# Statistics of Particles

In the microscopic treatment of matter, the particles, for example, molecules and elec­trons, are distributed in a system of energy levels. How the particles distribute among the energy levels determines the behavior of the system in a significant way. Regarding to applications with solar energy, an understanding of Maxwell-Boltzmann statistics and Fermi-Dirac statistics is essential. Maxwell-Boltzmann statistics is valid for a system of distinguishable particles in a system of energy levels allowing unlimited occu­pancy; whereas Fermi-Dirac statistics is valid for a system of indistinguishable particles in a system of energy levels allowing limited occupancy, that is, systems satisfying the Pauli exclusion principle.

The starting point of the derivation is the Boltzmann expression of entropy,

S = kB ln W, (D.1)

where kB is Boltzmann’s constant and W is the total number of configurations of the system.

Consider a situation of N particles in a system consisting of a series of energy levels. The occupancy number of the *th level is N^ The energy of the *th level is Ei, and the total energy of the system is E. We have the conditions

E = £* Ni Ei.

The condition of equilibrium is that the entropy, Eq. D.1, reaches maximum under the two constraints in Eq. D.2.

Intuitively, the more uniform the distribution, the greater the randomness, or the greater the entropy. However, the condition of constant total energy adds another condition: It is preferable to have more particles in the levels of lower energy and less particles in the levels of higher energy. The problem can be resolved using Fermat’s theorem together with the Lagrange multiplier method. Introducing two Lagrange multipliers a and в, the condition of equilibrium is

The distribution can be found by combining the solution of Eq. D.3 with the two constraints in Eqs. D.1 and D.2.

During the calculation, one needs an approximate value of the derivative of ln N! for large N. This can be simply taken as

dN in N! « ln N! – ln(N – 1)! « ln N. (D.4)

D. 1 Maxwell—Boltzmann Statistics

In the case of Maxwell-Boltzmann statistics, applicable to classical atomic systems, the particles are distinguishable, and the number of particles per energy level is unlimited. The number of possible configurations W can be determined as follows.

Suppose that N particles are placed into containers with occupation numbers N1, N2, … Ni, … and so on. Consider now the number of ways to place N1 particles into con­tainer 1. First, choose one of the N particles for the first position in container 1, that is, N ways. Next, choose one of the remaining N — 1 particles for the second position in container 1. This generates N — 1 ways. Altogether there are N(N — 1) ways. By continuing the process, the total number of ways is N(N — 1)(N — 2)…(N — N1 — 1). However, the placement of particles in the container is arbitrary. Therefore, there is a N1 !-fold redundancy. The net ways of placement are given as

W N (N — 1)(N — 2)…(N — ,¥, — 1) N! (D5)

Wl =————————- N!———————- = (N — N1)! N,!. (D’5)

Similarly, the number of ways of placing the remaining N — N1 particles into container 2 generates a factor

 (N — N1)! (N — N1 — N2)! N2!.

 (D.6)

 W2

Similarly,

 (N — N1 — N2)! (N — N1 — N2 — N3)! N3!.

 W3

 (D.7)

Continuing the process further, finally we find

N!

W = W1 W2 …Wi… = —

i Ni!

The entropy is

S = kB ln W = kB ln N! —J2 ln Ni! The condition of thermal equilibrium gives

 d dN

 S + Jn — ^ N + /ЗІЕ — J2 NiEi

 (D.10)

The result is

kB ln Ni = —a — ЗЕі, (D.11)

The meaning of the parameter З can be interpreted based on thermodynamics. Be­cause the system is under the condition of constant temperature and constant volume, according to Eq. 6.28, we have

dE = TdS. (D.12)

By treating S and E as variables, comparing Eq. D.3 and Eq. D.12, we find, heuristi – cally,

З = T. (D.13)

The constant a can be determined by the condition that the total number of particles is N. Equation D.11 can be rewritten as

 N (-ЕЛ ni = z ex4 k^) where Z is a constant determined by the condition

 (D.14) = N, (D.15) (D.16) Ni/N, Eq. D.16 can be written

 in other words,

, —Ei

pi = Z ехП kBBTT

Obviously, the sum of all probabilities is 1,

^ Pi =1

D. 2 Fermi—Dirac Statistics

Electrons are fermions obeying the Pauli exclusion principle. Each state can only be occupied by one electron. The electrons satisfies Fermi-Dirac statistics.

For each energy value, there can be multiple states. For example, each electron can have two spin states that have the same energy level. Let the degeneracy, that is, the number of states at energy Ei, be gi. The number of electrons staying at that energy level, Ni, should not exceed gi. The number of different ways of occupation in the gi states is

gi ■

(gi – Ni)! Ni! ’

which is a special case of Eq. D.5. Following Eq. D.3, we obtain

кв [ln(gi – Ni) – lnNi] = a + f3Ei,

or

 gi a + /3Eiкв

 (D.21)

 Ni

 exp

 + 1

Using Eq. D.13 and introducing the probability pi = Ni/gi,

 1

 (D.22)

 pi

 Ei — Ep квТ

 exp

 + 1

Equation D.22 is called the Fermi function. At the Fermi level Ep, the probability is 1/2. Apparently,

pi ^ t Ei ^ ep 5

pi ^ ° Ei ^ ep ■

At low temperature, where квТ ^ Ep, the Fermi-Dirac statistics becomes

At high temperatures, or at high energy levels (Ei — Ep)/квТ ^ 1, the Fermi-Dirac statistics reduces to Maxwell-Boltzmann statistics,

Ei

квТ

AM1.5 Reference Solar Spectrum

To facilitate testing of solar photovoltaic devices, American Society for Testing and Materials (ASTM) in conjenction with the photovoltaic (PV) industry and government research and development laboratories developed and defined two standard terrestrial solar spectral irradiance distributions: The standard direct normal spectral irradi – ance and the standard global spectral irradiance, incorporated into a single document, ASTM G-173-03; see Section 5.2.1.

The original data was presented in wavelength scale. The following table is pre­sented in photon energy scale. The first column is photon energy in eV. The second column is the extraterrestrial solar radiation spectrum, AM0. The third column is the direct normal solar radiation spectrum. The fourth column is the global solar radiation spectrum, including radiation scattered from the sky. All spectral data are presented in watts per square meter per eV (W/m2-eV).

Table E.1: AM1.5 Reference Solar Spectrum (W/m2 eV)

 c (eV) AM0 Direct Global c (eV) AM0 Direct Global 0.32 116 94 94 0.47 232 0 0 0.33 128 105 105 0.48 240 0 0 0.34 135 104 104 0.49 251 11 11 0.35 141 110 109 0.50 265 104 105 0.36 146 75 74 0.51 274 163 164 0.37 156 37 37 0.52 283 199 201 0.38 163 27 26 0.53 292 247 250 0.39 168 52 52 0.54 301 268 271 0.40 178 25 25 0.55 313 294 299 0.41 184 32 32 0.56 322 294 299 0.42 192 16 16 0.57 328 310 315 0.43 200 0 0 0.58 338 321 326 0.44 208 0 0 0.59 345 286 290 0.45 216 0 0 0.60 358 252 256 0.46 223 0 0 0.61 370 154 156

 e (eV) AM0 Direct Global e (eV) AM0 Direct Global 0.62 380 214 218 1.02 577 487 510 0.63 385 46 47 1.03 579 485 508 0.64 399 2 2 1.04 577 461 483 0.65 409 0 0 1.05 582 467 489 0.66 406 0 0 1.06 577 398 418 0.67 427 0 0 1.07 584 245 257 0.68 433 31 31 1.08 579 204 213 0.69 447 214 219 1.09 583 141 148 0.70 456 367 376 1.10 578 129 135 0.71 460 396 406 1.11 581 298 314 0.72 477 448 460 1.12 578 445 468 0.73 482 459 471 1.13 571 506 534 0.74 494 470 483 1.14 576 523 553 0.75 497 472 485 1.15 578 532 563 0.76 507 489 503 1.16 584 539 571 0.77 513 479 493 1.17 586 546 578 0.78 516 474 487 1.18 586 550 582 0.79 523 507 522 1.19 589 553 587 0.80 528 505 520 1.20 591 555 589 0.81 529 472 486 1.21 589 552 586 0.82 530 392 404 1.22 597 558 593 0.83 533 212 218 1.23 592 553 587 0.84 535 147 151 1.24 599 559 595 0.85 536 116 119 1.25 600 546 581 0.86 538 52 53 1.26 599 484 514 0.87 541 20 21 1.27 596 454 482 0.88 544 1 1 1.28 595 360 382 0.89 547 0 0 1.29 593 268 284 0.90 547 0 0 1.30 601 233 247 0.91 549 4 4 1.31 593 229 243 0.92 551 204 211 1.32 607 149 158 0.93 553 327 340 1.33 603 351 374 0.94 558 429 447 1.34 596 471 503 0.95 564 494 516 1.35 602 408 435 0.96 562 528 552 1.36 599 422 450 0.97 575 488 510 1.37 596 430 460 0.98 573 516 540 1.38 598 471 504 0.99 574 547 573 1.39 596 545 585 1.00 576 545 571 1.40 591 544 584 1.01 579 522 547 1.41 590 544 584
 e (eV) AM0 Direct Global e (eV) AM0 Direct Global 1.42 591 544 586 1.82 556 471 520 1.43 579 532 572 1.83 554 467 517 1.44 596 544 586 1.84 553 464 513 1.45 554 504 543 1.85 555 464 512 1.46 589 534 577 1.86 556 456 504 1.47 593 530 573 1.87 552 443 489 1.48 589 507 548 1.88 519 422 467 1.49 590 471 508 1.89 519 420 465 1.50 586 453 488 1.90 538 432 477 1.51 587 439 474 1.91 536 425 471 1.52 594 481 521 1.92 542 439 486 1.53 587 521 566 1.93 533 430 474 1.54 585 514 558 1.94 539 430 475 1.55 584 514 557 1.95 535 422 467 1.56 581 509 555 1.96 526 403 446 1.57 592 525 572 1.97 537 403 446 1.58 586 526 572 1.98 519 401 442 1.59 587 526 572 1.99 528 406 450 1.60 583 519 567 2.00 521 404 448 1.61 576 452 493 2.01 511 396 439 1.62 587 212 229 2.02 519 400 443 1.63 584 414 452 2.03 518 397 441 1.64 583 517 564 2.04 522 396 440 1.65 578 511 560 2.05 513 386 430 1.66 577 508 557 2.06 510 383 427 1.67 560 488 535 2.07 513 379 422 1.68 575 485 532 2.08 511 373 413 1.69 570 447 490 2.09 507 369 411 1.70 565 414 454 2.10 481 347 386 1.71 570 427 469 2.11 504 376 420 1.72 559 384 421 2.12 509 380 424 1.73 562 477 526 2.13 499 369 411 1.74 566 485 535 2.14 492 356 398 1.75 567 479 527 2.15 487 351 393 1.76 561 462 509 2.16 496 358 400 1.77 561 467 514 2.17 473 342 383 1.78 564 444 491 2.18 484 354 396 1.79 565 420 462 2.19 466 342 383 1.80 555 396 434 2.20 468 343 385 1.81 556 471 519 2.21 458 336 377

 e (eV) AM0 Direct Global e (eV) AM0 Direct Global 2.22 458 337 379 2.62 365 241 284 2.23 466 342 385 2.63 349 229 271 2.24 460 337 381 2.64 356 233 277 2.25 455 333 376 2.65 339 221 263 2.26 448 326 368 2.66 349 228 271 2.27 451 328 371 2.67 355 230 272 2.28 443 322 365 2.68 355 229 273 2.29 415 302 342 2.69 341 218 261 2.30 434 316 358 2.70 342 218 261 2.31 447 323 366 2.71 347 220 265 2.32 415 299 339 2.72 333 212 253 2.33 443 318 361 2.73 324 204 245 2.34 433 310 354 2.74 344 215 260 2.35 392 282 320 2.75 338 210 254 2.36 429 309 351 2.76 325 201 244 2.37 409 295 335 2.77 320 198 240 2.38 394 283 323 2.78 298 182 221 2.39 356 254 291 2.79 304 184 226 2.40 403 288 330 2.80 302 182 225 2.41 392 279 320 2.81 284 171 209 2.42 414 294 337 2.82 248 148 183 2.43 406 286 328 2.83 274 163 201 2.44 396 277 319 2.84 286 170 210 2.45 410 286 328 2.85 248 146 181 2.46 386 268 309 2.86 255 149 185 2.47 377 263 304 2.87 219 127 158 2.48 385 269 311 2.88 180 104 130 2.49 394 274 316 2.89 234 135 168 2.50 395 274 318 2.90 241 137 172 2.51 383 266 308 2.91 252 143 179 2.52 371 255 297 2.92 247 139 175 2.53 372 256 296 2.93 251 141 178 2.54 357 244 283 2.94 258 144 182 2.55 340 231 269 2.95 237 131 166 2.56 378 256 298 2.96 238 130 166 2.57 385 258 302 2.97 248 135 172 2.58 381 254 297 2.98 249 135 173 2.59 378 250 293 2.99 236 127 163 2.60 367 243 285 3.00 244 130 168 2.61 371 246 289 3.01 234 123 159
 e (eV) AM0 Direct Global e (eV) AM0 Direct Global 3.02 208 109 142 3.42 106 39 59 3.03 236 123 160 3.43 101 36 55 3.04 217 112 146 3.44 110 39 60 3.05 221 114 148 3.45 82 29 45 3.06 229 117 153 3.46 84 29 46 3.07 227 115 151 3.47 92 32 50 3.08 235 119 157 3.48 108 37 58 3.09 226 113 150 3.49 115 39 62 3.10 213 106 140 3.50 111 37 59 3.11 161 79 106 3.51 95 32 50 3.12 94 46 61 3.52 99 32 52 3.13 156 75 101 3.53 105 34 55 3.14 114 54 73 3.54 95 30 49 3.15 98 46 63 3.55 90 28 46 3.16 160 75 102 3.56 90 28 46 3.17 157 73 100 3.57 93 29 47 3.18 139 64 88 3.58 92 28 46 3.19 121 56 77 3.59 82 25 41 3.20 124 56 78 3.60 87 26 43 3.21 121 54 75 3.61 98 29 48 3.22 114 51 71 3.62 95 28 47 3.23 85 38 53 3.63 90 26 44 3.24 104 46 64 3.64 98 27 46 3.25 140 61 85 3.65 90 25 43 3.26 130 56 79 3.66 86 23 41 3.27 150 64 91 3.67 77 20 35 3.28 149 63 90 3.68 75 20 34 3.29 126 53 76 3.69 84 22 39 3.30 117 49 70 3.70 90 23 41 3.31 105 43 62 3.71 85 21 36 3.32 119 48 70 3.72 86 21 38 3.33 131 53 77 3.73 88 21 38 3.34 130 52 76 3.74 87 20 36 3.35 141 56 82 3.75 94 22 39 3.36 127 50 73 3.76 91 21 38 3.37 135 52 77 3.77 81 18 32 3.38 139 53 79 3.78 84 17 31 3.39 123 47 70 3.79 86 18 34 3.40 113 43 64 3.80 81 16 30 3.41 112 42 63 3.81 70 13 23

 e (eV) AM0 Direct Global e (eV) AM0 Direct Global 3.82 64 12 23 4.12 31 0 0 3.83 56 10 19 4.13 33 0 0 3.84 59 10 18 4.14 35 0 0 3.85 60 10 19 4.15 33 0 0 3.86 65 11 20 4.16 37 0 0 3.87 64 9 16 4.17 34 0 0 3.88 59 8 16 4.18 38 0 0 3.89 57 7 14 4.19 39 0 0 3.90 65 8 14 4.20 36 0 0 3.91 56 7 13 4.21 36 0 0 3.92 47 5 9 4.22 38 0 0 3.93 52 5 10 4.23 36 0 0 3.94 55 5 9 4.24 38 0 0 3.95 57 4 8 4.25 40 0 0 3.96 54 4 8 4.26 41 0 0 3.97 55 3 7 4.27 38 0 0 3.98 58 3 6 4.28 32 0 0 3.99 50 2 5 4.29 24 0 0 4.00 39 1 3 4.30 20 0 0 4.01 46 1 3 4.31 22 0 0 4.02 49 1 2 4.32 23 0 0 4.03 47 1 2 4.33 15 0 0 4.04 43 0 1 4.34 10 0 0 4.05 44 0 1 4.35 13 0 0 4.06 48 0 1 4.36 19 0 0 4.07 46 0 0 4.37 20 0 0 4.08 47 0 0 4.38 20 0 0 4.09 41 0 0 4.39 17 0 0 4.10 33 0 0 4.40 13 0 0 4.11 34 0 0 4.41 9 0 0

xThe mathematics of Hubbert’s theory is similar to the equations created by Pierre Francois Ver – hurst in 1838 to quantify Malthus’s theory on population growth [85].

[2]By definition, sechx = 1/cosh x = 2/(ex + e x).

[3]In the literature, the notation FF is often used to represent the fill factor, a factor of efficiency in the treatment of solar cells by Shockley and Queisser. To maintain consistency of notations, we use nf instead.

[4] , „ x. xc + ln C

nd — — In (C eXc) ———————– — 1,

xc xc

because the expression C varies much less than the exponential. Therefore, at very low cell temperatures, if there is no other recombination mechanism other than radiative recombination, the open-circuit voltage approaches the band gap energy in volts and the nominal efficiency approaches the ultimate efficiency.

The reverse saturation current in Eq. 9.38 is the absolute limit of the observed reverse saturation current (Eq. 9.29). In addition to radiative recombination, there are other types of recombination processes that increase the reverse saturation current and then reduce the efficiency of solar cells; see Section 9.3.

Updated: August 25, 2015 — 3:34 am