Fermi gas in harmonic oscillator potentials
低温甲醇洗技术及其在煤化工中的应用_赵鹏飞
CHEMICAL INDUSTRY AND ENGINEERING PROGRESS 2012年第31卷第11期・2442・化工进展低温甲醇洗技术及其在煤化工中的应用赵鹏飞，李水弟，王立志（中国神华煤制油化工有限公司北京工程分公司,北京100011）摘要：低温甲醇洗技术是利用冷甲醇作为溶剂，脱除CO2、H2S、COS等酸性气体的一种气体净化方法。
由于其具有优异的酸性气体选择性、吸收性以及运行稳定性等特点，已在石油化工、煤化工、化肥工业等领域得到了广泛的应用。
本文介绍了低温甲醇洗技术的特点及工业应用中存在的一些问题，综述了近年来低温甲醇洗技术在煤制甲醇、煤制合成氨、煤制天然气等煤化工领域的进展，并展望了低温甲醇洗技术的应用趋势：对于煤制甲醇，低温甲醇洗技术在合成气中CO2、H2S、COS等酸性气体的脱除具有独特优势；对于煤制合成氨，低温甲醇洗技术在粗煤气的净化和节能降耗方面具有重要意义；对于煤制天然气，低温甲醇洗技术在合成气脱除H2S、COS等净化处理方面具有很好的应用潜力。
最后指出今后应着重于开发具有独立自主知识产权的国际一流水平的大型低温甲醇洗技术。
关键词：低温甲醇洗; 煤化工; 甲醇; 合成氨; 合成天然气中图分类号：TQ 54 文献标志码：A 文章编号：1000–6613（2012）11–2442–07Rectisol technology and its application in coal chemical industryZHAO Pengfei，LI Shuidi，WANG Lizhi（Beijing Engineering Company, China Shenhua Coal to Liquid and Chemical Co., Ltd., Beijing 100011, China）Abstract：Rectisol technology licensed by both Linde AG and Lurgi AG is a gas purification method of removing acid gases including CO2，H2S，COS, by using cold methanol as solvent. Because of its excellent properties of selectivity, absorbency and operation stability, rectisol technology has been used in many fields, such as petrochemical, coal chemical, chemical fertilizer industries. In order to get comprehensive understanding of Rectisol technology, in this paper the performance characteristics of Rectisol technology and its existing problems in industrial applications are introduced. The progress in applications of rectisol technology is summarized in the coal chemical industry, especially in the area of coal to methanol, coal to synthetic ammonia and coal to synthetic natural gas. In addition, the development trends of rectisol technology are also proposed. As for coal to methanol, rectisol technology has unique advantage in removing acid gas from the synthesis gas. As for coal to synthetic ammonia, it has great significance for energy saving and consumption reduction. As for coal to synthetic natural gas, it has also a good application potential in purification process, such as removing H2S, COS from the synthesis gas. In the future, attentionshould be paid to developing China’s acid gas removal process with independent intellectual property right.Key words：rectisol; coal chemical industry; methanol; synthesizing ammonia; synthetic natural gas低温甲醇洗技术（rectisol）是以甲醇有机溶剂作为吸收剂，利用甲醇在低温条件下对CO2、H2S、COS等酸性气体溶解度大的物理特性，同时或分段脱除原料气中酸性气体的一种气体净化方法，已在石油工业、煤化工、化肥工业、城市煤气工业等领收稿日期：2012-04-06；收修改稿日期：2012-08-16。
Two-Dimensional Gas of Massless Dirac Fermions in Graphene
Two-Dimensional Gas of Massless Dirac Fermions in Graphene K.S. Novoselov1, A.K. Geim1, S.V. Morozov2, D. Jiang1, M.I. Katsnelson3, I.V. Grigorieva1, S.V. Dubonos2, A.A. Firsov21Manchester Centre for Mesoscience and Nanotechnology, University of Manchester, Manchester, M13 9PL, UK2Institute for Microelectronics Technology, 142432, Chernogolovka, Russia3Institute for Molecules and Materials, Radboud University of Nijmegen, Toernooiveld 1, 6525 ED Nijmegen, the NetherlandsElectronic properties of materials are commonly described by quasiparticles that behave as nonrelativistic electrons with a finite mass and obey the Schrödinger equation. Here we report a condensed matter system where electron transport is essentially governed by the Dirac equation and charge carriers mimic relativistic particles with zero mass and an effective “speed of light” c∗ ≈106m/s. Our studies of graphene – a single atomic layer of carbon – have revealed a variety of unusual phenomena characteristic of two-dimensional (2D) Dirac fermions. In particular, we have observed that a) the integer quantum Hall effect in graphene is anomalous in that it occurs at halfinteger filling factors; b) graphene’s conductivity never falls below a minimum value corresponding to the conductance quantum e2/h, even when carrier concentrations tend to zero; c) the cyclotron mass mc of massless carriers with energy E in graphene is described by equation E =mcc∗2; and d) Shubnikov-de Haas oscillations in graphene exhibit a phase shift of π due to Berry’s phase.Graphene is a monolayer of carbon atoms packed into a dense honeycomb crystal structure that can be viewed as either an individual atomic plane extracted from graphite or unrolled single-wall carbon nanotubes or as a giant flat fullerene molecule. This material was not studied experimentally before and, until recently [1,2], presumed not to exist. To obtain graphene samples, we used the original procedures described in [1], which involve micromechanical cleavage of graphite followed by identification and selection of monolayers using a combination of optical, scanning-electron and atomic-force microscopies. The selected graphene films were further processed into multi-terminal devices such as the one shown in Fig. 1, following standard microfabrication procedures [2]. Despite being only one atom thick and unprotected from the environment, our graphene devices remain stable under ambient conditions and exhibit high mobility of charge carriers. Below we focus on the physics of “ideal” (single-layer) graphene which has a different electronic structure and exhibits properties qualitatively different from those characteristic of either ultra-thin graphite films (which are semimetals and whose material properties were studied recently [2-5]) or even of our other devices consisting of just two layers of graphene (see further). Figure 1 shows the electric field effect [2-4] in graphene. Its conductivity σ increases linearly with increasing gate voltage Vg for both polarities and the Hall effect changes its sign at Vg ≈0. This behaviour shows that substantial concentrations of electrons (holes) are induced by positive (negative) gate voltages. Away from the transition region Vg ≈0, Hall coefficient RH = 1/ne varies as 1/Vg where n is the concentration of electrons or holes and e the electron charge. The linear dependence 1/RH ∝Vg yields n =α·Vg with α ≈7.3·1010cm-2/V, in agreement with the theoretical estimate n/Vg ≈7.2·1010cm-2/V for the surface charge density induced by the field effect (see Fig. 1’s caption). The agreement indicates that all the induced carriers are mobile and there are no trapped charges in graphene. From the linear dependence σ(Vg) we found carrier mobilities µ =σ/ne, whichreached up to 5,000 cm2/Vs for both electrons and holes, were independent of temperature T between 10 and 100K and probably still limited by defects in parent graphite. To characterise graphene further, we studied Shubnikov-de Haas oscillations (SdHO). Figure 2 shows examples of these oscillations for different magnetic fields B, gate voltages and temperatures. Unlike ultra-thin graphite [2], graphene exhibits only one set of SdHO for both electrons and holes. By using standard fan diagrams [2,3], we have determined the fundamental SdHO frequency BF for various Vg. The resulting dependence of BF as a function of n is plotted in Fig. 3a. Both carriers exhibit the same linear dependence BF = β·n with β ≈1.04·10-15 T·m2 (±2%). Theoretically, for any 2D system β is defined only by its degeneracy f so that BF =φ0n/f, where φ0 =4.14·10-15 T·m2 is the flux quantum. Comparison with the experiment yields f =4, in agreement with the double-spin and double-valley degeneracy expected for graphene [6,7] (cf. caption of Fig. 2). Note however an anomalous feature of SdHO in graphene, which is their phase. In contrast to conventional metals, graphene’s longitudinal resistance ρxx(B) exhibits maxima rather than minima at integer values of the Landau filling factor ν (Fig. 2a). Fig. 3b emphasizes this fact by comparing the phase of SdHO in graphene with that in a thin graphite film [2]. The origin of the “odd” phase is explained below. Another unusual feature of 2D transport in graphene clearly reveals itself in the T-dependence of SdHO (Fig. 2b). Indeed, with increasing T the oscillations at high Vg (high n) decay more rapidly. One can see that the last oscillation (Vg ≈100V) becomes practically invisible already at 80K whereas the first one (Vg <10V) clearly survives at 140K and, in fact, remains notable even at room temperature. To quantify this behaviour we measured the T-dependence of SdHO’s amplitude at various gate voltages and magnetic fields. The results could be fitted accurately (Fig. 3c) by the standard expression T/sinh(2π2kBTmc/heB), which yielded mc varying between ≈ 0.02 and 0.07m0 (m0 is the free electron mass). Changes in mc are well described by a square-root dependence mc ∝n1/2 (Fig. 3d). To explain the observed behaviour of mc, we refer to the semiclassical expressions BF = (h/2πe)S(E) and mc =(h2/2π)∂S(E)/∂E where S(E) =πk2 is the area in k-space of the orbits at the Fermi energy E(k) [8]. Combining these expressions with the experimentally-found dependences mc ∝n1/2 and BF =(h/4e)n it is straightforward to show that S must be proportional to E2 which yields E ∝k. Hence, the data in Fig. 3 unambiguously prove the linear dispersion E =hkc∗ for both electrons and holes with a common origin at E =0 [6,7]. Furthermore, the above equations also imply mc =E/c∗2 =(h2n/4πc∗2)1/2 and the best fit to our data yields c∗ ≈1⋅106 m/s, in agreement with band structure calculations [6,7]. The employed semiclassical model is fully justified by a recent theory for graphene [9], which shows that SdHO’s amplitude can indeed be described by the above expression T/sinh(2π2kBTmc/heB) with mc =E/c∗2. Note that, even though the linear spectrum of fermions in graphene (Fig. 3e) implies zero rest mass, their cyclotron mass is not zero. The unusual response of massless fermions to magnetic field is highlighted further by their behaviour in the high-field limit where SdHO evolve into the quantum Hall effect (QHE). Figure 4 shows Hall conductivity σxy of graphene plotted as a function of electron and hole concentrations in a constant field B. Pronounced QHE plateaux are clearly seen but, surprisingly, they do not occur in the expected sequence σxy =(4e2/h)N where N is integer. On the contrary, the plateaux correspond to half-integer ν so that the first plateau occurs at 2e2/h and the sequence is (4e2/h)(N + ½). Note that the transition from the lowest hole (ν =–½) to lowest electron (ν =+½) Landau level (LL) in graphene requires the same number of carriers (∆n =4B/φ0 ≈1.2·1012cm-2) as the transition between other nearest levels (cf. distances between minima in ρxx). This results in a ladder of equidistant steps in σxy which are not interrupted when passing through zero. To emphasize this highly unusual behaviour, Fig. 4 also shows σxy for a graphite film consisting of only two graphene layers where the sequence of plateaux returns to normal and the first plateau is at 4e2/h, as in the conventional QHE. We attribute this qualitative transition between graphene and its two-layer counterpart to the fact that fermions in the latter exhibit a finite mass near n ≈0 (as found experimentally; to be published elsewhere) and can no longer be described as massless Dirac particles. 2The half-integer QHE in graphene has recently been suggested by two theory groups [10,11], stimulated by our work on thin graphite films [2] but unaware of the present experiment. The effect is single-particle and intimately related to subtle properties of massless Dirac fermions, in particular, to the existence of both electron- and hole-like Landau states at exactly zero energy [912]. The latter can be viewed as a direct consequence of the Atiyah-Singer index theorem that plays an important role in quantum field theory and the theory of superstrings [13,14]. For the case of 2D massless Dirac fermions, the theorem guarantees the existence of Landau states at E=0 by relating the difference in the number of such states with opposite chiralities to the total flux through the system (note that magnetic field can also be inhomogeneous). To explain the half-integer QHE qualitatively, we invoke the formal expression [9-12] for the energy of massless relativistic fermions in quantized fields, EN =[2ehc∗2B(N +½ ±½)]1/2. In QED, sign ± describes two spins whereas in the case of graphene it refers to “pseudospins”. The latter have nothing to do with the real spin but are “built in” the Dirac-like spectrum of graphene, and their origin can be traced to the presence of two carbon sublattices. The above formula shows that the lowest LL (N =0) appears at E =0 (in agreement with the index theorem) and accommodates fermions with only one (minus) projection of the pseudospin. All other levels N ≥1 are occupied by fermions with both (±) pseudospins. This implies that for N =0 the degeneracy is half of that for any other N. Alternatively, one can say that all LL have the same “compound” degeneracy but zeroenergy LL is shared equally by electrons and holes. As a result the first Hall plateau occurs at half the normal filling and, oddly, both ν = –½ and +½ correspond to the same LL (N =0). All other levels have normal degeneracy 4B/φ0 and, therefore, remain shifted by the same ½ from the standard sequence. This explains the QHE at ν =N + ½ and, at the same time, the “odd” phase of SdHO (minima in ρxx correspond to plateaux in ρxy and, hence, occur at half-integer ν; see Figs. 2&3), in agreement with theory [9-12]. Note however that from another perspective the phase shift can be viewed as the direct manifestation of Berry’s phase acquired by Dirac fermions moving in magnetic field [15,16]. Finally, we return to zero-field behaviour and discuss another feature related to graphene’s relativistic-like spectrum. The spectrum implies vanishing concentrations of both carriers near the Dirac point E =0 (Fig. 3e), which suggests that low-T resistivity of the zero-gap semiconductor should diverge at Vg ≈0. However, neither of our devices showed such behaviour. On the contrary, in the transition region between holes and electrons graphene’s conductivity never falls below a well-defined value, practically independent of T between 4 and 100K. Fig. 1c plots values of the maximum resistivity ρmax(B =0) found in 15 different devices, which within an experimental error of ≈15% all exhibit ρmax ≈6.5kΩ, independent of their mobility that varies by a factor of 10. Given the quadruple degeneracy f, it is obvious to associate ρmax with h/fe2 =6.45kΩ where h/e2 is the resistance quantum. We emphasize that it is the resistivity (or conductivity) rather than resistance (or conductance), which is quantized in graphene (i.e., resistance R measured experimentally was not quantized but scaled in the usual manner as R =ρL/w with changing length L and width w of our devices). Thus, the effect is completely different from the conductance quantization observed previously in quantum transport experiments. However surprising, the minimum conductivity is an intrinsic property of electronic systems described by the Dirac equation [17-20]. It is due to the fact that, in the presence of disorder, localization effects in such systems are strongly suppressed and emerge only at exponentially large length scales. Assuming the absence of localization, the observed minimum conductivity can be explained qualitatively by invoking Mott’s argument [21] that mean-free-path l of charge carriers in a metal can never be shorter that their wavelength λF. Then, σ =neµ can be re-written as σ = (e2/h)kFl and, hence, σ cannot be smaller than ≈e2/h per each type of carriers. This argument is known to have failed for 2D systems with a parabolic spectrum where disorder leads to localization and eventually to insulating behaviour [17,18]. For the case of 2D Dirac fermions, no localization is expected [17-20] and, accordingly, Mott’s argument can be used. Although there is a broad theoretical consensus [18-23,10,11] that a 2D gas of Dirac fermions should exhibit a minimum 3conductivity of about e2/h, this quantization was not expected to be accurate and most theories suggest a value of ≈e2/πh, in disagreement with the experiment. In conclusion, graphene exhibits electronic properties distinctive for a 2D gas of particles described by the Dirac rather than Schrödinger equation. This 2D system is not only interesting in itself but also allows one to access – in a condensed matter experiment – the subtle and rich physics of quantum electrodynamics [24-27] and provides a bench-top setting for studies of phenomena relevant to cosmology and astrophysics [27,28].1. Novoselov, K.S. et al. PNAS 102, 10451 (2005). 2. Novoselov, K.S. et al. Science 306, 666 (2004); cond-mat/0505319. 3. Zhang, Y., Small, J.P., Amori, M.E.S. & Kim, P. Phys. Rev. Lett. 94, 176803 (2005). 4. Berger, C. et al. J. Phys. Chem. B, 108, 19912 (2004). 5. Bunch, J.S., Yaish, Y., Brink, M., Bolotin, K. & McEuen, P.L. Nanoletters 5, 287 (2005). 6. Dresselhaus, M.S. & Dresselhaus, G. Adv. Phys. 51, 1 (2002). 7. Brandt, N.B., Chudinov, S.M. & Ponomarev, Y.G. Semimetals 1: Graphite and Its Compounds (North-Holland, Amsterdam, 1988). 8. Vonsovsky, S.V. and Katsnelson, M.I. Quantum Solid State Physics (Springer, New York, 1989). 9. Gusynin, V.P. & Sharapov, S.G. Phys. Rev. B 71, 125124 (2005). 10. Gusynin, V.P. & Sharapov, S.G. cond-mat/0506575. 11. Peres, N.M.R., Guinea, F. & Castro Neto, A.H. cond-mat/0506709. 12. Zheng, Y. & Ando, T. Phys. Rev. B 65, 245420 (2002). 13. Kaku, M. Introduction to Superstrings (Springer, New York, 1988). 14. Nakahara, M. Geometry, Topology and Physics (IOP Publishing, Bristol, 1990). 15. Mikitik, G. P. & Sharlai, Yu.V. Phys. Rev. Lett. 82, 2147 (1999). 16. Luk’yanchuk, I.A. & Kopelevich, Y. Phys. Rev. Lett. 93, 166402 (2004). 17. Abrahams, E., Anderson, P.W., Licciardello, D.C. & Ramakrishnan, T.V. Phys. Rev. Lett. 42, 673 (1979). 18. Fradkin, E. Phys. Rev. B 33, 3263 (1986). 19. Lee, P.A. Phys. Rev. Lett. 71, 1887 (1993). 20. Ziegler, K. Phys. Rev. Lett. 80, 3113 (1998). 21. Mott, N.F. & Davis, E.A. Electron Processes in Non-Crystalline Materials (Clarendon Press, Oxford, 1979). 22. Morita, Y. & Hatsugai, Y. Phys. Rev. Lett. 79, 3728 (1997). 23. Nersesyan, A.A., Tsvelik, A.M. & Wenger, F. Phys. Rev. Lett. 72, 2628 (1997). 24. Rose, M.E. Relativistic Electron Theory (John Wiley, New York, 1961). 25. Berestetskii, V.B., Lifshitz, E.M. & Pitaevskii, L.P. Relativistic Quantum Theory (Pergamon Press, Oxford, 1971). 26. Lai, D. Rev. Mod. Phys. 73, 629 (2001). 27. Fradkin, E. Field Theories of Condensed Matter Systems (Westview Press, Oxford, 1997). 28. Volovik, G.E. The Universe in a Helium Droplet (Clarendon Press, Oxford, 2003).Acknowledgements This research was supported by the EPSRC (UK). We are most grateful to L. Glazman, V. Falko, S. Sharapov and A. Castro Netto for helpful discussions. K.S.N. was supported by Leverhulme Trust. S.V.M., S.V.D. and A.A.F. acknowledge support from the Russian Academy of Science and INTAS.43µ (m2/Vs)0.8c4P0.4 22 σ (1/kΩ)10K0 0 1/RH(T/kΩ) 1 2ρmax (h/4e2)1-5010 Vg (V) 50 -10ab 0 -100-500 Vg (V)50100Figure 1. Electric field effect in graphene. a, Scanning electron microscope image of one of our experimental devices (width of the central wire is 0.2µm). False colours are chosen to match real colours as seen in an optical microscope for larger areas of the same materials. Changes in graphene’s conductivity σ (main panel) and Hall coefficient RH (b) as a function of gate voltage Vg. σ and RH were measured in magnetic fields B =0 and 2T, respectively. The induced carrier concentrations n are described by [2] n/Vg =ε0ε/te where ε0 and ε are permittivities of free space and SiO2, respectively, and t ≈300 nm is the thickness of SiO2 on top of the Si wafer used as a substrate. RH = 1/ne is inverted to emphasize the linear dependence n ∝Vg. 1/RH diverges at small n because the Hall effect changes its sign around Vg =0 indicating a transition between electrons and holes. Note that the transition region (RH ≈ 0) was often shifted from zero Vg due to chemical doping [2] but annealing of our devices in vacuum normally allowed us to eliminate the shift. The extrapolation of the linear slopes σ(Vg) for electrons and holes results in their intersection at a value of σ indistinguishable from zero. c, Maximum values of resistivity ρ =1/σ (circles) exhibited by devices with different mobilites µ (left y-axis). The histogram (orange background) shows the number P of devices exhibiting ρmax within 10% intervals around the average value of ≈h/4e2. Several of the devices shown were made from 2 or 3 layers of graphene indicating that the quantized minimum conductivity is a robust effect and does not require “ideal” graphene.ρxx (kΩ)0.60 aVg = -60V4B (T)810K12∆σxx (1/kΩ)0.4 1ν=4 140K 80K B =12T0 b 0 25 50 Vg (V) 7520K100Figure 2. Quantum oscillations in graphene. SdHO at constant gate voltage Vg as a function of magnetic field B (a) and at constant B as a function of Vg (b). Because µ does not change much with Vg, the constant-B measurements (at a constant ωcτ =µB) were found more informative. Panel b illustrates that SdHO in graphene are more sensitive to T at high carrier concentrations. The ∆σxx-curves were obtained by subtracting a smooth (nearly linear) increase in σ with increasing Vg and are shifted for clarity. SdHO periodicity ∆Vg in a constant B is determined by the density of states at each Landau level (α∆Vg = fB/φ0) which for the observed periodicity of ≈15.8V at B =12T yields a quadruple degeneracy. Arrows in a indicate integer ν (e.g., ν =4 corresponds to 10.9T) as found from SdHO frequency BF ≈43.5T. Note the absence of any significant contribution of universal conductance fluctuations (see also Fig. 1) and weak localization magnetoresistance, which are normally intrinsic for 2D materials with so high resistivity.75 BF (T) 500.2 0.11/B (1/T)b5 10 N 1/2025 a 0 0.061dmc /m00.04∆0.02 0c0 0 T (K) 150n =0e-6-3036Figure 3. Dirac fermions of graphene. a, Dependence of BF on carrier concentration n (positive n correspond to electrons; negative to holes). b, Examples of fan diagrams used in our analysis [2] to find BF. N is the number associated with different minima of oscillations. Lower and upper curves are for graphene (sample of Fig. 2a) and a 5-nm-thick film of graphite with a similar value of BF, respectively. Note that the curves extrapolate to different origins; namely, to N = ½ and 0. In graphene, curves for all n extrapolate to N = ½ (cf. [2]). This indicates a phase shift of π with respect to the conventional Landau quantization in metals. The shift is due to Berry’s phase [9,15]. c, Examples of the behaviour of SdHO amplitude ∆ (symbols) as a function of T for mc ≈0.069 and 0.023m0; solid curves are best fits. d, Cyclotron mass mc of electrons and holes as a function of their concentration. Symbols are experimental data, solid curves the best fit to theory. e, Electronic spectrum of graphene, as inferred experimentally and in agreement with theory. This is the spectrum of a zero-gap 2D semiconductor that describes massless Dirac fermions with c∗ 300 times less than the speed of light.n (1012 cm-2)σxy (4e2/h)4 3 2 -2 1 -1 -2 -3 2 44Kn7/ 5/ 3/ 1/2 2 2 210 ρxx (kΩ)-4σxy (4e2/h)0-1/2 -3/2 -5/2514T0-7/2 -4 -2 0 2 4 n (1012 cm-2)Figure 4. Quantum Hall effect for massless Dirac fermions. Hall conductivity σxy and longitudinal resistivity ρxx of graphene as a function of their concentration at B =14T. σxy =(4e2/h)ν is calculated from the measured dependences of ρxy(Vg) and ρxx(Vg) as σxy = ρxy/(ρxy + ρxx)2. The behaviour of 1/ρxy is similar but exhibits a discontinuity at Vg ≈0, which is avoided by plotting σxy. Inset: σxy in “two-layer graphene” where the quantization sequence is normal and occurs at integer ν. The latter shows that the half-integer QHE is exclusive to “ideal” graphene.。
基于金纳米粒子局域表面等离子体共振吸收检测卡托普利
基于金纳米粒子局域表面等离子体共振吸收检测卡托普利X许 丹(西南大学化学化工学院,重庆 400715) 摘 要:柠檬酸根稳定的金胶在一定盐浓度下由于盐的电荷屏蔽效应而发生聚集。
加入一定浓度的卡托普利后,由于卡托普利分子中含有巯基和羧基,其分子中的巯基可以通过Au -S 键连在金纳米粒子表面,同时,在pH9.91的条件下,其分子中的羧基去质子化形成-COO -,导致金胶表面负电荷增多,纳米粒子之间的静电排斥力增大,金胶的聚集得到了抑制。
基于金胶由聚集到分散的现象,利用紫外-可见吸收光谱进行表征,建立了定量检测卡托普利含量的方法。
该方法的线性范围为0.04～1.2L M ,检出限为20nM 。
将此方法用于合成样的检测,回收率在86.3%～108.2%之间。
关键词:金纳米粒子;卡托普利;表面等离子体共振吸收 中图分类号:T Q460.7 文献标识码:A 文章编号:1006—7981(2012)03—0006—03 卡托普利(1-[(2S )-2-甲基-3-巯基-1-氧化丙基]-L -脯氨酸,Cap,结构如下图所示)是一种人工合成的血管紧张素转换酶抑制剂,目前广泛应用于治疗高血压及心力衰竭等疾病。
目前,定量测定卡托普利的分析方法有高效液相色谱法[1]、氧化还原滴定法[2]及化学发光法[3]等。
这些方法虽然灵敏度较高,但存在一些缺点如仪器设备昂贵、操作繁琐等。
因此,建立简便、快速、灵敏度高的检测卡托普利的方法仍然具有十分重要的意义。
图1 实验原理图近年来,金纳米粒子由于其独特的表面等离子体共振吸收性质被广泛用于色度传感。
13nm 柠檬酸根包被的金胶呈现酒红色,一旦发生聚集则呈现紫色或蓝色[4]。
这种颜色的改变很容易用肉眼捕获,不需要复杂的仪器。
金胶在一定浓度的NaCl 溶液中发生聚集,在本研究中,我们发现当体系中同时存在卡托普利后,金胶的聚集能得到抑制,基于此现象,建立了一种金胶由聚集到分散的状态来高灵敏检测卡托普利的新方法。
湿法磷酸选择性除杂制工业级磷酸二氢铵
Vol.53 No.4Apr.,2021第 53 卷 第 4 期2021 年 4 月无机盐工业INORGANIC CHEMICALS INDUSTRY湿法磷酸选择性除杂制工业级磷酸二氢铵王智娟（曲靖师范学院化学与环境科学学院，云南曲靖655000）摘 要：以湿法磷酸为原料，通过氟化钠选择性沉淀金属离子，使其以NaMgAl （F,OH ）6-H 2O 和XMgAlF e （X 二Na +、K +、NH 4+）非含磷沉淀析岀,再通氨中和制得工业级磷酸二氢铵。
分析了氟化钠加入量对杂质脱除（Na 、K 、Al 、Mg 、 Fe 和Ca 等金属阳离子）及液相氟残留的影响，结果表明氟化钠与磷酸质量比（m NaF /叫陀）为2.5%时，效果较好，此条件下制得的磷酸二氢铵纯度和五氧化二磷收率分别达到98.53%和86.2%o关键词:湿法磷酸;磷酸二氢铵；除杂中图分类号:TQ126.35 文献标识码:A 文章编号：1006-4990（2021）04-0048-04Fabrication of industrial grade ammonium dihydrogen phosphate via selective removal ofimpurities from wet-process phosphoric acidWang Zhijuan(Faculty of Chemistry and Environment Science , Qujing Normal University ,Qujing 655000, China)Abstract : Ammonium dihydrogen phosphate with high purity can be used as water-soluble fertilizer ,flame retardant ,feed andfood additives and recently it has been studied for preparation of phosphate optical glass , cathode materials for lithium batter ies and nonlinear optical materials.Due to its broad application prospect in the fields of agriculture , fire protection , food and materials , the demand for high purity ammonium dihydrogen phosphate is increasing.High purity ammonium dihydrogen phosphate can be fabricated via thermal phosphoric acid and extraction-purified wet-process phosphoric acid route.The former route is simple and can get high quality of product , but it is restricted to the influence of resources , energy and environment.While the latter route is complex and needs large investment.Hence , more researchers focus on the direct preparing high puri ty ammonium dihydrogen phosphate with raw wet-process phosphoric acid.But the lower P 2O 5 yield of product (25%~40%) inhibited its development due to the formation of a large amount of phosphorus-containing precipitates (e.g., metallic phospha te ) during ammoniation.Therefore , herein modified-process with sodium fluoride was used to selectively precipitate metalcations to compounds containing no phosphorus (e.g., (NH 4)x (Mg )y (Al )z F 6(OH )6・2H 2O ,NaMgAlF 6送出。
蒽醌／氢给体／氮氧自由的瞬态电子自旋极化
本文 利 用 时 间分 辨 Ｅ Ｒ波谱 仪 研 究 了光解 葸 Ｓ 醌 自由基 的化学诱 导动 态 电子极 化 ．考察 了加入 氮 氧 自由基后 体系极 化信号 的变 化情 况 ，探 讨 了此 种
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费米能级位置
什么是Fermi能级？为什么Fermi能级可以处于禁带中间？为什么本征半导体的Fermi能级位于禁带中央？为什么n型半导体的Fermi能级位于导带底附近？Fermi能级随着温度和掺杂浓度的改变而如何变化？Fermi能级（E F）是一个非常重要的物理概念，它在半导体电子学中起着极其重要的作用。
（1）Fermi能级的概念：在固体物理学中，Fermi能量(Fermi energy)是表示在无相互作用的Fermi粒子的体系中加入一个粒子所引起的基态能量的最小可能增量；也就是在绝对零度时，处于基态的Fermi粒子体系的化学势，或者是处于基态的单个Fermi粒子所具有的最大能量——Fermi粒子所占据的最高能级的能量。
另一方面，按照Fermi-Dirac统计，在能量为E的单电子量子态上的平均电子数为：式中的T为绝对温度，k为玻尔兹曼常数，E F是该Fermi-Dirac 分布函数的一个参量（称为化学势）。
在绝对零度下，所有能量小于E F的量子态都被电子占据，而所有能量大于E F的量子态都是空着的，则作为化学势的参量E F就是电子所占据的最高量子态的能量，因此这时系统的化学势也就与费米能量一致。
从而，往往就形象地把费米能量和化学势统称之为Fermi能级。
虽然严格说来，费米能级是指无相互作用的Fermi粒子系统在趋于绝对零度时的化学势，但是在半导体物理电子学领域中，费米能级则经常被当做电子或空穴的化学势来使用，所以也就不再区分费米能级和化学势了。
在非绝对零度时，电子可以占据高于E F的若干能级，则这时Fermi 能级将是占据几率等于50%的能级。
处于Fermi能级附近的电子（常称为传导电子）对固体的输运性质起着重要的作用。
（2）Fermi能级的含义：作为Fermi-Dirac分布函数中一个重要参量的Fermi能级EF，具有决定整个系统能量以及载流子分布的重要作用。
①在半导体中，由于Fermi能级（化学势）不是真正的能级，即不一定是允许的单电子能级（即不一定是公有化状态的能量），所以它可以像束缚状态的能级一样，可以处于能带的任何位置，当然也可以处于禁带之中。
升降算符在一维谐振子能级讨论中的应用
第36卷第12期2020年12月商丘师范学院学报JOURNAL OF SHANGQIU NORMAL UNIVERSITY Vol．36No．12Dec．2020收稿日期：2019－12－18基金项目：国家自然科学基金资助项目（51141009）；佳木斯大学科学技术研究面上项目（L2012－045）；黑龙江省省属本科高校基本科研业务费科研项目资助（2018－KYYWF －0956）；佳木斯大学大学生创新创业训练计划项目（2019xj25）作者简介：刘海宽（1998—），男，黑龙江同江人，佳木斯大学在读本科生，主要从事量子物理研究．通讯作者：马佳（1978—），女，黑龙江佳木斯人，佳木斯大学讲师，主要从事理论物理研究．升降算符在一维谐振子能级讨论中的应用刘海宽，张海丰，马佳（佳木斯大学理学院，黑龙江佳木斯，154007）摘要：利用升降算符方法研究了一维谐振子的能级分布问题，给出了占有数表象升降算符^a+、^a 作用下的线性谐振子本征波函数的递推关系，最后利用升降算符讨论了线性谐振子本征波函数的代数表示形式．结果表明升降算符方法在研究谐振子的能级分布和本征波函数方面简洁易懂．关键词：升降算符方法；一维谐振子；能级；本征波函数中图分类号：O431文献标识码：A文章编号：1672－3600（2020）12－0022－05Application of the raising and lowering operators in the discussion of one-dimensionalharmonic oscillator energy levels LIU Hɑikuɑn ，ZHANG Hɑifenɡ，MA Jiɑ（School of Science ，Jiɑmusi University ，Jiɑmusi 154007，China ）Abstract ：Using the raising and lowering operator method ，the energy level distribution of a one-dimensional harmonic oscillator is studied ，and the recurrence relation of the eigenwave function of a linear harmonic oscillator under the action of the elevating operator is given．Finally ，the algebraic representation of the eigenwave function of a linear harmonic oscillator is discussed．The results show that the method is concise and easy to understand in studying the energy level distribution and eigenwave function of harmonic oscillator．Key words ：raising and lowering operator method ；one-dimensional harmonic oscillator ；energy levels ；eigenwave function在量子力学中，谐振子模型是非常典型的物理模型之一，被广泛利用和研究，例如：肖奎等对一维线性谐振子波函数及概率分布的可视化进行分析［1］；张小伟分析了电场中线性谐振子本征值问题［2］；呼和满都拉等简单分析了一维谐振子升降算符的性质并对电子自旋角动量的升降算符进行了研究［3－5］；田杏霞等结合角动量升降算符分析矩阵表示和表象变换性质［6］；梁霄利用代数解法对升降算符进行了研究［7］；寻大毛研究了轨道角动量的升降算符［8］；王东方等在电子自旋和角动量耦合问题研究中使用了升降算符［9］；寻大毛等对轨道角动量量子数的升降算符进行了研究［10］．分析文献可以看出，对谐振子本征值问题求解多采用微分方程的解法，由于数学推导繁琐，理解起来较为困难，而占有数表象升降算符^a+、^a 在很多问题的研究中被广泛的应用，比较容易理解，所以本文旨在利用升降算符的方法研究谐振子的能量和波函数等．1用升降算符表示的哈密顿算符一维谐振子的哈密顿算符可以表示为H ^=12m ^p 2+12m ω2x 2（1）坐标和动量算符满足对易关系［x ，^p］=i （2）令Q ^=m ω槡x （3a ）P ^=1m 槡ω ^p （3b ）^a=1槡2（Q ^+iP ^）（4a ）^a+=1槡2（Q ^－iP ^）（4b ）则可以得到^a ，^a +，Q ^，P ^各个量之间的对易关系，进而将H ^利用^a ，^a +表示出来，并可以在H ^的表象中给出^a，^a +，Q ^，P ^，x ，^p 的矩阵元．根据式（2），易得［Q ^，P ^］=1［x ，p ］=i （5）［^a ，^a +］=12［Q ^+iP ^，Q ^－iP ^］=－i ［Q ^，P ^］=1（6）［^a ，^a +，^a ］=［^a ，^a +］^a =^a （7）［^a +，^a +，^a ］=^a +［^a +，^a ］=－^a +（8）由式（4）可得^a+^a =12（Q ^－iP ^）（Q ^+iP ^）=12（Q ^2+P ^2）+i 2（Q ^P ^－P ^Q ^）=12（Q ^2+P ^2－1）（9）所以H ^算符改写为H ^=p 22m +12m ω2x 2=12 ω（P ^2+Q ^2）= ω^a+^a +()12（10）2能量本征值的推导取任意波函数|ψ〉，在其上取^a+^a 的内积可得〈ψ|^a +^a |ψ〉=|^a |ψ〉|2≥0（11）所以H ^算符在|ψ〉上的内积为〈H ^〉=〈ψ|H ^|ψ〉≥12ω（12）当|ψ〉恰好为H ^算符的本征态时，则〈H ^〉=E ，则式（12）变为E ≥12 ω（13）说明能量值的最小值为12ω．按照式（7）和改写后的H ^算符易知［^a ，H ^］=［^a ，^a +，^a ］ ω=^aω（14）^a（H ^－ ω）=H ^^a （15）取本征矢|E'〉，并将其被算式（15）作用易得H ^^a|E'〉=^a （H ^－ ω）|E'〉=（E'－ ω）^a |E'〉（16）可见^a |E'〉也是H ^算符的本征矢，能量值为E'－ ω，于是可以得到由E'表示的等间距的各个能级E'，E'－ ω，E'－2 ω，…（17）对于基态的能量，根据式（10）H ^|E 0〉= ω^a+^a +()12|E 0〉= ω2|E 0〉（18）易得E 0= ω/2（19）根据式（8）和式（10）可得H ^^a+=^a +（H ^+ ω）（20）32第12期刘海宽，等：升降算符在一维谐振子能级讨论中的应用H ^^a+|E'〉=（E'+ ω）^a +|E'〉（21）可见即^a +|E'〉亦是哈密顿算符H ^的本征矢，相应的能量为E'+ ω，于是同样可以得到由E'表示的等间距的各个能级的另一种形式E'，E'+ ω，E'+2 ω， (22)另外，由式（6）可以对式（10）做进一步的处理为H ^= ω^a^a +－()12（23）当将式（23）作用在|E'〉上，易知E'≥ω2，可见式（23）中E'的取值没有最大值，所以总的能级分布为E n =n +()12ω，n =0，1，2， (24)3能量本征波函数的推导下面x 表象中利用升降算符^a+，^a 求解谐振子的本征函数．设谐振子基态波函数为ψ0（x ），由于^a |0〉=0（25）根据式（3）和式（4）可得（i ^p+m ωx ）ψ0（x ）= d d x+m ω()x ψ0（x ）=0（26）令α=m ω槡，则式（26）变为d d x ψ0+α2x ψ0=0（27）解得ψ0（x ）=N 0e－α2x 22（28）式中N 0是归一化常数，按照波函数的归一性∫+ɕ－ɕ|ψ0（x ）|2d x =1（29）所以N 0=槡απ14（30）另外，由于^a+|0〉=|1〉（31）则有ψ1（x ）=^a+ψ0（x ）=1槡2αx －1αd d ()x ψ0（x ）（32）所以ψ1槡=2αx ψ0=2槡απ1/4αxe－α2x 2/2（33）ψn =1槡n ^a +x ψn －1=12槡n αx －1αd d ()x ψn －1（34）令ξ=αx（35）则ψn =12槡n ξ－d d ()ξψn －1（36）所以利用基态波函数式（28）可以得到ψn =12nn ()！12ξ－d d ()ξnψ0=απ2n n 槡()！12ξ－d d ()ξne －ξ22=N n H n （ξ）e －ξ22（37）式（37）中N n =απ2n n 槡()！12（38）42商丘师范学院学报2020年4升降算符作用下本征波函数递推关系推导取谐振子能量本征矢为|n 〉，则正交归一化条件为〈n'|n 〉=δn'n（39）由于升降算符对|n 〉的作用满足^a|n 〉=λ（n ）|n －1〉（40a ）^a+|n 〉=υ（n ）|n +1〉（40b ）式中λ、υ待定，所以式（40）的共轭为〈n |^a +=λ*（n ）〈n －1|（41a ）〈n |^a =υ*（n ）〈n +1|（41b ）所以由式（40）和式（41）可得λ*λ=〈n |^a+^a |n 〉=n （42a ）υ*υ=〈n |^a+^a |n 〉=〈n |（^a +^a +1）|n 〉=n +1（42b ）可以解得λ（n ）=槡n （43a ）υ（n ）=n 槡+1（43b ）亦即^a |n 〉=槡n |n －1〉（44a ）^a +|n 〉=n 槡+1|n +1〉（44b ）所以^a和^a +不为零的矩阵元^a n －1，n =〈n －1|^a |n 〉=槡n （45a ）^a +n +1，n =〈n +1|^a +|n 〉=n 槡+1（45b ）^a n'n =〈n'|^a |n 〉=槡n δn'，n －1（46a ）^a +nn'=〈n |^a +|n'〉=^a n'n =槡n δn ，n'+1（46b ）下边给出坐标算符和动量算符作用在能量本征态上的递推关系．由于Q ^=1槡2（^a++^a ）（47a ）P ^=i 槡2（^a +－^a ）（47b ）所以Q ^|n 〉=n +1槡2|n +1〉+n槡2|n －1〉（48）P ^|n 〉=in +1槡2|n +1〉－i n 槡2|n －1〉（49）Q ^和P ^的非零矩阵元表示为Q ^n +1，n =Q ^n ，n +1=n +1槡2（50a ）P ^n +1，n =－P ^n ，n +1=in +1槡2（50b ）同理，由于x =m 槡ωQ ^= 2m 槡ω（^a ++^a ）（51a ）^p =m 槡ω P ^=im ω 槡2（^a+－^a）（51b ）所以x |n 〉=2m 槡ω（n 槡+1|n +1〉+槡n |n －1〉）（52a ）^p|n 〉=i m ω 槡2（n 槡+1|n +1〉－槡n |n －1〉）（52b ）易知x 、^p 的非零的矩阵元为52第12期刘海宽，等：升降算符在一维谐振子能级讨论中的应用x n +1，n =x n ，n +1=n +12 m ()ω12（53a ）p n +1，n =－p n ，n +1=i n +12m ()ω 12（53b ）5本征波函数递推关系的应用下边应用谐振子的能量本征态|n 〉的递推关系求解算符x ，^p ，x 2，^p 2的期望值以及不确定度△x ，△p ．根据式（39）和（52）易知珋x =〈n |x |n 〉=0（54a ）珋p =〈n |珋p |n 〉=0（54b ）由式（51a ）（51b ）易知x 2=2m ω（^a ++^a ）2= 2m ω（^a 2+^a +2+2^n+1）（55）^p2= 2m ω（^a +－^a ）2= 2m ω（2^n +1－^a 2－^a +2）（56）根据式（52）易知x 2=〈n |x 2|n 〉=m ωn +()12（57）p 2=〈n |p 2|n 〉=m ω n +()12（58）所以△x =（x 2－珋x2）12=m ωn +()[]1212（59）△p =（p 2－珋p 2）12=m ω n +()[]1212（60）△x ·△p =n +()12（61）6结论首先用升降算符把哈密顿算符表示出来，接着在x 表象中利用升降算符^a +，^a 求解谐振子的本征函数，推导出升降算符作用下本征波函数递推关系，并将本征波函数递推关系的应用下边应用谐振子的能量本征态|n 〉的递推关系求解算符x ，^p ，x 2，^p 2的期望值以及不确定度△x ，△p ．参考文献：［1］肖奎，罗子安．一维线性谐振子波函数及概率分布的可视演示［J ］．企业科技与发展，2019（03）：161－162．［2］张小伟．关于电场中线性谐振子问题的求解［J ］．黑龙江科学，2017，8（10）：178－180．［3］呼和满都拉．一维谐振子升降算符的性质及应用［J ］．集宁师范学院学报，2015，37（03）：97－101．［4］呼和满都拉．电子自旋角动量的升降算符［J ］．哈尔滨师范大学自然科学学报，2015，31（04）：79－81．［5］郭华．电子自旋角动量升降算符研究［J ］．贵州工程应用技术学院学报，2015，33（03）：132－141．［6］田杏霞．应用角动量升降算符分析角动量的矩阵表示和表象变换［J ］．通化师范学院学报，2014，35（08）：35－36+50．［7］梁霄．量子力学中代数解法之若干升降算符［J ］．大学物理，2013，32（09）：23－27．［8］寻大毛．轨道角动量量子数的升降算符［D ］．长沙：湖南大学，2010，6．［9］王东方．升降算符在电子自旋和角动量耦合中应用［J ］．佳木斯大学学报，2012，30（05）：749－750．［10］寻大毛．角动量量子数l 的升降算符和球谐函数的生成［J ］．大学物理，2011，30（01）：19－22．［责任编辑：王军］62商丘师范学院学报2020年。
微波灰化-电感耦合等离子体发射光谱法测定婴幼儿乳粉中的钙和磷
第 29 卷第 4 期分析测试技术与仪器Volume 29 Number 4 2023年12月ANALYSIS AND TESTING TECHNOLOGY AND INSTRUMENTS Dec. 2023分析测试经验介绍(407 ~ 413)微波灰化-电感耦合等离子体发射光谱法测定婴幼儿乳粉中的钙和磷陈丽梅，张 慧，白国涛，马彩霞，姚思雨，王 婧（呼和浩特海关技术中心，内蒙古呼和浩特 010020）摘要：采用微波灰化-电感耦合等离子体发射光谱法（ICP-OES）测定婴幼儿乳粉中钙、磷元素含量. 采用微波灰化法对婴幼儿乳粉进行前处理，正交试验方法确定微波灰化最佳条件，灰化后产物用2 mL硝酸溶液(体积比为1∶1)溶解后，用ICP-OES对钙、磷元素进行含量检测. 磷加标回收率为86%~104%，钙加标回收率为87%~96%.磷的相对标准偏差为2.5%~7.0%，钙的相对标准偏差为3.9%~10.0%，能够满足日常检测要求. 采用微波灰化法对婴幼儿乳粉中钙、磷元素进行样品前处理，相比微波消解方法，具有用时短、用酸量少、消解效果好、不需要进行赶酸处理等优点. 与干法灰化和湿法消解相比大大减少了样品处理时间. 采用微波灰化与ICP-OES结合对婴幼儿乳粉中的重要指标元素进行检测，在婴幼儿乳粉质量控制中有很好的应用价值.关键词：微波灰化；电感耦合等离子体发射光谱法；婴幼儿乳粉；钙；磷中图分类号：O657. 31 文献标志码：B 文章编号：1006-3757（2023）04-0407-07DOI：10.16495/j.1006-3757.2023.04.010Detemination of Calcium and Phosphorus in Infant Formula by Microwave Ashing- Inductive Coupled Plasma Emission SpectrometryCHEN Limei， ZHANG Hui， BAI Guotao， MA Caixia， YAO Siyu， WANG Jing（Technology Center of Hohhot Custom, Hohhot 010020, China）Abstract：The microwave ashing-inductive coupled plasma emission spectrometry (ICP-OES ) was used to determine the content of calcium and phosphorus in infant formula. The method of microwave ashing was used to pretreatment of the infant formula. The optimization conditions were determined by orthogonal test. After the microwave ashing, the ashes were dissolved with 2 mL nitric acid solution (volume ratio was 1∶1). The ICP-OES was used to determine the content of calcium and phosphorus. The recoveries of phosphorus and calcium were 86%~104% and 87%~96%, respectively. The relative standard deviation of phosphorus and calcium were 2.5%~7.0% and 3.9%~10.0%, respectively, which could meet the detection requirements. Compared with the method of microwave digestion, microwave ashing has the advantages of shorter time, less acid, better digestion effect and no need to drive acid treatment for calcium and phosphorus in infant formula. Compared with wet digestion and dry ashing, the sample processing time was greatly reduced. The detection of important index elements in infant formula by microwave ashing combined with ICP-OES has a good application value in the quality control of infant formula.Key words：microwave ashing；ICP-OES；infant formula；calcium；phosphorus收稿日期：2023−10−09； 修订日期：2023−12−12.基金项目：海关总署科研项目（批准号：2021HK186）[The Research Project of General Administration of Customs (2021HK186)]作者简介：陈丽梅（1981−），女，高级工程师，主要从事食品中元素检测工作，E-mail：******************.微波灰化技术是一种创新的样品前处理方法，利用抗热的密闭腔体及微波技术来加热，使得灰化速度提高，减少了能量损耗，增加了样品的处理量，工作环境清洁，使用安全. 不同于湿法酸消解，微波灰化的优点是处理过程比较简单，实验室日常工作非常通用，多用于过程及质量控制. 在石油工业、制药业、食品工业、塑料制品制造业、污染物治理等领域都有着广泛的应用. 微波灰化的原理是微波系统发射均匀的微波，穿透高温泡沫陶瓷护体，内置的耐高温泡沫瓷体内的碳化硅板将微波能量转化为热能，热量直接辐射到样品内，样品被均匀加热.微波的使用大大增加了加热效率，相较于传统马弗炉，加热速度大幅提高. 同时，微波灰化消解用酸量也非常小，既节约了酸的使用量，又减少了赶酸过程对环境的污染. 微波灰化消解法在元素检测前处理中主要应用于油料油品[1-5]、塑料 [6-7]、傣药[8]、食用菌[9]、小麦淀粉[10-11]、电泳材料[12]等样品.钙、磷都是人体必须的营养元素. 在人体吸收代谢过程中，钙和磷会相互影响. 根据GB 10765—2021《食品安全国家标准婴儿配方食品》[13]和GB 10767—2021《食品安全国家标准幼儿配方食品》[14]要求，钙磷元素比值范围分别为1∶1~2∶1和1.2∶1~2∶1，如果婴幼儿乳粉中钙磷比例失衡，婴幼儿对钙的有效吸收就会降低，造成膳食中钙营养元素的相对缺乏，影响骨骼和牙齿的发育，因此对婴幼儿乳粉中钙、磷元素进行检测十分必要.目前婴幼儿乳粉前处理方法主要有微波消解[13]、干法灰化[14]、高压罐消解[15]和湿法消解[16]. 国内未见微波灰化法在乳粉前处理中的应用研究. 由于乳粉基质比较复杂，添加物质较多，用传统的前处理方法对乳粉进行消解，有消解时间长、使用酸量大等不足. 本文以微波灰化为前处理方法，结合电感耦合等离子体发射光谱仪（ICP-OES）检测婴幼儿乳粉中钙、磷元素的含量，建立一种用时短、用酸量少的测定乳粉中钙、磷元素含量的检测方法.1 试验部分1.1 仪器与试剂电感耦合等离子体发射光谱仪（720型，安捷伦科技，美国），微波灰化仪（CEM，培安科技，美国），电子天平（Sartorius，塞多利斯公司，德国）（感量：0.001 g），瓷坩埚（50 mL）. 硝酸（默克，优级纯），钙、磷标准溶液1 000 mg/L（中国计量科学研究院）. 试验过程中所使用婴幼儿乳粉均购自超市，样品编号1、2、5样品为婴幼儿1段乳粉，样品编号3、4为纯奶粉，样品编号6为婴幼儿2段乳粉，样品编号7样品为婴幼儿3段乳粉. 试验过程中使用水均为去离子水. 所有使用的器具使用前均经20%硝酸浸泡过夜.1.2 试验方法1.2.1 标准溶液配制分别吸取钙、磷标准溶液1.0、2.0、3.0、4.0、5.0 mL于100.0 mL容量瓶中，使用2%硝酸定容至刻度，所配制溶液质量浓度分别为10、20、30、40、50 mg/L.硝酸溶液（硝酸∶水体积比为1∶1）配制：量取100 mL硝酸加入至100 mL去离子水中，混合均匀.1.2.2 试验方案在瓷坩埚中称量0.500 g乳粉，设定微波灰化程序进行试验，灰化完成后待微波灰化炉温冷却至150 ℃以下取出坩埚，晾至室温，用2.0 mL硝酸溶液（硝酸∶水体积比为1∶1）溶解灰分，将溶液转移至100.0 mL容量瓶中，用水清洗瓷坩埚，将清洗液转移至容量瓶中，用去离子水定容至刻度，混合均匀后进行钙、磷元素含量测定.1.3 ICP-OES仪器条件ICP-OES检测条件：功率1.2 kW，等离子体气气体流量：15.0 L/min，辅助器流量：1.50 L/min，雾化器流量：0.75 L/min，读数时间：1 s，进样时间：15 s，稳定时间：15 s. 各元素检测波长分别为钙317.933 nm、磷213.618 nm.1.4 微波灰化条件优化方案采用正交试验法对微波灰化条件进行优化，选取4因素4水平正交试验表，因素和水平如表1所列.表 1 正交试验的因素和水平Table 1 Factors and levels of orthogonal test因素因素A（灰化温度）/℃因素B（灰化时间）/min因素C（碳化温度）/℃因素D（碳化时间）/min 水平1450152002水平2500302504水平3550603008水平46009035012408分析测试技术与仪器第 29 卷2 结果与讨论2.1 微波灰化参数优化采用正交试验法对微波灰化的各个参数进行优化. 正交试验方案及结果如表2所列，正交试验结果分析如表3所列.从表3可以看出，磷元素4个因素极差从大到小的顺序为：A>C>B>D，钙元素4个因素极差从大到小的顺序为：A>B>C>D，因此灰化温度在整个灰化过程中起到重要的作用，碳化时间因素影响最小.2.1.1 微波灰化温度的优化正交试验因素的极差结果越大，代表该因素在试验条件中影响较大. 从正交试验极差分析结果可以看出，微波灰化温度极差最大，是影响灰化的最主要因素. 将微波灰化温度各个水平对应的k值为纵坐标，灰化温度为横坐标作因素水平图，如图1（a）所示. 从图1（a）可以看出，当灰化温度达到500 ℃时，钙和磷随着微波灰化温度的提高，所得到的含量没有明显的增加. 钙、磷两种元素最佳微波灰化温度虽然有所差别，但是差别不大. 综合以上结果，选择500 ℃作为微波灰化的最优温度.2.1.2 微波碳化温度的优化从表3可以看出，碳化温度是灰化效果的次主要因素. 碳化过程是将待处理的样品置于低温下使表 2 微波灰化正交试验方案及试验结果Table 2 Orthogonal test programs and results ofmicrowave ashing方案编号试验方案试验结果/（mg/kg）灰化温度/℃灰化时间/min碳化温度/℃碳化时间/min磷钙1450152002 2 605 3 514 2500153508 2 489 3 338 35501525012 2 545 3 411 4600153004 2 638 3 574 54503035012 2 199 2 928 6500302004 2 625 3 532 7550303002 2 615 3 504 8600302508 2 617 3 513 9450602504 2 490 3 334 105006030012 2 710 3 645 11550602008 2 542 3 414 12600603502 2 545 3 451 134******** 2 544 3 392 14500902502 2 650 3 565 155******** 2 721 3 664 166009020012 2 636 3 625表 3 灰化参数优化正交试验结果分析Table 3 Results analysis of orthogonal test/（mg/kg）元素 因素A因素B因素C因素D元素 因素A因素B因素C因素D 磷K19 83810 27710 40810 415钙K113 16813 83714 08514 034 K210 4741005610 30210 474K214 08013 47713 82314 104K310 4231028710 50710 192K313 99313 84414 11513 657K410 43610 5519 95410 090K414 16314 24613 38113 609k1 2 459 2 569 2 602 2 604k1 3 292 3 459 3 521 3 509k2 2 618 2 514 2 576 2 619k2 3 520 3 369 3 456 3 526k3 2 606 2 572 2 627 2 548k3 3 498 3 461 3 529 3 414k4 2 609 2 638 2 488 2 522k4 3 541 3 562 3 345 3 402极差159********极差249193184124极差顺序：A>C>B>D极差顺序：A>B>C>D最优水平：A2B4C3D2最优水平：A4B4C3D2注：K1、K2、K3、K4表示任意列上水平1、水平2、水平3、水平4所对应的试验结果之和；k1、k2、k3、k4表示水平1、水平2、水平3、水平4对应试验结果均值第 4 期陈丽梅，等：微波灰化-电感耦合等离子体发射光谱法测定婴幼儿乳粉中的钙和磷409其碳化，减少因温度上升过快而使得样品灰分的挥发. 以微波灰化碳化温度各个水平对应k 值为纵坐标，碳化温度为横坐标作因素水平图，如图1（b ）所示.从图1（b ）可以看出，过高的碳化温度并不利于微波灰化的结果. 随着碳化温度的提高，样品碳化过程过快，使得元素随着碳化的烟雾挥出，造成了元素含量的损失. 当碳化温度为300 ℃时，钙和磷含量均达到最佳水平. 因此选择最佳的碳化温度为300 ℃.2.1.3 微波灰化时间的优化以微波灰化时间各个水平对应k 值为纵坐标，微波灰化时间为横坐标作因素水平图，如图1（c ）所示.从图1(c ）中可看到并没有最优的灰化时间点出现，因此对微波灰化时间进行了单点优化，结果如图1(d)所示. 从图1(d) 中可以看出，当灰化时间达到90 min 后，延长灰化时间对于消解并没有明显的改善，综合考虑各元素的灰化时间，选择最优的灰化时间为90 min.2.1.4 碳化时间的优化以碳化时间各个水平对应k 值为纵坐标，碳化时间为横坐标作因素水平图，结果如图2所示. 从图2可以看出，当碳化时间达到4 min 时，钙和磷的含量水平达到最优，因此所选碳化时间为4 min.2.1.5 称样量的优化通过正交试验结果，选择最佳微波灰化条件：微波灰化温度500 ℃，灰化时间90 min ，碳化温度300 ℃，碳化时间4 min. 在最优的条件下对不同称样量（0.2、0.5、1.0、2.0、3.0 g ）样品进行前处理，采用ICP-OES 对其含量进行检测，以样品含量为纵坐标，样品称样量为横坐标作图3. 当称样量达到2.0 g 时，婴幼儿乳粉出现了灰化不完全的情况，定容样品中含有黑色沉淀. 从图3也可以看出，当称2 0002 5003 0003 5004 000元素质量分数/(m g /k g )T /℃1 5002 0002 5003 0003 5004 000T /℃元素质量分数/(m g /k g )1 5002 0002 5003 0003 5004 000元素质量分数/(mg /k g )t /min2 0002 5003 0003 5004 0004 5005 000元素质量分数/(m g /k g )t /min图1 （a ）灰化温度、（b ）碳化温度、（c ）微波灰化时间的因素水平图、（d ）微波灰化时间的单点优化Fig. 1 Factor level diagrams of (a) microwave ashing temperatures, (b) microwave carbonization temperatures and(c) microwave ashing times, (d) optimization of microwave ashing times1 5002 0002 5003 0003 5004 000元素质量分数/(m g /k g )t /min图2 碳化时间的因素水平图Fig. 2 Factor level diagram of microwave carbonizationtimes410分析测试技术与仪器第 29 卷样量达到2.0 g 时，检测到的样品含量出现下降的趋势，这是由于样品称样量过多时，样品没有被完全灰化，待测元素无法完全转移到溶液中，造成检测含量降低. 因此最佳称样量应小于2.0 g.2.2 各元素仪器方法学考察采用外标法对各元素进行检测，以元素浓度（X ）为横坐标，元素在ICP-OES 检测强度（Y ）为纵坐标拟合线性方程，得到线性方程和相对标准偏差（RSD ）. 各元素检出限取3倍样品空白标准偏差（n =11）计算，定量限取10倍样品空白标准偏差（n =11）计算. 所得结果如表4所列.表 4 线性方程、检出限和定量限Table 4 Linear regression equations, limits of detection and limits of quantitation元素线性方程线性范围/（mg/L ）相关系数RSD/%检出限/（mg/L ）定量限/（mg/L ）钙Y =37 341.809 + 55 064.512X 0~500.9990.860.023 70.079磷Y =602.582 + 1 742.811X0~500.9990.750.022 50.0752.3 加标回收试验在婴幼儿乳粉中添加标准溶液，采用最优灰化条件进行样品处理后采用ICP-OES 进行检测，结果如表5所列. 从表5可以看出，磷元素加标回收率为86%~104%，钙元素加标回收率为87%~96%. 磷和钙元素6次加标回收率的RSD 分别为2.5%~7.0%，3.9%~10.0%. 加标回收率结果能够满足日常检测要求.表 5 加标回收率Table 5 Results of recovery元素背景/(mg/kg)添加质量分数/(mg/kg)测得质量分数/(mg/kg)加标回收率/%RSD/（%，n =6）磷2 582500 3 0911027.01 000 3 623104 4.22 0004 30386 2.5钙4 1771 0005 0478710.02 0006 01792 4.73 0007 057963.92.4 实际样品的检测采用所建立的方法对市场采购的婴幼儿乳粉进行钙、磷元素含量测定，测定结果如表6所列. 从表中可以看出，所检测的样品编号1、2、5号样品钙磷比值分别为1.4、1.6、1.6，符合GB 10765—2021《食品安全国家标准 婴儿配方食品》[13]要求. 样品编号3、4 号样品是纯奶粉和甜奶粉，钙磷比值分别为1.2和1.1. 样品编号6、7号样品比值分别为1.7、1.3，符合GB 10767—2021《食品安全国家标准 幼儿配方食品》[14]要求.2.5 微波灰化与其他消解方法的比较将微波灰化法与其他前处理方法消解体系和所用时间进行比较，结果如表7所列. 微波灰化前处理方法的精密度和回收率能够满足日常检测的需求. 与干法灰化相比，由于微波的使用，大大增加了加热效率，从而使在进行婴幼儿乳粉前处理过程m /g1 5002 0002 5003 0003 5004 0004 500元素质量分数/(m g /k g )图3 样品称样量的选择Fig. 3 Optimization of sample weights第 4 期陈丽梅，等：微波灰化-电感耦合等离子体发射光谱法测定婴幼儿乳粉中的钙和磷411中所需要的灰化时间更短，总体处理时间与微波消解相当. 微波灰化前处理只需要在灰化完成后用少量酸溶解样品，因此用酸量与湿法消解相比更少.说明微波灰化用于婴幼儿乳粉中钙、磷检测的前处理，所用时间少，消解所用到的酸的种类和剂量也很少，减少了酸的使用对于实验环境和实验人员的危害.3 结论采用微波灰化对婴幼儿乳粉中钙、磷元素进行样品前处理，相比微波消解方法，具有用时短、用酸量少、消解效果好、不需要进行赶酸处理等优势. 与干法灰化和湿法消解相比大大减少了样品处理时间. 采用微波灰化与ICP-OES 结合对婴幼儿乳粉中重要的指标元素钙、磷元素进行检测，是一种快速、环保、准确度高的方法，在婴幼儿乳粉质量控制中有很好的应用价值.参考文献：俞晔, 乙小娟, 刘一军. 微波灰化-原子荧光光谱法测定植物油中砷[J ]. 现代科学仪器，2002（6）：48-50.[YU Ye, YI Xiaojuan, LIU Yijun. 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Studies of Trace Elements and Health ，2011，28 （3）：49-50.]［ 17 ］陈晓, 张晓文, 赵广才. 三种不同前处理方法对标准奶粉中锰含量测定的影响[J ]. 广东微量元素科学，2015，22（11）：27-29. [CHEN Xiao, ZHANG Xiao wen, ZHAO Guangcai, et al. The effect of manganese contentdeterminationbythreepretreatmentmethods [J ]. Guangdong Trace Element Science ，2015，22 （11）：27-29.]［ 18 ］第 4 期陈丽梅，等：微波灰化-电感耦合等离子体发射光谱法测定婴幼儿乳粉中的钙和磷413。
激光专业英语
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使烯烃催化气相氧化成不饱和醛的催化剂[发明专利]
![使烯烃催化气相氧化成不饱和醛的催化剂[发明专利]](https://img.taocdn.com/s3/m/ec23cdd10b4c2e3f56276326.png)
专利名称：使烯烃催化气相氧化成不饱和醛的催化剂
专利类型：发明专利
发明人：沃夫冈·玻克,多崔迟·阿那兹,古那特·朴日思彻,文那·玻克哈兹
申请号：CN90107664.3
申请日：19900913
公开号：CN1050340A
公开日：
19910403
专利内容由知识产权出版社提供
摘要：本发明涉及与含分子氧的气体进行氧化生产不 饱和醛所用的催化剂，其中以具有一定外表面积与体 积之比，空间尺寸、孔隙率、密度、比表面积、断裂强 度、磨耗和气体通流容积的催化剂单元的形式使用至 少含元素钼、铋、磷和氧的活性物质和含硅载体物 质。
可通过本发明涉及的特殊方法获得该催化剂。
申请人：底古萨有限公司
地址：联邦德国哈瑙
国籍：DE
代理机构：中国国际贸易促进委员会专利代理部
代理人：任宗华
更多信息请下载全文后查看。
合成的鞘脂类分子、药物、它们的合成方法及治疗方法[发明专利]
![合成的鞘脂类分子、药物、它们的合成方法及治疗方法[发明专利]](https://img.taocdn.com/s3/m/b0e5e5c9482fb4daa48d4bac.png)
专利名称：合成的鞘脂类分子、药物、它们的合成方法及治疗方法
专利类型：发明专利
发明人：艾梅·埃丁格,斯蒂芬·哈内斯伊恩
申请号：CN201680068271.8
申请日：20160926
公开号：CN108366990A
公开日：
20180803
专利内容由知识产权出版社提供
摘要：提供了包括氮杂环约束的鞘脂类化合物的小分子及它们的合成方法。
还提供了涉及治疗疾病例如肿瘤、癌症和其它疾病的制剂和药物。
还提供含有治疗有效剂量的一种或多种以药学上有效的盐或以纯的形式存在的小分子化合物的治疗剂，包括但不限于用于口服、静脉内或肌内给药的制剂。
申请人：加利福尼亚大学董事会,蒙特利尔大学
地址：美国加利福尼亚
国籍：US
代理机构：北京金思港知识产权代理有限公司
代理人：邵毓琴
更多信息请下载全文后查看。
使用钴催化剂的脱氢硅烷化以及交联[发明专利]
![使用钴催化剂的脱氢硅烷化以及交联[发明专利]](https://img.taocdn.com/s3/m/e1f748cabcd126fff6050b65.png)
专利名称：使用钴催化剂的脱氢硅烷化以及交联
专利类型：发明专利
发明人：C·C·H·阿蒂恩扎,P·J·奇里克,S·奈,K·M·刘易斯,K·J·韦勒,J·L·博耶,J·G·P·德利斯,A·罗伊,E·波尔
申请号：CN201310521264.7
申请日：20130816
公开号：CN103588804A
公开日：
20140219
专利内容由知识产权出版社提供
摘要：本发明为使用钴催化剂的脱氢硅烷化以及交联，文公开了含有三齿吡啶二亚胺配体的钴配合物及其作为高效和有选择性的脱氢甲硅烷基化和交联催化剂的用途。
申请人：莫门蒂夫性能材料股份有限公司,普林斯顿大学
地址：美国纽约州
国籍：US
代理机构：北京市柳沈律师事务所
代理人：封新琴
更多信息请下载全文后查看。
过渡金属π配位活化芳香环的催化研究
过渡金属π配位活化芳香环的催化研究
过渡金属π配位活化芳香环的催化研究是当前化学和材料科学领域的重要研究方向之一。
过渡金属具有丰富的电子构型和可调的氧化态，可以与π体系形成稳定的π配位络合物，活化芳香环，从而引发一系列的化学反应。
研究过渡金属π配位活化芳香环的催化作用，有助于深入理解芳香环的化学反应机制，探索新的催化反应途径，为开发高效、环保的催化体系提供理论支持。
同时，通过研究过渡金属π配位活化芳香环的催化作用，可以发现新的催化剂和催化反应，为工业生产和绿色化学的发展提供有力支持。
以上只是对过渡金属π配位活化芳香环的催化研究的简单介绍，如果需要更多详细信息，建议阅读相关论文或研究报告。
一种含氮杂环结构的硅桥联茂金属配合物及其应用
一种含氮杂环结构的硅桥联茂金属配合物及
其应用
本发明涉及一种含氮杂环结构的硅桥联茂金属配合物及其应用。
本发明提供了一种含氮杂环结构的硅桥联茂金属配合物，该配合物的化学式为：
其中，M表示金属中心，为Zr、Hf或Ti中的任意一种；R1、R2、R3和R4分别是不同的含有后缀“Me”的螯合基，其中Me表示甲基基团；L表示含氮杂环结构的硅桥联基；n表示茂基（Cp）的个数，为1或2。
该硅桥联茂金属配合物具有良好的热稳定性和可溶性，并且在催化乙烯聚合、丙烯聚合、乙烯-α-烯烃共聚合以及5-环己烯-1,2-环氧化反应等领域具有应用前景。
本发明的优点在于，所述硅桥联茂金属配合物中含有杂环结构，可以提高配合物的稳定性和反应活性，且可溶于非极性溶剂，更易于反应体系中的引入和分散。
同时，该硅桥联茂金属配合物在热稳定性方面也表现出色，可应用于高温催化反应中。
其应用领域范围广泛，具有实用性和经济性。
《中药化学B》第3次作业
A型题:请从备选答案中选取一个最佳答案1. 黄酮类化合物如有7—OH，则乙酸钠光谱带II向红位移 [1分]A。
5-20nmB.20—30nmC。
30-40nmD.40-50nmE.50—65nm2. 与盐酸镁粉反应、四氢硼钠反应均为阳性的是［1分]A。
黄酮B。
黄酮醇C.二氢黄酮D。
异黄酮E。
查耳酮3。
黄酮类成分甲醇钠光谱，如带I随时间延长而衰退，则说明可能有 [1分］A。
3—OH或4’-OHB。
3，4’二OH或3，3’,4’三OHC。
3,5二OH或3,5，7三OHD。
4’，5二OH或3,2’，4'三OHE。
5,7二OH或5,6二OH4。
Gibbs和Emerson反应的条件是［1分］A。
酚羟基B。
α—OHC。
β-OHD。
活泼氢E。
酚羟基对位有活泼氢5。
二氢黄酮的1HNMR谱中，H—3（ppm）常出现在 [1分]A。
2。
8 q 左右B.2.5 q左右C.2.8 d左右D.2。
5 t左右E.3。
0左右6。
鉴别黄酮有无邻二酚羟基的反应是 [1分]A。
HCl-Mg反应B。
FeCl3反应C.Mg(Ac）2反应D。
氨性SrCl2反应E.AlCl3反应7。
7-羟基香豆素的1HNMR谱中，母核上的质子化学位移在最低场的是 [1分]A。
3—HB.4-HC.5-HD。
6—HE.8—H8。
诊断试剂乙酸钠/硼酸主要是判断黄酮类成分是否含有（） [1分]A.3-OHB.5—OHC。
7—OHD.4’—OHE。
邻二酚OH9。
与四氢硼钠反应呈红色的是（）类成分. ［1分]A.黄酮B.黄酮醇C。
二氢黄酮D。
异黄酮E。
查耳酮10。
异羟肟酸铁反应的现象是： [1分］A。
兰色B.绿色C。
灰色D.红色E.黑色11。
与盐酸镁粉反应、四氢硼钠反应均为阴性的成分是 [1分]A。
黄酮类B.二氢黄酮类C。
异黄酮类D.黄酮醇类E。
黄酮醇苷12。
二氢黄酮、二氢黄酮醇的紫外光谱带II通常在( ) ［1分］A.200—270nmB.270-295nmC.300—320nmD.320—360nmE。
艾纳香中黄酮化合物的研究
艾纳香中黄酮化合物的研究
邓芹英;丁丛梅;张维汉
【期刊名称】《波谱学杂志》
【年(卷),期】1996(000)005
【摘要】从中药艾黄香中分离得到两种新的黄酮化合物，应用ＮＭＲ，ＭＳ，ＩＲ，ＵＶ等波谱数据，确定其化学结构３，５‘，３’－三竣工在－７，４’－二甲氧基黄酮和３，５，３‘，４’－四羟基－７－甲氧基黄酮。
【总页数】1页(P447)
【作者】邓芹英;丁丛梅;张维汉
【作者单位】中山大学化学系;中山大学化学系
【正文语种】中文
【中图分类】R931.6
【相关文献】
1.槲寄生中总黄酮化合物提取工艺的研究 [J], 李俶;李莉;倪永年
2.罗布麻中5种黄酮化合物抗氧化活性DFT研究 [J], 耿佳琳;焦玉苗;焉炳飞;方圣涛;李文佐
3.新鲜洋葱中总黄酮化合物的醇提工艺研究 [J], 刘阳;皮婷;覃彬华;郑丹;马明兰
4.红景天药材中两种黄酮化合物的提取工艺优化及含量测定研究 [J], 胡英婕
5.橘皮中类黄酮化合物提取工艺研究进展 [J], 李靓;朱涵彬;李长滨;潘春梅
因版权原因，仅展示原文概要，查看原文内容请购买。
美国由水制氢的新催化剂
美国由水制氢的新催化剂
曹刚
【期刊名称】《矿山环保》
【年(卷),期】2003(000)004
【总页数】1页(P48)
【作者】曹刚
【作者单位】无
【正文语种】中文
【中图分类】TQ116.21
【相关文献】
1.CuZnAl水滑石衍生催化剂上甲醇水蒸气重整制氢Ⅰ.催化剂焙烧温度的影响 [J], 汤颖;刘晔;路勇;朱萍;何鸣元
2.CuZnAl水滑石衍生催化剂上甲醇水蒸气重整制氢Ⅱ. 催化剂组成的影响 [J], 汤颖;刘晔;路勇;朱萍;何鸣元
3.美国研发将乙醇转化制氢的新催化剂 [J],
4.美国开发出应用于水气变换制氢反应的低成本高效新催化剂 [J], 章文
5.美国开发出低成本高效新催化剂应用于制氢的水气变换反应 [J], 钱伯章
因版权原因，仅展示原文概要，查看原文内容请购买。
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a rXiv:q uant-ph/033v18M ar2Fermi Gas in Harmonic oscillator potentials X.X.Yi 1,3J.C.Su 21Institute of Theoretical Physics,Northeast Normal University,Changchun 130024,China 2Department of Physics,Jilin University,Changchun 130023,China 3Institute of Theoretical Physics,Academia Sinica,Peking 100080,China Abstract Assuming the validity of grand canonical statistics,we study the properties of a spin-polarized Fermi gas in harmonic traps.Universal forms of Fermi temperature T F ,internal energy U and the specific heat per particle of the trapped Fermi gas are calculated as a function of particle number,and the results compared with those of infinite number particles.PACS numbers:03.75.Fi,05.30.Fk The ideal Fermi gas is an old and well-understood problem since the non-interaction Fermi gas is a good zeroth-order approximation for many familiar systems.Like the trapped degenerate atomic gases [1-3]that provide exciting opportunities for the manipulation and quantitative study of quantum statistical effects,the behavior of trapped Fermi gas also merits attention,both as a degenerate quantum system in its own right and as a possible precursor to a paired Fermi condensate at low temperature[4]though it perhaps is not as dramatic as the phase transition associated with bosons.The trapped atomic gases reported in Ref.[1-3]are dilute.The effects of predominantlyshort-range atom-atom interactions are therefore week.For dilute spin-polarized Fermi gases,the s-wave scattering amplitude vanishes due to the antisymmetry of the many-1Fermi wave function,and the p-wave scattering is small at low energy.Therefore the dilute Fermi gas is usually treated as an ideal Fermi gas.Unlike the usual system in most textbooks of statistical mechanics,e.g.[5],however,the number of trapped atoms in Ref.[1-3]isfinite(about109).Forfinite number of trapped fermions,the applicability of the usual thermodynamical calculations and the neglect of the ground energy are no longer available.For these relatively low numbers(compared to1023), the effects caused by the above approximation are nonvanishing.Harmonic traps provide a particularly simple realization of the confined Fermi system. In this trap,Butts and Rokhsar[6]calculated the spatial and momentum distributions for trapped fermions using Thomas-Fermi approximation.The spatial distribution of the trapped cloud provides an explicit visualization of a real-space”Fermi sea”and the mo-mentum distribution,unlike the spatial one,is isotropic.In this paper,we calculate the Fermi temperature,internal energy and the specific heat per particle of thefinite number trapped Fermi gas.We indeedfind marked differences from the usual treatments:a correction to the Fermi temperature,internal energy and the specific heat per particle of thefinite number trapped fermions,a constraint to the number of trapped fermions for given trap frequencies.Consider N spin-polarized fermions of mass m moving in anisotropic oscillator poten-tials.The single-particle levels are familiarE N=E(n x,n y,n z)= i=x,y,z¯hωi(n i+1(2)1−e−βωi2of such a system without interactions can be expanded as follows:Q(β)≃a2β−3+a1β−2+a0β−1+a−1+O(β)(3) witha2=12a2(ωx+ωy+ωz)(4) a0=18+1There are two additional terms b0and b1E in the density of states in comparison with the result of Butts[6],as we see in following,these additional terms will result in the shift of the thermodynamic quantities.Now,let us consider a system of N noninteracting fermions that the population N(E i) of a state with energy E i is given by the Fermi-Dirac distributionN(E i)=12b2E2F+3b0b2N′=0(12)with N′=N+12b1ε20+b0ε0.The terms with b0and b1are corrections fromfinite number effects to Fermi energy.In the large number particle limit,E F=(3b2N′)1/3=−1b2−1b2(13)in the situation of the number of trapped fermior satisfying N′<N max,N max=[−(pwherep=−3b21b2,q0=b312b22.(15)Otherwise,the eq.(12)have three real solution,but only the positive smallest one meets the requirements for Fermi energy from the physical point of view.We would like to point out that the results presented here are valid for k B T>>¯hωi(i=x,y,z).If this condition is broken,we should make use of a numerical method[8]to study the effects offinite particles.Forfinite temperature,neglecting the effect of zero-point energy,the eq.(11)gives:N=b2k3B T3f3(z)+b1k2B T2f2(z)+b0k B T f1(z)(16) where f n(z)stands for the Fermi integral,which is given thatf n(z)=1z−1x n+1dx,Γ(n)= ∞0e−x x n−1dx.z is a fugacity.The last two terms in eq.(16)are due to the effects offinite particles.If one introduces a temperatureT0F=1b2f3(z F))1/3,(17)which denotes the Fermi temperature of infinite number fermions trapped in anisotropic oscillator potentials[6],i.e.k B T0F=µ.Then the Fermi temperature T F defined as the same as in most textbooks forfinite number fermions in the case ofε0=0takes a formT F=T0F[1−1b21f3(z F)−2b21f3(z F)],(18)where z F=z|T=T0F.The last two terms in eq.(18)are correction to Fermi energy from thefinite number effect,it shows that the Fermi temperature forfinite trapped fermions is lower than that in the case[7]of infinite trapped fermions.Generally speaking,the specific heat is more interest from the experimental point of view,since the low-temperature behavior of specific heat c is generally treated as the hall-mark of onset of phase transition.It is well known that the specific heat can be derived from the internal energy,which is expressed byβU=β ∞0Eρ(E)Here we dropped the ground state energy,which is less than2ε0and remains constant. Substituting eq.(7)into eq.(19),one easilyfindsU=b2β3f3(z)+b0N∂UN{4b2k4B T3f4(z)+3b1k3B T2f3(z)+2b0k2B T f2(z)−3f1/2(z)−3f1/2(z)−3f1/2(z)}(22)It is evident that the specific heat per particle has increased due to thefinite particle effects. The results are illustrated in Fig.1.As state in most textbooks of statistical mechanics,for ideal Fermi gas in a3D box,the specific heat increases proportionally with temperature T for T<<T F,whereas it closes to a constant1.5k B for T→∞.For trapped fermions, however,the specific heat increases predominantly with T for T<<T F,and close to3k B with T→∞.Physically,the difference between the case of3D box and that of trapped fermions is the confinment.This leads to the difference in their energy levels.Consequently, different energy levels would result in different dependence of specific heat on temperature.In conclusion,we have discussed the behaviors of fermions trapped in an anisotropic oscillator potentials,it was shown that corrections due to the effect offinite fermions and the ground state energy are small,whether they are observable depends on the experimental parameters.We believe the results will be useful to them who is working experimentally in this area.ACKNOWLEDGMENTS:One of us(X.X.Yi)would like to thank prof.C.P.Sun for his helpful discussions.6References[1]M.H.Anderson,J.R.Ensher,M.R.Matthews,C.E.Wieman,and E.A.Cornell,Science269(1995)198.[2]C.C.Bradley,C.A.Sackett,J.J.Tollett and R.G.Hulet Phys.Rev.Lett.75(1995)1687.[3]K.B.Davis,M.O.Mewes,M.R.Andrews,N.J.Van Druten,D.S.Durfee,D.M.Kurn,andW.Ketterle,Phys.Rev.Lett.75(1995)3969.[4]H.T.C.Stoof,M.Houbiers,C.A.Sackett,and R.G.Hulet,Phys.Rev.Lett.76(1996)10.[5]F.London,Superfluids II(Wiley,New York,1954)K.Huang,Statistical mechanics(Wiley,New York,1987)ndau and E.M.Lifshitz,Course of Theoretical Physics,volume5:Statistical physics(pergamon,London,1959).[6]D.A.Butts and D.S.Rokhsar Phys.Rev.A55(1997)4346.[7]I.F.Silvera and J.T.M.Walraven J.Appl.Phys.52(1981)2304.[8]J.Schneider,H.Wallis,Phys.Rev.57(1998)1253.7Fig.1The specific heat per particle vs.temperature.The dots show the situation for a very large number of particles(N=1023),and the dashed line that for a smaller number(108).The trap frequencies areωx=500Hz,ωy=600Hz,ωz=800Hz.8。