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3 S1 − 3 D 1<br />

Im η<br />

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1<br />

−1<br />

−2<br />

−3<br />

Re η<br />

Λ = 10 fm -1<br />

Λ = 7 fm -1<br />

Λ = 5 fm -1<br />

Λ = 4 fm -1<br />

Λ = 3 fm -1<br />

Λ = 2 fm -1<br />

N 3 LO<br />

E t [MeV]<br />

0.0<br />

−2.0<br />

−4.0<br />

−6.0<br />

−8.0<br />

b osc = 1.5 fm<br />

Λ = 4.0 fm -1<br />

Λ = 3.0 fm -1<br />

Λ = 2.0 fm -1<br />

Λ = 1.6 fm -1<br />

0 10 20 30 40 50<br />

N max<br />

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Ù×�Ò� Ø�� ÔÖ�× Ö���� ��×�× �ÙÒ Ø�ÓÒ× Û�Ø� ��«�Ö�ÒØ � Ú�ÐÙ�× Ì�� Ö�×ÓÒ�Ò � ÔÓ×�Ø�ÓÒ<br />

�Ò �� ���ÒØ�¬�� �Ý Ó�×�ÖÚ�Ò� Ø�� ×ÐÓÛÐÝ Ú�ÖÝ�Ò� ����Ò �Ò�Ö���× �Ò Ø�� ×Ø���Ð�Þ�Ø�ÓÒ<br />

ÔÐ�Ø��Ù ���ÙÖ�× � ×�ÓÛ× Ø�� ¬ØØ�Ò� Ó� ��Ò×�ØÝ Ó� ×Ø�Ø�× Ù×�Ò� � ÄÓÖ�ÒØÞ��Ò �ÓÖÑ<br />

ÓÖÖ�×ÔÓÒ��Ò� ØÓ Ø�� ��×Ø ¬Ø Û�Ø� Ð��×Ø � �ÖÓÑ Ø�� �Ò�Ö�Ý Ð�Ú�Ð× Ó� ×Ø���Ð�Þ�Ø�ÓÒ<br />

����Ö�Ñ× � �ÖÓÑ ÓÙÖ �Ð ÙÐ�Ø�ÓÒ× Û� Ó�×�ÖÚ�� Ø��Ø Ø�� È Ó Ö�×ÓÒ�Ò � Û��Ø�×<br />

�× ×�ÓÛÒ �Ò ¬�ÙÖ� � �� Ö��×�× Û��Ö��× È Ó Ö�×ÓÒ�Ò � Û��Ø�× �× ×�ÓÛÒ �Ò ¬�ÙÖ�<br />

� �Ò Ö��×�× Û�Ø� Ø�� �Ò Ö��×� Ó� ÔÐ�×Ñ� ×ØÖ�Ò�Ø� ÇÒ� Ë � ÓÒ� È Ó �Ò� ØÛÓ È Ó<br />

Ö�×ÓÒ�Ò � �Ò�Ö���× �Ö �Ò� Û��Ø�× �ÐÓÒ� Û�Ø� Ø�� �ÖÓÙÒ� ×Ø�Ø� �Ò�Ö���× �ÓÖ Ø��<br />

Ú�Ö�ÓÙ× ���Ý� Ð�Ò�Ø�× Ö�Ò��Ò� �ÖÓÑ �Ò¬Ò�ØÝ ØÓ � ×Ñ�ÐÐ Ú�ÐÙ� ��Ú� ���Ò �Ð ÙÐ�Ø��<br />

Ê�×ÙÐØ× �Ö� Ù×��ÙÐ ØÓ Ø�� �ØÓÑ� Ô�Ý×� × ÔÐ�×Ñ� Ô�Ý×� × �Ò� �×ØÖÓÔ�Ý×� × Ö�×��Ö �<br />

ÓÑÑÙÒ�Ø��×<br />

Ê���Ö�Ò �×<br />

Ë Ã�Ö �Ò� � à ÀÓ È�Ý× Ê�Ú � � ��� � � ���Ñ È�Ý× Ä�ØØ �<br />

��� � � Æ�Û Â È�Ý× � � � � È�Ý× Ê�Ú � � � � �  È�Ý×<br />

� � �� � � Ö���Ö�Ò �× Ø��Ö��Ò<br />

� Ë��� �Ø �Ð ���Ñ È�Ý× Ä�ØØ � �<br />

� Ì��× ÛÓÖ� �× ×ÙÔÔÓÖØ�� �Ý Ø�� Æ�Ø�ÓÒ�Ð Ë ��Ò � �ÓÙÒ �Ð Ó� Ì��Û�Ò ÊÇ�<br />


Ì��Ð� � �ÖÓÙÒ� ×Ø�Ø� �Ò�Ö���× Ó� È× �Ò� È× �ÓÖ ��«�Ö�ÒØ ���Ý� Ô�Ö�Ñ�Ø�Ö×<br />

� �Ò�Ö���× Ó� È× �Ò�Ö���× Ó� È×<br />

ÈÖ�×�ÒØ ÛÓÖ� Ê�� � ℄ Ê�� � ℄ Ê�� � ℄<br />

� �� � � �� � � � � � � �<br />

� ���� �� ���� � � ������ � ������<br />

� � � ��� � � � � � � � � � � � ��<br />

E (a.u.)<br />

-0.038<br />

-0.040<br />

-0.042<br />

-0.044<br />

-0.046<br />

22<br />

-0.048<br />

0.3 0.4 0.5 0.6 0.7 0.8<br />

-1<br />

ω (a ) 0<br />

(a)<br />

16<br />

Density <strong>of</strong> states<br />

600<br />

500<br />

400<br />

300<br />

200<br />

100<br />

0<br />

E r =-0.0430748<br />

Γ =-1.15 x 10 -4<br />

-0.0436 -0.0432 -0.0428 -0.0424<br />

E (a.u.)<br />

���ÙÖ� � � ËØ���Ð�Þ�Ø�ÓÒ ÔÐÓØ× �ÓÖ Ø�� ÐÓÛ�×Ø È Ó ×Ø�Ø� Ó� È× �Ò ÔÐ�×Ñ�× �ÓÖ � �<br />

� ��Ð ÙÐ�Ø�� ��Ò×�ØÝ �Ö Ð�× �Ò� Ø�� ��×Ø ¬ØØ�� ÄÓÖ�ÒØÞ��Ò �ÓÖÑ ×ÓÐ�� Ð�Ò�<br />

ÓÖÖ�×ÔÓÒ��Ò� ØÓ Ø�� È Ó ×Ø�Ø� Ó� È× �ÓÖ Ø�� � Ø� �Ò�Ö�Ý Ð�Ú�Ð �Ò �<br />

Γ (a.u.)<br />

1.4x10 -4<br />

1.2x10 -4<br />

1.0x10 -4<br />

8.0x10 -5<br />

6.0x10 -5<br />

4.0x10 -5<br />

2.0x10 -5<br />

3 P o (1)<br />

(a)<br />

0.0<br />

0.00 0.02 0.04 0.06 0.08 0.10<br />

μ<br />

Γ (a.u.)<br />

1.5x10 -6<br />

1.4x10 -6<br />

1.3x10 -6<br />

1.2x10 -6<br />

1.1x10 -6<br />

1.0x10 -6<br />

9.0x10 -7<br />

(b)<br />

0.000 0.004 0.008 0.012 0.016<br />

���ÙÖ� � Ì�� ÐÓÛ�×Ø È Ó Ö�×ÓÒ�Ò �× Û��Ø�× �Ò � �Ò� Ø�� ÐÓÛ�×Ø È Ó Ö�×ÓÒ�Ò �×<br />

Û��Ø�× �Ò � �× �ÙÒ Ø�ÓÒ× Ó� ���Ý� Ô�Ö�Ñ�Ø�Ö× �<br />

Ë Ã�Ö �Ò� � à ÀÓ È�Ý× Ê�Ú � � � � � � È�Ý× Ê�Ú �� �<br />

� � Å �ÖÓÐÓÚ È�Ý× Ê�Ú � �� �� �<br />

� Î � Å�Ò��Ð×�Ø�Ñ �Ø �Ð È�Ý× Ê�Ú Ä�ØØ � � ��<br />

�<br />

1 P o (1)<br />

μ<br />

(b)


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ØÓØ�Ð Ó� Ø�Ö�� Ë � ¬Ú� Ë � �Ò� Ò�Ò� � È Ó Ö�×ÓÒ�Ò �× ��ÐÓÛ Ø��Ò � À� Ø�Ö�×�<br />

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�ÖÓÑ �Ø× �ÖÓÙÒ� ×Ø�Ø� ØÓ ×Ù � Ö�×ÓÒ�Ò � ×Ø�Ø�× �ÓÖ ��«�Ö�ÒØ ���Ý� Ð�Ò�Ø�× Û�ÐÐ �Ð×Ó ��<br />

ÔÖ�×�ÒØ�� ÇÙÖ Ö�×ÙÐØ× Û�ÐÐ ÔÖÓÚ��� Ù×��ÙÐ �Ò�ÓÖÑ�Ø�ÓÒ ØÓ Ø�� Ö�×��Ö � ÓÑÑÙÒ�Ø��× Ó�<br />

�ØÓÑ� Ô�Ý×� × ÔÐ�×Ñ� Ô�Ý×� × �Ò� �×ØÖÓÔ�Ý×� ×<br />

� Ì�� ÛÓÖ� �× ×ÙÔÔÓÖØ�� �Ý Ø�� Æ�Ø�ÓÒ�Ð Ë ��Ò � �ÓÙÒ �Ð Ó� Ì��Û�Ò Ê Ç �<br />


Ì��Ð� � �ÓÙÒ� �Ü �Ø�� ×Ø�Ø�× �Ò�Ö���× Ó� À� ÙÒ��Ö ���Ý� × Ö��Ò�Ò�<br />

�� × Ë × × Ë × Ô È × � � À� Ë<br />

� � � �� ����� �� � �� ���� ��� �<br />

� � � �� ����� �� � �� ��� ���<br />

� ��� � ���� �� ��� ��� �� �� ��� ������ ��<br />

� ��� ���� ��� � � � �� � �� � � � � �<br />

��×Ø Ö�×ÙÐØ×��℄<br />

E (a.u.)<br />

-0.436<br />

-0.438<br />

-0.440<br />

-0.442<br />

-0.444<br />

-0.446<br />

-0.448<br />

-0.450<br />

-0.452<br />

-0.454<br />

-0.456<br />

-0.458<br />

2s5p +<br />

2s5p −<br />

2p4d<br />

2s4p +<br />

2s4p −<br />

2p3d<br />

-0.460<br />

0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0<br />

-1<br />

ω (a ) 0<br />

(a)<br />

Density<br />

800<br />

700<br />

600<br />

500<br />

400<br />

300<br />

200<br />

100<br />

0<br />

E r =-0.446726<br />

Γ = 9.6 x 10 −5<br />

-0.448 -0.447 -0.446 -0.445<br />

���ÙÖ� � � ËØ���Ð�Þ�Ø�ÓÒ ÔÐÓØ× �ÓÖ Ø�� Ô � ×�Ô ×�Ô Ô�� ×�Ô ×�Ô È Ó<br />

×Ø�Ø�× Ó� À� �Ñ������ �Ò ���Ý� ÔÐ�×Ñ� �ÓÖ �� � � ��Ð ÙÐ�Ø�� ��Ò×�ØÝ �Ö Ð�×<br />

�Ò� Ø�� ��×Ø ¬ØØ�� ÄÓÖ�ÒØÞ��Ò �ÓÖÑ ×ÓÐ�� Ð�Ò� ÓÖÖ�×ÔÓÒ��Ò� ØÓ ×�Ô È Ó ×Ø�Ø�× Ó�<br />

À� �ÓÖ �� �<br />

Excitation energy (a.u.)<br />

0.92<br />

0.88<br />

0.84<br />

0.80<br />

0.76<br />

0.72<br />

Ionization limit<br />

1s4d 1 D<br />

1s3d 1 D<br />

1s2p 1 P<br />

1s2s 1 S<br />

(a)<br />

0.68<br />

0.00 0.04 0.08 0.12 0.16 0.20<br />

1/λ D<br />

Γ (a.u.)<br />

3.0x10 -4<br />

2.5x10 -4<br />

2.0x10 -4<br />

1.5x10 -4<br />

1.0x10 -4<br />

E(a.u.)<br />

5.0x10<br />

0.00 0.02 0.04 0.06 0.08 0.10<br />

-5<br />

1/λ D<br />

���ÙÖ� � �Ü �Ø�Ø�ÓÒ �Ò�Ö���× �Ò � Ó� Ø�� ��Û ×�Ò�Ð�Ø ×Ø�Ø�× Ó� À� �Ò� Ø�� Ö�×ÓÒ�Ò �<br />

Û��Ø�× �ÓÖ È Ó ×Ø�Ø�× Ó� À� �ÓÖ Ú�Ö�ÓÙ× ���Ý� ×���Ð��Ò� Ô�Ö�Ñ�Ø�Ö×<br />

Ê���Ö�Ò �×<br />

Ë Ã�Ö �Ò� � à ÀÓ È�Ý× Ê�Ú � � ��� � � ���Ñ È�Ý× Ä�ØØ �<br />

��� � � Æ�Û Â È�Ý× � � � � È�Ý× Ê�Ú � � � � � � È�Ý×<br />

Ê�Ú � � � Ê � �  È�Ý× � � �� � � ÁÒØ Â ÉÙ�ÒØÙÑ<br />

���Ñ � � � � � È�Ý× Ê�Ú � � �<br />

� Å �ÖÓÐÓÚ È�Ý× Ê�Ú � �� �� �<br />

Î � Å�Ò��Ð×�Ø�Ñ �Ø �Ð È�Ý× Ê�Ú Ä�ØØ � � ��<br />

� � Ï � �Ö��� ÁÒ ÄÓÒ�� Ê�Ò�� ��×�Ñ�Ö �ÓÖ �× ��× � Ë Ä�Ú�Ò � � Å� ��<br />

ÈÐ�ÒÙÑ� Æ�Û �ÓÖ� �� Ô<br />

��<br />

2s4p +<br />

2s3p +<br />

(b)<br />

(b)


ÇÒ� �Ó�Ý ��Ò×�ØÝ ÔÖÓÔ�ÖØ��× �ÓÖ ÓÖÖ�Ð�Ø�� ÓÒ��Ò×�Ø�×<br />

Ì Ã��Ö���Ö� Å Ì����Ö×�Ò �Î ���ÓÖÓÚ �Ò� � Ë Â�Ò×�Ò<br />

��Ô�ÖØÑ�ÒØ Ó�È�Ý×� × �Ò� �×ØÖÓÒÓÑÝ ÍÒ�Ú�Ö×�ØÝ Ó� ��Ö�Ù× �à � ��Ö�Ù× �<br />

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ØÓ ÙÒ�ØÝ Û�Ø� Æ �× � ÓÑÔ�Ò��� �Ý � �� Ö��×�Ò� �Ò�Ö�Ý Ô�Ö Ô��Ö Ï�Ø� ÓÒÐÝ ØÛÓ ÒÓÒ<br />

Ú�Ò�×��Ò� ����ÒÚ�ÐÙ�× Ø�� ÓÒ��Ò×�Ø� �Ö� Ø�ÓÒ ×�ÓÙÐ� �Ü ��� �� Û��Ö� ÓÒ� Ú�ÐÙ� �×<br />

Ø�Ö�� Ø�Ñ�× Ð�Ö��Ö Ø��Ò Ø�� ÓØ��Ö �ÓÖ Ö�Ð�Ø�Ú�ÐÝ Û��� Ö�ÔÙÐ×�ÓÒ Ø��× Ú�ÐÙ� �× �Ü �����<br />

�××�ÒØ��ÐÐÝ �ÓÖ �ÐÐ Æ �Ò� �Ú�Ò �ÓÖ � ×Ù�×Ø�ÒØ��Ð Ö�ÔÙÐ×�ÓÒ Æ � �× ×ÙÆ ��ÒØ ÁÒ<br />

Ø�� ÓÒØÖ��ÙØ�ÓÒ Û� ×��ÐÐ ��× Ù×× Ø�� Ñ�Ø�Ó� ×ØÖÙ ØÙÖ� Ó� Û�Ú��ÙÒ Ø�ÓÒ× ��Ø��Ð× Ó�<br />

ÓÖÖ�Ð�Ø�ÓÒ× �Ò�Ö���× �Ò� Ö����<br />

Condensate fraction<br />

1<br />

0.95<br />

0.9<br />

0.85<br />

0.8<br />

0.75<br />

0.7<br />

0.65<br />

0<br />

100<br />

a = 27.2a0<br />

a = 34.9a0<br />

Condensate fraction<br />

Pair energy<br />

200 300 400 500<br />

Number <strong>of</strong> particles N<br />

600<br />

���ÙÖ� � Ì�� ÓÒ��Ò×�Ø� �Ö� Ø�ÓÒ �Ò� �ÒØ�Ö� Ø�ÓÒ �Ò�Ö�Ý Ô�Ö Ô��Ö �× �ÙÒ Ø�ÓÒ× Ó� Æ �ÓÖ ��«�Ö�ÒØ<br />

× �ØØ�Ö�Ò� Ð�Ò�Ø� �Ò ÙÒ�Ø× Ó� Ø�� �Ó�Ö Ö���Ù× �<br />

Ê���Ö�Ò �×<br />

�  È�Ø�� � �Ò� Ä È È�Ø���Ú×��� È�Ý× Ê�Ú �� � �<br />

Ç Ë�Ö�Ò×�Ò � Î ���ÓÖÓÚ �Ò� � Ë Â�Ò×�Ò Â È�Ý× � � � �<br />

��<br />

1.6<br />

1.4<br />

1.2<br />

1<br />

0<br />

700<br />

0.8<br />

0.6<br />

0.4<br />

0.2<br />

N(N − 1) [�ω]<br />

Pair energy (E − 3 2 N)/ 1 2


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Cross section σ 00−20 [Α 2 ]<br />

o<br />

0.1<br />

0.01<br />

0.001<br />

0.0001<br />

1e-05<br />

Experiment: B. Mate, et al., JCP 122, 064313 (2005)<br />

Theory: this work, DJ PES<br />

Theory: this work, BMKP PES<br />

350 400 450 500 550 600 650 700 750 800<br />

Energy [cm -1 ]<br />

���ÙÖ� � �ÖÓ×× ×� Ø�ÓÒ �ÓÖ Ø�� � ÖÓØ�Ø�ÓÒ�Ð ØÖ�Ò×�Ø�ÓÒ �Ð ÙÐ�Ø�� Û�Ø� Ø�� � �Ò�<br />

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Û�Ø� Ø�� � ×ÙÖ�� � �Ò� ØÖ��Ò�Ð�× �ÓÛÒ Û�Ø� Ø�� �ÅÃÈ ÔÓØ�ÒØ��Ð<br />

k 00-22 (T) [cm -3 s -1 ]<br />

k 20-00 (T)[cm -3 s -1 ]<br />

1e-10<br />

1e-12<br />

1e-14<br />

1e-16<br />

1e-18<br />

1e-20<br />

1e-22<br />

0 500 1000<br />

T[K]<br />

1500 2000<br />

1e-11<br />

1e-12<br />

This work, DJ PES<br />

Flower, 1998<br />

DJ PES<br />

Danby et al., 1987<br />

1e-13<br />

0 500 1000<br />

T[K]<br />

1500 2000<br />

k 02-22 (T)[cm -1 s -1 ]<br />

k 11-13 [cm -3 s -1 ]<br />

1e-10<br />

1e-11<br />

1e-12<br />

1e-13<br />

1e-14<br />

1e-15<br />

1e-16<br />

0 500 1000<br />

T[K]<br />

1500 2000<br />

1e-10<br />

1e-12<br />

1e-14<br />

1e-16<br />

1e-18<br />

DJ PES<br />

Flower, 1998<br />

Flower, 1998<br />

DJ PES<br />

0 500 1000 1500 2000<br />

T[K]<br />

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