纵向分量要回家，横向分量要散伙。回家的慢，散伙得快。 物理上说，在磁场方向，平衡时候是有波尔兹曼分布，所以任何偏离平衡的情况，最后都要回到平衡，这个过程一般在100ms 到10s左右， x,y方向，自旋的相互关联最后要消失的，如同酒店墙上的很多钟表，最后时刻都是差别很大，没有关联的，这个过程大概在ms左右。 最后说一句，性质上，这个方程是维象方程，也就是说，从实验总结的，任何人愿意的话，都可以提出其他的形式来解释你自己的现象，如果能够解释的更好的话。 核磁共振布洛赫方程的稳态解与非稳态解 布洛赫方程是经典力学描述核磁共振现象最为重要的理论基础之一，是理解和做好核磁共振实验的必备知识。Das 曾用laplace变换求其解析解，但由于计算工具的限制，求取完整的解析解十分困难，以至于许多人不得不用数值方法求解布洛赫方程 NMR: Kinetics We have highlighted your search term bloch-mcconnell方程 for you. If you'd like to remove the search term, click here . Nuclear magnetic resonance (NMR) is an analytical technique used in chemistry to help identify chemical compounds, obtain information on the geometry and orientation of molecules, as well as to study chemical equilibrium of species undergoing physical changes of composition, among many others. Capitalizing on the ability to manipulate the magnetization through different pulse programs in NMR, allows for the study and understanding of the kinetics of a system. The exchange rates between two sites can be evaluated through dynamic nuclear magnetic resonance experiments (DNMR). 17 O is a common, NMR active nucleus that is used in the study of kinetics. 1. Introduction 2. 17O NMR for Kinetics Studies 2.1. Background and Equations 2.2. Bloch-McConnell Equations for Metal Site (Equations 1-3) 2.3. Bloch-McConnell Equations for Bulk Water Site (Equations 4-6) 2.4. Experiments 2.4.1. T2 Studies 2.4.2. T1 Studies 3. NMR Kinetic Studies 4. References 5. Outside Links 6. Problems 7. Solutions Introduction NMR uses radio frequency radiation to change the direction of nuclear spins that have been placed in a static magnetic field, and measures the change of magnetization as a function of time. Since its discovery, NMR has gone through many advancements that have enabled it to become a very useful analytical technique. The Fourier Transform NMR has enabled more complicated studies through the ability to create pulse programs that can manipulate the spectra, like saturate one species magnetization so no peak is produced. These pulse programs can also be used to tip the spin of certain nuclei, while keeping others along the z-axis. This is useful for many applications, including being able to quench signals, change the direction (positive or negative) of the signal, and track relaxation, to name a few examples. Using different pulse programs allows for the study of exchange rates between species. This is done by monitoring the changes in the environment of the NMR active nuclei as a result exchange between the sites. Because of the exchange, spins (magnetization) will be transferred, leading to changes in the bulk magnetization at both sites. Any NMR active nuclei can be used to study exchange rates, such as 13 C, 1 H, 17 O, but 17 O kinetic studies are often performed. This is done because 17 O enriched water can be used as one of the exchange sites, normally the bulk solvent site. 17 O NMR for Kinetics Studies Background and Equations Oxygen seventeen nuclei have a spin state of 5/2, making them susceptible to nuclear magnetic resonance. This isotope of oxygen is only 0.0373% naturally abundant, but using isotopically labeled oxygen compounds can result in useful information. Studying these nuclei in the presence of a magnetic field will provide information about the structure and environment of the oxygens in the molecule. Using dynamic NMR or DNMR, 17 O NMR experiments can be performed to understand chemical reactivity and kinetics of compounds. DNMR studies the effect of a chemical exchange between two sites that have either a different chemical shift or coupling constant. These studies are done by obtaining NMR spectra over time and analyzing the increase and/or decrease of the signals. Unlike other methods that are used to study kinetics, NMR studies can acquire information about the effects of the exchange on the molecules. To utilize NMR spectra to establish kinetic information, the Bloch equations must be adapted to include terms that take into account relaxation as a result of chemical reactivity. While investigating exchange reaction of 17 O water between two sites, the bulk water and water bound to a metal, it is assumed that the kinetics are 1 st order, such that: d u M d t = − k → M u M + k ← W u W d u W d t = − k ← W u W + k → M u M Where k ⃗ M k ⃗ W represent the rate of exchange between the bulk water and the bound water. These two sites can be said to be coupled because the isotopically enriched oxygen is exchanging between the metal site and bulk water site. As exchange occurs, the magnetization of the 17 O metal ensemble and 17 O water ensemble will change, not only due to magnetization relaxation, but also due to the exchange. The exchange rate terms can be added into the Bloch equations to take into account the relaxation. With the addition of this term, the equations are known as the Bloch-McConnell equations. Since there are two sites and three Bloch equations per site, there is a total of six equations for the change in magnetization of the system. Equations 1-3 are for the metal site, while Equations 4-6 are for the bulk water site. Bloch-McConnell Equations for Metal Site (Equations 1-3) d u M d t = v M ( ω r f − ω o ) − u M T 2 M − k → M u M + k ← W u W d v M d t = − u M ( ω r f − ω o ) − v M T 2 M − k → M v M + k ← W v W d m z M d t = v M ω 1 − ( m z M − m o ) T 1 M − k → M m z M + k ← W m z W Bloch-McConnell Equations for Bulk Water Site (Equations 4-6) d f r a c d u W d t = v W ( ω r f − ω o ) − u W T 2 W − k ← W u W + k → M u M d f r a c d v W d t = − u W ( ω r f − ω o ) − v W T 2 W − k ← W v W + k → M v M d f r a c d m z W d t = v W ω 1 − ( m z M − m o ) T 1 W − k ← W m z W + k → M m z M To analyze the NMR spectra, which is obtained by measuring the magnetization in the x-y plane, requires an equation that explains the magnetization change in the x-y plane as a function of time. In the rotating frame, the total magnetization in the x-y plane is comprised of two components the “real” and “imaginary” parts. Therefore, the total magnetization in the x-y plane can be expressed m x y = u + i v , or u = m x t − i v . Taking the derivative of this equation with respect to time leads to d m x y d t = d u d t + i d v d t . Using the previous relationships, the Bloch equations for the two sites can be simplified and rearranged to give the magnetization in the x-y place as a function of time. Invoking the law of detailed balance, which states that the exchange rate of the metal site times the amount of 17 O at this site is equal to the exchange rate of the bulk water site times the amount of 17 O at this site, will eliminate one of the rate coefficients, simplifying the equations even further gives Equation 7 . Equation 7: d = − where, L ¯ = R ¯ = k ¯ = L ¯ is the difference in the chemical shifts of the two sites signals, R ¯ is the relaxation in magnetization at each site without exchange, and k ¯ is the rate coefficients for the exchange. Taking the derivative of the equation, the magnetization of the bulk water signal and the magnetization of the metal site as a function of time results in Equation 8 . Equation 8: = e − t The m o x y is the initial magnetization along the x-y plane before relaxation. Using the Bloch-McConnell equation, the width and intensities of the peaks in the spectra become a function of the chemical exchange. By studying the change in the two peaks, the rate coefficients can be determined, which can be used to calculate other thermodynamic properties like entropy and enthalpy. Experiments T 2 Studies The most common way of studying chemical kinetics in NMR has been through the bandshape technique, which studies the change in the signals of the spectra as a result of exchange kinetics. Before any exchange occurs, two sharp signals are present, one for each of the two 17 O sites. As the exchange rate speeds up, the two peaks will begin to broaden and overlap. At an extremely fast exchange rate, the peaks will coalesce and be centered at the weighted average of the Larmor frequency of the bulk water and the bound water. This occurs because the two exchange sites will have two distinct peaks at slow exchange rates because the NMR active exchange species will be at each site for long enough time that detection of two separate sites will occur. As the exchange rate increases, the nucleus will be exchanging so quickly that the detection of the nucleus on either site becomes averaged out, creating one signal in the spectra. Figure 1 : Coalescing Peaks This figure shows the two peaks moving closer together and then coalescing as the exchange rate increases. The figure above shows peaks from a two-site exchange. The top spectrum depicts a slow exchange rate. As they move down, the rate constants are getting increasingly larger until the peaks coalesce. The width of the band at half height, or the full width at half max (FWHM) is used to track the rate coefficients of the exchange because it is proportional to both T 2 relaxation and the rate coefficient. During exchange rate experiments, Equation 9 can be used to calculate the exchange rate ( k ), which can then be used to determine other activation parameters such as Gibb's free energy, entropy, and enthalpy. Equation 9: k = π δ v 2 2 ( ω ∗ − ω o ) Swift and Connick used bandshape experiments to study the exchange of water from the bulk to a paramagnetic metal. They developed an equation that relates the band width of the signal to the mole fraction of each water site, the bulk P W and the metal P M . Equation 10 is the simplified Swift-Connick equation for the change in the paramagnetic water signal. ( Δ w is the change in the full width measured at half height) Equation 10: 1 P M ( Δ w ) = Δ w + Δ w 2 k 2 M By obtaining the NMR spectra and measuring the signal width, the exchange rate can be calculated. It can be seen that the peak broadness is a function of the T 2 M relaxation, or the transverse relaxation. It can be evaluated from the free-induction decay (FID). The FID is the time-domain signal of each frequency component. Each component results in a sine wave, which are then added together for the FID signal. At time zero, all the components are aligned, and over time they spread out and combine in a deconstructive manner, resulting in a decay of the total FID signal. This is an exponential decay and can be used to calculate the transverse relaxation with Equation 11 . Equation 11: M x y ( t ) = M o e − t T 2 The FID is obtained by using a simple π 2 pulse directed along the x-axis to tip the magnetization into the y-axis so it is processing in the x-y plane. The magnetization in the x-y plane is detected over time as the different frequency components process at different rates. This will result in the FID so T 2 M can be found. T 1 Studies A second method of observing the magnetization exchange between bulk solvent and the metal is by studying the \T_{1}\)relaxation. This method is completed by observing the intensity of one site's signal, while the other signal is saturated by the applied radio frequency pulse program. As the saturated site's spin is transferred to the second exchange site, the second site's magnetization intensity increases as result of the additional spin, while the saturated site's magnetization decreases. From the Bloch equations, the relaxation of magnetization in the z-axis is proportional to 1 T 1 M . Studying the T 1 relaxation tracks the magnetization change between the metal and bulk site through an inversion-recovery NMR experiment. This experiment is done using a two-pulse pulse sequence. A 180 o shape pulse with a radio frequency close to the bulk solvent's Larmor frequency is directed at the sample. Since a shape pulse is a selective pulse, it will only flip the magnetization of the bulk solvent. The solvent’s magnetization, having been flipped 180 degrees, will be aligned opposed to the applied magnetic field. Once probed, the magnetization will begin to relax along the z-axis until it reaches it equilibrium position. After some time (t), a 90 o square pulse is applied. A square pulse is not selective and excites a broader range of frequencies, resulting in both the metal and solvent magnetizations to be tipped into the x-y plane to be detected. At time zero, the magnetization signal for the solvent will be large and negative because it will tipped into the x-y plane in the negative direction. Figure 2: Magnetization as a result of the pulse program The bulk solvent magnetization is tipped to be against the static magnetic field, while the metal site magnetization remains inline with the static magnetic field. Then the short 90 0 pulse is applied to tip both site's magnetizations into the x-y plane for detection. The bulk solvent magnetization will result in a negative signal, and the metal site magnetization will be positive. As exchange occurs, the magnetization along the z-axis will become positive because of natural relaxation, and because the 17 O from the metal will be positively in the z-direction, helping speed up the relaxation. This can be tracked by varying the time between the 180 o and 90 o pulse, since the direction (positive or negative) and intensity of the peaks will change. Figure 3: Magnetization along z-axis The magnetization of the bulk water site, directed against the static magnetic field, will begin to relax back towards the initial condition (all magnetization direct with the static magnetic field). The relaxation will cause the the bulk water site's negative signal to decrease in size and then become positive as the oxygen on the metal site exchange. The above image shows that at t=0 the solvent peak is large and negative, since no magnetization will have had time to relax. Between t=1 and t=2, the magnetization along the z-axis has inverted back, and will produce a positive peak. The exponential line from the first negative peak to the last positive peak results in Equation 12. Equation 12: M z ( t ) = M o ( 1 − e t T 1 ) Figure 4: T 1 equation representation As the magnetization relaxes along the z-axis, it relaxes exponentially with Equation 12 T 1 can then be used to calculate the exchange rate through the Bloch-McConnell equations. A plot of the intensity of magnetization as a function of time would look like: Figure 5: Plot of intensity versus time for the bulk solvent site magnetization These are just two pulse sequences that can be used to study kinetics through NMR. More intricate pulse sequences can be used to perform kinetic studies on more complex systems. NMR Kinetic Studies NMR studies have been carried out to understand the kinetics water exchange with different compounds as a function of temperature and pressure. There also have been experiments that track the affects of pH on the exchange rates of chemical systems. Below is a list of some journal articles containing dynamic NMR experiments with 17 O. A quick search on DNMR experiments will produce many journal articles that have been published over the years. Phillips, Brian L., Susan Neugebauer Crawford, and William H. Casey. "Rate of water exchange between Al(C 2 O 4 )(H 2 O)4+(aq) complexes and aqueous solutions determined by 17O-NMR spectroscopy." Geochimica et Cosmochimica Acta 61.23 (1997): 4965-4973. Szab, Zolt N., Ingmar Grenthe. "On the mechanism of oxygen exchange between Uranyl(VI) Oxygen and water in strongly alkaline solution as studied by 17O NMR magnetization transfer." Inorganic Chemistry 49.11 (2010): 4928-4933. References Sandström, J. (1982). Dynamic NMR spectroscopy . London: Academic Press. H. M. McConnell, Journal of Chemical Physics 1958 , 28 , 430. T. J. Swift, G. M. Anderson, R. E. Connick, M. Yoshimine, Journal of Chemical Physics 1964 , 41 , 2553. T. J. Swift, R. E. Connick, Journal of Chemical Physics 1962 , 37 , 307. Outside Links http://en.wikipedia.org/wiki/Nuclear_magnetic_resonance Pulse Sequence Library: http://nmrwiki.org/wiki/index.php?title=Special:PulseSequenceDatabase FID: http://en.wikipedia.org/wiki/Free_induction_decay Problems Describe the difference between the T 1 and T 2 relaxation. Why are the Bloch-McConnell equations needed for kinetic NMR studies? Describe what would happen in an inversion-recovery experiment if the signals from the solvent and metal site occur at similar frequencies? What problems can arise from using the Swift-Connick equation and the bandshape technique? Draw the pulse sequence of the inversion-recovery experiment. (A schematic of the frequency versus time) Solutions The T 1 relaxation is relaxation of the magnetization in the z-axis and is known as longitudinal relaxation. The T 2 relaxation is relaxation in the x and y axis, or xy plane. It is known as transverse relaxation. The Bloch-McConnell equations are the extension to the Bloch equation. They include an extra term that takes into account exchange between two species. Without this relationship, kinetic studies would not be able to be studied using NMR. If the signals are too close together, the 180 degree shape pulse would flip both magnetizations, not separating the two to allow for analysis. More intricate pulse sequences can be used with more pulses to obtain the magnetizations in opposite directions along the xy plane. NMR signals are often small since its a very insensitive technique. Peaks can not only be easily lost in the noise, but the bandwidth may be extremely difficult to determine, making the Swift-Connick equation hard to use. Pulse sequence drawing for the inversion-recovery experiment
去盐湖城开会，犯懒，没带兔子。早上拉开旅馆房间的窗帘，见旭日在远处雪山的峰顶变幻着颜色，当时一个“悔”字涌上心头。 开会间偶然得到了个蹭车的机会，跟同事去了几处最漂亮的滑雪胜地。虽然已是暮春，满目仍是一片银白的童话世界，在高原直射的阳光下晶莹透亮。当时的心情：后悔大发了！ 晚上在犹他大学的体育馆参加一个活动，馆外就是冬奥会的冰场。两边落地窗的大厅，一面是日落盐湖，另一面是月出关山。当时……嗨！不提了，肠子都悔青了。 还是上两张同事拿全幅单反给拍的片子吧。 身边的这块牌子，赫然题着“通往锡安之路”（ The Road to Zion ），勾起了我的好奇心。 锡安，即锡安山（ Mount Zion ），所罗门圣殿的所在地，为犹太教圣地。公元前 586 年，所罗门圣殿毁于巴比伦人之手，希伯来人也因此沦为巴比伦之囚。“锡安”因此成了一个象征——异乡犹太人心目中神圣甜美的家园。《旧约·诗篇》第 137 章开篇这样悲叹道： By the rivers of Babylon we sat and wept when we remembered Zion. There on the poplars we hung our harps, for there our captors asked us for songs, our tormentors demanded songs of joy; they said, Sing us one of the songs of Zion! How can we sing the songs of the Lord while in a foreign land? 这段历史后来被威尔第改编成了歌剧，里面有一首著名的“希伯来奴隶合唱曲”，又名“飞吧！思想，乘着金色的翅膀”。歌中所表达的对家园的渴望是那样的真挚深切，在当时正谋求统一的意大利民众心中产生了强大的共鸣，以至于有人提议，要以它来作为统一后的意大利国歌。 大都会歌剧院演唱的“飞吧！思想，乘着金色的翅膀” 无独有偶，丧失了家园、被贩卖到美洲的黑奴们，也是唱着这样的歌来表达对故园的渴望与思念的。八十年代被成方圆唱红了大江南北的“巴比伦河”，几乎就是《旧约·诗篇》的逐字翻唱。锡安，成了精神家园的象征。 “巴比伦河” 而我身边的这块牌子，和犹太人、意大利人、黑奴都没有关系，它讲述了一段摩门教徒寻找家园的故事。创建于 19 世纪初的摩门教，因不能见容于传统的基督教，先是从东部被逼到了中西部，与那里的基督徒依然纷争频仍，在经历了“密苏里战争”、“伊利诺战争”后，被迫大规模地向更加地广人稀的西部迁徙。他们把这次迁徙看作是寻找家园的行动， the Road to Zion. 巧合的是，那时流离在欧洲的犹太人正在搞犹太复国运动，他们打出的口号是： Return to Zion. 犹太复国主义因此被称作是 Zionism. 只不过犹太人重返家园的旅途比摩门教徒要更加艰辛与坎坷。 通往锡安之路 迁徙途中的摩门教徒 摩门教徒们把这次西迁自比作犹太人出埃及，杨百翰是西迁的领导者，因此被称作是摩门教的摩西。 教徒们甫一在盐湖山谷落脚，杨百翰就提出了要在这高原的大荒川里营建一座圣殿。盐湖城附近的大山有不少是花岗岩材质的，于是教徒们开始了劈山造圣殿的工程。 摩门圣殿 圣殿旁的礼拜堂 耶稣受洗 渔夫彼得接受耶稣的感召 布道的耶稣 圣殿广场 圣殿广场一角 之所以带了个小 DC ，是为了开会时拍 poster 方便。我听报告时不太爱拍照，觉得比较扰民。不过听到有趣的段子时，还是忍不住要掏相机，好在这样的时候不多。 年会每年都以 Lauterbur Lecture 开场。 Lauterbur 是核磁共振成像技术的发明者，也是我们学会的创建者和第一任主席。 Lauterbur Lecture 始于 2002 年在夏威夷举行的年会，由 Lauterbur 本人作第一位演讲者。那年的 Lauterbur 已是 73 岁高龄， 他在演讲中详述了核磁共振成像技术发明的过程。从 Lauterbur 走上讲台到结束时离开讲台，两次全场起立，大家以长时间的鼓掌向这位老人表示敬意。次年， Lauterbur 和 Mansfield 共同获得诺贝尔医学奖。 Lauterbur Lecture 的开场白自然要和 Lauterbur 挂钩，今年的演讲者扯得更远，从 Ernst 的书开始讲起。 Ernst 是 1991 年诺贝尔化学奖得主，他与人合著的“ Principles of Nuclear Magnetic Resonance in One and Two Dimensions ”是经典。虽然里面涉及量子力学的内容我已经看不懂了，但这本书至今仍在我的书架上放着，有二十多年了。 Ernst 这本书的最后一章提到了核磁共振成像，相对于核磁共振谱仪在分析化学中的应用，核磁共振成像的概念很晚才被提出来。原因并不奇怪：电磁波成像的分辨率取决于波长，核磁共振辐射的是无线电波，波长比可见光的微米波段和 X 光的安米波段要长出许多。比如 100 兆赫的无线电波，波长有 3 米，见了大象都能绕过去。要是用无线电波的照相机给大象拍照，出来的结果肯定是“大象无形”。 Ernst 谈“大象无形” 然而 Lauterbur 偏偏不信这个邪，不但如此，在大家都在为提高核磁共振的灵敏度而努力，要把磁场调得均匀更均匀时，他偏偏要加上一个梯度磁场，把磁场变得哪儿跟哪儿都不一样，让一大堆原子核呕哑嘲哳地唱歌儿，发出不同的频率。这样做的结果如何呢？哈哈，大象现原形了！岂止是大象，现在的核磁共振技术，把分辨率做到 100 微米也是完全有可能的，能够看到细微的脑血管。 Lauterbur 让大象“原形毕露” 唉!写基金、审基金的季节又开始了，我打算休博数月了。 祝大家劳动节、青年节、儿童节、建党节、建军节快乐！家有小朋友的母亲节、父亲节快乐！
首先要感谢余昕老师推荐了一个非常实用的网络教程‘the basic of the NMR',解决了我在初期实验和科普（导师推荐读the spin dynamics一书)过程中遇到的几点困惑。其次，要感谢导师对我这个新手的耐心指导，让我在未知领域爬行过程中多了一份信心。 我要攻克的课题第一步是测量高温自扩散系数。利用自旋回升方法在恒定梯度的磁场中测量在不同脉冲间隔时间的回声强度，扩散系数可以通过解指数方程得到。在测量过程中，不仅要掌握测量的方法，更应该理解其中的物理机制以便解释扩散结果。其中最基本的物理机制就是扩散导致的相差（降低了自旋回升的强度）。将自旋相差看做是一个高斯分布。。。
标签: 核磁共振
