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We may note that silicon and gallium arsenide will absorb all of the visible spectrum, whereas gallium phosphide, for example, will be transparent to the red spectrum.
#### 14.1.2 Electron-Hole Pair Generation Rate
We have shown that photons with energy greater than $E_g$ can be absorbed in a semiconductor, there... | {
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Consider a silicon pn junction at T = 300 K with the following parameters:
$$N_a = 5 \times 10^{18} \text{ cm}^{-3}$$
$N_d = 10^{16} \text{ cm}^{-3}$
$D_n = 25 \text{ cm}^2/\text{s}$ $D_p = 10 \text{ cm}^2/\text{s}$
$\tau_{n0} = 5 \times 10^{-7} \text{ s}$ $\tau_{n0} = 10^{-7} \text{ s}$
Let the photocurrent ... | {
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Figure 14.4 shows the absorption coeffi cient as a function of wavelength for several semiconductor materials. As the absorption coeffi cient increases, more photon energy will be absorbed near the surface than deeper into the semiconductor. In this case, then, we will not have uniform excess carrier generation in a so... | {
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The pn junction is the basis of several photodetector devices including the photodiode and the phototransistor.
#### 14.3.1 Photoconductor
Figure 14.16 shows a bar of semiconductor material with ohmic contacts at each end and a voltage applied between the terminals. The initial thermal-equilibrium conductivity is ... | {
"Header 1": "**Optical Devices**",
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We may note that *JL*1 is in the reverse-biased direction through the pn junction. This component of photocurrent responds very quickly to the photon illumination and is known as the prompt photocurrent.
We may note, by comparing Equations (14.28) and (14.25), that the photodiode gain is unity. The speed of the photo... | {
"Header 1": "**Optical Devices**",
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#### **■ Solution**
We may calculate various parameters as follows:
$$L_n = \sqrt{D_n \tau_{n0}} = \sqrt{(25)(5 \times 10^{-7})} = 35.4 \,\mu\text{m}$$
$$L_p = \sqrt{D_p \tau_{p0}} = \sqrt{(10)(10^{-7})} = 10.0 \,\mu\text{m}$$
$$V_{bi} = V_t \ln \left(\frac{N_a N_d}{n_i^2}\right) = (0.0259) \ln \left[\frac{(1... | {
"Header 1": "**Optical Devices**",
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The phototransistor can have high gain through the transistor action. An npn bipolar phototransistor is shown in Figure 14.20a. This device has a large base–collector junction area and is usually operated with the base open circuited. Figure 14.20b shows the block diagram of the phototransistor. Electrons and holes gen... | {
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The peak photon energy decreases with temperature because the
Wavelength ( $\mu$ m)
1.0 0.95 0.9 0.85
100
0.036 eV
0.018 eV
10-3
Resolution $\rightarrow$ | $\rightarrow$ | $\rightarrow$ | $\rightarrow$ | $\rightarrow$ | $\rightarrow$ | $\rightarrow$ | $\righta... | {
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Figure 14.22 | GaAs diode emission spectra at T = 300 K and T = 77 K. (From Sze and Ng [17].)
bandgap energy decreases with temperature. We will show that the bandwidth of the emission spectra can be greatly reduced in a laser diode by using an optical resonator.
#### 14.4.2 Luminescent Efficiency
We have shown... | {
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These current densities can be written, respectively, as
$$J_n = \frac{eD_n n_{p0}}{L_n} \left[ \exp\left(\frac{eV}{kT}\right) - 1 \right]$$
(14.54a)
$$J_p = \frac{eD_p p_{n0}}{L_p} \left[ \exp\left(\frac{eV}{kT}\right) - 1 \right]$$
(14.54b)
and
$$J_R = \frac{en_i W}{2\tau_0} \left[ \exp\left(\frac{eV}{2kT}\ri... | {
"Header 1": "**Optical Devices**",
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\theta_c = 16.3^\circ)$$
Figure 14.28a shows the external quantum efficiency plotted as a function of the p-type doping concentration and Figure 14.28b is a plot of the external efficiency as a function of junction depth below the surface. Both figures show that the external quantum efficiency is in the range of 1 to... | {
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The gain factor in a pn homojunction diode is given by
$$\gamma(\nu) \propto \left\{ 1 - \exp\left[\frac{h\nu - (E_{Fn} - E_{Fp})}{kT}\right] \right\}$$
(14.66)
In order for $\gamma(\nu) > 1$ , we must have $h\nu < (E_{Fn} - E_{Fp})$ , which implies that the junction must be degenerately doped since we also have ... | {
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When the diode current becomes large, a single dominant mode with a narrow bandwidth is produced.
The performance of the laser diode can be further improved if a very narrow recombination region is used with a somewhat wider optical waveguide. Very complex structures using multilayers of compound semiconductor materi... | {
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(*b*) At what distance in the semiconductor does the generation rate drop to 10 percent of that at the surface?
- **14.6** Consider a silicon semiconductor that is illuminated with photons with energies *h-* 1.40 eV. (*a*) Determine the thickness of the material such that 90 percent of the energy is absorbed. (*b*) Det... | {
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For 3 volts applied to the photoconductor, determine (*a*) the thermal equilibrium current, (*b*) the steady-state excess carrier concentration, (*c*) the photoconductivity, (*d*) the steady-state photocurrent, and (*e*) the photocurrent gain.
- **14.20** Excess carriers are uniformly generated in a GaAs photoconductor... | {
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If $N \gg 1$ , show that the wavelength separation between two adjacent resonant modes is $\Delta \lambda = \lambda^2/2L$ .
- 14.35 If the photon output of a laser diode is equal to the bandgap energy, find the wavelength separation between adjacent resonant modes in a GaAs laser with $L = 75 \ \mu \text{m}$ .
###... | {
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n previous chapters, we have discussed the basic physics, operation, and characteristics of diodes and transistors. We have analyzed the frequency response as well as the current–voltage characteristics of these semiconductor devices. However, we have not specifically considered the generation of microwave signals usin... | {
"Header 1": "C H A P T E R",
"Header 3": "Semiconductor Microwave and Power Devices",
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As the domain grows (Figure 15.6a), the electric fi eld in this region increases which means that the electric fi eld in the remaining material decreases. The E fi eld in the material outside of the domain can drop below the critical value, as indicated in Figure 15.6b, while the E fi eld within the domain remains abov... | {
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Table 15.1 compares the parameters of a
| Parameter | Small-signal BJT<br>(2N2222A) | Power BJT<br>(2N3055) | Power BJT<br>(2N6078) |
|--------------------------------|-------------------------------|-----------------------|-----------------------|
| VCE (max) (V) | 40 ... | {
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(a)
$$R_L = 30 \Omega$$
, $I_C(max) = 2 \Lambda$ , $P(max) = 30 W$ ; (b) $R_L = 13.3 \Omega$ , $I_C(max) = 3 \Lambda$ , $P(max) = 3 \Omega$ , $P(max) = 3 \Omega$ , $P(max) = 3 \Omega$ , $P(max) = 3 \Omega$ , $P(max) = 3 \Omega$ , $P(max) = 3 \Omega$ , $P(max) = 3 \Omega$ , $P(max) = 3 \Omega$ , $P(max) = 3... | {
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#### TEST YOUR UNDERSTANDING
**TYU 15.1** Consider the vertical power silicon BJT shown in Figure 15.10. Assume that a reverse-biased voltage of 200 V is applied to the base–collector junction. Calculate the space charge width that extends into the (a) collector region and (b) base region.
$$[m_{4} \partial \part... | {
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Figure 15.24b is a plot of drain current versus gate-to-source voltage at three different temperatures. We may note that at high current, the current decreases with temperature at a constant gate-to-source voltage, providing the stability that has been discussed.
Power MOSFETs must operate in a SOA. As with power BJT... | {
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A.
$$A = A = A + A$$
, $A = A + A$ , $A = A + A$ , $A = A + A$ , $A = A + A$ , $A = A + A$ , $A = A + A$ , $A = A + A$ , $A = A + A$ , $A = A + A$ , $A = A + A$ , $A = A + A$ , $A = A + A$ , $A = A + A$ , $A = A + A$ , $A = A + A$ , $A = A + A$ , $A = A + A$ , $A = A + A$ , $A = A + A$ , $A = A + A$... | {
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#### **15.6 | THE THYRISTOR**
One of the important applications of electronic devices is in switching between an off or blocking state to an on or low-impedance state. Thyristor is the name given to a general class of semiconductor pnpn switching devices that exhibit bistable regenerative switching characteristics.... | {
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We can write
$$I_{C1} = \alpha_1 I_A + I_{C01} \tag{15.14a}$$
and
$$I_{C2} = \alpha_2 I_K + I_{C02} \tag{15.14b}$$
We now have $I_K = I_A + I_g$ and we can still write $I_{C1} + I_{C2} = I_A$ . Adding Equations (15.14a) and (15.14b), we find that
$$I_{C1} + I_{C2} = I_A = (\alpha_1 + \alpha_2)I_A + \alpha_... | {
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(From Ghandhi [7].)
Figure 15.39 | The current–voltage characteristics of the triac.
One particular gate control situation is shown in Figure 15.38b. Terminal 1 is positive with respect to terminal 2, and a negative gate voltage is applied with respect to terminal 1, so the gate curr... | {
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#### **CHECKPOINT**
After studying this chapter, the reader should have the ability to:
- Explain how a region of negative differential resistance is developed in the *I*–*V* characteristic of the tunnel diode.
- Discuss the concept of negative differential mobility in GaAs and discuss how this phenomenon leads t... | {
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"Header 3": "Semiconductor Microwave and Power Devices",
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(*a*) The on resistances of the three devices are *R*on1 1.8 , *R*on2 2 , and *R*on3 2.2 . Calculate the current in each device and the power dissipated in each device. (*b*) For some unknown reason, the on resistance of the second device increases to *R*on2 3.6 . Recalculate the current in each device and the power di... | {
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"Header 3": "Semiconductor Microwave and Power Devices",
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| a | Unit cell dimension (Å), potential well width, acceleration, gradient of impurity concentration, channel thickness of a |
|------------|-------------------------------------------------------------------------------------------------------------------------|
| | one-sided JFET (cm) ... | {
"Header 1": "**Selected List of Symbols**",
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**Table B.1** | International system of units\*
| Quantity | Unit | Symbol | Dimension |
|------------------------|----------|----------|-----------|
| Length | meter | m | |
| Mass | kilogram | kg | |
| Time |... | {
"Header 1": "**System of Units, Conversion Factors, and General Constants**",
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| $10^{-6}$ | micro- | $=\mu$ |
| $1 \text{ eV} = 1.6 \times 10^{-19} \text{ J}$ | $10^{-3}$ | milli- | = m |
| $1 J = 10^7 \text{ erg}$ | $10^{+3}$ | kilo- | = k |
| ... | {
"Header 1": "**System of Units, Conversion Factors, and General Constants**",
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| | Group I | Group II | Group III | Group IV | Group V | Group VI | Group VII | | Group VIII | |
|--------|-----------|-------------|-------------|-----------|-------------|---------------|----------------|---------|--------------------|--------|
| Period | a b ... | {
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"Header 2": "The Periodic Table",
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The electron volt (eV) is a unit of energy that is used constantly in the study of semiconductor physics and devices. This short discussion may help in "getting a feel" for the electron-volt.
Consider a parallel-plate capacitor with an applied voltage as shown in Figure D.1. Assume that an electron is released at x =... | {
"Header 1": "Unit of Energy—The Electron Volt",
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S chrodinger's wave equation is stated in Equation (2.6). The time-independent form of Schrodinger's wave equation is then developed and given by Equation (2.13). The time-independent Schrodinger's wave equation can also be developed from the classical wave equation. We may think of this development more in terms of a ... | {
"Header 1": "\"Derivation\" of Schrodinger's Wave Equation",
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In Chapter 3, we have discussed the relationship between the effective masses of electrons and holes and the E versus k diagrams. In that discussion, we have limited ourselves to a one-dimensional analysis in k space.
#### F.1 | ENERGY-BAND STRUCTURES
**GaAs Energy Bands:** The E versus k diagram for GaAs is given ... | {
"Header 1": "**Effective Mass Concepts**",
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For a simple electron gas, the electron kinetic energy can be written as
$$E = \frac{1}{2}m*(\langle v_d \rangle)^2 = \frac{p_x^2}{2m_{cn}^*} + \frac{p_y^2}{2m_{cn}^*} + \frac{p_z^2}{2m_{cn}^*}$$
For the case of silicon and the ellipsoid energy surface, we have
$$E = \frac{p_x^2}{2m_t} + \frac{p_y^2}{2m_t} + \f... | {
"Header 1": "**Effective Mass Concepts**",
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$$\operatorname{erf}(z) = \frac{2}{\sqrt{\pi}} \int_0^z e^{-t^2} dt$$
$$\operatorname{erf}(0) = 0 \qquad \operatorname{erf}(\infty) = 1$$
$$\operatorname{erfc}(z) = 1 - \operatorname{erf}(z)$$
| z | erf(z) | z | $\mathbf{erf}(z)$ |
|------|---------|------|-------------------|
| 0.00 | 0.00000 | 1.00 | 0.8... | {
"Header 1": "**Effective Mass Concepts**",
"Header 2": "**The Error Function**",
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#### ANSWERS TO SELECTED PROBLEMS
#### Chapter 1
- **1.1** (a) 4 atoms, (b) 2 atoms, (c) 8 atoms
- 1.3 (a) 2.35 Å, (b) $5 \times 10^{22}$ atoms/cm<sup>3</sup> (c) 2.33 gm/cm<sup>3</sup>
- **1.5** (a) 2.447 Å, (b) 3.995 Å
- **1.7** (a) 3.9 Å, (b) 5.515 Å, (c) 4.503 Å, (d) 9.007 Å
- **1.9** (a) 0.228 gm/cm<sup>3</s... | {
"Header 1": "A P P E N D I X",
"token_count": 3472,
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- (c) $n_i = 1.38 \text{ cm}^{-3}$ ; $3.28 \times 10^9 \text{ cm}^{-3}$ ; $5.72 \times 10^{12} \text{ cm}^{-3}$
- **4.3** (a) $T \cong 367.5 \text{ K}$ , (b) $T \cong 417.5 \text{ K}$
- **4.5** (a) $9.325 \times 10^{-6}$ , (b) $4.43 \times 10^{-4}$ , (c) $3.05 \times 10^{-3}$
- **4.7** 0.0854
- **4.11** For T... | {
"Header 1": "A P P E N D I X",
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0.5754 V
$10^{16} \text{ cm}^{-3}$ . 0.6946 V
$10^{17} \, \text{cm}^{-3}$ 0.8139 V
(b) For $N_a = N_d = 10^{14} \text{ cm}^{-3}$ , $V_{bi} = 0.9237 \text{ V}$
$10^{15} \text{ cm}^{-3}$ . 1.043 V
$10^{16} \text{ cm}^{-3}$ . 1.162 V
$10^{17} \text{ cm}^{-3}$ . 1.282 V
(c) Silicon:
For $N_a = N_d = 10^{... | {
"Header 1": "A P P E N D I X",
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(c) $V_R = 2.96 \text{ V}$
- **7.33** (a) $V_{bi} = V_t \ln \left[ \frac{N_{aO} N_{dO}}{n_i^2} \right],$
- (c) p region: $E = \frac{-eN_{a0}}{\epsilon}(x + x_p);$ n region: $0 < x < x_o$ , $E = \frac{eN_{dO}x}{2\epsilon} - \frac{eN_{dO}}{\epsilon}$ $\times (x_n - \frac{x_o}{2});$ $x_o < x < x_n, E = \frac{-eN_{... | {
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$1.45 \times 10^{-5} \,\mathrm{A}$ : 1.0 V: $1.99 \times 10^{-2} \text{ A}.$ 1.2 V: $6.90 \times 10^{-4} \,\mathrm{A}$
- **8.35** $J_{gen} = 1.5 \times 10^{-3} \text{ A/cm}^2$
- **8.37** (a) $r_d = 21.6 \Omega$ , $C_d = 11.6 \text{ nF}$ : (b) $r_d = 216 \Omega$ , $C_d = 1.16 \text{ nF}$
- **8.39** For 10 kHz, Z... | {
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0.422 mA
- **10.39** (a) $V_{GS} = 0 \text{ V}, I_D(\text{sat}) = 0.711 \text{ mA}$ 0.8 V, 2.84 mA 1.6 V, 6.40 mA
- **10.43** For $V_{SG} < 0.35$ V, $g_d = 0$ ;
For $V_{SG} > 0.35$ V, $g_d = 2(0.961)(V_{SG} - 0.35)$
**10.45** $V_T \approx 0.2 \text{ V}, \, \mu_n = 342 \text{ cm}^2/\text{V-s}$
**10.47** (a... | {
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| A | Aluminum phosphide (AlP), 2 |
|------------------------------------------------------|----------------------------------------------------|
| Abrupt junction approximation, 268 | Ambipolar diffusion, 201 ... | {
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"Header 3": "**INDEX**",
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See Schottky barrier height |
| heterojunction LED, 653–654 | Base current, 497–498, 554 |
| Base Gummel number, 541 | emitter bandgap narrowing, 526–528 |
|-----------------------------------------------------|-... | {
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See Common-base current gain; |
| Conduction bands, 74–75, 77, 81 | Common-emitter current gain |
| Conduction parameter, 410, 418, 431, 591, 610 | Current-voltage (C-V) relationship. See I-V relationship/ |
| Conduction-band edge ... | {
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See Glossary of terms |
| Critical electric fi eld, 268 | Degenerate n-type semiconductor, 130 |
| Crystal momentum, 72 | Degenerate p-type semiconductor, 130 |
| Crystal planes, 6–7 ... | {
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See also Allowed energy bands; |
| Electron and hole | Forbidden energy bands |
| concentrations, 107, 113, 123–124, 135–141 | Energy quanta, 26–27 |
| mobilities, 162–163 | Energy s... | {
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See also |
| nonuniform donor impurity concentration, 176 | Nonequilibrium excess carriers |
| npn bipolar transistor, 494 | Excess electron concentration, 212, 504–505 |
| npn bipolar transistor (punch-through), 532 | Excess electron p... | {
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See Gallium arsenide (GaAs) | Gamma function, 110 |
| GaAs-AlGaAs (gallium arsenide-aluminum gallium | GaP, 2, 165, 653 |
| arsenide) | Gate capacitance charging time (JFET), 600, 609 |
| HEMT, 604–60... | {
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See HEMT |
| optical devices, 662–663 | High injection, 524–526 |
| pn junction, 268 | High-level injection, 302–304, 322 |
| pn junction diode, 322 | High-speed logic circuits, 60... | {
"Header 1": "A P P E N D I X",
"Header 3": "**INDEX**",
"token_count": 1012,
"source_pdf": "datasets/websources/Physics_v1/Physics/Neamen.pdf"
} |
See also Energy-band | Infi nite surface recombination velocity, 231 |
| diagrams | Injection electroluminescence, 644, 648, 662 |
| metal-n-semiconductor junction, 332, 349 | InP (indium phosphide), 2, 621, 672 ... | {
"Header 1": "A P P E N D I X",
"Header 3": "**INDEX**",
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"source_pdf": "datasets/websources/Physics_v1/Physics/Neamen.pdf"
} |
See JFET | | |
| ideal I-V characteristic of a pn junction diode, | | | |
| 288–289 | K | | |
| JFET, 582–587 | Kinet... | {
"Header 1": "A P P E N D I X",
"Header 3": "**INDEX**",
"token_count": 1654,
"source_pdf": "datasets/websources/Physics_v1/Physics/Neamen.pdf"
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See Semiconductor/ | | |
| M | microwave power devices | | |
| Magnetic quantum number (m), 48 | Midgap, 402 | | |
| Majority carrier concentration, 182 ... | {
"Header 1": "A P P E N D I X",
"Header 3": "**INDEX**",
"token_count": 1062,
"source_pdf": "datasets/websources/Physics_v1/Physics/Neamen.pdf"
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See MOSFET | MOS turn-off thyristor, 700–701 | | |
| MOSFET (metal-oxide-semiconductor fi eld-effect | subthreshold current, 445–446, 478 |
|----------------------------------------------------|------------------------------------------... | {
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"Header 3": "**INDEX**",
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"source_pdf": "datasets/websources/Physics_v1/Physics/Neamen.pdf"
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See also Metal-semiconductor | P-n-i-n, 675 |
| ohmic contacts | Palladium (Pd), 333 |
| Ohm's law, 214, 410, 583 | Parabolic relationship between energy and momentum, |
| On resi... | {
"Header 1": "A P P E N D I X",
"Header 3": "**INDEX**",
"token_count": 2847,
"source_pdf": "datasets/websources/Physics_v1/Physics/Neamen.pdf"
} |
See Semiconductor/microwave | density of, 85–90, 94, 98 | | |
| power devices | Pauli exclusion principle, 60, 85 | | |
| Power MOSFET, 684–689, 701 | Quantum theory of solids, 58–105 ... | {
"Header 1": "A P P E N D I X",
"Header 3": "**INDEX**",
"token_count": 1144,
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See also SOA |
| Radiative effi ciency, 645–646 | Safety margin, 464 |
| Radiative recombination, 650, 663 | Saturation, 432, 499–500 |
| Radiative recombination rate, 646 | Saturation cond... | {
"Header 1": "A P P E N D I X",
"Header 3": "**INDEX**",
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See Heterojunctions | HEXFET, 685, 702 |
| Semiconductor in equilibrium, 106–155 | IMPATT diode, 675–677 |
| ca... | {
"Header 1": "A P P E N D I X",
"Header 3": "**INDEX**",
"token_count": 1981,
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} |
See also | |
| Silicon-silicon dioxide (Si-SiO2) interface, 471, 476–480 | Depletion region | |
| Silver (Ag), 333 | Space charge width, 249–254, 265, 268, 379 | |
| Simple cubic structure (sc), 4–5, 7 ... | {
"Header 1": "A P P E N D I X",
"Header 3": "**INDEX**",
"token_count": 1514,
"source_pdf": "datasets/websources/Physics_v1/Physics/Neamen.pdf"
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See Notations/symbols | Thyristor, 691–702 | | |
| Symbols, list of, 707–714 | avalanche breakdown, 693 | | |
| Symmetry effect, 96 | bilateral, 697–698 ... | {
"Header 1": "A P P E N D I X",
"Header 3": "**INDEX**",
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See also Ambipolar transport; | V-groove MOSFET (VMOS), 684–685, 702 | | |
| Carrier transport phenomenon | Visible spectrum, 622, 645 | | |
| Transverse electric fi eld, 451 | VLSI (very large scale integrated) circuits, ... | {
"Header 1": "A P P E N D I X",
"Header 3": "**INDEX**",
"token_count": 908,
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| 1 | | The story so far | 2 | | | | | | |
|---|-------------------|------------------------------------------------|-----|--|--|--|--|--|--|
| | 1.1 | Prequel: substructure and atoms | 2 | | | | | | |
| | 1.2 ... | {
"Header 1": "Contents",
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This section contains some preliminary background information needed to tell this story, and starts by summarizing the first indications that there might be an interesting story to tell.
#### 1.1 Prequel: substructure and atoms
The evidence that many of the properties of macroscopic things are best understood if th... | {
"Header 1": "1 The story so far...",
"token_count": 2010,
"source_pdf": "datasets/websources/Physics_v1/Physics/PPNotes.pdf"
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[\(1.1\)](#page-3-0) turn out to contain stillsmaller teeny-weeny atoms, each of which themselves satisfy Newtons 2nd law, then nothing in the above arguments need change at all (provided we assume the position x<sup>i</sup> to be an appropriately defined centre-of-mass coordinate).
Of course just because the interna... | {
"Header 1": "1 The story so far...",
"token_count": 596,
"source_pdf": "datasets/websources/Physics_v1/Physics/PPNotes.pdf"
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The first discovery of a particle we now still regard as elementary occurred during the closing years of the 19th century. Many of these developments arose as unintended consequences of the discovery of cathode rays that are the result of applying a high voltage to a small amount of gas inside an otherwise evacuated tu... | {
"Header 1": "1.1.1 More than just atoms",
"token_count": 325,
"source_pdf": "datasets/websources/Physics_v1/Physics/PPNotes.pdf"
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In 1896 Becquerel, in Paris, ran experiments seeking to determine whether fluorescent materials could be made to emit x-rays through exposure to sunlight. To this end he took a good fluorescent compound, wrapped it in dark paper and placed it next to a photographic plate, intending to expose it to sunlight. Although hi... | {
"Header 1": "1.1.2 X rays",
"Header 2": "1.1.3 Radioactivity",
"token_count": 600,
"source_pdf": "datasets/websources/Physics_v1/Physics/PPNotes.pdf"
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The nature of the cathode rays themselves was initially confused because of early experiments that incorrectly indicated that they were not deflected by magnetic fields (and so must be electrically neutral). (In retrospect these experiments were wrong because they were not able to get a good enough vacuum in the tube, ... | {
"Header 1": "1.1.2 X rays",
"Header 2": "1.1.4 The electron",
"token_count": 1426,
"source_pdf": "datasets/websources/Physics_v1/Physics/PPNotes.pdf"
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A separate benefit of these various early experiments was the road they opened up to detecting the presence of these various types of new particles. After all, radioactivity and x-rays had always been around but went undetected because they were invisible to our senses. Their presence was eventually found due to their ... | {
"Header 1": "1.1.2 X rays",
"Header 2": "1.1.5 Detection methods",
"token_count": 2034,
"source_pdf": "datasets/websources/Physics_v1/Physics/PPNotes.pdf"
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Middle panel: the spectrum of light emitted by a hot gas, showing the characteristic frequencies that are emitted by atomic transitions. Bottom panel: the absorption spectrum seen when light from a source emitting a continuous spectrum is viewed after being partially absorbed by passing through the same gas. Notice tha... | {
"Header 1": "1.1.2 X rays",
"Header 2": "1.1.5 Detection methods",
"token_count": 1915,
"source_pdf": "datasets/websources/Physics_v1/Physics/PPNotes.pdf"
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\tag{1.22}$$
Because each term in this equation involves only $\mathbf{x}_1$ or $\mathbf{x}_2$ (and not both at once) it is always satisfied by the solution
$$\Psi(\mathbf{x}_1, \mathbf{x}_1, t) = \psi(\mathbf{x}_1, t) \,\tilde{\psi}(\mathbf{x}_2, t), \qquad (1.23)$$
where $\psi(\mathbf{x})$ and $\tilde{\p... | {
"Header 1": "1.1.2 X rays",
"Header 2": "1.1.5 Detection methods",
"token_count": 2042,
"source_pdf": "datasets/websources/Physics_v1/Physics/PPNotes.pdf"
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Enrico Fermi found in 1934 that stable elements could be induced to become radioactive by bombarding them with neutrons, and by 1938 Otto Hahn, Lise Meitner and Fritz Strassmann discovered nuclear *fission* when they found that bombardment by neutrons could also split heavy nuclei into much much smaller pieces than hap... | {
"Header 1": "1.1.2 X rays",
"Header 2": "1.1.5 Detection methods",
"token_count": 1719,
"source_pdf": "datasets/websources/Physics_v1/Physics/PPNotes.pdf"
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A selection of scales known to arise in nature
| Measure in eV | Physical systems with these dimensions |
|------------------------------------|----------------------------------------------------------------------------------|
| 10−32 eV ... | {
"Header 1": "1.1.2 X rays",
"Header 2": "1.1.5 Detection methods",
"token_count": 400,
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It is particularly useful to combine the above choices and so both adopt fundamental units and express all remaining quantities in dimensions that are a power of energy, with energy measured in electron-Volts. This is very useful because the world around us is built from atoms and nuclei and so the scale of many phenom... | {
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"Header 2": "1.1.5 Detection methods",
"Header 3": "1.2.3 Hierarchies of scale",
"token_count": 1105,
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In reality any measurement has errors, and so any inference of physical properties (like particle masses, or energy levels, or decay rates) is uncertain inasmuch as repeated 'identical' measurements can return different values for the same quantity. This subsection provides a cartoon of some aspects of how these errors... | {
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"Header 2": "1.3 Lies, Damn Lies, and Measurement Errors",
"token_count": 2031,
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Similarly prove that $\sigma^2$ is its variance, inasmuch as $\langle (x - \mu)^2 \rangle = \langle x^2 \rangle - \mu^2 = \sigma^2$ .
In principle, p(x) can be used to extract the probability that a measurement of x returns a result in any given range of values. For instance $\int_a^b dx \, p(x) = P(a,b)$ is the... | {
"Header 1": "1.1.2 X rays",
"Header 2": "1.3 Lies, Damn Lies, and Measurement Errors",
"token_count": 848,
"source_pdf": "datasets/websources/Physics_v1/Physics/PPNotes.pdf"
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If errors are described by a Normal distribution, the value µ is the 'real' value the experimenter is trying to measure and σ is the size of the error forced on him/her by the measurement technique used. The goal is to design the experiment to minimize σ and thereby return measured values that are as close as possible ... | {
"Header 1": "1.1.2 X rays",
"Header 2": "Sampling and statistics",
"token_count": 2033,
"source_pdf": "datasets/websources/Physics_v1/Physics/PPNotes.pdf"
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On the same plot graph the Normal distribution with the same variance and mean and thereby see the Central Limit Theorem in action. A criterion for how big N must be in order to approximate the Binomial distribution with the Normal distribution is to ask for the 3-σ range of the Normal distribution, (µ − 3 σ, µ + 3 σ),... | {
"Header 1": "1.1.2 X rays",
"Header 2": "Sampling and statistics",
"token_count": 1045,
"source_pdf": "datasets/websources/Physics_v1/Physics/PPNotes.pdf"
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From a practitioner's perspective Special Relativity is the statement that the laws of physics (i.e. of nature) are invariant under a symmetry, so before diving in it is worth first reviewing how things work for a similar symmetry: the invariance of nature's laws under rotation of an observer's reference frame.
Laws ... | {
"Header 1": "1.1.2 X rays",
"Header 2": "1.4.1 Rotational invariance",
"token_count": 2016,
"source_pdf": "datasets/websources/Physics_v1/Physics/PPNotes.pdf"
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This shows that the matrix R is not an arbitrary one because it must satisfy the condition RR<sup>T</sup> = I where I is the unit matrix (whose components are δik); that is to say R must be an orthogonal matrix.[8](#page-32-0)
Since (RR<sup>T</sup> ) <sup>T</sup> = RR<sup>T</sup> is a 3 by 3 symmetric matrix, it has ... | {
"Header 1": "1.1.2 X rays",
"Header 2": "1.4.1 Rotational invariance",
"token_count": 2022,
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\tag{1.79}$$
Transformations that satisfy [\(1.79\)](#page-34-0) are called Lorentz transformations, and because Λ<sup>T</sup> η Λ is a symmetric 4 by 4 matrix they impose 10 conditions on the 16 components of Λ, leaving a 6-parameter family of symmetries. But three of these parameters are old friends, since when Λ i... | {
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"Header 2": "1.4.1 Rotational invariance",
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Since laws of physics are cast in terms of position, velocity, momentum and acceleration, we next seek to identify the 4-vectors containing each of these.
Consider for these purposes a particle moving along some trajectory r(t) in space, not necessarily with constant velocity. Such a particle sweeps out a world-line ... | {
"Header 1": "1.1.2 X rays",
"Header 2": "1.4.1 Rotational invariance",
"token_count": 1982,
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For instance, consider a trajectory describing a particle that accelerates along the x axis from rest at x = 0, until its speed reaches v = vmax at which point it then decelerates back to rest a distance ` away and then returns to x = 0, again at rest according to the specific rule
$$x^{\mu}(t) = \left\{ ct, x(t), y(... | {
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"Header 2": "1.4.1 Rotational invariance",
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Since much of what we know about subnuclear physics comes from studying collisions and decays, in this section we collect some useful tools for analyzing these types of processes.
Measurement of a decay or scattering rate carries two kinds of information: information following from conservation laws and information t... | {
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Charge assignments for a selection of common particles
| Particle (symbol) | rest mass | J | Q/e | В | L | $L_e$ | $L_{\mu}$ | T ... | {
"Header 1": "2 Calculational tools I",
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A decay process is a reaction in which a single particle transmutes itself into two or more other particles, such as the reaction
$$P \to D_1 + D_2 + D_3 + \dots + D_N$$
, (2.1)
<sup>&</sup>lt;sup>b</sup> Measured in tritium beta decay for $\overline{\nu}_e$ and inferred for $\nu_e$ using CPT.
<sup>&</sup>lt;... | {
"Header 1": "2.2 Decays: general properties",
"token_count": 2004,
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Baryon number also balances because $B_i = B_f = +1$ with $B_i = B(n)$ carried by the decaying neutron while $B_f = B(p) + B(e^-) + B(\overline{\nu}_e) = +1 + 0 + 0$ is carried purely by the final proton. Lepton number is balanced because the initial neutron has $L_i = L(n) = 0$ while the final lepton number is... | {
"Header 1": "2.2 Decays: general properties",
"token_count": 1645,
"source_pdf": "datasets/websources/Physics_v1/Physics/PPNotes.pdf"
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The rest-frame decay probability per unit time (or decay rate), Γ, for a given particle is a characteristic of that particle as intrinsic to it as is its mass or spin. The value of Γ depends on the details of the interactions responsible for the decay, and because of this measurements of Γ can be informative about thes... | {
"Header 1": "2.2.2 Decay rates",
"token_count": 1044,
"source_pdf": "datasets/websources/Physics_v1/Physics/PPNotes.pdf"
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The above methods for measuring Γ are fine if the decay is slow enough. Nuclear decays are seen with an enormous range of half-lives — running from lifetimes in the billions of years down to lifetimes measured in small fractions of a second — so for many of these the above methods suffice. But many other decays are muc... | {
"Header 1": "2.2.2 Decay rates",
"Header 2": "2.2.3 Line widths",
"token_count": 1968,
"source_pdf": "datasets/websources/Physics_v1/Physics/PPNotes.pdf"
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Consequently, the t- and $t_0$ -dependence of the decay contribution to the integrand in (2.16) is
$$\mathcal{A}_{AB|P}\mathcal{A}_{P|CD}\mathcal{A}_{P} e^{-i(E_A + E_B - E_C - E_D)t_0} e^{-i[(E_P - i\Gamma/2) - E_C - E_D](t - t_0)}, \qquad (2.19)$$
and this must be added to $\mathcal{A}_{pr}$ , whose $t_0$ -d... | {
"Header 1": "2.2.2 Decay rates",
"Header 2": "2.2.3 Line widths",
"token_count": 2018,
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Alternatively, the invariance of d3p/E can be seen by starting from the manifestly invariant starting point
$$\int d^4 p \, \delta(p_{\mu} p^{\mu} + m^2) \vartheta(p^0)(\cdots) = \int d^3 \mathbf{p} \, dp^0 \, \delta\left[-(p^0)^2 + E^2\right] \vartheta(p^0)(\cdots) = \int \frac{d^3 \mathbf{p}}{2E} (\cdots)|_{p^0 =... | {
"Header 1": "2.2.2 Decay rates",
"Header 2": "2.2.3 Line widths",
"token_count": 2003,
"source_pdf": "datasets/websources/Physics_v1/Physics/PPNotes.pdf"
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What is the ratio R<sup>π</sup> = Γ(π <sup>+</sup> → e <sup>+</sup>νe)/Γ(π <sup>+</sup> → µ <sup>+</sup>νµ) numerically? Compare your answer with the experimental value for this ratio, which can be found at the Particle Data Group [website](https://pdg.lbl.gov) (with information specifically about π <sup>±</sup> decays... | {
"Header 1": "2.2.2 Decay rates",
"Header 2": "2.2.3 Line widths",
"token_count": 1853,
"source_pdf": "datasets/websources/Physics_v1/Physics/PPNotes.pdf"
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One drawback of the previous section is that it is entirely phrased within the rest-frame of the target, and so the separation of the rate into a luminosity piece and a cross-section piece is not yet Lorentz invariant. This is a drawback because not all experiments are done with motionless targets (an example is a coll... | {
"Header 1": "2.2.2 Decay rates",
"Header 2": "2.3.2 Invariant and differential cross section",
"token_count": 2037,
"source_pdf": "datasets/websources/Physics_v1/Physics/PPNotes.pdf"
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(Figure source: https://commons.wikimedia.org/wiki/Spherical\_polar\_coordinates.png).
We now perform the integral over the delta functions explicitly. We start with the integral over one of the two final momenta, say $\mathbf{p}_D$ , whose integration with the momentum-conserving delta function amounts to everywher... | {
"Header 1": "2.2.2 Decay rates",
"Header 2": "2.3.2 Invariant and differential cross section",
"token_count": 2043,
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It is often true that the physics is invariant under such a rotation, and when this is so the cross section is independent of $\phi$ and so depends nontrivially only on $\theta$ . In this case the angular integral over $\phi$ amounts to multiplication of the result by $2\pi$ , leaving
$$\frac{\mathrm{d}\sigma}{... | {
"Header 1": "2.2.2 Decay rates",
"Header 2": "2.3.2 Invariant and differential cross section",
"token_count": 286,
"source_pdf": "datasets/websources/Physics_v1/Physics/PPNotes.pdf"
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Although formulae like (2.50) and (2.53) have the virtue of explicitness, they obscure Lorentz invariance and so make it more cumbersome to relate observables in different reference frames. For this purpose an alternative set of explicitly Lorentz-invariant variables, called *Mandelstam variables*, are often used inste... | {
"Header 1": "2.2.2 Decay rates",
"Header 2": "2.3.5 $2 \\rightarrow 2$ relativistic variables",
"token_count": 1953,
"source_pdf": "datasets/websources/Physics_v1/Physics/PPNotes.pdf"
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frame to derive the following useful expression for the differential Lorentz-invariant phase space
volume appearing in the cross section:
$$d\chi := (2\pi)^4 \delta^4(p_A + p_B - p_C - p_D) \frac{d^3 \mathbf{p}_C}{(2\pi)^3 2E_C} \frac{d^3 \mathbf{p}_D}{(2\pi)^3 2E_D}$$
$$= -\delta(s + t + u - m_A^2 - m_B^2 - m_C^... | {
"Header 1": "2.2.2 Decay rates",
"Header 2": "2.3.5 $2 \\rightarrow 2$ relativistic variables",
"token_count": 1904,
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We now turn to calculating a few cross sections from known interactions in order to see what measurements of cross sections can tell us about the underlying interactions at play.
#### 3.1 Classical two-body scattering
We start in this section with several examples calculated using classical Newtonian physics. Besid... | {
"Header 1": "2.2.2 Decay rates",
"Header 2": "3 Calculational tools II",
"token_count": 1986,
"source_pdf": "datasets/websources/Physics_v1/Physics/PPNotes.pdf"
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Writing the initial position as $\mathbf{r}(t_i) = r_i \mathbf{e}_r = b \, \mathbf{e}_x + y_i \mathbf{e}_y$ , the angular momentum is $\mathbf{L} = m \, \mathbf{r}_i \times \mathbf{v}_i = mbv_i \, \mathbf{e}_z$ and so points purely in the z direction, with magnitude $L = mbv_i$ that is independent of $y_i$ . If w... | {
"Header 1": "2.2.2 Decay rates",
"Header 2": "3 Calculational tools II",
"token_count": 2029,
"source_pdf": "datasets/websources/Physics_v1/Physics/PPNotes.pdf"
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#### 3.1.3 Scattering from a hard sphere
The simplest case is the case of a hard sphere, for which U(r) = 0 for r > R and $U = \infty$ for r < R. In this case an incoming particle experiences a purely normal force at the sphere's surface that requires the sign of the radial component of velocity to instantaneousl... | {
"Header 1": "2.2.2 Decay rates",
"Header 2": "3 Calculational tools II",
"token_count": 2013,
"source_pdf": "datasets/websources/Physics_v1/Physics/PPNotes.pdf"
} |
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