]  § X N ËP U ê sX N ËÅ k Ä ô p §; c  \ ¥ Œ £ ?’ Ò ×8 ý X N Ë “ Ó Þ ° ‚ Ç “ Ó ÞS ë s Å k Ä ô p §Ê Ý § Žô p §8 ý W Ä] K ¡

T

]  § X N ËP  U ê sX N ËÅ k Ä ô p §; c   \ ¥ Œ £ ?’ Ò ×8 ý X N Ë “ Ó Þ ° ‚ Ç “ Ó ÞS ë s Å k Ä ô p §Ê Ý § Žô p §8 ý W Ä] K ¡



™ »6 0# ” 

ô

 ǀ ª œ@ /† < Ɠ §  o† < Æ/ B N † < Æõ , " fÖ  ¦ 133-791

T

* å Ù • v

ô

 ǀ ª œ@ /† < Ɠ § Ó ü t o † < Æõ , " fÖ  ¦ 133-791



™ »? 4 w H

ô

 ǀ ª œ@ /† < Ɠ § l > / B N † < Æõ , " fÖ  ¦ 133-791

(2010¸   1 Z 4 20{ 9  ~ à Î6 £ §, 2010¸   6 Z 4 7{ 9  à º& ñ ‘ : r ~ à Î6 £ §, 2010¸   7 Z 4 13{ 9  > F  S X ‰& ñ )

ô

 Ç F K5 Å q Á º 8l (cluster) " é ¶  [ þ t \ " f [ þ t ä ¼t  · ú §“ É r [ j { 9  (epn: electron, proton, neutron [ þ t _  ' Í Qo  / å J  ) [ þ t _  þ j@ / à º(g

¹

= Z/n) \  @ /ô  Ç \ P  [ þ t> p u 1 p x“ : r‚  (isotherm) d ” [ þ t`  ¦ \  -t  ï  r 0 A[ þ t \    É r í

 H & h “   s † ½ Ó~ ½ Ó& ñ d ” [ þ t`  ¦ Y  L † < Ê\  _  # Œ “ : r • ¸_  † < Êà º– Ð Ä »• ¸ % i  . Õ ª[ þ t – РÒ'  ˜ Ð  ± ú “ É r \ P \  - t

 ï  r 0 A[ þ t _  l  ¨ î ç  H & ñ & h  \ P  6   x | ¾ Ó d ” [ þ t`  ¦ Y V– Ð
 Í Ç x . s  Ê ê [ þ t – РÒ'  5 \  -t  ï  r 0 A[ þ t \   

 É

r l  ¨ î ç  H & ñ & h  \ P  6   x | ¾ Ó(C

^ν

: J K

⁻¹

mol

⁻¹

) d ” , C

ν5

= √

⁵

C

l1

C

l2

C

l3

C

l4

C

l5

s  F K5 Å q[ þ t _  & ñ · ú š \ P  6   x

|

¾ Ó(C

p

: J K

⁻¹

mol

⁻¹

) _  z  ´+ « > X <s ' \   © œ ¸ ú ˜ ´ ú € Œ ¤ . Ag`  ¦ q 2 Ÿ © ô  Ç # Œ Q F K5 Å q[ þ t(Cu, B, Ni, Pd) _ 

\

 -t  ï  r 0 A[ þ t _  à º[ þ t“ É r 4 ˜ Ð   H ß ¼“ ¦ 5˜ Ð   H  Œ • . Ä »• ¸[ þ t \ " f epns  1 l x r & h “   \ P  [ þ t> p u s  F  g  _  l

‘ : r& h “   é ß –0 A  ) a    H & ñ `  ¦ % i Ü ¼ 9, > í ß –& h Ü ¼– Ð S X ‰ “   % i  .

^ 

¦ Þ Ôë ß –  © œÃ º, k

B

(J K

⁻¹

energy level

⁻¹

)  H F K5 Å q " é ¶  \  e ”   H ô  Ç \  -t  ï  r 0 A_  ¨ î ç  H q \ P ,

^Z_n

k

K

s 

 )

a  . Õ ªM : k

K

(J K

⁻¹

photon

⁻¹

)  H à º† < Æ& h Ü ¼– Ð F K5 Å q " é ¶  \  e ”   H — ¸Ž  H epn[ þ t \  @ /ô  Ç ô  Ç F  g  _  ¨ î ç  H q

\ P e ” `  ¦ µ 1 Ï|  % i  . y Œ • \ P  \  -t  ï  r 0 A  H F K5 Å q é # Qo  ë ß –[ þ t # Q| 9  M :, > á ¤ ° ú  “ É r > hà º_  [ þ t ä ¼t  · ú §`  ¦ C

• ¸ epn[ þ t(

^Z₅

)`  ¦ ° ú   H  . Õ ªM :  Ä » ×  æ$ í  [ þ t – Ð ë ß –[ þ t # Q”   5 Å q(core) epn[ þ t“ É r 1 ˜ Ð   Œ •“ É r  Ò& h “   \ 



-t  Y U6 \ š[ þ t – Ð  Œ •6   x % i  . \  -t  ï  r 0 A[ þ t“ É r Û ¼— 2 ;[ þ t _  + þ AI \     > á ¤ ° ú  “ É r 4 Õ ªÒ  ¨[ þ t`  ¦ s À Ò 9, Õ ª Û ¼

—

2 ;[ þ t _  + þ AI   H ←⇐, →⇐, ←⇒, →⇒ – Ð & ñ  ) a  . # Œl " f ←⇐ epn“ É r „   ü < € ª œ$ í  (+×  æ$ í  )_  Û ¼

—

2 ; ~ ½ ӆ ¾ Ó[ þ t s  ½ ¨(sphere)“   " é ¶  _  µ 1 ÚÜ ¼– Ð † ¾ Ó  9, →⇐ epn\ " f  H „   _  Û ¼— 2 ;“ É r î ß –Ü ¼– Ð € ª œ$ í  (+×  æ

$ í

 ) Û ¼— 2 ;“ É r µ 1 ÚÜ ¼– Ð † ¾ Ó  9, ←⇒ epn\ " f  H „   _  Û ¼— 2 ;“ É r µ 1 ÚÜ ¼– Ð € ª œ$ í  (+×  æ$ í  ) Û ¼— 2 ;“ É r î ß –Ü ¼

–

Ð † ¾ Ó  9, →⇒ epn\ " f  H „   ü < € ª œ$ í  (+×  æ$ í  )_  Û ¼— 2 ; ~ ½ ӆ ¾ Ó[ þ t s  î ß –Ü ¼– Ð † ¾ Óô  Ç . Õ ªM : ™ èà º& h “  

\

 -t  ï  r 0 A_  à º(n) X <s '  ´ ú Æ Ò# Q ˜ Ðl \ " f ) ‡6   x ÷ &% 3  . ‘ : r ƒ  ½ ¨\ " f ½ ¨ô  Ç F K5 Å q[ þ t _  Ÿ í o “ : r • ¸, T

s

s (K)[ þ t“ É r θ

D

s (Debye : £ ¤$ í “ : r • ¸: K)[ þ t ˜ Ð   s `›   ± ú l  M :ë  H \ , Õ ª[ þ t  ’  [ þ t s  ƒ  5 Å q ^ ‰ d ” [ þ t“   s 

†

½ Ó& ñ o  d ” [ þ t _  Y  L \  _ ô  Ç Ä »• ¸  H ƒ  5 Å q ^ ‰ s  : r \   8¹ ¡ ¤ ½ + Ë{ © œ % i  .

Ù þ

˜d ” # Q: : Ÿ x >  \ P % i † < Æ, F K5 Å q _  \ P 6   x | ¾ Ó, ƒ  5 Å q ^ ‰ Ó ü t o , Debye “ : r • ¸, s † ½ Ó& ñ o , ^  ¦ Þ Ôë ß –  © œÃ º, s x  ' p(epn)

Heat Capacity Equations of Metals at Constant Volume Based on Binomial Theorem Equations and Their Mechanics

Daekyoum Kim ^∗

Chemical Engineer Department, Hanyang University, Seoul 133-791

-729-

Youngpak Lee

Physics Department, Hanyang University, Seoul 133-791

Tae-won Kim

Mechanical Engineer Department, Hanyang University, Seoul 133-791 (Received 20 January, 2010 : revised 7 June, 2010 : accepted 13 July, 2010)

The thermal excitation isotherm equations for a maximum three particles (an electron(e), its proton(p) and its neutron(n); epn) number (g

1

= Z/n) of each energy level in metal atoms are derived as a function of temperature by multiplying sequential binomial theorem equations according to the energy levels. From those equations, the geometric mean heat capacity equations at constant volume of the lower thermal energy levels are expressed in sequence. Among these, the geometric mean heat capacity equation at constant volume for five energy levels, C

ν5

= √

⁵

C

11

C

12

C

13

C

14

C

15

, had the best fitting to the experimental heat capacity data at constant pressure(C

p

). The numbers of the energy levels of many metals (Cu, B, Ni, Pd), headed by Ag, are larger than four and smaller than five. In the derivations, we assumed and confirmed by computation that three particles (epn) in a metal atom is the basic unit for synchronized thermal excitation. The Boltzmann constant, k

B

, is found to be an average specific heat capacity,

^Z_n

k

K

, an energy level in the metal atom. Then, k

K

is mathematically found to be an average specific heat capacity of one photon for all epns in the metal atoms. Each thermal energy level has the same number of average unexcited orbital epns (Z/5) when the clusters are composed of metals. The core sets made of free neutrons react as if they are equal to or less than one additional energy level. The energy levels are grouped according to the forms of the spins, and the four kinds of the energy level spins are ←⇐, →⇐, ←⇒ and

→⇒. Here, for the ←⇐ epn, the direction of the spins of the electron and its proton (+neutron) is outward from the sphere of the atom. In the →⇐ epn, the direction of the spin of the electron is inward, and the spin of its proton (+neutron) is outward. In the ←⇒ epn, the direction of the spin of the electron is outward, and the spin of its proton (+neutron) is inward. In the →⇒ epn, the direction of the spins of the electron and its proton (+neutron) is inward. Then, a fractional energy level number is permitted in the data fitting. Because the saturation temperatures, T

s

s, of the metals obtained in the present study are much lower than the Debye temperatures, θ

D

s, the derivations through a multiplication of binomials that are themselves continuum equations satisfy the continuum theory.

PACS numbers: 05, 65.40.+g, 05.20.-y, 05.10.-a, 05.50.+q, 05.70.-a

Keywords: Statistical thermodynamics, Heat capacity of metal, Continuum physics, Debye temperature, Binomial theorem, Boltzmann constant, epn

I. ø m É Ã U Ø

l

/ B N(porous) “ ¦^ ‰ f  ¨ ‚ à Ì] j\ " f  © œ6   x Û ¼[ þ t _  f  ¨ ‚ à Ì1 p x

“

: r‚  `  ¦ : Ÿ x > \ P % i † < Æ& h Ü ¼– Ð ƒ  ½ ¨ô  Ç Ê ê, + þ AI  V_  ³ ð€   f  ¨

‚ Ã

Ì1 p x“ : r‚   [1]“ É r z  ´+ « >& h “   f  ¨ ‚ Ã Ì X <s ' [ þ t`  ¦ ´ ú Æ ҍ  H  כ ˜ Ð



, F K5 Å q[ þ t _  & ñ · ú š \ P  6   x | ¾ Ó X <s ' [ þ t`  ¦  s `›   ¸ ú ˜ ´ ú Æ ҍ  H

 כ

`  ¦ · ú ˜€ Œ ¤ .   " f ‘ : r ƒ  ½ ¨  H \ P s 
 F  g  \  ¦ €  $  µ 1 ß

∗

E-mail: [email protected]; Tel: +82-2-2612-9744

y

  H X <  s  >  ÷ &% 3  .

t

F K  t _  F K5 Å q _  \ P  6   x | ¾ Ó[ þ t \  @ /ô  Ç @ /³ ð& h “   z  ´+ « >[ þ t õ

 s  : r[ þ t`  ¦ ¶ ú ˜( R˜ Ѐ  , Dulong-Petit[ þ t“ É r “ : r • ¸ `  ¦  y Œ ™

\

    “ ¦^ ‰_  & ñ 6   x \ P  6   x | ¾ ӓ É r 3R \  ] X   H ô  Ç   H z  ´+ « >`  ¦

% i   [2]. # Œl " f R(JK ⁻¹ mol ⁻¹ ) “ É r ë ß –Ä » (universal)

Û ¼  © œÃ ºs  . 3Ü ¼– Ð Y  L   H s Ä »  H Ê ê\   7 H ô  Ç . Õ ª\ 

@

/ô  Ç s  : r& h “   ƒ  ½ ¨  H Einstein \  _  # Œ €  $  ' Ÿ  # Œ

&

’

  H X <, F K5 Å q[ þ t“ É r > á ¤ ° ú  “ É r = å S # Q! Qa Ë > (cut off) ”  1 l x à º, ν

(cycles sec ⁻¹ ) – Ð ”  1 l x   H " é ¶  [ þ t – Ð s À Ò# Q& ’  “ ¦ ´ ú ˜Ù þ ¡

Fig. 1. The thermal unexcitation and excitation mod- els of elections, protons and neutrons in a pseudo metal cluster.



 [3]. Õ ª_  s  : r d ” “ É r ×  æ ç ß –s 
 Z  }“ É r “ : r • ¸\ " f  H ¸ ú ˜ ´ ú  Ü

¼
, 0 - 10 K & ñ • ¸_  ± ú “ É r “ : r • ¸\ " f  H ¸ ú ˜ ´ ú t  · ú §  H



. Nernstü < Lindermann“ É r Õ ª כ \  @ /K  S X ‰ “     H z  ´+ « >

`

 ¦ €  $  # Œ, “ ¦^ ‰ " é ¶  [ þ t“ É r ¿ º Õ ªÒ  ¨ _  ”  1 l x à º– Ð ÷ &# Q e ”

 “ ¦ % i   H X <, 
  H 7 á x Ü ¼– Ð ν ”  1 l x à º\  ¦   É r Ñ ü t“ É r S Ü ¼– Ð 2ν ”  1 l x à º[ þ t`  ¦ ° ú   H  “ ¦ ´ ú ˜ % i   [2]. s \  @ / ô

 Ç s  : r d ” “ É r Debye  €  $  ½ ¨Ù þ ¡  [4]. Õ ª_  s  : r d ” “ É r

“

¦^ ‰ > _  \ P  6   x | ¾ ӓ É r “ : r • ¸ `  ¦  y Œ ™\     3R– Ð ÷ & 9,



 É r “ : r • ¸\ " f  H T ³ s   ) a  “ ¦ % i  . # Œl " f T (K)  H F

K5 Å q _  “ : r • ¸s  . Õ ª_  d ” “ É r   & ñ ^ ‰[ þ t _      › ¸ o ”  1 l x

`

 ¦ ¸ ú ˜ ¬ ¹   9, — ¸Ž  H “ : r • ¸\ " f z  ´+ « > X <s ' [ þ t \  ¸ ú ˜ ´ ú 



 H  “ ¦ ½ + É Ã º e ”  . Õ ª_  Ä »• ¸  H ƒ  5 Å q ^ ‰ s  : r (continuum theory) s  $ í w n ô  Ç   H  כ `  ¦ כ ¹½ ¨ô  Ç . Õ ª Q
 ƒ  5 Å q ^ ‰ s 



: r“ É r @ /| Ä Ì& h Ü ¼– Ð ^θ ₁₀

^D

\ " f ë ß –7 á ¤ ) a  “ ¦ ´ ú ˜ô  Ç  [5, 6]. ^θ ₁₀

^D

“

: r • ¸  H ƒ  5 Å q ^ ‰_  ô  Ç>  (continuum limit) ÷ & 9, s  “ : r

•

¸  A \ " f  H ™ èo  [ þ t o t  · ú §“ ¦ = å S l   H  כ Ü ¼– Ð  « Ñ

 )

a  .   " f Debye_  s  : r d ” “ É r ƒ  5 Å q ^ ‰ s  : r`  ¦ ë ß –7 á ¤ r  v

t  3 l w   H & h s  e ”  .

t

F K  Ò'  ô  Ç epn“ É r Fig. 1 \ 
  · p  ü < ° ú  s , ô  Ç > h _

 „   ü < Õ ª\  5 Å q ô  Ç € ª œ$ í  ü < Õ ª\  5 Å q ô  Ç ×  æ$ í  – Ð ½ ¨$ í

 )

a 3 { 9  [ þ t`  ¦ ´ ú ˜ô  Ç . Õ ªM : à º™ è" é ¶  _  epn“ É r ô  Ç „    ü

< Õ ª\  5 Å q ô  Ç € ª œ$ í    ) a  . 7 £ ¤ ep   ) a  . ô  Ç epn\  e ” 



 H { 9  [ þ t“ É r _ ” > r& h s   m  9, 1 l x r \  [ þ t›  H (excitation)



“ ¦ & ñ % i  . # Œl " f [ þ t> p u s   † < ʓ É r Fig. 1 \  e ”   H ô

 Ç > h_  epns  [ þ t ä ¼€   Fig. 2 + þ AI – Ð { 9  [ þ t s  ¹ ¡ §f ” # Œ F 

g   (photon)[ þ t`  ¦ µ 1 Ïô  Ç “ ¦ Ò q ty Œ • ½ + É Ã º e ”  . [ þ t> p u \ " f

„

   € ª œ$ í  ü < ×  æ$ í  \  ¦ Ÿ í† < Ê † < ʓ É r Õ ª F K5 Å q \  \  -t 

Fig. 2. Three resonances (photons) in an excited epn.

 „  ² ú ˜ | ¨ c M : €  $  \ P (\  -t )_  ² D G  Ò& h  ” > r F \  ¦ \ O  ± p



. Õ ªo “ ¦  8
  epn_   6   x _  ½ + Ë{ © œ$ í “ É r ~ ½ Ó  F  g

`

 ¦ ë ß –[ þ t l  0 A # Œ „   [ þ t`  ¦ 5 Å q r v   H X <,  $ 3 Ü ¼– Ð+ ‹ undulator
 wiggler`  ¦  6   x l  M :ë  H \ , Õ ª 5 Å q ÷ &  H „  



[ þ t“ É r € ª œ$ í  ü < ×  æ$ í  \  ¦ Ÿ í† < Ê “ ¦ e ”  “ ¦ Ò q ty Œ •  ) a  .

Õ

ª s Ä »  H Õ ª[ þ t s   $ 3 \   Òv 9  n = M :, epn[ þ t _  { 9  [ þ t s 

"

f– Ð % Á ° ú ˜o   H ~ ½ ӆ ¾ ÓÜ ¼– Ð
° ú ˜ M :ü <  r  ë ß –± ú ˜ M : F  g  [ þ t s


š ¸l  M :ë  H s  .

“

¦^ ‰ F K5 Å q[ þ t“ É r " é ¶   (valence) „   [ þ t _  F K5 Å q  ½ + ËÜ ¼

–

Ð ½ ¨$ í ÷ &# Q e ” Ü ¼ 9, s   H „  l „  • ¸• ¸ü < \ P „  • ¸• ¸ü < ° ú  

“ É

r  © œ ×  æ כ ¹ô  Ç : £ ¤$ í [ þ t \  % ò † ¾ Ó`  ¦ ŠҖ Ð p • 2 ; “ ¦ { 9 ì ø Í& h  Ü

¼– Ð · ú ˜ 9& ’  . Õ ª Q
 Õ ªM : " é ¶   „    epn[ þ t õ  " é ¶  

 „        epn[ þ t s  † < Êa  [ þ th þ t M :ü <  H ² ú ˜o , " é ¶  

„

  [ þ t ë ß – [ þ t›  H  €   ƒ  5 Å q ^ ‰ s  : r“ É r L :t l  M :ë  H \ , F K5 Å q _

 " é ¶   „      ½ + ˓ É r _ p  \ O  .

epn _  [ þ t> p u“ É r Û ¼— 2 ; 7 á x À Ó Z > – Ð [ þ t›  H  . — ¸Ž  H epn[ þ t“ É r 4 7

á

x À Ó(←⇐, →⇐, ←⇒, →⇒)_  Û ¼— 2 ;[ þ t – Ð ½ ¨$ í ÷ &# Q e ” 



.  © œ ± ú “ É r [ þ t> p u \  -t \  ¦ ° ú   H epn[ þ t s  €  $  [ þ t ä ¼ 9, Õ

ª[ þ t ë ß –s    H ç ß –_  epn[ þ t s   ) a  . 7 £ ¤ l ” > r \  ´ ú ˜   H " é ¶  

 „    % i ½ + É`  ¦   H  כ s  . Õ ª s Ä »  H : Ÿ x > † < Æ& h  > í ß –

\

" f ←⇐ Û ¼— 2 ; ë ß –s  ] j{ 9  ± ú “ É r \  -t  ï  r 0 A_  [ þ th þ t S X ‰ Ò

 ¦(P l1 )`  ¦  6   x # Œ d ” `  ¦ [ jÄ º“ ¦,
 Qt  ï  r 0 A[ þ t“ É r Z  }“ É r

\

 -t  ï  r 0 A[ þ t _  [ þ th þ t S X ‰Ò  ¦(P h1 )`  ¦  6   x # Œ d ” [ þ t`  ¦ [ j Ä

ºl  M :ë  H s  .  © œ ± ú “ É r \  -t  ï  r 0 A_  epn_  Û ¼— 2 ;“ É r

←⇐  “ ¦ Ò q ty Œ •  ) a  . Õ ª s Ä »  H ~ 1 >  ² ú ˜ ± ú ˜ à º e ”   H + þ A I

\  ¦ 2 [ “ ¦ e ” l  M :ë  H s  .

←⇐ epn Û ¼— 2 ;_  [ þ t> p u — ¸4 S q`  ¦ ¶ ú ˜( R˜ Ð . { 9 ì ø Í& h Ü ¼– Ð

· ú

˜ 9”   “ É r _  „   C • ¸\  ¦ \ V– Ð [ þ t€  , (Kr)4d ¹⁰ 5s ¹ \ " f 5s ¹ _  epn Û ¼— 2 ; ←⇐ s 
 4d ¹⁰ _  epn Û ¼— 2 ; ←⇐ 



© œ €  $  [ þ t›  H  “ ¦ ^  ¦ à º e ” Ü ¼
, s M : „   C • ¸ (Kr)\ 

•

¸ epn Û ¼— 2 ; ←⇐ e ” Ü ¼ 9, s • ¸ ] j{ 9  €  $  # Œl  | ¨ c à º e ”

 . # Œl " f (Kr)“ É r Krypton _  „   C • ¸\  ¦ ´ ú ˜ô  Ç .

II. —  Þ4 ] K ¡X ì Ä { ¢¨ |  S Ë

1. T ]  §X N ËP  { ¢¨ | 

Fig. 1“ É r { 9 & ñ ô  Ç > hà º_  " é ¶  [ þ t – Ð s À Ò# Q”   ô  Ç F K5 Å q

\

 @ /ô  Ç  © œ& h “   Á º 8l  (cluster)_  epn [ þ t`  ¦
  · p .

]

j{ 9  µ 1 Ú\  e ”   H \  -t  ï  r 0 A ] j{ 9  ± ú “ É r [ þ t> p u \  -t \  ¦

° ú

  H Û ¼— 2 ;(←⇐)\  @ / # Œ s  . Õ ª  6 £ § Z  }“ É r \  -t  ï  r 0

A  H →⇐ s  . Õ ª  6 £ § Z  }“ É r \  -t  ï  r 0 A  H ←⇒ s  .

Õ

ª  6 £ § Z  }“ É r \  -t  ï  r 0 A  H →⇒ s  . ] j{ 9  Z  }“ É r \  - t

 ï  r 0 A  H  6 £ § \ " f [ O " î ½ + É  Ä »×  æ$ í  [ þ t – Ð s À Ò# Q ”  



“ ¦  « Ñ  ) a  .

ô

 Ç Á º 8l \ " f epn[ þ t _  / B N ç ß –& h  C \ P s 
 ô  Ç Û ¼— 2 ; epn[ þ t \ " f { 9  [ þ t _  / B N ç ß –& h  C \ P “ É r ×  æ כ ¹ t  · ú § . Õ ª s

Ä »  H : Ÿ x > & h “   > í ß –\ " f epn[ þ t _  Û ¼— 2 ; à º[ þ t ë ß –`  ¦ “ ¦ 9

l  M :ë  H s  . ¢ ¸ ô  Ç s Ä »  H 8 £ ¤& ñ Û ¼[ þ t(H 2 or He) • ¸ F

K5 Å q õ  ° ú  s , Ä »  >  [ þ t ä ¼ 
 [ þ t ä ¼t   m ½ + É Ã º• ¸ e ”  l

 M :ë  H s  . 8 £ ¤& ñ Û ¼– Ð a % ~“ É r  כ “ É r ×  æ$ í   e ”   H ó ¡ š µ

¢

§ s  . s  — ¸4 S q“ É r ‰ & ³” > r   H „   & h  " é ¶  — ¸+ þ A`  ¦ „  ^ ‰& h  Ü

¼– Ð ^  ¦ M : # QÖ ¼ epn• ¸ # QÖ ¼ { 9  • ¸  8  
 N S 
   



o r v t • ¸ · ú §  H  .   " f l ” > r " é ¶  _  „   & h  — ¸4 S q (Schr¨ odinger model)`  ¦ “  & ñ €  " f, epn[ þ t _  ƒ  5 Å q ^ ‰ s 



: r`  ¦ ë ß –7 á ¤ r v l  0 A # Œ Fig. 1õ  ° ú  s  Õ ª§ 4  . Fig. 1õ  q

5 p w ô  Ç + þ AI  De Haas-van Alphen ´ òõ \  ¦ [ O " î l  0 A

# Œ ‚ à Г ¦" f [7]\  “ §¹ ¢ ¤& h  3 l q& h Ü ¼– Ð Õ ª 94 R e ”  .



 " f F K5 Å q" é ¶  [ þ t _  Á º 8l  — ¸4 S q`  ¦  © œ © œK  ˜ Ѐ  , — ¸

Ž

 H F K5 Å q[ þ t \  e ”   H ×  æ$ í  [ þ t“ É r „   [ þ t s 
 € ª œ$ í  [ þ t ˜ Ð



 ´ ú § .   " f C • ¸ epn[ þ t`  ¦ ë ß –[ þ t “ ¦ z Œ ™  H ×  æ$ í  [ þ t`  ¦



Ä » ×  æ$ í     . 0 A$ í [ þ t s  I € ª œ`  ¦ [  t “ ¦ e ”   H  כ % ƒ! 3 ,

 © œ Á º Ö  ¦  כ Ü ¼– Ð \ V8 £ ¤ ÷ &  H s   Ä » ×  æ$ í  [ þ t“ É r " é ¶   _

 ×  æd ”  (core)\  e ” Ü ¼ 9,  6 £ § Ü ¼– Ð Á º î  r C • ¸ ×  æ$ í   [

þ

t“ É r  Ä » ×  æ$ í  ü < C • ¸ € ª œ$ í  [ þ t  s \  e ” Ü ¼ 9,  © œ µ

1 Ú\   H C • ¸ „   [ þ t s  Û ¼Û ¼– Ð ½ ¨& h  (spherical)Ü ¼– Ð · ú ˜´ ú 

“ É

r  o \  ¦ Ä »t  €  " f, C • ¸ € ª œ$ í  \  · ¡ ­ # Q e ”  . „  ^ ‰& h 

“

  { 9  [ þ t“ É r C • ¸ Û ¼— 2 ;_  › ¸ o[ þ t \  _  # Œ   ½ + Ë÷ &# Q e ” 



“ ¦ Ò q ty Œ •  ) a  . " é ¶  _  ×  æd ” \  e ”   H  Ä » ×  æ$ í  [ þ t“ É r β y

Œ

™û Zü < % i  β y Œ ™û Z\  ¦ { 9 Ü ¼v  9, Õ ª\  Ò q t$ í  ) a „   [ þ t“ É r Ò q t

$ í

 ) a € ª œ$ í  ü < ×  æ$ í  [ þ t`  ¦ Ÿ í† < Ê # Œ 5 Å q (core) epn[ þ t`  ¦ + þ

A$ í ô  Ç “ ¦ & ñ  ) a  . Õ ª Q
 # Œl " f  Ä »×  æ$ í   s  À

ҍ  H ô  Ç \  -t  ï  r 0 A_  0 Au   H Z > – Ð ×  æ כ ¹ t  · ú § . ¢ ¸



 É r ƒ  ½ ¨\ " ft ë ß – s  5 Å q epn[ þ t _  Û ¼— 2 ;[ þ t“ É r C • ¸ Û ¼— 2 ; [

þ

t ë ß –
p u ×  æ כ ¹ >   Œ •6   x t  · ú §  H  . Õ ª Q
 ƒ  5 Å q ^ ‰ ô  Ç> 

“

: r • ¸ s  – Ð ? / 9€   5 Å q epn[ þ t s  ×  æ כ ¹ >   Œ •6   x   H  כ

° ú

 s  Ö ¼ ”   .  =
 €   0 K s  \ " f — ¸Ž  H " é ¶  [ þ t“ É r

% ò

& h  \  -t \  ¦ ° ú l  M :ë  H s  .  Ä »×  æ$ í  [ þ t s  Á º 8l  î

ß –\ " f # QÖ ¼ / B M \  0 Au  Ž  H t  1s  › ¸F K 3 l w ÷ &  H s † ½ Ó& ñ o

 d ” `  ¦ ° ú   H # Œì  r _  \  -t  ï  r 0 A\  ¦ s ê  r  “ ¦ Ò q ty Œ •  .

y

Œ

• \  -t  ï  r 0 A  H y Œ •y Œ • 
_  s † ½ Ó& ñ o  d ” Ü ¼– Ð ³ ð‰ & ³

 . Õ ª Q
 [ þ t> p u \  -t   Ø Ôl  M :ë  H \  ½ + Ë_  ½ ¨ç ß –“ É r



Ø Ô 9, y Œ • s † ½ Ó& ñ o  † ½ Ó[ þ t _  ½ + ˓ É r @ /| Ä Ì& h Ü ¼– Ð 1s 
, 1˜ Ð



 › ¸F K  Œ •“ É r # Q‹ "   © œÃ º  Ò q ty Œ •  ) a  . y Œ • \  -t  ï  r 0 A[ þ t _

 þ j@ / [ þ t ä ¼t  · ú §`  ¦ epn Û ¼— 2 ;[ þ t _  à º  H g 1 ( ^Z ₅ ) s   ) a  .

 © œ ± ú “ É r \  -t  ï  r 0 A_  [ þ t›  H epn[ þ t _  à º\  ¦ N 1 s 



  . s [ þ t“ É r P l1 = W l1 exp(−D l1 /k K T s )    H [ þ t

>

p

u S X ‰Ò  ¦`  ¦ t €  , [ þ t ä ¼t  · ú §`  ¦ S X ‰Ò  ¦“ É r 1 − P l1 = 1 − W _l1 exp(−D _l1 /k _K T _s ) s  . [ þ t> p u s   † < ʓ É r F K5 Å q \ " f 8 £ ¤

&

ñ Û ¼– Ð F K5 Å q F  g   „  ² ú ˜ (transfer) | ¨ c S X ‰Ò  ¦`  ¦ ´ ú ˜  9, [

þ

t ä ¼t  · ú §6 £ § s   † < ʓ É r 8 £ ¤& ñ Û ¼_  F K5 Å q F  g   F K5 Å q Ü ¼

–

Ð „  ² ú ˜ | ¨ c S X ‰Ò  ¦`  ¦ ´ ú ˜ô  Ç . # Œl " f W ^l1 õ  D ^l1 (J epn ⁻¹ energy level 1)“ É r ] j{ 9  ± ú “ É r \  -t  Y U6 \ š epn_  [ þ t> p u  © œ Ã

ºü < [ þ t> p u \  -t s  . ¢ ¸ T ^s (K)  H epn _  [ þ t ä ¼  H Ö  ¦ (rate) s  y Œ ™™ è l  r  Œ •   H   / B G& h  “ : r • ¸– Ð \ V8 £ ¤ ÷ &
,

‘

: r ƒ  ½ ¨\ " f  H & ñ · ú š \ P  6   x | ¾ Ó_  z  ´+ « >° ú כs  Ä »• ¸  ) a d ” [ þ t õ 

 © œ ¸ ú ˜ ´ ú   H “ : r • ¸– Ð % i  . N ¹ s  0 Ò'  g ¹  t    à º– Ð



Œ

•6   x   H s † ½ Ó& ñ o  d ” “ É r [8]

W ₁ (g ₁ , N ₁ ) = (p _l1 + 1 − p _l1 ) ^g

¹

= 1

≡

g

₁

X

N

1

≤g

1

g ₁ !

(g 1 − N 1 )!(N 1 )! p ^N _l1

¹

(1 − p l1 ) ^g

¹

^−N

¹

(1) Õ

ª  6 £ § Ñ ü t P : Y U6 \ šÂ Ò'  n  P : t  epn[ þ t _  [ þ t> p u S X ‰Ò  ¦“ É r p h1 = W h1 exp(−D h1 /k K T s ) s  ÷ &“ ¦, [ þ t ä ¼t  · ú §`  ¦ S X ‰Ò  ¦

“

É r 1 − P h1 = 1 − W _h1 exp(−D _h1 /k _K T _s )   ) a  . # Œl \ 

"

f W ^h1 ü < D ^h1 (J epn ⁻¹ for energy level 2 - n)“ É r 2   P : Â

Ò'  n  P : t _  \  -t  ï  r 0 A[ þ t \  @ /ô  Ç ô  Ç epn_  [ þ t> p u



© œÃ ºü < [ þ t> p u \  -t  s  . s [ þ t \  @ /ô  Ç s † ½ Ó& ñ o  d ” [ þ t“ É r



6 £ § õ  ° ú  s   ) a  .

W ₂ (N ₁ , N ₂ ) = (p _h1 + 1 − p _h1 ) ^N

¹

= 1 ∼ =

N

1

X

N

₂

≤N

1

(N ₁ )!

(N 1 − N 2 )!(N 2 )! p ^N _h1

²

(1 − p _h1 ) ^N

¹

^−N

²

.. .

W n−1 (N n−2 , N n−1 ) = (p h1 + 1 − p h1 ) ^N

ⁿ⁻²

= 1 ∼ =

N

n−2

X

N

n−1

≤N

n−2

(N n−2 )!

(N n−2 − N n−1 )!(N n−1 )! p ^N _h1

ⁿ⁻¹

(1 − p h1 ) ^N

ⁿ⁻²

^−N

ⁿ⁻¹

W n (N n−1 , N n ) = (p h1 + 1 − p h1 ) ^N

ⁿ⁻¹

= 1 ∼ =

N

n−1

X

N

n

≤N

n−1

(N n−1 )!

(N n−1 − N n )!(N n )! p ^N _h1

ⁿ

(1 − p h1 ) ^N

ⁿ⁻¹

^−N

ⁿ

(2) d ”

 (1) ü < d ”  (2)`  ¦   z o  Y  L  . Õ ªM : g 1 ≥ N 1 ≥ · · · ≥ N n−1 ≥ N n \  @ / # Œ W _T (N ₁ , N ₂ , · · · , N _n−1 , N _n , N ) = W ₁ W ₂ · · · W _n−1 W _n = 1

∼ =

N

n−1

X

N

n

≤N

n−1

· · ·

g

₁

X

N

1

≤g

1

g ₁ !p ^N _l1

¹

(1 − p _l1 ) ^g

¹

^−N

¹

p ^{N −N} _h1

¹

(1 − p _h1 ) ^N

¹

^−N

ⁿ

(g 1 − N 1 )!(N 1 − N 2 )! · · · (N n−1 − N n )!N n ! =

N

n−1

X

N

n

≤N

n−1

· · ·

g

₁

X

N

1

≤g

1

W t n = 2, 3, 4 · · · (3)

0 A d ” \ " f

W _t = g ₁ !p ^N _l1

¹

(1 − p _l1 ) ^g

¹

^−N

¹

p ^{N −N} _h1

¹

(1 − p _h1 ) ^N

¹

^−N

ⁿ

(g 1 − N 1 )!(N 1 − N 2 )! · · · (N n−1 − N n )!N !

(3-1)

N = N ₁ + N ₂ + · · · + N _n−1 + N _n (3-2) 0

A d ” [ þ t \ " f D l1 “ É r D h1 ˜ Ð   Œ •“ ¦ W l1 “ É r W h1 ˜ Ð   Œ • .

\

 -t  ï  r 0 A[ þ t \  › ' a >  \ O s  y Œ • [ þ t›  H epn[ þ t“ É r ° ú  “ É r € ª œ _

 \  -t (° ú  “ É r “ : r • ¸\ " f) / B N/ å L ÷ &l  M :ë  H \  — ¸Ž  H [ þ t

›

 H epn[ þ t \  @ /ô  Ç ? / Ò\  -t , U(J)  H  6 £ § õ  ° ú  s  ³ ð‰ & ³

½

+ É Ã º e ”  .

U = D _l1 N ₁ + D _h1 (N − N ₁ ) ∼ = u 1a N (4) d ”

 (4) \ " f u 1a (J epn ⁻¹ )  H — ¸Ž  H epn[ þ t \  @ /ô  Ç ¨ î ç  H [ þ t

>

p

u \  -t  s  . d ”  (3)\ " f ] j{ 9   H † ½ Ós  Õ ª d ”  „  ^ ‰\  ¦ t

C ô  Ç .   " f d ”  (3)-1_  „  p ì  rd ” _  y Œ • † ½ Ó_  > à º

0 s   ) a    H  כ `  ¦ & h 6   x “ ¦ Stirling’s   H  \  ¦  6   x # Œ Û  ¦

€

 . ' Í P : d ” “ É r g ₁ − N 1

β 1b

 g b

N _n N n−1 − N n



= N 1 − N 2 . (5) 0

A d ” \ " f

β _1b = W _h1 W l1



exp −(D _h1 − D _l1 ) k K T s











1 − W l1 exp ^−D _k

^l1

K

T

_s





1 − W h1 exp ^−D _k

^h1

K

T

_s







 (5-1)

g _b = 1 − W _h1 exp −D _h1 k K T s

= 1 − exp −D _h1 k K T s

for W _h1 = 1 (5-2)

 Qt  † ½ Ó[ þ t \  @ / # Œ  H (N 1 − N 2 )

 g b

N n

N n−1 − N n



= N 2 − N 3

.. .

(N _n−3 − N n−2 )



g _b N _n N n−1 − N n



= N _n−2 − N n−1

(N _n−2 − N _n−1 )



g _b N _n N n−1 − N n



= N _n−1 − N _n n = 2, 3, 4, · · · (6)

 )

a  . \ P % i † < Æ 1, 2 Z O g Ë :`  ¦ ½ + Ë$ í ô  Ç d ” “ É r & ñ & h \ " f  H T dS

= dU − µ E dN s  ÷ & 9, s  d ” \  d ” [ þ t (3)-1 ü < (4)\  ¦ @ / { 9

 €  , [ þ t›  H epn[ þ t _   o† < Æ Ÿ íJ $ ™[ >  (chemical potential), µ E (J excited epn ⁻¹ )  H

µ E

k _K T _s = u 1a

k _K T _s − ∂lnW t

∂N

= u 1a

k _K T _s − lnp h1 + ln(1 − p h1 )

−ln(N n−1 − N n ) + lnN n (7)

–

Ð  ) a  . [ þ t ä ¼t  · ú §“ É r epn[ þ t _  ¨ î ç  H  o† < Æ Ÿ íJ $ ™[ >  (chem- ical potential), µ U E (J unexcited epn ⁻¹ ) [5]  H

µ U E

k K T s

= µ ⁰ k K T s

+ ln T T s

(8) s

  ) a  . # Œl " f  7 Hë  H [1] _  d ”  (9)\  s  © œ l ^ ‰_   © œI  ~ ½ Ó

&

ñ d ” , P V = nRT (n: mol à º)   H d ” `  ¦ @ /{ 9  “ ¦, k ^B

@

/’  \  k K \  ¦  6   x €  , 0 A d ”  (8)`  ¦ % 3 `  ¦ à º e ”  . [ þ t›  H epn[ þ t _  \  -t \  ¦ 8 £ ¤& ñ   H Û ¼[ þ t s  s  © œ l ^ ‰  & ñ K

• ¸ ÷ &l  M :ë  H s  . ¨ î + þ A\ " f µ E = µ U E    H › ' a > \  ¦ & h  6

 

x €  ,  6 £ § õ  ° ú  “ É r Ÿ í o [ þ t> p u “ : r • ¸ “   (c ^s1 )`  ¦ & ñ _  ½ + É Ã

º e ”  .

c s1 x = c s1

T T s

= N n

N n−1 − N n

(9)

0 A d ” \ " f

C s1 = N ns

N n−1s − N ns

(9-1) d ”

 (9)-1 \ " f N n−1s ü < N ns   H Ÿ í o [ þ t> p u “ : r • ¸(T = T s ) \ 

"

f n − 1  P :ü < n  P : Y U6 \ š\ " f [ þ t›  H epn[ þ t _  à º[ þ t s  .

d ”

[ þ t, (6) \  (5)`  ¦ @ /{ 9  “ ¦   õ    z o   8ô  Ç Ê ê,   õ  d ”

\  d ”  (9)`  ¦ @ /{ 9 ô  Ç Ê ê & ñ o  €   d ”  (10)`  ¦ % 3   H  .

N ₁ = g ₁

z−z

ⁿ

1−z + ^z _g

ⁿ

b

β 1b + ^z−z _1−z

ⁿ

+ ^z _g

ⁿ

b

n = 2, 3, 4 · · · (10) 0

A d ” \ " f

z = c _s1 g _b T T s

(10-1) d ”

 (10)“ É r z  ´ 2 ; 8 l / B N f  ¨ ‚ à Ì] j\  f  ¨ ‚ Ã Ì  ) a Û ¼ ì  r  _  ³ ð

€

  f  ¨ ‚ Ã Ì 1 p x“ : r‚  õ  ° ú  “ É r + þ AI _  d ” s 
, Õ ª[ þ t“ É r €  •ç ß – " f

–

Ð  Ø Ô . „     H Langmuir + þ AI \  ¦ Õ ªo 
 Ê ê   H sig- moid(S   + þ A) + þ AI \  ¦ Õ ª 2 ; . d ”  (10)\ " f g ¹ “ É r F K5 Å q" é ¶



 1 mole\  @ / # Œ  H ZN A /n epn[ þ t s   ) a  .   " f g ¹

@

/’  \  ZN ^A /n`  ¦ @ /{ 9  “ ¦, Fig. 2\ " f ˜ Ѝ  H  ü < ° ú  s , ô

 Ç epn“ É r 3 > h_  F  g  [ þ t`  ¦ µ 1 Ï Ù ¼– Ð 3`  ¦ Y  L “ ¦, k ^K `  ¦ Y  L ô

 Ç Ê ê, (Z/n)k ^K \  ¦ k B – Ð u  ¨ 8 Š r v “ ¦, ¢ ¸ N ^A k B @ /’  \  R – Ð u  ¨ 8 Š r v €   ] j{ 9  ± ú “ É r Y U6 \ š_  & ñ & h  \ P  6   x | ¾ Ó d ” “ É r   6

£

§ õ  ° ú  s   ) a  .

C _ν1 = C _l1 = 3R 

z−z

ⁿ

1−z + ^z _g

ⁿ

b

 β _1b + ^z−z _1−z

ⁿ

+ ^z _g

ⁿ

b

n = 2, 3, 4 · · · (11) d ”

 (11)\ " f ' ‘   ν1 (or l1)“ É r F K5 Å q _  ] j{ 9  ± ú “ É r ï  r 0 A\ 

"

f & ñ & h  \ P  6   x | ¾ Ó d ” `  ¦
  · p . Einstein s  \ P  6   x | ¾ Ó d ” 

`

 ¦ ½ ¨½ + É M : { 9    H 3 " é ¶& h Ü ¼– Ð ¹ ¡ §f ” s Ù ¼– Ð 3`  ¦ Y  L K  

 )

a  “ ¦ % i   [2,8]. Debye   H 3 > h_  { 9  [ þ t ×  æ 
  H 7

á x& h Ü ¼– Ð ¹ ¡ §f ” s  9,   É r Ñ ü t“ É r S & h Ü ¼– Ð ¹ ¡ §f ” “    % i 



. Fig. 2\ " f ˜ Ѝ  H  ü < ° ú  s  C • ¸ „   ü < C • ¸ € ª œ$ í  

 ? /  H F  g    H  Å Ò  Œ •“ É r \ P `  ¦ ° ú   H y n Cs “ ¦,
 Qt  ¿ º

>

h_  F  g    H ×  æ$ í  _   Œ •6   x \  _ † < ÊÜ ¼– РŠҖ Ð \ P `  ¦ µ 1 Ï 



 H  כ Ü ¼– Ð  « Ñ ÷ &
, Õ ª ß ¼l   H " f– Ð ° ú   “ ¦ ˜ Ѐ Œ ¤ .

d ”

[ þ t (5) õ  (6)\  d ”  (9)ü < d ”  (10)-1[ þ t`  ¦ @ /{ 9  “ ¦    õ    z o  Y  L ô  Ç Ê ê, d ”  (10)õ  † < Êa  F & ñ § > = r v €  , n   P

: \  -t  Y U6 \ š_  [ þ t›  H à º, N ⁿ “ É r  6 £ § õ  ° ú  s   ) a   N n =  g ₁ − N 1

β 1b

 z ⁿ g b

= g 1

β _1b + 

z−z

ⁿ

1−z + ^z _g

ⁿ

b

 z ⁿ

g b

n = 2, 3, 4, · · · (12)

 Qt  \  -t  ï  r 0 A_  & ñ & h  \ P  6   x | ¾ Ó d ” [ þ t“ É r d ” [ þ t (5), (6), (10), (12)\  ¦  6   x # Œ ½ ¨ô  Ç .   " f N 2 , N ₃ , N ₄ , N ₅ , N ₆ \  @ /ô  Ç F K5 Å q _  & ñ & h  \ P  6   x | ¾ Ó d ” [ þ t“ É r  6 £ § õ  ° ú  s   ) a



.

C l2 = 3R

 z n − z β 1b + z n



n = 2, 3, 4, · · · (13-1)

C l3 = 3R  z n − z − z ² β _1b + z _n



n = 3, 4, · · · (13-2)

C l4 = 3R  z n − z − z ² − z ³ β _1b + z _n



n = 4, 5, · · · (13-3)

C l5 = 3R  z n − z − z ² − z ³ − z ⁴ β _1b + z _n



n = 5, 6, · · · (13-4)

C l6 = 3R  z n − z − z ² − z ³ − z ⁴ − z ⁵ β _1b + z _n



n = 6, 7, · · · (13-5) 0

A d ” [ þ t \ " f

z n = z − z ⁿ 1 − z + z ⁿ

g _b (13-5-1) 0

A d ” [ þ t`  ¦ í  H & h Ü ¼– Ð Y  L # Œ, l  ¨ î ç  H & ñ & h  \ P  6   x | ¾ Ó d ”  [

þ

t`  ¦ ½ ¨ €    6 £ § õ  ° ú   .

C _ν2 = p

C _l1 C _l2 (14-1)

C _ν3 = p

³

C _l1 C _l2 C _l3 (14-2)

C _ν4 = p

⁴

C _l1 C _l2 C _l3 C _l4 (14-3)

C ν5 = p

⁵

C l1 C l2 C l3 C l4 C l5 (14-4)

C ν6 = p

⁶

C l1 C l2 C l3 C l4 C l5 C l6 (14-5)

.. .

0

A d ” [ þ t \ " f l  ¨ î ç  H`  ¦ 2 [ô  Ç s Ä »  H y Œ • " f– Ð   É r \ 



-t  Y U6 \ š\  e ”   H epn[ þ t“ É r " f– Ð Y  L Ü ¼– Ð  Œ •6   x l  M :ë  H s

 9, ¢ ¸ > í ß – # Œ ˜ Ѐ Œ ¤`  ¦ M :,  © œ ¸ ú ˜ ´ ú   H ¨ î ç  H d ” [ þ t s 

÷

&l  M :ë  H s  . d ” [ þ t (5), (6), (9), (10), (12)\  ¦   ½ + Ë # Œ, g 1 \  @ /ô  Ç 8 ú x [ þ t›  H epn[ þ t _  à º_  q \  ¦
 ? /  H [ þ t> p u 1 p x

“

: r‚   d ” , ^N _g

1

“ É r  6 £ § õ  ° ú  s   ) a  .

N g 1

= N ₁ + N ₂ + · · · + N _n−1 + N _n g 1

= (N 1 − N 2 ) + 2(N 2 − N 3 ) + · · · + (n − 2)(N n−2 − N n−1 ) + (n − 1)(N n−1 − N n ) + (n − 1)N n + N n

g ₁

=

z−z

ⁿ

(1−z)

²

− ^(n−1)z _1−z

ⁿ

+ n ^z _g

ⁿ

b

β 1b + ^z−z _1−z

ⁿ

+ ^z _g

ⁿ

b

n = 2, 3, 4, · · · (15)

#

Œl " f [ þ t> p u 1 p x“ : r‚  s ê ø Í ´ ú ˜“ É r Ÿ í o # Œl  “ : r • ¸, T s \  ¦   H ç ß –Ü ¼– Ð % i l  M :ë  H \   6   x ½ + É Ã º e ”  . ô  Ç \  -t  ï  r 0 A\  27 á x À

Ó_  epn[ þ t s  e ”   H  â Ä º  H  6 £ § õ  ° ú  s   ) a  .

N

g ₁ + g ₂ = (1 + M )(N 11 + N 21 + · · · + N n−1 + N n1

g ₁ + g ₂

= a 1

 (g 1 − N 11 )(g 2 − M N 11 ) ^M M ^M β _2b



1+m¹

 z − z ⁿ

(1 − z) ² − (n − 1)z ⁿ 1 − z + n z ⁿ

g _bb



n = 2, 3, 4, · · · (16) 0

A d ” \ " f

N 11 =  (g 1 − N 11 )(g 2 − M N 11 ) ^M M ^M β _2b



1+M¹

 z − z ⁿ 1 − z + z ⁿ

g _bb



(16-1)

Z = c s2 T T s

g bb (16-2)

c s2 = N _n1s n n−11s − N n1s

(16-3)

a ₁ = 1 + M g 1 + g 2

(16-4)

β 2b =  W _h1 W l1

  W _h2 W l2

 M

exp  − {(D _h1 − D l1 ) + M (D _h2 − D l2 )}

k K T s



×







1 − W _l1 exp 

−D

l1

k

K

T

s

 1 − W _h1 exp 

−D

h1

k

K

T

s















1 − W _l2 exp 

−D

l2

k

K

T

s

 1 − W _h2 exp 

−D

l2

k

K

T

s









M

(16-5)

g _bb =



1 − W _h1 exp  −D _h1 k K T s



_1+M¹



1 − W _h2 exp  −D _h2 k K T s



_1+M^M

(16-6)

M = N 12

N 11

= N 22

N 21

= · · · = N n2

N n1

(16-7) d ”

 (16) \ " f g 1 = g ₂ , w _l1 = w _l2 , w _h1 = w _h2 , D _l1 = D _l2 , D _h1 = D _h2 , M = 1 s €   d ”  (16)  H d ”  (15)  ) a  .

III. + s ÇÊ Ýô p §Ê Ý ‚ º8 ý # Œ Q F K5 Å q[ þ t _  z  ´+ « > & ñ · ú š \ P  6   x | ¾ Ó X <s ' [ þ t`  ¦ ì ø Í z  ´+ « >

d ”

 (semi-empirical), C _p − C ν = AT C _p ² \  : Ÿ x õ r & , & ñ

Fig. 3. The semi-empirical heat capacity of silver [9] at the constant volume compared with the theoretical Eq.

(11) with standard errors (β _1b = .5112, T _s = 47.5 K, D _h1

= 355.3 cal average energy level (2-n) ⁻¹ , c s1 = .95, A = .0000388).

&

h

 \ P  6   x | ¾ Ó X <s ' \  ¦ % 3 # Q" f, ‘ : r ƒ  ½ ¨\ " f ½ ¨ô  Ç s  : r d ”  [

þ

t s  \ O  
 ¸ ú ˜ ´ ú   H t  · ú ˜  ˜ Ѐ Œ ¤ . — ¸Ž  H Table[ þ t s 
 Figure[ þ t \ " f “A”  H 0 A ì ø Í z  ´+ « >d ” _   © œÃ ºs  . Table 1\ 

"

f
  · p  ü < ° ú  s , Ag F K5 Å q \  @ /ô  Ç C p X <s '  [9]\  ¦ 0

A_  ì ø Í z  ´+ « >d ”  : Ÿ x õ  r & , ‘ : r ƒ  ½ ¨\ " f ½ ¨ô  Ç d ”  (14)-4ü <

Debye d ” Ü ¼– Ð ³ ðï  r \  Q (standard error = σ)[ þ t`  ¦ ½ ¨ 

#

Œ ˜ Ѐ Œ ¤ . „   \  @ /ô  Ç ³ ðï  r \  Q  H 0.0225 s “ ¦, Ê ê \ 

@

/ô  Ç ³ ðï  r \  Q  H 0.0304 s % 3  . A  H ° ú  “ É r à º(order)\  ¦

  · p . Õ ªM : ì ø Í z  ´+ « >d ” [ þ t _  & ñ & h  \ P  6   x | ¾ Ó[ þ t, C ν (∗) ü <

C _ν (∗∗) • ¸ ½ ¨Ù þ ¡ .

Einstein d ” s 
 Debye d ” “ É r  ˜ Ð× ¼– Ð > hà º(N ^A ) _  epn[ þ t \  @ /ô  Ç \ P  6   x | ¾ Ó`  ¦ ½ ¨   H d ” Ü ¼– Ð Ò q ty Œ •  ) a  .

Table 2 \ " f  H d ”  (14)-4`  ¦  6   x # Œ Ag F K5 Å q \  @ /ô  Ç y

Œ

• \  -t  ï  r 0 Aü < “ : r • ¸\    É r [ þ t> p u 1 p x“ : r‚  [ þ t`  ¦ ½ ¨ % i 



. s M : 6 \  -t  ï  r 0 A s  © œ\ • ¸ [ þ t> p u 1 p x“ : r‚  [ þ t s 
  è

ß – . s   H à ºd ”  © œ # Q~  ½ + É Ã º \ O  . s   H Table 2 \  ³ ðr 

t  · ú §€ Œ ¤ .

¢

¸   É r F K5 Å q[ þ t _  (Cu [10], B [11], Ni [12,13], Pd [14, 15]) & ñ · ú š \ P  6   x | ¾ Ó\  @ / # Œ• ¸ X <s '  ´ ú » ¡ §`  ¦ # Œ, Table 3 õ  4 s ü @\  Figs. 4, 5, 6, 7, 8\ 
 Í Ç x . Fig. 3 \ " f



 H d ”  (11)“ É r   / B G& h `  ¦ ° ú t  · ú §“ ¦, Langmuir + þ AI _  ‚  

`

 ¦
  · p . Õ ªA " f Õ ª ’   Û ¼Û ¼– Ѝ  H sigmoid + þ AI “   \ P  6

 

x | ¾ Ó`  ¦
 è ­ q à º \ O  . s   H  7 Hë  H [1] õ  q “ § % i `  ¦ M :, [

þ

t ä ¼t  · ú §`  ¦ epn S X ‰Ò  ¦`  ¦ “ ¦ 9 % i l  M :ë  H s  . Õ ª Q




 É r \  -t  ï  r 0 A_  \ P  6   x | ¾ Ó d ” [ þ t õ  Y  L # Œ l  ¨ î ç  H`  ¦ 2

[ €   sigmoid + þ AI _  \ P  6   x | ¾ Ó[ þ t`  ¦ ¸ ú ˜
 ? / 9, 0 A\ 

"

f [ O " î ô  Ç  ü < ° ú  s , ³ ðï  r \  Q  Å Ò  Œ •> • ¸  ) a  .

n: s  à º  H & ñ & h  \ P  6   x | ¾ Ó d ” `  ¦
 ? /l  0 Aô  Ç \  -t  ï  r 0

A[ þ t _  Ã º\  ¦
  · p . ™ èà º (fraction)[ þ t • ¸ 0 p x  . { 9 

Fig. 4. The semi-empirical heat capacity of silver [9] at the constant volume compared with the theoretical Eq.

(14)-2 (β 1b = 5.3, T s = 29 K, D h1 = 165 cal/mole, c s1

= .95, n = 3.09, A = .0000328, σ = .052), Eq. (14)-3 (β 1b = 1.8, T s = 39 K, D h1 = 185 cal/mole, c s1 = .95, n = 3.99, A = .0000368, σ = .028) and Eq. (14)-4 (β 1b

= .51, T s = 48 K, D h1 = 486 cal/mole, c s1 = .95, n = 4.99, A = .0000388, σ = .023).

Fig. 5. The semi-empirical heat capacity of copper [10]

at the constant volume compared with the theoretical Eq. (14)-2 (β 1b = 3.3, T s = 51.5 K, D h1 = 217 cal/mole, c s1 = .95, n = 3.42, A = .0000158, σ = .052), Eq. (14)-3 (β _1b = 1.4, T _s = 53.0 K, D _h1 = 156.9 cal/mole, c _s1 = .95, n = 4.02, A = .0000278, σ = .032) and Eq. (14)-4 (β _1b = .7, T _s = 59.0 K, D _h1 = 227 cal/mole, c _s1 = .95, n = 4.82, A = .0000378, σ = .025).

ì

ø Í ‚ à Г ¦" f[ þ t“ É r „    ¿ º 7 á x À Ó_  Û ¼— 2 ;[ þ t, ↑ (upward) õ 

↓ (downward) s  e ”  “ ¦ ´ ú ˜ “ ¦, € ª œ$ í  (+×  æ$ í  )\  @ /

# Œ" f• ¸ ⇑ spin õ  ⇓ spin s  e ”  “ ¦ ´ ú ˜   H  כ “ É r ½ + Ë{ © œ

t  · ú § . s  Qô  Ç ½ ¨ì  r“ É r 1 " é ¶& h s  . F K5 Å q[ þ t“ É r 3 " é ¶

½

¨› ¸\  ¦ ° ú “ ¦ e ”  . F K5 Å q[ þ t õ  ° ú  “ É r " é ¶  [ þ t“ É r Û ¼— 2 ; › ¸ o\  ¦

´ ú

2 X  l  M :ë  H \  ½ ¨ (sphere)– Ð Ò q ty Œ • K   ÷ & 9, „    ü

< € ª œ$ í  (+×  æ$ í  )[ þ t“ É r ½ ¨_  ×  æd ” \  @ / # Œ µ 1 ÚÜ ¼– Ð † ¾ Ó ô

 Ç Û ¼— 2 ;[ þ t ←, ⇐ õ  î ß –Ü ¼– Ð † ¾ Óô  Ç Û ¼— 2 ;[ þ t →, ⇒  e ”  “ ¦

½

+ É Ã º e ”  .

]

j{ 9  €  $  ô  Ç \ V– Ð+ ‹ " é ¶  _  í ß –ê ø Í z  ´+ « > é ß –€  & h ( a τ )“ É r

"

é

¶      ñ(Z)_  45 p x (power) ˜ Ð   H ß ¼ 9 55 p x ˜ Ð   H  Œ • 

Table 1. The experimental heat capacity data of silver at the constant pressure [9] compared with the semi-empirical heat capacity data at the constant volume by the different A values and Eq. (14)-4 (β _1b = 0.5112, T _s = 48 K, D _h1 = 485.3 cal epn ⁻¹ , c s1 = 0.95, n = 4.99, σ = 0.0225, T i = 12.5 K) and Debye (θ = 212 K, σ D = 0.0304).

Temperature(K) C

p

[9] C

ν

(*) C

ν

(Eq. (14)-4) C

ν

(**) C

ν

(Debye)

0. 0. 0. 0. 0. 0.

15 .1600 .1600 .2126 .1600 0.1643

25 .7470 .7465 .7606 .7470 0.7088

30 1.141 1.139 1.1383 1.140 1.113

40 2.005 1.998 1.9714 2.001 1.994

45 2.399 2.389 2.3775 2.393 2.410

60 3.420 3.393 3.3979 3.405 3.430

90 4.573 4.501 4.5424 4.532 4.587

120 5.162 5.039 5.0479 5.092 5.125

150 5.490 5.316 5.3047 5.391 5.406

170 5.644 5.436 5.4119 5.526 5.522

200 5.800 5.541 5.5220 5.653 5.639

230 5.911 5.602 5.5963 5.736 5.715

260 6.025 5.662 5.6496 5.819 5.768

290 6.080 5.668 5.6895 5.846 5.805

300 6.095 5.667 5.7006 5.852 5.815

* A = 3.88 × 10

⁻⁵

for C

p

− C

ν

= AT C

p²

to fit Eq. (14)-4.

* A = 2.58 × 10

⁻⁵

for C

p

− C

ν

= AT C

²p

to fit Debye Eq.

Table 2. The excitation isotherm of the excited epns(electron ⁺ proton ⁺ neutron) in each energy level as for the exper- imental heat capacity data of the silver at constant pressure [9] obtained from Eq. (14)-4 (β 1b = .5112, T s = 48 K, D _h1 , = 485.3 cal epn ⁻¹ for energy level 2-n, c _s1 = .95, n = 4.99, A = .0000384, σ = .0225, T _i = 12.5 K.)

Temperature(K) C

p

[9]

^N_g¹

1

N₂ g₁

N₃ g₁

N₄ g₁

N₅ g₁

0. 0. 0. 0. 0. 0. 0.

15 .160 .450 .132 .038 .011 .002

25 .747 .648 .309 .142 .060 .020

30 1.141 .723 .404 .216 .104 .039

40 2.005 .835 .580 .379 .222 .097

45 2.399 .873 .653 .460 .287 .134

60 3.420 .943 .811 .655 .472 .255

90 4.573 .987 .940 .858 .712 .454

120 5.162 .996 .977 .933 .828 .580

150 5.490 .999 .990 .964 .887 .661

170 5.544 .999 .994 .975 .911 .700

200 5.800 .999 .996 .984 .935 .744

230 5.911 .999 .998 .989 .951 .777

260 6.025 .999 .999 .993 .962 .802

290 6.080 .999 .999 .995 .969 .823

300 6.095 .999 .999 .995 .971 .829

“

¦ ´ ú ˜ô  Ç  [16,17]. # Œl " f 4˜ Ð  › ¸F K ß ¼   H  כ “ É r F K5 Å q [

þ

t“ É r 47 á x À Ó_  C • ¸ Û ¼— 2 ;[ þ t, ←⇐, →⇐, ←⇒, →⇒[ þ t`  ¦

t “ ¦ e ”    H  כ `  ¦ ´ ú ˜  9,
 Qt  1˜ Ð   Œ •“ É r ô  Ç \  - t

 ï  r 0 A  H  Ä »×  æ$ í  [ þ t – Ð s À Ò# Q”   5 Å q epn[ þ t • ¸ > á ¤ ° ú  

“

É r 4 7 á x À Ó_  Û ¼— 2 ;[ þ t`  ¦ ° ú “ ¦ e ”  “ ¦ Ò q ty Œ • K  ë ß – ÷ &
, z  ´ ]

j  H Õ ªX O t  · ú § (  É r ƒ  ½ ¨\ " f).

Table 3`  ¦ ¶ ú ˜( R˜ Ѐ  , 2> h_  \  -t  ï  r 0 A`  ¦
 ? /  H

&

ñ & h  \ P  6   x | ¾ Ó d ” , C ν2 = √

C l1 C l2 “ É r n _  ° ú כs  3˜ Ð   H 3   H % ƒ_  ° ú כ[ þ t s % 3  . s  כ “ É r 2 ˜ Ð   H  H \  -t  ï  r 0 A

`

 ¦ ° ú   H    H  כ `  ¦ _ p ô  Ç . Õ ª Q
 Õ ªM : r ' Ÿ  ‚ à ̚ ¸\ 

Table 3. The parameter values and σ values obtained by fitting Eq. (14)-1 to the experimental heat capacity data of metals at the constant pressure.

C

ν2

= √ C

l1

C

l2

Metals β

1b

T

s

(K) D

h1

(*) c

s1

n A σ

Ag 4.91 38.0 365.3 .95 3.19 .0000133 .095

Cu 23.2 26.0 948.3 .95 2.73 .0000432 .068

B 12.0 122.0 310.3 .95 3.00 .0000426 .063

Ni 35.2 28.0 258.3 .95 2.83 .0000432 .068

Pd 5.0 62.0 630.3 .95 3.19 .0000260 .068

* cal epn

⁻¹

for energy level 2-n

Table 4. The parameter values and σ values obtained by fitting Eq. (14)-5 to the experimental heat capacity data of metals at the constant pressure.

C

ν6

= √

⁶

C

l1

C

l2

C

l3

C

l4

C

l5

C

l6

Metals β

1b

T

s

(K) D

h1

(*) c

s1

n A σ

Ag 0.31 44.0 425.3 .95 5.79 .0000408 .034

Cu 0.20 56.0 176.9 .95 5.62 .0000438 .027

B 0.32 201.5 600.3 .95 5.20 .0000096 .015

Ni 0.31 73.0 318.3 .95 5.62 .0000827 .025

Pd 0.16 48.5 150.3 .95 5.19 .0000683 .041

* cal epn

⁻¹

for energy level 2-n

Fig. 6. The semi-empirical heat capacity of boron [11] at the constant volume compared with the theoretical Eq.

(14)-2 (β 1b = 3.1, T s = 182 K, D h1 = 450 cal/mole, c s1

= .95, n = 3.4, A = .000034, σ = .048), Eq. (14)-3 (β 1b

= 0.8, T _s = 197 K, D _h1 = 350 cal/mole, c _s1 = .95, n = 4.0, A = .000084, σ = .037) and Eq. (14)-4 (β _1b = 1.6, T _s = 184 K, D _h1 = 610 cal/mole, c _s1 = 95, n = 4.39, A

= -.0000796, σ = .007).

_

ô  Ç þ j& h  o  ) a ³ ðï  r \  Q  H  -Á º ß ¼l  M :ë  H \ ,  8 s  © œ

“

¦ 9_  # Œt  \ O  . Figs. 4, 5, 6, 7, 8[ þ t _  Å Ò$ 3 [ þ t`  ¦

¶ ú

˜( R˜ Ѐ  , 3> h_  \  -t  ï  r 0 A`  ¦
 ? /  H \ P  6   x | ¾ Ó d ” , C _ν3 = √

³

C _l1 C _l2 C _l3 “ É r n _  ° ú כ[ þ t“ É r 3 ˜ Ð   H ° ú כ`  ¦ t  9,

³

ðï  r \  Q• ¸ ß ¼ . 4> h_  \  -t Y U6 \ š`  ¦
 ? /  H \ P  6   x | ¾ Ó _

 d ” , C ν4 = √

⁴

C l1 C l2 C l3 C l4 “ É r n _  ° ú כs  4\   Å Ò   î

 r ° ú כ[ þ t s “ ¦, ³ ðï  r \  Q• ¸ B Ä º  Œ • . Õ ª Q
 d ”  n_  ° ú כ

Fig. 7. The semi-empirical heat capacity of nickel [12,13]

at the constant volume compared with the theoretical Eq. (14)-2 (β _1b = 6.5, T _s = 55.0 K, D _h1 = 398 cal/mole, c _s1 = .95, n = 3.32, A = .0000532, σ = .052), Eq. (14)-3 (β _1b = 1.7, T _s = 73 K, D _h1 = 368 cal/mole, c _s1 = .95, n = 4.12, A = .0000632, σ = .032) and Eq. (14)-4 (β 1b

= 0.7, T s = 69 K, D h1 = 218 cal/mole, c s1 = .95, n = 4.99, A = .0000732, σ = .022).

s

 4˜ Ð   H ß ¼“ ¦ 5 s  _  ° ú כ`  ¦ t  9 ³ ðï  r \  Q• ¸ ] j { 9

  Œ •“ É r \ P  6   x | ¾ Ó d ” “ É r C ν5 = √

⁵

C l1 C l2 C l3 C l4 C l5 s  . s 



 H ] j{ 9  a % ~“ É r d ” s  9, s  d ” “ É r & ñ S X ‰ >  n_  ° ú כ[ þ t`  ¦ ´ ú ˜ ô

 Ç  ½ + É Ã º e ”  .  8  H \  -t  Y U6 \ š (Table 4 ‚ à Л ¸)\  @ / ô

 Ç d ” “ É r ³ ðï  r \  Q & t l  M :ë  H \  “ ¦ 9_  # Œt  \ O 



. ¢ ¸ Fig. 6_  Å Ò$ 3 \ " f d ”  (14)-4\ 
  · p  ü < ° ú  s 

boron _  A_  ° ú כ“ É r 6 £ § (negative) _  ° ú כ`  ¦ ° ú   H  . s   H ì ø Í

Fig. 8. The semi-empirical heat capacity of palladium [14,15] at the constant volume compared with the the- oretical Eq. (14)-2 (β 1b = 5.0, T s = 41 K, D h1 = 490 cal/mole, c s1 = .95, n = 3.19, A = .0000460, σ = .029), Eq. (14)-3 (β _1b = 1.3, T _s = 58 K, D _h1 = 490 cal/mole, c _s1 = .95, n = 4.09, A = .0000560, σ = .020) and Eq.

(14)-4 (β _1b = .46, T _s = 62 K, D _h1 = 640 cal/mole, c _s1

= .95, n = 4.99, A = .0000560, σ = .023).

z 

´+ « > d ” s , C ^ν − C p = AT C _p ² s   ) a    H  כ `  ¦ _ p ô  Ç .

“

: r • ¸\  ¦ `  ¦ o €    Òx  ×  ¦ # Q Ž  H    H  כ `  ¦ _ p  l • ¸ ô  Ç



.

¿

º   P : \ V– Ð+ ‹ ˜ Ð: Ÿ x à º™ è" é ¶  [ þ t _  Zeeman ´ òõ  [18] \ " f à º™ è  H 47 á x À Ó[ þ t s  e ”  “ ¦ Ò q ty Œ •÷ &l  M :ë  H \ , 47 á x À

Ó[ þ t _  Û ¼& 7 ˜à Ô! 3  ‚  [ þ t`  ¦
 ? /    H X <, 3 7 á x À Ó[ þ t _  Û ¼

&

7 ˜à Ô! 3  ‚  [ þ t`  ¦
  · p . Õ ª s Ä »  H ←⇒ Û ¼— 2 ;õ  →⇐ Û ¼

—

2 ;Ü ¼– Ð  ) a à º™ è  H ° ú  “ É r Û ¼& 7 ˜à Ô! 3  ‚  `  ¦
 ? /l  M :ë  H s 



. ¢ ¸ô  Ç ˜ Ð: Ÿ x à º™ è " é ¶  [ þ t _  ) í Û ¼J $ ™ 111 €  \  f  ¨ ‚ Ã Ì z  ´+ « >

“

É r 47 á x À Ó[ þ t _  f  ¨ ‚ Ã Ì site[ þ t`  ¦ ° ú   H  כ Ü ¼– Ð ˜ Ð  [19], 47 á x À

Ó[ þ t _  à º™ è " é ¶  [ þ t s  ” > r F  # Œ 8 £ ¤& ñ Û ¼– Ð+ ‹ F K5 Å q " é ¶



_  4 ∼ 5 \  -t  ï  r 0 A\    É r [ þ t> p u s  ¸ ú ˜ ´ ú   H  כ Ü ¼– Ð Ò q

ty Œ •  ) a  .

\

 -t  ï  r 0 A 4˜ Ð   H ß ¼“ ¦ 5˜ Ð   H  Œ •“ É r   É r ô  Ç \ V



 H Ä ºo 
  6 £ § >   H 5\  ¦ ° ú   H  כ s  .

T _s (K): · ú ¡‚    7 Hë  H [1] \ " f P 0   H µ = µ ⁰ { 9  M :_  · ú š§ 4  s

% 3  .  ð ø Ít – Ð T s   H µ = µ ⁰ { 9  M :_  “ : r • ¸s  . # Œl 

"

f µ ⁰   H # Q‹ "  l ‘ : r& h “   & h \ " f, [ þ t ä ¼t  · ú §“ É r epn \  @ /ô  Ç



o† < Æ Ÿ íJ $ ™[ >  s  . “ É r \  @ /ô  Ç T _s   H 48 K s  . s  “ : r • ¸



 H θ D (=212 K) ˜ Ð   s `›   ± ú Ü ¼Ù ¼– Ð, ‘ : r ƒ  ½ ¨\ " f ½ ¨ô  Ç

&

ñ & h  \ P  6   x | ¾ Ó d ” [ þ t“ É r ƒ  5 Å q ^ ‰ s  : r`  ¦ ë ß –7 á ¤ r v “ ¦ e ”  .

s

  H s † ½ Ó ~ ½ Ó& ñ d ”   ^ ‰ q  # Œl  S X ‰Ò  ¦`  ¦ Ÿ í† < Ê “ ¦ e ” l  M

:ë  H \ , ƒ  5 Å q ^ ‰ s  : r`  ¦ % ƒ6 £ § Ä »• ¸ d ”  Ò'  ë ß –7 á ¤ r †   “ ¦

^

 ¦ à º e ”  .

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[1] D. Kim, Korean J. Chem. Eng. 27, 730 (2010).

[2] F. W. Sears and G. I. Salinger, Thermodynam- ics, Kinetics, Theory and Statistical Thermodynam- ics, 3rd ed. (Addison-Wesley Publishing Co., 1975), Chap. 3, p. 81-96; Chap. 13, p. 386-416.

[3] A. Einstein, Annalen Der Physik 22, 180 (1907).

[4] P. Debye, Annalen Der Physik 39, 789 (1912).

[5] E. D. Knuth, Introduction To Statistical Thermody- namics (McGraw Hill, 1966), Chap. 14, p. 162-173 [6] C. Gorter, J. Progess in Low Temperature Physics

(Interscience publishers INC., New York, 1955),

Chap. 11.

[7] C. Kittel, Introduction to Solid State Physics, 6th ed. (John Wiley & Sons Inc, 1991), Chap. 9, p. 243- 252.

[8] Reif and Federick, Fundamentals of Statistical and Thermal Physics (McGraw-Hill International Ed., 1985), Chap. 1, p. 10-46; Chap. 7, p. 253-256 [9] P. F. Meads, W. R. Forsythe and W. F. Giauque, J.

Am. Chem. Soc. 63, 1902 (1941).

[10] W. F. Giauque and P. F. Meads, J. Am. Chem. Soc.

63, 1897 (1941).

[11] H. L. Johnston, H. N. Hersh and E. C. Kerr, J. Am.

Chem. Soc. 73, 1112 (1951).

[12] V. A. Eucken and H. Werth, Z. Anorg Allgemeine Chem. 188, 152 (1930).

[13] R. H. Busey and W. F. Giauque, J. Am. Chem. Soc.

74, 3157 (1952).

[14] P. Mitacek. Jr. and J. G. Aston, J. Am. Chem. Soc.

85, 137 (1963).

[15] G. L. Pickard and F. E. Simon, The Proceedings of The Physical Society 61, 1 (1948).

[16] O. Oldenberg, Rasmussen and C. Norman, Modern Physics for Engineers (MacGraw-Hill Book Com- pany, 1966), Chap. 17, p. 376-404.

[17] E. Bleuler and G. J. Goldsmith, Experimental Nu- cleonics, 2nd ed. (Rinethart & Company Inc., New York, 1956), Chap. 1, p. 4.

[18] F. L. Pilar, Elementary Quantum Chemistry (McGraw-Hill Co., 1990), Chap. 5, p. 144-146.

[19] P. W. Tamm and L. D. Schmidt, J. Chem. Phys. 55,

4253 (1971).