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.

Подпись: d dNi Подпись: S + a image850 Подпись: 0. Подпись: (D.3)

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

image847

(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

Подпись: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

 

image854

in other words,

 

image855

image856

Подпись: (D.17), —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

Подпись:Подпись:Подпись: Wi =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 + /3Ei
кв

 

(D.21)

 

Ni

 

exp

 

+ 1

 

image686

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

1

 

(D.22)

 

pi

 

Ei — Ep

квТ

 

exp

 

+ 1

 

image861

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

Подпись: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.

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