Di r ecci ó n:Di r ecci ó n: Biblioteca Central Dr. Luis F. Leloir, Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires. Intendente Güiraldes 2160 - C1428EGA - Tel. (++54 +11) 4789-9293

Co nta cto :Co nta cto : digital@bl.fcen.uba.ar

Tesis Doctoral

Adsorción de fluidos sobre sustratosAdsorción de fluidos sobre sustratossólidos, estudiada en el marco desólidos, estudiada en el marco de

funcionales de la densidadfuncionales de la densidad

Sartarelli, Salvador Andres

2012

Este documento forma parte de la colección de tesis doctorales y de maestría de la BibliotecaCentral Dr. Luis Federico Leloir, disponible en digital.bl.fcen.uba.ar. Su utilización debe seracompañada por la cita bibliográfica con reconocimiento de la fuente.

This document is part of the doctoral theses collection of the Central Library Dr. Luis FedericoLeloir, available in digital.bl.fcen.uba.ar. It should be used accompanied by the correspondingcitation acknowledging the source.

Cita tipo APA:

Sartarelli, Salvador Andres. (2012). Adsorción de fluidos sobre sustratos sólidos, estudiada enel marco de funcionales de la densidad. Facultad de Ciencias Exactas y Naturales. Universidadde Buenos Aires.

Cita tipo Chicago:

Sartarelli, Salvador Andres. "Adsorción de fluidos sobre sustratos sólidos, estudiada en elmarco de funcionales de la densidad". Facultad de Ciencias Exactas y Naturales. Universidadde Buenos Aires. 2012.

❯❱ ❯

t ♥s ①ts ② trs

♣rt♠♥t♦ ís

s♦ró♥ ♦s s♦r sstrt♦s só♦s

st ♥ ♠r♦ ♥♦♥s ♥s

ss ♣rs♥t ♣r ♦♣tr ♣♦r tít♦ ♦t♦r

❯♥rs ♥♦s rs ♥ ♥s íss

♦r ♥rs rtr

rt♦r ss r s③ ③②s③

♦♥sr♦ st♦s r rt♥ ③ ③

r r♦

♥tr♦ tó♠♦ ♦♥stt②♥ts ♦♠só♥ ♦♥ ♥rí tó♠ ♥♦s rs

❯♥rs ♦♥ ♥r r♠♥t♦ ♥♦s rs

♥♦s rs

s♦ró♥ ♦s s♦r sstrt♦s só♦sst ♥ ♠r♦ ♥♦♥s ♥s

s♠♥

s♦ró♥ ss ♥♦s ás♦s r ② ❳ s♦r sstrt♦s só♦s ♠ts ♥♦s s ②

② ♥♦s ♦tr♦s ♦♠♣st♦s s ♥st ♥ ♠r♦ ♦rí ♥♦♥ ♥s st♠♦s t♥t♦

s♦ró♥ s♦r ♥ s♦ ♣r ♣♥ ♦♠♦ ♥♦ r♥rs ♣♥s s♠étrs ♥ ♣r♠r r ♣r♦♣♦♥♠♦s

♥ ♦r♠ ♣r ♥tró♥ ♣rs ♦ q ♣ t③rs ♥ t♦♦ r♥♦ t♠♣rtrs T ♥tr

♣♥t♦ tr♣ ② rít♦ Tt ≤ T ≤ Tc ❱r♠♦s q ♣ró♥ ♣r t♥só♥ s♣r ♥trs

íq♦♣♦r r ② ❳ s ♦rrt ♠♦s s♦tr♠s s♦ró♥ ② ♣rs ♥s

♥③♠♦s í♥s ♣r♠♦♦ ② tr♠♥♠♦s t♠♣rtrs ♠♦♦ Tw ② ríts ♣r♠♦♦ Tcpw ♥

♥♦ r♥rs s♠étrs ♥♦♥tr♠♦s q ♦ rts ♦♥♦♥s ♦s st♦s ♥♠♥ts t♥♥ ♣rs

♥s q ♦♥ s♠trí ♣♦t♥ r♦ ♣♦r s ♣rs ② ♥ ♦rró♥ ♥tr ♣ró♥

st♦s ♣rs s♠étr♦s ② ①st♥ í♥s ♣r♠♦♦ s♦♠♦s ♦s sst♠s ♣♥♦ s stísts

♦rrs♣♦♥♥ts ♦s ♥s♠s ♥ó♥♦ sst♠ rr♦ ② r♥ ♥ó♥♦ sst♠ rt♦ ♦str♠♦s q

♣r sst♠s rt♦s t♠é♥ s ♦t♥ s♦♦♥s s♠étrs ♦♦s ♦s t♠s stá♥ st♦s ♥ t

Prs s s♦ró♥ ♦s r♠♥t♦ ♣ís ♥trss ② ♥ó♠♥♦s ♠♦♦ ♣tr s♣ó♥

t♥ s♠trí

s♦r♣t♦♥ ♦ s ♦♥ s♦ sstrts st ♥t r♠ ♦ t ♥st② ♥t♦♥s

strt

s♦r♣t♦♥ ♦ ♥♦ ss ss r ② ❳ ♦♥ s♦ sstrts ♦ ♠ts ♥ ♦tr ♦♠♣♦sts r

♥②③ ♥ t r♠ ♦ t ♥st② ♥t♦♥ ♦r② ❲ ♥stt ♦t t s♦r♣t♦♥ ♦♥ s♥ ♥

t ♥ ♦ s②♠♠tr ♣♦r sts ♥ rst st♣ ♣r♦♣♦s ♥ ♥②t ♦r♠ ♦r ♣r ♥trt♦♥ t♥

t♦♠s ♥ s ♦r t ♥tr r♥ ♦ t♠♣rtrs T t♥ tr♣ ♣♦♥t ♥ t rt ♦♥ Tt ≤ T ≤ Tc

❲ r tt ♣rt♦♥s ♦r t sr t♥s♦♥ ♦ t q♣♦r ♥tr ♦ r ♥ ❳ r ♦rrt

❲ t s♦r♣t♦♥ s♦tr♠s ♥ ♥st② ♣r♦s ❲ ♥②③ t ♣rtt♥ ♥s ♥ tr♠♥ tt♥

t♠♣rtrs Tw ♥ rt ♣rtt♥ ♦♥s Tcpw ❲ ♦♥ tt ♥r rt♥ ♦♥t♦♥s tr r r♦♥

stts t ♥st② ♣r♦s tt r t s②♠♠tr② ♦ t ♣♦t♥t ①rt ② t s r s ♦rrt♦♥

t♥ t ♦rr♥ ♦ s②♠♠tr ♣r♦s ♥ t ①st♥ ♦ ♣rtt♥ ♥s ♦r s♦♥ t qt♦♥s

♣♣ ♦t t ♥♦♥ ♦s s②st♠ ♥ r♥ ♥♦♥ ♦♣♥ s②st♠ ♥s♠s ❲ s♦ tt ♦r ♦♣♥

s②st♠ ♦♥ s♦ ts s②♠♠tr s♦t♦♥s

②♦rs s♦r♣t♦♥ ♦ s r♦t ♦ ♠s ♥trs ♥ tt♥ ♣♥♦♠♥ ♣♦♥t♥♦s s②♠♠tr② r

♥

r♠♥t♦s

r♦ rr t♦s s ♣rs♦♥s q ♥ ♠♥r ♦tr ♠ ②r♦♥

♦♥rtr st sñ♦

r s③ ③②s③ ♥♦ s♦♦ ♣♦r r♠ ♣t♦ ♣r trr ♥t♦ é s♥♦

t♠é♥ ♣♦r sr③♦ q ♣s♦ ♣r q t♦♦ ♥ ♣rt♦ ♣♦r s s♣♦só♥

② ♣♦r t♦♦ ♦ q rs s ①♥ ♣r♦s♦♥ ♣r♥í ❨ t♠é♥ ♣♦r s

♣♥

r③♦ t♠é♥ s ①♥ts ♣rs♦♥s q t♥♦ ♦♠♦ ♦♠♣ñr♦s tr♦

♦♦♦ rr r♥st♦ ②rs ② r♦ ❱r q♥s ♥ ♠ás ♥ ♦♣♦rt♥

trr♦♥ ♠ás ♥t ♣r q ②♦ ♣r r ♠ás t♠♣♦ st ss

rs ♦♥③ás ♣♦r ♣rstr♠ t ♦③ ♦♠♣t♦r ② t♠é♥ ♣♦r ♦s

♠♦♠♥t♦s st♥só♥

s♣♠♥t rs ♠ ♠ ♣♦r t♦♦ ♣♦②♦ ② ♥t

r♦ t♠é♥ rr ♠② s♣♠♥t ♦s srs ♠♥♦s ①♣♦♥s

rs ♥♦ ② st♥ ♣♦r t♦♦ ♦ q ②r♦♥ ② ♣♦rq ♠ s♥

②♥♦

rs ♥ ② ♥ ♣♦r t♦ q sts ♣rs♦♥ts s♥ ♥trr

♦

♦s srs q qr♦ ♠♦

② q ② ♥♦ stá♥

❮♥ ♥r

srr♦♦ ss

♥tr♦ó♥ ♥r

♦s ♠♣s

♦rí ♥♦♥ ♥s

r ♥tr♦ó♥ stór

t♦rí ♣ ♦s s♣t♦s ♥tí♦s ② t♥♦ó♦s

trs rr♠♥ts tórs

ét♦♦s ♠ó♥

♦rís ♦♥s ♥trs

♦rís ♣rtr♦♥s

♦rí ♥♦♥ ♥s

♥tr♦ó♥ ♦r♠s♠♦

♦s ♦♠♦é♥♦s

♥ó♥ ♣rtó♥ ♦rrs♣♦♥♥t s

♥ó♥ ♣rtó♥ s♦ ♦♥ s ♥tr♦♥s

♥♦♥s ♦rró♥

♦s ♥♦ ♦♠♦é♥♦s ♦rís ♦s ② ♥♦ ♦s

♦r♠s♠♦ r③♦♥♥s♥rs♦♥

♦rí s s ♥♠♥ts

s♣t♦s ♦r♠s t♦rí

♣t♠③ó♥ ♥♦♥ ♣r♦♣st♦ ♥ st ss

♥tr♦ó♥

str ♣r♦♣st

st ♥♦♥

st♦s ♦t♥♦s

st ♦s ♣rá♠tr♦s ♦rrs♣♦♥♥ts ó♥

❱ró♥ á♠tr♦ ♠♦r ② ♥t♥s ♣♦t♥ ♣rs

❱ró♥ ♦♥t s ② ♣♦t♥ ♣rs ♥ ♥ó♥ st♥

rs ♦①st♥ íq♦♣♦r ó♥

á♦s ♦rrs♣♦♥♥ts ró♥

❱ró♥ á♠tr♦ ♠♦r ② ♥t♥s ♣♦t♥ ♣rs

❱ró♥ ♦♥t s ② ♣♦t♥ ♣rs ♥ ♥ó♥ st♥

rs ♦①st♥ íq♦♣♦r

á♦s ♦rrs♣♦♥♥ts ❳♥ó♥

á♠tr♦ ♠♦r ② ró♥ ♣r♦♥ ♣♦t♥ ♣rs

❱ró♥ ♦♥t s ② ♣♦t♥ ♣rs ♥ ♥ó♥ st♥

rs ♦①st♥ íq♦♣♦r

♥só♥ s♣r

r ♥tr♦ó♥

♥só♥ s♣r ② ♣♦t♥ tr♠♦♥á♠♦

♥só♥ s♣r ♥trs íq♦♣♦r ó♥

♥só♥ s♣r ♥trs íq♦♣♦r ró♥

♥só♥ s♣r ♥trs íq♦♣♦r ❳♥ó♥

s♦ró♥ s♦r sstrt♦s ♣♥♦s

í♥tss ♥tr♦t♦r

♦♥s rr♥

P♦t♥s s♦ró♥

P♦t♥

P♦t♥

P♦t♥ ❩

♦♦ ② í♥ ♣r♠♦♦

♦♦ s♠♣ ♣r tr♥só♥ ♠♦♦

st♦s ♣r♦♣♦s s♦r s♦ró♥

s♦ró♥ ó♥ ♥s♠ ♥ó♥♦

s♦ró♥ s♦r s ②

s♦ró♥ s♦r ②

s♦ró♥ s♦r

s♦ró♥ ró♥ ♥s♠ ♥ó♥♦

s♦ró♥ r s♦r ♠ts ♥♦s

s♦ró♥ r s♦r ② 2

s♦ró♥ r s♦r

s♦ró♥ ❳♥ó♥ ♥s♠ ♥ó♥♦

s♦ró♥ ❳ s♦r ②

s♦ró♥ ❳ s♦r ②

s♦ró♥ ❳♥ó♥ ♥s♠ r♥ ♥ó♥♦

♦♥stró♥ ① s t♥♥ts

s♠♥ s♦r s♦ró♥

s♦ró♥

s♦ró♥ r

s♦ró♥ ❳

♥♦ r♥rs ♣♥s

s♠♥ ♥t♥ts

♥st♦♥s ♣r♦♣s

♦♥♥♠♥t♦ ró♥ ♥ r♥rs ♣♥s rrs

r ♥ r♥rs ♥ír♦ ró♥♦

r ♥ r♥rs ts ♥♦s

♠♣rtr í♠t r♣tr s♠trí

♦♥♥♠♥t♦ ❳♥ó♥

❳ ♥ r♥rs rrs ts ♥♦s

❳ ♥ r♥rs rrs ♥s♦

❳ ♥ r♥rs rts ♦♦

♦♥♥♠♥t♦ ó♥

♦♦♥s ♦♥ ♣rs s♠étr♦s ó♥

♦♦♥s ♦♥ ♣rs s♠étr♦s ó♥

♥áss s♦ró♥ s♦r ♦♦

s♠♥

♣é♥

P♦t♥ ❲

♦♥s♦♥s

♣ít♦

srr♦♦ ss

r♥t ♦s út♠♦s ñ♦s ♦ r♥♦ ♥trés t♥t♦ ♣rt ís ♦♠♦ qí♠ ♣♦r st♦

s ♣r♦♣s tr♠♦♥á♠s ② strtr ♠r♦só♣ ♦s s♦r♦s s♦r sstrt♦s só♦s

s ♥st♦♥s srr♦s ♥ st ss stá♥ r♦♥s ♦♥ ♥ t♠ q s♣rtó ♥str r♦s

♥♦ ①♠♥á♠♦s ♣r♦♣s ♥♦ r♥rs ♦♥ ♦s s♠♣s

② ♠ás ♦s és ♥ ♥ ② ss ♦♦r♦rs str ♥♦ r♥rs ♣♥s s♠étrs

♠♥t té♥ ♥á♠ ♦r ♥♦♥trr♦♥ q ♦ rts ♦♥♦♥s ♦s ♣rs ♥s ♥♦

t♥♥ s♠trí ♣♦t♥ q r ♣r ♣rs é♥ts s♦r ♦ ❬ ❪ s♣♦♥ ♥r♥

♥ ♣♥♦ xy s ♣rs r♥r ② q ♣r ♥s s ♥ó♥ ♦♦r♥ z ♣r♣♥r

s ♣rs st ♠♥r ♣rí ♥ó♠♥♦ r♣tr s♣♦♥tá♥ s♠trí st t♦ ♣r

♥♦ ♥tró♥ ♥tr sstrt♦ ② ♥ ♣rtí ♦ ♥♦ s ♠♦ ♠ás rt q ♥tró♥ ♥tr

♣rtís ♦ ♥trés ♣♦r t♠ s ♥r♠♥t♦ ♥ ♦s út♠♦s ñ♦s ♥t♠♥t r♠ ② ♥st♥ ❬❪

♥③r♦♥ ♥♦ r♥rs 2 ♦♥ r ♣♥♦ ♦rí ♥♦♥ ♥s ♥♦♥tr♥♦

q ♣r rts t♠♣rtrs ♣r t♦ ♦ s q ♣rs s♠étr♦s t♥♥ ♠♥♦r ♥rí r q

♦s s♠étr♦s q♦s á♦s r♦♥ r③♦s ♠♥t♥♥♦ ♦s t♠♣rtr T ② ♥ú♠r♦ ♣rtís

♣♦r ♥ ár ♥ ♣r n = N/A st♦ s s ♦♥srr♦♥ r♥rs rrs q ♦rrs♣♦♥♥ ♥s♠

♥ó♥♦

s♥♦ ♦s á♦s r♠ ② ♥st♥ ♦♥stt♠♦s q st♦s t♦rs tr♦♥ ♥tró♥ ♥tr ♦

② s ♣rs ♣♥tá♥♦s ♦str♠♦s q ♠♥♥♦ ♣♥t♠♥t♦ s♣r♥ ♦s ♣rs s♠étr♦s

❬❪

st♦s út♠♦s rst♦s ♥♦s ♥r♦♥ sr ♦s sst♠s ♠ás ♣r♦♣♦♥s ♣r ♥ ♣♦s r③ó♥

♥♦♥tr♠♦s q ♦s ss ♥♦s r r ② ❳ s♦r♦s s♦r sstrt♦s ♠ts ♥♦s s

② s♦♥ ♦s sst♠s ♠ás ♦s ♥r♠♦s st♦ st♦s sst♠s t③♥♦ ♥ rsó♥

♣r♦♣st ♣♦r ♥♦tt♦ ② ♦♦r♦rs ❬ ❪ ② ♦s ♣♦t♥s s♦ró♥ rsts ♣r♦♣st♦s ♣♦r ③♠s②

♦ ② ❩r♠ ❬❪ ♥ ♠r♦ ♥♦♥tr♠♦s q st ♦r♠ó♥ t♥ ♣r♦♠s ♥ r♥♦ t♠♣rtrs

r♥♦ tr♣ Tt ♣♦r ♠♣♦ ♠♣♦s r t♥só♥ s♣r ♣r Tt ≤ T <

❬❪ rró♥♦ rst♦ ♣r s♦tr♠ ♥ sst♠ r2 ❬❪ ♥t♠♦s q ♥♦♥♥♥t ♦

♣r♦♦ ♣♦t♥ ♥tró♥ ♥tr ♣rtís ♦ Pr♦♣s♠♦s ♦tr ♦r♠ ♣r s ♥tró♥

♣rs ② r♠♦s q ♦t♥♥ s♦♦♥s ♥ t♦♦ r♥♦ t♠♣rtrs ♥tr ♣♥t♦ tr♣ ② rít♦

Tt ≤ T ≤ Tc ♥ ♣rtr ♣r♠♦s ♦rrt♠♥t s t♥s♦♥s s♣rs ② r ❬ ❪

♥ts ♥③r ♥♦ r♥rs st♠♦s s♦ró♥ ss ♥♦s s♦r ♥ s♦ ♣r ♣♥ ♠ts

♥♦s ② ♥♦s ♦tr♦s sstrt♦s ❬ ❪ ♠♦s s♦tr♠s s♦ró♥ ② ♣rs ♥s ♣r

s♦s sst♠s ♥ ♦s s♦s ♦♥ í r♦s ♥ r ❲s ♠♦s s ♦♥str♦♥s ① ♣rt♥♥ts

② ♦t♠♦s í♥s ♣r♠♦♦ ♦ r♦ sts í♥s q ♦♠♥③♥ t♠♣rtr ♠♦♦ Tw

♦①st♥ ♥ ♣í ♥ ♦♥ ♦tr rs q ♠ q T ♠♥t s ♣r♥ ③ ♠ás st rs

s rít ♣r♠♦♦ Tcpw ①st♥ sts í♥s ♣r♠♦♦ ♥ q ② tr♥s♦♥s

s ♣r♠r ♦r♥ ♥③á♥♦s tr♠♥♠♦s Tw ② Tcpw ♦♠♣ró♥ ♦♥ t♦s ①♣r♠♥ts ② ♣rs

♥s ♦s ♦♥ s♠♦♥s ♦♥t r♦ s ①t♦s

♦♥t♥ó♥ ♥③♠♦s ♥♦ r♥rs ♣rs é♥ts q ♦♠♦ ② ♦ í♠♦s sñ♦ ♥tr♦r

♠♥t ♣r♦♥ s♦r ♦ ♥ ♣♦t♥ s♠étr♦ rs♣t♦ ♣♥♦ ♣r♦ s ♣rs q ♣s ♣♦r ♥tr♦

r♥r ♥♦♥tr♠♦s q s s♦♦♥s q ♥ st♦s ♥♠♥ts ♦♥ ♣rs ♥s s♠étr♦s

♣r♥ ♥ sst♠s q ①♥ í♥s ♣r♠♦♦ ❬ ❪ t♦ ♦rr ♥tr♦ ♥ tr

♠♥♦ r♥♦ n ② s♣r ♣♦r ♦♠♣t♦ ♥♦ T s♣r ♥ ♦r rít♦ Tsb ❯♥ ♥áss t♦ ♠str

q t♦ ♦♥t♥ú ♣r T > Tw st ♥③r Tsb q rst sr Tcpw ♦rrs♣♦♥♥t

♦♠♥ó♥ sstrt♦♦ st st ♠♥r tr♠♥♠♦s q ② ♥ ♦rrs♣♦♥♥ ♥tr í♥s

♣r♠♦♦ ② ♣rs ♥♠♥ts s♠étr♦s ♥ ♥ rsr♦r♦ rr♦ st♦s st♦s s♠étr♦s ♥ ♣r♥♣♦

♣♦rí♥ st③rs

♥ ♠r♦ ♦s ①♣r♠♥t♦s ♥r♠♥t s r③♥ ♥ ♦♥tt♦ ♦♥ ♥ rsr♦r♦ q ♣r♠t ♥ rrstrt♦

♥tr♠♦ ♣rtís ♥♦ ♣rsó♥ ♦ s q♥t ♣♦t♥ qí♠♦ µ ♥ sts ♦♥♦♥s T ② µ

♦♥st♥ts ② q str sst♠ ♥ ♠r♦ ♥s♠ r♥ ♥ó♥♦ s♦♠♦s s ♦♥s

♥ st stíst ② ♦t♠♦s s♦tr♠s s♦ró♥ ♦♥ ♦s ♠s♠♦s st♦s sts ② ♠tsts q ♦♥

♥ sq♠ ♦s st♦s ♥♠♥ts s♦♥ q♦s q t♥♥ ♠í♥♠♦ r♥ ♣♦t♥ ♥♦

♥rí r ①r st ♦♥ó♥ q ♥ s♦♦ st♦ s♠étr♦ st q q t♥ ♦s ♥

♣r ♣í ♥ ② ♦s ♦tr ♣r ♣í rs q ♦①st♥ ♥ s♦ s♦ró♥ s♦r

♥ s♦ ♣r st st♦ t♥ ♠s♠♦ r♥ ♣♦t♥ q ♦s ♣rs s♠étr♦s ♦♥ ② ♥ ♣í ♥

♦s ♥ s ♣rs ♦ ♥ ② ♥ ♣í rs ♦s ♥ s ♣rs ❬ ❪

♥áss sst♠s rsts ♦ ♦ ♥ ♠r♦ s t♦t♠♥t ♦r♥

♣ít♦

♥tr♦ó♥ ♥r

♦s ♠♣s

♦s ♦s s♠♣s s♦♥ q♦s ♦♥stt♦s ♣♦r ♣rtís át♦♠♦s ♦ ♠♦és s♥t♠♥t s♠étrs ♦♠♦

♣r q ss ú♥♦s r♦s rt s♥ ♦s trs♦♥s ❯♥ ♠♣♦ ♦ s♦♥ ♦s ♦s ♦♥stt♦s ♣♦r

ss ♥♦s st♦s stá♥ ♦♥♦r♠♦s ♣♦r át♦♠♦s ♣♦rs s♥ ♠♦♠♥t♦ ♣♦r ♣r♠♥♥t ♦♥ s r③s

♥tró♥ s♦♥ s ♣ró♥ ♠♦♠♥t♦s ♣♦rs tr♥st♦r♦s ♦s ♠♦♠♥t♦s tr♥st♦r♦s sr♥ ♦♠♦

rst♦ s t♦♥s ♥ s str♦♥s ♦s tr♦♥s q ♦♥♦r♠♥ s ♥s tró♥s ♦s

át♦♠♦s ♦ ♠♦és ♦♠♦ ♦♥s♥ st tó♥ ♥ ♥♦ ♦s át♦♠♦s ♣ ♣rr ♥ ♠♦♠♥t♦

♣♦r tr♥st♦r♦ q ♥ ♦tr♦ ♠♦♠♥t♦ ♣♦r ♥ át♦♠♦ ♥♦ ♥ó♠♥♦ ♦♥♦♦ ♦♠♦ ♥tró♥

s♣rsó♥ ♦ ♦♥♦♥ ❯♥ rsó♥ ♠② t r st t♠át ♣ ♦♥strs ♥ ❬❪

s t♦♥s ♥ ♠♦♠♥t♦ ♣♦r s♦♥ ts q ♥ ♣r♦♠♦ ♥ ♥ ♦r ♥♦ ♥ ♠r♦ s r

♦r♥tó♥ ♣♦r q s ♥r ♥tró♥ q s st ♦♥ át♦♠♦ ♠♦é ♥♦ ♥♦ s ♦♥sr

q ♣rt♣♥ ♥♦ ♠ás ♦s át♦♠♦s ♠♦és s s♠♣r trt

♣♦r③ ♣♥ t♣♦ sst♥ ♣r♦ ♥ st♠ó♥ s ♥rís ♥♦rs ♥ sts ♥tr

♦♥s s ♠t ♥ ♦r ♥♦ s♣r♦r ♥ ♠♦ ♥ ♠②♦rí ♦s s♦s P♦r ♠♣♦ ♣r

s♦ r ♥ s s íq t♠♣rtrs ♥tr tr♣ ② rít ♥rí ♥tró♥ ♥tr ♦s ♣♦♦s

♥♦s s ♠♦ r ♥ ③ s ♥r♦r s ♥rís ♥♦rs ♥ ♣♥ts r♦♥♦

♦♠♦ ♦s q s ♥r♥ ♥ s♦ ② ♥s ♥ s ♠♥♦r s ♦s ♦♥s ♥ só♦s rst♥♦s

P♦r ♦tr♦ ♦ ♥♦ ♦s ♦♥stt②♥ts sts sst♥s s ♥ t ♠♥r q s st♥ s rt♠♥t

♣qñ s r③s r♣ss q ♣r♥ ♥tr ss rs♣ts ♥s tró♥s ② ♥tr ♦s ♥ú♦s ♣r♦♠♥♥

♣♦r s♦r s trts ② ♥rí ♥♦r r ♥ r♦s ór♥s ♠♥t

á♦ ♠♥t ♣♦t♥s r♣s♦s s ♠② ♦♠♣♦ ② rqr rs♦r ♣r♦♠ ♦rrs♣♦♥♥t

strtr tró♥ sst♠ ♥♦ s ♣r♦ s♦♣♠♥t♦ ♥tr át♦♠♦s ♦ ♠♦és s t♥r ♥

♥t rátr r♠ó♥♦ ♦s tr♦♥s t♦r q r ♦♠♣ ♣r♦♠ ♥ ♠r♦ ♣r ♠♦s

♣r♦♣óst♦s st sr q r♣só♥ ♠♥t rá♣♠♥t s♠♥r st♥ ② q ♣r s sr♣ó♥

♣♥ ♠♣rs ①♣rs♦♥s ts ♦♠♦ Bexp(−αr) ♦♥ r s st♥ ♥tr át♦♠♦s ♦ ♠♦és ② ♦s

♣rá♠tr♦s B ② α ♣♥♥ ♥tr ♦trs ♦ss t♣♦ sst♥ st ①♣rsó♥ sr ♥ ♠á♥ á♥t

❬❪ ♣ rrs B/rn ♦♥ r♦ ♦♥ ♦s t♦s ①♣r♠♥ts n ♣ t♦♠r ♥ ♦r ♥tr ②

Pr ♥tr♦♥s ♥tr át♦♠♦s q ♦♠♣♦♥♥ ♦s ss ♥♦s n = 12 s ①♣♦♥♥t q ♠♦r ♣r ♦s t♦s

♠♣ír♦s

♥ss rt♠♥t ts ♦♠♦ s q ♣r♥ ♥ ♥ s íq ♥ ♦♥♦♥s ♥♦r♠s s♦♥ s r③s

r♣ss s q tr♠♥♥ strtr ♦ st ♠♥r s s ♦♥♦ strtr ② tr♠♦♥á♠

♥ ♦ ♣r♠♥t r♣s♦ s r③s trts ♣♥ trtrs ♦♠♦ ♥ tér♠♥♦ ♥♥♠♥t ♣rtrt♦

♣r♦♠

sí ♥ st ♦♥t①t♦ ♥ ♠♦♦ ♦ ♣r str ♦s s♠♣s ♦♥sst ♥ ♦♥srr♦s ♦♠♦ ♥ ♦♥♥t♦

♣rtís ♠♦♦ srs rs ♦♥ ♥tr♦♥s trts

♦rí ♥♦♥ ♥s

r ♥tr♦ó♥ stór

♦rí ♥♦♥ ♥s ♦♥stt② ♥ rr♠♥t ♦s ♣r str ♦♠♣♦rt♠♥t♦

♦s ♦s ♥ s srr♦ó ♥ ♦♠♦ ♣r♦♠♥t♦ tr♥t♦ rs♦ó♥ ó♥

rö♥r ♣r ♣♦r ♣r♠r ③ ♠♥r ♠② ♣r♠♥r ♥ ♥♦ ♦s tr♦s ♥ r ❲s

❬❪ r♦♥♦ ♣r♦♠s ♥trss ♥♦ ♦♠♦é♥s ♥ sst♠s síq♦

tr♦ ♥ r ❲s ♥♦ t♦ ♠②♦r trs♥♥ ② ♦ q s♣rr st é ♦♥

♦r ♥t♦ s ♣r♠rs ♥♦♦♥s ♥ t♦rí ♥♦♥ ♥s r♦♥ srr♦s ♥

♣♦r ♦♠s ② r♠ st♦s t♦rs rr♦♥ ♥ ♣r♠r rsó♥ t♦rí ♥♦♥ ♥s ♥

♦♥t①t♦ ís tó♠ ♥♦ st♥ ♣r♦♠ r ♥rí ♥ át♦♠♦ r♣rs♥t♥♦

s ♥rí ♥ét ♦♠♦ ♥ ♥ó♥ ♥s tró♥ ② ♦♠♥♥♦ st♦ ♦♥ s ①♣rs♦♥s áss

s ♥tr♦♥s ♥ú♦tró♥ ② tró♥tró♥ q t♠é♥ s ♣♥ r♣rs♥tr ♥ tér♠♥♦s ♥s

tró♥ ❯♥ ñ♦ ♠s tr ♠♦♦ ♦♠sr♠ ♠♦r♦ ♣♦r r q♥ ñó ♥ ♥♦♥

♦rrs♣♦♥♥t ♥rí ♥tr♠♦

Ps ♦s ♦r♦s ♥③♦s ♥ s ♠♦♠♥t♦ t♦rí ♦♠sr♠r ♥♦ rst s♥t♠♥t ♣rs

♣r ♠②♦rí s ♣♦♥s st t ♣rsó♥ st ♦♥ ♠♥r ♣♦♦ r♣rs♥tr

tér♠♥♦ ♦rrs♣♦♥♥t ♥rí ♥ét ♦♠♦ ♥ ♥♦♥ ♥s tró♥

s tór ♣r t ré♥ ♥ ♣♦r ♦♥r ② ♦♥ q♥s ♠♦strr♦♥ q

♥rí ♣♦í ①♣rsrs ♦♠♦ ♥ ♥♦♥ ♥s ② q ♠ás ♥s rr sst♠ s q

♠♥♠③ st ♥♦♥ t♦r♠ ♦♥r ② ♦♥ ❬❪ s á♦ t♥t♦ ♣r ♦s♦♥s ♦♠♦ ♣r r♠♦♥s

♣ó♥ ♠ás ♠♣♦rt♥t t♦rí ♣ró ñ♦ s♥t ♥♦ ♦♥ ② ♠ ♠♦strr♦♥ q ♣rtr

s ♣♦s srr ♥ ó♥ ♣r ♦rts ♥ ♣rtí ♦s s s ♦t♥ ♥s tró♥

st ♦r♠ó♥ ♥ ♥t♦♥s ♦♠♦ ♥ t♦rí ♥♦♥ ♥s tró♥ s♥ ♠r♦ ♥♦ st

♠♦s ♦s ñ♦s ♥♦ s ♦♥tr♦♥s ♦♥r ♦♥ ② ♠ str♦♥ ♦r♠s♠♦

tór♦ ♥ q s s ♠ét♦♦ s♦ t♠♥t

♥ ♦♥ ró ♣r♠♦ ó í♠ ♣♦r ss ♣♦rts srr♦♦

ás tr ♦ q ♦r♥♠♥t s srr♦ó ♥ ♠r♦ t♦rí á♥t ♥♦ rtst ♦♥ át♦♠♦s

st♦♥r♦s s ①t♥ó ♦♠♥♦ ♠á♥ á♥t ♣♥♥t t♠♣♦ ② st♦ ♠tr

♦♥♥s ♥ ♥ ② r♦ss ♣r♦♥ ♥r③ó♥ t♦r♠ ♦♥r ② ♦♥ ♣r s♦

♣r♦♠s ♣♥♥ts t♠♣♦ ♦ ♦♥♦♥s s♥t♠♥t ♥rs ♦ q s♥tó s ss ♣r

♥♦♠♥ ♦rí ♥♦♥ ♥s ♣♥♥t ♠♣♦ ♦ ♥ ss ss ♥ ♥s ②

♦② í ② s ♥s ♣ó♥ st ♦r♠s♠♦ ♥ ♦♠♥♦ rtst

♥ t ♦s ♠ét♦♦s á♦ r♦s ♦rí ♥♦♥ ♥s s♦♥ ♦s ♠ás t③

♦s t♥t♦ ♥ ♣r♦♠s r♦♥♦s ♦♥ strtr tró♥ ♠tr ♦♠♦ ♥ ís ♠tr

♦♥♥s ♥ ♥r ② ♥ qí♠ á♥t

♥ rs♦ó♥ ó♥ rö♥r ♣r♠t srr ♦r♠ ①t ♦♠♣♦rt♠♥t♦ sst♠s

♠r♦só♣♦s s ♣ ♣ró♥ s ♠t ♣♦r ♦ q ss ♦♥s s♦♥ ♠s♦ ♦♠♣s

rs♦r ♥♠ér♠♥t ♦ ♠♥♦s ú♥ ♥ít♠♥t ♦♥ s ♣♦ ♦rr st ♣r♦♠ r♦r♠á♥♦♦

♣r sr ♣③ ♦t♥r ♣♦r ♠♣♦ ♥rí ② stró♥ tró♥ st♦ ♥♠♥t ♣rs♥♥♦

♥ó♥ ♦♥ sst♠ ❯♥ ♥t s♥ ♦r á♦ r ♥ q ♥s ♥ sst♠

N ♣rtís s ♥ ♠♥t ♠♦ ♠ás s♠♣ q ♥ó♥ ♦♥ ♣s ♥ ♣r♥♣♦ r♥ q

♣♥ s♦♦ rs s ♦♦r♥s ♣♥t♦

❯♥ s♥t s q s♦ ♥ ♦s s♦s ♠ás s♠♣s ♥♦ s ♦♥♦ ♠♥r ①t ♥♦♥ q r♦♥

♥s ♦♥ ♥rí sst♠

♣rtr ♦s ñ♦s ♦r♠s♠♦ ♠♣♠♥t t③♦ ♣r á♦s ís st♦

só♦ s♥ ♠r♦ s ♦♥sr q ♥♦ r ♦ st♥t ♣rs♦ ♥ á♦s t③♦s ♥ ♦♥t①t♦ qí♠

á♥t

♠é♥ ♥ s♦s ñ♦s st ♦r♠ó♥ s ♣♦ s♦ ♦s ♦♠♦é♥♦s ♦♥sr♦s ♦♠♦ ♥ ♦♥♥t♦

srs rs ♦♥ ♥tr♦♥s t♣♦ trts ♦♥ rs♣t♦ ♦♥tró♥ s s í♥ ♣r♦♣st♦

rs ①♣rs♦♥s rrs ♣r r♣rs♥tr s♠ ♦s tér♠♥♦s ①♣♥só♥ ♦rrs♣♦♥♥t ♥rí

r ♦♠♦ ♣♦r ♠♣♦ ♦r♠ s ①t r♥♥ ② tr♥ ❬❪

♥s é s ♦♠♥③ó str sst♠s ♥♦ ♦♠♦é♥♦s ♦♠♦ ♦s q ♣r♥ ♥♦ ♥

♦ s s♦r s♦r s♣r ♥ só♦

♥ s és ♦s ② ♦s s ♥tr♦r♦♥ ♠♦rs ♥ ①♣rsó♥ ①s♦ ♥rí r s

♦♠♣♦♥♥t s♥ t♦rí Pr♠r♦ s s♣s♦ q s tér♠♥♦ í ♣♥r ♥ ♥s ♣r♦♠

s♦r ♥ sr á♠tr♦ s srs rs q ♦♠♣♦♥í♥ ♦ st ♣r♦♠♥t♦ q stá

♣♦ ♥ ♥ tr♦ r③♦♥ ❬❪ ♦♥t♥í ♣rá♠tr♦s q s st♥ ♣r r♣r♦r ♥ó♥

♦rró♥ ♦s r♣♦s

♥ s ♣r♦♣s♦ ♦tr ①♣rsó♥ ♣r ①s♦ ♥rí r s q ♥♦ ♦♥t♥í ♣rá♠tr♦s rs

♠ás ♦rrs♣♦♥♥t á♠tr♦ s srs rs q r♣rs♥t♥ át♦♠♦s ♦ ♠♦és ♦

st ♣r♦①♠ó♥ ♣r♦♣st ♣♦r ♦s♥ ❬❪ ② r♦r♠ ♣♦r r ② ♦s♥r ❬❪ s ♦♥♦ ♦♠♦ ♦rí

s s ♥♠♥ts

t♦rí ♣ ♦s s♣t♦s ♥tí♦s ② t♥♦ó♦s

s ♣♥t♦ st ♥tí♦ st♦ st♦ íq♦ ♣♥t ♥♦s sí♦s s♥♦ st♦ ♠tr

q ♣rs♥t ♠②♦r t ♣r s ♦♠♣r♥só♥ ♥trs q ①st♥ ♥♦s ♠♦♦s tór♦s ♣r só♦s ②

ss ♥ ♥♦ s ♣♦s ♦t♥r ♥♦s ♠♦♦s ♣r rátr ♥tr♠♦ ♥tr st♦s ♦s st♦s ♠tr

r♥t st s♦ s ♥ tr♠♥♦ ①♣r♠♥t♠♥t ♦♥ ♥ ♣rsó♥ s ♣r♦♣s tr♠♦♥á♠s

♥ r♥ ♥ú♠r♦ íq♦s ♦ q ♣r♠t r ♦s rs♦s ♠♦♦s srr♦♦s ♦♥trst♥♦ ♦s rst♦s

q ♣r♥ st♦s ♠♦♦s ♦♥ ♦s t♦s ♠♣ír♦s

st♦ st♦ íq♦ t♠é♥ rst r♥ ♠♣♦rt♥ s ♥ ♣♥t♦ st t♥♦ó♦ ♣s st

r♦rr q ♠②♦r ♣rt ♦s ♣r♦s♦s ♥strs t♥♥ r ♥ st♦ íq♦ ♥♥rí í♠

♠♦r♥ s ♥r♥t r♦ ♣r♦♠s q rqr♥ ♦♥♦♠♥t♦ ó♥ st♦ ♥ íq♦

♣r♦ ♦ ♥ ♠③ íq ♥t♣í ♣♦r③ó♥ ♣ ♦rí t Pr tr♠♥r sts

♣r♦♣s s t③♥ ♥r♠♥t ♦♥s st♦ ♠♣írs

s ♦♥s st♦ ús ♦♥stt②♥ s♥ s ♠ás ♣♦♣rs ♦ s s♠♣ ♥ r ❲s

♣r♠r♦ ♥ ♣r♦♣♦♥r st t♣♦ ♦♥s s ♥t♦♥s ♥ ♣r♦ ♦trs s♥ s♣r♦rs ♦♠♦

s♦♥ ó♥ ♦♥♦ ② P♥♦♥s♦♥ ♦s sts ♦♥s ♦♥s ♥ ♥ ♣r♥♣♦

♣r sst♥s ♣rs ♥ s♦ ①t♥s ♠③s t③♥♦ s ♥♦♠♥s rs ♠③ s ♣♥t♦

st t♥♦ó♦ s rts st t♣♦ ①♣rs♦♥s s r♦♥♥ ♥♦ s♦♦ ♦♥ s s♥③ s♥♦ ♠ás ♦♥

r♣③ ♦♥ q ♣r♠t♥ r ♦ ♦s á♦s rtrísts s♣♠♥t ss ♥♦ á♦

♣r♦♣s tr♠♦♥á♠s ♦♣r s♦♦ ♥ ♣qñ ♣rt t♠♣♦ ♠♣♦ ♥ sñ♦ ♣r♦s♦

♥str P♦r ♦tr♦ ♦ ♦ ♥♠♥t♠♥t s é♥ss s ♣♦ss ♠♦rr s ♣r♦♥s q

s ①tr♥ s s♦♥ ♠② ♠ts s s♦♥s rs♣t♦ ♦s rst♦s ①♣r♠♥ts q s ♦t♥♥

♣rtr sts ♦♥s ♠str♥ ♥ ♦♠♣♦rt♠♥t♦ rrát♦ q ♦ rátr ♠♣ír♦ s ♠s♠s

♠♣♦s tr tr♠♥r ♦♥ ♣rsó♥ ♦r♥ ts ♥s

♥ s♦ s ♠③s rátr ♣rt♦ s ♦♥s st♦ ús ♠♣írs s ♠② ♠t♦

Ps q t③ó♥ ①♣rs♦♥s ♠♣írs s ♦♣♥♦ ♥ r♦ ♠♣♦rt♥t ♥ t rst ♥t

q ♥ ♠♦r sst♥ ♥ ♣rsó♥ ♦s rst♦s ♦t♥♦s t♥t♦ ♣r sst♥s ♣rs ♦♠♦ ♣r ♠③s

só♦ s ♦♥srá t♥♥♦ ♥tr③ ♠♦r s ♥tr♦♥s ♥tr ♣rtís ② t③♥♦ ♠ét♦♦s

q s ♣♦②♥ s♦r s r♠ r ♣♦r á♥ stíst

P♦r ♦tr♦ ♦ ♥ st♦s út♠♦s ñ♦s ♦ ♥ ♣rt srr♦♦ ♥♥♦t♥♦♦í ♦r♠s♠♦ ♦ró ♥

♥♦ ♠♣s♦ ♥ s ♣ó♥ sst♠s ♦♥stt♦s ♣♦r ♦s s♦r♦s s♦r sstrt♦s só♦s ♣

t ♠♥♣ó♥ ♠tr ♥ s ♠♦r ② tó♠ ♣r♠t sñ♦ ② ró♥ ♦s ♠ás

rs♦s s♣♦st♦s q ♦♥tr♦♥ ♠♥♠♥t♦ ró♥ ② ró♥ rs♦s ♦s ♥tr sí ② ♦♥

♠trs só♦s ② ♥♦s ♦♥t♥♦ ♦♥ ♥s ás ② ♦♠♣rts ♥ ss q ♥ s ♠ró♠tr♦s

st ♥♥ó♠tr♦s ❯♥ ♣♥♦r♠ ♦s st♥t♦s t♣♦s ♣♦♥s ♣♦ss s s♣♦st♦s ♥strs st

♣rt♦s sr t③♦s ♥ ♦♦í ② ♠♥ stá rs♠♦ ♥ ♥ tr♦ qrs ② ❬❪

tr ♣ó♥ t♥♦ó s♠ ♠♣♦rt♥ st r♦♥ ♦♥ t③ó♥ ró♥♦ ♦♠♦ ♥t

♥rí ♥ ♣rtr ♠♥♠♥t♦ ró♥♦ ♠♦r ♦♥♥♦ ♥ r♣♥♦ stá trt♦ ♥

❬❪

P♦r t♦♦ ♦ ♦ ♥ts s ♦♥♥♥t ♦♠♣r♥r ♦s ♥ó♠♥♦s ♦s ♦♥ s♦ró♥ ♦s s♦r

s♣rs strtr ♦ s♦r♦ ♥ ♥ só♦ q tr♠♥ ♣♦r ♥ ♥ s ♥tr♦♥s

só♦♦ ② ♦♦ ♦ s♦r♦ ♣ ♣rs♥trs ♦♠♦ íq♦ ♦ ♣♦r ♦ ♥s ♥ s só

♣♥♥♦ ♥s ♦ ♥r♠♥t ♦ ♠ás r♥♦ sstrt♦ stá ♥ s íq ♦ ♥s

só ♠♥trs q ♠ás ♦ s ♥♥tr ♥ s s♦s st♦s sst♠s s♦r♦s ♣rs♥t♥ ♥trss

íq♦♣♦r ②s ♣r♦♣s strtrs ② tr♠♦♥á♠s s♦♥ ♣r♦s♠♥t ♥③s ♥ ♦r♠ tór

trs rr♠♥ts tórs

♥ ♥r ♦s trt♠♥t♦s ♥trss s♦♥ ♥trí♥s♠♥t t♦s♦s ♣♦rq ♣rs♥t♥ ♣rs ♥s

q ♣♥♥ ♣♦só♥ ♦♥ ♠ás ♦s r♥ts ♣♥ sr ♠② ♥t♥s♦s Pr rs♦r st s

sst♠s s ♣♥ ♣r ♠ás t♦rí ♥♦♥ ♥s ♠ét♦♦s s♦s ♥ s♠♦♥s

♦♥t r♦ ♦ ♦rt♠♦s á♦ q t③♥ ♦r♠s♠♦ ♥á♠ ♦r ♥tr ♦tr♦s ♥ r♦s

r♦s r♥t ♣ó♥ s ♣♥ ♥♦♥trr rsú♠♥s ♥rs ♦ó♥ sts ♦r♠♦♥s r

♣♦r ♠♣♦ s rr♥s ❬ ❪

♦♠♥③r♠♦s st ♣rt♦ ♦♥ ♥ r rs♠♥ r sts ♦trs rr♠♥ts

s t♦rís ♠♦r♥s st♦ íq♦ q ♦♠♥③r♦♥ ♦♠♦ ♥ ♥t♥t♦ srr ♦♠♣♦rt♠♥t♦ tr♠♦

♥á♠♦ íq♦s s♠♣s ♦♠♦ ♦s ♦♥stt♦s ♣♦r ss ♥♦s ♥ st♦ íq♦ ❬❪ ♣♥ r♣rs ♥ trs

r♥s ♠s

ét♦♦s ♠ó♥

♦rís ♦♥s ♥trs

♦rís ♣rtr♦♥s

ét♦♦s ♠ó♥

rs ♥♦r♠ ♥ ♦r♦ ♥ srr♦♦ ♦s sst♠s ó♠♣t♦ s ♦ ♣♦s tr♠♥r s

♣r♦♣s tr♠♦♥á♠s rs♦s ♦s ♠♥t s♠♦♥s ♦♠♣t♦♥s t③♥♦ ♦s srr♦♦s

tór♦s q s s♣r♥♥ á♥ stíst r ♣♦r ♠♣♦ ❬❪

s té♥s s♠ó♥ ♦♠♣t♦♥ ♦♠♥③r♦♥ srr♦rs ♣rtr ♦s ñ♦s ♦♥ ♥♠♥t♦

♠♦r♥s ② ♣♦r♦ss ♦♠♣t♦rs

s s♠♦♥s ♦♠♣t♦♥s rqr♥ ♦♠♦ ♥♦r♠ó♥ ♥tr ♣♦t♥ ♥tr♠♦r P♦r ♦tr♦ ♦

tr♠♥ó♥ ♣♦t♥ ♥tró♥ ♥tr át♦♠♦s ♦ ♠♦és s ♥ ♣r♦♠ ♥♦ t♦♦ rst♦ q

♥♦ ♥ á♠t♦ á♥ á♥t ♦s ♣♦t♥s ♥tr♠♦rs ♦t♥♦s trés á♦s

♠á♥♦á♥t♦s s♦♥ t♦í ♣♦♦ ♣rs♦s ♦ rt♠♥t ♥①st♥ts st♦ s ñ t r♦♥

♦♥ ♥tró♥ ♥tr trs ♦ ♠s r♣♦s q ♥ ♠♥r s ss♥ t③♥♦ ♣♦t♥s t♦s Pr

tr♠♥r s ♣r♦♣s tr♠♦♥á♠s st♦ íq♦ ♣ sr ♥sr♦ ♥ ♦♥♦♠♥t♦ ♣♦t♥

q s ♦r♥ ♥tr ♠ás ♦s r♣♦s ♣s ♥ st st♦ át♦♠♦s ♦ ♠♦és ♣♥ ♥trtr ♥♦ s♦♦

♣rs ♥ t s ♦♥♦ ♠② ♣♦♦ ♥ rr♥ ♣♦t♥s ♥tró♥ q ♥♦r♥ ♠ás

♦s r♣♦s P♦r st r③ó♥ t♦♦s ♦s ♠ét♦♦s s♠ó♥ ts s♦s ♥ r♠♦♥á♠ stíst

t③♥ ♦♠♦ ♣♥t♦ ♣rt ♣♦t♥s ♣rs ♠♣ír♦s ♦ s♠♠♣ír♦s q ♣♥ ♦♥srrs ♦♠♦

♣♦t♥s ♣rs t♦s q ♥ ♥t ♥t♦♥s ♣rs♥ st t♣♦ ♥tr♦♥s

s té♥s s♠ó♥ t♠♥t t③s s♦♥ ás♠♥t ♦s s♠ó♥ ♦♥t r♦

❬❪ ② ♥á♠ ♦r ❬❪ ♥ ♠s s tr ♦♥ ♥♦s ♣♦♦s ♠s ♠♦és ♣r tr ♥ t♦

♦st♦ ♦♠♣t♦♥ ♣♦só♥ ② ♦ sts ♠♦és ♦♦♥♥ ♦r tr♠♥s ♦♥s

♠♦♠♥t♦

♥ s♦ s♠ó♥ ♦♥t r♦ s s q s r♥ts ♦♥r♦♥s q s ♥ ♥r♥♦ rst♥

♣r♦♣♦r♦♥s t♦r ♦t③♠♥♥ st t♣♦ s♠♦♥s ♣r♦♣♦r♦♥♥ s♦♦ ♣r♦♣s qr♦

sst♠

♥ s♦ ♥á♠ ♦r s rs♥ ♣r ♦♥ró♥ ♥r s ♦♥s ♠♦♠♥t♦

♦♥♥t♦ át♦♠♦s ♦ ♠♦és t♥♥♦ ♥ ♥t s r③s ② ♦s t♦rqs r♦s s♦r ss ♣rtís ♥

st út♠♦ s♦ s ♦t♥♥ t♠é♥ ♣r♦♣s ♥á♠s sst♠ ♥ stó♥

Pr ♠s té♥s ♦s t♠♣♦s ó♠♣t♦ s♦♥ ♥r♠♥t r♥s ② ♠♥t♥ rá♣♠♥t ♦♥ ♥ú♠r♦

♣rtís ♥♦rs

♦rís ♦♥s ♥trs

s t♦rís s♦r ♦♥s ♥trs ♥r♦♥ ♣r♥♣♦s st s♦ ♠r ♦s tr♦s r♥tí♥ ②

❩r♥t q ♦♥r♦♥ ♥♦♠♥ ó♥ r♥st♥❩r♥ ❩ ❬❪ st ó♥ s ♥ r

♥ ó♥ ♥tr q r♦♥ ♥ó♥ ♦rró♥ t♦t ♦♥ ♥♦♠♥ ♥ó♥ ♦rró♥ rt

♣r♦♠ ♥♠♥t s ♦♥s ♥trs s ♥♦♥trr ♥ s♥ ró♥ ♥tr sts ♥♦♥s

♥♦♠♥ ró♥ rr st ró♥ ♣r♠t rs♦r ♦rrs♣♦♥♥t sst♠ ♦♥s r♦

♦r♠s♠♦ s ♣r♦①♠♦♥s t♦rí stá♥ ♣s ♦♥t♥s ♥ ró♥ rr ①st♥ ás♠♥t s

s♥ts r♦♥s rr

ró♥ rr ♦r♠ ♣♦r Prs ② ❨ P❨ ❬❪

ró♥ rr ♥ í♣r♦♥t

ró♥ rr ♥ í♣r♦♥t rr♥ ❬❪

♥ ♥ tr♦ rsó♥ ♥s s ♣ ♥♦♥trr ♥ ró♥ ó♥ ❩ ② ♥s ss

r♦♥s rr s ♣rtr t♦rí ♥♦♥ ♥s ❬❪

♦s sts ♦♥s ♥trs s ♥ ♣♦ st♦ íq♦s ♦♥ r♥ts ♣♦t♥s ♣rs ♦♥

st♦s ♣♦t♥s ♥tró♥ ♥r♠♥t ♣♦s♥ s♠trí sér

♥ ♠②♦r ♣rt ♦s s♦s s ♥sr ♥ rs♦ó♥ ♥♠ér ó♥ ♥tr ② ♥ ♠② ♣♦♦s s♦s

s♦ó♥ s ♥ít

Pr ♠♦♦ srs rs ❲rt♠ ❬❪ ♥♦♥tró ♥ s♦ó♥ ♥ít ó♥ ❩ t③♥♦

ró♥ rr P❨ st♦ ♣r♠tó ♦t♥r ♦rs ♥ít♦s ♥ó♥ stró♥ r

P♦str♦r♠♥t ❱rt ② ❲ss ❬❪ ♦rrr♦♥ ♠♣ír♠♥t s♦ó♥ ó♥ ❩ ♦♥ ró♥

rr P❨ ② ♣r♦♣♦r♦♥r♦♥ ór♠s ♥íts ♠ás ♣rss ♣r tr♠♥ó♥ strtr ♥ ♦

srs rs

♦rís ♣rtr♦♥s

s t♦rís ♣rtr♦♥s ♥♥ ♥ ♦s ñ♦s ② s ♣♥ ♦♥ ♥♦s rst♦s r♥t é ♦s

s♥ ♥ ♦ q ♥rí r ♠♦t③ ♥ íq♦ q ♥tr♦♥ sú♥ ♥ ♣♦t♥ U

♣ srr♦rs ♥ t♦r♥♦ ♥ sst♠ rr♥ q ♦♥sst ♥ ♥ ♠♦♦ ♦ ②s ♠♦és

♥trtú♥ sú♥ ♥ ♣♦t♥ U0 ♥♦♠♥♦ ♣♦t♥ rr♥ ♥ s♥t♦ q U = U0 + δU s♥♦

‖δU‖ << ‖U0‖♦s tér♠♥♦s srr♦♦ ♥ sr ♥♦♠♥♦s tér♠♥♦s ♣rtrt♦s ♠♣♥ ♣r♦♠♦s ② t♦♥s

♣♦t♥ ♣rtró♥ ♦ rr♥

ó♥ sst♠ rr♥ stsr ♦s rqst♦s ♥♠♥ts

r♥t③r ♥ rá♣ ♦♥r♥ srr♦♦ ♣rtr♦♥s

Pr♠tr q strtr ② s ♣r♦♣s tr♠♦♥á♠s sst♠ rr♥ s♥ á♠♥t tr♠

♥s

sí ♣r íq♦s s♠♣s ♥♦♥tr♠♦s só♥ ♣♦t♥ rr♥ ② ♣rtró♥ ♣♦r rr ②

♥rs♦♥ ❬❪ ② ♣♦r ♣♦r ❲s ♥r ② ♥rs♥ ❲ ❬❪ ② ♣é♥

♣r♠r ♣♦t♥ sú♥ s s♥♦ ② s♥ sú♥ s♥♦ s ♣r♠r r st♦ s s

r③s ♥tr♠♦rs ♦ ♥trtó♠s

sr ♣rtrt ♦♥r ♦♥ ♠②♦r r♣③ ♥♦ s t③ só♥ ♣♦t♥ sú♥ rtr♦ ❲

q ♥♦ s t③ só♥ ♦♠♦ ♥ ♥ ♣st♦ ♠♥st♦ ♦s st♦s s♠ó♥ s♦♥ s r③s

r♣ss s q tr♠♥♥ ts ♥ss strtr ♦ ❬❪ ♣r♦♦♥♦ ♥ rá♣ ♦♥r♥

srr♦♦ ♣rtrt♦ ♥♦ s ♥②♥ t♦t♠♥t ♥ sst♠ rr♥

sst♠ rr♥ ❲ ♣ r♦♥rs á♠♥t ♦♥ ♥ ♦ srs rs ♣r q s s♣♦♥

♥ ó♥ st♦ t ♦♠♦ r♥♥tr♥ ❬❪ ② ♥ ♣r♠tr③ó♥ ♠② ♣rs ♥ó♥

stró♥ r ❬❪

sí ♣s t③ó♥ ♣♦t♥ ❲ ♥tr♦ ♦s sq♠s ♣rtrt♦s ♣r♠t ♣rr rst♦s q

stá♥ ♥ ♥ r♦ ♦♥ ♦s ①♣r♠♥ts ♦♥ ♥t ♦♣rt rr ♦s t♠♣♦s ♦♠♣t♦ ♥ ró♥

♦s ♠♣♦s ♥ rs♦ó♥ s ♦♥s ♥trs ♦ s s♠♦♥s ♥♠érs

P♦r ♦ t♥t♦ ♣♦♠♦s r♠r q ♣rtr ♦s ñ♦s ② ♣r♠r ♠t ♦s s ♣r♦♥ r rs

♣r♦♣s tr♠♦♥á♠s ♣rtr ♥ ♣♦t♥ ♦ ❬❪

♣ít♦

♦rí ♥♦♥ ♥s

♥tr♦ó♥ ♦r♠s♠♦

❯♥ ♠♦♦ ♦ s♠♣ ♦♥sst ♥ ♥ sst♠ N ♣rtís ♠s m q ♥trtú♥ ♥tr s ♥ ♣r♥♣♦

♣rs ♠♥t ♥ ♣♦t♥ U ♦♥sr♠♦s t♠é♥ q ♥ sts ♣rtís s ♥ s

♣♦s♦♥s ri ♣♥ str s♦♠ts ♥ ♣♦t♥ ①tr♥♦ Uext P♦♠♦s srr st sst♠ ♠♥t ♥

♠t♦♥♥♦ ♦r♠

HT =∑

i

− ~2

2m∇2

i +∑

i,j

U (ri − rj) + Uext(r).

s rtrísts íss sst♠ stá♥ ♦♥t♥s ♥ ♥ó♥ st♦ ψ s♦ó♥ ó♥ rö

♥r

HTψ = Eψ.

E r♣rs♥t ♥rí st♦ ψ srá ♥ ♥ó♥ ♣♥♥t ♣r♦r N+1 rs ψ = ψ(r1,r2,........rN,t)

♦ ♥s ♥ó♥ ♥ r♣♦ ♣ sr ♦t♥ ♣rtr ①♣rsó♥

ρ(r1) = N

´

dr2dr3.......drNψ2(r1, r2, ........rN)

´

dr1dr2.......drNψ2(r1, r2, ........rN).

①st ♠ás ♥ sq♠ ♣r♦①♠♦♥s ♣r ♦rr st ♣r♦♠ ♠♦s r♣♦s

❯♥ ♣♦s ♠♥♦ ♣r s trt♠♥t♦ ♣r♦♥ st♠♥t t♦r♠ ♦♥r ♦♥ ❬❪ ② r♠♥ ❬❪

q ♠str ①st♥ ♥ ró♥ ♥í♦ ♥tr ♥s ρ ② ♣♦t♥ ①tr♥♦ ♣♦ sst♠

Uext

Uext(r) ⇐⇒ ρ(r).

Pr ♥str♦ sst♠ N ♣rtís ♦♥ ♥tr♦♥s s♦♠t♦ ♥ ♣♦t♥ ①tr♥♦ ♣♦♠♦s ♦♥strr r♥

♣♦t♥ Ω ♦♠♦ ♥ ♥♦♥ ♥s sst♠

Ω[ρ, Uext] = Fint[ρ] +

ˆ

drρ(r)Uext(r)− µN.

♥ st ①♣rsó♥ µ s ♣♦t♥ qí♠♦ ② Fint r♣rs♥t ♥rí r ♥trí♥s ♠♦t③ st s ♥

♦♠♦ ♥rí r sst♠ N ♣rtís ♥♦ ♥♦ ♥trtú ♦♥ ♥♥ú♥ ♠♦ ①tr♥♦

♥ú♠r♦ ♣rtís s ♣ ①♣rsr ♥ ♥ó♥ ♥s trés ①♣rsó♥

N =

ˆ

ρ(r)dr,

♥t r♥ ♣♦t♥ s ♣ ♦t♥r ♥s rs♦♥♦ s♥t ♣r♦♠ r♦♥

δΩ[ρ, Uext]

δρ(r)= 0,

♦ q♥t♠♥t

δFint[ρ]

δρ(r)= µ− Uext(r).

♥rí r ♥trí♥s ♠♦t③ sst♠ s ♣ ♦♥strr ♣rtr ♥ó♥ ♣rtó♥ ♥ó♥

QN (V, T )

Fint(V, T ) = −kBT lnQN (V, T ).

st ♥rí s ♣ srr ♦♠♦ s♠ ♥ ♥rí q ♦rrs♣♦♥ ♥ s Fid) ♦♥ ♥

♥rí s ♥tr♦♥s ♥tr s ♣rtís ♦♥ s ③ ♣r ♥str♦ ♠♦♦ st t♠ s ♣

♣♥sr ♦♠♣st ♣♦r ♦s tér♠♥♦s ♥♦ ♦ ♦♥ ♥ sst♠ srs rís ♦♥ ♥tr♦♥s ♣r♠♥t

r♣ss FHS ② ♦tr♦ s♦♦ s ♥tr♦♥s trts ♥ ♦♥♥t♦ ♠♦és ♣rt♠♥t sérs

Fatt s

Fint = Fid + FHS + Fatt.

♦s ♦♠♦é♥♦s

♦s ♦s ♦♠♦é♥♦s s♦♥ q♦s ♦♥ ♥s ♦♥stt② ♥ ♠♣♦ sr ♥♦r♠

♥ó♥ ♣rtó♥ ♥ó♥ ❬❪ ♦♥t♥ ♠t♦♥♥♦ srt♦ ♥ tér♠♥♦s ♥ ♣♦t♥ ♥tró♥

♣rs U(rij) ♦♥ rij = ri − rj ② ♥rí ♥ét s N ♣rtís p2i /2m

H =

N∑

i=1

p2i2m

+

N∑

i=1

N∑

j>1

U(ri,j).

♦ ♥ s♦ ás♦

QN (V, T ) =1

N !h3N

ˆ

..

ˆ

exp

−β

N∑

i=1

p2i2m

+N∑

i=1

N∑

j>1

U(ri,j)

dp1...dpN =

=1

N !h3N

ˆ

..

ˆ

exp

[

−β∑

i

p2i2m

]

dp1..dpN .

ˆ

..

ˆ

exp

−βN∑

i=1

N∑

i>j

U(ri,j)

dr1..drN =

=1

N !

[

(

2πm

βh2

)1

2

]3Nˆ

..

ˆ

exp

−βN∑

i=1

N∑

i>j

U(ri,j)

dr1...drN =

=1

N !

[

(

2πm

βh2

)1

2

]3N

ZN (V, T ).

♥ st ①♣rsó♥ β = 1/kBT ② kB s ♦♥st♥t ♦t③♠♥♥

♥ s ♥tr♦♥s ♥tr s ♣rtís stá♥ ♦♥t♥s ♥ ♥♦♠♥ ♥tr ♦♥ró♥

r♣rs♥t ♣♦r

ZN (V, T ) =

ˆ

..

ˆ

exp

−βN∑

i=1

N∑

i>j

U(ri,j)

dr1...drN .

rst♦ st ①♣rsó♥ s ♣ ♥tr ♦♥ ♥rí ♥ét ♥ sst♠ N ♣rtís ♦♥ s ♣♥

s♣rr ♦s t♦s á♥t♦s

♥ó♥ ♣rtó♥ ♥ó♥ s t♦r③ ♥t♦♥s ♥ tér♠♥♦s q ①♣rs♥ ♦♠♣♦rt♠♥t♦ sst♠

② ♦tr♦s tér♠♥♦s q ♥ ♥t s ♥tr♦♥s ♥tr s ♣rtís s r

QN−int(V, T ) = QN−id(V, T )ZN (V, T )

V N= QN−idQN−inter.

♠♥t ①♣rs♠♦s ♥♦ st♦s t♦rs ♣r ♦ srr s ♦rrs♣♦♥♥ts ♥rís rs s♦s

♦♥ ♦s

♥ó♥ ♣rtó♥ ♦rrs♣♦♥♥t s

s ás♦

t♦r s♦♦ s s ♦t♥ ♦♥sr♥♦ q ♦s tér♠♥♦s r♦♥♦s ♦♥ s ♥tr♦♥s Uij s♦♥

é♥t♠♥t ♥♦s st ♠♥r ♥ó♥ ♣rtó♥ s sr ♦♠♦

QN−id(V, T ) =1

N !

[

(

2πm

βh2

)1/2]3N

Z0N (V, T ) =

1

N !Λ3NZ0N (V, T ) =

V N

N !Λ3N,

♦♥ Λ s ♦♥t ♦♥ ❵r♦ ♥ ♣♦r

Λ = h/(2πmkBT )1/2.

♥ qr♦ tr♠♦♥á♠♦ ♥rí r ♥trí♥s ♠♦t③ s s r♦♥ ♦♥ ♥ó♥

♣rtó♥ ♥ó♥ ♠♥t st ♠♥r

FN,id(V, T ) = −kBT[

Nln

(

Λ3

V

)

+NlnN −N

]

= kBTN

[

ln

(

Λ3N

V

)

− 1

]

,

sí ♥rí r ♣♦r ♥ ♦♠♥ s ①♣rs ♦♠♦

FN,id(V, T )

V= kBTρ

[

ln(

Λ3ρ)

− 1]

,

♦♥

ρ =N

V,

r♣rs♥t ♥s ♥ s♦ ♥ ♦ ♦♠♦é♥♦

s á♥t♦

♥rí r ♣r ♥ s ♦s♦♥s s ♣ srr ♦♠♦

FN,id(V, T )

V= kBTρ

[

ln(

Λ3ρ)

− 1

Λ3ρg5/2(Λ

3ρ)

]

,

♦♥ ♥íó♥

g5/2(ρΛ3) =

∞∑

s=1

(ρΛ3)s

s5/2.

óts q s ρΛ3 ≪ 1 ♦ s s s ♠♦és stá♥ ♠② st♥s ♥ ♦♠♣ró♥ ♦♥ ♦♥t ♦♥

r♦ q t♥♥ s♦s s r♣r ①♣rsó♥ ás ♥rí r

♥ó♥ ♣rtó♥ s♦ ♦♥ s ♥tr♦♥s

♥ó♥ ♣rtó♥ s♦ ♥ sst♠ N ♣rtís q ♥trtú♥ ♥tr sí ② q qí ♠♦s ♥♦t♦

♣♦r QN,inter s sr ♥ ♥ó♥ ♥tr ♦♥ró♥

QN,inter(V, T ) =ZN (V, T )

V N.

♣rtr st út♠ ①♣rsó♥ s ♣ ♦t♥r ♥rí r ♠♦t③ s♦ s ♥tr♦♥s ♥tr

s ♣rtís q ♦♥stt②♥ ♦

FN,inter(V, T ) = −KBT ln [QN,inter(V, T )] =

= −KBT ln

[

ZN (V, T )

V N

]

.

♦ ♥rí r ♣♦r ♥ ♦♠♥ s sr ♦♠♦

FN,inter(V, T )

V= −KBT

1

Vln

(

ZN (V, T )

V N

)

.

P♦r ♦tr♦ ♦ ♣♦♠♦s ①♣♥r ①♣♦♥♥ q ♦♠♣♦♥ ZN (V, T ) ♦♥sr♥♦ q s ♥tr♦♥s s

s♥ ♠②♦r♠♥t ♣rs ♥tr♥♦ s ♣ rsrr s♥t ♠♥r

exp

−βN∑

i=1

N∑

i>j

U(ri,j)

=∏

pares

exp [−βU(ri,j)] .

♦ ♦♥sr♠♦s q ♣♦t♥ ♥tró♥ ♥tr ♦s ♣rtís ♣♥ s♦♦ st♥ q s♣r

s ♣rtís

s s

U(ri,j) = U (‖ri − rj‖) = U(rij)

♣r♦①♠ó♥ ♥ ♥ ♠♦♦ ♦ ♦♥ s ♦♥sr q s ♠♦és q ♦ ♦♠♣♦♥♥ s♦♥ ♣rt♠♥t

sérs ♣♦♠♦s srr t♦♦ ♥ tér♠♥♦s ♥♦♠♥s ♥♦♥s ②r fij ❬❪

exp [−βU(ri,j)] = 1− βU(ri,j) +1

2β2U2(ri,j) + ............ = 1 + fij .

♦ ♥ s♦ ♥ q s ♣rtís sté♥ ♠② rs ♦♠♦ ♦rr ♣r ♦s ♠② ♥s♦s fij s ♣qñ ② ♥t♦♥s

s♥t srr♦♦ ♦♥ s rt♥♥ s♦♠♥t ♦s tér♠♥♦s q ♦rrs♣♦♥♥ r♠s rrs ❬❪

r♣r♦ ♠♥t ♥tr♥♦

exp

−βN∑

i=1

N∑

i>j

U(ri,j)

=∏

pares

exp [−βU(ri,j)] =∏

pares

[1 + fij ] =

=∑

ij

fij +∑

ijk

fijfjkfki + ...........,

♦

ZN (V, T )

V N=

1

V N

ˆ

....

ˆ

∑

ij

fij +∑

ijk

fijfjkfki + ...........

dr1dr2.......drN =

= 1 +N(N − 1)

2

1

V 2

ˆ ˆ

f12dr1dr2 +N(N − 1)(N − 2)

6

1

V 3

ˆ ˆ ˆ

f12f13f23dr1dr2dr3 + ..... .

② ♥ í♠t tr♠♦♥á♠♦

N

V≃ N − 1

V≃ N − 2

V≃ ... ≃ ρ.

s t♥ ♥ ♥t ♠ás q ♦s tér♠♥♦s ♥trs s♦♥ tér♠♥♦s ♦rró♥ ♣♦r ♥ ♣qñ♦s ♥t♦♥s

s ♣ ♣r ♣r♦①♠ó♥ ln(1 + x) ≃ x

♦FN,inter(V, T )

V= −KBT

1

Vln

(

ZN (V, T )

V N

)

=

= −kBT1

Vln

[

1 +1

2ρ2ˆ ˆ

f12dr1dr2 +1

6ρ3ˆ ˆ ˆ

f12f13f23dr1dr2dr3 + ......

]

=

= −kBT1

V

[

1

2ρ2ˆ ˆ

f12dr1dr2 +1

6ρ3ˆ ˆ ˆ

f12f13f23dr1dr2dr3 + ......

]

.

st♦ út♠♦ ♦♥stt② ♥ srr♦♦ ♥ ♣♦t♥s ♥s srr♦♦ r

♥áss s ♦rrs♣♦♥♥ts ♦♥tr♦♥s ♥rí

♥ts srr s ♦♥tr♦♥s ♥rí r ♠♦t③ ♥trí♥s q s r♥ srr♦♦

♥tr ♦♥ró♥ ♠♦s ♥s ♦♥sr♦♥s rs♣t♦ ♣♦t♥ ♥tró♥ s ♣rtís

q ♦♥stt②♥ ♦ Pr ♦ q s s♠r♠♦s q

st ♣♦t♥ s ♣ s♦♠♣♦♥r ♦♠♦ s♠ ♥ ♣♦t♥ r♣s♦ Urep q tú st ♥ st♥

r0 ② ♦tr♦ trt♦ Uatt q tú ♣r st♥s ♠②♦rs r0 ❱ r

U =

Urep r < r0

Uatt r ≥ r0.

♦♥sr q sst♠ ♦♥st N srs rs r♦ r0 ♦

Urep ≡ ∞ r < r0.

♥♥♦ ♥ ♥t t♦♦ st♦ s ♣♥ r s ♦♥tr♦♥s r♥ts ór♥s

♦♥tró♥ ♦r♥ rát♦F

(2)N,inter(V, T )

V= −kBT

1

2Vρ2ˆ ˆ

f12dr1dr2 = −kBT1

2ρ24π

ˆ

∞

0

f12(r)r2dr =

−2πkBTρ2

[ˆ r0

0

(−1)r2dr +

ˆ

∞

0

[

exp

(

Uatt(r)

kBT

)

− 1

]

r2dr

]

⇒

F(2)N,inter(V, T )

V=

2π

3kBTr

30ρ

2 + 2πρ2ˆ

∞

r0

Uatt(r)r2dr.

♣r♠r tér♠♥♦ r ♥ út♠ ①♣rsó♥ st r♦♥♦ ♦♥ ♥rí r ♥ sst♠ srs

rs ♠♥trs q s♥♦ st ♦ ♥ sst♠ N ♣rtís ♦♥ ♥tr♦♥s t♣♦ trts

♦♥tró♥ ór♥s s♣r♦rs

♥ s ♦♥tr♦♥s ♠②♦r ♦r♥ ♥ ♥s s♦♦ s rt♥ ♣♦t♥ r♣s♦ srs rs sí ♣♦r

♠♣♦ trr ♦r♥ ♦rró♥ s sr ♦♠♦

F(3)N,inter(V, T )

V= −kBT

1

6Vρ3ˆ ˆ ˆ

f12f13f23dr1dr2dr3 = kBT5

16

(

2π

3r30

)2

ρ3.

♦♥

η =π

6r30ρ,

s ró♥ ♦♠♥ ♦♣♦ ♣♦r s srs rs ró♥ ♠♣qt♠♥t♦

♦ ♦rró♥ trr ♦r♥ s ♣ srr ♦♠♦

F(3)N,inter(V, T )

V= kBTρ4η.

♥á♦♠♥t s ♣♦s ①♣rsr s ♦rr♦♥s rt♦ ② q♥t♦ ♦r♥

F(4)N,inter(V, T )

V= kBTρ5η

2,

F(5)N,inter(V, T )

V= kBTρ

3673

600η3.

♥♦s st♦s tér♠♥♦s r♦♥ ♦s ♣♦r ② ♦♦r ❬❪

st ♦r♠ ①♣rsó♥ ♥rí r ♥trí♥s q srt ♦♠♦Fint(V, T )

V= kBTρ

[

ln(

Λ3ρ)

− 1]

+ kBTρ

[

4η + 5η2 +3673

600η3 + .....

]

+

+2πρ2ˆ

∞

r0

Uatt(r)r2dr.

s♠ tér♠♥♦s ♦rrs♣♦♥♥ts ♦♥tró♥ srs rs ♣r ♥ ♥ú♠r♦ ♥♥t♦ tér♠♥♦s

♠♥r ♣r♦①♠ ♣♦r r♥♥ ② tr♥ ❬❪ rst♦ s

4η + 5η2 +3673

600η3 + ..... ≃ 4η − 3η2

(1− η)2 = fHS [ρ]

tér♠♥♦ fHS [ρ] s ♥s ①s♦ ♥rí r ♣r ♥ s srs rís

♦ t♥♥♦ ♦♠♦ s sr tér♠♥♦ ♦rrs♣♦♥♥t s ♣♦♠♦s st♥r ♦s st♦♥s

s♦ ás♦

FN,int(V, T )

V= kBTρ

[

ln(

Λ3ρ)

− 1]

+ kBTρfHS [ρ] + 2πρ2ˆ

∞

r0

Uatt(r)r2dr.

s♦ á♥t♦

FN,int(V, T )

V= kBTρ

[

ln(

Λ3ρ)

− 1

Λ3ρg5/2(Λ

3ρ)

]

+ kBTρfHS [ρ] + 2πρ2ˆ

∞

r0

Uatt(r)r2dr.

♥♦♥s ♦rró♥

s ♥♦♥s ♦rró♥ ♥♦ ② ♦s r♣♦s s ♣♥ ♦t♥r ♣rtr ①s♦ ♥rí r Fexc

❬❪ ♥ s♦ ♥ q ♦ ♥trtú ♦♥ ♠♣♦s ①tr♥♦s st tér♠♥♦ ♦♥t♥ ♠ás ♥rí

♥tr♦♥s ♥tr♥s q ♦rrs♣♦♥ ♦s ♠♣♦s ①tr♥♦s sts ♥♦♥s s ♦t♥♥ ♥t♦♥s

♠♥t ró♥ ♥♦♥ ①s♦ ♥rí r s ② ❬❪

c(1)(r) = −δ(

1kBT Fexc[ρ]

)

δρ(r),

c(2)(r, r′) = −δ2(

1kBT Fexc[ρ]

)

δρ(r)δρ(r′).

♦s ♥♦ ♦♠♦é♥♦s ♦rís ♦s ② ♥♦ ♦s

s ♣r♠rs t♦rís srr♦s ♥ ♦♥t①t♦ ♥♦♥ ♥s r♦♥ s ♥♦♠♥s t♦rís

♣r♦①♠ó♥ ♦ ♥s

s s ♥ tr♠♦♥á♠ ♥ ♦ ♥♦r♠ ρ(r) = ρ0 ♣r

F [ρ] = Nf(ρ0),

s♥♦ f(ρ0) ♥rí r ♠♦t③ ♣♦r ♣rtí

st ♠♥r ♦r♠ó♥ ♦♥sst s♠♣♠♥t ♥ ♥ ♥r③ó♥ ♣r s♦ ♥♦

♥♦r♠ ♥♥♦ ♥ ♥t ①♣rsó♥ ♣r ♥ú♠r♦ ♣rtís ❬ ❪

FLDA =

ˆ

ρ(r)f(ρr)dr.

st ♣r♦①♠ó♥ s ♣ str s ♦♥sr♠♦s ♥ ♦ q ♣rs♥t r♦♥s s♣s ♠② ss

♥s ♥ ♠♥r s ♦♠♦ trtr ♦ ♥♦ ♦♠♦é♥♦ ♦♠♦ tr♦③♦s ♥♣♥♥ts ♦

♦♠♦é♥♦

③ ♥ sst♠s ás♦s s ♠② ♠t ♥♦ sí ♥ sst♠s á♥t♦s ♦♥ ♠♥♦r ♥ s

♦rr♦♥s ♦♥rt ♥ ♥ ♣r♦①♠ó♥ r③♦♥♠♥t ♥ ♥ ♠s st♦♥s

♦r♠ó♥ s ①t ♣r s ♦ s ú♥♦ sst♠ q s sr ♠♥t ♦♥

st ♣r♦①♠ó♥ Pr ♦rr♦♦rr st♦ st ♦♥ ♥tr♦r ♥ rst♦ s

c(2)(r) ≈ δ(r).

s r st ♣r♦①♠ó♥ ♥♦ t♥ ♥ ♥t s ♥tr♦♥s ♥ ♥ ♥t♦r♥♦ ♣♥t♦ s♣r♥♦ ♣♦r ♦ t♥t♦

♦s t♦s s ♦rr♦♥s

♠♦♦ s ♣ ♠♦rr ♦♥ ♥ srr♦♦ ♥ r♥ts ♥s ♦r♠s♠♦ ♥♦♠♥♦

sr♥♦ ♥ tér♠♥♦s ♥rí r ♣♦r ♥ ♦♠♥

f(ρ(r)) = ρ(r)ψ(ρ(r)),

② tr♥♥♦ st srr♦♦ ♥ s♥♦ tér♠♥♦ ①♣♥só♥ ❬❪ st ♥t♦♥s

FSGA[ρ] =

ˆ

dr[

f(ρ(r)) + f2(ρ(r))(∇ρ(r))2 + .........]

,

♦♥ f(ρ(r)) ♦rrs♣♦♥ ♥rí r sst♠ ♥♦r♠ ② tr♠♥♦ f2(ρ(r)) ② ♦s ss♦s tér♠♥♦s

♥ s♦ ♦♥t♥r ①♣♥só♥ t♥ ♥ ♥t s r♦♥s s♣s ♥s

Pr ♦t♥r ①♣rsó♥ f2(ρ(r)) t♥♠♦s q ♠♣♦♥r ♥ ♦♥ó♥ ♦♥ ♥str♦ ♥♦♥ ♦♠♦

♣♦r ♠♣♦ q r♣r♦③ rs♣st ♥ s♣r ♥♦ ♣rtr♠♦s sst♠ ♣r♦①♠ó♥

s á ♥s♦ ♥ st♦♥s ♦♥ ♥s rí rt♠♥t ♣r♦ ♦♥ strt rstró♥ q ♦

♠♦ ♣r♦rs ♦ r♦ ♥ st♥ ♠② r♥ ♥ ♦♠♣ró♥ r♥♦ ró♥ ♣♦t♥

♥tr♠♦r ♥ ♣rát st rstró♥ ♠t ♠♦ s♦ s ♥ s s♦ ♦♥ é①t♦ ♣r srr

s rtrísts ♥trs íq♦♣♦r r ♥ ♣♥t♦ rít♦ ♣r♦♠s ♦♥ ♥trss íq♦íq♦

♥ ♠③s ♣r♦♠s ♠♦♦ t

s t♦rís ♥♦ ♦s ♥♦♥ ♥s r♥ s ♦s t♥♥ ♥ ♥t s

♦rr♦♥s ♠♦rs ♦rt♦ r♥♦ ② ♥t♦♥s ♣r♦♥ ♥ rr♠♥t ♠♣rs♥ ♣r str sst♠s

♥♦♠♦é♥♦s ♦♥ rts r♦♥s ♥ ♣r ♥s st♦s ♣r♦♠s ♣r♥ ♣♦r ♠♣♦ ♥

st♦ s♦ró♥ ♦s s♠♣s s♦r sstrt♦s só♦s ♦s ♠ét♦♦s r♦s sr♥

♣r♦♣♠♥t ♥ó♠♥♦s ts ♦♠♦ tr♥s♦♥s s ♦♥♥só♥ ♣r ♥ó♠♥♦s ♠♦♦ t

♦s ♦r♠s♠♦s ♥♦ ♦s ♥ ♥t t♠é♥ s ♦s♦♥s q ♣r♥ ♥ ♣r ♥s ♥♦

♦ s s♦r s♦r ♥ ♣r rst♦ ♥♦ t♥t♦ ♥ s♠♦♥s ♦♥tr♦ ♦♠♦ ♥á♠

♦r q st s♥t ♥ ♦s á♦s ♦s ♥t ♠♥t ♦s ♦r♠s♠♦s ♦s

♥ s♦ s st♥ ♦s ♦r♠♦♥s ♥ s r③♦♥ ♥s ② ♥rs♦♥

q ♠♦s ♥♦♠♥♦ s♠♦♦t ♥st② ♣♣r♦①♠t♦♥ t③ ♥ ♠♣♦ ♥ss ♣r♦♠♦ ♦♥

rt♦s t♦rs ♣s♦ q ♦♥stt②♥ ♣rá♠tr♦s rs t♦rí

♦tr♦ ♦r♠s♠♦ ♥♦♠♥♦ ♦rí s s ♥♠♥ts srr♦♦ ♣♦r ♦s♥ r

② ♦s♥r ♥♦ ♦♥t♥ ♣rá♠tr♦s rs

trs ♦♥sr♦♥s rrs ♦s ♦r♠s♠♦s ② s♦♥ s s♥ts ❬❪

♥♦ s st♥ tr♥s♦♥s ♠♦♦ tt♥ s t♦rís ♦s r♥ s♦rst♠♥

á♥♦ ♦♥tt♦ ♥tr ♣r ② ♦ ♥ ♥ stó♥ ♠♦♦ ♣r

♥ ♦♥trst ♦♥ s ♣r♦♥s st ♦r♠s♠♦ sr ♠♥t ♦①st♥ íq♦s ♥

sst♠s ♦s ♠♥s♦♥s ♦♥♥♦s ♥ ♣♦r♦s ♠② str♦s t♠♣rtrs ♥r♦rs rít

P♦r ♦tr♦ ♦ ♥♦ s st tr♥só♥ ♣r ♦s ♦♥♥♦s ♥tr ♣rs tr♥só♥ ♦♥ s

♣s ♥ íq♦ ♥s♦ ♥ s ♦ ♣rs♦♥s ♠②♦rs stró♥ s í♥s ♦①st♥ ♣r

s ♠♥t ♥♦ r♥ ♠s♦ s ♦t♥s ♠♥t ♦ st♦ sr q s

♦rr♦♥s ♦rt♦ ♥ ♥♦ s♦♥ ♥ st s♦ ♠♣♦rt♥ts

Pr ♥ stó♥ ♠♦♦ ♣r ♣rsó♥ ♦♥♥só♥ ♦t♥ ♠♥t st ♥ ♥ r♦

♦♥ s ♣r♦♥s q ② ♣ ♦ ♥ ♣r s♦ró♥ ♥ ♣♦r♦s str♦s ♦♥ r♦s s♣r♦rs

♦s ♥♦ á♠tr♦s ♠♦rs ♦ ♣rs st♥s ♥ ♥♦s ♣♦♦s á♠tr♦s ♠♦rs

♥ st tr♦ t③♠♦s ♥♠♥t♠♥t ♠♦♦ ♥q ♣rs♥t♠♦s ♥ r sr♣ó♥ ♠♦s

♦r♠s♠♦s

♦r♠s♠♦ r③♦♥♥s♥rs♦♥

trt♠♥t♦ sst♠s ♥♦ ♦♠♦é♥♦s ♦♥ ♦♥ ♦s ♥tr♦rs ♠ét♦♦s ó♥ f(ρr) ❬ ❪

♥ st♦♥s ♦♥ ρ t♦♠ ♦rs ♠② t♦s ♥ ts r♥st♥s ♦♠♣♦rt♠♥t♦ ♥ sst♠ ♥♦r♠

srá ♣rs♠♠♥t ♠② r♥t q ♣rs♥tr sst♠ r ❯♥ ♠♣♦ st♦ út♠♦ ♦ ♦♥stt②

s♦ ♥ ♦ srs rs ♥ st s♦ ♥♦ ♣♦rí♠♦s trtr st♦♥s ♥ ♦♥ ♥s ♦ s♣r

í♠t ♠♣qt♠♥t♦ ♠á①♠♦

♦s ♥♦♥s ♣s trtr ♦s sst♠s s srr♦r♦♥ ♣r♥♣♠♥t ♥ é ♦s s

❬ ❪ t③r♦♥ ♣r ♦ té♥s ♦♠♦ srr♦♦ ①s♦ ♥rí r Fexc[ρ] r♦r

♥s ♦ ♥♦r♠ srr♦♦ ♣rtrt♦ r♦r ♥ ♦ rr♥ ♥ ♦♥♦♦ t

♥♦ s s ♦r♦♠ ❬❪ r③♦♥ ♣r♦♣s♦ ♥tr♦r ♦rr♦♥s ♥♦ ♦s ♥♦♥ ♠♥t

s♦ ♥ stró♥ ♥s ♣s ♦rs r♥ ρ(r) q s ♥ ♥ó♥ ♥♦ ♦ ρ(r) ♥ ♣♥t♦

r sst♠ ♦ ♣ ♠♥r ρ(r) ♦♠♦ s r ♥ s♣ ♥s ♣r♦♠ ♥ ♥ ♦♠♥

r♦♥♦ ♦♥ ♥ s ♥tr♦♥s ♦s ♣♦s ♥ ♥s ♦ s♦♥ s③♦s ♥ ρ(r) ② ♥♦ s ♦rr

rs♦ r ♥rí r ♥ ♣♥t♦s ♦♥ ♠♣qt♠♥t♦ ♥③ ♦rs r♥ts s♥t♦ ís♦

♥ st ♣r♦①♠ó♥ ♥rí r ①s♦ ♦♥ s srs rs s sr ♦♠♦

Fexc−HS [ρ] =

ˆ

drρ(r)fHS [ρ(r)],

♦♥ fHS s ♥s ①s♦ ♥rí r ♣♦r ♣rtí q ♦rrs♣♦♥ sst♠ ♥♦r♠ Pr

sst♠ srs rs á♠tr♦ σ ♣♦♠♦s sr ó♥ st♦ r♥♠tr♥ ❬❪

Pr ♣♦r ♦♣rr ♦♥ s ♥sr ♥ sr♣ó♥ ρ(r) ❯♥ ♣♦s s s ❬ ❪

ρ(r) =

ˆ

d3r′ρ(r′)w (|r− r′|) ,

♦♥ w(r) s ♥ ♥ó♥ ♣s♦

Pr tr♠♥r s ♥♦♥s ♣s♦ s ♠♣♦♥♥ ♦s ♦♥♦♥s

♥ó♥ ♣s♦ str ♥♦r♠③

♥♦♥ r♣rr ♦r♠ r③♦♥ í♠t ♦rrs♣♦♥♥t ♥ ♦ ♥♦r♠

s r ♥ ♠♥r ♦ ♣♦s ♥ó♥ ♦rró♥ rt q rst r♥ó♥ ♥♦♥

❬ ❪ str ♥ ♥ r♦ ♦♥ ♣r♦①♠ó♥ Prs❨ ❬❪ q ♦♥stt② ♥ ♣r♦①♠ó♥

rr♥t ♥ t♦rí íq♦s ♦♠♦é♥♦s

ó♥ ♠ás s♥ ♣♦s ♣r w(r) s ♥ó♥ só♥

w(r) =

34πσ

−3 r < σ

0 r > σ.

st ó♥ r♣r♦ ♥ rtríst ♠♣♦rt♥t ♥ó♥ ♦rró♥ rt ♦♠♦ s♦♥t♥

♥ r = σ ♣r♦ s♦rst♠ ♥ ♠s♠ ♥ sí s ♦t♥ ♥ ♥♦♥ q ♣r♦♣♦r♦♥ rst♦s

tt♠♥t ♥♦s ♥ ♣r♦♠s ♦♥ s st♥ ♥trss ♥tr ♦s srs rs ② ♥ ♣r r

♦ tr♥s♦♥s ♦só♦ ❬ ❪

r③♦♥ ❬❪ ♣r♦♣s♦ ♥ ♥ó♥ ♣s♦ ♦♥ ♥ ♣♥♥ ♥ ♥s ♣r ♠♦rr sr♣ó♥

♥ó♥ ♦rró♥ rt c(r) q s s♣r♥ ♥♦♥ ♦♥ st ♥ ♥ó♥ ♥s ♣s

rst

ρ(r) =

ˆ

d3r′ρ(r′)w (|r− r′| ; ρ(r)) .

st ♥ ♥ó♥ ♥tr♦ r r ♥♦♥ ♥♦s tér♠♥♦s ♥ c(r) ② ♣♦r t♥t♦ s ♣ ♦t♥r

♥ ♣s♦ q r♣r♦③ ♠♦r s ♦rr♦♥s ♦♠♥

s♠ q w(r, ρ) s ♣ srr♦r ♥ ♥ sr ♣♦t♥s ♥s

w(r; ρ) = w0(r) + w1(r)ρ+ w2(r)ρ2 + ....... .

♦♥ó♥ ♥♦r♠③ó♥ s♦r w(r, ρ) ♠♣ ♥sr♠♥t

ˆ

drwi(r) =

1 i = 0

0 i = 1, 2, ....

❯s♥♦ srr♦♦ ♥ ♣♦t♥s ♥s w(r) ♦ ♣♦r ② r♥♥♦ ♥♦♥ sú♥

s ♦t♥ ♥ srr♦♦ ♥ ♣♦t♥s ♥s ♥ó♥ ♦rró♥ rt P♦♠♦s

♦♠♣rr ♦ srr♦♦ ♦♥ q s ♦t♥ ♥ ①♣♥só♥ r c(r) ② s ♦r♠ rr

♥♦r♠ó♥ ♥sr ♣r r s ♥♦♥s ♣s♦ wi

st ♠♥r w0(r) s ♥♦ ♥ó♥ só♥ ♣♦r ② ♣r w1(r) s ♦t♥ ♥ ó♥

♥tr q rs♦rs ♥♠ér♠♥t

Pr rst♦ ♦♥ts wi, i ≥ 2 s ♥st♥ t♠é♥ ♦s tér♠♥♦s srr♦♦ fHS(ρ)

óts q ♥ st ♣♥t♦ ② ♥ ♣qñ ♥♦♥sst♥ ♣♦r ♥ ♦ s t③ ó♥ st♦ r♥♠

tr♥ ② ♣♦r ♦tr♦ s ♣rt♥ r♣rr s ♦rr♦♥s ♣r♦①♠ó♥ Prs❨ ♥ ♠r♦ s

♦rr♦♥s ② ♥rí ①s♦ ♣♦r ♣rtí stá♥ r♦♥s ♣♦r ♠♦ ♥tr ♦♠♥ c(r)

♦ st ♥♦♥sst♥ ♥♦ s ♣♦s ♥tó♥ tér♠♥♦ tér♠♥♦ wi ♣r i ≥ 2 ❯♥ tr♥t

♣r ♦rr ♦♥sst♥ s tr♥r srr♦♦ ♥ i = 2 r③♥♦ ♥ st ♥ó♥

♣s♦ w2(r) ♣r q s ♦t♥ ♥ ♥ sr♣ó♥ c(r) ♥ t♦♦ r♥♦ ♥ss s s tr♥

srr♦♦ ♥ i = 1 ú♥♠♥t s r♣r♦ ♦♠♣♦rt♠♥t♦ c(r) ♥ ♥ r♥♦ ♥ss ♣qñ♦ ♥

sr ♥rs♦♥♥♦ ♥ srr♦♦ ♠ét♦♦ tr♥sr♠♦s ♦♥t♥ó♥ ♦s rst♦s q s ♦t♥♥ ♣r

s ♥♦♥s ♣s♦

w0(r) =

34πσ3 r < σ

0 r > σ,

w1(r) =

0,475− 0,648(

rσ

)

+ 0,113(

rσ

)2r < σ

0,288(

σr

)

− 0,924 + 0,764(

rσ

)

− 0,187(

rσ

)2σ < r < 2σ

0 r > 2σ

,

w2(r) =

5πσ3

144 (6− 12)(

rσ

)

+ 0,113(

rσ

)2r < σ

0,288(

σr

)

− 0,924 + 0,764(

rσ

)

− 0,187(

rσ

)2σ < r < 2σ

0 r > 2σ

.

♦ ♦♥ ρ(r) sí ♦♥str s r♣r c(r) Prs❨ ♥ ♥ r♥♦ ♥ss ♠② ♠♣♦

♥♦♥ q s ♦t♥ ♥ ♣r♦①♠ó♥ ♦♥str♦ st ♠♥r s ♣③ srr stst♦r♠♥t

st♦♥s s♣♠♥t ♠② ♥♦♠♦é♥s ♦♠♦ s q ♣r♥ ♥ ♣r♦♠s só♦s srs rs ♦

♥trss ♥tr srs rs ② ♣rs rs ❬❪

①st♥ ♦trs t♦rís ♦♥ s tr ♦♥ ♥ss ♣r♦♠s ♣♦♠♦s tr rs♣t♦ t♦rí rt♥ ②

sr♦t ❬❪ ♥ st ♦r♠ó♥ ♥ r r ♥ ①♣♥só♥ ♥ ♥s ♣r r s ♥♦♥s ♣s♦

sts s ♥ ♥♠ér♠♥t ♣r q ♥♦♥ r♣r ♦rró♥ Prs❨ ♥ t♦♦ r♥♦

♥ss ♥ st út♠♦ s♦ ♦st♦ ♦♠♣t♦♥ s ♠♦ ♠②♦r

♠é♥ ♠r♥ ♥ ♠♥ó♥ s t♦rís ss ♥ ♥ ♣s♦ q s ♥♣♥♥t ♣♦só♥ ♣♦r ♠♣♦

♦ ❲ ♥t♦♥ ② sr♦t ❬❪ ♦ ts♦ ② s ❬❪

♠♣♠♥tó♥ sts ♣r♦①♠♦♥s s ♠ás s♥ ♦st ♥ sr♣ó♥ ♠ás r♠♥tr sst♠

♦rí s s ♥♠♥ts

♥ út♠ é s♦ ❳❳ ♦s♥ ♣r♦♣s♦ ♦ q ♥♦♠♥♦ ♦rí s s ♥♠♥ts

② ♥ s ♦t♦ ♥ ♥♦ t♣♦ ♥♦♥s ❬ ❪ st♦s ♥♦♥s s♦♥ s♣í♦s ♣r

r♣♦s r♦s s♦s srs ♥r♦s t ② t♥♥ ♥ strtr ♦♠♣t♠♥t r♥t ♦s r♦s

♣r♦①♠ó♥ s ♥ ♦♠trí ♠♦r ♥ r ♦♠♥ ①♦ ♥tr ♦s ♣rtís st

t♦rí srr♦ ♥ ♣r♥♣♦ ♣r trtr ♣r♦♠s ♠③s ♦s

♥ s ♦r♠ó♥ ♦r♥ ♥rí ①s♦ ♣r ♥ ♠③ srs rs s ♣ srr ♦♠♦

Fexc−HS [ρi(r)]

kBT=

ˆ

dr′fHS [nα(r′)] ,

♦♥ ρi s ♥s s♣ i ② fHS s sr ♦♠♦ ♥ó♥ s nα q ♦♥stt②♥ ♣r♦♠♦s s

♠s ♥♠♥ts s ♣rtís r♦ só♥ ♦♠♥ t

nα(r′) =

∑

i

ˆ

dr′ρi(r′)ω

(α)i r− r

′

s ♥♦♥s ♣s♦ ω(α)i stá♥ r♦♥s ♦♥ ♦♠trí s ♣rtís q ♦♥stt②♥ ♦ át♦♠♦s ♦

♠♦és ② s♦♥ ♣♥t♦ ♥ t♦rí ♦r α st ♦ ♦♥ ♠♥s♦♥ s♣♦ q

♥♦♠♥r♠♦s D 0 < α < D st♦s ♣s♦s s ♦t♥♥ ♥♦ ♦ ♥ s♦♠♣♦só♥ ♥ó♥

②r ❬❪ ♥ tér♠♥♦s s♠s ♦♥♦♦♥s ♥♦♥s ♣s♦ ♥ s♦ srs rs ①st ♥ ú♥

s♦♠♣♦só♥ st t♣♦ q ♥♦r ♥♦♥s ω(α)i ♥ s♦ ♣rtr ♥ q ♠♥só♥ s♣♦

s ♦s ♥♦♥s ♣s♦ s♦♥ srs ② r♣rs♥t♥ ♦♠♥ ② s♣r sr ② ♥ trr s ♥

t♦r s♣r s rst♥ts s♦♥ ♣r♦♣♦r♦♥s sts trs ♥♦ ♥ ♥áss ♠♥s♦♥ ② t♥♥♦

♥ ♥t s r♦♥s q s ♦t♥♥ t♦rí ♣rtí s ❬ ❪ ♥ ♣rtr q rr ♥ ♥

♦ ♥♦r♠ ♥ sér r♦ R → ∞ s♣♦♥ ♥ tr♦ q PV ♦♥ P s ♣rsó♥

② V ♦♠♥ s ♦t♥ á♠♥t ♥ ①♣rsó♥ ♣r ♥rí ①s♦ ♥♦♥ sí

♦♥str♦ ♣r ♠♥t ♦s rst♦s q ♣r♥ ♥ ♣r♦♠s ♦♥ ♠♣♦s ♥s t♠♥t

♥♦ ♥♦r♠s ♣r♦♠s ♦♥ s st s♦ró♥ srs rs s♦r ♥ ♣r r

trs ♥ts ♠♣♦rt♥ts stá♥ r♦♥s ♦♥ ♦ q s ♥♦♥s ♦rró♥ s ♦t♥♥

rá♣♠♥t ♣rtr ♦s ♣s♦s ♣r ♥ ♦ ♥♦r♠ ♠♦♥♦♦♠♣♦♥♥t ♥♦♥ ♦s♥

r♣r ó♥ st♦ ② ♦rró♥ Prs❨ ♥óts r♥ ♦♥ s t♦rís ♦♥strs

①♣ít♠♥t ♣r q s ♦t♥ st ♦♠♣♦rt♠♥t♦

♣sr sts ♥ts ♥♦♥ ♣rs♥t rt♦s ♣r♦♠s ♥ st♦ só♦ srs rs ♣s

♣r q s s♠♣r ♥st r♥t íq♦ ♥ ♥ s só ♣rtí stá ♦♥♥ ♥ ♥

s ♦♥ ♥ ♣r♦♠♦ ♦♣ó♥ rí ♥tr ② ♥♦ s st í♠t ♦♥ ♥♦♥

♦s♥ s ♦t♥ ♥ r♥ ♥ ♥rí r q ♥ s♥ s ♦r♥ ♦ t♦rí st

♥tr♣rtó♥ ♣r♦♣ó srr♦♦ ♥♦♥s s♦s ♥ s ♦tr♦ ♣♥t♦ st ♦♥♦♦ ♦♠♦

ró♥ ♠♥s♦♥ ♠♥s♦♥ r♦ss♦r ♥ st s♦ s ♠♣♦♥ ♥♦♥ q r♣r í♠t ①t♦

♥ s s ❬ ❪ rst♦ s♦♥ ♥♦♥s q ♣r♦s ♥ ♦trs ♠♥s♦♥s r♣r♥

s♦ó♥ ①t Prs❨ ♥ ❬❪ ② ♥ s♦ó♥ q rr♦ t♦rí ♣rtí s

♥ trs ♠♥s♦♥s s♥ ♠r♦ s ♦t♥ ♥ ó♥ st♦ q ♥♦ ♣r♦ rst♦s ♣ts Pr

rs♦r st ♥♦♥♥♥t r③♦♥ ❬❪ ♣r♦♣s♦ ♥ ♠♦ó♥ ♥ ♥♦ ♦s tér♠♥♦s ♥♦♥ ♦

s♥ ♥♦r♣♦r♥♦ ♥ ♣s♦ t♥s♦r ♦ ♥♦♥ ♥♦ t♥ r♥s ♥ í♠t r♣r rst♦

①t♦ ♥ ② r♣r♦ ♥ó♥ ♦rró♥ ② ó♥ st♦ Prs❨ ♣r ♦ ♥♦r

♠ s♣t♦ s só sr ♠♥t strtr ♥q ♥ ♣ró♥ ♥ss

♦①st♥ ♦ q ♦ s sr ♦♥ ó♥ st♦ Prs❨ ♣♥ ♠♦rr

s ♣r♦♥s ♣r ♦①st♥ s s ♠♦ ♥♦♥ ♦r♠ q s ♦t♥ ó♥ st♦

r♥♥tr♥ ❬❪ ♣r♦ ♥t♦♥s ♠♣♦r sr♣ó♥ só♦

s♣t♦s ♦r♠s t♦rí

♦♠♦ s ①♣rs♦ ♥ts ♦r♠ó♥ stá ♥s♣r ①st♥ ♥♦ís ♥tr s t♦rís ♣rtí

s ② Prs❨ ♣r srs rís t♦rí ♣rtí s ♥ st ♦♥t①t♦ s♦♦ ♣♦rt

tr♠♦♥á♠ ♣r♦♠

ró♥ t♦rí ❬ ❪ ♣rt ♥ ♦sró♥ ♥r ①♣rsó♥ Prs❨

r♦♥ ♦♥ ♥ó♥ ♦rró♥ rt ♦s ♣rtís ♦rrs♣♦♥♥t ♥ ♦ srs rs

P♦♠♦s ①♣rsr ♥ó♥ ♦rró♥ ♦♠♦

c(x) =

−λ1 − 6ηλ2x− 12ηλ1x

3 x < 1

0 x > 1,

r ♦♠trí ♦s srs rís q s ♥trs♥ t♦rí s s ♥ ♦♥t♥♦ ♦♠étr♦ ♥ó♥ ♦rró♥ ♦rrs♣♦♥♥t ♥ st s♦ ♦s srs rís q s ♥trs♥ ♥ r R ♦rrs♣♦♥ r♦ s srs ② r s st♥ ♥tr ss ♥tr♦s st♦r ∆S ♠r ár tr♥srs ♥trsó♥

♦♥ x = r/d, η s ró♥ ♠♣qt♠♥t♦ ②

λ1 =(1 + 2η)

2

(1− η)4 ,

λ2 =− (2 + η)

2

4 (1− η)4 .

♣ sr rsrt ♥ ♥ó♥ rts ♥ts q rtr③♥ ♦♠trí ♦s srs q

s ♥trs♥ sts srs t♥♥ r♦ R ② stá♥ s♣rs ♥ st♥ r ♦♥

r < 2R

♥ s ♣ ♦srr ♥ sq♠ stó♥

s ♥ts ♠♥♦♥s s♦♥ ♦♠♥ ♥trsó♥ V só♥ tr♥srs ♥trsó♥∆S(r)

② r♦ ♠s♠ ∆R(r) = 2R−R ♦♥ R = R+ r/4

①♣rs ♥ ♥ó♥ sts ♥ts ♣r♠♥t ♦♠étrs ♥ó♥ ♦rró♥ s sr ♦♠♦

−c(r) = χ(3)∆V (r) + χ(2)∆S(r) + χ(1)∆R(r) + χ(0)Θ(|r| − 2R),

♦♥

χ(0) =1

1− ξ3,

χ(1) =ξ2

(1− ξ3)2 ,

χ(2) =ξ1

(1− ξ3)2 +

ξ22

4π (1− ξ3)3 ,

χ(3) =ξ0

(1− ξ3)2 +

2ξ1ξ2

(1− ξ3)3 +

ξ32

4π (1− ξ3)4 ,

♦♥

ξα = ρR(α).

qí s R(α) ♦♥stt②♥ s ♠♥ts ♦♠étrs ♥♠♥ts ♣r♦♠ ♥trsó♥

R(0) = 1,

R(1) = R,

R(2) = 4πR2,

R(3) =4

3πR3,

♥ s♦ ♥ ♦ ♦♠♦é♥♦ sts ♥ts ♦♥♥ ♦♥ ♦s t♦rs ♣s♦ q ♣r♥ ♥ t♦rí

♣rtí s ❬❪

s ♥ts r♦♥s ♦r♣ ♦♠♥ ár ② r♦ ♣♥ sr ①♣rss ♥ tér♠♥♦s ♦♥♦♦♥s

♥♦♥s ♦♥ s q s♣és s ♥ ♣sr s ♥ss s♦s ♦♥ ♦♠trí s srs ♥♠♦s

♥t♦♥s s s♥ts ♥♦♥s ♣s♦ srs q s♦♦ ♥♦r♥ ♥ts ♦♠étrs ♥♠♥ts

ω(3)(r) = Θ(|r| −R),

ω(2)(r) = δ(|r| −R),

ω(1)(r) =ω(2)(r)

4πR,

ω(0)(r) =ω(2)(r)

4πR2,

Pr ♦tr♦ ♦ t♥♠♦s s s♥ts ♥♦♥s ♣s♦ t♦rs

ω(V 2)(r) = −∇ω(3)(r) =r

Rδ(|r| −R),

ω(V 1)(r) =ω(V 2)(r)

4πR,

♥t♠♥t r♦♥ ♣r♦♣sts ♦trs ♥ts t♥s♦rs ♣r ♦♥sr ♣r♦①♠♦♥s ♠ás ♣rss sts

♥ts ♥♦ srá♥ t♥s ♥ ♥t ♥ ♣rs♥t tr♦

♥t ♠♣♦ sts ♥♦♥s ♣s♦ ♣♦♠♦s ①♣rsr ♦♠♥ ② ár ♥trsó♥ ♦♠♦

∆V (r) = ω(3) ⊗ ω(3) =

ˆ

Θ(|r′| −R)Θ(|r− r′| −R)dr′ =

=2

3π(

2R3 − 3R2r + r3)

Θ(|r′| − 2R),

∆S(r) = 2ω(3) ⊗ ω(2) = 2

ˆ

Θ(|r′| −R)δ(|r− r′| −R)dr′ =

= 4πR2Θ(|r′| − 2R)(

1− r

2R

)

,

∆R(r) =∆S(r)

8πR+

1

2RΘ(|r| − 2R) =

(

R− r

4

)

Θ(|r| − 2R),

P♦r ♦tr♦ ♦ ①s♦ ♥rí r ♦ srs rís s ♣ ①♣rsr ♦♠♦ s ♥ ♥

♦♥ nα(r) ♦rrs♣♦♥í ♥s ♣r♦♠ q ♥ s♦ ♥ ♦ ♦♥ ♥s ♥♦r♠

♦♥ ♦♥ s rs ♥ t♦rí ♣rtí s

n0 = ρ,

n1 = Rρ,

n2 = 4πR2ρ,

n3 =4

3πR3ρ,

♥s ♣r♦♠ ♣r ♥ ú♥ s♣ ♣rtís s ①♣rs ♦♠♦

nα(r) =

ˆ

dr′ρr′ω(α)r− r′.

♥ó♥ ♦rró♥ rt ♥ ♣rtí s ♣ r trés ró♥ ♥♦♥ ♥rí

r

c(1)(r) = − 1

kBT

δFHS [ρ]

δρ(r)=

ˆ

dr′∑

α

∂fHS

∂nαω(α)r− r

′,

② ♥s ♥rí r s ♣ r♦♥r ♦♥ s ♥ss ♣r♦♠s ♠♥t ①♣rsó♥

fHS = n0ln(1− n3) +n1n2 − nV1 nV2

1− n3+

1/3n32 − n2(nV2 nV2)

8π(1− n3)2.

qí nV1 ② nV2 s♦♥ t♦rs

st ♠♥r ♥ í♠t ♦s ♦♥ ♥s ♥♦r♠ s r♥♥tr ①♣rsó♥ Prs❨ ♣r

♥rí r ♦rrs♣♦♥♥t ♦ srs rs

fHS [ρ] = −ln(1− η) +1

2

6η − 3η2

(1− η)2.

♣ít♦

♣t♠③ó♥ ♥♦♥ ♣r♦♣st♦ ♥

st ss

♥ st ♣ít♦ ♣rs♥t♠♦s ♥ ♥ ♣r♦♣st ♣r ♦♣t♠③r ♥♦♥ ♥s ♥ ♦ s♦r♦

♥♦♥ sí ♦♣t♠③♦ ♦ trs q ♣r♠tó r♣r♦r ♦♥ ♣rsó♥ t♥só♥ s♣r

♣r♦♣s r♦♥s ♦♥ ♠♦♦ s♣rs t♠♣rtrs ♠♦♦ ② ♣r♠♦♦ ② ♦trs ♣r♦♣s

s ♦♥ s♦ró♥ ♥ ♣rs ② r♥rs strs ts ♦♠♦ ♥ó♠♥♦s r♣tr s♠trí ♥ ♦s ♣rs

♥s q ♣♥ ♣rr ♥♦ ♦ s s♦r ♥ ♥ r♥r ♣♥ ♣rs é♥ts

♥tr♦ó♥

❯♥ t♦rí ♠♦r ♦s sr ♣③ ♣rr ♠♥t s ♣r♦♣s qr♦ t ♦♠♦

♦①st♥ s ss íq♦♣♦r ♥♦ ♦ ♥♦ s ♥♥tr s♦♠t♦ ♥♥ú♥ ♣♦t♥ ①tr♥♦

♥ ♦s á♦s t♦s ♣r ♦s ♦♥♦r♠♦s ♣♦r ss ♥♦s ♥tr♦ sq♠ t♥t♦ ♥á♠

♦r ♦♠♦ ♦s ♠ét♦♦s s♠ó♥ ♦♥t r♦ ♦ r♥ ♥ó♥♦ ♦♥t r♦ ②

s s♠ q ♥tró♥ ♥tr ♠♦és ♣♦rs ♦♠♦ s q ♦♥♦r♠♥ st s sst♥s

s ♣ r♣rs♥tr ♠♥t ♥ ♣♦t♥ Φ t♣♦ ♥♥r♦♥s ♦ ú♥ ♦tr♦ ♣♦t♥ q

r ♥s ss rtrísts s♥s ❬❪ ①♣rsó♥ ♣r st ♣♦t♥ s

Φ(r) = ΦLJ(r) = 4εff

[

(σffr

)12

−(σffr

)6]

,

♦♥ εff s ♥t♥s ♣♦t♥ ♦♥ q ♥trtú♥ s ♣rtís át♦♠♦s ♦ ♠♦és σff s s

á♠tr♦ ② r s st♥ ♥tr s

♦s ♣rá♠tr♦s εff ② σff ♣♥♥ t♣♦ sst♥ r r♦ ♥ st st♦ ♥♦s♦tr♦s tr♠♦s

♦♥ ♦s q s ♥♥tr♥ ♣rs♦♥s q ♥♦ s♦♥ ①tr♠s ♥ st s♦ ♠②♦r ♦♥tró♥ ♥tró♥

♥tr♠♦r ♦ ♥trtó♠ ♣r♦♥ ♠♦és ♦ át♦♠♦s q ♥trtú♥ ♣rs ♣♦r ♦ ♦s

r s ♥tr♦♥s ♦s r♣♦s s♦♥ s ♦♠♥♥ts ❬❪

♥t s♠♦♥s ♥♠érs ♣rtr ♦ s ♦r r♣r♦r ♦♥ ♥ ♣rsó♥ r

♦①st♥ ①♣r♠♥t ♥♦ s ♠♣ ♥ st ♣♦t♥ ♦rs t♣♦s ♣r εff ② σff

0 10 20 30 4025

30

35

40

45

50

ρ [nm−3]

T [K

]

Tc

Ne

r rs ♦①st♥ ss íq♦♣♦r ♣r ♦s ír♦s ♦rrs♣♦♥♥ ♦rs ①♣r♠♥ts ② r ♥r♦r ♦s ♦rs ♦s ♥♦ s t③ ♦♠♦ ♣♦t♥ ♣rs ♦r Tc rr♥ t♠♣rtr rít

♥ ♦r♠s♠♦ ♥r♠♥t ♣r ♥ó♥ stró♥ ♦s r♣♦s ρ(2) s t♦♠ ♥ ♥t

s♥t ♣r♦①♠ó♥

ρ(2)(r, r′) = g(r, r′)ρ(r)ρ(r′) ∼ ρ(r)ρ(r′) ,

r q s s♣r strtr ♦♥t♥ ♥ g ♥ó♥ ♦rró♥ ♣rs ♥ st ♣r♦①♠ó♥

t③♥♦ ♦s ♦rs εff ② σff ♥ ♣♦t♥ q ♣r♠t♥ r♣r♦r ♠♥t r♠

s ♥ s s♠♦♥s ♥♠érs ♥♦ s ♦r r♣r♦r ♠♥r ó♣t♠ r ♦①st♥ íq♦♣♦r

♥ s ♣♥ ♦srr ♦s rst♦s q s ♦t♥♥ ♠♥t ♥♦ s t③ ♦♠♦ ♣♦t♥

♣rs ♣r ♥ ♦ ♦♠♦ Pr rst♦ ♦s ♦s q qí st♠♦s ♥♦ s ♦r♥ ♠♦rs

sts r s s ②

♦rs rst♦s s ♣♥ ♦♥sr s s tr ♦♥ ♣♦t♥ srr♦♦ ♣♦r ❲s ♥r ② ♥rs♥

♦ ❲ r ❬❪ ② ♣é♥ ② ①♣rsó♥ s

ΦWCA =

4εff

[

(σff

r

)12 −(σff

r

)6]

si r ≥ 21/6σff

−εff si r ≤ 21/6σff.

rá♦ ♣ ♦srrs ♥ ♦♥ s ♦ ♦♠♣r ♦♥ ♥ ♥ st s♦ s ♦♥sr♦ q

á♠tr♦ sr rí s ♣rá♠tr♦ σff ❲ ♣rsr ♦♠♣♦rt♠♥t♦ ♣r

r♦ r♥♦ ♣♦t♥ ♦ ♦♥ tér♠♥♦ trt♦ ♠s♠♦ s r ♦♥ tér♠♥♦ ♦rrs♣♦♥♥t

r−6 ♦ q s r♥t ♣r st♦ ♥ó♠♥♦s s♦ró♥

♥ q ♦rrs♣♦♥ ❬❪ s ♣ ♦srr r ♦①st♥ íq♦

♣♦r ♣r ♥ ♦ ♥♥r♦♥s ♠♥t st t♣♦ ♣♦t♥s t♥♥♦ ♥ ♥t tr♠♥s

♣r♦①♠♦♥s ♥ st rá♦ ♦s trá♥♦s s r♥ ó♥ st♦ ♣r♦♣st ♣♦r ♦♥s♦♥ ② ♦♦r

L−J

WCA

dHS

r P♦t♥ ♣rs ♥ ♥ó♥ st♥ ♥tr♠♦r ♥ ♥s rtrrs r ♣♥t♦s ② r②s♦rrs♣♦♥ ♣♦t♥ ❲ r tr③♦ só♦ r♣rs♥t ♣♦t♥ ♥ st s♦ s ♦♥sr♦ q dHS = σff

♦rs ❬❪ q st ♣rt♠♥t ♦s ♦rs ①♣r♠♥ts ♦s r♦s ♦rrs♣♦♥♥ á♦s r③♦s

♠♥t s♥♦ ♦♠♦ ♣♦t♥ ♣rs ♦rrs♣♦♥♥t srs rs ♦♥ ♥tr♦♥s trts

r♣rs♥ts ♣♦r ♣♦t♥ ❲ ♦♥ á♠tr♦ ♠♦r dHS s srs s t♦♠♦ ①♣rsó♥

rr♥rs♦♥ ❬❪

í♥ ♣♥t♦s ♦rrs♣♦♥ á♦s r③♦s t♠é♥ ♠♥t ② ♦♥ ♣♦t♥ ♣rs ❲ ♣r♦

♦♥ s t③ó ♠♦♦ ② ♦♦r♦rs ❬❪ ♥ á♠tr♦ ♠♦r ♣♥

t♠♣rtr ♥ s♦ í♥ só s ♦♥sr♦ á♠tr♦ ♠♦r ♦♠♦ ♥ ♦♥st♥t

σff

Pr ♠♦rr s ♣r♦♥s r♦♥s ♦①st♥ ss r♥♦ ♠♦ ② r③♦♥ ♥

st♦s s♦r s♦ró♥ r ♥ O2 ♥ ♥s②♦ ♦tr♦s ♣r♦♠♥t♦s ♥②♥♦ ♥ ss á♦s ①♣rs♦♥s

rs ♦r♠♦♥s ♥♦ ♦s ♦♠♦ ♣♦r ♠♣♦ r ♣ít♦ ♥♦ ♦rrt♦

rr♥ ❬❪ ♦①st♥ ss ♣ sr tr♠♥ s♥♦ ♣r ♣♦t♥ qí♠♦ ② ♣rsó♥

①♣rs♦♥s q r♥ ♥rí r ♠♦t③

F [ρ(r)] = FHS [ρ(r)] +1

2

ˆ

dr

ˆ

dr′ρ(r)ρ(r′)Φatt(|r− r′|) ,

♦♥ FHS ♦rrs♣♦♥ ♥rí r ♠♦t③ ♥ s srs rs ② Φatt s ♣rt trt

♣♦t♥ ♥tr♠♦r

Pr s♦ ♥ ♦ ♦♠♦é♥♦ ♥s ρ s s♣r♥♥ s ♦♥s st♦ ♥r③s

♥ r ❲s

P = PCSHS +

1

2bρ2,

µ = µCSHS + bρ,

0 0.3 0.6 0.90.7

0.8

0.9

1

1.1

1.2

1.3

1.4

ρσ3

kT/ε

r r♠ ♦①st♥ ss íq♦♣♦r ♦rrs♣♦♥♥t ♥ ♦ st rá♦ ♦rrs♣♦♥ ❬❪ ♦s trá♥♦s s ♦t♥♥ ó♥ st♦ ♦♥s♦♥ ❬❪ ♦s r♦s í♥ ♣♥t♦s ② í♥ ♥ ♦rrs♣♦♥♥ á♦s r③♦s ♠♥t s♥♦ ♦♠♦ ♣♦t♥ ♣rs ❲♦♥ s ♥ t③♦ trs ♣r♦①♠♦♥s st♥ts ♣r á♠tr♦ q s s♥ s ♠♦és r t①t♦

♦♥

b =

ˆ

drΦattr,

s♥♦ PCSHS ② µCS

HS ♣rsó♥ ② ♣♦t♥ qí♠♦ rs♣t♠♥t ♦rrs♣♦♥♥ts ó♥ st♦

r♥♥tr♥ ♣r ♥ s srs rs ❬❪

sts ♦♥s s r t♠♣rtr rít Tc Pr s♦ r

kBTc =−0,09010

d3HS

b.

♥ tr♦s ♣r♦s ♥s ② r③♦♥ ❬ ❪ ♦♥ s ♥③ r s ♦♥sr ♦♠♦ ♣♦t♥ trt♦

Φatt t♦ ♣rt ♥t ♣♦t♥ t♠♣rtr rít q s r st♦s ♠♦♦s st s♥

t♠♥t ♣♦r ♦ t♠♣rtr rít ♠ ② trés ♦♥s ♥trs ♦ s

s♠♦♥s ♦♥t r♦

r③ó♥ st♦ s q ♣♦t♥ q ♣rt♣ ♥ ♥♦♥ ♥rí r ♠♦t③ r♣rs♥t

♣♦t♥ ♥tró♥ r ♣s♦ ♦♥ ♥ó♥ ♦rró♥ ♣rs ♣♦s ♥ ♣r♠r ♣♦ ♠②

♣r♦♥♥♦ r♦r ♠í♥♠♦ ♣♦t♥ ❱ r st ♠♥r s ♥r♠♥t r♦ s ♥tr♦♥s

trts ♣r♦ ♥ st ♠♦♦ ♠♣♦ ♠♦ s st ♦♥sr♥♦ q g = 1 ♥ t♦♦ ♣♥t♦ ♦ ♦ ♣r

♥tr♦r ♥ ♠♥r ♦s t♦s g s♥ ♣rr st q ♠♣rs

Φatt(r) = ΦL−J(r → ∞) = −4ε(σffr

)6

,

♣r ♣rr ♦rrt♠♥t s ♣r♦♣s ♠♦♦ ♦s t♦s t♦rs r♥ ♥ ♣rá♠tr♦ λ ♥ tér♠♥♦

r♣s♦ ② sí ♦♣t♥ s♥t ♣♦t♥ ♥tró♥ ♣rs

0 0.3 0.61.1

1.2

1.3

ρ*

T*

Ar

r r♠ ♦①st♥ ss ♦rrs♣♦♥♥t r st r ♦rrs♣♦♥ rr♥ ❬❪♥ rá♦ s ♦♠♣r♥ s rs ♦t♥s ♠♥t t③♥♦ ♣♥t♦s í♥ só r♥t q♦t♥♥ ♦s t♦rs í♥ r② ♠♥t ♦r♠s♠♦

ΦBCT =

4εff

[

λ(σff

r

)12 −(σff

r

)6]

si r > λ1/6σff

0 si r ≤ λ1/6σff.

st♥♦ ♦r ♣rá♠tr♦ λ ♠♥r t ♦t♥r ♣r t♠♣rtr rít ♠s♠♦ ♦r q s

♦t♥ ♠♥t ♦r♠s♠♦ ♥s ♣r♦♥ts ♠♦♦ q ♣r♦♣♦r♦♥ ♥ ♠② ♥

st ♦s t♦s ①♣r♠♥ts

♥ s ♠str r ♦①st♥ r trés ♠♣♠♥tó♥ st ♣r♦♠♥t♦

st ② s ♦♠♣r ♦♥ ♦t♥ s t③♥♦ ♣♥t♦s st r ♦rrs♣♦♥

❬❪ óts q s ♦t♥ ♥ st rt♠♥t ♣t s♦♦ ♣r s ♣♦r

P♦str♦r♠♥t ♥♦tt♦ ② ♦♦ ❬❪ ♥ ♦♥t①t♦ ♥ t♦rí ♠♣♦ ♠♦ ♣rt♥♦

♣r♦♣st ♦♥sr♥ q t♠ñ♦ t♦ ♣r á♠tr♦ ♠♦r σff ♣♥ t♠♣rtr

♦ ♣♦t♥ ♣rs ΦBCT−AT ♦♣t♦ ♣♦r st♦s t♦rs s é♥t♦ ♦♥ s q

σff = σff (T )

ΦBCT−AT =

4εff

[

λ(

σff

r

)12

−(

σff

r

)6]

si r > λ1/6σff

0 si r ≤ λ1/6σff

.

♥ st s♦ ♦s ♣rá♠tr♦s λ ② σff (T ) s st♥ ♣r t♠♣rtr ♣r r♣r♦r ♠♥t s

♦♥♦♥s ♦①st♥ ♥tr íq♦ ② s ♣♦r ♥ qr♦ tr♠♦♥á♠♦ ② r ♥tr♦♥s ①tr♥s

s

P (ρv) = P (ρl) = Pcoex,

♦♥ ρv ② ρl r♣rs♥t♥ ♥ss ♣♦r ② íq♦ rs♣t♠♥t ② Pcoex s ♣rsó♥ ♦①st♥

♠s ss ♦s ♦rs ①♣r♠♥ts ♦①st♥ ♣♥ ①trrs ♣♦r ♠♣♦ ❬❪

♥ ♥ ♥áss ♣♦str♦r ♥♦tt♦ ② ♦♦r♦rs ❬❪ t③♥ ♠s♠♦ ♣♦t♥ ♣r♦

r♥ ♥ ♣rá♠tr♦ ♦♥ ♥♦♥ tér♠♥♦ q ♦rrs♣♦♥ s st♦s trs ♣rá♠tr♦s ♦

s♦♥ st♦s s♠tá♥♠♥t ♣r r♣r♦r s ♦♥♦♥s ♦①st♥ ② st r

P (ρv) = P (ρl) = Pcoex

µ(ρv) = µ(ρl)

st ♠♦♦ ♣rs♥t ♣r♦♠s ♥♦ s ♥t♥t str ♦tr s ♠♣♦rt♥ts ♣r♦♣s qr♦ ♥

♦ ♥♦s rr♠♦s t♥só♥ s♣r s♣t♦ q ♥③r♠♦s ♥ ♣ít♦

str ♣r♦♣st

♦s♦tr♦s ♠♦s ♦♣t♦ ♦♠♦ ♣♥t♦ ♣rt ♣r♦①♠ó♥ r ♣ít♦ ♦♥ s ♥♦♥s

♣s♦ r♥ ♦ q ♦rr ♥ stá♥ ss ♥ ♦♠trí s ♣rtís q ♦♥stt②♥

♦

♥ ♣r♥♣♦ t③r♠♦s ♥ rsó♥ ♣♦t♥ ♣rs Φ q ♦♥t♥ ♥ tér♠♥♦ ♦rrs♣♦♥♥t ♥

♣♦t♥ srs rs ② ♥ ♦♠♣♦♥♥t trt s♦ ♣♦t♥ ❲

r♥♦ ♥ ♦♥t①t♦ ♠♥t t③ó♥ st ♣♦t♥ ♠ás ♦r♦ t♠♣♦♦ s ♦r

♦♥sr ♥ ♥ st ♦s t♦s ①♣r♠♥ts ♦rrs♣♦♥♥ts ♦①st♥ ss ♦♠♦ ♣

♣rrs ♥ s s

♣r♦♣st st tss ♦♥sst ♥t♦♥s ♥ ♠♣r ♦♠♦ ♣♥t♦ ♣rt ♥ ♥♦♥ ♥s ♦♥ s

s♥ts rtrísts

♦♠♦ s ♦♠♥t♦ ♥ts ♦♠♣♦♥♥t trt ♣♦t♥ ♣rs srá ❲ ♣r♦ ♦♥ ♣rá♠tr♦s εff (T )

② σff (T ) st♦s s εff ② σff ♠♥r t r♣r♦r ♦s ♦rs ①♣r♠♥ts ♦①st♥ íq♦

♣♦r ② ♦rrs♣♦♥♥t ♥ ♦ ♥ qr♦ tr♠♦♥á♠♦ r ♥tr♦♥s ①tr♥s

♣rt r♣s st ♣♦t♥ ♦rrs♣♦♥rá srs rs á♠tr♦ dHS = σff (T )

s r

Φ =

∞ si r < σff

ΦWCA si r ≥ σff

tér♠♥♦ ♥♦♥ ♥s ♦ ♦♠♣♦♥♥t s ♣♦rá ♦♥t♥r ♦tr♦ ♣rá♠tr♦

♦♥ q t♠é♥ s st ♥ s♠tá♥♦ ♦♥ ♦s ♦s ♥tr♦rs ♣r r♣r♦r r ♦①st♥ ②

♦♥ó♥ st

♥ s♦ ♣♦r ♠♣♦ ♠♦tó♥ t③r ♥ t♦r ♦rró♥ ♥ tér♠♥♦ ♦rrs♣♦♥♥t

s q ♣♦r ♣rs♥ t♦s á♥t♦s Pr s♣rr st♦s t♦s ♥ s♦ ♦s

tó♠♦s rrs q ❬❪

Λ ≪ l

♦♥ Λ s ♦♥t ♦♥ tér♠ r♦ ② l r♣rs♥t s♣♦ ♥tr♣rtí r

l ≈ (V/N)1/3

= ρ−1/3

r♦

t♥ ♣r ♥ s sst♥s t♠♣rtr tr♣ ♥ ♥s ♦♥st♥t ♦t③♠♥ ② t♦r Λ/l q ♥ ♠♥t ♦s t♦s á♥t♦s ♥t♦ ♠ás r♥♦ ♥♦ s ♦r st ró♥♠ás ♠♣♦rt♥ts s♦♥ ♦s t♦s á♥t♦s

íq♦ Tt ❬❪ Λ/l

r♦♥♦ 2 ó♥ ró♥ ❳♥ó♥ Λ = 0,01194x10−9m

s ♥s s íq s tí♣ q ♣♦s ♣♦ró♥ ♥ ♣rsó♥ q ♦rrs♣♦♥

♦①st♥ ♦♥ s s ♣♦r s t♠♣rtr ♥ t s t♥ ♦s ♦rs ró♥ Λ/l

t♠♣rtr tr♣ ♣r ♦s ♦s ♦♥ ♦s q trt♠♦s ♥ st tr♦ ② s ♦♠♣r♥ ♦♥ ♦rrs♣♦♥♥t

ró♥♦ ♦♥ ró♥ ♣rát♠♥t ♥

st ♠♥r ♥ tér♠♥♦ ♦rró♥ s ás♦ s st t♥t♦ ♣r s♦ ♦♠♦ ♥ ♠♥♦r

r♦ ♣r r

P♦r ♦tr ♣rt ♣r t♠♣rtrs r♥s rít ♦♥ s t♦♥s s ♥ ♠♣♦rt♥ts ♣♦rí

strs ♣rs♥ st tér♠♥♦ ♦ ♣rt♠♥t♦ ♦♥ó♥ s ♦♠♦ ♦♥s♥

sts t♦♥s

st ♥♦♥

♥♥♦ ♥ ♥t ♦s ♣r♥♣♦s tr♠♦♥á♠ r♥ ♥rí ♥tr♥ s ♣ ①♣rsr ♥ tér♠♥♦s

r♥ ♥rí r

dE = dF + d(TS) =⇒ dF = dE − TdS − SdT = −PdV + µdN − SdT.

♣rsó♥ ② t♠é♥ ♣♦t♥ qí♠♦ ♣♥ rrs ♥rí r ♠♦t③

P = −(

∂F

∂V

)

T,N

= ρ2∂f

∂ρ,

µ =

(

∂F

∂N

)

T,V

= f + ρ∂f

∂ρ.

♥ sts ①♣rs♦♥s f = F/N s ♥rí r ♠♦t③ ♣♦r ♣rtí

st ♥rí s q ♦t♥r♠♦s s ♦♥s q sr♥ ♥str♦ ♦r♠s♠♦ ② ♠r♠♦s fDF

Pr s♦ ♥ ♦ ♦♠♦é♥♦ ♦♥ ♥tr♦♥s trts ♥tr s ♣rtís q ♦ ♦♥stt②♥ ♥rí

r s ♣ ①♣rsr ♦♠♦

fDF = fid +∆fHS +1

2bρ+ αkBT

[

ln(

Λ3ρ)

− 1]

=

= fHS +1

2bρ+ αkBT

[

ln(

Λ3ρ)

− 1]

♦♥ ♦♠♦ ♥ts s ♥ b =´

Φattrdr Pr ♣♦t♥ b = −(32π/9)εffσ3ff ♠♥trs q ♣r

♣♦t♥ ♣r♦♣st♦ srá

b = −(32π√2/9)εff (T )σ

3ff (T )

tér♠♥♦ q ♦rrs♣♦♥ s ♠♦s r♦ ♥ ♣rá♠tr♦ ♦♥ q ♠♠♦s α q ♦rr

só♥ ♦♠♣♦rt♠♥t♦ ♦ ♦s t♦s á♥t♦s ♦ ♣r str ♠♦r ♦s ♦rs ♦①st♥

♥♥♦ ♥ ♥t s s ②

P (T ) = PHS(T ) + αkBTρ− (32π√2/18)εff (T )σ

3ff (T )ρ

2

µ(T ) = µHS(T ) + αkBT ln(

Λ3ρ)

− (32π√2/9)εff (T )σ

3ff (T )ρ

♥ ♥str♦ ♦♥t①t♦ tr♦ t♥t♦ ♣r ♣rsó♥ ♦♠♦ ♣r ♣♦t♥ qí♠♦ ♠♦♦ srs rs

PHS ② µHS ♦♣t♠♦s s q ♦rrs♣♦♥♥ ♦r♠ó♥ Prs❨ ❬❪ ♦♠♣t ♦♥ ♦r♠s♠♦

s r

PHS(T ) = PPYHS = kBTρ

(

1 + η + η2

(1− η)3

)

µHS(T ) = µPYHS = kBT

η

2

(

14− 13η + 5η2

(1− η)3

)

− ln(1− η) + ln(Λ3ρ)

♦♥ η s t♦r ♠♣qt♠♥t♦

η =π

6ρσ3

ff (T )

♦ t♥♥♦ ♥ ♥t ♦ ♥tr♦r ♣r t♠♣rtr s ♠♣♦♥ st ♦s ♦rs ♦①st♥

r♦s á♦ ♦s ♦rs ①♣r♠♥ts

P (ρl) = P (ρv) = P0

♥t♦ ♦♥ ♦♥ó♥ st

µ(ρv) = µ(ρl)

st ♠♣♦só♥ ♦s ♥♦s ♦rs εff σff q ♠♦s r♥♦♠r♦ ♦♥ εff (T ) ② σff (T ) ② ♦r α

♣♦st q st♦s ♦rs ♦s ♣r ♥ ♦ ♦♠♦é♥♦ s♦♥ ♦s ♣r♦♣♦s ♣r str ♣r♦♠s

♦s ♥♦♠♦é♥♦s ♦♠♦ ♦s q s ♣rs♥t♥ ♥♦ st♦s s♦♥ s♦r♦s s♦r sstrt♦s só♦s

st♦s ♦t♥♦s

♦♥t♥ó♥ ♥ ♦s r♥ts rá♦s s ①♥ ② s ♦♠♣r♥ ♦s rst♦s ♦t♥♦s ♣r ♦r♠s♠♦

♦♥ ♥♦♥ ♠♦♦ sú♥ ♥str ♣r♦♣st ♣r r ② ❳ ♥ r♥♦ t♠♣rtrs q

r♦

♥ st t s ♣♥ ♣rr ♦s ♦rs ♦rrs♣♦♥♥ts ♥t♥s ♥tró♥ ♦♦ á♠tr♦ ♠♦r ② s t♠♣rtrs tr♣s ② ríts r ② ❳ ♥t♦ s rr♥s ♦♥ r♦♥①trí♦s

♦ εff/kB ❬❪ σff ❬♥♠❪ Tt ❬❪ Tc ❬❪ ❬❪r ❬ ❪❳ ❬ ❪

♥ s ♣♥t♦ tr♣ rít♦ ♠♥t♦ ♥ st♦s rá♦s s ♣♥ ♦srr ♦s sts r

♦①st♥ ♥♦ s t③♥ ♣♦t♥s t♣♦ ♦ ❲ t♦♠♥♦ ♣r ♦s ♣rá♠tr♦s σff ② εff ♦s ♦rs

rr♥ q r♥ ♥ t

P♦r ♦tr♦ ♦ ♥ ró♥ ♣r♦♣st st tss s rr♦♥ s s♦♥s ♣♦r♥ts ♥ ♦r ♦s

♣rá♠tr♦s ❲ á♠tr♦ ♠♦r ♥t♥s ♣♦t♥ ♣rs ② ♦♥t s♦♦ s

♥ ♥ó♥ t♠♣rtr q ♣r♠t ♦rrt♦ st ♦s ♦rs ①♣r♠♥ts ♦①st♥

óts q s r♦♥s ♦s ♣rá♠tr♦s s♦♥ ♠á①♠s ♣r♦①♠rs t♠♣rtr rt ♦♥ ♦♠

♣rs s♦tér♠ χT r ② ♣♦r ♥ s t♦♥s ♥ ♥s s♦♥ ♠♣♦rt♥ts ♣s

< N2 > − < N >

< N >= ρkBTχT

♠é♥ s ♠str ♦♠♦ s ♠♦ ♣♦t♥ ♣rs ♥ ♥ó♥ st♥ r ♥tr ♠♦és

st ♦s ♣rá♠tr♦s ♦rrs♣♦♥♥ts ó♥

❱ró♥ á♠tr♦ ♠♦r ② ♥t♥s ♣♦t♥ ♣rs

♥ ♦s rá♦s s ♣♥ ♦srr ♦♠♦ s ♠♦♥ ♦s ♦rs ♦s ♣rá♠tr♦s ♣r str ♦s

t♦s ①♣r♠♥ts ♦①st♥ ♥ s♦ st ♦ í s ♠str♥ ró♥ ♣♦r♥t á♠tr♦

♠♦r tí♣♦ δσ ② ♥t♥s ♣♦t♥ ♣rs δε

s♦ á♠tr♦ ♠♦r sts s♦♥s s ♠♥t♥♥ ♣♦r ♦ ♠♥trs q ♣r ♣♦t♥

♥♦ s♣r♥ ♣r t♦♦ r♥♦ t♠♣rtrs ♥tr tr♣ ② rít

❱ró♥ ♦♥t s ② ♣♦t♥ ♣rs ♥ ♥ó♥ st♥

♥ ♦s rá♦s s ♣♥ ♦srr ♦♠♦ ♠ ♦♥ t♠♣rtr tér♠♥♦ ♦rró♥ s

♣rá♠tr♦ ♥♦t♦ ♣♦r α ② t♠é♥ ♦♠♦ s ♠♦ ♠♥t ♣♦t♥ ♣r s t♠♣rtrs tr♣

② rít

t♦r q ♦rr tér♠♥♦ s s r ♥♦ t♠♣rtr st ♣rt t♠♣rtr

rít ♥ ♠♥♦s ♥ ❱ r st t♦r s ♠♥t♥ ♣rát♠♥t ♣qñ♦ s♦ ♥♦ t♠♣rtr

s r ♠s♦ rít ♥ ♥ st t♠♣rtr α s ♠♥t♥ ♣♦r ♦

♥ s♦ ♣♦t♥ ♣rs s s♦♥s rs♣t♦ ♣♦t♥ ❲ ♦ ♦♥ ♣rá♠tr♦s σ ② ε ♦s

♥♦ s♣r♥ ♥ t♦♦ r♥♦ t♠♣rtrs ♥tr tr♣ ② rít

25 30 35 40 450

10

20

30

40

50

T [K]

δσ(%

)

Tc

Tt

Ne

25 30 35 40 45

10

20

30

40

50

T [K]

δε

Tt

Tc

Ne

r ♠♦s rá♦s ♠str♥ ♦♠♦ ♥ ♠♦rs ♦s ♣rá♠tr♦s ♣♦t♥ ♥ ♥ó♥ t♠♣rtr ♣rstr r ♦①st♥ íq♦♣♦r ♥ rá♦ ③qr s ♣ ♦srr ó♠♦ ♠♦rs á♠tr♦ ♠♦r r s ♣ ♦srr ♠♦ ♥sr♦ ♥ ♥t♥s ♣♦t♥ ♣rs s ♠á①♠s s♦♥s ♥srs s♦♥ ♣r t♠♣rtrs r♥s rít

25 30 35 40 450

0.1

0.2

0.3

0.4

0.5

T [K]

Tt

Ne

Tc

α

0 0.2 0.4 0.6−50

−40

−30

−20

−10

r [nm]

Φa

ttr [

k BT

]

Ne

WCA

Tc

Tt

L−Jd

HS

r ③qr s ♦sr ♦r ♦♥t s ♥ ♥ó♥ t♠♣rtr ♦r ♣♦r♥t r ♣♦t♥ ♣rs ♦ ♣r t♠♣rtr rít r ♠r ♦♥ Tc ② tr♣ rsñ ♦♥ Tt s r ♥t♦ ❲ r r② rst♥t r ♦rrs♣♦♥ ♥

0 10 20 30 4025

30

35

40

45

50

ρ [nm−3]

T [

K]

Ne

WCA

L−J

r ♦①st♥ ss íq♦♣♦r ♦sr ♥ í♥ ♥ r ♦①st♥ ♦rrs♣♦♥♥t t♦s ①♣r♠♥ts ♦s ír♦s ♥r③♦s ♥ s í♥ ♦rrs♣♦♥♥ ♦s ♦rs ♦s t③♥♦ ♥str♣r♦♣st P♦r ♦ ♥ í♥ r②s s ♠str♥ ♦s rst♦s á♦ ♠♥t ♦r♠s♠♦ ♥♦♥ ♦♠♦ ♣♦t♥ ♣rs s t③ó ♥ ♥ í♥ r②s ② ♣♥t♦s s ♠str♥ ♦s rst♦s ♦s♥tr♦ ♠s♠♦ ♦r♠s♠♦ ♣r♦ t③♥♦ ♦♠♦ ♣♦t♥ ♣rs ❲

rs ♦①st♥ íq♦♣♦r ó♥

♥ s ♠str ♦♠♦ ♥♦s ♠♦♦s ♣♦t♥ t③♦s ♥tr♦ ♥ ♦r♠s♠♦ st♥

r ♦①st♥ íq♦♣♦r óts ♦s ♠r♦s rst♦s q s ♦♥s♥ s s t③ ♥ ♣♦t♥

♦♥ ♥ ♣♦t♥ ♦♠♦ ❲ st s ♠ás ♣rs♦ ♥ ♥ r♥♦ ♥♦ ♠② ♠♣♦ ♥ss

á♦s ♦rrs♣♦♥♥ts ró♥

❱ró♥ á♠tr♦ ♠♦r ② ♥t♥s ♣♦t♥ ♣rs

♥ st s♦ ró♥ ♣♦r♥t á♠tr♦ ♠♦r s ♠♥t♥ ♣♦r ♦ ♣r

♣rát♠♥t t♦♦ r♥♦ t♠♣rtr ♥t♥s ♣♦t♥ ♣rs s sí ♥ ♥♦ ♠ás

♠♥t♥é♥♦s ♣♦r ♦ st ♦s s r st ♥ t♠♣rtr ♦rrs♣♦♥♥t

rít

❱ró♥ ♦♥t s ② ♣♦t♥ ♣rs ♥ ♥ó♥ st♥

♦r ♦♥t α s ♠♥t♥ ♣♦r ♦ st ♥ t♠♣rtr ♦rrs♣♦♥♥t rít

♦r ♠á①♠♦ q ♥③ s ♥ t♠♣rtr rít

♥ s♦ ♣♦t♥ ♣rs s s♦♥s rs♣t♦ ♣♦t♥ ❲ ♦ ♦♥ ♣rá♠tr♦s σ ② ε ♦s

♥♦ s♣r♥ ♥ t♦♦ r♥♦ t♠♣rtrs ♥tr tr♣ ② rít

rs ♦①st♥ íq♦♣♦r

80 100 120 140 1600

10

20

30

40

50

T [K]

δσ(%

) Ar

Tt T

c

80 100 120 140 1600

10

20

30

40

50

T [K]

δε

ArT

cT

t

r ♠♦s rá♦s ♠str♥ ♦♠♦ ♥ ♠♦rs ♦s ♣rá♠tr♦s ♣♦t♥ ♥ ♥ó♥ t♠♣rtr ♣rstr r ♦①st♥ íq♦♣♦r ♥ rá♦ ③qr s ♣ ♦srr ♦♠♦ rrs á♠tr♦ ♠♦r r s ♣ ♦srr ♠♦ ♥sr♦ ♥ ♥t♥s ♣♦t♥ ♣rs s ♠á①♠s s♦♥s ♥srs s♦♥ ♣r t♠♣rtrs r♥s rít

80 100 120 140 1600

0.1

0.2

0.3

0.4

0.5

T [K]

α

ArT

cT

t

0 0.2 0.4 0.6 0.8−150

−125

−100

−75

−50

−25

r [nm]

Φa

ttr [

k B T

]

dHS Ar

WCA

Tc

Tt

r ③qr s ♦sr ♦r ♦♥t s q ♣r ♥ ♥str♦ ♥♦♥ ♥ ♥ó♥ t♠♣rtr r ♣♦t♥ ♣rs ♦ ♣r t♠♣rtr rít í♥ ♠r ♦♥ Tc ② tr♣í♥ sñ ♦♥ Tt s r ♥t♦ ❲ í♥ r② rst♥t r ♦rrs♣♦♥ ♥

0 5 10 15 20 2580

100

120

140

160

ρ [nm−3]

T [

K]

Ar

WCA

L−J

r ♦①st♥ ss íq♦♣♦r r ♦sr r ♦①st♥ ♦rrs♣♦♥♥t t♦s ①♣r♠♥ts í♥ ♥ ♦s ír♦s ♥r③♦s ♥ s í♥ ♦rrs♣♦♥♥ ♦s ♦rs st♦s t③♥♦ ♥str♦♥♦♥ ♥s P♦r ♦ ♥ í♥ r②s s ♠str♥ ♦s rst♦s á♦ ♠♥t ♦r♠s♠♦ ♥ ♦♥ ♦♠♦ ♣♦t♥ ♣rs s t③ó ♥ ♥ í♥ r②s ② ♣♥t♦s s ♠str♥ ♦srst♦s ♦s ♥tr♦ ♠s♠♦ ♦r♠s♠♦ ♣r♦ t③♥♦ ♦♠♦ ♣♦t♥ ♣rs ❲

♥ s ♣ ♦srr r ♦①st♥ íq♦♣♦r r óts q q ♥

s♦ ♥tr♦r ♥♦ s ♦♥s♥ ♥♦s rst♦s ♥♦ s t③ ♦♠♦ ♣♦t♥ ♣rs ♥ ♠♥trs q

♣♦t♥ ❲ s ♥ ♦♥s ♥ ♠♦r rst♦ t♥♥ r ♥t ♥ st ♥♦ ♥♦r♠

♥♠♥t♠♥t ♣r r♥♦ ♥s q ♦rrs♣♦♥ s íq

á♦s ♦rrs♣♦♥♥ts ❳♥ó♥

á♠tr♦ ♠♦r ② ró♥ ♣r♦♥ ♣♦t♥ ♣rs

♥ ♦s rá♦s s ♠str♥ ♦♠♦ ♥ ♠♦rs ♦r st♦s ♣rá♠tr♦s ♣r ♦♥sr str

r ♦①st♥ íq♦♣♦r ró♥ ♥ á♠tr♦ ♠♦r s ♠♥t♥ ♣♦r ♦ st

♥ t♠♣rtr ♦rrs♣♦♥♥t rít ♠á①♠ só♥ s ♣r♦① ♥ ② s ♥③ ♥

t♠♣rtr rít ♥ ♥t♦ ♥t♥s ♣♦t♥ ♥tró♥ s ró♥ s ♠♥t♥ ♣♦r ♦

st ♥ t♠♣rtr s rít ♥ st ♦r t♠♣rtr st ró♥ ♥③

♥ ♠á①♠♦

❱ró♥ ♦♥t s ② ♣♦t♥ ♣rs ♥ ♥ó♥ st♥

Pr ❳ ♦♥t α s ♠♥t♥ ♣♦r ♦ st ♣r♦①♠♠♥t ♥ t♠♣rtr

s r st ♣r♦① rít ♥③♥♦ ♦r ♣rát♠♥t ♥ t♠♣rtr rít

♥ s♦ ♣♦t♥ ♣rs s s♦♥s rs♣t♦ ♣♦t♥ ❲ ♦ ♦♥ ♣rá♠tr♦s σ ② ε ♦s

♥♦ s♣r♥ ♥ t♦♦ r♥♦ t♠♣rtrs ♥tr tr♣ ② rít

rs ♦①st♥ íq♦♣♦r

150 200 250 3000

10

20

30

40

50

T [K]

δσ(%

) XeT

tT

c

150 200 250 3000

10

20

30

40

50

T [K]

δε

Tt

Tc

Xe

r rá♦ ③qr ♠str ó♠♦ ♠r á♠tr♦ ♠♦r ♣r ♦♥sr ♥ ♥ st r ♦①st♥ íq♦♣♦r r s ♣ ♦srr ♠♦ ♥sr♦ ♥ ♥t♥s ♣♦t♥ ♣rs s ♠á①♠s s♦♥s ♥srs ♦rrs♣♦♥♥ t♠♣rtrs r♥s rít

150 200 250 3000

0.1

0.2

0.3

0.4

0.5

T [K]

α

Tc

Tt

Xe

0 0.2 0.4 0.6 0.8−300

−250

−200

−150

−100

−50

r [nm]

Φa

ttr [

k BT

]

Xe

L−J

TcT

t

WCA

dHS

r ③qr s ♦sr ♦r ♦♥t s q ♣r ♥ ♥str♦ ♥♦♥ ♥ ♥ó♥ t♠♣rtr r ♣♦t♥ ♣rs ♦ ♣r t♠♣rtr rít í♥ ♠r ♦♥ Tc ② tr♣í♥ sñ ♦♥ Tt s r ♥t♦ ❲ í♥ r② rst♥t r ♦rrs♣♦♥ ♥

0 5 10 15150

200

250

300

ρ [nm−3]

T [

K]

Xe

WCA

L−J

r ♦①st♥ ss íq♦♣♦r ❳ ♦sr r ♦①st♥ ♦rrs♣♦♥♥t t♦s ①♣r♠♥ts í♥ ♥ ♦s ír♦s ♥r③♦s ♥ s í♥ ♦rrs♣♦♥♥ ♦s ♦rs st♦s t③♥♦ ♥str♦♥♦♥ ♥s P♦r ♦ ♥ í♥ r②s s ♠str♥ ♦s rst♦s á♦ ♠♥t ♦r♠s♠♦ ♥ ♦♥ ♦♠♦ ♣♦t♥ ♣rs s t③ó ♥ ♥ í♥ r②s ② ♣♥t♦s s ♠str♥ ♦srst♦s ♦s ♥tr♦ ♠s♠♦ ♦r♠s♠♦ ♣r♦ t③♥♦ ♦♠♦ ♣♦t♥ ♣rs ❲

♥ s ♣♥ ♦srr ♦s sts ♦r♦s ♥♦ ♥ s t③♥ ♦♠♦ ♣♦t♥s ♣rs

♦ ❲ s♦ s t♥ tr♦ ♦♠♦ ♣r ② r ♥ ♥t♦ ❲ ♥ s♦ s

♣♦r s ♦t♥ ♥ st r③♦♥ ♣r ♥ ♣r♦① r♥♦ ♥s q ♦rrs♣♦♥ s s

♣ít♦

♥só♥ s♣r

♥t ♣r♦♠♥t♦ ♦♣t♠③ó♥ ♥♦♥ ♣r♦♣st♦ ♥ st ss s ó t♥só♥ s♣r

♥trs íq♦♣♦r ♥ ♥ó♥ t♠♣rtr ♣r r ② ❳ ♦s ♦rs ①♣r♠♥ts

r♦♥ ①trí♦s ❬❪ ♣r♦♠♥t♦ ♣r♦♣st♦ ♥ st ss ♣r ♦♣t♠③r ♥♦♥ ♥♦s ♣r♠tó

♣rr ♦♥ ♣rsó♥ ♦s ♦rs t♥só♥ s♣r ♣r ♦s ♦s q qí st♠♦s só♥

rs♣t♦ ♦s t♦s ①♣r♠♥ts ♥♦ s♣r ♥ t♦♦ r♥♦ t♠♣rtrs ♥tr tr♣ ② rít

r ♥tr♦ó♥

t♥só♥ s♣r ♥trs íq♦♣♦r ♦rrs♣♦♥♥t ♥ ♦ ♥ qr♦ tr♠♦♥á♠♦ s

♦♥s♥ ♥ r③s ♥tr♠♦rs ♦ ♥trtó♠s ② ♣♥ s t♠♣rtr

st t♥só♥ γlv ♣ r♦♥rs ♦♥ t♠♣rtr rít ♦ ♠♥t ♥ ①♣rsó♥ sr t♦rí

♥ó♠♥♦s rít♦s

γlv = γ0lv

[

1− T

Tc

]p

.

♥ st ①♣rsó♥ γ0lv ♣♥ s rtrísts ♦ ② p s ♥ ①♣♦♥♥t rít♦ ②♦ ♦r s♦ st♠♦

♣♦r r♥ts t♦rs ♦ r♥ts ♦♥sr♦♥s ❬❪

P♦r ♠♣♦ ❲♦♠ ❬❪ ♦ st♠ó ♥t♥♦ s♣s♦r ♥trs íq♦♣♦r ♦♥ ♦♥t

♦rró♥ q ♣r♦♥ ♦rí ♦rró♥ ♦r srr♦ ♣♦r r♥st♥ ❩r♥ ② ②

r♦ ♠♦♦ ♦♣t♦ s st♦s ♦ ♦ ♦♥t♥♦ ♦s ♦rs q ♦t♦ r♦♥ ♦

♥ ♥ tr♦ ♣♦str♦r ♠t ② ♦♦r♦rs ❬❪ s ♥③ó ♥ ♠str ❳ t ♣r③ ♦♥t♠♥♦

♦ s♠♦ ♦♥ ♣♣♠ ♥ ♦♠♥ ♥tró♥♦ ② ♣♣♠ r Pr st ♠str s ♥♦♥tró q ♦s ♣rá♠tr♦s

ó♣t♠♦s ♥

γ0lv = 54.6± 0.1 dyn cm−1

p = 1.287± 0.017

r rt♥r t③ ♣r rs♦r s ♦♥s ♥ ♦♥t①t♦ s ♣rs ♣rs ♥trs stá♥ s♣rs ♥ st♥ ② ♥♦ ♥trtú♥ ♦♥ ♦ q s ♥♥tr ♥tr s Pr á♦ s♦♥sr q ①t♥só♥ ♥ sts ♣rs ♥ ♥♥t ♥ st sq♠ z r♣rs♥t ♦♦r♥ ♣♦só♥ t♦♠♥♦ ♦♠♦ ♦r♥ ♦♦r♥s ♣r ③qr

♥ ♥ t♦♦ r♥♦ t♠♣rtrs ♥tr tr♣ ② rít st♦s ♦rs ♣r♠t♥ ♣rr ♦♥

♣rsó♥ t♥só♥ s♣r ♥ ♥ r♥♦ str♦ t♠♣rtrs ♥ t♦r♥♦ rít ♦s ♦rs q ♣r

st ♠♦♦ ♦♥ ♦s ♥tr♦rs ♣rá♠tr♦s s sí♥ ♦s ①♣r♠♥ts ♥ ♠ás ♥ t♥só♥

s♣r ♥ st r♥♦ str♦ ♠ ♣r ❳ ♣♦r ❩♦ ② ♦♦r♦rs ❬❪ t③♥♦ té♥s

s♣tr♦s♦♣ ó♣t ♠③ ♥ st té♥ s ♥③ s♣tr♦ ③ q s ♥ ♥ást♠♥t ♥

♥trs íq♦♣♦r ♦ s ①t♦♥s tér♠s ❬ ❪

♥só♥ s♣r ② ♣♦t♥ tr♠♦♥á♠♦

t♥só♥ s♣r ♥trs íq♦♣♦r s ♣ r trés r♥ ♣♦t♥ Ω

γlv =1

2A[Ω + P0V ] =

1

2[Ω/A+ P0L] .

♦♥

Ω = F − µN .

♥ P0 r♣rs♥t ♣rsó♥ stró♥ ♣♦r ♦rrs♣♦♥♥t ♥ tr♠♥ t♠♣rtr

N s ♥ú♠r♦ ♠♦és ♦ ② L s ♥♦ r ♥tr♦ s rs♥ s

♦♥s ♥ ♦♥t①t♦ t③♥♦ ♣r ♦ ♣r♦♣st ♣rs♥t ♣r ♥♦♥ ❬ ❪

♥ ♦s á♦s s t♦♠ ♥ ♦r ♦ ♣r N r ♦s á♦s s♦♥ r③♦s ♥tr♦ ♦r♠s♠♦ ♥ó♥♦

s ♣rs ♥♦ ♥trtú♥ ♦♥ s ♠♦és ♥q ♥ ♦♥♦♥s ♦♥t♦r♥♦ s♦r ♥s s

♠♦és ♦ ♥trtú♥ ♥tr sí ♠♥t ♥ ♣♦t♥ Uff (r) = Φ (|r− r′|)

s s♣♦♥ q ♣r♦♠ s ♥♠♥s♦♥ s♦ ♦♥srr q ár s ♣rs t♥ ①t♥só♥

♥♥t ♥♥♦ z ♦♠♦ r ♣♦só♥ s ♠♣♦♥

ρ(z = 0) = ρ(z = L) = 0 .

Pr rs♦r s ♦♥s ♦♥ s s ♦♥♦♥s ♦♥t♦r♥♦ ♣rós s ♦ s♥t♠♥t

♥ ♦♠♦ ♣r srr q ♦ s ♥♥tr ♥ s íq ♥ ♥tr♦ r♦♦ s ♣♦r

♥ s ♣rs ♠s♠ s r

ρ(z ≃ L/2) = ρl(T ) .

♥ ♥str♦s á♦s ♠♦s ♦♣t♦ ♥♦s ♦ 60σff

s ♦♥s s ♣♥ srr ♦♠♦

νidkBT ln[

Λ3ρ(z)]

+Q(z) = µ .

♦♥

Q(z) = fHS [ρ(z); dHS ] +

ˆ L

0

dz′ρ(z′)δfHS [ρ(z′); dHS ]

δρ(z′)

δρ(z′)

δρ(z)

+

ˆ L

0

dz′ρ(z′)Φatt (|z − z′|) + Usf (z) .

♦♠♦ s ♦♥sr q ♥♦ ①st ♥tró♥ ♥tr ♦ ② s ♣rs r♣♥t q ♦ ♦♥t♥♥ Usf (z) = 0

Pr rs♦r sts ♦♥s s út rsrrs s♥t ♠♥r

ρ(z) = ρ0 exp

( −Q(z)

νidkBT

)

,

♦♥

ρ0 =1

Λ3exp

(

µ

νidkBT

)

,

♣♦r ♦tr♦ ♦ s ♥ ♥ú♠r♦ ♣rtís ♣♦r ♥ ár

n =1

A

ˆ L

0

ρ(z)dz .

ró♥ ♥tr ♣♦t♥ qí♠♦ ♦ ♠t♣♦r r♥ s t♥♠♦s ♥ ♥t q tr♠♦s ♥

♦r♠s♠♦ ♥ó♥♦ ② n q ♦♥stt② ♥ t♦ ♥str♦ ♣r♦♠ s ♦t♥ sstt②♥♦ ♥

st ♠♥r

µ = −νidkBT ln[

1

nΛ3

ˆ L

0

dz exp

( −Q(z)

νidkBT

)

]

.

st sst♠ s rs trt♠♥t ♥♥♦ ♦♠♦ s♠♥t s rs ♠♥s♦♥s ts ♦♠♦

0 10 20 30 40 50 600

0.2

0.4

0.6

0.8

z*

Ne

ρ*

r ♥s ♥ ♥ó♥ st♥ ♥ s ♣rs ♣r r♥ts ♦rs t♠♣rtr ♦st ♦♥♥♦ ♥ ♥ rt♥r ♥ ♥♦ q♥t 60σff s t♠♣rtrs ♥ í♥ ♣♥t♦s í♥ r②s ② í♥ só

z∗ =z

σff,

ρ∗ = ρσ3ff ,

L∗ =L

σff.

♥ s ♣ ♦srr ♠♦♦ ♠♣♦ ♥♦s ♣rs ♥s q s ♦t♥♥ ♦ rs♦r

sts ♦♥s ♣r s♦ ♣r r♥ts ♦rs t♠♣rtr ♥ st s♦ s t③ó ♥

60σff ♥♦ ♦♥ σff = ♥♠

♥só♥ s♣r ♥trs íq♦♣♦r ó♥

♥ ♥ tr♦ ♥♦tt♦ rtr♦♦ ♦♦ ② ♦ ❬❪ s t♥só♥ s♣r st ♦ ♥

♦♥t①t♦ ♥ t♦rí ♠♣♦ ♠♦ t③♥♦ ♣r ♦ ♥ ♣♦t♥ ♣rs t♣♦ ♦♥ ♣rá♠tr♦s

♣♥♥ts t♠♣rtr r ♣ít♦ ♦s ♦rs st♦s ♣rá♠tr♦s r♦♥ ♦s ♣rtr ♥

♣r♦♣st r♥♦ ♠♦ ② r③♦♥ ❬❪ ♦s t♦s t♦rs ♦♥str②♥ ♣♦t♥ ♠♥r t

s♠r t♦ ♥ó♥ ♦rró♥ ♣rs g(|r− r′|) ♥♦ ♠ás ♣r♦♥♦ ♠í♥♠♦ q ♣♦s

♣♦t♥ ♥ s ♦♠♣r ♣rt trt ♣♦t♥ ♦♥ ❲ ② ♦♥ ♣r♦♣st

♦rrs♣♦♥♥t

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8−2.5

−2

−1.5

−1

−0.5

0

0.5

1

r [nm]

Φa

ttr(r

)/k B

T T=30 K

dHS

Ne

r P♦t♥ ♣rs ♥ ♥ó♥ st♥ ♥trtó♠ ♣r r só ♦rrs♣♦♥ ♣♦t♥ r ♥ í♥ r② s ♣♦t♥ ♣r♦♣st♦ ♣♦r ♥♦tt♦ rtr♦♦ ♦♦ ② ♦ ❬❪♦ ♥ t♠♣rtr í♥ ♣♥t♦s ② r②s ♦rrs♣♦♥ ♣♦t♥ ❲ á♠tr♦ ssrs rís s ♥ ♦♥ dHS

♥ ♣♥ ♦srrs ♦s rst♦s ♦s á♦s ♥t♦ ♦s rst♦s q ♥♦s♦tr♦s ♠♦s

♦ ♠♣♥♦ ♥str ♣r♦♣st st ♥♦♥ ♥ s rs s ♣ ♣rr ♥

t♥♥ r♥ t♥só♥ s♣r ♥♦ t♠♣rtr s ♣r♦①♠ tr♣ ♦s t♦rs

♥♦ ♠str♥ rst♦ ss á♦s ♣r ss t♠♣rtrs s ♠♥t ♠á①♠ q ♥ s r♥♦ ①st♥

t♦s ①♣r♠♥ts ♣r ❬❪

♥ s ①♥ rrs q ♦rrs♣♦♥♥ s s♦♥s ♣♦r♥ts ♦s rst♦s ② s

♦♠♣r♥ ♦♥ s s♦♥s ♦s rst♦s q ♥♦s♦tr♦s ♠♦s ♦t♥♦

♥só♥ s♣r ♥trs íq♦♣♦r ró♥

♥ s ①♥ ♦s ♦rs t♥só♥ s♣r r q ♠♦s ♦t♥♦ ② t♠é♥ ♦s ♦rs

♦s ♠♥t t♦rí ①♣♦♥♥ts rít♦s ② s♠♦♥s q ♠♣♥ s ♥t ♣♦r r♦②♠

①♥r ❬❪ Pr r③r s s♠♦♥s ♦s t♦rs ♥ ♦♠♦ r♦ ♦rt ♣♦t♥

st♥s ♥trtó♠s s♣r♦rs ♦s ② á♠tr♦s ♠♦rs s r ♣♦t♥ s t♦♠ ♦♠♦ ♥♦

♣r st♥s s♣r♦rs st♦s ♦rs

♥ s ♥ r♦ s s♦♥s ♣♦r♥ts ♥tr ♦s ♦rs ♦s ♣♦r ♦s rs♦s ♠ét♦♦s ②

♦s t♦s ①♣r♠♥ts

20 25 30 35 40 450

1

2

3

4

5

T [K]

γ lv/k

B

Tt T

c

Ne

r ♥só♥ s♣r ♥ ♥ó♥ t♠♣rtr rtr ③②s③ ② ❯rrt ❬❪ ♦s r♦s ♥r♦s♦rrs♣♦♥♥ t♦s ①♣r♠♥ts í♥ ♣♥t s ♦t♥ ♠♥t t♦rí ♥ó♠♥♦s rít♦s ♦♥ γ0lv = 1135 ♥♠2 ② p = ♦s trá♥♦s s♦♥ ♦s ♦rs ♦s ♣♦r ♥♦tt♦ rtr♦♦ ♦♦ ②♦ ❬❪ óts q ♦s ♠♥♦♥♦s t♦rs ♥♦ ①♥ á♦s ♥ ③♦♥ ♦♥ ② ♦rs ①♣r♠♥ts ♦sír♦s ♦rrs♣♦♥♥ ♥str♦s á♦s

25 30 35 40 45−10

0

10

20

30

40

T [K]

∆γ

Tt

Tc

Ne

r s♦♥s ♣♦r♥ts ♥tr ♦s ♦rs ♦s t♥só♥ s♣r ② ♦s sts t♦s ①♣r♠♥ts s ♦♠♥s ③s ♦rrs♣♦♥♥ ♦s á♦s s ♦♠♥s r♦s ♦rrs♣♦♥♥ ♥str♦sá♦s óts t♥♥ s s♦♥s ♦rrs♣♦♥♥ts ♦s á♦s ♥♦ t♠♣rtr sr tr♣ P♦r ♦tr♦ ♦ ♥ s r♥ís ♣♥t♦ rít♦ ♥strs s♦♥s s ♠♥t♥♥ ♣♦r ♦ ② ♥♦ s♣r♥ ♥ ♠♣♦ r♥♦ st ♦r ♠♥trs q ♦s t♦s t♦rs ♥③

80 100 120 140 1600

2

4

6

8

10

12

T [K]

γ lv/k

B [1

02 K/n

m2 ]

Tt

Tc

Ar

r ♥só♥ s♣r r ♥ ♥ó♥ t♠♣rtr rtr ② ③②s③ ❬❪ r r②s ♦rrs♣♦♥ t♦rí ♥ó♠♥♦s rít♦s ♦s r♦s r♥♦s s♦♥ t♦s ①♣r♠♥ts ❬❪ ♥t♦ trá♥♦s ♦♠♦♦s ♠♥ts ♦rrs♣♦♥♥ á♦s ♥á♠ ♦r ♦♥ r♥ts r♦s ♦rt t♦ rc ♣r ♣♦t♥ ♣rs ♦s trá♥♦s s s♦♥ ♥ r♦ ♦rt rc = σff ② ♦s ♠♥ts rc = σff ❬❪♦s ír♦s ♦rrs♣♦♥♥ ♥str♦s á♦s

80 100 120 140 1600

10

20

30

40

50

60

T [K]

∆γ

Tt

Tc

Ar

r s♦♥s ♣♦r♥ts ♥tr ♦s ♦rs ♦s ② ♦s t♦s ①♣r♠♥ts ♦rrs♣♦♥♥ts t♥só♥s♣r ♥trs íq♦♣♦r r s ♦♠♥s ③s ♦rrs♣♦♥♥ só♥ q ♣r♦♥♥t ♦♠♣rr ♦s t♦s ①♣r♠♥ts ♦♥ ♦s ♦t♥♦s ♠♥t s♠♦♥s ♥á♠ ♦r ♣r r♦s ♦rt rc = σff ❬❪ s ♦♠♥s rs ♦s á♦s r♦s ♠s♠♦ ♦r♠s♠♦ ♣r♦ ♦♥ rc = σff❬❪ s ♦♠♥s r♦s ♦rrs♣♦♥♥ ♦♠♣ró♥ ♦rs ①♣r♠♥ts ♦♥ ♥str♦s á♦s

160 180 200 220 240 260 280 3000

2

4

6

8

10

12

14

16

T [K]

γ lv/k

B [1

02 K /n

m2 ] Xe

Tt T

c

r ♥só♥ s♣r ❳ ♦♠♦ ♥ó♥ t♠♣rtr ♦s trá♥♦s r♥♦s ♦rrs♣♦♥♥ t♦s ①♣r♠♥ts ❬❪ ♦s r♦s r♥♦s ♦rrs♣♦♥♥ t♠é♥ t♦s ①♣r♠♥ts ♦s trés ♣r♦♠♥t♦s♠ás s♦st♦s ♦♠♦ ♦ ♦♥stt②♥ s té♥s s♣tr♦s♦♣ ó♣t ♠③ ❬❪ s rs r②s ② ♦♥r②s ② ♣♥t♦s s♦♥ rst♦s ♦t♥♦s s ♥ t♦rí ①♣♦♥♥ts rít♦s ❬ ❪ ♦s ♠♥ts ② strs♦rrs♣♦♥♥ s♠♦♥s ♦♥ s t③ó ♥ ♣♦t♥ ♣rs ♦♥ r♥ts ♦rs r♦ ♦rt❬ ❪ ♦s ír♦s ♦rrs♣♦♥♥ ♥str♦s á♦s

♥só♥ s♣r ♥trs íq♦♣♦r ❳♥ó♥

♠str t♥só♥ s♣r ❳ ♥ ♥ó♥ t♠♣rtr í s ♦r♦♥ ♦s ♦rs ♦t♥♦s

s♥♦ ♥♦♥ ♥str♦ srr♦♦ ♥ ♣ít♦ ♥t♦ ♦rs ①♣r♠♥ts ② ♦s ♦s trés

té♥s s♣tr♦s♦♣ ó♣t ♠③ ❬❪

r♥ t♠é♥ ♦rs ♦t♥♦s ♠♥t ♥ t♦rí ①♣♦♥♥ts rít♦s ② ♠♥t s♠♦♥s

♥á♠ ♦r ♥ rá♦ s rs r②s ② ♦♥ r②s ② ♣♥t♦s s♦♥ rst♦s ♦t♥♦s s ♥

t♦rí ①♣♦♥♥ts rít♦s ❬ ❪ r♦s st♦s t♦s s ♦t♥♥ ♦s ♦rs γ0lv = 3950 ♥♠2 ②

p = ♣r r ♦♥ r②s ② γ0lv = 4460 ♥♠2 ② p = ♣r rst♥t

♥ s ♣ ♣rr ♠♦♥t♦ s s♦♥s ♣♦r♥ts ♥tr ♦s r♥ts ♦rs ♥ st

s♦ s ♦♠♣r♥ ♦s ♦rs ①♣r♠♥ts ♦♥ ♦s q s ♥ ♠♥t ♥á♠ ♦r ② Pr

s♦ s t③♥ r♦s ♦rt ② σff

150 200 250 300−20

−10

0

10

20

30

T [K]

∆γ

XeT

tT

c

r s♦♥s ♣♦r♥ts ♥ ♥ó♥ t♠♣rtr ♥tr ♦s ♦rs ①♣r♠♥ts ② ♦s ♦s ♠♥t♥á♠ ♠♦r ② ♥ r s♦♥s ♦s ♦rs ♦t♥♦s ♥ s♠♦♥s q ♥ s♦ ♥á♠ ♦r ♣r ♥ r♦ ♦rt σff q ♣r s ♦♠♥s rs ♣r♦ ♦♥ ♥ r♦ ♦rt σff ♥ r♦♦ s♦♥s ♦s ♦rs q ♣r♥ ♦s á♦s r③♦s ♦♥ ♥str♦ ♥♦♥

♣ít♦

s♦ró♥ s♦r sstrt♦s ♣♥♦s

♣rs♥t♥ ② ♥③♥ ♦s rst♦s ♦t♥♦s st♦ s♦ró♥ r ② ❳ s♦r sstrt♦s ♣♥♦s

♦♥♦r♠♦s ♣♦r r♥ts ♠♥t♦s ♥♦s sr s ② ♥ ♥♦ st♦s sstr

t♦s s ♠♥st♥ ♥ó♠♥♦s ♦♥ rtrísts ♣r♦♣s ♦ ás♠♥t ♥t♥s ♣♦t♥ ♦♥

q ♥trtú♥ ♦♥ ♦ q s s♦r s♦r ♦s ♦s ♣♦t♥s t③♦s t♥t♦ ♣♦t♥ ♣rs

♦♠♦ ♦rrs♣♦♥♥t ♥tró♥ sstrt♦♦ s♦♥ ♣♦t♥s rsts ♥ s♥t♦ q ♣r♠t♥ ♣rr

♠♥t ♦s ♦rs ①♣r♠♥ts q t♦♠♥ rts ♠♥ts r♥ts ts ♦♠♦ ♣rsó♥ ♦①s

t♥ ♥s ② ♣♦t♥ qí♠♦ ♦rrs♣♦♥♥ts ♥ ♦ ♥ qr♦ tr♠♦♥á♠♦ ② ♦trs ♣r♦♣s

r♦♥s ♦♥ ♠♦♦ ② ♣r♠♦♦ s s♣rs s♦r s q s s♦r♥

í♥tss ♥tr♦t♦r

♥ ♦s út♠♦s ñ♦s ♦ r♥♦ ♥trés t♥t♦ ♣♦r ♣rt ís ♦♠♦ qí♠ ♣♦r st♦

s ♣r♦♣s tr♠♦♥á♠s ② strtr ♠r♦só♣ ♦s s♦r♦s s♦r sstrt♦s só♦s

s ♣♥t♦ st tór♦ ♦s ♣r♦♠s ♥trs s s♦♥ ♥trí♥s♠♥t t♦s♦s ② q ♦♥t♥♥

♥sr♠♥t r♦♥s s♣s ♥ ♣r ♥s ② ♥♦♥s ♦rró♥ ♥sótr♦♣s

♥ st ss s tr♥s♦♥s s ♥ ♥trs s ② ♥ ♦s s♦r♦s s♦r ♣rs ♦ ♦♥♥♦s

♥ r♥rs s ♠♦s ♥st♦ t③♥♦ ♦r♠s♠♦ ♥♦♥ ♥s ♥tr♦♦ ♥ ♦s

♣ít♦s ♥tr♦rs

♦s sstrt♦s só♦s s♦r ♦s s s s♦r ♦ s♦♥ ♠♦♦s ♣♦r ♣rs s♣r♦sts strtr

q s♦♦ ♣♦rt♥ ♣r♦♠ ♥ tr♠♥♦ ♣♦t♥ ♥♦ ést ♥ ♥ó♥ st♥ ♣r♣♥r z

♥tr át♦♠♦ ♦ ♠♦é ♦ ② ♣♥♦ sstrt♦

♥ s ♦s út♠s és srr♦♦ ♥♦♥s s♦ ♥ ♠♣♦ ♠② t♦ ♥tr♦ ís

♠tr ♦♥♥s ♦tr♦♥ ss♦s r♥♠♥t♦s s♦r s ♣r♦①♠♦♥s ♠ás s♠♣s ♥♦♥

♥s ♥♦♥s ♥s ♥ s srs rs q ♣r♠tr♦♥ ♠♣♦rt♥ts ♥s ♥

♦♠♣r♥só♥ ② sr♣ó♥ ♦s ♣r♦s♦s s♦ró♥ ♦s s♦r só♦s st♦s ♦r♠ó♥ ♣s

♣r♦s♦s s♦ró♥ ♥ ♣♦r♦s str♦s ② ♥ ♥♥♦strtrs t

Pr ♦♥t♥r ♣♦♠♦s s♥tt③r s st♦♥s s♥s q s ♥ rt♦ ♥ ♣ít♦ ② ♠♥♦♥r ♥s

♦trs

r♦rr ♥t♦♥s q sr♣ó♥ ♦♠♣♦rt♠♥t♦ ♥ ♦ s♦r♦ ♥ ♥ ♣r q s r

s ♦♥s ♣r♦ r♥t♥❩r♥ ❩ ♣rs♥t ♥s ts r♦♥s ♦♥ á♦

♥♦ s ♣ st♦♥s ♥ ♦♥ s r③s ♥tr ♠♦és ♦ ♥②♥ t♥t♦ tér♠♥♦s trt♦s

♦♠♦ r♣s♦s ♦ ♥ s♦s ♦♥ ♣r ♥s s ♠② ♥♦♠♦é♥♦ ♦ r♥♦ ♥ ♣r ♦ ♦♥

♦ st ♠② ♦♥♥♦ ♦s ♥tr♦ ♣♦r♦s ♦ r♥rs strs ❬ ❪ ♥ st út♠ rr♥

s ♦♠♣r♥ rst♦s ♣r♦♥♥ts ♣r st t♣♦ ♣r♦①♠♦♥s ♦♥ ♦s ♦t♥♦s ♠♥t ♦s ♠ét♦♦s

♦♥t r♦

♥ ♥t♦ t♦rís ♦♠♦ ♠♦s ♥ r r♦♥t♦ ♦ ♦♠♥t♦ ♥ ♣ít♦ ♣r r♦rr q

①st♥ ás♠♥t s s♥ts ♣r♦①♠♦♥s

♣r♦①♠ó♥ ♦ ♥s ♥ s ♦♠t♥ s ♦rr♦♥s ♠♦rs ♦rt♦ r♥♦

s ♣r♦①♠♦♥s rátr ♥♦ ♦ ♦♠♦ ♣♦r ♠♣♦ ♥s s③ ♦ q ♦rrs♣♦♥

t♦rí s ♠s ♥♠♥ts

♥ ♣r♦①♠ó♥ srr♦ ♠♦s é ❬ ❪ s t♥ ♥ ♥t st t♣♦

♦rr♦♥s trr ♦♥ ♥ ♥s ♣r♦♠♦ ♣s ♦♥ ú♥ t♣♦ ♥ó♥ ♣r♦♣ tr ♦♥

ró♥ ♠♣qt♠♥t♦ r♦♥ ♦♥ t♠ñ♦ ♠♦r

♥ ♣r♦①♠ó♥ ♣r♦♣st ♣♦r ♦s♥ ♥ ❬❪ ♥r♥t r s♦♥ s ♠s ♥

♠♥ts ♦♠étrs s ♠♦és ♦ s r♦ s s♣r ② s ♦♠♥

♦s ♦r♠s♠♦s ② r♥ ♣r♠t♥ ♦rr ♣r♦♠s rt♠♥t ♥♦♠♦é♥♦s ♥tr♦

♥♦ s ♥tr♦♥s ♦rt♦ ♥ st ♠♥r ♦s sq♠s ♥♦ ♦s ♥ ♥t ♥tr ♦trs ♦ss

s ♦s♦♥s ♥ s á♠tr♦ ♠♦r ♣r ♥s q ♣r ♥ ♥ ♦ ♥♦ s

s♦r s♦r ♥ sstrt♦ só♦

♥ st♦s út♠♦s ñ♦s s ♥ ♣r♦♣st♦ ♦tr♦s r♥♠♥t♦s sts ♦r♠♦♥s ♦♥ ♦s q s ♦r♦ ♦rr

♦♥ st♥t é①t♦ ♣r♦♠s ♦①st♥ ss íq♦rst♥s ♥ ♦s srs rs ♦ ♥r♦s

rí♦s ♥ ♥ ♠♥só♥ s♦r♦s s♦r rt♦s sstrt♦s ♥ rr♥ ❬❪ r③♦♥ r③ ♥ r r♦♥t♦

t♠

P♦r ♦tr♦ ♦ ♥ tr♦ ♦t ❱s♥②♦ ② ♠r ❬❪ s ♣♦♥♥ r s♠ts ② r♥s

♥ s ♣r♦♥s q rr♦♥ s ♣r♦①♠♦♥s ② ♥♦ s♦♥ ♣s st♦ ♥ s♦ró♥

♦s s♦r s♣rs sós ♣♥s ♥ q ♦rrs♣♦♥ ❬❪ s ♥

♦♠♣ró♥ ♥tr ♦s rst♦s ♦t♥♦s ♠♥t ② ♥ s♠ó♥ ♦♥t r♦ ❬❪ ♣r

s♦ ♥ ♦ srs rs ♦♠♦ s ♣ ♣rr ♥ st r s r♥s ♦s r♦♥s ♦♥

♦r♠ ♣r s♦♥ ♠í♥♠s ♠ás ♦s t♦rs ❬❪ ♣r♦♥ rst♦s s♦ró♥ 2 s♦r

rt♦ sst♠ ②♦ ♣♦t♥ s♦ró♥ s ♠ás ♥t♥s♦ q ♦s t③♦s ♥ ♥str♦s á♦s ♥

t rr♥ s ♠str q ①st ♥ ♥ r♦ ♥tr ♦s ♣rs ♦t♥♦s ♣r s sst♠ ♦♥ ♦s rs♦s

♦r♠s♠♦s

♦s♦tr♦s trr♠♦s ♠②♦r♠♥t ♥tr♦ ♦r♠s♠♦ ♦♥ t♥t♦ ♦♠♣♦♥♥t ♣rsó♥ ♦♠♦

♣♦t♥ qí♠♦ s♦♦ srs rs s ♦rrs♣♦♥♥ ♣r♦♣st Prs❨ ❬❪

♦♥s rr♥

0 0.5 1 1.5 2 2.5 30

5

10

15

z*

ρ*

−MC (Diamantes)−SDA (línea sólida)−FMT (línea de puntos)

r ♥s ♥ ♥ó♥ st♥ ♥ ♣r só ♣r ♥ ♦ srs rs ♦♥ ηbulk = ② T = 150 st rá♦ ♦rrs♣♦♥ ❬❪ ♣♦t♥ q r ♣r s♦r ♦ s ♥ ♦s♠♥ts ♦rrs♣♦♥♥ s♠♦♥s ♦♥t r♦ s ♦ ♣♦r ♦♦♦s ② sr ❬❪ s í♥ssós s♦♥ á♦s ♦s t③♥♦ ♦r♠s♠♦ ♥♦ ♦ ♥♦♥ ♥s ♥ ♣r♦①♠ó♥ í♥ ♣♥t♦s ♦rrs♣♦♥ á♦s r③♦s ♦♥ ♣r♦①♠ó♥

Pr ♦s ♣r♦♠s s♦ró♥ s♦r ♥ ♣r ♥r♦s ♥ st ss s♣♦♥r♠♦s q ♦s ♣rs ♥s

s♦♦ ♣♥♥ ♦♦r♥ z ♣r♣♥r sstrt♦ ❬ρ(r) ≡ ρ(z)❪ ♦ s q s♦♥ ♥r♥ts ♥ ♣♥♦ xy

♣r♦ sstrt♦ s♦r♠♦s s ♦♥s ❬♣ít♦ s ② ♥tr♦ ♥

tr♠♥♦ ♥♦ ♦♥ s ♦♥♦♥s ♦♥t♦r♥♦ ρ(z > L) = ρ(z → ∞) = ρB ♦♥ ρB s ♥s

qr♦ q ♣♦s ♦ ♣♦r r ♥tró♥ ♦♥ ♣r ♥ ①♣rs♦♥ Q ♣♦r

s ♥tr♦rá ♣♦t♥ Usf (z) ♦rrs♣♦♥♥t

sts ♦♥s s rs♥ ♥ ♦♥t①t♦ ♦r♠s♠♦ ♥ó♥♦ sí ♦♠♦ r♥ ♥ó♥♦ s s♦♦♥s

q s♦♥ sts ♦ ♠tsts ♣r s s dµdn > 0 q s ♦t♥♥ s♥♦ st♦s ♦s ♦r♠s♠♦s s♦♥ é♥ts

s♦ ♥ s♦s ①♣♦♥s ♥ ❬❪ ♣ ♥♦♥trrs ♥ ♥áss tór♦ r♦♥♦ ♦♥ s ♣r♦♥s

q s s♣r♥♥ ♠♦s ♦r♠s♠♦s í s ♠str q s♦♦ ♣r sst♠s ♠② ♦♥♥♦s ♦♥ ♥ ♣qñ♦

♥ú♠r♦ ♣rtís ♣♥ ①str r♥s s s tr ♦♥ ♥♦ ♦tr♦ sq♠

r r♥♦ st ♦ ♠tst r♥ s ♣♥t♦ st s s♦♦♥s ♦t♥s

r s♠♣♠♥t ♥ q ♦r♠s♠♦ r♥ ♥ó♥♦ ♥♦ ♣r♠t ♦t♥r s♦♦♥s ♥sts P♦r ♦tr ♣rt

s ♣♥t♦ st á♦ ♦st♦ ♦♠♣t♦♥ s ♠♥♦r ♥♦ s tr ♦♥ ♦r♠s♠♦ ♥ó♥♦

♦ q ♥ ♣s♦ tró♥ s♦ó♥ ♦t♥ ♥tr♦ st ♣r♦①♠ó♥ stá ♥♦r♠③ ②

q ♥ú♠r♦ ♣rtís ♣r♠♥ ♦♥st♥t

♥ts ♣rs♥tr ♥str♦s rst♦s ♦rrs♣♦♥♥ts s♦ró♥ r ② ❳ s♦r r♥ts sstrt♦s

srr♠♦s ♦s ♣♦t♥s sstrt♦♦ ♠ás t③♦s ② rs♠r♠♦s ♦s ♦♥♣t♦s ás♦s r♦♥♦s ♦♥

♠♦♦ s♣rs

P♦t♥s s♦ró♥

♥ trtr ①st♥ r♦s ♣♦t♥s ♣r♦♣st♦s ♣r srr ♥tró♥ ♥ át♦♠♦ ♦ ♠♦é

♦ ♦♥ ♥ sstrt♦ ♦s ♠ás s♠♣s s♦♥ ② q s ♦t♥♥ s♣♦♥♥♦ q ♥ át♦♠♦ ♦ ♠♦é

♦ ♥trtú ♦♥ ♥ át♦♠♦ ♦ ♠♦é sstrt♦ ♠♥t ♥ ♣♦t♥ sótr♦♣♦

V paressf (r) = 4 εsf

[

(σsfr

)12

−(σsfr

)6]

.

qí sí♥ s rr♥ sstrt♦ ② f ♦ εsf r♣rs♥t ♠ó♦ ♦r ♠á①♠♦

♥tró♥ ♥tr ♥ át♦♠♦ ♦ ♦♥ ♦tr♦ sstrt♦ ② σsf s r♥♦ s ♥tró♥

♥t♦ εsf ♦♠♦ σsf r♥ ♥♦♠r ♣rá♠tr♦s r ② ♥ ♣r♠r ♣r♦①♠ó♥ s ♣♥ r

s♥♦ s rs ♦r♥t③rt♦t

εsf = (εff εss)1/2

,

σsf =σff + σss

2.

t♦ t♦t s tr♠♥ ♥tr♥♦ s♦r t♦t ♦s át♦♠♦s ♦ ♠♦és ♣rtís sstrt♦

♠é♥ ② ♣♦t♥s ♠ás ♦r♦s ♣♦r ♠♣♦ ♦s srr♦♦s ♣rtr ♣r♠r♦s ♣r♥♣♦s ♦

r♦ ss sr♠♦s st♥t♦s t♣♦s ♣♦t♥ ♣♥♥♦ sstrt♦ ♦♥t♥ó♥ ♣r♦r♠♦s

srr♦s ♦♥ ú♥ t

P♦t♥

st ♣♦t♥ s ♠ás s♠♣ r ♣♦r ♠♣♦ ❬❪ ② s ♦t♥ s♣♦♥♥♦ q sstrt♦ s ♥ ♦♥t♥♦

tr♠♥s♦♥ ♦♥ ♥ stró♥ ♥♦r♠ át♦♠♦s ♦ ♠♦és ② ♥s s ρs st♥

♣r♣♥r ♥ ♣rtí ♣♥♦ sstrt♦ s z ♥trr ♣♦t♥ V paressf (r) ♦ sts ♦♥♦♥s

s ♦t♥

Usf (z) =

2π3 εeff

[

215

(σsf

z

)9 −(σsf

z

)3]

si z > 0

0 si z ≤ 0,

♦♥

εeff = εsfρsσ3sf .

st ♣♦t♥ ♦ sr♠♦s ♣r str s♦ró♥ r s♦r ♥ sstrt♦ 2 ♦♥ s ♥ sr♠♦s ♦s

♦rs ♦s ♣rá♠tr♦s sr♦s ♣♦r ♥r ② ♠ ❬ ❪

♥ trtr ♣♦t♥ ♦ ♣♦r t♠é♥ ♣r srt♦ ♣r z > 0 ♥ ♠♥r tr♥t

❬❪

Usf (z) =4C3

3

27W 2sf

(

1

z

)9

− C3

(

1

z

)3

,

♦♥ Wsf s ♣r♦♥ ♣♦③♦ ♣♦t♥ ② C3 s ♠rs ♣rá♠tr♦ ♥ r ❲s ♠♦s

♠♥♦♥r q ❩r♠ ② ♦♥ ❬❪ r♦♥ ♦rs Wsf ② C3 ♠♦r♥♦ s♠♣ ♣r♦①♠ó♥ ♦r♥t③

rt♦t

P♦t♥

♥ r ♦♥srr q sstrt♦ s ♥ ♦♥t♥♦ ♥ s ♣ s♣♦♥r q stá ♦♠♣st♦ ♣♦r ♥ ssó♥

♣♥♦s ♣r♦s ♥♦r♠s t ❬❪ st♥ ♣r♣♥r ♥ ♣rtí ♦ ♥ ♣♥♦

sstrt♦ s z ♣r♦s♦ ♥tró♥ V paressf (r) s♦r s ♣♥♦

Usf (z) = 4π εsf ρs σ2sf

[

1

5

(σsfz

)10

− 1

2

(σsfz

)4]

,

♦♥ ρs s ♥s s♣r ♣rtís sstrt♦ st ♣♦t♥ s ♦r♠

sstrt♦ ♥♦ s s♦♠♥t ♥ ♠♦♥♦♣ só s♥♦ q stá ♦♥stt♦ ♣♦r r♦s ♣♥♦s ② q s♠r s♦r

t♦♦s ♦s ♦t♥é♥♦s

Usf (z) = 4π εsf ρs σ2sf

np∑

j=0

[

1

5

(

σsfz + jd

)10

− 1

2

(

σsfz + jd

)4]

,

♦♥ d s st♥ ♥tr ♦s ♣♥♦s sstrt♦ ② np ♥t ♦s ♣♥♦s st ♣♦t♥ s ♦

♥♦♠♥ ♣♦r s♠♠ ♥ ❬❪ ♣♦♠♦s r st ♣♦t♥ srt♦ s♥t

♠♥r

Usf (z) = ε1s

3∑

j=0

[

2

3

(

σsfz + jd

)10

− 5

3

(

σsfz + jd

)4]

,

s♥♦ ♣r♦♥ st ♣♦t♥ Wsf/kB ≃ ε1s/kB

P♦t♥ ❩

♦♥ ♣♦str♦r tr♦ ❩r♠ ② ♦♥ ❬❪ r♦s t♦rs ♠♦rr♦♥ trt♠♥t♦ t♦

♥ tró♥ sstrt♦ ♦ r♦ st í♥ ③♠s② ♦ ② ❩r♠ ❬❪ srr♦r♦♥ s ♣r♠r♦s

♣r♥♣♦s ♥ ♣♦t♥ ♥tró♥ sstrt♦♦ ② ①♣rsó♥ s

Usf (z) = V0 (1 + αz) e−αz − f [(z − zvdW )β(z − zvdW )]CvdW

(z − zvdW )3,

♦♥

f(x) = 1− e−x(1 + x+x2

2) ,

r♦ ❱♦rs ♦s ♣rá♠tr♦s ♦rrs♣♦♥♥ts ♣♦t♥ ❩ ♣r r ② ❳ ♥ ♥tró♥ ♦♥ sstrt♦s ♥♦st♦♠♦s rt♠♥t r♦ ❬❪ V0❱ α−1

0 0 = CvdW ❱30 ② zvdW 0

V0 α CvdW zvdW

s

r V0 α CvdW zvdW

s

❳ V0 α CvdW zvdW

s

β(z) =α2z

(1 + αz).

♦s ♦rs ♦s ♣rá♠tr♦s V0 α CvdW ② zvdW ♣♥♥ t♥t♦ ♦ ♦♠♦ sstrt♦ ♣r♠r tér♠♥♦

st ①♣rsó♥ st r♦♥♦ ♦♥ tér♠♥♦ r♣s♦ rtr♦ ② r♠♣③ ♦♥tró♥ ♣r♦♣♦r♦♥

1/z9 s♥♦ tér♠♥♦ ♦♥stt② ♦♠♣♦♥♥t trt ♥ r ❲s q ♣r z >> zvdW

t♥ −C3/z3 ♥ó♥ ♠♦rt♠♥t♦ f(x) ♥t s♦♣♠♥t♦ s ♥♦♥s

♦♥ ♦rrs♣♦♥♥ts ♦s át♦♠♦s sstrt♦ ♥ó♥ β(z) ♦rr ♣r♦♠ r♥ q st

♣♦t♥ ♣♦s ♥ z = zvdW ② q ♥♦ s ♦rrs♣♦♥ ♦♥ ♥♥ú♥ ♦ ís♦ ❬❪

st ♣♦t♥ ❩ ♦ ♠♦s t③r ♣r ♠②♦rí ♦s sst♠s q ♥str♠♦s ♥ r♦ s

♠str♥ ♦s ♦rs q t♦♠♥ ♦s st♥t♦s ♣rá♠tr♦s ♦rrs♣♦♥♥ts ♦s ♠♥t♦s ♥③♦s ♥ st tr♦

♦♦ ② í♥ ♣r♠♦♦

♥③♠♦s ♥s st♦♥s rr♥ts s tr♥s♦♥s ♠♦♦ ② ♣r♠♦♦ ♣r sst♠s íq♦♣♦r

s♦r♦s s♦r s♣rs

♦♥srr s♦ró♥ ♥ sstrt♦s ♣♥♦s ♣r♥ ♥ó♠♥♦s ts ♦♠♦ ♠♦♦ tt♥ ② ♣r♠♦♦ ♣r

tt♥ ♣r str st♦s ♥ó♠♥♦s s ♥sr♦ ♦♥srr ♥ ♣r♦♠ s♦ró♥ ♥ ♠ás ♥ ♠♥só♥

s tr♥s♦♥s tt♥ r♦♥ sts ♣♦r ♣r♠r ③ ♥ ♥ ♦s tr♦s ♥♣♥♥ts ♥ ❬❪

② ♥r ② ♠ ❬❪ ♥ st só♥ ♠♦s ♦♠♥tr ♥♦s ♦s s♣t♦s ♠ás r♥ts ♥ trtr

s ♣♥ ♥♦♥trr ts rs♦♥s tórs s♦r t♠ ♦♠♦ ♣♦r ♠♣♦ ♥♥s ❬❪

s ♣r♦♣s ♠♦♦ ♥ s♣r ♣♥♥ ás♠♥t t♠♣rtr ② ♦♥t ♥tr

♠á①♠ ♥t♥s ♥tró♥ q ①st ♥tr sstrt♦ ② ♦ Wsf ② ♠á①♠ ♥t♥s

r③s ♥tr ♠♦és ♠s♠♦ ♦ εff

W ∗

sf =Wsf

εff.

①st♥ ♠tt sst♠s q ①♥ tr♥s♦♥s ♠♦♦ ♦s ♠♦s ②r ♥♦ ♠② s♥♦ ♣r

r s rtrísts ♥rs trt ♥ sst♠ ♦♥stt♦ ♣♦r ♥ ♦ ♥ ♦♥tt♦ ♦♥ ♥ s♣r

❯♥ sq♠ st stó♥ stá r♣rs♥t♦ ♥ ♥ st s♦ r♣rs♥t s íq ② ❱

s ♣♦r

í s r♦♥ s t♥s♦♥s s♣rs sstrt♦íq♦ γsl sstrt♦♣♦r γsv ② íq♦♣♦r γlv ♥

qr♦ ♦rrr q

γsv = γsl + γlv cosθ ,

♦♥ θ s ♦ ♥♦♠♥ s♠♥t á♥♦ ♦♥tt♦ st ①♣rsó♥ s ♦♥♦ ♦♥ ♥♦♠r ó♥

❨♦♥ ② tr♠♥ st♦ ♥ sst♠ ♥♦ θ = 0 s r ♥ ♥ stó♥ ♠♦♦ ♦♠♣t♦ s

t♥ q

γsv = γsl + γlv

② stó♥ ♥rét♠♥t ♠ás ♦r s ♥tr♣♦♥r ♥ ♣ ♠r♦só♣ íq♦ ♥tr sstrt♦ ②

s ♣♦r ♥t♦♥s q ①st ♠♦♦ t♦t ♦ ♦♠♣t♦ ♦ s♠♣♠♥t ♠♦♦ ♥ ♦ ♦♣st♦ s

t♥ θ = π ♦♥ó♥ ♥♦ ♠♦♦ s ♥ s t♠é♥ ♦♠ú♥ rrrs é ♦♠♦ tt♥ ♦♠♣t♦ ♣♦r s

♣♦r ② ♣♦r ♦ t♥t♦

γsl = γsv + γlv

♥ st s♦ ♥ ♣ ♠r♦só♣ ♣♦r r ♥tr sstrt♦ ② íq♦ rst♦ st♦s ♥tr♠♦s

♦♥ 0 < θ < π ♦rrs♣♦♥♥ st♦♥s ♠♦♦ ♣r ♥ st♦s st♦s ♣ s♦r s♦r

s♣r t♥ ♥ s♣s♦r ♠r♦só♣♦ s ♣ srr ♦♠♦

|γsv − γsl| ≤ γlv

♠♦♦ ♦♠♣t♦ s ♥③ ♥♦ s r ①♣rs st ♠♦♦ ♥♦s ♥ s

♦♥♦♥s ♥srs ♣r q s é ♥ tr♥só♥ ♠♦♦ s r ♣s♦ ♥ stó♥ ♠♦♦ ♣r

♦♠♣t♦ ♣♦♥♠♦s q ♥♦s st♠♦s s♦r í♥ ♦①st♥ íq♦♣♦r ② r♠♦s t♠♣rtr

rá♥♦♥♦s ♣♥t♦ rít♦ st♦ q γlv s ③ ♠ás ♣qñ ② t♥ r♦ ♦♠♦

γlv ∼(

Tc − T

T

)

−α

♦♥ ①♣rsó♥ ♣ré♥tss ♦rrs♣♦♥ t♠♣rtr r ② α ①♣♦♥♥t rít♦

♥ r♠♥tó q γsv − γsl t♥í r♦ ♦♠♦ r♥ ♥ ♥ss s ss íq ② ♣♦r

γsv − γsl = [ρv − ρl]

(

Tc − T

T

)β

r ♥ r s ♦sr♥ s trs st♦♥s q s ♣♥ ♠♥str ♥♦ ♥ íq♦ stá ♥ ♦♥tt♦ ♦♥ ♥s♣r s trs ❱ ② ♥t♥ s ss só ♣♦r ② íq rs♣t♠♥t ② θ r♣rs♥t á♥♦ ♦♥tt♦ ♥ ♦ ♣rt s♣r♦r ③qr íq♦ s ①♣♥ r♥♦ t♦ s♣r ♦r♠♥♦♥ ♣ s♣s♦r ♥♥♦ q ♣♥ ♥t íq♦ s♣♦♥ ♥ st s♦ s q íq♦♠♦ ♦♠♣t♠♥t s♣r ♥ ♦tr♦ ①tr♠♦ ♦ s♣r♦r r♦ s ♥tr♣♦♥ ♥ ♣ ♣♦r ♥tr s♣r ② íq♦ st ♠♥r íq♦ ♥♦ ♥tr ♥ ♦♥tt♦ ♦♥ só♦ q ♦♥stt② s♣r st ♦ s ♦ ♦♥♦ ♦♥ ♥♦♠r s♦ ♦♠♣t♦ r②♥ stó♥ ♥tr♠ sq♠t③ ♥ ♣rt ♥r♦r r ♥♦♠r ♠♦♦ ♣r ♥ s ♠s♠♦ ♦ s ♥ sq♠t③♦ s r③s q sr♥ ♥tr só♦ íq♦ ② ♣♦r γsl r♣rs♥t r③ q só♦ r s♦r íq♦ γsv s r③ q só♦ r s♦r ♣♦r ② γlv q íq♦ r s♦r ♣♦r

r ♠♦s ♥ ♣♦t♥ qí♠♦ t♦r r♠♥t♦ ② t♥só♥ s♣r ♦rrs♣♦♥♥ts tr♥s♦♥s ♠♦♦ ♣r♠r ② s♥♦ ♦r♥ ♦ ♠♦♦ ♦♥t♥♦ ♥ r♦ s♣r♦r ♣♦♠♦s ♦srr ♦♠♦ ♠♥sts rs ♣r ♥ tr♥só♥ s♥♦ ♦r♥ ② ♥ ♥r♦r rs♣t♦ ♠♦ ♣r ♥ tr♥só♥ ♠♦♦ ♣r♠r ♦r♥ r t①t♦

♦♥ ρv ② ρl s♦♥ s ♥ss ♣♦r ② íq♦ rs♣t♠♥t ② ♦♥ ♠ás β < α st ♠♥r

s♥t♠♥t r ♣♥t♦ rít♦ s ♠♣r q γsv −γsl = γlv ② ♣♦r ♦ t♥t♦ t♥ r ♥ tr♥só♥

♠♦♦ ♥q r③♦♥♠♥t♦ s ♥ ♣rt rró♥♦ ♣s r♥ ♥tr s t♥s♦♥s s♣rs t♥

r♦ ♦r♠ ♠ás ♦♠♣ ♦ q s♥t♠♥t r ♥ ♣♥t♦ rít♦ t♥ r ♥ tr♥só♥

♠♦♦ s ♥ ♦♥só♥ só

♥♦ r③ sstrt♦ s ♠♦r ♦♠♣r ♦♥ ♥t♥s ♥tró♥ s ♠♦és

♦ ①strá ♥ tr♥só♥ ♠♦♦ ♣r♠r ♦r♥ ♣r t♠♣rtrs ♥r♦rs rít

♥ ♥r s tr♥s♦♥s ♠♦♦ s♦♥ ♣r♠r ♦r♥ ♣r♦ t♠é♥ s ♥ó♠♥♦s ♠♦♦ ♦♥t♥♦ ♦

tr♥só♥ ♠♦♦ s♥♦ ♦r♥

❱r♥♦ ♣♦t♥ ♥tró♥ ♥tr ♦ ② sstrt♦ s ♣♦s ♦t♥r ♦s ♦s ♦♠♣♦rt♠♥t♦s ♥

♠♦s s rtrísts ♠s tr♥s♦♥s ♦s rá♦s s♣r♦rs ♦rrs♣♦♥♥ ♥ ♣♦tét

stó♥ ♠♦♦ ♦♥t♥♦ ② ♦s ♥r♦rs ♥ tr♥só♥ ♠♦♦ ♣r♠r ♦r♥

♥ ② ♠♦s r♠ ss ♥ ♣♥♦ ♣♦t♥ qí♠♦t♠♣rtr qí µ0 s ♣♦t♥

qí♠♦ ♦①st♥ íq♦♣♦r ♦rrs♣♦♥♥t ♥ íq♦ q ♥♦ st s♦♠t♦ ♥♥ú♥ ♣♦t♥

①tr♥♦ ♥ts r t♠♣rtr rít Tc s ♣r♦ tr♥só♥ ♠♦♦ Tw tr♥só♥ s

♣r♠r ♦r♥ ② s♦ ♥ í♥ ♣r♠♦♦ r ♦①st♥ q rr♥ t♥♥♠♥t ♥ Tw② ♥③ ♥ ♥ ♣♥t♦ rít♦ rátr ♠♥s♦♥ ❯♥ ♣rá♠tr♦ q ♣ rstr ♠② út ♣r str

sts tr♥s♦♥s s r♠♥t♦ Γℓ ♥ó♥ ♠ás ♣r♦♣ ♣♥ sst♠ ♥ ♥str♦ ♠♣♦

s q r s íq ② ♣♦♠♦s ♥r♦ ♦♠♦

Γℓ =

ˆ

[ρ(z)− ρv] dz

st ♥tr s ①t♥ s♦r t♦♦ s♠s♣♦ ♦♣♦ ♣♦r ♦ ♠♥tr ♥ú♠r♦ ♣rtís

t♠♣rtr ♠♦♦ Tw s ♣r♦ ♠②♦r st♦ ♥ ♦r r♠♥t♦ q tr♠♥ s♣s♦r

♣í s♦r ♣sá♥♦s ♥ ♣í ♣♦♦ s♣s♦r ♥ ♠②♦r s♣s♦r st s♦♥t♥

♥ r♠♥t♦ ♣rr r♥t t♦♦ ♥♦♠♥♦ r♠♥ ♣r♠♦♦ q s ①t♥ ♥tr Tw ②

t♠♣rtr rít ♣r♠♦♦ Tcpw P♦r ♥♠ st út♠ t♠♣rtr ♥ ♥r♠♥t♦ ♥ ♥ú♠r♦

♣rtís q r♠♥t♦ r③ ♦♥ ♦♥t♥ r s s ❬ ❪

sí ♥ s♦ ♥ tr♥só♥ ♣r♠r ♦r♥ s♦ró♥ ♣rs♥t ♥ s♦♥t♥ ♥♦ trs♠♦s

r♠♥ ♣r♠♦♦ ② r ♥ ♦①st♥ ♦♠♦ ♣ ♦srrs ♥ ♣♦♥♠♦s ♦r

q ér♠♦s ♣s str t♥só♥ s♣r ♥ ♥ó♥ s♣s♦r ♣ íq♦ s♦r

strt♠♥t st♦ ♥♦ s ♣♦s ♣s s ♥s ♦♥♦♥s tr♠♦♥á♠s ①st ♥ s♣s♦r q ♠♥♠③

t♥só♥ s♣r s♥♦ ♥sts rst♦ ♦s st♦s ♦ ♦st♥t s♣s♦r ♣ íq♦ s♦r

s ♥ r ♠② ♥t s♥♦ ♥ ♣rát t r ♥ st♦ st t♣♦ ♦♥ t♦rí ♥♦♥

♥s r ❬❪ ♥ ♠♦s ♦♠♣♦rt♠♥t♦ r♠♥t♦ ♥ s♦ ♠♦♦

♦♥t♥♦ sú♥ ♠♥t♠♦s ♣♦t♥ qí♠♦ s tr②t♦rs ♠rs ♣♦r s ♥ Pr T < Tw

í♥ ♦♥t♥ s♦ró♥ r ② s ♠♥t♥ ♥t ♥ ♦①st♥ íq♦♣♦r Pr T > Tw í♥

s♦♥t♥ ♣ s♦r r ♦r♠ ♦♥t♥ ② r ♥♦ s ♥③ ♦①st♥ st r♥

s ♦rít♠ ♣r ♣♦t♥s ♦rt♦ ♥ ♥♠♦s ♦r t♥só♥ s♣r ♥♦s st♠♦s s♦r

í♥ ♦①st♥ íq♦♣♦r ♥ s♦ ♠♦♦ ♦♥t♥♦ rí♠♦s ♦ ♣r♦ ♣r T < Tw

í♥ ♦♥t♥ ② ♥ ♠í♥♠♦ ♣r ♥ ♦r ♥t♦ Γℓ q s s♣③ r sú♥ ♠♥t♠♦s

t♠♣rtr st q ♥♠♥t r ♣r ♦rs T ≥ Tw í♥ s♦♥t♥ P♦r ♦♥trr♦ s

tr♥só♥ s ♣r♠r ♦r♥ t♥só♥ s♣r ♣rs♥t ♦s ♠í♥♠♦s óts q ♥ ①st♥ ♦s

♠í♥♠♦s s♦t♦s ♣r Tw ♥♦ ♣r ♦rs ♥t♦s Γℓ ② ♦tr♦ ♣r Γℓ → ∞

♣ rr ♥ ó♥ q ♣r♠t tr♠♥r ♦r Tw r ♣♦r ♠♣♦ ❬❪ ♣rt

s♥tót ♣♦t♥ s♦ró♥ tér♠♥♦ ♦ ♦♥ ♦♠♣♦♥♥t trt st ♣♦t♥ t♥ ♦r♠

♣♦t♥ t♣♦ ♥ r ❲s s r s

Usf (z) ∼ − C3

z3,

♠♥t r♠♥t♦s tr♠♦♥á♠♦s s

∆µpw(T ) = apw(T − Tw)3/2 .

♥ st ①♣rsó♥ ∆µpw s ♣♦t♥ qí♠♦ r♦ s r q ② rr♥ s ♦①st♥

íq♦♣♦r ♥ ♦ ♥ qr♦ tr♠♦♥á♠♦ s♥ ♥tr♦♥s ①tr♥s ♥t st ①♣rsó♥ s

♣ tr③r í♥ ♣r♠♦♦ q ♠♦r st ♦s ♦rs ∆µpw − T ② ♦t♥r ♦r Tw

♠ q ♠♥t ♥t♥s rt ♣♦t♥ trt♦ sstrt♦ Tw s r♥♦ ♣♥t♦

tr♣ ♣♥♦ ①str ♦ q s ♦♥♦ ♦♠♦ ♠♦♦ ♣♥t♦ tr♣ Tw = Tt

s tr♥s♦♥s ♣r♠r ♦r♥ ♥ s♦ ♦srs ♣r r♦s ♦s s♦r♦s ② ①st ♥ ♥♥t ♥t

♠♦♥s ①♣r♠♥ts r♦♥s ♦♥ st ♥ó♠♥♦ ❯♥ ♠♣♦ ♦ ♦♥stt② s♦ 4 s♦r♦

s♦r rt♦s sstrt♦s só♦s r ♣♦r ♠♣♦ ❬❪

P♦r ♦tr♦ ♦ s ♦srr♦♥ tr♥s♦♥s ♣r♠♦♦ ♣r ró♥♦ ♠♦r 2 ♥♦ s s♦rí s♦r

② s ❬ ❪

s♦ró♥ s♦r ② s st ♥tr ♦tr♦s ♣♦r ss t♥ ② ♥ ❬❪ Pr st♦s

t♦rs ♥♥tr♥ ♠♦♦ ♥♦♠♣t♦ ♣r t♠♣rtrs ♠♥♦rs Tc = ② ♠♦♦ ♦♠♣t♦ ♣r

t♠♣rtrs s♣r♦rs ♥ s♦ s ♥♦ s ♦t♦ ♦♥ó♥ ♠♦♦ ♣r ♥♥♥ t♠♣rtr

♥tr♦ ♥tr tr♣ ② rít

♠♦♦ ♣♥t♦ tr♣ stá ♥st♦ ①♣r♠♥t♠♥t ♥ ♦s s♦s s♦ró♥ r ② ❳ s♦r

❬ ❪

♦s st♦s ♠r♦só♣♦s s r③r♦♥ ♠♥t ② s♠♦♥s ♦♠♣t♦♥s q t③♥ ♦♥t r♦ ②

♥á♠ ♦r ♥ s s ❬ ❪ ♣♥ ♥♦♥trrs st♦s r♦♥♦s ♦♥ s ♣r♦♥s

q s ♦tr♦♥ ♠♥t st♦s ♠ét♦♦s á♦

sr♥♦ strtr ♦♥♥t♦ s♦tr♠s s♦ró♥ ♣♦♠♦s ♥rr ♥s st♦♥s r♦♥s

♦♥ ♦s ♥ó♠♥♦s ♠♦♦ ② ♣r♠♦♦

❯♥ tr♥só♥ ♣r♠♦♦ s ♠♥st ♥♦ strtr s♦tr♠s s t q s ♣s ♥ s♦tr♠

q ♣rs♥t ♥ ♦♦♣ ♥ r ❲s ♥ q ♥♦ ♦ ♣♦s ♥ st s♦ t♠♣rtr ♦rr st

tr♥só♥ ♦rrs♣♦♥ t♠♣rtr rít ♣r♠♦♦ Tcpw í♥ ♣r♠♦♦ s r q ♥

♦s ♣♥t♦s ♥ ♥ r♠ s ∆µpw−T q s ①t♥ s Tw st Tcpw ❯♥ ♣♥t♦ s♦r st r t♥

♦♠♦ ♦♦r♥s ♣♦t♥ qí♠♦ q ♥ ♦♥stró♥ ① ② t♠♣rtr s♦tr♠

♦rrs♣♦♥♥t st ♠♥r strtr ♦♥♥t♦ s♦tr♠s r ♣rs♥ ♥ tr♥só♥

♣r♠♦♦ ② ♣r♠t r Tw ② t♠é♥ Tcpw

♥♦ r③ sstrt♦ s ♠♦r ♦♠♣r ♦♥ ♥t♥s ♥tró♥ s ♠♦és

♦ ①strá ♥ tr♥só♥ ♠♦♦ tt♥ ♣r♠r ♦r♥ ♣r t♠♣rtrs ♥r♦rs rít

t♠♣rtr ♠♦♦ Tw ♥ ♣í ♦ s♦r♦ ♦①st ♦♥ ♥ ♣í rs ♠s♠♦

❯♥ tr♥só♥ ♣r♠♦♦ ♣rtt♥ st rtr③ ♣♦r s♥ s♦♥t♥ ♥ ♦r t♦r

r♠♥t♦ ❬❪ ♥ s♥t♦ q t♠♣rtr ♠♦♦ s ♣r♦ ♠②♦r st♦ ♥ ♦r t♦r

r♠♥t♦ ② st s♦♥t♥ s♣r t♠♣rtr rít ♣r♠♦♦ ❬❪ Pr t♠♣rtrs

♠②♦rs Tcpw s♣s♦r ♣í s♦r r ♦♥t♥♠♥t

♦♦ s♠♣ ♣r tr♥só♥ ♠♦♦

P♦r ♦tr♦ ♦ ♥tr♦ ♥♦♠♥♦ ♦♦ ♠♣ ❬❪ s ♣ str ♥ ró♥ ♥tr ♠♦♦

♥ s♣r só ② s t♥s♦♥s s♣rs ♥ s ♥trss só♦íq♦ ② íq♦♣♦r r♦ ♦♥

❬❪ ♦♥ó♥ ♠♦♦ ♣ sr srt ♦♠♦ ♥ ♦♥ó♥ s♦r ♥tr ♣♦t♥

s♦ró♥ Iu

Iu = −ˆ

∞

zmin

Usf (z)dz ≥2γlv(T )

[ρl(T )− ρv(T )]=Wγρ(T ) .

s♣♦♥ q ♦♥ó♥ ♠♦♦ r ♥ s ♣ s♦ ♦s s♠♣s s♦r♦s s♦r

sstrt♦s trt♦s s♠s② ♦ ② ❩r♠ r♦♥ Iu ♥tr♥♦ s ♣r♦♣♦ ♣♦t♥ s♦ró♥

❩ ♦♥t♥ó♥ tr♠♥r♦♥ Tw ♥♦ Wγρ(T ) ♦♥ ♦rs ①♣r♠♥ts γlv(T ) ρl(T ) ② ρv(T )

♥②r♦♥ ♦s rst♦s ♣r Tw ♥ r♦ ❬❪

♦♠♦ ♠♣♦ ♦s ♦rs Iu ♣r ♦s ♣r r♥ts sstrt♦s stá♥ st♦s ♥ r♦ ②

♠r♦s ♦♥ í♥s ♦r③♦♥ts ♥ ♥ st r t♠é♥ stá♥ ♥♦s ♦s ♦rs Wγρ(T )

r♦ ❱♦rs ♦s ♦♥ts Iu ♥ ♥s ❬ ♥♠❪ ♣r ♥♦s ♠ts ♥♦s ② ♣r ❬❪ st ♦♥ts ♥tr♥♦ ♣♦t♥ ❩

s

20 25 30 35 40 450

5

10

15

20

25

T [K]

Wγρ

e

Iu

[Knm

] Ne T

t

Tc

Mg

Li

Na

K

Cs, Rb

r ❱♦rs Wγρ Iu ♥ ♥ó♥ t♠♣rtr ♣r s♦r♦ s♦r rs♦s sstrt♦s rtr ③②s③ ②❯rrt ❬❪ sts rs s♦♥ ♥srs ♣r ♥③r s ♣r♦♣s ♠♦♦ ♥ ♦♥t①t♦ ♥♦♠♥♦♦♦ ♠♣ s í♥s ♦r③♦♥ts ♦rrs♣♦♥♥ ♥tr Iu ♦s ír♦s s♦♥ Wγρ s t♠♣rtrs tr♣② rít s ♥♥ ♦♥ Tt ② Tc rs♣t♠♥t

♥ s r s ♣ ♦rr♦♦rr q ♦♥ ♦s ♦rs Tw ♣♦s ♣♦r s♠s② ♦

② ❩r♠ ❬❪ ♣r st♥t ♥ Tw ♣r ② q s♦♥ ♦rs ♦s ♥ ③♦♥ ♥tr

♥tr♦ Tt ↔ Tc ♣r♦ ss rst♦s ♣r ② ♠str♥ sr♣♥s ♦♥ ♦s t♦s ①♣r♠♥ts

Pr s♦r♦ s♦r t♠♣rtr ♠♦♦ ♦t♥ ♦♥ Tw = 38 s s♥t♠♥t

♠ás q ①♣r♠♥t Tw =

sí ♦♠♦ stá ♦r♠♦ tér♠♥♦ Wγρ(T ) ♥r ♥ ♣r♦♠ ♥ s ❬ ❪ r♠♦s ♦♠♣♦rt♠♥t♦

s♥tót♦ Wγρ(T ) ♣r T r♥s Tc ♥♥♦ ♥ ♥t ①♣rsó♥ ♣r t♥só♥ s♣r íq♦♣♦r

♣♦r

γlv(T ) ≃ γ0lv (1− T/Tc)5/4

,

② ①♣rsó♥ ♥♠ ♣r r♥ ♥ss íq♦ ② ♣♦r ♥ ♦①st♥ ❬

❬❪❪

ρl(T )− ρv(T ) ≃2

7ρc (1− T/Tc)

1/3,

♦♥ ρc s ♥s ♦ Tc s ♣ r Wγρ(T ) ② rsrr ♦♥ó♥ ♠♦♦ ①♣rsá♥♦

♦♠♦

Iu = −ˆ

∞

zmin

Usf (z)dz ≥Wγρ(T ) ≃4γ0lv7ρc

(

1− T

Tc

)11/12

.

♦♠♦ Wγρ(T → Tc) → 0 ♣♦r ♠ás é q s tró♥ ♥ sstrt♦ s♦r ♥ ♦ ♥tr Iu srá

♣♦st ♥♥♦ q ♦ s♠♣r ♠♦r sstrt♦ ♣r ♥ T < Tc ♥ ♠r♦ st ♣ró♥ ♥♦ s

♦rrt ♣és s ♦sró q ♣r ♥tr♦♥s sstrt♦♦ és ② ♦s q ♥♦ ♠♦♥ sstrt♦ ♥ ú♥

t♠♣rtr rít st s s♦ ♥♦ s s♦r s♦r s ❬❪

tr ♥ st ♠♦♦ ♣r ♥ s♦ s♦r♦ s♦r q ♣rs♥t ♥ Tw r♥ Tt

t③rs ♣r♦①♠ó♥

−ˆ

∞

zmin

ρ(z)Usf (z)dz ≃ − ρl

ˆ

∞

zmin

Usf (z)dz ,

s ♣r ♥ ♣rt ♠♣♦rt♥t ♦♥tró♥ q ♦rrs♣♦♥ ♠á①♠♦ ♣r ♥s ♦ ♦rr

st♦ ♥ ♠í♥♠♦ ♣♦t♥ s♦ró♥ ♥só♥ st t♦ tr ♦♠♦ ♦♥s♥ ♥ s♥s♦ ♥

♦r ♦t♥♦ ♣r Tw

P♦r ♦ ①♣st♦ ♥ st só♥ ♣♦♠♦s r♠r q s s♦♦ ♥ ♣r♦①♠ó♥ r ♣r♦♠ ♠♦♦

st♦s ♣r♦♣♦s s♦r s♦ró♥

♥ st só♥ srr♠♦s s♦ró♥ r ② ❳ s♦r rs♦s sstrt♦s ♥ ♦s s♦s ② r ♣rs♥

tr♠♦s rst♦s ♦t♥♦s ♥ ♠r♦ ♥s♠ ♥ó♥♦ ♥ ♠♦ ♣r ❳ ♣rs♥tr♠♦s rst♦s

♣r♦st♦s ♣♦r ♠♦s ♥s♠s ♥ó♥♦ ② r♥ ♥ó♥♦

Pr♠íts♥♦s ♥t③r q ♥str ♣r♦♣st ♣r ♣r♠t ♣rr ♥ó♠♥♦s r♦♥♦s ♦♥ ♠♦♦

s♣rs ♥ t♦♦ r♥♦ t♠♣rtrs ♥tr tr♣ Tt ② rít Tc ♦ s♦r♦ ñ♥♦

s ③ q q ♥♦r ♦r♠s♠♦ r ② ♦s♥r ❬❪ ② ♦s ♣♦t♥s ♣rs q sr♥

♥♦tt♦ ② ♦♦r♦rs ❬ ❪ ♥ r♥♦ t♠♣rtrs r♥♦ Tt st ♥♦♥♥♥t ② s ♠♦stró

①♠♥r γlv(T ) ♣r ♥ ♣ít♦

s♦ró♥ ó♥ ♥s♠ ♥ó♥♦

♥ rá♦ ③qr s ♠str ♣♦t♥ ❩ ♦rrs♣♦♥♥t ♥tró♥ ♦♥

trs ♠ts ♥♦s ② ♥ rá♦ r st ♠s♠ r s ♥ r♦ ♥ ♥ s ♠ás

−0.1 0 0.1 0.2 0.3−100

−80

−60

−40

−20

0

z−zmin

[nm]

Usf/k

B [K

]

Rb

Na

Li

Mg

Ne

−0.1 0 0.1 0.2 0.3 0.4−60

−50

−40

−30

−20

−10

0

z−zmin

[nm]

Usf

[K]

Ne

Cs, Rb

K

Na

Li

r P♦t♥s ♥tró♥ ❩ ♥tr ② rs♦s sstrt♦s rtr ③②s③ ② ❯rrt ❬❪ ♥ rá♦ ③qr s ♠str♥ ♥♦s ♦s sstrt♦s t③♦s qí ♥ r r s ♦sr ♥ ♠♣ó♥ st r ♦♥ s ♥ r♦ ♦s ♣♦t♥s ♥tró♥ ♦♥ s ② ♥ í♥ r②s ② r②s ②♣♥t♦s ♥ ♣♦t♥ t♣♦ ♣r Wsf/kB = ② rs♣t♠♥t ♥ ♠s rs r♦ ssss s ♦♥ ♦s ♣♦t♥s ♥③♥ s ♦r ♠í♥♠♦

♦♥♥♥t s rs ♦rrs♣♦♥♥ts ♥tró♥ ♦♥ s ② óts q ♦s ♣♦t♥s ♣r

② s s♦♥ ♠② s♠rs ♥ st rá♦ t♠é♥ s rr♦♥ rs ♣♦t♥ ♣r sst♠

s ♦♥ ♣r ♦s ♥t♥ss Wsf/kB = 48 ② s♥♦ ♦s ♣rá♠tr♦s σsf = ♥♠ ②

d = ♥♠ t♦♠♦s ❬❪ óts q ♥ s♦ Wsf/kB = 50 ♣♦t♥ rst♥t s ♠②

s♠r ❩

s♦ró♥ s♦r s ②

s ♣ ♦♥r q ♦s sstrt♦s s ② tr♥ é♠♥t ♥ trtr

s r♣♦rt t♦s ①♣r♠♥ts q ♥♥ q s ♠♦♦ ♣♦r ❬❪ t♠♣rtrs r♥s

rít ② q s ♥♦ ♦ s s st♠♦♥s tórs s♦♥ ♦♥trt♦rs rst♦s ♦t♥♦s ♦♥ s♠♦♥s

♦♥t r♦ ❬❪ sr♥ q ♥♦ ♠♦ ♠♥trs q á♦s ♦♥ ❬❪ ♥♥ q s ♥ ♠r♦

♦s t♦rs st út♠♦ tr♦ ♥♦ ♣r♦♥ tr♠♥r ♥ ré♠♥ ♣r♠♦♦ ♦♠♦ sí ♦ ♣♠♦s r

♥♦s♦tr♦s

♥ s ró ♥ ♦♥♥t♦ s♦tr♠s s♦ró♥ s♦r strtr ♥ q

t♠♣rtr ♠♦♦ Tw st ♣♦r ♦ ♦s r sñ ♦♥ trá♥♦s ♦sr♠♦s sts

rs ♠ás t♠♥t ♠♦s q ①st ♥ st♦ ♥ t♦r r♠♥t♦ Γl ♥ s rs trá♥♦s ②

r♦s ♠á①♠♦ ♥ r trá♥♦s ② q st st♦ s♣r ♥③rs t♠♣rtr

♦rrs♣♦♥♥t s♦tr♠ sñ ♦♥ ír♦s st♦ ♥ q Tcpw st ♦♠♣r♥ ♥tr ②

í♥ ♣r♠♦♦ ♦♥str ♦♥ ♦s t♦s sts s♦tr♠s s ♠str ♥ st ♦♥

rr♦ ♥ ♦r Tw = st rst♦ stá ♥ ♥ r♦ ♦♥ ♦s ♦rs ①♣r♠♥ts t♦s

♥ s s ❬ ❪ ♦♥ s r♣♦rt q st tr♥só♥ ♦rr ♦s

−0.25 −0.2 −0.15 −0.1 −0.05 00

2

4

6

8

10

12

∆µ/kB [K]

Γ l

Ne/Rb

r s♦tr♠s s♦ró♥ s♦r rtr ③②s③ ② ❯rrt ❬❪ r sñ ♦♥ trá♥♦s ♦rrs♣♦♥ ♥ t♠♣rtr sñ ♦♥ r♦s s s♦tr♠ ② q ♣♦s ír♦s♦rrs♣♦♥ strtr ♦♥♥t♦ s♦tr♠s ♣♦♥ ♥ í♠t Tcpw st t♠♣rtr rístr ♥ ♥tr♦ t♠♣rtrs q ♥ ♥tr ♦s ②

35 37 39 41 43 45−1

−0.8

−0.6

−0.4

−0.2

0

T [K]

∆µpw

/kB [K

]

Li

Na

K

Rb

Tc

Ne

r í♥s ♣r♠♦♦ ♦rrs♣♦♥♥ts s♦r♦ s♦r r♥ts sstrt♦s rtr ③②s③ ② ❯rrt ❬❪Pr s ② ♥t♥s tró♥ s s é♥t ♦♠♦ ♣ ♦srrs ♥ ♦rrs♣♦♥♥t í♥ ♣r♠♦♦ s ♥st♥s

−0.25 −0.2 −0.15 −0.1 −0.05 00

5

10

15

20

25

30

∆µ/kB [K]

Γ l

Ne/K

r s♦tr♠ s♦ró♥ s♦r rtr ③②s③ ② ❯rrt ❬❪ r sñ ♦♥ strs ♦rrs♣♦♥ ♥ t♠♣rtr r sñ ♦♥ trá♥♦s s s♦tr♠ q q ♣♦s r♦ss s♦ t♠♣rtr ② ír♦s ♥ óts st♦ ♥ ♦r t♦r r♠♥t♦ ∆Γl = ♥ t♠♣rtr ② ∆Γl = 25 ♣r

Pr♠íts♥♦s ♦♠♥tr s s♠♦♥s ♥♠érs s ♦ ♣♦r rtr♦♦ ② ♦♦r♦rs ♥ ♠r♦

❬❪ s ♠s♠s r♦♥ r③s ♥tr♦♥♦ ♥ ♦rt ♥ ♣♦t♥ ♣rs ♠♥r t q

♣r ♥ st♥ ♥trtó♠ s♣r♦r ♦s 5σff st ♣♦t♥ s ♦♥sr ♥♦ ♥ ♥r♠♥t ♦s

♦rs r♣♦rt♦s ♣♦r s♠♦♥s s♦♥ t♦♠♦s ♦♠♦ rr♥ ♦ q ♠♥t st t♣♦

s♠♦♥s ♥♦ ♣ ①♣rs ♠♦♦ ♣♦r s ♥ ♥ ♠♣♦rt♥t rí q rr s

t♠ñ♦ ♥ t③ ♥ ♣♥♦ xy 10σff ×10σff s s♥t♠♥t r♥ ♣r srr

♦♥t ♦rr♦♥ sst♠ ♦♥ ♣rsó♥ rqr r Tc

P♦r ♦tr♦ ♦ ♥ s♦ s ② ♦s ♣♦t♥s ❩ s♦♥ ♣rát♠♥t é♥t♦s s r ♥tr♦ ♥str♦

sq♠ ♣r♦①♠♦♥s s t♥ s ♠s♠s ♣r♦♣s q P♦r ♦ t♥t♦ t♠é♥ ♠♦rí

s Pr ♦t♥r ♥♦♠♦♦ rí q tr tró♥ s st ♣♦s ② s♦ ♣r♦♣st ②

st ♣♦r ♥♦tt♦ ② ♦♦r♦rs ❬❪ r♣♦ ♦♥ ♥♦ ♦s ♦♦r♦rs s t♦r ♣♦t♥ ❩

♥ s ♠str♥ s♦tr♠s ♦rrs♣♦♥♥ts s♦ró♥ s♦r st sstrt♦ tr ♠ás

♥t♥s♠♥t q ♣♦r s r③ó♥ ♣r q ♣s♦ t♠♣rtrs s ♦t♥ ♠ás s♦tr♠s ♥tr Tw ②

Tc ♣ ♦srr q ①st ♥ st♦ ♠② ♣qñ♦ ♥ ♦r t♦r r♠♥t♦ ♣r♦①♠♠♥t

∆Γl = ♥ t♠♣rtr ♠♥trs q ♦s st st♦ ♥③ ♥ ♦r ♠②♦r ♣r♦①♠♠♥t

∆Γl = 25 st ♦♠♣♦rt♠♥t♦ t tr♠♥ó♥ t♥t♦ Tw ♦♠♦ Tcpw q s stú ♥tr ♦s ②

♦s

í♥ ♣r♠♦♦ ♣r sst♠ stá ♥ ♥ st ♦s t♦s ♦♥

Tw =

P♦r út♠♦ ♥ r♦ s ♥ ♠♦♦ rs♠♥ ♦ sñ♦ st qí ♦s rst♦s ♥str♦s

á♦s ② s ♦s ♦♠♣r ♦♥ ♦s q ♣r♥ ♥ ♦rí

r♦ Pr♦♣s ♠♦♦ ♦rrs♣♦♥♥ts s♦r♦ s♦r sstrt♦s ♣♥♦s ♦♥♦r♠♦s ♣♦r ♠ts ♥♦s② ♦♥ st♦s W ∗

sf

s t♠♣rtrs ♠♦♦ Tw ② rít ♣r♠♦♦ Tcpw ①♣rss ♥ ♥ s♦ Tcpw s♥ ♦t s♣r♦r ♣r st t♠♣rtr ♣ ♠♦r ♦ ♥♦ s♣r ♥ st s♦ ❲ ♥♦♥tt♥ ♥ q ♥♦ ♠♦ sstrt♦② r②♥ ♥ s♦ t♦r apw q ♦rrs♣♦♥ ó♥ st ❬ ❪P❲ rr♥ ♦s rst♦s ♦t♥♦s ♠♥t ♥str♦s á♦s ② ♦s rst♦s q s ♦t♥♥s♥♦ ♦tr♦s ♥♦♥s ♥tr ♣ré♥tss s ♥ rr♥ ♦♥ r♦♥ t♦♠♦s st♦s t♦s

Pr♦♣ s W ∗

sf Tw①♣r♠♥t ❬❪ ❲

Tw ❬❪ Tw ❬ ❪ ❲ ❲

Tw ❬❪ Tw ❬P❲❪

apw/kB [−1/2]❬P❲❪

Tcpw ❬❪ Tcpw ❬❪ Tcpw ❬P❲❪

0 1 2 3 4−1.75

−1.5

−1.25

−1

−0.75

−0.5

−0.25

0

Γl

∆µ/k

B [K

]

Ne/Li T=38 K

r s♦tr♠s s♦ró♥ ♦rrs♣♦♥♥ts s♦r♦ s♦r ♥ sstrt♦ rtr ③②s③ ② ❯rrt ❬❪♥ rá s ♦sr ♦♠♦ ♠ ♣♦t♥ qí♠♦ r♦ ♥ ♥ó♥ t♦r r♠♥t♦ ♣r sst♠ ♥ t♠♣rtr r só ♦♥ trá♥♦s ♥srt♦s ♦rrs♣♦♥ á♦s ♦♥s t③ó ♣♦t♥ ❩ ♠♥trs q s rs ♦♥ r♦s ② ír♦s r♦♥ ♦t♥s t③♥♦ ♣♦t♥ t♦♠♥♦ Wsf/kB = 48 ② rs♣t♠♥t ♥ s♦ s í♥s r②s ♥♥ s ♦♥str♦♥s ①

s♦ró♥ s♦r ②

♥ s♦ s♦r♦ ♥ ② ♥ ♣r♥ ♥s ♦♥tr♦♥s ♥ trtr á♦s r③♦s ♣♦r

♥♦tt♦ ② ♦♦ ♥ s rst♦s ❬❪ sr♥ ①st♥ ♠♦♦ ♣♦r ♦ t♠♣rtr

rít ♠♥trs q ♦♥ ② ♦♦r♦rs ❬❪ s♦r s s♠♦♥s ♥♠érs ♦♥ r③s ♣r

♣♦t♥ rr♥ ♦♥só♥ ♦♣st

♠♦s ♥③♦ s ♣r♦♣s ♠♦♦ ♦♠♣t♥♦ s s♦tr♠s s♦ró♥ s♦r ♠♦s sstrt♦s s♥♦

♦s ♦rrs♣♦♥♥ts ♣♦t♥s ❩ ♦s rst♦s ♥str♦s á♦s stá♥ r♦ ♦♥ s ♦♥s♦♥s

♥♦tt♦ ② ♦♦r♦rs Pr ♠♦s sstrt♦s ② ♠♦s ♥♦♥tr♦ ①st♥ tr♥s♦♥s

♠♦♦ ss ♣♦r ♥ ré♠♥ ♣r♠♦♦

Pr ♦♠♣tr st♦ t♠é♥ ♠♦s ♦♥ ♥str s♦tr♠s s♦ró♥ ♣r T = 38 s♥♦

♣♦t♥ ♦ ♣♦r ♠♦s ♦rrs ♣r ♦s ♦s ♦rs Wsf ♦♥sr♦s ♥

♦♥s♥t♠♥t ♣rs♥t♠♦s ♥ rst♦s ♣r s♦r♦ s♦r ♦t♥♦s t③♥♦

♣♦t♥ ❩ ② ♣r ♦s ♦rs Wsf/kB = 48 ② ♦s sts s♦tr♠s ♦rrs♣♦♥♥ ♥

t♠♣rtr P♦♠♦s ♣r♦r ♦♣♦rt♥ ♣r srr ♣r♦♠♥t♦ q s s ♣r r

s ♦♥tr♦♥s ① árs s q tr♠♥♥ s í♥s ♦r③♦♥ts r②s ♥

♥s♠ ♥ó♥♦ ♥♦s ♣r♠t r ∆µ(T ) = µ(T )−µ0(T ) ♣r t♦♦ ♦r n ♥♦ ♥ st ♦♥t①t♦

r ① ♥ ♦♦♣ ♥ r ❲s ♦s ♦s st♦s ♥ ♦①st♥ st ♦③♦s ♥ nmin ② nmax sts♥

µ(nmin) = µ(nmax) = µe(Ωe/A) .

st♦s st♦s s♦♥ tr♠♥♦s trés ♦♥stró♥ ① árs s

ˆ nmax

nmin

[µ(T )− µ0(T )] dn = µpw(T ) [nmax − nmin] .

strtr s s♦tr♠s ♥t ①st♥ rí♠♥s ♣r♠♦♦ ♣r t♦♦s ♦s

♣♦t♥s ♥③♦s st ♠♥r ♣♦♠♦s ♦♥r q t♥t♦ ♣r ♣♦t♥ ❩ ♦♠♦ ♣r

♠♦ ♥tr♦ ♦r♠s♠♦

♥♥♦ ♥ ♥t q s s♦tr♠s s♦r ② ♥♦ ♣rs♥t♥ r♥s ♥♠♥ts ♦♥ s

s♦r ♥♦ ♣rs♥t♠♦s rs ♦♥ s ♦♥str♦♥s ① rt♠♥t ♥♠♦s s ♦rrs♣♦♥♥ts

í♥s ♣r♠♦♦ ♥

st♥♦ ♦s t♦s ♣r ♦♥ ♦t♠♦s Tw = ♣♦r ♦tr♦ ♦ ♦t s♣r♦r ♣r

t♠♣rtr rít ♣r♠♦♦ s Tmaxcpw = 42 Pr ♦s ♦rrs♣♦♥♥ts ♦rs ♦t♥♦s s♦♥ Tw =

② Tmaxcpw = 39

♦s rst♦s ♥str♦s á♦s stá♥ ♥♦s ② s ♦♠♣r♥ ♦♥ ♦s s♣♦♥s ♥ ♦rí ♥ r♦

♥ st s♦ r♥ ♦♥ rs♣t♦ ♦s ♦rs ♦t♥♦s ♥ tr♦ ♥♦tt♦ ② ♦♦ ❬❪ s tr

t♥t♦ s♦ r♥ts ♣♦t♥s ♣rs ♦ sí ♦♠♦ t♠é♥ t③ó♥ r♥ts ♦rs ♣r

♦♥t q ♦rr tér♠♥♦ s ♥ ♥♦♥

0 2 4 6 80

20

40

60

80

100

z*

ρ [n

m

]−3

Ne/Mg T=28 K

ρl

ρv

r Prs ♥s s♦ró♥ s♦r ♣r ♦s r♥ts ♥ú♠r♦s ♣rtís rtr ③②s③ ②❯rrt ❬❪ ♣♦t♥ ♥tró♥ ♦rrs♣♦♥ ❩ t♠♣rtr s ♦ ♥ ♦s sí♠♦♦sρl ② ρv ♥♥ ♥s ♣r♦♠♦ ♣♦ró♥ íq ② ♦rrs♣♦♥♥t ♣♦r rs♣t♠♥t ♥ sts♦ ρl = ♥♠−3 ② ρv = ♥♠−3 ♣♥ ♦srr ♥ rá♦ ♦s ♣rs s♦♦s st♥t♦s t♦rs r♠♥t♦ r ♥r♦r ♦rrs♣♦♥ Γl = ② rst♥t Γl = ♥ ♠s♠ rá s ♦♠♣r♥♥str♦s rst♦s ♦♥ ♦s ♦rs ♦s ♠♥t ír♦s ①trí♦s ❬❪

−8 −7 −6 −5 −4 −3 −2 −1 00

0.5

1

1.5

2

2.5

3

3.5

Γ l

−8 −7 −6 −5 −4 −3 −2 −1 020

22

24

26

28

30

∆ µ /k B [K]

T [K

]

(a) Ne/Mg

(b) T

w

r s♦tr♠s s♦ró♥ ♣r s♦r♦ s♦r ♥ sstrt♦ rtr ③②s③ ② ❯rrt ❬❪ ♦♠♥③♥♦ s r ③qr s t♠♣rtrs s♦♥ ♠r ♦♥ ír♦s í♦s ② ♠str ♦rrs♣♦♥♥t í♥ ♣r♠♦♦

s♦ró♥ s♦r

♥ s♦ s♦ró♥ s♦r ♣♦t♥ s♦ró♥ ♦♥Wsf/kB = 95 s rt♠♥t trt♦

♦ ♥tró♥ s s ♦ ♠ás ♥t♥s q ♦♥ ♣♦r ♦ t♥t♦ s s♣r ♥ Tw r♥ t♠♣rtr

tr♣ ♦♥ ♥ ré♠♥ ♣r♠♦♦ ♦r st ①♣tt ♦♥r♠ ♣♦r s♠♦♥s ♦♥t r♦ q

sr♥ ♥ tr♥só♥ ♣r♠♦♦ r♥ t♠♣rtr tr♣ ♠♥trs q ♥♦ ①st♥ r♣♦rts á♦s

r③♦s ♥ ♠r♦ s ♥ s s♠♦♥s ♦rrs ♣♦r ♦♥ ② ♦♦r♦rs ❬❪ s♥♦

♥ ♣♦t♥ t♣♦ s tr♠♥ó q ♠♦♦ s♣r s ♣r♦ ♥ t♠♣rtr

♥ ♠♦ ♥♦ s ♣♦ r③r ♣r st sst♠ á♦s ♥ ♦♥t①t♦ ♦r♠ ♦♥ ♣r♦♣st

r ② ♦s♥r ② ♦s ♦♠♣♠♥t♦s sr♦s ♣♦r ♥♦tt♦ ♦♦ rtr♦♦ ② ♦ ♥ s s ❬ ❪

st ♠♣♦s s q ♥ s ♣r♦①♠ó♥ s ♣r♦ ♥ s♦ó♥ t♠♣r♥ ♣r T > Tt ②

s s♦♦♥s r♥ ♥♦ t♠♣rtr s ♣r♦①♠ tr♣ st ♥♦♥♥♥t ♥♦ s ♣rs♥t ♥♦

tr♠♦s ♦♥ sr ♣♦r ♥♦s♦tr♦s

♥ s ♠str♥ ♦s ♣rs ♥s ♣r s♦r♦ s♦r ♦s ♦♥ ♥str ♣r♦♣st

♣r ② s♥♦ ♣♦t♥ ❩ ② s ♦s ♦♠♣r ♦♥ ♦s ♦t♥♦s ♠♥t s♠♦♥s ♦♥t r♦

❬❪ ♥ ♥ s s♠♦♥s ♥♦ s t③r♦♥ ①t♠♥t ♦s ♠s♠♦s ♣♦t♥s q s♠♦s ♥♦s♦tr♦s ♦s

♣rs r♥ ♠♥t ♣r ♥ r♥ rt♠♥t ♣♦♦ ♣r ♥ ♦r ♠á①♠♦ q ♥③

♥s ♦rrs♣♦♥♥t r s♦ ♦♥ ♠②♦r ♥ú♠r♦ ♣rtís st r♥ ♦rrs♣♦♥

♥ só♥ ≃ 13% rs♣t♦ ♦s ♦rs q rr♦♥ ♦s á♦s r③♦s ♦♥ s♠♦♥s s

♥rís t♠♣♦♦ r♥ sst♥♠♥t s s♦♥s ♥♦ s♣r♥

s s♦tr♠s s♦ró♥ stá♥ s ♥ ♦♠♦ s ♣ ♦srr s♦tr♠ ♥♦

♣rs♥t ♥♥ú♥ ♦♦♣ ♥ r ❲s ♣♦r ♥ ♥♦ s ♥sr♦ tr③r ♥♥♥ ♦♥stró♥ ① st♦

s s♣r♥ q t♠♣rtr rít ♣r♠♦♦ r♦♥rá r♦r ♦s í♥ ♣r♠♦♦

♦♥str ♦♥ ♦s ♦rs ∆µpw ♥ ♥ó♥ t♠♣rtr stá r ♥ st♥♦ st♦s

t♦s ♦♥ ♦t♠♦s q t♠♣rtr ♠♦♦ stá ♠♥t ♣♦r ♦ tr♣ Tw =

s r ♠♦ qr t♠♣rtr T > Tt ♦s s t♠♣rtrs tr♠♥s ♥ st ♥áss

stá♥ ♥s ♥ r♦ str♦ ♦r Tw s s♠r Tw = 22 tr♠♥♦ ♣♦r ♦♥ ② ♦♦r♦rs

❬❪ ♠ás r♥ Tcpw − Tw ♦t♥ ♣♦r ♥♦s♦tr♦s ♦♥ ♦♥ ♠♥t ss s♠♦♥s

s♦ró♥ ró♥ ♥s♠ ♥ó♥♦

♦r ♦♥ ❬❪ r ♠♦ t♦s s s♣rs ♥s ♠♥trs q ♦s á♦s r③♦s ♠♥t

♦r♠s♠♦ ❬❪ sr♥ ♠♦♦ s♦♠♥t ♣r sstrt♦s ♦♠♦ ② P♦r ♦tr♦ ♦ s♠♦♥s

♥♠érs ❬❪ ♥ ♦♠♦ rst♦ q r ♠♦ ♣r♦ ♥♦ ♥

♥ s♦ 2 str ② ♦♦r♦rs ❬❪ sr♥ q r♠♥t♦ ♣í s♦r s ♦♥t♥♦

♣st♦ ♠♥st♦ ♥ ♦s ①♣r♠♥t♦s ② q ♣ srrs ♣rt♠♥t ♥ ♦♥t①t♦ ss á♦s

r③♦s ♣♥♦ st♦s t♦rs srr♦r♦♥ ♥ ♣♦t♥ s♥♦ rst♦s á♦s q ♥ s♦

r♦s ♣r♠r♦s ♣r♥♣♦s ♦s ♥t ♣♦r rs ② ♦♦r♦rs ❬❪ ♥ ♠r♦ ♦s r♣♦rt♥

s♦tr♠s ♣r T = 105 ② q s♦♥ ♥♦♥sst♥ts ♥tr sí ♦♠♦ r♠♦s ♠ás ♥t

−0.1−0.05 0 0.05 0.1 0.15 0.2 0.25 0.3−500

−400

−300

−200

−100

0

z − zmin

[nm]

Usf

/kB [K

]

Ar

Cs Rb

K

Na

Li

Mg

CO2

r P♦t♥s ♥tró♥ ♥tr r ② rs♦s sstrt♦s ♥ ♥ó♥ st♥ sstrt♦ rtr ② ③②s③❬❪ ♦♦s ♦s ♣♦t♥s ①♣t♦ 2 ♦rrs♣♦♥♥ ♣♦t♥s t♣♦ ❩

P♦r ♦tr♦ ♦ ♦♠♦ ② ♠♥♦♥♠♦s ♥tr♦r♠♥t ♥ s♦ s♦ró♥ r s♦r s ♦sr♦ ①♣r

♠♥t♠♥t ♠♦♦ ♣♥t♦ tr♣ ❬❪

♦♦s st♦s s♦s ♦s ①♠♥♠♦s ♥ st ss ♣♥♦ rsó♥ ♣rs♥t ♥ ♣ít♦s ♥tr♦rs

q ♥ s♦ ♣r str s♦ró♥ r s♦r t♦♦s ♦s sstrt♦s ♥♦s ② ♣r

♠♦s t③♦ ♣♦t♥ ❩ ♦s ♦rs ♦s ♣rá♠tr♦s q ♥tr♥♥ ♥ st ♣♦t♥ stá♥ st♦s ♥

r♦

♥ s ♣ ♦srr ♦♠♦ ♣♥♥ st♦s ♣♦t♥s st♥ sstrt♦ Pr 2 ♠♦s

t③♦ ♠s♠♦ ♣♦t♥ q sr♦♥ str ② ♦♦r♦rs ❬❪ ♥ rá♦ t♦s sts rs stá♥ ♥trs

♥ ♦r z ♦rrs♣♦♥♥t ♠í♥♠♦ ♣♦t♥ ♣♦t♥ r s♦r q r s

st r

s♦ró♥ r s♦r ♠ts ♥♦s

♣ ♦rr♦♦rr ♥ trtr q r s♦r♦ s♦r s ♥♦ ♦s sst♠s ♠ás st♦s

♠str s s♦tr♠s ♦t♥s ♦♥ ♥str♦s á♦s ♥ rá♦ ③qr st r ② s ♥

♦ s rs♣ts ♦♥str♦♥s ① í♥ ♣r♠♦♦ q í s r s ♠str ♥ rá♦

r

♥ s ♣ ♦srr ♣rs ♥s ♦t♥♦s ♣r T = 114 ♠ q ♥ú♠r♦

♣rtís ♠♥t s ♣s♥♦ s ♦s ♣rs ♥r♦rs ♠♥♦r s♣s♦r ♥ í♥s r②s ♦s ♣rs

♠ás rs♦s q stá♥ ♦s ♦♥ í♥s ♦♥t♥s st♦ ♥ s♣s♦r s ♣ís s stá ♦

st♦ ♥ ♦r t♦r r♠♥t♦ ♠♦str♦ ♣♦r s♦tr♠ ♦rrs♣♦♥♥t ♥

0 1 2 3 4 5 6 7−5

−4

−3

−2

−1

0

Γl

∆µ/k

B [K

]

Ar/Li

108 110 112 114 116 118 120−4

−3.5

−3

−2.5

−2

−1.5

−1

−0.5

0

T [K]

∆µpw

/kB [K

]

Ar/Li

Tw

r s♦tr♠s s♦ró♥ r s♦r rá♦ ③qr ♦rrs♣♦♥♥t í♥ ♣r♠♦♦ s ♣♦srr ♥ rá♦ r rtr ② ③②s③ ❬❪ Pr rá♦ ③qr ♦s trá♥♦s rr ♦rrs♣♦♥♥ ♦s ír♦s ♦s ♠♥ts r♦s trá♥♦s ♦ ② s strs ♥ r r ♦s ír♦s ♦rrs♣♦♥♥ ♥str♦s á♦s ② r st ♦rrs♣♦♥

st♥♦ ♦s t♦s í♥ ♣r♠♦♦ ♦♥ ♦t♠♦s Tw = 110 ♣♦r

♦♠♣tt st í♥ ♣r♠♦♦ t♠é♥ ♥♠♦s ♥ ♦t s♣r♦r ♣r Tcpw s

♦ ♠♦str♠♦s rá♦s ♦♥ s s♦tr♠s s♦rsó♥ r s♦r ② rt♠♥t ♥♠♦s s ♦rrs♣♦♥

♥ts í♥s ♣r♠♦♦ ♥ s♦ s s♥ts t♠♣rtrs ♠♦♦

Tw = ♣r ② Tw = ♣r Pr st út♠♦ sst♠ ♥♦ s ♥ r♣♦rt♦ st

♠♦♠♥t♦ ♦tr♦s st♦s ♠♦♦ ② ♣r♠♦♦ ♦♥ ♦r♠s♠♦

♥ ♠♦ ♠♦str♠♦s s s♦tr♠s s ♣r s♦ró♥ r s♦r ♥ strtr

sts rs ♥t ①st♥ ♥ í♥ ♣r♠♦♦ ♦♥ ♥ tr♥só♥ ♠♦♦ ♣r ♥

t♠♣rtr ♠♥t ♥r♦r ♦s st í♥ ♣r♠♦♦ Tw = ♠s♠♦ rá♦

s ♥r q Tcpw t♥ ♦♠♦ ♦t s♣r♦r ♦r

♥ q ♠str s í♥s ♣r♠♦♦ s ♣r r s♦r♦ s♦r r♥ts sstrt♦s

♠ás ♥str♦s t♦s ♠♥ts r♦s trá♥♦s ② ír♦s s♥ r♥♦ ♠♦s ♥♦ í♥s ♣r

② ♠♥ts ② r♦s r♥♦s ♦s s♥♦ ♦tr ♥ ❬❪ ♣ ♦srr q s ♦s

í♥s r♣♦rts ♥ st út♠ rr♥ ①♥ ♥ ♦rr♠♥t♦ Tc

♥ r♦ r♥♠♦s ♦s rst♦s ♦t♥♦s ♦♥ ♥str♦s á♦s ♣r s t♠♣rtrs ♠♦♦ ②

♣r♠♦♦ ♣rá♠tr♦ st apw ♥t♦ ♦♥ t♦s q ♣♦s ♥ ♦rí

♦♦s ♦s á♦s sr♥ q r ♠♦ s ♥ ♦s ♦rs Tw ♥♦ ♦♥♥ s r♥s ♥tr

♥str♦s rst♦s ② ♦s ♥♦tt♦ ② ♦♦ ❬❪ ♣r r ② t♠é♥ ♣r r s ♣♥ trr ♦♠♦ ♥

s♦ s♦ r♥ts ♣♦t♥s ♣rs f f ② t③ó♥ r♥ts ♦rs ♣r ♦♥t

tér♠♥♦ s ♥♦♥

♦tr♦ rst♦ ♣r r ♦ ♣♦r ❩♥ ② ❩♦♥ ❬❪ ♣♦str♦r♠♥t ♥str♦ tr♦ ❬❪

t③ ♣♦r s♦s t♦rs ♦♥t♥ ♥ ♦rró♥ s ♥ ♥ tr♥s♦r♠ r♣♦ r♥♦r♠③ó♥

❬ ❪ q rí sr ♣r trt♠♥t♦ ♥ó♠♥♦s rít♦s st ♥♦ ♦r Tw s s♦♠♥t

0 1 2 3 4 5 6 7 8 9 100

0.2

0.4

0.6

0.8

1

1.2

1.4

z*

ρ*

Ar/Li T=114 K

r Prs ♥s ♥ s♦ró♥ r s♦r ♣r r♥ts ♥ú♠r♦s ♣rtís rtr ②③②s③ ❬❪ s rs r②s ♦rrs♣♦♥♥ ♣ís ♥s ♠♥trs q s♦♥ ♣ís rss st♦ ♥ strtr ♣rs stá ♦ ♦♥ st♦ q ♠str s♦tr♠ ♥

105 115 125 135 145 155

−3.5

−3

−2.5

−2

−1.5

−1

−0.5

0

T [K]

∆µpw

/kB [K

]

Ar

Tc

Li

Li Na

Na Rb K

r í♥s ♣r♠♦♦ ♦rrs♣♦♥♥ts r s♦r♦ s♦r ♠ts ♥♦s rtr ② ③②s③ ❬❪ ♦srs♦s sí♠♦♦s s♥ r♥♦ ♦rrs♣♦♥♥ á♦s r③♦s t③♥♦ ♥str♦ ♥♦♥ ♦s t♦s sñ♦s♦♥ sí♠♦♦s r♥♦s ♦♥ ♥r♦ r♦♥ ①trí♦s ❬❪ s í♥s sós s♦♥ s q st♥ st♦s t♦s♠♥t

r♦ Pr♦♣s ♠♦♦ ♦rrs♣♦♥♥ts r s♦r♦ s♦r sstrt♦s ♣♥♦s ♦♥♦r♠♦s ♣♦r ♠ts ♥♦s 2 ② ♦♥ st♦s W ∗

sf

s t♠♣rtrs ♠♦♦ Tw ② rít ♣r♠♦♦ Tcpw ①♣rss ♥ r♦s ♥ ♥ s♦ Tcpw s ♥ ♦t s♣r♦r ♣r st t♠♣rtr ♣ ♠♦r ♦ ♥♦ s♣r ♥ st s♦ ❲ ♥ q r ♥♦ ♠♦ sstrt♦ ② ♥♠♦♦ ♦♥t♥♦ ♣rtr ♥ t♠♣rtr ♠②♦r ♦ tr♣ t♦r apw q ♦rrs♣♦♥ ó♥ st ❬ ❪P❲ rr♥ ♦s rst♦s ♦t♥♦s ♠♥t ♥str♦s á♦s ② ♦s rst♦s q s ♦t♥♥s♥♦ ♦tr♦s ♥♦♥s ♥tr ♣ré♥tss s ♥ rr♥ ♦♥ r♦♥ t♦♠♦s st♦s t♦s

Pr♦♣s s 2 W ∗

sf Tw ①♣r♠♥t ❬ ❪

Tw ❬❪ Tw ❬❪ ❲ ❲ ± Tw ❬ ❪ Tw ❬❪ Tw ❬P❲❪ Tt Tt Tt

apw/kB [−1/2]❬P❲❪

Tcpw ❬❪ Tcpw ❬❪ Tcpw❬P❲❪

0 2 4 6 8 10 12 14−1

−0.8

−0.6

−0.4

−0.2

0

Γl

∆µ/k

B [K

]

Ar/Rb

r s♦tr♠s ♦rrs♣♦♥♥ts r s♦r♦ s♦r rtr ② ③②s③ ❬❪ r sñ ♦♥ ír♦s♦rrs♣♦♥ ♥ t♠♣rtr q ♣♦s ♠♥ts ♥ ♥ ♦♥ r♦s♦rrs♣♦♥ ② ♠r ♦♥ trá♥♦s ♥

0 1 2 3 4 5 6 7

−40

−30

−20

−10

0

Γl

∆µ/

k B [K

]

Ar/Mg

r s♦tr♠s s♦ró♥ r s♦r ♣r rs t♠♣rtrs rtr ② ③②s③ ❬❪ í♥ ♦♥ ír♦s♦rrs♣♦♥ T = 105 q ♣♦s ♠♥ts í♥ ♦♥ r♦s Tnb = ② trá♥♦s Tt =

♥ ∼ 15% ♠ás ♦ q ♥str♦

P♦r ♦tr♦ ♦ ♦♠♦ s ♥ ♥ r♦ s s♠♦♥s r③s ♦♥ sr♥ ♦♥trr♠♥t

♦ ♥♦♥tr♦ ♣♦r ♥♦s♦tr♦s q r ♥♦ ♠♦ ♥ ❨ ♠♦s ♥tr♦r♠♥t q st♦s rst♦s

♣♦s ♣♦r rtr♦♦ ② ♦♦r♦rs ❬❪ t♠♣♦♦ ♥ ♥t ♠♦♦ ♣♦r r♦

①♣r♠♥t♠♥t ❯♥ ♣♦s s st ② ♠♥♦♥♠♦s ♥ ó♥

s♦ró♥ r s♦r ② 2

♥ s ♠str♥ s♦tr♠s ♣r r s♦r ② ♥ s ♦rrs♣♦♥♥ts s♦ró♥

r s♦r 2 ♥ st út♠♦ s♦ s ró ♥ í♥ ♣♥t♦s ② r②s ② ♥t♦ s rs rs ♥str♦s

á♦s r q ♦tr♦♥ str ② ♦♦r♦rs ♣s ♥ ♥ P ❬❪ ② q ♥♦s♦tr♦s

♣rs♥t♠♦s ♥ í s ♠str♥ ♦s s♦tr♠s ♣r s s♥♦ ♥♦ ♣♦t♥

í♥ ♦♥ ír♦s q ♣r♦♣♦♥♥ ♦s t♦rs ❬ ❬❪❪ ② ♣♦t♥ í♥ ♦♥ trá♥♦s

♦ ♣♦r ♦♥ ♦s ♣rá♠tr♦s ♣r♦♣st♦s ♣♦r ♥r ② ♠ εeff/kB = (εff εss)1/2

/kB =

σsf = (σss + σff )/2 = ♥♠ ② ρsσ3sf = ♣r♦♠ s ♦♠♦ s ♣ ♦srr ♥ q

s♦tr♠ t♠♣rtr r♥ tr♣ s♥♦ ♣♦t♥ s♦ró♥ q ♣r♦♣♦♥♥ ♥ s

tr♦ ♣rs♥t ♥ ♦r♠ ♠s♦ ♦st♦r ② ♥ rró♥♦ r ♦♥ s♦tr♠ ♣r ♥ ♦r

t♦r r♠♥t♦ Γl ≈ 3 st ♥ s q ♣♦t♥ ♣rs ♦ ♦ ♣♦r

♣r♦ ♥ s♦ó♥ ♣r♠tr ♣r T > Tt

0 1 2 3 4 5 6 7

−50

−40

−30

−20

−10

0

Γl

∆µ/

k B [K

]Ar/CO

2

r s♦tr♠s s♦ró♥ r s♦r 2 ♣r ♥s t♠♣rtrs rtr ② ③②s③ ❬❪ í♥ ♦♥ ír♦s♦rrs♣♦♥ T = 105 ♠♥ts r♦s ② trá♥♦s s rs r②s ② ♣♥t♦s ② r②s s♦♥ s♦tr♠s s ♣♦r str ② ♦♦r♦rs ♥ ❬❪

0 1 2 3 4 5 6 7 8−25

−20

−15

−10

−5

0

5

10

Γl

µ [K

]

r s♦tr♠s s♦ró♥ r s♦r 2 s ♣♦r str ② ♦♦r♦rs st rá♦ ♦rrs♣♦♥ ❬❪ í♥ ♦♥ trá♥♦s s s♦tr♠ ♦♥ ♣♦t♥ s s♦tr♠ss ♦♥ ♣♦t♥ s♦ró♥ ♣r♦♣st♦ ♣♦r st♦s t♦rs stá♥ ♥s ♣♦r rs ♦♥ ír♦s ② ♦♥r♦s ♦rrs♣♦♥♥♦ rs♣t♠♥t T = 105 ②

0 1 2 3 4 5 6 7−700

−600

−500

−400

−300

−200

−100

0

Γl

∆µ/k

B [K

]

Ar/Au

r s♦tr♠s s♦ró♥ r s♦r rtr ② ③②s③ ❬❪ ♠str♥ s s♦tr♠s ♦rrs♣♦♥♥ts st♠♣rtrs trá♥♦s ② ír♦s óts strtr s♠♦♥♦♣ q ♣r ♥ sts♦ ♣r s♦tr♠

s♦ró♥ r s♦r

♦♥stt② ♥ sstrt♦ rt♠♥t trt♦ ♣r r ♥ st s♦ t③ó ♥ ♣♦t♥ srt♦ sú♥

♦♥ ♦s ♣rá♠tr♦s Wsf/kB = 987 ② C3 = ♥♠3 st♦s ♦rs r♦♥ ①trí♦s ♦s r♦s

② ❬❪

str♦ st♦ r q ♥♦ r s s♦r s♦r st sstrt♦ s ♦♥♥s ♥ sss ♣s ② ♣r ♥

strtr s♠♦♥♦♣ s♦tr♠ ♠r ♦♥ ♥ trá♥♦ ♥ q ♦rrs♣♦♥ ♥ t♠♣rtr

st ♥ó♠♥♦ s s♠r r♣♦rt♦ ♥ rr♥ ❬❪ ♦♥ s st s♦ró♥ r s♦r

♥ sstrt♦ rt♦ r t rr♥ s♠t s q ♣r ♠♦s sst♠s s

♥t♥ss s ♥tr♦♥s ♥tr ♦ ② sstrt♦ s♦♥ ♣rs

s♦ró♥ ❳♥ó♥ ♥s♠ ♥ó♥♦

♥st♠♦s s♦ró♥ ❳ s♦r sstrt♦s ♣♥♦s ♠ts ♥♦s ② s♦r ♥ st út♠♦ s♦ ①st♥

t♦s ♠♣ír♦s ♦t♥♦s ♥ ①♣r♠♥t♦s s♦s t♥t♦ ♥ só♥ át♦♠♦s ♦ ♦♠♦ ♥ ró♥

tr♦♥s ♥rí ❬❪

♥ ♥str♦ st♦ t③♠♦s ♦s ♣♦t♥s ❩ ♦♥ ♦s ♣rá♠tr♦s st♦s ♥ r♦ ♥ s

♠str♥ ♦s ♣♦t♥s s♦ró♥ ♥ ♥ó♥ st♥ sstrt♦

s♦ró♥ ❳ s♦r st♦s sstrt♦s ♣rs♥t ♥ ♠♣ r st♦♥s r♦♥s ás♠♥t ♦♥

♣♦s ♣♦r r♦rrr ♥ ♠♣♦ r♥♦ W ∗

sf

♥ s♦ s tr♥s♦♥s ♣r♠♦♦ ❳ ① rts rtrísts q ♥♦ stá♥ ♣rs♥ts ♥ ♥

♥ r ♥♦ s s♦r♥ s♦r ♦s ♠s♠♦s sstrt♦s

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8−6

−5

−4

−3

−2

−1

0

1

2

z [nm]

Usf

/εff

Xe

Mg

Li

Na

Cs

r P♦t♥s ♥tró♥ ❩ ♣r ❳ ♦♥ r♥ts sstrt♦s ♥ ♥ó♥ st♥ sstrt♦ rtr② ③②s③ ❬❪

♣r♦♣st♦ ♥ st ss ♥♦s ♣r♠t♦ ①♠♥r t♠♥t rs sts rtrísts

s s♦♦♥s s tr♠♥r♦♥ rs♦♥♦ s ♦♥s t♥t♦ ♣r r♥rs rrs ♦r♠s♠♦ ♥ó♥♦

♦♠♦ ♣r r♥rs rts r♥ ♥ó♥♦ ♦♠♥③r♠♦s sr♥♦ rst♦s ♦t♥♦s ♥ ♠r♦

♥s♠ ♥ó♥♦

♥ s ♣♥ ♦srr ♦s ♣rs ❳ s♦r♦ s♦r sstrt♦s s ② ♥ ♥♦

st♦s rá♦s s ♠str r♠♥t♦ ♣r ♥ss ♥♦ ♠♥t ♥ú♠r♦ ♣rtís ♣♦r

♥ ár ♥ s♦ t♠♣rtr ♦rrs♣♦♥ ó♥ ♥♦r♠ Tnb =

❳ ♣♦st♦ s♦r st♦s sstrt♦s ① s rtrísts tí♣s ♥ ♦ s♦r♦ ♣rs♥t♥♦ ♦s

♦♥s ♥ ♣rt ♣r r♥ ♣r ♦ sts ♦s♦♥s s♠♥②♥ ♥③á♥♦s ♥ ♦r s

♦♥st♥t ♣r ♥s q ♦rrs♣♦♥ s íq ρl ás ♦s ♣r ♦ ♣rs♥t t

♥trs íq♦♣♦r ② ♥♠♥t s ♥s s st③ ♥ ρv

s♦ró♥ ❳ s♦r ②

♥ s ♠str♥ s♦tr♠s ② í♥ ♣r♠♦♦ ♦rrs♣♦♥♥ts s♦ró♥ ❳ s♦r

♦r Tw tr♠♥♦ ♣♦r st ♦♥ s ♦t s♣r♦r ♣r Tcpw s

s♦ró♥ s♦r ♥r ♥ sr s♦tr♠s q ♣♥ sr ♣rs ♥ strtr st

♦♥♥t♦ s♦tr♠s ♥t ①st♥ ♥ tr♥só♥ ♠♦♦ ♦♥ ♥ Tw ♣♦r

0 5 10 15 20 250

2

4

6

8

10

12

14

z*

ρ [n

m−3

]

ρv

ρl

↑

↓

Xe/Cs T=270 K

0 2 4 6 8 10 12 14 16 180

5

10

15

20

25

z*

ρ [n

m−3

]

Xe/Na T=246 K

ρl

↑

ρv

↓

0 5 10 15 200

5

10

15

20

25

30

35

z*

ρ [n

m−3

]

↑ ρl

↓

ρv

Xe/Li T=209 K

0 3 6 9 12 150

10

20

30

40

50

60

70

80

z*

ρ [n

m−3

] Xe/Mg T=Tnb

↓

ρl

r Prs ♥s ♦rrs♣♦♥♥ts ❳ s♦r♦ s♦r ♣rs ♣♥s sós s ② r♦♦♥ ♦ ♥♦ ♥ rá♦ ♦s á♦s r♦♥ r③♦s s♥♦ ♣♦t♥s t♣♦ ❩ ♣r ♥tró♥♥tr sstrt♦ ② ♦ ♥ s♦ t♠♣rtr s ó♥ ♥♦r♠ Tnb s ♥ts ρl ②ρv s♦♥ s ♥ss s ss íq ② ♣♦r ♥ ♦①st♥ s t♠♣rtrs ♦♥srs

0 1 2 3 4 5 6 7−1.2

−1

−0.8

−0.6

−0.4

−0.2

0

Γl

∆µ/k

B [K

]

Xe/Na

(a) 241 243 245

−1.2

−1

−0.8

−0.6

−0.4

−0.2

0

T [K]

∆µ/k

B [K

]

Xe/Na

Tw

Cpw

(b)

r rá♦ ③qr s♦tr♠s s♦ró♥ ❳ s♦r ③②s③ ② rtr ❬❪ s t♠♣rtrs t♦♠♥ s♥t r♥♦ ♦rs trá♥♦s ♦ st ír♦s ♥ ♣s♦s ♥ r♦ rá♦ r í♥ ♣r♠♦♦ ♦rrs♣♦♥♥t ♥ s rá♦ s♦tr♠s t♠♣rtr ♠♦♦ q rr♦♥ st♦s rst♦s s ② rít ♣r♠♦♦ t♥ ♦♠♦ í♠t s♣r♦r ♦s

0 1 2 3 4 5 6−0.7

−0.6

−0.5

−0.4

−0.3

−0.2

−0.1

0

Γl

∆µ/k

B [K

]Xe/K

r s♦tr♠s s♦ró♥ ❳ s♦r s t♠♣rtrs t♦♠♥ ♦s ♦rs trá♥♦s rr st ír♦s ♥ ♣s♦s ♥ r♦ t♠♣rtr ♠♦♦ q rr♦♥ st♦s rst♦s s ② rít ♣r♠♦♦ t♥ ♦♠♦ í♠t s♣r♦r ♦s

st ♦r s ♠②♦r q q s ♦t♥ ♥ ♣r♦①♠ó♥ q ♣r ♥ t♠♣rtr

♠♦♦ t♠♣rtr rít ♣r♠♦♦ t♥ ♦♠♦ ♦t s♣r♦r ♦r

♦s ♦rs ♦t♥♦s Tw ② Tcpw stá♥ st♦s ♥ r♦

s♦ró♥ ❳ s♦r ②

♥ ♦s rá♦s s ② s ♠str♥ s♦tr♠s s♦ró♥ q ♠♦s ♦ ♣r ♦s sst♠s

❳ ② ❳ ♥t♦ ♥♦ s♦s rá♦s r s ♥♥tr♥ s rs♣ts í♥s ♣r♠♦♦

♥ s♦ s s♦tr♠s s♦ró♥ s♦r s ♥ tr③♦ s rs♣ts ♦♥str♦♥s ① árs

s

♥ st♦s sst♠s ♦♥ ♥tró♥ sstrt♦♦ s ♠ás ♥t♥s q ♥ s♦ ② s ♠♥st ♥

strtr s♦tr♠s ♠út♣s ♦♠♦ s ♠♥♦♥s ♥ ♥ tr♦ P♥t ② ♦♦r♦rs s ❬ ❪

♥ st s♦ ♥♦s♦tr♦s ♠♦s ♥♦♥tr♦ sts ♠s♠s strtrs tr♥♦ ♦♥ ♦s ♣♦t♥s rsts ú♥

♥ s♦ rt♠♥t trt♦ s ♣♦ ♥r ♥ strtr í♥s ♣r♠♦♦ ♦♠♦ s ♣

♦srr ♥ rá♦ r

Pr ♥③r ♦♥ st ♥áss q s r③ó ♣r r♥rs rrs ♥ ♦♥t①t♦ ♦r♠s♠♦ ♥ó♥♦ ♥

s ♠str♥ ♥ ♥ s♦♦ rá♦ ♠♦♦ ♦♠♣ró♥ s í♥s ♣r♠♦♦ ♠út♣s q s ♦t♥♥

♥♦ s ♥③ s♦ró♥ ❳ s♦r ♦s rs♦s sstrt♦s st♦s qí

♥ r♦ ①♣♦♥♠♦s r♦s ♦s rst♦s q ♠♦s ♦t♥♦ r♦♥♦s ♦♥ s ♣r♦♣s

♠♦♦ ❳ s♦r ♥♦s sstrt♦s ② ♠♦str♠♦s t♠é♥ ♦s rst♦s ①trí♦s s rr♥s q í s

♥♥

0 1 2 3 4 5 6 7−6

−5

−4

−3

−2

−1

0

Γl

∆µ/k

B [

K]

Xe/Li

(c) 198 201 204 207 210

−6

−5

−4

−3

−2

−1

0

T [K]∆µ

/kB [

K]

Xe/Li

(d)

↑T

w

Cpw1

Cpw2

Cpw3

Cpw4

r s♦tr♠s s♦ró♥ ② í♥ ♣r♠♦♦ ♣r ❳ s♦r♦ s♦r ③②s③ ② rtr ❬❪ rá♦ ③qr s♦tr♠s s♦ró♥ ❳ s♦r ♥ ♦♥ s ♠str♥ s ♦♥str♦♥s ① ♣r ♥ s rs s t♠♣rtrs t♦♠♥ ♦s ♦rs trá♥♦s ♦ st ír♦s ♥♣s♦s ♥ r♦ rá♦ r strtr í♥s ♣r♠♦♦ ♦rrs♣♦♥♥t

0 1 2 3 4 5 6 7 8

−100

−80

−60

−40

−20

0

Γl

∆µ/k

B [K

]

Xe/Mg

161 165 169 173 177 181−120

−100

−80

−60

−40

−20

0

T [K]

∆µ/k

B [

K]

Xe/Mg

(f)

← Tt

Cw4

Cw3

C

w2

Cw1

r s♦tr♠s s♦ró♥ ② í♥ ♣r♠♦♦ ♣r ❳ s♦r♦ s♦r ③②s③ ② rtr ❬❪ rá♦ ③qr s♦tr♠s s♦ró♥ ❳ s♦r ♥♦ s ♠str♥ ♥ st s♦ s ♦♥str♦♥s ①♦rrs♣♦♥♥ts ♣rs♥t♥ rs ♣r t♠♣rtr tr♣ ② ♣r ♥s ♦trs t♠♣rtrs st ♦s rá♦ r í♥ ♣r♠♦♦ ♦rrs♣♦♥♥t

r♦ Pr♦♣s ♠♦♦ ♦rrs♣♦♥♥ts ❳ s♦r♦ s♦r sstrt♦s ♣♥♦s ♦♥♦r♠♦s ♣♦r ♠ts ♥♦s ② ♦♥ st♦s W ∗

sf

s t♠♣rtrs ♠♦♦ Tw ② rít ♣r♠♦♦ Tcpw ①♣rss ♥ ♥ s♦ Tcpw s ♥ ♦t s♣r♦r ♣r st t♠♣rtr ♣ ♠♦r ♦ ♥♦ s♣r ♥ st s♦ ❲ ♥ q ❳ ♥♦ ♠♦ sstrt♦ ② ♥q ❳ r ♦♥t♥♠♥t s♦r sstrt♦ ♣rtr ♥ t♠♣rtr ♠②♦r ♦ tr♣ t♦r apw q ♦rrs♣♦♥ ó♥ st ❬ ❪P❲ rr♥ ♦s rst♦s ♦t♥♦s ♠♥t ♥str♦s á♦s ② ♦s rst♦s q s ♦t♥♥s♥♦ ♦tr♦s ♥♦♥s ♥tr ♣ré♥tss s ♥ rr♥ ♦♥ r♦♥ t♦♠♦s st♦s t♦s

Pr♦♣ s W ∗

sf Tw ❬❪

Tw ❬ ❪ ❲ ❲ ❲ Tw ❬P❲❪ Tt

apw/kB [−1/2]❬P❲❪

Tcpw ❬P❲❪

s♦ró♥ ❳♥ó♥ ♥s♠ r♥ ♥ó♥♦

♦♦s ♦s rst♦s ♣r♦s ♥ s♦ ♦t♥♦s rs♦♥♦ s ♦♥s ♥ ♥ rt♥r rr

♦♥ s ♠♥t♥ ♦ ♥ú♠r♦ ♣rtís ♣♦r ♥ ár ♠ás t♠♣rtr

P♦r ♦tr♦ ♦ ♥ ♦s ①♣r♠♥t♦s ♦♥ ♥ ♦ ♣♦s rt♦ r♦ ♦♥♥♠♥t♦ st♥♦ ♥ ♦♥tt♦ ♦♥

♥ ♥t♦r♥♦ ♣ rs t♥t♦ t♠♣rtr ♦♠♦ ♣rsó♥ ♦ ♥ s t♦ ♣♦t♥ qí♠♦ s♥ ♠♣♦♥r

♥ rstró♥ s♦r n ♣r♠t♥♦ ♥ ♦ rrstrt♦ ♣rtís s ♦ s ♥t♦r♥♦ ♥ st út♠♦

s♦ s♦ st♦♥s t♦ ♦♥♥♠♥t♦ ♦ r♦ ♦r n ♥ú♠r♦ ♠♦ ♣rtís ♦

♦rrs♣♦♥rs ♦♥ q ♣♦s ♠s♠♦ sst♠ ♥♦ st rr♦ s r < n >≡ n = N/A

♥ st ♥♦ sq♠ tr♦ ♥ sst♠s rt♦s ♦♥ s ♦r ♣♦t♥ qí♠♦ s ♦♥s

tró♥ rs s ❬s ② ♣ ❪ s♦♥ rsts ♣♥♦ ♦r♠s♠♦ r♥

♥ó♥♦ st ♥③r s s♦♦♥s ♣r ρ(z) ② n ó♣t♠s

st ♠♥r ♣r ♥ ♦ ♦r r ♥♣♥♥t µ ♣♦t♥ qí♠♦ ♥s ♣ sr

♥rt♥♦

ρ(z) =1

Λ3exp

(

µ−Q(z)

νidkBT

)

.

s s♦♦♥s t♦♦♥sst♥ts s♦♥ ♦t♥s tr♥♦ st ①♣rsó♥ st ♥③r r♦ ♦♥r♥

rqr♦

♦ ♥ú♠r♦ ♣rtís ♣♦r ♥ ár s ♦t♥♦ ♠♥t ①♣rsó♥

150 180 210 240 270 300−12

−10

−8

−6

−4

−2

0

T [K]

∆µ

/kB

[K]

← Tt

Tc

Cw3

Cw4

Cw5

Cpw4

C

pw3

Cpw2

Cpw1

Xe

r í♥s ♣r♠♦♦ ♦rrs♣♦♥♥ts ❳ s♦r♦ s♦r rs♦s sstrt♦s rtr ② ③②s③ ❬❪ ♥ rá s ♣ ♦srr s í♥s ♦rrs♣♦♥♥ts s♦ró♥ s♦r s ♥ ♦♥ ír♦s s s ♥ sstrt♦ ♥s ♦♥ trá♥♦s s s♦s ♠rs ♦♥ r♦s ② s ♦rrs♣♦♥♥ts sñs ♦♥ r♦♠♦s ♦♦s st♦s á♦s r♦♥ r③♦s ♥ ♥tr♦ t♠♣rtr q s tr♣ sñ ♣♦r Tt st t♠♣rtr rít ♥ ♦♥ Tc

n =1

Λ3exp

(

µ

νidkBT

)ˆ L

0

dz exp

( −Q(z)

νidkBT

)

.

♦♥ µ sr ♣r♦ ♠♦♦ ♠♣♦ s ①♥ ♥♦s rst♦s ♦rrs♣♦♥♥ts s♦ró♥

❳ s♦r ♥ ♣r ♣♥ t♠♣rtr ♥ ♥ s ♣ ♦srr ♥

s♦tr♠ s♦ró♥ ♣r st t♠♣rtr í♥ só ♦rrs♣♦♥ s s♦♦♥s q s ♦t♥♥ t③♥♦

♦r♠s♠♦ r♥ ♥ó♥♦ ♠♥trs q r②s ♦rrs♣♦♥ s s♦♦♥s ♦t♥s ♠♥t ♦r♠s♠♦

♥ó♥♦ ♦s st♦s C1 ② C2 s♦♥ ♦s st♦s sts ♥ér♦s ♦rrs♣♦♥♥ts ♣ís s♦rs r♥ts

s♣s♦rs C1 st s♦♦ ♣í ♠ás

♠s rs s s♣r♣♦♥♥ s♦ ♥ ró♥ q ♥tr ♦s ♣♥t♦s s♣♥♦s S1 ② S2 ♦♥ dµ/dn = 0 s

r ♦♥ s s♦♦♥s s♦♥ ♥sts

r♥ ♥ó♥♦ ♣♦rt s♦♦♥s ♦♥ st r♥t③ ♦♥① ♥rí r FDF st ♣r♦♣

t♠é♥ s ① ♣♦r á♦s ♠r♦só♣♦s r♦♥s ♦s ♥ ♦♥t①t♦ ♥áss ♦♥♦♥s ♣r♦s

♥ ♦♥♥ó♥ ♦♥ ♥s ♣r♦♥ts PP r ♣♦r ♠♣♦ s s ❬ ❪

♥ t r í♥ rt ♦r③♦♥t ♦rrs♣♦♥ ♥ ♦♥stró♥ ① árs s ❬ ❪

st ♦♥stró♥ s ♣ r s♦♦ ♣r r ♣♥t♦s ♣r r ♦①st♥ ♥♦ s tr ♦♥

♦s rst♦s q s ♦t♥♥ s♥♦ ♥s♠ r♥ ♥ó♥♦ ①st ♦tr♦ ♠ét♦♦ ♥♦♠♥♦ ♦♥stró♥

① s t♥♥ts q ♠ás ♥t s rr♥

❯t③♥♦ s s♦♦♥s s ♠♥t r♥ ♥ó♥♦ s ♠♣♦s r s árs ♥♦rs ♥

♦ q ♥♦ ①st♥ t♦s ♥ ró♥ ♥♦ ♦♥① ♥tr S1 ② S2 ♣♦r s♦ ② q t③r ú♥ ♠ét♦♦

tr♥t♦

0 1 2 3 4 5−1

−0.9

−0.8

−0.7

−0.6

−0.5

−0.4

Γl

∆µ/k

B [K

]

Xe/Na T=245 K

C1

S1

S2

C2

M1 M

2

r s♦tr♠ s♦ró♥ ♦rrs♣♦♥♥t ❳ ♣♦st♦ s♦r ♥ ♣r ♣♥ ♥ t♠♣rtr ③②s③ ② rtr ❬❪ r só s♦♥ rst♦s ♦t♥♦s s♥♦ ♦s ♦r♠s♠♦s ♥ó♥♦ ② r♥♥ó♥♦ ♠♥trs q r r② s♦♥ rst♦s q s ♦t♥♥ s♦♠♥t ♦♥ ♥s♠ ♥ó♥♦ ♦s♣♥t♦s s♣♥♦s stá♥ ♠r♦s ♦♥ S1 ② S2 ♠♥trs q C1 ② C2 s♦♥ ♦s st♦s sts ♥ér♦s rt r②s ② ♣♥t♦s q ♥ ♦s st♦s M1 ② M2 ♥♦t ♦♥stró♥ ① árs s

♦♥stró♥ ① s t♥♥ts

♦s ♠ét♦♦s tr♥t♦s ♣r r③r ♥ ♦♥stró♥ ① stá♥ sr♣t♦s ♥ r♦s t①t♦ ♣♦r ♠♣♦

♥ ❬❪ ② ♥tt ❬❪ st út♠ rr♥ ♣rs♥t ♥ sí♥tss s ♣♦ss ♦♦s

♦s ♣r♦♠♥t♦s s s♥ ♥ ♦♥sr♦♥s ♥réts ♦♥t♥ó♥ ♠♦s ♥r ♠ét♦♦ t③♦

♥ st ss

♥rí r s s♦r s ♣ srr ♦♠♦

dF = d(E − TS) = −SdT + γdA+ µdN,

♦♥ E s ♥rí ♥tr♥ t♦t ② γ r♣rs♥t t♥só♥ s♣r ♦ ♥rí s♣r ♣♦r ♥ ár

t♠♣rtr T ② ár A ♦♥st♥t ♣♦♠♦s srr ♣♦t♥ qí♠♦ ♦♠♦

µ =

(

∂F

∂N

)

T,A

= f + n∂f

∂n,

st ♠♥r ♣♦♠♦s ①♣rsr t♥só♥ s♣r ♥ ♦r♠

γ =

(

∂F

∂N

)

T,N

=∂f

∂n−1= −n2 ∂f

∂n= n(f − µ) =

F − µN

A=

Ω

A= ω,

♦♥ Ω s ♣♦t♥ r♥ ♥ó♥♦

s ♦♥♦♥s qr♦ ♣r ♥ ♦ ♥ s♦♦ ♦♠♣♦♥♥t q ♣♦s ♥ t♠♣rtr ♥♦r♠ s♦♥ q

t♥t♦ t♥só♥ s♣r ♦♠♦ ♣♦t♥ qí♠♦ s♥ ♠♣♦s ♥♦r♠s P♦r ♦ t♥t♦ s ss ♦rrs♣♦♥♥ts

íq♦ ② ♣♦r ♣♥ ♦①str ♥ t♠♣rtr s ② s♦♦ s ♦♠♦ ♦ rqr ♣r

ú♥ ♣♦t♥ qí♠♦ qr♦ µe ♠s ss t♥♥ ♠s♠ t♥só♥ s♣r γe ♥ ♦trs ♣rs s

♠s ss ♣♦s♥ ♠s♠♦ ♦r ω sí s ♥ ♦♥♥t♦ (n, µ, fDF ) ♦t♥♦ ♥ st♦r ♦♥①♦ FDF

♦ r♦ ♦s st♦rs C1 − S1 ② S2 −C2 r s ♣ r ω ② ♦♥ st♦ tr③r ♦♥stró♥

① s t♥♥ts ❬ ❪ ♠ét♦♦ q ♣s♠♦s srr r♠♥t

♠ét♦♦ s t♥♥ts s s ♥ r s í♥s y t♥♥ts s rs f(n−1) ♣r ♠♦s st♦rs

C1 − S1 ② S2 − C2

♦r t♥♥t ∂f/∂n−1 s ω ♦①st♥

ωe =

(

∂f

∂n−1

)

nmin

=

(

∂f

∂n−1

)

nmax

,

♦

y − f (nmin)

n−1 − n−1min

=y − f (nmax)

n−1 − n−1max

.

sts ♦♥s ♦♥♥

y = f (nmin) +

(

∂f

∂n−1

)

nmin

(

n−1 − n−1min

)

= f (nmax) +

(

∂f

∂n−1

)

nmax

(

n−1 − n−1max

)

,

s♥♦ s s ② ♣♦♠♦s srr

y = µ (nmin) + ω (nmin)n−1 = µ (nmax) + ω (nmax)n

−1 ,

s ♦♠♥ó♥ ♣♦t♥ qí♠♦ ② ♣♦t♥ r♥ ♥ó♥♦ ω ♦rrr ♣r s ♦s ss ♥

♦①st♥ ♥♦ s í♥s t♥♥ts ♦♥♥ q

y = µe + ωen−1 .

st ♠♥r s ♣♥ ♦t♥r t♥t♦ ♣♦t♥ qí♠♦ ♦♠♦ r♥ ♣♦t♥ ♦rrs♣♦♥♥ts qr♦

µe ② ωe s ♦♠♦ s ♥ss ♦rrs♣♦♥♥ts ♥ s ss ♥ ♦①st♥ nmin ② nmax

♥ ♠r♦ st ♣r♦♠♥t♦ ♥♦ s ♠ás ♣rs♦ ♣r tr♠♥r tr♥só♥ s st ♦ s♦

♥③♦ ♣♦r s♦ ② ♦♦r♦rs ♥♦ st♥ s♦r ❬❪

❯♥ ♠♥♦ tr♥t♦ ♣r tr♠♥r ♦①st♥ ♦♥sst ♥ ♣rtr µ ② ω ♦♠♦ ♥♦♥s ♥ú♠r♦

♣rtís ♣♦r ♥ ár ♦ rr ω s µ ② st ♠♥r tr♠♥r ♦r sts rs

♦rrs♣♦♥♥ts ♦①st♥ ss ♥ qr♦ µe ② ωe r ♣♦r ♠♣♦ ❬❪ s

♦♥str♦♥s ♠♦strs ♥ ❬❪ ② ❬❪

♥ ♣♥t♦ r ♠r♦ ♦♠♦ Mcross tr♠♥ t♥t♦ µpw(T ) ♦♠♦ ωpw ♦rrs♣♦♥♥t

♦①st♥ ss ♥ st ♣♥t♦ r ② ♥ qr rtríst♦ ♥ tr♥só♥ ♣r♠r ♦r♥

♦♠♦ sr s s♣rr ♦s ♦rs µpw(T ) ♥ s s ② ♦♥♥

♦s st♦s ♦ r♦ í♥ C1 −Mcross − C2 s♦♥ sts ♠♥trs q ♦s ♦rrs♣♦♥♥ts M1 − S1 ②

S2 −M2 s♦♥ ♠tsts ú♥ ♥s♥ ② ♦♥ ♣ít♦ ❬❪ ♦s st♦s ♠tsts ♣♥

♣rr ♥ ①♣r♠♥t♦ s s ♣r♦ ♦♥ s♥t ♦ ♠♥r ♥♦ ♣r♠tr q sst♠ ♥

♥ st♦ qr♦ tr♠♦♥á♠♦ st

−0.9 −0.8 −0.7 −0.6 −0.5 −0.4

−1.498

−1.496

−1.494

∆µ/kB [K]

ω/k B

[K/n

m2 ]

Xe/Na T=245 K

C1

S1

S2

C2

Mcross

r r♥ ♣♦t♥ ♣♦r ♥ ár ♥ ♥ó♥ ♣♦t♥ qí♠♦ r♦ ♣r ❳ s♦r♦ s♦r ♥ ♣r ♥ t♠♣rtr ③②s③ ② rtr ❬❪ ♦♦s ♦s t♦s rá♦ r♦♥ ♦s ♥ ♦r♠s♠♦ r♥ ♥ó♥♦ r q ♦♥t♥ ♦s ♣♥t♦s C1 ② S1 ♦rrs♣♦♥ ♣ís s ♦s♦rs s♦r ♣r ♠♥trs q q ♦♥t♥ C2 ② S2 st tr♠♥ ♣♦r ♣ís rss ♣♥t♦ r ♥♦ ♦♥ Mcross ♥t ♦①st♥ ♠s ♣ís

s♠♥ s♦r s♦ró♥

s♦ró♥

str♦s á♦s ♣r♥ ♦♦ ♣♦r ♥ ♥ r♦ ♦♥ ♦s rst♦s ①♣r♠♥ts ♦t♥♦s

♣♦r ss t♥ ② ♥ ❬❪ q ♦♥tr♥ s ♦♥s♦♥s s q s ♥♦ s ♠♣♥ s♠♦♥s

♥♠érs r♣♦rts ♥ r♦ s s ❬ ❪

♠♦s ♣♦♦ ♣♦♥r ♥ ♥ ♥ tr♥só♥ ♣r♠♦♦ ♥ s♦ sst♠ ♥ st s♣t♦

①st♥ t♦s ♠♣ír♦s ② st♠♦♥s tórs q ♦♥r♠♥ ♠♦♦ ♣♦r ♣r♦ st st ♠♦♠♥t♦

♥♦ s r♣♦rtr♦♥ á♦s q ♣♦♥♥ ♥ ♥ ①st♥ ♥ tr♥só♥ ♣r♠♦♦

Pr sst♠ r♥ ♦ q s ♦tr♦♥ ♥ st♦s ♦♥ t③♥♦ ♥♦♥s ts

♥str ♣r♦♣st ♦♥r♠ ①st♥ ♥ tr♥só♥ ♣r♠♦♦ t ♦ ♣r♥ á♦s

r ♣♦r ♠♣♦ ❬❪

♠é♥ ♦t♠♦s ♥ t♠♣rtr ♠♦♦ r♥ tr♣ ♥ ♦♥ t③ó♥ ♦s ♥♦♥s

ts ♣r♦♦ r♥s á♦ ♥♠ér♦ ♦r st t♠♣rtr st ♥ ♥ r♦ ♦♥

♦t♥ ♣♦r rtr♦♦ ② ♦♦r♦rs ❬❪

♣♦rt♠♦s á♦s s♦r sst♠ st ♠♦♠♥t♦ ♥♦ ①st♥ ♦tr♦s rst♦s s♦r st sst♠

t♠♦s ♥ ♥ st ♣r ♥s s♦r s♦r ♥ sstrt♦ t♥ rt ♦♠♦ ♥ st

s♦ ♦s ♦rs q ♦t♠♦s s ♦♠♣rr♦♥ ♦♥ ♦s ♦s ♥ s♠♦♥s r③s ♣♦r ♦♥ ②

♦♦r♦rs ❬❪

s ♠♣♦rt♥t ♥♦tr q ♠♥t s♠♦♥s ♥♦ s ♥ ♣♦♦ ♦t♥r r♠♥s ♣r♠♦♦ ♣r

t♠♣rtrs r♥s rít ❬❪

s♦ró♥ r

♥ st♦ s♦ró♥ r s♦r 2 ♥♦♥tr♠♦s strtrs s♦tr♠s s♦ró♥ ②s ♦r♠s

s♦♥ s s♣rs r s ②

Pr sst♠ r ♦♥ ♥♦ ①st♥ ♣rát♠♥t r♣♦rts s♦r st♦s ♠♦♦ ♥♦♥tr♠♦s q ①st

♠♦♦ r♥ ♦ q sñ♥ ♥♦s rst♦s ♦t♥♦s ♠♥t s♠♦♥s ♥♠érs

s ♥ ♦s ♦rs q rr♦♥ s s♠♦♥s s♦♥ t♦♠♦s ♦♠♦ rr♥ ♥ ♥♦s s♦s ♦

♠♦ ♣♦r ♠♣♦ ♥ sst♠ ❬ ❪ s ♣r♦♥s ♥♦ st♥ t♦♦ ♥ ♦s t♦s

①♣r♠♥ts

str♦s rst♦s s st♥ ♦s ♦rs ①♣r♠♥ts ♦s s ♣r♦♣s ♠♦♦ t♥t♦ ♣r 2

❬❪ ♦♠♦ ♣r ❬❪

♦sr ♥ s♦ ♣ró♥ ♥ strtr s♠♦♥♦♣s ♦♠♣♦rt♠♥t♦ s♠r r♣♦rt♦

♥ trtr ❬❪ ♥♦ s st s♦ró♥ r s♦r rt♦ ♦♥ ♥t♥s ♥tró♥ s t♥

♥t♥s ♦♠♦ sst♠ r

s♦ró♥ ❳

str♦ ♥♦♥ ♣r♠tó r♣r♦r ♥♦ s♦ ♣♦t♥s rsts strtr í♥s ♣r♠♦♦

s♦r s q s t♦r③♦ ♥ ♦tr♦s st♦s s♦r t♠ s ❬ ❪

♥♦♥tr♠♦s ♥ tr♥só♥ ♠♦♦ ♣r sst♠ ❳ ♣r ♥ t♠♣rtr ♠♥t ♥r♦r ♦s

② ♥ í♠t t♠♣rtr rít ♣r♠♦♦ ♥ ♦s

♥ s♦ ❳ ♦t♠♦s ♥ t♠♣rtr ♠♦♦ ♦r♥ ② ♥ í♠t ♣r Tcpw

Pr sst♠ ❳ t♠♣rtr ♠♦♦ q ♦t♥♠♦s s st♥t r♥♦ q ♣r

Tw = 201K ♦♠♦ s ♣ r ♥ r♦ ❬❪ st ♦r st ♣♦r ♦ ♦t♥♦ trés

s♠♦♥s ♥♠érs ♦♥ ♥ sts s♠♦♥s s ♦t♦

Pr sst♠ ❳ ♠♦s ♥♦♥tr♦ r♠s s♦tr♠s s♠rs s r♣♦rts ♥

❬❪

r♥♦ ♦♥ sst♠s rt♦s ♦♥ s ♥ ♣♦t♥ qí♠♦ ② ♥ t♠♣rtr s ♦s ♠s♠♦s

rst♦s q ♥♦ s tr ♦♥ sst♠s rr♦s ♦♥ n ♣r♠♥ ♦ s ♣♥t♦ st ♣r

t♦ ♦ ♠♥♦r ♦st♦ ♦♠♣t♦♥ q ♥s♠♥ ♦s á♦s s ♦♥♥♥t trr ♦♥ sst♠s rr♦s

rs♦♥♦ s s ♥ ♠r♦ ♦r♠s♠♦ ♥ó♥♦

♣ít♦

♥♦ r♥rs ♣♥s

♥③ s♦ró♥ r ② ❳ ♥ r♥rs ♣♥s ♦♥♦r♠s ♣♦r sstrt♦s ♠ts ♥♦s

② 2 q ♣r s♦ ♣rs s ♣rs♥t♥ ♦s rst♦s rs♦r s ♦♥s

♣r sst♠s t♠♣rtr ♦♥st♥t t♥t♦ rr♦s ♦♥ ♥ú♠r♦ ♣rtís ♦ ♦♠♦ rt♦s sst♠s q

♣♥ ♥tr♠r ♣rtís ♦♥ ♥ rsr♦r♦ ② ♣r ♦s s s ♠♥t♥ ♦ ♣♦t♥ qí♠♦

st♥ s ♣r♦♣s r♦♥s ♠♦♦ ② ♣r♠♦♦ s s♣rs q ♦♥♦r♠♥ r♥r s

♣r♦♣s s ♣ís s♦rs ② r♣tr s♠trí s s♦♦♥s

♣♥t ①st♥ ♥ ró♥ ♥tr s♠trí ♣r♠♦♦ ② ♠♦♦ ♥♥tr q ♦ rts

♦♥♦♥s ♥t♦ ♦s st♦s s♠étr♦s ♣♥ ♦①str st♦s s♠étr♦s

♦s á♦s r♦♥ r③♦s ♥tr♦ ♦♥t①t♦ ♥ ♣r♦①♠ó♥ ② ♥tró♥ ♦s

♦♥ sstrt♦s stá r♣rs♥t ♣♦r ♣♦t♥s rsts sr♣t♦s ♥ ♣ít♦s ♥tr♦rs s♦ ♣r s②st♠

r2 q ró ♥ trt♠♥t♦ ♣rtr

s♠♥ ♥t♥ts

s ♣r♠rs ♥st♦♥s s♦ró♥ ♦s s♠♣s ♥ r♥rs ♣♥s r♦♥ r③♦s ♣♦r ♥ ♥ ②

♦♦r♦rs q♥s ♥③r♦♥ st s sst♠s ♥♦ ♦ s♠♦♥s q í♥ s♦ ♥á♠

♦r r ♣♦r ♠♣♦ s s ❬ ❪ st♦s á♦s r♦♥ r③♦s ♥ t ♦r ②♥♠s

Pr♦ss♦r P s♣♠♥t ♦♥str♦ ♣r t ♥

♥tr ♦s ñ♦s ② ♥ t ❬ ❪ ② ♠r t ❬ ❪ ♣♥ ♦s rst♦s ss st♦s

♥ s♦s ♥áss s s♠í q s ♠♦és ♦ ♥trtú♥ ♥tr s ② ♦♥ ♦s át♦♠♦s q ♦♥stt②♥ s

♣rs r♥r ♠♥t ♣♦t♥s t♣♦ s ♣rs s r♥rs st♥ st♥s ♥

á♠tr♦s ♠♦rs st♥ q s ♦♥sr s♥t ♦♠♦ ♣r s♣rr s ♥tr♦♥s ♥tr s

♠♦és ♦ s♦rs ♥ s ♣rs ♦♣sts r♥r

♥ s s♠♦♥s t③s ♣♦r st♦s t♦rs s ♥③ó ♥ sst♠ ♦♥♦r♠♦ ♣♦r ♦s t♣♦s ♣rtís

t ♠♥r q rt ♥t ♣rtís ♥ s♣ ♦r♠♥ sstrt♦ só♦ ② ♥ ♠♥♦r ♥t

−8 0 8

0.4

0.8

1.2

z*

ρ*

−8 0 8

0.4

0.8

1.2

z*

ρ*

r ♥ ♠s rs s ♦sr ♦♠♦ s ♠♦ ♥s ♥ ♦ s♦r♦ ♥ ♥ó♥ st♥ s ♣rs st♦s rá♦s ♦rrs♣♦♥♥ s s ② ❬❪ ♦♥sr q s ♠♦és ♦♥trtú♥ ♥tr s ② ♦♥ ♦s át♦♠♦s q ♦♥stt②♥ s ♣rs r♥r ♠♥t ♣♦t♥s t♣♦ st♥ st ♠♥s♦♥③ ♦♥ á♠tr♦ ♠♦r t♠ñ♦ s ♠♦és ♦ σff ♥ rá♦ ③qr ♦s rst♦s q ♦rrs♣♦♥♥ W ∗

sf = ♠str♥ ♥ s♦ó♥ st q ♥ st s♦ ♣r

s♣és 104 ♣s♦s á♦ r③♦s ♦♥ ♣♦str♦r qr♦ s í♥s rts ♠ás rss♦rrs♣♦♥♥ sstrt♦ ② s s ♥♥ ♦♥♦♥s ♦♥t♦r♥♦ ♣rós ♥ rá♦ r s♦sr ♥ s♦ó♥ r♠♥t s♠étr ♦t♥ ♣r W ∗

sf = st ♣r st ♣r s♣és 104

♣s♦s á♦ ♣♦str♦rs qr♦

♦tr s♣ ♦♥♦r♠♥ ♦ s♦r♦ st ♦r♠ s str♦♥ ♥ó♠♥♦s ♠♦♦ r♥♦

ró♥W ∗

sf =Wsf/εff ♥ ♥tr♦r♠♥t ♥r♠♥t♥♦ ♣t♥♠♥t ♦r W ∗

sf ♥ s s♠♦♥s

♥♠érs s ♥♦♥trr♦♥ s s♥ts st♦♥s

Pr ♦rs ♦s W ∗

sf ♣r ♥s ♣rs♥t ♠s♠ s♠trí q ♣♦t♥ ♥tró♥

sstrt♦♦ s r ♣r r s♠étr♦ ♥ st s♦ ♣rí♥ ♦s ♥trss só♦♣♦r ❱ ② ♦s

íq♦♣♦r ❱ ♦rrs♣♦♥♥ts s♥ tr♥só♥ ♠♦♦

♥ s♦ ♦rs ♥tr♠♦s W ∗

sf s♦ó♥ ♣♦í r♦♠♣r ♦♥ s♠trí ♣♦t♥ ①tr♥♦ ♣r

♥♦ ♥t♦♥s ♥ ♣r s♠étr♦ stó♥ r t q s ♣rs♥t ♥ ♥trs só♦íq♦ ♦tr

❱ ② ♥ trr ❱ ♠♦♦ r s♦♦ ♣r ♦♥sst♥t ♥ ♣ró♥ ♥ ♣óst♦ ♦ s♣s♦r

♣r s♦r ♥ s ♣rs ♠♥trs q ♣♦r s ♦♥♥tr s♦r ♦tr ♣r

s ♦♥t♥ ♠♥t♥♦ ♦r W ∗

sf ♦s ♣rs ♦í♥ ♣rs♥tr s♠trí ♣♦t♥ ♥tró♥

sstrt♦♦ ② ♣rí♥ ♦s ♥trss ② ♦s ❱ ♥ st út♠♦ s♦ ♦ ♠♦ ♦♠♣t♠♥t ♠s

♣rs

♥ s ♠str♥ ♠♦♦ ♠♣♦ ♦r♠ st s ♣rs ♦t♥♦s ♦♥ ♦s á♦s

♠♥♦♥♦s ♥á♠ ♦r r ❬❪

t♦r s♠trí ♣♥ ás♠♥t ♥ s t♥s♦♥s s♣rs q ♣r♥ ♥♦ íq♦

♥trtú ♦♥ s ♣rs ❬❪ ú♥ ② ❨♦♥ ♣♦♠♦s srr ①s♦ ♥rí r ♣r s♠étr♦

♦♠♦

γtot = γSL + γLV + γSV = 2γSL + γLV (1 + cosθ) ,

r ♥s ♦ s♦r♦ ♥ ♥ó♥ st♥ ♥ s ♣rs ♥ r♥r ♦r♠ ♣♦r ♣rsé♥ts st rá♦ ♦rrs♣♦♥ ❬❪ st s♦ ♦♥ W ∗

sf = ♦rrs♣♦♥ ♥stó♥ ♠♦♦ ♣r ♦s t♦rs ♥ rá♦ ♥r♦r ♦♠♣r♥ ss rst♦s ♦t♥♦s ♠♥t ♦r♠s♠♦ r í♥ ♥ ♦♥ ♦s ♦t♥♦s ♥ ♦tr♦s tr♦s ♠♥t s♠♦♥s ♦♠♣t♦♥s r③s ♠♥t ♥á♠ ♦r r s s ❬ ❪ ♥ ♣rt s♣r♦r rá♦ s ♣♦srr ♣r ♥ s t♦t

♦♥

cosθ =γSV − γSL

γLV< 1.

s ♥r♠♥t tró♥ s ♣rs

γSV − γSL → γLV =⇒ cosθ → 1 ,

♦♥ ♦ sst♠ r③ ♥ tr♥só♥ ♣s♥♦ s♦ó♥ s♠étr s♠étr ♦♥

γtot = 2(γSL + γLV ) .

♥ st s♦ ♠s ♣rs r♥r s♦♥ ♠♦s ♣♦r íq♦

P♦str♦r♠♥t ♦s ♥áss ♥ ♥ st♦s ❱③♦ ② r③♦♥ ❬❪ ♥ ♥t ①st♥

♣rs s♠étr♦s ♦t♥♦s ♥ ♦♥t①t♦ ♥ ♣r♦①♠ó♥ ♥♦♠♥ q ② ♥♦s

♠♦s rr♦ ♥ ♣ít♦ ♦rrs♣♦♥ t rr♥ ♥ ♦♥ s ♦♠♣r♥

♦s ♣rs ♦t♥♦s ♠♥t s♠♦♥s ♥♠érs ♦♥ ♦s q s r♥ t③ó♥ ♦r♠s♠♦

st♦s á♦s ♦rrs♣♦♥♥ W ∗

sf = ♣r ♦t♥♦ ♣♦r ♠♦s ♠ét♦♦s s r♠♥t s♠étr♦ ②

♦rrs♣♦♥ ♥ stó♥ ♠♦♦ ♣r ♦♥ ♥ ♥trs ② ♦tr ❱ ♥ ♣r r r♥r

♦s á♦s r♦♥ r③♦s ♥ ♦♥t①t♦ ♦r♠s♠♦ ♥ó♥♦

♥ ♥ tr♦ ñ♦ r ② ö♥ ❬❪ r③♥♦ s♠♦♥s ♦♥t r♦ ② á♦s ♥♦♥

trr♦♥ q ♥ s♦ r♥rs rts ♥s ♦♥ ♦s t♠é♥ ♣♥ ♣rs♥trs ♣rs s♠étr♦s

♥t♠♥t r♠ ② ♥st♥ ❬❪ ♠♦strr♦♥ q ♦ rts r♥st♥s s♦♦♥s s♠étrs ♣♦rí♥

♦rrr ♥♦ r s s♦r♦ ♥ r♥rs rrs 2 ♠ás ②s♦ Ptr② ② ♦♦♦s ❬❪

s♦♥ ② ♦♥s♦♥ ❬❪ r♣♦rtr♦♥ ①st♥ ts s♦♦♥s ♦t♥s ♠♥t s♠♦♥s ♦♥t r♦

② ♥tr♦ ♠r♦ ♦rí ♥á♠ ♠♣♦ ♦ rs♣t♠♥t

♥st♦♥s ♣r♦♣s

♠♦s ♥③♦ s♦ró♥ rs♦s ♦s ♥♦s r ② ❳ t♥t♦ ♥ r♥rs rrs s r ♥s

♦♥ ♥ ♥t ♠♦és ♦♠♦ rts ♦♥ r♥r s ♠♥t♥ ♥ ♦♥tt♦ ♦♥ ♥ rsr♦r♦

♣rtís ♦ ♠♥r t q ♣♦t♥ qí♠♦ s ♠♥t♥ ♦ ♦♥s♥t♠♥ts ♦♥s

r♦♥ rsts ♥tr♦ ♦♥t①t♦ ♦s ♦r♠s♠♦s ♥ó♥♦ ② r♥ ♥ó♥♦ sr♣t♦s ♥tr♦r♠♥t

♠♥t ①♣♦♥♠♦s ♦s rst♦s st♦ s♦ró♥ r ❳ ② ♥ r♥rs ♣♥s rrs

♥ srr♠♦s ♦♠♣♦rt♠♥t♦ ❳ ② ♥ sst♠s rt♦s

♦ ①♣rs♦ ♥ ♣ít♦ s ♦♥s ♣r ♣♦t♥ qí♠♦ ② ♥s ❬s ② ❪

s ♦t♥♥ ♥srt♥♦ ♥ Q ❬ ❪ ♣♦t♥ ♥tró♥ sstrt♦♦ ♦ r L q ♥

s♦ ♥ ♣r r t♠ñ♦ ♦♥ s rs♦í sst♠ ♦r s ♥♦ r♥r ♦♠♦

♥ts s♦ó♥ s tr♠♥ tr♥♦ s ♦♥s st ♥③r ♥ tr♠♥♦ ♦r ♦♥r♥

♦♥♥♠♥t♦ ró♥ ♥ r♥rs ♣♥s rrs

♥ st só♥ ♣r♠r♦ ♥③♠♦s r ♥ s ♦♥s♦♥s ♣rs♥ts ♣♦r r♠ ② ♥st♥

♥ s tr♦ r♥t s♦r ♥♦ r♥rs 2 ❬❪ ♦♥ st ♥ ♣♠♦s ♠s♠ q t③r♦♥

♥ s st♦ st♦ s ♦♥ ♣♦t♥s é♥t♦s ♦s ♦♣t♦s ♣♦r s♦s t♦rs ♦ ♣rs♥t♠♦s

♥ rst♦ ♣r sst♠ r2 ♦t♥♦ ♦♥ ② ♥ ♣♦t♥ Usf (z) rst

♦♥t♥ó♥ ①♠♥♠♦s ♣♦s ♦t♥r s♦♦♥s s♠étrs ♣r ♣rs r s♦r♦s ♥

r♥rs ♠ts ♥♦s t③♥♦ t♦rí ② ♣♦t♥s rsts ❩

r ♥ r♥rs ♥ír♦ ró♥♦

♥ s rtí♦ r♠ ② ♥st♥ ❬❪ r♣♦rt♥ ①st♥ ♣rs ♥ss q r♦♠♣♥ s♠trí

♣♦t♥ q r♥ s ♣rs s♦r ♦ ♣♦t♥ ♥tró♥ sstrt♦♦ q t③♥ st♦s

t♦rs s q ♦♣♦rt♥♠♥t ♥♦s ♠♦s rr♦ ♥ ♣ít♦ ♥tr♦r ❬ ❪

♥ tr♦ ❬❪ ♦s t♦rs ♥③r♦♥ ♥ r♥r ♣♥ rr ♥♦ L ♦♥♦r♠ ♣♦r ♦s ♣rs

é♥ts 2 ♣sr♦♥ s♥t ♣♦t♥ ♣r ♥tró♥ ♦ ♦♥ s ♣rs

r ♦r♠ r♥r t③ ♣♦r r♠ ② ♥st♥ st r ♦rrs♣♦♥ ❬❪ qíW1 ② W2 ♦♥stt②♥ ♦s ♣rs sós 2 s♣rs ♣♦r ♥ st♥ L ♥ st rá♦ q ♦♦r♥ ③ ♥ st ♦ σfw1 ② σfw2 r♣rs♥t♥ ♦s á♠tr♦s s ♠♦és ♣r ♦♥srs ♦♠♦srs rís ♠♥trs q σff r♣rs♥t á♠tr♦ tó♠♦ ♦ ♦ s ♠s♠s ♦♥sr♦♥s r♥r s ♥♥tr rr ♥ ss ①tr♠♦s

Usf (z) = Usf1(z + σsf ) + Usf2(lw − z + σsf ) ,

♦♥ lw = L− 2σsf s ♥♦ t♦ r♥r ② s ♦♥tr♦♥s ♣r t♥♥ ♦r♠

Usfi(zi) =2π

3εeff

[

2

15

(

σsfzi

)9

−(

σsfzi

)3]

.

sí ♥ st♥ σsf s ♣rs rs q ♦♥♦r♠♥ r♥r ♥tr♦í♥ ♥ ♣♦t♥ ♥♥t♦ r♣s♦

♥ tr♥sr♠♦s tr♦ st♦s t♦rs ♦s ♣rá♠tr♦s r sstrt♦♦ q

♣r♥ ♥ st ♣♦t♥ εeff =Wsfρsσ3sf ② σsf ♥ Wsf/kB = 153 ρ∗s = ρsσ

3sf = ② σsf = ♥♠

♥ st ♦♥t①t♦ ♠♦s ♥♦tr q ♥ ♥ tr♦ s♦♥ ② rts ❬❪ s ♣r♦♣♦♥ q st ♣♦t♥

t♦♠ ♦r♠

Usf (z) = Usf1(z +σsf2

) + Usf2(lw − z +σsf2

)

st♦ s ♦③♥♦ r♣só♥ ♥♥t ♥ st♥ ♦rrs♣♦♥♥t σsf/2 s ♣r r r♥r

s♥♦ ♥t♦♥s ♥♦ t♦ r♥r lw = L− σsf

♥ s ♠str♥ ♠♦s ♣♦t♥s ♥ ♠s♠ rá ♣rs♥t♥ ♦s st♦♥s q ♦rrs♣♦♥♥

♦s ♣♦ss ♠♦♦s D r♣rs♥t st♥ s ♠♦és ♦ át♦♠♦s ♦ ♥ s ♣rs

♦t♥♠♦s

D =

z + σsf (BR)

z +σsf

2 (NG) ,

0 1 2 3 4 5 6 7 8 9 10−400

−300

−200

−100

0

100

200

ξ/10 [nm]

Usf

[K]

Ar/CO2

o minimum

σsf

σsf

/2

< >

< >

r P♦t♥s ♥tró♥ ♥tr r ② 2 ♥ ♥ó♥ st♥ ♥tr♠♦r ξ ③②s③ ② rtr❬❪ r só s ♦rrs♣♦♥ ♦♥ ♠♦♦ t③♦ ♣♦r r♠ ② ♥st♥ ♥ st s♦ s ♦♦ ♥st♥ σsf ♣r r r♥r ♥ r♣só♥ ♥♥t ♣r t ♠♣♥♦ st ♠♥r q ♣♦t♥ ♥ s ♠í♥♠♦ ♥ ♠♦♦ s♦♥ ② rts r r② ♣♦t♥ ♥♥t♦ ♦♦♦♥ σsf/2 ♣r♠t ♥③r s ♠í♥♠♦

q r♠♦s q ♦♥♥ r♥ts ♣r♦♣s ♠♦♦

st rs♣t♦ s ♣ ♦♥sttr q t♥ ♥ ♥t ♦♠♣♦♥♥t s♦t r♣só♥

tér♠♥♦ (σsf/z)9 ♣♦t♥ ♥trs q ♠♦♦ ♦rt ♣♦t♥ ♥ts q st

♠í♥♠♦ s ♥③♦ st♦ ♣r♦ r♥s ♠♣♦rt♥ts ♥ ♣r ♥s ♦ ♠♦♦

q ♥s ♥♦ s ♥ ♥ ♥ z = 0 ♥ ♥ z = lw ♦ q ♥ q ♦ st ♥ ♦♥tt♦ ♦♥

♣r rí q ♦♥♦r♠ r♥r ♠♥trs q ♠♦♦ ♦♥ q ♦ s ♦♥♥tr ♦r♠♥♦

♥ ♣r♠r ♣ q s ♠♥t♥ s♣r ♣r

♦ q st♦s ♦s ♠♦♦s s ♣rs♥t♥ ♦♠♦ ♣♦ss ♥♦s♦tr♦s ♥③r♠♦s ♠♥t ♥str ♣r♦♣st ♥

sq♠ q ♦♥t♠♣ s ♦s st♦♥s Pr ♦ ♣♦t♥ ♦♥ q trr♠♦s srá srt♦ ♦r♠

Usf (z) = Usf1(z +σsfν

) + Usf2(lw − z +σsfν

)

♦♥ ♣rá♠tr♦ ν t♦♠rá ♥t♦♥s ♦rs ♥tr ②

s♦♠♦s s ♦♥s ♥tr♦ ♦r♠s♠♦ ♥ó♥♦ tr♥♦ ♦♥ s rs ♠♥s♦♥s

z∗ = z/σff ρ∗ = ρσ3ff ② T ∗ = kBT/εff

Pr♠r♦ st♠♦s ♣r♦♠ ♦♥ ♣♦t♥ q t③r♦♥ rs♦♥♦ s ♦♥s ♥ ♥ r♥r

15σff ♥♦ ♥ t♠♣rtr t♠♣rtr ó♥ ♥♦r♠ T ∗ = ♣r r♥ts ♦rs

♥s ♠ ρ∗av = nσ3ff/lw = n∗/l∗w ♦s ♣rá♠tr♦s ♣♦t♥ ♥tró♥ rr s♦♥

εff = ② σsf = ♥♠ ♥♦ st r♥r ♥③♦ ♣♦st♥♦ ♥ ♥ú♠r♦ ♦ át♦♠♦s

r

ró♥ ♥tró♥ ♥tr♦ [0, l∗w] s ó ♥ ♥ qs♣♦ q ♣r♠t s♦♥s ♣♦r á♠tr♦

tó♠♦ st♦ s ♥♦ s ♠ás ♥♦ q t③♦ ♥ ♠♦♦ s s♦♦♥s s ♣t♥ s♦♦ s ♥♦r♠

rt r♥ ♥tr ♦s s♦♦♥s ♦♥sts s ♠♥t♥ ♣♦r ♦ 10−15

♥♦♥tr♠♦s q ♣♦r ♦ ♥ tr♠♥♦ ♦r ♥s ♠ ♥ st s♦ ♣r♦① ♦r q

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8−12.2

−12

−11.8

−11.6

−11.4

−11.2

−11

−10.8

ρ*av

µ/k BT

T = 87 K lw

= 15 σff

Ar/CO2

r P♦t♥ qí♠♦ ♥ ♥ó♥ ♥s ♠ ③②s③ ② rtr ❬❪ r♥r 15σff ♥♦ st♦♥stt ♣♦r ♦s ♣rs é♥ts 2 ♦ s r ♥ t♠♣rtr r só ♦rrs♣♦♥ st♦s ♥♠♥ts q ♣♦s♥ s♠trí ♣♦t♥ ①tr♥♦ ♣♦t♥ sstrt♦♦ í♥ r②♦rrs♣♦♥ st♦s ♥♠♥ts q ♦♥ s s♠trí ♦♠♦ s ♦♠♥t ♥ t①t♦ s s♦♦♥s s♠étrss ①t♥♥ ♥ ♥ r♥♦ ♥ss ρ∗sb1 < ρ∗sb < ρ∗sb2 ② ①t♥só♥ ♣♥ t♠♣rtr r ♣♥t♦s ② r②s st s♦ s♥ ♦♠♣t ♠♦♦ ② ♦rrs♣♦♥ ♥ tr♥só♥ s ♣r

♥♦♠♥r♠♦s ρ∗sb1 ♦s ♣rs ♥s ♠♥t♥♥ s♠trí ♣♦t♥ ①tr♥♦ ♣♦t♥ ♣r♦ ②

♠ás sr sts ♦rrs♣♦♥♥ st♦s ♥♠♥ts

s ♠♥t ♦r ♥s ♠ ♣r♥ ♦♠♦ s♦♦♥s ♥♠♥ts ♣rs q r♦♠♣♥ ♦♥

s♠trí ♣♦t♥ ①tr♥♦ st út♠ stó♥ s s♦st♥ st q s ♥③ ♥ ♦r tr♠♥♦

♥s ♠ q ♥ ♥str♦ s♦ ρ∗sb2 = ♣r♦ st ♦r st♦ ♥♠♥t r♦r

s♠trí ♣♦t♥ ①tr♥♦ qr s t♠♣rtr

❱ r q ♥♦♥tr♠♦s ♥ r♥♦ ♥ss ♠s

ρ∗sb1 < ρ∗sb < ρ∗sb2

t q ♥ t♠♣rtr s♦ó♥ ♥♠♥t r♦♠♣ ♦♥ s♠trí ♣♦t♥ ①tr♥♦ q s ♣

♦ ♠é♥ ♦♠♣r♦♠♦s q s ♠♥t♠♦s t♠♣rtr st r♥♦ s t♦r♥ ♠ás str♦

♥ s ♠str ♦♠♦ ♠ ♣♦t♥ qí♠♦ ♥ ♥ó♥ ♥s ♠ sts s♦♦♥s

♦rrs♣♦♥♥ts ♥ st♦ ♥♠♥t ♦♥ s♠trí r♦t ♣r♥ ♦♥ ♣♦t♥ qí♠♦ ♣rs♥t

♥ró♥ ♦♠♦ s ♣ r ♥ rá ♥ r♥♦ s♠trí ♥ ú♥♦ ♦r ♣♦t♥ qí♠♦ q♥

s♥♦s ♠ás ♥ ♦r ♥s ♠

♥ s ♣♥ ♦srr ♣rs ♥s ♦rrs♣♦♥♥ts r ♥ r♥rs ♣rs 2 ♦♥

ss r♥ts s♠trís st♦s ♣rs ♦rrs♣♦♥♥ ♥ stó♥ ♥♦ ♠♦♦ ♠♦♦ ♥ s♦ ♣r ②

♠♦♦ s ♦s ♣rs q ♦♥♦r♠♥ r♥r ② s ♦rrs♣♦♥♥ ♦♥ ♦s r♣♦rt♦s ♥ s s ②

❬❪ ♦t♥♦s ♠♥t ♥á♠ ♦r ♦♠♦ s ♠str ♥ r ♣r ♦ r♦♠♣ ♦♥

s♠trí ♣♦t♥ ①tr♥♦

♥ t s ♠str♥ ♣r r♦s ♦rs ♥s ♠ s ♦rrs♣♦♥♥ts ♥rís rs q ♠♦s

♦t♥♦ sí ♦♠♦ s ♦t♥s ♣♦r r♠ ② ♥st♥ sr♥♦ ♦♠♥ q ♦♥t♥ s s♦♥s

0 2.5 5 7.5 10 12.5 150

0.25

0.5

0.75

1

1.25

z*

ρ*

T = 87 K lw

= 15 σff

ρ*av

= 0.3865

r ♥s ♦ s♦r♦ ♥ ♥ó♥ st♥ ♥ s ♣rs ♥ r♥r ♦♥♦r♠ ♣♦r ♣rsé♥ts 2 ③②s③ ② rtr ❬❪ s r♥ts st♦♥s s♦♥ s ♠♥♦♥s ♥ t①t♦ r ♣♥t♦s ② r②s ♦rrs♣♦♥ ♥ ♣r s♠étr♦ ♦♥ s♥ ♠♦♦ ♥ ♠s ♣rs s♦ r r②s s ♥ ♣♦s s♦ó♥ s♠étr ② ♥t ♥ ♠♦♦ ♣r s♦ r só s ♥♠♥t♥ s♦ó♥ s♠étr q ♣r♦ ♥ ♠♦♦ t♦t r♥r s♦ ♦s á♦s s r③r♦♥ ♣r ♥ r♥r 15σff ♥♦ t♠♣rtr ♥ ② ♥s ♠ ♥ ρ∗av =

♣♦r♥ts s q s r♥s s♦♥ ♣qñs

♥ s ♠str ♦ó♥ ♦s ♣rs ♥s ♦t♥♦s ♥r♠♥t♥♦ ♥ú♠r♦ ♣rtís q

s ♣♦st ♥tr♦ r♥r 15σff ♥♦ ♦♦s ♦s ♣rs s♦ sñ♦ ♦♥ ♥ú♠r♦ ♦rrs♣♦♥♥

s♦♦♥s s♠étrs r sñ ♦♥ ♦rrs♣♦♥ ♥ ♥s ♠ ρ∗av = ② r

♥ ♦♥ ♣♦s ♥ ♥s ♠ q ρ∗av = ♥tr sts ♦s rs ♣r ♥ sr rs

s♦s ♥ss ♥tr♠s ♣ ♦srr q t♦s sts rs ♣rs♥t♥ ♦s♦♥s ♠② rts

r ♥ s ♣rs r♥r ② q sts ♦s♦♥s ♥r♠♥t♥ ♦r ss ♠á①♠♦s ♠ q

♠♥t ♥s ♠

♥ ♣r ♠r♦ ♦♥ ♦rrs♣♦♥♥t s♦ó♥ s♠étr ♠②♦r ♥s ρ∗av = ♦s ♣♦s

r♦ ❱♦rs ♥rí r ♠♦t③ ♣r ♥s ♥ss ♠s ③②s③ ② rtr ❬❪ ♦♠♣r♥ ♦s♦rs ♥rí r t♥t♦ ♣r s s♦♦♥s s♠étrs ♦♠♦ ♣r s s♠étrs sí ♦♠♦ s s♦♥s♣♦r♥ts ♥tr ♦s rst♦s ② ♥str♦s rst♦s P❲ ♥ út♠ ♦♠♥ Fcap ♦rrs♣♦♥ ♥rí r ♥ s♦ ♦♥♥só♥ ♣r ♥ t♦♦s ♦s s♦s s t③ó ♥ r♥r 15σff ♥♦ t♠♣rtr s ó ♥ ② ν = 1 s ♥ss ♣r♦♠♦ ♥ ♥s kBT/σ

2ff ♣rt♥♥ ♥tr♦

♥ ♦♥ ①st♥ s♦♦♥s s♠étrs

♦♦♥s s♠étrs ♦♦♥s s♠étrsρ∗av P❲ r♥ P❲ r♥ Fcap

0 2.5 5 7.5 10 12.5 150

0.25

0.5

0.75

1

1.25

z*

ρ*

T = 87 K lw

= 15 σff

1 2 3

4

Ar/CO2

r ♥s ♥ ♥ó♥ st♥ ♥ s ♣rs ♣r r s♦r♦ ♥ ♥ r♥r ♣♥ ♥♦st 15σff ♦♥♦r♠ ♣♦r ♦s ♣rs é♥ts 2 ③②s③ ② rtr ❬❪ ♥ rá♦ s ♠str♥ ♣rs s♠étr♦s ♦rrs♣♦♥♥ts st♦s ♥♠♥ts q s ♦t♥♥ rs♦♥♦ s ♦♥s ♣r r♥ts♥ss ♠s s r sñ ♦♥ st sñ ♦♥ ♦s ♦rs ♥s s♦♥ ② ♣r ♠r♦ ♦♥ ♦rrs♣♦♥ ♥♣r ♦♥ ♥♠♥t s rst♦ s♠trí

sts ♦s♦♥s ♦♠♥③♥ rr s r s♦♥ ♥r♦rs ♣♦r ♠♣♦ ♦s ♦rrs♣♦♥♥ts ♣r

♣rtr qí s s ♥r♠♥t ♥s s rr ♥ s♦ó♥ s♠trí ♣♦t♥ ①tr♥♦

♣r ♦rrs♣♦♥ ♥ ♣r s♠étr♦ s♦♦ ♥ ♥s ♠ ρ∗av = óts ♥ st s♦

q ♥ ♥r♠♥t♦ ♣♥s ♥ ♦r ♥s ♠ ♥r ♥ ♠♦ r♣t♦ ♥ s♠trí

s♦ó♥ s ♣s ♥ s♦ó♥ s♠étr ♥ s♠étr

♣s ♥ ♣r ♦♠♦ ♦tr♦ ♦♠♦ ♥♦ ♦♥ ♦rrs♣♦♥ ♥ r♥r q s ♥♦

íq♦ st tr♥só♥ s ♦♥♦ ♦♥ ♥♦♠r tr♥só♥ s ♣r ♦rrs♣♦♥♥t st♦

♥ ♣♦t♥ qí♠♦ q s ♣r♦ ♥ st stó♥ s ♣rs♠♥t ♥ ♠♥stó♥ ♠ás st t♣♦

tr♥s♦♥s

r♦ s♠trí s ♥t ♠♥t ♦♥t s♠trí ∆N q s ♥ ♦♠♦

∆N =1

2Ns

ˆ lw

0

|ρ(z)− ρ(lw − z)| dz

stó ♦♠♦ s ♠♦ s♠trí ♥ ♥ó♥ L r♥r ♣r T = 87 ♥ rá♦ ③qr

s ♠str♥ ♦s ♦rs ♦♥t s♠trí ♣r r♥rs r♥ts ♥♦s ♥ ♥ó♥

♥s ♠ ρ∗av ♣ ♦srr q s rs s♠trí ♦rrs♣♦♥♥ts s r♥rs ② σff ♥♦ ♥♦ r♥ sst♥♠♥t ♥ ♥t♦ s ♣r♥ r♥♦ ρ∗av ♦♥ ② s♦♦♥s s♠étrs

s♠♥② s♠♥r ♥♦ L∗ st t♥♥ q ♣r r♥rs σff ♥♦ s ♥♥tr♥ s♦♦♥s

s♠étrs ♣r t♠♣rtrs ♥tr tr♣ ② rít ♥ r r s ♠str ♦♠♦ s ♠♦

t♦r s♠trí ♣r r♥r σff ♦♥ t♠♣rtr ♦s rst♦s s♦♥ ♦♠♣ts ♦♥ ♦s q

♦tr♦♥ r♠ ② ♥st♥ ❬❪

0 0.1 0.2 0.3 0.4 0.5 0.6 0.70

0.2

0.4

0.6

0.8

1

ρ*av

∆N

30

15

12

7.5

6

T = 87 K

Ar/CO2

0 0.1 0.2 0.3 0.4 0.5 0.60

0.2

0.4

0.6

0.8

1

ρav

*

∆N

Ar/CO2

Iw

=15σff

T=104 K

T=96 K

T=87 K

r s rs ♦rrs♣♦♥♥ ♣rá♠tr♦ s♠trí ♥ ♥ó♥ ♥s ♠ ♣r r s♦r♦ ♥ ♥r♥r q st ♦♥stt ♣♦r ♦s ♣rs é♥ts 2 ③②s③ ② rtr ❬❪ r ③qr♦rrs♣♦♥ ♥str♦s á♦s ② ♠str ♦♠♦ ♠ st ♣rá♠tr♦ t♥t♦ ♠♦r ♥♦ r♥r♦♠♦ ♠r ♦r ♥s ♠ ♥ st s♦ ♠♦s ♦ ró♥ t♦r s♠trí ♣rr♥rs ♦♥ ♥♦s ② σff ♣r ♥ t♠♣rtr ♠é♥ s r♦♥ á♦s♣r ♥ r♥r σff ② í ♥♦ s ♥♦♥trr♦♥ s♦♦♥s s♠étrs ♣r ♥♥ú♥ r♥♦ ♥ss ♥ t♠♣rtrs ♥ r r s ♠str♥ ♦s rst♦s ♣r t♦r s♠trí ♦ ♣r r♥r σff r♥♦ t♠♣rtr t♥♥ s s♠r ♦t♥ s♠♥r ♥♦

P♦str♦r♠♥t s ♥③ó s♠trí ♥ ♥ó♥ ♣rá♠tr♦ ν q ♠♦ ♣r♥♣♠♥t ♣r♦♥

♣♦t♥ ♥tró♥ ♠♥♦ ♣♦só♥ ♥ ♣r t ♥ st♥ ♥tr σsf/2 ② σsf♦♠♦ s ♣ ♦srr ♥

Pr ♥ r♥r 15σff ♥♦ ♥♦♥tr♠♦s q ♥r♠♥t♦ ♥ ♦r ν ♣r♦ ♥ ró♥ ♥

♥tr♦ [ρ∗sb1, ρ∗

sb2] ♥ s ♠♥st r♣tr s♠trí s♣t♦ st út♠♦ t♦ s ♣

♦srr ♥ ♦♠♦ ♠ s♠trí ♥ ♥ó♥ ♥s ♠ ♣r ♥♦s ♦rs ν

♥ st♠♦s ♦ó♥ s♠trí ♥ ♥ó♥ ν ♣r ♥ ♥s ♠ ρ∗av =

str♦s á♦s rr♦♥ q s s♦♦♥s s♠étrs s♣r♥ ♥♦ ν >

P♦r út♠♦ ♥ s ♠str♥ ♦♠♦ ♦♦♥♥ ♦s ♣rs ♥s ♣r ♥♦s ♦rs ν ♥

rá s ♣ ♦srr ♦♠♦ s ♥♥♦ r ♦♥♥tr♦ ♥ ♣r ③qr ♥ r♥r 2

♦rrs♣♦♥♥t s♦ó♥ ♠②♦r s♠trí ♣r r ♠ q r ♦r t♦r ν

♣♦r ♥ ♠ q s♠♥② s♠trí ♥ s♦ ♥ q ♣rá♠tr♦ ♥③ ♦r ν = s♦ó♥

♣s sr ♦♠♣t♠♥t s♠étr ② ♦ ♠♦ ♠s ♣rs r♥r

♥♠♦s ♦r ♦ q ♦rr ♥ t♦r♥♦ ♦r ♣rá♠tr♦ ν q ♣♦rí♠♦s ♥♦♠♥r ♦♠♦ rt♦ ♣♦r

♥♠ ♥♦ ①st♥ s♦♦♥s s♠étrs ♣♦só♥ ♣r rí q s♠ ♣♦t♥ ♥♥t♦ q

♥tr♦♥ ♥ s tr♦ r♠ ② ♥st♥ st ♦③ ♥ ν = 1

♦♠♦ s ♠str ♥ ♣♦só♥ ♣r ③qr ♣r ♥tr♦r ♥ st r rs♣t♦

♣r r s

ξHW =σsfν

Pr ♥ r♥r ♦♥ ♥ ♥♦ t♦ 15σff ♦r ♠í♥♠♦ ♣♦t♥ ♥tró♥ sstrt♦♦ ♦

0 0.1 0.2 0.3 0.4 0.5 0.6 0.70

0.2

0.4

0.6

0.8

1

ρ*av

∆ N

T = 87 K

lw

= 15 σff

Ar/CO2

r t♦r s♠trí ♥ ♥ó♥ ♥s ♠ ♣r r♥ts ♦rs ♣rá♠tr♦ ν ♣r r s♦r♦♥ ♥ r♥r ♣rs é♥ts 2 ③②s③ ② rtr ❬❪ ♥ st r ♣ ♦srrs r♥♦ ♥ss ♣r ♦s s s♦ó♥ s s♠étr s r ♠ás ①tr♥ ♠ás ♥tr♥ ν t♦♠ ♦ss♥ts ♦rs ② r♥r ♣♦s ♥ ♥♦ σff ② t♠♣rtr st ♥

1 1.04 1.08 1.12 1.16 1.20

0.2

0.4

0.6

0.8

0

1

ν

∆ N

T = 87 K

lw

= 15 σff

ρ*av

= 0.1932

Ar/CO2

r t♦r s♠trí ♥ ♥ó♥ ♣rá♠tr♦ ν ♣r ♥ sst♠ ♦♥ ♥s ♠ ♦♥st♥t ③②s③ ② rtr❬❪ ♥ st s♦ ♥s ♠ t♦♠ ♦r ρ∗av = ♠♦s ♥♦♥tr♦ s♦♦♥s s♠étrs s♦♦ ♣r♦rs ν st r♥r ♣♦s ♥ ♥♦ σff ② st ♦♥stt ♣♦r ♦s ♣rs é♥ts 2 ♦ s r ♥ t♠♣rtr

0 2.5 5 7.5 10 12.5 150

0.2

0.4

0.6

0.8

1

1.2

z*

ρ* av

T = 87 K

lw

= 15 σff

ρ*av

= 0.1932

Ar/CO2

r Prs ♥s ♥ ♥ó♥ st♥ ♥ s ♣rs ♥ r♥r ♣rs é♥ts 2 σff ♥♦ ③②s③ ② rtr ❬❪ rá♦ ♠str ♦ó♥ ♦s ♣rs ♥s ♣r ♥♦s♦rs ♣rá♠tr♦ ν r só q ♦rrs♣♦♥ s♦ó♥ ♠②♦r s♠trí s s♦ ♦♥ ♦rν = ♥ r r② ν = Pr r ♣♥t♦s ② r②s ν = ② ♣r ♣♥t♦s ν = ♦r ♣r ♣r s ♦♠♣t♠♥t s♠étr♦ ♥ st út♠♦ s♦ ♦ s ♥♥tr ♥ s s íq♠♦♥♦ ♠s ♣rs

♣♦r s t♠é♥ ♥ ♠② ♥ ♣r♦①♠ó♥ ♠í♥♠♦ q ♥ ♣♦t♥ ♦rrs♣♦♥♥t

♥tró♥ ♦♥ ♣r ③qr r♥r st ♠í♥♠♦ st ♦③♦ ♥

ξmin =

(

2

5

)1/6

σsf

♦♥t sts st♥s ξmin

ξHW=

(

2

5

)1/6

νc ≃ 1,013

♦♠♦ ♦r st ♦♥t s♣r ♥ ♠í♥♠♦ ♣♦t♥ ♣r st νc s ♥③ ♥tr♦

r♥r Pr t♠♣rtrs ♠②♦rs ♦s ② ♦rs ν ♥tr ② ♥♦♥tr♠♦s q s s♦♦♥s s♠étrs

s♣r♥ ♣r ♦tr♦ ♦r νc q ♥♦♠♥♠♦s rít♦

♠é♥ s ♥ ♦ á♦s ♥ r♥rs ♠② ♥s ② st 120σff ♥♦ ♥ s s ♣s

♥ stó♥ s♠trí ♥ s♠trí s ♣♦í ♣r♦r ♥ ♠♥r ♠② rs ♥♦ s r③

♥ ♠♦ ♠② ♣qñ♦ ♥ ♥ú♠r♦ ♣rtís

Pr ♥③r ♥áss sst♠ r2 ♣♠♦s t♦rí ♦♥ ♣♦t♥ Usf (z) rst ♦♥tr♦

♣♦r str t ❬❪ q ② ♥tr♦♠♦s ♥ ♣ít♦ ♥ s ♠str q st ♣♦t♥ rst

s ♠ás ♥t♥s♦ q st rtríst ♥♦ ♦r ♣ró♥ ♣rs s♠étr♦s ♥ ♥ r♥r ♦r♠

♣♦r ♦s ♣rs é♥ts 2

♥ s ♣ ♦srr ♥ sr ♣rs s♠étr♦s q ♣r♥ ♥ ♥ r♥r 60σff ♥♦ s

♥♥♦ ♦♥ r t♠♣rtr s ♣♥t♦ tr♣ Pr t♦♦s ♦s ♣rs ♥s ♠

íq♦ ♦♠♦ ② ♦ ♠♦s ♦ ♥ ♣ít♦ ♣r st sst♠ ♦♥ 2 ♦♥stt② ♥ sstrt♦

rt♠♥t trt♦ ♠♦s ♥♦♥tr♦ tr♥só♥ ♠♦♦ s t♠♣rtr ② ♣♦r ♦ t♥t♦ ♥♦ ② í♥s

0 0.2 0.4 0.6 0.8 1 1.2−600

−400

−200

0

200

z [nm]

V[K

] Ar/CO2

r P♦t♥s ♥tró♥ r ♦♥ 2 ♥ ♥ó♥ st♥ ♦rrs♣♦♥ ❬❪ r ♥ í♥ r② ♦rrs♣♦♥ ♣♦t♥ ♠♥trs q r só s ♣♦t♥ t③♦ ♣♦r ♦st♦rs t rr♥ ♦♠♦ ♣ ♦srrs st ♣♦t♥ s ♠ás ♥t♥s♦ q ♦ ♥♦♦r ♣ró♥ s♦♦♥s s♠étrs

♣r♠♦♦ ❬❪ ♦♥s♥t♠♥t s r③♦♥ q ♥♦ s ♦t♥♥ ♣rs s♠étr♦s r ♣r ♥♥♥

t♠♣rtr

r ♥ r♥rs ts ♥♦s

♥ st ó♥ ♥st♠♦s ♥♦ r♥rs ② ♦♥ r ♦s á♦s s r③r♦♥ s♥♦

② ♣♦t♥s rsts ❩ ♠♦s ♥♦♥tr♦ ♥ ♦♥①ó♥ ♥tr ♣r♦♣s s♠trí ② ♠♦♦

ú♥ ♣♦r ♥♠ t♠♣rtr ♠♦♦ Tw ♦t♠♦s s♦♦♥s s ♦♥s ♦rrs♣♦♥♥ts

st♦ ♥♠♥t q r♦♠♣♥ ♦♥ s♠trí ♣♦t♥ sstrt♦♦ sts s♦♦♥s ♣rsst♥ ♣r

t♠♣rtrs T > Tw st q s ♥③ t♠♣rtr rít ♣r♠♦♦ Tcpw ♣rtr í ♥♦ s

♥♦♥trr♦♥ ♥tr♦s ♥s ♥ ♦♥ s s♦♦♥s ♥♠♥ts s♥ s♠étrs r♠♦s ♥t♦♥s q

t♠♣rtr s rst s♠trí st♦ ♥♠♥t s Tcpw

ú♥ ♥str♦ sr ② ♥t♥r ♥♦ ② ♥ trtr ♦tr♦s tr♦s ♦♥ s r♣♦rt st ♦ ♥ sst♠s

ís♦s ♦♥ ♣♦t♥s rsts

♥ ♥str♦ st♦ s♦ró♥ s♦r ①♣♦r♠♦s ró♥ ♥tr s♠trí ♦s ♣rs ② ♥♦ s

r♥rs Pr st ♣r♦♣óst♦ tr♠♦s ♦♥ r♥rs ② 60σff ♥♦ ♦♠♦ ♠♣♦ ♣♦♠♦s tr

q ♦ ♥s ② ❲♥ ❬❪ ♥ tr♦ ♦♥ r♥rs ② 40σff ♣r r ♥t ss st♦s

s♦r ♦♥♥só♥ ♣r íq♦s ♥tr♦ ♦r♠s♠♦

s r♥rs ♥s r♥t③♥ q ♥tró♥ ♥tr s ♠♦és s s♦rs ♥ s ♣rs ♦♣sts s

♣rát♠♥t s♣r ❯♥ ♥♦ 40σff ♣r st ♦ s♦r♦ ♥ ♦s sstrt♦s st♦s s ♠ás q

0 10 20 30 40 50 600

1

2

3

4

z*

ρ*ρ*

l

Ar/CO2 T=83.78 K

r ♥s ♥ ♥ó♥ st♥ ♥ s ♣rs ♣r r s♦r♦ ♥ ♥ r♥r ♣♥ ♥ 60σff ♦♥♦r♠ ♣♦r ♦s ♣rs é♥ts 2 rtr ② ③②s③ ❬❪ t♠♣rtr s ♣♥t♦ tr♣ ♥ r s ♠str♥ r♦s ♣rs ♦rrs♣♦♥♥ts ♥ r♥♦ ♥ss q ♥ s st ♦♥ ♥ ♣s♦ ♦♥st♥t í♥ ♦r③♦♥t ♥ ♦♥ ρ∗l ♦rrs♣♦♥ ♥s ♠ s q ② Pr st sst♠ ♦♥ s ♣rs 2 ♦♥stt②♥ ♥ sstrt♦ rt♠♥t trt♦ ♥♦ ♠♦s♥♦♥tr♦ s♦♦♥s s♠étrs

s♥t ♣r r♥t③r st♦

s ♦♥♥♥t str q ♦s á♦s r③♦s ♠♥t s♠♦♥s q ♠♣♥ ♥á♠ ♦r ♦rt♥

♣♦t♥ ♥tró♥ ♣rs ♦ ♥ ú♥ ♦r tr♠♥♦ r♦s tó♠♦s ♠ás ♥r♠♥t

♥ sts s♠♦♥s s tr ♦♥ r♥rs ♥♦ ♠s♦ ♥s ② ú♥ ♣r r♥rs ♥♦sts 10σff ♥♦

♥ ♥♦s s♦s s ♦♥sr♦ q ♥tró♥ ♥tr s ♣♦r♦♥s s ♦ss s ♣rs ♦♣sts

r♥r s s♣r

♦s á♦s ♥str♦s r③♦s ♥ ♦♥t①t♦ ♥♦ t♥♥ ♦rts ♥ ♣♦t♥ ② ♦st♦ ♦♠♣t♦♥

s r③♦♥ ú♥ tr♥♦ ♦♥ r♥rs ①tr♠♠♥t ♥s st 120σff ♥♦

r♦ ♦ sñ♦ ♣r♠♥t ♦t♠♦s s♦♦♥s s♠étrs ♦rrs♣♦♥♥ts st♦ ♥♠♥t

t♠♣rtrs ♠ás r♥s q Tw q ♥ s♦ r s♦r♦ ♥ sú♥ ♥str♦s á♦s s Pr

st sst♠ í♥ ♣r♠♦♦ t♥ ♥ ♦♥t ♠♣♦rt♥t ①t♥é♥♦s st s r♦s ♣♦r ♥♠

t♠♣rtr ♠♦♦ t ♣ ♦srrs ♥ rá♦ r ♣ít♦

♥ s ♠str♥ ♥♦s ♦s ♣rs ♦t♥♦s ♥♦ r s s♦r ♥ ♥ r♥r ♥ 40σff

♦♥♦r♠ ♣♦r ♣rs é♥ts T = 115 Tw í ♣♥ ♦srrs ♣rs s♠étr♦s q

♦rrs♣♦♥♥ st♦s ♥♠♥ts sñ♦s ♣♦r ② ♥t♦ ♥ ♣r s♠étr♦ ♠②♦r ♥s ♠

♦♠♦

s ♠♣♦rt♥t str q ①st♥ ♦s s♦♦♥s s♠étrs ♥rs ♥ r s rs ② ♦rrs

♣♦♥♥ ♣rs s♠étr♦s ♦♥ r st s♦r♦ ♠②♦r♠♥t ♥ ♣r ③qr r♥r s♦ó♥

q ♥♦♠♥r♠♦s ♣é♥♦s ♦t♥r ♥ ♣r t♦t♠♥t q♥t s♣r st ♦♥ r

st s♦r♦ ♠②♦r♠♥t ♥ ♣r r s♦ó♥ ♥♦♠♥ ♦♠♦ ♠♥♦♥♠♦s ♦s ♣rs

② s♦♥ ♥r♦s

♥ s ♠str ♥í r s♦ s s♦♦♥s t♥t♦ s♠étrs ♦♠♦ s♠étrs ♣r T = 115

♥ st r ♣♦♠♦s ♦srr ♥ r♥♦ st♥t ♠♣♦ ♥ss ♠s ♥ ♦♥ s s♦♦♥s q r♦♠♣♥

s♠trí ♣♦t♥ ①tr♥♦ ♣♦s♥ ♠♥♦r ♥rí q s s♦♦♥s q ♠♥t♥♥ s s♠trí ♦s♦tr♦s ♠♦s

0 5 10 15 20 25 30 35 400

0.2

0.4

0.6

0.8

1

1.2

1.4

z*

ρ*

Ar/Li T=115 K

1

2 3

r ♥s r s♦r♦ ♥ ♥ r♥r ♥ ♥ó♥ st♥ ♥ s ♣rs r♥rrtr ② ③②s③ ❬❪ r♥r ♣♦s ♥ ♥♦ 40σff ② s ♦♥s s rs♦r♦♥ ♣r ♥t♠♣rtr s rs ♦rrs♣♦♥♥ r♥ts ♦rs ♥s ♠ r sñ ♦♥ ♦♥stt② ♥ ♣r s♠étr♦ ♥♦st♦ ② ♥s ♠ s ρ∗av = ♣r sñ♦ ♦♥ ♣rρ∗av = s t♠é♥ s♠étr♦ ② s s♠étr♦ ♦♥ ρ∗av =

0 0.05 0.1 0.15 0.2 0.25−11.5

−11.4

−11.3

−11.2

−11.1

−11

ρ*av

f DF/k

BT

Ar/Li T=115 K

r ♥rí r ♣♦r ♣rtí ♥ ♥ó♥ ♥s ♠ ♣r r s♦r♦ ♥ ♥ r♥r ♣rs é♥ts ♥ t♠♣rtr rtr ② ③②s③ ❬❪ ♥ st s♦ r♥r ♣♦s ♥ ♥♦ 40σff r ♥ ♦♥ ír♦s ♦rrs♣♦♥ s♦♦♥s s♠étrs ♠♥trs q trá♥♦s s s♦ s s♦♦♥s s♠étrs ♦♠♦ ♠str rá♦ ①st ♥ r♥♦ st♥t ♠♣♦ ♦♥ st♦ ♥♠♥tr♦♠♣ ♦♥ s♠trí ♣♦t♥ ①tr♥♦ st r♥♦ s ①t♥ st ♥ ♦r ρ∗av =

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.80

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

ρ*av

∆ N

Ar/Li

L*=10

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.80

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

ρ*av

∆ N

Ar/Li

L*=20

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.80

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

ρ*av

∆ N

Ar/Li

L*=40

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.80

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

ρ*av

∆ N

L*=60

Ar/Li

r ♦♥t s♠trí ♥ ♥ó♥ ♥s ♠ ♣r r s♦r♦ ♥ r♥rs ♣rs é♥ts ♦ r♥ts t♠♣rtrs rtr ② ③②s③ ❬❪ ♠str♥ rst♦s ♣r L∗ = 10, 20, 40 ② 60σff s s rs ♠ás ①tr♥s s ♠ás ♥tr♥s s t♠♣rtrs ♥ ②

♦t♥♦ q st r♥♦ s ①t♥ s ρ∗av = st ρ∗av = ♣r r s♦r♦ s♦r ♥♦

t♠♣rtr s Pr t♠♣rtrs ♠ás ts r♥♦ s♠♥② ② ♣♦r ♥♠ ♦s ♥♦ s

♥♦♥trr♦♥ s♦♦♥s s♠étrs

♥ ♠♦str♠♦s rst♦s ♣r t♦r s♠trí ∆N ♥ ♥ó♥ ♥s ♠ ♣r r♥rs

r♥ts ♥♦s Prs♥t♠♦s rá♦s ♣r ♥ r♥r ♥♦st 10σff ♥ ♦♥ ♥♦ s ♣♥ s♣r

r ♦♠♣t♠♥t ♦s t♦s s ♥tr♦♥s ♥tr ♠♦és s s♦rs ♥ s ♣rs ♦♣sts q

♦♥stt②♥ r♥r ② ♠ás ♥s ② 60σff

♥ rá♦ ② rs s ♣r r♥ts t♠♣rtrs ♦♠♦ ♣ ♦srrs ♥ t♦♦s ♦s rá♦s

♣r ♥ r♥♦ ♥ss ♠s ♥ st♦ ♥♠♥t r♦♠♣ s♠trí ♣♦t♥ ①tr♥♦ ②

s r♥♦ s r ♠ q t♠♣rtr ♠♥t P♦r ♥♠ í♠t s♣r♦r Tcpw s s♦♦♥s

s♠étrs s♣r♥

♠é♥ s ♦sr q r♥♦ s♠trí ♥ t♠♣rtr s ♠ás str♦ ② s ♥tr ♥ ♦rs ♠s

♣qñ♦s ρ∗av ♠ q s ♥r♠♥t ♥♦ r♥r

st ♦♠♣♦rt♠♥t♦ ①♣♦r♦ t♠é♥ ♥ s♦ró♥ s♦r sstrt♦s ♠ás és ② ♥ í♥s

♥rs s ♥♦♥tr♦ ♣r ♦s rtrísts t♦t♠♥t s♠rs ♠♦♦ ♠♣♦s s♦ró♥ r

♥ r♥rs ♦♥♦r♠s ♣♦r sstrt♦s ② 60σff ♥♦ s ♠str ♥ s s ②

Pr s♦ s st♦ ♦♠♣♦rt♠♥t♦ ♣r ♦rs t♠♣rtr tr♣ t♠♣rtr

q st ♣♥s ♣♦r ♦ t♠♣rtr ♠♦♦ Tw = ② ♥♦ ♦rs ♠ás ♣♦r ♥♠

t♠♣rtr ♠♦♦ s♠trí s♣r ♥♦ s s♣r ♥ t♠♣rtr q st ♣♦r ♥♠

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.80

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

ρ*av

∆ N

Ar/Na

L*=60

r ♦♥t s♠trí ♥ ♥ó♥ ♥s ♠ ♦rrs♣♦♥♥t r s♦r♦ ♥ rtr ② ③②s③❬❪ ♥ st s♦ s r♥rs s♦♥ s♥t♠♥t ♥s 60σff ♣é♥♦s s♣rr ♦♠♣t♠♥t ♦s t♦s s ♥tr♦♥s ♥tr ♠♦és s s♦rs ♥ s ♣rs ♦♣sts s s rs ♠ás ①tr♥s s♠ás ♥tr♥s s t♠♣rtrs t♦♠♥ ♦s s♥ts ♦rs ②

Tw

♥ s♦ s ♥③ó s♠trí ♣r t♠♣rtr tr♣ ♥ t♠♣rtr q st ♣♥s ♣♦r ♦

♠♦♦ Tw = ② trs ♦rs q stá♥ ♣♦r ♥♠ st t♠♣rtr ♥ st s♦ ♠♦s

♥♦♥tr♦ ♥ í♠t s♣r♦r ♣r Tcpw

Pr s♦ ♠♦s ♥♦♥tr♦ rtrísts ♠♦♦ t♦t♠♥t ♥á♦s

♠♣rtr í♠t r♣tr s♠trí

tr♠♥♠♦s ♦r ♠♥r sst♠át ♣r s♦ s♦ró♥ ♥ r♥rs t♠♣rtr ♠á①♠

♣♦r ♥♠ s♣r♥ s s♦♦♥s s♠étrs ♦♠♦ s♦♦♥s ♠ás ♥rí st ♥áss

♦ ♠♦s t③♥♦ ♦s ♦rs ♦s q r♦♥ ♦♦s ♥ ♦s rá♦s

Pr t♦♦s ♦s t♠ñ♦s r♥r t③♦s s s♦♦♥s s♠étrs s♣r♥ t♠♣rtrs ♠②♦rs ♦s

tr♠♥ ♥t♦♥s ♦r ♥s rt ♠ ρ∗av(crit) q s ♦rrs♣♦♥ ♦♥ ♠á①♠♦ ∆N

♣r L∗ st ♦r ♣ sr ①♣rs♦ ♦♠♦

ρ∗av(crit) =A

Bσ2ff

♦♥ σff ①♣rsrs ♥ ♥♥ó♠tr♦s Pr L∗ = 60 ♦t♥♠♦s A/B = 9/16 ② ♣r L∗ = 40 ♦t♥♠♦s

A/B = 17/24 Pr r♥rs 20σff st ♦♥t 55/48 ② ♣r r♥rs t♥ ♥♦sts ♦♠♦ 10σff 97/48

♠str ♦s ♦rs q t♦♠ ♦♥t s♠trí ♦rrs♣♦♥♥ts ρ∗av(crit) ♦s t♥t♦

♣r s♦♦♥s t♣♦ ♦♠♦ t♠♣rtrs ♠②♦rs s ♠♦♦

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.80

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

ρ*av

∆ N

Ar/Rb

L*=60

r ♦♥t s♠trí ♥ ♥ó♥ ♥s ♠ ♦rrs♣♦♥♥t r s♦r♦ ♥ ♣r r♥rs♠② ♥s rtr ② ③②s③ ❬❪ r♥r ♣♦s 60σff ♥♦ s s rs ♠ás ①tr♥s s ♠ás♥tr♥s s t♠♣rtrs t♦♠♥ ♦s s♥ts ♦rs ②

−0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8110

112

114

116

118

120

∆N

T [K

]

Tcpw

Ar/Li

r ♠♣rtr ♥ ♥ó♥ ♣rá♠tr♦ s♠trí ♣r s♦♦♥s ② ♦rrs♣♦♥♥t r s♦r♦ ♥r♥rs rtr ② ③②s③ ❬❪ st♦s á♦s ♥ s♦ r③♦s ♣r r♥rs ♥♦s 10σff trá♥♦s rr 20σff r♦s 40σff trá♥♦s ♦ ② 60σff ír♦s ♥s ♦rrs♣♦♥ ρ∗av = ρ∗av(crit) s rs sós ♦rrs♣♦♥♥ st s♦s t♦s ♠♥t ♣♦♥♦♠♦

2 3 4 5 6 7 8−0.85

−0.8

−0.75

−0.7

−0.65

−0.6

−0.55

−0.5

Γl

∆µ/k

B [K

]Xe/Na T=245 K

CS1

SS1

SS2

SA2

CS2

MS1

MA

MS2

SA1

r s♦tr♠s s♦ró♥ ❳ ♥ ♥ r♥r ♣♥ ♣rs é♥ts ③②s③ ② rtr ❬❪ t♠♣rtr T = st á♦ r③♦ t♥t♦ ♣r ♥ r♥r rr ♦♠♦ rt ♦s ♦rs s♦♦s s s♦♦♥s s♠étrs stá♥ ♦♥t♥♦s ♥ r q ♥ ♦s ♣♥t♦s ♠r♦s ♦♥ CS1 SS1 SS2 ② CS2 ss♦♦♥s s♠étrs stá♥ ♦♥t♥s ♥ r q ♥ SS1 SA1 SA2 ② SS2 ♦s tr♠♦s ♥♦s ♦♥ í♥s tr③♦s ♦rrs♣♦♥♥ s♦♦♥s ♦♥ dµ/dn < 0 rt ♦r③♦♥t ♣♥t♦s ② r②s s ♦♥stró♥ ① árs ♦s st♦s MS1 ② MS2 s♦♥ s♠étr♦s ② MA s ♥ st♦ s♠étr♦ st♦s st♦s s♦♥ sts

♦s t♦s r♦♥ st♦s ♦♥ s♥t ♣♦♥♦♠♦

T = Tsb + a2∆2N + a4∆

4N + a6∆

6N

s ♣ ♦srr q t♦s sts rs st s♦♥ ♦♥sst♥ts ♦♥ ♦r t♠♣rtr ♣♦r ♥♠

♥♦ ①st♥ s♦♦♥s s♠étrs Tsb =

♦♥♥♠♥t♦ ❳♥ó♥

♥ s♦ st s ♥♦ ♣rs♥t♠♦s ♣r♠r♦ ♥áss ♥♦ r♥rs rrs ② P♦r

út♠♦ ♦♠♦ ♠♣♦ r♥rs rts sr♠♦s ♦s rst♦s ♣r

❳ ♥ r♥rs rrs ts ♥♦s

♥ s rr♦♥ s♦tr♠s s♦ró♥ ♣r ❳ ♣♦st♦ ♥ ♥ r♥r q stá T = 245

st t♠♣rtr stá st♦ ♦ Tcpw ♦♠♦ ♣ ♦srrs ♥

♥ rá♦ s ♥♥ ♦rs ♣♦t♥ qí♠♦ r♦ ♦rrs♣♦♥♥ts t♥t♦ s♦♦♥s s♠étrs ♦♠♦

s♠étrs s s♠étrs stá♥ ♦♥t♥s ♥ r q ♥ ♦s ♣♥t♦s sñ♦s ♣♦r CS1 SS1 SS2 ② CS2

0 4 8 32 36 400

5

10

15

20

25

z*

ρ [n

m−3

]

Xe/Na T=245 K

r Prs ♥s ♦rrs♣♦♥♥ts ❳ s♦r♦ ♥ ♥ r♥r rr ♣rs é♥ts ♥t♠♣rtr ③②s③ ② rtr ❬❪ rá ♠str ♣rs s♠étr♦s ♦rrs♣♦♥♥ts tr♦♦rs t♦r r♠♥t♦ ② r ♥ í♥ rs Γℓ = ♦rrs♣♦♥ st♦♠r♦ ♦♥ MA ♥ ♠♦s ♥♦tr q st♦s ♣rs t♠é♥ s ♦t♥♥ ♣r r♥r rt

4 8 12 160

20

40

60

80

Γl

∆N

(%)

Xe/Na

r t♦r s♠trí ♥ ♥ó♥ t♦r r♠♥t♦ ♣r ❳ s♦r♦ ♥ ♥ r♥r rr ③②s③② rtr ❬❪ s t♠♣rtrs ♥ s r ♠r ♦♥ ♥ trá♥♦ st t♠♣rtr r sñ ♦♥ ♥ ír♦ ♥③♥♦ ♥ r♦ sts t♠♣rtrs stá♥ ♥ ♥tr♦ q s t♠♣rtr ♠♦♦ st t♠♣rtr rít ♣r♠♦♦

0 1 2 3 4 5 6 7−6

−5

−4

−3

−2

−1

0

Γl

∆µ/k

B [

K]

Xe/Li

2 3 4 5 60

5

10

15

20

25

Γl

∆ N (

%)

Xe/Li

r s♦tr♠s s♦ró♥ ❳ ② t♦r s♠trí ③②s③ ② rtr ❬❪ r ③qr ♦rrs♣♦♥ s s♦tr♠s s♦ró♥ ❳ ♥ ♥ r♥r ♥ st r ♣♥ ♦srrs s sss ♦♥str♦♥s ① ♣r ♥ s s♦tr♠s s t♠♣rtrs s♥♥ ♥ ♣s♦s ♥ r♦ s trá♥♦ ír♦ ♥ r r s ♠str t♦r s♠trí ♥ ♥ó♥ t♦r r♠♥t♦ ♦♠♦ s ♣ ♦srr ♣r ♠ás ♥ r♥♦ ♥ss ♣r ♦s s s s♦♦♥s r♦♠♣♥♦♥ s♠trí ♣♦t♥ ①tr♥♦ ♥ ♦s trs r♥♦s s t♠♣rtrs ♥ s trá♥♦s st ír♦s

♠♥trs q s s♠étrs stá♥ ♥s ♦ r♦ r q ♥ SS1 SA1 SA2 ② SS2 ♦s tr♠♦s

♥♦s ♦♥ í♥s ♦♥t♥s ♦rrs♣♦♥♥ st♦s sts ② ♠tsts ♦♥ dµ/dn > 0 ♥ ♠♦ s í♥s

tr③♦s ♦rrs♣♦♥♥ s♦♦♥s ♦♥ dµ/dn < 0 ♥ s ♠str♥ ♣rs s♠étr♦s ♣r r♦s

♦rs r♠♥t♦

♦♥t s♠trí ♣r st sst♠ rr♦ ♦ ♣r rs t♠♣rtrs stá ♦ ♥

í s ♣ ♦srr ♥ ♦♥t♥♦ s♦♦♥s s♠étrs

♥ ♠♦str♠♦s strtr s s♦tr♠s s♦ró♥ ❳ ♥ ♥ r♥r 40σff

♥♦ st r ♦♥ ♦♥ ♠♦str ♥tr♦r♠♥t ♣r ❳ s♦r♦ s♦r ♥ ♣r Pr

♥ sts s♦tr♠s s ♥ ♦ s ♦rrs♣♦♥♥ts ♦♥str♦♥s ① árs s

♣♦str ❳ ♥ r♥rs r♥ s♦ ♥♦ r♥rs ♠♦s ♥♦♥tr♦ q ♥

t♠♣rtr ♣♥ ①str ♠ás ♥ ♥tr♦ ♥ss ♣r ♦s s s ♣rs♥t♥ r♣trs ♥

s♠trí s s♦♦♥s

♠str ♦♥t s♠trí í s ♦sr♥ trs ♥tr♦s ♥s ♥ r♥♦s ♣r

♦s s s s♦♦♥s ♦t♥s ♣rs♥t♥ s♠trí ♥ st s♦ r♥ ♦s st♦s ♥tr♦r♠♥t

s♦r ♠s ♣rs ♦ s s♦r ♥ s íq s ♠♣♦rt♥t rstr t♠é♥ q q ♥ ♦s s♦s

①♣st♦s ♥tr♦r♠♥t ♣♦r ♥♠ rt ♥s s♦ó♥ r♣r s♠trí ♣♦t♥ ①tr♥♦

♥ ♦s rá♦s ② s ♠str♥ ♣rs ♥ss q ♦rrs♣♦♥♥ ♥♦

♦s ♥tr♦s ♥s ♠♦str♦s ♥ í s ♦♠♣r♥ ♥ ♣r s♠étr♦ ♦♥ s♠étr♦ st

♠♥♦r r♠♥t♦ óts q ♥ t♦♦s ♦s s♦s s♠trí ♦s ♣rs s♠♥② ♥♦ ♠♦s ♣s♥♦

♣r♠r ♥tr♦ út♠♦

♥ s ♣ ♦srr t s♦tr♠ ♦♥str t♥t♦ ♦♥ s♦♦♥s s♠étrs r

tr③♦ ♦♥t♥♦ ♦♠♦ s♠étrs tr③♦ s♦♥t♥♦ s♦ ♦s ♣rs q s ♠str♥ ♥ s

0 3 6 34 37 400

10

20

30

40

z*

ρ [n

m−3

] Xe/Li T=205 K

(a) 0 3 6 34 37 400

10

20

30

40

z*

ρ [n

m−3

] Xe/Li T=205 K

(b)

0 3 6 34 37 400

10

20

30

40

z*

ρ [n

m−3

] Xe/Li T=205 K

(c)

r ♥s ♦rrs♣♦♥♥t ❳ s♦r♦ ♥ ♥ r♥r ♣rs é♥ts ③②s③ ② rtr ❬❪ r♥r ♣♦s ♥ ♥♦ 40σff ♦s rá♦s ② stá♥ s♦♦s ♣r♠r s♥♦ ② trr ♥tr♦ ♥s rs♣t♠♥t ♣r ♦s s ①st♥ s♦♦♥s s♠étrs r ♥ t♦♦s ♦s rá♦ss ♦♠♣r ♣r s♠étr♦ ♦♥ s♠étr♦ ♠♥♦r r♠♥t♦ ♥ s♦ rá♦ ♣r s♠étr♦♣♦s ♥ ♦r Γ = ② s♠étr♦ Γ = ♥ rá♦ t♦r r♠♥t♦ t♦♠ ♦s ♦rs ② ♥ rá♦ ♣r s♠étr♦ ♣♦s ♥ ♦r Γ = ② ♣r s♠étr♦ Γ = ótsq st s♦ ♦♥t s♠trí ♥♦ s♣r 5%

5.4 6−1.69

−1.68

−1.67

∆µ/k

B [K]

2 3 4−4

−3

−2

Γl

∆µ/k

B [K]

Xe/Li

T=205 K

r P♦t♥ qí♠♦ r♦ ♥ ♥ó♥ t♦r r♠♥t♦ ♣r ❳ s♦r♦ ♥ ♥ r♥r ♣rsé♥ts ♥ t♠♣rtr ③②s③ ② rtr ❬❪ rá♦ r t♦♦ r♥♦ ♥ss♥ ♦♥ s ♠♥st♥ s s♦♦♥s s♠étrs s rs sós ♦rrs♣♦♥♥ s s♠étrs ② s r②s s s♠étrs ♥ st rá♦ ♥♦ s ♠str♥ s ♦♥str♦♥s ① ♦rrs♣♦♥♥ts sts s♦tr♠s

−20 −10 0 10 20−120

−100

−80

−60

−40

−20

0

∆N

(%)

∆ f D

F/kB [

mK

]Xe/Li

1

2

3 4

r ♥rí r ♣♦r ♣rtí ♥ ♥ó♥ t♦r s♠trí ♣r ❳ s♦r♦ ♥ ♥ r♥r ♣rs é♥ts ③②s③ ② rtr ❬❪ ♥ r s ♣♥ ♦srr s rrrs ♥rí ♥tr s s♦♦♥s s♠étrsír♦s ② s s♠étrs trá♥♦s ♣r s t♠♣rtrs ② s q s s♥r♦♥ ♦s♥ú♠r♦s ② rs♣t♠♥t ♦s ♥♦ s ♥♦♥tró ♥♥♥ rrr

s s♦♦♥s s♠étrs ♣r♥ ♥ ♥ ♠á①♠♦ rt♦ ♣♦t♥ qí♠♦ ♦rrs♣♦♥♥t s s♦♦♥s

s♠étrs q ♠r♠♦s µsym ② s♣r♥ ♥ s♥t ♠í♥♠♦ rt♦ st ♠♥r s s♦♦♥s

s♠étrs ♣r♥ s♦♦ ♥ ró♥ ♦♥ ♣♥♥t µsym s ♥t ♦ q ♠♣ ♥st s

s♦♦♥s s♠étrs ♥rs♠♥t ♣♥♥t ♣♦t♥ qí♠♦ s♦ó♥ s♠étr µasym s ♥t

s♦r ♥ r♥ ♣♦ró♥ ró♥ ♥ ♦♥ s sts♥ s ♦♥♦♥s st s s♦♦♥s s♠étrs

P♦r ♦tr♦ ♦ ♣♦♠♦s r ♥ st♠ó♥ rs rrr ♥rét q s♣r ♠s s♣s

s♦♦♥s s♠♥♦ ♥ t♠♣rtr T ② ♥ ♥ú♠r♦ ♦ ♣rtís ♣♦r ♥ ár ①♣rsó♥

ρ(z) = (1− α)ρasym(z) + αρsym(z)

♥♦s ♣r♠t ♣sr s ♦s st♦s s♠étr♦s ♦s s♠étr♦s ♠♦♥♦ ♥tr ② ♦r ♣rá♠tr♦ α

♥t♦ ♥rí r ♣♦r ♣rtí fDF = FDF /N ♦♠♦ ♦♥t s♠trí ∆N s♦♥ ♦s r♥♦

♦r st ♣rá♠tr♦

♥ s ♣ ♦srr ♥ ♦♠♣ró♥ s♦r ♠s♠ s ♥rí ♣r rs t♠♣rtrs

í s ♠str♥ s rrrs ♥réts ♦rrs♣♦♥♥ts s♦♦♥s s♠étrs ② s♠étrs ♦♠♦ ♥ ♥ó♥

t♦r s♠trí ♣r ♥ ♦r ♦ n q ♥ st s♦ s st ♦r ♦rrs♣♦♥ ♣r

♥s ♠á①♠ s♠trí

Pr ♦♦r t♦♦ ♥ ♠s♠ s ♥rí s rst♦ ♥ ♦♥st♥t q ♠♠♦s f0DF st ♠♥r

♥ ♦r♥s s t♥ ♦s ♦rs ∆fDF = fDF − f0DF

str♦s á♦s rr♦♥ q rrr ♥rí s♠♥② ♦♥ ♥r♠♥t♦ t♠♣rtr ② s♣r ♥

t♠♣rtr

❳ ♥ r♥rs rrs ♥s♦

2 3 4 5 6 70

3

6

9

12

15

Γl

∆ N (

%)

Xe/Mg

r t♦r s♠trí ♣r sst♠ ❳ ♥ ♥ó♥ t♦r r♠♥t♦ ③②s③ ② rtr ❬❪ ♦♠♦s ♣ ♦srr ♣r ♠ás ♥ r♥♦ ♥ss ♣r ♦s s s s♦♦♥s r♦♠♣♥ ♦♥ s♠trí ♣♦t♥ ①tr♥♦ ♦s trá♥♦s ♦ ♦rrs♣♦♥♥ t♠♣rtr tr♣ ♦s ír♦s Tnb ♦s trá♥♦s rr ② ♦s r♦s

r♥sr♠♦s s♦♦ ♥♦s rst♦s ♥str♦ st♦ s♦ró♥ s♦r ♥ sstrt♦ t♥ rt ♦♠♦

♣s s ♣rs♥t♥ rtrísts s♠rs ❳ ♠♦ ♠♥r ♦♥t♥ ♣rtr

t♠♣rtr ♣♥t♦ tr♣

str♦s á♦s ♥♦s ♣r♠tr♦♥ ♥♦♥trr ♣r rss t♠♣rtrs ② ♣r sstrt♦s rt♠♥t rts

♦♠♦ ② st♦♥s ♥ ♦♥ ♣r ♠ás ♥ ♥tr♦ ♥s ♣r♥ s♦♦♥s s♠étrs

r s♦s ♥tr♦s st♦ ♥♠♥t r s♠trí ♣♦t♥ ①tr♥♦ Pr♥♣♠♥t ♣♦♠♦s

rstr ♥áss st sst♠ q t♠é♥ ♣r♥ ♠út♣s ♥tr♦s ♦♥ ♣♥ ♣rs♥trs st♦s

s♠étr♦s s ♥á♦ ♥ s ♣♥ ♦srr trs ♥tr♦s ♠② str♦s ♦♥

♣♥ ♣rs♥trs st♦s s♠étr♦s ♦♠♦ ♣r ♦s ♠ás sstrt♦s st♦s ♥tr♦s s♦♥ ♠ás ♥♦st♦s ♠

q s t♠♣rtr ♥ ♥á♦ s ♣♥ ♦srr ♦s ♣rs ♥♦ ♦s

♣♥s ♣♦r ♥♠ ♦tr♦ ♦s t♠♣rtr ♣♥t♦ tr♣ s♦♦s r♠♥t♦s q ♥

② ♣r r s♣r♦r

❳ ♥ r♥rs rts ♦♦

♥③♠♦s s♦ró♥ ❳ ② ♥ r♥rs rts ♦♠♣r♥♦ ♥ ♥♦s s♦s ♦♥ ♦s rst♦s ♦t♥♦s

♥ r♥rs rrs ♥ st s♦ r♥r s ♦♦ ♥ ♦♥tt♦ ♦♥ ♥ rsr♦r♦ ♦ ♦♥ ♣

♥tr♠r ♥ ♦ rrstrt♦ ♣rtís ♣r♦ t ♦r♠ q t♦♦ sst♠ q rtr③♦ ♣♦r ♥

♦r ♣r♦ ♣♦t♥ qí♠♦ µ

♦♠♦ ② ♠♦s ♥ ♣ít♦ ♥s♠ ♠ás ♦ ♣r trtr st t♣♦ sst♠s ♦♥ µ T ② V

0 1 2 3 37 38 39 400

20

40

60

80

z*

ρ [n

m−3

] Xe/Mg T=Tt

r Prs ♥s s♠étr♦s ❳ s♦r♦ ♥ ♥ r♥r ♣rs é♥ts t♠♣rtr ♣♥t♦ tr♣ ❳ ③②s③ ② rtr ❬❪ ♦s t♦rs r♠♥t♦ ♥ ② r s♣r♦r

♣r♠♥♥ ♦♥st♥ts s r♥ ♥ó♥♦

♦♠♦ sí ♥ s♦ s♦ró♥ s♦r sstrt♦s ♣♥♦s s s♦♦♥s q s ♦t♥♥ ♥ st s♦

♥♦ r♥ sst♥♠♥t s ♦t♥s ♥♦ s tr ♦♥ r♥rs rrs s♦ ♣♦rq s s♦♦♥s

♦t♥s s♦♥ qs ♦♥ dµ/dn > 0 q ♦rrs♣♦♥♥ st♦s sts ♦ ♠tsts

♥ s ♣ ♦srr ♥ s♦tr♠ s♦ró♥ ♣r st t♠♣rtr ♥ rá♦ s ♥♥ ♦rs

♣♦t♥ qí♠♦ r♦ ♦rrs♣♦♥♥t t♥t♦ s♦♦♥s s♠étrs ♦♠♦ s♠étrs ♦t♥s ♣r

r♥r rt sí ♦♠♦ ♣r ♠s♠ r♥r rr ♥ s ♠s♠s ♦♥♦♥s

♦s st♦s ♦t♥♦s ♠♥t ♠♣♦ r♥ ♥ó♥♦ s♦♥ ♠♦str♦s ♥ í♥ só ♥ í♥ r②s s

♠str♥ ♦s rst♦s q s ♦tr♦♥ ♣r ♠s♠ r♥r rr ♦♥ ♥ú♠r♦ ♣rtís s ♦♦

♥ú♠r♦ ♠♦ ♣rtís q rr♦♥ ♦s á♦s ♣r r♥r rt ② ♦♥ ♠ás rs

♦rrs♣♦♥♥t ♦r ♣♦t♥ qí♠♦ ♦♠♦ s ♣ ♦srr ♥ rá ①st♥ ♥tr♦s ♦♥ ♠s

rs s s♣r♣♦♥♥ ♦♠♦ ② s ♦♠♥tó ♥♦ s st♦ s♦ró♥ s♦r ♣rs ♦r♠s♠♦ r♥ ♥ó♥♦

s♦♦ ♣r♠t ♥♦♥trr rst♦s ♦rrs♣♦♥♥ts s♦♦♥s sts ♦ ♠tsts dµ/dn > 0 í♥

r②s ♦r③♦♥t ♥ st r ♦rrs♣♦♥ ♦♥stró♥ ① árs s st í♥ ♥ ♦s st♦s

sts ♠r♦s ♦♥MS1 ②MS2 ♣r♦ t♠é♥ ♣s ①t♠♥t ♣♦rMA q ♦rrs♣♦♥ ♥ st♦ s♠étr♦

st

♥ rá s ♣♦♥ ♥ ♥ q ♥ ♠♦s s♦s r♥rs rts ♦ rrs ①st♥ s♦♦♥s q r♦♠♣♥

s♠trí ♣♦t♥ ①tr♥♦ ♣♦ ♦ ② q ♣r st sst♠ ♥ s ♦♥♦♥s ♣rs t♠♣rtr

② ♣♦t♥ qí♠♦ ♥♦ r♥r s rt ①st♥ trs st♦s ♥ ♦①st♥ ♥♦ ♦s s s s♠étr♦

❱ r ♣r st sst♠ ♥♦ stá rr♦ ♦♠♦ s ♠str ♥ ♣♥ ①str ♥ ♦♥t♥♦

s♦♦♥s s♠étrs ♥♦ r♥r s r s♦♦ ♥ s♦ó♥ s♠étr st q s ♥ ♦♥ MA

♥ t♥ s♦♦ ♠s♠♦ ♣♦t♥ qí♠♦ q s s♠étrs MS1 ② MS2 q ♦①st♥ ♥tr♦

r♥r

♥ s ♣♥ ♦srr ♦♥str♦♥s ① tr♠♥s ♠♥t ♠ét♦♦ s t♥♥ts q

♦♥sr s♦♦♥s sts ② ♠tsts q ♣r♠t♥ ♦t♥r ♦s ♦rs ♦①st♥ ♥♦ s tr

−0.8 −0.7 −0.6 −0.5

−1.664

−1.662

−1.66

∆µ/kB [K]

ω/k B

K/n

m2 ]

Xe/Na T=245 K

CS1

SS2

SS1

CS2

SA1

SA2

Mcross

r r♥ ♥rí r ♣♦r ♥ ár ♥ ♥ó♥ ♣♦t♥ qí♠♦ r♦ ♦rrs♣♦♥♥t ❳ s♦r♦♥ ♥ r♥r ♣rs é♥ts ③②s③ ② rtr ❬❪ st rá♦ r③♦ t♥♥♦ ♥ ♥tt♥t♦ s s♦♦♥s s♠étrs ♦♠♦ s s♠étrs ♦♥ ♦s ♦rs r♥ ♥rí r ② ♣♦t♥ qí♠♦ s s♦♦♥s s♠étrs s ♦♥♦♥♥ s rs q ♥♥ ♦s ♣♥t♦s CS1 ♦♥ SS1 ② SS2 ♦♥ CS2 ♦♥ ss♦♦♥s s♠étrs s r♠ó r q ♥ SA1 ♦♥ SA2 ♦s sts rs s ♦rt♥ ♥ ♥ s♦♦ ♣♥t♦♠r♦ ♦♥ Mcross st♦ ♠♣rí q ♦ rts ♦♥♦♥s ♣♥ ♦①str trs st♦s ♥ qr♦ ♦s ♦s s♠étr♦s ② ♥♦ s♠étr♦ ♦r ♣♦t♥ qí♠♦ s♦♦ Mcross s ♠s♠♦ q s ♥♦ s ♠♣♠♥t ♦♥stró♥ ① s árs ♣r s s♦♦♥s s♠étrs

♦♥ s s♦♦♥s s ♠♥t r♥ ♥ó♥♦ ♦♥ stá♥ s♥ts ♦s st♦s ♥sts ♦s rs ♥

♣♥t♦Mcross s í♥s ②♦s ①tr♠♦s ♥ s♦ ♠r♦s ♣♦r CS1−SS1 ② SS2−CS2 tr♠♥♥ ♦s ♦rs

♦①st♥ ♦rrs♣♦♥♥ts s s♦♦♥s s♠étrs q ♥ ♥ s♦ sñ♦s ♣♦r MS1 ② MS2 s

♠♣♦rt♥t r♠rr q ♦s t♦s r♦s ♥ st r ♥♥ q ♣r ♦r µ ♦s st♦s s♠étr♦s

♠tsts s♠♣r t♥♥ r♥ ♣♦t♥ ω ♠ás ♦ q ♦s ♠tsts s♠étr♦s s♦ ♥ ♣♥t♦ Mcross

í♥ SA1 − SA2 st s♦ s s♦♦♥s s♠étrs q ♦♠♦ ♣ ♦srrs ♥ rá s ♦rt ♦♥

s ♦trs í♥s ♥ ♠s♠♦ ♣♥t♦

s r ♦ q ♦t♦r s♥t♦ r♠ó♥ q ♠♦s ♥ st tr♦ s♦r ♦①st♥ trs st♦s

sts ♦s ♦s s♠étr♦s ② rst♥t s♠étr♦ s st♠♥t q ♦r ♦①st♥ ♦

♣rtr s s♦♦♥s s♠étrs rr♦ ①t♠♥t ♠s♠♦ ♦r ♥tr♦ ♣rsó♥ ♥♠ér ♦♥ q s

tr q ♦t♥♦ ♣r st♦ s♠étr♦

♣r ♥s ♦rrs♣♦♥♥t st♦ MA s♦ ♠r♦ ♦♥ ♥ í♥ rs ♥ ♦♥

♣r♥ ♠ás ♣rs ♦rrs♣♦♥♥ts ♥♦s ♦rs r♠♥t♦ ♦s ♣rs q ♣r♥ ♥ st r

r♦♥ ♦s ♣r ♥ r♥r rr ♥q ♥ s♦ st♦ MA t♠é♥ s ó ♣r ♠s♠

r♥r rt ♥♦ ♦♠♦ rst♦ ♠s♠ r

♠str ♣rs ♦s ♣r r♥rs rts ♦rrs♣♦♥♥ts ♦s s♥ts ♦rs r

♠♥t♦ ② í ♥♠♥t s ♠str st♦ MA r ♥tr é♥t♦ ♦ ♣r

r♥rs rrs ♥t♦ ♦s st♦s s♠étr♦s q ♣♥ ♦①str ♦♥ é MS1 ② MS2 ♦s st♦s MA

♦s t♥t♦ ♣r r♥rs rrs ♦♠♦ rts s♥ é♥t♦s s ♥ ♦♥s♥ q♥ ♥tr

0 10 20 30 400

10

20

z*

0

10

20

ρ [n

m−3

] 0

10

20 Xe/Na T=245 K MS1

Γl = 2.72

MA Γ

l = 4.53

MS2

Γl = 6.35

r Prs ♥s ♦rrs♣♦♥♥ts ❳ s♦r♦ ♥ ♥ r♥r rt ♦♥stt ♣♦r ♣rs é♥ts t♠♣rtr ③②s③ rtr ❬❪ st♦s ♣rs ♦rrs♣♦♥♥ s trs s♦♦♥s q ♣♥♦①str ♥ r♥r r♦ ♥tr sñ♦ ♦♥ MA ♦rrs♣♦♥ s♦ó♥ s♠étr ♦s ♣rs MS1 ②MS2 s♦♥ s♠étr♦s ♦♦s st♦s ♣rs t♥♥ s♦♦s ♥ tr♠♥♦ ♥ú♠r♦ ♠♦ ♣rtís ♣♦r ♥ ár ♣r s♠étr♦ ♣ ♣♥srs ♦♥stt♦ ♥ ♥ s ♣rs ♣♦r ♦s ♣rs s♠étr♦ss♦r♦s ♥ ♥ s

♦s ♥s♠s ♥ó♥♦ ② r♥ ♥ó♥♦ s s♦♦♥s ♦t♥s ♥tr♦ ♦r♠s♠♦ ♥ó♥♦ ♣rát♠♥t ♥♦

r♥ s q s ♦t♥♥ ♥ r♥ ♥ó♥♦ s♦ ♥ s♦ sst♠s ♠② ♦♥♥♦s q ♠ás ♣♦s♥

♣♦s ♣rtís ♠é♥ s ♦sró q s♦ó♥ s♠étr s s ♣♥s ♦♠♦ ♦♥♦r♠ ♣♦r ♦s ♣♦r♦♥s

♦ s♦rs ♥ ♥ s ♣rs s ♣ ♦t♥r ♦♠♥♥♦ s ♣♦r♦♥s ♦ s♦rs

♥ ♣r ♦rrs♣♦♥♥ts s s♦♦♥s s♠étrs ♦①st♥

♥ s st ♥rí r ♣♦r ♣rtí s s♦♦♥s s♠étrs ♥ ró♥ ♦♥ ♦rrs♣♦♥♥t

s s♠étrs ② s ①♥ rst♦s ♦s t♥t♦ ♣r r♥rs rrs ♦♠♦ ♣r r♥rs rts ♥

♦r♥s s ♦♦ r♥ ♥tr ♥rí r ♣♦r ♣rtí s♦ s♦ó♥ s♠étr ②

s♦ó♥ s♠étr ♥ st rá♦ ♦s ♦rs ♦t♥♦s ♣r r♥rs rrs stá♥ ♥♦s ♦♥ í♥

r②s ② ♦s ♦t♥♦s ♣r s ♠s♠s r♥rs rts ♦♥ í♥ ♣♥t♦s ♥ ♥ st♦r ♦♥ s s♦♦♥s s♦♥

sts ♠s rs s s♣r♣♦♥♥ r♠♥t s ♣ ♦srr ①st♥ ♥ ♥tr♦ s SA1 st

SA2) ♦♥ s s♦♦♥s s♠étrs ♣♦s♥ ♠♥♦r ♥rí q s ♦rrs♣♦♥♥ts s♠étrs ♥ st r s

♠r♦ ♥ ♣♥t♦ q ♦rrs♣♦♥ st♠♥t st♦ ♦①st♥ s♠étr♦ MA

♥ s♦ ♦s st♦s s♠étr♦s MS1 ② MS2 ♣í ♦r♠ s♦r ♣r r♥r ♦♥

rs♣t♠♥t ♦♥ s strtrs ② rs q ♣r♥ ♥ ♦①st♥ ♥♦ s st ♥ó♠♥♦

s♦ró♥ s♦r ♥ ♣r P♦r ♦tr♦ ♦ ♣r MA s ♣ ♦srr ♦①st♥ ♥ ♣í rs ♥

♣r r ♦♥ ♥ s♦r ♣r ③qr

P♦r ♦tr♦ ♦ rr t♠♣rtr ♠♦♦ ♣r ♥ ♦ ♦r ♣♦t♥ qí♠♦ ♦s st♦s s♦♦s

♦♥ ♦r ♠ás ♦ r♥ ♣♦t♥ s ♥ ♣♦r s♦r r ♥♦t ♣♦r CS1 − Mcross − CS2 ② s

strtr ♦rrs♣♦♥ ♥ ♣í ♥♦r♠ q s ①t♥ s♦r ♣♥♦ ♣r♣♥r ③ P♦r ♦ t♥t♦

3 3.5 4 4.5 5 5.5 6−0.6

−0.5

−0.4

−0.3

−0.2

−0.1

0

Γl

[f DF(a

s)−f

DF(s

)]/k B

[10−4

K]

Xe/Na T=245 K S

S1

SA1

MA

SA2

r r♥ ♥tr s ♥rís rs ♣♦r ♣rtí ♥ ♥ó♥ t♦r r♠♥t♦ ♦rrs♣♦♥♥t s♦♦♥ss♠étrs ② s♠étrs s t♥t♦ ♣r r♥rs rrs ♦♠♦ rts ③②s③ ② rtr ❬❪ ♥ str fD(as) r♣rs♥t ♥rí r ♣♦r ♣rtí ♦rrs♣♦♥♥t s s♦♦♥s s♠étrs ♠♥trs qfD(s) r♣rs♥t ♠s♠ ♠♥t ♣r♦ ♣r s♦♦♥s s♠étrs ♥ ré♠♥ ♠t♦ ♣♦r SA1 ② SA2 ♣♦t♥ qí♠♦ s s♦♦♥s s♠étrs ① rtr ♣♦st ② ♣♦r ♥ st♦s rst♦s ♦rrs♣♦♥♥ts♥t♦♥s s♦♦♥s sts t♠é♥ s♦♥ ♦t♥♦s t③♥♦ ♦r♠s♠♦ r♥ ♥ó♥♦ r♥rs rts♥ rá♦ s ♥♦ ♥rí st♦ s♠étr♦ MA ú♥♦ st♦ s♠étr♦ ♣♦s ♦♥ s♥t♦ ís♦ q♣r ♥♦ s rs ♣r♦♠ r♥r rt

0 5 10 15 20 25

241

242

243

244

245

246

Γl

T [K

]

Xe/Na

Tw

Tcpw

r ♦①st♥ st♦s ❳ s♦r♦ ♥ ♥ r♥r ♣rs é♥ts ③②s③ ② rtr ❬❪s t♠♣rtrs rí♥ ♥tr ♠♦♦ ② rít ♣r♠♦♦ st♦s ♦rs r♦♥ ♦s t♥t♦ ♣rr♥rs rrs ♦♠♦ rts ♦s ír♦s sñ♥ st♦s s♠étr♦s ♠♥trs q ♦s ♠♥ts ♦rrs♣♦♥♥ st♦s s♠étr♦s í♥ ♣♥t♦s r ♠♥♦ ③ s♦♦ ♣r ♠rr t♥♥ q s♥♦s ♦rs ♠ q t♠♣rtr ♠♥t st ♥③r t♠♣rtr rít ♣r♠♦♦ t♦♦s ♦sst♦s ♦♥r♥ ♥♦ s♦♦ q ♣♦s s♠trí ♣♦t♥ ①tr♥♦

♥♦ ② r♣tr s♠trí s♦r s ♣♥♦

♠str ♦ó♥ ♦♥ t♠♣rtr ♦s st♦s ♣rs♥ts ♥ ♦①st♥ ♣r ♥s

t♠♣rtrs ♥tr ♠♦♦ ② rít ♣r♠♦♦ st♦s ♦rs r♦♥ ♦s t♥t♦ ♣r r♥rs

rrs ♦♠♦ rts st ♠♥r ♦ ♣♦st♦ ♥ ♥ s ♣rs ♦rrs♣♦♥♥t st♦s

s♠étr♦s s r♥♥ ♥ s s♣s♦r st q t♠♣rtr ♥③ rít ♣r♠♦♦ ♦♥ s

s♦♦♥s s♠étrs s♣r♥ q♥♦ ♥t♦♥s ♥ s♦♦ st♦ q r s♠trí ♣♦t♥ ①tr♥♦

s♦♦ ♥ ♣♦ró♥ ♦♥ s♣s♦r tr♠♥♦

♦s rst♦s ♦t♥♦s ♣r s s♦tr♠s sst♠ ❳ ♥③♦ rr ♣♥ ①t♥rs ♣rt♠♥t

♣r ♦tr♦s sst♠s ts ♦♠♦ ❳ ♦ ❳ ♦♥ ♣r♥ ♦tr sr ♦♦♣s ♥ r ❲s

♦♥♥♠♥t♦ ó♥

♥ st s♦ ♦♠♥③♠♦s ♥③♥♦ ♥♠♥t ♥s ♣r♦♣s ♦♥r♥♥ts ♦♥ ♠♦♦ ② ♣r♠♦♦

♠♥t st♦ s♦ró♥ ♥ r♥rs ♥ ③ ♣r♦r ♦♠♦ ♥ ♣ít♦ ♦♥ s ♥③ó s♦ró♥

s♦r ♣rs ♥ sí s s♦tr♠s s♦ró♥ ♠♥t s s♦♦♥s s♠étrs q ♣r♥ ♥ r♥rs

② t♠♣rtrs q stá♥ ♣♦r ♥♠ t♠♣rtr ♠♦♦

Pr ② s t③r♦♥ r♥rs 40σff ♥♦ ♥ s♦ s t③r♦♥ r♥rs ♠ás ♥s

60σff ♣r tr ♦s t♦s s ♥tr♦♥s ♥tr s ♣ís s s♦rs ♥ s ♣rs ♦♣sts

r♥r

♦♦♥s ♦♥ ♣rs s♠étr♦s ó♥

♥ s ♣ ♦srr ♥ s♦tr♠ s♦ró♥ ♣r sst♠ q ♦rrs♣♦♥ t♠♣rtr

st t♠♣rtr s ♠ás t ♣r ♥ s♦tr♠ st sst♠ ♣rs♥t ♥ ③♦ ♥ r

❲s í s ♠str♥ s s♦♦♥s s♠étrs ♦t♥s t♥t♦ ♣r r♥r rr tr③♦ r②s ♦♠♦ ♣r

♠s♠ r♥r rt tr③♦ ♦♥t♥♦ ♥tr ♦s ♣♥t♦s CS1 ② SS1 sí ♦♠♦ ♥tr SS2 ② CS2 sts rs s

s♣r♣♦♥♥ ró♥ ♥st s ①t♥ ♥tr ♦s ♣♥t♦s s♣♥♦s SS1 ② SS2 ♦s st♦s MS1 ② MS2 stá♥

s♦♦s ♦♥ ♣ís s st♥t♦s s♣s♦rs q ♦①st♥ ♥ qr♦ tr♠♦♥á♠♦ ♥tr♦ r♥r

♦s st♦s ♥tr MS1 ② SS1 ② ♦s ♣rs♥ts ♥tr SS2 ② MS2 s♦♥ ♠tsts ♥ ♥♦ r♥r ♣♥

sr ♥③♦s qr ♦s st♦s ♥tr MS1 ② SS1 s♠♣r ② ♥♦ ♥ ♥♦ r♥r sst♠

♥♦ ♥ st♦ st MS1 ♥á♦♠♥t ♥ ♦ r♥r ♣♥ sr ♥③♦s qr

♦s st♦s ♥tr SS2 ② MS2 s ♥ st ♣r♦s♦ s t♥ ♦ ♣r q sst♠ ♥♦ ♥ st♦ MS2

Pr s♦ r♥r rr ♣♦t♥ qí♠♦ ♦①st♥ s ♦t♥ tr③♥♦ ♥ ♦♥stró♥

① árs ♥ ♦♥ ♥ rt rt ♥ tr③♦ r②s ② ♣♥t♦s ♥ st s♦ ♦r ♣♦t♥

qí♠♦ r♦ s s♦♦ qr ♦s st♦s sts MS1 ② MS2 sñ♦s ♥ rá♦

♥ s♦ ♠s♠ r♥r ♣r♦ ♦r rt ♦s st♦s ♦①st♥ st s ♦t♥♥ ♥♦ st

♠s♠♦ ♦r ♣♦t♥ qí♠♦ r♦ ♥ st s♦ ♣r ♦t♥r ♣♦t♥ ♦①st♥ s rá♦

s♦tr♠ s r③ ♥ ♦♥stró♥ ① t♥♥ts ♦s rst♦s s ♠str♥ ♥

♦♠♦ ♣ ♦srrs st ♦r ♦♥ ♦♥ ♦r ♣♦t♥ qí♠♦ r♦ s♦♦ ♣♥t♦ Mcross

q ♣r ♥ ♥ ♦♥só♥ ♦s st♦s ♥ ♦①st♥ st ♥tr♦ r♥r rr ♦ rt

♣♦s♥ ♠s♠♦ ♦r ♣♦t♥ qí♠♦ r♦

♥ s ró ♣r sst♠ ♥tr♦r r♥ ♥tr ♥rí r ♣♦r ♣rtí fDF ) ② ♦r

♦♦r♥ y ♦rrs♣♦♥♥t ♥ ♥ó♥ n∗ = nσ2ff ♥ ♦①st♥ sts r♥s

♥ r r♦ t s ♣ ♣rr ♥ rá

♥ s ♠str♥ s♦tr♠s ♦rrs♣♦♥♥ts s♦♦♥s s♠étrs ♦t♥s ♥ ♥♦ r♥rs

rrs ② ♦♥ ♦s t♦s s r♦♥ ♣r s t♠♣rtrs ♥s ♥ ♥♦ ♦s

rs♣t♦s rá♦s ♥ ♦s s í♥s rts ♦rrs♣♦♥♥ ♦♥str♦♥s ① árs ♥ ♥♦

st♦s rá♦s s ♣ ♦srr ♦ó♥ ♥ó♠♥♦ ♣r♠♦♦ rtr③♦ ♣♦r ♥ st♦ ♥ ♦r

r♠♥t♦ st ♦r♠ ∆Γl s ♣ ♥tr ♦♥ ♣rá♠tr♦ ♦r♥ q rtr③ st tr♥só♥

s♣r ♥t♦♥s q st ♣rá♠tr♦ ♠♣ ♥ ② ♣♦t♥s t♣♦

∆Γl(T ) = bpw (Tcpw − T )1/2

.

♥ t s ①♣♦♥♥ ♦s ♦rs ♦t♥♦s t♠♣rtrs ♠♦♦ ② rít ♣r♠♦♦ ♦s

♠♥t sts r♥rs ❬♣r r t♠♣rtr ♠♦♦ s t③ó ❪ s t♠♣rtrs ♠♦♦

s st♥ ♣rt♠♥t s q s ♦tr♦♥ tr♥♦ ♦♥ s♦ró♥ s♦r ♣rs ♣ít♦ ♥ ♥t♦

−1.2 −1.1 −1 −0.9 −0.8 −0.70

1

2

3

4

5

6

7

∆µ/kB [K]

Γ l

Ne/Li T=38.5 K

CS1

MS1

SS1

SS2

MS2

CS2

r s♦tr♠ s♦ró♥ ♦♥str ♦♥ s♦♦♥s s♠étrs q ♦rrs♣♦♥ s♦r♦ ♥ ♥ r♥r 40σff ♥♦ t♠♣rtr r só s ♦t♥ t♥t♦ ♣r r♥r rr ♦♠♦ ♣r ♠s♠ r♥r rt r r② ♦rrs♣♦♥ s♦♦♥s ♦t♥s s♦♠♥t ♥ r♥rs rrs ♦s♣♥t♦s ♠r♦s ♦♥ SS1 ② SS2 s♦♥ ♣♥t♦s s♣♥♦s ♠♥trs CS1 ② CS1 s♦♥ ♦s st♦s ♥ér♦s s♦♦s ♣ís s r♥ts s♣s♦rs ♦s ♣♥t♦s ♠r♦s ♦♥ MS1 ② MS2 ♦rrs♣♦♥♥ ♦s ♣ís s♥ ♦①st♥ s♣s♦rs r♥ts í♥ rt r②s ② ♣♥t♦s ♦rrs♣♦♥ ♥ ♦♥stró♥ ① árs

−1.2 −1.1 −1 −0.9 −0.8 −0.7−1.81

−1.8

−1.79

−1.78

−1.77

−1.76

∆µ/kB [K]

ω+ /kBT

Ne/Li T=38.5 K

CS1

SS2

Mcross

SS1

CS2

r r♥ ♣♦t♥ ♥ ♥ó♥ ♣♦t♥ qí♠♦ ♣r s♦r♦ ♥ ♥ r♥r rt ♦♥stt ♣♦r ♣rsé♥ts ♥ t♠♣rtr st rá♦ ♦rrs♣♦♥ ♥ ♦♥stró♥ ① t♥♥ts ♣♥t♦ r ♠r♦ ♦♥Mcross ♥ ♦①st♥ ♠s ♣ís r t①t♦ ♥ st rá♦ ω+ = ωσ2

ff

0.1 0.15 0.2 0.25 0.3 0.35 0.4

−0.1

0

0.1

0.2

0.3

0.4

1/n*

(f DF−y

)/kBT

[10−3

] Ne/Li T=38.5 K

MS2

MS1

CS2

S

S2 S

S1

CS1

r r♥ ♥tr ♥rí r ② rt y ❬r ♣ ❪ s♦ ♦♥stró♥ ① st♥♥ts ♥ ♥ó♥ 1/n∗ r♥ s ♥ ♥ ♦s ♦rs 1/nmin ② 1/nmax t ♦ rqr

r♦ ♥ ♣rs♥t t s ♥♥ ♦rs t♠♣rtrs ♠♦♦ ② ríts ♣r♠♦♦ ♣r s♦ s♦r♦ ♥ r♥rs ♥s ② Pr s t③r♦♥ r♥rs 60σff ♥♦ ♠♥trs q ♣r ② s ♠♣r♦♥ r♥rs 40σff ♠é♥ s ♠str♥ ♦s ♦rs q t♦♠♥ ♦s ♣rá♠tr♦s apw ② bpw ♦s ♠♦♦s r♣rs♥t♦s ♥ s s ②

Tw [] Tcpw []

apw/kB [−1/2] bpw [−1/2]

s t♠♣rtrs ríts ♣r♠♦♦ ② q t♥r ♥ ♥t q ♥ ♥áss s♦r ♣rs s♦♦ s ♥♦

♥ ♠t s♣r♦r ♣r s r ♣ít♦

♦♦♥s ♦♥ ♣rs s♠étr♦s ó♥

♦♠♥③r♠♦s ♥③♥♦ ♥ ♦r♠ ♥ér ♦s ♣rs s♠étr♦s q ♣r♥ ♥ s♦ró♥ ♥ r♥rs

② ② ♣♦str♦r♠♥t r♠♦s ♥ ♥áss t♦ ♣r ♥ r♥r

①♠♥r♠♦s ♣r♠r♠♥t ♦s s♦tr♠s s♣rs ♣♥s ♥ ♥ r♦ ♦rrs♣♦♥♥ts s t♠♣rtrs

② ♣r ♥ r♥r ♥ s ♠str♥ sts s♦tr♠s q ♥ s♦ ♦♥strs

t♥t♦ ♦♥ s♦♦♥s s♠étrs ♥s ♣♦r í♥s sós ♦♠♦ ♦♥ s♠étrs í♥ r②s ♥ ♠s

s♦tr♠s s s♦♦♥s s♠étrs ♥♥ ♥ ♦s ♣♥t♦s s♣♥♦s q ♦rrs♣♦♥♥ ♦s ♠á①♠♦s rt♦s

sts rs ♥ s♦ s♦tr♠ sts s♦♦♥s s♠étrs s ①t♥♥ ♥ t♦ ♥ r♦♥ ♦♥

−0.25 −0.2 −0.15 −0.1 −0.05 00

4

8

12

16

∆µ/kB [K]

Γ l

Ne/Rb

(a)

−0.8 −0.6 −0.4 −0.2 0 0

4

8

12

16

∆µ/kB [K]

Γ l

Ne/Na

(b)

−1.4 −1.2 −1 −0.8 −0.6 −0.4 −0.2 0 0

4

8

12

16

∆µ/kB [K]

Γ l

(c)

Ne/Li

36 38 40 42 44−1.2

−1

−0.8

−0.6

−0.4

−0.2

0

T [K]

∆µ/k

B [K

]

(d)

Ne/Alkali Li

Na

Rb Tc

r s♦tr♠s s♦ró♥ r♠♥t♦ s ♣♦t♥ qí♠♦ r♦ ♦rrs♣♦♥♥ts s♦r♦ ♥ r♥rs t♠♣rtrs q ♥ s ♦s trá♥♦s ♦ st ♦s ír♦s ♦♥ ♥ ♣s♦ Pr s t③ó ♥ r♥r 60σff ♥♦ ♦ ♠s♠♦ q ♣r♦ ♣r sst♠ t♠♣rtrs q ♥ s ♦s trá♥♦s ♦st ír♦s ♦♥ ♥ ♣s♦ Pr s ♦♥sró ♥ r♥r 40σff ♦ ♠s♠♦ q ♣r♦ ♣r sst♠ t♠♣rtrs q ♥ s ♦s trá♥♦s ♦st ír♦s ♦♥ ♥ ♣s♦ Pr s ♦♥sró ♥ r♥r 40σff í♥s ♣r♠♦♦ ♣r ♦s sstrt♦s ♥③♦s

2 3 4 5 6 7 8

−1.2

−0.8

−0.4

0

0.4

n*

∆µ/k

B [K

]

Ne/Li

(a)

SS1

SA1

SS2

SA2

SA1

SS1

SA2

SS2

2 3 4 5 6 7−4

−3

−2

−1

0

1

2

n*

[f DF(a

s)−f

DF(s

)]/k

BT

[10

−3]

Ne/Li

(b)

SS1

SA1

SS2

SA2

SS1

SA1

SA2

SS2

r rá♦ ♦rrs♣♦♥ ♦s s♦tr♠s sst♠ ♣r s t♠♣rtrs r s♣r♦r ② r ♥r♦r s í♥s ♣♥t♦s rts ♥♥ ♠á①♠♦ ♦r n∗ ♦ s rst s♠trí s s♦♦♥s s♥♦ ♦s rst♥ts ♣♥t♦s s r ♥ t①t♦ ♥ rá♦ s ♠str r♥ ♥tr ♥rí r ♣♦r ♣rtí s s♦♦♥s s♠étrs ② s s♠étrs ♥ ♥ó♥ n∗

♣r ♠s t♠♣rtrs r ♥r♦r ♦rrs♣♦♥ ♥ t♠♣rtr ② s♣r♦r ♥ ♥ ré♠♥ ♠t♦ ♣♦r SS1 ② SA1 ♣♦t♥ qí♠♦ s♦♦ s s♦♦♥s s♠étrs ① ♥rtr ♥t ♣♦r ♥♠ n∗(SA1) st rtr ♥ ♣♦st st♦s rst♦s q s ♦tr♦♥ ♣rr♥rs rrs s ♦t♥♥ t♠é♥ ♥ s♦ r♥rs rts

s s♠étrs s♦♥ ♥sts ♥tr ♦s ♣♥t♦s SS1 ② SS2 Pr st♦ s♣♥♦ ♠á①♠♦ st s♦tr♠

♣♦t♥ qí♠♦ r♦ s ♥t♦ st♦ ♣r♦ q ré♠♥ s♠étr♦ s♣r③ ♥♦ ∆µasym ♥

♦r q ∆µsym t♥ ♥ SS1 Pr ♦tr s♦tr♠ ①st ♥ r♥ tt ♠♣♦rt♥t ♣♦t♥

qí♠♦ r♦ st♦ s♣♥♦ SS1 s ♣♦st♦ ② ♥t♦♥s s s♦♦♥s s♠étrs s ①t♥♥ ♠♥r

t q ∆µasym s ♣r♦①♠ r♦ ② s♠trí ♣rr st q s ♣r♦ ♥ ♦♥♥só♥ s ♣r

♥ s ♣ ♦srr q ♥rí r ♣♦r ♣rtí s s♦♦♥s s♠étrs s ♥r♦r

s s♠étrs ♥ r♥♦ ♥♦ ♣♦r SS1SA1SA2 ♣r ♠s t♠♣rtrs ❱ r s s♦♦♥s s♠étrs

♦♥stt②♥ ♦s st♦s ♥♠♥ts s♦♦s ♦s r♥ts ♦rs n∗

s ♦♣♦rt♥♦ sñr t♠é♥ q ♦ r♦ r r② ♥tr SS1 ② SA1 ♣r ♠s s♦tr♠s s s♦♦♥s

s♦♥ ♥sts dµasym/dn < 0 ♥ ♠r♦ t♦s sts s♦♦♥s ②s ♥rís s♦♥ ♥r♦rs s ♥rís

s s♦♦♥s s♠étrs s♦♥ st③s r ♥ú♠r♦ ♣rtís ♦♠♦ ♦rr ♥ s♦ ♥ r♥r

rr ❬❪

♥ ♦s rá♦s ② s ♠str♥ r♦s ♣rs s♠étr♦s ♦rrs♣♦♥♥ts s♦r♦

♥ r♥rs ② rs♣t♠♥t ♦♥ s ♣ ♦srr ♦♠♦ ♦♦♥♥ sts s♦♦♥s ♥♦ s

♥ ♥♥♦ s r♥rs ♦♥ ♦ ♥ sts rs stá♥ s♦s r♥t ♥ú♠r♦ ♣rtís

♥ ♥♦ st♦s rá♦s s rs tr③♦ rs♦ ♦rrs♣♦♥♥ st♦ s♠étr♦ q s q ♦①st

♦♥ ♦s ♦s st♦s s♠étr♦s rs♣t♦s t♠♣rtr sñ ♥ ♥♦ ♦s

s rs s ♥ í♥ r②s s s♦♥ s s♦♦♥s s♠étrs ♠ás r♥s ♠②♦r ♥ú♠r♦

♣rtís ♦rrs♣♦♥♥ts ♣rs s♦r♦s ♣♦♦ s♣s♦r ♦s ♣rs s♠étr♦s q ♥ ♣rr

♦ st♦s st♦s s♠étr♦s s s s ♦♥♥♦ ♣rtís r♥r stá♥ s♦♦s ♣ís

♠②♦r s♣s♦r q s q ♥♥ sts rs

♥ ♦s rá♦s r s ♠str♥ ♦♠♦ ♠ t♦r s♠trí ♣r ♥s t♠♣rtrs ♦♠♦

♥ó♥ r♠♥t♦ ♦sr♠♦s ♣♦r ♠♣♦ rá♦ í s ♥ ♦ rs s♠trí

♥ s s ♦rrs♣♦♥♥ t♠♣rtrs q ♥ s r ♠ás t ♥ ♣s♦s

♠♦ r♦ st ♠♥r ♣♦r ♠♣♦ ♣r s♦ ♦s ♣rs s♦♦s q s ♠str♥ ♥ s♦

st♦ s♠étr♦ r r② s♦♥ ♥♦s ♦s ♣♥t♦s q ♦♠♣♦♥♥ s♥ r ♦♥t♥♦ s ♦

rr q s ①t♥ ♥tr ♦s st♦s ♣rs♥ts ♥tr SA1② SA2 ♦s st♦s SA1② SA2 s♦♥ ♦s ♠s♠♦s q

stá♥ sñ♦s ♥ tr③♦ rs♦ r r♣rs♥t st♦s sts s♦♦♥s q s ♦t♥♥

t♥t♦ ♠♥t ♦r♠s♠♦ ♥ó♥♦ ♦♠♦ ♦♥ r♥ ♥ó♥♦

s rtrísts ♦s sst♠s ② ♥ ♥t♦ strtr ♦s ♣rs ♥s s♦♥

s♠rs s sst♠ Pr sst♠ r♥ ♥♠♥t r ♥ q ♦

♠♥♦r ♥t♥s ♥tró♥ sstrt♦♦ ♦s ♣rs ♥♦ ♣rs♥t♥ ♦s ♣♦s q ♣r♥ ♥ s♦

♦ rtríst♦s ♦r♠ó♥ ♥ strtr ♣s q s ♠♥st ♥ ♦ s♦r♦

r♥♦ ♣r

♥áss s♦ró♥ s♦r ♦♦

♥③ ♦♥ t ♥ s♦tr♠ s♦ró♥ s♦r st s♦tr♠ ♦rrs♣♦♥ t♠♣rtr

st t♠♣rtr s ♠ás t ♣r st sst♠ ① ♥ tr♥só♥ ♣r♠r ♦r♥ s s♦tr♠s

s♦ró♥ s ♥ t♥t♦ ♣r r♥r rt ♦♠♦ ♣r ♠s♠ r♥r rr st♦s rst♦s s

♠str♥ ♥ ♦s rá♦s ♥ rá♦ r q r♦rr ♦s ♣♥t♦s CS1 MS1 SS1 SS2 MS2

② CS2 st s♦ s♦♦♥s s♠étrs ♠♥trs q ♣s ♣♦r SS1 SA1 ② SA2 s ♦t♥ ♣rtr s♦♦♥s

s♠étrs Pr ♥ ♦r r♠♥t♦ s♣r♦r s♦♦ SA2 ♥♦ ①st♥ s♦♦♥s q r♦♠♣♥ s♠trí

♣♦t♥ sstrt♦♦ í♥ rt r②s ② ♣♥t♦s s ♥ ♦♥stró♥ ① ♣rtr

s s♦♦♥s s♠étrs ♦t♥s ♦♥ r♥r rr

♦s st♦s r♣rs♥t♦s ♥ s rs í♥ só ♦rrs♣♦♥♥ ♦s q ♣r♥ ♥ s♦ ♠s♠

r♥r rt Pr r ♥ s♦ r♥r rt ♦s st♦s s♠étr♦s q ♦①st♥ t♠♣rtr

♥ q ♥ r stá♥ r♣rs♥t♦s ♣♦r MS1 ② MS2 s ♥sr♦ rrrr ♥ ♦♥stró♥ ①

t♥♥ts ♥ s ♠str♥ rs ♦rrs♣♦♥♥ts st ♠ét♦♦ ♥ r q s

①t♥ s CS1 st SS1 s ♦rt ♥ ♣♥t♦ ♠r♦ ♦♥ Mcross ♦♥ r q s ①t♥ ♥tr CS2 ②

SS2 s s♦r st ♠s♠♦ rá♦ s tr③ r q ♦rrs♣♦♥ ♦♥stró♥ ① t♥♥ts ♣r

s s♦♦♥s s♠étrs r q s ①t♥ ♥tr SA1 ② SA2 s ♦t♥ q st r ♦rt s ♥tr♦rs

♥ ♠s♠♦ ♣♥t♦ Mcross st ♠♥r st♦ ♠r♦ ♦♥ MA ♦rrs♣♦♥♥t ♥ s♦ó♥ s♠étr

t♥ s♦♦ ♠s♠♦ ♦r ♣♦t♥ qí♠♦ s s♦♦♥s s♠étrs MS1 ② MS2 ♦ ♥ q

♣r s♦ r♥r q ♥ Mcross ♦①st♥ trs st♦s ♥ qr♦ tr♠♦♥á♠♦ ♦s s♠étr♦s

② ♥♦ s♠étr♦ st♦s trs st♦s sts♥

∆µS1(T, ωe) = ∆µS2(T, ωe) = ∆µA(T, ωe = ∆µe(T, ωe) ,

♦♥ µe(T, ωe) = −0,38991 ♦♥ ωe/kBT = −2,22737 ♥♠−2

♣r ♥s ♦ s♦r♦ ♦rrs♣♦♥♥t st♦MA ♣ ♦srrs ♠r♦ ♦♥ tr③♦ rs♦

♥ ② ♥ ♥ rá♦ ♥tr

♠str q s Mcross ♣r ♦r µ ♦s st♦s ♠tsts s♠étr♦s s♠♣r t♥♥

r♥ ♣♦t♥ ♠ás ♦ q ♦s s♠étr♦s P♦r ♦tr♦ ♦ ♣r t♠♣rtrs s♣r♦rs ♠♦♦ ♣r ♥

♦ µ ♦s st♦s q ♣♦s♥ r♥ ♣♦t♥ ♠ás ♦ stá♥ r♣rs♥t♦s ♣♦r ♣♥t♦s ♥ í♥ tr♠♥ ♣♦r

CS1 −Mcross − CS2 ② ♠♦♦ ♠♣ q ♦rrs♣♦♥♥t strtr st♦s st♦s ♦♥sst ♥ ♣ís

s ♥♦r♠s s♦r ♣♥♦ xy P♦r ♦ t♥t♦ ♥♦ s r♦♠♣ s♠trí ♥ st ♣♥♦

s ♥ rá♦ s♠r ♥ st s♦ s ♠str♥ s r♥s fDF −y ♦rrs♣♦♥♥tst♥t♦ st♦s s♠étr♦s ♦♠♦ s♠étr♦s qí µe/kBT = −6,97235 [µ0/kBT (T = = −6,96284] ② ω+

e /kBT =

ωeσ2ff/kBT = −2,22737

0 10 20 30 400

10

20

30

40

z*

ρ [n

m−3

]

(a)

Ne/Li T=38 K

0 2 4 6 8 10 120

15

30

45

60

75

90

Γl

∆ N (

%)

(b)

Ne/Li

SA1

SA2

SA2

SA1

0 10 20 30 400

5

10

15

20

25

30

z*

ρ [n

m−3

]

(c)

Ne/Na T=41 K

0 2 4 6 8 10 120

20

40

60

80

Γl

∆ N (

%)

Ne/Na

(d)

SA1

SA1

SA2

SA2

0 10 20 30 40 50 600

5

10

15

20

25

z*

ρ [n

m−3

]

(e)

Ne/Rb T=42.75 K

0 2 4 6 8 10 120

10

20

30

40

50

Γl

∆ N (

%)

(f)

Ne/Rb

SA1

SA2

SA2

SA1

r Prs ♥s s♠étr♦s ♥ ♥ r♥r ♥ t♠♣rtr ♣r rt♦s ♦rs Γl r tr③♦ rs♦ ♦rrs♣♦♥ ♣r s♠étr♦ q ♦①st ♦♥ ♦s ♦s s♠étr♦s r r②sst s♦ ♦♥ ♣r s♠étr♦ ♠tst ♠♥♦r s♣s♦r r t①t♦ t♦r s♠trí ♣r sst♠ ♥ ♥ó♥ t♦r r♠♥t♦ ♣r t♠♣rtrs ♥tr ♦s r s♣r♦r ② ♦s r ♥r♦r ♥ ♣s♦s ♠♦ r♦ s♥♦ ♦s ♣♥t♦s SA1 ② SA2 s ♠s♠♦ q ♣r ♦ ♠s♠♦ q ♣r♦ ♣r sst♠ ♥ t♠♣rtr ♦ ♠s♠♦ q ♣r♦ ♣r sst♠ ♣r ♥ r♥♦ t♠♣rtrs q ♥ s ♦s rs♣r♦r ♦s r ♥r♦r ♥ ♣s♦s ♠♦ r♦ ♦ ♠s♠♦ q ♣r♦ ♣r sst♠ ♥ t♠♣rtr ♦ ♠s♠♦ q ♣r♦ ♣r sst♠ ♣r ♥ r♥♦ t♠♣rtrs q ♥ s ♦s rs♣r♦r ♦s r ♥r♦r ♥ ♣s♦s rt♦ r♦

−0.5 −0.45 −0.4 −0.35 −0.3

2

4

6

8

∆µ/kB [K]

Γ l

(a)

Ne/Na T=41 K

CS1

CS2

SS1

SS2

SA1

SA2

MS1

MS2

MA

−0.5 −0.45 −0.4 −0.35 −0.3

−2.24

−2.23

−2.22

∆µ/kB [K]

ω+/k

BT

(b)

Ne/Na T=41 K

CS1

CS2

Mcross

SS1

SA2

SS2

SA1

r s♦tr♠ s♦ró♥ ② r♥ ♥rí r ♥ ♥ó♥ r♠♥t♦ ♣r sst♠ t♥t♦ ♣r s♦♦♥ss♠étrs ♦♠♦ s♠étrs ♥ rá♦ s ♠str ♥ s♦tr♠ ♦rrs♣♦♥♥t ♥ t♠♣rtr í♥ só st ♦♥stt ♣♦r s s♦♦♥s ♦t♥s ♥ ♥ r♥r rt ② r② q ♦rrs♣♦♥ ♠s♠ r♥r ♣r♦ rr s s♦♦♥s s♠étrs ♦rrs♣♦♥♥ r q ♥ ♦s ♣♥t♦s CS1 SS1SS2 CS2 ♠♥trs q s s♠étrs sts s♦s r q ♥ SS1 SA1 SA2 SS2 í♥ rt♥ ♣♥t♦s ② r②s s ♥ ♦♥stró♥ ① q ♦♥t ♦s st♦s s♠étr♦s MS1 ② MS2 ♥ ♦①st♥♦♥ ♥ st♦ s♠étr♦ MA ♥ rá♦ s♥♦ ♦s ♣♥t♦s s ♠s♠♦ q ♣r rá♦ srs q ♦♥t♥♥ CS1SS1 ② SS2CS2 ♦rrs♣♦♥♥ s♦♦♥s s♠étrs ② s q ♦♥t♥♥ SA1SA2

stá♥ s♦s s s♠étrs ♣♥t♦ Mcross ♥ ♦③ó♥ r s trs í♥s ② tr♠♥ ♦①st♥ ♦s s♦♦♥s s♠étrs ② ♥ s♠étr MS1 MS2 ② MA ♠♦strs ♥

♣ ♦srr ♥ rá q s trs rs t♥♥ s ért ♥ r♦ s ♦r♥s ♦♥ ♦①st♥ ♦s

st♦s MS1 MS2 ② MA ♦s ♦rrs♣♦♥♥ts ♣rs s ♠str♥ ♥

♥ s ♠str♥ r♠s s ♦①st♥ st♦s ♣r ♦s sst♠s st♦s í s

tr③r♦♥ trs í♥s rts ♦r③♦♥ts q ♥♥ s t♠♣rtrs ♠♦♦ ② Pr ♥♦

st♦s ♠ts s ♣ ♦srr q ♥ s♦ ♣ ♠ás rs s♣s♦r r t♥t♦ ♥

s♦ s♠étr♦ ♦♠♦ ♥ s♠étr♦ ♥♦ t♠♣rtr s ♣r♦①♠ ♠♦♦ ♥♦ s ♥③

t♠♣rtr rít ♣r♠♦♦ ♦s trs st♦s ♥ ♦①st♥ ♦♥r♥ ♥♦ s♦♦ ♣♥t♦ q ♥ rá♦

s♦ ♠r♦ ♦♥ ♥ str

P♦r út♠♦ ♥ s ♠str ♦ó♥ ♣rs s♠étr♦s ♦♠♦ ♥ó♥ t♠♣rtr ♣r

♥♦s ♦rs ♥ r♥♦ Tw < T < Tssb ♣ ♦srr ♥ s♦s rá♦s ♦♠♦ ♠ q ♠♥t

t♠♣rtr r t♠é♥ s♣s♦r ♣í ♥ ♥ tr♠♥t♦ ♣í ♠ás rs ♣♦st ♥

st s♦ s♦r ♣r r r♥r

s♠♥

♥③♠♦s s♦ró♥ ❳ ② ♥ r♥rs rts ♦♠♣r♥♦ ♥ ♥♦s s♦s ♦♥ ♦s rst♦s ♦t♥♦s

♥ r♥rs rrs

P♥t♠♦s s♦r s ♥str♦s á♦s ①st♥ ♥ ró♥ rt ♥tr ♦s st♦♥s q ♥

♣r♥♣♦ ♣r♥ ♦♠♦ t♦t♠♥t s♦♥①s ♦s rr♠♦s ♣♦r ♥ ♦ s♠trí s s♦♦♥s ♥ s♦

r♥rs ② ♣♦r ♦tr♦ s ♣r♦♣s ♠♦♦ sstrt♦ ♥ stó♥

sí ♠♥t ♥áss ♥str♦s rst♦s ♥♦♥tr♠♦s q s s♦♦♥s st♦ ♥♠♥t ♣♥

0.1 0.13 0.16 0.19 0.22 0.25 0.28

0

0.04

0.08

0.12

0.16

1/n*

(f DF−y

)/kBT

[10−3

] Ne/Na T=41 K

MS2

MA M

S1

CS2

CS1

SS2

SS1

SA1

SA2

(a)

0 5 10 15 20 25 30 35 400

10

20

z*

0

10

20

ρ [n

m−3

] 0

10

20

30 Ne/Na T=41 K M

S1 Γ

l=1.83

MA Γ

l=3.76

MS2

Γl=5.70

(b)

r ♥ rá♦ s ♣ ♦srr r♥ fDF − y ♦♥ y = µe + ωen

−1 ♥ ♥ó♥ 1/n∗ ♣r ♥r♥r ♥ ♦♥ ♥ t♠♣rtr s♥♦ ♦s ♣♥t♦s s ♠s♠♦ q st r♥ s ♥ ♥ ♦s ♦rs Γ s♦♦s ♦s st♦s q ♦①st♥ ♥ qr♦ tr♠♦♥á♠♦s s♠étr♦s ② ♥♦ s♠étr♦ ♥ rá♦ ♥r♦r s ♠str♥ ♣rs ♥s ♦rrs♣♦♥♥ts ♠s♠♦sst♠ ② ♠s♠ t♠♣rtr ♦s ♠r♦s ♦♥ MS1 ② MS2 ♦rrs♣♦♥♥ s♦♦♥s s♠étrs ② ♠r♦♦♥ MA ♥ s♦ó♥ s♠étr Pr r③r st♦s á♦s s t③ó t♥t♦ ♦r♠s♠♦ ♥ó♥♦ ♦♠♦ r♥♥ó♥♦

0 4 8 12 1635

37

39

41

43

45

Γl

T [K

]T

w(Rb)

Tcpw

Rb

Tcpw

Tw(Na)

Na

Tcpw

Tw(Li)

Li

Ne Tc

r r♠ s ♦rrs♣♦♥♥t ♦①st♥ st♦s s♦r♦ ♥ r♥rs ② t♠♣rtrs Tw < T < Tcpw ♦s ír♦s stá♥ s♦♦s st♦s s♠étr♦s ♠♥trs q ♦s ♠♥ts st♦s s♠étr♦s s í♥s ♣♥t♦s ♥ s♦ s ♠♥♦ ③ ♦♥ ♥ ♥r t♥♥ ♦♥♥t♦ ♣♥t♦s ♦s st♦s t♥♥ ♥ ú♥♦ st♦ ♥♦ t♠♣rtr s r rít ♣r♠♦♦♦♥ s rstr s♠trí s s♦♦♥s st ú♥♦ st♦ s♦ ♥♦ ♦♥ ♥ strs í♥s ♦r③♦♥ts ♠r♥ s ♦rrs♣♦♥♥ts t♠♣rtrs ♠♦♦ ♣r ♥♦ ♦s sstrt♦s

010

2030

40

36.537

37.538

38.539

0

10

20

30

40

50

z*T [K]

ρ [n

m−

3 ] Ne/Li

r Prs ♥s s♦r♦ ♥ ♥ r♥r ♦♠♦ ♥ ♥ó♥ st♥ z∗ ♥ s ♣rs♣r rt♦s ♦rs t♠♣rtr st♦s t♦s ♦rrs♣♦♥♥ á♦s r③♦s ♠♣♥♦ t♥t♦ r♥rs rts♦♠♦ rrs

r♦♠♣r s♠trí ♥ ♣♦r ♥♠ t♠♣rtr ♠♦♦ ♣r ♥ tr♠♥♦ r♥♦ ♥ss ② q

♣♦s st r♣tr s ①t♥ ♥♦ ♠ás á t♠♣rtr rít ♣r♠♦♦

trr ♦♥ r♥rs rts ♥♦♥tr♠♦s q ♥t♦ s ts s♦♦♥s s♠étrs ♣ ♦①str ♥

st♦ s♠étr♦ st rst♦ st ♠♦♠♥t♦ ♥♦ s♦ r♣♦rt♦

Pr sst♠s rsts ♦t♠♦s ♦s r♠♥s ♠♦♦ r♣♦rt♦s ♥ trtr ♦♥ ♠♥t ♠♦

ó♥ s ♥tr♦♥s ♦sstrt♦ s r♣r♦í rt strtr í♥s ♣r♠♦♦

♥♦♥tr♠♦s s♦♦♥s sts s♠étrs tr♥♦ ♦♥ r♥rs ♥♦sts ♣♥s 15σff ♥♦

t♠♦s s trs ss ♣rs q s r♣♦rt♥ ♥ ♦rí s ❬ ❪

str♦s rst♦s ♠str♥ q ♦s t♦s ♣rs ♦♠♥③♥ sr ♠♣♦rt♥ts ♣r r♥rs ♥♦ ♠♥♦r

♦s 12σff

♥③♠♦s r♦ ♥♥ q t♥ ♥♦ r♥r ♥ s♠trí ♣r ♥s ♣r ♦

s r♦♥ ♦ á♦s ♥ r♥rs rs♦s ♥♦s

♣ít♦

♣é♥

P♦t♥ ❲

strtr ♥ ♦ ♥♦r♠ st s♥♠♥t tr♠♥ ♣♦r ♣rt r♣s ♣♦t♥ ♥tr♠♦

r P♦♠♦s ♠♥r ♦ ♦♠♦ ♦r♠♦ ♣♦r srs rs q ♥trtú♥ ♠♥t ♥ ♣♦t♥ tr♠♥♦

♣♦r ♠♣♦ ♥ ♣♦t♥ ♥♥r♦♥s ♦ st ♣♦t♥ trt ♦ r♣s ñ s♦♠♥t ♣

qñs ♦rr♦♥s ♦♠♣♦rt♠♥t♦ ♥r♦ ♣♦r ♥ú♦ r♣s♦ P♦r ♦♥s♥t sr ♣♦s trtr

♦ ♣♦t♥ ♦♠♦ ♥ ♣rtró♥ q tú s♦r ♦ q ♠r♠♦s sst♠ rr♥ q s q

♥ q ♣♦t♥ ♥tró♥ ♥tr ♣rtís st ♦r♠♦ s♦♠♥t ♣♦r ♥ú♦ r♣s♦ sst♠

rr♥ ♣ str ♥t♦♥s ♦♥stt♦ ♣♦r ♠♣♦ ♣♦r ♥ ♦ srs rs ♦ q s s ①tr♦

rr♥ ❬❪

r♦ st ♠♦♦ s♣♦♥♠♦s q ♥rí ♥tró♥ ♥tr s N ♣rtís q ♠r♠♦s

Φ(r1, r2, .......rN ) s ♣ ①♣rsr ♦♠♦

Φ(r1, r2, .......rN ) = Φ0(r1, r2, .......rN ) + Φ1(r1, r2, .......rN )

♦♥ Φ0 s ♦♥tró♥ ♣r♦♥t ♥ú♦ r♣s♦ ♣♦t♥ ② Φ1 s ♦rrs♣♦♥♥t ♦♥tró♥

♦ ♣♦t♥

tr♠♦s ♥tr♦ ♦♥t①t♦ ♥ó♥♦ ♦♠♦ ♠♦s ♥ ♣t♦ ♣rt ♦♥r♦♥ ♥rí

r ♠♦t③ FN (V, T ) q r♦♥ ♥ó♥ ♣rtó♥ ♥ó♥ ZN (V, T ) ♠♥t s♥t

①♣rsó♥

e−βFN (V,T ) =ZN (V, T )

N !=

1

N !

ˆ

dr1dr2dr3.......drN exp(βΦ(r1, r2, .......rN ))

s ♦ ♠♠♦s Z0N (V, T ) ② F 0

N (V, T ) ♥ó♥ ♣rtó♥ ♥ó♥ ② ♥rí s♦ sst♠

rr♥ rs♣t♠♥t ♣♦♠♦s srr

e−βF 0

N (V,T ) =Z0N (V, T )

N !=

1

N !

ˆ

dr1dr2dr3.......drN exp(βΦ0(r1, r2, .......rN ))

♥ó♥ stró♥ N r♣♦s ♥♦r♠③ q sr sst♠ rr♥ srá ♠ f0N (r1, r2, .......rN )

st ♥ó♥ s q ♥ ♣r♦♠♦ ♥ r ♥♠ qr A ♥ s♥t♦ q

〈A〉 =ˆ

dr1dr2dr3.......drN f 0N(r1, r2, .......rN)

♦ st ♥♦♥ str♦♥ s ♣ srr ♦♠♦

f0N (r1, r2, .......rN ) =1

N !0N (r1, r2, .......rN )

e−βΦ0(r1,r2,.......rN )

Z0N (V, T )

♦♥ 0N r♣rs♥t ♣r♦ ♥♦♥trr s ♣rtís st s♦ sst♠ rr♥ ♥ ♥t♦r♥♦s

s ♣♦s♦♥s r1, r2, .......rN st ①♣rsó♥ s q

e−βΦ0(r1,r2,.......rN ) = Z0N (V, T )f0N (r1r2rN )

② t♥♥♦ ♥ ♥t ♣♦♠♦s srr

e−βFN (V,T ) = e−βF 0

N (V,T )

ˆ

dr1dr2dr3.......drN f 0N(r1, r2, .......rN)e−βΦ1(r1,r2,.......rN ) =

= e−βF 0

N (V,T )⟨

e−βΦ1(r1,r2,.......rN )⟩

♦♥ ♠♦s ♥♦ ♣r♦♠♦ ♠♥t stró♥ ♥♦r♠③ f0N (r1, r2, .......rN )

P♦r ♦tr♦ ♦ ♣♦♠♦s srr⟨

e−βΦ1(r1,r2,.......rN )⟩

= exp∞∑

n=1

(−β)nn!

Cn

♦♥ Cn s ♥♦♠♥♦ ♠♥t ♦r♥ ♥ t♦rí str♦♥s ①st♥ ①♣rs♦♥s ①♣íts

st♦s ♠♥ts ♥ tér♠♥♦s r t♦r Φ1 ♥ ♥str♦ s♦ P♦r ♠♣♦

C1 = 〈Φ1〉

C2 =⟨

Φ21

⟩

− 〈Φ1〉2

C3 =⟨

Φ31

⟩

− 3 〈Φ1〉⟨

Φ21

⟩

+ 2⟨

Φ21

⟩

s ♦t♥ sí s♣és t♦♠r ♦rt♠♦s

FN (V, T ) = F 0N (V, T ) + Φ1(r1, r2, .......rN )− β

2

[

⟨

Φ21

⟩

− 〈Φ1〉2]

+ .............

♣♦t♥ s t♦ ♣rs s r s s s♣r♥ s ♥tr♦♥s ♠ás ♦s r♣♦s ♥t♦♥s s

♣♦t♥ ♥tró♥ ♦s ♣rtís q s♦♠♣s♠♦s ♥ ♥ ♣rt s♦ sst♠ rr♥

φ0(r12) ② ♦tr s♦ ♣rtró♥ φ1(r12) r ♦ ♣♦t♥ ♥ st s♦ s♥♦

s♠♥♦ ó♥ ♥tr♦r s ♣ ①♣rsr ♥ tér♠♥♦s ♥ó♥ stró♥ r sst♠

rr♥ Pr r♦ st t♥r ♥ ♥t q ♥trr ♦♥ ♥ ♦t♥r ♣r♦♠♦ Φ1(r1, r2, .......rN )

♥♦s ♣rrá s ♦♥tr♦♥s t♦s s N(N −1)/2 ♣rs ♣rtís q ♦♥♦r♠♥ ♦ st

♠♥r ♦t♥♠♦s

〈Φ1(r1, r2, .......rN )〉 = 1

2N(N − 1)

ˆ

r1r2φ(r12)ˆ

dr1dr2dr3.......drN f 0N(r1, r2, .......rN) =

=1

2

ˆ

dr1dr2φ1(r12)20(r12) =

1

2Nρ

ˆ

drφ1(r)g0 (r)

♥ st ①♣rsó♥ ρ s ♥s ♦ ② g0(r) s ♥ó♥ stró♥ r s♦ ♦

rr♥ st út♠ ①♣rsó♥ ♥♦ s ♠s q ♦♥tró♥ ♣r♠r ♦r♥ ♥rí r íq♦

♥♦r♠ ①♣rsó♥ s á♠♥t ♥tr♣rt ② q st♠♦s ♣rt♥♦ s♣♦só♥ q strtr

♦ st tr♠♥ ♣♦r ♣rt r♣s ♣♦t♥ ♣♦r ♦ t♥t♦ ♦ q st ♥ s♣♦♥♠♦s s st♠♥t

q ♥ó♥ stró♥ r ♦ s ♣rs♠♥t g0(r) ①♣rsó♥ ♥tr♦r ♥♦ s ♥t♦♥s ♦tr

♦s q ♥rí ♥tró♥ ♦t♥ s♠♥♦ s♦r t♦♦s ♦s ♣rs ♣rtís ♥rí ♣r

♣s♥♦ ♦♥ ♣r♦ q t ♣r ①st s♥♦ ♦r♥ ♣rtr♦♥s qí r♣rs♥t♦ ♣♦r

trr s♠♥♦ s ♠② ♦♠♣♦ ♣r sr ①♣rs ♥ tér♠♥♦s ♥ó♥ ♦rró♥ ♥♦s q♠♦s

♣r♠r ♦r♥ ♣♦♠♦s srr ♥rí r ♦♠♦

FN (V, T ) = F 0N (V, T ) +

1

2Nρ

ˆ

drφ1(r)g0 (r)

st ó♥ s ①tr♠♠♥t út ♦r tr♠♥r s ♣r♦♣s tr♠♦♥á♠s ♥ íq♦ ♣rtr

♦♥♦♠♥t♦ ♣♦t♥ ♥tró♥ ♥tr s ♣rtís q ♦ ♦♥stt②♥ ♥ s♦ s♣♦♥r

♥rí r ② ♥ó♥ stró♥ r sst♠ rr♥ P♦r st r③ó♥ ♥ sst♠ rr♥

♦♥♥♥t s ♥ ♠♦♦ s srs rs ♣s ♠s ♣r♦♣s s ♦♥♦♥ ♠② ♥ ♣r st sst♠

rt♦s ♠♦♦s ♣♦t♥ ♦♥♥ ♠♥r ♥tr st stó♥ st s s♦ ♣♦t♥ ♣♦③♦

r♦ ♦ ♥ ♣♦t♥ sr r ♦♥ ♦ trt ♣s ♥ st♦s s♦s s ♥tr t♦♠r ♣rt sr

r ♦♠♦ ♣♦t♥ rr♥ ② ♦ ♦♠♦ ♣rtró♥ ♥ ♠r♦ ♠♦♦s ♣♦t♥ ♠ás rsts

♦♠♦ ♥♥r♦♥s r♥ ♥ ♥ú♦ r♣s♦ r♦ ♠♦♦ q ♥♦ ①st ♥ s♣ró♥ ♥tr ②

ú♥ ♣♦t♥ ♥t♦♥s ♣

rqr ♦s ♣s♦s ♦♥s ♥♣♥♥ts ♥tr s ♥♣♥♥ts ♣r♦①♠ó♥ q s♥

s♦ st ①♣rsó♥

③r ♥ ♣rtó♥ rtrr ♣♦t♥ ♥ ♥ ♣rt rr♥ φ0(r) ② ♥ ♣rtró♥ φ1(r)

♣r♦①♠r ♣♦t♥ rr♥ φ0(r) ♣♦r ♥♦ sr r ♦♥ á♠tr♦ t♦ d q sr

♦t♥♦ ♥ ♠♥r ♠♦♦ q ♥rí r ② t♠é♥ ♥ó♥ stró♥ r sst♠

rr♥ s ♣r♦①♠♥ ♥ ♦ srs rs á♠tr♦ d

st♥ts ♠♥rs r③r ♦s ♣s♦s ② ♦♥♥ st♥ts t♦rís q ♥t♠♥t ♥ st♥t♦s rst♦s

♥♦ s ♣♥ ♠s♠♦ ♠♦♦ ♣♦t♥ ♥tr♠♦r ♦s ♦s ♠♦♦s ♠s r♥ts s♦♥ s t♦rís

rr ② ♥rs♦♥ ② ❲s♥r♥rs♥ ❲ r rr♥s ❬❪ ② ❬❪

♥③♠♦s ♦r st út♠♦ ♣♥t♦ ♦♥sr♠♦s ♥t♦♥s ♥ ♣♦t♥ φ0(r) q s rt♠♥t r♣s♦ ♣r♦

♦♥t♥♦ ♥ st s♦ t♦r ♦t③♠♥♥ e−βφ0(r) q ♥♦♠♥r♠♦s e0(r) t♥ ♦r♠ q s

♠str ♥ rá♦ s♣r♦r ② ♥♦ s ♠② r♥t ♥ ♣♦t♥ q ♠♠♦s ed(r) ♦rrs♣♦♥♥t

♥ ♣♦t♥ srs rs ♦♥ á♠tr♦ s srs s d rá♦ ♥tr ♠s♠ r s♠♣r

② ♥♦ ♦r d st ♦♥♥♥t♠♥t ♦ P♦r ♦ t♥t♦ s ♠♦s ♣r♦♣♠♥t ♦r st

á♠tr♦ t♥r♠♦s q ♥t

∆e(r) = e0(r)− ed(r)

srá st♥t r♦ ♦♠♦ s ♠str ♥ r s♦♦ ♥ ♥ ♣qñ♦ ♥tr♦ q ♠r♠♦s ξd ♥ó♥

∆e(r) s ♦♥♦ ♦♥ ♥♦♠r ♥ó♥ ♣ ♥t♦♥s s ♦♠♦ s♣♦♥♠♦s ξd s ♣qñ♦ ♣♦♠♦s ♣♥sr

♥ r ♥ srr♦♦ ♥ ♣♦t♥s ♥ tér♠♥♦s ξd ♦ ♠s ♦♥♥♥t s srr♦r

A =−βFexc

V

♦♥ ♥♠♦s ①s♦ ♥rí r ♦♠♦

Fexc = −kBT lnZN (V, T )

V N

st ♦r♠ srr♦♦ s ♣ srr s♥t ♠♥r

r ♥ rá♦ s♣r♦r s ♠str ♦r♠ q ♣♦s t♦r ♦t③♠♥♥ e0(r) ♦rrs♣♦♥♥t ♥ ♣♦t♥rt♠♥t r♣s♦ ♣r♦ ♦♥t♥♦ ♥ó♥ ♥♦ s ♠② r♥t ♥ ♣♦t♥ q ♠♠♦s ed(r) ♦rrs♣♦♥♥t sr r ♦♥ á♠tr♦ sr s d rá♦ ♥tr ♠s♠ r út♠♦ rá♦♠str ♦r ∆e(r) = e0(r)− ed(r)

A = Ad +

ˆ

drδA

δe(r)|e=ed ∆e(r)+

+1

2

ˆ ˆ

drdr′δ2A

δe(r)δe(r′)|e=ed ∆e(r)∆e(r′) + ............

♦♥ Ad♦rrs♣♦♥ sst♠ srs rs ♥♥♦ ♥ ♥t ♥ó♥ ①s♦ ♥rí r

♠ás ♦♥sr♥♦ q ♦r♠s♠♦ ♥♦♥ ♥s s s♣r♥ qδFex

δφ(r)=

1

2ρ2g(r)

s δA

δe(r)=

1

2ρ2gN (r)eβφ(r)

♣♦r ♦ t♥t♦ s ♣ rsrr ♦♠♦

A = Ad +1

2ρ2ˆ

dr g0N (r)eβφ0(r)∆e(r) + ........

♦♥ s♣r♥ ♥ q sts ♥ts stá♥ s♦s sst♠ rr♥

♥ st út♠ ①♣rsó♥ s ♥ s♣r♦ tér♠♥♦s s♣r♦rs ♣r♠r ♦r♥ ♣s ♥②♥ ♥♦♥s

stró♥ ♠ás ♦s ♣rtís q ♥ ♥ á♦ tér♠♥♦ ♣r♠r ♦r♥ ó♥

s ♦r♥ ξd ♣s ∆e(r) ♦ s ♣r ♥t♦♥s ♥ ♥ ó♥ t♦♠r ♦r d ♣r q st

tér♠♥♦ s ♥ st ♦♥stt② rtr♦ t♦rí ❲ ♣r á♠tr♦ t♦ sr r P♦r ♦ t♥t♦

♥ st t♦rí á♠tr♦ s ♦t♥ rs♦rˆ

dr g0N (r)eβφ0(r)∆e(r) = 0

♥rí r ♥ ♥ ♣♦r

A0 = Ad

ó♥ ❲ ♥♦ s ú♥ ♣♦s Pst♦ q ∆e(r) s ♥♦ ♥ s♦♦ ♥ ♥ ró♥ ♠② str r♦r

r = d ♣♦♠♦s r ♥ srr♦♦ ②♦r t♦r r2g0N (r)eβφ0(r) q ♣r ♥ ♥tr♥♦ tér♠♥♦

r♦

r2g0N (r)eβφ0(r) = σ0 + σ1

( r

d− 1)

+ σ2

( r

d− 1)2

+ ...........

♦♥σmdm

=

(

dm

drmr2g0N (r)eβφ

0(r)

)

r=d

sstt②♥♦ ♥ ♦t♥♠♦s∞∑

m=0

σmm!

Im = 0

♦♥

Im =

ˆ

∞

0

d( r

d

)

∆e(r)( r

d− 1)m

=

− 1

m+ 1

ˆ

∞

0

dr( r

d− 1)m+1 d

dr

(

e−βφ0(r))

φ0(r) r rá♣♠♥t ♦♥ r r q ♣r ♥ ♥tr♥♦ st ♥ó♥ s ♥ó♥ t

♥tr ♥ r = d ② st ♠♥r sr q ♣r ♥ ♦♥r rá♣♠♥t t♥♠♦s ♥t♦♥s

s♦♦ ♣r♠r tér♠♥♦ ♠♦♦ q ♦♥ó♥ ♣r ♦t♥r á♠tr♦ t♦ sr r s I0 = 0 ♦

d =

ˆ

∞

0

dr(

1− e−βφ0(r))

st út♠ s ①♣rsó♥ á♠tr♦ t♦ ♣r t♦rí ♥ s♦ ♣rtr ♣r♦ ♠② s q

♣♦t♥ rr♥ t♥ ♦r♠ ♣♦t♥ ♥rs st♥ ♥tr♠♦r ♦ s

φ0(r) = ε(σ

r

)n

♦♥ ε s ♥ ♦♥st♥t ♥tr t♥ ①♣rsó♥ ♥ít rst♦ s

d = σ

(

ε

kBT

)1/n

Γ

(

n− 1

n

)

= σ

(

ε

kBT

)1/n

(1 +γ

n+ .......)

♦♥ γ = 0,5772.... s ♥♦♠♥ ♦♥st♥t r

♣r♥♣ r♥ ♥tr s ♣rsr♣♦♥s ❲ ❬ ❪ ② ❬ ❪ s q st út♠ r ♥

á♠tr♦ sr r q ♣♥ s♦♠♥t t♠♣rtr ♠♥trs q á♠tr♦ ❲ ♣♥ t♥t♦

t♠♣rtr ♦♠♦ ♥s st ♠②♦r ① ♥ ♥ ♥r q t♦rí ❲ s ♥ ú♥

s♥t♦ ♠ás ♦♠♣t q

Ps♠♦s ♦r ♥③r ♦♠♦ s♣rr ♣♦t♥ φ(r) ♥ ♥ ♣rt rr♥ φ0(r) ② ♥ ♣rtró♥ φ1(r)

♦ ♥ ♣♦t♥ ♥♦ ①st ♥ ú♥ ♠♥r r st♦ ♥ ♣r♥♣♦ ♣rrí q st rtrr ♥tr♦

♥ t ♦♥ ♥ t♦rí ♣r♦ ♣♦r ♦♥trr♦ ① ①tr q ♥tr♦ ♦r♠ rtrr

♣♥tr st ♣rtó♥ ♣r♠t ♠♥t ♥ ó♥ ♣r♦♣ ♣♦r ♠♦rr sst♥♠♥t ♦♥r♥

sr ♣rtrt ♦♠♠♦s ♠♦♦ ♠♣♦ s♦ ♣♦t♥ ♥♥r♦♥s st ♣♦t♥ r

♥ ♥ú♦ ♥♥t♠♥t r♣s♦ ♣♦r ♦ t♥t♦ s ss♣t sr s♣r♦ ♦♥ t♦t rtrr ♦r

①♣rsr♦ ♥ ♦s tér♠♥♦s ♣r♦s

r P♦t♥ ♥♥r♦♥s ♦rt♦ trs ♦r♠s r♥ts r s ♥ ♣r♦♣st rr② t③ sñ ♦♥ s ♣rtó♥ q ♣r♦♣♦♥♥ rr ② ♥rs♦♥ ② s q ♦rrs♣♦♥ t♦rí ❲ ♥ ♠♦♦ s tér♠♥♦ r−12 ♦♠♦ ♣♦t♥ rr♥ ② ♦ q s♥ ♥ó♥ r−6 q s t♦♠ ♦♠♦ ♣rtró♥ ♣r♦♣st ♦♥sst ♥ t♦♠r ♦♠♦ ♣♦t♥ rr♥ ♣rt ♣♦st ♣♦t♥ ♥♥r♦♥s r r < σ ♠♥trs q ♣rtró♥ ♦♥sst ♥ ♣rt♥t ♦ ♣♦t♥ r > σ ♥ ♣r♦♣st ❲ ♣rtó♥ s ♣♦r r ♣♦t♥ ♦♥st ♣rtó♥ s ♦♥s q ♣rtró♥ rí ♠♦ ♠ás ♥t♠♥t í ♦♥ ♥ó♥ stró♥r ♣rs♥t ♥ ♠á①♠♦ ♦

♥ s r♣rs♥t ♦♥ ♥ í♥ ♦♥t♥ ó♥ ♣r ♣♦t♥ rr♥ ② ♦♥ ♥ tr③♦s

ó♥ ♣r ♣rtró♥ ♥ ♥ s trs ♣♦ss q s ♣rs♥t♥

♦♣ó♥ s rr ② t③ ♦♣ó♥ s ♣rtó♥ ② s q ♦rrs♣♦♥

❲ ♥ ♣r♠r s s tér♠♥♦ r−12 ♦♠♦ ♣♦t♥ rr♥ ② ♦ q s♥

♥ó♥ r−6 q s t♦♠ ♦♠♦ ♣rtró♥ ♦s rst♦s ♣r ó♥ st♦ ♦ ♥♥r♦♥s

♦t♥♦s ♦♥ st ♠♦♦ s♦♥ ♣t♠♥t ♥♦s ♦♠♣r♦s ♦♥ s s♠♦♥s ♥♦ t♠♣rtr s

s♥t♠♥t t T∗ = kBT/ε ≥ 3 ♣r♦ ♠♣♦r♥ ♥♦t♠♥t ♠ q t♠♣rtr s♥ st♦

s ♥t♥ ♣s ♦♠♦ s ♦sr ♥ r ♣♦t♥ rr♥ s ♥♦t♠♥t ♠ás ♥♦ ② ♣♦r ♦

t♥t♦ r♥t ♥ sr r q q ♦rrs♣♦♥ ♣♦t♥ r ♥ ró♥ ♠ás ♠♣♦rt♥t s r

r♥ ♠í♥♠♦ ♣♦t♥ ♦♥ ♦♥tró♥ ♥rí r s ♠②♦r

♣r♦♣st ♦♥sst ♥ t♦♠r ♦♠♦ ♣♦t♥ rr♥ ♣rt ♣♦st ♣♦t♥ ♥♥r♦♥s

r r < σ ♠♥trs q ♣rtró♥ ♦♥sst ♥ ♣rt ♥t ♦ ♣♦t♥ r > σ á♠tr♦

s s♥♦ ♣rsr♣ó♥ ①♣ ♥ts ♦♥tró♥ ♣rtró♥ ♥rí r s ♦t♥

♣rtr ♦♥ g0(r) = gd(r)

♥♦ s t③ st t♦rí ♣r ♦t♥r s ♣r♦♣s tr♠♦♥á♠s ♦ ♥♥r♦♥s rst♦

s ♠② ♥♦ ♥q ♠♣♦r ♥ss ② t♠♣rtrs r♥s ♣♥t♦ tr♣ s s♦♦♥s ♠♦r♥

♥♦t♠♥t s s r ♥ tr♠♥♦ ♣rtrt♦ s♥♦ ♦r♥ st ♦ t♠é♥ s ♣ ♥t♥r ♥tt

♠♥t s t♥♠♦s ♥ ♥t q ♣rtó♥ ♥② ♥ ♣rtró♥ ♣rt rá♣♠♥t ♠♥t

♣♦t♥ q s r = σ st ♠í♥♠♦ ♥ rm = 21/6σ ♥ st r♥♦ st♥s s st♠♥t ♦♥

♥ó♥ stró♥ r srs rs ♥③ s ♦r ♠á①♠♦ ② ♣♦r ♦♥s♥t s t♦♥s

♥rí ♣rtrt t♦t s♦♥ r♥s P♦r ♦ t♥t♦ s♥♦ ♦r♥ ♣rtr♦♥s sr ♠♣♦rt♥t ❬

s♥♦ ♦r♥ ♣r♦♣♦r♦♥ ts t♦♥s ♦rrs♣♦♥ trr s♠♥♦ ❪ st ♠♥r

♣r♦♣st s ♥ ♥♦ s ♦♠♣r ♦♥ ♦s rst♦s s♠♦♥s ♣r♦ s♦♦ s s ♥② ♥

s♥♦ ♦r♥ ♣rtrt♦ ♥s ♥r ♥ s♥♦ ♦r♥ s ♠♥ s s ♠♦♦ ❲ ♥

st s♦ ♣♦t♥ s ♥ r = rm ② s t♦♠

φ0(r) =

φLJ(r) r < rm

0 r > rm

φ1(r) =

−ε r < rm

φLJ(r) r > rm

♦♥ st ♣rtó♥ s ♦♥s q ♣rtró♥ rí ♠♦ ♠ás ♥t♠♥t í ♦♥ ♥ó♥ stró♥

r ♣rs♥t ♥ ♠á①♠♦ ♦ st ♦r♠ ♥ st sq♠ ♣r♠r ♦r♥ ♣rtrt♦ s ♦♥s♥

rst♦s t♥ ♥♦s ♦♠♦ ♦s q s ♦t♥♥ ♥ ♠♦♦ ♦♥ s♥♦ ♦r♥ ♣rtr♦♥s ♥s

♣r st♦s tr♠♦♥á♠♦s ♣ró①♠♦s ♣♥t♦ tr♣

♣ít♦

♦♥s♦♥s

♠♦s ♣rs♥t♦ ♥ ♣r♦♠♥t♦ ♥♦♦s♦ st ♥♦♥ ♥s q ♥♦s ♣r♠tó ♣rr

♥ ♦♥t①t♦ s ♦rís ♥♦♥ ♥s rss ♣r♦♣s s♦s ♦♥ ♦s s♠♣s

♦♥stt♦s ♣♦r ss ♥♦s rs ♦ s♦r♦s s♦r sstrt♦s só♦s ♣rs ♣♥s ♦ r♥rs ♦♥stts ♣♦r

♠♥t♦s ♥♦s ② ♥♦s ♦tr♦s ♠♥t♦s ♦ ♦♠♣st♦s

♥ st tss ♣r♦♣♦♥♠♦s ♥ ♣r♦♠♥t♦ st ♥♦♥ ♦♥ r♦ ♥ tér♠♥♦ ♦♥

♥♦ ♥♦♥ s ♦♥str② ♥t♦♥s t♥♥♦ ♥ ♥t ♦ s♥t

P♦r ♥ ♦ s t③ó ♥ ♣♦t♥ ♣rs t♦ q s ♦t♥ ♦♣t♠③♥♦ ♦s ♣rá♠tr♦s ♣♦t♥

♣rs ❲s ♥r ② ♥rs♥ r ❬❪ ② ♣é♥ st st ♣♦t♥ ♣rs q

ss ♣rá♠tr♦s ♣r♦♥ ♣♦t♥ ② á♠tr♦ r♦③♦ át♦♠♦s ♦ ♠♦és ♣s♥ ♣♥r

t♠♣rtr ② s r③ s♦r ♦ ♦♠♦é♥♦ ♣r r♣r♦r s ♣r♦♣s ♦①st♥ s ss

íq♦♣♦r

♣♦t♥ ♣rs t♦ sí ♦t♥♦ s t③ó ♣r srr ♦ t♥t♦ ♥ ♦♥♦♥s ♦♠♦é♥s

♦♠♦ ♥♦♠♦é♥s

r♦ ♥♦♥ ♥ ♣rá♠tr♦ ♦♥ tér♠♥♦ s q s st ♥ s♠t♥♦ ♦s rst♥ts

♣rá♠tr♦s ♣♦t♥ ❲ stó♥ st ♦♥t st r♦♥ ♣r ♥♦s ♦s ♦♥

♣rs♥ t♦s á♥t♦s ♦♠♦ s♦ ó♥ ② ♥ ♠♥♦r r♦ ♣r ró♥ Pr♦ t♠é♥ s t③ó♥

♣♦rí strs ♣♥♦ só♥ ♦♥ó♥ q ♣rs♥t♥ ♦s ♦s ♥♦ s t♠♣rtr

s r ♣♥t♦ rít♦ ♦♥ s t♦♥s ♥ ♥s r♥ ② ♦♥ s r♥ rs ♦trs

♣r♦♣s

♣r♦♣st♦ ♦♥♦ r♥♦ q ♣ró♥ ♣r t♥só♥ s♣r ♥trs ♦♥tt♦

♥tr s íq ♦♠♦é♥ ② ♣♦r ♦♠♦é♥ ♥ ♦①st♥ tr♠♦♥á♠ ♥ t♠♣rtr s

♦rrt ♥óts q ♥trs íq♦♣♦r s ♥♦♠♦é♥ str ♣r♦♣st ♣r♠t ♦t♥r rst♦s ♥

t♦♦ r♥♦ t♠♣rtrs ♥tr Tt ② Tc ♦ st ♠♥r ♠♦s rst♦ ♥♦♥♥♥t q ♣rs♥t

♣r♦♣st ♥♦tt♦ ② ♦♦r♦rs ❬ ❪ q ♦r♥ r♥s q ♥♦ ♣r♠t♥ ♦t♥r s♦♦♥s

♥♠ér♠♥t sts ♣r t♠♣rtrs r♥s Tt

♥ s♦ ♦s s♦r♦s s♦r ♣rs ♣♥s ♦♥ ♦ r♥♦ ♣r s rt♠♥t ♥♦♠♦é♥♦

s r♣r♦r♦♥ ♦rrt♠♥t ♣rs ♥s tr♠♥♦s t♥t♦ ♠♥t s♠♦♥s ♦♥t r♦

❬ ❪ ♦♠♦ ♣♦r ♠♦ t♦rís ♥♦ ♦s ♥ ♦♥t①t♦ ♣r t♠♣rtrs s Tt

♥③r♦♥ t♠é♥ ♣r♦♣s ♠♦♦ s♣rs ② st♦♥s rr♥ts tr♥só♥ ♠♦♦ sts

♣r♦♣s r♦♥ ①trís ♥áss strtr s s♦tr♠s s♦ró♥ ♥ st s♦ ♥♦s ♦s

rst♦s ♠ás r♥ts q ♠♦s ♦t♥♦ r♦♥

Pr sst♠ r2 s ♣♦ ♥r s♦tr♠s ② strtr s ♠ás ♣s q s r♣♦rts ♥ ♥♦s

tr♦s ❬❪

Pr sst♠ r ♦♥ ♥♦ ①st♥ ♣rát♠♥t r♣♦rts s♦r st♦s ♠♦♦ st sst♠ t

③♥♦ ♥♦♥tr♠♦s q ①st ♠♦♦ r♥ ♦ q sñ♥ ♥♦s rst♦s ♦s ♠♥t

s♠♦♥s ♥♠érs

♥ sst♠ r s ♦sr ♣ró♥ ♥ strtr s♠♦♥♦♣s st ♦♠♣♦rt♠♥t♦ s

s♠r r♣♦rt♦ ♥ trtr ❬❪

♥♦♥tr♠♦s ♦rs ♠♦♦ ② ♣r♠♦♦ ♣r sst♠s ♦♠♦ ❳ ❳ ❳

♦♠♣♠♥tr♦ ♦s tr♦s ♦♠♦ s rr♥s ❬❪ ②❬❪ ♥♦s♦tr♦s ♠♦s ♦r♦ r♣r♦r ♣r

sst♠s rs ♦♠♦ ♦s ♦♥stt♦s ♣♦r ❳♥ó♥ s♦r♦ s♦r rs♦s sstrt♦s s strtrs í♥s

♣r♠♦♦ ♠út♣s s♦r s q s t♦r③ ♥ st♦s tr♦s

♥ t♦♦s ♦s s♦s s ♦♣ró ♥tr♦ ♥s♠ ♥ó♥♦ s r ♦♥ ♥ sst♠ ♦♥ s ♠♥t♥í ♦♥st♥t

♥ú♠r♦ ♣rtís ♦ ♠ás t♠♣rtr ② ♦♠♥ ♥q ♥♦s rst♦s r♦♥

r♦t♥♦s tr♥♦ ♥tr♦ ♥s♠ r♥ ♥ó♥♦ ♠ás r♣rs♥tt♦ ♦ q ♦♥stt② ♥ sst♠

r

♥③r♦♥ ♥s ♣r♦♣s st♦s ♦s s♦r♦s ♥ r♥rs ♦♥♦r♠s ♣♦r ♣rs é♥ts s

♣t♦ ♣r s♦r♦ s ♥♦♥trr♦♥ t♥t♦ ♣rs q r♥ s♠trí ♣♦t♥ ①tr♥♦ ♣♦ sí

♦♠♦ ♣rs q r♦♠♣♥ st s♠trí sts st♦♥s stá♥ ♥ ♣rt♦ ♦r ♦♥ s r♣♦rts ♥ trtr

s ❬ ❪

♥ts rst♦s r♦♥♦s ♦♥ r♣tr s♠trí ♣r ♥s ♦t♥♦s ♥ ♦♥t①t♦

♥ s♦ r♣♦rt♦s ♣♦r r♠ ② ♥st♥ ❬❪ s♣t♦ st♦ ♥♦s♦tr♦s ♠♦s ♣♦♦ r♣r♦r ♣♦r

♠♦ ♥str♦ ♥♦♥ s♦s rst♦s ② ♠ás st♠♦s ♥ ♦rró♥ ♥tr ♥s t♠♣rtr ②

r♣tr s♠trí ♠♦s ♦ q r♣tr s♠trí ♥ ♣r q♥ ♦♥stt② st♦ ♥♠♥t

s ♣r♦ ♣r rt♦ r♥♦ ♥ss st r♥♦ ♦♥stt② ♥ ♦♥♥t♦ ♦♥t♥♦ ♦♥ s s♦♦♥s s♦♥

sts ♦ ♠tsts s ③ st r♥♦ ♣♥ t♠♣rtr t ♠♥r q ♠ q st

♠♥t r♥♦ ♥ss ♣r st♦ ♠ás ♥rí s s♠étr♦ s t♦r♥ ♠ás str♦

P♦r ♦tr♦ ♦ ♦♥tr♠♦s r ♣♦s ró♥ ①st♥t ♥tr ♥ó♠♥♦ r♣tr ② s ♣r♦♣s

♠♦♦ s♦r ♥ ♣r ♠♦s ♥♦♥tr♦ q ♠á①♠ t♠♣rtr ♣r ♥ ①st♥ s♦♦♥s

s♠étrs ♥tr♦ r♥r ♦♥♦r♠ ♣♦r ♦s ♣rs é♥ts s ①t♥ ♣♦r ♥♠ ♦rrs♣♦♥♥t

t♠♣rtr ♠♦♦ ♥ s ♣rs ♣r♦ ♥♦ ♠s á t♠♣rtr rt ♣r♠♦♦ q

♦rrs♣♦♥ s ♣r s r t♠♣rtr r♣tr s♠trí q ♠♦s ♥♦ ♦♠♦ q ♣♦r

♥♠ ♥♦ ①st♥ s♦♦♥s s♠étrs ♣r ♥♥ú♥ r♥♦ ♥ss ♦♥ ♦♥ t♠♣rtr

rít ♣r♠♦♦ sté♥♦s st ♠♥r ♥ ró♥ ♥tr ♦s ♥ó♠♥♦s q ♥ ♣r♥♣♦ ♥♦ ♣r♥

r♦♥♦s

P♦r t♠♦ s str♦♥ r♥rs rts r sst♠s ♥ ♦♥tt♦ ♦♥ ♥ rsr♦r♦ ♣rtís s

Pr ♦ s tr♦ ♥ ♥s♠ r♥ ♥ó♥♦

♥ st s♦ s s♦♦♥s q s ♦t♥♥ s♦♥ qs q s♦♥ sts ♦ ♠tsts Pr sts r♥rs t♠é♥

s ♦tr♦♥ s♦♦♥s s♦s st♦s ♥♠♥ts q r♦♠♣♥ s♠trí ♣♦t♥ ①tr♥♦ ♣r♦

r♥ r♥r rr s♦♦ ♥ s♦ó♥ s♠étr ♣♦s ♦r ♣♦t♥ qí♠♦ ♦①st♥

❱ r ♥ st t♠♦ s♦ ♠ás ♦s ♦s st♦s s♠étr♦s sts q ♦①st♥ s♦♦s ♥ ú♥♦

♦r ♣♦t♥ qí♠♦ ♣r ♥ trr st♦ s♠étr♦ t♠é♥ st q ♣♦s ♠s♠♦ ♦r

♣♦t♥ qí♠♦ ② q ♦ ♣ ♣♥srs q ♦①st ♦♥ ♦s ♦s s♠étr♦s st ♦ ♥♦ r♣♦rt♦ st

♠♦♠♥t♦ ♥ ♥♥ú♥ ♦tr♦ tr♦ s♦r t♠

s ♥st♦♥s r③s ♥ ♠r♦ ♣rs♥t ss r♦♥ ♦r♥ ♦s s♥ts rtí♦s

♥st② ♣r♦s ♦ r s♦r ♥ sts ♦ 2 ♣♦♥t♥♦s s②♠♠tr② r♥

③②s③ ♥ rtr

♦r♥ ♦ ♠ P②ss

s♦r♣t♦♥ ♦ ♦♥ srs st t ♥st② ♥t♦♥ t♦r②

rtr ③②s③ ♥ ❯rrt

P②s

♦rrt♦♥ t♥ s②♠♠tr ♣r♦s ♥ sts ♥ st♥r ♣rtt♥ ♥s

rtr ♥ ③②s③

P♣rs ♥ P②ss

s♦r♣t♦♥ ♦ r ♦♥ ♣♥r srs st t ♥st② ♥t♦♥ t♦r②

rtr ♥ ③②s③

P②s

♣♦♥t♥♦s ②♠♠tr② r♥ ♥ rst♦rr ♣s tr♥st♦♥s ♦ s♦r s

rtr ③②s③ ♥ ❯rrt

♥tr♥t♦♥ ♦r♥ ♦ rt♦♥ ♥ ♦s

♦♥♥♠♥t ♦ r t♥ t♦ ♥t ♣r s♠♥♥t s

rtr ♥ ③②s③

♦r♥ ♦ ♠ P②ss

♦rrs♣♦♥♥ t♥ s②♠♠tr ♥ ♦ sts ♥ rst♦rr ♣s tr♥st♦♥ ♥s

③②s③ ♥ rtr

P ♥s

s♦r♣t♦♥ ♦ s ♦♥ s♦ srs r♦t t♦r r② ♥s ②rs

rtr ♥ ③②s③

P②s rt ♥ ♣rss ♦♣②s

♥ ♦ ♣♥r sts ♦ ♠ts t ♦rrs♣♦♥♥ ♦ ♣rtt♥ ♥s ♥ t

♣♣r♥ ♦ s②♠♠tr ♣r♦s

rtr ♥ ③②s③

P②s s♦♠t♦ rrt♦

♦rí

❬❪ ♥ ♥ ♥ ♥ ♥ ❱♦ss♥ P②s tt

❬❪ ♥ ♥ ♥ ♥ ❱♦ss♥ ♥ r tt P②s

❬❪ r♠ ♥ ♥st♥ ♠ P②s

❬❪ ③②s③ ♥ rtr ♠ P②s

❬❪ ♥♦tt♦ ♥ ♦♦ P②s

❬❪ ♥♦tt♦ rtr♦♦ ♦♦ ♥ ❲ ♦ P②s tt

❬❪ ③♠s② ❲ ♦ ♥ ❩r♠ ♦ ♠♣ P②s

❬❪ str ♥♦tt♦ rs ♥ ♦♦ P②s tt

❬❪ rtr ③②s③ ♥ ❯rrt P②s

❬❪ rtr ♥ ③②s③ P♣rs ♥ P②ss

❬❪ rtr ♥ ③②s③ P②s

❬❪ rtr ③②s③ ♥ ❯rrt ♥tr ♥ ♦s

❬❪ rtr ♥ ③②s③ P②s ♦♣②s

❬❪ ③②s③ ♥ rtr P ♥s

❬❪ rtr ♥ ③②s③ ♠ P②s

❬❪ rtr ♥ ③②s③ P②s s♦♠t♦ rrt♦

❬❪ r♥ ♦ P②s

❬❪ r♠s♦♥ P②s

❬❪ ♥ ❲s ❩ P②s ♠

❬❪ P ♦♥r ♥ ❲ ♦♥ P②s

❬❪ r♥♥ ♥ tr♥ ♠ P②s

❬❪ P r③♦♥ P②s rrt

❬❪ P r③♦♥ ❯ r♦♥ ♥ ♥s ♦ P②s

❬❪ ♦s♥ P②s tt

❬❪ r ♥ ♦s♥r P②s

❬❪ qrs ♥ ♦ P②s

❬❪ Pt♦s s ❨r♥♦ ❩♦ ♥ ♥ rt Pr♦♥s ♦ t t♦♥

♠② ♦ ♥s

❬❪ P ♥♥s r♦r❲②rt ♥ r ♣r② ♥ ❲tt♥ P♥♦♠♥ r♦♣s s

Prs ❲s ♣r♥r❱r ❨♦r

❬❪ ❲ r ❲ ♦ ♥ ❩r♠ P②s s♦r♣t♦♥ ♦rs ♥ P♥♦♠♥ ♦r Ps♥

♥♦ ❨♦r

❬❪ P ♥s♥ ♦♥ ♦r② ♦ s♠♣ qs sr ♦♥♦♥

❬❪ tr♦♣♦s ❲ ♦s♥t ♦s♥t r ♥ r ♠ P②s

❬❪ r ♥ ❲♥rt ♠ P②s

❬❪ Prs ♥ ❨ P②s

❬❪ ♦ P②s

❬❪ ♥s P②s

❬❪ ❲rt♠ P②s tt

❬❪ ❱rt ♥ ❲ss P②s

❬❪ rr ♥ ♥rs♦♥ ♠ P②s

❬❪ ❲s ♥r ♥ ♥rs♥ ♠ P②s

❬❪ rr ttst ♥s r♣r ♥ ♦ ❨♦r

❬❪ r♠♥ P②s

❬❪ Ptr ttst ♥s sr ♠str♠

❬❪ ♥ ttst ♠♥s ❲② ❨♦r

❬❪ P ♥tt tst qs ♦♥♣ts ♥ Pr♥♣s Pr♥t♦♥ ♠ rs②

❬❪ ②r ♠ P②s

❬❪ rr P ♦♥r ♥ P♦♠♣ ♠ P②s

❬❪ ♥ ❲ ♦♦r ♠ P②s

❬❪ P r③♦♥ ♥ ♥s ♦ P②s

❬❪ P r③♦♥ ♦ P②s

❬❪ ♥s ♥ ♥♠♥ts ♦ ♥♦♠♦♥♦s s ♥rs♦♥ r ❨♦r

❬❪ ♦r♦♠ ♦♥s♦♥ ♥ rsr st P②s

❬❪ P r③♦♥ ♥ ♥s ♦ P②s

❬❪ ❲ rt♥ ♥ ❲ sr♦t P②s

❬❪ ♥t♦♥ ♥ ❲ sr♦t P②s

❬❪ ts♦ ♥ s P②s

❬❪ ❨ ♦s♥ ♠ P②s

❬❪ ss rs ♥ ♦t③ ♠ P②s

❬❪ ♦t③ ♥ ♦♥s♦♥ ♠ P②s

❬❪ ❨ ♦s♥ ♠t ö♥ ♥ P r③♦♥ P②s

❬❪ P r③♦♥ ♥ ❨ ♦s♥ P②s

❬❪ Prs tt P②s

❬❪ P r③♦♥ P②s tt

❬❪ P r③♦♥ P②s

❬❪ P ♦t ❱s♥②♦ ♥ ❱ ♠r P②s

❬❪ rr ♥ ♥rs♦♥ ♦ P②s

❬❪ ♦♥s♦♥ ❩ö ♥ ♥s ♦ P②s

❬❪ ♥s ♥ ♠ ♦ P②s

❬❪ r♥♦ ♠♦ ♥ P r③♦♥ P②s

❬❪ r♥♥ ♥ tr♥ P②s

❬❪ ♥s ♥ P r③♦♥ P②s

❬❪ ❱ ♥♦ ❱ssr♠♥ ❱ ♦st♣ ♥ ❱sr r♠♦♣②s Pr♦♣rts ♦ ♦♥

r♦♥ r②♣t♦♥ ♥ ❳♥♦♥ ♠s♣r ❲s♥t♦♥

❬❪ r ♥ ♦s♥r P②s

❬❪ t rs ❲ ♦ ♥ P②s

❬❪ rtr♦♦ ❲ ♦ ♥ P②s

❬❪ ♥♠ ♠ P②s

❬❪ ❲ ♠♠♦♥ ♥♥ ♥ r♥ ♥ ♠str② ❲♦♦ t♥r r♥

ts ♠r ② P ♥str♦♠ ❲ r t♦♥ ♥sttt ♦ t♥rs ♥ ♥♦♦②

trsr tt♣♦♦♥st♦

❬❪ ❲ ♥ ❨♥ ♠ P②s

❬❪ ♠t P r♥r ♥ Prr ♠ P②s

❬❪ ❲♦♠ ♠ P②s

❬❪ ❩♦ ♥s ♥ ♥ P②s tt

❬❪ t② ♥ ❯ ♥r P②s tt

❬❪ ♦t ♥ t P②s

❬❪ r♦②♠ ♥ ①♥r ♠ P②s

❬❪ ❱r s ♥ ss ♦ P②s

❬❪ ♥ ♥ t ♠ P②s

❬❪ ♥s P r③♦♥ ♥ ❯ r♦♥ ♦ P②s

❬❪ ♥ sq ♥ ❲s ♠ P②s

❬❪ ♦♦♦s ♥ sr ♦ P②s

❬❪ ❲t ♦♥③á③ ♦♠á♥ ♥ ❱s♦ P②s tt

❬❪ ♦♦ ♠ ♣♣ ❱

❬❪ ♥r ❲ ♠ P②s tt

❬❪ ❲ ♠ ♥ ♥r P②s

❬❪ ♥ ❲ ♦ ❲ ♠ ♥ r♥r P②s

❬❪ ❩r♠ ♥ ❲ ♦♥ P②s

❬❪ ❲ t ♥trt♦♥ ♦ ss t s♦ ss Pr♠♦♥ ①♦r

❬❪ ♦♥ t♥ rtr♦♦ ❲ t ♥ ❲ ♦ P②s

❬❪ ❲ ♥ ♠ P②s

❬❪ P ♥♥s ♦ P②s

❬❪ P♥t ♥ ❲♦rts P②s

❬❪ ♥ ❲ ♦ ♣♦♥t♦ ❲ ♠ ♥ r♥r ♦ P②s

❬❪ P r③♦♥ ♥ ♥s ♦ P②s

❬❪ P♥t ♥ sr P②s tt

❬❪ ③②s③ P②s

❬❪ ♥ str ❲ ♥ ❲ ♦ rrr♦ ❲ ♠ ♥ ♦♦ P②s

tt

❬❪ str ♥ ❲ ♥ ♦ ♠♣ P②s

❬❪ ♦ss P ♦r ♥ t P②s

❬❪ ss t♥ ♥ ❲ ♥ P②s tt

❬❪ r♠ s ♥ ③♥♥ P②s tt

❬❪ t♠ t ♥ P ♦r P②s tt

❬❪ P ♠r r♥ r ♥ ♥ ♥ P②s

❬❪ ♥rs♦♥ P r③♦♥ ♥ ♦ ♥ ❱s♦ ♠ P②s

❬❪ ❱ ♥♦② ♥ ❲ ♠ P②s tt

❬❪ rtr♦♦ t♥ ♦♥ ❲ ♦ ♥ ❲ t P②s

❬❪ P rs ③③♥ sñs tr ♦rst ♥ ♠s♦♥ ♠

P②s

❬❪ ❩♥ ♥ ❩♦♥ P②s

❬❪ ❲s♦♥ P②s

❬❪ ❲t ♥ ❩♥ ♠ P②s

❬❪ ❲t ❱ ♥ trss ♥ ♠ P②s

❬❪ ③②s③ ♥ st P②s

❬❪ ♠♥ts r♦ts ♥ tr P②s tt

❬❪ ③②s③ P②s

❬❪ ❱ ♠r P ♦t ♥ ❱s♥②♦ P②s

❬❪ ❲ s♦ ♥♦t ♠♣ ♠♥ts ♥ r♦ts P②s

❬❪ s♦♥ ♥ P ♦♥s♦♥ ♦ ♠♣ P②s

❬❪ r ss ♥s ❯♥rstt

❬❪ r ♥ r♥ ♥ ♣ Pr♣♦s ♦♠♣trs r ♠ ♦♥♦♥

❬❪ P ♠r r♥ r ♥ ♥ ♥ P②s

❬❪ P ♠r r♥ r ♥ ♥ ♥ P②s

❬❪ ❱s♦ ♥ P r③♦♥ ♠ P②s

❬❪ r ♥ ö♥ P②s

❬❪ ❲ ②s♦ Ptr② ♥ ♦♦♦s P②s

❬❪ s♦♥ ♥ rts ♠ P②s

❬❪ ♦ ♥s ♥ ❲♥ ♠ P②s

❬❪ r♦s ís íq♦s ♥sttt♦ ♥s trs r ♦♥s♦ ♣r♦r ♥st

♦♥s ♥tís