Pulsed NQR: an example of selective excitation

97

Journalofh~olecuhrStmcture,111(1983)97-102 ElsevierSciencePublishers B.V.. Amsterdam-Printedin

PULSED

B.C.

AN

NQR:

SANCTUARY

Department

OF

F.0.

and

2K6,

SELECTIVE

EXCITATION

TFXHE

of Chemistry,

~34

Quebec,

EXAMPLX

The Netherlands

McGill

University,

801

Sherbrooke

St.

West,

Montreal,

Canada

INTRODUCTION In pulsed

?XUR the

to bs preferable

are

all

developed

where

(7) to cover

the

to pulse

large

effects

of rf guises

betveen

-,,2,

correspond be a cotmon

with

field

-1

and

consider

The

the

EXCITATION

multi_wle

(ref.81

in terms

rotation

group.

spherical

NQR

levels,

are

(1.2) all

pulses

and

has

that

found

should

frequency

recently

spectral

ensure

are

various

range

been

frequencies

is particularly it

are

pertinent

to NQR

is rarely

possible

pure pulses.

which

advantages

no

gives

rise

longer

the

NHR

initial

with that

integer

'--'

1/2,

selective

spin

there

(e.g. fra

or,

will

as in I = 1 both

the

rf

rotations.

formulation

case.

In

to transitions 3/2

to effects

purely

the

condition.

such

frequencies

of a multiplet

with

and

for

spectral

This

any

of I = 3/2

to describe

magnitude;

corresponding

quadrupole

In contrast,

are

for

frequencies

by two

is used

of arbitrary

predicted

by the

shared

excitations

spins

as in case

affected

The

of selective

theory

pulse

is extended

to

in NQR.

IN NHR

formulation

of NUR

of a multipole Physically

k

simply

the

(ref.3-7)

operator describes

corn_ponent, q = A m defines

polarizations operator,

single

pure

starting

applications

SELECTIVE

that

level

outlined

not

constant

bs completely

of

quadrupole

paper

are

to

duration

NMR

of selective

<---> 0 I 0 c---> 1).

the

In this theory

theory

will

pairs

always

in which

&functions

of the

infinitely-short

pulse

irradiation

involving

are'not

pulses

case

NMR

it is shovn

isolated

<-->

ideal the

pulsed

coupling

to

rotations

simultaneously.

experimznts

particular,

-3/2

quadrupolar

the

cases

Selective

frequencies

paper

NQR

such both

selectively

those

pulsed.

all

In this pulsed

for

corresponding

prodzce

in principle

A theory

sizaultaneously

pulses

simply

in practice

frequencies,

finite.

of pure

they

whilst

polarizations. cover

use

since

the

expectation

writes

basis the

vhich

multipole

the

spin

density

is irreducible character

and

multiquantum

coherences

value

corresponding

of the

operator under

the

the

(ref.6).

The

nulti_wle

viz: (1)

0022-2860/83/$03.00

0 1983 Elsevier Science Publi&ersB.V.

98

.. ,.

Such

kuouledge

The

spin

of

density

the

nuclear

opekatoz 21

=A( 21+1

UIW

where

the

au,(t)

is the

the

nultipole

spin

basis

state.

as,

(2)

+)~kqtI,i

pro&aI-y,yktZ standard

vkq,

has

been

used.

The

tima

evolution

is

&a,

Hamiltonian

In solving one

dif f exantial

exp(

U =

commutator equation

(3)

for

it will

(31,

of form

Instead,

describes

in the

[)rlrul(t)l_

=

at

operators

completely

be written

kolq=-k

by the

j’.

can

+k 2

z

adjoi+

determined

vherex

EI+

polarizations

oI(t)

is

be

the

system.

seen

-Ok),

that

exponentiation

in

as

calculated

and

a set

of

has

(ref.13).

anduse

been

of unitary

completely

or' coupled

first

avoided.

order

obtained,

(4)

irhereeisa of the

(2)

vector

spanning

indices

k and

has

q, and

a matrix

representation

form

(5)

kq,k'q'

It is clear derived

that

in part

is necessary Tbi

the from

the

initiaI

maltipole

theory

of

selective

irradiated

features.

For l/2;

transitiors level

tion

formalism share

of the

Such

an

v-rite fonaaI operator

the

the

mlti$let~&xept

equations

NQR

of adjacent

these

levels

spin

are

(ref.7)

will

act

on a

analagous

on the

l/2:

a multiplet 1.

transitions subspac+2

(l>-(5).

The

two

within spin

straight and

allw

multiplet

q =

a zultiplet

When

Fuxther is

subset

strongest

define

excited

in Fig.

(2Ir$?)

to

to

(41.

quimtum

the

(ref.5).

It

of

multiplet

single are

be used

(5) are

op&ators.

a soluticm

only

simultaneously

to multiquantum spin

HKR

elsewhere

is not

(4) and

to obtain

frequencies

may

illustrated

in

Liouville

f ocusses

discussed

level

level'construction

I, multiplot

k(O)

excitation

has..been

multiple=

with

In conventional for

a co-n

implicit

dealing

conditions

pulse-

observed:

q = fl pairs the

that

with

by the are

submanifold,

applies

(ref.91

s_=cify

Art = fl cohereno&

of

representations

ideas

to

of levels

s#_u

and

concepts earlier

of

a three 1 nov

generalizaforward. one spin

to density

is then

(6)

Vhicbis

particularly

simple

for

IH=

l/2,

or

1.

For

a Hamiltoni&

99

L-1 (a)

Selective excitation of (a) one transition defining an quantum transitions defining an k=', multiplet spin.

Fig.1. single

{31~-I(Ifl)}-%xw,co~(ti-~)

where

Q is the phase

selective

w, eff=

J(I+m)(I-mtl)

and

the

quadrupolr

for

In=

1

,

Geff =

the

and

the

w

= yHo,

rf amplitude

it is found

is replaced

and

(b) two

sin(k.-0)

that

for

(ref.3,10)

the

(7)

single

quantum

by

(8)

"1 term

G

effective

is replaced

term

For

by ceff(ref.3).

IH=

l/2,

Qeff=

0 and

is

2Q

(9)

1(21-l)

It is significant of z,' the result

angle

excitation,

+ fiIyw,

I&,

for

that

effect

multiplet

of an rf pulse

is a pure

rotation

spins

is solely

iM = j/2,

regardless

determined

by the

of the

rf Zeeman

magnitude term-

The

(ref.6,7),

(10)

The

angles

in the origin

may

arising being

be

fram spread

ligner

For

&I,,

and

2w,

factors

of the

(21+1)

levels are

lnplies .

the

pulse

quadru_pole

both

physically

cntrices

a selective

for

on and

w, is replaced

localization over

and*;:

rf and

given

interpreted

rotation

(ref.12)

B are

context,

of the

these

both

a and

present

found as

IH
use

frame

on a multiplet

interactions

mast

of

the

(21n+l)

the

defined spin

that

processes

in y and

non-selective

with

conditions

It is seen

in'selective

increases

rf to the

defined

off-resonance

U, eff-

as in the

rotating

acting

by

subspace, In

convention , -iq = e

1,=-l,

be considered.

(ref.lO.11);

effective

case.

by.%

the

This

(ref.6); (8) is the

rf power

rather (70)

of Ednds uot 0; combination

has

been

than

the

of

solved

by

100 perturbation

theory

rf amplitude

is larger

is less

6.

than

selective

than

clearly

in the limits

lw,

w, whilst

effktive

result

the

!0,~>!5:

1-t

the

The

pulses.

&L?+,:

Qefl

(ref.31

jGeff i.

effl’

Since

quadrupole

is particularly

the

effective

coupling

constant

easily

satisfied

by

is,

(12)

eff

,

Aw/Q

co9

8=

and

vhere

is gives

(13)

the

effective

quadnpole

occurring

in the

development

matrices

H =

(t)

by*

&ff = The

(14)

forms

of the H rratricw for

tref.6);

g(t)

single

for

quantum

I < 9/2

and

it

process

for

IH=

1 have

hen

given

elsewhere

is simply,

=

(15) .

Clearly

a, pure

to visualize simplifies The within

In=

correspontig

for

outlined

done

in NQR

of In=

case use

at

l/2

is physically

resonance,

easier

(eq- 11)

b2=0

1P_=l.

here

21 +1 subset of H Sensity operator

the

to

1, alth0ugi-1 in actuzl

corsiderably

treatment

mfitiplet easily

rotation

than

describes

single

levels.

To

from

full

problems,

the

expecially

density vhen

selective

excitations

it is necessary

to extract

quantum

be of use,

the

This

matrix

o_perator.

system

is initially

the

is

at

equilibrium.

PUISm

NQR

Asan

example,

principle The

axis

equilibrium

o,,,(O)

where,

=

the

density

k[E3,2

rs

Fn

case'of

convention

and matrix

an axially in the

vtric

absence

qua&pole

for

of an external

field

paper

operator,is

(16)

0

-1

01

a

III,

0 0

within

+ 9;(O>y20(I)}

010 0

I=3/2

is treated.

.(17)

and

20 =

0

Y

..o i

0

0

1 0 0. 1 0

O-l

0

0 0

(18) 1

101

The

matrix

easily

decomposed

a,,,,(O) The

representations

t{E

=

into

3/Z

of

%2(3/2)

4:

levels

0

=$

0

-i

equation

An

(18)

is

l/2 tc give,

(l/2

bl) refer

to an

the

up_pe-st,

and

H,

&=

l/2 multiplet

m-l,

represented

ZI'O &l/2

(-l/2)

spin

of two

in matrix

(6)

/a

o\

H

!j

(2G'

it follows

important to the

(22)

aspect

of

this

<-- > l/2

3/2

degenerate

states

needed

the

and

1 1 -1 4oOy y)

and

picture

is that

transition

and

inply

the

clearly

-18

O

0

0

o\

!

H _si

(2')

that

Oi($ $1 = i+;(O) refer

of

Equation

by I1

From

spins

(ref.5).

(19)

(M) and

composed

notation

multiplet

basis

Px'

-

notation

states

to the

refer

that

the

the

-3/2

nultiplet

-i+:(O)

=

(23)

trrO multiplet <-->

-l/2

spins

of

transition.

spin

(l/2

l/2)

l/2 The

is not

Y" and

-l/2

<---> l/2

two of the three

only

I = 3/2

at zero

on each

multiplet

gives

the

full

modification Eq.

(10)

spin

t,

Chwsing

as

invoking

in NQR NMR

duration

of

for

the

(8)

the

([cK~~~),

formalism the

Eq.

(10)

the

+3/2

are

is precisely

excited,

describes results after

iS the

Zeeman

It

excited.

processes Eq.

c3,2

from

w.

is not

substituting

matrix,

the

frequency

and

l/2;

density

for

quantum

is selective.

required

quadru__wle (LW~,

single

field

derived

transition

the frcu

the

zbsence

that

the

effects Eq.

of the

on

of a pulse

10 into The

pulse.

because

pulse

Eq.

19

only

rotating

fr-

in

frequency.

pulse,

at resonance

<---> +

l/2 transition,

with

the

pure

so that

APO

and

gives

@:(t,) = pIv11(I)a3,2(t)] =i

Since the

+i(tu)

pulse..

9,(t,) for

=

phase

ref.11

r/3/10 0; sin(J3w,tw)

a

the

-JZ

ok

angle

with

,

Hence

of

Q:(O)

the

two

result

sin

Q = O= -1.

sin

ei@

w t ow

corn_pcnents of

(24)

0: correspond

to



and



after

is simply

(r'3wltw) This

agrees

sin

w OWt

with

(25)

= 2/r/5 the

expression

Eq.(13)

calculated

in

'102 CON(ILUSION

.&D

Provided inter-

the

z Q,,=

of necessity

0, even

Ucxover.

necessary

operdtor total

irradiation

for

spin

ls only

pertuibing

system can be described

spin

constants.

only

GEX&.ALIBATIONS

the

spin

I, as

the in

operator

of

presence

kf amplitude

.to extract

density

in the

a single

spins

use

(ref,6); aI(

large

Cl01

The

?/at

fraa

and

reinsert

X=3/2

the

for

any

half-

these

all

have

coupling

according

l/2 of

case

of

qua-polar

is increased

multiplet

resonance,

spins

by &tiplet

(8).

It is

spin

density

result

In the

to

total the NKR

is treated

in III

(ref.51. For

except and

integer

the

share

common

multiplet by

Financial Council

the

(14).

by

sane

-1

(11)

sup~rt of Cauada

as

E'urther, of

0.

ths

this

holds

These the

in III. effect

qud.rupolar

provided!

for

treatxzent

<-->

therefore

level; of &=l,

a combination

described

8 9 10 11 12 13

spin

(9) and

rather

spins

1 C---> 0 and

two

,

vork

uas

for

all

are

both

single

transitions

Iu this of the and

w, eff!

tw

case

must

pulse

ti not that

a

zero

pure

may

trzzsitions

in frequency

be treated

is not

geff

rf evolution

quantum

coincident

but

by a Fs given

rotation

but

approxfcately

>> Izeffl.

pravided

by Natural

Sciences

and

Engineering

(NSIRC).

G- Bodenhausen, Prog- NWR S_pectroscopy, :4 (1981). 137. A. Xokaun and R-R: Ernst, J. Chen. Phys., 67(1977), 1752. B.C. Sanctuzry, (Paper V), Wol. Phys., in press (19831. Ides., (Paper I), J. Churn., Phys., 73 (19761, 1048. Id-., (Paper III), Bol. Phys., 48 (19831, 1155. B.C. Sanctuary, T-K. Halstead and P. Osment, ibid., .in press (1983). B.C. Sanctuary and T-K- Balstead, (paper IX), J- Bag. Resonance, 53(19831. 187. u. Faso, Rev. nod. Ehys., 29 (1957), 74. C-N- Banweil and H. RiPas, Mol. Phys-, 6 (19631, 225. S. vega, J. Che!a. Phys., 68 (19771, 5586. H. Blocn, B. Hekzw.and E-L. Hahn, Phys. Rev., 97 (1955) 1699. A-R- Edeon&, 1960, Angular #omentm in Quantum Mechanics, (Princeton Univ.). T-P. Das and A-RS&a, Phys- Rev-, 98 (1955). 316.