Chapter 7, Fermi and Bose Gases Video Solutions, Thermal Physics

Problem 1

Density of orbitals in one and tivo diutensions. (a) Show that the density of orbilats of a frec eiectron in one dimension is
$$
\mathcal{D}_{1}(\varepsilon)=(L / \pi)\left(2 \operatorname{m} / h^{2} \varepsilon\right)^{/ 1},
$$
where $L$ is the length of the line. (b) Show that in two dimensions, for a square of area $A$,
$$
\mathrm{q}_{2}(\varepsilon)=A \mathrm{~m} / \pi h^{2}
$$
independent of $\mathrm{k}$

Suzanne W.

Numerade Educator

22:10

Problem 2

Energy of refativisic Feruti gas. For electrons with an energy $\varepsilon$ o $a$ where $u_{1}$ is the rest mass of the electron, the eaergy is given by $\tau \simeq p c$, where $p$ is the momentum. For electrons in a cube of volume $V=L^{3}$ the momentum is of the fortn $(\pi h / L)$, multiplied by $\left(1 x_{x}^{2}+u_{y}^{2}+n_{z}^{2}\right)^{y / 2}$, exacty as for the non relativistic limit, (a) Show that in this extrente relativistic limit the Fermi energy of a gas of $N$ electroas is given by
$$
\varepsilon_{\boldsymbol{r}}=\operatorname{Arc}(3 \mathrm{n} / \mathrm{\pi})^{1 / 2} \text { . }
$$
where $n-N / V$. (b) Show that the total eneryy of the ground state of lhe gas is
$$
U_{0}=\frac{3}{4} N_{F}_{F}
$$
The general problem is trealed by F.Jüliner, Zeitschrifl für Physik $47,542+1928$ ).

Aditya Jain

Numerade Educator

07:18

Problem 3

Pressure and entropy of degenerate Fermigas. (a) Show that a Fermi electron gas in the ground state cxerts a pressure
$$
p=\frac{\left(3 \pi^{2}\right)^{2 \cdot 3}}{5} \cdot \frac{h^{2}}{m}\left(\frac{N}{V}\right)^{3 / 3}
$$
In a uniform decrease of the volume of a cube cvery orbital has its energy ralsed: The energy of an orbithl is proportionat $101 / t^{2}$ or to $1 / V^{2 i 3}$, (b) Find
Notice that $\sigma \rightarrow 0$ as $t \rightarrow 0$.

Uma Kumari

Numerade Educator

01:58

Problem 4

Chemical porential rersus teuperature. Explain graplically why the initial curvalurc of $\mu$ versus $\tau$ is upward for a fermion gas in one dincnsion and are differen1, where $\Phi_{1}$ is given in Problem $1 .$ It wil be found useful to set op the integral for $N$, the number of particles, and to consider from the graphs the behavior of the integrand between zero tempera: tre and a finite temperature.

Manik Pulyani

Numerade Educator

09:34

Problem 5

Liqutd ${ }^{3} \mathrm{He}$ as a Fermi gas. The atom ${ }^{3}$ He has spin $I \mathrm{~m} \frac{1}{2}$ ald is a fermion.
(a) Calculate as in Table $7.1$ the Fermi sphere parameters $0_{F}, \varepsilon_{F}$, and $I_{F}$ for ${ }^{\text {'He at absolute zero, viewed as a gas of noninteracting fermions. The densily }}$ of the tiquid is $0.081 \mathrm{~g} \mathrm{~cm}^{-3}$, (b) Calculate the heat capacity at low temperatures $T \propto T_{F}$ and compare with the experimental value $C_{r}=2.89 \mathrm{Nk}_{\mathrm{B}} T$ as observed for $T<0.1 \mathrm{~K}$ by A. C. Anderson, W. Recse, and $\mathrm{J} . \mathrm{C}$. Wheatiey, Phys. Rer. 130, $495(1963)$; sce also Figure $7.18$. Excellent surveys of the properties of liquid ${ }^{3} \mathrm{He}$ arc given by $\mathrm{J}$. Wilks, Properties of liquid and solid heliutel, Oxford, 1967 , and by $\mathrm{J}$. C. Wheatley, "Dilute solutions of 'He in ${ }^{4}$ He at low temperatures" Ancrican Journat of Physics $36,181-210(1968)$. The principtes of refrigerators based on ${ }^{3} \mathrm{He}-{ }^{4}$ He mixtures are reviewed in Chapter 12 on cryogenics; such refrigerators produce steady temperatures down to $0.01 \mathrm{~K}$ in continuously acting operation.

Eduard Sanchez

Numerade Educator

03:13

Problem 6

Mass-ratius relationslip for white dwarfs. Consider a white dwarfofmass $M$ and radius $R$. Let the electrons be degenerate but nonrelativistic; the protons are nondcgenerate. (a) Show that the order of magnitude of the gra vitational sel[-energy is $-G M^{2} / R$, where $G$ is the gravitalional constant. (If the mass density is constant wíthin the sphere of radtus $R$, the exact potential energy is
Figure 7,18 Heat capacity of liquid ${ }^{3}$ tic and of a 5 percent solution of ${ }^{3} \mathrm{He}$ in liquid ${ }^{4}$ He. The quantity plottect on the verlical axis is $C / T$, and Hic horizontal axis is $T$. Thus lor a Fermi gas in the degenerale temperalure segion the theorelical cut ves of $C_{i} T$ at contan! voiune are horizontal. The curve for pure 'He is taken at constant pressure, which accounts for the slight s'ope. The curve for the solution of ' Ihal the 'He in solution acts as a Ferm? gas; the degenerate region at !ow lemperature gots over to the nondegenetate region 21 higher temperature. The solid line through the experimental points for the solution is drawn for $T_{8}^{\prime}=0.33 I \mathrm{~K}$, which agrces wilh the calculation for free atoms if the effective ruass is taken as $2.33$ times the mass of an atem of 'He. Curves after J. C. Wheatley, Amer. J. Physics $36(1968)$.
$-3 G M^{2} / 5 R$. (b) Show that the order of magnitude of the kinctic energy of the clectrons in the ground state is where $m$ is the mass of an clcctron and $M_{1} u$ is the mass of a proton. (c) Slaw that if lhe gravitational and kittctic energies are of the same order of magnitede (as required by dic vitial theoten of mechanics). $M^{1 / 3} R \approx 10^{20} g^{1 / 3} \mathrm{~cm} .$ (d) Ifthe
mass is equal 10 that of the $\operatorname{Stn}\left(2 \times 10^{33} \mathrm{~g}\right)$, what is the density of the white dwar? (e) 11 is believed that pu!sats ate stars composed of a cold degenerate gas of ncutfons. Show that fot a neutron star $M^{1 / 3} R \approx 10^{17} \mathrm{~g}^{\mathrm{t} / \mathrm{s}} \mathrm{cm}$. What is the value of the tadius for a neutron star with a mass equal to that of the Sua? Express the result in $\mathrm{km}$.

Hubert Agamasu

Numerade Educator

09:09

Problem 7

Plroton comdeusutiou. Considet a science fiction untucrse in which the number of photons $N$ is constant, al a concentralion o $\left\{\mathrm{I} 0^{20} \mathrm{~cm}^{-3}\right.$. The numben of thermally excited pliotons we assume is given by the resull of Problem 4.I, which is $N_{e}=2.404 V_{\tau}^{3} / \pi^{2} h^{3} c^{3}$. Find the critical lemperature in $K$ below which $N_{e}<N$. The excess $N-N_{e}$ will be in tite plioton mode of lowest frequency; the excess night be described as a photon condensate in which there is a large concentration of photons in the lowest mode. In reality there is no sucl principle iniat the total number of photons be constatu, thence there is no photon condensatc.

Guilherme Barros

Numerade Educator

01:58

Problem 8

Fnerg', beat cepacity', and cutrapy of degencale bison gas. Find expres-
eapacily, ind entopy of a gis of 1o a valume $V$. Put the defiatite integral in dimeasionhess form; it ned not be evatuated, The cilcalated heat capacity above thd below $t_{E}$ is shown in between the tivo curtes is marked: $1 t$ is ascribed to the effect of interactions botween the ato?ns.

Manik Pulyani

Numerade Educator

01:26

Problem 9

Bosan gas in one dimension. Calculate the integral for $N_{c}(\tau)$ for a oncdimensiontal gas of noninteracting bosons, and show that the integral ?ocs not convergc. This result suggests that a boson ground state condensate does not form in one dimension. Take $i=1$ for the calculation. (The problen should really be treated by means of a sum over ortitals on a finite line.)

Chai Santi

Numerade Educator

06:50

Problem 10

Relatiristie ubite drarf stars, Consicicr a Fcrni gas of $N$ electrons each of rest mass $m$ in a sphere of radius $R$. Conditions in certain white dwarfs are such that the greal majority of electrons have extreme relativistic kinetic energies $\varepsilon=p c$, where $p$ is the momentum. The de Broglic relation remains $\lambda=2 \mathrm{nh} / \mathrm{p} .$ Problem 2 gives the ground stale kinetic energy of the $\mathrm{N}$ electrons on the assumption that $\varepsilon=p c$ for all elecrons. Treat the sphere as a cube of equal votume. (a) Use the standard virial theorem argument to predict the value of $N$. Assurne that the whole star is ionized hychogen, but ncelect the kinetic energy of the protons compared to that of the electrons. (b) Estimate the value of $N .$ A carefu! treatment by Chandrasekhar leads not to a single value of $N$, but to a limit above which a stable whive dwarí cannot exist: see
D. D. Clayton, Principles of suellar ecolution and nucleosynthesis, MoGraw.Hi!!, 1968, p. 161; M. Hamit, Astrophysical concepts, Wilcy, $1973 .$

Mahnoor Amin

Numerade Educator

02:50

Problem 11

Fluctuations in a Fermigas. Show for a single or bita! of a fermion system that
$$
\left\langle(\Delta N)^{2}\right\rangle w\langle N\rangle\{1-\langle N\rangle\}
$$
if $\langle N\rangle$ is the average number of fermions in that orbita!. Notice that the fluctualion vanishes for orbitals with energies decp enougl below the Femi cncrgy so that $\langle N\rangle=1$. By definition, $\Delta N$ M $N-\langle N\rangle$.

Penny Riley

Numerade Educator

02:50

Problem 12

Fluctuations in a Base gas. If $\langle N\rangle$ as in $(11)$ is the average occtpancy of a single orbital of a boson system, then from $(5.83)$ show that
$$
\left\langle(\Delta N)^{2}\right\rangle \approx\langle N\rangle\langle 1+\langle N\rangle)
$$
Thus if the occupancy is large, with $\langle N\rangle \gg 1$, the fractional fluetualions are of the order of unity: $\left\langle(\Delta N)^{2}\right\rangle /\langle N\rangle^{2} \approx 1$, so that the actual fluctuations can be enormous. It has been said that "bosons trave! in flocks." The first edilion of this text has an elementary discussion of the flucluations of photons.

Penny Riley

Numerade Educator

12:36

Problem 13

(a) Skeich carefully the chemical potentia! versus the number of paricles for a. boson gas in volume $V$ at
temperature $\tau$, lnclude both classical and quanture tegimes. (b) Do the same for a system of fermions.

Mahnoor Amin

Numerade Educator

03:43

Problem 14

Two orbilal boson systeut. Cousider a sysiem of $N$ basons of spin zero, with orbitals at the single partiche energics 0 and $\varepsilon$. The chenical polential is $\mu$, and the temperalure is $\tau$. Find $r$ stuch thal the thet mal averagc population of the lowest orbital is twice the populaion of the orbital a! $\varepsilon$. Assunse $N \gg 1$ and make what approxonalions are reasonibic.
If the atons in a gas have integtal spin (coucting lite sunt of clectronic and nuclear spins), they can fotm a boson condensale when the gas is coolcu below the Einstein condensation temperature $\tau_{\text {E }}$ given by $(72)$ :
$$
\tau_{\mathrm{E}}=\left(2 \pi h^{2} M t\right)(N I 2.612 \mathrm{~V})^{2 / 3}
$$
For itoms in the vapor phase the Einsem' condensalion temperalure is very low because the number uensties are very luw: In $(1995)$ eatly successful experiments were carricd out al Boulder, MIT, und elsewhere. Such experimcnts, which arc exiraordinarily complex, mark the exciting forefront of the quantum gas ficld. A large fiterature on BEC experiments and theory is on she Web.

One set of experinients (MIT) starled with a beam of sodium atoms exiting an oven at $600 \mathrm{~K}$ at a concentration $\mathrm{NiV}$ of $\mathrm{HO}^{\mathrm{H}} \mathrm{cm}^{-3}$. What hapetis next is the result of a number of clever tricks with laser beants directed on one part or another of the beaul of atons. First the atoms are slowed by ore
siow cnougl for $10^{10}$ utonts to be tripped wititin a ntagncto-opical trap. Fusther tricks, includitle eviporation, reduced the leinperature of the gas to $2 \mu K$, the uitralow tenueralure $\tau_{E}$ at witicl the condensate was formed. The coitcentration al $\tau_{\mathrm{E}}$ was tgain $10^{14}$ atoms $\mathrm{cm}^{3}$
sitly slowby once released from die trap. The atoms in excited states move relaively rapidly out of their steady-siate positions. The positions of the atons can be recorded as a function of tine after relense, using a laser beam. The rumber of aroms in excited orbitals is in good agreement with the $\tau^{3}$ law, (73). With this iechrique the signature of Bose- Einsicin condenstion is the sudden appearance of a sharp peak of atoms as the iemperature is decreased ithrough $\tau_{\mathrm{E}} .$ The penk comes frotn light scattered by atoms in thc condensate; the wings of the line from bight scaltered by atorns in excitcd orbials.

Salamat Ali

Numerade Educator