Nuclear magnetic resonance in condensed matter

Transcription

1 University of Ljubljana Faculty of mathematics and physics Physics department SEMINAR Nuclear magnetic resonance in condensed matter Author: Miha Bratkovič Mentor: prof. dr. Janez Dolinšek Ljubljana, October 2012 Abstract The seminar outlines basic principles that are important in nuclear magnetic resonance spectroscopy. Essential model of pulsed NMR is described along with relaxations after. Various interactions with nucleus influence the spectrum. Special focus is being put on quadrupole effect in first and second order corrections. Dipolar coupling, J-coupling and chemical shift are briefly described to give an overview of major interactions of NMR sprectroscopy. 1

2 Contents 1 Introduction 2 2 NMR: Basic principles 2 3 Quadrupole effect 5 4 Other NMR interactions 12 5 Conclusion 14 References 15 1 Introduction Nuclear magnetic resonance, (NMR), was discovered in 1945 by Bloch and Purcell, who received Nobel prise for this discovery. Since then the development of NMR as a technique parallels the development of electromagnetic technology and advanced electronics [1, 2]. NMR principles and applications are today fundamental tool in medicine, spectroscopy and material science. 2 NMR: basic principles 2.1 Nucleus in homogenous magnetic field (classical view) All nuclei have the intrinsic quantum property of spin. The overall spin of the nucleus is conventionally determined by the spin quantum number I. Beside the angular momentum,, nucleus also possesses magnetic moment which is oriented in the same direction and determined by =, (1) where γ is the gyromagnetic ratio (depending on the nucleus). In the magnetic field, which will be aligned with the z-axis, there is a torque imposed on magnetic moment = =. (2) The torque is proportional to the time derivative of angular momentum =. (3) Equation (2) has a solution in the form of precession. The Larmor frequency is precession frequency of the nuclear magnetic moment around the magnetic field =. (4) Magnetic moment per unit volume is called magnetization. 2

3 thus it follows =, (5) =. (6) When we impose short magnetic field, of Larmor frequency, in the x direction (perpendicular on ), the precession occurs and there is an angle,, between directions of magnetization and (Fig. 1). The azimuth angle of magnetization is dependent upon intensity of impulse, and its duration, T. Typically the signal will be such, that the starting angle of precession will take place at = /2 or =. This is the reason the impulses are called /2 pulse or pulse. Let us consider /2 example. After the pulse is over the magnetic moment is affected only by the outer magnetic field. One would expect for precession to be going on forever under such conditions. However, precession of single magnetic moment is also affected by inner randomly changing magnetic fields of magnetic moments of other nuclei and electrons. That is why the magnetization direction is returning to its thermodynamic equilibrium direction along the external field. 2.2 Relaxation principles, and It is useful to take a look from a rotating (with Larmor frequency) polar coordinate system on Fig.2. In the ideal precession case, the magnetic moment would have a static direction from Larmor rotating systems perspective. It turns out that magnetic moments loose their polar orientation, and eventually they randomly disperse in all directions. Projection of magnetization in! plane is exponentially decreasing with a time constant " #, which is called transverse relaxation time. Synchronic precession can be considered to be Fig. 2: Rotating polar coordinate system. just one of possible states with same 3 Fig. 1: Precession around static field is a result of impulse of radio frequency magnetic field [3].

4 energy. In time, system will occupy all possible states. Because magnetic moments loose their phase coherence, we introduce another pulse which turns them around so they are now gathering back together. Once they are again aligned, we can measure maximum signal of magnetization in the z-direction. The purpose of the second,, pulse is in the fact that the first signal happens right after initial /2 pulse, so it can not be properly measured. The second pulse also has to occur soon enough, so there is still phase coherence present. The signal measured is called the spin echo (Fig. 3). Fig. 3: Signals in radio frequency solenoid; spin echo amplitude is dependent of time delay, $. Beside the transverse relaxation, there is also spin lattice or longitudinal relaxation present, i.e. returning the nuclei to its thermodynamic equilibrium state. Magnetization can be described accordingly % & =%'1 exp ', -... (7) In this case, the whole energy of the magnetic moments of nuclei changes, so there has to be interactions nuclei-electron present, therefore spin lattice interaction [4]. 2.3 Simple quantum description Classical view on NMR is appropriate for intuitive understanding of precession. However, we must turn to quantum mechanics for any further calculations. First reason for this is that energy states of nuclei in magnetic field are discrete. The application of a magnetic field produces Zeeman energy of the nucleus of amount. We have therefore a Hamiltonian H=. (8) Taking the field to be of magnitude B along the z-direction, we get H= ħ0 &, (9) 4

5 where 0 & is the z-direction component of a dimensionless spin operator I, defined by the equation: =ħ1. (10) The allowed energies are 2=ħ3, (11) where m can take any of the values 3=0,0 1,, 0. Zeeman levels are illustrated in Fig. 4 for the case I = 3/2. as is the case for the nuclei of Na or Cu. The levels are equally spaced. The operator 0 & has matrix elements between states 3 and 3, 3 0 & 3, which vanish unless 3 9 =3:1. Consequently the allowed transitions are between levels adjacent in energy, giving Fig. 4: Energy levels ;2=ħ=ħ, (12) which is again expression for Larmor frequency, mentioned before (4). Since that is the case, we can expect spectral line to be very narrow, positioned exactly at Larmor frequency [4]. Magnetization as we said fades out with time constant " #. Signal is therefore proportional to exp' </" #.=>?' <.. In a frequency domain (Fourier transform of signal) the corresponding term is Lorentzian shaped line, 1/" #. 3 Quadrupole effect So far we have not considered any electrical effects on the energy of the nucleus. That such effects do exist can be seen by considering a non-spherical nucleus. Suppose it is somewhat elongated and is acted on by the charges shown in (Fig. 5). We see that Fig.5 (b) will correspond to a lower energy, since it has put the tips of the positive nuclear charge closer to the negative external charges [5]. There is, therefore, an electrostatic energy that varies with the nuclear charge distribution orientation. 5

6 Fig. 5: Oval shaped nucleus in the field of four charges, DE on -axis, -E on!- axis. Configuration (b) is energetically more favorable, because it puts the positive charge at the tips of nucleus closer to negative charges E. To present a quantitative theory, we begin with a description in terms of the classical charge density of the nucleus, ρ. We shall obtain a quantum mechanical answer by replacing the classical ρ by its quantum mechanical operator. Classically, the interaction energy E of a charge distribution of density ρ with a potential V(r) due to external sources is 2=FG'H.I'H.JH. (13) Potential can be expanded in Taylor series about the origin (in center of nucleus): M I'H.=I'0.D L D MN O PQR M # L T S V D... MN O MN U QR (14) Index α and β stand for,! and W. Next we define I L = M Interaction energy can be now written in the form MN O PQR and I LS = MT V. MN O MN U QR 2=I FG'H.JHD I L F L GJHD #! I LSF L S GJH. (15) Choosing the origin at the mass center of the nucleus, we have for the first term the electrostatic energy of the nucleus taken as a point charge. The second term involves the electrical dipole moment of the nucleus. It vanishes, since the center of mass and center of charge coincide. Moreover, a nucleus in equilibrium experiences zero average electric field I L. It is interesting to note that even if the dipole moment were not zero, the tendency of a nucleus to be at a point of vanishing electric field would make the dipole term hard to see. The third term is the so called electrical quadrupole term. We note at this point that one can always find principal axes of the potential V such that I LS =0 if YZ[ (16) I must also satisfy Laplace s equation \ # I= I LL =0. In the case of cubic symmetry all derivatives are zero, the quadrupole coupling then vanishes. This situation arises, for example, with ]^#_ in ]^ metal. It is convenient to consider the quantities `LS, defined by the equation 6

7 `LS =ab3 L S d LS H # egjh. (17) This will be useful turned around as a L S GJH= 1 3 (`LS+ad LS H # GJH). (18) We continue with writing expression for quadrupole energy 2 (#), 2 (#) = f (I LS`LS +I LS d LS FH # GJH), (19) Since V satisfies Laplace's equation, the second term on the right of (10.11) vanishes, giving us 2 (#) = f I LS`LS. (20) This term is independent of nuclear orientation. For proper quantum mechanical expression for the quadrupole coupling, we simply replace the classical G and `LS by their quantum mechanical operators, With the help of Wigner-Eckart theorem, we can continue B g = f I LS`h LS. (21) 0,3i`LS i0,3 9 = =j 0,3P _ # b0 L0 S 0 S 0 L e d LS 0 # P0,3 9. (22) We will express constant, j, with matrix element for 3=3 =0 and Y=[=W. k`= 0,0 `&& 0,0 =j 0,0i30 & # 1 # i0,0 = =j 0,0 0(20 1) 0,0. (23) Constant is therefore j= lg m(#mn). (24) Quadrupole Hamiltonian is rewritten in quantum mechanical form as B g = lg fm(#mn) I LSo _ # b0 L0 S 0 S 0 L e d LS 0 # p. (25) 7

8 If we express operators 0 N and 0 q with operators 0 ± and apply interaction (25) in eigenspace where I LS is diagonal, then we get B g = lt rg sm(#mn) o30 & # 1 # + t # b0 u # +0 n # ep, (26) Where ke=i && and v, asymmetric parameter, defined by v= wwn xx yy. (27) These two parameters are the ones which determine the shape of the spectrum. 3.1 First order corrections of quadrupole interaction When the quadrupole interaction is small compared to interaction nucleus-external magnetic field, perturbation theory can be applied. However it is often strong enough that the second order correction is of significant importance. That is why they are called first and second order quadrupole interactions. Perturbation theory corrections [6] of Zeeman energy for m-th energy level are 2 z =2 () z + 3iB g i3 + i zi{ # i} i +. (28) ~ n~ Let us consider now just the first order correction 2 z ( ƒq ) = 3iB g i3 for the m-th energy level of nucleus with the spin 0: 2 z = ħ 3+ 3iB g i3 = ħ 3+ lt rg sm(#mn) (33# 0(0+1)). (29) When transition between 3 and 3 1 levels occurs, the frequency we detect with nuclear magnetic spectroscopy is z,zu = ~ n~ - ħ = + _lt rg (23+1)). (30) sm(#mn)ħ The 3=1/2 1/2 quantum transition is called central transition, which is unaffected by the quadrupole anisotropy to first order. Transitions between other levels are called satellite transitions (Fig. 6). 8

9 Fig. 6: Spectrum of quadrupole interaction of first order for I=5/2. Dotted line presents theoretic calculation, and full line presents the actual spectrum. It is apparent that spectrum does not consist from narrow, separated lines, but from connected peaks with allowed frequencies even between peak positions. This is a consequence of angular dependence of quadrupole interaction, described by angle, which is the angle between external field and interaction principle axis. Quadruple interaction is heavily dependent upon such orientation [7]. In other words, the first order term splits the spectrum into 2I components of N 3 #, (31) is intensity at frequency ˆz () away [4]. We can take a look at asymmetry parameter influence on (Fig.7). As v varies, the lineshapes of both the central and satellite transitions change, which can provide useful structural information as v is related to the local symmetry. Fig. 7: The effects of the asymmetry parameter (v) on the first order satellite without the central peak [7]. Peak symmetry, however is conserved in first order corrections. 9

10 3.2 Second order quadrupole interaction In order to get the second order correction 2 z ( l ) = i zi{ i} i ~ n~ # (the third term on the right side of (28)), we have to calculate all matrix elements 3iB g i?. In this case the central transition changes too, as do all other transitions. The change of frequency is z,zu = + 2 n # 2 # ħ. (32) A comprehensive energy diagram is presented in Fig.8. Zeeman energy levels are equidistantly apart. After applying first and second order corrections, levels change accordingly. Fig. 8: Energy level diagram for I=5/2 nucleus for Zeeman energy levels and corresponding first and second order corrections. Here θ is again the angle between the principal axis of the interaction and the magnetic field. The first order interaction has an angular dependency with respect to the magnetic field of 3cos # ( 1) (the P2 Legendre polynomial), the second order interaction depends on the P4 Legendre polynomial [7]. 10

11 Beside energy levels, line shapes are affected by v, similar to first order case. On Fig.9 we see the central transition peak will be split in two peaks with small value of asymmetric parameter. In real experiments such two peaks can be positioned very close and are blurred by other effects. They often appear as one asymmetric wide peak. Comparison between measured and theoretical prediction was made for I=5/2 nuclei (Fig.10). Asymmetric parameter is set to v=0 [7]. Fig. 9: The effects of asymmetry parameter on second order central transition lineshapes [7]. Fig. 10: Central transition for nucleus of spin I = 5/2 of second order quadrupole interaction. Dotted line is theoretical prediction, full line presents what would be measured. Again transition frequency is dependent upon relative interaction orientation (angle ). [7] 11

12 We can see from (30), that for spin I=½, the quadrupolar correction of first (consequently the second) order vanishes. Since the coupling of nucleus with electric field gradient takes place only with half-integer spin larger than ½, all elements are not subject to such interaction. On Fig. (11), we can see both kinds of elements marked on periodic table. Fig. 11: Periodic table; most elements nuclei have spin larger than ½ [8]. 4 Other NMR interactions 4.1 Direct dipole coupling The direct dipole coupling is spin-spin interaction of each spin influencing on their neighbor through magnetic field. Interaction energy of two magnetic moments Ž and is 2 = s o Q _( )( ) Q p, (31) where is radius vector from Ž to. The dipole coupling is very useful for molecular structural studies, because it is dependent only on intermolecular distance (H n_ ). 12

13 4.2 J-coupling (indirect coupling, scalar coupling) J-coupling is the coupling between two nuclear spins due to the influence of bonding electrons (with spin š) on the magnetic field running between the two nuclei with interaction energy 2 l } 1š. Each nucleus weakly magnetizes electrons, which generate a magnetic field at the site of the neighboring nuclei and vice versa. In a magnetic field a nuclear spin is oriented in one possible eigenstate. An electron nearby tends to be antiparallel to the nuclear spin, owing to the Fermi interaction between the two particles (Fig. 12). The bond's second electron must be of opposite spin following Pauli's exclusion principle. The second electron defines the preferred orientation of the bound nucleus and gives rise to a small excess of antiparallel oriented nuclear spins that are directly bond. Coupling over more bonds can be explained by Hund's rule which states that electron spins close to a nucleus tend to be ordered in parallel. The information is thus transported over to the next bond. Since only s-electrons have finite probability to be near the nucleus the J- coupling increases with increasing s- character of the chemical bond [9]. Fig. 12: A simple model of J-coupling [10]. 4.3 Chemical shift The signal frequency that is detected in nuclear magnetic resonance would be a pure Larmor frequency if the only magnetic field acting on the nucleus was the externally applied field. However when the magnetic field is applied, it induces currents in the electron clouds in the molecule. The circulating electrical currents in turn generate a magnetic field and the nuclear spins sense the sum of the applied external field and the induced field generated by the molecular electrons. This change in the effective field on the nuclear spin causes the NMR signal frequency to shift (Fig.13). Larmor frequency of nucleus is always diminished by atomic electrons. The magnitude of the shift depends upon the type of nucleus and the details of the electron the nearby atoms and molecules. Fig. 13: Energy diagram (between bare and atomic nucleus) of chemical shift. Rate of change in magnetic motion field, in œ, is called shielding factor [11]. 13

14 4.4 Spin Hamiltonian overview In general Hamitonian is the sum of different terms representing different physical interactions B=B +B # +B _ +. These can be divided in magnetic and electric interactions. It is convenient if each B ƒ is time independent. Terms that depend on spatial orientation may average to zero with rapid molecular tumbling. On (Fig.14) we can see relative importance of different interactions for solids and liquids [12]. Fig. 14: Overview of different interactions; anisotropic liquids and solids have similar proportions of NMR effects. Hamiltonian of isotropic liquids is without contribution of quadrupole effect and direct dipole interaction [12]. Interaction where magnetic moments of electrons are oriented in approximately the same direction is not included. 5 Conclusion Although we have mentioned all the major interactions, there are also additional physical influences on shape and position of spectral lines, for example Knight shift. Different interactions dominate for different molecules, the level of anisotropy is often of significant importance. It is necessary to have all different interactions in mind when dealing with NMR experiments. 14

15 References [1] ( ) [2] ( ) [3] ( ) [4] ( ) [5] C.P. Slichter, (Principles of Magnetic Resonance, Springer 1996) [6] ( ) [7] M.E.Smith, E.R.H. van Eck, Prog. Nucl. Ma. Res. Sp: 34, p159, (1999). [8] ( ). [9] ( ) [10] ( ) [11] ( ) [12] ( ). 15