Experimental, Pitfalls and Suggested Solutions

Abstract

The most important distortions that may affect the PFG NMR measurements in a destructive way will be discussed; unwanted coherence transfer pathways, eddy current transients, internal magnetic field gradients, convection, vibration and the effect of finite gradient pulse lengths. All the distortions discussed may be present on high field equipment (i.e. superconducting magnets) as well as on low resolution equipment (i.e. permanent magnets). Although, the magnitude and nature of the effects may vary depending on the type of equipment used. Ways to circumvent or suppress the effect from the abovementioned artefacts will also be discussed, as the application of phase sequences, crusher gradients, preemphasis adjustment, bipolar pulsed field gradients, convection compensated PFG NMR sequences, the second cumulant approximation and corrected effective diffusion/observation times.

3.6 Appendix: Generating a Linear Magnetic Field Gradient Along an Axis

To be able to build a set of coils that generates a magnetic field gradient , one must evaluate the magnetic field, using the following two approaches

• Starting with the law of Biot-Savart [28], and make a Taylor expansion of the magnetic field

• $$ \vec{H} = \frac{I}{4\pi }\int {\frac{{d\vec{s} \times \vec{r}}}{{r^{3} }}} $$

  (3.25)

  where ds is a small vector element of the conducting rod, and I is the current passing through it

• Starting with the magnetic vector potential

  $$ \vec{H} = \frac{1}{{\mu_{0} }}\nabla \times \vec{A} $$

  (3.26)

The first approach will be used to evaluate the magnetic field from a quadropolar coil (applied in low field NMR systems comprising permanent magnets), while the second approach will be used to evaluate the field from a set of circular coils (anti-Hemlholz coils used in high field NMR comprising superconducting magnets)

3.1.1 3.6.1 Magnetic Field Gradients Generated by a Quadropolar Coil

For an infinitely long conducting rod, the magnetic field in the x-direction is written

$$ H_{x} = \frac{I}{4\pi }\left[ {\int\limits_{ - \infty }^{\infty } {\frac{(b - y)dz - zdy}{{\left[ {z^{2} + (a - x)^{2} + (b - y)^{2} } \right]^{3/2} }}} } \right] $$

(3.27)

The conductor is located along the z-axis, thus dy = 0. By substituting c = [(a − x)² + (b − y)²]^1/2 and x = c tanθ, and recognizing that equation is an even function of z, the magnetic field is found to be [49]

$$ H_{x} = \frac{I}{4\pi }{Re}\left[ {\frac{1}{(a + ib) - (x + iz)}} \right] $$

(3.28)

Here i denotes the imaginary number, and Re means that the real part of the expression yields the magnetic field. When positioning 4 infinitely long conducting rods as shown in Fig. 3.30, a quadropolar coil is produced. Substitution of ς = x + iy and re(iϕ) = a + ib leads to the following relation for one conducting rod

$$ H_{x} = \frac{I}{2\pi }{Re}\left[ {\frac{1}{{re^{(i\phi )} - \zeta }}} \right] $$

(3.29)

Then applying a Taylor expansion ((ς/r) < 1)

$$ H_{x} = \frac{I}{2\pi r}{Re}\sum\limits_{n = 0}^{\infty } {\left( {\frac{\zeta }{r}} \right)}^{n} e^{( - i(n + 1)\phi )} $$

(3.30)

For the 4 conductors with current polarity as shown in Fig. 3.28, the resulting magnetic field is written

$$ H_{x} = \frac{I}{2\pi r}{Re}\sum\limits_{n = 0}^{\infty } {\left( {\frac{\zeta }{r}} \right)}^{n} \left[ {e^{ - i(n + 1)(\phi )} - e^{ - i(n + 1)(\phi + \pi /2)} + e^{ - i(n + 1)(\phi + \pi )} - e^{ - i(n + 1)(\phi + 3\pi /2)} } \right] $$

(3.31)

When performing the summation, the nonzero contributions are for n = 1, 5, 9 …

$$ H_{x} = \frac{I}{2\pi r}{Re}\left[ {4 \cdot \left( {\left( {\frac{\zeta }{r}} \right)e^{ - 2i\phi } + \left( {\frac{\zeta }{r}} \right)^{5} e^{ - 6i\phi } + \left( {\frac{\zeta }{r}} \right)^{9} e^{ - 10i\phi } + \cdots } \right)} \right] $$

(3.32)

In the area where ((ς/r) ≪ 1), the dominating contribution will be the linear one. Substitution of ς with x + iy and e^−2iϕ with cos2ϕ − i sin2ϕ, and neglecting the imaginary terms, the magnetic field is written

$$ H_{x} = \frac{2I}{{\pi r^{2} }}(x\cos 2\phi + y\sin 2\phi ) $$

(3.33)

By choosing the right ϕ, the desired linear magnetic field gradient can be set to be either in the x or y-direction [49].

Fig. 3.30

A quadropolar coil as seen from above

3.1.2 3.6.2 Magnetic Field Gradients Generated by an Anti-helmholz Coil

When calculating the magnetic field from a circular current loop, it is convenient to use spherical coordinate system. In the following the spherical coordinates will be used to evaluated the magnetic field from an anti-Helmholz coil , starting with the vector potential B = ∇ × A. For a spherical system the vector potential reduces to one component, A_ϕ, which is parallel with the direction of the current [50]. The area in which we are interested in, close to the z-axis and the junction of the axes, is without any current sources, so the differential equation for A_ϕ is written (Fig. 3.31)

$$ \nabla \times \nabla \times A_{\phi } = \frac{1}{r}\left( {\frac{{\partial^{2} }}{{\partial r^{2} }}(rA_{\phi } ) + \frac{\partial }{r\partial \theta }\left[ {\frac{{\partial \sin \theta A_{\phi } }}{\sin \theta \partial \theta }} \right]} \right) = 0 $$

(3.34)

The solution to this equation involves separation of the variables

$$ A_{\phi } (r,\theta ) = R(r){\Theta} (\theta ) $$

(3.35)

The solution of the radial part is written

$$ R(r) = \sum\limits_{n} {\left[ {C_{1n} r^{n} + C_{2n} r^{ - (n + 1)} } \right]} $$

(3.36)

The differential equation for the angular part is written

$$ \frac{\partial }{\partial \theta }\left[ {\frac{\partial }{\sin \theta \partial \theta }(\sin \theta {\Theta} )} \right] = - n(n + 1){\Theta} $$

(3.37)

Performing the partial derivation and making the substitution u = cosθ the equation is on the form of associated Legendre functions of the order n and degree 1 [51].

$$ \frac{\partial }{\partial u}\left[ {(1 - u^{2} )\frac{{\partial {\Theta} }}{\partial u}} \right] + \left[ {n(n + 1) - (1 - u^{2} )^{ - 1} } \right]{\Theta} = 0 $$

(3.38)

Thus the general solution to the vector potential A_ϕ is written

$$ A_{\phi } = \sum\limits_{n} {\left( {C_{1n} r^{n} + C_{2n} r^{ - (n + 1)} } \right)} P_{n\cos \theta }^{1} $$

(3.39)

The magnetic field can now be found

$$ H_{r} = \frac{1}{{\mu_{0} }}\frac{{\partial \sin \theta A_{\phi } }}{r\sin \theta \partial \theta }\sum\limits_{n} {n(n + 1)(C_{1n} r^{n - 1} + C_{2n} r^{ - (n + 2)} )} P_{n\cos \theta } $$

(3.40)

$$ H_{\theta } = - \frac{1}{{\mu_{0} }}\frac{{\partial (rA_{\phi } )}}{r\partial r}\sum\limits_{n} {\left( {(n + 1)C_{1n} r^{n - 1} - nC_{2n} r^{ - (n + 2)} } \right)} P_{n\cos \theta }^{1} $$

(3.41)

These equations are generally valid for a symmetric magnetic field, i.e. A_ϕ parallel to the direction of the current. To find the magnetic field from a current loop it is convenient to start with the expressions in (3.40) and (3.41) and the constraints at r = c (see Fig. 3.32).

Fig. 3.31

Lines of force from a current loop

Fig. 3.32

Boundary conditions

If the total current in the loop is I, then K_ϕ = I/(c dθ). The distance from the centre to the loop is c, a is the radius of the current loop, and dθ is an infinitesimal change in θ (the current loop is assumed to be infinitely thin). The first condition (H _r1  = H _r2 ) produces a relation between C_1n and C_2n

$$ C_{1n} c^{n - 1} = - C_{2n} c^{ - (n + 2)} $$

(3.42)

For the second condition, H _θ1 − H _θ2 = μK _ϕ , the K _ϕ is expanded in an orthogonal set of associated Legendre functions $P_{n\cos \theta }^{1}$

$$ K_{\phi } = \sum\limits_{n} {D_{n} P_{n\cos \theta }^{1} } $$

(3.43)

The coefficients D_n are found using

$$ N_{n} (1)D_{n} = \int\limits_{\theta = 0}^{\pi } {K_{\phi } } P_{n\cos \theta }^{1} \sin \theta d\theta \approx \frac{{IP_{n\cos \alpha }^{1} \sin \alpha }}{c} $$

(3.44)

where

$$ N_{n} (1) = \int\limits_{\cos \theta = - 1}^{\cos \theta = + 1} {P_{n\cos \theta }^{1} } P_{n\cos \theta }^{1} d\cos \theta = \frac{2(n + 1)!}{(2n + 1)(n - 1)!} $$

(3.45)

C_1n and C_2n are then found using the second condition, H _θ1 − H _θ2 = μK _ϕ , and the magnetic field is finally written

$$ \begin{array}{*{20}l} {H_{r} = \frac{I}{2c}\sin \alpha \sum\limits_{n = 1}^{\infty } {\left( \frac{r}{c} \right)^{n - 1} P_{n\cos \alpha }^{1} P_{n\cos \theta } } ,} \hfill & {r < c} \hfill \\ {H_{r} = - \frac{I}{2c}\sin \alpha \sum\limits_{n = 1}^{\infty } {\left( \frac{c}{r} \right)^{n + 2} P_{n\cos \alpha }^{1} P_{n\cos \theta } } ,} \hfill & {r > c} \hfill \\ {H_{\theta } = - \frac{I}{2c}\sin \alpha \sum\limits_{n = 1}^{\infty } {\left( \frac{r}{c} \right)^{n - 1} \frac{1}{n}P_{n\cos \alpha }^{1} P_{n\cos \theta }^{1} } ,} \hfill & {r < c} \hfill \\ {H_{\theta } = - \frac{I}{2c}\sin \alpha \sum\limits_{n = 1}^{\infty } {\left( \frac{c}{r} \right)^{n + 2} \frac{1}{n - 1}P_{n\cos \alpha }^{1} P_{n\cos \theta }^{1} } ,} \hfill & {r > c} \hfill \\ \end{array} $$

(3.46)

These equations are the starting point when aiming at generating a linear magnetic field in the centre of a coil configuration. The easiest way to generate such a field is to use two current loops separated by a certain distance and having current of opposite polarity. For c < r the magnetic field from the coil configuration shown in Fig. 3.33 is written

$$ \begin{array}{*{20}l} {H_{r} = + \frac{I}{2c}\sin \alpha \sum\limits_{n = 1}^{\infty } {\left( \frac{r}{c} \right)^{n - 1} P_{n\cos \alpha }^{1} P_{n\cos \theta } } \left( {1 - ( - 1)^{(n + 1)} } \right),} & {r < c} \\ {H_{\theta } = - \frac{I}{2c}\sin \alpha \sum\limits_{n = 1}^{\infty } {\left( \frac{r}{c} \right)^{n - 1} \frac{1}{n}P_{n\cos \alpha }^{1} P_{n\cos \theta }^{1} \left( {1 - ( - 1)^{n + 1)} } \right)} ,} & {r < c} \\ \end{array} $$

(3.47)

Due to the symmetry of the current loops, every second term in the summation will cancel as

$$ P_{n\cos (\pi - \alpha )}^{1} = ( - 1)^{(n + 1)} P_{n\cos \alpha }^{1} $$

(3.48)

The summation may then start from n = 2, and the first part of the magnetic field will then be proportional to r, while the second and first nonlinear part will go as r ³. This first nonlinear term can be eliminated by choosing cosα = (3/7)^1/2. This is equivalent to letting the two current loops be separated by a distance ((3)^1/2 · a). Then the expression for the magnetic field along the z-direction can be written

$$ H_{z} = + \frac{4I}{7a}\sum\limits_{n = 1}^{\infty } {\left( \frac{r}{a} \right)^{2n - 1} \frac{4}{7}^{{\frac{2n - 1}{2}}} P_{{2n\sqrt \frac{3}{7} }}^{1} } \left[ {P_{2n\cos \theta } \cos \theta - \frac{1}{2n}P_{2n\cos \theta}^{1} \sin \theta} \right] $$

(3.49)

The first order correction to H _z along the z-axis (θ = 0) will go as (z/a)⁵.

Fig. 3.33

The anti-Helmholz coil