Midterm 2 Solutions 09

# L r e c b 10 points from the partition fnnction

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Unformatted text preview: .~l -r e ..&quot;C. := = b) (10 points) From the partition fnnction o f a single spin, deduce that the total entropy o f the system is (at small mB/'t) as ~ N s(1og3-~-2)' You may want to remember that for small 10g(1 + E) m 2B2 - e+ 0(e2 ) , 3r E, 8 2 c osh(e) - 1 + -+ 0(£3). 2 + o.--~-~--'~---;:::\M.~ l l -+ T&quot;&quot; e. ~B y v. .Jr. e. £' 3 ~l ~~ ~ - ~ tt. ~) ~f}~ r -+ 0 e,'a?&gt; - (~ -~) 1- o C(~1:~]J ­ o.l&lt;o \vv. ~ s.L- \,~ :\llV&gt; &lt;:I ~s cS Nos -::; l QD~ - ~ (~f J~J '3 so~id c) (5 points) The spin system is part o f a larger system (e.g .• the containing the spins) that we suppose isolated from the outside. We decrease reversibly the magnetic field to zero. Use the conservation o f entropy to show that the solid temperature decreases. This method is used in adiabatic demagnetization refrigerators, which use salt pills in superconducting solenoids and typically reach lOOmK temperatures. w~ l,~ c:1.£ ( /y ~ 'V0&quot;&lt; ~ ... ~~ f~J v '-'- ~ 5 ~ J -: 0:&gt;-1..0 .. h.~~ 3)tl... f l \.IV\. 'S --RvW' go~.A ~v.. d..\$ ~ l~ ~v-.-~&quot;,-ht.,.-e... ­ -F ((I. &lt; 2&quot; C~·&quot;'v-(:' t.. .. cl c= LJ,;;. J.S c ..J\ -;:; 1­ AU &quot;\ CJ e o'\ ? S :: ~:t '3&gt; S~-L ~ 'Tg '?&gt; 1- ­ . --CD \~ . J, tJ&gt; ('(5 (~t&gt; 0 3 '3 \~~...
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## This note was uploaded on 01/23/2014 for the course PHYSICS 112 taught by Professor Steveng.louie during the Spring '06 term at University of California, Berkeley.

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