Detecting body cavity bombs with Nuclear Quadrupole Resonance336688/... · DETECTING BODY CAVITY...

DETECTING BODY CAVITY BOMBS WITH

NUCLEAR QUADRUPOLE RESONANCE

A Thesis Presented

By

Michael London Collins

to

The Department of Electrical and Computer Engineering

in partial fulfillment of the requirements

for the degree of

Master of Science

in the field of

Electrical Engineering

Northeastern University

Boston, Massachusetts

August, 2014

1 ABSTRACT

1 Abstract

Nuclear Quadrupole Resonance (NQR) is a technology with great potential for detectinghidden explosives. Past NQR research has studied the detection of land mines and bombsconcealed within luggage and packages. This thesis focuses on an NQR application thathas received less attention and little or no publicly available research: detecting body cavitybombs (BCBs). BCBs include explosives that have been ingested, inserted into orifices, orsurgically implanted. BCBs present a threat to aviation and secure facilities. They areextremely di�cult to detect with the technology currently employed at security checkpoints.

To evaluate whether or not NQR can be used to detect BCBs, a computational model isdeveloped to assess how the dielectric properties of biological tissue a↵ect the radio frequencymagnetic field employed in NQR (0.5-5MHz). The relative permittivity of some biologicaltissue is very high (over 1,000 at 1MHz), making it conceivable that there is a significant e↵ecton the electromagnetic field. To study this e↵ect, the low-frequency approximation known asthe Darwin model is employed. First, the electromagnetic field of a coil is calculated in freespace. Second, a dielectric object or set of objects is introduced, and the free-space electricfield is modified to accommodate the dielectric object ensuring that the relevant boundaryconditions are obeyed. Finally, the magnetic field associated with the corrected electric fieldis calculated. This corrected magnetic field is evaluated with an NQR simulation to estimatethe impact of dielectric tissue on NQR measurements. The e↵ect of dielectric tissue is shownto be small, thus obviating a potential barrier to BCB detection. The NQR model presentedmay assist those designing excitation and detection coils for NQR. Some general coil designconsiderations and strategies are discussed.

ii

CONTENTS

Contents

1 Abstract ii

2 Introduction 1

3 Basic NQR Theory 2

3.1 Purpose . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

3.2 Quadrupole Moment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

3.3 Energy Splitting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

3.4 Exciting and Detecting Transitions . . . . . . . . . . . . . . . . . . . . . . . 7

3.4.1 Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

3.4.2 Echoes and Relaxation Times . . . . . . . . . . . . . . . . . . . . . . 9

4 NQR as a Detection Tool 11

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

4.2 The Potential of NQR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

4.3 NQR Challenges and Current Research . . . . . . . . . . . . . . . . . . . . . 12

5 Free Space NQR Model 16

5.1 Shinohara’s Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

5.1.1 Powdered Samples . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

5.1.2 Calculating the NQR Signal . . . . . . . . . . . . . . . . . . . . . . . 18

5.2 An improved method for calculating NQR signal . . . . . . . . . . . . . . . . 19

5.3 Programming the Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

5.4 Validating the Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

5.4.1 Experimental Hardware - NQR Spectrometer . . . . . . . . . . . . . 21

5.4.2 The Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

5.4.3 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . 25

5.5 Utility of the Free Space Model . . . . . . . . . . . . . . . . . . . . . . . . . 26

5.5.1 Sensitivity Map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

5.5.2 Signal from example targets . . . . . . . . . . . . . . . . . . . . . . . 28

6 Coil Design Considerations 32

6.0.3 Desirable qualities . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

6.0.4 Constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

6.0.5 Methodology of Design . . . . . . . . . . . . . . . . . . . . . . . . . . 34

7 NQR in Biological Tissue - Background 35

iii

LIST OF FIGURES

8 Correcting for Dielectric Media 37

8.1 The Darwin Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

8.2 The Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

8.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

9 Calculating Electric Field - Jacobi Method 41

10 Calculating the E↵ect of Dielectric Media with Axial Symmetry 45

10.1 Calculating the initial electromagnetic field . . . . . . . . . . . . . . . . . . . 45

10.2 Defining the Dielectrics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

10.3 Finding the correction charges and current density . . . . . . . . . . . . . . . 50

10.4 Calculating the Correction to Magnetic Field . . . . . . . . . . . . . . . . . . 52

10.5 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

11 Generalization to Arbitrary Geometry 59

11.1 Concept . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

11.2 Free Space Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

11.3 Defining Objects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

11.4 Correction Charge and Current . . . . . . . . . . . . . . . . . . . . . . . . . 61

11.5 Correction Magnetic Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

11.6 The e↵ect of frequency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

11.7 Other Coil Designs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

11.8 Nonuniform Grid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

12 Conclusion 69

12.1 Implications for BCB detection by NQR . . . . . . . . . . . . . . . . . . . . 69

A Code for calculating correction electric field 80

B Code for calculating correction magnetic field 85

List of Figures

1 Nuclear Orientation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2 Quadrupole Visualization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

3 Energy Level Diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

4 NQR Spectra of Various Explosives . . . . . . . . . . . . . . . . . . . . . . . 12

5 Nuclear Magnetization - Bessel Function . . . . . . . . . . . . . . . . . . . . 18

6 Spectrometer Front End . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

iv

LIST OF FIGURES

7 Solenoid and NaNO2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

8 Nutation Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

9 Plane of Interest . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

10 Gradiometer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

11 Sensitivity Maps for a Gradiometer . . . . . . . . . . . . . . . . . . . . . . . 29

12 Diagram of Cylindrical Target in Two Possible Orientations . . . . . . . . . 30

13 Nutation Curves for Two Cylinder Orientations . . . . . . . . . . . . . . . . 31

14 Permittivity of Biological Tissue . . . . . . . . . . . . . . . . . . . . . . . . . 36

15 Correcting Electric Field - Part 1 . . . . . . . . . . . . . . . . . . . . . . . . 43

16 Correcting Electric Field - Part 2 . . . . . . . . . . . . . . . . . . . . . . . . 44

17 Magnetic Flux Density of a Circular Coil . . . . . . . . . . . . . . . . . . . . 46

18 Electric Field of a Circular Coil . . . . . . . . . . . . . . . . . . . . . . . . . 47

19 Magnetic Flux Density of a Circular Coil - Planar View . . . . . . . . . . . . 48

20 Sample Object . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

21 Edge Detection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

22 Correction Current Density . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

23 Simulated Liver Tissue . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

24 Correction Magnetic Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

25 Relative Correction Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

26 Magnetization Response to Changes in Nutation Angle . . . . . . . . . . . . 58

27 Magnetic Flux Density - Cartesian . . . . . . . . . . . . . . . . . . . . . . . 60

28 Defining Dielectric Objects . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

29 Current Density - Cartesian . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

30 Correction Magnetic Field - Cartesian . . . . . . . . . . . . . . . . . . . . . . 64

31 E↵ect of Frequency on Correction Magnetic Field . . . . . . . . . . . . . . . 65

32 Logarithmic Spiral Coil . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

33 Gradiometer Made of Compressed Linear Spirals . . . . . . . . . . . . . . . . 67

34 Nonuniform Grid Construction . . . . . . . . . . . . . . . . . . . . . . . . . . 68

v

2 INTRODUCTION

2 Introduction

At airports, o�cials are responsible for detecting contraband such as knives, firearms,

explosives, and illegal drugs. They employ an array of technologies in this e↵ort, each with

advantages and drawbacks concerning cost, speed, and the kind of contraband which each

is capable of detecting.

Metal detectors are good at detecting large metal objects such as conventional knives

and firearms but are incapable of finding ceramic or plastic weapons, explosives, or drugs.

More recently, whole body imaging (WBI) technology has been implemented. WBI systems

use X-rays, mm waves, or terahertz waves to penetrate clothing and form an image of the

person being screened. Computer algorithms or trained operators then examine these images

to identify anomalies which indicate the presence of concealed objects. WBI systems are

capable of detecting and localizing threats of all kinds that are concealed beneath clothing.

WBI systems are the focus of current research aiming to improve their e�cacy and lower their

cost. However, the electromagnetic radiation used in WBI is not capable of penetrating skin

deeply, and so any contraband concealed inside of a person will remain invisible to these

methods despite ongoing improvements. The final technology in widespread use is trace

detection. Trace detectors are sensitive to the tiny amounts of explosive that are likely to

be present near bombs or the humans who handled them. While the technology has a lot of

promise, it may be ine↵ective against hermetically sealed explosives that are handled with

high care.

Currently, the largest gap in personal screening systems is the ability to detect non-

metallic objects that have been swallowed, surgically implanted, or inserted into orifices.

While illegal drugs have been smuggled in this manner for a long time, the problem of de-

tecting objects concealed inside the body has taken on new urgency with the rise of terrorism.

Terrorists may use Body Cavity Bombs (BCBs) to attack individuals or airplanes [9,10]. For

example, a Europol report concluded that the attempted assassination of Muhammad bin

Nayef in 2009 may have used a BCB.

While some known search methods could be e↵ective at detecting BCBs, they are inap-

propriate for use at checkpoint security. For example, transmission X-ray systems produce

too much radiation, and cavity searches are a violation of privacy. Nuclear quadrupole

resonance (NQR) may be capable of detecting BCBs safely and non-invasively.

This thesis begins by introducing NQR from a theoretical standpoint. It then considers

NQR from a practical point of view and examines its potential for detecting BCBs. Finally,

it analyzes how body tissue may a↵ect NQR measurements.

1

3 BASIC NQR THEORY

3 Basic NQR Theory

3.1 Purpose

Nuclear quadrupole resonance is a quantum phenomenon in which a nucleus transitions

between discrete energy levels, absorbing and emitting photons in the process. While a

complete description of NQR at the level of quantum mechanics is beyond the scope of

this thesis, this section introduces the main concepts that are necessary to use NQR for

material detection and identification. Examining the nucleus and its electric environment

using classical means goes a long way toward describing NQR. We will begin with the classical

description to develop an intuitive understanding of NQR and then declare the conclusions

of the full quantum mechanical treatment when necessary. Further discussion of NQR theory

can be found in the excellent works of [15, 46, 88, 90, 98].

3.2 Quadrupole Moment

This discussion of NQR begins with an understanding of the quadrupole which is at the

heart of the phenomenon. Consider a nucleus to be a distribution of positive charge with

density ⇢, in space which has electric field potential �. � is external to and independent of

the nucleus. Classically, the electrostatic energy, V, is given by integrating over the volume

of the nucleus:

V =

Z�⇢ d⌧ (1)

The field potential varies slowly over the scale of a nucleus, so it may be well represented

using a Taylor’s series expansion in Cartesian coordinates:

V = �|0Z

⇢ d⌧ +3X

i=1

@�

@x

i

��0

Zr

i

⇢ d⌧ +1

2

3Xi=1

3Xj=1

@

2�

@x

i

@x

j

��0

Zr

i

r

j

⇢ d⌧ + . . . (2)

Here, (x1, x2, x3) = (x, y, z), (r1, r2, r3) are the distances from the origin along (x, y, z), and

the symbol |0 indicates evaluation at the origin. The origin is chosen to be center of the

nucleus. The three terms of Eq. 2 represent the electric monopole, dipole, and quadrupole

moments’ contributions to the nucleus’ energy. Each will be considered separately.

Experiments and quantum theory have shown the dipole contribution to be equal to

zero [41, 90]. Intuitively, this can be explained through a concept known as definite parity

wherein protons are just as likely to appear on one side of the nucleus as the opposite side,

canceling any potential dipole terms [18]. Definite parity also precludes the existence of

2

3 BASIC NQR THEORY

any higher multipole terms with 2n-poles where n is odd. In general, other high multipole

terms do exist, but are so weak, that they do not need to be considered in simple NQR

experiments [98]. For the nucleus which will be the focus of this thesis, 14N, there are no

multipole moments higher than quadrupole [41]. Accordingly, we drop these terms from Eq.

2.

The quadrupole moment term of Eq. 2 can be simplified by choosing a coordinate

system in which the cross terms, i 6= j, vanish. One such coordinate system is defined by

the electric field gradient (EFG) and is called the EFG principal axis system, an idea which

will be returned to shortly. The second coordinate system, which we shall use presently, is

defined by the nucleus’ orientation. This nuclear coordinate system is referred to here as

(x0, y

0, z

0), where the nucleus is symmetric about the z0 axis. This symmetry causes the cross

terms to cancel [41]. Adopting these coordinates, Eq. 2 becomes:

V = �|0Z

⇢ d⌧ +1

2

✓@

2�

@x

02

��0

Zr

2x

0⇢ d⌧ +@

2�

@y

02

��0

Zr

2y

0⇢ d⌧ +@

2�

@z

02

��0

Zr

2z

0⇢ d⌧

◆(3)

Since potential varies slowly over space, the monopole term of Eq. 3 is dominant. Energy

due to the monopole term is minimized when the nucleus occupies a position where potential,

�, is lowest. A nucleus within a solid or molecule will come to rest at an equilibrium position

where the electric field is zero [88]. Since the monopole term is constant for any one location,

it is dropped in order to focus on the quadrupole term:

V =1

2

✓@

2�

@x

02

��0

Zr

2x

0⇢ d⌧ +@

2�

@y

02

��0

Zr

2y

0⇢ d⌧ +@

2�

@z

02

��0

Zr

2z

0⇢ d⌧

◆(4)

In Eq. 4, it is clear that energy depends on the second spatial derivative of potential.

Since potential is the integral of electric field, its second derivative is simply the electric field

gradient (EFG). To interpret the EFG’s e↵ect on the nucleus, consider the case of a spherical

nucleus. In that case, the integrals of Eq. 4 would equal one another:

V =1

2

✓@

2�

@x

02

��0

+@

2�

@y

02

��0

+@

2�

@z

02

��0

◆Zr

2z

0⇢ d⌧ (5)

Since the EFG is established by fields external to the nucleus, and the nuclear charge distri-

bution is very small, LaPlace’s equation can be used. Note that in special cases, Poisson’s

equation is used instead [88], but here we apply LaPlace’s equation:

r2� = 0 (6)

3

3 BASIC NQR THEORY

Then Eq. 5 reduces to zero and the nucleus’ energy is independent of the EFG. However,

some nuclei are not spherically symmetric, but instead possess an oblate or prolate distribu-

tion of charge. The energy of these nuclei depends upon their orientation to the EFG. For

a simplified view, consider Figure 1 in which a prolate nucleus is located at a point where

the electric field is zero but the gradient is nonzero. Note how the nucleus’ energy depends

upon its orientation.

Figure 1: A cigar-shaped charge distribution represents a nucleus in the presence of positiveand negative charges +q and �q. If the nucleus is positive, orientation b has lower energythan orientation a even though the externally defined electric field and nucleus’ position isthe same in both cases. Figure from Slichter [88].

It is also helpful to visualize the decomposition of a nuclear charge distribution into its

monopole and quadrupole terms as shown in Figure 2.

Figure 2: An oblate nucleus (left) is visually decomposed into its monopole term (right) andquadrupole term (center). Figure from Miller and Barrall [58].

The nucleus’ quadrupole moment can be classified by a single scalar parameter. To do

so, we again rely on nuclear symmetry about the z

0 axis, which gives:Zr

2x

0⇢ d⌧ =

Zr

2y

0⇢ d⌧ (7)

so Eq. 4 becomes:

V =1

2

✓@

2�

@z

02

��0

Zr

2z

0⇢ d⌧ +

@

2�

@x

02

��0

+@

2�

@y

02

��0

� Zr

2x

0⇢ d⌧

◆(8)

4

3 BASIC NQR THEORY

and via LaPlace’s equation:

V =1

2

✓@

2�

@z

02

��0

Zr

2z

0⇢ d⌧ � @

2�

@z

02

��0

Zr

2x

0⇢ d⌧

◆

V =1

2

@

2�

@z

02

��0

Z(r2

z

0 � r

2x

0)⇢ d⌧ (9)

To arrive at the standard result, it is necessary to convert distance along the x

0-axis, rx

0 ,

into the total distance from the origin, r0:

r

0 = r

2x

0 + r

2y

0 + r

2z

0 = 2r2x

0 + r

2z

0 (10)

V =1

4

@

2�

@z

02

��0

Z(3r2

z

0 � r

02)⇢ d⌧ (11)

Normalizing by the charge of a proton, e, and dropping the EFG term, we arrive at the

scalar nuclear quadrupole moment, Q, a property of the nucleus which measures how oblate

(squashed) or prolate (stretched) the nucleus is about its axis of symmetry:

Q =1

e

Z(3r2

z

0 � r

02)⇢ d⌧ (12)

3.3 Energy Splitting

EFG Principal Axis System (PAS): So far, we have been using axes defined by the

nucleus. Another coordinate system which is useful is defined by the EFG and is called

the EFG principal axis system (PAS), or sometimes the “molecular” or “crystal” coordinate

system.

The EFG’s physical origin is the collection of charges that are external to the nucleus.

This includes the electrons immediately surrounding the nucleus, other nuclei and electrons

within the molecule, and even adjacent molecules. As such, the EFG is highly specific not

only to the chemical bonds of an atom but to the overall crystalline structure. The EFG

is therefore a↵ected by factors such as temperature and crystalline inhomogeneities. The

EFG and its interaction with the quadrupole moment are partially explainable through both

valence bond theory and molecular orbital theory [41].

To define the EFG PAS quantitatively, we once again examine the components of the

EFG, evaluated at a particular nuclear site. In general, for a cartesian coordinate system

5

3 BASIC NQR THEORY

(x1, x2, x3),

�

ij

=@

2�

@x

i

x

j

6= 0 i 2 {1, 2, 3}, j 2 {1, 2, 3} (13)

For i 6= j, a particular coordinate system can always be found in which �

ij

= 0 [46,64]. This

is is the PAS, and its axes are designated (x, y, z). By convention, (x, y, z) are named such

that |�zz

| � |�yy

| � |�xx

|. �

zz

is given the special name eq. The asymmetry of the PAS is

defined by ⌘ [108]:

⌘ =�

xx

� �

yy

eq

(14)

Including Quantum E↵ects: So far, our description has been entirely classical and is

useful for understanding how the nucleus’ orientation to the EFG a↵ects its energy. However,

the description is incomplete because quantum e↵ects dictate that only certain orientations

are permitted. This leads to discrete states with quantized energy. To identify these states,

it is necessary to formulate the system’s Hamiltonian, from which the discrete energy levels,

or eigenvalues, can be taken. When the EFG is axially symmetric (⌘ = 0), the Hamiltonian

is given by substituting the spin angular momentum operator into the classical expression

for energy [41]. In this case, the �

xx

and �

yy

terms can be ignored. This leads to energy

eigenvalues:

E

m

=eQ�

zz

��0

4I(2I � 1)

�3m2 � I(I + 1)

�=

e

2qQ

4I(2I � 1)

�3m2 � I(I + 1)

�(15)

where eQ is given by Eq. 12, I is the spin quantum number of the nucleus, and (m =

�I,�I + 1, ..., I � 1, I). Spin quantum number is a property of the nucleus. The above

equations are only valid for spin quantum numbers I > 1/2 since those with I 1/2 do

not possess quadrupole moments directly observable by NQR (they may, however, possess

intrinsic quadrupole moments observable by other experiments [46]). It is interesting that

even after taking quantum mechanics into consideration, the result is expressible in terms

of classically derived scalars eQ, eq, and ⌘ [88].

Transition Frequencies: NQR involves inducing and observing transitions between eigen-

states. These transitions are associated with discrete amounts of energy, �E = E

m1 �E

m2 .

Transition energies are related to frequency via Planck’s relation, E = hv, where E is energy,

h is the Planck constant, and v is frequency. These are the NQR frequencies for a particular

nucleus in a particular EFG. The next section will discuss the mechanisms of excitation and

detection, but first, we take a closer look at transition frequencies.

6

3 BASIC NQR THEORY

For the remainder of this thesis, we will focus on one nucleus, 14N, since it is ubiquitous

in conventional explosives such as TNT, RDX, and PETN, and has been the subject of

much research. 14N has I = 1, and so we expect three eigenstates (m = �1, 0, 1) as well as

three transition frequencies associated with the di↵erences between states. Calculating the

eigenstates with Eq. 15, we see that the m = �1 and m = 1 states are degenerate, leaving

only a single transition frequency associated with the transition from m = 0 to either m = 1

or m = �1 and vice versa. This transition frequency is called ⌫

Q

:

⌫

Q

=3e2qQ

4I(2I � 1)h(16)

Eq. 16 is only valid when ⌘ = 0 and ⌫

Q

depends only on �

zz

as a result (recall that this

assumption was made when obtaining Eq. 15). To account for the asymmetrical nature of

EFGs encountered in real life, a quantum mechanical derivation must be used. Closed form

solutions are known only for I = 1, and I = 3/2. While derivations are beyond the scope of

this thesis, their result is satisfyingly simple and can be written in terms of the parameters

already attained.

We look at the I = 1 solution since it applies to 14N. For I = 1, EFG asymmetry breaks

the degeneracy of the m = 1 and m = �1 levels. This gives three states and three transition

frequencies:

⌫0 =2

3⌘⌫

Q

, ⌫± = (1± ⌘

3)⌫

Q

(17)

Figure 3a depicts the three energy levels for 14N and associated transition frequencies.

Note that transition frequencies are sometimes expressed in angular form, ⌫i

= 2⇡!i

.

3.4 Exciting and Detecting Transitions

3.4.1 Principle

Coupling to the nucleus: In NQR, energy is applied to a nucleus in order to excite a

transition to a higher energy state. In principle, the applied energy could take the form of

an artificially generated EFG that interacts with the quadrupole moment directly. The large

EFG that would be required makes this an impractical approach [17]. Some work has been

done generating an EFG indirectly with acoustic energy, a technique known as NAR [98].

In NQR, however, the energy is applied as a radio frequency (RF) magnetic field.

Nuclei spin about their axis of symmetry, z0. This essentially forms a current loop since

the nucleus is made of positively charged particles. A magnetic dipole moment therefore

7

3 BASIC NQR THEORY

Figure 3: For an I = 1 nucleus in an EFG with ⌘ = 0, there are only two energy statesand a single associated transition frequency, as shown in b. When asymmetry is introduced,the degeneracy is broken, and three energy states become possible. The angular transitionfrequencies !

x

, !y

, and !

z

are associated with the di↵erence between energy levels. Figurefrom Lee [46].

arises along z

0. Nuclei with spin quantum number greater than 0 exhibit this property.

When an external magnetic field is applied, the nucleus experiences torque through its

dipole moment. To excite nuclei to higher energy states, magnetic fields can be applied

at the proper transition frequencies which couple to the dipole moment.

In a simple implementation, an RF magnetic field is applied by a transmitting coil as

a short pulse of length ⌧ which excites nuclei to a higher energy state. After the pulse has

turned o↵, nuclei reorient themselves to their ground state. During this reorientation, the

moving magnetic dipole moment generates a field which may be detected by a receiving coil

(typically the same one used for excitation).

For an axially symmetric EFG, Bloom et al. provide a geometric description of the

nucleus’ motion (precession) following excitation by a magnetic pulse [6]. However, for the

general case, no simple visualization exists that can be explained in classical terms [15].

Nevertheless, there are some simple geometric principles critical to a basic understanding of

NQR detection systems.

Direction of applied RF field and nuclear magnetization: To predict the e↵ect of

an RF pulse on a nucleus, physicists add a Hamiltonian term (due to the RF field) to the

original Hamiltonian due to the quadrupole term [15, 46]. A result of this analysis is that

the direction of the applied RF pulse is critical to its e↵ect on the nucleus. The applied field

may be broken down into its components along the PAS axes (x, y, z). Each component has

the potential to excite a single one of the three possible transitions discussed above. If we

call the applied field B, Bz

may excite ⌫0, By

may excite ⌫�, and Bx

may excite ⌫+ [46, 82].

8

3 BASIC NQR THEORY

Excitation will only occur if the applied field frequency su�ciently matches its associated

transition. After a resonant pulse has ceased, the nucleus behaves as a magnetic dipole

oscillating linearly along this same direction. So for example, if a magnetic field pulse is

applied at ⌫+ along the x direction of the PAS and then ceases, the nucleus will temporarily

behave as a dipole oscillating along the x axis. If the original pulse were applied at the

same frequency yet had both an x and y component, only the x component would a↵ect the

nucleus.

Nuclear magnetization in a single crystal: The total oscillating magnetic dipole may

be quantified as an ensemble of the dipoles due to each nucleus being excited. Take the

example of a single crystal composed of many identical molecules aligned in the same direc-

tion. Let us select a particular nuclear site within the molecule (a specific 14N nucleus, for

example). The PAS of that nucleus will align with the PAS of the same nuclear site within

every other molecule in the crystal. In this case, the nuclei’s net magnetization, Ms

, can be

easily written as the sum of each nucleus’ magnetic dipole moment [86]. Assuming that a

magnetic field of amplitude B is applied directly along an axis of the PAS at the associated

resonant frequency:

M

s

= Csin(�B⌧) (18)

where � is the gyromagnetic ratio of the nuclear spin (intrinsic property of the nucleus), ⌧

is the length of time that the pulse is applied, and C is given by:

C = N

�h

3

h⌫

kT

(19)

where N is the number of nuclei in the sample, k is the Boltzmann factor, and T is the

absolute temperature. The described magnetization oscillates linearly following the pulse,

allowing a receiving coil to detect its signal. As time progresses, the system relaxes to

thermal equilibrium and the net dipole moment decays.

3.4.2 Echoes and Relaxation Times

Much of the NQR research that has been done attempts to identify the best methods

to excite nuclei and detect their return signal for detection applications. Consider a simple

example in which an investigator is attempting to determine whether a single, specific ex-

plosive or other “material of interest (MOI)” is present within an unknown volume. First, in

a laboratory setting, an MOI’s NQR frequencies are characterized along with their depen-

9

3 BASIC NQR THEORY

dence upon environmental factors such as temperature [98]. Armed with this knowledge, an

investigator attempts to excite one of the MOI’s NQR frequencies in an unknown sample. If

the MOI is present within the sample, a return signal will be observed.

In the simplest case, the investigator would apply a single pulse to the sample with a

transceiver coil, then detect the oscillating magnetic dipole produced as a result. The return

signal often decays very quickly (S / e

�t/T

⇤2 ) where T

⇤2 is known as the e↵ective relaxation

time. The signal’s exponential decay immediately following an excitation pulse is known as

Free Induction Decay (FID). This presents a problem to the investigator since there is only

a a very short time window in which to detect the signal. Furthermore, the transceiver coil

is typically part of a tuned circuit and is likely to experience “ringing” even after the applied

pulse has been turned o↵. It can therefore be di�cult to distinguish between an FID signal

and residual ringing in the coil.

The signal decay during FID is due to individual spins losing coherence [24,36,58]. It is

possible to refocus the spins with an additional pulse so that the signal re-emerges at a later

time. This means that the signal emerges after “ringing” e↵ects have subsided. Refocusing

pulses can be linked consecutively in a train so that the signal returns many times. This is

important because the signal observed in real-life detection applications is often very low. By

increasing the number of measurements made, N, the signal to noise ratio (SNR) improves

(SNR /pN). The strength of each successive echo decreases exponentially according to

the spin-lattice relaxation time, T 1. T

1 is the time constant associated with the return of

the system to thermal equilibrium. It establishes the wait time necessary between successive

excitations.

In addition to the pulsed (transient) methods described above, work has been done with

“continuous wave” methods where the sample is exposed to a continual AC magnetic field

that is either fixed at a single frequency or slowly sweeps across frequencies. While common

in early work and spectroscopy [50], it tends to provide a smaller signal than is ideal for

real-world detection problems [16]. That said, modern variations on non-pulsed methods

(such as adiabatic half passage) show potential for field detectors [56].

10

4 NQR AS A DETECTION TOOL

4 NQR as a Detection Tool

4.1 Introduction

An ideal explosives detector would be capable of instantly scanning a given region and

then report the presence, identity, location, and quantity of explosives found. It would be

sensitive to all known explosives, work in real-life conditions, have no adverse e↵ects on its

surroundings, and be impossible to thwart. It would always find any explosive present (high

probability of detection) and not give false alarms (low false positives). In addition all this, it

must be a↵ordable and accepted by societal standards. While no single technique is capable

of meeting all criteria of this wish list, they are helpful to keep in mind while evaluating a

given technology.

This section briefly considers NQR in light of these criteria. We will first discuss the

capabilities that NQR has to o↵er. We will then look at the challenges that researchers have

encountered and the work which has been done to overcome these di�culties.

4.2 The Potential of NQR

Pure NQR was first observed experimentally in 1946 by Nierenberg et al. [61, 62]. It

was described theoretically in 1950 by Pound, whose work coincided with experiments by

Dehmelt and Kruger [15, 70]. Through his work with the British military, Pound became

interested in using NQR to detect land mines, making the idea of using NQR as an explosives

detector almost as old as the technology itself [58]. In the twentieth century, NQR was never

deployed widely for explosives detection, but it did become an important laboratory tool for

chemists who use it to understand molecular structure [4, 91]. Recent advances in theory,

electronics, signal processing, and computational power have once again made NQR’s utility

as a detection tool an active research topic.

As an explosives detector, NQR has some prominent advantages. Nitrogen is highly

ubiquitous in solid explosives including TNT, RDX, PETN, and HMX. In principle, NQR

can be used to detect any of these since the 14N isotope is 99.63% naturally abundant [7]. In

general, there are three NQR frequencies for each Nitrogen nucleus found within a molecule

(TNT, for example has the chemical formula C7H5N3O6; thus, three nitrogens and nine

total NQR frequencies). Even nitrogen atoms with similar valence bonds may have distinct

spectra since the EFG is a function not only of chemical bonding, but also overall molecular

and crystalline structure. For example, TNT possesses two NO2 groups, with distinct NQR

frequencies for the nitrogen contained in each. The spectra for various compounds is spread

11


over a wide range (0 to 6MHz) and the line width of each line is typically just 3kHz [8],

making the spectra of each substance very unlikely to match any other substance. This

specificity enables NQR to uniquely identify particular explosives. Furthermore, by observing

resonance lines unique to a particular substance, it makes very high detection rates and low

false positive rates possible [16, 24, 29, 38, 39, 101]. High specificity also qualifies NQR for

use detecting counterfeit pharmaceuticals, pharmaceutical quality control, and detecting

concealed narcotics [3, 4, 19, 23, 37, 85].

Figure 4: Each solid containing nitrogen has a unique NQR spectra. This allows uniqueidentification of many explosives. Figure from Miller and Barrall [58].

As shown in Figure 4, NQR frequencies for the common explosives RDX, TNT, PETN,

and tetryl are below 6MHz. The electromagnetic radiation that must be generated to excite

nuclei is therefore very low energy and non-ionizing. At su�ciently low power levels, there

is no danger presented by the AC magnetic fields typically employed in NQR. Because these

frequencies overlap with those used in AM, maritime, and amateur radio communications,

organizations such as the FCC have compiled ample data on their safety [13].

NQR frequencies of 0.5-5MHz correspond to wavelengths of 60-600m in air. These fields

pass easily through most dielectric media. This gives NQR the ability to “see” through

materials such as earth, luggage, or tissue in order to detect deeply concealed MOI. This

thesis will give special attention to how the fields pass through biological tissue for use

detecting explosives concealed within people.

4.3 NQR Challenges and Current Research

While having potential for high detection rates, low false positives, and unique identifi-

cation, NQR has been hampered by long detection times that are the consequence of peren-

12


nially low signal to noise ratio. As noted in the previous section, nuclei act as oscillating

magnetic dipoles during NQR. Unfortunately, their signal is very weak. In field applica-

tions, it is common for the NQR signal to be about the same intensity as the transceiver

coil’s thermal noise, making the signal to noise ratio (SNR) very low [16,27,38]. Since SNR

scales with the square root of the number of measurements taken, (SNR /pN), a decisive

measurement may often be possible, yet take too long for field applications as a method for

primary screening. For this reason, it is often suggested that NQR is used as a secondary

screening technology to confirm or clear detections made by a faster, yet less discriminate

technology [5, 48]. Improving the SNR and finding other ways to lower detection time have

been active fields of NQR research.

The previous section introduced the concept of a pulse sequence that produces spin

echoes. Di↵erent patterns of pulse sequences have been studied to identify those which

enable most e�cient detection for various MOI depending upon their T1, T2, and NQR

frequencies [53, 55, 76, 79, 80, 84]. Another excitation method is to apply the magnetic field

at a frequency slightly o↵-resonance [53].

Another active area of research is to improve the SNR through advanced signal process-

ing techniques [37–39,95, 101].

NQR frequencies are generally temperature dependent, adding a complication to detec-

tion scenarios where the region being scanned has an unknown temperature. Temperature

dependence can be so strong that detailed characterization even allows NQR to act as a

sensitive thermometer [105]. In detection applications, methods to deal with temperature

dependent spectra shift may be as simple as estimating the target’s temperature or selecting

a specra line with low temperature dependence. A di↵erent approach is to exploit tempera-

ture dependence as an identifying MOI characteristic. A recent algorithm claims to improve

detection this way, even when sample temperature is unknown [38].

The frequencies used in NQR often overlap with those used in A.M., shortwave, and

maritime radio communications. While this is a good thing in terms of the fields’ safety near

organisms, it adds the issue of radio frequency interference (RFI). One way to avoid RFI

is to operate the NQR detection in a shielded environment. This strategy is more feasible

in controlled applications such as checkpoint screening than in landmine detection. When

unavoidable, a strategy for compensating for RFI is to add reference antennas designed to

measure RFI so that it can be subtracted from the detection coil’s signal [5,48,49,100,102].

Alternatively, a gradiometer can be used as the transceiver coil. Gradiometers are

insensitive to large-scale ambient signals [35, 96]. At first, it could seem as though using a

gradiometer could reduce the fraction or strength of the resonating nuclei which the coil is

13


sensitive to. However, if used for both transmission and detection, reciprocity guarantees

that all coil designs (including gradiometers) are inherently optimal for receiving any signal

from the nuclei which they are responsible for exciting. The limiting condition then becomes

that the coil be capable of generating a field of su�ciently uniform magnitude. This is

discussed in more detail in sections 5.2 and 6.0.3.

In addition to minimizing RFI, coils are designed to maximize sensitivity to small signals

and to generate as little thermal noise as possible. Coil designs which have been studied

include simple loops, uniform and non-uniform spiral coils, birdcage coils, and planar and

axial gradiometers [26, 47,54,60,86, 96].

In addition to conventional coils, researchers have investigated Superconducting Quan-

tum Interference Devices (SQUIDs) [1, 34], high temperature superconductors [109], and

atomic magnetometers [45, 83]. For field applications, conventional coils seem to perform

best due to their simplicity and cost.

Cross-polarization techniques o↵er particular promise for enhancing NQR signals [42,

59, 63, 71, 78]. In these methods, a pulsed, static magnetic field is applied before NQR

measurements are taken. This pre-polarizes the sample, raising the energy of the system

by exciting protons (hydrogen nuclei) before 14N nuclei are excited. Energy is transferred

from the protons to 14N via the coupling of magnetic dipoles. Work by Kim and Rudakov

shows that the resulting increase in NQR signal can be an order of magnitude [42,78]. This

technique may not require magnetic fields as powerful or well-characterized as those that are

necessary in Nuclear Magnetic Resonance (NMR) experiments, making field use feasible.

If technology were to succeed at boosting the SNR to levels adequate for field use, NQR

would still have a few drawbacks. In the standard detection schemes, MOI are searched

for one-at-a-time. While this specificity has the advantages described earlier, it introduces

its own challenges. It requires setting the hardware’s excitation and detection frequencies

for each spectra line that is searched for. This may be accomplished using orthogonal coils

capable of detecting multiple frequency lines at the same time or switching hardware that

changes the characteristic frequency of coils between searching for one MOI and the next.

However, it may still take an appreciable amount of time to attempt to resonate the spectra

lines of many di↵erent explosives which may be hidden within an area.

Another issue has to do with characterization. Much of NQR research to-date has

focused on detecting explosives within land mines in which the MOI is either known be-

forehand, or is one of just a few di↵erent kinds of well-characterized military explosives. In

applications where the threat is unknown, a larger number of MOI must be searched for.

Especially when searching for terrorist’s bombs, one would also have to characterize home-

14


made explosives and all of the variability that their non-commercial manufacture involves.

For example, impurities are known to e↵ect NQR characteristics such as signal intensity, T ⇤2 ,

and line-width [8,40,68,89], and small impurities in TNT are known to reduce the signal by

a factor of ten [27]. Even the physical form of a material a↵ects NQR measurements (such

as the particle size in a powdered sample) [8, 89]. While most of these e↵ects are not large

enough to interfere with a detector’s operation, some are important to consider. Adding

NQR characteristics to explosives databases would help this e↵ort.

The other major disadvantage of NQR is its inability to detect liquids. In liquids, the

NQR signal disappears due to motional averaging [46]. This prevents NQR from being able

to detect either liquid explosives or liquid precursors to improvised bombs. With its ability

to detect liquid explosives, NMR is a complementary technology to NQR [2,20,21]. Much of

the hardware used in NMR is also useful in NQR, so it may be convenient to package the two

technologies together so that they share use of electromagnetic shielding from background

noise, detection coils, and even polarizing coils (allowing use of NQR with cross polarization

techniques).

Finally, it is worth considering the e↵ect of metal on NQR measurements. An MOI

shielded in metal may become invisible to an NQR detector. Experiments have shown NMR

(which operates at similar frequencies) to be e↵ective even through shielded containers such

as aluminum cans [52]. This is because the skin depth of low frequency fields is so large.

Therefore, it is possible that NQR could confront the problem of light shielding directly.

However, it may be more robust to plan on using NQR in conjunction with a metal detector

to detect shielding; the NQR hardware itself could even be used as a metal detector.

Another issue with metal is that it may introduce spurious signals that complicate NQR

measurements (such as paper clips within a piece of luggage) [72]. These “ringing” e↵ects

are mostly transient, however, so can be dealt with by using a pulse sequence that makes

the NQR signal observable after transients from ringing metal have subsided.

15

5 FREE SPACE NQR MODEL

5 Free Space NQR Model

Before examining how biological tissue a↵ects NQR measurements, we begin with a

model of NQR in free space. Some analytical and computational models have been developed

by physicists to explain how complicated interactions between nuclei a↵ect NQR in systems

with more than a single atom. For example, there is coupling between the magnetic dipole

moments of various nuclei undergoing quadrupole resonance that has been accounted for [11].

These models are useful for constructing optimal pulse sequences and in processing the NQR

signal [38].

However, such models do not account for the overall geometry of the problem or the

inhomogeneities in the magnetic field that are expected in real life. An analytic model has

been developed by Shinohara et al. that is capable of predicting the e�cacy of various coil

designs at detecting targets in various positions via NQR [86]. The group that developed it

demonstrated a basic and successful implementation of the model in Microsoft Excel VBA.

This section introduces the model and then details our implementation and evaluation of

the algorithm. The model provides a solid understanding of how geometry a↵ects NQR

measurements in free space. The next section examines how magnetic fields are altered by

biological tissue and how this model can be used to evaluate the significance of that e↵ect.

5.1 Shinohara’s Model

5.1.1 Powdered Samples

In section 3.4.1, we noted that following a pulse delivered exactly on resonance at fre-

quency !, a single crystal of material will act as an oscillating magnetic dipole with magne-

tization M

s

given by Eq. 18.

Note that this equation assumes that the applied magnetic field is identically aligned

with the single axis of the principle axes system (PAS) which corresponds to the NQR

frequency being excited. If the applied field were orthogonal to this axis, zero magnetization

would be induced. In real life, explosives and other MOI are typically found as powders or

in other “poly-crystalline” forms composed of many tiny crystals with random orientation.

Within such an arrangement, only a tiny fraction will be aligned such that they are maximally

sensitive to the applied field while a similar fraction will be almost entirely invisible.

We can account for the arbitrary alignment of a given crystal’s PAS with respect to

the applied field. When the applied field and relevant axis of the PAS form an angle ↵, we

simply substitute the projection of the magnetic field into Eq. 18. M

s

continues to point

16


along the relevant axis of the PAS:

M

s

= Csin(�⌧Bcos↵) (20)

As an example, consider the case where the ⌘0 transition in a single crystal is excited.

This transition is associated with the z

0 axis of the EFG PAS. A magnetic field is applied

along the laboratory frame axis z. z and z

0 form an angle ↵. Then M

s

points along z

0, and

may contain components in each of (x, y, z).

To account for the myriad crystals within a sample, one could sum the contributions

M

s

due to each. However, it is much easier to integrate M

s

due to a single crystal over all

positions that it is capable of attaining. In doing so, we are assuming that a given small

volume of material can be expressed as a single magnetization M

T

. This integration is easiest

to visualize if we refer to each crystal’s z0 axis by its angular displacement from lab frame z

in polar coordinates ✓0 and �

0. If a magnetic field is applied along z, the z component of Ms

is then:

M

sz = Csin(✓0) sin(�⌧Bcos ✓0) (21)

We then integrate this expression over all possible values of ✓0 and �

0 to obtain M

Tz :

M

Tz = C

Z 2⇡

0

d�

0Z

⇡

0

sin(✓0) sin(�⌧Bcos ✓0)d✓0 (22)

Using the same procedure to integrate the x and y components leads to their self-

cancellation, so M

TZ = M

T

. The expression becomes dominated by a Bessel function of

order 3/2, which is a well-known result:

M

T

= C

r⇡

2�B⌧

J3/2(�B⌧) (23)

Eq. 23 means that a small volume of powdered material will act as a magnetic dipole

during NQR. MT

points in the same direction as the applied field. This result has been

experimentally verified [65]. The maximum signal that can be expected from a powdered

sample is 43.6% of that which is attainable from a single crystal with the same number of

nuclei. Although smaller, this signal is independent of coil orientation, unlike the case of a

single crystal in which certain orientations create blind spots. The product �B⌧ is referred

to as the nutation angle [56]. The nutation angle which maximizes M

T

is 119�, but pulses

which meet this nutation angle are often referred to as 90� pulses. A normalized plot of Eq.

23 is shown in Figure 5.

17


Figure 5: Nuclear magnetization as a function of nutation angle is dominated by the 3/2order Bessel Function. The nutation angle is directly proportional to the applied magneticfield’s magnitude as well as pulse length. The ⇡/2 pulse occurs at 2.08 rad while the ⇡ pulseoccurs at 4.49 rad.

5.1.2 Calculating the NQR Signal

In [86], the authors discretize an NQR target (such as a block of explosive) into voxels.

Each voxel is treated as a magnetic dipole, with M

T

calculated using Eq. 23. This requires

knowing the magnetic field at that point. The magnetic field of a coil can be calculated

using the assumption that because NQR frequencies are so low, the problem is essentially

magnetostatic. The Biot-Savart law can then be employed:

B =µ0

4⇡

ZC

Idl⇥ r

|r|3 (24)

where µ0 is the permeability of free space, dl is the vector describing the length and

direction of a segment of current-carrying wire, I is the current, and r is the vector pointing

from dl to the observation point. The coil is numerically discretized into many small elements

dl to employ Eq. 24.

Once Eq. 23 has been used to obtain the magnetization at various points within the

target, the task turns to calculating the signal induced in the coil. The authors do so by

employing Faraday’s law:

IE · dl = �

ZS

@B

@t

· dS (25)

18


whereHE · dl is the voltage induced in the coil and �

RS

@B@t

· dS is the time derivative

of magnetic flux over the coil’s surface. To use Faraday’s law directly, one must calculate

the magnetic flux over the coil. Treating each voxel of the target as a magnetic dipole,

the magnetic field due to each can be found over the surface of the coil. Summing these

contributions, the total magnetic flux if obtained. At some position r, away from a magnetic

dipole, the dipole’s magnetic field, Bd, is given by [12]:

Bd = r⇥ µ0

4⇡

MT ⇥ r

|r|3 (26)

and the flux due to a dipole over the coil’s surface is:

ZS

Bd · dS =µ0

4⇡

ZS

r⇥ MT ⇥ r

|r|3 · dS (27)

Since the nuclei resonate sinusoidally, phasor notation can be used. Multiplying flux

by j! then gives its time derivative and the NQR signal’s peak voltage due to each dipole.

Summing the contributions due to each dipole gives the total signal. Considering N total

dipoles, each with magnetization MTi

, the total signal received is V:

V = �j!

µ0

4⇡

NXi=1

ZS

r⇥ MTi

⇥ r

|r|3 · dS (28)

5.2 An improved method for calculating NQR signal

While conceptually simple, complicated coil designs may bound surfaces which are much

more di�cult to define than in the case of a single loop coil. When the coil used to excite

the nuclei is the same as that used for detection, reciprocity can be used to calculate the

received signal much faster. This method was introduced to us by Dr. Peter Volegov of Los

Alamos National Laboratory who uses it for modeling NMR problems. For a vector field,

A, the Stokes’ theorem states:

Il

A · dl =ZS

r⇥A · dS (29)

where l is a contour enclosing the surface S with infinitesimal normal elements dS.

Applying Stokes’ theorem to Eq. 27, we obtain:

ZS

Bd · dS =µ0

4⇡

IMT ⇥ r

|r|3 · dl (30)

19


then using the scalar triple product, (B⇥C) ·A = B · (C⇥A), this expression becomes:

ZS

Bd · dS =µ0

4⇡

IMT · r⇥ dl

|r|3ZS

Bd · dS = �MT · µ0

4⇡

Idl⇥ r

|r|3 (31)

Recall that we are using the same coil as both a transmitter and receiver. In this case,

part of the part of the right hand side of Eq. 31 is identical to the Biot-Savart law (Eq. 24)

when current equals unity. The expression then reduces to:

ZS

Bd · dS = �MT ·Bu (32)

where Bu is the magnetic field at the location of the magnetic dipole due to the

transceiver coil when unit current is running through it. Earlier, we discussed how the

magnetic field of the coil and induced magnetization are perfectly aligned in the case of

powdered samples. The dot product then reduces to multiplication. Again treating the

dipole as time harmonic, the signal amplitude (Vd) induced in a transceiver coil by a single

dipole can be derived from Eq. 32 by applying Faraday’s Law along the contour defined by

the coil wires, and

Vd = j!MTBu (33)

Since the computational algorithm originally begins by calculating the magnetic field

due to the coil, Bu is already known (or attainable by a scaling factor). Finally, the total

signal (V) is the sum due to the N dipoles which the target is being modeled as:

V =NXi=1

j!MTBu (34)

This method is both simple and unambiguous. It is also a good demonstration of how

the principle of reciprocity applies to NQR. A coil that establishes nuclear magnetization

within a sample is ideally situated to detect that magnetization.

5.3 Programming the Model

We worked in conjunction with a group at Los Alamos National Laboratory that works

principally on Nuclear Magnetic Resonance (NMR). In NMR, energy level splitting in nuclei

20


occurs as a result of their magnetic dipoles’ interaction with a static magnetic field, as

opposed to the quadrupole interaction in NQR’s case. Like in NQR, NMR uses an RF

magnetic field to excite nuclei and establish magnetization. From the point of view of

electromagnetic modeling, NMR and NQR are quite similar. The Los Alamos group had

previously developed a toolbox in MATLAB to 1) calculate the magnetic field produced

by a coil in the magnetostatic approximation, 2) calculate the magnetization formed by

excited nuclei due to NMR in response to the field, and 3) calculate the return signal due to

this magnetization. We wrote additional functions for this toolbox that employ Eq. 23 to

calculate magnetization due to NQR. This function replaces step 2 of the described sequence,

allowing NQR problems to be modeled. The code that we used and extended at Los Alamos

cannot be released to the general public and is not included here.

Our own implementation of the free space model in MATLAB is conceptually identical

to that employed by [86] except for own method of final signal calculation described in 5.2.

Our contributions to the basic model include computation of nutation curves which were

compared against experimental data (section 5.4) and the use of 2D sensitivity maps to

visualize the data (section 5.5.1).

5.4 Validating the Model

Before using the model in any critical work, it is important to check it against experi-

mental data. Shinohara et al. programmed the model in Microsoft Excel VBA and used it

to generate a one dimensional sensitivity profile [86]. That is, they used the model to predict

how the NQR signal generated by a small cube of HMT would change as its position was

translated within a plane parallel to the coil. They did this with both a circular loop coil

and a planar gradiometer. They compared their program’s results against experiment and

the data agree well. To compare our implementation of the model against experiment, we

used data taken previously by Los Alamos researchers. This section describes the physical

experiment then compares its result to the simulation’s prediction.

5.4.1 Experimental Hardware - NQR Spectrometer

Dr. Young Jin Kim and Dr. Todor Karaulanov of Los Alamos built an NQR spectrome-

ter, then tested it on the 4.64 MHz line of sodium nitrite NaNO2. The spectrometer is based

on the design described by Hiblot et al. [30]. Grandinetti provides an excellent description

of spectrometry hardware in his NMR book [25], while other NQR-specific spectrometer

design considerations can be found elsewhere [24,75,99]. This subsection introduces how the

spectrometer used in our experiment functions.

21


The RF pulse or pulse sequence is programmed into a PC. The PC is connected to a

Tecmag Apollo console which generates the specified pulse train. Its output passes through

an amplifier capable of boosting the signal to approximately 350W and 2.75A peak current.

The amplifier’s output is connected to the “front end” of the spectrometer. The front end

includes the transceiver coil and matching and tuning components. After the pulse has

passed through the coil, the coil must be used in receive mode. Therefore, the front end is

also connected to a preamplifier. The signal from the preamp passes through a secondary

amplification stage (again, using the Apollo console) and back to the PC in order to record

the NQR signal.

The front end serves multiple functions and is generally home-built whereas the other

components are available as commercial, o↵-the-shelf technology. It consists of all the compo-

nents between the power amplifier (transmit mode) and preamplifier (receive mode). Figure

6 shows the front end Dr. Karaulanov and Dr. Kim chose to use in their spectrometer [30].

Using the nomenclature of this figure, L is the transceiver coil, while CM and CT are the

matching and tuning capacitors respectively. Taking into account the transceiver coil’s in-

ductance L, the values of CM, and CT are carefully selected to form a circuit which is resonant

at the particular NQR frequency chosen. That is the primary purpose of CT. Meanwhile,

the primary purpose of CM is to match the coil’s impedance to that of the preamplifier in

order to maximize the amount of power received by the preamp. The spectrometer used in

our experiment uses vacuum variable capacitors for CM and CT to allow for fine tuning (and

to switch to a di↵erent NQR frequency for another experiment).

Care must be taken to ensure that power from the amplifier flows into the coil, yet does

not damage the pre-amp during transmission. However, the power of the NQR signal must

be maximally transmitted to the preamp during receive mode. The remaining components

accomplish this. The components in the blue box (L0,C1,C2) are chosen to form a �/4

circuit at the NQR frequency being used. The �/4 circuit allows optimal transmission to

the preamplifier from the coil during receive mode. During transmit mode, the pair of cross

diodes near the pre-amplifier allows the transmit pulse to see a path to ground - making the

�/4 circuit behave as an open circuit. This blocks the transmit pulse from reaching (and

possibly damaging) the preamplifier. Since the signals used in NQR are very small (on the

order of nV), it is important to limit sources of noise. A second pair of cross diodes next

to the amplifier blocks small signals present in the amplifier from entering the rest of the

circuit during receive mode.

A solenoidal coil is used because it creates a strong, fairly uniform field through the

sample. It is made of 40 turns of 21AWG wire wrapped around a machined form. It is 3.5cm

in diameter and 7cm long.

22


Figure 6: The front end of the NQR spectrometer used is shown between the power amplifierand preamplifier. The elements in the blue box form a quarter wavelength circuit. Theelements in the red box provide tuning and matching. Figure from [30]

23


5.4.2 The Experiment

Measurements: A 100g sample of powdered sodium nitrite, NaNO2, was placed inside of a

cylindrical container and into the solenoidal transceiver coil of the spectrometer. A sinusoidal

pulse with frequency 4.64MHz was applied to the sample for a time ⌧ . The free induction

decay (FID) signal was measured 250µs after applying the pulse to allow ringing e↵ects in

the coil to subside prior to measurment. ⌧ ranged from 5 to 240µs. For all measurements,

the spectrometer’s power amplifier was set to 250W, which we estimate creates 2.22A peak

current through the coil. The experiment was conducted at room temperature (295K).

Simulation: We simulated this experiment using the NQR software described above. The

coil was modeled as a solenoid discretized into 1280 sections. The sodium nitrite was modeled

as a cylinder centered within the solenoid. We used the estimate of 2.22A for peak current

flowing through the coil. The characteristics of sodium nitrite (density, molar mass, NQR

frequency) were programmed along with the gyromagentic ratio of 14N and the sample’s

temperature. The maximum FID signal was calculated using Eqs. 24 , 23, and 31. Pulse

lengths covered the same range as the measured data and were incremented 1µs at a time.

Figure 7: A 100g cylinder of sodium nitrite (green) is simulated inside of a solenoidal coilof 40 turns of wire (blue) to replicate the physical experiment.

24


5.4.3 Results and Discussion

The measured and simulated results are plotted alongside one another in Figure 8.

This nutation curve describes signal intensity as a function of pulse length. For both the

experiment and simulation, we plot the absolute value of the received signal.

Figure 8: The measured and simulated nutation curves for a 100g sample of sodium nitrite.The received NQR signal is plotted as a function of pulse length while applied power is heldconstant.

The simulated and measured data agree fairly well. There are two notable di↵erences

between the data sets. First, the simulated data predicts that the length of the 90� and

180� pulses are slightly shorter than those which were measured (the plot of measured data

appears to be shifted ahead of the simulated data). For example, the 90� pulse (that which

maximizes signal) was predicted by simulation to be 80µs, whereas experimentally, it was

observed to be closer to 90µs. The second notable discrepancy is that the measured signal

did not return all the way to zero for the 180� pulse (that which brings the signal to zero).

There are many possible explanations for these discrepancies:

Both discrepancies could be partially attributed to variations in the sample’s NQR

characteristics across its volume. These variations can be created by subtle factors such as

temperature gradients, particle size, and the powder’s physical packing as discussed in 4.3. If

these variations were present within the sample, ine�cient excitation could result - thereby

requiring longer pulse times than predicted. The variations could also cause the net signal

to not average to zero during the 180� pulse since frequency shifts. These e↵ects would take

25


place due to C (Eq. 19) of the magnetization expression (Eq. 23) acting as a weighting

operator during summation (Eq. 31).

The fact that a non-zero NQR signal was recorded for the 180� pulse could be due to the

noise floor of the experimental apparatus. Measured NQR signal intensity is no more than

the FFT of the received signal. So it is very possible that the lowest values that were recorded

(0.08 in the normalized plot) are simply the values of noise in the system. Unfortunately,

we do not have any baseline measurements available with which to check whether this were

actually the case.

Another unknown is the time which is required for the applied pulse to reach its maxi-

mum amplitude. The simulation assumes that the pulse reaches its maximum value instan-

taneously and the antenna has no ringing. In real life, there is both a ring-up and ring-down

time. This could cause a discrepancy between commanded and e↵ective pulse lengths. We

do not have measurements of the ring up and ring down times for this particular experiment,

but the same spectrometer has displayed ringing in the past.

The peak to peak current of the applied pulse was estimated. If it was estimated to be

too high, that would explain the apparent shift of the measured data ahead of simulated in

the plot. A di↵erent error source could be the complex interactions between nuclei such as

dipole-dipole interactions. These e↵ects are not considered in the simple model.

In spite of these potential sources for error, the simulation and measured data agree

closely enough to demonstrate the simulation’s basic e↵ectiveness. Before being used in

critical work, it should be compared more rigorously against experiment to better understand

why and how it deviates from reality. For now, we consider our results and those taken by

Shinohara et al. to have demonstrated that the simulation is e↵ective in free space [86].

5.5 Utility of the Free Space Model

5.5.1 Sensitivity Map

The free space model can be used as a tool to optimize coil design. One way to evaluate

coils is with sensitivity maps. A sensitivity map is a plot showing how sensitive an NQR

coil is to material located at di↵erent points in space. To the best of our knowledge, this is

the first time that 2D sensitivity maps have been created for NQR. To generate a sensitivity

map, a plane is chosen a certain distance away from the coil. It is colorized according to the

NQR signal that would result from each voxel within that plane if a particular MOI were

present there. Sensitivity maps can be used to examine how changes in coil design, MOI,

pulse strength, and pulse length a↵ect a detector’s sensitivity.

26


Figure 9: Plane of interest 10cm away from a planar gradiometer coil. Detector sensitivityto material within this plane is mapped.

For our example, we design a planar gradiometer coil comprised of two circular coils.

Each circular coil has five loops of wire, spaced 5mm apart from one another. The inner and

outer diameters of each circular coil are 10cm and 14cm as shown in Figure 10.

Figure 10: The planar gradiometer used in our example. ID= 10cm, OD=14cm

The coil is set to detect the 3.5MHz line of RDX. Its peak current is set to 10A. Two

planes of interest are defined; one is 3.5cm from the coil and the other 7.0cm. In Figure

11a, the magnetic field magnitude is plotted for these two planes. We choose magnitude

as opposed to any particular component of the magnetic field since magnitude is what

27


determines the degree of excitation.

We then choose choose ⌧ = 285µs, and generate sensitivity maps for the two planes, as

shown in Figure 11b. We observe that the 285µs pulse results in a high return signal from

RDX that is located 3.5cm from the coil and within its central region. However, the signal

at 7cm is about four times weaker.

Next, we generate sensitivity maps for ⌧ = 600µs as shown in Figure 11c. In this case,

sensitivity in the central region at 3.5cm has plummeted, and may not be usable at all.

However, the peripheral region at 3.5cm returns a much stronger signal than was observed

for ⌧ = 285µs. Likewise, detection in the central region at 7.0cm has improved significantly

from the case of the shorter pulse. The return signal for this region is over twice as strong

as was observed for 285µs.

It is important to observe that the ⌧ = 600µs creates magnetization within the RDX

at 7.0cm that is roughly the same as the magnetization created by ⌧ = 285µs at 3.5cm.

However, since the RDX is further from the coil at 7.0cm, the received signal is weaker

despite magnetization being the same.

We return to this example in the next section, 6 - Coil Design Considerations. First,

consider another way to use the model to evaluate a coil design.

5.5.2 Signal from example targets

Sensitivity maps are best suited for qualitatively evaluating which regions a coil is most

sensitive to. It is also possible to study sensitivity to particular objects, as we did to create

the nutation curve in section 5.4.2. This section reviews how total signal from an example

object is calculated and illustrates how that capability may be useful in designing a coil.

First, an object’s material and resonant frequency are defined as with the making of

sensitivity maps. The object is then given a particular size, shape, and position near the

coil. The voxels which the object occupies are identified. Let these N voxels be indexed by

the index i.

The simulation is then run, calculating the sensitivity of each voxel over a large volume

just as was done to generate sensitivity maps. Once again, sensitivity is the peak voltage that

would be returned from that voxel if an MOI was present there. By summing the sensitivities

for each of the N voxels that comprise the test object, the total signal received from that

object is obtained. That is, st

=PN

i=1 si. The total signal, st

, is the initial amplitude of the

FID signal received from the object.

To demonstrate this, we use the same gradiometer coil as in the previous example with

28


(a) Magnetic field

(b) Sensitivity Maps for ⌧ = 285µs

(c) Sensitivity Maps for ⌧ = 600µs

Figure 11: The left column shows results for the plane 3.5cm from the coil. The right columnshows results at 7.0cm. In part a, the magnetic field magnitude is shown. Parts b and cshow sensitivity maps to RDX for pulse lengths of 285µs and 600µs respectively when the peaktransmit current through the coil is 10A. The shorter pulse length is superior at detectingobjects where the magnetic field is greatest. The longer pulse length is superior at detectionobjects where the magnetic field is weaker.

29


peak current of 10A. We create at cylindrical target to simulate 60g of RDX. Its height is

5.55cm and diameter is 2.75cm. We o↵set the cylinder from the coil’s center line and place

it so that its nearest point is 2.0cm from the coil as shown in Figure 12a. We calculate st

for

various pulse lengths from 100µs to 360µs to generate a nutation curve. We then rotate the

cylinder so that its end now faces the coil as shown in Figure 12b. Again its closest point

is 2cm away from the coil, but its orientation causes its center of mass to be moved further

from the coil. We repeat the experiment. Nutation curves for the cylinder’s two orientations

are shown in Figure 13. The end-on orientation resulted in a weaker signal that required a

slightly longer pulse for maximum excitation.

This modeling capability may be very useful when designing a coil. An array of test

objects can be created representing the spectrum of possible threats that a coil is designed

to detect. Any given coil can then be quickly tested against these test objects.

(a) cylinder oriented with its length parallel to coil

(b) cylinder oriented with its end facing the coil

Figure 12: A 60g cylinder of RDX is simulated in two di↵erent orientations with respect tothe transceiver coil.

30


Figure 13: Normalized nutation curves for the two simulated orientations of the cylinder.The cylinder with its end facing the coil has a lower signal than the cylinder parallel to thecoil.

31

6 COIL DESIGN CONSIDERATIONS

6 Coil Design Considerations

The previous examples suggest that the computational model can be a useful tool for

optimizing coil design and pulse length. For illustration, let us return to the example of

section 5.5.1 where it was shown that a much longer pulse length is required for optimal

stimulation 7cm from the coil than at 3.5cm. Suppose an operator is interested in detecting

potential objects at both those ranges. The operator could choose to use the short pulse

(⌧ = 285µs) to excite both ranges if a sensitivity of about 2⇥ 10�12 Volts/Voxel is enough

to overcome the noise present in the detector. Whether this sensitivity is high enough

depends on the type, quality, and bandwidth of the coil and circuits used. It also depends

on how many detections the operator has time to average in order to su�ciently improve

the SNR (recall, SNR /pN , where N is the number of measurements). Time constraints

(such as minimum required throughput for an airport detector) may limit the number of

measurements. This is especially the case for materials with long T1 such as TNT since this

increases the rest time necessary between successive measurements [24,27,36]. If ⌧ = 285µs

provides insu�cient sensitivity at 7.0cm, it may be worth finding a compromise value of

pulse length between 285µs and 600µs that provides su�cient sensitivity at all ranges. If no

such compromise value exists, the operator may be forced to conduct two separate searches

- one at the short pulse length and one at the long. However, it may be possible to eliminate

this issue altogether by a better coil design. This section introduces desirable qualities in a

coil as well as constraints that a designer will encounter.

6.0.3 Desirable qualities

Uniform Magnetic Field Magnitude: To overcome the di�culty of two or more optimal

pulse lengths for various detection regions, a coil should generate a magnetic field of uniform

magnitude. In the case of a field with perfectly uniform magnitude, a single pulse length

would optimally excite the MOI in any position. For this reason, coils should be designed

to project a field of uniform magnitude over the region being investigated. While it may be

obvious, it is important to note that uniform magnitude is not the same as uniform field.

Dependence on magnitude rather than field is a result of reciprocity, as discussed in section

4.3. This is the reason why planar gradiometers may be suitable coils despite creating a very

inhomogenous field.

Sensitivity: Coils that maximize the amount of magnetic flux received from the excited

sample maximize the received signal. This is accomplished by increasing the coil’s surface

area through increase in radius or the number of turns. Of course, doing so increases the

32


Q-factor of the coil, Q. This relationship between sensitivity and Q is mentioned in [24]:

SNR / Q1/2.

If these were the only considerations, one could simply construct large coils made of

thousands of turns of wire. Unfortunately, there are both operational constraints and those

due to physical e↵ects that limit the length of wire within a coil and its overall size.

6.0.4 Constraints

Size Constrained by Application: Certain applications may require small coils inca-

pable of projecting a uniform field for very far. The field of surface coils falls o↵ rapidly after

about half a radius from the coil [24].

Coil inductance: For the simple circuitry described in 5.4.1, there is a maximum accept-

able value for coil inductance. Exceeding this value will cause unacceptably long ring times.

Furthermore, the bandwidth of the coil may become too narrow to cover the range of fre-

quencies being examined or to accommodate unknown shifts in NQR frequency associated

with unknown temperature or impurities. More advanced circuits could be used to facilitate

use of higher Q coils [109].

Johnson-Nyquist noise: Another constraint is the Johnson-Nyquist noise generated by

the coil. The RMS thermal noise, vn

, of a length of wire with AC resistance R is given by:

v

n

=p

4kB

TR�f (35)

where, as before, k

B

is the Boltzmann constant and T is temperature. �f is the

bandwidth in hertz over which the noise is being measured - in our case, we would use

the bandwidth of the coil. Since noise scales with resistance (which in turn scales with

wire length), larger coils produce more thermal noise than smaller ones. When a coil is

su�ciently large, its thermal noise may dominate over that of the pre-amp and become

the limiting factor of the SNR. Thermal noise may be exacerbated by the skin e↵ect and

proximity e↵ect. These e↵ects can be mitigated by using litz wire [83] and hollow wire (such

as the outer shield of a coaxial cable) [35]. Thermal noise in NMR coils (which are very

similar to NQR coils) has also been studied [33].

33


6.0.5 Methodology of Design

If designing a coil from the beginning, physical size and coil inductance are the first

parameters to be limited. From there, obtaining an optimal design is a matter of balancing

uniformity of magnetization, sensitivity to various regions, and thermal noise.

If thermal noise of the coil is determined to be a significant problem for a given appli-

cation, it may be worth the e↵ort to include its e↵ect in the NQR model. For wire with

well-characterized noise properties, approximating thermal noise is not hard. The larger

di�culty would lie in combining this noise approximation with received NQR signal in order

to estimate SNR. Up to this point, we have been calculating the peak value of the FID

signal. To translate peak FID into a good estimate of SNR, one would have to account for

the decay of the FID signal according to the e↵ective relaxation time, T2⇤. More broadly,

one would also have to consider other relaxation times and the specific pulse sequence that

is being employed. Since a lot of NQR research has focused on signal processing, it may be

worthwhile to combine that knowledge with an electromagnetic-based NQR model such as

the ones discussed in this thesis.

34

7 NQR IN BIOLOGICAL TISSUE - BACKGROUND

7 NQR in Biological Tissue - Background

To our knowledge, no one has studied how biological tissue a↵ects the electromagnetic

fields used in NQR. The remainder of this thesis discusses a model that we developed to

address this question.

There are three principle ways in which biological tissue may e↵ect the fields used in

NQR: 1) field interaction with tissue’s magnetic permeability, 2) field interaction with tissue’s

electric permittivity, and 3) losses due to conduction. One can estimate that e↵ects due to

tissue’s magnetic permeability are negligible since the relative permeability of biological

tissue is one [87]. The problems of electric permittivity and conduction losses deserve more

attention.

The e↵ect of losses due to conductive media is well demonstrated both theoretically and

experimentally by Suits et al. [96]; additional mention of lossy media includes [26,29,31,33,

76]. Lossy media reduces the e↵ective Q factor of the coil by introducing ohmic losses. The

practical result of this is that better SNR may be obtained by keeping the coil at a certain

“lifto↵” distance away from the lossy media. Suits et al. characterize these ohmic losses and

identified the optimal lifto↵ distance for media with conductivity equal to and greater than

that found in human tissue and fluid [22, 96]. We therefore consider the problem of lossy

media for NQR to be solved. It should be noted that the eddy currents established in lossy

media are considered too weak to otherwise impact the magnetic field.

The final unknown for doing NQR near biological tissue is the e↵ect of dielectrics.

The relative permittivity of biological tissue is very high at NQR frequencies. For NQR

frequencies, permittivity typically increases as frequency decreases. Some examples for the

upper and lower end of the NQR range are given in Figure 14.

As far as we know, the e↵ect of dielectrics has never been studied for NQR although Suits

et al. do mention it as a potential problem. The question has been discussed for Nuclear

Magnetic Resonance (NMR) and Magnetic Resonance Imaging (MRI) which is based on

NMR. Since MRI is used on human patients, any e↵ects observed due to the dielectric

properties of the body help inform whether NQR can be successfully used to search for

explosives concealed within tissue.

NMR is very similar to NQR except that the energy splitting of the nucleus occurs due

to an applied static magnetic field rather than the EFG established by atomic structure.

Like in NQR, an RF magnetic pulse is applied to excite nuclei and the nuclei’s subsequent

radiation is detected. NMR is the phenomenon at the core of Magnetic Resonance Imaging

(MRI). While studying dielectric e↵ects, the crucial di↵erence between NMR and NQR is

that NMR frequencies are typically much higher. For protons (hydrogen nuclei), NMR

35

7 NQR IN BIOLOGICAL TISSUE - BACKGROUND

Figure 14: Approximate values of relative permittivity for various biological tissue. Datataken from the graphs compiled by Gabriel and Gabriel [22]

frequencies typically range from 20-500 MHz depending on the strength of the static field

applied. High field and low field experiments extend this range.

One very recent NMR paper by Harimoto et al. examined the e↵ect of dielectrics such

as saline on NMR measurements and the resulting MRI image [28]. They found that at

63.5MHz, the dielectric properties of tissue have a significant e↵ect on MRI images. They

further used finite element modeling to show that this was because of distortion of the

magnetic field. Other research has shown both analytically and experimentally that MRI

measurements can be a↵ected by dielectrics [32,33,43,103,104,106,107]. However, the NMR

research also suggests that at low frequencies, dielectric losses may be insignificant. For

example, an analysis by Redpath and Hutchison modeled dielectric losses as stray capacitance

[73]. They found dielectrics to be insignificant at the studied frequencies (less than 16MHz).

While their experiment was specific to NMR rather than NQR, the frequencies used are

similar.

Our goal is to determine the significance of dielectric e↵ects on NQR measurements.

To do so, we develop a computational algorithm appropriate for low frequencies. Rather

than analyzing dielectric losses as stray capacitance, we study how the magnetic field itself

is a↵ected. Following the timeline of actual development, we first present an algorithm

designed to work in two dimensions by exploiting cylindrical symmetry. We then present a

generalization to completely arbitrary coils and targets in three dimensions.

36

8 CORRECTING FOR DIELECTRIC MEDIA

8 Correcting for Dielectric Media

8.1 The Darwin Model

In the free space model described previously, the electric field can be ignored since the

magnetic field is determined solely by the distribution of current through the coil. However,

when dielectrics are introduced, the electric field will reconfigure itself in response to them,

changing the magnetic field in the process. To account for dielectrics, we must adopt a model

that recognizes the coupling of electric and magnetic fields.

Maxwell’s equations are di�cult to directly implement computationally for this problem.

Techniques that our lab specializes in such as Finite Di↵erence Frequency Domain (FDFD)

and Finite Di↵erence Time Domain (FDTD) could possibly be used but are better suited

for higher frequencies. Furthermore, an FDFD or FDTD solution in three dimensions is

very computationally intensive, and it can be di�cult to specify arbitrary coils using these

methods.

The distinguishing feature of NQR systems is their low frequency, so quasistatic mod-

els can be used to simplify the problem. The electroquasistatic model (EQS) calculates

the electric and magnetic fields directly from sources (the original charge distribution and

current). This means that it fails to capture the link between electric and magnetic fields

and is therefore unsuitable for our purpose. The magnetoquasistatic model (MQS) does a

better job since it includes dependence of the electric field upon the time derivative of the

magnetic field. However, it does not provide a reciprocal means of calculating the magnetic

field from the electric. This is because it is only capable of handling current distributions

with zero divergence (r ·J = 0, where J is current density). Although, the current distribu-

tion through the coil is divergenceless, it becomes necessary to describe the correction field

due to dielectrics in terms of a diverging current distribution. E↵ectively, the requirement

that r · J = 0 prevents dielectrics from a↵ecting the magnetic field in our application. An

excellent overview of the quasistatic models discussed here is given by Larsson [44].

The Darwin model is very similar to MQS except that it uses the continuity equation

for charge density (⇢) and current density (J) [44]. That is,

@⇢

@t

+r · J = 0 (36)

As will be seen, this relationship is crucial to our method of calculating the e↵ect of

dielectrics on the magnetic field. The other expressions that make up the Darwin model

can be expressed in integral form. Magnetic field is calculated with the Biot-Savart law as

37


expressed in terms of time-dependent current density,

B(r, t) =µ0

4⇡

Z Z ZJ(r0, t)⇥R

R

3d⌧ 0 (37)

where r is field position, r0 is source position, R is the vector from the source to field

position, R is that vector’s magnitude, and ⌧

0 is the region which includes all sources.

The electric field is expressed in two separate parts:

E = EF

+ EC

(38)

The Coulomb electric field, EC

, comes from charge density:

EC

(r, t) =1

4⇡✏0

Z Z Z⇢(r0, t)R

R

3d⌧ 0 (39)

The Faraday electric field, EF

, comes from the magnetic field:

IE

F

· dl =ZS

(�@B

@t

) · dS (40)

EF

(r, t) = � 1

4⇡

Z Z Z@B(r0, t)

@t

⇥ R

R3d⌧0 (41)

8.2 The Algorithm

In the case of NQR detection, there are no free charges, and so EC

is initially zero.

However, the Faraday portion of the electric field can be calculated either using Faraday’s

law (Eq. 40) or Eq. 41. Doing so, it will be found that EF

is continuous. However, there is

a boundary condition on the electric field at the interface of any two dielectric media. The

perpendicular component of electric flux density across any boundary is equal to the surface

charge density present on that boundary:

E?1✏1 � E?2✏2 = � (42)

where E?1 and E?2 are the components of E perpendicular to the interface, directly

above and below the boundary. ✏1 and ✏2 represent respective dielectric constants. � is the

surface charge density that exists on the boundary between media and is equal to zero in

the case of NQR problems. The Faraday portion of the electric field, EF

, is a function of

the magnetic field and is determined without any consideration of dielectric media. Since

38


� = 0, EF

does not satisfy the boundary condition in general. Therefore, we must introduce

a correction term which allows the total electric field, E, to satisfy the boundary condition.

We can decompose the boundary condition into its Faraday and Coulomb components. The

charge density is set to zero to reflect physical truth:

EF?1

✏1 � EF?2

✏2 + EC?1

✏1 � EC?2

✏2 = 0 (43)

Next, we explicitly identify the amount by which EF

fails to meet the boundary con-

dition. This term can be thought of as a fictitious correction charge; we call it �

C

for its

connection to the Coulomb component of the electric field (EC

). It is attained by rearranging

Eq. 43:

EF?1

✏1 � EF?2

✏2 = �EC?1

✏1 + EC?2

✏2

EF?1

✏1 � EF?2

✏2 = �

C

(44)

�EC?1

✏1 + EC?2

✏2 = �

C

(45)

Noting the similarity in form of Eq. 44 and Eq. 45 to the boundary condition, we can

treat �C

as a fictitious surface charge density. Its value is calculated from Eq. 44. Then, via

Eq. 45, �C

creates EC

. That is, EC

is the correction term to EF

. When EC

is calculated

this way and summed with EF

, the original boundary condition (Eq. 43) is satisfied.

At this point, we have identified the proper electric field in the presence of dielectric

media. The task then turns to finding the associated magnetic field. At first, this may seem

like a di�cult task since (in our model) the magnetic field is dependent only upon current

density. It is possible because we have altered the current density from its original value by

introducing the correction charge, �C

. Rather than computing the magnetic field directly

from the electric, we can convert �C

into a current density then apply Eq. 37. Integrating

both sides of Eq. 36 and employing the divergence theorem, we obtain:

Z Z Z@⇢

@t

d⌧ 0 = �{

J · s dS (46)

where the right hand side integrates over that surface which encloses the volume over

which the left hand side is integrated. We are interested in finding the current density due

to the correction charges. We call this JC

to distinguish it from the current density of the

coil, JI

. We select a small portion of the boundary between two dielectric media with area,

A. The charge on that boundary is then qC

= �

C

A. By selecting A to be su�ciently small,

we can treat QC

as a point charge. Then, employing Eq. 46 with spherical symmetry, we

39


obtain an expression for current density. We assume sinusoidal excitation, and obtain

JC

=�j!Q

C

4⇡r2r (47)

where r is the distance from the charge to the point at which JC

is being evaluated and

r is the unit vector pointing from the charge to the point of evaluation.

8.3 Summary

The initial current density of the coil is used with the Biot-Savart Law to calculate the

initial magnetic field. The electric field associated with the initial magnetic field is called

EF

and fails to take account of the presence of dielectrics. The fictitious correction charge

surface density �

C

is calculated from EF

at each interface where permittivity changes. The

continuity relation is used to calculate the correction charge density, JC

from �

C

. The

correction charge density is used with the Biot-Savart Law to calculate the correction to the

magnetic field that exists as a result of nonuniform permittivity. Finally, the NQR model of

section 5.1 can be applied with the corrected magnetic field.

40

9 CALCULATING ELECTRIC FIELD - JACOBI METHOD

9 Calculating Electric Field - Jacobi Method

Before employing the described correction algorithm in an NQR application, we wish to

test that it does a suitable job of correcting the electric field. We start with a two dimensional

test. The code is given in appendix A.

First, a square domain is defined in Cartesian coordinates that is 1.2m on each side. The

domain is discretized into pixels 0.25mm on each side. There are approximately 16 million

total pixels.

A rectangular dielectric object is then defined in the middle of the grid. It is small

compared to the grid, only 8cm x 20cm to keep it far from the domain’s edges. It is assigned

relative permittivity of 8. The edges of the dielectric object are a gradient with permittivity

decreasing to unity over the span of 3.025mm. This is done to facilitate convergence of the

solution.

Next, an electric field is defined to exist with a uniform strength of one V/m, pointing

in the x direction over the entire domain. This field represents the free space field created

by a coil. It is physically unrealizable with the dielectric present. To correct the electric

field, correction charges are identified on the edge of the object (in the gradient zone where

permittivity decreases from 8 to 1). These charges were calculated with equation 44, which

is the discontinuity in the normal component of electric flux density across a boundary.

For our algorithm, the correction charges are then treated as free charges and the dielec-

tric object is temporarily ignored. The electric potential � is related to free charge density,

⇢

f

, by Poisson’s equation:

r2� = ⇢

f

/✏ (48)

where ✏ is the permittivity of the medium.

This equation can be solved using the Jacobi method [81]. We use the Jacobi method

rather than a relaxation technique in part for its conceptual simplicity. With the Jacobi

method, it is easy to exploit the vectorized method of calculation that MATLAB excels

at. The potential took about nine hours to compute on a MacBook with 2.6GHz Intel i7

processor and 16GB of 1600MHz DDR3 RAM.

Finally, the electric field was calculated from potential. This field is the correction field

which must be subtracted from the original field in order to find the corrected field.

Plots of the final, corrected electric field and electric flux density were examined and

found to satisfy physical boundary conditions. That is, the component of electric field

tangential to the boundary was continuous while the component of electric flux density

normal to the boundary was continuous. Thus, the procedure of introducing correction

41


charges and subtracting the field that they produce corrects for the presence of dielectric

objects.

The various steps of the procedure are shown in figures 15 and 16.

42


1. 2.

3. 4.

Figure 15: 1) A dielectric block is defined (shown in red). 2) An expanded view of the block.3) The initial electric flux density appears identical to the block. This is because the electricfield is initially uniform everywhere. 4) The correction charges are identified. Since theelectric field points downward along the vertical direction of the plot, they appear along thetop and bottom edge of the dielectric. In all figures of fields and flux density, only the x

component is shown. The x component points down vertically in the figure. All units are SI.

43


1. 2.

3.

Figure 16: 1) The correction electric field is calculated from the correction charge density.It strengthens the electric field within the dielectric and weakens it in free space. 2) Thetotal electric field is calculated by adding the correction field to the original. Note that it isappropriately continuous for the boundary parallel to its direction (downward vertical). 3)The total electric flux density now obeys the boundary condition that its normal componentbe continuous across boundaries. Compare this figure with the initial flux density in 3 ofFigure 15. In all figures of fields and flux density, only the x component is shown. The x

component points down vertically in the figure. All units are SI.

44

10 CALCULATING THE EFFECT OF DIELECTRIC MEDIA WITH AXIALSYMMETRY

10 Calculating the E↵ect of Dielectric Media with Ax-

ial Symmetry

10.1 Calculating the initial electromagnetic field

For simplicity, the first simulation to employ the algorithm of section 9 exploited cylin-

drical symmetry to calculate the initial electromagnetic field established by the coil. This

section describes that implementation.

A coil made of a single, circular loop is studied. Cylindrical coordinates are used where

⇢ points away from the coil’s center, � points in the direction of the coil’s curve, and the z

axis passes through the coil’s center. The coil is centered about the z axis at z = 3.25cm.

The loop has a radius of 10cm and is discretized into three three hundred sections. A

current of 1A is specified and the Biot-Savart Law is used to calculate the magnetic flux

density. The field values were calculated at points along a plane orthogonal to the plane

occupied by the coil (the plane’s axes point in the z and ⇢ directions). Magnetic flux density

is shown in Figure 17. The plane covers a region of 15cm along the z axis and 20cm along

the ⇢ axis. Sampled every 0.5mm, this creates a grid of 401x301 points. Via symmetry,

B�

= 0.

For a coil of radius a, the analytic expression for the axial component of magnetic field

at ⇢ = 0 is:

Bz

=µ0

4⇡

2⇡a2I

(z2 + a

2)3/2(49)

The simulation and analytic expression agree very well for all points along the z axis

(percent error less than 0.8%). We conclude that discretizing the coil into three hundred

points is su�cient.

The circumferential component of EF

can be calculated using Faraday’s Law, while the

other components remain unknown. This is a satisfactory approximation for our purpose

since the circumferential component dominates for a circular loop. Obtaining the electric

field requires that we introduce time variation to the problem. A frequency of 5MHz was

chosen.

To employ Faraday’s law, we compute the magnetic flux passing through planes parallel

to that occupied the coil. The result is plotted in Figure 18.

45


Figure 17: The magnetic flux density generated by a coil with 10cm radius and 1A of currentis plotted. This view shows a cross-section of the coil. As a result, the coil is only visibleat two points within the plot (located at ⇢ = ± 10cm, z = 3.25cm). The region that coil iscentered over is the region used in NQR detection. This region is strongly dominated by thez component of the field.

46


Figure 18: The circumferential component of the electric field, E�

, generated by the circularcoil.

47


10.2 Defining the Dielectrics

The boundary condition (Eq. 43) pertains to the component of the electric field that is

normal to the boundary between dielectric media. In this example, the initial electric field

has only a circumferential component. Therefore, to study the e↵ect of dielectric objects,

it is necessary to simulate objects with discontinuities along the � direction. This requires

breaking the axial symmetry that has been used so far.

To extend the volume being studied to three dimensions, we introduce the � dimension.

The 2-D solutions for initial electric and magnetic field are replicated across � to attain the

free-space 3-D solution. With this new 3-D representation, it is possible to generate plots

of the electric and magnetic field in planes defined parallel to the coil as we did in section

5.5.1. An example is Figure 19, which shows the z component of the magnetic field.

Some care must be taken in handling the data now that it is stored in Cartesian co-

ordinates. The data is initially stored as a cylindrical 3-D array which must be converted

into Cartesian coordinates. The resulting array is not monotonic as is the usual case when

a Cartesian grid is made from the standard meshgrid.m and ndgrid.m functions.

Figure 19: The z component of the magnetic flux density is plotted for a plane defined 1cmaway from a coil. The coil has a radius of 10cm and 1A of current.

48


With the new 3-D coordinates, it is possible to define a geometry array, herein referred

as the GT array. This is an array which contains information about the position of physical

objects and their electric properties. For this example, we choose to make a very simple

object. Our array containing information about the volume is discretized into voxels that

have a chockstone shape. Voxels are defined to lie between radii ⇢1 and ⇢2, angles �1 and

�2, and between axial positions z1 and z2. The GT array contains information about what

substance each voxel in the volume is made of. Objects are constructed by defining the

material of a certain set of voxels. For simplicity, we choose to construct objects that have

the same basic chockstone shape as an individual voxel.

To visualize the constructed object, several MATLAB functions initially appear suitable

such as isosurface.m and isonormals.m. However, these functions rely on the underlying

coordinate system being Cartesian and monotonic. While we have converted our coordinate

system to Cartesian coordinates from cylindrical, it is still not monotonic. Therefore, we

must write our own function to visualize objects defined in cylindrical coordinates. It is

powered by the low-level patch.m function. This function takes the eight corner points of

each voxel and uses them to define its six faces. Each face is broken into two triangles which

patch.m can be used to plot. In reality, the two voxel faces that have normal components

along the radial direction are curved. However, since each voxel is small, we can approximate

them as being planar for visualization purposes. A sample object is shown in Figure 20.

Figure 20: Simple chockstone shaped objects are created by defining adjacent voxels to bemade of the same material.

49


10.3 Finding the correction charges and current density

With the dielectric objects defined, the correction charge can be calculated. First, the

electric flux density, DF

is calculated by multiplying EF

at each voxel by the dielectric

constant of the material occupied by that voxel (as defined by the GT array).

Next, it is necessary to identify discontinuities in DF

. The boundary condition (Eq. 43)

specifies that only the component of DF

normal to the boundary between media must be

considered. In this case, only the � component of DF

is known. Therefore, the boundaries

that matter are those for which permittivity changes in the � direction. Discontinuities in

DF

along the radial and axial directions can be ignored.

To calculate discontinuities in the circumferential direction, we shift DF

by one index

position along �, taking care that the last value stored in the array (at � = (2⇡/N)(N � 1))

properly wraps back to the first index position (at � = 0). The shiftedDF

array is subtracted

from the original DF

array in order to reveal discontinuities; the resulting array contains

information about the correction charge density (�C

) present at each point in the volume.

This shift-and-subtract operation is also useful for identifying locations where permittiv-

ity changes. For example, the operation can be applied to the GT array so that visualization

shows only the boundaries between media in the � direction. This operation is performed

in Figure 21.

Since the computer simulation is discretized, �C

calculated by the shift-and-subtract

method will generally be nonzero between all voxels, even including those for which permit-

tivity is constant and DF

is continuous in reality. With su�ciently fine discretization, �C

will approach zero for the points where DF

is continuous. To make the simulation more

e�cient, we manually set �C

= 0 where permittivity is known to be constant. This is done

by doing a shift-and-subtract operation on the GT array to identify boundaries. �

C

is set

equal to zero for all locations which do not coincide with a boundary.

It is important to note that that the indices of the new �

C

array do not align with the

coordinate arrays. Instead, �C

exists shifted 1/2 an index position along the circumferential

direction. We calculate QC

by multiplying �

C

by the area of the boundary between voxels.

QC

can then be treated as a point source.

The correction current density due to each correction charge is calculated using Eq. 47.

First, the current density at all points in the volume due to a single correction charge is

calculated in a single vector operation. The current density due to each subsequent charge

is calculated the same way and summed withf the prior contributions. Current density is

calculated this way since attempting to fully vectorize the calculation (the contributions due

to each point being calculated simultaneously) would overwhelm the computer’s RAM. The

50


(a) Objects Defined

(b) Edges Detected

Figure 21: Top: Two objects are shown. Bottom: The shift-and-subtract operation is per-formed on the objects in the top of the figure. The result reveals the faces of the objects whichhave surfaces normal to �. These are the faces for which �

C

will be nonzero.

51


current density is best visualized with a quiver plot as shown in Figure 22.

As expected, current density points from the boundary where the correction charge is

positive to where it is negative. Current density is strongest close to the boundary.

10.4 Calculating the Correction to Magnetic Field

Finally, the corrected magnetic field can be calculated from current density using Eq. 37.

Using the low-frequency approximation, the time dependence of Eq. 37 can be ignored. Eq.

37 calls for current density to be integrated over the source volume. We accomplish this by

multiplying each voxel’s current density by the source voxel’s volume. The computational

procedure is similar to that used to find the current density: the magnetic field at all

voxels due to a single voxel is calculated in a single operation and then summed with the

contributions due to all other voxels. The magnetic field is output in Cartesian coordinates.

Since this function must take N2 cross products where N is the number of voxels in the

volume, it can take a long time to run. For a volume discretized into a 25x50x25 grid, the

function takes about eight minutes to run on a Windows PC with a 3.0GHz dual core intel

processor and 4GB RAM.

To perform our first assessment of biological tissue on magnetic fields, we choose to

study liver tissue. With a dielectric constant of about 425 at 5MHz, it has one of the highest

permittivities of any tissue found in the human body. It is also a fairly large organ, weighing

about 1.5kg. We thus use the liver as an example of the “worst case” field distortion that

the body may create upon the magnetic fields used in NQR.

A chockstone shaped object with permittivity of 425 is created to simulate a piece of

liver tissue. Its length in the z dimension is 5.5cm and it has a total volume of 231cm3. It is

positioned with its nearest face 0.8cm from the plane of the coil. The simulated tissue and

its position are visualized in Figure 23. The domain of the simulation is discretized into a

68x70x39 grid for 185,640 total voxels. The correction magnetic field due to the simulated

liver tissue is calculated. The calculation takes 1.9 hours on a MacBook with 2.6GHz Intel

i7 processor and 16GB of 1600MHz DDR3 RAM.

10.5 Results and Discussion

The change in magnetic field: The z component of the magnetic field is studied since it

is strongly dominant for a circular coil, as is shown by Figure 17. The e↵ect of the simulated

tissue on the magnetic field is shown in Figure 24. The left column shows the initial magnetic

field created by the coil. This is the field that would exist in free space. The right column

52


(a) Edges Detected

(b) Edges Detected

Figure 22: Top left: An object (red) is shown in a plane in front of the coil. Top right: Thecorrection current density created in response to the object. Bottom: An enlarged view of thecurrent density plot. Note how the current density originates at the boundaries of the objectnormal to electric field.

53


Figure 23: Left: the simulated liver tissue viewed in a 2D plane parallel to the coil. Theplane is located 2cm from the coil. Right: a 3D view of the simulated liver from an angle.

shows the correction to the original magnetic field. The three rows show the field at planes

2cm, 4cm, and 6cm from the coil.

Qualitatively, the e↵ect of the dielectric object is as expected. The correction field is

strongest within and just outside of the dielectric object. However, much more important

than the shape of the correction field is its strength. It is far weaker than the free space

field. Note that the color axis used for the initial field is three orders of magnitude larger

than that used to depict the corrected plots.

To quantify the di↵erence in strength between the initial and correction field, we divide

the correction field by initial field. The result is plotted for the plane 2cm from the coil

(Figure 25). The value of the correction field is never greater than 4/10,000 of the maximum

initial field.

The e↵ect on nuclear magnetization: To estimate whether the correction field would

have any e↵ect on an NQR measurement, we recall Equation 23 which determines nuclear

magnetization as a function of field strength and pulse length. It is also helpful to review

Figure 5.

The nutation angle which maximizes nuclear magnetization is �B⌧ = 2.08 rad and this

is referred to as the ⇡/2 pulse. This pulse is used to achieve maximum FID signal.

Intuitively, it is apparent from the equation and Figure 5 that a correction field that is

4/10,000 the strength of the original will not have a significant e↵ect on magnetization for

the ⇡/2 pulse. This is because the derivative of Eq. 23 is zero at the function’s maximum.

A fractional change of 4/10,000 will result in a fractional change in magnetization that is

54


Figure 24: The left column shows the initial magnetic flux created by the coil. This is theflux that would exist in free space. The right column shows the correction to the originalmagnetic flux due to simulated liver tissue. The three rows show the field at planes 2cm,4cm, and 6cm from the coil from top to bottom.

55


Figure 25: The correction flux is divided by 9 ⇥ 10�6, the highest flux of the initial patternfor the plane 2cm from the coil (see the left column of Figure 24). The correction flux issmall compared to initial flux.

56


even smaller.

The e↵ect of departing from this optimal value is examined in Figure 26. It shows that

the percent change in magnetization is less than the percent change in nutation angle for

small changes. For example, a nutation angle 5% less than is optimum results in magneti-

zation that is 99.71% of the maximum value. Even with a 15% decrease in nutation angle,

magnetization remains over 97% its maximum value.

Nutation angle is directly proportional to the applied magnetic field. If pulse length is

held constant, the x-axis of Figure 26 could simply be relabeled “percent change in magnetic

field”. Therefore, the correction field calculated in our example for liver tissue would have

an unnoticeable e↵ect on maximum magnetization.

The correction field calculated for liver tissue was 4/10,000 the strength of the free space

field. Liver has one of the highest electrical permittivities in the body, so this is a “worst-

case” simulation. For argument, suppose that the correction field that exists in real life is

fifty times stronger than calculated. It would then be 2% of the free space field. Referring

to Figure 26, the nuclear magnetization would still be 99.95% of its maximum value. We

therefore expect the correction field to have a negligible e↵ect on NQR measurements for

any field application that uses 90� pulses.

However, many modern pulse sequences also make use of ⇡ pulses to refocus the spins

and create echoes. The ⇡ pulse is that which creates the first zero of nuclear magnetization

and occurs at 4.49 rad. Here, the derivative of magnetization is not zero, and so changes in

field strength will have larger e↵ect, plotted in Figure 26. A five percent change from the

⇡ pulse creates nuclear magnetization that is 10% of its maximum value when it should be

zero. For a pulse sequence using both ⇡/2 and ⇡ pulses, inhomogeneities in the magnetic

field will have larger impact on the ⇡ pulse’s e↵ect.

Although the derivative of the magnetization expression is not zero for the ⇡ pulse,

nor is it particularly large. For example, a 1% departure from the ⇡ pulse would create

magnetization that is 2% of its maximum value.

The correction field’s total e↵ect on a series of echoes may be larger than the e↵ect of an

individual pulse on the FID signal since there is the chance for error to accumulate during

each refocusing pulse.

Nevertheless, a correction field that is 4/10,000 the strength of the original field is very

small. This change is much smaller than the di↵erence in initial field strength that is likely to

exist between two points of a given target. Therefore, even in the case of refocusing pulses,

dielectrics are unlikely to have a noticeable e↵ect on the NQR signal.

57


Figure 26: Top: If the nutation angle experience a small change from its ⇡/2 value, nuclearmagnetization falls o↵ very slowly from its maximum. Magnetization is plotted as a functionof the percent change in nutation angle from the ⇡/2 pulse. Bottom: small deviations fromthe ⇡ pulse have larger consequences. A 5% change from the ⇡ pulse results in nuclearmagnetization decreasing from 0 to -10% of its maximum value.

58

11 GENERALIZATION TO ARBITRARY GEOMETRY

11 Generalization to Arbitrary Geometry

11.1 Concept

The simulation of the previous section exploited axial symmetry. This method was

conceptually simple and allowed Faraday’s Law to be employed directly to calculate the

initial (free space) electric field. The drawbacks of this approach are that only circular coils

can be studied, and that the radial and axial components of the electric field are ignored.

This section describes a more general approach that allows arbitrary coil geometries. The

code is given in Appendix B.

Without being able to employ the integral form of Faraday’s Law, we need a new way

to compute the electric field. Equation 41 from the Darwin model provides this means.

First, the magnetic field is calculated over a large region surrounding the coil. Equation

41 is then applied to calculate the Faraday component of electric field. Due to the close simi-

larity in form between this equation and the Biot-Savart law, the problem is computationally

identical to calculating the magnetic field from correction current density. Therefore, the

same semi-vectorized approach discussed in the last section is applied.

Once the initial (free space) electric field is known, the algorithm can proceed in a

manner identical to the previous section. The correction charges due to dielectrics and the

associated current density are identified. The correction magnetic field is then calculated.

11.2 Free Space Fields

Cartesian coordinates are chosen for the generalized program. A circular coil is defined

with a radius of 10cm. We simulate 5MHz frequency and 1A peak current. The z component

of the magnetic field is shown in Figure 27. The electric field is calculated with equation 41

as discussed in section 11.1.

The grid used in this example is made of 101x101x25 points spanning 0.3 x 0.3 x 0.072m

in 3mm increments. There are N total points and the time to compute the free space electric

field from magnetic field is three hours on a MacBook with 2.6GHz Intel i7 processor and

16GB of 1600MHz DDR3 RAM. Since the computation of correction magnetic field from

correction current density uses the same function, it also takes three hours for the N point

grid.

59


Figure 27: The z component of magnetic field is shown for a plane located 1.0cm from thecoil.

60


11.3 Defining Objects

As in the previous section, objects are created by choosing the permittivity of individual

voxels. In principle, any shape of object can be created this way. In this example, we

construct two di↵erent rectangular prisms of dielectric material as shown in Figure 28. They

are each assigned relative permittivity of 500 to simulate kidney tissue, which has one of the

highest permittivities of biological tissue. They are positioned with their nearest face 1cm

from the plane of coil.

Figure 28: A pair of objects with the same permittivity as kidney tissue are defined in carte-sian coordinates. The coil is shown in blue.

11.4 Correction Charge and Current

In the case of axial symmetry, only the faces of dielectric objects that were normal to

the circumferential direction were considered. With the generalized program, all sides of the

object are considered. The shift-and-subtract scheme of section 10.3 is applied for all three

dimensions to find the correction charges. In this scheme, the x component of electric field is

used to find charges that exist where media changes along the x direction. Likewise for the

y and z directions. The resulting current density from all charges is then calculated. Figure

29 shows the current density created by the pair of dielectric objects. The result is as we

expect. As was the case for axial symmetry (Figure 22), a net current density is established

within the object to compensate for the object’s presence.

An interesting e↵ect is that the current density curls about some of the sharp corners

61


Figure 29: Top left: The dielectric prisms as viewed in a plane 2cm from the coil. The coilis shown in black. Top Right: The current density 2cm from the coil is plotted. The coil’sposition is shown in black. Bottom: A larger view of the current density formed by the lowerleft object.

62


of the object. These create half-loops of current that are expected to create strong magnetic

field in their centers. This e↵ect may play a dominant role wherever dielectric objects present

sharp corners.

11.5 Correction Magnetic Field

The correction magnetic field is calculated from the current density using the Biot-

Savart Law. The calculation takes three hours since each voxel of the domain contributes to

the magnetic field at every other voxel. The results are shown in Figure 30.

The magnetic field is strongest at the corners of the objects. At these points, the

correction current either curls to form half loops of current or diverges strongly.

As was the case for axial symmetry, the correction field is much smaller than the original

field and is unlikely to have an e↵ect on NQR measurements. The correction field is on the

order of 10�10Wb compared with 10�6Wb for the initial field.

11.6 The e↵ect of frequency

Permittivity increases as frequency decreases, so it is worth investigating the e↵ect of

frequency on the correction field. As shown in Figure 14, the permittivity of biological tissue

at 1MHz is generally two to four times the permittivity at 5MHz. However, the electric

field is directly proportional to frequency, so the decrease from 5MHz to 1MHz results in a

field one fifth as strong. This is expected to create proportionally weaker correction charges.

Therefore, we expect the correction magnetic field to be smaller at 1MHz than at 5MHz.

Meanwhile, the magnetic field has constant strength regardless of frequency. Therefore, the

correction field is expected to be smaller in both absolute terms and relative to the initial

field at 1MHz than at 5MHz. We test this hypothesis with the same geometry as described

in the previous section. The permittivity of the objects is increased from 500 to 1100. The

resulting correction magnetic field is shown in Figure 31. As expected, the field is very

similar to the 5MHz case, but weaker all around.

11.7 Other Coil Designs

Many coils popular for NQR use are not simply circular loops. The generalized program

is capable of handling arbitrary coil shapes. Coils are discretized into a set of discrete

sections. Each section is defined by its position in space, direction, and length of the wire

which it represents. This can be thought of as a series of vectors lined up head to tail,

63


Figure 30: Top: The z component of the correction magnetic field 2cm from the coil. Bottom:The z component of the initial field. In both top and bottom, units are Wb.

64


Figure 31: The correction magnetic field at 1MHz. The color axis is kept the same as wasused in Figure 30 for 5MHz. The basic shape is the same of the field is the same as in the5MHz case, but it is weaker.

connecting consecutive positions sampled along the coil. Using this very general formulation,

many coil shapes are possible and there is no requirement of symmetry.

Mozzhukhin discusses using a coil with logarithmically spaced windings [60]. Such a

coil is shown in Figure 32. As another example, a gradiometer is made with linear spiral

windings and then compressed towards its center axis. It is shown in Figure 33.

11.8 Nonuniform Grid

The preceding calculations were performed with a single Cartesian grid with regular

spacing between voxels. It may be useful to use a nonuniform grid in some cases. For

example, in the region very close to the coil, the magnetic field changes much more rapidly

than it does far away from it. In order to extract as much information as possible from the

magnetic field in this region, fine discretization is necessary. However, fine discretization

over the entire volume of interest would lead to excessively long computation times. A

nonuniform grid may be used to achieve a high sampling rate near the coil while using a

lower sampling rate far away to keep computation time low.

We chose to employ a very simple nonuniform grid in which several uniform grids with

65


Figure 32: Top: A spiral coil with logarithmically spaced windings. Bottom left: The z

component of magnetic field 1cm from the coil. Bottom right: The magnitude of magneticfield 1cm from the coil.

66


Figure 33: Top: A gradiometer coil with linearly spaced spiral windings. Bottom left: Thez component of magnetic field 1cm from the coil. Bottom right: The magnitude of magneticfield 1cm from the coil.

67


di↵erent voxel size are placed adjacent to one another. The method is analogous to “stack-

ing slabs” where each slab has di↵erent spatial resolution. For example, a narrow box is

defined just large enough to contain the coil. This region is discretized with high resolution.

Next, grids are defined on either side of the original box. These secondary grids have lower

resolution than the first. A third grid is then added in the same manner. This style of grid

construction is visualized in Figure 34.

When a nonuniform grid is used, information in one grid section must be used to compute

fields in another. This capability has been programmed. Vector fields such as current density

or magnetic field in one grid can be used to determine magnetic field or electric field in a

di↵erent grid.

Figure 34: The blocks of this figure represent the di↵erent individual grids that contribute tothe total grid. Each prism (color) has di↵erent spatial resolution. In this visualization, thebright blue slab would contain the coil and be discretized with fine resolution.

68

12 CONCLUSION

12 Conclusion

12.1 Implications for BCB detection by NQR

As first discussed in section 10.5, dielectric e↵ects are unlikely to a↵ect NQR signals.

The tests shown above were designed to maximize the dielectric’s e↵ect on the NQR

signal. The permittivities tested represented the organs with the highest permittivities

within the body. Most other features of anatomy (skin, bone, muscle, fat, etc.) have similar

or lower permittivities. The objects were tested in free space to maximize the discontinuity

in permittivity at their boundaries. Within a body, the jumps in permittivity from one tissue

to another will be much smaller.

The correction field generated in these “worst case” scenarios was extremely small com-

pared with the field generated in free space. Even the largest were under 1/1,000 the strength

of the initial field. This was true even right at the edges and corners of dielectrics. For most

coils in field applications, the error introduced by natural inhomogeneity of the coil’s ini-

tial (free space) field over the span of a target will be much larger than the error due to

dielectrics. At the nutation angle which maximizes nuclear magnetization (90� pulse), the

partial derivative of nuclear magnetization with respect to magnetic field is zero. Therefore,

small changes in magnetic field result in even smaller changes in nuclear magnetization. It is

extremely unlikely for the dielectric properties of biological tissue to have an e↵ect on FID

measurements made with a 90� pulse. The e↵ect may be more significant in pulse sequences,

but is still expected to be very small.

The detection of body cavity bombs is unlikely to be a↵ected by the dielectric properties

of the body. Any losses due to the body are likely to be dominated by tissue’s conductive

properties. Existing literature indicates the e↵ect of lossy media to be manageable.

69

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A CODE FOR CALCULATING CORRECTION ELECTRIC FIELD

A Code for calculating correction electric field

This code is described and used in section 9. The main code and its supporting functions

are shown.

1 %% This code demonstrates how correction charges can be used to correct the

2 % electric field.

3 % (c) Michael Collins, 2014

4 % [email protected]

5

6 %% Define Domain, it is square.

7 clear all; close all;

8 tic;

9 h=.00027; L=1.2; % For fine discretization, try using h=.00027; L=1.2;

10 x=[0:h:L]; y=x;

11 M=length(x); N=M;

12

13 %% Define the dielectric

14 eps0=8.854e-12;

15 dm=100; dn=170;

16 eps=ones(M,N); epsM=8; k=11; deps=(epsM-1)/k;

17 %eps is the relative permittivity of each point

18 eps(floor(M/2)-dm:floor(M/2)+dm,floor(N/2)-dn:floor(N/2)+dn)=epsM;

19 for n=[1:k]; % Creates a gradient boundary to facilitate convergence

20 eps(floor(M/2)-dm-n:floor(M/2)+dm+n,floor(N/2)-dn-n)=epsM-n*deps;

21 eps(floor(M/2)-dm-n:floor(M/2)+dm+n,floor(N/2)+dn+n)=epsM-n*deps;

22 eps(floor(M/2)-dm-n,floor(N/2)-dn-n:floor(N/2)+dn+n)=epsM-n*deps;

23 eps(floor(M/2)+dm+n,floor(N/2)-dn-n:floor(N/2)+dn+n)=epsM-n*deps;

24 end

25 eps=eps.*eps0;

26

27 %%

28 dedx=diverg(eps,1,h); dedy=diverg(eps,2,h); dedx=zeroedge(dedx); ...

dedy=zeroedge(dedy);

29

30 %% Define charge distribution

31 %The following creates correction charges. We start with an applied field

32 %in x direction

33

34 EIx=ones(M,N); EIy=zeros(M,N); DIy=EIy.*eps; DIx=EIx.*eps;

35 DIx=EIx.*eps;

36 % Shifts Dx forwards:

80


37 idx = repmat({':'}, ndims(DIx), 1); % initialize subscripts

38 n = size(DIx, 1);

39 idx{1} = [ n 1:n-1 ];

40 Dx2 = DIx(idx{:});41

42 rho=(DIx-Dx2)./(2*h);

43 rho=zeroedge(rho);

44

45 idx = repmat({':'}, ndims(rho), 1);

46 n = size(rho, 1);

47 idx{1} = [ 2:n 1 ];

48 rho2= rho(idx{:});49 rho=rho+rho2;

50 rho=zeroedge(rho);

51

52

53 %% Calculate Potential, V, using Jacobi Method

54 V=zeros(M,N);

55 NCOUNT=0; Vold=zeros(M,N); Vsum=Vold; G=zeros(M,N);

56 loop=1;

57 while loop == 1;

58 Vold=V;

59 Vxm=shiftup(Vold,1); Vxp=shiftdown(Vold,1); Vym=shiftup(Vold,2); ...

Vyp=shiftdown(Vold,2);

60 Vxm=zeroedge(Vxm); Vym=zeroedge(Vym); Vxp=zeroedge(Vxp); Vyp=zeroedge(Vyp);

61

62 G=((eps+dedx*h/2).*Vxp)+((eps-dedx*h/2).*Vxm)...

63 +((eps+dedy*h/2).*Vyp)+((eps-dedy*h/2).*Vym)...

64 +rho*hˆ2;

65 V=G./(4.*eps);

66

67 dif=V-Vold; dif=abs(dif);

68 difsum=sum(sum(dif));

69 Vsum=sum(sum(abs(V)));

70 test=100.*difsum/Vsum;

71

72 if(test�.0000000001)

73 NCOUNT=NCOUNT+1;

74 if(NCOUNT>24000)

75 loop=0;

76 disp('Sol does not converge in 24000 iterations')

77 end

78 else

79 loop=0;

81


80 disp(['converges in ',num2str(NCOUNT),' iterations']);

81 end

82 end

83

84 %% Calculate E and D

85 % Correction Field

86 ECx=-diverg(V,1,h); ECy=-diverg(V,2,h);

87 DCx=ECx.*eps; DCy=ECy.*eps;

88 % Total Field

89 ETx=EIx-ECx; ETy=EIy-ECy;

90 DTx=DIx-DCx; DCy=DIy-DCy;

91

92 toc;

93

94 %% Plot results

95 % rho, eps, and initial electric flux density

96 figure;

97 imagesc([x(1),x(end)],[y(1),y(end)],rho(:,:)); axis image; colorbar;

98 title('rho','FontSize', 16); xlabel('Y axis (m)','FontSize', 14); ...

ylabel('X axis (m)','FontSize', 14);

99 figure;

100 imagesc([x(1),x(end)],[y(1),y(end)],eps(:,:)); axis image; %colorbar;

101 title('epsilon','FontSize', 16); xlabel('Y axis (m)','FontSize', 14); ...


102 figure;

103 imagesc([x(1),x(end)],[y(1),y(end)],DIx(:,:)); axis image; colorbar;

104 title('Initial Flux Density','FontSize', 16); xlabel('Y axis ...

(m)','FontSize', 14); ylabel('X axis (m)','FontSize', 14);

105

106 % Potential

107 figure;

108 imagesc(x,y,V(:,:)); axis image; colorbar;

109 title('Potential','FontSize', 16); xlabel('Y axis (m)','FontSize', 14); ...


110 % Correction E field & D field (x components)

111 figure;

112 imagesc([x(1),x(end)],[y(1),y(end)],-ECx(:,:)); axis image; colorbar; ...

caxis([-.6 1.1]);

113 title('Ex Correction','FontSize', 16); xlabel('Y axis (m)','FontSize', ...

14); ylabel('X axis (m)','FontSize', 14);

114 figure;

115 imagesc([x(1),x(end)],[y(1),y(end)],DCx(:,:)); axis image; colorbar; ...

%caxis([-10e5 10e5]);

116 title('Dx Correction','FontSize', 16); xlabel('Y axis (m)','FontSize', ...

82



117 % Total E field & D field (x components)

118 figure;

119 imagesc([x(1),x(end)],[y(1),y(end)],ETx(:,:)); axis image; ...

colorbar;%caxis([-10e5 10e5]);

120 title('Ex Total','FontSize', 16); xlabel('Y axis (m)','FontSize', 14); ...


121 figure;

122 imagesc([x(1),x(end)],[y(1),y(end)],DTx(:,:)); axis image; colorbar; ...

%caxis([-10e5 10e5]);

123 title('Dx Total','FontSize', 16); xlabel('Y axis (m)','FontSize', 14); ...


124

125 % Correction E field & D field (y components)

126 figure;

127 imagesc([x(1),x(end)],[y(1),y(end)],ECy(:,:)); axis image; colorbar; ...

%caxis([-1e-10 1e-10]);

128 title('Ey Correction','FontSize', 16); xlabel('Y axis (m)','FontSize', ...


129 figure;

130 imagesc([x(1),x(end)],[y(1),y(end)],DCy(:,:)); axis image; colorbar; ...

%caxis([-10e5 10e5]);

131 title('Dy Correction','FontSize', 16); xlabel('Y axis (m)','FontSize', ...


diverg.m:

1 %% Calculates 1-D Divergence.

2 %% NOTE: Border points are erroneous. They should be discarded.

3

4 function [DPx]=diverg(Px,dim,dX)

5 %Shift forward

6 idx = repmat({':'}, ndims(Px), 1); % initialize subscripts

7 n = size(Px, dim);

8 idx{dim} = [ n 1:n-1 ];

9 Px2 = Px(idx{:});10

11 % Shift backward



14 idx{dim} = [ 2:n 1 ];

15 Px3 = Px(idx{:});16

83


17 % Find difference and divide by 2*dX

18 % Backward shifted minus forward shifted

19 DPx=(Px3-Px2)/(2*dX);

20 end

shiftup.m:

1 function [Sx]=shiftup(Px,dim)

2 %Shift forward



5 idx{dim} = [ n 1:n-1 ];

6 Sx = Px(idx{:});7 end

shiftdown.m:

1 function [Sx]=shiftdown(Px,dim)

2

3 % Shift backward



6 idx{dim} = [ 2:n 1 ];

7 Sx = Px(idx{:});8

9 end

zeroedge.m:

1 %% Sets border values of 2D matrix to zero

2 function [out]=zeroedge(test)

3 n1 = size(test, 1); n2 = size(test, 2);

4 test(1,:)=0; test(n1,:)=0;

5 test(:,1)=0; test(:,n2)=0;

6 out=test;

7 end

84

B CODE FOR CALCULATING CORRECTION MAGNETIC FIELD

B Code for calculating correction magnetic field

This appendix contains the code used to calculate the correction magnetic field. It works

in Cartesian coordinates and does not rely on symmetry. Examples of its implementation

are shown in section 11. The main file is followed by the functions which it relies upon.

With the default demonstration values, it takes about 1,150 seconds to run on a MacBook

Pro with 2.6GHz Intel i7 processor and 16GB of 1600MHz DDR3 RAM.

1 % This code calculates the magnetic field due to a coil in the presence of

2 % dielectric media at low frequency.

3 % (c) Michael Collins 2014

4 % [email protected]

5

6 clear all; close all;

7 %% Declare Globals:

8 u=pi*4e-7; eps0=8.854e-12; f=5e6; w=2*pi*f;

9

10

11 %% Create the Grid.

12 % An array is constructed for each axis. It contains the discrete points

13 % on each axis. Input length units in meters.

14 dX=.02; Xmax= 0.15; % Xmax is 1/2 approximate domain length; dX is step ...

size

15 nX=ceil(Xmax/dX)+1; X=[0:dX:(nX-1)*dX]; Xzero=length(X);

16 temp=-fliplr(X); X=[temp(1:length(temp)-1),X];

17

18 dY=.02; Ymax= 0.15;

19 nY=ceil(Ymax/dY)+1; Y=[0:dY:(nY-1)*dY]; Yzero=length(Y);

20 temp=-fliplr(Y); Y=[temp(1:length(temp)-1),Y];

21

22 dZ=.01; Zmax= 0.035;

23 nZ=ceil(Zmax/dZ)+1; Z=[0:dZ:(nZ-1)*dZ]; Zzero=length(Z);

24 temp=-fliplr(Z); Z=[temp(1:length(temp)-1),Z];

25

26 [X3D,Y3D,Z3D]=ndgrid(X,Y,Z);

27

28

29 %% The following section may be uncommented to enable the nonuniform grid.

30 % This feature allows find discretization near the coil (original grid).

31 % It is used to calculate free space electric field. Support could be added

32 % to calculate the correction magnetic field as well by calculating

33 % correction charge density across the grids.

85


34

35 % Create the 2nd Grid

36 %dX2=.002; nX2=ceil(Xmax/dX2)+1; X2=[0:dX2:(nX2-1)*dX2];

37 %temp=-fliplr(X2); X2=[temp(1:length(temp)-1),X2];

38 %dY2=.002; nY2=ceil(Ymax/dY2)+1; Y2=[0:dY2:(nY2-1)*dY2];

39 %temp=-fliplr(Y2); Y2=[temp(1:length(temp)-1),Y2];

40 %dZ2=.003; Zmax2=.05;

41 %nZ2=ceil((Zmax2-max(Z))/dZ2); Z2=[max(Z)+dZ2:dZ2:max(Z)+nZ2*dZ2];

42 %temp=-fliplr(Z2); Z2=[temp,Z2];

43 %[X3D2,Y3D2,Z3D2]=ndgrid(X2,Y2,Z2);

44

45 % Create the 3rd Grid

46 %dX3=.01; nX3=ceil(Xmax/dX3)+1; X3=[0:dX3:(nX3-1)*dX3];

47 %temp=-fliplr(X3); X3=[temp(1:length(temp)-1),X3];

48 %dY3=.01;

49 %nY3=ceil(Ymax/dY3)+1; Y3=[0:dY3:(nY3-1)*dY3];

50 %temp=-fliplr(Y3); Y3=[temp(1:length(temp)-1),Y3];

51 %dZ3=.01; Zmax3=.09;

52 %nZ3=ceil((Zmax3-max(Z2))/dZ3); Z3=[max(Z2)+dZ3:dZ3:max(Z2)+nZ3*dZ3];

53 %temp=-fliplr(Z3); Z3=[temp(ceil(nX3/2):length(temp)),Z3];

54 %[X3D3,Y3D3,Z3D3]=ndgrid(X3,Y3,Z3);

55 %Ng=prod(size(X3D2));

56 %Nt=Ng+prod(size(X3D3))+prod(size(X3D));

57

58

59 %% Create a single turn, circular coil, radius cR, current cI

60 cz=-.025; % coil position along z axis.

61 cR=0.1; cI=1; cN=600; n=[1:cN]; % cN = number of sample positions

62 % Creates 3xN array of source points (CS) (sampled positions)

63 [CS(1,:),CS(2,:),CS(3,:)]=pol2cart(2*pi*n/cN,cR,cz);

64 % Creates vectors corresponding to direction of each section (CJ)

65 CJ(2,:)=CS(1,:)/cR; CJ(1,:)=-CS(2,:)/cR; CJ(3,:)=zeros(1,cN);

66 Ilength=cR*2*pi/cN; %Length of each coil arc.

67 CJ=CJ.*Ilength; clear Ilength;

68 % For coil visualization:

69 xL = 0; yL = 0;

70 th = 0:pi/50:2*pi;

71 xunit = cR * cos(th) + xL;

72 yunit = cR * sin(th) + yL;

73

74

75 %% Calculate the Magnetic Field

76 % The Biot Savart function is used to calculate magnetic field.

77 tic; k=u/4/pi;

86


78 [B,Bx,By,Bz]=biotsavartCS(dX,dY,dZ,X3D,Y3D,Z3D,CJ,CS,k);

79 B=B*cI; Bx=Bx*cI; By=By*cI; Bz=Bz*cI; % The magnetic field components

80 toc;

81 % This section can be uncommented for support of nonuniform grid

82 %[B2,B2x,B2y,B2z]=biotsavartCS(dX,dY,dZ,X3D2,Y3D2,Z3D2,CJ,CS,k);

83 %B2=B2*cI; B2x=B2x*cI; B2y=B2y*cI; B2z=B2z*cI;

84 %[B3,B3x,B3y,B3z]=biotsavartCS(dX,dY,dZ,X3D3,Y3D3,Z3D3,CJ,CS,k);

85 %B3=B3*cI; B3x=B3x*cI; B3y=B3y*cI; B3z=B3z*cI;

86

87

88 %% Plot the Magnetic Field

89 layer = 5; % specifies the plane of interest

90 figure;

91 set(gcf, 'renderer', 'zbuffer');

92 h=pcolor(X3D(:,:,layer),Y3D(:,:,layer),Bz(:,:,layer)); colorbar; ...

%caxis([-2e-5 5e-5]);

93 xlabel('x (m)','FontSize', 14); ylabel('y (m)','FontSize', 14); axis image;

94 set(h, 'EdgeColor', 'none');

95 title('Magnetic Field - z (Wb) ','FontSize',16); set(gca,'fontsize',15)

96 hold on;

97 p = plot(xunit, yunit,'k'); % plots coil's shadow

98

99

100 %% Calculate the Initial Electric Field

101 tic;

102 k=-w/(4*pi);

103 [EI,EIx,EIy,EIz]=biotsavartCD(dX,dY,dZ,X3D,Y3D,Z3D,B,k);

104 toc;

105 % This section can be uncommented for nonuniform grid support

106 % Grid 2's effect on Grid 1

107 %[EItemp,EIxtemp,EIytemp,EIztemp]=biotsavartCD234(dX2,dY2,dZ2,X3D2,Y3D2,Z3D2,B2,k,X3D,Y3D,Z3D);

108 %EI=EI+EItemp; EIx=EIx+EIxtemp; EIy=EIy+EIytemp; EIz=EIz+EIztemp;


110 %[EItemp,EIxtemp,EIytemp,EIztemp]=biotsavartCD234(dX3,dY3,dZ3,X3D3,Y3D3,Z3D3,B3,k,X3D,Y3D,Z3D);

111 %EI=EI+EItemp; EIx=EIx+EIxtemp; EIy=EIy+EIytemp; EIz=EIz+EIztemp;


113 %[EI2,EIx2,EIy2,EIz2]=biotsavartCD234(dX,dY,dZ,X3D,Y3D,Z3D,B,k,X3D2,Y3D2,Z3D2);


115 %[EItemp,EIxtemp,EIytemp,EIztemp]=biotsavartCD(dX2,dY2,dZ2,X3D2,Y3D2,Z3D2,B2,k);

116 %EI2=EI2+EItemp; EIx2=EIx2+EIxtemp; EIy2=EIy2+EIytemp; EIz2=EIz2+EIztemp;


118 %[EItemp,EIxtemp,EIytemp,EIztemp]=biotsavartCD234(dX3,dY3,dZ3,X3D3,Y3D3,Z3D3,B3,k,X3D2,Y3D2,Z3D2);

119 %EI2=EI2+EItemp; EIx2=EIx2+EIxtemp; EIy2=EIy2+EIytemp; EIz2=EIz2+EIztemp;

120 % Results aren't calculated within grid 3 since it's only used as a source

87


121 % to capture effects of magnetic far field.

122 EIphi=GenAx(EIx,EIy,EIz,X3D,Y3D); % Finds circumferential component of ...

field

123 %EIphi2=GenAx(EIx2,EIy2,EIz2,X3D2,Y3D2);

124

125

126 %% Plot the Electric Field


128 figure;


130 h=pcolor(X3D(:,:,layer),Y3D(:,:,layer),EIphi(:,:,layer)); colorbar; ...

%caxis([-2e-5 5e-5]);



133 title('Electric Field - circumferential (V/m) ','FontSize',16); ...

set(gca,'fontsize',15)

134 hold on;


136

137

138 %% Create the GT array (define objects)

139 GT=zeros(size(X3D));

140 GT(3:5,6:10,2:3)=2; % object defined in this position

141 %GT(28:38,28:38,8:25)=2; % a second object could be created ...

142 eps=ones(size(GT)); % GT stores info of which object is where. Eps stores

143 % the relative permittivity of those objects and free space.

144 ind=find(GT==2); eps(ind)=1100; % Objects set to kidney permittivity @ 5MHz

145 % Other objects could be created with different permittivities:

146 %GT(19:33,19:33,14:20)=3;

147 %eps=ones(size(GT));

148 %ind=find(GT==3); eps(ind)=425; % Liver @ 5MHz

149

150

151 %% Visualize the GT file

152 ind=find(GT==2);


154 [h]=visvox(X3D,Y3D,Z3D,dX,dY,dZ,ind)

155 hold on; zunit=zeros(size(xunit)); zunit(:)=cz; % Plots coil

156 p = plot3(xunit, yunit,zunit);

157 set(gca,'fontsize',13); set(gcf, 'Color', 'w');

158

159

160 %% Calculate Correction Charges

161 [qx]=charge(EIx,GT,eps,1,eps0,dY,dZ);

88


162 [qy]=charge(EIy,GT,eps,2,eps0,dX,dZ);

163 [qz]=charge(EIz,GT,eps,3,eps0,dX,dY);

164 qM=zeros([size(qx) 3]);

165 qM(:,:,:,1)=qx; qM(:,:,:,2)=qy; qM(:,:,:,3)=qz;

166 % Visualize correction charges associated with x axis

167 ind=find(qx);

168 [h]=visvox(X3D,Y3D,Z3D,dX,dY,dZ,ind);

169

170

171 %% Calculate Correction Current

172 Jx=zeros(size(X3D)); Jy=Jx; Jz=Jx;

173 % We must account for the qc being a half grid point advanced in its

174 % respective dimension from the position indicated in grid arrays.

175 for dim=[1 2 3];

176 qnow=qM(:,:,:,dim);

177 ind=find(qnow); l=zeros(1,3); l(dim)=1;

178 X1=X3D(ind)-l(1)*dX/2;

179 Y1=Y3D(ind)-l(2)*dY/2;

180 Z1=Z3D(ind)-l(3)*dZ/2;

181

182 Jxs=zeros(size(X3D)); Jys=Jxs; Jzs=Jxs;

183 for p=1:size(X1)

184 vx=X3D-X1(p); vy=Y3D-Y1(p); vz=Z3D-Z1(p);

185 mag=(vx.ˆ2+vy.ˆ2+vz.ˆ2).ˆ0.5;

186 J1x=(w/(4*pi))*qnow(ind(p))*vx./mag.ˆ3;

187 J1y=(w/(4*pi))*qnow(ind(p))*vy./mag.ˆ3;

188 J1z=(w/(4*pi))*qnow(ind(p))*vz./mag.ˆ3;

189 Jxs=Jxs+J1x; Jys=Jys+J1y; Jzs=Jzs+J1z;

190 end

191 Jx=Jx+Jxs; Jy=Jy+Jys; Jz=Jz+Jzs;

192 end

193 J(:,:,:,1)=Jx; J(:,:,:,2)=Jy; J(:,:,:,3)=Jz;

194

195

196 %% Visualize Corection Current

197 layer=3;

198 figure;

199 quiver(X3D(:,:,layer),Y3D(:,:,layer),Jx(:,:,layer),Jy(:,:,layer));

200 xlabel('x (m)','FontSize',14); ylabel('y (m)','FontSize',14); axis image;

201 hold on; h = plot(xunit, yunit,'k');

202 set(gca,'fontsize',15); set(gcf, 'Color', 'w');

203

204

205 %% Calculate the magnetic flux density due to correction charges

89


206 k=u/(4*pi); tic;

207 [BC,BCx,BCy,BCz]=biotsavartCD(dX,dY,dZ,X3D,Y3D,Z3D,J,k);

208 toc;

209

210 %% Calculate size of the correction field relative to the original field

211 dif=BC./B;

212

213 %% Plot correction magnetic field


215 figure; set(gcf, 'renderer', 'zbuffer');

216 h=pcolor(X3D(:,:,layer),Y3D(:,:,layer),BCz(:,:,layer)); colorbar; ...

%caxis([-2e-5 5e-5]);



219 title('Magnetic Correction Field - z component ','FontSize',16); ...

set(gca,'fontsize',15)

220 hold on;


biotsavartCS.m:

1 %% Biot-Savart Function - For a Coil

2 % This function operates in Cartesian Coordinates. It accepts a vector

3 % field, J, as well as the positions that J occupies, S.

4 % J is an 3xN array, corresponding to x,y,z values of J.

5 % S is an 3xN array, corresponding to x,y,z values of S.

6 % J and S are the same length.

7

8 % The Biot-Savart law is applied to J to calculate the field values, A, at

9 % points specified by X3D, Y3D, and Z3D.

10 % X3D,Y3D, and Z3D have the same size. Ax, Ay, and Az each have the same

11 % size as X3D, Y3D, and Z3D. A is a 4-D array made up of Ax,Ay, and Az

12 % lying atop one another in the fourth dimension.

13

14 % dX,dY,dZ, represent the step size in the respective directions

15

16 % The constant k is applied to A.

17

18 function [A,Ax,Ay,Az]=biotsavartCS(dX,dY,dZ,X3D,Y3D,Z3D,J,S,k)

19 clear R; R=zeros([size(X3D) 3]); A=R; A1=R; Jtemp=R;

20 V=1; %dX*dY*dZ; %Biot Savart law is to integrate over SOURCE, NOT VOXEL

21 length=size(S,2);

22 for f = 1:length;

90


23 R(:,:,:,1)=X3D-S(1,f);

24 R(:,:,:,2)=Y3D-S(2,f);

25 R(:,:,:,3)=Z3D-S(3,f);

26 Rmag=(R(:,:,:,1).ˆ2+R(:,:,:,2).ˆ2+R(:,:,:,3).ˆ2).ˆ0.5;

27 Rmag=repmat(Rmag,[1 1 1 3]);

28

29 %Jtemp=repmat(J(:,f),[size(X3D),1]); % Is this right?

30 Jtemp(:,:,:,1)=repmat(J(1,f),[size(X3D)]);



33

34 v=cross(Jtemp,R,4);

35 A1=V.*v./(Rmag.ˆ3);

36 A=A+A1;

37 end

38 A=k.*A; Ax=A(:,:,:,1); Ay=A(:,:,:,2); Az=A(:,:,:,3);

39 end

biotsavartCD.m:

1 %% Biot-Savart Function - Single Grid

2 % This function takes a vector field and integrates it over a volume in

3 % accordance with the Biot-Savart Law. It can be used to calcualte B as a

4 % result of J or E f as a result of B. This function calculates the ...

relevant field

5 % within the same grid in which the source is defined.

6

7 function [A,Ax,Ay,Az]=biotsavartCD(dX,dY,dZ,X3D,Y3D,Z3D,J,k)

8 clear R; R=zeros([size(X3D) 3]); A=R; A1=R; Volume=R; Jtemp=R;

9 Volume=dX*dY*dZ;

10 for f = 1:size(X3D,1)

11 for g= 1:(size(X3D,2))

12 for h=1:size(X3D,3)

13 R(:,:,:,1)=X3D-X3D(f,g,h);

14 R(:,:,:,2)=Y3D-Y3D(f,g,h);

15 R(:,:,:,3)=Z3D-Z3D(f,g,h);

16 Rmag=(R(:,:,:,1).ˆ2+R(:,:,:,2).ˆ2+R(:,:,:,3).ˆ2).ˆ0.5;


18 Jtemp=repmat(J(f,g,h,:),[size(X3D),1]);


20 A1=Volume.*v./(Rmag.ˆ3); A1(f,g,h,:)=[0,0,0];

21 A=A+A1;

22 end

91


23 end

24 end

25 A=k.*A; Ax=A(:,:,:,1); Ay=A(:,:,:,2); Az=A(:,:,:,3);

26 end

biotsavartCD234.m:

1 %% Biot-Savart Function - Multiple Grids

2 % This function takes a vector field and integrates it over a volume in

3 % accordance with the Biot-Savart Law. It can be used to calcualte B as a

4 % result of J or E f as a result of B. This function can calculate ...

final field at

5 % a grid different from the source grid.

6 function ...

[A,Ax,Ay,Az]=biotsavartCD234(dX,dY,dZ,X3D,Y3D,Z3D,J,k,X3D2,Y3D2,Z3D2)

7 clear R; R=zeros([size(X3D2) 3]); A=R; A1=R; Volume=R; Jtemp=R;

8 Volume=dX*dY*dZ;


10 for g= 1:(size(X3D,2))

11 for h=1:size(X3D,3)

12 R(:,:,:,1)=X3D2-X3D(f,g,h);

13 R(:,:,:,2)=Y3D2-Y3D(f,g,h);

14 R(:,:,:,3)=Z3D2-Z3D(f,g,h);

15 Rmag=(R(:,:,:,1).ˆ2+R(:,:,:,2).ˆ2+R(:,:,:,3).ˆ2).ˆ0.5;


17 Jtemp=repmat(J(f,g,h,:),[size(X3D2),1]);


19 A1=Volume.*v./(Rmag.ˆ3);

20 A=A+A1;

21 end

22 end

23 end

24 A=A.*k;

25 Ax=A(:,:,:,1); Ay=A(:,:,:,2); Az=A(:,:,:,3);

26 end

GenAx.m:

1 % Takes x and y components of a vector in a cartesian grid and converts

2 % them to the circumferential component.

3

4 function [EIphi] = GenAx(EIx,EIy,EIz,X3D,Y3D)

92


5

6 Phi3D=zeros(size(X3D));

7 [Etphi,Etp,Etz]=cart2pol(EIx,EIy,EIz);

8


10 for g = 1:size(X3D,2)

11 for h = 1:size(X3D,3)

12

13 if X3D(f,g,h) > 0 && Y3D(f,g,h) > 0

14 Phi3D(f,g,h)=atan(Y3D(f,g,h)./X3D(f,g,h));

15 elseif X3D(f,g,h) 0 && Y3D(f,g,h) � 0

16 Phi3D(f,g,h)=atan(abs(X3D(f,g,h))./Y3D(f,g,h))+(pi/2);

17 elseif X3D(f,g,h) 0 && Y3D(f,g,h) < 0

18 Phi3D(f,g,h)=atan(abs(Y3D(f,g,h))./abs(X3D(f,g,h)))+(pi);

19 elseif X3D(f,g,h) > 0 && Y3D(f,g,h) < 0

20 Phi3D(f,g,h)=atan(X3D(f,g,h)./abs(Y3D(f,g,h)))+(3*pi/2);

21 end

22

23 end

24 end

25 end

26

27 alhpa=Etphi-Phi3D+(pi/2);

28 beta=(EIx.ˆ2+EIy.ˆ2).ˆ.5;

29 EIphi=beta.*cos(alhpa);

30 end

charge.m:

1 %% Works on 3D Arrays to calculate correction charge

2 function [qx]=charge(Ex,GT,eps,dim,eps0,dA,dB)

3 Dx=Ex.*eps.*eps0;

4

5 idx = repmat({':'}, ndims(Dx), 1); % initialize subscripts

6 n = size(Dx, dim);

7 idx{dim} = [ n 1:n-1 ];

8 Dx2 = Dx(idx{:});9 qx=(Dx2-Dx)*dA*dB;

10

11 GT2 = GT(idx{:});12 GTdif=(GT2-GT);

13 ind=find(GTdif == 0); qx(ind)=0;

14

93


15 n1 = size(Dx, 1); n2 = size(Dx, 2); n3 = size(Dx, 3);

16 qx(1,:,:)=0; qx(n1,:,:)=0;

17 qx(:,1,:)=0; qx(:,n2,:)=0;

18 qx(:,:,1)=0; qx(:,:,n3)=0;

19 end

visvox.m:

1 function [h]=visvox(X3D,Y3D,Z3D,dX,dY,dZ,ind)

2 h=figure;

3 for i = 1:length(ind)

4 x0=X3D(ind(i)); y0=Y3D(ind(i)); z0=Z3D(ind(i));

5 x1=x0-dX/2; x2=x0+dX/2; y1=y0-dY/2; y2=y0+dY/2; z1=z0-dZ/2; z2=z0+dZ/2;

6 x(1)=x1; y(1)=y1; z(1)=z1;

7 x(2)=x2; y(2)=y1; z(2)=z1;

8 x(3)=x1; y(3)=y2; z(3)=z1;

9 x(4)=x2; y(4)=y2; z(4)=z1;

10 x(5)=x1; y(5)=y1; z(5)=z2;

11 x(6)=x2; y(6)=y1; z(6)=z2;

12 x(7)=x1; y(7)=y2; z(7)=z2;

13 x(8)=x2; y(8)=y2; z(8)=z2;

14 V=[x',y',z'];

15 F=[1 2 3; 2 3 4; 1 5 7; 1 3 7; 2 6 8; 2 4 8; 1 2 5; 2 5 6; 5 6 7; 6 ...

7 8;];

16 p = patch('Faces',F,'Vertices',V,'FaceColor','g');

17 set(p, 'EdgeColor', 'none');

18 end

19 daspect([1,1,1])

20 view(3); axis tight

21 camlight

22 lighting gouraud

23 xlabel('x axis (m)'); ylabel('y axis (m)'); zlabel('z axis (m)');

24 end

94

Date post:	20-Jul-2018
Category:	Documents
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