Implications of Pseudospin Symmetry on Relativistic Magnetic Properties and Gamow - Teller Transitions in Nuclei

20 Pages·0.16 MB·English

by  Joseph N. Ginocchio

Implications of Pseudospin Symmetry on Relativistic Magnetic
Properties and Gamow - Teller Transitions in Nuclei
Joseph N. Ginocchio
Theoretical Division, Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA
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9 Abstract
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n
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J Recently it has been shown that pseudospin symmetry has its origins in

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2 a relativistic symmetry of the Dirac Hamiltonian. Using this symmetry we

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relate single - nucleon relativistic magnetic moments of states in a pseudospin
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2 doublettotherelativisticmagneticdipoletransitionsbetweenthestatesinthe
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1 doublet, and we relate single - nucleon relativistic Gamow - Teller transitions
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within states in the doublet. We apply these relationships to the Gamow -
/
h
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- Teller transitions from 39Ca to its mirror nucleus 39K.
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I. INTRODUCTION
For nucleons moving in a relativistic mean field with scalar V and vector potentials V ,
S V
an SU(2) symmetry exists for the case for which V = V [1]. This symmetry manifests
S V
−
itself in nuclei as a slightly broken symmetry [2–5] since VS+VV is small for realistic mean
|VS−VV |
fields [6–10], and, in fact, gives rise to what has been called “pseudospin symmetry”. The
original observations that led to the coining of the word “pseudospin symmetry” were quasi-
degeneracies in spherical shell model orbitals with non - relativistic quantum numbers (n ,
r
ℓ, j = ℓ+1/2) and (n 1,ℓ+2, j = ℓ+3/2) where n , ℓ, and j are the single-nucleon radial,
r r
−
orbital, and total angular momentum quantum numbers, respectively [11,12]. This doublet
˜
structure is expressed in terms of a “pseudo” orbital angular momentum ℓ = ℓ + 1, the
average of the orbital angular momentum of the two states in doublet, and “pseudo” spin,
˜
s˜ = 1/2. For example, (n s ,(n 1)d ) will have ℓ = 1 , (n p ,(n 1)f ) will have
r 1/2 r 3/2 r 3/2 r 5/2
− −
˜ ˜
ℓ = 2, etc. These doublets are almost degenerate with respect to pseudospin, since j = ℓ s˜
±
for the two states in the doublet; examples are shown in Figure 1. Pseudospin “symmetry”
was shown to exist in deformed nuclei as well [13,14] and has been used to explain features
of deformed nuclei, including superdeformation [15] and identical bands [16,17]. However,
the origin of pseudospin symmetry remained a mystery and “no deeper understanding of
the origin of these (approximate) degeneracies” existed [18]. A few years ago it was shown
that relativistic mean field theories gave approximately the correct spin orbit splitting to
produce the pseudospin doublets [19]. Finally the source of pseudospin symmetry as a
broken symmetry of the Dirac Hamiltonian related to V V was pointed out [2–5]. For
S V
≈ −
˜
spherical nuclei, pseudo-orbital angular momentum ℓ is also conserved and physically is the
“orbital angular momentum” of the lower component of the Dirac wavefunction.
One consequence of this relativistic SU(2) pseudospin symmetry is that the spatial wave-
function for the lower component of the Dirac wavefunctions will be equal in shape and
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magnitude for the two states in the doublet [3–5]. For spherical nuclei, this means that
the radial wavefunctions for the lower components in the doublet will have the same num-
ber of nodes, so we label these states with pseudo-radial quantum number (i.e.; the radial
quantum number of the lower component (n˜ = 0,1,...)). Furthermore,the pseudo-orbital
angular momentum will be a conserved quantum number for spherical symmetric scalar and
˜
vector potentials and so we label the states with the pseudo-orbital angular momentum ℓ
[4]. Finally, the total angular momentum j (~j =~ℓ˜+1~˜/2), and projection m, are conserved
as well. The Dirac wavefunction for the two states in the doublet are
Ψ = (g [Y χ]j=ℓ˜+1/2,if [Y χ]j=ℓ˜+1/2),
n˜,ℓ˜,j=ℓ˜+1/2,m n˜−1,ℓ˜,j ℓ˜+1 m n˜,ℓ˜,j ℓ˜ m
Ψ = (g [Y χ](j=ℓ˜−1/2),if [Y χ](j=ℓ˜−1/2)), (1)
n˜,ℓ˜,j=ℓ˜−1/2,m n˜,ℓ˜,j ℓ˜−1 m n˜,ℓ˜,j ℓ˜ m
whereg,f aretheradialwavefunctions, Y arethesphericalharmonics,χisatwo-component
ℓ˜
Pauli spinor, and [...](j) means coupled to angular momentum j. We note that the upper
˜
component of the j = ℓ 1/2 wavefunction has the same radial quantum number as the
−
˜
lower component, whereas the upper component of the j = ℓ + 1/2 wavefunction has ra-
dial quantum number one unit less than the lower component. The normalization of the
wavefunction gives
∞
[g2 +f2 ]r2dr = 1;
n˜′,ℓ˜,j n˜,ℓ˜,j
Z0
˜ ′ ˜ ′
j = ℓ+1/2, n˜ = n˜ 1; j = ℓ 1/2, n˜ = n˜. (2)
− −
For a square well potential, the overall phase between the two amplitudes will be a minus
sign [2] so we expect that, in the symmetry limit for realistic potentials, f (r) =
n˜,ℓ˜,j=ℓ˜+1/2
f (r) = f (r). For the relativistic mean field approximation to relativistic La-
− n˜,ℓ˜,j=ℓ˜−1/2 n˜,ℓ˜
grangrians with realistic zero range interactions and to nuclear field theory with meson
exchanges it was indeed shown that, f (r) f (r) [3,10].
n˜,ℓ˜,j=ℓ˜+1/2 ≈ − n˜,ℓ˜,j=ℓ˜−1/2
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However, to date, the effect of pseudospin symmetry on the relativistic wavefunction
has not been tested empirically. Since the lower component of the Dirac wavefunction
is small [3,5,10] this effect will be difficult to detect except perhaps in certain forbidden
transitions. For example, single - nucleon magnetic dipole and Gamow-Teller transitions
between pseudospin doublets are forbidden non-relativistically (i.e., “ℓ forbidden” [20,21])
because the orbitalangular momenta of the two states differ by two units. However, they are
not forbidden relativistically. In this paper we shall use approximate pseudospin symmetry
in the wavefunction to derive relations between single-nucleon relativistic magnetic moments
and magnetic dipole transtions within a pseudospin doublet on the one hand, and between
single-nucleonrelativisticGamow-Tellertransitionswithinapseudospindoubletontheother
hand. These relationships provide a test for the influence of pseudospin symmetry on the
single - nucleon wavefunctions.
II. MAGNETIC MOMENTS AND TRANSITIONS
The relativistic magnetic dipole operator for a particle with charge e is given by [22,23],
e
µˆ = g (α~ ~r) +µ σ , (3)
i ρ i A,ρ i
−2 ×
where α~ is the usual Dirac matrix, ~r is the three space vector, ρ = π for a proton and
ν for a neutron, g is the orbital gyromagnetic ratio, g = 1,g = 0, and µ is the
ρ π ν A,ρ
anamolous magnetic moment, µ = 1.793µ , µ = 1.913µ , where µ = e¯h is the
A,π 0 A,ν − 0 0 2Mc
nuclear magneton. The magnetic moment is given in terms of the matrix element of this
operator with m = j,
µ = Ψ µˆ Ψ , (4)
j,ρ h n˜,ℓ˜,j,m=j,ρ| | n˜,ℓ˜,j,m=j,ρi
and the square root of the magnetic transition probability between two states in the doublet
is given in terms of the reduced matrix element of this operator,
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1
B(M1 : n˜,ℓ˜,j′ n˜,ℓ˜,j) = Ψ µˆ Ψ (5)
→ ρ (2j′ +1)h n˜′,ℓ˜,j′,ρ|| || n˜,ℓ˜,j,ρi
q
Using the Dirac wavefunction (1), this results in
˜
j = ℓ 1/2
−
e g (j +1/2) ∞ (2j +1) ∞
µ = − ρ g f r3 dr +µ (1 f2 r2 dr) , (6)
j,ρ 2(j +1) n˜,ℓ˜,j n˜,ℓ˜,j,ρ A,ρ − (j +1) n˜,ℓ˜,j,ρ
Z0 Z0
˜
j = ℓ+1/2
e g (j +1/2) ∞ µ ∞
µ = ρ g f r3 dr A,ρ (j (2j +1) f2 r2 dr) ,
j,ρ 2(j +1) n˜−1,ℓ˜,j,ρ n˜,ℓ˜,j,ρ − (j +1) − n˜,ℓ˜,j,ρ
Z0 Z0
(7)
′ ˜ ˜
j = ℓ+1/2, j = ℓ 1/2
−
(2j +1)
B(M1 : n˜,ℓ˜,j′ n˜,ℓ˜,j) = B(M1 : n˜,ℓ˜,j n˜,ℓ˜,j′) =
→ ρ −v(2j +3) → ρ
q u q
u
t
1 (2j +1) e g ∞ ∞
[ ρ [g f +g f ] r3 dr +4µ f f r2 dr ].
− 4v (j +1) 2 n˜−1,ℓ˜,j′,ρ n˜,ℓ˜,j,ρ n˜,ℓ˜,j,ρ n˜,ℓ˜,j′,ρ A,ρ n˜,ℓ˜,j′,ρ n˜,ℓ˜,j,ρ
u Z0 Z0
u
t
(8)
A. Non-relativistic Limit
The Dirac equation with speherically symmetric potentials reduces to two coupled one -
dimensional radial equations for the upper and lower components, (g,f) [2],
d 1+κ
h¯ c[ + ]g = [2Mc2 E +V V ] f , (9)
dr r n˜′,ℓ˜,j,ρ − S − V n˜,ℓ˜,j,ρ
d 1 κ
h¯ c[ + − ]f = [E +V +V ] g , (10)
dr r n˜,ℓ˜,j,ρ S V n˜′,ℓ˜,j,ρ
where
˜ ˜ ˜ ˜
κ = ℓ,j = ℓ 1/2; κ = ℓ+1,j = ℓ+1/2, (11)
− −
M is the nucleon mass, and E is the binding energy. In order to determine ∞gfr3 dr we
0
R
use (9, 10) to derive [22]:
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g f =
n˜′,ℓ˜,j′,ρ n˜,ℓ˜,j,ρ
h¯c d d 1+κ 1 κ
[g g +f f + g g + − f f ]
2Mc2 +2V n˜′,ℓ˜,j′,ρdr n˜′,ℓ˜,j,ρ n˜,ℓ˜,j′,ρdr n˜,ℓ˜,j,ρ r n˜′,ℓ˜,j′,ρ n˜′,ℓ˜,j,ρ r n˜,ℓ˜,j′,ρ n˜,ℓ˜,j,ρ
S
(12)
In the non-relativistic limit, the potentials are ignored with respect to the nucleon mass,
although VS .48 in the interior of the nucleus. Also terms quadratic in f are ignored.
Mc2 ≈
This gives
∞ h¯ ∞
r3 dr[g f +g f ] = (κ+κ′ 1) r2 drg g . (13)
n˜′,ℓ˜,j′,ρ n˜,ℓ˜,j,ρ n˜′,ℓ˜,j,ρ n˜,ℓ˜,j′,ρ 2Mc − n˜′,ℓ˜,j′,ρ n˜′,ℓ˜,j,ρ
Z0 Z0
For j′ = j, ∞r3 drg g =1 from the normalization condition (2). Therefore in
0 n˜′,ℓ˜,j,ρ n˜′,ℓ˜,j,ρ
R
the non-relativistic limit, the magnetic moments become,
˜
µ = (j +1/2) g µ +µ ; j = ℓ 1/2, (14)
j,ρ ρ 0 A,ρ
−
j
˜
µ = ((j +1/2) g µ µ ); j = ℓ+1/2. (15)
j,ρ ρ 0 A,ρ
(j +1) −
Thenon-relativisticlimitsforthemagneticmomentsin(14,15)areequivalent totheSchmidt
values [24].
′ ′
However, for j = j, it follows from (11) that κ+κ 1 = 0 and therefore,
6 −
˜ ′ ˜ ′
B(M1 : n˜,ℓ,j n˜,ℓ,j) = 0; j = j, (16)
ρ
→ 6
Thus the non-relativistic limit of the B(M1) is zero which is as it should be since the
transition is from ℓ to ℓ 2 as stated in the Introduction.
±
B. Pseudospin Symmetry
Instead of looking at the non-relativistic limit, we examine the pseudospin limit which
assumes that the spatial wave functions of the lower components of the doublet are equal
and opposite in sign,
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f (r) = f (r) = f (r). (17)
n˜,ℓ˜,j=ℓ˜+1/2,ρ − n˜,ℓ˜,j=ℓ˜−1/2,ρ n˜,ℓ˜,ρ
Inserting this relation into (6, 7, 8) we obtain,
˜
j = ℓ 1/2
−
e g (j +1/2) ∞ (2j +1) ∞
µ = ρ g f r3 dr +µ (1 f2 r2 dr) , (18)
j,ρ 2(j +1) n˜,ℓ˜,j,ρ n˜,ℓ˜,ρ A,ρ − (j +1) n˜,ℓ˜,ρ
Z0 Z0
˜
j = ℓ+1/2
e g (j +1/2) ∞ µ ∞
µ = ρ g f r3 dr A,ρ (j (2j +1) f2 r2 dr) , (19)
j,ρ 2(j +1) n˜−1,ℓ˜,j,ρ n˜,ℓ˜,ρ − (j +1) − n˜,ℓ˜,ρ
Z0 Z0
′ ˜ ˜
j = ℓ+1/2, j = ℓ 1/2
−
(2j +1)
B(M1 : n˜,ℓ˜,j′ n˜,ℓ˜,j) = B(M1 : n˜,ℓ˜,j n˜,ℓ˜,j′) =
→ ρ −v(2j +3) → ρ
q u q
u
t
1 (2j +1) e g ∞ ∞
[ ρ [ g +g ]f ] r3 dr 4µ f2 r2 dr ]. (20)
− 4v (j +1) 2 − n˜−1,ℓ˜,j′,ρ n˜,ℓ˜,j,ρ n˜,ℓ˜,ρ − A,ρ n˜,ℓ˜,ρ
u Z0 Z0
u
t
For neutrons g = 0, and hence we have one unkown quantity, ∞f2 r2 dr. Therefore,
ν 0 n˜,ℓ˜,ρ
R
if we know one magnetic quantity, we can predict two others,
j +1
B(M1 : n˜,ℓ˜,j′ n˜,ℓ˜,j) = (µ µ ), (21)
ν j,ν A,ν
→ −s2j +1 −
q
j +2 2j +1 j +1
B(M1 : n˜,ℓ˜,j′ n˜,ℓ˜,j)ν = (µj′,ν + µA,ν). (22)
→ 2j +3 s j +1 j +2
q
For protons there are three unkown integrals, and so we can only derive one relationship
between the three magnetic quantities,
B(M1 : n˜,ℓ˜,j′ n˜,ℓ˜,j) = ((j +2)(2j +1)µj′,π −(2j +3)(j +1)µj,π +4 (j +1)2 µA,π);
π
→ 2 (2j +3) (j +1)(2j +1)
q
q
′ ˜ ˜
j = ℓ+1/2, j = ℓ 1/2. (23)
−
IfthemagneticmomentsaregivenbytheSchmidtvaluesasin(14,15),thenthemagnetic
transitionsin(21,22,23)willbeidentically zero, which isconsistent withthenon-relativistic
limit.
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The relativistic mean field overestimates the isoscalar magnetic moments of nuclei [23].
However, when the response of the spectator nucleons is included, the relativistic isoscalar
magneticmomentsagreebetterwithexperiment [25]. Theresponseofthespectatornucleons
do not significantly affect isovector magnetic moments since the dominant mesons in the
relativistic field theory are isoscalar. If we define the isoscalar and vector operators as
1 1 1 1
µ = (µ +µ );µ = (µ µ );µ = (µ +µ );µ = (µ µ );
j,S j,ν j,π j,V j,ν j,π A,S A,ν A,π A,V A,ν A,π
2 2 − 2 2 −
1
B(M1 : n˜,ℓ˜,j′ n˜,ℓ˜,j) = ( B(M1 : n˜,ℓ˜,j′ n˜,ℓ˜,j) + B(M1 : n˜,ℓ˜,j′ n˜,ℓ˜,j) );
S ν π
→ 2 → →
q q q
1
B(M1 : n˜,ℓ˜,j′ n˜,ℓ˜,j) = ( B(M1 : n˜,ℓ˜,j′ n˜,ℓ˜,j) B(M1 : n˜,ℓ˜,j′ n˜,ℓ˜,j) ),
V ν π
→ 2 → − →
q q q
(24)
then the relations are separated into relations among the isoscalar and isovector magnetic
properties:
B(M1 : n˜,ℓ˜,j′ n˜,ℓ˜,j) = ((j +2)(2j +1)µj′,S/V −(2j +3)(j +1)µj,S/V +4 (j +1)2 µA,S/V);
S/V
→ 2 (2j +3) (j +1)(2j +1)
q
q
′ ˜ ˜
j = ℓ+1/2, j = ℓ 1/2. (25)
−
III. GAMOW - TELLER TANSITIONS
The Gamow - Teller operator is given by
g
A
GT = στ±, (26)
√2
where gA is the axial vector coupling constant (= 1.2670 (35)) and τ± are the isospin raising
and lowering operator. Thus this operator is a pure isovector operator. Using the Dirac
wavefunction (1), this results in
˜
j = ℓ 1/2
−
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(j +1) (2j +1) ∞
B(GT : n˜,ℓ˜,j,ρ n˜,ℓ˜,j,ρ¯) = g (1 f f r2 dr) , (27)
q → s j A − (j +1) Z0 n˜,ℓ˜,j,ρ n˜,ℓ˜,j,ρ¯
˜
j = ℓ+1/2
g ∞
B(GT : n˜,ℓ˜,j,ρ n˜,ℓ˜,j,ρ¯) = A (j (2j +1) f f r2 dr) , (28)
q → − j(j +1) − Z0 n˜,ℓ˜,j,ρ n˜,ℓ˜,j,ρ¯
q
′ ˜ ˜
j = ℓ+1/2, j = ℓ 1/2
−
(2j +1)
B(GT : n˜,ℓ˜,j′,ρ n˜,ℓ˜,j,ρ¯) = B(GT : n˜,ℓ˜,j,ρ¯ n˜,ℓ˜,j′,ρ) =
→ −v(2j +3) →
q u q
u
t
(2j +1) ∞
g f f r2 dr . (29)
−s j +1 A Z0 n˜,ℓ˜,j′,ρ n˜,ℓ˜,j,ρ¯
where ρ¯= π if ρ = ν and ρ¯= ν if ρ = π.
We notice that
˜ ˜ ˜ ˜
B(GT : n˜,ℓ,j,ρ n˜,ℓ,j,ρ¯) = B(GT : n˜,ℓ,j,ρ¯ n˜,ℓ,j,ρ), (30)
→ →
q q
but, in general,
B(GT : n˜,ℓ˜,j′,ρ n˜,ℓ˜,j,ρ¯) = B(GT : n˜,ℓ˜,j′,ρ¯ n˜,ℓ˜,j,ρ), (31)
→ 6 →
q q
A. Non-Relativistic Limit of the Gamow - Teller Transitions
Since terms quadratic in f are ignored in the non- relatvistic limit, we get the usual
results,
(j +1)
˜ ˜ ˜
B(GT : n˜,ℓ,j,ρ n˜,ℓ,j,ρ¯) = g ; j = ℓ 1/2, (32)
A
→ s j −
q
j
˜ ˜ ˜
B(GT : n˜,ℓ,j,ρ n˜,ℓ,j,ρ¯) = g ; j = ℓ+1/2 (33)
A
→ −s(j +1)
q
B(GT : n˜,ℓ˜,j′,ρ n˜,ℓ˜,j,ρ¯) = 0; j′ = j (34)
→ 6
q
9
B. Pseudospin Symmetry
Using pseudospin symmetry, (17), there is only one unkown for the Gamow - Teller
transtions and hence each transition is related to the other,
′ ˜ ˜
j = ℓ+1/2, j = ℓ 1/2.
−
j j +1
B(GT : n˜,ℓ˜,j′,ρ n˜,ℓ˜,j,ρ¯) = ( B(GT : n˜,ℓ˜,j,ρ n˜,ℓ˜,j,ρ¯) g ),
A
→ −s2j +1 → −s j
q q
(35)
(j +2)(2j +1) j +1
B(GT : n˜,ℓ˜,j′,ρ n˜,ℓ˜,j,ρ¯) = ( B(GT : n˜,ℓ˜,j′,ρ n˜,ℓ˜,j′,ρ¯)+ g ),
→ q 2j +3 → sj +2 A
q q
(36)
˜ ˜
B(GT : n˜,ℓ,j,ρ n˜,ℓ,j,ρ¯) =
→
q
(2j +1) j +2 2 j +1
( B(GT : n˜,ℓ˜,j′,ρ n˜,ℓ˜,j′,ρ¯) g ), (37)
A
− (2j +3) s j → − (2j +1) sj +2
q
B(GT : n˜,ℓ˜,j′,ρ n˜,ℓ˜,j,ρ¯) = B(GT : n˜,ℓ˜,j′,ρ¯ n˜,ℓ˜,j,ρ). (38)
→ →
q q
This last relation, (38), also follows from isospin symmetry as well, but if pseudospin
symmetry is conserved than the relation holds even though isospin may be violated; i.e.,
f = f .
n˜,ℓ˜,π 6 n˜,ℓ˜,ν
IV. AN EXAMPLE: 39K, 39CA
The nuclei 39K and 39Ca are mirror nuclei. The ground state and first excited state
19 20 20 19
of 39K are interpreted as a 0d and 1s proton hole respectively, while the ground
19 20 3/2 1/2
state and first excited state of 39Ca are interpreted as a 0d and 1s neutron hole
20 19 3/2 1/2
˜
respectively. These states are members of the n˜ = 1,ℓ = 1 pseudospin doublet. The M1
10