CURICULUM VITAE A. DATA DIRI 01. N a m a : Dr. H. Muris, M.Si 02. Tempat/Tanggal Lahir : Tinggas, 1965 03. Jenis Kelamin : Laki-laki 04. Fakultas/Jurusan : FMIPA/Fisika 05. Pangkat/Golongan/NIP : Lektor Kepala/IV/a/131925820 06. Bidang Keahlian : Fisika Material 07. Alamat Rumah : BTN Minasa Upa G20/14 Makassar. 90224. Telp. (0411) 886307 HP. 081342403676 08. Alamat Kantor : Jurusan Fisika FMIPA UNM Kampus Parangtambung Makassar Tlp/Fax. (0411)840622, HP. 081342403676 09. e-mail : murisfmipaunm@yahoo.com 10. Riwayat Pendidikan Tinggi : Jenis Pendidikan Tempat Tahun lulus Spesialisasi Sarjana (S1) Pra Magister (Pra S2) Magister (S2) Doktor (S3) IKIP Ujung Pandang ITB Bandung Université de la Méditerranée Marseille, Prancis 1989 1992 1994 2001 Pendidikan Fisika Fisika Fisika Material B. Riwayat Pekerjaan Dosen Tetap Jurusan Fisika FMIPA Universitas Negeri Makassar, 1990 - sekarang. Ketua Program Studi Fisika FMIPA Universitas Negeri Makassar, 2003 - 2004. Pembantu Dekan Bidang Akademik FMIPA Universitas Negeri Makassar, 2004 - sekarang. Dosen Program Pascasarjana UNM Makassar, 2006 - sekarang

Fisika Statistik Rujukan Utama : Rujukan Tambahan : Introdution to Statistical Physics for Students by Pointon Longman, England Rujukan Tambahan : Buku Buku Fisika Zat Padat, Fisika Kuantum dan Fisika Modern yang relevan

Pokok Bahasan Pengantar Statistik Maxwell Boltzmann Aplikasi Statistik Maxwell Boltzmann Statistik Bose Einstein Statistik Fermi Dirac Temperatur dan Entropy Aplikasi Statistik Termodinamika Ensemble Kanonik Grand Ensemble Kanonik

Pokok Bahasan Pengantar Statistik Maxwell Boltzmann Aplikasi Statistik Maxwell Boltzmann Statistik Bose Einstein Statistik Fermi Dirac Temperatur dan Entropy Aplikasi Statistik Termodinamika Ensemble Kanonik Grand Ensemble Kanonik

Sistim Termodinamika, Parameter Makroskopik Sistim terbuka dimana dimungkinkan terjadi pertukanan energi dan materi dengan lingkungan. Sistim tertutup terjadi pertukaran energi maupun materi dengan lingkungannya Isolated systems tidak memungkinkan terjadinya pertukaran energi maupu materi dengan lingkungannya Paramater internal dan external : temperatur, volume, tekanan, energi, medan magnet, dll. (nilai rata-rata, fluktuasi diabaikan).

Pengertian Dasar Statistik Mean : Rata-rata Mode : yang paling mungkin Median : Titik tengah Varians : Ragam, Lebar Distribusi

Pengertian Dasar Statistik Misalkan suatu variabel yang diselidiki : 3,4,4,3,5,3,4

Pengertian Dasar Statistik Rata-rata dengan fungsi probabilitas xi f f(xi) xi f(xi) 3 3/7 9/7 4 12/7 5 1 1/7 5/7 7 28/7 = 4 Ternyata diperoleh hasil rata-rata yang sama yakni 4

Pengertian Dasar Statistik kontinyu Hasil ini diperoleh dari pengembangan bentuk diskrit Jika fungsinya kontinyu maka : Bagaimana anda mengartikan parameter statistik berikut ?

Pengertian Dasar Statistik

Fungsi Gaussian Fungsi seperti akan banyak dijumpai dalam pembahasan statistik partikel

Ruang Euclid dan Ruang Fase dV z y dz x dx dy

Ruang Euclid dan Ruang Fase Ruang fase Ruang momentum

Rata Rata Sifat Assembly Misalkan dalam assembly terdapat sejumlah N molekul dengan energi total E dan berada dalam volume V. p(N) menyatakan koordinat momentum x(N) menyatakan koordinat posisi p(N) x(N)

Rata Rata Sifat Assembly Jika X adalah perilaku yang ingin dicari rata-ratanya dalam ruang fase tersebut Normalisasi terhadap ruang

Rata Rata Sifat Assembly Jika X merupakan fungsi yang diskrit, maka perata-rataan fungsi X dapat dinyatakan dengan : Normalisasi probabilitas menghasilkan

Assembli Klasik dan Kuantum - Terbedakan antara satu dengan lainnya (distinguishable) - Energi kontinu - Tak memenuhi prinsip larangan Pauli Kuantum : Terdapat dua tipe Tipe I (fermion) : - Tak terbedakan antara satu dengan lainnya (indistinguishable) - Energi disktrit - Memenuhi prinsip larangan Pauli Misalnya : elektron dalam zat padat

Assembli Klasik dan Kuantum Kuantum : Terdapat dua tipe Tipe II (boson) : - Tak terbedakan antara satu dengan lainnya (indistinguishable) - Energi disktrit - Tidak memenuhi prinsip larangan Pauli Misalnya : foton atau partikel alpha

Statistik Maxwell Boltzmann Distribusi Energi Misalkan dalam sistim yang ditinjau terdapat N sistim : Sistem 1 dengan energi ε1 Sistem 2 dengan energi ε2 ……………………. Sistem i dengan energi εi Sistem N dengan energi εN

Statistik Maxwell Boltzmann Distribusi Energi Misalkan dalam sistim yang ditinjau terdapat N sistim : Sistem 1 dengan energi ε1 Sistem 2 dengan energi ε2 ……………………. Sistem i dengan energi εi Sistem N dengan energi εN

Statistik Maxwell Boltzmann Prinsip Kekekalan

Statistik Maxwell Boltzmann Jumlah pilihan jika memilih sejumlah N1 di antara N partikel Jika g1 menyatakan bobot, maka jumlah pilihan yang ada adalah :

Statistik Maxwell Boltzmann Perluas lagi dengan mengambil sejumlah N2 dari N-N1 Perluas lagi dengan mengambil sampai n kali

Statistik Maxwell Boltzmann Secara umum dapat ditulis :

Contoh Pemakaian Empat partikel dengan notasi a,b,c dan d didistribusi pada dua pita energi 2 pada pita 1 dan 2 pada sistim 2. Bobot masing-masing adalah 3 dan 4. Jadi : N1 = N2 = 2 g1 = 3 , g2 = 4

Contoh Pemakaian a b a b c,a c d c d d b Ini hanyalah 3 contoh gambar dari 864 kemungkinan yang ada. Sekarang adalah giliran anda untuk melengkapinya.

Statistik Maxwell Boltzmann Peluang terbesar diperoleh dengan mengambil dw/dn = 0 Rumus Stirling

Distribusi Maxwell Boltzmann   g() = P()

Aplikasi Statistik Maxwell Boltzmann ky Untuk partikel kuantum dalam kotak 2D (e.g., electron pd FET): 2D k kx - Tak bergantung pd  # states within ¼ of a circle of radius k 3D kz g() 3D 2D kx 1D ky  Thus, for 3D electrons (2s+1=2):

Distribusi Kecepatan Maxwell vy Nampak bahwa persamaan ini merupakan perkalian antara faktor Boltzmann dengan sebuah tetapan. Tetapan tersebut dapat diperoleh dari normalisasi v vx vz v P(v) Distribusi energi, N – the total # of particles speed distribution (distribusi kecepatan) P(vx) vx Distrbusi kecepatan dalam arah x, vx

Karakteristik Nilai Kecepatan Lihat bahwa distribusi ini tidak simetrik, sehingga perlu dicari perata-rataan sebagai berikut P(v) The root-mean-square speed is proportional to the square root of the average energy: v Harga kec.maksimum : Kelajuan rata-rata :

Soal (Maxwell distr.) Consider a mixture of Hydrogen and Helium at T=300 K. Find the speed at which the Maxwell distributions for these gases have the same value.

Soal (Maxwell distr.) Find the temperature at which the number of molecules in an ideal Boltzmann gas with the values of speed within the range v - v+dv is a maximum. maximum: At home: Find the temperature T at which the rms speed of Hydrogen molecules exceeds their most probable speed by 400 m/s. Answer: 380K

Pelebaran Garis Spektrum Doppler Bagian ini adalah salah satu contoh penerapan distribusi laju dari statistik Maxwell Boltzmann, yakni pelebaran spektrum akibat efek Doppler. Misalkan molekul gas melakukan radiasi dengan panjang gelombang dalam arah x dengan kecepatan vx menuju kepada seorang pengamat. Pengamat akan menerima radiasi dengan panjang gelombang.

Pelebaran Garis Spektrum Doppler Karena efek Doppler, maka panjang gelombang yang diamati pengamat adalah :

Pelebaran Garis Spektrum Doppler Dari distribusi Maxwell Boltzamann Ubah sebagai fungsi panjang gelombang

Pelebaran Garis Spektrum Doppler Intensitas radiasi : Dengan mengukur intensitas radiasi maka dapat ditentukan temperatur gas emisi

Prinsip Ekipartisi Energi Jika energi sistem dinyatakan dalam bentuk kuadrat posisi dan momentum maka tiap bentuk kuadrat tersebut akan memberikan energi rata-rata ½ kT Contoh molekul gas dengan massa m, energinya dapat dinyatakan dengan Maka energi rata-ratanya adalah :

Prinsip Ekipartisi Energi Nyatakan energi sebagai dan Misalkan = u2 maka

Prinsip Ekipartisi Energi Hasilnya memberikan : Maka : Karena ada satu bentuk kuadrat maka memberikan energi rata-rata ½ kT Contoh 2 : Osilator harmonik dengan dua jenis energi

Prinsip Ekipartisi Energi Maka : dpxdx Ubah ke koordinat polar :

Prinsip Ekipartisi Energi Maka : Karena terdiri dari dua bentuk kuadrat maka energinya adalah 2 x ½ kT = kT Untuk osilator harmonik 3D maka :

Prinsip Ekipartisi Energi Energi rata-rata untuk osilator harmonik 3 D. Jadi dalam hal ini ada 6 derajat kebebasan ( f = 6) dimana tiap derajat kebebasan memberikan kontribusi energi sebesar ½ kT

Prinsip Ekipartisi Energi Jika terdapat NA (bil. Avogadro) molekul gas dan berlaku sebagai osilator harmonik 3D, maka, terdapat 6 derajat kebebasan,maka : Panas jenis per gram atom zat padat :

Panas jenis gas Jika terdapat NA (bil. Avogadro) molekul gas dan berlaku sebagai osilator harmonik 3D, maka, terdapat 6 derajat kebebasan,maka : Panas jenis per gram atom zat padat :

STATISTIK BOSE-EINSTEIN

STATISTIK BOSE-EINSTEIN

STATISTIK BOSE-EINSTEIN

STATISTIK BOSE-EINSTEIN

STATISTIK BOSE-EINSTEIN

STATISTIK BOSE-EINSTEIN

STATISTIK BOSE-EINSTEIN

STATISTIK FERMI-DIRAC Jumlah untuk semua kemungkinan susunan yang berbeda Jumlah untuk semua kemungkinan susunan yang berbeda untuk satu tingkatan energi

STATISTIK FERMI-DIRAC Gunakan rumus Stirling

STATISTIK FERMI-DIRAC

STATISTIK FERMI-DIRAC ~ kBT  =  (with respect to )

STATISTIK FERMI-DIRAC Distribusi jumlah partikel partikel Melalui normalisasi gs = 1 diperoleh fungsi distribusi. Maka f(e) merupakan probabilitas sebagai fungsi energi Sebagai fungsi probabilitas maka harga fungsi ini maksimum 1 dan minimum 0

Radiasi Benda Hitam Two types of bosons: Composite particles which contain an even number of fermions. These number of these particles is conserved if the energy does not exceed the dissociation energy (~ MeV in the case of the nucleus). (b) particles associated with a field, of which the most important example is the photon. These particles are not conserved: if the total energy of the field changes, particles appear and disappear. We’ll see that the chemical potential of such particles is zero in equilibrium, regardless of density.

Radiation in Equilibrium with Matter Typically, radiation emitted by a hot body, or from a laser is not in equilibrium: energy is flowing outwards and must be replenished from some source. The first step towards understanding of radiation being in equilibrium with matter was made by Kirchhoff, who considered a cavity filled with radiation, the walls can be regarded as a heat bath for radiation. The walls emit and absorb e.-m. waves. In equilibrium, the walls and radiation must have the same temperature T. The energy of radiation is spread over a range of frequencies, and we define uS (,T) d as the energy density (per unit volume) of the radiation with frequencies between  and +d. uS(,T) is the spectral energy density. The internal energy of the photon gas: In equilibrium, uS (,T) is the same everywhere in the cavity, and is a function of frequency and temperature only. If the cavity volume increases at T=const, the internal energy U = u (T) V also increases. The essential difference between the photon gas and the ideal gas of molecules: for an ideal gas, an isothermal expansion would conserve the gas energy, whereas for the photon gas, it is the energy density which is unchanged, the number of photons is not conserved, but proportional to volume in an isothermal change. A real surface absorbs only a fraction of the radiation falling on it. The absorptivity  is a function of  and T; a surface for which ( ) =1 for all frequencies is called a black body.

Photons Apa Itu ? The electromagnetic field has an infinite number of modes (standing waves) in the cavity. Any radiation field is a superposition of plane waves of different frequencies. The characteristic feature of the radiation is that a mode may be excited only in units of the quantum of energy hf (similar to a harmonic oscillators) : T This fact leads to the concept of photons as quanta of the electromagnetic field. The state of the el.-mag. field is specified by the number n for each of the modes, or, in other words, by enumerating the number of photons with each frequency. According to the quantum theory of radiation, photons are massless bosons of spin 1 (in units ħ). They move with the speed of light : The linearity of Maxwell equations implies that the photons do not interact with each other. (Non-linear optical phenomena are observed when a large-intensity radiation interacts with matter). Presence of a small amount of matter is essential for establishing equilibrium in the photon gas. We’ll treat a system of photons as an ideal photon gas, and, in particular, we’ll apply the BE statistics to this system. The mechanism of establishing equilibrium in a photon gas is absorption and emission of photons by matter.

Potensial Kimia Foton = 0 The mechanism of establishing equilibrium in a photon gas is absorption and emission of photons by matter. The textbook suggests that N can be found from the equilibrium condition: Thus, in equilibrium, the chemical potential for a photon gas is zero: On the other hand, However, we cannot use the usual expression for the chemical potential, because one cannot increase N (i.e., add photons to the system) at constant volume and at the same time keep the temperature constant: - does not exist for the photon gas Instead, we can use - by increasing the volume at T=const, we proportionally scale F - the Gibbs free energy of an equilibrium photon gas is 0 ! Thus, For  = 0, the BE distribution reduces to the Planck’s distribution: Planck’s distribution provides the average number of photons in a single mode of frequency  = /h.

Rapat Keadaan Foton The average energy in the mode: In the classical (high temperature) limit: In order to calculate the average number of photons per small energy interval d, the average energy of photons per small energy interval d, etc., as well as the total average number of photons in a photon gas and its total energy, we need to know the density of states for photons as a function of photon energy. Rapat Keadaan Foton kz kx ky extra factor of 2: two polarizations

Spektrum Radiasi Benda Hitam Rata-rata jumlah foton per satuan volume denga frekwensi  dan +d: - Rapat Spektrum (hukum Radiasi Planck) u adalahfungsi energi: Radiasi spektrum benda hitam u(,T) - the energy density per unit photon energy for a photon gas in equilibrium with a blackbody at temperature T.

Pendekatan Klasik (f kecil ,  besar), Hkm Rayleigh-Jeans Pd frekwensi rendah dan temp. tinggi - purely classical result (no h), can be obtained directly from equipartition Hukum Rayleigh-Jeans This equation predicts the so-called ultraviolet catastrophe – an infinite amount of energy being radiated at high frequencies or short wavelengths.

Hukum Rayleigh-Jeans u sebagai fungsi dari panjang gelombang In the limit of large :

 frekwensi tinggi , Hukum Pergeseran Wien’s At high frequencies: - Ditemukan secara eksperimen oleh Wien Wien Nobel 1911 Maksimum u() berfeser ke frekwensi tinggi ketika temperatur naik. Hukum Pergeseran Wien u(,T) - the “most likely” frequency of a photon in a blackbody radiation with temperature T Numerous applications (e.g., non-contact radiation thermometry) 

“night vision” devices max  max - does this mean that ? Wrong!   T = 300 K max  10 m “night vision” devices

Radiasi Sinar Matahari Temperatur permukaan- 5800K As a function of energy, the spectrum of sunlight peaks at a photon energy of   (umax)  0.88 m, near infrared - close to the energy gap in Si, 1.2 eV, which has been so far the best material for photovoltaic devices (solar cells) Spectral sensitivity of the eye:

Hukum Radiasi Stefan-Boltzmann Jumlah total foton persatuan volume : - increases as T 3 Energi total foton per satuan volume : (apat energi gas foton) Tetapan Stefan-Boltzmann Hukum Stefan-Boltzmann Energi rata-rata per foton : (just slightly less than the “most” probable energy)

Daya yang dipancarkan oleh Benda Hitam For the “uni-directional” motion, the flux of energy per unit area energy density u 1m2 c  1s Integration over all angles provides a factor of ¼: (the hole size must be >> the wavelength) Thus, the power emitted by a unit-area surface at temperature T in all directions: The total power emitted by a sphere of radius R: T

Beberapa Contoh Dewar The value of the Stefan-Boltzmann constant: Consider a human body at 310K. The power emitted by the body: While the emissivity of skin is considerably less than 1, it emits sufficient infrared radiation to be easily detectable by modern techniques (night vision). Radiative transfer: Liquid nitrogen is stored in a vacuum or Dewar flask, a container surrounded by a thin evacuated jacket. While the thermal conductivity of gas at very low pressure is small, energy can still be transferred by radiation. Both surfaces, cold and warm, radiate at a rate: i=a for the outer (hot) wall, i=b for the inner (cold) wall, r – the coefficient of reflection, (1-r) – the coefficient of emission Let the total ingoing flux be J, and the total outgoing flux be J’: Dewar The net ingoing flux: If r=0.98 (walls are covered with silver mirror), the net flux is reduced to 1% of the value it would have if the surfaces were black bodies (r=0).

Efek Rumah Kaca Absorption: the flux of the solar radiation energy received by the Earth ~ 1370 W/m2 Emission: Rorbit = 1.5·1011 m Transmittance of the Earth atmosphere RSun = 7·108 m  = 1 – TEarth = 280K However, in reality  = 0.7 – TEarth = 256K To maintain a comfortable temperature on the Earth, we need the Greenhouse Effect ! The complicated issue of global worming: adding CO2 (and other “greenhouse” gases) to the atmosphere tends in itself to raise the earth’s average temperature, but also may increase cloudiness, which lowers it. One thing is clear: since climate is largely determined by the heat balance in the atmosphere, anything that changes the atmospheric absorption is bound to have climatic consequences.

Pengurangan Massa Matahari The spectrum of the Sun radiation is close to the black body spectrum with the maximum at a wavelength  = 0.5 m. Find the mass loss for the Sun in one second. How long it takes for the Sun to loose 1% of its mass due to radiation? Radius of the Sun: 7·108 m, mass - 2 ·1030 kg. max = 0.5 m  This result is consistent with the flux of the solar radiation energy received by the Earth (1370 W/m2) being multiplied by the area of a sphere with radius 1.5·1011 m (Sun-Earth distance). the mass loss per one second 1% of Sun’s mass will be lost in

Fungsi Distribusi untuk gas Fermi Ideal The probability of the i-state with energy i to be occupied by ni particles (the total energy of this state ni i) : The grand partition function for all particles in the ith single-particle state (the sum is taken over all possible values of ni) : If the particles are fermions, n can only be 0 or 1: The mean number of particles in this state: - the Fermi-Dirac distribution T =0 ~ kBT  =  (with respect to ) At T = 0, all the states with  <  have the average # of particles 1 (i.e., they are occupied with 100% probability), all the states with  >  have the average # of particles 0 (i.e., they are unoccupied). With increasing T, the step-like function is “smeared” over the energy range ~ kBT.

Fungsi Distribusi Gas Bose Ideal The grand partition function for all particles in the ith single-particle state: (the sum is taken over the possible values of ni) If the particles are bosons, n can any integer  0: The mean number of particles in this state: Distribusi Bose Einstein The mean number of particles in a given state for the BEG can exceed unity, it diverges as   , and is nonexistent for  > .

Probabilitas, Fungsi Distribusi, Rapat Keadaan ….  The probability that the system is in state s with energy E and N particles U(x) x The macrostate of such system is completely defined if we know the mean occupancy for all energy levels, which is often called the distribution function: While f(E) is often less than unity (much less in the case of an ideal gas), it is not a probability. (e.g., it can exceed unity in a Bose gas). where n=N/V – the density of particles If we can neglect the spectrum discreteness: where g() is the density of states

Kaitan Termodinamika, Potensial Kimia Consider the grand potential which is a generalization of F=-kBT lnZ the appearance of μ as a variable, while computationally very convenient for the grand canonical ensemble, is not natural. Thermodynamic properties of systems are eventually measured with a given density of particles. However, in the grand canonical ensemble, quantities like pressure or N are given as functions of the “natural” variables T,V and μ. Thus, we need to use to eliminate μ in terms of T and n=N/V. μ for an ideal gas is negative: when you add a particle to a system and want to keep S fixed, you typically have to remove some energy from the system. Boltzmann Gas MB < 0: - the occupancy cannot be negative for any 

Potensial Kimia untuk Gas Fermi Fermi Gas When the average number of fermions in a system (their density) is known, this equation can be considered as an implicit integral equation for (T,n). It also shows that  determines the mean number of particles in the system just as T determines the mean energy. However, solving the eq. is a non-trivial task.  / EF 1 depending on n and T,  for fermions may be either positive or negative. 1 kBT/EF The limit T0: adding one fermion to the system at T=0 increases its energy U by EF. In this situation F = U-TS = U (S is also 0: all the fermions are packed into the lowest-energy states), so that the chemical potential, which is the change in F produced by the addition of one particle, is EF: The change of sign of (n,T) indicates the crossover from the degenerate Fermi system (low T, high n) to the Boltzmann statistics. The condition kBT << EF is equivalent to n >> nQ: The crossover occurs at n~nQ When n<<nQ the chemical potential becomes negative:

Potensial Kimia untuk Gas Bose Bose Gas The occupancy cannot be negative for any , thus, for bosons,   0 ( varies from 0 to ). Also, as T0,   0  T For bosons, the chemical potential is a non-trivial function of the density and temperature (for details, see the lecture on BE condensation).

the Maxwell-Boltzmann distribution Pendekatan Klasik The Fermi-Dirac and Bose-Einstein distributions must reduce to the Maxwell-Boltzmann distribution in the classical limit, for all i. Hence, the Maxwell-Boltzmann distribution and The same result, of course, we would get if we start from the equation for the average nk in Boltzmann statistics: Comparison of the MB, FD, and BE distributions plotted for the same value of . Note that the MB distribution makes no sense when the average # of particle in a given state becomes comparable to 1 (violation of the dilute limit).  = 

Pendekatan Klasik (cont.) In terms of the density, the classical limit corresponds to n << the quantum density: We can also rewrite this condition as T>>TC where TC is the so-called degeneracy temperature of the gas, which corresponds to the condition n~ nQ. More accurately: For the FD gas, TC ~ EF/kB where EF is the Fermi energy (Lect. 24) , for the BE gas TC is the temperature of BE condensation (Lect. 26). Critical density for bosons: Since   0, the maximum possible value of n is obtained when  = 0, and where nQ is the quantum concentration, which varies as T 3/2

Pendekatan Ketiga Distribusi 1 2 3 Fermi-Dirac Maxwell-Boltzmann Bose-Einstein 3 2 zero-point energy, Pauli principle 1 1 2 3

Comparison between Distributions CV /NkB Fermi-Dirac Maxwell-Boltzmann Bose-Einstein 2 1.5 1 T/TC

Comparison between Distributions Maxwell Boltzmann Bose Einstein Fermi Dirac distinguishable Z=(Z1)N/N! nK<<1 spin doesn’t matter localized particles  don’t overlap gas molecules at low densities “unlimited” number of particles per state indistinguishable integer spin 0,1,2 … bosons wavefunctions overlap total  symmetric photons 4He atoms unlimited number of particles per state indistinguishable half-integer spin 1/2,3/2,5/2 … fermions wavefunctions overlap total  anti-symmetric free electrons in metals electrons in white dwarfs never more than 1 particle per state

Aplikasi Statistik Termodinamika Paramagnetism Fungsi Partisi

Aplikasi Statistik Termodinamika Momen magnet rata-rata Fungsi Partisi

Aplikasi Statistik Termodinamika Kapasitas panas magnetik

Aplikasi Statistik Termodinamika Untuk temperatur rendah

Aplikasi Statistik Termodinamika Jika dideferensial terhadap B

Aplikasi Statistik Termodinamika