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Competition Between Paramagnetism and Diamagnetism in Charged

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Competition Between Paramagnetism and Diamagnetism in Charged Competition between paramagnetism and diamagnetism in charged Fermi gases Xiaoling Jian, Jihong Qin, and Qiang Gu∗ Department of Physics, University of Science and Technology Beijing, Beijing 100083, China (Dated: November 10, 2018) The charged Fermi gas with a small Lande-factor g is expected to be diamagnetic, while that with a larger g could be paramagnetic. We calculate the critical value of the g-factor which separates the dia- and para-magnetic regions. In the weak-field limit, gc has the same value both at high and low temperatures, gc = 1/√12. Nevertheless, gc increases with the temperature reducing in finite magnetic fields. We also compare the gc value of Fermi gases with those of Boltzmann and Bose gases, supposing the particle has three Zeeman levels σ = 1, 0, and find that gc of Bose and Fermi gases is larger and smaller than that of Boltzmann gases,± respectively. PACS numbers: 05.30.Fk, 51.60.+a, 75.10.Lp, 75.20.-g I. INTRODUCTION mK and 10 µK, respectively. The ions can be regarded as charged Bose or Fermi gases. Once the quantum gas Magnetism of electron gases has been considerably has both the spin and charge degrees of freedom, there studied in condensed matter physics. In magnetic field, arises the competition between paramagnetism and dia- the magnetization of a free electron gas consists of two magnetism, as in electrons. Different from electrons, the independent parts. The spin magnetic moment of elec- g-factor for different magnetic ions is diverse, ranging trons results in the paramagnetic part (the Pauli para- from 0 to 2. magnetism), while the orbital motion due to charge de- The g-factor is a characteristic parameter to measure gree of freedom in magnetic field induces the diamagnetic the intensity of paramagnetic effect. It is expected that part (the Landau diamagnetism)[1]. The Pauli paramag- the quantum gas displays diamagnetism in small g region, netism and the Landau diamagnetism compete with each but paramagnetism in large g region. The main purpose other. For electrons whose Lande-factor g = 2, the zero- of this paper is to calculate the critical value of g. We field paramagnetic susceptibility is two times stronger also present a comparison with results of charged spin-1 than the diamagnetic susceptibility, so the free electron Bose gases which have been obtained previously[12]. gas exhibits paramagnetism altogether. Magnetic properties of relativistic Fermi gases have II. THE MODEL also been under extensive investigation. Daicic et al. developed statistical mechanics for the magnetized pair- fermion gases and found that the intrinsic spin causes We consider a charged Fermi gas of N particles, with important effects upon the relativistic para- and dia- the effective Hamiltonian written as magnetism[2]. H¯ µN = D (ǫ + ǫ µ) n , (1) − L jkz σ − jkz σ The study of ultracold atoms has stimulated renewed j,kXz,σ research interest in the magnetism of quantum gases[3– where µ is the chemical potential and ǫ is the quan- 6]. When atomic gases are confined in optical traps[3, 4], jkz tized Landau energy in the magnetic field B, their spin degree of freedom becomes active, leading to the manifestation of magnetism. Theoretically, the 1 ~2k2 ǫ = ( + j)~ω + z (2) paramagnetism[7] and ferromagnetism[8–10] in a neutral jkz 2 2m∗ spin-1 Bose gas have ever been studied. In experiments, with j = 0, 1, 2,... labeling different Landau levels and 87 magnetic domains have been directly observed in Rb ω = qB/(m∗c) being the gyromagnetic frequency of a ∗ condensate, a typical ferromagnetic spinor condensate[5]. fermion with charge q and mass m . DL = qBSxy/(2π~c) Very recently, the exploration of magnetism in quantum marks the degeneracy of each Landau level with Sxy the arXiv:1006.5179v1 [cond-mat.stat-mech] 27 Jun 2010 gases has been extended to the Fermi gas. It is already total section area in x, y directions of the system. ǫσ observed that an ultracold two-component Fermi gas of denotes the Zeeman energy, neutral 6Li atoms exhibits ferromagnetism caused by re- ~q pulsive interactions between atoms[6]. ~ ǫσ = g ∗ σB = gσ ω, (3) Furthermore, it is possible to realize charged quantum − m c − gases from neutral ultracold atoms. So far, cold plasma where g is the Lande-factor and σ refers to the spin-z index of Zeeman state F, σ . has been created by photoionization of cold atoms[11]. | i The temperatures of electrons and ions are as low as 100 The grand thermodynamic potential of the Fermi gas is expressed as 1 −β(ǫjk +ǫσ−µ) ΩT =06 = DL ln[1 + e z ], (4) ∗ −β Corresponding author: [email protected] j,kXz ,σ 2 −1 0.50 where β = (kBT ) . Performing Taylor expansions and (a) then integrating out kz, Eq. (4) becomes ωV m∗ 3/2 Ω = T =06 2 0.00 − ~ 2πβ M ∞ 3 ~ω ~ ( 1)l+1l− 2 e−lβ( 2 −gσ ω−µ) − , (5) × 1 e−lβ~ω Xl=1 Xσ − -0.50 where V is the volume of the system. For simplicity, the (b) following notation is introduced, 0.80 ∞ p l+1 α/2 −lx(ησ+δ) σ ( 1) l e Fτ [α, δ]= − , (6) M (1 e−lx)τ 0.40 Xl=1 − ~ ~ ~ where x = β ω and η = ( ω/2 µ + ǫ )/( ω). Then σ − σ Eq. (5) is rewritten as 0.00 (c) ∗ 3/2 ωV m -0.44 Ω = F σ[ 3, 0]. (7) T =06 ~2 1 − 2πβ − d Xσ M For a system with the given particle density n = N/V , the chemical potential is obtained according to the fol- -0.48 lowing equation ∗ 3/2 0.0 0.4 0.8 1 ∂Ω m σ n = = x 2 F1 [ 1, 0]. (8) −V ∂µ 2πβ~ − g T,V Xσ A similar treatment has been employed to study diamag- FIG. 1: (a) The total magnetization density (M), (b) the netism of the charged spinless Bose gas[13] and extended paramagnetization density (M p), and (c) the diamagnetiza- to the study of competition of diamagnetism and param- tion density (M d) as a function of g at t = 0.2. The dotted agnetism in charged spin-1 Bose gases[12]. This method and dashed lines correspond to ω = 5 and 10, respectively. is more applicable at relatively high temperatures. To determine gc, we need calculate the magnetization and density M as a function of the magnetic field B and tem- perature T . The system is paramagnetic when M > 0 3/2 σ 1 M 6 =t F [ 3, 0]+ x(gσ ) while diamagnetic when M < 0. M can be derived from T =0 1 − − 2 Xσ the thermodynamic potential by the standard procedure 1 ∂Ω M 6 = , which yields σ σ T =0 − V ∂B T,V F1 [ 1, 0] xF2 [ 1, 1] . (11) × − − − ~ ∗ 3/2 q m σ In following discussions, ω is used to indicate the magni- M 6 = F [ 3, 0] T =0 m∗c 2πβ~2 1 − tude of magnetic field since it is proportional to B. Xσ 1 + x(gσ )F σ[ 1, 0] xF σ[ 1, 1] . (9) − 2 1 − − 2 − III. RESULTS AND DISCUSSIONS For carrying out numerical calculations, it is useful 1 First, we look at a spin- 2 Fermi gas, setting σ = 1 to make the parameters dimensionless, such as M = to present the two Zeeman levels. The dimensionless± ∗ ~ ~ ∗ ∗ ∗ m cM/(n q), ω = ω/(kBT ), and t = T/T . Here T magnetization density M as a function of g is shown in is the characteristic temperature of the system, which is 2 Fig. 1(a). As expected, M is negative in the small g ∗ 2 ∗ given by kB T = 2π~ n 3 /m . Then we have x = ω/t, ′ ′ ∗ region, which means that the diamagnetism dominates. ησ =1/2 µ /ω gσ and µ = µ/(kBT ) in the notation For each given value of ω, M grows with g and changes σ − − Fτ [α, δ]. Equations (8) and (9) can be re-expressed as its sign from negative to positive in the larger g region, indicating that the paramagnetic effect is enhanced due 1= ωt1/2 F σ[ 1, 0] (10) to increase of g. This phenomenon is also observed in the 1 − Xσ Bose system[12]. gc is just the value of g where M = 0. 3 0.50 = 10 0.49 0.45 0.42 = 5 c c g g = 20 0.40 = 10 0.35 = 5 0.28 0.35 0 5 10 15 20 0 5 10 15 20 1/t 1/t FIG. 2: Plots of the critical value of g-factor as a function of FIG. 3: The gc 1/t curves for charged spin-1 gases obey- 1/t. The magnetic field is chosen as ω = 20 (solid line), 10 ing the Bose-Einstein− (BE, dotted line), Maxwell-Boltzmann (dashed line) and 5 (dotted line), respectively. (MB, solid line) and Fermi-Dirac (FD, dashed line) statistics, respectively. The magnetic field is chosen as ω = 10 and 5. The total magnetization M shown in Fig. 1(a) con- 1 1 sists of both the paramagnetic and diamagnetic contri- where λ = hβ 2 /(2πm∗) 2 , z = eβ(µ−ǫσ) and the Fermi- butions. Figure 1(b) depicts the pure paramagnetic con- Dirac integral fn(z) is normally defined as tribution to M, which is calculated by M p = gm where m = n n− . Figure 1(c) plots the diamagnetic con- ∞ n−1 1 − 1 1 x tribution to M, M d = M M p. For each fixed value fn(z) −1 x dx, (14) − ≡ Γ(n) Z0 z e +1 of ω, the diamagnetization M d is slightly weakened with increasing g.
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