Boltzmann's constant k= 1.38066E-23 J/K
k= 0.00008617 eV/K
Atomic mass unit u= 1.66E-27 kg
umev= 931.5 MeV/c^2
Wien constant 2.90E-03
Planck's constant h= 4.14E-15 eV s
h= 6.63E-34 J s
h/ 2 pi hbar 1.05E-34 J s
h/ 2 pi hbev 6.59E-16 eV s
Speed of light c= 3.00E+08 m/s
hc= 1.24E+03 eV nm
hbarc= 1.97E+02 eV nm
Electron charge e= 1.60E-19 C
electron volt eV= 1.60E-19 joule
electron mass mel 9.11E-31 kg
mel 5.11E-01 MeV/c^2
proton mass mp 1.67E-27 kg
mpev 9.38E+02 MeV/c^2
mpg 0.93827231 GeV/c^2
neutron mass mn 1.67E-27 kg
mnev 9.40E+02 MeV/c^2
mng 0.93956563 GeV/c^2
Stefan's constant= sigma= 5.67E-08 watt/m^2K^4
Nuclear distance unit fermi 1.00E-15 m (a femtometer)
cross section unit barn= 1.00E-28 m^2
barn= 100 fm^2
common combo for scattering ke^2 1.44 MeV fm
First Bohr orbit radius a0 a0 0.0529 nm
Bohr magnetion mub 5.79E-05 eV/T
muba 9.27E-24 A m^2 A=ampere
#5 Given a Fermi level of 9.39 eV for Zinc, calculate the density of
Equation 14-10, p376 relates the number density to the Fermi level.
Density of electrons = n = 8*sqrt(2)*pi*m^(3/2)*((2/3)*EF^(3/2)/h^3
n = 8*sqrt(2)*pi*(mc^2)^(3/2)*((2/3)*(9.39 eV)^(3/2)/(hc)^3 = 130.625595 /nm^3
The density of zinc is 7.13E3 kg/m^3 and its atomic mass is 65.4. The number of zinc atoms per cubic meter is
n' = (6.02E23)*(7.13E3 kg/m^3)/(65.4E-3 kg) = 6.56309E+28 atom/m^3
This gives 1.990306717 electrons per atom, and clearly indicates two conduction electrons per atom in zinc.
This is consistent with the zinc electron configuration 4s2.
#9 Given an aluminum Fermi level of 11.6 eV, calculate the Fermi speed
and the electron heat capacity.
a. The Fermi speed is given by vF=c*sqrt(2*EF/mc^2) = 2021411.802 m/s
Equation 14.63, pg 384.
b. The electron contribution to the specific heat is given by eq 14.53, pg 383.
Cel = pi^2*k^2*NA*T/(2*EF)= 5.70482E+17 eV/K = 0.0913 J/K mole
#10 Find temperature where electron and phonon contributions to specific
heat are equal in copper. Given EF=7.0 eV and Debye temperature 343K
From eq. 14.53 on pg 383 we can set the contributions equal. This gives
k/(2*EF)= 12*pi^2*T^2/(5*TD^3), so that T=sqrt(5*k*TD^3/(24*pi^2*EF)
T=sqrt(5*k*343^3/(24*pi^2*7)= 3.238 K
#12 Typical phonon and electron energies in copper at room temperature.
The typical phonon energy is given by kTD/2. Using the value of Debye temperature in #10
Phonon energy = k*343/2= 0.015 eV
The typical electron energy is given by 3*EF/5 = 3*7 eV/5 = 4.2 eV
#13 Melting point and density as indicator of Debye temperature
Figure 14-10 on page 381 shows the correlation between Debye temperature and melting point.
Given a melting point of 934K for aluminum and a density of 2.7E3 kg/m^3
Since there are two curves, need to know which to extrapolate for aluminum.
From pg 372 we see that Al has a face centered cubic structure, so extrapolation from copper
seems in order. The Debye temperature of Cu is 343K, its density is 8.9 kg/m^3
Use constant = (Td(Cu)/d(Cu))*sqrt(M(Cu)/Tm(Cu))=(Td(Al)/d(Al))*sqrt(M(Al)/Tm(Al))
Then use d(Al)=(A/density)^(1/3) to express everything in terms of mass number and density.
Mass numbers for Al = 27 and Cu= 63.6.
Melting temperature for Cu = 1348K.
Td(Al) = Td(Cu)*(d(Al)/d(Cu))*sqrt(A(Cu)*Tm(Al)/(Tm(Cu)*A(Al))
Td(Al)= 391.76 K
The actual value is 428 K.
#14. Given that the mean free path of electrons in copper at 4K is about
Calculate the resistivity of copper.
From eq 14.68 on pg 385, the resistivity is given by rho= m*vF/(n*e^2*d)
From example 14-7 we see that the Fermi speed for Cu is 1.6E6 m/s
From example 14-8 the electron density of copper is 8.5E28/m^3
Then rho=9.11E-31 kg*1.6E6 m/s/(8.5E28/m^3*(1.6E-19 C)^2*3E-3)= 2.23284E-13 Ohm m
#29 Density of conductions electrons in silver and B field for
Given density 1.05E4 kg/m^3, A=108. Silver has one conduction electron per atom.
a. Density of electrons n=rho*NA/(A*1E-3 kg)=1.05E4 kg/m^3*6.02E23/(108*1E-3 kg) = 5.85278E+28 /m^3
b. Magnetic field for a 1 microvolt Hall voltage if J=1E6 Amp/m^2 flows in sheet of width y=1E-2 m.
The Hall voltage is given by VH= JBy/ne
The required magnetic field is then B= neVH/Jy = (5.85E28 /m^3*1.6E-19 C*1E-6 V)/(1E6 Amp/m^2*1E-2 m)
Required magnetic field = 0.936 Tesla.