ARTCC RADAR CTLR DSNDED A B737 THROUGH ALT OF B727 WITH LTSS. CTLR KNEW IT WOULD BE TIGHT AND NEEDED TO WATCH THE ACFT CLOSELY; BUT GOT DISTR WITH INTRAFAC COORD. CONFLICT ALERT ACTIVATED; BUT CTLR'S INTERVENTION WAS TOO LATE TO PREVENT A SYS ERROR.

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I CLRED THE ACR Y B737 DIRECT TO APE FOR 2 REASONS. I WANTED TO GET HIM DOWN LNDG CVG AND GET HIM OUT OF THE PATH OF THE PIT DEPS SO I COULD CLB THEM. I GOT LOWER (FL310); HE WAS AT FL350; AND DSNDED HIM. I REALIZED THERE WAS A POTENTIAL PROB SO I PLANNED TO WATCH HIS RATE OF DSCNT TO SEE IF I NEEDED EVASIVE ACTION OR AN EXPEDITE CLRNC. RIGHT AFTER I DSNDED THE B737; I NOTICED A SIT BTWN 2 ACFT AT FL370 BEING HANDED OFF TO ME. AFTER REVIEWING BOTH ACFT'S RTE OF FLT; I HAD TO CALL THE SECTOR HANDING ME THESE ACFT AND TELL HIM TO SEPARATE THEM. AFTER NUMEROUS ATTEMPTS TO GET A HOLD OF THIS SECTOR; I GAVE UP. WITHIN SECONDS OF HANGING UP THE LINE; THEY ANSWERED. MY ASSISTANT ASKED; 'WHAT DO YOU WANT TO TELL THEM?' SO I SAID I WOULD GET THE LINE. AFTER TELLING THEM TO SEPARATE THESE 2 ACFT; I HUNG UP AND WAS IMMEDIATELY REDIRECTED TO THE B737 AND B727 INCIDENT BECAUSE OF CONFLICT ALERT. UNFORTUNATELY; CONFLICT ALERT WENT OFF WHEN THE ACFT WERE ONLY 6 MI APART. IT WAS IMPOSSIBLE AT THAT POINT TO KEEP STANDARD SEPARATION OF 5 MI; BUT I CERTAINLY TRIED. I EXPEDITED THE DSNDING ACFT; TURNED THE ACR X B727 APPROX 80 DEGS AND TURNED THE B737 APPROX 50 DEGS. THE CLOSEST THEY GOT WAS 4 MI. ONE OTHER FACTOR MAY BE THAT ACFT THAT DON'T WANT TO START DOWN INTO AN ARPT; DSND AT A VERY SLOW RATE. I BELIEVE THIS WAS THE CASE.