Number of Spin $I$ States of Identical Particles

0.09 MB·

by  Y. M. Zhao

Number of Spin I States of Identical Particles
Y. M. Zhao1,2,3,4,∗ and A. Arima5
1Department of physics, Shanghai Jiao Tong University, Shanghai 200030, China
2Cyclotron Center, Institute of Physical Chemical Research (RIKEN),
Hirosawa 2-1, Wako-shi, Saitama 351-0198, Japan
3Center of Theoretical Nuclear Physics, National Laboratory of Heavy Ion Accelerator, Lanzhou 730000, China
4Department of Physics, Southeast University, Nanjing 210018, China
5Science Museum, Japan Science Foundation, 2-1 Kitanomaru-koen, Chiyodaku, Tokyo 102-0091, Japan
5 (Dated: February 9, 2008)
0
0 In this paper we study the enumeration of number (denoted as DI) of spin I states for fermions
2 in a single-j shell and bosons with spin l. We show that DI can be enumerated by the reduction
from SU(n+1) to SO(3). New regularities of DI are discerned.
n
a
PACSnumbers: 05.30.Fk,05.45.-a,21.60Cs,24.60.Lz
J
Keywords:
7
2
The enumeration of number of spin I states (denoted and I = nl for bosons with spin l. One defines
max
1 as D ) for fermions in a single-j shell or bosons with P(n,0) = D = 1 for I = 0. Then one has
v I I=Imax 0
0 spin l (We use a convention that j is a half integer DI=Imax−I0 =P(n,I0)−P(n,I0−1).
and l is an integer) is a very common practice in nu- Now we look at D for n¯ “bosons” of spin L = n,
7 I 2
0 clear structure theory. One usually obtains this num- with n¯ = 2l for bosons or n¯ = 2j +1−n for fermions.
1 ber by subtracting the combinatorial number of angu- I of these n¯ “bosons” with spin L equals that of n
max
0 lar momentum projection M = I + 1 from that with bosons with spin l or that of n fermions in a single-j
5 M = I [1]. More specifically, DI equals to the com- shell. Furthermore, P(n,I0) of I0 =i1+i2+···in¯ with
0
binatorial number of M = I subtracted by that of the requirement0≤i ≤i ≤···≤i ≤2L=n,always
/ 1 2 n¯
h M = I + 1, where M = m1 + m2 + ··· + mn, with equals that of I0 = i1+i2+···in with the requirement
-t the requirement that m1 ≥ m2 ≥ ··· ≥ mn for bosons that 0 ≤ i1 ≤ i2 ≤ ··· ≤ in ≤ 2j+1−n for n fermions
cl and m1 > m2 > ··· > mn for fermions, where n is the or 0 ≤ i1 ≤ i2 ≤ ··· ≤ in ≤ 2l for n bosons. This
u number of particles (This procedure is called Process A result can be explained from the fact as follows. The
n in this paper.). The combinatorial numbers of different P(n,I0)ofn¯ “bosons”withspinLcorrespondstoYoung
: M’s look irregular, and such an enumeration would be diagrams up to n rows, and 2l columns for bosons or
v
prohibitively tedious when j and l are very large. The 2j+1−ncolumns forfermions. The conjugatesofthese
i
X number of states of a few nucleons in a single-j shell is Young diagrams are those up to 2l rows for bosons or
r usually tabulated in textbooks, for sake of convenience. 2j+1−nrowsforfermions,anduptoncolumns,which
a
Another well-known solution was given by Racah [2] correspond to partitions in Process B for n fermions in
in terms of the seniority scheme, where one has to intro- a single-j shell or bosons with spin l. Therefore, Process
duce (usually by computer choice) additional quantum B for n¯ bosons with spin L = n/2 provides us with an
numbers. More than one decade ago, a third route was alternativeto constructDI for n bosons with spin l or n
studied by Katriel et al. [3] and Sunko et al. [4], who fermions in a single-j shell.
constructedgenerating functions ofthe number of states This alternative (Process B for n¯ bosons with spin L)
for fermions in a single-j shell or bosons with spin l. suggeststhe followingidentity. Ifl=(2j+1−n)/2(nis
There were two efforts in constructing analytical for- even), i.e., Imax of bosons equals that of fermions, then
mulas of DI. In Ref. [5], D0 for n = 4 was obtained DI for bosons equals that of fermions. This identity can
analytically. In Ref. [6], DI was constructed empirically be easily confirmed. It means that one can obtain DI of
for n = 3 and 4, and some D ’s for n=5. It is there- n fermions in a single-j shell by using that of n bosons
I
foredesirabletoobtainadeeperinsightintothisdifficult with spin l=(2j+1−n)/2, or vice versa.
problem. ProcessBforn¯ bosonswithspinL=n/2isalsouseful
EquivalenttoProcessA,weproposehereanotherpro- in constructing formulas of DI. One can see this point
cedure, called process B and explained as follows. Let from the fact that Process B involves SU(n+1) symme-
P(n,I )bethenumberofpartitionsofI =i +i +···i , try, which is independent of j and l, while in Process A
0 0 1 2 n
with 0 ≤ i ≤ i ≤ ··· ≤ i ≤ 2j +1−n for fermions differentj shellforfermionsandspinl forbosonsinvolve
1 2 n
or 0 ≤ i ≤ i ≤ ··· ≤ i ≤ 2l for bosons. Here different symmetries (SU(2j+1) and SU(2l+1) ).
1 2 n
I = nj − n(n−1) for fermions in a single-j shell, Below we exemplify our idea by n = 4. The relevant
max 2
symmetryforProcessBofn¯ bosonswithspinLisSU(5)
(i.e., L=n/2=2, d bosons). n¯ equals 2l and 2j−3, for
four bosons and four fermions, respectively.
∗Electronicaddress: [email protected] Our first result is that DI of four bosons with spin
2
l always equals that of four fermions in a single j shell valuesofI andI =2I +3foroddvaluesofI. ForI ≤l,
0 0
when l = (2j −3)/2. Our second result is that we can
deriveD offourbosonswithspinl bythis newmethod. 2I0
I
D = f . (1)
HereoneneedsDI ofdbosonswithn¯ =2l. Thisproblem I=2I0 λ
wasstudiedintheinteractingbosonmodel,suggestedby λX=I0
Arima and Iachello [7] in seventies. Below we revisit the
For κ=0 and I ≤l (I =2I ≤2l),
enumeration of D for d bosons with particle number 0 0
I
n¯ =2l.
D =(I +1)k
I=2I0 0
Let us follow the notation of Ref. [7] and define
n¯ = 2l = 2ν +v = 2ν + 3n + λ. D of n¯ d bosons − 9K2−K+3KK+(2K−5)θ(2K−5) +δK0 ,
δ I
is enumerated via the procedure as follows. (1) v (cid:0) (cid:1) (2)
takes value 2l,2l−2,2l−4,···,0, which corresponds to
where K = I0 , K=(I mod 6). θ(x)=1 if x>0 and
ν = 0,1,2,···,n/2 = l, respectively. (2) For each 6 0
value ofv, n takes valuefrom0 to v . (3) Foreach zero otherwi(cid:2)se.(cid:3)One can repeat the same procedure for
∆ 3 κ=2 and 4. We list these results as below:
set of v and n∆, λ is determined by(cid:2)v−(cid:3) 3n∆. (4) For For κ=2 and I ≤l,
each λ obtained in step (3), the allowed spin is given by 0
λ, λ+1,λ+2, ···, 2λ−3, 2λ−2, 2λ. Note that there
D =(I +1)k
is no state with 2λ−1. One easily sees that there is no I=2I0 0
I = 1 states for d bosons, because λ = 1 presents I = 2 − 9K2−K+3KK+(2K−5)θ(2K−5)
state (2λ−1 is missing). (cid:0) I +3 I +5 (cid:1)
InordertoobtainDI,itisnecessarytoknowthenum- +(cid:20) 06 (cid:21)+(cid:20) 06 (cid:21)+δK0−δK3 . (3)
berofλappearingintheaboveprocessforeachI. Letus
callthisnumberfλanddefinen¯ =2l=6k+κ,κ=0,2,4, For κ=4 and I0 ≤l,
and k ≥1. Below we exemplify how we obtainf by the
λ
case of κ=0. We have the following hierarchy: DI=2I0 =(I0+1)(k+1)
− 9K2−K+3KK+(2K−5)θ(2K−5)
λ0 kf+λ1 v0,6,12,···,6k (cid:0) −(cid:20)I06+3(cid:21)−(cid:20)I06+4(cid:21)+δK4 . (cid:1) (4)
1 k 4,10,16,···,6k−2
For I is odd and I ≤ 2l, we use a relation D −
2 k 2,8,14,···,6k−4 D = I0 +1. This relationwas obtainedIe=m2Ip0iri-
I=2I0+3 2
3 k 6,12,18,···,6k cally in Ref.(cid:2)[6](cid:3)and can be obtained mathematically by
4 k 4,10,16,···,6k−2 calculating
5 k−1 8,14,20,···,6k−4
2I0+3
6 k 6,12,18,···,6k D = f
I=2I0+3 λ
7 k−1 10,16,22,···,6k−2 λ=XI0+3
8 k−1 8,14,20,···,6k−4 and compare with D .
I=2I0
9 k−1 12,18,24,···,6k For the case with I ≥ 2l, we define I = I −2I
max 0
10 k−1 10,16,22,···,6k−2 for even I and I = Imax −2I0 −3 for odd I. fλ=I0 =
I0 −δ . We obtain that
11 k−2 14,20,26,···,6k−4 6 (I0 mod 6),0
(cid:2) (cid:3)
12 k−1 12,18,24,···,6k I I I
13 k−2 16,22,28,···,6k−2 DImax−2I0 =DImax−2I0−3 =3(cid:20) 60(cid:21)((cid:20) 60(cid:21)+1)−(cid:20) 60(cid:21)
14 k−2 14,20,26,···,6k−4 I
+ ( 0 +1)((I mod 6)+1)+δ −1 . (5)
15 k−2 18,24,30,···,6k (cid:20) 6 (cid:21) 0 (I0 mod 6),0
16 k−2 16,22,28,···,6k−2
Thus we solve the problem of enumeration of D for
I
17 k−3 20,26,32,···,6k−4 four bosons with spin l or four fermions in a single-
.. .. .. j shell by using the new enumeration procedure. One
. . . .
may obtain D of other n (n is even) cases by ap-
I
plying this method similarly, if the reduction rule of
From this tabulation we have that fλ =k+δm0−δm5− SU(n+1)→SO(3) is available.
λ , where m is equal to λ mod 6 when κ = 0, and [ ] A question arises when we apply this method to odd
6
m(cid:2) e(cid:3)anstotakethelargestintegernotexceedingthe value n cases,for whichspinLofn¯ bosonsinvolvedinProcess
inside. B is not an integer (L = n/2). These bosons are there-
For the sake of simplicity we define I = 2I for even fore not “realistic”. For such cases I of n bosons with
0
3
spin l cannot equal that of n fermions in a single j shell. D , for n fermions in a single j shell or n bosons with
I
Namely,thereisnosimilarcorrespondenceofD between spinl. We provedthatD ofn bosonswithspinl equals
I I
bosons and fermions when n is odd [8]. However, D of thatofnfermionsinasingle-j shellwhen2l=2j+1−n,
I
n¯ fictitious bosons with spin n/2 (n is odd) obtained by where n is even. We have alsoexemplified the usefulness
Process A equals that of n bosons with spin l or that of of this new method in constructing analytical formulas
n fermions in a single-j shell, where n¯ = 2l (even value) of D by n=4.
I
and 2j+1−n (odd value) for bosons and fermions, re- For odd n, the procedure of our new method involves
spectively. Inotherwords,D ofn¯ fictitious bosonswith half integer spin L for “bosons”. Further consideration
I
spin n/2 equals that of n bosons with spin l if n¯ =2l or of this fictitious situation is necessary.
that of n fermions in a single-j shell if n¯ = 2j+1−n, We would like to thank Professors K. T. Hecht and
here n is odd. Further discussion is warranted on this I. Talmi for their reading and constructive comments of
problem. this manuscript.
To summarize, We have presented in this paper an
alternative to enumerate the number of spin I states,
[1] Forexample,R.D.Lawson,TheoryofNuclearShellModel Phys. Rev.C35, 1936 (1987).
(Clarendon, Oxford, 1980), P. 8-20. [5] J.N.GinocchioandW.C.Haxton,SymmetriesinScience
[2] G. Racah, Phys. Rev. 63, 367 (1943); A. de-Shalit and I. VI, Edited by B. Gruber and M. Ramek (Plenum Press,
Talmi,NuclearShellModelTheory(Academic,NewYork, New York,1993), p. 263.
1963). [6] Y.M.ZhaoandA.Arima,Phys.Rev.C68,044310(2003).
[3] J. Katriel, R. Pauncz, and J. J. C. Mulder, Int. J. Quan- [7] F. Iachello and A. Arima, The Interacting Boson Model
tumChem.23,1855(1983);J.KatrielandA.Novoselsky, (Cambridge University Press, England, 1987), P. 38.
J. Phys.A22, 1245 (1989). [8] AcorrespondenceofDI wasnotedinSec.IIofRef.[6]for
[4] D.K.SunkoandD.Svrtan,Phys.Rev.C31,1929(1985); large I cases.
D. K.Sunko,Phys.Rev. C33, 1811 (1986); D. K.Sunko,