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On the magnetic interaction in the deuteron
November 1936
Phycisa III, no . 9
ON THE l\IAGNETIC INTERACTION IN THE DEUTERON by H. B. G. CASIMIR lnstituut voor tbeoretischc natuurkunde der Rijksunivcrsit cit t e Lcid cn.
Summary It is shown, that the magnetic interaction between proton and neutron would give rise to an energy difference of roughly 1
It is generally assumed, that the fundamental state of the deuteron is a 35-state, i.e. that the spins of the proton and the neutron are parallel to each other and that there is no orbital an gular momentum. The energy difference between this state and the lowest 15_ state will depend on the nature of the interaction between proton and neutron. If we assume that this interaction is an exchange interaction of the Majorana type 1), then this difference will be entirely due to magnetic interaction. The object of the present note is to calculate this magnetic en ergy. Let r p and r., be the coordinates of the proton and the neutron and 'I'" (r p• r,,) the wavefunction of the deuteron, then the magnetic interaction energy will be given by: E 1II
= -
ff
_1_ Y plI
-r- (i p • ill) 'I'" d,p d,,,,
where i p and i., are the operators of the current of the proton and the neutron and where YplI = I r p - r; I. In our case the current is only due to the spin; neglecting relativistic correction terms we can write : \Tr f * 'Ip 'If' -_ ro t p (IT' r' " * tLp 'V) I , '1"* i n '¥ -- rot n ('F* II. 'F), rn wh ere
(J.p
and tLlI are the operators of magnetic moment. -
936-
CASnlIR, ON THE ~[AGNETIC INTERACTION IN THE DEUTERO~
937
It is sufficient to calculate E", for the -triplet state: since the interaction is of the cosine type we shall have E", (IS) = - 3E",(3S) and the energy difference D between '5- and 3S-sta t e will be given by 4E",. \Ve consider a ~S-state in which the z-component of the angular momentum is I, then only \F*(tip). \1" and \1"* ([!,,). \1' are different from zero. Writing (yp/2) (etz/2M c) and (y,,/2) (etz/2Mc) for the numerical values of the z-component of the magnetic moments we find:
+
E",= -(y,,/2) (yp/2) (2~;c)2ff} (o~• p0:" + oOyP o~, p"
)llJJ'j2 d-;pd-;,..
JlI
Introducing relative coordinates I;
=
xp -
eft )2f -1 (&2 E",=+(y,,/2) (yp/2) ( 2'{ -'w2 1. c p o ;
md since \Y depends only on P = E", =
+ (y,,/2) (yp/2) = -
XII
we obtain
(J2 ) I\YI2d,s +--;;;-z 0-'1
V(~2 + ''1 2 + ~2)
(2:J ~f ~
this reduces to
2
(y,./2) (yp/2) (2~;:t
L\ 1'1"1 2 d.-s =
~' I 'Y(O) 12 •
(I)
The value of 1'1"(0)f will depend on the assumptions made about he interaction force. In order to estimate the order of magnitude t is sufficient to consider the case of a very narrow rectangular iotcntial hole. One finds
1'1"(0) 12 = _I 2VME ME. , 4...
li
tz2
vhere E is the binding energy and E. the depth of the potential
.ole. It follows that D = 4E", (35) = -
4
3" (yp/2) (y,./2)
e2 vl1,fE
tic
1'ffc E •.
Putting (yp/2) = 2,9, (y,,/2) = -2,05 and E = 2 X IC6 eV we find 1 D = 370 E;,
+
which will be equal to 0,5 or 1 X lCs eV. This value is radically different from that given by Bet he and B a c her 2) (viz. 00 eV). These authors do not start from the expression for the
938
CASDIIR, ON THE MAGNETIC INTERACTION IN THE DEUTERON
magnetic interaction between two currents but from the expression for the interaction between two magnetic dipoles. This is only legitimate if the wavefunctions of the particles do not overlap as is well known from investigations on hyperfine structure. As a matter of fact our formula (I) is identical with Fer m i's well known formula for the hyperfine structure splitting caused by an s-electron 3). Our result shows, that if the exchange interaction were exactly of the Majorana type the lowest state of the deuteron would be a singlet state *). So unless one wants to assume that the angular momentum is due to orbital motion, the fact that the deuteron has a spin of 1.1i shows, that the interaction is not of the Majorana type. It seems to us to be of some interest, that one can arrive at this conclusion without making use of the results of experiments on the scattering of neutrons by protons. Received September 7, 1936.
REFERENCES I) 2) 3)
E. :11 a j 0 ran a, Z. Phys. 82, 137, 1933. H. Bet h e and R. F. B a c her, Rev. mod. Phys. 8, 1936. E. Fer m i, Z. Phys. GO, 320, 1930.
*) In the last section of a recent paper (Verh. Teylers Twecdc Genootsch. dec1 II; Haarlern 1936) the author made an opposite statement.
Phycisa III, no . 9
ON THE l\IAGNETIC INTERACTION IN THE DEUTERON by H. B. G. CASIMIR lnstituut voor tbeoretischc natuurkunde der Rijksunivcrsit cit t e Lcid cn.
Summary It is shown, that the magnetic interaction between proton and neutron would give rise to an energy difference of roughly 1
It is generally assumed, that the fundamental state of the deuteron is a 35-state, i.e. that the spins of the proton and the neutron are parallel to each other and that there is no orbital an gular momentum. The energy difference between this state and the lowest 15_ state will depend on the nature of the interaction between proton and neutron. If we assume that this interaction is an exchange interaction of the Majorana type 1), then this difference will be entirely due to magnetic interaction. The object of the present note is to calculate this magnetic en ergy. Let r p and r., be the coordinates of the proton and the neutron and 'I'" (r p• r,,) the wavefunction of the deuteron, then the magnetic interaction energy will be given by: E 1II
= -
ff
_1_ Y plI
-r- (i p • ill) 'I'" d,p d,,,,
where i p and i., are the operators of the current of the proton and the neutron and where YplI = I r p - r; I. In our case the current is only due to the spin; neglecting relativistic correction terms we can write : \Tr f * 'Ip 'If' -_ ro t p (IT' r' " * tLp 'V) I , '1"* i n '¥ -- rot n ('F* II. 'F), rn wh ere
(J.p
and tLlI are the operators of magnetic moment. -
936-
CASnlIR, ON THE ~[AGNETIC INTERACTION IN THE DEUTERO~
937
It is sufficient to calculate E", for the -triplet state: since the interaction is of the cosine type we shall have E", (IS) = - 3E",(3S) and the energy difference D between '5- and 3S-sta t e will be given by 4E",. \Ve consider a ~S-state in which the z-component of the angular momentum is I, then only \F*(tip). \1" and \1"* ([!,,). \1' are different from zero. Writing (yp/2) (etz/2M c) and (y,,/2) (etz/2Mc) for the numerical values of the z-component of the magnetic moments we find:
+
E",= -(y,,/2) (yp/2) (2~;c)2ff} (o~• p0:" + oOyP o~, p"
)llJJ'j2 d-;pd-;,..
JlI
Introducing relative coordinates I;
=
xp -
eft )2f -1 (&2 E",=+(y,,/2) (yp/2) ( 2'{ -'w2 1. c p o ;
md since \Y depends only on P = E", =
+ (y,,/2) (yp/2) = -
XII
we obtain
(J2 ) I\YI2d,s +--;;;-z 0-'1
V(~2 + ''1 2 + ~2)
(2:J ~f ~
this reduces to
2
(y,./2) (yp/2) (2~;:t
L\ 1'1"1 2 d.-s =
~' I 'Y(O) 12 •
(I)
The value of 1'1"(0)f will depend on the assumptions made about he interaction force. In order to estimate the order of magnitude t is sufficient to consider the case of a very narrow rectangular iotcntial hole. One finds
1'1"(0) 12 = _I 2VME ME. , 4...
li
tz2
vhere E is the binding energy and E. the depth of the potential
.ole. It follows that D = 4E", (35) = -
4
3" (yp/2) (y,./2)
e2 vl1,fE
tic
1'ffc E •.
Putting (yp/2) = 2,9, (y,,/2) = -2,05 and E = 2 X IC6 eV we find 1 D = 370 E;,
+
which will be equal to 0,5 or 1 X lCs eV. This value is radically different from that given by Bet he and B a c her 2) (viz. 00 eV). These authors do not start from the expression for the
938
CASDIIR, ON THE MAGNETIC INTERACTION IN THE DEUTERON
magnetic interaction between two currents but from the expression for the interaction between two magnetic dipoles. This is only legitimate if the wavefunctions of the particles do not overlap as is well known from investigations on hyperfine structure. As a matter of fact our formula (I) is identical with Fer m i's well known formula for the hyperfine structure splitting caused by an s-electron 3). Our result shows, that if the exchange interaction were exactly of the Majorana type the lowest state of the deuteron would be a singlet state *). So unless one wants to assume that the angular momentum is due to orbital motion, the fact that the deuteron has a spin of 1.1i shows, that the interaction is not of the Majorana type. It seems to us to be of some interest, that one can arrive at this conclusion without making use of the results of experiments on the scattering of neutrons by protons. Received September 7, 1936.
REFERENCES I) 2) 3)
E. :11 a j 0 ran a, Z. Phys. 82, 137, 1933. H. Bet h e and R. F. B a c her, Rev. mod. Phys. 8, 1936. E. Fer m i, Z. Phys. GO, 320, 1930.
*) In the last section of a recent paper (Verh. Teylers Twecdc Genootsch. dec1 II; Haarlern 1936) the author made an opposite statement.