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\begin{center}
	2015�N�x���v�͊wII �h��11 (7��2���o��A7��9����o)
	��
\end  {center}
\begin{flushright}
	�S�� �g�X ��
\end  {flushright}

\noindent [���1.]
(a) �����I�ȋ��E�����Ō��q��$D(\omega)$��$E$�����߂Ȃ����B

(b) �يj�Q���q���q�P�̕��z�֐���
$Z_1=Z_G j_{rot}Z_S$�̎��A�P���q������̔�M��
$C_V=C_{V,G}+C_{V,rot}+C_{V,S}$ �ƂȂ邱�Ƃ������Ȃ����B
 �Y����$G,rot,S$ �͏d�S�A��]�A�X�s����\���B

(c) ���q�̊������[�����g��$I$�Ƃ���ƁA
  $C_{V,rot}$�ɂ��Ēቷ�̋Ɍ��ŉ��x$T$�̈ˑ�����\���������߂Ȃ����B
 $x<<1$ ��$\ln (1+x) \simeq x$ ���g���A$\ell>1$�͖������ėǂ��B

\bigskip

\noindent[��] (���ۂ̉𓚂ɂ͂����܂Ōv�Z�𒚔J�ɏ����Ȃ��ėǂ��B)

(a)  $D(\omega)$�̌v�Z�́A�h��4�Ƃقړ����悤�Ɍv�Z�ł���B
 �g����Ԃ��l����̂͑��̕��ʂ̗��q�Ɠ����ŁA���̏�Ԃ�g����Ԃ�1�_�ɑΉ�������B
 �������A$g=2$�Ȃ̂ŁA�g����Ԃ�1�_��2�̏�ԂƑΉ����Ă���B
 
  �O����$\omega$�ɂ����Ԃ̐�$N(\omega)$�́A
   �g����Ԃ̋��̒��ɓ����Ă���_�̐����瓾����B
 ���a$\omega/c$�̑̐ς�_�̊Ԋu��3��Ŋ���΁A$N(\omega)$���v�Z�ł���B
 �_�̊Ԋu��$2\pi/L$������
\begin{equation}
	N(\omega)
	 = \frac{4\pi}
		{3}\left(\frac{\omega}
					{c}\right)^3\left(\frac{V}
					{8\pi^3}\right)	\times 2
\end   {equation}
 ���������āA
\begin{equation}
	D(\omega) = \frac{dN(\omega)}
			{d\omega}
	= \frac{8\pi}
		{c^3}\omega^2\left(\frac{V}
					{8\pi^3}\right)
	= \frac{V}
		{\pi^2 c^3}\omega^2							\label{Deps}
\end  {equation}
 �������A$\omega<0$��$D(\omega)=0$�ƂȂ�B
 
 \bigskip
 
 $E$�́A
\begin{equation}
	E = \int^\infty_{-\infty}b(\hbar\omega)\hbar\omega
		D(\omega)d\omega
\end  {equation}
 �����ŁA$b(\hbar\omega)$�̓{�[�X���z��$\varepsilon=\hbar\omega$����ꂽ���́B
  $b(\hbar\omega)$��$D(\omega)$�̋�̓I�Ȍ`��������ƁA
\begin{align}
	E	&= \int^\infty_0 \frac{\hbar\omega}
			{\exp[\hbar\omega/\kB T]-1}
		\frac{V\omega^{2}}
			{\pi^2c^3}d\omega		\\
\intertext{$\hbar\omega/\kB T=x$�Ƃ��ĕϐ��ϊ�����ƁA}
	&= \frac{V(\kB T)^{4}}
		{\pi^2c^3\hbar^3}\int^\infty_0 \frac{x^3dx}
							{\exp[x]-1}
\end{align}
 �����ŁA
\begin{equation}
	I = \int^\infty_0 \frac{x^3dx}
				{\exp[x]-1}
\end{equation}
  �Ƃ����΁A
\begin{equation}
	E	= \frac{V(\kB T)^{4}}
		{\pi^2c^3\hbar^3}I
\end{equation}
  �Ɠ��������܂�B
 $I$�́A�����ʂɂ��Ȃ��萔�ŁA����\footnote{\www
 ��g��b�����w�V���[�Y 7 �u���v�͊w�v�A�����m�� ���A��g���X}P278(A11)��
  �ɂ��΁A$\pi^4/15$�̒l�ɂȂ�B

\bigskip

(b)�G�l���M�[�ƕ��z�֐��͎��̊֌W���Ō��΂��(���v�͊wI�Q��: ����P74(3.17)���A
  ��K�m�[�g\footnote{%
\large http://www.cmt.phys.kyushu-u.ac.jp/%
\texttt{\symbol{"7E}}A.Yoshimori/hoshu15.pdf
}P10(21)��)�B
\begin{equation}
	\fcolorbox[rgb]{0,0,0}{1,0.9,0.9}{$\displaystyle
	E = -\round{\beta}\ln Z_1
	$}
				\label{Ekousiki}
\end   {equation}
 �����ŁA$\beta=1/(\kB T)$�Ƃ����B
$Z_1=Z_G j_{rot} Z_S$������
\begin{align}
	E &= -\round{\beta}\ln(Z_G j_{rot} Z_S)
												\\
\intertext{�ΐ��̌�������A}
	&=  -\round{\beta}\ln Z_G
	 -\round{\beta}\ln	j_{rot}
		-\round{\beta}\ln Z_S			\label{tesei} \\
\intertext{$\eps_G\equiv-\round{\beta}\ln Z_G$�A
  $\eps_{\text rot}\equiv-\round{\beta}\ln j_{rot}$�A
  $\eps_S\equiv-\round{\beta}\ln Z_S$�Ƃ���ƁA}
	&= \eps_G +\eps_{\text rot} +\eps_S	\label{energy}
\end   {align}
  ����́A�G�l���M�[��3�̎��R�x�̊�^�̘a�ŏ����鎖��\���Ă���B

 ����A��M�́A (����P57(2.87)���A��K�m�[�gP9(13)��)
\begin{equation}
	\fcolorbox[rgb]{0,0,0}{1,0.9,0.9}{$\displaystyle
	C_v = \left(\frac{\pd E}
		   {\pd T}\right)_V
	$}				\label{hentsukousiki}
\end   {equation}
 \eqref{energy}����������ƁA
\begin{equation}
	= \frac{\pd\eps_G}
		   {\pd T} +\frac{\pd\eps_{\text rot}}
		   			{\pd T}+\frac{\pd\eps_S}
		   								{\pd T}
\end   {equation}
 $C_{VG}=\pd\eps_G/\pd T$�A
 $C_{V\text rot}=\pd\eps_{\text rot}/\pd T$�A$C_{VS}=\pd\eps_S/\pd T$�Ƃ���΁A
  ������������B

\bigskip

\maru{2} ��]�̕��z�֐��́A���ƂŐ��������ʂ�A
 $\Theta=\hbar^2/(2I\kB)$�Ƃ���ƁA
\begin{equation}
	j_{rot}(T) = \sum^\infty_{l=0}(2l+1)\exp[-l(l+1)\frac{\Theta}
						{T}]			\label{8-5}
\end    {equation}
 �����ŁA$T<<\Theta$���l����ƁA$l$�̑傫�����͖����ł���(���2�Q��)�B

 $l>1$�𖳎�����ƁA
\begin{equation}
	j_{rot}(T) = 1 +3\exp[-2\frac{\Theta}
			{T}]
\end    {equation}
 \siki{Ekousiki}����A
\begin{align}
	\eps_{\text rot} &= -\round{\beta}\ln j(T)		\\
	&= -\round{\beta}\ln\left(1 +3\exp[-2\frac{\Theta}
					{T}] \right)	\\
\intertext{$x<<1$�̎��̌���$\ln(1+x)=x+\dotsb$����A}
	&= -\round{\beta}\left(3\exp[-2\frac{\Theta}
					{T}]
		  +\dotsb \right)	\\
	&= -\round{\beta}\left(3\exp[-2\beta\kB\Theta]
		  +\dotsb \right)	\\
	&= 6\kB\Theta\exp[-2\beta\kB\Theta]
		  +\dotsb
\end   {align}
 \siki{hentsukousiki}����A
\begin{align}
	C_{V,rot} &= \frac{\pd\eps_{\text rot}}
		   		{\pd T}	\\
	&= \round{T}
	\left(
		6\kB\Theta\exp[-2\beta\kB\Theta]
		  +\dotsb		\right)	\\
	&= \round{T}
	\left(
		6\kB\Theta\exp[-2\frac{\Theta}
					{T}]
		  +\dotsb		\right)	\\
	&= 6\kB\Theta\frac{2\Theta}
			{T^2}e^{-2\Theta/T} 
				+\dotsb	
\end   {align}


\noindent [���2.] �يj2���q���q��$j_{rot}$�ɂ��āA�ቷ�ő傫��$\ell$�͖����ł��邱�Ƃ������B


\bigskip

\noindent[��] \siki{8-5}��$T<<\Theta$���l����B
\begin{gather}
	\frac{\Theta}
		{T} >> 1 ������AX\equiv\exp[-\frac{\Theta}
					{T}]�́A�������B
	�܂�AX<<1		\\
	\exp[-l(l+1)\frac{\Theta}
			{T}] = X^{l(l+1)}�ŁA
	���AX<<1������A		\\
	X^{l(l+1)}�́A	\fcolorbox[rgb]{0,0,0}{1,0.9,0.9}{$l$���傫���قǏ������B}
\end   {gather}
 �܂�A$T<<\Theta$�̎��́A$l$�̑傫�����͖����ł���B
\end  {document}