CLIMBING GOV-LTT CONVERGED WITH CLIMBING ACR-MLG RESULTING IN LESS THAN STANDARD SEPARATION.

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I WAS WORKING THE LCH HI ALT SECTOR WHICH HANDLES ACFT BTWN NEW ORLEANS; LA; AND HOU; TX. I HAD JUST TAKEN OVER THE SECTOR DURING THE BEGINNING STAGES OF THE NOON DEP RUSH WHEN I HAD A GOV LTT X COME ON MY FREQ. THE LTT X HAD DEPARTED NBG AND WAS CLBING PRETTY FAST TO FL230 AND HAD TO BE POINTED OUT TO THE SECTOR JUST E OF MINE. THAT BOUNDARY IS ONLY 10 MI W OF MSY. THE LTT X CALLED ON LEAVING FL225 AND I ISSUED FL350; HE HAD REQUESTED FL390 FROM NBG TO EFD. SOON AFTER I GAVE THE LTT FL350 I ACCEPTED A HDOF ON AN MLG THAT HAD DEPARTED MSY. AT THAT TIME HE WAS LEAVING FL188. BY THIS TIME THE HOU DEP RUSH HAD MATERIALIZED AND I WAS VERY BUSY WITH SOME DEPS AND ARRS AT HOU. THIS IS ON THE FAR LEFT SIDE OF MY SCOPE AND THE LTT AND MLG ARE ON THE FAR RIGHT SIDE OF MY SCOPE WHICH IS BEING OPERATED ON THE 150 MI RANGE BECAUSE OF THE SECTOR SIZE. IN ADDITION; I HAD APPROX 15-17 ACFT ON FREQ. WHEN THE MLG CAME OVER HE WAS LEVEL AT FL230 AND I ISSUED HIM FL280 FOR SOME POSSIBLE OPP DIR TFC. THE MLG WAS ABOUT 7 MI TO THE N OF THE LTT; BUT WHAT I DID NOT KNOW WAS THAT THE LOW ALT CTLR HAD ASSIGNED THE LTT A HDG OF 270 BUT DID NOT ASSIGN THE MLG A HDG. THE MLG WAS APPARENTLY FLYING A HDG OF ABOUT 255 DEGS. I EXPECTED THE LTT TO CONTINUE CLBING FAST FOR 2 REASONS: 1) BECAUSE OF THE TYPE ACFT AND 2) BECAUSE HE HAD FILED FOR FL390 FOR A 200 MI TRIP. BUT; THIS WAS NOT THE CASE. THE LTT INSTEAD REDUCED HIS RATE OF CLB TO 600 FPM. THE MLG WAS DOING ABOUT 300 FPM. AS I SAID EARLIER; I WAS VERY BUSY ON THE LEFT SIDE OF MY SCOPE AND BY THE TIME I LOOKED BACK AND SAW THE SITUATION; ABOUT 1 MIN; THE LTT WAS LEAVING FL266 AND THE MLG WAS LEAVING FL255. I ASSIGNED THE MLG FL260 AND TOLD THE LTT TO EXPEDITE HIS CLB WHICH HE DID NOT DO. THE MLG WENT AS HIGH AS FL262 AT THE TIME THE LTT WAS AT FL268 AND THEY WERE 3.5 MI APART. IT WAS A COMBINATION OF EVENTS THAT CAUSED THIS SITUATION. 1) THE LTT PLT SHOULD HAVE ADVISED ME OF HIS REDUCED RATE OF CLB. 2) THE LOW ALT CTLR SHOULD HAVE ADVISED ME OF THE ASSIGNMENT/NON ASSIGNMENT OF HDGS THAT CAUSED THE TFC TO CONVERGE. 3) BECAUSE OF THE EXPECTED RATE OF CLB; I LET MYSELF BE LULLED INTO THINKING THE SITUATION WOULD NOT BE A PROB AND THEREFORE DIVERTED MY ATTN TO WHAT I FELT WERE GREATER PROBS AT THE TIME.