Larmor equation

An NMR spectrum is effectively a graph of the intensity of absorption of Rf radiation (y-axis) against the frequency of the Rf radiation (x-axis). Since frequency and magnetic field strength are linked by the Larmor equation, the x-axis could also be calibrated in units of magnetic field strength. In a CW NMR spectrometer, the x-axis of the output device (usually a pen plotter) is coupled to the frequency sweep so that the response of the sample is displayed as the frequency of the Rf transmitter varies. 

If the frame rotates at the Larmor frequency rotational field term of eq. (1.28) reaches co0jy. Since the effective field is zero, the Larmor equation (1.8 a) is obtained ... 

M precesses about the x axis (Fig. 1.6(b)). The precession frequency [Pg.12]

According to the Larmor equation (1.8), chemical shifts can be related to field differences zlBs, measurable in millitesla. 

Also because of the Larmor equation (1.8), the frequency or field differences /1vs or ABS are proportional to the swept radio frequency Vj (in MHz) or the field strength of B0 (in T). Therefore, chemical shifts dvs (or ABS) obtained at different radio frequencies v, (or field strengths B0) have to be adjusted to the same radio frequency (or field) before comparison. In order to get chemical shift values which are independent of the frequency or field strength used, the d scale of chemical shifts is introduced. <5 values are obtained by dividing the frequency differences Avs (in Hz) by the frequency iq used (in MHz = 106 Hz). 

Specifically, we analyze the following problem a spin S is coupled to a controlled magnetic field B (stationary for now, but to be varied slowly in a Berry-phase experiment) and a randomly fluctuating field -X (f), which we treat as a random variable with the correlation function given by Sxit). Its dynamics is governed by the Larmor equation ... 

Transitions between the two energy states, spin up and spin down, can occur by absorption or emission of electromagnetic radiation of frequency, vL, which is given by the Larmor equation [44] ... 

Equation 2.45, often called the Larmor equation, is the basic mathematical expression for NMR. The precession (or Larmor) frequency is directly proportional to the applied magnetic field and is also proportional to y (or (Ji/I), which varies from one nuclide to another. One significant feature of the Larmor equation is that the angle 0 does not appear. Hence the magnetization precesses at a frequency governed by its own characteristic properties and that of the magnetic field. On the other hand, the energy of this spin system does depend on d, because... 

Suppose that a gradient Gx is imposed along the x axis in an otherwise uniform magnetic field B0. From the Larmor equation (2.42), the resonance frequency of a sample at a specific value of x is then... 

As AE = A V, the frequency of the associated electromagnetic radiation is given by the Larmor equation... 

All of the gradient-based methods can be understood from the Larmor equation and thus for a spin at location r after the application of a gradient pulse g of duration 8, the phase will be given by... 

This is the Larmor equation. Since co = Ittv, this expression is equivalent to Equation 15.7. The two approaches are different one leads to the frequency of the quanta which separates both energy states and the other to the mechanical precession frequency. These two frequencies have the same value. 

So far as NMR is concerned, molecules present in dilute solutions form independent entities that do not interact noticeably between themselves. Alternatively, within a given molecule, the electronic and steric environment of each nucleus creates a very weak local magnetic field which shields it more or less from the action of the external field Bq. Each atom has a particular environment - at least if there are no specific elements of symmetry in the molecule. According to the Larmor equation, these very weak local variations in the intensity of the field will affect the frequency of resonance as compared to what would be seen in a vacuum. This effect is called shielding or deshielding of the nuclei. 

All variations in a affect the resonant frequency of the corresponding nucleus. This phenomenon is called a chemical shift. As many chemical shifts are observed as there are molecules containing different shielding constants a. Consequently large molecules lead to complex spectra as a result of their numerous <7 values. For a nucleus i for which 7=1/2, the Larmor equation becomes, on the introduction... 

The equation describes the manner in which the nuclear magnetization, M, at position r and time t processes about the magnetic flux density, B, in which it is found. The constant y is the magnetogyric ratio of the nuclides under study. The processional frequency, co, is given by the Larmor equation. 

To summarize some nuclei, notably protons, have magnetic moments. Spin-flip nuclear-magnetic transitions of these nuclei can be observed at frequencies predicted by the Larmor equation, using complicated and expensive apparatus. The strength of the signal is directly proportional to the number of nuclei involved. 

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