{n})}
2642:
2519:
2243:
2197:
2078:
2030:
1974:
1937:
1610:
1554:
1340:
1076:
1035:
963:
888:
632:
569:
539:
517:
475:
310:
132:
52:
The multislice method has found wide application in
4016:Peng, L.-M.; Dudarev, S. L.; Whelan, M. J. (2011).
614:{\displaystyle \Psi (r)=\exp(i\mathbf {k\cdot r} )}
3984:"Theorie der Beugung von Elektronen an Kristallen"
3744:
3534:
3511:
3473:
3447:
3424:
3401:
3364:
3344:
3320:
3190:
3170:
3143:
2977:
2933:
2893:
2746:
2706:
2626:
2501:
2227:
2180:
2061:
2016:
1960:
1920:
1591:
1538:
1321:
1054:
1021:
949:
903:
858:
613:
552:
525:
503:
459:
284:
4138:A Coherence Function Approach to Image Simulation
4117:
4015:
3968:Electron Microscopy in Materials Science: Part II
3409:). In the case of the small-angle approximation (
789:
770:
758:
739:
4171:
3949:
3432:100 mRad) we can approximate the phase shift as
2754:defines the transmission function of the slice.
28:algorithm is a method for the simulation of the
4058:(4. impression ed.). London: Butterworth.
4019:High energy electron diffraction and microscopy
3679:Table 2 - Timeline of various Multislice Codes
1968:is the wavelength of the electrons with energy
3913:"FFT Multislice Method—The Silver Anniversary"
3895:
563:Hence for an incident plane wave of the form
4135:
2884:
2799:
4123:High-Resolution Electron Microscopy, 4th Ed
3105:
621:the Schrödinger equation can be written as
4111:
3964:
3952:Advanced Computing in Electron Microscopy
3843:is under development as of October 2019.
3910:
3876:
3874:
3872:
3109:
15:
4078:
3943:
3889:
1556:
4172:
4129:
4051:
2017:{\displaystyle E=\hbar ^{2}k^{2}/{2m}}
4154:and J. M. Zuo, Plenum, New York, 1992
3981:
3869:
3855:is a multi-GPU enabled code based on
3549:
2747:{\displaystyle q_{n}({\mathbf {X} })}
1055:{\displaystyle \phi ({\mathbf {r} })}
299:images of aperiodic systems as well.
4055:Electron microscopy of thin crystals
4011:
4009:
2228:{\displaystyle V({\mathbf {X'} },z)}
623:
4085:Proceedings of the Physical Society
3900:. North Holland Publishing Company.
13:
3526:
3439:
2978:{\displaystyle \mathbf {log_{2}} }
1680:
964:
957:. Now we consider a wave-function
817:
637:
570:
504:{\displaystyle G(\mathbf {r,r'} )}
416:
339:
315:
260:
232:
195:
175:
166:
14:
4196:
4006:
2934:{\displaystyle V(\mathbf {X} ,z)}
2636:where, * represents convolution,
950:{\textstyle \mathbf {k} =(0,0,k)}
3402:{\displaystyle k=2\pi /\lambda }
3338:
3302:
3263:
3231:
2969:
2965:
2961:
2958:
2918:
2854:
2782:
2736:
2664:
2551:
2445:
2360:
2308:
2297:
2256:
2207:
2148:
2091:
1961:{\displaystyle \lambda =2\pi /k}
1894:
1834:
1830:
1827:
1742:
1710:
1623:
1483:
1479:
1476:
1427:
1423:
1420:
1367:
1363:
1360:
1347:
1290:
1270:
1244:
1240:
1237:
1213:
1209:
1206:
1193:
1171:
1167:
1164:
1089:
1044:
1012:
1006:
986:
919:
826:
805:
781:
777:
774:
750:
746:
743:
676:
670:
645:
604:
598:
542:
519:
490:
486:
483:
443:
425:
404:
383:
379:
376:
353:
323:
98:Transmission electron microscopy
42:transmission electron microscopy
4157:
4144:
4072:
3512:{\displaystyle \cos(0.1)=0.995}
4045:
4036:
3975:
3958:
3904:
3683:
3500:
3494:
3481:is on the order of 0.5% since
3315:
3288:
3279:
3259:
3253:
3241:
3227:
3221:
2928:
2914:
2870:
2849:
2787:
2777:
2741:
2731:
2701:
2659:
2604:
2581:
2575:
2546:
2465:
2439:
2378:
2355:
2349:
2292:
2280:
2251:
2222:
2201:
2102:
2086:
1877:
1860:
1845:
1821:
1794:
1777:
1762:
1736:
1730:
1704:
1628:
1618:
1528:
1511:
1493:
1471:
1465:
1448:
1437:
1414:
1406:
1389:
1376:
1355:
1299:
1284:
1278:
1264:
1254:
1231:
1225:
1222:
1201:
1181:
1158:
1148:
1094:
1084:
1049:
1039:
1016:
999:
990:
982:
973:
967:
944:
926:
895:
835:
820:
814:
799:
764:
724:
681:
662:
650:
640:
608:
591:
579:
573:
498:
479:
434:
419:
413:
398:
392:
371:
358:
348:
328:
318:
275:
263:
247:
235:
229:
217:
190:
178:
100:(TEM) image simulations - the
1:
3862:
3788:Humboldt University of Berlin
1062:for the amplitude. Equation (
553:{\displaystyle \mathbf {r'} }
47:
3917:Microscopy and Microanalysis
3810:
3797:
3756:
3702:of Northwestern University.
3425:{\displaystyle \theta \sim }
3345:{\displaystyle \mathbf {u} }
526:{\displaystyle \mathbf {r} }
7:
4150:Electron Microdiffraction,
4052:Hirsch, P. B., ed. (1971).
3898:Diffraction Physics, 3rd Ed
3560:
2947:
1064:
1029:with a modulation function
872:
10:
4201:
4125:. Oxford University Press.
4097:10.1088/0370-1328/80/2/301
3769:
3713:
904:{\displaystyle {\hat {z}}}
4079:Sturkey, Lorenzo (1962).
3965:Metherell, A. J. (1975).
3929:10.1017/S1431927604040292
3846:
3692:
3455:. For 100 mRad the error
533:due to a source at point
115:
4000:10.1002/andp.19283921704
3911:Ishizuka, Kazuo (2004).
3830:
3781:
3709:
3535:{\displaystyle \Delta z}
3448:{\displaystyle \Delta z}
3365:{\displaystyle \lambda }
3106:Practical considerations
20:Multislice to Microscopy
3896:John M. Cowley (1995).
3144:{\displaystyle N\log N}
3120:fast Fourier transforms
2942:computational expense.
3950:Dr. Earl J. Kirkland.
3883:Acta Crystallographica
3821:JĂĽlich Research Centre
3746:
3536:
3513:
3475:
3449:
3426:
3403:
3366:
3346:
3322:
3192:
3172:
3145:
3115:
2979:
2935:
2895:
2845:
2748:
2708:
2628:
2503:
2435:
2229:
2182:
2063:
2018:
1962:
1922:
1700:
1593:
1540:
1323:
1056:
1023:
951:
905:
860:
615:
554:
527:
505:
461:
286:
21:
4185:Mathematical modeling
4136:Heiko Muller (2000).
3837:University of Warwick
3747:
3745:{\displaystyle C_{s}}
3614:O’Keefe & Kilaas
3537:
3514:
3476:
3450:
3427:
3404:
3367:
3347:
3323:
3193:
3173:
3171:{\displaystyle N^{2}}
3146:
3113:
2980:
2936:
2896:
2811:
2749:
2709:
2629:
2504:
2401:
2230:
2183:
2064:
2019:
1963:
1923:
1661:
1594:
1541:
1324:
1057:
1024:
952:
906:
861:
616:
555:
528:
506:
462:
287:
84:, Sergei Dudarev and
60:. The mapping from a
19:
3729:
3704:Debye-Waller factors
3523:
3485:
3459:
3436:
3413:
3376:
3356:
3334:
3212:
3182:
3155:
3126:
2954:
2908:
2760:
2718:
2640:
2517:
2241:
2195:
2076:
2028:
1972:
1935:
1608:
1552:
1338:
1074:
1033:
961:
915:
886:
630:
567:
537:
515:
473:
308:
130:
122:Schrödinger equation
94:Schrödinger equation
66:electron diffraction
3680:
3474:{\displaystyle d-S}
3102:
54:electron microscopy
3988:Annalen der Physik
3982:Bethe, H. (1928).
3817:Ernst Ruska-Centre
3742:
3678:
3550:Available software
3532:
3509:
3471:
3445:
3422:
3399:
3362:
3342:
3318:
3188:
3168:
3141:
3116:
3100:
2975:
2931:
2891:
2889:
2744:
2704:
2624:
2622:
2499:
2497:
2225:
2178:
2176:
2059:
2057:
2014:
1958:
1918:
1916:
1589:
1583:
1536:
1534:
1319:
1317:
1052:
1019:
947:
911:-direction, i.e.,
901:
856:
854:
611:
550:
523:
501:
457:
455:
282:
280:
30:elastic scattering
22:
4119:John C. H. Spence
4065:978-0-408-18550-9
4029:978-0-19-960224-7
3776:John C. H. Spence
3700:Laurence D. Marks
3681:
3677:
3256:
3224:
3191:{\displaystyle N}
3114:Simulation of GaN
3103:
3099:
2167:
2124:
1881:
1798:
1656:
1259:
1133:
898:
880:
879:
794:
710:
209:
160:
62:crystal structure
4192:
4164:
4161:
4155:
4148:
4142:
4141:
4133:
4127:
4126:
4115:
4109:
4108:
4076:
4070:
4069:
4049:
4043:
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4013:
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4003:
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3480:
3478:
3477:
3472:
3454:
3452:
3451:
3446:
3431:
3429:
3428:
3423:
3408:
3406:
3405:
3400:
3395:
3371:
3369:
3368:
3363:
3351:
3349:
3348:
3343:
3341:
3327:
3325:
3324:
3319:
3311:
3310:
3305:
3266:
3258:
3257:
3249:
3234:
3226:
3225:
3217:
3197:
3195:
3194:
3189:
3177:
3175:
3174:
3169:
3167:
3166:
3150:
3148:
3147:
3142:
2984:
2982:
2981:
2976:
2974:
2973:
2972:
2945:
2944:
2940:
2938:
2937:
2932:
2921:
2900:
2898:
2897:
2892:
2890:
2883:
2869:
2858:
2857:
2844:
2843:
2842:
2826:
2825:
2824:
2786:
2785:
2776:
2775:
2753:
2751:
2750:
2745:
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2713:
2711:
2710:
2705:
2700:
2699:
2687:
2686:
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2652:
2651:
2633:
2631:
2630:
2625:
2623:
2619:
2618:
2603:
2602:
2593:
2592:
2574:
2573:
2555:
2554:
2539:
2538:
2508:
2506:
2505:
2500:
2498:
2494:
2483:
2479:
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2453:
2452:
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2347:
2335:
2334:
2316:
2315:
2314:
2301:
2300:
2279:
2278:
2260:
2259:
2234:
2232:
2231:
2226:
2215:
2214:
2213:
2187:
2185:
2184:
2179:
2177:
2173:
2169:
2168:
2166:
2158:
2157:
2152:
2151:
2144:
2125:
2123:
2109:
2095:
2094:
2068:
2066:
2065:
2060:
2058:
2048:
2023:
2021:
2020:
2015:
2013:
2005:
2000:
1999:
1990:
1989:
1967:
1965:
1964:
1959:
1954:
1927:
1925:
1924:
1919:
1917:
1913:
1902:
1901:
1900:
1887:
1883:
1882:
1880:
1876:
1855:
1854:
1853:
1848:
1842:
1841:
1840:
1824:
1818:
1799:
1797:
1793:
1766:
1761:
1750:
1749:
1748:
1729:
1718:
1717:
1716:
1699:
1692:
1683:
1673:
1657:
1655:
1644:
1627:
1626:
1598:
1596:
1595:
1590:
1588:
1587:
1559:
1545:
1543:
1542:
1537:
1535:
1531:
1527:
1506:
1501:
1500:
1491:
1490:
1489:
1464:
1440:
1435:
1434:
1433:
1417:
1405:
1375:
1374:
1373:
1351:
1350:
1328:
1326:
1325:
1320:
1318:
1314:
1313:
1302:
1298:
1297:
1296:
1277:
1276:
1260:
1258:
1257:
1252:
1251:
1250:
1234:
1228:
1221:
1220:
1219:
1197:
1196:
1184:
1179:
1178:
1177:
1161:
1140:
1134:
1132:
1131:
1130:
1111:
1093:
1092:
1061:
1059:
1058:
1053:
1048:
1047:
1028:
1026:
1025:
1020:
1015:
989:
956:
954:
953:
948:
922:
910:
908:
907:
902:
900:
899:
891:
874:
865:
863:
862:
857:
855:
848:
834:
833:
832:
813:
812:
811:
795:
793:
792:
787:
773:
767:
763:
762:
761:
756:
742:
716:
711:
709:
708:
707:
688:
680:
679:
649:
648:
624:
620:
618:
617:
612:
607:
559:
557:
556:
551:
549:
548:
532:
530:
529:
524:
522:
510:
508:
507:
502:
497:
496:
466:
464:
463:
458:
456:
452:
451:
450:
449:
433:
432:
431:
412:
411:
410:
391:
390:
389:
357:
356:
347:
346:
327:
326:
291:
289:
288:
283:
281:
210:
208:
207:
206:
193:
174:
173:
163:
161:
159:
151:
150:
141:
64:to its image or
4200:
4199:
4195:
4194:
4193:
4191:
4190:
4189:
4170:
4169:
4168:
4167:
4162:
4158:
4149:
4145:
4134:
4130:
4116:
4112:
4077:
4073:
4066:
4050:
4046:
4041:
4037:
4030:
4014:
4007:
3980:
3976:
3963:
3959:
3948:
3944:
3909:
3905:
3894:
3890:
3885:. Vol. 10.
3879:
3870:
3865:
3849:
3833:
3813:
3800:
3784:
3772:
3759:
3736:
3732:
3730:
3727:
3726:
3719:
3712:
3695:
3686:
3552:
3524:
3521:
3520:
3486:
3483:
3482:
3460:
3457:
3456:
3437:
3434:
3433:
3414:
3411:
3410:
3391:
3377:
3374:
3373:
3357:
3354:
3353:
3337:
3335:
3332:
3331:
3306:
3301:
3300:
3262:
3248:
3247:
3230:
3216:
3215:
3213:
3210:
3209:
3183:
3180:
3179:
3162:
3158:
3156:
3153:
3152:
3127:
3124:
3123:
3108:
2968:
2964:
2957:
2955:
2952:
2951:
2917:
2909:
2906:
2905:
2888:
2887:
2876:
2862:
2853:
2852:
2832:
2828:
2827:
2820:
2816:
2815:
2781:
2780:
2771:
2767:
2763:
2761:
2758:
2757:
2735:
2734:
2725:
2721:
2719:
2716:
2715:
2695:
2691:
2676:
2672:
2663:
2662:
2647:
2643:
2641:
2638:
2637:
2621:
2620:
2614:
2610:
2598:
2594:
2588:
2584:
2563:
2559:
2550:
2549:
2528:
2524:
2520:
2518:
2515:
2514:
2496:
2495:
2487:
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2457:
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2418:
2417:
2410:
2406:
2405:
2391:
2387:
2372:
2368:
2359:
2358:
2343:
2339:
2324:
2320:
2307:
2306:
2305:
2296:
2295:
2268:
2264:
2255:
2254:
2244:
2242:
2239:
2238:
2206:
2205:
2204:
2196:
2193:
2192:
2175:
2174:
2159:
2153:
2147:
2146:
2145:
2143:
2136:
2132:
2113:
2108:
2090:
2089:
2079:
2077:
2074:
2073:
2056:
2055:
2044:
2031:
2029:
2026:
2025:
2006:
2001:
1995:
1991:
1985:
1981:
1973:
1970:
1969:
1950:
1936:
1933:
1932:
1915:
1914:
1906:
1893:
1892:
1891:
1869:
1856:
1849:
1844:
1843:
1833:
1826:
1825:
1820:
1819:
1817:
1810:
1806:
1786:
1770:
1765:
1754:
1741:
1740:
1739:
1722:
1709:
1708:
1707:
1685:
1684:
1666:
1665:
1648:
1643:
1622:
1621:
1611:
1609:
1606:
1605:
1582:
1581:
1575:
1574:
1564:
1563:
1555:
1553:
1550:
1549:
1533:
1532:
1520:
1507:
1502:
1496:
1492:
1482:
1475:
1474:
1457:
1441:
1436:
1426:
1419:
1418:
1413:
1410:
1409:
1398:
1379:
1366:
1359:
1358:
1346:
1345:
1341:
1339:
1336:
1335:
1316:
1315:
1306:
1289:
1288:
1287:
1269:
1268:
1267:
1253:
1243:
1236:
1235:
1230:
1229:
1212:
1205:
1204:
1192:
1191:
1180:
1170:
1163:
1162:
1157:
1141:
1139:
1138:
1126:
1122:
1115:
1110:
1097:
1088:
1087:
1077:
1075:
1072:
1071:
1043:
1042:
1034:
1031:
1030:
1005:
985:
962:
959:
958:
918:
916:
913:
912:
890:
889:
887:
884:
883:
853:
852:
841:
825:
824:
823:
804:
803:
802:
788:
780:
769:
768:
757:
749:
738:
737:
736:
717:
715:
703:
699:
692:
687:
669:
668:
644:
643:
633:
631:
628:
627:
597:
568:
565:
564:
541:
540:
538:
535:
534:
518:
516:
513:
512:
489:
482:
474:
471:
470:
454:
453:
442:
441:
440:
424:
423:
422:
403:
402:
401:
382:
375:
374:
367:
352:
351:
342:
338:
331:
322:
321:
311:
309:
306:
305:
279:
278:
250:
202:
198:
194:
169:
165:
164:
162:
152:
146:
142:
140:
133:
131:
128:
127:
118:
58:crystallography
50:
12:
11:
5:
4198:
4188:
4187:
4182:
4166:
4165:
4156:
4152:J.C. H. Spence
4143:
4128:
4110:
4091:(2): 321–354.
4071:
4064:
4044:
4035:
4028:
4005:
3994:(17): 55–129.
3974:
3957:
3942:
3903:
3888:
3867:
3866:
3864:
3861:
3848:
3845:
3832:
3829:
3812:
3809:
3799:
3796:
3783:
3780:
3771:
3768:
3758:
3755:
3739:
3735:
3711:
3708:
3694:
3691:
3685:
3682:
3676:
3675:
3672:
3669:
3665:
3664:
3661:
3658:
3654:
3653:
3650:
3647:
3643:
3642:
3639:
3636:
3632:
3631:
3628:
3625:
3619:
3618:
3615:
3612:
3606:
3605:
3602:
3597:
3591:
3590:
3587:
3584:
3580:
3579:
3576:
3573:
3569:
3568:
3567:Year Released
3565:
3562:
3551:
3548:
3531:
3528:
3508:
3505:
3502:
3499:
3496:
3493:
3490:
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3444:
3441:
3421:
3418:
3398:
3394:
3390:
3387:
3384:
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3361:
3340:
3317:
3314:
3309:
3304:
3299:
3296:
3293:
3290:
3287:
3284:
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3278:
3275:
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3269:
3265:
3261:
3255:
3252:
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3237:
3233:
3229:
3223:
3220:
3187:
3165:
3161:
3140:
3137:
3134:
3131:
3107:
3104:
3098:
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3023:
3020:
3017:
3013:
3012:
3009:
3006:
3003:
3000:
2996:
2995:
2992:
2989:
2986:
2971:
2967:
2963:
2960:
2949:
2930:
2927:
2924:
2920:
2916:
2913:
2886:
2882:
2879:
2875:
2872:
2868:
2865:
2861:
2856:
2851:
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2835:
2831:
2823:
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2807:
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2801:
2798:
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2792:
2789:
2784:
2779:
2774:
2770:
2766:
2765:
2743:
2738:
2733:
2728:
2724:
2703:
2698:
2694:
2690:
2685:
2682:
2679:
2675:
2671:
2666:
2661:
2658:
2655:
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2646:
2617:
2613:
2609:
2606:
2601:
2597:
2591:
2587:
2583:
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2577:
2572:
2569:
2566:
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2534:
2531:
2527:
2523:
2522:
2493:
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2362:
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2327:
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2258:
2253:
2250:
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2224:
2221:
2218:
2212:
2209:
2203:
2200:
2172:
2165:
2162:
2156:
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2142:
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2131:
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2116:
2112:
2107:
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2098:
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2088:
2085:
2082:
2081:
2054:
2051:
2047:
2043:
2040:
2037:
2034:
2033:
2012:
2009:
2004:
1998:
1994:
1988:
1984:
1980:
1977:
1957:
1953:
1949:
1946:
1943:
1940:
1912:
1909:
1905:
1899:
1896:
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1397:
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1380:
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1372:
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1362:
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1354:
1349:
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1343:
1312:
1309:
1305:
1301:
1295:
1292:
1286:
1283:
1280:
1275:
1272:
1266:
1263:
1256:
1249:
1246:
1242:
1239:
1233:
1227:
1224:
1218:
1215:
1211:
1208:
1203:
1200:
1195:
1190:
1187:
1183:
1176:
1173:
1169:
1166:
1160:
1156:
1153:
1150:
1147:
1144:
1137:
1129:
1125:
1121:
1118:
1114:
1109:
1106:
1103:
1100:
1098:
1096:
1091:
1086:
1083:
1080:
1079:
1051:
1046:
1041:
1038:
1018:
1014:
1011:
1008:
1004:
1001:
998:
995:
992:
988:
984:
981:
978:
975:
972:
969:
966:
946:
943:
940:
937:
934:
931:
928:
925:
921:
897:
894:
878:
877:
868:
866:
851:
847:
844:
840:
837:
831:
828:
822:
819:
816:
810:
807:
801:
798:
791:
786:
783:
779:
776:
772:
766:
760:
755:
752:
748:
745:
741:
735:
732:
729:
726:
723:
720:
714:
706:
702:
698:
695:
691:
686:
683:
678:
675:
672:
667:
664:
661:
658:
655:
652:
647:
642:
639:
636:
635:
610:
606:
603:
600:
596:
593:
590:
587:
584:
581:
578:
575:
572:
547:
544:
521:
500:
495:
492:
488:
485:
481:
478:
448:
445:
439:
436:
430:
427:
421:
418:
415:
409:
406:
400:
397:
394:
388:
385:
381:
378:
373:
370:
366:
363:
360:
355:
350:
345:
341:
337:
334:
332:
330:
325:
320:
317:
314:
313:
277:
274:
271:
268:
265:
262:
259:
256:
253:
251:
249:
246:
243:
240:
237:
234:
231:
228:
225:
222:
219:
216:
213:
205:
201:
197:
192:
189:
186:
183:
180:
177:
172:
168:
158:
155:
149:
145:
139:
136:
135:
117:
114:
86:Michael Whelan
49:
46:
38:John M. Cowley
9:
6:
4:
3:
2:
4197:
4186:
4183:
4181:
4178:
4177:
4175:
4160:
4153:
4147:
4139:
4132:
4124:
4120:
4114:
4106:
4102:
4098:
4094:
4090:
4086:
4082:
4075:
4067:
4061:
4057:
4056:
4048:
4039:
4031:
4025:
4021:
4020:
4012:
4010:
4001:
3997:
3993:
3990:(in German).
3989:
3985:
3978:
3970:
3969:
3961:
3953:
3946:
3938:
3934:
3930:
3926:
3922:
3918:
3914:
3907:
3899:
3892:
3884:
3877:
3875:
3873:
3868:
3860:
3858:
3854:
3844:
3842:
3838:
3828:
3826:
3822:
3818:
3808:
3806:
3795:
3793:
3789:
3779:
3777:
3767:
3765:
3754:
3737:
3733:
3724:
3717:
3707:
3705:
3701:
3690:
3673:
3670:
3667:
3666:
3662:
3659:
3657:HREMResearch
3656:
3655:
3651:
3648:
3645:
3644:
3640:
3637:
3634:
3633:
3629:
3626:
3624:
3621:
3620:
3616:
3613:
3611:
3608:
3607:
3603:
3601:
3598:
3596:
3593:
3592:
3588:
3585:
3582:
3581:
3577:
3574:
3571:
3570:
3566:
3563:
3559:
3556:
3547:
3543:
3529:
3506:
3503:
3497:
3491:
3488:
3468:
3465:
3462:
3442:
3419:
3416:
3396:
3392:
3388:
3385:
3382:
3379:
3359:
3328:
3312:
3307:
3297:
3294:
3291:
3285:
3282:
3276:
3273:
3270:
3267:
3250:
3244:
3238:
3235:
3218:
3207:
3203:
3199:
3185:
3163:
3159:
3138:
3135:
3132:
3129:
3121:
3112:
3095:
3092:
3089:
3086:
3083:
3082:
3078:
3075:
3072:
3069:
3066:
3065:
3061:
3058:
3055:
3052:
3049:
3048:
3044:
3041:
3038:
3035:
3032:
3031:
3027:
3024:
3021:
3018:
3015:
3014:
3010:
3007:
3004:
3001:
2998:
2997:
2993:
2990:
2987:
2950:
2946:
2943:
2925:
2922:
2911:
2901:
2880:
2877:
2873:
2866:
2863:
2859:
2846:
2839:
2836:
2833:
2829:
2821:
2817:
2812:
2808:
2805:
2802:
2796:
2793:
2790:
2772:
2768:
2755:
2726:
2722:
2696:
2692:
2688:
2683:
2680:
2677:
2673:
2669:
2656:
2653:
2648:
2644:
2634:
2615:
2611:
2607:
2599:
2595:
2589:
2585:
2578:
2570:
2567:
2564:
2560:
2556:
2543:
2540:
2535:
2532:
2529:
2525:
2512:
2509:
2491:
2488:
2484:
2480:
2475:
2472:
2468:
2461:
2458:
2454:
2448:
2436:
2429:
2426:
2423:
2419:
2411:
2407:
2402:
2398:
2395:
2392:
2388:
2384:
2381:
2373:
2369:
2365:
2352:
2344:
2340:
2336:
2331:
2328:
2325:
2321:
2317:
2311:
2302:
2289:
2286:
2283:
2275:
2272:
2269:
2265:
2261:
2248:
2236:
2219:
2216:
2210:
2198:
2189:
2170:
2163:
2160:
2154:
2140:
2137:
2133:
2129:
2126:
2120:
2117:
2114:
2110:
2105:
2099:
2096:
2083:
2071:
2052:
2049:
2045:
2041:
2038:
2035:
2010:
2007:
2002:
1996:
1992:
1986:
1982:
1978:
1975:
1955:
1951:
1947:
1944:
1941:
1938:
1929:
1910:
1907:
1903:
1897:
1888:
1884:
1873:
1870:
1866:
1863:
1857:
1850:
1837:
1814:
1811:
1807:
1803:
1800:
1790:
1787:
1783:
1780:
1774:
1771:
1767:
1758:
1755:
1751:
1745:
1733:
1726:
1723:
1719:
1713:
1701:
1696:
1693:
1689:
1686:
1677:
1674:
1670:
1667:
1662:
1658:
1652:
1649:
1645:
1640:
1637:
1634:
1631:
1615:
1603:
1600:
1584:
1578:
1571:
1565:
1560:
1546:
1524:
1521:
1517:
1514:
1508:
1503:
1497:
1486:
1468:
1461:
1458:
1454:
1451:
1445:
1443:
1430:
1402:
1399:
1395:
1392:
1386:
1383:
1381:
1370:
1352:
1333:
1330:
1310:
1307:
1303:
1293:
1281:
1273:
1261:
1247:
1216:
1198:
1188:
1185:
1174:
1154:
1151:
1145:
1142:
1135:
1127:
1123:
1119:
1116:
1112:
1107:
1104:
1101:
1099:
1081:
1069:
1067:
1066:
1036:
1009:
1002:
996:
993:
979:
976:
970:
941:
938:
935:
932:
929:
923:
892:
876:
869:
867:
849:
845:
842:
838:
829:
808:
796:
784:
753:
733:
730:
727:
721:
718:
712:
704:
700:
696:
693:
689:
684:
673:
665:
659:
656:
653:
626:
625:
622:
601:
594:
588:
585:
582:
576:
561:
545:
493:
476:
467:
446:
437:
428:
407:
395:
386:
368:
364:
361:
343:
335:
333:
303:
300:
298:
292:
272:
269:
266:
257:
254:
252:
244:
241:
238:
226:
223:
220:
214:
211:
203:
199:
187:
184:
181:
170:
156:
153:
147:
143:
137:
125:
123:
113:
111:
107:
103:
99:
95:
89:
87:
83:
77:
75:
70:
67:
63:
59:
55:
45:
43:
39:
35:
34:electron beam
31:
27:
18:
4159:
4146:
4137:
4131:
4122:
4113:
4088:
4084:
4074:
4054:
4047:
4038:
4018:
3991:
3987:
3977:
3967:
3960:
3951:
3945:
3923:(1): 34–40.
3920:
3916:
3906:
3897:
3891:
3882:
3850:
3834:
3814:
3801:
3785:
3773:
3760:
3720:
3696:
3687:
3553:
3544:
3329:
3208:
3204:
3200:
3117:
2988:Discrete FT
2948:Data Points
2902:
2756:
2635:
2513:
2510:
2237:
2190:
2072:
1930:
1604:
1601:
1547:
1334:
1331:
1070:
1063:
881:
870:
562:
468:
304:
301:
293:
126:
119:
90:
82:Lianmao Peng
78:
71:
51:
25:
23:
3684:ACEM/JCSTEM
3671:Stadelmann
110:Born series
4180:Microscopy
4174:Categories
3863:References
3561:Code Name
3090:4,194,304
3073:1,048,576
74:Hans Bethe
48:Background
26:multislice
4105:0370-1328
3937:1431-9276
3851:The code
3811:DR. PROBE
3798:STEM-CELL
3757:MacTempas
3660:Ishizuka
3638:Kirkland
3623:MacTEMPAS
3527:Δ
3492:
3466:−
3440:Δ
3420:∼
3417:θ
3397:λ
3389:π
3360:λ
3298:λ
3292:π
3286:
3254:~
3251:ϕ
3222:~
3219:ϕ
3136:
2813:∫
2809:σ
2803:−
2797:
2689:−
2608:∗
2596:ϕ
2544:ϕ
2526:ϕ
2403:∫
2399:σ
2393:−
2385:
2353:ϕ
2337:−
2303:−
2287:∫
2249:ϕ
2130:
2121:λ
2053:λ
2042:π
2036:σ
1983:ℏ
1948:π
1939:λ
1867:−
1831:−
1804:
1784:−
1775:λ
1734:ϕ
1681:∞
1678:−
1663:∫
1659:∫
1653:λ
1646:π
1638:−
1616:ϕ
1518:−
1480:−
1455:−
1446:≈
1424:−
1396:−
1364:−
1353:⋅
1282:ϕ
1241:−
1210:−
1199:⋅
1186:−
1168:−
1146:
1136:∫
1124:ℏ
1120:π
1108:−
1082:ϕ
1037:ϕ
1010:⋅
997:
980:ϕ
965:Ψ
896:^
818:Ψ
778:−
747:−
734:⋅
722:
713:∫
701:ℏ
697:π
685:−
674:⋅
660:
638:Ψ
602:⋅
589:
571:Ψ
417:Ψ
365:∫
340:Ψ
316:Ψ
261:Ψ
233:Ψ
196:∂
176:Ψ
167:∂
144:ℏ
138:−
4121:(2013).
3646:JMULTIS
3575:O’Keefe
3056:262,144
2991:Fast FT
2881:′
2867:′
2492:′
2476:′
2462:′
2449:′
2312:′
2211:′
1911:′
1898:′
1874:′
1838:′
1791:′
1759:′
1746:′
1727:′
1714:′
1690:′
1671:′
1525:′
1487:′
1462:′
1431:′
1403:′
1371:′
1311:′
1294:′
1274:′
1248:′
1217:′
1175:′
846:′
830:′
809:′
785:′
754:′
546:′
494:′
447:′
429:′
408:′
387:′
3825:website
3819:at the
3805:website
3792:website
3770:JMULTIS
3635:TEMSIM
3627:Kilaas
3586:Kilaas
3583:TEMPAS
3564:Author
3093:22,528
3076:10,240
3039:65,536
3022:16,384
106:Feynman
4103:
4062:
4026:
3935:
3853:cudaEM
3847:cudaEM
3693:NCEMSS
3610:NCEMSS
3572:SHRLI
3330:Where
3178:where
3096:186.2
3084:2,048
3079:102.4
3067:1,024
3059:4,608
3042:2,048
2994:Ratio
1931:where
1548:where
469:where
116:Theory
32:of an
3841:clTEM
3831:clTEM
3782:QSTEM
3723:NUMIS
3716:Numis
3710:NUMIS
3674:2004
3668:JEMS
3663:2001
3652:1990
3641:1988
3630:1978
3617:1988
3604:1987
3600:Marks
3595:NUMIS
3589:1987
3578:1978
3507:0.995
3062:56.9
3028:18.3
3011:10.7
3005:4,096
1602:Thus
102:Howie
4101:ISSN
4060:ISBN
4024:ISBN
3933:ISSN
3857:CUDA
3764:here
3649:Zuo
3050:512
3033:256
3025:896
3016:128
2714:and
2024:and
297:HREM
56:and
24:The
4093:doi
3996:doi
3992:392
3925:doi
3498:0.1
3489:cos
3283:exp
3133:log
3087:11
3070:10
3045:32
3008:384
2794:exp
2382:exp
2127:exp
1801:exp
1143:exp
994:exp
719:exp
657:exp
586:exp
4176::
4099:.
4089:80
4087:.
4083:.
4008:^
3986:.
3931:.
3921:10
3919:.
3915:.
3871:^
3839:.
3827:.
3807:.
3794:.
3766:.
3053:9
3036:8
3019:7
2999:64
2985:N
2188:.
1928:,
1599:.
1329:.
560:.
124::
88:.
4107:.
4095::
4068:.
4032:.
4002:.
3998::
3954:.
3939:.
3927::
3738:s
3734:C
3718:.
3530:z
3504:=
3501:)
3495:(
3469:S
3463:d
3443:z
3393:/
3386:2
3383:=
3380:k
3339:u
3316:)
3313:z
3308:2
3303:u
3295:i
3289:(
3280:)
3277:0
3274:=
3271:z
3268:,
3264:u
3260:(
3245:=
3242:)
3239:z
3236:,
3232:u
3228:(
3186:N
3164:2
3160:N
3139:N
3130:N
3002:6
2970:2
2966:g
2962:o
2959:l
2929:)
2926:z
2923:,
2919:X
2915:(
2912:V
2885:}
2878:z
2874:d
2871:)
2864:z
2860:,
2855:X
2850:(
2847:V
2840:1
2837:+
2834:n
2830:z
2822:n
2818:z
2806:i
2800:{
2791:=
2788:)
2783:X
2778:(
2773:n
2769:q
2742:)
2737:X
2732:(
2727:n
2723:q
2702:)
2697:n
2693:z
2684:1
2681:+
2678:n
2674:z
2670:,
2665:X
2660:(
2657:p
2654:=
2649:n
2645:p
2616:n
2612:p
2605:]
2600:n
2590:n
2586:q
2582:[
2579:=
2576:)
2571:1
2568:+
2565:n
2561:z
2557:,
2552:X
2547:(
2541:=
2536:1
2533:+
2530:n
2489:X
2485:d
2481:)
2473:z
2469:d
2466:)
2459:z
2455:,
2446:X
2440:(
2437:V
2430:1
2427:+
2424:n
2420:z
2412:n
2408:z
2396:i
2389:(
2379:)
2374:n
2370:z
2366:,
2361:X
2356:(
2350:)
2345:n
2341:z
2332:1
2329:+
2326:n
2322:z
2318:,
2309:X
2298:X
2293:(
2290:p
2284:=
2281:)
2276:1
2273:+
2270:n
2266:z
2262:,
2257:X
2252:(
2223:)
2220:z
2217:,
2208:X
2202:(
2199:V
2171:)
2164:z
2161:2
2155:2
2149:X
2141:k
2138:i
2134:(
2118:z
2115:i
2111:1
2106:=
2103:)
2100:z
2097:,
2092:X
2087:(
2084:p
2050:E
2046:/
2039:=
2011:m
2008:2
2003:/
1997:2
1993:k
1987:2
1979:=
1976:E
1956:k
1952:/
1945:2
1942:=
1908:z
1904:d
1895:X
1889:d
1885:)
1878:)
1871:z
1864:z
1861:(
1858:2
1851:2
1846:|
1835:X
1828:X
1822:|
1815:k
1812:i
1808:(
1795:)
1788:z
1781:z
1778:(
1772:i
1768:1
1763:)
1756:z
1752:,
1743:X
1737:(
1731:)
1724:z
1720:,
1711:X
1705:(
1702:V
1697:z
1694:=
1687:z
1675:=
1668:z
1650:E
1641:i
1635:1
1632:=
1629:)
1624:r
1619:(
1585:)
1579:y
1572:x
1566:(
1561:=
1557:X
1529:)
1522:z
1515:z
1512:(
1509:2
1504:/
1498:2
1494:)
1484:X
1477:X
1472:(
1469:+
1466:)
1459:z
1452:z
1449:(
1438:|
1428:r
1421:r
1415:|
1407:)
1400:z
1393:z
1390:(
1387:k
1384:=
1377:)
1368:r
1361:r
1356:(
1348:k
1308:r
1304:d
1300:)
1291:r
1285:(
1279:)
1271:r
1265:(
1262:V
1255:|
1245:r
1238:r
1232:|
1226:]
1223:)
1214:r
1207:r
1202:(
1194:k
1189:i
1182:|
1172:r
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1159:|
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