Magnetic resonance with nonuniform radiofrequency field

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Magnetic resonance with nonuniform radiofrequency field

J. Stepišnik

To cite this version:

J. Stepišnik. Magnetic resonance with nonuniform radiofrequency field. Journal de Physique, 1978,

39 (6), pp.689-692. �10.1051/jphys:01978003906068900�. �jpa-00208801�

MAGNETIC RESONANCE WITH NONUNIFORM RADIOFREQUENCY ^FIELD

J.

STEPI0160NIK

Physics Department

and J. Stefan

Institute, University

^of

Ljubljana,

⁶¹⁰⁰⁰

Ljubljana, Yugoslavia

(Reçu

^{le 5} avril 1977, révisé le 1 er

février

^1978,

accepté

^{le 8}

février 1978)

Résumé. ²⁰¹⁴ On étudie la forme de raies RMN de

spins

nucléaires se

déplaçant

^{dans un} champ

de

radiofréquence

^{non uniforme. On montre} que l’élargissement ^observé, interprété

précédemment

comme un raccourcissement du temps ^de relaxation, peut aussi être analysé ^{en termes du mouvement}

des spins excités de façon ^non uniforme. On présente des résultats dans les cas d’auto-diffusion et d’écoulements de fluides.

Abstract. ²⁰¹⁴

Migrating

spins subjected ^{to a} nonuniform radiofrequency ^field bring ^{about an}

unusual magnetic ^resonance response. The article shows that the observed [1], ^{NMR linewidth} broadening ^{in the} inhomogeneous ^rf field,

interpreted

ⁱⁿ previous publications ^{as due to a} shortening

of the relaxation time, ^can also be understood in terms of movement of

nonuniformly

^excited spins

across the

receiving

coils. Results for the self-diffusion and the liquid ^{flow are} presented.

Classification Physics ^Abstracts

76.60

In

magnetic

^resonance

experiments,

the radio-

frequency (rf)

field with a

magnetic

^field

component perpendicular

^to the static

magnetic

^{is used to induce}

the resonant transitions between

spin

^{states. Its}

coupling

^{with the}

microscopic magnetic

^{moments has}

been studied with

regard

^{to various} aspects ^else- where

[3],

^but

usually

the rf field

magnitude

^{has been}

assumed to be uniform

along

^the

sample,

whereas its eventual

inhomogeneity

^{has been}

regarded

^{as an}

obstacle to achieve _proper

experimental conditions, especially

^for

pulsed

^NMR

[4].

In this work the

application

of the nonuniform rf field to self-diffusion measurements and flow

velocity

determination are considered. As in

experiments

^with

a static

magnetic

^field

gradient [6],

^{it seems that the}

labelling

^{of the}

spin position by

^the

spatial dependence

of the rf field

might modify

^{the resonance}

signal

^and

provide

information about molecular

migration.

^In

the

following

^{treatment the NMR of the}

self-diffusing spins

^{and the}

spins

ⁱⁿ

flowing liquid

will be treated

by using

^the

ordinary Bloch-Torrey equations [5]

ⁱⁿ

which the rf terms

depend

upon the local coordinate.

This

approach

differs from

previous

treatments

[1, 2]

in which the NMR

line-broadening

^{has been studied}

in terms

of spin

relaxation time

shortening.

1. Self-diffusion and nonuniform RF field. ^- In the frame of the coordinates

revolving

around the direction of the static

magnetic field-z-axis,

^with

frequency

^{0153, the}

Bloch-Torrey equations

^{are :}

Tl

describes the relaxation time of the

longitudinal magnetization density M_,(r), whereas T2

^denotes

relaxation of the transverse

component

of the magne- tization

density

^’

which are

revolving

^with

frequency

^{d - co -} mo around z-axis. Here (1)0 is the Larmor

frequency

^{and the}

frequency

of the rf field. Terms

describing

^the

spin

interaction with the rf field include

where

H,

^{is the}

amplitude

^{of the} nonuniform rf

field, supposed

^{to be}

parallel

^to the x-axis over the whole

sample ( 1 ).

The

magnetization

transfer due to the molecular self-diffusion describes the terms with the self- diffusion tensor D.

By knowing

^the

boundary

conditions the

partial

differential

equations (1)

^{can be} solved. If the

specimen

is confined within the test cell without

magnetization

outflow

through

^the

walls,

^the

boundary

condition is

e) ^In practice, ^this hypothesis ^{is not} realistic since the equations VH, ^{= V x} Hl = 0 would then exclude _{any wl} dependence ^{on r.}

However, ^{we shall} accept ^it for the sake of simplicity. ^{It would be}

easy to generalize ^our calculations to more complicated situations.

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:01978003906068900

690

where n indicates the unit vector

perpendicular

^{to the}

wall. The

stationary

solution of

(la)

^{can be found}

by setting

^its

right

^{side with zero, and}

integrating

^{it over}

the

specimen

^volume.

By taking

^{into account condi-}

tion

(4)

^the

steady

^state

magnetization along

^{z-axis is}

with,Xo

⁼

Mo

^V.

When

solving equation ( 1 b),

elaborate calculations

can be avoided

by assuming

^the

magnetization M,,

constant. This condition is

fulfilled,

^{even under}

saturation conditions if the rf field does not

change abruptly,

and if the molecular

migration

^{is fast}

enough

to

keep MZ spatially

^uniform.

^Thus,

^the

steady

^state

solution of

(1b)

^{for the}

sample

^{in the}

rectangular

test-cell is

with

In the above

expressions

^{n =}

(nx,

ny,

nz)

^with

na ⁼

1, 2, 3, 4,

^...

and g

⁼

(gx’

gy,

gz) with ga

⁼

il/da, where da

denotes the cell dimension

along a-axis,

a ⁼ x, y ^{or z.} an is the coefficient of the Fourier

expansion

with

The

expression (6)

^{can be}

simplified by assuming

^{the rf}

field to be nonuniform

along

the x-axis but homo- geneous

along y

^{and z-axis}

(see

^footnote

(l)) :

with

When

dealing

^{with the cw}

NMR,

^{where the same} coils are used in order to excite the

spins

^{and to detect}

their response, the NMR

signal

^can be obtained

by taking

^{into account} the mutual

spin-coils

^inductance

according

^{to which the} contribution of the volume element of the

sample

^to the induced

voltage

^is pro-

portional

^{to the}

product

^{of its}

magnetization density

and the rf field

magnitude

^at this location.

Thus,

^the total

signal

^is

proportional

^to

where S is the area of the

sample

^{in the}

plane

per-

pendicular

^to the x-axis.

By substituting (8)

^and

(7)

into

(9)

^{the NMR} response ^can be found to be :

The resonance curve

(10)

^{exhibits an} unusual line-

shape,

which is the

superposition of various

^{lines with}

different linewidths. The linewidth of the narrowest line in

(10)

^is

equal

^to

1/T2,

whereas the other lines have the widths

IIT2. (8).

^{Similar resonance curves}

have been obtained with a NMR

experiment

^with

nonuniform rf field

performed by

^R.

Barbé,

^{M. Leduc} and F. Laloë

[1, 2] where, by using optical pumping,

the

magnitude

^{of the}

longitudinal magnetization (5)

of

’He spins

has been observed. The value of the

longitudinal magnetization

^can be found if the ima-

ginary

part

of ms(x), (10),

which is in fact the y-compo- nent of the

magnetization density,

is substituted into

(5). Thus,

^the

magnetization along

the z-axis is obtained :

with

This saturation curve

(Fig. 1)

is similar to that one

obtained with

’He

NMR

experiment [2],

^{where the}

FIG. 1. ^- Saturation of longitudinal magnetization for rf field with different magnitudes ^{and constant} gradient

experimental

results have been

explained

^{in terms of}

the

spin-relaxation

^time

shortening

^{due to the}

magnetic

field fluctuations

acting

^{on the}

spins

^{which are}

diffusing

across the nonuniform rf field

[1].

The authors have

accepted

^this

approach

after the similar

explanation

of the

spin

relaxation of the _gas molecules in the nonuniform static

magnetic

^field

[7].

^The

point

^{of view}

adopted

^{here is to}

^consider,

instead of

spin

^relaxation

rate

changes,

the nonuniform

spin

excitations induced

by

^the

inhomogeneous

rf field and assisted

by

^the

magnetization

^{transfer. In this}

approach

^{the Bloch-}

Torrey equations (1)

^without any ^new relaxation term

completely

describe the effect.

In order to get ^{a better}

understanding

^{of the}

phe-

nomenon one can

imagine

^{a NMR}

pulse experiment

with an

applied

nonuniform rf

pulse.

^{If a weak and}

short rf

pulse

^is

applied

^{at time} zero, the transverse

magnetization

time evolution is

governed by :

where à

depends

upon the width and

strength

^{of the rf}

pulse. Expression (12)

demonstrates that the self- diffusion tends to reestablish the uniform

magneti-

zation distribution

again.

^{The cw NMR}

signal (10)

^is

related to the transient

signal (12) through

the Fourier transformation :

This

correspondence

^{is valid}

only

^{for weak}

pulse spin

excitations and loses its

meaning

^when strong ^rf

pulses

are

applied.

2. NMR

Signal dependence

^{on fluid}

velocity. - Liquid flow, through

^a

glass

^{tube with rf} coils wound

on

it, brings

^{about two}

competing

^NMR effects : the influx of the new nuclei into the coils reduces the

signal

saturation

and, simultaneously,

^the

increasing

^flow

velocity

decreases the

signal

because the nuclei

spend

insufficient time in the

magnetic

^{field to be}

fully polarized

^before

entering

the rf coils

[8, 9].

^{But in}

addition to

this,

^{there is also a} third effect which is due to the rf field

inhomogeneity

and has been observed when

performing

atomic beam

experiments [10].

This effect has been considered

by

J. P. Barrat and J. M. Winter

[11]

^{in terms of the} transition _pro- babilities in the rf field with the time

dependent magnitude.

^{Here we shall}

employ

^the

Bloch-Torrey equations

^{to treat this}

phenomena.

By assuming

^{that the}

spins

^are

completely polarized

before

entering

^{the rf}

coil,

^and the rf field to be

weak,

we can remove the saturation and the

polarization

reduction

[9]

from further consideration.

Therefore, only

^the

spin

^flow

through

the nonunifflrm rf field

might modify

^{the NMR}

signal. Now,

the fluid volume is not restricted to a certain

region,

and extends

beyond

^{the coils}

dimensions, therefore,

^the

spatially dependent

^{rf field}

amplitude

^{can be}

expanded

^{into a}

Fourier

integral

^-

The time evolution of the transverse

magnetization

^is

governed by equation (1b)

^{in which the} self-diffusion term is

replaced by

^{the term}

describing ^the

magne- tization transfer due to the flow of a fluid :

If the flow

velocity

^{V is assumed to} be uniform inside the rf coils and directed

along

^the

gradient,

^{then the}

steady

^{state solution}

of(15)is:

^‘

Here the rf field is nonuniform

along

^x-axis.

By substituting

the above

expression

^into

(9)

^{the cw}

induction

signal

^becomes

The above

expression

is similar to the

expression (10)

where the effects of self-diffusion have been

considered,

but here the

integral

^over q-space

replaces

^{the summa-}

tion in

(10).

^The

particular q-line

ⁱⁿ

(17)

retains the linewidth

1/T2,

but is shifted from the

origin

^{of the}

FIG. 2. ^- Resonance curve for stationary (v ⁼ 0) ^and flowing liquid (v . T2 -Id ⁼ 0.5) in the solenoidal rf coils.

692

spectra ^{for qv. Thus the net NMR curve} which is the

superposition

of various

q-lines differently

shifted from the resonance, results into a broad NMR

signal [10]

(Fig. 2).

Usually

the rf field is set up

by

solenoidal coils in which the

magnetic

^{field can}

approximated

^as

being

uniform inside the solenoid of the

length d,

^{but zero}

outside of it. For such a nonuniform rf

field,

^the Fourier components

(14)

^are

equal

^to

The substitution of

(18)

^into

(17) gives

the result-

ing

^NMR

signal :

Figure

^{2 shows the} resonance curve for the

stationary

and

flowing liquid

evaluated from

(19).

^It demonstrates that the fluid flow

through

^{the rf}

field,

^the

magnitude

of which is nonzero

only

^{inside some} restricted

region,

reduces the line

intensity

^and modifies its

shape.

In this case the weak rf

pulse

^will induce the _magne- tization with the

following

^{time evolution}

By substituting (20)

^into

(13)

^{one obtains} the expres- sion

(19) again.

^It shows that the

steady

^{state NMR}

signal (19)

is the volume _average over the

spectral

^of

the correlation function

m(x, 0) m(x, t).

3. Conclusion. ^- The molecular nonlocalized motion causes a considerable modification of the NMR

resonance curve when a

spatially

nonuniform rf field is.

applied.

^{This can be}

employed

for the self-diffusion coefficient determination in _gases as well as for the measurements of flow velocities. In the

following

article we shall demonstrate how this

technique

^{can be}

used for

velocity profile mappings.

Acknowledgments.

^- This work was initiated

during

the author’s visit to the

University of Washing-

ton as a

Fulbright fellow,

^{and was later}

supported by

^a

Boris Kidric Foundation research grant.

References [1] BARBÉ, R., LEDUC, M. and LALOË, F., Lett. Nuovo Cimento,

8 (1973) 915; ^J. Physique ³⁵ (1974), ^699.

[2] BARBÉ, R., LEDUC, ^{M. and} LALOË, F., ^J. Physique ³⁵ (1974)

935.

[3] ABRAGAM, A., ^The Principles ^{of Nuclear} Magnetism, ^Oxford (University Press) ^1961.

[4] GARROWAY, ^A. N., MANSFIELD, ^{P. and} STALKER, ^D. C., Phys.

Rev. B 11 (1975) ^121.

[5] TORREY, ^H. C., Phys. ^{Rev. 104} (1965) ^3595.

[6] HAHN, E. L., Phys. ^{Rev. 80} (1950) ^580.

[7] KLEPPNER, D., GOLDENBERG, ^{H. M. and} RAMSEY, ^N. F., Phys. ^{Rev. 126} (1962) ^603.

[8] JONES, ^{D. W. and} CHILD, T. F., in Advances in Magnetic

resonance vol. 8 (J. Waugh) ^{1976, p. 123.}

[9] ZHERNEVOI, ^{A. T. and} LATISHEV, G. D., Nuclear Magnetic

Resonance in a Flowing Liquid (Consultant Bureau, New York) (1965).

[10] ^{KuscH, P.,} Phys. ^{Rev. 101} (1956) ^627.

[11] BARRAT, J. P. and WINTER, ^J. W., ^J. Physique ¹⁷ (1956) ^833.