UBC_1965_A1 H35.pdf - cIRcle - University of British Columbia

The U n i v e r s i t y

of B r i t i s h

Columbia

FACULTY OF GRADUATE STUDIES

PROGRAMME OF THE FINAL ORAL EXAMINATION FOR THE

DEGREE OF

DOCTOR OF PHILOSOPHY

of

WALTER NEWBOLD HARDY

B.Sc.j The U n i v e r s i t y

of B r i t i s h

Columbia, 1961

MONDAY, OCTOBER 26, 1964, AT 9:30 A.M. IN ROOM 303, HENNINGS, ( P h y s i c s )

COMMITTEE IN CHARGE Chairman;

W. H. Gage C. A. McDowell * P. R a s t a l l R. F . S n i d e r

M. Bloom K. L . Erdman R. Howard External

Examiner:

Department Massachusetts

J . S. Waugh

o f Chemistry

I n s t i t u t e o f Technology

NUCLEAR SPIN RELAXATION IN GASEOUS H , HD and D 2

2

ABSTRACT The l o n g i t u d i n a l and t r a n s v e r s e n u c l e a r r e l a x a t i o n times, Ti and T , have been measured i n normal H2 gas a t 2

77.5°K i n the p r e s s u r e range 0.05 t o 2 atmospheres. t h i s r e g i o n Ti goes through a minimum, and T significantly

2

In

deviates

from a l i n e a r dependence on t h e d e n s i t y .

Comparison of t h e experimental

data with e x i s t i n g

theory

e s t a b l i s h e s t h e f o l l o w i n g r e s u l t s f o r t h e J = l s t a t e of orthohydrogen: i. ii.

iii.

Ti

-

a u t o c o r r e l a t i o n f u n c t i o n s of t h e m o l e c u l a r a n g u l a r momentum o p e r a t o r s a r e e x p o n e n t i a l or n e a r l y so, the r a t i o of the c o r r e l a t i o n times ^"1 , "^2 which are a s s o c i a t e d w i t h o p e r a t o r s of t h e form J+, and J$. r e s p e c t i v e l y l i e s w i t h i n t h e l i m i t s 0.6 ^ t l / t 2 ^ 1, and t h e s p l i t t i n g of the m o l e c u l a r Zeeman l e v e l s cannot be n e g l e c t e d as i n t h e o r i g i n a l Schwinger theory. f o r t h e p r o t o n and deuteron

deuterons

i n normal D

temperature

2

i n HD gas and f o r t h e

gas was measured as a f u n c t i o n of

and p r e s s u r e i n t h e range 20 t o 37 3°K and

0 t o 8 atmospheres.

To w i t h i n experimental

dependence of T-^ on t h e d e n s i t y ^ low

e r r o r the

i s linear.

I n HD be-

65 K, when only the J=0 and J = l s t a t e s of t h e 0

molecule

a r e a p p r e c i a b l y populated,

pendence o f T-^/^> deuteron,

i s identical

de-

f o r both p r o t o n and

l e a d i n g t o a v a l u e of f

f o r the J = l s t a t e of HD.

the temperature

/ T

2

Above 100°K, T

= 1.07 t 15% ±

f f o r

the

p r o t o n i s i n v e r s e l y p r o p o r t i o n a l t o the temperature, whereas f o r t h e deuteron T^/ ^ independent.

The experimental

i s almost

temperature

results are interpreted

as evidence t h a t i n HD gas the process o f m o l e c u l a r r e o r i e n t a t i o n i s dominated by t h e a n i s o t r o p i c

inter-

molecular force a r i s i n g

from, the s e p a r a t i o n of the. c e n t r e s

of mass and charge, of the m o l e c u l e .

In

gas two

relaxa-

t i o n times were found, one a s s o c i a t e d w i t h the S=l. s p i n s t a t e of paradeuterium and the other a s s o c i a t e d w i t h the S=2

s p i n s t a t e of orthodeuterium.

appears in

to go through a minimum;

the analogous

measured by p r e v i o u s workers

minimum , but. at 80°K, ;

ting

At 40°K (T, / £ )

a l s o goes through a

This i s consistent with interpre-

the minimum as a quantum, mechanical

effect.

quantity

The J«2 component of (T-^/ £

not. go through a minimum., which

diffraction

)g_2

J

however, does

suggests t h a t the

m o l e c u l a r i n t e r a c t i o n s are s i g n i f i c a n t l y d i f f e r e n t the J = l and J=2

interfor

s t a t e s of the m o l e c u l e .

GRADUATE STUDIES

Field

of Study:

Physics

Elementary Quantum Mechanics Waves E l e c t r o m a g n e t i c Theory Nuclear P h y s i c s Solid State Physics Special R e l a t i v i t y Magnetism S t a t i s t i c a l Mechanics Advanced Quantum Mechanics

Related

F.A. Kaempffer R.W, Stewart G.M. V o l k o f f J.B, Warren R. B a r r i e P. R a s t a l l M. Bloom R. B a r r i e F.A. Kaempffer

Studies:

Electronics Quantum Chemistry

W.A.G. Voss J'.A.R. . Coope.

PUBLICATION

N u c l e a r Spin R e l a x a t i o n Times i n Pressures, 467

(1963).

Gas a t Low

B u l l . Amer. Phys. Soc. 8_,

NUCLEAR SPIN RELAXATION IN GASEOUS H , HD and D 2

2

by

WALTER N^ HARDY B.Sc., The U n i v e r s i t y of B r i t i s h Columbia^

1961

A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY i n the Department of PHYSICS

We accept t h i s

t h e s i s as conforming to the

r e q u i r e d standard

.

THE UNIVERSITY OF BRITISH COLUMBIA October, 1964

In the

r e q u i r e m e n t s f o r an

British

mission

advanced

for reference

for extensive

p u r p o s e s may

be

of

and

written

Department

of

the

Head o f my

V-*\





38

MEASUREMENT OF RELAXATION TIMES

45

3- 1

Pulsed N.M.R. Techniques

45

3-2

Apparatus

47

i ) General i i ) Pulsed

47 spectrometer

i i i ) Temperature measurement and c o n t r o l i v ) Metal

dewar

v) Gas h a n d l i n g system v i ) Sample h o l d e r s

49 60 66 70 74

-v-

Chapter IV

V

Page EXPERIMENTAL RESULTS AND DISCUSSION

79

4-1

The T

79

4-2

R e l a x a t i o n Measurements i n HD

93

4-3

R e l a x a t i o n Measurements i n Dg

102

Minimum i n E

Q

SUGGESTIONS FOR FURTHER EXPERIMENTS

107

Appendix A

B

C i r c u i t D e t a i l s of Pulsed Spectrometer and Temperature C o n t r o l Unit

109

Theory of Boxcar I n t e g r a t o r

118

Bibliography

136

LIST OF TABLES

Table Page 1

Values of P j ( x )

15

2

Some p r o p e r t i e s of the H^, HD and

3

M o l e c u l a r constants f o r H^, HD and

29

4

Values of Ca/C. at room temperature

74

molecules

16

-vii-

LIST OF ILLUSTRATIONS

Figure 1

Page Nature of the c o u p l i n g between the n u c l e a r and

spins

the l a t t i c e

19

2

T ( D ) / T ( P ) vs f A ;

31

3

Energy l e v e l s of a molecule i n the J = l s t a t e

42

4

Block diagram of 30 Mc pulsed spectrometer

48

5

Schematic of method

2

2

a

f o r o b t a i n i n g t i m i n g sequence

f o r T j measurement

51

6

R e s i s t a n c e thermometer c i r c u i t

64

7

Metal dewar

67

8

Schematic of HD gas h a n d l i n g system

72

9

Diaphragm pump

72

10

30 Mc sample h o l d e r

75

11

7 Mc sample holder

78

12

T y p i c a l t r a c e s f o r T^ measurement

i n Hg gas

81

13

Typical

i n Hg gas

81

14

T

15 16

T v s p i n Hg at 77.5°K ^l^min ^ l b

17

2

t r a c e s f o r Tg measurement

\s j? i n Hg at 77.5°K

85 87

x

a

(X/pV

n

d

vs

( 1

o T

)

a

s

a

f u n c t

of < / r ^

t,/K

T

vs pressure f o r proton i n HD at 77.5°K

19

^\/f>

20

^\/p

21

T,/p vs 1/T f o r S=l and S=2

v

s

IA

f°

r

90 91

18

x

ion

protons and deuterons i n HD gas

vs 1/T f o r protons i n HD gas above 100°K s p i n s t a t e s of Dp gas

94 96 98 104

-viii-

Figure

Page

IA

Coherent gated o s c i l l a t o r and t r i p l e r

110

2A

Gated power a m p l i f i e r

111

3A

Tripler

112

4A

Phase s h i f t e r

113

5A

Pulse mixer and a m p l i f i e r

114

6A

7 Mc p r e a m p l i f i e r

115

7A

Boxcar i n t e g r a t o r

116

8A

Current r e g u l a t o r f o r temperature c o n t r o l

117

IB

E q u i v a l e n t c i r c u i t of boxcar i n t e g r a t o r

118

2B

General appearance of boxcar output from noise input

121

3B

H y p o t h e t i c a l low-pass f i l t e r

126

4B

Input waveform;

127

5B 6B 7B

e x p o n e n t i a l decay

vs U / t Input waveform; Sin X

vs

135 e x p o n e n t i a l l y damped sine wave

x = uots

130 135

-ix-

ACKNO WL EDGEME NT S I would l i k e to express my s i n c e r e g r a t i t u d e to Dr. Myer Bloom f o r h i s guidance and encouragement throughout the work. The

considerable

e f f o r t s of Dr. D. Llewelyn

Williams

i n the c a p a c i t y of temporary s u p e r v i s o r are g r a t e f u l l y acknowledged.

Dr, Kenneth Gray has read parts of the t h e s i s and

o f f e r e d v a l u a b l e comment. Of

the many members of the Physics Department who

have c o n t r i b u t e d t e c h n i c a l a s s i s t a n c e to the work, I e x p e c i a l l y want to thank Mr. John Lees, our glassblower, Morrison

and Mr. Peter Haas of the workshop. I would l i k e to thank my p r i n c i p a l

student

and Mr. W i l l i a m

critic

and f e l l o w

John Noble f o r the encouragement and h e l p f u l c r i t i c i s m

he has o f f e r e d during

the course of t h i s work.

I am indebted

to Frank Bridges

f o r h i s assistance

i n t a k i n g many of the measurements. Thanks are due the N.M.R. group of the Chemistry Department who k i n d l y loaned

the s e t of V a r i a n

homogeneity

c o i l s used i n the experiments. My wife has c o n t r i b u t e d an i n c a l c u l a b l e amount to the w r i t i n g of the t h e s i s .

Without her constant

moral

support

t h i s t h e s i s would never have been completed. I express my s i n c e r e a p p r e c i a t i o n to my law Mrs.

P a t r i c i a Hardy who d i d the f i n a l The

acknowledged.

t y p i n g of the t h e s i s .

f i n a n c i a l a s s i s t a n c e provided

Research C o u n c i l over the past

three years

sister-in-

by the N a t i o n a l i s gratefully

-1CHAPTER I

INTRODUCTION The

techniques of Nuclear Magnetic

Resonance (N.M.R.)

allow us to study the c o l l e c t i v e behaviour of the n u c l e a r spins i n a bulk sample; i n p a r t i c u l a r , we can measure the r a t e at which the n u c l e a r spins approach surroundings.

thermal e q u i l i b r i u m with

In most diatomic and polyatomic gases

their

the mechan-

ism of r e l a x a t i o n i s the s o - c a l l e d Schwinger mechanism, whereby the spins are r e l a x e d by the i n t e r n a l magnetic

f i e l d s of the

molecule which f l u c t u a t e as a r e s u l t of molecular I f the i n t e r n a l

collisions.

i n t e r a c t i o n s are known, as they are i n hydrogen

and i t s deuterated m o d i f i c a t i o n s , n u c l e a r s p i n r e l a x a t i o n

times

can provide d e t a i l e d i n f o r m a t i o n on the process of molecular r e o r i e n t a t i o n , which i n t u r n gives i n f o r m a t i o n on the a n i s o t r o p i c i n t e r a c t i o n s between molecules.

The aforementioned

possibil-

i t y was the m o t i v a t i o n f o r a systematic study of r e l a x a t i o n i n gases begun by L i p s i c a s and Bloom*, of which t h i s t h e s i s i s a continuation. magnetic

We now give a b r i e f sketch of the elements of

resonance.

(See Abragam^ f o r a complete d i s c u s s i o n . )

An i s o l a t e d nucleus with s p i n angular momentum t i l and magnetic

moment p = #~ftl c o l l i n e a r with i t , has energy

l e v e l s i n a magnetic

field H

Q

given by

-2-

where mj

i s the p r o j e c t i o n of I along

ternal f i e l d , system.

u s u a l l y taken to be the z-axis

Let us now

a c t i n g with

the d i r e c t i o n of the

consider

of the

coordinate

an assembly of s p i n s weakly

t h e i r environment.

ex-

inter

A f t e r thermal e q u i l i b r i u m

has

been e s t a b l i s h e d the ensemble of s p i n systems can be

described

by the p o p u l a t i o n

given

d e n s i t i e s of the energy l e v e l s E , m

the Boltzmann d i s t r i b u t i o n

by

function

(1.2)

ir»=-x T i s the temperature of the matter i n which the nuclear are embedded, normally

c a l l e d the " l a t t i c e "

temperature,

i r r e s p e c t i v e of whether the sample i s a s o l i d , In a gas

the " l a t t i c e "

l i q u i d or

c o n s i s t s of the molecule on which

s p i n i s s i t u a t e d which i s coupled the molecules i n the gas.

by c o l l i s i o n s

to the

The

net magnetization

spins i s t h e r e f o r e given I

gas. the

r e s t of

(For a d i l u t e hon-monatomic gas,

i n t e r a c t i o n between spins on d i f f e r e n t molecules can be lected.)

spins

of a bulk

the

neg-

sample c o n t a i n i n g N

by

I

3kT (1.3)

-3-

For H

= IO

0

species,so

IO"

T »

?K f o r most n u c l e a r

-4

t h a t the high temperature approximation

now

i n v e s t i g a t e the behaviour

of the

i n the presence of a r o t a t i n g r . f . f i e l d

Neglecting

relaxation effects

of the s p i n s w i t h —> M

is valid for

°K.

4

We It

HQ i s of the order I O

gauss

4

magnetization

perpendicular

( i . e . n e g l e c t i n g the

t h e i r surroundings)

to H .

interaction

the equation

of motion f o r

is

AH d

=

K n x H (1.4)

t

—i* where H i s the t o t a l magnetic f i e l d T h i s f o l l o w s from the f a c t t a t i o n value

reference to

sample.

that the quantum mechanical expec-

of the magnetic moment of a s i n g l e s p i n obeys

corresponding To

a p p l i e d to the

c l a s s i c a l equation s o l v e (1.4)

of motion.

i t i s convenient

frame r o t a t i n g with

the l a b o r a t o r y frame.

the

angular

to transform

v e l o c i t y OJ with

In the r o t a t i n g frame the

to a respect

equation

of motion f o r M becomes:

_

^ ?] +

^ M x

(1.5) I f H i s only the s t a t i c f i e l d H choosing we

get ^13=

frame. H

0

^=

-Xl^Tc^lk^is

0

along

the 2!-axis

f

then

a u n i t v e c t o r i n the Z - d i r e c t i o n )

0 so that I^fis f i x e d i n the r o t a t i n g r e f e r e n c e

T h e r e f o r e ^ i n the l a b o r a t o r y frame M precesses

at the frequency

w

-X H , c a l l e d

~-

Suppose now frequency

by

Q

we

UL> p e r p e n d i c u l a r

add

to H

to H

Q

.

Q

the Larmor

a field

about

frequency.

r o t a t i n g at a

I f the r o t a t i n g r e f e r e n c e

-4-

f r a m e i s c h o s e n t o have i t s x - a x i s c o i n c i d e n t w i t h H,,

the

equa-

i

t i o n of motion f o r M i n t h i s

K

_

IM

frame i s

now

rW

X

(1.6) where

( i.

i s a unit vector

ence f r a m e . )

+ u;) +

] * .

l

field

field

has

Q

+ .x+0

as a l l of the molecules r e v e r t to the

ground s t a t e J = 0,

I = 0,

ever,

I t i s an experimental

that the time taken f o r the establishment

between the ortho and order of y e a r s ,

f o r pure hydrogen at N.T.P.

T h i s i s due

t r a n s i t i o n can be induced

that i s d i f f e r e n t

molecules can cause such t r a n s i t i o n s The

mixture of two

net r e s u l t

i s that

and

no N.M.R. s i g n a l ,

Ortho Hg,

an I = 1 s p i n system.

different

must be t r e a t e d as a

( r = 3,

Para Hg

resonance the

two

has

gives

It is interesting

corresponding

wave f u n c t i o n s .

the

the r e s t r i c t i o n

the n u c l e a r s i g n a l the 2N

p r o p o r t i o n a l to 2N

75% ortho, 25% para mixture,

from

T h i s f o l l o w s from the r e s u l t

For N molecules of Hg,

give a s i g n a l

resonance

that would r e s u l t

t h a t , e v e r y t h i n g e l s e being equal,

s p i n V> p a r t i c l e s

like

to note t h a t f o r

same c o l l e c t i o n of s p i n H> p a r t i c l e s without

p o r t i o n a l to 1(1+1).

and

to an e q u i l i b r i u m mixture at

i s the same as the s i g n a l

of anti-symmetric

1=0

on the other hand, behaves

high temperatures) the magnitude of the n u c l e a r

The

pro-

d i s t i n c t gases.

m o d i f i c a t i o n s are q u i t e d i f f e r e n t .

signal

only by a

t h i s mechanism i s

From the p o i n t of view of n u c l e a r

normal H

the to the

at the s i t e s of the two

tons; hence only the magnetic c o u p l i n g between

very weak.

of e q u i l i b r i u m

para m o d i f i c a t i o n s i s very long, of

f a c t that an ortho-para magnetic f i e l d

f a c t , how-

i s pro-

uncoupled

1(1+1) = - N.

on the other hand, gives a

p r o p o r t i o n a l to jN 1(1+1) = 2 N s i n c e 1 = 1 for H . 4 2 T h i s r e s u l t i s a m a n i f e s t a t i o n of the correspondence p r i n c i p l e . signal

-13-

The

deuteron has

intrinsic

s p i n S = 1 and

i s therefore

a Boson, r e q u i r i n g the t o t a l molecular wave f u n c t i o n to be symm e t r i c with r e s p e c t to interchange of the two simplify

the n o t a t i o n S i s used

number and I i s used

nuclei.

f o r the deuteron

(To

s p i n quantum

f o r the proton s p i n quantum number.)

The

even J s t a t e s must t h e r e f o r e combine w i t h the even n u c l e a r s p i n —>

s t a t e s S = 0,2 with the odd

(S =

—>>

—>

+ S ^ ) , and

the odd J s t a t e s must combine

n u c l e a r s p i n s t a t e S = 1.

(J = 1,3,-—) to ortho D

(J = 0, 2,

The

r a t i o of para

) at high

Dg

temperatures

2 i s given by 3UT-H)

_

(2J77) -h5-(p.TH)

so that a normal mixture ortho D

.

j_ 2.

of Dg

c o n s i s t s of 1/3

Conversion between the two

para Dg

and

2/3

deuterium m o d i f i c a t i o n s

i s even slower than the c o n v e r s i o n between ortho and

para

hydrogen because of the s m a l l e r magnetic moment of the

deutron.

In c o n t r a s t with hydrogen, both deuterium m o d i f i c a t i o n s have n u c l e a r s p i n s t a t e s t h a t g i v e N . M . R .

signals.

The

amplitude

of the n u c l e a r s i g n a l i n normal,D^ i s p r o p o r t i o n a l to (2S+1)S(S+1) = 6 f o r the S = 1 s p i n s t a t e and

(2S+1) = 30 f o r the S = 2 s p i n

s t a t e , whence the r a t i o of the n u c l e a r s i g n a l Dg molecules 30 - ~ = 5. 6

to that from the para molecules

At low temperatures

from the ortho i s g i v e n by

the two m o d i f i c a t i o n s are

ex-

pected to have very d i f f e r e n t n u c l e a r s p i n r e l a x a t i o n

times.

In the ground s t a t e (J = 0) of the ortho molecule

intra-

molecular i n t e r a c t i o n s are absent

and

the

the r e l a x a t i o n time

should

-14-

be a c c o r d i n g l y

long.

On the other hand, the

i n t e r a c t i o n s i n para

are present

s i n c e the ground s t a t e has It

has

J =

been pointed

communication) that when the taken i n t o account the S = 0,

even at low

temperatures

1. out by N.F.

Ramsey ( p r i v a t e

intramolecular '- l-n\te'Taetlon;s\ .g'r'.e :

;

s t a t e s of the ortho molecule denoted

S ~ 2 are not the proper s p i n e i g e n s t a t e s .

a p p l i e d magnetic f i e l d

For

a t i o n times we

assume that 1/9

simply

become a

the purpose of c a l c u l a t i n g r e l a x -

are i n the S = 0 s t a t e and

c)

When the

i s l a r g e , however, S„ should

good quantum number.

5/9

of the ortho

molecules

are i n the S = 2 s t a t e .

HD HD

i s a mixed molecule and

symmetry requirements.

The

J s t a t e s i s given

by

P

where P

intramolecular

simply

=

are

no

e q u i l i b r i u m d i s t r i b u t i o n of

e.

(3 3>u

hence there

the

T

i s the p r o b a b i l i t y of f i n d i n g a molecule i n a J

state. Table 1 gives values

of

^ for

various

values

0 e

of x = QR,

,

T

Table 2 l i s t s and

D

some of the p r o p e r t i e s of the H

molecules r e l e v e n t

to the present

discussion.

, HD

by

It

J

T (°K) for

% =

64.3°K

•

T

0

1

2

3

4

5

6

7

16.1

4.00

1.00

1.01X10"

23.0

2.80

0.99

i.ioxio"

32.2

2.00

0.948

5.22X10~

45.9

1.40

0.845

0.154

9.50*10

64.3

1.00

0.705

0.286

8„75*10**

3

s.osxio"

90.0

0.714

0.559

0.402

3.85X10"

2

7.42X10"

4

3.13X10"

6

128.6

o.sa

0.422

0.466

0.105

7.32X10"

3

1.7 3X10~

4

1.42X10"

180

0.357

0.316

0.465

0.186

3.05X10"

2

2.26X10"

3

7.75X10""

5

1.26X10""

257

0.25

0.230

0.414

0.256

8.01X10*

2

1.39X10"

2

1.40X10"

3

8.22X10** 2.86X10"

360

0.179

0.168

0.353

0.288

0.138

4.25X10~

2

8.70X10"

3

1.21X10"* 1.14X10"

T a b l e 1. Values of P j ( x ) =

3

2

2

2.91X1Q"

paHi) e " ^

5

=4

1

^

•

5

X = ®R

.

1

5

6

5

3

6

4

Molecule

(a.m.u.)

H

2

Spin, and 6

InterRoHard Nuclear t a tional Sphere Distance Constant Mol. Wt. Diameter

2.016

(I)

~

2.90

(1)

0.742

i

(sec gauss )

e (°K> R

-

10

I =V 3.023

^

2.90

0.742

64.26

2

4.030

^2.90

0.742

1 = 0

1=0,2,—

1 = 1

J=l,3,

2J+1

1= 2 S

3(2J+1)

2

J=0,l,2 —

2J+1

S = 0

= 1

J=0,2,—

43.03 0.410 1 0

4

S=2

S = 1

Table 2.

Statistical Weight of J , I (or J,S) State

4

S = 1

S

D

Molecular Rotational States

85.34

2.67

HD

Nuclear Spin States

Some p r o p e r t i e s of the Hg, HD and D

2

molecules.

5(2J-KL)

J=l,3,

3(2J+1)

-17

2.2 Theory

of R e l a x a t i o n i n Gases

i) Introduction As a l r e a d y e x p l a i n e d i n Chapter I the n u c l e a r spins i n hydrogen are r e l a x e d by the f l u c t u a t i n g i n t r a m o l e c u l a r fields

at the s i t e of the n u c l e i .

mechanism of r e l a x a t i o n was f i r s t subsequently extended

and improved

A formal theory f o r t h i s developed

extended

i&hd Opechowski'.,

by Needier

2 R Abragam , Johnson and Waugh , and o t h e r s .

here c l o s e l y f o l l o w s the treatment

by Schwinger^ and

The theory presented

given by Abragam,

suitably

where necessary using the r e s u l t s of Johnson and

Waugh, and Needier and Opechowski... i i ) Relaxation i n

9 From the molecular beams work of Rabi et a l . , i t i s known that an i s o l a t e d

molecule

i n a magnetic f i e l d can

be d e s c r i b e d by a Hamiltonian H

s

-->

where I

LOj

—>

and I - ^f.

—>

—>

are the proton spins with I i

s

t

field

n

e

(2.1)

—>

+ 1 = 1 ,

ppoton Larmor frequency i n the a p p l i e d

H ,

Uiy i s the Larmor frequency of the r o t a t i o n a l magnetic moment of the molecule

(Huij. = / i j . H ) , 0

-18-

2

T

H

i s the r o t a t i o n a l energy of the molecule 0

quency u n i t s ) , 27 gauss i s the magnetic f i e l d

H

sites "tt

(in fre-

—-

at the proton

produced by the r o t a t i o n of the molecule,

" 34 gauss i s the d i p o l a r c o u p l i n g constant between the two protons which are separated by a d i s t a n c e b,

H of

i s the u n i t v e c t o r

the molecular

Jb^ . b

See Table 3 f o r values

constants.

In the absence of c o l l i s i o n s r a d i o - f r e q u e n c y spectrum of lines, and

between molecules the

c o n s i s t s of two sets o f d i s c r e t e

one s e t centered about " f c ^ , the n u c l e a r Zeeman

the others atfvuly , the molecular Zeeman lines,,

example Ramsey, "Molecular Beams" .) 10

lines,

(See f o r

In a gas where

under-

goes r a p i d r e o r i e n t a t i o n due to the a n i s o t r o p i c f o r c e s a c t i n g on the molecule

during c o l l i s i o n s ,

the f i n e s t r u c t u r e of the

n u c l e a r Zeeman l i n e gives way to a s i n g l e narrow l i n e c o r r e s ponding to that p a r t of the n u c l e a r s p i n Hamiltonian

unaffected

by the o r i e n t a t i o n of J .

( &

The molecular Zeeman l i n e s

m

3

t r a n s i t i o n s ) on the other hand are broadened by the c o l l i s i o n s . In other words, molecular

c o l l i s i o n s b r i n g about r a p i d

rium among the ffVj l e v e l s ,

whereas the n u c l e a r spins are slowly

brought to e q u i l i b r i u m through

a weak c o u p l i n g to J .

equilib-

Fig. 1

i l l u s t r a t e s the nature of the c o u p l i n g between the n u c l e a r spins on a s i n g l e molecule of N.M.R.

and the " l a t t i c e " .

The f i r s t

task

theory i s to r e l a t e the p r o p e r t i e s of the n u c l e a r •• •

Zeeman l i n e

( i . e . the p r o p e r t i e s of the n u c l e a r m a g n e t i z a t i o n < i > )

LATTICE ENERGY SINK consists of the t r a n s n a t i o n a l and r o t a t i o n a l energy of the molecules. Remains at temperature \ because of l a r g e thermal capacity.

Fig.

Molecular Zeeman Very strong coupling

Energy-

Nuclear Zeeman Weak coupling

Energy

1 . Block diagram i l l u s t r a t i n g the nature of the c o u p l i n g between the nuclear spins of a s i n g l e molecule and the " l a t t i c e " .

- 2 0 -

to the p r o p e r t i e s of the molecular Zeeman l i n e s To c a l c u l a t e the equation of motion

(properties

f o r 0.

Since t r a n s i t i o n s

m o d i f i c a t i o n s are h i g h l y f o r b i d d e n , two

relax-

a t i o n times appear. We the present

now

t u r n to a d i s c u s s i o n of the

experiments the rate of e f f e c t i v e

Cje/u.^T) .

In

collisions

( i . e . c o l l i s i o n s e f f e c t i v e i n r e o r i e n t i n g J ) i s much g r e a t e r than the Larmor frequency

of the r o t a t i o n a l magnetic moment of

the molecule,**^- .

been pointed out by Johnson

I t has

Waugh^ that under these

c o n d i t i o n s the

s t a t e are independent o f ^ M . and are r e q u i r e d .

above c o n d i t i o n s the c o l l i s i o n s and lost.

Since

^e^jX) f o r a given J

only two

To see t h i s , one

c o r r e l a t i o n functions

need only note that under the

vector J i s e s s e n t i a l l y f i x e d between

the uniqueness of the f i e l d

the

and

KI/A.

d i f f e r from one

r o t a t i o n of the coordinate

direction is

another only by a

system (as to the

^^/A. ), then

t h e i r s t a t i s t i c a l p r o p e r t i e s must be n e a r l y independent of At t h i s stage

p.

i n the c a l c u l a t i o n the assumption i s

u s u a l l y made that the c o r r e l a t i o n f u n c t i o n s are (This assumption i s d i s c u s s e d

exponential.

i n d e t a i l i n s e c t i o n v.)

i s not q u i t e s u f f i c i e n t , however, to put

It

-26-

A more c o r r e c t e x p r e s s i o n i s :

(V^M^ The

(M^\e

(2

.

4)

1 1

We

have a l r e a d y assumed that

the

e f f e c t of g ^ * ^ ^

the

argument of Johnson

on

and Waugh.

Zeeman s p l i t t i n g a n d }

so that

However, the formulas

a l s o be used i n the r e g i o n where

and Opechowski^ f i r s t

-Ly

^ ^ ( f ) i s s m a l l , as r e q u i r e d by

l a r g e r than u^- and here the e f f e c t

the

e #.

^ V * ^ takes i n t o account the molecular Zeeman

factor

splitting.

will

W

i s only a few times

i s not n e g l i g i b l e . Needier

suggested i n c l u d i n g the molecular the term a l s o appears independently i n

treatment by Abragam. Using equation (2.4) i n (2.3a) and

(2.3b) one o b t a i n s

>*^-^TK *

3

X

(2.5a) and

^£*J^fiT^ _«^5L. +

^._^fl_ J) 3

(2.5b)

with

^ If

-

f *6M p



the c o r r e l a t i o n f u n c t i o n s are assumed to be

Gaussian,

properties

of J .

This

aspect

o f t h e t h e o r y was s k e t c h e d i n

sections

i i t o i v and c a n be c o n s i d e r e d

However,

the second

task

t o be c o m p l e t e l y

o f N.M.R. t h e o r y ,

to relate

solved.

the prop-

—>

erties first

o f J t o the i n t e r a c t i o n s principles,

i s a problem

between m o l e c u l e s

starting

from

o f a n o t h e r m a g n i t u d e and e x c e p t 11 13

for

t h e work

accomplished

o f Bloom and Oppenheim i nthis

culates

correlation

actions

using

then

relates

direction. functions

the"constant

'

very

little

T h e Bloom-Oppenheim

of the i n t e r m o l e c u l a r

acceleration

them t o t h e c o r r e l a t i o n

h a s been theory inter-

a p p r o x i m a t i o n " and

functions

for intra— >

molecular There w i l l

interactions

(i.e. correlation

be no a t t e m p t

t o give

a full

functions

of J ) .

i n t e r p r e t a t i o n of

cal-

-39-

the present

experimental r e s u l t s i n terms of the Bloom-

Oppenheim theory,

since many of the

r e s u l t s of the

t i n e n t to t h i s work are as yet unpublished. t h i s s e c t i o n i s to d i s c u s s

c o l l i s i o n processes.

ence of the n u c l e a r extremely weak and

considerations

per-

purpose of

those p r o p e r t i e s of the

can be deduced from general the d e t a i l s of the

The

theory

that

without going i n t o

Note that the

influ-

spins on the time dependence of J i s i n the

following discussion i t w i l l

be

neglected. In s e c t i o n i i some of the p r o p e r t i e s of have a l r e a d y

been d i s c u s s e d .

I t was

eight possible c o r r e l a t i o n functions K

0

•'

mal

f o r a given J s t a t e only experimental c o n d i t i o n s .

Symmetry about the z-axis g

and

The

g . = g

D

f a c t t h a t the

two The

the

concluded that of of the

and

the the

were d i s t i n c t under c o n d i t i o n of

r e q u i r e s that g j j

nor-

cylindrical i ' ^22

=

,, which reduces the number to

=

five.

c o r r e l a t i o n times are much s h o r t e r than

Larmor p e r i o d of the

the

r o t a t i o n a l magnetic moment of the mole-

cule f u r t h e r reduces the number of d i s t i n c t c o r r e l a t i o n functions We

now

to two,

one

f o r the

one

f o r the K j j ^ p .

wish to i n v e s t i g a t e the assumption u s u a l l y made

that the c o r r e l a t i o n f u n c t i o n s first

and

describe

are e x p o n e n t i a l ;

to do t h i s

we

the behaviour of a quantum molecule i n a d i l u t e

gas. Quantum Mechanical Random Walk In a d i l u t e gas

a molecule spends e s s e n t i a l l y a l l

of i t s time i n a s t a t e of i s o l a t i o n

(except

f o r the

presence

40-

of the e x t e r n a l l y a p p l i e d s t a t e of the molecule a s h o r t time t in

coll. i n

cult

During a c o l l i s i o n , the

c h a n g e s a b r u p t l y t o some o t h e r s t a t e i n

: after

t h i s new s t a t e u n t i l

on t h e a v e r a g e

field).

the c o l l i s i o n

remains

the next c o l l i s i o n , which w i l l

a t i m e %. later,£c » and ti

to define t

the molecule

t

c

o

^ ^ .

Now i t i s d i f f i -

f o r a r e a l gas, as t h e i n t e r -

coll.

y

molecular forces are not f i n i t e i n t e r a c t i o n between m o l e c u l e s g i b l e when t h e m o l e c u l e s

ranged.

1

However, i f the

c a n be c o n s i d e r e d t o be n e g l i -

a r e s e p a r a t e d by a d i s t a n c e g r e a t e r

t h a n some c h a r a c t e r i s t i c d i s t a n c e , t h e n f o r l o w enough t h e c o n d i t i o n Zi ^^coll

c

a

°e

n

i s as f o l l o w s :

density

satisfied.

A n a t u r a l way o f d e s c r i b i n g cally

occur

the process

the state of the molecule

mathemati-

a t any t i m e

t c a n be e x p r e s s e d as

where t h e / a r e t h e e i g e n s t a t e s o f t h e i s o l a t e d with eigenvalues E . n 3

stant.

The e f f e c t

of a s i n g l e

by an N - d i m e n s i o n a l a

n

Between c o l l i s i o n s

linear

before the c o l l i s i o n

probability

c a n be r e p r e s e n t e d

n

, r e l a t i n g the

after the c o l l i s i o n .

The

d i s t r i b u t i o n p(G^) s h o u l d c o n t a i n a l l o f t h e

Stated i n this

random w a l k

the c o l l i s i o n

language

o f t h e random w a l k

s t a t e s has s t r i k i n g tational

the a ( t ) are conn

transformation

to the a

n e c e s s a r y i n f o r m a t i o n about

problem

collision

molecule,

t h e quantum

mechanical

o f t h e f r e e r o t a t o r among i t s

similarities

w i t h the problem

of a c l a s s i c a l

solved recently i n f u l l

processes.

of the r o -

p a r t i c l e which

g e n e r a l i t y by I v a n o v . 1 4

has been

For a

-41

c l a s s i c a l p a r t i c l e the e f f e c t of a s i n g l e r o t a t i o n can be represented by a r o t a t i o n matrix g^.

Ivanov solves f o r

P(g,n), the p r o b a b i l i t y d e n s i t y

f o r the o r i e n t a t i o n g of

the molecule a f t e r n r o t a t i o n s ,

i n terms of the p r o b a b i l i t y

of i n d i v i d u a l r o t a t i o n s

The p r o b a b i l i t y

p(gj).

W ( g t ) f o r the o r i e n t a t i o n t

density

g of the molecule at time t can

then be obtained i f the p r o b a b i l i t y W ^ ( t )

f o r n r o t a t i o n s to

take place i n a time t i s known:

The form of lA/ (t) i s the same f o r e i t h e r quantum mechanical n

or c l a s s i c a l p a r t i c l e s i n a d i l u t e gas and i s given by the Poisson d i s t r i b u t i o n

hi where 7^ i s the average time between c o l l i s i o n s . that

I t appears

a quantum theory of Brownian motion f o r a f r e e

has not been c o r r e c t l y solved

as yet

.

rotator

However, one might

hope f o r a s o l u t i o n by methods s i m i l a r to those used by Ivanov, T r a n s i t i o n P r o b a b i l i t y Method a (t) n are

a~7 n

To make the problem t r a c t a b l e i t i s assumed that (t = 0 unless n'=. n . Since the a ( t ) a , ( t ) n n

the elements of the d e n s i t y

representation

t h i s amounts to n e g l e c t i n g

elements of the d e n s i t y treated

matrix i n the i n t e r a c t i o n the

off-diagonal

m a t r i x , and the problem can then be

by the t r a n s i t i o n p r o b a b i l i t y method. We f i r s t

t r e a t the case of a molecule that

remains

-42-

i n the J = 1 s t a t e . l e v e l s mj = 1, 0,

The

p o p u l a t i o n s of the three energy

-1 are denoted by p^, P

q

and

p

respect-

ively. -

"V -I

F i g . 3.

Energy l e v e l s of a molecule i n the J = 1 state.

A p p l y i n g the t r a n s i t i o n p r o b a b i l i t y three

and

=W

01

1 0

=W_

=W _

10

Q

of high temperatures and

Jl ° ^

whence

o b t a i n the

equations

Assuming t h a t V V (limit

method we

dL(p, -

=

" N ^ * ^ ,

=

1

low

=1^

andVV-^

f i e l d s ) we

obtain

-(yO.+^yAJ^-h)

£

5 5

Will

-43-

so that q,„ and 10 20 y

^2.

=3w— t

y

#

V V1, i s

are e x p o n e n t i a l with

and Wj-faW* the p r o b a b i l i t y per second f o r a t r a n s e

i t i o n A nij = + 1 and Hg

i s the p r o b a b i l i t y per second f o r a

t r a n s i t i o n A mj = + 2.

I f there are no s e l e c t i o n r u l e s

(W

x

=W ) 2

then

% - tt.

For a r b i t r a r y W , X

W

t

2

h

e

ratio

^ / T ^

can vary between the l i m i t s 0 and 3. I f the t r a n s i t i o n p r o b a b i l i t y method

i s a p p l i e d to

a molecule having J > 1 then i n general the

r e o r i e n t i n g J.) T h i s i s supported

by the work of Johnson and

Q

IVaugh

who s t u d i e d a d u l t e r a t e d Hg at room temperature.

If

C i s small compared to C, then the only e r r o r i n t r o d u c e d 9 h

-74-

by the i m p u r i t y i s an e r r o r i n the value of the pressure (equal to the p a r t i a l pressure of the i m p u r i t y ) . other hand, i f C

G

i s l a r g e compared to

can be a p p r e c i a b l e .

the e r r o r i n

A t a b l e of values f o r C

Johnson and Waugh i s given below.

taken

I t i s reasonable

that these values w i l l be the same f o r for

On the

from

to assume

and roughly the same

HD.

c /c v h .62 .69 1.00 1.34 2.80 3.24 3.66 5.30 5.94 7.94

Molecule He Ar H

2

Oo

CF°

No

cd

NO

CClFo

Table 4.

Values room

of C_/C. at g n temperature.

18.2 21 , 6 29.6

N0 9

COo CHF3

For the measurements only i m p u r i t i e s l i k e l y

taken at 77°K and below, the

to be present are He, Ne, Ar, Og and

Ng, and of these Ng has the s t r o n g e s t e f f e c t . the extent of 1% w i l l

r e s u l t i n a 2.7%

N^ present to

e r r o r i n T^.

however, a gross impurity and one expects

This i s ,

the percentage

of

Ng to be w e l l below 1%. vi)

Sample

Holders;

For the measurements

on H^ gas a wide v a r i e t y of

sample holders was used; a t y p i c a l one i s shown i n F i g . 10. The primary

o b j e c t i v e i n d e s i g n i n g a sample h o l d e r and c o i l f o r

-75-

nn Output to Receiver

To T r a n s m i t t e r

are

by:

(4.1)

(4.2) where

and

M

=

N =

^

^

ft-

J_L ^

~2.

H

H

HH

=

3.41

=

For a given value of n =

3.94

X 10

X

10

11

1 1

-2 -2 s e c " gauss

sec"

2

gauss"

2

- k^/k^, the values of

-89-

k^ and kg are u n i q u e l y determined (

(y\ - lOj)

k /p«

by the high d e n s i t y

1) value of T ^ p

1

= Tg/p

.

From

a s e r i e s of measurements at about 8 atmospheres p r e s s u r e , Tj/p

at 77.5°K was

found to be 0.122

(This agrees with the value of 0.120

msec/amagat -

3%.

msec/amagat i n normal 22

hydrogen obtained by L i p s i c a s and H a r t l a n d .

)

From the

values of k^ and kg obtained, T^ and Tg were computed as a f u n c t i o n of p

f o r v a r i o u s values of X, jXx. between, the

l i m i t s of 0 and 3.

The

expected

r e s u l t s f o r s e l e c t e d values of

appear i n F i g s . 14 and 15.

Note that the absolute values of

k^ and kg do not a f f e c t e i t h e r the shape or the value of the T j minimum; o n l y the p o s i t i o n of the minimum with respect to the gas d e n s i t y can be a f f e c t e d .

In F i g . 16 the value of T^

at the minimum and a l s o the asymptotic value of 1 / j the minimum i s p l o t t e d as a f u n c t i o n of X /X\. . K

the r a t i o of Tg/p of t ^ / t V

.

a

'

3 0 v e

Together,

a n
T

) 1

a b

s

a

.1

.I 1 2.0

i

function

of%/z^.

i L_ 1 3.0

I



min

gas at the minimum, and ( l ^ > T ) b i s

value of l / ^ T ^ below the T

1

minimum

-91-

F i g . 17.

( T ^ o ) b vs 2

T,/T . ( T / p ) a and (T /p)b z

9

2 ^ denote the asymptotic ' the

C ^t- r) t

u J

H

values of T^/p above

), and below C?,^ ^ ( j A . - ^

minimum r e s p e c f l i v e l y .

1

)

-92-

I f \A)J i s i n c l u d e d i n eq. (4.1) the theory f i t s the experimental %/Vx.

- 0.6

data extremely w e l l .

X

is

Tf/X-x. ^

almost

independent

quence the

tf/Vil

3>

3 °

x

the upper l i m i t We l i m i t s on

T

n

(see F i g . 16).

u

now

l°

e

is fairly

X,/Vj,.

limit

w e r

minimum

As a

conse-

appeal to the Tg Although

i s q u i t e sharp whereas

results

to impose

there i s s u b s t a n t i a l

a v a l u e of %I~C± between 0 and experiment.

on

diffuse.

the Tg data i n the r e g i o n of i n t e r e s t ,

theory and

Unfortunately,

r e s u l t s can set only the r a t h e r wide l i m i t s

0*k>^-X. fT ^= x

data.

the shape and value of the of % /Z

near

the theory l i e s w e l l

o u t s i d e the s c a t t e r of the experimental 1 ^

For

the theory j u s t skims the top of the data

the minimum, whereas f o r V,/T ~ 0.4

for

then f o r

further

scatter in

i t would appear t h a t

1 gives the best f i t between

Note t h a t although Tg/p

at low

pres-

sures does not depend very s t r o n g l y on the v a l u e ofXjXx. t h i s dependence i s monotonic (see F i g . 17). improving

the l i m i t s t a i n 0.6

As a r e s u l t ,

the accuracy of the Tg measurements may

e f f e c t i v e way

of narrowing

the l i m i t s on

imposed on Z,/Xx. by the T 1

4z X,jXx, £ 1.0.

and T £

The ment was of

based

be the most Combining

r e s u l t s we

0

ob-

I t i s i n t e r e s t i n g to note t h a t Bloom

( p r i v a t e communication) has p r e d i c t e d t h a t X/X orthohydrogen

,

x

- 0.6

for

i n the J = 1 s t a t e . f o r e g o i n g comparison on the assumption

J were e x p o n e n t i a l .

tween theory and

between theory and e x p e r i -

t h a t the c o r r e l a t i o n f u n c t i o n s

However, the e x c e l l e n t

experiment

agreement

be-

w i t h r e s p e c t to the shape of the

93-

minimum demonstrates e x p e r i m e n t a l l y that the

correlation

f u n c t i o n s are at l e a s t c l o s e to being e x p o n e n t i a l . p a r i s o n , the t h e o r e t i c a l behaviour

of

For com-

when gaussian c o r -

r e l a t i o n f u n c t i o n s are employed i s shown i n F i g . 15 f o r f, The

obvious

disagreement

with the experimental

be s i g n i f i c a n t l y lessened by changing 4.2

results

=

T . x

cannot

the value of T, j

.

R e l a x a t i o n Measurements i n HD Experimental R e s u l t s T

f o r the proton and deuteron

as a f u n c t i o n of temperature

i n HD

gas was

measured

and pressure f o r temperatures

tween 30 and 373°K and pressures up to 8 atmospheres. ready mentioned i n chapter I I , T^ f o r e i t h e r n u c l e a r s p e c i e s i n HD

because of the cross r e l a x -

to the accuracy of these experiments,

out by the experiments

Hqwever,

the recovery of

should be e x p o n e n t i a l with a time

the corresponding value of T

As a l -

i s not s t r i c t l y d e f i n e d

a t i o n i n t r o d u c e d by the d i p o l e - d i p o l e i n t e r a c t i o n .

or

be-

constant equal to

(see chapter I I ) .

where the observed

^X-^)

T h i s was

r e c o v e r i e s of

borne {X-i}

o r ^ S j ^ were always e x p o n e n t i a l . To w i t h i n experimental l i n e a r l y on the d e n s i t y f

throughout

For a few of the temperatures was

e r r o r T^ was

found

to depend

the temperature

range.

where the s i g n a l / n o i s e r a t i o

f a v o u r a b l e , measurements were made down to the

lowest

d e n s i t i e s p o s s i b l e i n a search f o r the T j minimum. The 18.

r e s u l t s of one

T^ f o r the proton i n HD

such attempt at 77.5°K was

are shown i n F i g . measured as a

h70

h60

T| (msec) h50

h40

h30

HD

775 °K

H20

10

10

20

30

PRESSURE F i g . 18.

40

50

(psi)

as a f u n c t i o n of pressure f o r the proton i n HD

gas at 7 7 . 5

-96-

f u n c t i o n of pressure from 60 p s i down to 1 p s i . At the time when the measurements were made, the s l i g h t d e v i a t i o n from l i n e a r i t y at the lowest pressures was nificant.

However, a subsequent

T = 77.5°K

the value of

thought to be

insig-

c a l c u l a t i o n showed that f o r

at the minimum should be a p p r o x i -

mately 1.2 msec, to be compared with 1„7 msec, the lowest value of

measured.

The beginning of the T^ minimum

may

i n f a c t have been observed. For the range of temperatures and p r e s s u r e s encountered i n the experiments the d e n s i t y f> i s given with s u f f i c i e n t accuracy by c o m p r e s s i b i l i t y t a b l e s f o r Hg.

To

o b t a i n g r e a t e r accuracy one can use the r e s u l t s of Knaap 23 et al„

who

have measured the second v i r i a l

of the hydrogen

i s o t o p e s between 20 and 70°K.

The values of ^ / p

f°

T

obtained from p l o t s of T^ vs j> 1/T

i n F i g . 19.

coefficients

r

the proton and deuteron

are p l o t t e d as f u n c t i o n of

A s i n g l e value of T^ f o r the proton i n HD

at 20.5°K obtained by B l o o m ( T ^ / p ) proton vs 1/T

24

i s also included.

In F i g . 9,

i s shown f o r temperatures above

100°K

i n order to e x h i b i t the temperature dependence more c l e a r l y . The g e n e r a l f e a t u r e s of the r e s u l t s can be b r i e f l y i ) below 65°K^ T^^C>

f o r both n u c l e a r s p e c i e s

r a p i d l y as the temperature i s lowered.

To w i t h i n

stated: increases experi-

mental e r r o r the temperature dependencies are almost c a l , with (T-j^/jO ) p r o t o o /CT i i ) above 1 0 0 ° K , ( T , / p

deuteron

^

identi-

1.33,

)deuteron i s almost

temperature

-96-

T

(°K)

2.0

3.0 100/T

(°K ) _I

F i g . 19. T^/p vs 1/T f o r protons and deuterons i n HD gas. Pj-i i s the p r o b a b i l i t y that an HD molecule i s i n the J=l s t a t e .

-97-

independent to

whereas ( T ^ / p ) p r o t o n i s very n e a r l y p r o p o r t i o n a l

1/T. Interpretation Below 65°K only the J = 0 and J = 1 s t a t e of the

molecule

are a p p r e c i a b l y populated.

In the J = 0 s t a t e the i n -

t r a m o l e c u l a r i n t e r a c t i o n s are absent, hence the strong temperature dependence of of the J = 1 s t a t e .

Now we expect that

T

T, so that below 65°K

i s due to the change i n p o p u l a t i o n

vT, IT _L

X £±=^ ^

.

m u l t i p l y i n g the experimental values of T,/p obtains ^ T ^ / p ) j _ ^ .

by P

i s almost

temperature

T h i s i s i n sharp c o n t r a s t with the temperature in H

0

one

The r e s u l t s are shown i n F i g . 19 and

i n d i c a t e t h a t (T^/p ) j _ j

of ( T , / p )

T h e r e f o r e by

below 80° observed

independent. dependence

by L i p s i c a s and

22 „ Here ( T ^ / p ) ortho-para i n c r e a s e s very s h a r p l y

Hartland. below 80°K

(Note

o

that i n HD below 65°K, most of the mole-

cules are i n the J = 0 s t a t e and the molecular are analogous

to the ortho-para type c o l l i s i o n s

collisions i n H^.

HD r e s u l t s should t h e r e f o r e be compared with the Hg

The

results

f o r low ortho c o n c e n t r a t i o n . ) The very d i f f e r e n t

temperature

dependencies

of

( T ^ / p ) J - J i n Hg and HD can p o s s i b l y be e x p l a i n e d by the f a c t that an a d d i t i o n a l i n t e r a c t i o n i s present between HD that i s absent Hg molecule

f o r Hg molecules.

molecules

R e f l e c t i o n symmetry

r e q u i r e s that terms having the symmetry

of the

of the

F i g . 20. T ^ o vs 1/T f o r the protons i n HD gas above 100°K. 1

-99

odd

order spherical

intermolecular molecules

harmonics

interactions.

are almost

molecule

metrical

This

centre. and

An

temperatures energy J

a term

without

and

g r e a t e r t h a n kT

one

might

are very always onant from

expect

rare.

collides collision

o f t h e f o r m /m

conserved.

The

J

(AJ

(the energy o f HD

that

transitions

a molecule

A more d i r e c t

geo-

present. induce low

between i s 128

i n d u c e d by

i n the J = 0

the K),

t h e Vim

state.

energy play

i s automatically an

important

that

an

molecules

paring

the

f o r changes i n J w i t h k i n e t i c

8 0 ° K an a v e r a g e between e a c h is ing

t o be

o f about

change

can be

additional

i s p r e s e n t f o r HD

A simple

calculation 8 hard

i n t h e J,m

sphere state

o b t a i n e d by

shows t h a t

T

state

intercomtheory

f o r HD

at

type c o l l i s i o n s

occur

o f the m o l e c u l e .

This

compared w i t h 25 — > 3 0 0 h a r d s p h e r e

p e r change i n t h e J,m

goes

f o r the s m a l l

action

cross-sections.

res-

reorientation.

indication

cross-section

terms almost

A

p o s s i b l e where J Except

of

a change i n

Kelvin

therefore

i n the p r o c e s s o f m o l e c u l a r

HD

i n the J = 1 s t a t e

of the J = 1 l e v e l , t e r m may

the

At

differency

i n degrees

and

symmetry

can be

= - 1),

the

the c e n t r e

with

involves

process i s therefore

V/m

shape,

o f the f o r m Vitr> c a n n o t

However, a m o l e c u l e with

and

coincident

1—T>0 and j ' goes f r o m 0 — > 1 .

Zeeman s p l i t t i n g

role

i n size

i s not

changing

from

a l t h o u g h t h e Hg

a change i n the J s t a t e

= 0 and J = 1 s t a t e

absent

d e s t r o y s the r e f l e c t i o n

interaction

changes i n mj

must be

Now,

identical

o f mass o f t h e HD

the m o l e c u l e

Yg^

o f an Hp

collisions

molecule

at

occurthe

100-

same temperature.

The numbers 25

o r t h o - o r t h o , and ortho-para

and 300

collisions

are a p p r o p r i a t e f o r

respectively.

It i s

evident that the i n t e r a c t i o n s r e s p o n s i b l e f o r molecular r e o r i e n t a t i o n are q u i t e d i f f e r e n t f o r the two molecules.

The

p r e d i c t i o n s of the Bloom-Oppenheim theory f o r Hg are t h e r e f o r e not r e l e v a n t to the HD

molecule.

The c h i e f m o t i v a t i o n f o r the study of T that a value of T (D)/T (P). 1

1

Xj/z,.

i n HD was

could be obtained from a measurement of 1.33

The value of T < D ) / T ( P ) = T ( D ) / T ( P ) = 1

1

2

2

obtained e x p e r i m e n t a l l y (with approximate l i m i t s of 1.22 1.55),.together with the r e s u l t s of chapter I I y i e l d s = 1.07

with approximate l i m i t s of 0.9

and 1.18.

l i s h e s that there are probably no c o l l i s i o n a l f o r the J = 1 s t a t e of HD, and i n p a r t i c u l a r as p r e d i c t e d f o r the J = 1 s t a t e of Hg.

and ^/z^

This estabselection

7f Jf f

v

^

rules .0.6

U n f o r t u n a t e l y the

Bloom-Oppenheim theory has not yet been a p p l i e d to s i t u a t i o n s where J can change and a t h e o r e t i c a l value f o r %/V-j, i n HD i s not a v a i l a b l e at p r e s e n t . The high temperature dependence of T^/p ton or deuteron

i s an extremely

cuss p r o p e r l y .

At room temperature there are s e v e r a l J s t a t e s

a p p r e c i a b l y populated

complicated

f o r the p r o -

phenomena to d i s -

but not enough that the sum

S

can be approximated by an i n t e g r a l ; t h i s g r e a t l y hinders calculations.

In a d d i t i o n , %

and Y

%

i n general depend

on both J and T, n e i t h e r of which dependencies i s known. added c o m p l i c a t i o n to the temperature dependence of the

An

-101-

i s that the "composition" of the gas depends on

temperature:

an average must be performed over the v a r i o u s J s t a t e s of the molecule with which

the molecule of i n t e r e s t i s c o l l i d i n g .

The net r e s u l t of these v a r i o u s c o m p l i c a t i o n s i s that i t i s virtually

i m p o s s i b l e to e x t r a c t e i t h e r the J dependence or

the T dependence of the full

from the experimental data.

d i s c u s s i o n of the temperature

awaits f u r t h e r t h e o r e t i c a l

"^/p

dependence of

results.

For Hg Bloom and Oppenheim^ have p r e d i c t e d ^

o< J

I ''•for high J .

At s u f f i c i e n t l y high

t h i s r e s u l t s i n T a n d T> dependencies and d i p o l a r p a r t s of T^/p proton T^ i s dominated

respectively.

Note that i n HD

then ( T ^ / p ) p

(Tj/p ) j (

e

u

t

e

r

o

interaction,

t

r o

I f the r e s u l t f o r would be propor-

o n

p r o p o r t i o n a l to T ~ i .

n

On the other hand a strong c o l l i s i o n model p r e d i c t s (T

l/p

>proton^

dependencies

a

n

deuteron

oC

d

I f i t i s assumed that HD

*

T

h

that

^served

e

results.

can be t r e a t e d by a

theory such as the Bloom-Oppenheim theory

then the ie temperature 7^ oi T

T

do not agree with e i t h e r of these

"weak c o l l i s i o n "

the

the deuteron T-^ i s

by the quadrupolar i n t e r a c t i o n .

Hg were a p p l i e d to HD

spin-rotational

by the s p i n r o t a t i o n a l

whereas f o r not too high temperatures

and

temperatures

f o r the

a

t i o n a l to T~i

that

%

£•

dominated

A

dependence of c"> i s expected to be

at high temperatures.

dependence of (T^/p

)p ton

7£ would have to be ^

o

r o

) ~ S

2

^ i / p ^ j ^ £ > obtained T

by p|, the p r o b a b i l i t y that an Ortho

molecule i s i n the s t a t e J - 2, i s a l s o shown i n F i g . 10* The

temperature

rriately by T ~ ° »

dependence of ( T / )*ZA 1

4 5

Or very n e a r l y

I

s

given approxi-

T ^. J

At 32°K, ( T ) ? d i d riot have a l i n e a r dependence on the d e n s i t y arid t h i s was a t t r i b u t e d to w a l l r e l a x a t i o n s

1

2

500

200

-10465

100

27 ~~r~

33

40

50

T (°K)

1000

- ^\/p (msec/amagat) NORMAL

D

100

T\/p for S = l

spin state

l/jb

spin state

* j-

•

T

for S=2 IO

2

10

-a-

1.0

2.0

3.0

100/T (°K"')

F i g . 21. vs 1/T f o r S=l s p i n s t a t e of para Dg and S=2 spin s t a t e of ortho Dg as measured i n normal Dg. P| * p r o b a b i l i t y that an ortho molecule i s i n the s t a t e 3-2. s

t

n

e

-105-

effects.

The

values of T

measured were of the order of

sec f o r p i n the region of 40 amagats: shows that at these

a simple c a l c u l a t i o n

d e n s i t i e s a molecule takes

only a

seconds to d i f f u s e to the w a l l s of the sample holder, c a t i n g that w a l l r e l a x a t i o n may

80

be important.

The

few indi-

value

of

(T,/p) at 32°K given i n F i g . 10 i s t h e r e f o r e only a lower ' J=2 l i m i t on the true value of ( T j / j o ) ^ " ^ . To o b t a i n meaningful S = 2

1

r e s u l t s below 40°K i t would be necessary s u r f a c e s i n contact with

to ensure that a l l

the gas were made of some substance

i n e f f e c t i v e i n f l i p p i n g the nuclear s p i n s . are known to have t h i s

T e f l o n and Nylon

property.

Interpretation With respect to i n t e r m o l e c u l a r i n t e r a c t i o n s and sequent time dependence of ~f f o r a molecule i n a gas,

con-

the

Dg

molecule i s e s s e n t i a l l y i d e n t i c a l to the Hg molecule i n every respect but mass. should

Hence any

be able to c o r r e l a t e r e l a x a t i o n phenomena i n the

molecules without any of the present should any

d e t a i l e d theory of r e l a x a t i o n

a d j u s t a b l e parameters.

r e s u l t s f o r Dg with

t h e r e f o r e be f r u i t f u l

theory.

A comparison

the a v a i l a b l e data f o r Hg

i n e s t a b l i s h i n g the c o r r e c t n e s s of

With t h i s i n mind we

f e a t u r e s of the experimental

two

p o i n t out two

results for

significant

D^:

i ) ( ^ / j ^ " ! appears to go through a minimum at 40°K and 6 O not at 80 K as does the analagous q u a n t i t y i n H^. 3

(Note that normal D para Hg

mixture.)

i s e q u i v a l e n t to a 33% ortho

67%

-106-

ii)

( T ^ / ^ o ) d o e s not have the same temperature depenS~ 1 dence as (Tj/jP^j-i* through a minimum.

a n (

* *

n

p a r t i c u l a r does not go

As y e t the occurence of the T^/jo minimum f o r o r t h o 22 para c o l l i s i o n s i n Hg has not been f u l l y e x p l a i n e d . we note that i f the p o s i t i o n

of the minimum i s governed by 1, 22

quantum mechanical d i f f r a c t i o n e f f e c t s in D

2

i s shifted

However,

then the minimum

i n the r i g h t d i r e c t i o n by the r i g h t amount.

The average de B r o g l i e wavelength ^

o f a molecule of mass M

i n a gas at temperature T i s g i v e n by One expects the onset of d i f f r a c t i o n e f f e c t s by the r a t i o of A

to be governed

to the s i z e of the molecule.

For molecules

that d i f f e r only i n mass, the temperature at which d i f f r a c t i o n e f f e c t s become important i s t h e r e f o r e i n v e r s e l y

proportional

to the mass of the molecule. The

f a c t that ( T ^ ) ^

2

and ( T ^ ) ^ have d i f f e r e n t

temperature dependencies seems to i n d i c a t e that the i n t e r molecular i n t e r a c t i o n s are s i g n i f i c a n t l y d i f f e r e n t f o r molecules of d i f f e r e n t J even f o r J ^ to s p e c i a l

cases Bloom and Oppenheim

communication) ular

1.

In a p p l y i n g t h e i r theory J and Bloom ( p r i v a t e

have assumed the same form f o r the i n t e r m o l e c -

interaction

f o r a l l molecules with 3 ^ 1 .

The absence

of a minimum i n (T^/^OjLg suggests that at low temperatures at l e a s t , t h i s assumption may be u n j u s t i f i e d .

•107-

CHAPTER V

SUGGESTIONS FOR FURTHER EXPERIMENTS

At present the only method a v a i l a b l e f o r experiment a l l y determining T, and T

0

the r a t i o

i n the r e g i o n X X n

x

f,/ziin ^

H !—.

9

gas i s to measure The measurements r e -

ported i n t h i s t h e s i s have e s t a b l i s h e d the r a t h e r wide l i m i t s :

on

0.6 ^

^ 1.0.

Since the only e x i s t i n g de-

t a i l e d theory of r e l a x a t i o n i n H

gas

^

predicts

%jZT

X

- 0 . 6 f o r the J = 1 s t a t e of ortho hydrogen, i t i s o b v i o u s l y worthwhile

t r y i n g to improve the accuracy of the measurements.

In p r i n c i p l e the accuracy of such measurements would be i n creased by u s i n g higher magnetic f i e l d s : the N.M.R. s i g n a l at a given d e n s i t y i n c r e a s e s with H , and a l s o the T^ m i n i 0

mum occurs at a higher d e n s i t y .

However, these gains would

probably be p a r t l y o f f s e t by the poorer noise f i g u r e s of a m p l i f i e r s above 3 0 Mc and a l s o by the l a r g e r f i e l d g e n e i t i e s that would have to be t o l e r a t e d .

inhomo-

I t i s doubtful,

f o r example, that the same method of measuring T^ could be used at f i e l d s s i g n i f i c a n t l y higher than 7 k i l o g a u s s . I t i s the author's guess that c a r e f u l measurements of T^ and Tg at

20°K

for H

q

i n the

10

k i l o g a u s s range w i l l

the most accurate values of X. JX-^, (

provide

A complete study would

i n c l u d e measuring X, jXx as a f u n c t i o n of temperature as a f u n c t i o n of ortho-para c o n c e n t r a t i o n , although not expected

to depend on e i t h e r of these v a r i a b l e s .

and a l s o

T)/c^is I t would

-108

a l s o be of i n t e r e s t to study %/'Ci. i n order to s y s t e m a t i c a l l y check

i n adulterated

hydrogen

the d e t a i l s of the Bloom-

Oppenheim t h e o r y . Except f o r i n c r e a s i n g the accuracy and extending the temperature

range of the present measurements on HD,

experimental problem of r e l a x a t i o n i n pure HD complete.

A study of the

the

is essentially

minimum f o r both n u c l e a r s p e c i e s

would be v a l u a b l e as a check on the i n t e r n a l c o n s i s t e n c y of the theory, but i t i s not l i k e l y to provide any new mation.

The

next obvious step i s to i n v e s t i g a t e

i n a d u l t e r a t e d HD;

relaxation

a most i n t e r e s t i n g study would be to

a d u l t e r a t e HD with pure para Hg ortho Dg(J = 0,2,

).

and at low temperatures s p h e r i c a l l y symmetric of i n t e r e s t

infor-

(J = 0 , 2 , — ) and a l s o

These molecules d i f f e r o n l y i n mass most of the molecules are i n the

J = 0 state.

One

of the main p o i n t s

i n the study i s that c o l l i s i o n s i n v o l v i n g

changes i n J should be suppressed.

The

first

gas

task of f u t u r e

s t u d i e s would be to extend the temperature measurements on T,/p

HD

interest.

present r e l a x a t i o n measurements on Dg

are of a p r e l i m i n a r y nature.

resonant

A d u l t e r a t i o n of the

with helium would a l s o be of fundamental The

pure

range of the

f o r the S = 1 s p i n s t a t e of

paradeuterium.

-109-

APPENDIX A

CIRCUIT DETAILS OF PULSED SPECTROMETER and TEMPERATURE CONTROL UNIT

Re-ference

(30 Mc>

out"

100 >

,UjJ

,

Output t

0

~ amb/ifier

Spqn c o p p e r

Resislbrs m

con

Cu|xiLCitors.

III F i g . IA.

Coherent gated o s c i l l a t o r and

tripler

III

in

p>E

\^>l

Ferrite bewis

3 0 Mc Ou"l"pu"l"

fo

sample, coil F r o m tripler (^ollovA/ino, gc,r

1i"-h ^

oscillator;

2/A

3, Wns. on 1 w/df res \ star

30yu.h.

-170 V

-r750 V

Hem'ie bec«is

i i

Fig.

2A.

Gated power

amplifier

+ iooo V

-112

u rH

a •H

U

H

CO

•rt

fa*

0

q

ft o

00

+2£5 V

In

p u t ('from

coherent . acted oscillator)

(to r e t V \ A A — i — < 0 input of main " QmpliTiGr)

D e l a y line : Bel Fuse

F i g . 4A.

Clipper

^VS-2S0

Phase s h i f t e r (as used at 7 Mc)

F i g . 5A.

Pulse mixer and a m p l i f i e r

F i g . 6A.

7 Mc

preamplifier

+ ISO V

P

+ZZSV

o7266's 220K W V - o / tA - A ' W - o l^OK -AAAr-o IOOK

-AAA/—o^

= 0)

5^ -"-

+

e

Q

c

+

- - - ) (B.2)

One

should

are t a k i n g p l a c e . the sample time r

according

note that two

The t

s

:

d i s t i n c t averaging

input wave form i s f i r s t e.(t)—>e I

.

The

n

to the r e c i p e given i n E q . B 2 ,

e n

processes

averaged

during

are then averaged *

the value of ^ being

a measure of the e f f e c t i v e n e s s of t h i s averaging

processs.

Now

1 (e.g. f o r

i t i s sometimes i m p r a c t i c a l to have r = t « RC

T very l o n g ) . respondingly

c

One

i s then a u t o m a t i c a l l y committed to a c o r -

small gain i n the s i g n a l to noise r a t i o from

the second of the two

averaging

processes.

companying impairment of the f i r s t

However, the

type of averaging

is

•'•See Morse and Feshbach, "Methods of T h e o r e t i c a l P h y s i c s " (McGraw-Hill, 1953) Part I, p. 693.

ac-

-121-

unnecessary

and

r e s u l t s from u s i n g a simple RC c i r c u i t

to

do both types of averaging.

Under these c o n d i t i o n s a true

i n t e g r a t i n g device should be

used,

Eg.B2 gives a g e n e r a l s o l u t i o n f o r any e ^ t ) . the boxcar i s a l i n e a r device during samples,

Since

the output r e -

s u l t i n g from a n o i s y s i g n a l i s simply equal to the sum the outputs f o r the s i g n a l and noise s e p a r a t e l y . We

of

first

i n v e s t i g a t e the response of the boxcar to a noise i n p u t . Response of Boxcar to Noise The output v o l t a g e i s constant between hence the g e n e r a l appearance

samples,

of the boxcar output i s as

shown i n F i g . 2B.

k — T — H

^N-

F i g . 2 B . General appearance

The

of boxcar output from noise i n p u t .

q u a n t i t i e s of i n t e r e s t a r e : i)

the mean square value of the output v o l t a g e fluctuations

ii)

E^,

the frequency spectrum

To get at these q u a n t i t i e s we FT n

.

n+p E

From Eq.

first

of these

c a l c u l a t e the f u n c t i o n

B.2

= r(e~ + e , e n n n-1

- r

+ e

„e~ n-2

fluctuations.

2 r

)

-122-

and

E

a r(e ^ n+p

n+p

+ cr e~ n+p_i

so that i f the average

= ( s i n c e (e" ) n

P"

i s independent

= £HlliSnL -r

2

i s zero f o r

+ e of

)

2 r

0

of v^e^,

c^r-(i^/j

c -

+ e , e" n+p-2

r

2 r

+

n' , then

e- 4

]

, r

n)

^£.(JzSe^

for

> r

(B.3)

e e i s zero f o r n ^ n' i f the c o r r e l a t i o n time n n ' ing the input noise i s much s h o r t e r than T. f o r a l l cases of Now

r--iL«i

characteriz-

This i s s a t i s f i e d

interest.

the q u a n t i t y ^ ^ n

n +

p i s very n e a r l y equal to the

c o r r e l a t i o n f u n c t i o n E ( t ) E ( t + t r ) evaluated at the time f =• pT ( s i n c e r i s small the c o r r e l a t i o n changes very l i t t l e a time T and the above i s a good approximation). Q T E(t).

c l o s e l y approximates

the reduced

but the above e x p r e s s i o n i s e a s i e r to work with.) F o u r i e r transform of

Therefore

c o r r e l a t i o n f u n c t i o n of

(The exact c o r r e l a t i o n f u n c t i o n i s e a s i l y

—2"

during

calculated Taking

the

—-t-7*

e T ^

of the output f l u c t u a t i o n s

J(uJ):

we

o b t a i n the s p e c t r a l d e n s i t y

-123-

"&. the rms

noise out-

s put

i s proportional

to i

.

The output s i g n a l / n o i s e

ratio i s

s

z

therefore

p r o p o r t i o n a l to

S i n X i s p l o t t e d i n Fig.7B. f o r X = idds

=

1.2

The maximum S/N

r a t i o i s obtained

where T

or t C : 0 . 4 L

M

f

= 2^L

Note that here a l s o S/N does not go to zero f o r very t„ but to a small s We our is the

now

hypothetical

small

but f i n i t e v a l u e (see s e c t i o n I ) , c a l c u l a t e the e f f e c t of passing low pass f i l t e r .

swept l i n e a r l y with time we

Since

e ^ ( t ) through

the sample p o s i t i o n

can put t' = s t where t i s

a c t u a l time and t' i s the i n t e r v a l between the s t a r t of

each waveform

and the sample gate.

Hence

(#) and

therefore,

putting

21

t '

,

S U) =

u/

t

we get

- 1 3 3 -

which can be w r i t t e n i n the form:

r

where