Magnetic Resonance Imaging for Groundwater (Legchenko/Magnetic Resonance Imaging for Groundwater) || The Basics of NMR

2

The Basics of NMR

2.1. NMR phenomenon

2.1.1. Precession of free spins

The phenomenon of Nuclear Magnetic Resonance (NMR) could be observed bystudying nuclei which possess both a magnetic moment and angular momentum.The hydrogen nucleus, a proton, has a spin angular momentum J and an associatedmagnetic moment μ . Since they are parallel to each other, we can write

Jγμ = , [2.1]

with γ being the gyromagnetic ratio. It is a constant, which has different values for

different nuclei. For protons 710257707.42/ ×=πγp

Hz/Tesla.

When a proton is placed in an external magnetic field B , the magnetic field willproduce a torque on the magnetic moment of amount B×μ . Hydrogen nuclei alsopossess an angular momentum and hence the nucleus acts as a gyroscope andprecesses about the external magnetic field. The equation of motion for thisprecession is

BJ ×= μdtd

. [2.2]

Since Jγμ = , we obtain

)( Bγμμ ×=dtd

. [2.3]

Magnetic Resonance Imaging for Groundwater Anatoly Legchenko© 2013 ISTE Ltd. Published 2013 by ISTE Ltd.

16 Mag

Assucorrespothe z -ax

Fig

A spwith the

f

The Lgyromagthe Larmemit ene

2.1.2.M

Whenoriented,

macrosc

dynamicof the bephysics [allows si

gnetic Resonanc

uming a laboonding unit vexis, precession

gure 2.1. Prece

in’s magneticLarmor frequ

γπω 2/ Bf ==

Larmor frequegnetic ratio γmor frequencyergy only at th

Macroscopic sp

n there is no, resulting in

opic magnetiz

cs of a singleehavior of the[ABR 61]. Thimplifying the

ce Imaging for G

oratory refereectors ( kji ,, )n of the magne

ession of the ma

c moment μ iuency

π2/B .

ency is the reis a constant

y is also a propheir Larmor fre

pin magnetiza

o external man zero macros

zation gradua

spin needs ae macroscopiche classical tree treatment of

Groundwater

ence frameand that the

etic moment is

agnetic moment

is precessing a

sonance frequt, which has dperty of the nuequency.

ation

agnetic fieldscopic magne

ally builds an

quantum mecc spin magneteatment is an af the NMR phe

with the axemagnetic fiels shown in Fig

t about an exter

about the app

uency of the mdifferent valuuclei. Magnet

B , the protoetization. Wh

equilibrium v

chanical explatization can beapproximationenomenon wit

es ( zyx ,, )ld B is appligure 2.1.

rnal magnetic fi

plied magnetic

magnetic nuclees for differentic nuclei can

on spins are ren B is app

value0

M . W

anation, the dee given usingn, but in manyth sufficient a

with theied along

field

c field B

[2.4]

ei. As thent nuclei,absorb or

randomlyplied, the

Whilst the

escriptiong classicaly cases, itccuracy.

The Basics of NMR 17

We assume that in the investigated area, the equilibrium spin magnetization for

protons per unit volume0

M is constant and that dVMM0

== M .0

M is

described by the Curie equation

T4N

22

0 kBM

γ= , [2.5]

where N is the number of protons per unit volume, T is the absolute temperature,is the Planck constant and the Boltzmann constant 23103805.1 −⋅=k [J/°]. Since

2810692.6N ⋅= [/m3] then0

30

10287.3 BM −⋅= at 293K (20°C).

Using the classical model, the NMR phenomenon can be described by a set ofequations proposed in 1946 by Felix Bloch [BLO 46]. The Bloch equations haveprovided, for liquid samples at least, in most cases a correct quantitative descriptionof the behavior of the magnetic resonance phenomenon.

In an arbitrary homogeneous magnetic field B , the equation of motion of themacroscopic spin magnetization M is

)( BMM ×= γdtd . [2.6]

Assuming that the applied magnetic field is the sum of a static field0

B and a

much smaller radio frequency field1

B , in the laboratory frame the Bloch equations

can be written as

,)(

,)(

,)(

2

2

1

0

T

M

dt

dM

T

M

dt

dM

T

MM

dt

dM

yy

y

xx

x

zz

z

+×=

+×=

−+×=

BM

BM

BM

γ

γ

γ

[2.7]

18 Magnetic Resonance Imaging for Groundwater

where10

BBB += ;1

T is called the longitudinal, or spin-lattice relaxation time;

and2

T the transverse, or spin-spin relaxation time. In the laboratory frame the

vector M is rotating about the static field0

B with the Larmor frequency

0 0Bω = −γ (clockwise). A linearly polarized, oscillating magnetic field applied in

the x -direction )cos(21

tBx

ω=B can be presented as two rotating components with

the angular frequency ω+ and ω− . Only co-rotating with the spin system part

(clockwise) with the amplitude1

B has an effect on the spin magnetization. The

field1

B will be the co-rotating component of the applied oscillating field. The

effect of the counter-rotating component is usually neglected.

In a reference frame, rotating around0

B with the angular frequency ω there is

an effective static field

)/(01

γω−+= BB kiBeff

, [2.8]

where γω/ component called the fictitious field.

In the rotating frame the Bloch equations can be rewritten as

.)(

,)(

,)(

2

2

1

0

T

M

dt

dM

T

M

dt

dM

T

MM

dt

dM

yyeff

y

xxeff

x

zzeff

z

+×=

+×=

−+×=

BM

BM

BM

γ

γ

γ

[2.9]

At th

and cons

[2.8] it fo

B

thus simp

d

d

d

In th

angle θas shown

Fig

he exact reson

sidering equa

follows that

1BiB

eff= ,

plifying the B

(

(

(

dt

dM

dt

dM

dt

dM

y

x

z

×=

×=

×=

M

M

M

γ

γ

γ

e equilibrium

between then in Figure 2.2

gure 2.2. Rotati

a) equilib

nance (0

ωω =

ations [2.4] we

Bloch equation

.)

,)

)

21

21

1

01

T

M

T

M

T

M

yy

xx

z

+×

+

−+

B

B

B

m, M is orient

spin magneti2a.

ion of the spin m

brium position;

0) the fictitio

e can write B

ns to

,

1

Mz

ted along the

zation and the

magnetization u

b) when a mag

ous field cance

0/0

=− γωB

external magn

e geomagnetic

under exact reso

gnetic field1

B

The Basics of

els the static

. Then, from

netic field0

B

c field is equa

onance ( =Δω

is applied

NMR 19

field0

B

equation

[2.10]

[2.11]

0and the

al to zero

0 ):

20 Mag

At th

with a c

M to ro

time t , t

M

M

M

where th

magnetiz

When

in the

(Figure 2

the frequ

rotating

Fi

gnetic Resonanc

he exact reson

arrier frequen

otate about B

the three comp

)cos(

)sin(

,0

MM

MM

M

z

y

x

θ

θ

=

=

=

he angle θ =

zation is sketc

n the resonan

rotating fram

2.3). The angu

uency offset (

frame frequen

igure 2.3. Rotaa) case Δω

ce Imaging for G

nance, when w

ncy equal to th

1in the yz ,

ponents of the

,

,

0

0

M

M

tBt11

γω ==

ched in Figure

nce conditions

me follows a

ular frequency

ωωω −=Δ0

ncy (ω ) and t

ation of the spinω> 0; b) commo

Groundwater

we apply a m

he Larmor fre

plane with t

e spin magnet

is called th

e 2.2b.

s are not respe

a precession

y of the preces

) between the

the applied ma

n magnetizationon case for any

magnetic field

equency it cau

the angular fre

tization are

he flip angle.

ected (0

ωω ≠

cone aroun

ssion ( 2 ωω =eff

e Larmor frequ

agnetic field f

under off-resonsign of Δω; c) c

1B along th

uses the magn

equency1

ω =

. The rotatio

0), the motion

d the effect2

12 ωω Δ+ ) de

uency (0

γω =

frequency (ω

nance conditioncase Δω< 0

he x -axis

netization

1Bγ= . At

[2.12]

on of the

n of spins

tive field

epends on

0Bγ ), the

11Bγ= ).

ns:


The total precessional angle is given by tBtteffeff

γωθ ==)( . At time τ=t ,

the spin magnetization will, in general, have three components, which are [MAN 79]

.)cos(

,)sin(

,)cos(1

02

122

0

1

02

1 )(

MM

MM

MM

eff

eff

z

eff

eff

y

eff

eff

x

ω

τωωω

τωω

ω

τωω

ωω

+Δ=

=

−Δ

=

[2.13]

The magnetic resonance signal depends on the component of the nuclearmagnetization perpendicular to the static magnetic field (x,y plane). The

)Re(⊥

= MMy

component corresponds to the real part of the NMR signal and

)Im(⊥

= MMx

- to the imaginary part. The position of the magnetization vector

0M and its projection M

⊥after a magnetic field

1B being applied during time τ

)( τωθeff

= is shown in Figure 2.4. When 0=Δω , thex

M component is equal to

zero and hence the signal, which is thus proportional only toy

M has no phase shift

(Figure 2.4b). If 0≠Δω , then the signal will be a vector in the complex plane. Thephase shift depends on the sign and magnitude of the frequency offset. It isillustrated in Figure 2.4a for a positive frequency offset and in Figure 2.4c for anegative one. Obviously, the magnitude of the total spin magnetization is constant

zyxMMMM 222

02 ++= .

22 Mag

Formagnetizfact, whifrom the

2.2. NM

2.2.1. Lo

At thto a Bochangingfield), th

relaxatio

relaxatio

as

M

d

gnetic Resonanc

Figure 2.4.referen

demonstrationzation under tilst spins, whie resonance wi

MR relaxation

ongitudinal re

hermal equilibltzmann distrg the static mhe nuclear sp

on process cha

on time. In Bl

1(

(

0z

z

MM

dt

dM

−=

×= Mγ

ce Imaging for G

Precession of tnce frame cons

n purposes tthe resonanceich are at theill be only slig

elaxation

brium, nuclei aribution. Afteagnetic fieldpin system ret

aracterized by

och equations

).

)

1/

0

1

Tt

z

e

T

M

−

−+×B

Groundwater

the spin magnetsidering differen

the proportionand off-resonexact resonanghtly perturbe

are distributeder this equilibor irradiatingturns to equil

y a time1

T , c

s [2.7] the cha

,

1

zM−

tization in the tint frequency off

nality betweenance conditionce, will respoed at their equi

d among the ebrium distribusamples by alibrium with

called the spin

ange of zM

ilted rotatingffsets Δω

en magnitudeons is not respond strongly,ilibrium positi

energy levels aution is disruan oscillatingits surroundi

n-lattice or lon

with time is d

es of thepected. Inthose farion.

accordingupted (bymagneticings by a

ngitudinal

described

[2.14]

1T r

system a

environm

In or

magnetiz

that allcoherenc

the static

system

magnetizthe spin-

equilibri

disappea

Figur

reflects efficie

and its surrou

ment is weak

rder to illustr

zation M afte

the magneticce. Immediate

c field0

B , i

approaches

zation M is-lattice relaxa

ium along

aring without c

re 2.5. The spin

(duration τ):longitudinal r

ency of the ex

undings. When

1T will be lar

rate the effect

er application

c moments kely after the pu

is turned awa

equilibrium

correspondingation as a rotat

0B . The n

changing the a

magnetization

a) immediatelyrelaxation on th

xchange of the

n the couplin

rge, but strong

t of the longi

of a pulse of

keep the samulse is termin

ay at the angl

with the ti

gly decreasingtion of the ve

non-equilibrium

angle θ .

after applicatio

after the pulsehe spin magneti

e magnetic en

ng between th

g coupling wi

itudinal relax

f a magnetic fi

me precessingnated, M , wh

le θ (Figure

ime constant

g. It would bector M , in th

m magnetiza

on of a pulse of

is terminated;ization after a t

The Basics of

nergy between

he spin system

ll render1

T

xation1

T on

ield1

B , let u

frequency anich was orien

2.5a). Then,

t1

T and

e erroneous the yz , plane

ation M is

f a magnetic fie

b) effect of thetime interval td

NMR 23

n the spin

m and the

small.

n the spin

us assume

nd phasented along

, the spin

the spin

to presente, back to

s simply

eld1

B

24 Mag

2.2.2. Tr

The

magnetic

is originrandomly

magnetiz

magnetic

such an

( 0>⊥

M

nuclei to

iμ lose

yx, pla

constant

Figur

(durrelax

gnetic Resonanc

ransverse rela

macroscopic

c moments of

nally no phasy oriented a

zation onto

c field1

B is

orientation th

0 ) as shown

o exchange en

the phase co

ane as shown

2T , called th

re 2.6. The spin

ration τ): a) immxation with the

ce Imaging for G

axation

spin magneti

f individual sp

se coherenceand consequ

yx, plane i

applied, the m

hat a non-zero

in Figure 2.6

nergy with eac

oherence after

in Figure 2.6

he spin-spin o

magnetization

mediately aftertime constant T

Groundwater

ization M is

pins that also

of the maguently a proj

is equal to z

momentsi

μ

component o

6a. Because

ch other and

r each interact

6b, which cau

r transversal r

after applicatio

the pulse is terT2 on the spin m

a vector com

are vectors M

gnetic momenjection of th

zero ( 0=⊥

M

will have the

of M is gener

of the natura

with the envi

tion and begi

uses M⊥to d

relaxation tim

on of a pulse of

rminated; b) effmagnetization af

mposed of th

∑=i

iμM . Si

nts of spins,he macrosco

0 ). However

e same phase a

rated in the x

al processes th

ronment, the

in to spread a

diminish with

e.

f a magnetic fie

fect of the transvfter a time inter

e sum of

ince there

they areopic spin

r, after a

and lie in

y, plane

hat cause

moments

across the

h the time

eld1

B

verserval td

In Bl

M

d

Notezero (M

be at non

respectedrelaxatio

Since

different

precess a

slower (

phase co

character

Figur

(durrelax

loch equations

1(0

2

xx

MM

T

M

dt

dM

⊥−=

−=

that when th0⊥ →M ) the N

n-equilibrium

d. This is a fuons.

e the static m

t portions of th

at slightly dif

(i

ωωω Δ+=0

oherence (Fig

rized by the ti

re 2.7. The spin

ration τ): a) immxation with the t

s [2.7] the dec

).

,

2/Tt

y

e

dt

dM

−−

=

he transverseNMR signal c

with the envi

undamental di

magnetic field

he sample exp

fferent frequen

iω ). This pro

gure 2.7) and

ime constant T

magnetization

mediately aftertime constant T

cay of M⊥ is d

,

2

y

T

M−=

component oannot be mea

ironment so th

fference betw

d may not be

perience sligh

ncies, some f

ocess also ca

consequently

*2

T .

after applicatio

the pulse is terT2* on the spin m

described as

of the spin masured, but the

hat the limitat

ween the transv

e perfectly ho

tly different v

faster than the

auses the mom

y M⊥

will d

on of a pulse of

rminated; b) effmagnetization af

The Basics of

magnetizatione spin system

tion2 1

T T≤

versal and lon

omogeneous,

values0

B . He

e rotating fram

mentsi

μ to

diminish with

f a magnetic fie

fect of the transvafter a time inter

NMR 25

[2.15]

becomesmay still

is always

ngitudinal

nuclei in

ence they

me, some

lose the

h the rate

eld1

B

verserval td


Considering both the longitudinal and transverse relaxations,*2

T can be

approximated as [FUK81]

0

212*

1

2

11 BTTT

Δ++≅ γ , [2.16]

where0

BΔ characterizes inhomogeneity of the static magnetic field0

B and γ is

the gyromagnetic ratio.*2

T is the shortest relaxation time and122

* TTT ≤≤ . Often,

the effect of the longitudinal relaxation on*2

T may be neglected and*2

T can be

considered as

0

22*

11 BTT

Δ+≅ γ . [2.17]

During the NMR experiment (usually <2 s) distribution of the static magnetic

field0

B within the investigated sample does not change. Consequently, the

magnetic moments iμ are located in the same magnetic field, they have a constant

Larmor frequency0 0i i

constω ω ω= + Δ = and dephasing is regular tii

ωϕϕ Δ+=0

(Figure 2.7). Note that2

T characterizes dephasing caused by a random energy

exchange between spins and their environment what cause random dephasing

0i irandomφ φ φ= + Δ = (Figure 2.6). It is a notable difference between

*2

T and2

T

that is used for measuring2

T with the spin echo (SE) method.

2.2.3. Diffusion in non-homogeneous magnetic field

In the previous sections we assumed that nuclear spins do not change theirposition within the investigated sample. However, in liquids, spins can move by


diffusive motion of the atoms. Hence, if the static magnetic field0

B is not

homogeneous or a field gradient is imposed then such a movement will cause the

magnetic momentsi

μ to change their Larmor frequency during one measuring

cycle, thus introducing additional dephasing. Such a dephasing will accelerate

relaxation of the NMR signal with the enhanced relaxation rate2D

T . In an infinite

fluid medium,2D

T is given by the following expression [CAR 54]

12/1 222

2

DtGT

D

γ= , [2.18]

where G is the spatial magnetic field gradient, D is the diffusion coefficient and tis the time during which diffusion will occur. Thus, the transverse relaxation rateenhanced by the effect of spin diffusion is given by

⎟⎠⎞⎜

⎝⎛ −−=

⊥ DTtTtMM

220//exp . [2.19]

The diffusion coefficient depends on temperature and it can be calculated forwater using empirical function based on results reported by Sengers and Watson[SEN 86] (Figure 2.8).

Figure 2.8. Diffusion coefficient of water as a function of temperature


Equation [2.18] is valid only for an infinite fluid medium and for water in rocksit is only approximately correct. However, assuming that most of the spins do notexperience the presence of pore walls, we can use equation [2.18] to demonstrate the

relationship between2D

T and a spatial magnetic field gradient G . In porous

media, G may be imposed by magnetic particles that perturb the static magnetic

field0

B . In this case, the distribution of the magnetic field gradient within the

investigated sample may be non-uniform and2D

T can be calculated only

approximately. Let us assume that G is a first order approximation of an equivalent

inhomogeneity of the static magnetic field. Figure 2.9a shows estimated2D

T

versus a magnetic field gradient considering three values of the diffusion time.

Figure 2.9b presents dependence of the enhanced relaxation2D

T on the diffusion

time for different values of the field gradient G .

Figure 2.9. The enhanced relaxation rate T2D computed considering an inhomogeneous staticmagnetic field: a) T2D versus the spatial magnetic field gradient G assuming the diffusiontime t of 200, 400 and 800 ms; b) T2D versus diffusion time and considering three values of themagnetic field gradient G

When the distribution of the static magnetic field gradient G is unknown it can

be roughly estimated assuming22

* TT << in equation [2.17] as

2*0

1

TBG

γ≅Δ≅ , [2.20]


where*2

T is an experimental value and GB ≈Δ0

is an estimate of a static magnetic

field inhomogeneity. The relationship between G and*2

T calculated using equation

[2.20] is shown in Figure 2.10. Figures 2.9 and 2.10 allow qualitative estimationof the enhanced relaxation rate caused by diffusion of spins. For example,

assuming*

7.5 ms2

T = , we obtain the corresponding value of the field gradient

005.0=G Gauss/cm (Figure 2.10). Then, Figure 2.9b allows the estimation of

2DT for, e.g., the diffusion time of 2000 ms as

28500 ms

DT = .

Figure 2.10. Estimated dependence of the relaxation timeconstant T2* on the magnetic field gradient G

Now, let us estimate the possible effect of2D

T on the transverse relaxation. In

equation [2.19] the transverse relaxation term is given by

)/11(/1/1/1222

xTTTD

+=+ , [2.21]


where22

/TTxD

= . Figure 2.11 shows that for obtaining less than 10% error in

the transverse relaxation term,2D

T must be ten times longer than2

T . This

condition can be respected either for a short diffusion time or for small values of thefield gradient (Figure 2.9b).

Figure 2.11. The relative error in the transverse relaxation termcaused by the enhanced relaxation versus T2D/T2 ratio

The numerical example presented in Figure 2.12a shows relaxation of thenuclear magnetization (equation [2.19]) as a function of diffusion time. For

computing, we assumed the following input: 52 10−= ×D cm2/s,2

400 msT = and

42.6752 10γ = × rad×s-1×Gauss-1. The results demonstrate shortening of therelaxation caused by spin diffusion and when relaxation graphs were fitted by anexponential function we obtain estimates of the amplitude and time constant.Figure 2.12b shows the relative error of estimation in comparison with the known

2T value. We can see that diffusion principally affects estimation of the relaxation

time2

T and the initial amplitude was estimated with a much smaller error.


Figure 2.12. Demonstration of the effect of a field gradient on the relaxation rate:a) transverse relaxation versus diffusion time in different field gradients; b) relative errorof estimation of the amplitude and relaxation rate as a function of the field gradient

Note that for rocks, the estimates presented above should be considered only asqualitative because the internal field gradients of porous media are induced by themagnetic susceptibility variations that depend not only on the magnitude of thesevariations, but also on the pore and grain geometries. Understanding of the effect ofstrong internal field gradients on NMR measurements is very important, but it is avery complex problem that is beyond the scope of this book. Numerous attempts ofquantification of these effects have been reported [NEU 74; BEN 90; HÜR 98;ZHA 01; SUN 02; HÜR 02; FRE 06; POM 08; CHO 08; CHO 09; GRU 12], but itis recognized that quantification of the internal field gradients’ distribution andanalysis of experimental data remains very difficult.

2.3. NMR measurements

2.3.1. Free induction decay (FID)

For measuring the NMR signal, one or more coils are positioned around the

investigated sample. A static magnetic field0

B is usually created by a special

magnet but the Earth’s magnetic field can also be used. The simplest NMRexperiment consists of observing the free induction decay signal (FID). A pulse of

alternating current energizes the induction coil thus creating a magnetic field1

B

oscillating with the Larmor frequency00

Bγω = (Figure 2.13a). The pulse causes


rotation of the nuclear magnetization M around the x axis with the angularfrequency

11Bγω = . When the pulse is terminated M is turned at a flip angle

pτωθ1

= and the transverse component of the nuclear magnetization

0sin( )

yM M= θ produces measurable NMR signal (FID). The shape of the signal

envelope depends on the distribution of internal magnetic field gradients and mayhave exponential or Gaussian shape [SUK 01; GRU 11] as shown in Figure 2.13b.Oscillating with the Larmor frequency 0ω , the FID signal decays with the time

constant*2

T . For the exponential envelope

)cos()/exp()(002

*0

ϕω +−= tTtete , [2.22]

where0e and

0ϕ are the initial amplitude and initial phase respectively. For the

Gaussian envelope it may be approximated as

)cos()/(5.0exp)(00

22*

0ϕω +⎟

⎠⎞⎜

⎝⎛−= tTtete . [2.23]

In both cases the initial amplitude0e and relaxation time

*2

T can be obtained

directly from the FID signal.

Figure 2.13. Schematic presentation of the FID measuring procedure: a) nuclearmagnetization excitation pulse being applied; b) corresponding FID signal proportional

to the My component is drawn considering exponential and Gaussian envelopes

The

popularmethodscomponeapplying

Theequilibri

magnetic

(Figure

compone

starts de

compone

Figure 2recoveryin the –zd) a 90°nuclear m

FID signal is

methods are care based onent of nucleag different pul

inversion recium, the nuc

c field0

B (F

2.14b). The n

ent M⊥is ze

ecreasing with

ent 0M M− i

2.14. Evolutionmeasuring schz direction; c)pulse (duratio

magnetization |M

s also used fo

called the inven measuring thar magnetizatse sequences.

covery measuclear magneti

Figure 2.14a).

nuclear magn

ero and hence

h the time c

increases. Aft

n of the nucleaeme: a) equilibrelaxation of Mon τp /2) turns MM0-M| is turned

or measuring

ersion recoverhe FID signaltion, separate

uring schemezation M v

The first puls

netization Me no signal can

onstant1

T

ter a time dela

ar magnetizatibrium position;M with the timM in the –y dird along the +y a

the relaxation

ry and the satl generated ond from the e

e is presentevector is orie

e turns M at

is fully excit

n be measure

(Figure 2.14c

ay dτ the seco

ion M correspb) a 180° puls

me constant T1rection and a raxis

The Basics of

n time1

T . T

turation recovnly by the eqexcited comp

ed in Figureented along t

the flip angle

ted, but its tr

d (FID1=0). T

c) and an eq

ond pulse is ap

ponding to these of duration τduring time inrelaxed compon

NMR 33

The most

ery. Bothquilibriumonent by

2.14. Inthe static

e °=180θ

ransversal

Then, M

quilibrium

pplied.

inversionτp turns Mnterval τd;nent of the

34 Mag

The p

turns nu180°,90°compone

(equilibr

NMR sig

compone

between

measurin

For e

The FID

of1

T u

e

gnetic Resonanc

pulse has a v

clear magneti° sequence. Aent M is ori

rium)0

M M−

gnal (FID2) is

ents of the nu

the pulses τ

ng procedure i

Figure 2.15.

each value of

D2 amplitude p

sing the follow

⎜⎜

⎝

⎛−= 21)(

0ee

dτ

ce Imaging for G

value twice as

ization at theAfter the seconented in the

M turns in

s proportional

uclear magnet

dτ and measu

is shown in Fi

A typical seque

fd

τ position

plotted agains

wing equation

⎟⎟

⎠

⎞− )/exp(

1T

dτ

Groundwater

small as the

flip angle =θnd pulse is te–y direction

the +y direct

to the differe

tization.1

T

uring the cor

igure 2.15.

ence of inversio

of the pulses

st delayd

τ (

n

.

productp

τω1

°= 90 . This puerminated, theand the alrea

tion (Figure

ence between t

can be estim

rresponding F

on recovery for

s, the signal i

(bottom graph

p, and conseq

ulse sequencee non-relaxedady-relaxed co

2.14d). The m

the excited an

mated by varyi

FID2 signal. A

measuring T1

is plotted agai

h) allow the e

quently it

e is called(excited)omponent

measured

nd relaxed

ing delay

A typical

inst time.

estimation

[2.24]

The smethod

direction

measure

non-relaxsignal th

Figure 2recoverythe +y dirpulse (dumagnetiza

saturation recthe 90°,90° s

n and the FID

d (Figure 2.16

xed componehat is proportio

2.16. Evolutionmeasuring scherection; c) relaxuration τp ) turation |M0-M| is

covery measurequence is us

D1 signal cor

6b). The seco

nt M along tonal only to th

of the nucleaeme: a) equilibxation ofM witrns M in the –turned along th

ring scheme ised. After the

rresponding t

ond pulse is ap

the −z axis whhe relaxed com

ar magnetizatiobrium position;th the time cons–z direction anhe +y axis

s illustrated ine first pulse, M

to the initial

pplied after a

hich allows umponent M −

0

on M correspob) a 90° pulsestant T1 duringnd a relaxed co

The Basics of

n Figure 2.16M is turned

amplitude0e

delayd

τ . It

us to measureM− (Figure 2

onding to theof duration τp ttime interval τomponent of th

NMR 35

6. For thisin the +y

0can be

turns the

the FID22.16d).

saturationturns M in

τd; d) a 90°he nuclear

36 Mag

1T c

delay τ

amplitud

as

e

The

magnetic

common

obtain thprocedurresults.

TheMeasurevalue, buinverted.

the sampdifferent

gnetic Resonanc

Figure 2.17. A

can be estima

d. A typical

de plotted aga

⎜⎜

⎝

⎛−= ex1)(

0ee

dτ

saturation re

c field1

B w

n case for the

he same flip ares based on t

modified saements are carut it is prefer. The pulses p

ple. After thet for different

ce Imaging for G

A typical seque

ated by measu

l measuring p

ainst a time de

⎟⎟

⎠

⎞− )/xp(

1T

dτ

.

covery metho

within the inv

e surface NM

angle for diffethe specific fli

aturation recrried out usingrred that bothproduce the s

e first pulse, Mt portions of

Groundwater

ence of saturati

uring the FID

procedure is

elayd

τ (botto

od can be m

vestigated vol

MR. A non-un

rent portionsip angles (90°

overy schemg a θ,-θ sequepulses be ide

same flip angl

M is turnedthe sample (

on recovery for

D2 signal for

shown in Fi

om graph) allo

modified for a

lume is non-u

iform1

B re

of the sample° or 180°) may

me is presenence. The flipentical. The sle and hence

through the a(Figure 2.18b)

r measuring T1

different valu

igure 2.17. T

ows us to esti

applications w

uniform, whi

enders it impo

e and hence, my produce err

nted in Figuangle θ maysecond pulse h

21θθ −= in an

angle θ, whic). During a d

ues of the

The FID2

imate1

T

[2.25]

when the

ich is the

ossible to

measuringors in the

ure 2.18.have anyhas to beny part of

ch can bedelay dτ ,

relaxatioangle −θentire saOnly thevalues of

A typ

plotted a

the equa

e

where x

FID2 sig

Figure 2saturationτp turns Mduring athe +z dirflip angle

on occurs andθ and thus, thample is oriene relaxed comf the delay dτ

pical measuri

against a time

ation

⎜⎜

⎝

⎛−−= 1)(

0ee

dτ

000/ eex

ττ= i

gnal measured

2.18. Evolutionn recovery meaM in the z-y platime interval τdrection and a re -θ

then the secohe non-relaxednted along themponent produ

d .

ing procedure

e delay dτ (b

−−− exp()1(0

xτ

is due to possi

d when /dT⎛τ⎜

⎝

n of the nucleasuring schemeane at a flip and ; d) the seconrelaxed compon

ond pulse is ad componente +z axis andces the FID2

e is shown in

ottom graph)

⎟⎟

⎠

⎞− )/

1T

dτ ,

ible difference

10T ⎞→⎟

⎠.

ar magnetizati: a) equilibriumngle +θ ; c) red pulse (durationent of the nucle

applied. The pof the nuclea

d does not prosignal that is

n Figure 2.19

allows the es

es between the

ion M correspm position; b) tlaxation of M won τp but of invear magnetizat

The Basics of

pulse producear magnetizatoduce an NMmeasured for

. The FID2 a

stimation of

e pulses and e

ponding to thethe first pulse ofwith the time cverse polarity) ttion |M0-M| is t

NMR 37

es the fliptionin the

MR signal.r different

amplitude

1T from

[2.26]

0τe is the

e modifiedof durationconstant T1turns M inturned at a

38 Mag

Fig

2.3.2. Sp

The

heteroge

necessar

T

An Sa 90°,90allows u

Figursequence(Figuresrotatingfaster tha

gnetic Resonanc

gure 2.19. A typ

pin echo (SE)

SE techniqu

eneous static m

ry to respect th

*2 2 1

T T T<

SE can be obs0° sequence.us to obtain the

re 2.20 scheme. The first 902.20b, 2.16bclockwise wian the rotating

ce Imaging for G

pical sequence o

ue was devel

magnetic field

he following c

.

erved with diHowever, a 9e maximum am

matically show0° pulse sets tb) and the FIDith the Larmog frame and h

Groundwater

of modified satu

loped by Ha

d0

B [HAH 5

condition

fferent pulse90°,180° sequmplitude of th

ws the formatithe nuclear mD1 signal canor frequency ωhence they rota

uration recover

ahn for NMR

50]. For obser

sequences. Fouence is oftenhe SE signal.

ion of an SE cmagnetizationn be observed0ω some mag

ate clockwise

ry for measuring

R measureme

rving an SE si

or example, Hn preferred b

considering aM in the +yd. In a referengnetic momentwith a relativ

g T1

ents in a

ignal it is

[2.27]

Hahn usedbecause it

90°,180°directionnce framets iμ areve angular


frequencyi

ωΔ+ . Some other magnetic moments are slower and rotate

counterclockwise with a relative angular frequencyi

ωΔ− . The local difference in

the Larmor frequency causes dephasing of the nuclear magnetization with the time

constant*2

T . Dephasing during the timed

τ (Figure 2.20c) causes the FID1 signal to

decrease. As*2 1

T T , M remains excited even when FID1 becomes zero, but

relaxation with the time constant1

T also takes place and an equilibrium component

0M M− grows (Figure 2.16c). The second pulse performs two actions. It sets the

non-relaxed component M along the –y axis as shown in Figure 2.20d and sets the

relaxed component0

M M− at the –z direction. In both cases the FID2 signal is

absent because M is oriented along the –y axis, remains dephased and the

transversal component of0

M M− is zero. However, if a sequence other than 90°,

180° is used or the second pulse is not perfect and produces an angle other than the180° flip angle, then the FID2 signal will appear. After the second pulse, magneticmoments iμ that are faster ( +Δωi ) are set behind the rotating frame but continue

rotating clockwise, and those that are slower (i

ωΔ− ) are set before the rotating

frame and rotate counterclockwise. At time τd after the second pulse, the dephasedcomponents of M will meet again along the –y axis thus producing the refocusing

effect (Figure 2.20e). The transverse relaxation with the time constant2

T cause

random dephasing and somei

μ will not refocus, which attenuates SE amplitude.

The SE measuring scheme (Figure 2.21) consists of transmitting two consecutive

pulses separated by a time intervald

τ . The SE signal can be observed at timed

τ

after the second pulse is terminated. The SE signal is composed of two FID signalscorresponding to the refocusing and following dephasing of the nuclearmagnetization as shown in Figure 2.21. The FID signal can also be observed.

40 Mag

Figure 2.b) the madurationthrough te) refocumacrosco

Figure 2.of two puimmediatrefocusing

gnetic Resonanc

.20. Graphicalacroscopic spinτp ; c) dephasinthe flip angle θusing of the mopic spin magne

.21. A schematiulses separatedtely after the cog pulse

ce Imaging for G

illustration of tn magnetizationng of the macro

θ=180° by a refmacroscopic maetizationM atte

ic presentationd by a time interresponding pu

Groundwater

the formation on M is turnedoscopic magneefocusing pulseagnetization duenuated by the t

of the SE measerval τd. The FIulse. The SE sig

of an SE signal:at the flip angtization in the xapplied at timuring a time itransversal rela

suring proceduFID1 and FID2gnal can be mea

: a) equilibriumgle θ=90°, byx-y plain; d) Mme τd after the finterval τd; f)axation

ure consisting ofsignals can beasured at time τ

m position;a pulse of

M is turnedfirst pulse;refocused

of injectione observedτd after the


Formation of an SE is a phenomenon that depends on internal magnetic field

gradients and the shape of the signal may be complex. In a simple case, an SE

envelope can be considered as Gaussian as is given by [HAH 50]

⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜

⎝

⎛ −−=

22*

2

2)(2

)2(exp)(

T

tete d

dSE

ττ

. [2.28]

For practical use it is convenient to present it as

⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜

⎝

⎛

Δ

−−=

5.02

2

236.0

)2(exp)(

t

tete d

dSE

ττ

, [2.29]

where5.0

tΔ is the half-width of the echo signal (the width of the echo envelope at

the level of one half of the maximum echo amplitude). Thus the relaxation time*2

T

can be easily estimated as*

0.520.424T t≈ ×Δ .

For calculating the SE amplitude we assume the mathematical and geometrical

interpretations presented in Hahn’s paper [HAH 50]. Under near-resonance

conditions, for pulses of equal intensity1

B and neglecting the relaxation and

molecular diffusion, M⊥can be calculated as [BLO 55]

20 1 2sin sin 0.5M M= −

⊥⎛ ⎞ ⎛ ⎞θ × θ⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

, [2.30]

where111 p

B τγθ = and212 p

B τγθ = are the flip angles for corresponding pulses. If a

90°,180° pulse sequence is used then θθθ ==12

2 and consequently the SE

amplitude can be calculated as

( )30sinM M= −

⊥θ . [2.31]


Note, that after equation [2.30], the SE signal calculated using equation [2.31] isinsensitive to the polarity of the second pulse.

In practice, the SE signal can be measured after attenuation by relaxation and

molecular diffusion. Thus, the SE signal at a timed

τ2 is

⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜

⎝

⎛

+−=⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛

312exp

222

2

02d

d

GD

Tee

d

τγττ

, [2.32]

where G is the spatial magnetic field gradient and D is the diffusion coefficient.

If21222 −<< TGD

dτγ then the diffusion effect can be neglected and SEs can be

used for measuring the transverse relaxation time constant2

T . For that, SE

amplitude is measured for different values ofd

τ as shown in Figure 2.22a. Each run

consists of two pulses and one SE signal. Plotting SE amplitudes versus time

(bottom graph) one can estimate2

T from

)/2exp(202 Tee

ddττ −= . [2.33]

The Hahn’s SE is sensitive to the diffusion effect but it is also a time consuming

procedure. For that reason, more efficient methods for measuring2

T have been

developed. One of the most common is the CPMG method named after the authors:Carr and Purcell [CAR 54], and Meiboom and Gill [MEI 58]. The methodsupposes the application of a 90° pulse followed by a 180° pulse delayed at a time

dτ . The first echo measured at a time

dτ2 is the Hahn’s echo. Then, 180° pulses

separated by timed

τ2 are applied. Each pulse causes refocusing of the nuclear

magnetization and the formation of SEs as shown in Figure 2.22b.

F

The e

e

where n=

Note

time, an

the CPM

number

significais the timof the ecmethod r

Figure 2.22. A sT2: a) Hah(90°, τd , 18

envelope of a

⎜⎜⎜⎜

⎝

⎛

−= 2exp0n

ne τ

=1,2,…,N is th

, that for mea

d hence dτ ,

MG method (

of echoes in

antly attenuateme efficiencycho train in onrequires N cyc

schematic drawhn’s echoes with80°, 2τd , 180°,

train of echo

+⎜⎜⎜⎜⎜⎜

⎝

⎛

31

2

2

d

GD

T

γτ

he number of

asuring2

T us

are progressi

(equations [2

the train N i

e the diffusionsince the metne measuring ccles.

wing showing typh (90°, 180° seq2τd ,…, 180° se

amplitudes al

⎟⎟⎟⎟

⎠

⎞

⎟⎟⎟⎟⎟⎟

⎠

⎞22d

G τ ,

echoes.

sing the Hahn

ively increase

.34]) dτ is

is increased.

n term. Anoththod allows thcycle. For the

ypical procedurquence); b) CPMequence) with a

llows the estim

n’s SE (equati

ed ( 2d

t = τ ).

constant ( t =

Thus, renderi

her advantagehe obtainmentsame number

The Basics of

res for measurinMG methoda train of SE

mation of2

T

ons [2.32] an

However, wh

2d

n= τ ) and

ing dτ small

of the CPMGt of the entirer of echoes, th

NMR 43

ng

as

[2.34]

d [2.33]),

hen using

only the

l, we can

G methodenvelopehe Hahn’s


For practical realization of the CPMG method, we should wait for a fulldephasing between the pulses. Thus,

*2

3d

Tτ > . [2.35]

At time2

2T relaxation is completed at approximately 85% and we can assume

the measuring cycle to be *2 2

2 6 2d

N N T Tτ = = . Hence the number of echoes N

will be given by

2*2

3

TN

T< . [2.36]

The CPMG method is also sensitive to homogeneity of the static magnetic fieldand to the imperfection of the excitation pulses. For example, if the pulses are

slightly different from 180° then they will cause an error in the2

T estimation.

These items have been discussed in detail in the literature, for example by Majumdar[MAJ 86a; MAJ 86b].

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Magnetic Resonance Imaging for Groundwater (Legchenko/Magnetic Resonance Imaging for Groundwater) || The Basics of NMR