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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:

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Presentasi berjudul: "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:"— Transcript presentasi:

1 CURICULUM VITAE A. DATA DIRI 01. N a m a : Dr. H. Muris, M.Si 02. Tempat/Tanggal Lahir: Tinggas, Jenis Kelamin: Laki-laki 04. Fakultas/Jurusan: FMIPA/Fisika 05. Pangkat/Golongan/NIP: Lektor Kepala/IV/a/ Bidang Keahlian: Fisika Material 07. Alamat Rumah: BTN Minasa Upa G20/14 Makassar Telp. (0411) HP Alamat Kantor : Jurusan Fisika FMIPA UNM Kampus Parangtambung Makassar Tlp/Fax. (0411)840622, HP Riwayat Pendidikan Tinggi: Jenis PendidikanTempatTahun lulusSpesialisasi Sarjana (S1) Pra Magister (Pra S2) Magister (S2) Doktor (S3) IKIP Ujung Pandang ITB Bandung Université de la Méditerranée Marseille, Prancis Pendidikan Fisika Fisika Fisika Material B. Riwayat Pekerjaan 1.Dosen Tetap Jurusan Fisika FMIPA Universitas Negeri Makassar, sekarang. 2.Ketua Program Studi Fisika FMIPA Universitas Negeri Makassar, Pembantu Dekan Bidang Akademik FMIPA Universitas Negeri Makassar, sekarang. 4.Dosen Program Pascasarjana UNM Makassar, sekarang

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

3 Pokok Bahasan 1.Pengantar 2.Statistik Maxwell Boltzmann 3.Aplikasi Statistik Maxwell Boltzmann 4.Statistik Bose Einstein 5.Statistik Fermi Dirac 6.Temperatur dan Entropy 7.Aplikasi Statistik Termodinamika 8.Ensemble Kanonik 9.Grand Ensemble Kanonik

4 Pokok Bahasan 1.Pengantar 2.Statistik Maxwell Boltzmann 3.Aplikasi Statistik Maxwell Boltzmann 4.Statistik Bose Einstein 5.Statistik Fermi Dirac 6.Temperatur dan Entropy 7.Aplikasi Statistik Termodinamika 8.Ensemble Kanonik 9.Grand Ensemble Kanonik

5 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).

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

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

8 Pengertian Dasar Statistik Rata-rata dengan fungsi probabilitas xixi ff(x i )x i f(x i ) 333/79/7 433/712/7 511/75/7 7128/7 = 4 Ternyata diperoleh hasil rata-rata yang sama yakni 4

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

10 Pengertian Dasar Statistik

11 Fungsi Gaussian Fungsi seperti akan banyak dijumpai dalam pembahasan statistik partikel

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

13 Ruang Euclid dan Ruang Fase Ruang fase Ruang momentum

14 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)

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

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

17 Assembli Klasik dan Kuantum b.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 a.Klasik - Terbedakan antara satu dengan lainnya (distinguishable) - Energi kontinu - Tak memenuhi prinsip larangan Pauli

18 Assembli Klasik dan Kuantum b.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

19 Statistik Maxwell Boltzmann Distribusi Energi Misalkan dalam sistim yang ditinjau terdapat N sistim : Sistem 1 dengan energi ε 1 Sistem 2 dengan energi ε 2 ……………………. Sistem i dengan energi ε i ……………………. Sistem N dengan energi ε N

20 Statistik Maxwell Boltzmann Distribusi Energi Misalkan dalam sistim yang ditinjau terdapat N sistim : Sistem 1 dengan energi ε 1 Sistem 2 dengan energi ε 2 ……………………. Sistem i dengan energi ε i ……………………. Sistem N dengan energi ε N

21 Statistik Maxwell Boltzmann Prinsip Kekekalan

22 Statistik Maxwell Boltzmann Jumlah pilihan jika memilih sejumlah N 1 di antara N partikel Jika g 1 menyatakan bobot, maka jumlah pilihan yang ada adalah :

23 Statistik Maxwell Boltzmann Perluas lagi dengan mengambil sejumlah N 2 dari N-N 1 Perluas lagi dengan mengambil sampai n kali

24 Statistik Maxwell Boltzmann Secara umum dapat ditulis :

25 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 : N 1 = N 2 = 2g 1 = 3, g 2 = 4

26 Contoh Pemakaian abab c,a cdcd db Ini hanyalah 3 contoh gambar dari 864 kemungkinan yang ada. Sekarang adalah giliran anda untuk melengkapinya.

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

28 Distribusi Maxwell Boltzmann  0   0 g()g() =  0 P()P()

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

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

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

32 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.

33 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: Find the temperature T at which the rms speed of Hydrogen molecules exceeds their most probable speed by 400 m/s. Answer: 380K At home:

34 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 v x menuju kepada seorang pengamat. Pengamat akan menerima radiasi dengan panjang gelombang.

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

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

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

38 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 :

39 Prinsip Ekipartisi Energi Nyatakan energi sebagai dan Misalkan = u 2 maka

40 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

41 Prinsip Ekipartisi Energi Maka : Ubah ke koordinat polar : dp x dx

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

43 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

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

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

46 STATISTIK BOSE-EINSTEIN

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

54 STATISTIK FERMI-DIRAC Gunakan rumus Stirling

55 STATISTIK FERMI-DIRAC

56 T =0 ~ k B T  =  ( with respect to  )

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

58 Radiasi Benda Hitam Two types of bosons: (a)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.

59 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 u S (,T) d as the energy density (per unit volume) of the radiation with frequencies between and +d. u S (,T) is the spectral energy density. The internal energy of the photon gas: In equilibrium, u S (,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.

60 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) : 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). T The mechanism of establishing equilibrium in a photon gas is absorption and emission of photons by 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.

61 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 Thus, - the Gibbs free energy of an equilibrium photon gas is 0 ! 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.

62 Rapat Keadaan Foton kyky kxkx kzkz extra factor of 2: two polarizations In the classical (high temperature) limit: The average energy in the mode: 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.

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

64 Pendekatan Klasik (f kecil, besar), Hkm 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 Pd frekwensi rendah dan temp. tinggi - purely classical result (no h), can be obtained directly from equipartition

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

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

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

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

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

70 Daya yang dipancarkan oleh Benda Hitam For the “uni-directional” motion, the flux of energy per unit area c  1s energy density u 1m 2 T Integration over all angles provides a factor of ¼: 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: (the hole size must be >> the wavelength)

71 Beberapa Contoh 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: Dewar 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: The net ingoing flux: Let the total ingoing flux be J, and the total outgoing flux be J’: 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 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).

72 Efek Rumah Kaca Transmittance of the Earth atmosphere Absorption: Emission: the flux of the solar radiation energy received by the Earth ~ 1370 W/m 2  = 1 – T Earth = 280K R orbit = 1.5·10 11 m R Sun = 7·10 8 m However, in reality  = 0.7 – T Earth = 256K To maintain a comfortable temperature on the Earth, we need the Greenhouse Effect ! The complicated issue of global worming: adding CO 2 (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.

73 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: 7·10 8 m, mass - 2 ·10 30 kg. max = 0.5  m  This result is consistent with the flux of the solar radiation energy received by the Earth (1370 W/m 2 ) being multiplied by the area of a sphere with radius 1.5·10 11 m (Sun-Earth distance). the mass loss per one second 1% of Sun’s mass will be lost in

74 Fungsi Distribusi untuk gas Fermi Ideal The probability of the i-state with energy  i to be occupied by n i particles (the total energy of this state n i  i ) : If the particles are fermions, n can only be 0 or 1: The grand partition function for all particles in the i th single- particle state (the sum is taken over all possible values of n i ) : - the Fermi-Dirac distribution At T = 0, 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 ~ k B T. The mean number of particles in this state: T =0 ~ k B T  =  ( with respect to  )

75 Fungsi Distribusi Gas Bose Ideal The grand partition function for all particles in the i th single-particle state: (the sum is taken over the possible values of n i ) 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  > .

76 Probabilitas, Fungsi Distribusi, Rapat Keadaan …. 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). x U(x)U(x)  The probability that the system is in state s with energy E and N particles where n=N/V – the density of particles If we can neglect the spectrum discreteness: where g(  ) is the density of states

77 Kaitan Termodinamika, Potensial Kimia cannot be negative for any  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 Consider the grand potentialwhich is a generalization of F=-k B T 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.  MB < 0: - the occupancy

78 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. The limit T  0: adding one fermion to the system at T=0 increases its energy U by E F. 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 E F : depending on n and T,  for fermions may be either positive or negative. k B T/E F 1 1  / EF / 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 k B T > n Q : The crossover occurs at n~n Q When n<

79 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).

80 Pendekatan Klasik The Fermi-Dirac and Bose-Einstein distributions must reduce to the Maxwell- Boltzmann distribution in the classical limit, for all i. Hence, and the Maxwell- Boltzmann distribution The same result, of course, we would get if we start from the equation for the average n k 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).

81 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>>T C where T C is the so-called degeneracy temperature of the gas, which corresponds to the condition n~ n Q. More accurately: For the FD gas, T C ~ E F /k B where E F is the Fermi energy (Lect. 24), for the BE gas T C 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 n Q is the quantum concentration, which varies as T 3/2

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

83 Comparison between Distributions T/T C C V /Nk B Fermi-Dirac Maxwell-Boltzmann Bose-Einstein

84 Comparison between Distributions Maxwell Boltzmann Fermi Dirac Bose Einstein distinguishable Z=(Z 1 ) N /N! n K <<1 spin doesn’t matter localized particles  don’t overlap gas molecules at low densities “unlimited” number of particles per state n K <<1 indistinguishable integer spin 0,1,2 … bosons wavefunctions overlap total  symmetric photons 4 He 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

85 Aplikasi Statistik Termodinamika Paramagnetism Fungsi Partisi

86 Aplikasi Statistik Termodinamika Momen magnet rata-rata Fungsi Partisi

87 Aplikasi Statistik Termodinamika Kapasitas panas magnetik

88 Aplikasi Statistik Termodinamika Untuk temperatur rendah

89 Aplikasi Statistik Termodinamika Jika dideferensial terhadap B

90 Aplikasi Statistik Termodinamika


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