An estimate of the uncertainty due to counting statistics in a novel type of neutron EDM experiment

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An estimate of the uncertainty due to counting statistics in a novel type of neutron EDM experiment

D. Thompson

To cite this version:

D. Thompson. An estimate of the uncertainty due to counting statistics in a novel type of neutron EDM experiment. Journal de Physique I, EDP Sciences, 1992, 2 (3), pp.241-245. �10.1051/jp1:1992139�.

�jpa-00246478�

Classification Physics Abstracts

B.40F 14.20 29.25

An estimate of the uncertainty due to counting statistics in

_a

novel type ^of neutron EDM experiment

D.

Thompson (*)

Ruthe~ord

Appleton Laboratory,

Didcot, OX11 OQX, G-B- (Recejved 5 July 1991, accepted in final form 8 November 199I)

Abstract. A novel type ^{of EDM}

experiment using

ultracold neutrons

produced

in _a

supertherrnal

_source,

coupled

with the dressed neutron

technique

has been

proposed.

Here _an estimate is made of the

uncertainty

in the electric

dipole

moment of neutron that would arise from

neutron

counting

statistics in such _an

experiment.

Golub

[I]

has

proposed

_a novel method of

searching

for the electric

dipole

moment

(EDM)

of the _neutron. Instead of

using Ramsey's separated

fields method

[2-5], polarized

ultra cold

neutrons

(UCN)

_are to be

produced

in _a

superthermal

UCN _source

[6, 7] containing

_a small

quantity

of

polarized

3He.

Absorption

of neutrons

by

^3He takes

place

almost

entirely through

the channel in which the _neutron and 3He

spins

are

anti-parallel [8].

The presence of _an electric field the EDM of the _neutron will alter the neutron

precession frequency changing

the

angle

between the _neutron

spins

and the 3He

spins,

^{and hence}

changing

the

absorption

rate.

A refinement _to this idea

[9]

_uses the dressed _neutron

technique [10]

to

give

the 3He _atoms

and the _neutrons the _same effective

magnetic

_moment,

and, therefore,

the _same

precession

frequency

in _a

magnetic

field. In this paper, we calculate the

uncertainty

in the

EDM,

due _to neutron

counting statistics,

for _a

particularly simple

form of this

experiment.

Let there be _an electric

Eo

and _a

magnetic Bo

^field

along

the _z

axis,

and the initial conditions be 100fb neutron

polarization

and 100fb 3He

polarization,

both

along

_+z.

Apply

_a

w/2

tipping pulse

to rotate both the neutron

spins

and the 3He

spins

into the x-y

plane.

In the presence of the

dressing field,

and

ignoring

_any EDM both the _neutrons and the 3He will precess around _z with the _same

frequency.

Now

interrupt

the

dressing

field for

just long enough

to allow _an

angle

of w/2 to build up between the _neutrons and the 3He. In _a reference

frame

rotating

at the _common

precession frequency

the neutrons will _now be

aligned along

the

y'

axis and the 3He

along

the x' axis.

Including

the effects of the EDM of the neutron, the

absorption probability

for the neutrons will be

P~(t)

=

( _{1«~ (l}

+ sin

wt)

+ «_

(I

sin

wt)i, (1)

(*)

Present address _:

Department

of

Physics, University

of Zambia, PO Box 32379, Lusaka, Zambia.

242 JOURNAL DE PHYSIQUE I N° 3

where _:

n~ ^{is the} 3He

density,

assumed constant

v is the relative

velocity

between the 3He and the neutrons

cr~ is the

absorption

cross section for

parallel spins

cr_ is the

absorption

_cross section for

anti-parallel spins

and w =

2 d*

Eo/h

is the

precession frequency

due to the EDM.

We write d* for the EDM because the

dressing

field also reduces the effective EDM. For the

dressing

field needed _{to set} the

magnetic

moments

equal

d*

=

0.7 d~

[9].

We also define _:

T+ =

(n3

_"+

v)~

the

lifetime,

due _to

absorption by

the

3He,

that the _neutrons would have if the _neutron and 3He

spins

were

parallel

_;

T- =

(n3

_"-

v)~

the lifetime the neutrons would have if the

spins

were

anti-parallel

_;

r~ the lifetime of the _neutrons in the storage ^{volume in the absence of}

3He.

The evolution of the _neutron

density n(t)

is then

dn _{n n} I I _n I I

I

To 2 _r~ ^~ _{r_} 2 r~ r_

~~~ °'~

=

nri nrj

^~/2

(nrj ~/2)

sin _wt

(2)

where

r/

₌

rj

+ rj and

ri

=

rj

~j For small _wt the

absorption

of the neutrons

by

the 3He includes _a term linear in d*.

The solution _to

equation (2),

with _an initial

density

_no, is

n(t)

= no

exp[- (rp

+

r/ ~/2)

t +

rj ~(cos

wt

1)/(2

_w

)]. (3)

The

absorption

of the _neutrons is _to be monitored

by detecting

the scintillations

produced by

the reaction

products [I]

n +

~He

- ~H _{+ p +} 746 kev

(4)

The detection rate

dN~/dt

is the rate at which the _neutrons in _a volume V _are absorbed

by

the

3He

)

=

Vn

(t) P~(t)

=

Vno

_exp

[- (ri

+

rj ~/2)

t +

ri ~(cos

_wt

1)/(2

_w

)]

_x

[r/

~/2 ₊

(ri ~/2)

sin

wt] (5)

For small _{wt we can}

expand

the

trigonometrical

functions and _we find that the number of

absorptions

detected in _a storage ^{time T is}

Vno

jT

^~

N ~ ⁼ exp

[- (rj

_{+ rj}

/2) t] [rj

+ w ri (t rj t

/4)

dt 2

~

where _x

=

(rp

+

r/ ~/2)

T.

When the direction of the electric field is reversed _w

changes sign

and _so there is _a

change

in

counts between the situation with

E~

and Bo

parallel

and that with

Eo

and

Bo antiparallel.

If this

change

in counts is

AN,

then the effective EDM is

~

~ j

(rp

₊

r/

~/2)~ _j

~ ~

2

Eo Vno _ri (rj ~[l (I

+

x) e~~]

₊

_r/ ~x~e~~/4)

^~~~

Because the

change

in counts due to _an EDM is

small,

the counts, ^for both

E~

^and

Bo

parallel

and

Eo

^and

Bo antiparallel,

will be

approximately equal

and

given by

~ ~

'~~

(rj r/ ~)/2

~ ~~ ~~~

and the

uncertainty

in

N~

will be

=

fi

_{and that in AN}

=

fi. _Bearing

_{in mind that}

two measurement

cycles

_are

required

_to deduce

d*,

the

uncertainty

in d* per measurement

cycle

_cr~~ is

5

~

( rp

+

r)

^{~/2 ~}

(

l e~ ^~

)

^~

~~~

fi _{ri ~/Vno (rp ~[l (I}

₊

_{x) e~~]}

₊

_{r/ ~x~e~~/4)}

^~~~

When

comparing

different

experimental

methods it is _more useful to look at the

uncertainty

per

day.

If _{we assume} that

filling

the

storage

^{volume takes} a

negligible

time

compared

to

T,

and _we let _s

= 86 400 the number of seconds in _one

day,

the

uncertainty

_per

day

is

«~~

~~~

fi

~ i

~(rp

₊

_r/

_~/2)~

_{ix (I}

_e~ ~

)]

$ Eo ~/Vno' ri (rp ~[l (I

₊

x) e~~]

_{+ rj}

~x~e~~/4)

^~~~~

It is the

uncertainty

_per

day crj~

that _must be minimized in order to

optimize

the

experiment. Setting

the

derivative,

with respect to x, of

equation (10) equal

to zero we have

r) 4[(1 e~~)~ (l

+

x~) x~]

=

(11)

rj~ x~e~~[x(2 _{e~~) 3(1 e~~)]}

USIIIg Ti~

=

r/I

_~r~

~ / _{~ ~}

~~ ~~~~~

~ ~ ~~~ ~~~~~~~~~ °~

~~~~~1°II (lo)

with

respect

to

rj

~/rj

=

3

[2(e~

I

x)/x~- Ii

_±

(9[2(e~

I

x)/x~- _1]~

₊

_8(e~

_I

_x)/x~)

^~

₍₁₂₎

Equations (11)

and

(12)

were solved

simultaneously, using

_a Hewlett Packard 48SX

calculator,

to

give

rj ~/rp

= l.007 and _x

=

2.247

From this _we deduce that the

optimal ^~He density

is that which

gives

rj

₌ n~

v(cr~

+ cr_ =

rp

_~, ^but we note that for the

experiment

described the lifetime of

the neutrons due _to

absorption by

the

~He

is 2 r~. At the same time the

optimal

_storage time T

is

= 1.5 To.

244 JOURNAL DE

PHYSIQUE

I N° 3

In the _more

general

case where the neutron

polarization

is _not 100

fb,

but

P~, equation (2)

should be

replaced by

~ o 2~~)Pnsinwt

~ ~

(j~~

so that the

uncertainty

_per measurement

cycle

and per

day

_are increased

by

_a factor of

[P~[~ Similarly

for _an

arbitrary ^~He polarization P~

the

uncertainty

is increased

by '~3'

Using

the

optimal

values r~ = To and _x

= 2.247 the dressed _neutron

technique gives

_an

uncertainty

in the effective

EDM,

_per measurement

cycle,

of

h I

("+

+

"-~,

2.33

('~)

~~~

Eo' /~

_{~o "+ "-}

(~n ~3'

and _an

uncertainty

_per

day

of

~~~ /~''0 fi lo' _{))) )- l'} ~n~~31

_{'~'~~ ~~~~}

By

_way of

comparison Ramsey's separated

fields

method,

with the

storage

^time T

= To

optimized

for the minimum

uncertainty

_per

day

has _an

uncertainty

_per measurement

cycle

of

"~

2~o' fi ^{o ~}

~~~~

o

where _a

m

P(

and is

reported

to be 0.64

by

Smith et al.

[4].

The

uncertainty

_per

day

is

~~ ,~ ^o fi' _~ro'

_{~ ~~~ ~~~~}

A detailed

comparison

of the two methods is difficult without _{a concrete}

experimental design

for the dressed _neutron

method,

since

Golub,

Richardson and Lamoreaux

[9]

anticipate

substantial

improvements

in both _neutron

density

and electric field

strength.

If _we

simply

take the _same values of

Eo,

_no, V and To for the two cases, use the ideal

figures

of

P~

₌

P~

_{= a}

=

I and «~ = 0 and include _a factor of 0.7 for the

lowering

of the effective EDM in the dressed _{neutron case,} then the

uncertainty

in the EDM per

day

is

approximately

4.8 times

higher

for the dressed _neutron method than for

Ramsey

method. The

anticipated

neutron

density

and electric field

strength [9]

should _more than compensate ^{for this.}

References

[Ii GOLUB R., J.

Phys.

France 44 (1983) L321.

[2] RAMSEY N. F., Molecular Beams (Oxford

University

Press, 1956).

[3] ALTAREV I. S. _et al., JETP Lett. 44 (1986) 460.

[4] SMITH K. F. _et al., Phys. Lett. 8234 (1990) 191.

[5] SMITH K. F. et al., to be

published.

[6] GOLUB R. and PENDLEBURY J. M., Phys. Lett. 62A (1977) 337.

[7] GOLUB R. _et al., Z.

Phys.

851(1983) 187.

[8] PASSELL L. and SCHERMER R. I.,

Phys.

Rev. 150 (1966) 146.

[9] GOLUB R., RICHARDSON D. J. and LAMOREAUX S. K., Ultra Cold Neutrons (Adam

Hilger,

Bristol, 1991).

[10] MUSKAT E., DUBBERS D. and SCHARPF O.,

Phys.

Rev. Lett. 58 (1987) 2047.

Proofs

not corrected

by

the author