1# Modified Embedded Atom Method (MEAM) potential 2 3This directory contains the modified embedded atom method (MEAM) potential model 4driver [[1,2,3,4,5,6,7,8](#references)]. 5 6## MEAM model driver 7 8This KIM model driver is written in C++ and implements three styles of 9modified embedded atom method (MEAM) potentials: 10 11- ['meam/c'](#meamc-style) 12- ['meam/spline'](#meamspline-style) 13- ['meam/sw/spline'](#meamswspline-style) 14 15The style of the potential is automatically detected based on the input 16files to the driver. The input files are ASCII text files formatted to be 17consistent with the other MD codes that implement MEAM potentials, such as 18LAMMPS, serial DYNAMO code, and Warp. 19 20This driver has a design to mimic and reproduce the behavior of the 21[LAMMPS](https://lammps.sandia.gov) 22['meam/c'](https://lammps.sandia.gov/doc/pair_meamc.html), 23['meam/spline'](https://lammps.sandia.gov/doc/pair_meam_spline.html), and 24['meam/sw/spline'](https://lammps.sandia.gov/doc/pair_meam_sw_spline.html) pair 25styles.\ 26We adapted the LAMMPS format from the 27[LAMMPS documentation](https://lammps.sandia.gov/doc/) as of `Oct 15, 2020`. 28 29For any of the three styles mentioned above, the driver expects an element file. 30Depending on the specific potential style, other files may be required/supplied 31and will be outlined in their respective sections below. 32 33### Element file format 34 35The element file contains a unique list of chemical element names separated 36by spaces: 37 38```bash 39 Elem1 Elem2 Elem3 ... 40``` 41 42For example, for an Si and C alloy system the element file may look like this: 43 44```bash 45 Si C 46``` 47 48For an alloy system containing Al, Si, Mg, Cu, and Fe, it could look like this: 49 50```bash 51 Al Si Mg Cu Fe 52``` 53 54**NOTE:** 55 56There is no limit on the maximum number of elements in the element file. 57 58**NOTE:** 59 60The order in which elements are listed in the element file defines the 61integral indices used to identify each element in the [MEAM parameter 62file](#meam-parameter-file-format) (if you choose to specify one). 63Therefore, care must be taken to ensure that the indices used to assign 64parameter values in the MEAM parameter file are consistent with the order of 65the elements in the element file. (See details 66[below](#meam-parameter-file-format) on the syntax for settings in the 67'meam/c' parameter file.) 68 69## MEAM/C style 70 71In the 'meam/c' style, the driver computes pairwise interactions for a variety 72of materials using modified embedded atom method (MEAM) potentials 73[[1,2,3,4,5,6,7](#references)]. 74Conceptually, it is an extension to the original EAM potentials that adds 75angular forces. It is thus suitable for modeling metals and alloys with `fcc`, 76`bcc`, `hcp` and `diamond` cubic structures, as well as covalently bonded 77materials like silicon and carbon. 78 79In the MEAM formulation, the total energy Ei of an atom I is given by: 80 81<img src="https://latex.codecogs.com/svg.latex?E_i%20=%20U_\alpha%20(\Bar{\rho}_i)+\frac{1}{2}\sum\limits_{j\ne%20i}%20\phi_{\alpha%20\beta}(r_{ij}),"/> 82 83```tex 84 E_i = U_\alpha (\Bar{\rho}_i) 85 + \frac{1}{2}\sum\limits_{j \ne i} \phi_{\alpha \beta}(r_{ij}), 86``` 87 88where `U` is the embedding energy which is a function of the atomic electron 89density `rho bar`, and `phi` is a pair potential interaction. The pair 90interaction is summed over all neighbors J of atom I within the cutoff distance. 91 92As with EAM, the multi-body nature of the MEAM potential is a result of the 93embedding energy term. Details of the computation of the embedding and pair 94energies, as implemented in this model driver, are given in 95[[9](#references)] and references therein. 96 97The various parameters in the MEAM formulas are listed in two files. 98 991) The MEAM library file, usually `library.meam`. 1002) The MEAM parameter file (alloy parameter file). The name usually contains the 101 elements' names and ends in the `.meam` extension. For example, for an 102 alloy containing Si and C, the `SiC.meam` file (alloy parameter file) 103 contains the specific parameter settings. 104 105### MEAM library file format 106 107This file has generic MEAM settings for a variety of elements. The MEAM library 108file is an input file by some MD codes (e.g., LAMMPS, serial DYNAMO code, and 109Warp). Aside from blank and comment lines (which start with #), which can 110appear anywhere, its format is a series of entries, each of which has 19 111parameters and can span multiple lines [[11](#references)]. 112 113```txt 114 elt, lat, z, ielement, atwt, alpha, b0, b1, b2, b3, alat, esub, asub, t0, t1, t2, t3, rozero, ibar 115``` 116 117The `elt` and `lat` parameters are text strings, such as `elt = Si` or 118`elt = Cu` and `lat = bcc, dia, ... or fcc`. Because Fortran MD codes use the 119library file, these strings may include single quotes [[11](#references)]. 120 121To choose and identify the library file's proper settings, we use the value of 122the `elt` string. Since there can be multiple entries in the library file with 123the same `elt` value, we follow the LAMMPS code [[10](#references)], reading 124the first matching entry and ignoring the rest. 125 126In the MEAM library file, the parameters 127`lat, z, ielement, atwt, alat, esub, asub` correspond to single-element 128potential parameters. The meaning of these paramters is as follows: 129 130```txt 131 lat = lattice structure of reference configuration 132 z = number of nearest neighbors in the reference structure 133 ielement = atomic number 134 atwt = atomic weight 135 alat = lattice constant of reference structure 136 esub = energy per atom (eV) in the reference structure at equilibrium 137 asub = "A" parameter for the MEAM (see e.g., [1,2]) 138``` 139 140The `alpha, b0, b1, b2, b3, t0, t1, t2, t3` parameters correspond to the 141literature's standard MEAM parameters [[1,2](#references)] (the `b*` 142parameters are the standard beta parameters). 143Note that only parameters normalized to `t0 = 1.0` are supported. 144The `rozero` parameter is an element-dependent density scaling that weights the 145reference background density (see, e.g., equation 4.5 in [[9](#references)]) and 146is typically 1.0 for single-element systems. The `ibar` parameter selects the 147form of the function `G(Gamma)` used to compute the electron density. Different 148values of the `ibar` parameter correspond to the following forms: 149 150```txt 151 0 => G = sqrt(1+Gamma) 152 1 => G = exp(Gamma/2) 153 2 => not implemented 154 3 => G = 2/(1+exp(-Gamma)) 155 4 => G = sqrt(1+Gamma) 156 -5 => G = +-sqrt(abs(1+Gamma)) 157``` 158 159An example MEAM library file for an SiC alloy system is shown below: 160 161```bash 162 # DATE: 2012-06-29 UNITS: metal DATE: 2007-06-11 163 # CONTRIBUTOR: Greg Wagner, gjwagne@sandia.gov 164 # CITATION: Baskes, Phys Rev B, 46, 2727-2742 (1992) 165 # meam data from vax files fcc,bcc,dia 11/4/92 166 # elt lat z ielement atwt 167 # alpha b0 b1 b2 b3 alat esub asub 168 # t0 t1 t2 t3 rozero ibar 169 'Si' 'dia' 4. 14 28.086 170 4.87 4.8 4.8 4.8 4.8 5.431 4.63 1. 171 1.0 3.30 5.105 -0.80 1. 1 172 'C' 'dia' 4. 6 12.0111 173 4.38 4.10 4.200 5.00 3.00 3.567 7.37 1.000 174 1.0 5.0 9.34 -1.00 2.25 1 175``` 176 177**NOTE:** 178 179There is no limit on the maximum number of elements that can be read from the 180MEAM library file. 181 182### MEAM parameter file format 183 184The MEAM parameter file is an optional file containing settings that 185override or complement the library file settings. The format is the same as 186is read by other Fortran MD codes. Aside from blank and comment lines (start 187with `#`), which can appear anywhere, each line has one of the following 188forms: 189 190```txt 191 keyword = value 192 keyword(I) = value 193 keyword(I,J) = value 194 keyword(I,J,K) = value 195``` 196 197The driver also ignores anything after a trailing comment within a line 198(starting with `#`). 199 200The indices I, J, K correspond to the elements listed in the [MEAM element 201file](#element-file-format) in the order in which they appear. These indices 202start from 1 to be compatible with LAMMPS and the other Fortran MD codes. 203For example, if one were modeling an alloy containing Al, Si, Mg, Cu, and 204Fe, and used a MEAM element file containing the following: 205 206```bash 207 Al Si Mg Cu Fe 208``` 209 210then an index of 1 would refer to Al, an index of 2 to Si, an index of 3 to 211Mg, an index of 4 to Cu, and an index of 5 to Fe. 212 213The recognized keywords for the parameter file are as follows: 214 215```txt 216 rc = cutoff radius for cutoff function. (default = 4.0) 217 delr = length of smoothing distance for cutoff function. 218 (default = 0.1) 219 rho0(I) = relative density for element I 220 Ec(I,J) = cohesive energy of reference structure for I-J mixture 221 delta(I,J) = heat of formation for I-J alloy. If Ec_IJ is input as 222 zero, then driver sets 223 Ec_IJ = (Ec_II + Ec_JJ)/2 - delta_IJ 224 alpha(I,J) = alpha parameter for pair potential between I and J (can 225 be computed from bulk modulus of reference structure 226 re(I,J) = equilibrium distance between I and J in the reference 227 structure 228 Cmax(I,J,K) = Cmax screening parameter when I-J pair is screened 229 by K (I<=J); default = 2.8 230 Cmin(I,J,K) = Cmin screening parameter when I-J pair is screened 231 by K (I<=J); default = 2.0 232 lattce(I,J) = lattice structure of I-J reference structure: 233 fcc = face centered cubic 234 bcc = body centered cubic 235 hcp = hexagonal close-packed 236 dim = dimer 237 dia = diamond (interlaced fcc for alloy) 238 dia3 = diamond structure with primary 1NN and secondary 239 3NN interaction 240 b1 = rock salt (NaCl structure) 241 c11 = MoSi2 structure 242 l12 = Cu3Au structure (lower case L, followed by 12) 243 b2 = CsCl structure (interpenetrating simple cubic) 244 ch4 = methane-like structure, only for binary system 245 lin = linear structure (180 degree angle) 246 zig = zigzag structure with a uniform angle 247 tri = H2O-like structure that has an angle 248 nn2(I,J) = turn on second-nearest neighbor MEAM formulation for I-J 249 pair (see for example []()). (default = 0) 250 0 = second-nearest neighbor formulation off 251 1 = second-nearest neighbor formulation on 252 attrac(I,J) = additional cubic attraction term in Rose energy I-J pair 253 potential (default = 0) 254 repuls(I,J) = additional cubic repulsive term in Rose energy I-J pair 255 potential (default = 0) 256 zbl(I,J) = blend the MEAM I-J pair potential with the ZBL potential 257 for small atom separations (ZBL) (default = 1) 258 theta(I,J) = angle between three atoms in line, zigzag, and trimer 259 reference structures in degrees (default = 180) 260 gsmooth_factor = factor determining the length of the G-function smoothing 261 region. (default = 99.0) 262 Only significant for ibar=0 or ibar=4. 263 99.0 = short smoothing region, sharp step 264 0.5 = long smoothing region, smooth step 265 augt1 = integer flag for whether to augment t1 parameter by 266 3/5*t3 to account for old vs. new meam formulations, 267 (default = 1) 268 0 = don't augment t1 269 1 = augment t1 270 ialloy = integer flag to use alternative averaging rule for t 271 parameters, for comparison with the DYNAMO MEAM code 272 (default = 0) 273 0 = standard averaging (matches ialloy=0 in DYNAMO) 274 1 = alternative averaging (matches ialloy=1 in DYNAMO) 275 2 = no averaging of t (use single-element values) 276 mixture_ref_t = integer flag to use mixture average of t to compute the 277 background reference density for alloys, instead of the 278 single-element values (see description and warning 279 elsewhere in this doc page) (default = 0) 280 0 = do not use mixture averaging for t in the reference 281 density 282 1 = use mixture averaging for t in the reference density 283 erose_form = integer value to select the form of the Rose energy 284 function (see description below) (default = 0). 285 emb_lin_neg = integer value to select embedding function for negative 286 densities (default = 0) 287 0 = F(rho)=0 288 1 = F(rho) = -asub*esub*rho (linear in rho, matches 289 DYNAMO) 290 bkgd_dyn = integer value to select background density formula 291 (default = 0) 292 0 = rho_bkgd = rho_ref_meam(a) (as in the reference 293 structure) 294 1 = rho_bkgd = rho0_meam(a)*Z_meam(a) (matches DYNAMO) 295``` 296 297`Rc, delr, re` are in distance units (`Angstroms` in the case of metal units). 298`Ec` and `delta` are in the energy units (`eV` in the case of metal units). 299 300Each keyword represents a quantity which is either a scalar, vector, 2d array, 301or 3d array and must be specified with the correct corresponding array syntax. 302The indices I,J,K each run from 1 to N where N is the number of MEAM elements 303being used. 304 305Thus, these lines, 306 307```txt 308 rho0(4) = 3.0 309 alpha(1,5) = 4.2 310``` 311 312set `rho0` for the fourth element to the value 3.0 and set the `alpha` for 313the cross-species interaction between elements 1 and 5 to 4.2. 314 315The `augt1` parameter is related to modifications in the MEAM formulation of the 316partial electron density function. The augmentation is the default option. But 317when the parameter values are fit using the modified density function, `augt1` 318is 0. 319 320The `mixture_ref_t` parameter is related to the background reference density 321computation. Setting `mixture_ref_t=1` causes the driver to use an alloy mixture 322averaged value of t to compute the background reference density. The proper way 323to calculate the background reference density is to use the single-element 324values of the t parameters. 325 326**NOTE:** 327 328One should avoid using `mixture_ref_t=1` whenever possible [[11](#references)]. 329Using `mixture_ref_t=1` will give incorrect results for the second-neighbor 330MEAM, and it is non-standard for the first-neighbor MEAM. 331 332The parameters `attrac` and `repuls,` along with the integer selection parameter 333`erose_form,` can be used to modify the Rose energy function used to compute the 334pair potential. This function gives the energy of the reference state as a 335function of interatomic spacing. The form of this function is: 336 337<img src="https://latex.codecogs.com/svg.latex?\begin{align}%20\nonumber%20&a^*%20=%20\alpha%20(\frac{r}{r_e}%20-%201)\\%20\nonumber%20\\%20\nonumber%20&\text{form}=0\\%20\nonumber%20&E=\left\{\begin{matrix}-E_c\left(1+a^*+%20{a_\text{repuls}}~%20\frac{{a^*}^3}{r/re}%20\right)exp(-a^*)&%20a^*%20<%200\\%20-E_c\left(1+a^*+%20{a_\text{attrac}}~%20\frac{{a^*}^3}{r/re}\right)exp(-a^*)%20&%20a^*%20\geqslant%200%20\end{matrix}\right.%20\\\nonumber\\%20\nonumber%20&\text{form}=1\\%20\nonumber%20&E=-E_c\left(1+a^*+%20\left(-{a_\text{attrac}}~+\frac{{a_\text{repuls}}}{r}\right){a^*}^3%20\right)exp(-a^*)%20\\%20\nonumber%20\\%20\nonumber%20&\text{form}=2\\%20\nonumber%20&E=\left\{\begin{matrix}-E_c\left(1+a^*+%20{a_\text{repuls}}~%20{a^*}^3%20\right)exp(-a^*)%20&%20a^*%20<%200\\-E_c\left(1+a^*+%20{a_\text{attrac}}~%20{a^*}^3%20\right)exp(-a^*)%20&%20a^*\geqslant%200%20\end{matrix}\right.%20\end{align}"/> 338 339```tex 340 a^* = \alpha (\frac{r}{r_e} - 1) 341 342 form=0 343 344 E=\left\{\begin{matrix} 345 -E_c\left(1+a^*+ {a_\text{repuls}}~ \frac{{a^*}^3}{r/re} \right)exp(-a^*) 346 & a^* < 0\\ -E_c\left(1+a^*+ {a_\text{attrac}}~ \frac{{a^*}^3}{r/re} 347 \right)exp(-a^*) & a^* \geqslant 0 \end{matrix}\right. 348 349 form=1 350 351 E=-E_c\left(1+a^*+ \left(-{a_\text{attrac}}~+ 352 \frac{{a_\text{repuls}}}{r}\right){a^*}^3 \right)exp(-a^*) 353 354 form=2 355 356 E=\left\{\begin{matrix} 357 -E_c\left(1+a^*+ {a_\text{repuls}}~ {a^*}^3 \right)exp(-a^*) & a^* < 0\\ 358 -E_c\left(1+a^*+ {a_\text{attrac}}~ {a^*}^3 \right)exp(-a^*) & a^* 359 \geqslant 0 \end{matrix}\right. 360``` 361 362For example, below is an example MEAM parameter file for a SiC alloy system: 363 364```bash 365 # DATE: 2007-06-11 UNITS: metal CONTRIBUTOR: Greg Wagner, gjwagne@sandia.gov CITATION: Unknown 366 lattce(1,2) = 'dia' 367 Ec(1,2) = 6.4325 368 alpha(1,2) = 4.37 369 re(1,2) = 1.8878 370 rho0(2) = 2.25 371 rc = 4.0 372 delr = 0.1 373 Cmax(1,2,1) = 4.0 374 Cmax(1,2,2) = 4.0 375 Cmax(2,2,1) = 4.0 376 Cmax(1,1,2) = 4.0 377``` 378 379### MEAM/C style example 380 381As an example, a KIM Portable Model (PM) for pure `Al` systems that uses 382this model driver with the `meam/c` style might contain the following three 383input files: 384 385```bash 386 elems.meam 387 library.meam 388 Al.meam 389``` 390 391where `elems.meam` element file would contain 392 393```bash 394 # list of elements 395 # Elem1, Elem2 396 Al 397``` 398 399the `library.meam` library file might consist of 400 401```bash 402 # References: 403 # elt lat z ielement atwt 404 # alpha b0 b1 b2 b3 alat esub asub 405 # t0 t1 t2 t3 rozero ibar 406 # 407 'Al' 'fcc' 12. 13 26.9815 408 4.68604 1.56205 5.39270 5.29601 -1.00047 4.05000 3.35999 1.06859 409 1.0 -1.54917 -1.28508 10.01041 1.0 0 410``` 411 412and the `Al.meam` MEAM parameter file could contain 413 414```bash 415 re(1,1) = 2.86378 416 attrac(1,1) = 0.39558 417 repuls(1,1) = 0.09806 418 Cmin(1,1,1) = 1.00769 419 Cmax(1,1,1) = 2.31407 420 rc = 5.0 421 delr = 0.1 422 augt1 = 1 423``` 424 425## MEAM/SPLINE style 426 427The `meam/spline` style supported by this model driver stems from code 428primarily intended for modeling metals [[8,16](#references)]. 429 430This driver supports both a single species (also called "old-style" MEAM) and 431multi-component species (also called "new-style" MEAM). In this style, the atomic 432electron density is, 433 434<img src="https://latex.codecogs.com/svg.latex?\Bar{\rho}_i=\sum\limits_{j%20\ne%20i}\rho_j(r_{ij})%20+%20\sum\limits_{\substack{j%20<%20k,%20\\%20j%20\ne%20i}}%20f_j(r_{ij})%20f_k(r_{ik})%20g_{jk}{\left(\cos(\theta_{jik})\right)},"/> 435 436```tex 437 \Bar{\rho}_i=\sum\limits_{j \ne i}\rho_j(r_{ij}) + \sum\limits_{\substack{j < k, \\ j \ne i}} f_j(r_{ij}) f_k(r_{ik}) g_{jk}{\left(\cos(\theta_{jik})\right)}, 438``` 439 440where `rho` is the density, and `theta` is the angle between atoms. The five 441functions `phi`, `U`, `rho`, `f`, and `g` depend on the atoms in the 442interaction. Specifically, if there are N species, then there are N 443different functions `U`, `rho`, and `f`, and there are N(N+1)/2 functions 444`phi` and `g`. Each of these functions is tabulated as a cubic spline. 445 446While the list of elements is in the element file, the potential file indicates 447the cutoffs and the coefficients for these spline functions. 448 449The MEAM potential file name usually contains the elements' names and ends in 450the `.meam.spline` extension. For example, for an SiC alloy system, the 451`SiC.meam.spline` file contains all of the specific parameter settings. 452 453### MEAM potential file format 454 455In the new-style potential file, the second line begins with the string 456`meam/spline` followed by the number of unique species and their names. The 457names of the elements and the order in which they appear should match the 458MEAM element file. 459 460### MEAM/SPLINE style example 461 462As an example, a KIM Portable Model (PM) for pure `Si` systems that uses 463this model driver with the `meam/spline` style might contain the following two 464input files: 465 466```bash 467 elems.meam 468 Si.meam.spline 469``` 470 471where `elems.meam` element file would contain 472 473```bash 474 # list of elements 475 # Elem1, Elem2 476 Si 477``` 478 479and the `Si.meam.spline` potential file could contain 480 481```bash 482 Spline-based MEAM potential for Si. Reference: T. J. Lenosky, B. Sadigh, E. Alonso, V. V. Bulatov, T. D. de la Rubia, J. Kim, A. F. Voter, and J. D. Kress, Modell. Simul. Mater. Sci. Eng. 8, 825 (2000) 483 10 484 -4.266966781858503300e+01 0.000000000000000000e+00 485 1 0 1 0 486 1.500000000000000000e+00 6.929943430771341000e+00 1.653321602557917600e+02 487 1.833333333333333300e+00 -4.399503747408950400e-01 3.941543472528634600e+01 488 2.166666666666666500e+00 -1.701233725061446700e+00 6.871065423413908100e+00 489 2.500000000000000000e+00 -1.624732919215791800e+00 5.340648014033163800e+00 490 2.833333333333333000e+00 -9.969641728342462100e-01 1.534811309391571000e+00 491 3.166666666666667000e+00 -2.739141845072665100e-01 -6.334706186546093900e+00 492 3.500000000000000000e+00 -2.499156963774082700e-02 -1.798864729909626500e+00 493 3.833333333333333500e+00 -1.784331481529976400e-02 4.743496636420091500e-01 494 4.166666666666666100e+00 -9.612303290166881000e-03 -4.006506271304824400e-02 495 4.500000000000000000e+00 0.000000000000000000e+00 -2.394996574779807200e-01 496 11 497 -1.000000000000000000e+00 0.000000000000000000e+00 498 1 0 0 0 499 1.500000000000000000e+00 1.374674212682983900e-01 -3.227795813279568500e+00 500 1.700000000000000000e+00 -1.483141815327918000e-01 -6.411648793604404900e+00 501 1.899999999999999900e+00 -5.597204896096039700e-01 1.003068519633888300e+01 502 2.100000000000000100e+00 -7.310964379372824100e-01 2.293461970618954700e+00 503 2.299999999999999800e+00 -7.628287071954063000e-01 1.742018781618444500e+00 504 2.500000000000000000e+00 -7.291769685066557000e-01 5.460640949384478700e-01 505 2.700000000000000200e+00 -6.662022220044453400e-01 4.721760106467195500e-01 506 2.899999999999999900e+00 -5.732830582550895200e-01 2.056894449546524200e+00 507 3.100000000000000100e+00 -4.069014309729406300e-01 2.319615721086100800e+00 508 3.299999999999999800e+00 -1.666155295956388300e-01 -2.497162196179187900e-01 509 3.500000000000000000e+00 0.000000000000000000e+00 -1.237130660986393100e+01 510 8 511 7.351364478015182100e-01 6.165217237728655200e-01 512 1 1 1 1 513 -1.770934559908718700e+00 -1.074925682941420000e+00 -1.482768170233858500e-01 514 -3.881557649503457600e-01 -2.004503493658201000e-01 -1.492100354067345500e-01 515 9.946230300080272100e-01 4.142241371345077300e-01 -7.012475119623896900e-02 516 2.377401824966400000e+00 8.793892953828742500e-01 -3.944355024164965900e-02 517 3.760180619924772900e+00 1.266888024536562100e+00 -1.581431192239436000e-02 518 5.142959414883146800e+00 1.629979548834614900e+00 2.611224310900800400e-02 519 6.525738209841518900e+00 1.977379549636293600e+00 -1.378738550324104500e-01 520 7.908517004799891800e+00 2.396177220616657200e+00 7.494253977092666400e-01 521 10 522 -3.618936018538757300e+00 0.000000000000000000e+00 523 1 0 1 0 524 1.500000000000000000e+00 1.250311510312851300e+00 2.790400588857243500e+01 525 1.722222222222222300e+00 8.682060369372680600e-01 -4.522554291731776900e+00 526 1.944444444444444400e+00 6.084604017544847900e-01 5.052931618779816800e+00 527 2.166666666666666500e+00 4.875624808097850400e-01 1.180825096539679600e+00 528 2.388888888888888800e+00 4.416345603457190700e-01 -6.673769465415171400e-01 529 2.611111111111111200e+00 3.760976313325982700e-01 -8.938118490837722000e-01 530 2.833333333333333000e+00 2.714524157414608400e-01 -5.090324763524399800e-01 531 3.055555555555555400e+00 1.481440300150710900e-01 6.623665830603995300e-01 532 3.277777777777777700e+00 4.854596610856590900e-02 7.403702452268122700e-01 533 3.500000000000000000e+00 0.000000000000000000e+00 2.578982318481970500e+00 534 8 535 -1.395041572145673000e+01 1.134616739799360700e+00 536 1 1 1 1 537 -1.000000000000000900e+00 5.254163992149617700e+00 1.582685381253900500e+01 538 -7.428367052748285900e-01 2.359149452448745100e+00 3.117611233789983400e+01 539 -4.856734105496561800e-01 1.195946960915646100e+00 1.658962813584905800e+01 540 -2.285101158244838800e-01 1.229952028074150000e+00 1.108360928564026400e+01 541 2.865317890068852500e-02 2.035650777568434500e+00 9.088861456447702400e+00 542 2.858164736258610400e-01 3.424741418405580000e+00 5.489943377538379500e+00 543 5.429797683510331200e-01 4.948585892304984100e+00 -1.882291580187675700e+01 544 8.001430630762056400e-01 5.617988713941801200e+00 -7.718625571850646200e+00 545``` 546 547## MEAM/SW/SPLINE style 548 549The `meam/sw/spline` style supported by the driver is similar to the 550`meam/spline` style except for an additional Stillinger-Weber (SW) term 551that's added onto the energy. 552 553For this MEAM style, only a single species is supported. The total energy Ei 554of an atom I is given by, 555 556<img src="https://latex.codecogs.com/svg.latex?E_i%20=%20U_\alpha%20(\Bar{\rho}_i)%20+%20\frac{1}{2}\sum\limits_{j%20\ne%20i}%20\phi_{\alpha%20\beta}(r_{ij})%20+%20{E_{SW}}_i,"/> 557 558```tex 559 E_i = U_\alpha (\Bar{\rho}_i) 560 + \frac{1}{2}\sum\limits_{j \ne i} \phi_{\alpha \beta}(r_{ij}) + {E_{SW}}_i, 561``` 562 563where, 564 565<img src="https://latex.codecogs.com/svg.latex?{E_{SW}}_i%20=%20\sum\limits_{\substack{j%20<%20k,%20\\%20j%20\ne%20i}}%20F_j(r_{ij})%20F_k(r_{ik})%20G_{jk}{\left(\cos(\theta_{jik})\right)}."/> 566 567```tex 568 {E_{SW}}_i = \sum\limits_{\substack{j < k, \\ j \ne i}} F_j(r_{ij}) F_k(r_{ik}) G_{jk}{\left(\cos(\theta_{jik})\right)}. 569``` 570 571The atomic electron density is, 572 573<img src="https://latex.codecogs.com/svg.latex?\Bar{\rho}_i=\sum\limits_{j%20\ne%20i}\rho_j(r_{ij})%20+%20\sum\limits_{\substack{j%20<%20k,%20\\%20j%20\ne%20i}}%20f_j(r_{ij})%20f_k(r_{ik})%20g_{jk}{\left(\cos(\theta_{jik})\right)},"/> 574 575```tex 576 \Bar{\rho}_i=\sum\limits_{j \ne i}\rho_j(r_{ij}) + \sum\limits_{\substack{j < k, \\ j \ne i}} f_j(r_{ij}) f_k(r_{ik}) g_{jk}{\left(\cos(\theta_{jik})\right)}, 577``` 578 579where `rho` is the density, and `theta` is the angle between atoms. The seven 580functions `phi`, `U`, `F`, `G`, `rho`, `f`, and `g`, are represented by cubic 581splines. 582 583While the species name is in the element file, the potential file indicates 584the cutoffs and the coefficients for these spline functions. 585 586The MEAM potential file name usually contains the element's name and ends in 587the `.meam.sw.spline` extension. 588 589For example, for an Ti system, the `Ti.meam.sw.spline` file contains the 590specific parameter settings. 591 592### MEAM/SW/SPLINE style example 593 594As an example, a KIM Portable Model (PM) for pure `Ti` system that uses this 595model driver with the `meam/sw/spline` style might contain the following two 596input files: 597 598```bash 599 elems.meam 600 Ti.meam.sw.spline 601``` 602 603where `elems.meam` element file would contain 604 605```bash 606 # list of elements 607 # Elem1, Elem2 608 Ti 609``` 610 611and the `Ti.meam.sw.spline` potential file might contain 612 613```bash 614 # DATE: 2012-02-01 UNITS: metal CONTRIBUTOR: Robert Rudd, robert.rudd@llnl.gov CITATION: Hennig, Lenosky, Trinkle, Rudin, and Wilkins, Phys Rev B, 78, 054121 (2008) COMMENT: Spline-based MEAM+SW potential for Ti (R.G. Hennig et al., Phys. Rev. B 78, 054121 (2008)). 615 13 616 -20.0 0.0 617 1 0 1 0 618 1.742692837163632546e+00 3.744277175966389349e+00 0.0e+00 619 2.055801767399996649e+00 9.108397309055012991e-01 1.087025233010940539e+01 620 2.368910697636360752e+00 3.880458966341450711e-01 -1.553224188359566327e+00 621 2.682019627872724410e+00 -1.884090653346989427e-02 2.436300413714785229e+00 622 2.995128558109088512e+00 -2.480989296390785637e-01 2.679127139950259640e+00 623 3.308237488345452615e+00 -2.644895502965277645e-01 -1.250563846088559861e-01 624 3.621346418581816273e+00 -2.271961892825972718e-01 1.106625553693794339e+00 625 3.934455348818180376e+00 -1.292930901760148965e-01 -5.920536767960294933e-01 626 4.247564279054544478e+00 -5.968536693320936753e-02 -4.701234146338039155e-01 627 4.560673209290908581e+00 -3.110002556084055444e-02 -3.807399733552661869e-02 628 4.873782139527271795e+00 -1.384736320244591042e-02 -7.115479604519592272e-02 629 5.186891069763635898e+00 -3.203412727856397215e-03 -8.176829245009833991e-02 630 5.500000000000000000e+00 0.000000000000000000e+00 -5.714229647754404118e-02 631 2 632 0.000000000000000000e+00 0.000000000000000000e+00 633 1 0 0 0 634 2.055801767399996649e+00 0.000000000000000000e+00 0.000000000000000000e+00 635 4.410000000000000142e+00 0.000000000000000000e+00 0.000000000000000000e+00 636 2 637 0.000000000000000000e+00 0.000000000000000000e+00 638 1 0 0 0 639 -1.000000000000000000e+00 0.000000000000000000e+00 0.000000000000000000e+00 640 9.284366193603559303e-01 0.000000000000000000e+00 0.000000000000000000e+00 641 11 642 -1.000000000000000000e+00 0.000000000000000000e+00 643 1 0 0 0 644 2.055801767399996649e+00 1.747527966078008976e+00 -5.258697870855910423e+02 645 2.291221590659997087e+00 -5.867796394513962177e+00 2.527963170143783884e+02 646 2.526641413919997525e+00 -8.337628873718182732e+00 7.173183889436359095e+01 647 2.762061237179997519e+00 -5.839871284245343297e+00 -1.935877422943027337e+00 648 2.997481060439997957e+00 -3.114064823053663389e+00 -3.929991928613325314e+01 649 3.232900883699998396e+00 -1.725724506500649014e+00 1.434241360892369954e+01 650 3.468320706959998834e+00 -4.428977016930342181e-01 -2.949255346943084177e+01 651 3.703740530219999272e+00 -1.466643003340316054e-01 -3.180105342079309949e+00 652 3.939160353479999266e+00 -2.095507944774975262e-01 3.334908384292152750e+00 653 4.174580176740000148e+00 -1.442384563223014227e-01 3.719186918672943154e+00 654 4.410000000000000142e+00 0.000000000000000000e+00 -9.667170204974903314e+00 655 4 656 7.769349460453946719e-03 1.051977061603435737e-01 657 1 1 1 1 658 -5.514233164927543385e+01 -2.974556800785587707e-01 1.528706039584923782e-03 659 -4.474098990370920603e+01 -1.544945872174040002e-01 3.893372250824005966e-04 660 -3.433964815814297822e+01 5.098657168053199323e-02 3.812492703804592413e-04 661 -2.393830641257674685e+01 5.734269470406565539e-01 1.566392648865140222e-02 662 10 663 2.773275116566607856e+00 0.000000000000000000e+00 664 1 0 1 0 665 2.055801767399996649e+00 -1.485215264177146000e-01 7.220108674930609993e+01 666 2.317379348799997185e+00 1.684530491819494546e+00 -4.727446892749223650e+01 667 2.578956930199997277e+00 2.011336597660079661e+00 -1.518595783989647252e+01 668 2.840534511599997813e+00 1.144409274729116577e+00 3.339782043718398263e+00 669 3.102112092999998350e+00 2.861606802608402389e-01 2.587867610190767831e+00 670 3.363689674399998442e+00 -3.459281125712435068e-01 6.140706938696685491e+00 671 3.625267255799998978e+00 -6.257480601030769307e-01 3.739769675678216831e+00 672 3.886844837199999514e+00 -6.119510825772711549e-01 4.647490848815326814e+00 673 4.148422418600000050e+00 -3.112059650865105498e-01 2.832757465268814556e+00 674 4.410000000000000142e+00 0.000000000000000000e+00 -1.506120868737344587e+01 675 8 676 8.336422748105723812e+00 -6.040245747365641193e+01 677 1 1 1 1 678 -1.000000000000001998e+00 7.651409192994791664e-02 -1.106523212957412881e+02 679 -7.245090543770936753e-01 1.415582454149486025e-01 4.488534055085228403e+01 680 -4.490181087541854632e-01 7.578869734074348274e-01 -2.530651153464005532e+01 681 -1.735271631312771679e-01 6.301157037831857100e-01 -2.485101449562865383e+00 682 1.019637824916310997e-01 9.049597305054708773e-02 2.687693869168511718e+00 683 3.774547281145393951e-01 -3.574158665670761348e-01 -1.015585700534352709e+00 684 6.529456737374476072e-01 -6.529321764728414079e-01 1.342247859683878808e+01 685 9.284366193603559303e-01 -6.009121906529053980e+00 -4.527525426982541035e+02 686``` 687 688## References 689 690[1] M.I., Baskes, J.S., Nelson, and A.F., Wright, "Semiempirical modified 691 embedded atom potentials for silicon and germanium," Phys. Rev.B, 692 40:6085--6100, Sep 1989. 693 694[2] M.I., Baskes, "Modified embedded atom potentials for cubic materials 695 and impurities," Phys. Rev.B, 46:2727--2742, Aug 1992. 696 697[3] M.I., Baskes, "Determination of modified embedded atom method parameters 698 for nickel," Materials Chemistry and Physics, 50(2):152--158, 1997. 699 700[4] B.J., Lee, and M.I., Baskes, "Second nearest-neighbor modified embedded 701 atom method potential," Phys. Rev.B, 62:8564, Oct 2000. 702 703[5] B.J., Lee, M.I., Baskes, H., Kim, and Y.K., Cho, "Second nearest-neighbor 704 modified embedded atom method potentials for bcc transition metals," 705 Phys. Rev.B, 64:184102, Oct 2001. 706 707[6] S.M., Valone, M.I., Baskes, and R.L., Martin, "Atomistic model of helium 708 bubbles in gallium-stabilized plutonium alloys," Phys. Rev.B 73:214209 June 709 2006. 710 711[7] H., Huang, N.M., Ghoniem, J.K., Wong, and M., Baskes, "Molecular dynamics 712 determination of defect energetics in beta-SiC using three representative 713 empirical potentials," Modelling and Simulation in Materials Science and 714 Engineering, 3(5):615--627, Sep 1995. 715 716[8] T.J., Lenosky, B., Sadigh, E., Alonso, V.V., Bulatov, T.D., Rubia, J., Kim, 717 A.F., Voter, and J.D., Kress, "Highly optimized empirical potential model 718 of silicon," Modelling and Simulation in Materials Science and Engineering, 719 8(6):825--841, Oct 2000. 720 721[9] H., Fang, P.M., Gullett, A., Slepoy, M.F., Horstemeyer, M., Baskes, 722 G.J., Wagner, and M., Li, "Numerical tools for atomistic simulations," 723 Report 918395, Sandia National Laboratories, Jan 2004. 724 725[10] [https://lammps.sandia.gov](https://lammps.sandia.gov) 726 727[11] [https://docs.lammps.org/pair_meamc.html](https://docs.lammps.org/pair_meamc.html) 728 729[12] [https://docs.lammps.org/pair_meam_spline.html](https://docs.lammps.org/pair_meam_spline.html) 730 731[13] [https://docs.lammps.org/pair_meam_sw_spline.html](https://docs.lammps.org/pair_meam_sw_spline.html) 732 733[14] B.J., Thijsse, "Relationship between the modified embedded atom method and 734 stillinger-weber potentials in calculating the structure of silicon," 735 Phys. Rev. B, 65:195207, May 2002. 736 737[15] B., Jelinek, S., Groh, M.F., Horstemeyer, J., Houze, S.G., Kim, G.J., 738 Wagner, A., Moitra, and M.I., Baskes, "Modified embedded atom method 739 potential for Al, Si, Mg, Cu, and Fe alloys," Phys. Rev. B, 85:245102, 740 Jun 2012. 741 742[16] M.R., Fellinger, "First principles-based interatomic potentials for 743 modeling the body-bentered cubic metals V, Nb, Ta, Mo, and W," Ph.D. 744 thesis, The Ohio State University, Jan 2013. 745 746[17] P., Zhang and D.R., Trinkle, "A modified embedded atom method potential 747 for interstitial oxygen in titanium," Computational Materials Science, 748 124:204--210, Nov 2016. 749 750## License 751 752[LGPLv2.1](https://www.gnu.org/licenses/old-licenses/lgpl-2.1.html) 753 754## Copyright 755 756```txt 757LAMMPS - Large-scale Atomic/Molecular Massively Parallel Simulator 758http://lammps.sandia.gov, Sandia National Laboratories 759Steve Plimpton, sjplimp@sandia.gov 760Copyright (2003) Sandia Corporation. Under the terms of Contract 761DE-AC04-94AL85000 with Sandia Corporation, the U.S. Government retains 762certain rights in this software. This software is distributed under 763the GNU General Public License. 764``` 765 766```txt 767Copyright (2011) Lawrence Livermore National Security, LLC. 768Produced at the Lawrence Livermore National Laboratory. 769Written by Alexander Stukowski (<alex@stukowski.com>). 770LLNL-CODE-525797 All rights reserved. 771 772This program is free software; you can redistribute it and/or modify it under 773the terms of the GNU General Public License (as published by the Free 774Software Foundation) version 2, dated June 1991. 775 776This program is distributed in the hope that it will be useful, but 777WITHOUT ANY WARRANTY; without even the IMPLIED WARRANTY OF MERCHANTABILITY 778or FITNESS FOR A PARTICULAR PURPOSE. See the terms and conditions of the 779GNU General Public License for more details. 780 781Our Preamble Notice 782A. This notice is required to be provided under our contract with the 783U.S. Department of Energy (DOE). This work was produced at the 784Lawrence Livermore National Laboratory under Contract No. 785DE-AC52-07NA27344 with the DOE. 786 787B. Neither the United States Government nor Lawrence Livermore National 788Security, LLC nor any of their employees, makes any warranty, express or 789implied, or assumes any liability or responsibility for the accuracy, 790completeness, or usefulness of any information, apparatus, product, or 791process disclosed, or represents that its use would not infringe 792privately-owned rights. 793 794C. Also, reference herein to any specific commercial products, process, 795or services by trade name, trademark, manufacturer or otherwise does not 796necessarily constitute or imply its endorsement, recommendation, or 797favoring by the United States Government or Lawrence Livermore National 798Security, LLC. The views and opinions of authors expressed herein do not 799necessarily state or reflect those of the United States Government or 800Lawrence Livermore National Security, LLC, and shall not be used for 801advertising or product endorsement purposes. 802``` 803 804```txt 805Copyright (2012) Lawrence Livermore National Security, LLC. 806Produced at the Lawrence Livermore National Laboratory. 807Written by Robert E. Rudd (<robert.rudd@llnl.gov>). 808Based on the spline MEAM routine written by Alexander Stukowski 809(<alex@stukowski.com>). 810LLNL-CODE-588032 All rights reserved. 811 812The spline-based MEAM+SW format was first devised and used to develop 813potentials for bcc transition metals by Jeremy Nicklas, Michael Fellinger, 814and Hyoungki Park at The Ohio State University. 815 816This program is free software; you can redistribute it and/or modify it under 817the terms of the GNU General Public License (as published by the Free 818Software Foundation) version 2, dated June 1991. 819 820This program is distributed in the hope that it will be useful, but 821WITHOUT ANY WARRANTY; without even the IMPLIED WARRANTY OF MERCHANTABILITY 822or FITNESS FOR A PARTICULAR PURPOSE. See the terms and conditions of the 823GNU General Public License for more details. 824 825Our Preamble Notice 826A. This notice is required to be provided under our contract with the 827U.S. Department of Energy (DOE). This work was produced at the 828Lawrence Livermore National Laboratory under Contract No. 829DE-AC52-07NA27344 with the DOE. 830 831B. Neither the United States Government nor Lawrence Livermore National 832Security, LLC nor any of their employees, makes any warranty, express or 833implied, or assumes any liability or responsibility for the accuracy, 834completeness, or usefulness of any information, apparatus, product, or 835process disclosed, or represents that its use would not infringe 836privately-owned rights. 837 838C. Also, reference herein to any specific commercial products, process, 839or services by trade name, trademark, manufacturer or otherwise does not 840necessarily constitute or imply its endorsement, recommendation, or 841favoring by the United States Government or Lawrence Livermore National 842Security, LLC. The views and opinions of authors expressed herein do not 843necessarily state or reflect those of the United States Government or 844Lawrence Livermore National Security, LLC, and shall not be used for 845advertising or product endorsement purposes. 846 ``` 847 848```txt 849Copyright (c) 2020, Regents of the University of Minnesota. 850All rights reserved. 851``` 852 853## Contributing 854 855Contributors:\ 856 Yaser Afshar (UMN),\ 857 Daniel S. Karls (UMN) 858 859Contributing authors, original code in the LAMMPS USER-MEAMC (MEAM/C):\ 860 Greg Wagner (SNL),\ 861 Sebastian Hütter (OvGU) 862 863Contributing authors, original code in the LAMMPS USER-MISC (MEAM/SPLINE):\ 864 Alexander Stukowski (LLNL),\ 865 Will Tipton (Cornell),\ 866 Dallas R. Trinkle (UIUC),\ 867 Pinchao Zhang (UIUC),\ 868 Robert Rudd (LLNL) 869 870Contributing authors, original code in the LAMMPS USER-MISC (MEAM/SW/SPLINE):\ 871 Robert Rudd (LLNL) 872