Physics Oct Reading

Transcription

1 Physics Oct Readig Fiish K&K chapter 7 ad start o chapter 8. Also, I m passig out several Physics Today articles. The first is by Graham P. Collis, August, 1995, vol. 48, o. 8, p. 17, Gaseous Bose-Eistei Codesate Fially Observed. This describes the research leadig up the first observatio of a BE codesate that s ot a superfluid or supercoductor. The secod is by Barbara Goss Levi, March, 1997, vol. 50, o. 3, p. 17, Bose Codesates are Coheret Iside ad Outside a Atom Trap, describig the first atom laser which was based o a BE codesate. The third is also by Levi, October, 1998, vol. 51, o. 10, p. 17, At Log Last, a Bose-Eistei Codesate is Formed i Hydroge, describig eve more progress o BE codesates. I additio, there is a recet Sciece report o a atomic Fermi Gas, DeMarco, B., ad Ji, D. S., September 10, 1999, vol. 285, p. 1703, Oset of Fermi Degeeracy i a Trapped Atomic Gas. More o Fermi Gases So far, we ve cosidered the zero temperature Fermi gas ad doe a approximate treatmet of the low temperature heat capacity of Fermi gases. The zero temperature Fermi gas was straightforward. We simply said that all states, startig from the lowest eergy state, are filled util we ru out of particles. The eergy at which this happes is called the Fermi eergy ad is the same as the chemical potetial at 0 temperature, ɛ F = µ(τ = 0). Basically, all we had to do was determie the desity of states, a problem we ve dealt with before. Workig o the low temperature heat capacity required a approximate calculatio of the eergy versus temperature for a cold Fermi gas. I this calculatio we assumed that the desity of states ear the Fermi eergy is costat ad this allows oe to pull the desity of states out of the itegral ad also to set the chemical potetial to its 0 temperature value. These approximatios work quite well for the electro gas i metals at room temperature because the Fermi temperature for these electro is typically several tes of thousads of Kelvis. To calculate the eergy, etc., at arbitrary temperatures, oe must umerically itegrate the Fermi-Dirac distributio times the desity of states to obtai the umber of particles. The the chemical potetial is varied util the desired umber of particles is obtaied. Kowig the chemical potetial, oe ca itegrate the desity of states times the Fermi-Dirac distributio times the eergy to get the eergy at a give temperature. All of this requires umerical itegratio or approximate techiques.

2 Physics Oct Figure 7.9 ad tables 7.2 ad 7.3 of K&K demostrate that the low temperature heat capacities (low eough that the Debye lattice vibratios are accurately followig a T 3 heat capacity) have a compoet proportioal to the temperature ad list the proportioality costats for various metals. Oe thig you will otice is that the proportioality costats agree with the calculatios to oly 30% ad up to a factor of 2 i at least oe case. This is most likely due to the fact that the electros are ot really a o-iteractig gas. Also, there are effects due to the crystal structure such as eergy bads ad gaps. Other Fermi Gases I additio to the coductio electro gas i metals, Fermi gases occur i other situatios. I heavy elemets, the umber of electros per atom becomes large eough that a statistical treatmet is a reasoable approximatio. This kid of treatmet is called the Thomas-Fermi (or sometimes the Fermi-Thomas) model of the atom. Also i heavy elemets, the umber of ucleos (eutros ad protos) i the ucleus is large ad, agai, a statistical treatmet is a reasoable approximatio. The radius of a ucleus is R ( cm) A 1/3, where A is the umber of ucleos. The coefficiet i this relatioship ca vary by a teth or so depedig o just how oe measures the size scatterig by charged particles, scatterig by eutros, effects o atomic structure, etc. Aside: the uit of legth cm which is oe femtometer is called the Fermi i uclear physics. The volume of a ucleus is ad the umber desity or cocetratio is V = 4π A cm 3, uc = A V = cm 3. The uclear desity (with this umber of sigificat digits, the mass differece betwee eutros ad protos is egligible) is ρ uc = gcm 3. Basically, all uclei have the same desity. Of course, this is ot quite true. Nuclei have a shell structure ad full shell uclei are more tightly boud tha partially full shell uclei. Also, the very lightest uclei show some deviatios. Nevertheless, the desity variatios are t large ad it s reasoable to speak of the uclear desity.

3 Physics Oct The eutro to proto ratio i uclei is about 1 : 1 for light uclei up to about 1.5 :1 for heavier uclei. Assumig the latter value, the it is the eutros whose Fermi eergy is importat. ɛ F = h2 ( 3π 2 (0.6 uc ) ) 2/3 = erg = 32 MeV. 2m This is a little larger tha K&K s umber because it s computed for a ucleus with 40% protos ad 60% eutros, istead of equal umbers. Sice the average kietic eergy i a Fermi gas is 3ɛ F /5, the average kietic eergy is about 19 MeV i a heavy ucleus. The experimetally determied bidig eergy per ucleo is about 8 MeV. This varies somewhat, especially for light uclei; it reaches a peak at 56 Fe. To the extet that the bidig eergy per ucleo ad the kietic eergy per ucleo are costat, the potetial eergy per ucleo is also costat. This reflects the fact that the uclear force is the short rage strog force ad uclei oly see their earest eighbors. The strog force is about the same betwee eutros ad protos, betwee protos ad protos ad betwee eutros ad eutros. But, the protos have a log rage electromagetic iteractio. As the umber of particles goes up the ati-bidig eergy of the protos goes up faster tha the umber of protos (ca you figure out the expoet?) so the equilibrium shifts to favor eutros i spite of the fact that they are slightly more massive tha protos. The Fermi temperature for eutros i a heavy ucleus is T F = ɛ F /k = K, so uclei (which are usually i their groud state) are very cold! I a star like the Su, gravity is balaced by the pressure of a very hot, but classical, ideal gas. The Su has a mass about 300,000 times that of the Earth ad a radius about 100 times that of the Earth, so the average desity of the Su is somewhat less tha that of the Earth (it s about the desity of water!). The temperature varies from about 20 millio Kelvis at the ceter to about 6000 K at the surface. So it s completely gaseous ad the electros are o-degeerate throughout. Sice the su is radiatig, it is coolig. Eergy is supplied by uclear reactios i the Su s core. A typical white dwarf star has about the mass of the Su but the radius of the Earth. It s the degeeracy pressure of the electros that balaces gravity i a white dwarf. White dwarves shie by coolig. There are o uclear reactios i the core, so after they cool eough, they become ivisible. White dwarves are discussed i K&K, so let s move o to eutro stars which are ot discussed i K&K. I a eutro star it s the degeeracy pressure of the eutros that balaces gravity. A typical eutro star has a mass like the Su (M = g) but a radius smaller tha New Jersey, let s say R 10 km. Let s assume that the mass i a eutro star is uiformly distributed. What s the desity? ρ = M/V = gcm 3,

4 Physics Oct about three times uclear desity. (Of course, the desity i a star is ot uiform ad it may exceed 10 times uclear desity i the ceter, but we re just tryig to do a back of the evelope calculatio here.) I terms of the cocetratio of eutros, this correspods to 0 = cm 3. The Fermi eergy for these eutros is ad the Fermi temperature is ɛ F,0 86 MeV, T F =10 12 K. Neutro stars are owhere ear this hot. Otherwise they would be very strog sources of gamma rays. Istead they are thought to have temperatures of millios of degrees ad radiate X-rays. Ibelieve there are some observatios which idicate this. Also, due to havig a magetic field ad rotatig, they ca radiate electromagetic eergy ad are observed as pulsars. I ay case, the eutros i a eutro star are cold! A iterestig questio is why is a eutro star made of eutros? (Well, if it were t, we probably would t call it a eutro star, but besides that?) I particular, what s wrog with the followig? Let the star be made of protos ad electros, each with the cocetratio we ve just calculated. The the star is electrically eutral because there is a sea of positively charged protos ad a sea of egatively charged electros. But the protos have a slightly lower mass tha the eutros ad this is true eve if oe adds i the mass of the electro, so this cofiguratio would seem to be eergetically favored over the eutro cofiguratio. I fact, a free eutro decays accordig to p + e + ν e, where ν e is the electro ati-eutrio. The half life is about 15 miutes. Neutrios are massless (or very early so) ad for our purposes we ca igore them. That is, we ca assume that the eutros are i equilibrium with the protos ad electros. If we eed to chage a eutro ito a proto ad electro, the above reactio will do it. If we eed to chage a proto ad electro ito a eutro, there is p + e + ν e. What would be the Fermi eergies of the protos ad electros i our hypothetical star? The Fermi eergy for the protos would be very early the same as that for the eutros above (because the cocetratio would be the same ad the mass is early the same). O the other had, the Fermi eergy of the electros would be larger by the ratio of the eutro mass to the electro mass, a factor of 1838, so the electro Fermi eergy would be about 160,000 MeV, eough to make about 170 ucleos! Remember that the

5 Physics Oct chemical potetial (the Fermi eergy sice all the gases are cold) is the eergy required to add a particle to a system. If eutros are i equilibrium with protos ad electros, the the chemical potetial of the eutros equals the chemical potetial of the protos plus the chemical potetial of the electros mius the eergy differece betwee a eutro ad a proto plus electro. I other words ɛ F, = ɛ F,p + ɛ F,e (m m p m e )c 2. Deote the cocetratios of the eutros, protos, ad electros by, p,ad e.the for charge eutrality ad p = e, p + =, where is the cocetratio of the ucleos, which is ot chaged by the reactios above. To simplify the otatio a bit, let x = p = e, 1 x =. Each of the Fermi eergies ca be writte i terms of the cocetratios ɛ F, = ɛ F,p = ɛ F,e = ( h2 (3π 2 ) 2/3 = ɛ F,0 2m h2 2m p (3π 2 p ) 2/3 = ɛ F,0 h2 2m e (3π 2 e ) 2/3 = ɛ F,0 0 0 ) 2/3 (1 x) 2/3, ( ) 2/3 m x 2/3, 0 m p ( ) 2/3 m x 2/3, m e where ɛ F,0 = h2 (3π 2 0 ) 2/3, 2m is the Fermi eergy for a pure eutro gas at the cocetratio 0 we calculated previously. We plug these eergies ito the eergy equatio to obtai ( ) 2/3 ( ) 2/3 ( ɛ F,0 (1 x) 2/3 m = ɛ F,0 + m ) x 2/3 E, 0 0 m p m e where E =(m m p m e )c 2 =0.783 MeV is the mass eergy excess of a eutro over a proto ad electro. If we rearrage slightly, we obtai (1 x) 2/3 = ( m + m ) x 2/3 E ( 0 ) 2/3, m p m e ɛ F,0

6 Physics Oct or or (1 x) 2/3 =( ) x 2/ ( x 2/3 = (1 x) 2/ ( 0 ) 2/3, ( 0 ) ) 2/3 If is i the eighborhood of 0,thex is small, we ca igore x o the right had side, ad we fially obtai x At higher cocetratios x will get slightly smaller ad at lower cocetratios x will grow slowly. The cocetratio of eutros, protos, ad electros are equal (x =0.5) whe = = cm 3. Such low cocetratios will be attaied oly very ear the surface of the eutro star. Caveats: (1) The Fermi eergy of the electros works out to be about 87 MeV, so the electros are extremely relativistic, so we really should t be usig our o-relativistic formula for the electro Fermi eergy. Oe of this week s homework problems gives you a chace to modify the treatmet to allow for relativistic electros. (2) With a electro ad proto istead of a eutro, the pressure chages, so the equilibrium coditio that we set up is ot quite right. Nevertheless, this calculatio gives the flavor of what s ivolved ad poits to the correct coclusio: For most of its volume a eutro star is almost pure eutros! Of course, we ca tur the earlier questio aroud: how is it that uclei have ay protos??? Have t we just show that at uclear desities, the ucleos must exist as eutros, ot protos???.