The Hanle effect and its use for the measurements of very small magnetic fields

NUCLEAR

INSTRUMENTS

AND

METHODS

IIO

(i973) 2 5 9 - 2 6 5 ;

©

NORTH-HOLLAND

PUBLISHING

CO.

P a r t I V . H a n l e effect 259 - 300

T H E H A N L E E F F E C T AND ITS USE F O R T H E M E A S U R E M E N T S OF VERY S M A L L M A G N E T I C FIELDS ALFRED

KASTLER

Laboratoire de Physique, l~cole Normale Sup~rieure, Universitd de Paris, 24, Rue Lhomond, Paris (5e), le, France T h e Hanle effect is briefly reviewed. It is s h o w n that optical p u m p i n g can be used to create alignment o f states. T h e c o m b i n e d use o f H'anle effect a n d optical p u m p i n g then p e r m i t s one to m e a s u r e magnetic fields as small as 10_9 G.

I thank Prof. Bashkin for giving me the opportunity to speak at this beam-foil conference. | must apologize that my lecture is not directly on beam-foil spectroscopy, but I may show you a connection between the Hanle effect and BFS. As some contributed papers after my talk are also on the Hanle effect, let me remind you just what this effect is. The original Hanle experiment was made about half a century ago 1) when Hanle was a young assistant of James Franck at the University of G6ttingen. The scheme of his experiment appears in fig. 1. Consider the right-handed reference frame x, y, z. A scattering cell, R, contains mercury vapor at very low pressure. A mercury lamp emits resonance light (2 2537 A) which travels in the y-direction and is linearly polarized with the electric vector, E, in the z-direction. The resonance light is scattered in all directions by the mercury vapor at R, and we observe the intensity of that part of the light which is scattered in the x-direction. In the experiment, the local magnetic field (due to the earth or parts of the laboratory) must be rigorously compensated. The scattered light is highly polarized, for natural

mercury, the degree of polarization being about 83%. We know, today, that if R is filled with an even isotope of mercury, the degree of polarization approaches 100% when the vapor pressure is very small. Now Hanle measured both the intensity and the degree of polarization of the scattered light as a function of a magnetic field H 0, which he applied in the x-direction, that is to say, in the direction of observation. By the intensity, I mean the intensity of the light which is linearly polarized in the z-direction. His results appear in fig. 2. There you see that, as Ho increases, the degree of polarization goes down (fig. 2b), and that the plane of polarization rotates 4 5' 40 = 30* 20* I0"

.I

I

I

2

5

I I

I 2

3

I00 90 80 70 60

E

G

by

50 40 30 20 I0

Fig. 1. T h e scheme o f H a n l e ' s original experiment. R is a bulb which contains m e r c u r y vapor. It is irradiated by linearlypolarized m e r c u r y light (2 2537A) from a m e r c u r y lamp (not shown). T h e E-vector is in the z-direction. H0 is a u n i f o r m magnetic field which m a y be applied in the x-direction, which is also the direction o f observation.

H o GAUSS

Fig. 2. H a n l e ' s data on resonant scattering. (a) R o t a t i o n o f the plane o f polarization. (b) Depolarization o f the light.

259 IV. H A N L E E F F E C T

260

ALFRED KASTLER

(fig. 2a). In zero field, the direction of polarization of the observed light is the same as that of the incident light, but when the magnetic field is applied, the plane of polarization is rotated through the angle ~, 10 °, 20 °, etc., up to, in this case, the asymptotic value of 45 ° . The experiment shows that the Hanle curve, that is, the curve of depolarization, is of Lorentz shape, for which one can conveniently define a field, H1/2, for which the degree of polarization is reduced to half the initial value. We call this field the field of the halfwidth of the curve. The Hanle curve may be represented by the equation p =

ei l + aH~ '

(1)

a magnetic field which is perpendicular to this induced moment Mo, that moment will precess in the equatcrial plane (the plane perpendicular to the magnetic field). The angle, O, of this Larmor precession is given by 0 = ogoZ,

(2)

where o90 is the circular frequency of the Larmor precession of the dipole moment. That frequency is proportional to the applied field, H o. The factor of proportionality, 3', is called the gyromagnetic factor of the excited state of the atom, and is related to the Land6 g-factor of the state by the formula e

where Pi is the initial degree of polarization (for H o = 0), and a is a constant. Now Hanle tried to work out a theory of his experiment. Please remember that quantum mechanics did not yet exist, so Hanle constructed a semi-classical theory. When we send 2 2537 A photons into the mercury vapor, the light is absorbed by ground-state mercury atoms. Thus the atoms are lifted to the 63p1 state (fig. 3), from which they decay back to the ground state. Hanle assumed that the E-vector of the incident light (see fig. 1), that is, the oscillating electric field of the incident light wave, induces an oscillating electric dipole moment in the excited state, that moment being parallel to the E-vector and oscillating at the same frequency as the incident light wave. It is this dipole moment which radiates in all directions and defines the intensity and polarization of the scattered resonance light. We call this dipole moment Mo. Now if we turn off the incident light, this dipole moment will decay exponentially with the time-decay parameter z which is identified with the lifetime of the excited state. But, now, if we simultaneously apply

- - 6

~; = g 2 m 0

where e/m o is the ratio of charge to rest mass of the electron. Now, in the Hanle experiment, these two phenomena - the exponential decay of the dipole moment and its precession in the equatorial plane - come in simultaneously, and fig. 4 may show you the scheme of what happens. In fig. 4, z is the coordinate of the induced dipole moment, Mo, and the plane of the figure is the equatorial plane in which the precession of M o occurs. The moment begins to rotate, but it also decays, and the effect of both of these events is that there is a sort of fan. If the applied field is small, so that COoZis small, you get the shape of fig. 4a. If the field is stronger, fig. 4b results, and, finally, if COoTbecomes very large, the figure (see fig. 4c) becomes isotropic in the y - z plane. In the case of fig. 4b, where ~o z -- 1, the dipole moment is reduced to half its initial value, which corresponds to the half width of the Hanle curve. Now if you make this Hanle experiment on a vapor, then you can only see the mean effect of this precession and decay. You see the fan with all its vectors and

3~ .... 6

z

z

3Pi°

6 3Po°

y ......

(o)

:. . . .

(~T<< I Mz ~

Mo

My Small 6 IS o

Fig. 3. The low-lying states of Hg.

(3)

y..

(b)

, ....

COT=I

y .

.

.

{c)

.

.

.

.

.

w ~" >> I

Mo MZ=-"~

M z Small

My =M__.D 2

My Small

Fig. 4. L a r m o r precession of the induced transverse m o m e n t Mz for three different cases.

M E A S U R E M E N T S OF V E R Y S M A L L M A G N E T I C

you have to integrate to get the mean polarization plane and to get the depolarization. You see that not only does M z decay but there is a component Mr which is created at right angles to M~. Of course, the theory of the Hanle effect can be worked out. The semi-classical theory shows that when the magnetic field is applied, the moment in the z-direction decays according to

261

FIELDS

the rotation of the vector of the emitted light. Of course, to see these things, you must induce some anisotropy in the excited state of the atom. While Hanle did this with the absorption of polarized light, the beam-foil source appears to have such anisotropy as an intrinsic feature of the excitation process. The Hanle experiment I have described was done Z

M:=

M o

1 + (COo'C)2

,

(4)

so that the whole figure depends only on the product tOoZ. Now the nice thing if you work in beam-foil spectroscopy is that then you see not only the integral effect of this fan, but you can also follow, path by path, the individual vectors which rotate because the time scale t is converted into the space scale y. Thus you can see independently the decay, whence you can measure the characteristic decay time, and you can see

Circularly Polarized Licjht

x~

Fig. 5. The transfer o f angular m o m e n t u m from a beam o f light to gas atoms by resonance absorption o f circularly polarized light.

Fig. 6. Apparatus used by Cohen-Tannoudji and Haroche for measuring small magnetic fields. IV. H A N L E E F F E C T

262

ALFRED

on an excited state, but the experiment can be repeated on a g r o u n d state. Here we take advantage o f the fact that we can create an anisotropy in the ground state, that is, we can create an anisotropic Mo vector, by optical pumping2). I will not develop here the theory of optical pumping, but only say that if we bave a m o n a t o m i c vapor - take an alkali vapor - and we make an optical resonance experiment using exciting light which is circularly polarized, that light carries angular m o m e n t u m . By the absorption prGcess, the angular m o m e n t u m is transferred to the absorber atom. In this way one can orient the magnetic axis of the atoms in the direction o f the light beam which, to be consistent with m y previous discussion, I take to be y-direction (see fig. 5). By this means we create in the vapor a bulk magnetic m o m e n t M o which is in the direction of the light beam. Now, if we apply a magnetic field perpendicular to the vector M o, the vector begins a L a r m o r precession, and, as in the previous case, we again have a characteristic decay time z. Here, however, where we are considering an atomic ground state, this time z is identical with the paramagnetic relaxation time of the atoms in their g r o u n d state. N o w coming back to the original Hanle experiment, y for an excited state, i.e., the gyromagnetic ratio or the L a r m o r frequency in a field of 1 G, is o f the order of 10 7 s -~. Also, the lifetime z is o f the order o f 10 - 7 S, SO the characteristic field which corresponds to the half width of the Hanle curve is

H1/2 =

~- 1Z- l ,

(5)

or 1 G, as you saw earlier in fig. 2. In fact, it was from measuring H1/2 that Hanle determined (yz) -1. He t o o k the g-factor f r o m the formula for Russell-Saunders coupling - it is 3/2 for the case of H g - and he thereby obtained the lifetime of the 3P 1 level o f Hg, his result being z = 1.2×10-7s.

(6)

Later on, this g-factor was measured accurately by Brossel 3) in his double-resonance experiment, and he showed that g is not exactly 3/2 but is 1.48, so the difference is not very large. Hence the lifetime value which Hanle got was already a very g o o d value. N o w what is different in the ground state o f an alkali atom is this: the magnetic m o m e n t is the magnetic m o m e n t o f the valence electron, and y is o f the order o f 10 v, just as for the excited state o f Hg, but the lifetime of the alkali atoms corresponds to the relaxation time in the vapor. If we use a paraffin

KASTLER

coating on the wall of the bulb, that time is o f the order o f 0.1 s. It is a very long relaxation time. Thus we see that in this case, the Hx/z value o f the Hanle curve is o f the order of 1 0 - 6 G so measuring the Hanle curve is a way of measuring very small magnetic fields. The basis for measuring the magnetic m o m e n t which is induced in the vapor by optical pumping is this: theory shows and experiment confirms that the vapor becomes increasingly transparent as the atomic orientation is built up. Moreover, the change in transparency of the vapor is proportional to the magnetic m o m e n t which is induced in the vapor. Therefore the optical signal gives a direct measurement of the induced moment, and we are then able to study its decay and the Hanle curve. O f course, before we do such an experiment, we must not only compensate for the earth's field, but also guard against all parasitic fields in the laboratory. This was done 4) in our laboratory by C o h e n - T a n noudji and his research students D u p o n t - R o c and Haroche, and I m a y just show you the device used for magnetic screening (fig. 6). The experiment was built inside of a magnetic screening device. We had to use five concentric cylinders of mu-metal, each cylinder having a length of a b o u t 2 m, a diameter about 1 m, and apertures at the ends just large enough so the light can get in and out. The light passes through the bulb, which contains R b vapor, and goes on to the photomultiplier. Inside the magnetic shield we also have Helmholtz coils which enable us to apply small magnetic fields to the vapor. I must say that in spite of the five-fold screening we had some trouble when the elevator (ascenseur) was going up and down in the laboratory and we had to build a device to cancel out the elevator effect. In fig. 7, we see the Hanle curve observed by D u p o n t R o c and Haroche 4) on Rb vapor. The abscissa is the scale of the applied field, H 0. The range of 5/aG is shown and you see that the half width is actually of the order of one microgauss for this Hanle curve. In fig. 7, you see that there is still some noise. One can ABSORBED

J

o

Fig. 7. Hanle effect on the ground state of 87Rb.

MEASUREMENTS

OF V E R Y S M A L L M A G N E T I C

263

FIELDS

RECORDER

SYNCHRONOUS'-~

[

=

LAMP 87Rb

PLASTIC LIGHT PIPE

/

Rb \MAGNETIC ~

[

SHIELD 5 CONCENTRICCYLINDERS

Fig. 8. L a b o r a t o r y a r r a n g e m e n t for the m e a s u r e m e n t o f very small m a g n e t i c fields.

show that the experiment can be much improved if, on the steady field H0 which gives rise to the Hanle effect, we superimpose an oscillating field H 1 cos cot, where co is much higher than the coo of the Larmor precession in the field H o. In this case, when we observe the modulation of the signal at frequency co (i.e., the frequency of the oscillating field) we get a curve which is the derivative of the Hanle curve, and which will be shown in fig. 9. First I show (fig. 8) the laboratory arrangement. Only one of the shielding cylinders is shown. The 87Rb light source and the circular polarizer, DI plus the ½-wave plate, which together cause the optical

0

NOISE (SENSITIVITYx IOOI

pumping and produce the moment in the direction of the light beam, can also be seen. The Helmholtz coils giving the steady field H 0 and the oscillatory field H1 cos cot are shown schematically. By detecting the light in the photomultiplier, we can measure the transparency of the 87Rb vapor in the bulb at the center of the apparatus. Since we have an ac signal, we can use synchronous detection, which gives an excellent signalto-noise ratio. Fig. 9 illustrates the results. It shows the derivative of the Hanle curve. You see that the noise is gone. We have to enhance the sensitivity by a factor of 100 in order to see the noise. The abscissa scale is in #G, and it is clear that the slope of the curve at zero field is very steep so a small change in Ho gives a very large light signal. This is shown in fig. 10. In fig. 10, we see the signals obtained when the Helmholtz coils are used to change the magnetic field by 2 × 10-9 G. The field change is produced by a square pulse. With the noise as shown, we can measure fields of the order of 10-9 G. Let me now show you an interesting application of this sensitive method of measuring very small magnetic fields. The idea, which was proposed by Cohen-

2'IO-gGI ~

0

Fig. 9. Derivative o f Hanle curve.

L ,

,

J ,

Time > (Minu~es)

Fig. 10. T h e signal f r o m small changes o f 2.1 × 10-9 G in magnetic field; time c o n s t a n t o f detection = 3 s. IV. H A N L E

EFFECT

264

ALFRED KASTLER

Tannoudji and Haroche, was to use optical pumping in 3He vapor. This experiment was actually done first by Colegrove, Schearer, and WaltersS). We may understand the experiment as follows. In helium the ground state (see fig. 11) is 1So; the aS I state is metastable. If we use an electrodeless discharge, it is easy to populate the metastable aS1 state. There are lines in the near infrared which connect the 3S, and 3po levels. Suppose we use circularly polarized, infrared light to populate 2p 3P1,2, which then decays back to the metastable 3S 1 state. This orients the metastable state. If, further, we use 3He for the gas, the electronic orientation of the 3S, state also produces a nuclear orientation because of the hyperfine coupling. Now in the helium gas, there are collisions between the metastable and ground-state atoms. In these collisions, there is exchange of orientation, which leads to nuclear orientation in the 3He ground state. This is quite easy to accomplish. In the experiment I describe6), the pressure in the glass bulb (6 cm diameter) was about 3 torr, and the degree of nuclear polarization obtained was of the order of 5%. The point of this is that each aHe nucleus is a tiny magnet. Hence orientation of the 3He nuclei in the bulb produces a macroscopic external magnetic field. It can be calcu_ _ 2 p

Is

3pg

- - I - - 2

2sISo

lated that for the pressure of 3 torr and a 5% degree of nuclear orientation, that external field must be of the order of 10 -8 G. Since we can measure such fields with our Rb cell, we can use such a cell to detect this nuclear field. The experimental arrangement appears in fig. 12. There is a helium cell in which the atoms are optically pumped to produce a bulk magnetic moment in the y-direction. Next to it is a rubidium vapor cell which is optically pumped to produce a bulk magnetic moment in the z-direction. An oscillating magnetic field, Ht cos ~ot, is also applied in the y-direction. Thus we see that the steady field H o is provided entirely by the nuclear magnetism of the aligned 3He nuclei. An important feature to keep in mind is that the a7Rb magnetic moments are sensitive only to a field Ho which is parallel to the oscillating field; they are not affected by a field which is perpendicular to the oscillating field. In fig. 13, we display the building up of the nuclear magnetic field, as monitored by the precession of the aTRb magnetic moments. You see that after half an hour or so, we reach a stationary field of the order of 5 x 1 0 -8 G. This is a very slow pumping process. Fortunately, the relaxation time of the oriented 3He atoms is of the order of two hours, so we have enough time to do the experiment. Fig. 14 shows the whole arrangement. Inside the magnetic shield we have a circular polarizer through AH

(Gauss)

Is 2s 3St

.5xlO-s f

J ~ 0

Is2 tSo Fig. 11. Low-lying levels of He.

I0 '

2'0

30 '

~rne (mini

Fig. 13. Magnetic field from aligned 3He nuclei as a function of time. The field is monitored by S7Rb~yapor.

3H: ) "~

~

H

'

C°6°JIx;

87~

~

~.r

~

LAMP

~

Bz

j Fig. 12. Arrangement for detecting the nuclear magnetism of oriented 3He nuclei,

LAMP He

Fig. 14. Arrangement for detecting the magnetic field from aligned 3He nuclei.

MEASUREMENTS

I

ifoH-7 gauss)

OF V E R Y S M A L L M A G N E T I C

N

(b)

(a)

265

FIELDS

N

k.~.4Rotation Of Earth

+ Rotation Of Earth

I

w

•

w

Sun

Sun

s

$

(a)

(b)

Time

' I0

o

v

' 185

' 190

(

mYoI

(c]

I0"9gaussT

~ Time

6~o

6~5

6~o

(m'7.)

Fig. 15. Magnetic field f r o m aligned 3He nuclei as a function o f time after the exciting lamp is turned off.

which light from a rubidium lamp is passed so that it can pump the 87Rb vapor. We show the photomultiplier which serves to monitor the transparency of the 87Rb vapor. There is also a 3He cell which, with the help of a He lamp and a mirror, is pumped so as to produce a magnetic moment, hence a magnetic field, in the vertical direction. It is this field which is sensed by the S7Rb vapor. Now we apply a small field, H, to the 3He atoms, this field being parallel to the magnetic moment in the 87Rb vapor, and an oscillating magnetic field H 1coscot which is perpendicular to the 87Rb magnetic moment. But, as we said a moment ago, the small field H has no effect on the 87Rb atoms. However, this field, which is of the order of 10 -6 G, causes the 3He nuclei to precess. This means that the direction of the nuclear field oscillates, and this oscillation does affect the orientation of the S7Rb magnetic moment. In fact, what happens is illustrated in fig. 15, where we plot the magnetic field which is sensed by the transparency of the S7Rb vapor as a function of time. The time scale is in minutes. If, after the 3He nuclei are oriented to saturation we turn off the helium lamp, the nuclear magnetic field decays in time, and this is also exhibited in fig. 15. The field variation can be follewed, as you see, for ten hours. This is a good method of measuring the relaxation time which, as I mentioned earlier, turns out to be about two hours. Finally, let me outline an experiment which we have not done, but which we would like to do, more for pedagogical than for scientific reasons. Fig. 16a shows

Fig. 16. A n alternative to the F o u c a u l t p e n d u l u m .

the earth and sun in the evening. On the equator, which I choose only for simplicity, we have a bulb of 3He with the nuclei oriented as shown in fig. 16a. The vector points to the sky. After saturation is achieved, the aligning light is turned off. Now the 3He nuclei do not know that the earth is rotating so their orientation is unaffected by the change from night to day. However, what we expect to find in the morning, as seen in fig. 16b, is that the orientation has been reversed, namely, the vector points to the ground. Thus one can prove that the earth is rotating, and our experiment is equivalent to the Foucault pendulum in the Smithsonian Institution. I thank Physical Review and Physical Review Letters for permission to reproduce several figures from articles in those journals. References 1) W. Hanle, Ergeb. Exact. N a t u r w . 4 (1925) 214. 2) A. Kastler, in New directions in atomic physics, vol. lI (eds. E. U. C o n d o n a n d O. Sinano~lu; Yale University Press, New H a v e n , Conn., 1972) ch. 1, 2. a) j. Brossel, A n n . Phys. 7 (1952) 622; J. Brossel and F. Bitter, Phys. Rev. 86 (1952) 308. 4) j. D u p o n t - R o c , S. H a r o c h e a n d C. C o h e n - T a n n o u d j i , Phys. Letters 28A (1969) 638. 5) F. D. Colegrove, L. D. Schearer a n d G. K. W a k e r s , Phys. Rev. 132 (1963) 2561. o) C. C o h e n - T a n n o u d j i , J. D u p o n t - R o c , S. H a r o c h e a n d F. Laloe, Phys. Rev. Letters 22 (1969) 758.

Discussion BICKEL: HOW do y o u get a very h o m o g e n e o u s magnetic field inside y o u r c h a m b e r when y o u are aligning the h e l i u m ? KASTLER: Oh, it does n o t have to be very h o m o g e n e o u s . There a r e two limitations with the H e l m h o l t z coils. T h e coils inside m u s t be large e n o u g h that the field in the bulb is rather h o m o g e n e o u s b u t n o t too near the walls. Otherwise the current in the Helmholtz coils would induce mirror images in the m u - m e t a l a n d distort the field. Therefore we m u s t c o m p r o m i s e , b u t we do get a field which is sufficiently h o m o g e n e o u s . IV. H A N L E ' E F F E C T